kirchoff plate theory

Department of Precision and Microsystems Engineering

Dynamic Performance Modeling of Optical Element Curvature

Manipulation

Author: J.B. van Wuijckhuijse

Report no: EM 10.031

Coach: ir. C.L. Valentin

Professor: prof. dr. ir. D.J. Rixen

Specialisation: Engineering Mechanics

Type of report: Masters Thesis

Date: Delft, June 25, 2010

Abstract

A modern day society would be unthinkable without the existence of integrated circuits

(or IC’s). These IC’s can be found in almost every electronic device and have revolu-

tionized the world of electronics. To keep up with the ever growing demand for smaller

and faster IC’s, the semiconductor industry is continuously exploring new techniques

in order to reduce the size and optimize the efficiency of their products.

This thesis focuses on a new methodology to enhance the accuracy of the lithographic

production step in semiconductor manufacturing, which is the most influential step

in obtaining smaller and more efficient IC’s. During this fabrication step, three di-

mensional patterns are created on semiconductor material which subsequently form

the structure of the IC. These patterns are created via patterned image projection on

photoresistant material, deposited on the semiconductor substrate.

The performance of the lithographic step is measured with several performance param-

eters. This thesis focuses on the improvement of one of these performance parameters,

referred to as depth of focus. Depth of focus describes the out of plane imaging error of

lithographic systems. In order to reduce this error, an additional correction method to

the current leveling strategies is proposed which reduces the image to substrate non-

conformities by bending of the photomask, which holds the pattern that is projected

onto the substrate. This creates the possibility to correct for field curvature of the

substrate topology and subsequently improve the depth of focus.

The realization of a curvature correction mechanism in lithographic machinery does

have a number of issues that need to be solved. First of all, in order to correct for

substrate unflatness, it is required to obtain curvature setpoints from substrate topology

which serve as input specification for the photomask bending actuator. Therefore an

anticlastic curvature correction fitting function is proposed, which is able to derive

curvature setpoints from measured substrate topology data.

i

ABSTRACT

Using the anticlastic curvature augmented fitting function, a study is performed on

state of the art substrate flatness measures and the identification of correctable field

curvature using a photomask bending actuator. During this analysis, an ideal actuator

is assumed.

Thereafter a study is performed on the focus budget improvement of curvature cor-

rection using a dynamic model of the photomask bending actuator. The design of the

photomask bending actuator is assumed to be given. Due to the complexity of this de-

sign, numerical modeling requires the use of model reduction techniques. Therefore, the

most common reduction techniques are discussed, such as Guyan and Craig-Bampton

reduction. Because it is essential that, after reduction, the numerical model is able

to accurately describe curvature deformation of the photomask, an additional mode

selection scheme is proposed for constructing the reduction basis of the Craig-Bampton

technique. This mode selection scheme is based on the theory of observability Grami-

ans.

After obtaining an experimentally validated numerical model of the photomask bending

actuator, a dynamic analysis is performed on the curvature correction performance of

the photomask bending actuator. For this analysis, the identified curvature setpoints

serve as input specification of the numerical model. During this analysis, also the

influence of model reduction techniques on the model accuracy is studied.

From the obtained results, it can be concluded that field curvature correction using the

photomask bending actuator does introduce improvements on the focus budgets with

respect to current leveling strategies. Furthermore, the accuracy lowering effect of the

model reduction techniques is proven to be neglectably small for the specific analyses

set out in this thesis. After performing this research, several challenges remain, such

as nonlinearity of actuator components and the enhancement of the proposed Gramian

theory based mode selection procedure. Furthermore, the obtained numerical model of

the bending actuator is able to serve future research topics, such as actuator control

design.

ii

Acknowledgements

From April 2009 onwards, I had the opportunity to be part of the ASML Research

Mechatronics team and perform my internship and MSc. project. ASML provided

me with the possibility to challenge my mind and reach beyond the ordinary, which

improved my skills and knowledge in many ways. The work presented in this thesis is

the outcome of an interesting and educating period of my life and could not be presented

without the support of colleagues, university staff, family and friends. I would therefore

like to express my gratitude for those who have guided me during the past 14 months.

First of all, I would like to express my sincere gratitude to Ir. Chris Valentin, who gave

me the opportunity to write my thesis and be part of the ASML Research Mechatronics

team. The motivation, inspiration, creativity and enormous enthusiasm of Chris gave

me the ability to perform at higher level. Besides from being a very pleasant colleague

he also became a good friend.

Many thanks go to Prof. Dr. Ir. D.J. Rixen. His impressive professional knowledge and

inspirational teachings gave me the possibility to fulfill my Masters track with great

enthusiasm.

Furthermore I would like to thank my former colleagues at the ASML Research Mecha-

tronics department for a pleasant working atmosphere and great support.

Last but not least, I would like to thank my family and friends for their support. I would

especially like to thank my girlfriend Mariska for her immense support, encouragement

and loving throughout my academic career.

Breda,

June 2010

Bas van Wuijckhuijse

iii

ACKNOWLEDGEMENTS

iv

Contents

Abstract i

Acknowledgements iii

Nomenclature ix

Introduction 1

Semiconductor lithography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Lithography performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Limiting factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Photomask curvature manipulation . . . . . . . . . . . . . . . . . . . . . . . . 5

Assignment definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

I General Theory 11

1 Photomask Curvature modeling 13

1.1 Kirchoff plate theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.1.1 Stress and strain . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.1.2 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.1.3 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.2 Curvature fitting functions . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.2.1 Anticlastic curvature augmented fitting function . . . . . . . . . 23

1.2.2 Coefficient extraction . . . . . . . . . . . . . . . . . . . . . . . . 25

v

CONTENTS

2 Model reduction techniques 27

2.1 Dynamics of undamped discretized structures . . . . . . . . . . . . . . . 28

2.2 Component model reduction for structural problems . . . . . . . . . . . 30

2.2.1 Guyan’s reduction . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.2.2 Mode displacement method . . . . . . . . . . . . . . . . . . . . . 32

2.2.3 Craig-Bampton mode component synthesis . . . . . . . . . . . . 33

2.2.4 Reduction by rigid interface projection . . . . . . . . . . . . . . . 35

2.3 Dynamic substructuring in the nodal domain . . . . . . . . . . . . . . . 37

2.3.1 Primal assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.3.2 Dual assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3 Gramian theory 41

3.1 System state space description . . . . . . . . . . . . . . . . . . . . . . . 42

3.1.1 Constrained beam case: state space formulation . . . . . . . . . 45

3.2 Time response of a modal system . . . . . . . . . . . . . . . . . . . . . . 50

3.2.1 Constrained beam case: time response . . . . . . . . . . . . . . . 53

3.3 Observability Gramian . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.3.1 Constrained beam case: observability Gramian . . . . . . . . . . 59

3.4 Controllability Gramian . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.4.1 Constrained beam case: controllability Gramian . . . . . . . . . 66

3.5 Field curvature mode selection . . . . . . . . . . . . . . . . . . . . . . . 67

3.5.1 Specific output realization . . . . . . . . . . . . . . . . . . . . . . 68

3.5.2 Observability Gramian . . . . . . . . . . . . . . . . . . . . . . . . 69

3.5.3 Field curvature output based Gramians . . . . . . . . . . . . . . 71

3.5.4 Constrained beam case: curvature mode selection . . . . . . . . . 74

3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

II Photomask curvature analysis methodology 81

4 Curvature analysis procedure 83

4.1 Specification derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.1.1 Intradie leveling . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.1.2 Interdie leveling . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.1.3 Mechanical resonance specification . . . . . . . . . . . . . . . . . 90

vi

CONTENTS

4.2 Performance evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5 Specification derivation procedure 97

5.1 Wafer topology measurements . . . . . . . . . . . . . . . . . . . . . . . . 97

5.1.1 Raw data acquisition . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.1.2 Data postprocessing . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.2 Field curvature extraction . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.2.1 Intradie curvature correction . . . . . . . . . . . . . . . . . . . . 104

5.2.2 Interdie curvature correction . . . . . . . . . . . . . . . . . . . . 107

5.2.3 Mechanical resonance specification . . . . . . . . . . . . . . . . . 109

5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

III Results 115

6 Numerical modeling and validation 117

6.1 Numerical modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.1.1 Photomask . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

6.1.2 Bending clamps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

6.1.3 Rods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.1.4 Z-supports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

6.1.5 Preload springs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

6.1.6 Piezoelectric actuators . . . . . . . . . . . . . . . . . . . . . . . . 124

6.2 Experimental validation . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

6.2.1 Correlation measures . . . . . . . . . . . . . . . . . . . . . . . . . 126

6.2.2 Photomask validation . . . . . . . . . . . . . . . . . . . . . . . . 127

6.2.3 Bending clamp validation . . . . . . . . . . . . . . . . . . . . . . 138

6.3 Reduced coupled model . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

6.3.1 Model coupling procedure . . . . . . . . . . . . . . . . . . . . . . 144

6.3.2 Reduced model accuracy . . . . . . . . . . . . . . . . . . . . . . . 146

6.3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

7 Photomask curvature focus improvement 155

7.1 Analysis procedure and settings . . . . . . . . . . . . . . . . . . . . . . . 156

7.1.1 Model setpoint implementation . . . . . . . . . . . . . . . . . . . 157

7.1.2 Newmark time integration . . . . . . . . . . . . . . . . . . . . . . 158

vii

CONTENTS

7.1.3 Performance tracking . . . . . . . . . . . . . . . . . . . . . . . . 159

7.2 Intradie performance tracking . . . . . . . . . . . . . . . . . . . . . . . . 160

7.3 Filtered intradie performance tracking . . . . . . . . . . . . . . . . . . . 163

7.4 Interdie performance tracking . . . . . . . . . . . . . . . . . . . . . . . . 164

7.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

8 Conclusions and recommendations 167

8.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

8.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

Bibliography 175

A ASML Twinscan system architecture 179

B Linear least squares methodology 183

C Derivation of the observability Gramian 185

D Gram-Schmidt orthonormalization 191

E Specifications of measurement equipment 193

E.1 Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

E.2 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

F Newmark time integration 197

Index 201

viii

Nomenclature

As State space, state matrix

Bs State space, input matrix

C Damping matrix

Cs State space, output matrix

Ds State space, feedthrough ma-

trix

G Gramian matrix

g Interface force vector [N ]

h Field topology [m]

p External force vector [N ]

S Static mode matrix

u State space, input vector

x State space, state vector

y State space, output vector

q Reduced set of DoF

qr Rigid interface DoF

B Signed boolean compatibility

matrix

K Stiffness matrix

L Boolean equilibrium matrix

M Mass matrix

q Nodal DoF

qf General interface DoF

qi Internal DoF

qr Stiff interface DoF

Tr Rigid interface transformation

matrix

T Reduction matrix

Λ Eigenvalue matrix

φi Interface fixed mode

Υ State space, output projection

matrix

γ Shear strain vector [−]

λ Lagrange multipliers

φ Eigenmode

σ Stress vector[Nm2

]ε Principal strain vector [−]

Δ Laplace operator

η modal DoF

γr Generalized stiffness of mode r

κ Curvature[1m

]λ Imaging wavelength [m]

μr Generalized mass of mode r

ν Poisson ratio [−]

ix

CONTENTS

ω Eigenfrequency[rads

]ρ Density

[kgm3

]ζ Modal damping ratio [%]

A Area[m2]

dp Force to voltage ratio [−]

E Young’s modulus[Nm2

]e Focus error [m]

ef MA+MSD focus error [m]

ew Wafer focus error [m]

Fb Blocking force [N ]

I Area moment of inertia[m4]

ka Axial stiffness[nm

]

kr Radial stiffness[nm

]M Bending moment [Nm]

oz Offset [m]

P Distributed load [N ]

Rx x-axis rotation[1m

]Ry y-axis rotation

[1m

]u Displacement in x-direction [m]

Um Maximum piezoelectric voltage

[V ]

v Displacement in y-direction [m]

w Displacement in z-direction [m]

Wc Controllability Gramian

Wo Observability Gramian

x

Introduction

The modern world of today would not be as it is without the existence of integrated

circuits. Integrated circuits (or IC’s) can be found in almost every electronic device

and have enabled the immense technological growth the world experienced over the

last decades. The mobile phone for instance, who made his entrance in the 1950’s,

has made through immense innovative changes since its arrival and became one of the

most widely used products in the world. To keep up with the ever growing demand

for smaller and faster mobile devices, the IC industry (or semiconductor industry) is

constantly exploring new techniques to reduce the size and improve the efficiency of

their products.

The demand for technological enhancement of semiconductor products requires the

improvement of the manufacturing process. The most influential production step in

the semiconductor industry is performed by lithographic systems, of which the basics

are explained in the next section. ASML is the world’s leading provider of lithographic

systems. The company is continuously developing systems with improved accuracy and

speed, enabling the creation of smaller and more efficient IC’s.

This thesis focuses on a new method to enhance the accuracy of the lithographic

step in semiconductor manufacturing. The next section explains the basics of semi-

conductor lithography. Thereafter lithographic performance measures are set out. The

section Photomask curvature manipulation proposes an accuracy enhancement method

in lithographic processes from which the thesis assignment is derived in the following

section. The introduction ends with an outline of the thesis.

Semiconductor lithography

Fabrication of an integrated circuit requires a variety of physical and chemical process

steps on a semiconductor substrate (also called wafer). The fundamental process step

is called lithography, in which three dimensional patterns are created on the substrate.

1

INTRODUCTION Company-secret

Semiconductors are materials that have conducting properties and are used to create

state of the art electronics that can be found at the heart of every integrated circuit. The

term lithography originates from the Greek words λιθoσ (lithos) and γραφια (graphia),

meaning ’writing on stones’. A typical IC production process is graphically represented

in figure 1 and can be decomposed into 5 process steps:

1. deposition of a thin layer of photoresist onto the substrate,

2. projection of a pattern onto the photoresist,

3. removal of the exposed photoresist, creating the desired negative pattern,

4. using a process called etching , exposed area’s are cleared of material,

5. doping of the etched area’s, making them conductive or insulating.

This process is repeated for a multitude of times with different projection patterns to

create a full three dimensional structure. The lithography step, the second step in

Figure 1: IC production process [1]

figure 1, is performed in lithography systems. State of the art systems make use of a

scanning process and are subsequently called scanners. In these scanners, a light slit is

passed onto a patterned photomask, leading to a projection of the photomask pattern

on the wafer. In order to scale down the image by a factor 4, a double telecentric lens

is used (see figure 2). To achieve the scanning motion, two stages respectively holding

the reticle and wafer move in opposite direction because the lens mirrors the image.

2

LITHOGRAPHY PERFORMANCE

y

z

x

Backward

Backward

Forward

Forward

Lens

Photomask on reticle stage

Light slit

Wafer on wafer stage

(De)acceleration

Figure 2: Step and scan procedure.

The exposed substrate area on which the photomask pattern is imaged is called die.

During the scanning process of a die, the stages move at constant speed in y-direction.

After exposing a die to the light source, the wafer stage moves in x-direction to the next

projection location. In the process of scanning a pattern onto the substrate, there are

several factors influencing the performance of the process. These performance factors

will be discussed in the next section.

Lithography performance

The performance of lithographic processes is measured with several performance param-

eters. These parameters are introduced in section Parameters. Also, the performance

of the lithographic process is bounded by several factors which are discussed in the

section Limiting factors.

Parameters

To produce IC’s at a minimum cost and optimize their performance, lithographic equip-

ment is continuously improved. In general, performance of lithographic equipment is

often characterized by the following four parameters [2] (see also figure 3) :

• Throughput, number of wafers processed per hour.

• Overlay, inplane measure for repeated positioning error. Overlay is the average

horizontal alignment error between subsequent layers, produced by several litho-

3


graphic steps performed on one substrate. The 3σ variation around the overlay

value is called fading.

• Yield, percentage of produced devices which function correctly.

• Depth of focus, or DOF. Out of plane measure for error between focal plane of

the lens and substrate position. If the substrate is out of the focal plane of the

lens, the aerial image becomes ’blurred’ at wafer level. Two important parameters

determining DOF are:

– Critical dimension, or CD. Average line width, defined as CD = k1λ

NA [3],

where k1 is a process dependent constant, λ the imaging wavelength and

NA the numerical aperture. Numerical aperture is a dimensional number

that characterizes the range of angles over which the double telecentric lens

can emit light.

– Critical dimension uniformity, or CDU. CDU is defined as the 3σ variation

around the CD value [4].

2nd layer

1st layer

substrate

CD

overlay

CDU

Figure 3: Lithographic performance parameters.

Improvement on depth of focus results in smaller and more accurate features on sub-

strates, enabling the production of smaller and more efficient IC’s. Several performance

limiting factors can be found which bound the DOF value. These performance limiting

factors are discussed in the next section.

Limiting factors

Depth of focus is dependent on several factors. The improvement of DOF is obtained

by improvement on these factors. The main performance limiting factors are:

• wafer flatness,

4

PHOTOMASK CURVATURE MANIPULATION

• photomask flatness,

• lens element uncertainties,

• stage positioning errors in z-direction,

• calibration errors in machine elements,

• photomask pattern deviations.

All contributing factors in DOF limitations are bundled in a budget specification (also

known as focus budget). The drive to improve on DOF constantly requires the budget to

be lowered, as specified by the International Technology Roadmap for Semiconductors

[5]. Improvement on wafer flatness is one of the most influential factors which can cause

significant budget reduction [6]. One way to improve wafer flatness is by introducing

new wafer polishing techniques. Another method is to measure wafer topology and

account for image to wafer plane nonconformities during the lithography process. This

thesis focuses on the second option and proposes a new methodology for image to wafer

plane nonconformity correction called photomask curvature manipulation.

Photomask curvature manipulation

Due to the continuous drive to manufacture smaller features, wafer unflatness is be-

coming a performance limiting factor. State of the art image to wafer nonconformity

corrections are no longer able to meet imaging requirements. The conventional correc-

tion strategy (also known as leveling) positions the wafer in the optimal focal plane

of the lens using translational and rotational setpoints applied to the wafer stage. To

extend the conventional correction methods and further reduce influences of image to

wafer plane nonconformities on focus budgets, augmentation of the conventional level-

ing strategy with a curvature correction is proposed (see figure 4). This field curvature

correction can be achieved by bending the photomask, resulting in a curved aerial im-

age proportional to the applied bending moment. The wafer is subsequently positioned

in the optimal focal plane of the lens with respect to translation and rotation whilst

being exposed by the curved aerial image.

Although a curvature correction can be achieved by application of a bending mo-

ment on the photomask, as will be shown in chapter 1, realization of such a correction

mechanism inside lithography systems does have a number of issues that need to be

5


Conventional leveling Field curvature augmented leveling

Photomask unflatnessPhotomask unflatness

Lens systemLens system

Aerial image Aerial image

Wafer unflatnessWafer unflatness

Unconformity

Aerial image exposureAerial image exposure

Substrate topologySubstrate topology

(a) (b)

Figure 4: Difference in leveling strategies with (a) Conventional leveling and (b) Field

curvature augmented leveling.

solved. Considering the mechatronic system overview of the curvature manipulation

process in figure 5, the following must be investigated to achieve a curvature correction

mechanism:

• Specification derivation (r in figure 5): analysis of wafer unflatness and identifi-

cation of the required amount of curvature correction.

• Mechatronic design (all building blocks in figure 5): design of a photomask bend-

ing actuator, able to produce a bending moment on a photomask while respecting

boundary conditions.

• Performance evaluation (y in figure 5): analysis of focus budget improvement

with the designed system.

This thesis will focus on the first and third fields of research. The specific research

question is presented in the next section.

Assignment definition

As mentioned in the previous section, field curvature augmented leveling provides the

possibility to reduce influences of image to wafer plane nonconformities on focus bud-

6

ASSIGNMENT DEFINITION

r R

Cff

Cfb A P y

dmS

dp

++

++

+−

R: input shaping

Cff : feedforward controller

Cfb: feedback controller

A: actuator

P : mechanical plant

S: sensor

r: reference signal

dp: plant disturbance

dm: measurement noise

y: plant output

Figure 5: Mechatronic system overview of the curvature manipulation process.

gets. This thesis concentrates on the first and third research topics stated in the

previous section which has led to the following thesis assignment:

Dynamic system analysis of field curvature correction performance in

lithography systems using curvature manipulation of an optical element.

Considering figure 5, the first and third research topic stated at the previous sec-

tion concentrate on the system analysis of the mechanical plant P with its input r

and output y. The mechanical plant is a distributed parameter system and the geo-

metrical complexity complicates the use of standard lumped mass models. Therefore

this research requires the usage of numerical Finite Element Models (FEM) to obtain

valuable results.

Performance evaluation requires the knowledge of the amount of correctable curva-

ture determined in the identification procedure. Identifying the amount of correctable

field curvature in state of the art wafers is dependent on several factors:

• Wafer topology measurements: field topology identifying measurements where

the amount of field curvature is obtained.

• Fitting functions: field curvature setpoints obtained using fitting functions.

7


• Boundary conditions: field curvature is corrected by shaping the photomask and

is bounded by mechanical and geometrical specifications.

The identified correctable curvature values serve as input setpoints (r in figure 5) for

a numerical model of the mechanical plant from which curvature tracking performance

can be obtained. Modeling the curvature tracking has dependency on several factors:

• Model geometry knowledge.

• Model accuracy knowledge, provided for by experimental modal analysis.

• Model size efficiency. Modeling of the physical photomask bending actuator re-

quires model reduction techniques in order to keep computational efficiency and

numerical accuracy within limits.

• Field curvature calculation. The tracking performance is obtained from realized

curvature, gained using fitting functions which generate fit versus model topology

errors.

Thesis outline

This thesis consists of three parts. Part I discusses the general theory applied through-

out this thesis. Part II covers the analysis methodology of obtaining photomask cur-

vature for both identification as well as modeling purposes. In Part III the developed

analysis methodology is applied in practice. The different parts are arranged as follows:

Part I

• Chapter 1 discusses the modeling of photomask curvature using plate theory and

obtaining field curvature using fitting functions.

• Chapter 2 gives an introduction into model reduction techniques and dynamic

substructuring.

• Chapter 3 discusses an additional model reduction technique in which dynamic

model energy characteristics are the foundation of the reduction technique. The

study on model energy characteristics is called Gramian theory .

Part II

8

THESIS OUTLINE

• Chapter 4 discusses the curvature identification and tracking procedure. This

chapter sets out the steps to perform in acquiring focus budget improvements

using field curvature correcting augmented leveling strategies.

• Chapter 5 covers the identification of correctable field curvature, serving as a

study on required field curvature corrections as well as an identification procedure

of curvature setpoint inputs for the numerical model.

Part III

• Chapter 6 sets out the numerical modeling of the physical bending actuator as-

sembly and the experimental validation procedure of those models. Finally the

reduced order model is presented.

• Chapter 7 discusses the analysis on the model focus improvement study. First

a methodology is discussed for obtaining realized focus improvement considering

the reduced model. Field curvature setpoints of the analysis are provided for by

the study on correctable field curvature in chapter 5.

• Chapter 8 finalizes the thesis with conclusions and recommendations.

9


10

Part I

General Theory

11

Chapter 1

Photomask Curvature modeling

Lithography focus budget improvements can be obtained by applying a correction on

the curvature of the substrate topology. As pointed out in the introduction, this correc-

tion is achieved by manipulation of the photomask, shaping the aerial image projected

on the substrate. The photomask can be considered as a plate structure and curvature

is achieved by applying bending moments along the edges, as will be explained in this

chapter. The obtained mechanical deformation of the photomask after bending is sub-

sequently used in the formulation of a fitting function. This fitting function is then used

to derive setpoints for a photomask bending actuator from measured substrate topolo-

gies. This chapter introduces mechanical modeling of plate structures using Kirchoff’s

assumptions and Green strain expressions in section 1.1. Section 1.2 uses the plate

theory to formulate a curvature fitting function suited for curvature determination of

a bended photomask.

1.1 Kirchoff plate theory

Consider a plate like structure defined in a Cartesian coordinate system (x, y, z). The

deformation field can be described as:

u(x, y, z)v(x, y, z)w(x, y, z).

(1.1)

Figure 1.1 displays a half cross section of the plate with its inertia axis equal to the

principal axis. The plate is deformed by a distributed load p and a bending moment

M . For plate structures, the following kinematic assumptions can be applied [7]:

• The material is linear elastic, homogeneous and isotropic [8];

13

CHAPTER 1. PHOTOMASK CURVATURE MODELING Company-secret

12h

z, w

x, u

u0

u1

M

px

−12h

θ w

Figure 1.1: The cross section of a plate subjected to a distributed load P and bending

moment M.

• Cross sections remain straight, unstretched (εzz = 0) and orthogonal to the mid-

plane of the beam;

• No shear deflection (γxz = γyz = 0).

These assumptions hold for the photomask because the thickness is small compared to

the planar dimensions. The ratio of the typical in-plane dimension to the thickness,

which is 24, places the photomask in the group called thin plates, see also [9]. The

material assumptions hold for the photomask model since it is made from fused-silica,

which is a homogeneous and isotropic material. The above assumptions are founded

by the classical or Kirchhoff plate theory [9]. From the assumptions it follows directly

that:

θ = ∂w∂x , u1 = −z ∂w

∂x , w = w, (1.2)

with θ the angle between reference and deformed configuration, u1(x, y) the displace-

ment in x-direction due to the moment M and w the displacement in z-direction due to

M . Equation 1.2 holds for displacements in both the x and y directions. The displace-

ment field for three dimensional plates can therefore be described using (1.2). Indeed,

the displacements due to the distributed load p are equal to:

u(x, y, z) = u0(x, y)− z∂w(x, y)

∂x(1.3)

v(x, y, z) = v0(x, y)− z∂w(x, y)

∂y(1.4)

w(x, y) = w(x, y), (1.5)

with u0(x, y) and v0(x, y) the extensions in the x− and y−direction due to the dis-

tributed load p. Stress and strain relations can be formulated using the above defined

14

1.1. KIRCHOFF PLATE THEORY

displacement field. These relations are presented in the next section. In the remain-

der of this chapter, coordinate dependencies in equations are omitted for the sake of

simplicity.

1.1.1 Stress and strain

The derivation of plate curvature relations requires knowledge about plate stresses and

strains due to the applied loads. Strain relations for the plate, or any mechanical

structure, are derived using Green’s strain relations. Indeed, considering Green’s strain

formulation [7]:

εij =1

2

(∂ui∂xj

+∂uj∂xi

+∂um∂xi

∂um∂xj

). (1.6)

In non-linear analysis of structures subjected to bending, often the Green strain tensor

is submitted to the large displacement approach by taking into account large displace-

ments by rotation of the system while keeping an assumption of small strains. When

in plane deformations of the plate remain small it holds that:

∂u

∂x<

∂w

∂x,∂u

∂y<

∂w

∂y(1.7)

∂v

∂x<

∂w

∂x,∂v

∂y<

∂w

∂y(1.8)

∂u

∂x� 1,

∂u

∂y� 1 (1.9)

∂v

∂x� 1,

∂v

∂y� 1 (1.10)

Using the large displacement approach, equations (1.7)-(1.10), for the following strain

components of the plate are obtained:

εxx =1

2

(∂u

∂x+

∂u

∂x+

(∂u

∂x

)2

+

(∂v

∂x

)2

+

(∂w

∂x

)2)

=∂u

∂x+

1

2

(∂w

∂x

)2

(1.11)

εyy =1

2

(∂v

∂y+

∂v

∂y+

(∂u

∂y

)2

+

(∂v

∂y

)2

+

(∂w

∂y

)2)

=∂v

∂y+

1

2

(∂w

∂y

)2

(1.12)

εzz =1

2

(∂w

∂z+

∂w

∂z+

(∂u

∂z

)2

+

(∂v

∂z

)2

+

(∂w

∂z

)2)

(1.13)

15


γxy = γyx = 2εxy =∂u

∂y+

∂v

∂x+

∂u

∂x

∂u

∂y+

∂v

∂x

∂v

∂y+

∂w

∂x

∂w

∂y

=∂u

∂y+

∂v

∂x+

∂w

∂x

∂w

∂y(1.14)

γxz = γzx = 2εxz =∂u

∂z+

∂w

∂x+

∂u

∂x

∂u

∂z+

∂v

∂x

∂v

∂z+

∂w

∂x

∂w

∂z

=∂x

∂z+

∂w

∂x(1.15)

γyz = γzy = 2εyz =∂v

∂z+

∂w

∂y+

∂u

∂y

∂u

∂z+

∂v

∂y

∂v

∂z+

∂w

∂y

∂w

∂z

=∂v

∂z+

∂w

∂y. (1.16)

The strain expressions for a thin plate are found by substituting the displacement ex-

pressions for a plate (1.3)-(1.5) into equations (1.11)-(1.16) and applying the kinematic

assumptions:

εxx =∂u0∂x

− z∂2w

∂x2+

1

2

(∂w

∂x

)2

(1.17)

εyy =∂v0∂y

− z∂2w

∂y2+

1

2

(∂w

∂y

)2

(1.18)

εzz = 0 (1.19)

γxy =∂u0∂y

− 2z∂2w

∂x∂y+

∂v0∂x

+1

2

∂w

∂x

∂w

∂y(1.20)

(1.21)

γxz =∂u

∂z+

∂w

∂x=

∂u0∂z

− ∂w

∂x+

∂w

∂x= 0 (1.22)

γyz =∂v

∂z+

∂w

∂y=

∂v0∂z

− ∂w

∂y+

∂w

∂y= 0. (1.23)

Leaving out the zero terms, the strain vector becomes:

ε =[εxx εyy γxy

]T. (1.24)

The relation between stress and strain is described by Hooke’s law and reads [10]:

σ = Cε, (1.25)

where σ indicates the stress vector and C the stiffness tensor. The stress vector and

stiffness tensor are defined by:

σ =

⎡⎣ σxx

σyyτxy

⎤⎦ C = E

1−ν2

⎡⎣ 1 ν 0

ν 1 00 0 1−ν

2

⎤⎦, (1.26)

16


where ν represent the Poisson ratio. Having introduced the relations for the stresses

and strains, it is possible to use these relations to derive the induced stresses and strains

due to a distributed load an bending moment. The next section discusses the resulting

stresses and strains due to bending moment application.

1.1.2 Curvature

In the previous section, the stress relations were derived. Let us now focus on the

stresses that are caused by bending moment application as is shown in figure 1.2.

Relations for the bending moments are obtained by integration of these stresses over

x

z

y

σyy

τxy

σxxτxy

Mxx Mxy

Mxy

Myy

−12h

12h

b

l

Figure 1.2: Plate stresses induced by the applied bending moments.

the thickness and length of the plate (assuming a square plate), i.e.:

Mxx =

b∫0

h2∫

−h2

zσxxdzdy (1.27)

Myy =

l∫0

h2∫

−h2

zσyydzdx (1.28)

17


Mxy =

b∫0

h2∫

−h2

zτxydzdy (1.29)

where b and l are the length of the plate in x and y direction respectively. The

components of ε that do not have dependencies on z, do not influence curvature defor-

mation within the plate. Therefore, inserting (1.25) into (1.27)-(1.29), only z dependent

components of ε results in

Mxx =

∫ b

0

∫ h2

−h2

z2E

1− ν2

(−∂2w

∂x2− ν

∂2w

∂y2

)dzdy (1.30)

Myy =

∫ l

0

∫ h2

−h2

z2E

1− ν2

(−ν

∂2w

∂x2− ∂2w

∂y2

)dzdx (1.31)

Mxy =

b∫0

h2∫

−h2

z2E

1− ν2(1− ν)

(− ∂2w

∂x∂y

)dzdy. (1.32)

Working out the integrals yields:

Mxx =Ebh3

12 (1− ν2)

(−∂2w

∂x2− ν

∂2w

∂y2

)(1.33)

Myy =Elh3

12 (1− ν2)

(−ν

∂2w

∂x2− ∂2w

∂y2

)(1.34)

Mxy =Elh3

12 (1− ν2)(1− ν)

(− ∂2w

∂x∂y

). (1.35)

Note that the terms concerning area moments of inertia can clearly be pointed out in

the equations. Furthermore, considering the definitions for curvature [7],

κxx = −∂2w

∂x2(1.36)

κyy = −∂2w

∂y2(1.37)

κxy = −2∂2w

∂x∂y, (1.38)

it is clear that bending moments result in plate curvature.

Let us now derive the plate deformation for the case when only the Mxx bending

moment is applied. Due to the bending moment, the structural deformation shows both

curvatures in x as well as y direction, as shown in figure 1.3. The relation between

these principal curvatures can be extracted by inserting equations (1.36)-(1.38) into

18


κyy y

x

z

κxx

Mxx

Mxx

Figure 1.3: Plate deformed by a uniform applied moment Mxx

(1.33)-(1.35) and applying the boundary conditions Mxx �= 0, Myy = 0 and Mxy = 0:

Ebh3

12 (1− ν2)(κxx + νκyy) = Mxx (1.39)

Elh3

12 (1− ν2)(νκxx + κyy) = 0 (1.40)

Elh3

12 (1− ν2)(1− ν)

(1

2κxy

)= 0. (1.41)

From (1.40) it follows directly that

κyy = −νκxx. (1.42)

Equation (1.42) shows that for a thin plate, on which the kinematic assumptions of

section 1.1 hold, the principle curvatures are related by the Poisson ratio. This phe-

nomenon is referred to as anticlastic curvature. The definition of anticlastic is ’having

opposite curvatures’. In case of a uniform bending moment Mxx along the edges of the

plate, a uniform curvature exists along the length and width.

1.1.3 Equations of motion

Having introduced the relationships for stress, strain and bending moments in the

previous section, it is possible to use these relations to formulate the equations of

motion of plate structures. Consider the equilibrium of an infinitesimal small plate

element, as presented in figure 1.4. The vertical translational equilibrium reads

mwdxdy = qdxdy − Vydx− Vxdy + (Vy + dVy) dx+ (Vx + dVx) dy, (1.43)

ordVy

dy+

dVx

dx−mw + q = 0. (1.44)

19


Myy

Vy Mxy

z

Mxy

Vx

Mxx

dy Vy + dVy

Mxy + dMxy

Myy + dMyy

y

q (x, y)

dx

Vx + dVx

Mxy + dMxyMxx + dMxx

x

Figure 1.4: Equilibrium of a plate element.

The moment equilibria around the x and y-axis are respectively

−dMxydy − dMydx+ Vydxdy +HOT = 0 (1.45)

dMxydx+ dMxdy − Vxdydx+HOT = 0, (1.46)

or

Vx ≈ ∂Mxy

∂y+

∂Mx

∂x(1.47)

Vy ≈ ∂Mxy

∂x+

∂My

∂y(1.48)

by assuming a first order approximation, i.e./ when dx and dy are going to zero. The

vertical translational equilibrium can be expressed by inserting (1.47) and (1.48) into

(1.44):

∂2Mx

∂x2+

∂2My

∂y2+ 2

∂2Mxy

∂x∂y−mw + q = 0. (1.49)

The expressions for the bending moments derived in section 1.1.2 can be substituted

into equation (1.49) leading to the differential equations for translational deformation

of plates. Normalizing per unit of length this differential equation reads:

Eh3

12 (1− ν2)

(∂4w

∂x4+

∂4w

∂y4+ 2

∂4w

∂x∂y

)+ mw − q = 0, (1.50)

orEh3

12 (1− v2)Δ2w + mw − q = 0, (1.51)

20


where Δ the Laplace operator and m and q the mass and external forces per unit of

length respectively.

Let us now assume the case when no inertia and external forces exist, i.e.:

w = 0 (1.52)

q = 0. (1.53)

The displacement field satisfying equation (1.51) then yields [9]:

w =1

2

(C1x

2 + C2y2). (1.54)

Substituting (1.54) into (1.36)-(1.38) shows that the integration constants are the cur-

vature expressions of a plate subjected to a bending moment Mxx:

C1 = κxx (1.55)

C2 = κyy. (1.56)

Substitution of the Poisson ratio relation (1.42) into (1.54) yields the description of the

displacement field:

w =1

2κxx(νy2 − x2

). (1.57)

It is easy to see that the plate deflection represents an anticlastic curved surface de-

pendent on the Poison ratio. It can easily be shown that this anticlastic curvature is

dependent on the applied bending moment Mxx and thickness of the plate. Let us

therefore derive the curvatures from equations (1.39)-(1.41):

Mxx = M = − Eh3

12 (1− ν2)(κxx + νκyy) (1.58)

Myy = 0 = − Eh3

12 (1− ν2)(νκxx + κyy) (1.59)

Mxy = 0, (1.60)

Working out (1.58) and (1.59) for κxx and κyy yields:

κxx = −12M

Eh3(1.61)

κyy =12νM

Eh3. (1.62)

Indeed, equation (1.42) is automatically satisfied since:

κyy = −∂2w

∂y2= −12νM

Eh3= −νκxx. (1.63)

This section introduced the fundamental plate relations. The next section continues

with the derived relations, where they are used in a fitting function for the identification

of substrate curvature values, which is one of the objectives of this thesis.

21


1.2 Curvature fitting functions

In the introduction of this thesis, the importance of focus budget improvement mod-

eling was discussed. In order to identify focus budget improvement, knowledge of the

amount of correctable curvature in state of the art wafer topologies is required. Because

these curvature corrections are achieved by manipulation of photomask curvature, it is

possible to use the curvature relations derived in the previous section as the basis of a

curvature fitting function for wafer topologies. The derived curvature setpoints from

the fitting function will serve as reference signals for the mechatronic design (r in figure

5).

For the derivation of curvature setpoints, wafer topology measurement data is used.

This is data which is obtained from substrates of the lithographic process inside litho-

graphic machines. These substrates, or wafers, have a certain flatness due to the

manufacturing process and static deflection due to their constraints. The topology

measurement is obtained by optical level sensors, scanning the surface of the wafer.

The fitting algorithm is then used to derive the leveling strategy setpoints from the

measurement data. These leveling setpoints are used to position the wafer in the opti-

mal focal plane of the lens by adapting the following parameters:

• wafer stage:

– translations,

– rotations, both over the x and y axis,

• photomask:

– curvature setpoints.

These values change for every position of the scanning slit on the wafer.

Besides the determination of curvature setpoints the fitting algorithm is used in

other parts of this thesis, namely:

• Focus budget improvement for the case when ideal curvature setpoints are real-

ized,

• performance evaluation of mechatronic system with respect to curvature setpoint

tracking.

22

1.2. CURVATURE FITTING FUNCTIONS

Section 1.2.1 describes the fitting function for the determination of leveling strategy

setpoints. This function is also known as the anticlastic curvature augmented fitting

function. Section 1.2.2 deals with the extraction of the model coefficients in case of an

overdetermined system description.

1.2.1 Anticlastic curvature augmented fitting function

Photomask manipulation is used to correct for image to wafer nonconformities, there-

fore a fitting function is needed to describe the required setpoints for correction of

these deviations. The fitting function must on the one hand satisfy the mechanics of

a photomask and on the other hand find a suitable solution to an overdetermined set

of discrete measurement datapoints, obtained from the wafer field topology measure-

ments. Before defining the anticlastic curvature augmented fitting function, a global

reference system is defined. All datapoints from the level sensor are described with

respect to this reference frame. The global reference system is defined as displayed in

figure 1.5. As mentioned in the introduction, the photomask is subjected to a light

z

x

y (scan direction)

Die

Slit

Wafer

Slit center position xs, ys

o

x

yq

z

Topology

Figure 1.5: Global reference system during scanning step at time t.

source in a scanning motion (see also figure 2). This light slit is projected onto the

substrate die field through the double telecentric lens. During a scanning period, the

light slit passes over the die. At each time instant, the field topology of the slit posi-

tion area can be approximated by the anticlastic curvature augmented fitting function.

23


Figure 1.5 displays the slitposition at a time instant t on wafer and the corresponding

measured field topology by the level sensor with respect to the global reference system.

As can be seen in the figure, the global reference system x lies in the center of the wafer.

Inside the global reference system, the field topology of the slit position is described

with respect to the local reference system x, which is positioned at the center of the

slit position (xs, ys) and has an offset oz. The local reference system is translated and

rotated with respect to the global coordinate system.

In order to obtain a fitting function that includes curvature, tilt and offset variables

and also satisfies the photomask physics, equation (1.57) is used. Using (1.57) an arbi-

trary point q, figure 1.5, can be described in global coordinates. Using the offset o(z)

point q is a superposition of global coordinates of the local frame and the local coordi-

nates of q transformed back to the global frame. In vector notation this superposition

reads:

q(x) = o(x) + q(x) + r (xs, ys)× q(x), (1.64)

where o (x) is the origin of the local system described in global coordinates and r

is the rotation vector describing the rotations around the principal axes of the local

system. Assuming that the rotation around the z-axis is constrained, this rotation

vector becomes [11]:

r (xs, ys) =[Rx (xs, ys) Ry (xs, ys) 0

]T. (1.65)

Note that the origin of the local system in the global space is in fact the slit position

with an offset o(z). Considering figure 1.5, equation (1.65) and the physical description

of plate bending, equation (1.57), (1.64) can be written as

⎡⎣ qx

qyqz

⎤⎦ =

⎡⎣ ox

oyoz

⎤⎦+

⎡⎢⎣

qxqy

12κxx

(νq2y − q2x

)⎤⎥⎦+

⎡⎣ Rx

Ry

0

⎤⎦×

⎡⎢⎣

qxqy

12κxx

(νq2y − q2x

)⎤⎥⎦ . (1.66)

Assuming Rx · 12κxx

(νq2y − q2x

)and Ry · 1

2κxx

(νq2y − q2x

)to be small with respect to

qx and qy, it follows directly from equation (1.66) that:[qxqy

]≈[oxoy

]+

[qxqy

](1.67)

The height of the arbitrary point q can now be described using equations (1.66) and

(1.67):

qz (x, y) = oz (xs, ys)−Ry(xs, ys)qx +Rx(xs, ys)qy +1

2κxx(xs, ys)

(νq2y − q2x

). (1.68)

24

1.2. CURVATURE FITTING FUNCTIONS

Equation (1.68) is also referred to as the anticlastic curvature augmented fitting func-

tion.

If all the discrete measurement points of a slit position field are considered, the field

can be described using (1.68). Section 1.2.2 discusses the extraction of the anticlastic

curvature augmented fitting function parameters for a set of discrete measurement

points within the exposed part of the die using a least squares approximation.

1.2.2 Coefficient extraction

When the fitting function derived in the previous section is used to describe a field

of discrete measurement points, the solution becomes an approximation when there

are more measurement points than parameters to be identified. The introduced error

between model and measurement points can, for any point q, be described as:

e (x, y) =

(oz (xs, ys)−Ry (xs, ys) qx +Rx (xs, ys) qy +

1

2κxx (xs, ys)

(νq2y − q2x

))−h (x, y) ,

(1.69)

with topology data h (x, y), local frame x, global reference frame x and slit position

(xs, ys)1. Equation (1.69) represents the focus error between the anticlastic curvature

augmented fitting function and field topology. To find the optimal fit between the

measured data and the fitting model, i.e. the fit with the smallest focus error, the

anticlastic curvature augmented fitting function (1.68) is rewritten into

Xα ≈ h. (1.70)

Considering an equation space of Rm×n where m represents the amount of equations

and n the amount of parameters, the components of (1.70) for each slit position read

(m > n): ⎡⎢⎢⎢⎢⎢⎣

1 −x1 y112

(νy21 − x21

)1 −x2 y2

12

(νy22 − x22

)1 −x3 y3

12

(νy23 − x23

)...

......

...1 −xm ym

12

(νy2m − x2m

)

⎤⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎣

ozRy

Rx

κxx

⎤⎥⎥⎦ ≈

⎡⎢⎢⎢⎢⎢⎣

h1h2h3...

hm

⎤⎥⎥⎥⎥⎥⎦ (1.71)

1The offset, rotational and curvature parameters are identified at wafer level. Due to properties of

the double telecentric lens system, curvature at wafer level is equal to curvature at reticle level.

25


The optimal solution to the overdetermined problem, (1.71), is found using the linear

least squares methodology (see appendix B):

α = minα∈R

‖Xα− h‖2

= minα∈R

m∑j=1

e2j (x, y) , (1.72)

which is equal to

α =(XTX

)−1XTh. (1.73)

In summary, determining the parameters for the anticlastic curvature augmented

fitting function requires the use of linear least squares solving. The solution is a repre-

sentation of the most optimal fit in terms of translation, rotation and curvature of the

field topology. The values are used as leveling strategy setpoints and can be used for

analysis regarding focus improvement potential.

26

Chapter 2

Model reduction techniques

Advanced modeling techniques as Finite Elements are the most common analysis tools

in mechanical engineering. Although very accurate models can be developed with this

technology, analysis speed is limited by the state of the art computational power. Per-

forming a dynamic system analysis on field curvature correction performance requires

a model of the mechanical plant, as discussed in the introduction (see figure 5). To

make quick iterations during the analysis there is a demand for ’simple’ models, such

as lumped mass models. Though due to the geometric complexity and the distributed

parameter nature of the plant, the use of lumped mass models for the analysis is com-

plicated. Therefore this research requires the use of Finite Element Models (FEM) to

model the mechanical system.

In case of photomask curvature manipulation, complex designs are modeled with

FE, leading to large numerical models. The required accuracy of the dynamic system

analysis would cause a demand for large computational power which causes analysis

iterations to have slow progress. A proposed solution to preserve analysis accuracy but

reduce the computational effort of mechanical models is component model reduction.

Component reduction methodology uses the properties of discrete and linear struc-

tural models to reduce the number of generalized coordinates while retaining nodal

descriptions of the system.

Section 2.1 discusses the properties of discrete and linear structural dynamic prob-

lems used in component model reduction. Section 2.2 treats three different methodolo-

gies in component model reduction, Guyan’s reduction, modal truncation and Craig-

Bampton mode component synthesis, followed by a section treating reduction of com-

ponent interfaces. Finally, section 2.3 sets out the theory for dynamic substructuring,

27

CHAPTER 2. MODEL REDUCTION TECHNIQUES Company-secret

used to couple multiple reduced order system components to obtain the dynamic de-

scription of a system consisting of multiple subsystems.

2.1 Dynamics of undamped discretized structures

A discrete undamped system of N generalized coordinates, or Degrees of Freedom

(DoF), is represented by

Mq(t) +Kq(t) = p(t) q(t) =

⎡⎢⎣

q1(t)...

qN (t)

⎤⎥⎦. (2.1)

Equation (2.1) is used to describe the dynamic properties of the system, with vector q(t)

holding the displacements of the DoF and q(t) their accelerations. Vector p(t) contains

the externally applied loads. The square matrices M and K are respectively the mass

and stiffness matrices of size N × N respectively. The mass matrix is by definition

symmetric and positive definite, whilst the stiffness matrix is symmetric and positive

definite in the absence of rigid body modes and stable systems [12]. Stability requires

the potential energy of the system to have a minimum at an equilibrium position.

Therefore in the vicinity of that equilibrium, displacements remain small and create

the possibility to regard the system as linear, having a positive definite stiffness matrix.

Dynamic properties of the system can be obtained in the case when no external

forces are applied and are found in states where the system response is harmonic over

time. This harmonic (or particular) solution of the system describes the response of

the generalized coordinates using one common scale factor, or

q(t) = φx (t) , (2.2)

where φ represents the scale factor and x (t) a unique time function. The scale factor is

independent on time and only describes the ratio between the generalized coordinates.

This ratio is called eigenmode or eigenshape. Inserting (2.2) into (2.1) leads to:

x (t)Mφ+ x (t)Kφ = 0

− x (t)

x (t)Mφ = Kφ

−s2Mφ = Kφ. (2.3)

28

2.1. DYNAMICS OF UNDAMPED DISCRETIZED STRUCTURES

Because x2 is a real and negative quantity, x can be replaced with an imaginary quantity

jω (where j is the imaginary unit and ω is a real number) resulting in:

Kφ − ω2Mφ = 0(K − ω2M

)φ = 0, (2.4)

Since the system is described by N equations and the mass and stiffness matrix are

by construction nonsingular, there are N independent non-trivial solutions (x �= 0) to

equation (2.4). This leads to:

(K − ω2

nM)φ(n) = 0 n = 1, 2, .., N (2.5)

Each solution holds an ωn, or eigenfrequency , and a particular ’shape’ or eigenmode

φ(n). An under-constrained system also inhabits Rigid Body Modes. The mathematical

result of rigid body modes is that the stiffness matrix K becomes singular, resulting

in eigenfrequencies of zero[rads

]. Therefore these modes are not harmonic and de-

scribe rigid body displacements of the system. Rigid body modes thus describe the

relationship

Kφ(m) = 0m = 1, 2, ..,M

M < N(2.6)

An under-constrained system thus has M rigid body modes and N flexible body modes.

A very useful property of discrete system with light or no damping and distinct

eigenfrequencies is mode orthogonality [12]. Mode orthogonality provides that different

modes of a system are ’M -’ and ’K-orthogonal’. For example, consider two eigenmodes

denoting φ(r) and φ(s) associated with eigenfrequencies ωr and ωs such that ωr �= ωs.

The orthogonality relationships between the eigenmodes with distinct eigenfrequencies,

with respect to the mass and stiffness matrices, read:

φ(s)TMφ(r) = 0 (2.7)

and

φ(s)TKφ(r) = 0, (2.8)

implying that the work produced by inertia and elastic forces of mode r on a displace-

ment described by mode s is zero. The orthogonality relationships provide a basis for

modal superposition. Modal superposition uses mode orthogonality to describe any

dynamic response of a system with a complete and orthogonal basis. This description

is the starting point of reduction techniques as modal truncation and Craig-Bampton

mode component synthesis which are introduced in the next section.

29


2.2 Component model reduction for structural problems

In the introduction it was pointed out that model reduction is a useful tool in describ-

ing a model with less DoF while maintaining analysis accuracy. The complexity of

the photomask manipulation assembly requires the use of a very fine meshing in the

FE model. Dynamic analysis of such large sets of DoF places high demands on com-

putational power. The approach in this section is not only to reduce the DoF of the

photomask manipulation design but also treat the assembly parts as individual compo-

nents (or substructures) while reducing them. After reduction of the components, an

assembled model is built using dynamic substructuring (DS), discussed in section 2.3.

The advantage of reducing individual components of the photomask manipulation as-

sembly is that they demand less computational efficiency. Furthermore during dynamic

analysis of the assembly it is easier to determine the contributions of each component

in the response.

The basic principle of reduction techniques is to replace the DoF by a smaller set of

generalized coordinates representing amplitudes of possible displacement modes. Figure

2.1 represents two general substructures acted upon by external and interface forces.

The linear and undamped equations of motion of substructure (s) read (see also section

p(I)

g(II)1g

(I)2

I

g(I)1

II

p(II)

Figure 2.1: Two general substructures with external and interface forces.

2.1):

M (s)q(s)(t) +K(s)q(s)(t) = p(s)(t) + g(s)(t), (2.9)

where p(s)(t) and g(s)(t) are the vectors representing the time dependent external and

interface forces for each substructure respectively. Reduction techniques create a new

representation of the system by replacing the DoF q(s) (size N) with a smaller set of

generalized coordinates using

q(s)(t) = T (s)q(s)

(t) + r(s) (t) , (2.10)

30

2.2. COMPONENT MODEL REDUCTION FOR STRUCTURAL PROBLEMS

where T (s) is a reduction matrix (size N × D,D < N), q a set of reduced degrees of

freedom and r(s) (t) a residual load vector satisfying the dynamic equations. Assuming

r(s) (t) to be negligible and inserting (2.10) into (2.9) with premultiplication of T (s)T

generates a dynamic system description for the reduced basis q:

M (s) ¨q(s)

(t) + K(s)q(s)(t) = p(s)(t) + g(s)(t), (2.11)

where the matrices M (s) and K(s) are the reduced mass and stiffness matrices

M (s) = T (s)TM (s)T (s) (2.12)

K(s) = T (s)TK(s)T (s) (2.13)

and where p(s)(t) and g(s)(t) are the reduced external and interface loads

p(s)(t) = T (s)Tp(s)(t) (2.14)

g(s)(t) = T (s)Tg(s)(t) (2.15)

respectively. Note that equation (2.11) is an approximation. Premultiplication with

the reduction matrix and subsequent formulation of equation (2.11) will therefore only

be allowed when assuming that the residual load produces no work on the reduction

space stored in T (s), ie. T (s)Tr(s)(t) = 0.

The next section starts with an introduction to component model reduction and

the methodology of a static reduction method called Guyan’s reduction.

2.2.1 Guyan’s reduction

The most basic reduction technique is the static condensation of systems, known as

Guyan-Irons reduction, or simply Guyan reduction [13, 12]. If the governing undamped

dynamic equations of a general substructure (s) (figure 2.1) are represented by equation

(2.9), a distinction can be made between internal DoF (q(s)i ) and interface DoF (q

(s)f )

on which external and interface forces operate. In the remainder of this thesis, the

subscript indicating the substructure number and the explicit time dependency of the

DoF and forces is omitted for ease of notation. Using the distinction in DoF results in[Mff Mfi

Mif Mii

] [qfqi

]+

[Kff Kfi

Kif Kii

] [qfqi

]=

[p0

]+

[g0

]. (2.16)

The response of the internal DoF can be divided into an inertial part and an elastic

part. Neglecting the inertia forces in the response of the internal DoF gives:

Kifqf +Kiiqi = 0

qi = −K−1ii Kifqf . (2.17)

31


Equation (2.17) is used by the Guyan reduction as reduction basis since −K−1ii Kif

holds static modes of the internal degrees of freedom. The internal DoF therefore can

be condensed on the interface DoF using[qfqi

]=

[I

−K−1ii Kif

]qf (2.18)

q = T q (2.19)

where T is the reduction matrix, q the original generalized coordinates and q the

reduced set of DoF. The system description on the reduced basis is created by apply-

ing the reduction matrix in equations (2.11), (2.12) and (2.14) after rearranging the

generalized coordinates to the form:

q =

[qfqi

](2.20)

The reduced system description is exact in case of static problems but since the

inertia contributions of the internal DoF are neglected, dynamic problems will result

in an approximate solution. This solution holds its validity when the highest squared

eigenfrequency of interest is much lower than the first squared eigenfrequency of the

internal system description, i.e. the system when all interface DoF are constrained.

The approximate solution of the reduced system will always show overestimation of the

eigenfrequencies because the introduced restrictions of the reduction results in a stiffer

system.

2.2.2 Mode displacement method

Orthogonality of eigenmodes is a very useful tool in dynamic system analysis. The N

eigenmodes of a N DoF light- or undamped system as described by equation (2.9) are

able to exactly describe any solution with an orthogonal set of equations. Therefore,

any vector of size N can be described as a linear combination of the modes of the

system, or

q =N∑

n=1

ηn (t)φn, (2.21)

where ηn (t) are time dependent modal amplitudes and φn the modes of the system.

This approach is also known as modal superposition [14]. Replacing q in equation (2.9)

and premultiplying by each eigenmode φ(r)T (r = 1, 2, . . . , N) in order to project the

solution onto the modal directions leads to:

φ(r)T

(M

N∑n=1

ηn (t)φ(n) +K

N∑n=1

ηn (t)φ(n)

)= φ(r)Tp+ φ(r)Tg. (2.22)

32


Due to orthogonality of the eigenmodes, equation (2.22) provides N uncoupled equa-

tions, or normal equations:

μnηn (t) + γnηn (t) = φ(n)Tp+ φ(n)Tg n = 1, . . . , N (2.23)

ηn (t) + ω2nηn (t) = pn + gn, (2.24)

with modal mass μn, modal stiffness γn, eigenfrequency ωn and external and interface

loads pn (t) and gn (t) defined as:

μn = φ(n)TMφ(n)

γn = φ(n)TKφ(n)

ω2n =

γnμn

pn =φ(n)Tp

μn

gn =φ(n)Tg

μn.

(2.25)

The mode displacement method uses modal superposition to describe a dynamic

system in uncoupled equations but reduces the number of modal coordinates by trun-

cating the set of orthogonal modes describing the system. For instance, the first R

modes can be used to describe the dynamic response of a system. In case of the mode

displacement method, equation (2.21) becomes

q ≈R∑

n=1

ηn (t)φ(n) R < N. (2.26)

It can be shown that the accuracy of the approximation is dependent on both the spatial

representation accuracy of the basis as well as the frequency content of the excitation

and the eigenspectrum of the system [12]. Therefore, the modal truncation should be

implemented with care.

2.2.3 Craig-Bampton mode component synthesis

The Craig-Bampton reduction method [12, 15] is an extension of the Guyan reduction.

The dynamic response of a system, divided into its internal and interface components as

introduced by equation (2.16) can fully be described using the static modes of forces on

the interfaces and the internal vibration modes of the interface fixed system. Recalling

33


the static condensation of the Guyan reduction in section 2.2.1, the internal degrees of

freedom are condensed into the interface DoF using the static modes (S):

qi = −K−1ii Kifqf (2.27)

= Sqf (2.28)

The internal vibration modes are found by considering the modes of the interface fixed

system. The dynamics of interior DoF can then be described using the static modes S

and the matrix holding the mass normalized interface fixed vibration modes Φi1:

qi = Sqf +Φiηi, (2.29)

where ηi are the modal amplitudes of the interface fixed modes. To create a reduction

basis, the interface fixed vibration modes are truncated to a degree suiting for the anal-

ysis type. Note that the accuracy criteria for the reduction basis is dependent on spatial

and spectral parameters of the system and the spectral parameters of the excitation

force. The reduction matrix T is formulated using the newly formed description of the

interface DoF:

[qfqi

]=

[I 0S Φi

] [qfηi

]= T q (2.30)

The system description on the reduced basis is created by applying the reduction matrix

in equations (2.11), (2.12) and (2.14) after rearranging the generalized coordinates to

the form:

q =

[qfqi

](2.31)

In case of static analysis, the Craig Bampton is exact since the static representation

of all interior DoF is condensed into the interface DoF. If the analysis is dynamic

however, the reduced system solution will become an approximation. The accuracy of

the approximation with respect to the exact solution is determined by the truncation

level of the interface fixed modal representation of the interior DoF.

1Φi is set such that ΦTi MiiΦi = I and ΦT

i KiiΦi = diag(ω2i,1, . . . , ω

2i,N

)= Ωi,N

34


2.2.4 Reduction by rigid interface projection

The flexible interface responses of substructures can often be neglected when an in-

terface area is situated at a position of high stiffness [16]. It is therefore possible to

approximate the interface by a rigid area by condensing the DoF into new local trans-

lational and rotational DoF (see figure 2.2).

Consider the linear and undamped discrete equations of motion of a general sub-

structure presented in equation (2.17). The DoF can be arranged into internal dof qi,

general interface DoF qf and interface DoF on a rigid area qr:

⎡⎣ Mff Mfr Mfi

Mrf Mrr Mri

Mif Mir Mii

⎤⎦⎡⎣ qf

qrqi

⎤⎦+

⎡⎣ Kff Kfr Kfi

Krf Krr Kri

Kif Kir Kii

⎤⎦⎡⎣ qf

qrqi

⎤⎦ =

⎡⎣ pf

pr

0

⎤⎦+

⎡⎣ gf

gr0

⎤⎦

(2.32)

Figure 2.2(a) represents a general interface suited rigid interface projection. To describe

the area as rigid, the DoF of figure 2.2(b) are adopted. This section only considers

interface descriptions with 3 translational DoF per node. The geometric transformation

qr,1z

qr,1yqr,1x

qr,2z

qr,2yqr,2x

qr,3z

qr,3y

qr,3x

qRz

qz

qRyqy

qRx

qx

(a) (b)

z

x

y

Figure 2.2: Interface DoF on a stiff area in (a) generalized coordinates qr and (b) rigid

interface DoF q.

35


between the 3 node interface description and the rigid body modes reads:

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

qr,1xqr,1yqr,1zqr,2xqr,2yqr,2zqr,3xqr,3yqr,3z

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 0 0 0 −d1,z d1,y0 1 0 d1,z 0 −d1,x0 0 1 −d1,y d1,x 01 0 0 0 −d1,z d1,y0 1 0 d1,z 0 −d1,x0 0 1 −d1,y d1,x 01 0 0 0 −d1,z d1,y0 1 0 d1,z 0 −d1,x0 0 1 −d1,y d1,x 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎣

qxqyqzqRx

qRy

qRz

⎤⎥⎥⎥⎥⎥⎥⎦+ qe

qr = Trqr + qe (2.33)

where qe is the residual vector satisfying the (small) flexible responses of the interface

and dn,j the relative distance of node n to the rigid interface node in direction j. If qe

is sufficiently small to be neglected, the reduced set of system parameters with rigid

interface description is found by:

⎡⎣ qf

qrqi

⎤⎦ ≈

⎡⎣ I 0 0

0 Tr 00 0 I

⎤⎦⎡⎣ qf

qrqi

⎤⎦

q ≈ T q (2.34)

The system description on the reduced interface basis is next created by applying the

reduction matrix in equations (2.11), (2.12) and (2.14) after arranging the generalized

coordinates into the form

q =

⎡⎣ qf

qrqi

⎤⎦ . (2.35)

A check can be performed on the actual rigidness of the interface, i.e. the orthogo-

nality of the flexibility residual qe on the reduced space [16, 17]. This check is performed

by first considering the approximated response of the rigid interface in terms of gener-

alized interface coordinates qr:

qr ≈ Trqr (2.36)

qr ≈(T Tr Tr

)−1T Tr qr. (2.37)

Note that equation (2.36) is an overdetermined set of equations and (2.37) is in fact the

least squares solution of qr, minimizing the error between solution and equation space

(see also appendix B). To compare the rigid and actual interface response, the rigid

36

2.3. DYNAMIC SUBSTRUCTURING IN THE NODAL DOMAIN

interface response is projected onto the generalized interface coordinates by substituting

(2.37) into (2.36):

qr ≈ Tr

(T Tr Tr

)−1T Tr qr. (2.38)

Subsequently the ’completeness’ of the rigid interface approximation can be determined,

which is referred to as the rigidness of the interface:

rigidness =

∥∥∥Tr

(T Tr Tr

)−1T Tr qr

∥∥∥‖qr‖ 100%. (2.39)

2.3 Dynamic substructuring in the nodal domain

Dynamic substructuring in general is a method used to divide large structures into

smaller substructures, requiring less computational effort in analyzing them [12, 18, 19].

Therefore, dynamic substructuring is a very useful tool in combining different reduced

substructures into a new, reduced system description of the assembled model. The

coupling of substructures can be performed in different domains, the physical (or nodal),

modal and frequency domain. This thesis characterizes its system models by their mass,

stiffness and damping parameters and therefore requires substructure coupling in the

nodal domain. This will be discussed in the remainder of this section.

Recall from section 2.2.1 that the linear and undamped equations of motion of a

general substructure (s) read:

M (s)q(s) +K(s)q(s) = p(s) + g(s), (2.9)

After reduction of the substructure, the substructure is described with a reduced set of

generalized coordinates:

M (s) ¨q(s)

+ K(s)q(s) = p(s) + g(s), (2.11)

Suppose S substructures exist, the assembly requires the formulation of the substruc-

tures system matrices in block diagonal form:

M ¨q + Kq = p+ g, (2.40)

with

M = diag(M (1),M (2), . . . ,M (S)

)(2.41)

K = diag(K(1), K(2), . . . , K(S)

)(2.42)

37


and

q =

⎡⎢⎢⎢⎣

q(1)

q(2)

...

q(S)

⎤⎥⎥⎥⎦ p =

⎡⎢⎢⎢⎣

p(1)

p(2)

...

p(S)

⎤⎥⎥⎥⎦ g =

⎡⎢⎢⎢⎣

g(1)

g(2)

...

g(S)

⎤⎥⎥⎥⎦ (2.43)

The coupling requirements of the substructures lie in the design specifications and

can be formatted into compatibility and equilibrium conditions. The compatibility

condition matches the degrees of freedom on coupled interfaces, given that the interface

DoF on both substructures are matching. For instance, if an interface on substructure

i is coupled to an interface on substructure j, the compatibility condition assigns:

qif − qjf = 0 (2.44)

This is only true if the original set of nodal interface DoF are maintained after the

reduction, i.e. q(s)f = q

(s)f . Indeed this is true for Guyan and Craig-Bampton reduction

because they have a reduction basis which holds the identity matrix for the interface

DoF. For the complete set of substructures, the compatibility conditions can be written

as:

Bq = 0, (2.45)

Where B is a signed boolean matrix, acting on the interface DoF of the substructures.

The equilibrium condition demands that the interface forces gf of coupled interfaces

are opposite and equal, conform Newton’s third law. This condition states that

LT g = 0, (2.46)

where L is a boolean matrix coupling the interface forces between coupled interfaces.

Again this is only true if the interface forces are preserved after reduction, i.e. g = g.

Equations (2.40), (2.45) and (2.46) describe the coupling between the S substruc-

tures, containing an arbitrary amount of coupling interfaces. This set of equations is

used to obtain the coupled system description and can be found using primal or dual

assembly techniques. Primal assembly defines a unique set of interface DoF, eliminating

the interface forces, while dual assembly retains the full set of generalized coordinates

in addition to the interface forces in the form of Lagrange multipliers. The assembly

techniques are discussed in section 2.3.1 and 2.3.2.

38

2.3. DYNAMIC SUBSTRUCTURING IN THE NODAL DOMAIN

2.3.1 Primal assembly

In the primal assembly procedure, a unique set of interface DoF is formulated for

the assembled structure. The interface forces are therefore eliminated, automatically

satisfying the equilibrium condition. The new set of generalized coordinates is related

to the original DoF through L:

q = Lqp (2.47)

Due to the unique interface DoF, the compatibility condition is automatically satisfied

for any set qp:

Bq = BLqp = 0, (2.48)

from which follows that L represents the nullspace of B. Inserting (2.47) into (2.40)

results in:

ML ¨qp + KLqp = p+ g, (2.49)

Premultiplication by LT satisfies the equilibrium condition (equation (2.46)) and leads

to the primal assembled system description:

Mp¨qp + Kpqp = pp, (2.50)

with the system parameters defined as

Mp = LTML (2.51)

Kp = LT KL (2.52)

pp = LT p. (2.53)

(2.54)

2.3.2 Dual assembly

Dual assembly retains all interface DoF, i.e. q is not changed. The system assembly

is obtained by satisfying the equilibrium condition. The equilibrium condition can be

met by stating that

g = −BTλ, (2.55)

where λ are Lagrange multipliers, measuring the intensity of the interface forces. In-

serting (2.55) into the equilibrium condition reads

LT g = −LTBTλ = 0, (2.56)

39


which is always satisfied because BT is in the nullspace of LT .

Having satisfied the equilibrium condition, the dual assembled system description

can be written as:M ¨q + Kq +BTλ = p

Bq = 0,(2.57)

or, in matrix notation[M 00 0

] [¨qλ

]+

[K BT

B 0

] [qλ

]=

[p0

]. (2.58)

2.4 Summary

This chapter treated the techniques of component model reduction, required for accu-

rately describing responses of large or complex structures. For photomask manipulation

design, component model reduction is required since the mechanical plant is has high

geometric complexity. Three reduction methods were introduced where internal DoF

where condensed into interface DoF using a basis of static and/or dynamic component

modes. In the case when interfaces have high stiffness with respect to the component

compliance, an additional interface reduction technique can be applied. In the final

section, a method for assembling the reduced components was introduced, assembling

the substructures in the nodal domain.

The next chapter proposes an extension to the Craig-Bampton reduction method.

The scope of this thesis is to analyze curvature manipulations of the photomask bending

actuator. Therefore curvature of the photomask is of high importance in the dynamic

system description of the photomask manipulation assembly and must be present in the

reduction basis of any applied reduction scheme discussed in the previous section. The

reduced model dynamic accuracy of the Craig-Bampton mode component synthesis

is defined by the set of the interface fixed modes in the reduction basis. It is not

automatically satisfied that the obtained reduction basis is able to preserve describe

curvature deformation properties of a system. The next section therefore introduces a

selection procedure for the interface fixed modes, based on Gramian theory.

40

Chapter 3

Gramian theory

The scope of this thesis is to analyze curvature manipulations of the photomask bend-

ing actuator. The curvature of the photomask is therefore of high importance in the

dynamic system description of the photomask manipulation assembly. Subsequently

the reduction basis used to obtain the reduced order model of the manipulation assem-

bly must obtain curvature information. This chapter proposes a selection criteria for

creating a reduction basis for a Craig-Bampton model reduction which contains this

information.

Reduced model accuracy of Craig-Bampton reduction is defined by the truncation

of the interface fixed modes present in the reduction basis. It is not automatically sat-

isfied that the truncated set of modes is able to fully describe curvature deformation.

A method to describe the influences of modes on curvature deformation is the calcula-

tion of modal energy contribution to the deformation. Observability and controllability

Gramians provide a ’measure’ of energy put into a system by external forces or en-

ergy observed at the response, based on modal system description [20]. By using these

Gramians as a selection method for the Craig-Bampton reduction basis, it is possible

to identify which interface fixed modes have high energy contributions in the system.

If curvature is obtained from the system response, it is then possible to identify which

modes contribute the highest energy to the curvature deformation. With these modes a

reduction basis is obtained, ensuring a good approximation of curvature information af-

ter reduction of the system. Throughout this chapter, the following model assumptions

are applied:

• The system is interface fixed.

• No rigid body modes are present in the system.

41

CHAPTER 3. GRAMIAN THEORY Company-secret

• The system is linear time invariant, also referred to as LTI system.

• The system is asymptotically stable.

Having set out the general idea of observability and controllability Gramian calculation

for reduction basis selection, section 3.1 sets out the structural dynamic structural sys-

tem description used in Gramian calculation. This description is based on a first order

system, namely state space description. The theory on observability and controllability

Gramians is presented respectively in sections 3.3 and 3.4.

3.1 System state space description

In linear algebra, a Gramian matrix, named after Jørgen Pedersen Gram, is a matrix

of all possible inner products of a set of vectors. An important application of the

Gramian matrix is the computation of linear independency of the set of vectors. In

control theory the observability and controllability Gramians are based on the Gramian

theory. They give information about observability and controllability properties of

mechanical structures. Observability is a measure of how well internal states of a

system are visible in the system output. Controllability determines the ability to bring

a system to a desired state by only a certain amount of admissible manipulations.

Modal description of a system allows observability and controllability of structures

to be found as combinations of individual modes. This is useful information in ob-

taining a reduction basis which is fully controllable and observable. Secondly, modal

description is required in obtaining the mass normalized interface fixed modes for the

Craig-Bampton reduction scheme. Let us consider the following structural model with

N DoF:

Mq +Cq +Kq = p. (3.1)

Since this theory is defined for selecting interface fixed modes for the Craig-Bampton

reduction basis, no rigid body motions are assumed to be present within the system’s

eigenmodes. Assuming well separated eigenfrequencies and small damping terms, mode

orthogonality (see section 2.1) is still valid. Let Φ =[φ(1) φ(2) . . . φ(N)

]repre-

sent the N orthogonal modes and let C be defined as structural damping. The normal

equations are then found as (also see section 2.2.2):

ηn + 2ζnωnηn + ω2nηn = pn n = 1, . . . , N (3.2)

42

3.1. SYSTEM STATE SPACE DESCRIPTION

with:

μn = φ(n)TMφ(n) (3.3)

γn = φ(n)TKφ(n) (3.4)

ω2n =

γnμn

(3.5)

pn =φ(n)Tp

μn. (3.6)

The modal damping ζn is found by noting that:

dn = φ(n)TCφ(n) (3.7)

2ζnωn = dn

ζn =1

2μ− 1

2n γ

− 12

n dn. (3.8)

Observability and controllability of a system is not only defined by the system

parameters, but also by its input and output locations. Therefore it is convenient to

describe the system in a state space form using input, output and state variables:

x = Asx+Bsu (3.9)

y = Csx+Dsu, (3.10)

where x is the state vector, y the output vector, u the input vector, As the state

matrix, Bs the input matrix, Cs the output matrix and Ds the feedthrough matrix.

The feedthrough matrix is dependent direct information from the input and is not

present throughout this thesis.

The most natural state vector is obtained when considering:

x =[η1 η1 η2 η2 · · · ηN ηN

]T. (3.11)

Proper choice of the state vector representing the normal equations of the system (3.2)

influences the condition of the state matrix [20, 21]. When subjected to a large number

of matrix computations, an ill-conditioned state matrix could experience a buildup

of numerical errors, leading to erroneous results. To overcome these problems, the

following state vector can be adopted:

x =[η1 ω1η1 η2 ω2η2 · · · ηN ωNηN

]T. (3.12)

The state vector (3.12) ensures better matrix conditioning than the common state

vector (3.11) because addition of ωn to ηn has the similar effect as differentiation of ηn

43


to ηn. Therefore (3.12) returns all states corresponding to each mode in the same order

of magnitude, without dependency on units [21].

Suppose the input is defined at R points and the state vector is given by (3.12), the

state space parameters for the normal equations (3.2) become:

As = diag(A(n)

s

)(3.13)

A(n)s =

[ −2ζnωn −ωn

ωn 0

]n = 1, 2, . . . , N

Bs =

⎡⎢⎢⎢⎢⎢⎢⎢⎣

φ(1)1μ1

φ(1)2μ1

· · · φ(1)Rμ1

0 0 · · · 0...

.... . .

...φ(N)1μN

φ(N)2μN

· · · φ(N)RμN

0 0 · · · 0

⎤⎥⎥⎥⎥⎥⎥⎥⎦

(3.14)

u =[p1 p2 . . . pR

]T. (3.15)

The output matrix Cs can be defined for both position and velocity measurements. As-

suming position is measured at S points, i.e. the observed DoF are[q1 q2 · · · qS

]T,

the output matrix becomes:

Cs =

⎡⎢⎢⎢⎢⎢⎢⎣

0φ(1)1ω1

· · · 0φ(N)1ωN

0φ(1)2ω1

· · · 0φ(N)2ωN

......

. . ....

...

0φ(1)Sω1

· · · 0φ(N)SωN

⎤⎥⎥⎥⎥⎥⎥⎦. (3.16)

In case of velocity measurements, i.e. the observed DoF are[q1 q2 · · · qS

]T, the

output matrix reads:

Cs =

⎡⎢⎢⎢⎢⎣

φ(1)1 0 · · · φ

(N)1 0

φ(1)2 0 · · · φ

(N)2 0

......

. . ....

...

φ(1)S 0 · · · φ

(N)S 0

⎤⎥⎥⎥⎥⎦ (3.17)

Finally, a combination of position and velocity measurements returns an output matrix

built as a combination of (3.16) and (3.17).

To illustrate the theory set out in this chapter, a simple constrained beam case is

adopted. The next section formulates the state space description of the constrained

beam case.

44


3.1.1 Constrained beam case: state space formulation

In this example, the state space formulation of a simple beam structure is determined.

This structure is presented in figure 3.1 and consists of two finite beam elements.

The interfaces of the beam are modeled as translational constraints at both ends.

Furthermore, structural damping is assumed, making it possible to model the damping

θ1

y2

θ2 θ3

ll

y

x

Figure 3.1: Translational constrained beam with two finite elements.

as a linear combination of the mass and stiffness matrices [22, 20]:

C = αM + βK, (3.18)

with C, M andK respectively the damping, mass and stiffness matrix of the structure.

The nonnegative scalars α and β determine the amount of structural damping where

α influences the global structural damping and β the frequency dependent structural

damping. These parameters are set to:

α = 1 · 10−2 (3.19)

β = 1 · 10−7 (3.20)

Implementing the finite element system descriptions for beam elements, the homoge-

neous system of equations reads in matrix notation:

Mq +Cq +Kq = 0, (3.21)

45


with

q =[θ1 y2 θ2 θ3

]T(3.22)

M =ml

420

⎡⎢⎢⎣

4l2 13l −3l2 013l 312 0 −13l−3l2 0 8l2 −3l2

0 −13l −3l2 4l2

⎤⎥⎥⎦ (3.23)

K =EI

l3

⎡⎢⎢⎣

4l2 −6l 2l2 0−6l 24 0 6l2l2 0 8l2 2l2

0 6l 2l2 4l2

⎤⎥⎥⎦ , (3.24)

and

C =ml

420 · 102

⎡⎢⎢⎣

4l2 13l −3l2 013l 312 0 −13l−3l2 0 8l2 −3l2

0 −13l −3l2 4l2

⎤⎥⎥⎦+

EI

1 · 107l3

⎡⎢⎢⎣

4l2 −6l 2l2 0−6l 24 0 6l2l2 0 8l2 2l2

0 6l 2l2 4l2

⎤⎥⎥⎦

(3.25)

The constants m, l, E and I are defined as the mass, length, Young’s modulus and

moment of inertia of an element respectively.

The squared eigenfrequencies of the undamped system, i.e. the solution to the

eigenvalue problem, yield:

ω21 = −24

(−207 + 2√10371

)13

EI

ml4(3.26)

ω22 = 120

EI

ml4(3.27)

ω23 =

24(207 + 2

√10371

)13

EI

ml4(3.28)

ω24 = 2520

EI

ml4(3.29)

The corresponding eigenmodes φ are graphically displayed in figure 3.2. To give the

example a more physical meaning the remainder of this chapter uses aluminum as beam

material. Several properties of the beam are:

• total length: L = 1 [m],

• height and width: b = h = 3 · 10−2 [m],

• Young’s modulus: E = 70 [GPa],

• density: ρ = 2700[

kgm3

],

46


φ(1) φ(2)

φ(3) φ(4)

Figure 3.2: Eigenmodes of the constrain beam.

• total mass: m = Lbhρ = 1.22 [kg]

• area moment of inertia: I = bh3

12 = 6.75 · 10−8[m4].

Implementing the properties into equations (3.26) to (3.29) and taking the square root

yields the undamped eigenfrequencies for the aluminum beam:

⎡⎢⎢⎣

ω1

ω2

ω3

ω4

⎤⎥⎥⎦ =

⎡⎢⎢⎣

6.18 · 1021.73 · 1036.87 · 1031.25 · 104

⎤⎥⎥⎦[rad

s

](3.30)

The (mass normalized) eigenmodes Φ of the constrained aluminum beam are found as:

Φ =[φ(1) φ(2) φ(3) φ(4)

]

=

⎡⎢⎢⎣

4.06 9.94 21.39 26.291.29 0 −1.17 00 −9.94 0 26.29

−4.06 9.94 −21.39 26.29

⎤⎥⎥⎦ . (3.31)

obtained by application of equations (3.3) to (3.5). The normal equations are (external

47


forces are considered later):

⎡⎢⎢⎣

μ1

μ2

μ3

μ4

⎤⎥⎥⎦ =

⎡⎢⎢⎣

1111

⎤⎥⎥⎦ (3.32)

⎡⎢⎢⎣

γ1γ2γ3γ4

⎤⎥⎥⎦ =

⎡⎢⎢⎣

3.82 · 1057.47 · 1054.72 · 1071.57 · 108

⎤⎥⎥⎦ =

⎡⎢⎢⎣

ω21

ω22

ω23

ω24

⎤⎥⎥⎦ . (3.33)

The modal damping is obtained by calculating1 (3.8):

⎡⎢⎢⎣

ζ1ζ2ζ3ζ4

⎤⎥⎥⎦ =

⎡⎢⎢⎣

3.90 · 10−5

1.38 · 10−4

3.44 · 10−4

6.27 · 10−4

⎤⎥⎥⎦ . (3.34)

Using the eigenmodes and modal damping terms, the state matrix As can be con-structed. When the state vector is defined as (3.12), the state matrix reads:

As =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

−1.82 · 10−2 −617.91 0 0 0 0 0 0

617.91 0 0 0 0 0 0 0

0 0 −7.57 · 10−1 −2.73 · 103 0 0 0 0

0 0 2.73 · 103 0 0 0 0 0

0 0 0 0 −4.73 −6.87 · 103 0 0

0 0 0 0 6.87 · 103 0 0 0

0 0 0 0 0 0 −15.59 −1.25 · 1040 0 0 0 0 0 1.25 · 104 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

.

(3.35)

The input matrix is defined by the forces acting on the system. Let the torques pθ1 and

pθ3 act on the system. The force vector in the equations of motion is then equal to:

p =

⎡⎢⎢⎢⎢⎣

pθ10

0

pθ3

⎤⎥⎥⎥⎥⎦ =

⎡⎢⎢⎢⎢⎣

1 0

0 0

0 0

0 1

⎤⎥⎥⎥⎥⎦u (3.36)

with input vector u defined by:

u =

[pθ1pθ3

]. (3.37)

1For the sake of explanation modal damping is very low in this example. Usually, damping ratios

are 10−3 or 10−2, this however does not compromise the theory set out in this chapter.

48


Conversion to the normal equations representations results in (see (3.6))

p =

⎡⎢⎢⎢⎢⎣

μ1 0 0 0

0 μ2 0 0

0 0 μ3 0

0 0 0 μ4

⎤⎥⎥⎥⎥⎦

−1 ⎡⎢⎢⎢⎢⎣

φ(1)1 φ

(2)1 φ

(3)1 φ

(4)1

φ(1)2 φ

(2)2 φ

(3)2 φ

(4)2

φ(1)3 φ

(2)3 φ

(3)3 φ

(4)3

φ(1)4 φ

(2)4 φ

(3)4 φ

(4)4

⎤⎥⎥⎥⎥⎦

T ⎡⎢⎢⎢⎢⎣

1 0

0 0

0 0

0 1

⎤⎥⎥⎥⎥⎦u

=

⎡⎢⎢⎢⎢⎣

4.06 −4.06

9.94 9.94

21.39 −21.39

26.29 26.29

⎤⎥⎥⎥⎥⎦u, (3.38)

which can be used to formulate input matrix Bs:

Bs =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

4.06 −4.06

0 0

9.94 9.94

0 0

21.39 −21.39

0 0

26.29 26.29

0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

. (3.39)

The output y is defined by the output matrix. Suppose the output y2 is measured, the

output is then written as:

y =[0 1 0 0

]⎡⎢⎢⎢⎢⎣

θ1

y2

θ2

θ3

⎤⎥⎥⎥⎥⎦ . (3.40)

Conversion to the modal domain yields:

y =[0 1 0 0

]⎡⎢⎢⎢⎢⎣

φ(1)1 φ

(2)1 φ

(3)1 φ

(4)1

φ(1)2 φ

(2)2 φ

(3)2 φ

(4)2

φ(1)3 φ

(2)3 φ

(3)3 φ

(4)3

φ(1)4 φ

(2)4 φ

(3)4 φ

(4)4

⎤⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎣

η1

η2

η3

η4

⎤⎥⎥⎥⎥⎦

=[1.29 0 −1.17 0

]⎡⎢⎢⎢⎢⎣

η1

η2

η3

η4

⎤⎥⎥⎥⎥⎦ (3.41)

49


The output matrix Cs is dependent on the state vector. Since the states regarding

displacement are premultiplied by an eigenfrequency ωn, the output matrix should be

changed accordingly. Therefore, using (3.41), the output matrix reads:

Cs =[0 1.29

ω10 0 0 −1.17

ω20 0

]=[0 2.10 · 10−3 0 0 0 −1.70 · 10−4 0 0

]. (3.42)

In this section a dynamic system description was defined by use of the state space

representation of a second order modal model. The time response of the modal system

is extracted in the next section.

3.2 Time response of a modal system

The solution to the state space description (3.9) consists of a homogeneous and par-

ticular part. The homogeneous part is dependent on the system properties and the

particular part is induced by the driving forces exerted on the system. The exerted

forces are defined at Bsu. Consider the state space reorganized into the general first

order linear equation form [23]:

x (t)−Asx (t) = Bsu (t) (3.43)

To solve this equation, an integrating factor2 χ (t) is used which has the following form:

χ (t) = e−Ast. (3.44)

Premultiplying (3.43) with (3.44) yields:

e−Astx (t)− e−AstAsx (t) = e−AstBsu (t). (3.45)

Equation (3.45) becomes integrable by considering the series expansion of e−AstAs:

e−AstAs =

(I +

Ast

1!+

(Ast)2

2!+ . . .

)As

=

(As +

A2st

1!+

A3st

2

2!+ . . .

)

= As

(I +

Ast

1!+

(Ast)2

2!+ . . .

)

= Ase−Ast, (3.46)

2Function multiplied with the differential equation, chosen so that the resulting equation is directly

integrable [23].

50

3.2. TIME RESPONSE OF A MODAL SYSTEM

which, implemented into (3.45) yields:

e−Astx (t)−Ase−Astx (t) = e−AstBsu (t)

d

dte−Astx (t) = e−AstBsu (t). (3.47)

The time dependency of the system allows the system to be integrated between times

τ = 0 and τ = t:

t∫0

d

dτe−Asτx (τ)dτ =

t∫0

e−AsτBu (τ)dτ

[e−Asτx (τ)

]t0=

t∫0

e−AsτBu (τ)dτ

e−Astx (t)− x (0) =

t∫0

e−AsτBu (τ)dτ

x (t) = eAstx (0) +

t∫0

eAs(t−τ)Bu (τ)dτ. (3.48)

The first part of (3.48) is the homogeneous solution of the system found in the absence

of external forces. The second part represents the particular solution of the system,

also known as the Duhamel integral [12, 24, 25]. Equation (3.48) can also be written

as:

x (t) = xh (t, t = 0) + xp (t, τ) , (3.49)

with xh the homogeneous solution and xp the particular solution. The homogeneous

solution is trivial for a diagonal matrix As, but unfortunately this is not the case.

Diagonalization of As is however possible by use of its eigenvectors and eigensolutions

[26, 27, 28]. Consider the linear transformation ofAs to its eigenvalues and eigenvectors:

AsΦs = ΦsΛs, (3.50)

or

As = ΦsΛsΦ−1s , (3.51)

where Λs is the diagonal matrix holding the eigenfrequencies. By acknowledging that

Ans =

(ΦsΛsΦ

−1s

)(1)

(ΦsΛsΦ

−1s

)(2)

. . .(ΦsΛsΦ

−1s

)(n)

= ΦsΛnsΦ

−1s (3.52)

51


the exponential eAst of the homogeneous solution can be written as:

eAst = (Ast)0 +

(Ast)1

1!+

(Ast)2

2!+ . . .

= ΦsΛ0sΦ

−1s t+

ΦsΛ1sΦ

−1s t

1!+

ΦsΛ2sΦ

−1s t2

2!+ . . .

= Φs

((Λst)

0 + (Λst)1 + (Λst)

2 + . . .)Φ−1

s

= ΦseΛstΦ−1

s , (3.53)

using the series expansion of eAst. Since As consists of block diagonal matrices (see

(3.13)), each block term can be treated individually, therefore the homogeneous solution

can be written as:

xh (t, t = 0) = eAstx (0)

= diag(eA

(n)s t)x (0)

= diag

(Φ(n)

s eΛ(n)s t(Φ(n)

s

)−1)x (0) n = 1, . . . , N, (3.54)

(3.55)

where diag means block diagonals in this case. Let x(n) (0) be the boundary conditions

for each block diagonal part of As:

x(n) (0) = x(n)0 =

[η(n)0

ωnη(n)0

]. (3.56)

The eigenvalue matrix Λ(n)s of A

(n)s read:

Λ(n)s =

[−ζnωn + ωn

√ζ2n − 1 0

0 −ζnωn − ωn

√ζ2n − 1

]

=

[−ζnωn + jωn

√1− ζ2n 0

0 −ζnωn − jωn

√1− ζ2n

](3.57)

j �√−1

when ζn fulfills the demand of small damping, i.e. 0 < ζn � 1. The corresponding

eigenvectors of A(n)s are:

Φ(n)s =

[−ζn + j

√1− ζ2n −ζn − j

√1− ζ2n

1 1

]. (3.58)

Using equations (3.57) and (3.58), the homogeneous solution of the block diagonal part

n reads:[ηn

ωnηn

]= Φ(n)

s

[e−ζnωnt+jωn

√1−ζ2nt 0

0 e−ζnωnt−jωn

√1−ζ2nt

](Φ(n)

s

)−1[

η(n)0

ωnη(n)0

],

(3.59)

52

3.2. TIME RESPONSE OF A MODAL SYSTEM

or[ηn

ωnηn

]= e−ζnωnt

[cos (Ωnt)− ζnωn

Ωnsin (Ωnt) − ωn

Ωnsin (Ωnt)

ωnΩn

sin (Ωnt) cos (Ωnt) +ζnωn

Ωnsin (Ωnt)

][η(n)0

ωnη(n)0

]

(3.60)

Ωn = ωn

√1− ζ2n,

when decomposed into its goniometric parts by application of Euler’s formula [25] (given

for an arbitrary variable ξ):

ejξ = cos (ξ) + j sin (ξ) . (3.61)

3.2.1 Constrained beam case: time response

In the previous section the state space representation was calculated for the constrained

aluminum beam of figure 3.1. As input forces, the torques pθ1 and pθ3 were chosen

(equation (3.36)). The output is y2, the height of the beam in the middle.

Let us now calculate the time response of the system, released from an initial

position q0, which is defined as the equilibrium configuration of the system under

influence of the force vector (3.37), adopted as:

u =

[100

−100

][Nm] . (3.62)

The spatial equilibrium position q0 is found by (see (3.22) and (3.24)):

q0 = K−1p

=

⎡⎢⎢⎢⎢⎣

3.78 · 104 −1.13 · 105 1.89 · 104 0

−1.13 · 105 9.07 · 105 0 1.13 · 1051.89 · 104 0 7.56 · 104 1.89 · 104

0 1.13 · 105 1.89 · 104 3.78 · 104

⎤⎥⎥⎥⎥⎦

−1 ⎡⎢⎢⎢⎢⎣

100

0

0

−100

⎤⎥⎥⎥⎥⎦ (3.63)

=

⎡⎢⎢⎢⎢⎣

1.06 · 10−2 [rad]

2.65 · 10−3 [m]

0 [rad]

1.06 · 10−2 [rad]

⎤⎥⎥⎥⎥⎦ (3.64)

The initial configuration is presented in figure 3.3. In state space format the calcu-

lation of the time response of the system, released from the initial configuration q0,

requires equation (3.64) to be written in terms of state vector boundary conditions x0,

determined by equation (3.56). The initial state vector x0 is found by assuming that

53


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

3×10−3

Figure 3.3: Initial configuration of the constrained beam.

the initial positions do not experience any speed and acceleration. Implementing (3.62)

into (3.9) therefore yields:

0 = Asx0 +Bsu (3.65)

x0 = −A−1s Bsu

=[0 1.31 0 0 0 0.62 0 0

]T(3.66)

The homogeneous time response of y2 can be found by calculating (3.60) for n =

1, . . . , 4, followed by premultiplication of the output matrix Cs (see (3.42)):

y2 = Csx

= CseAtx0

=[0 2.10 · 10−3 0 0 0 −1.70 · 10−4 0 0

]eAt

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0

1.31

0

0

0

0.62

0

0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(3.67)

The time response over time t is graphically represented by figure 3.4.

This section explained the derivation of the timeresponse of a modal system. The

next section uses this derivation and the state space description to formulate the ob-

servability Gramian, which is based on output energy.

3.3 Observability Gramian

Given a state space model (3.9)-(3.10) with absence of the feedthrough matrix Ds. For

the solution of control problems it is important to know whether the behavior of the

54

3.3. OBSERVABILITY GRAMIAN

t [s]

y 2[m

]

0 50 100 150 200 250-3

-2

-1

0

1

2

3×10−3

Figure 3.4: Time response of y2 due to initial conditions x0.

states can be described by the output y (t). This property of the system is described

using the concept of observability, which is defined by [26, 29]:

Definition 3.3.1. Let y (t; t0,x0,u) denote the response of the output variable y (t) of

the state space system

x (t) = Asx (t) +Bsu (t)

y (t) = Csx (t) ,

to the initial state x (t0). When Bsu (t) is known, its effect can be eliminated, so that

without loss of generality one can assume u (t) ≡ 0. Such a realization is observable on

[t0, T ] if x (t0) can be deduced from the knowledge of {y (t) , t0 ≤ t ≤ T}.

Using modal coordinates, observability gives valuable information about which se-

lection of modes contribute in the responses of the system. Given the time response,

formulated by (3.48), the full output y (i.e. homogeneous and particular) can be written

as:

y = yh + yp

= CseAstx0 +Cs

t∫0

eAs(t−τ )Bsu (τ) dτ, (3.68)

where x0 = x (0) is the initial condition at t = 0 = t0. Given (3.68), the following

theorem holds [30]:

55


Theorem 3.3.1. The system (3.9) is observable on [t0, T ] if and only if the 2N columns

of the [S × 2N ] matrix function CseAstx0 are linear independent on [t0, T ].

In the study of observability, y, Cs, Bs and u are assumed to be known. The question

to be dealt with is whether x0 can be described. Therefore equation (3.68) may be

shortened to its homogeneous part:

yh = CseAstx0 (3.69)

= y −Cs

t∫0

eAs(t−τ)Bsu (τ) dτ .

The observability problem is now to determine x0 in (3.69) with the knowledge of y

and Cs. Once x0 is known the state x (T ) can be calculated and observed from yh.

Therefore the system is said to be fully observable when every initial state can be found

from (3.69). This is only possible when CseAst is full column rank.

To determine if CseAst is full rank, a simple test can be performed. The Taylor

expansion of yh reads:

yh =∞∑k=0

CsAks

tk

k!x0. (3.70)

Equation (3.70) shows that yh is a linear combination of block rows⎡⎢⎢⎢⎢⎣

Cs

CsAs

...

CsA∞s

⎤⎥⎥⎥⎥⎦ , (3.71)

from which easily can be seen that full observability requires the linear independency

of the columns of (3.71). In other words, the column space of (3.71) for a system with

2N states is 2N . Since (3.71) holds powers of As up to infinity, it is not easy to find a

solution in practice. Therefore the Cayley-Hamilton theorem is used [28]:

Theorem 3.3.2. For a square matrix As ∈ R2N and k � 0, the k-th power Ak

s is a

linear combination of the powers up to order 2N − 1, i.e.:

Aks = g2NA2N−1

s + g2N−1A2N−2s + . . .+ g3A

2s + g2A

1s + g1I, (3.72)

where gn are scalars.

Using the theorem, the column space of⎡⎢⎢⎢⎢⎣

Cs

CsAs

...

CsA2N−1s

⎤⎥⎥⎥⎥⎦ (3.73)

56


is equal to the column space of (3.71). The system is fully observable if (3.73) has

2N independent columns, i.e. is full rank, and can therefore be used to check the

observability of a system. Equation (3.73) is also known as the observability matrix .

To check if a system is fully observable a more general method is known. This

method uses an observability Gramian to see which states are observable by use of

energy calculations. Consider a model released from its initial state at t = 0 in the

absence of external forces. The homogeneous time response of that system can be

measured in terms of output strain energy. In state space format, this output energy

yields [31]:

E =

t=∞∫t=0

yhTyhdτ . (3.74)

Substitution of (3.69) into (3.74) yields

E =

t=∞∫t=0

((Cse

Asτ)x0

)TCse

Asτx0dτ

=

t=∞∫t=0

xT0

(Cse

Asτ)T

CseAsτx0dτ

= xT0

⎡⎣ t=∞∫t=0

eATs τCT

s CseAsτdτ

⎤⎦x0 (3.75)

= xT0 Wo (t = ∞)x0, (3.76)

where Wo is defined as the observability Gramian [21]. More general, the definition for

the observability Gramian reads:

Wo (t) �t∫

t=0

eATs τCTCeA

Ts τdτ (3.77)

Because the state matrixAs is asymptotically stable, the observability Gramian reaches

a steady state when t goes to infinity3. To express Wo (t) at t = ∞, the general first

and second derivative of the Gramian are examined (see also (3.46)):

Wo (t) = eATs tCTCeAst (3.78)

Wo (t) = ATs e

ATs tCTCeA

T t + eATs tCTCAse

Ast

= ATs e

ATs tCTCeA

T t + eATs tCTCeAstAs. (3.79)

3The eigenvalues have a negative real part due to subcritical damping [24], therefore the exponential

eAst goes to 0 when t goes to infinity.

57


Next the second derivative is integrated on [0, t]:

Wo (t) =

t∫t=0

(AT

s eAT

s τCTCeAT τ + eA

Ts τCTCeAsτAs

)dτ + Wo (t = 0) ,

= ATs Wo (t) +Wo (t)As + Wo (t = 0) (3.80)

where Wo (t = 0) is an integration constant and yields (using (3.78)):

Wo (t = 0) = CTC. (3.81)

If now Wo (t) is calculated for t = ∞, equation (3.78) returns:

Wo (t = ∞) = 0. (3.82)

which is indeed true since the observability Gramian reaches steady state when t goes

to ∞. By finally inserting (3.81) and (3.82) into (3.80), the following equation is found

for the observability Gramian at t = ∞:

ATs Wo (t = ∞) +Wo (t = ∞)As +CTC = 0. (3.83)

Equation (3.83) is also known as the Lyapunov equation [32]. Note that in case of a

mechanical model described by N DoF, Wo is a matrix holding 2N by 2N terms, i.e.:

Wo =

⎡⎢⎢⎣

W(1,1)o . . . W

(1,(2N))o

.... . .

...

W((2N),1)o . . . W

((2N),(2N))o

⎤⎥⎥⎦ (3.84)

Equation (3.83) can be solved in close form for the observability Gramian using the

block diagonal matrix As. Consider the case of measuring displacements, i.e. Cs is

defined by (3.16), the closed form solution of Wo then reads [21] (see also C):

Wo,(m−1:m)(j−1:j) =ckldkl

[2ωkωlekl ωk

(4ζlωlekl −

(ω2l − ω2

k

))ωl

(4ζkωkekl +

(ω2l − ω2

k

))2(ζlω

3l + ζkω

3k

)+ 8ζkζlωkωlekl

]

(3.85)

dkl = 4ωkωl (ζkωk + ζlωl) (ζkωl + ζlωk) +(ω2k − ω2

l

)2ckl =

S∑s=1

φ(k)s φ

(l)s

ωkωl

ekl = ζkωk + ζlωl

For each block element of Wo the indices m, j, k and l are defined as:

m and j = 2, 4, 6 . . . 2N

k =m

2

l =j

2.

58


In case of small damping terms (0 < ζk � 1) equation (3.84) can be simplified. Indeed,

the diagonal block terms (m = j) become:

Wo,(m−1:m)(m−1:m) ≈ diag

⎛⎜⎜⎜⎝

S∑s=1

(φ(k)s

)24ζkω

3k

,

S∑s=1

(φ(k)s

)24ζkω

3k

⎞⎟⎟⎟⎠. (3.86)

The off diagonal terms Wo (m �= j) become negligible when damping is considered to be

low. Furthermore, if the eigenfrequencies of the system are well spaced, the denominator

dkl in equation (3.84) becomes large compared to the nominator, ensuring a small and

negligible entry in Wo. It can therefore be concluded that, in case of small damping,

the observability Gramian can be approximated by equation (3.86). The eigenvalues

of the observability Gramian are then equal to equations (3.86) [21], which is useful

in obtaining an error indication of the approximation. The two-norm relative error

between the eigenvalues and the approximation of Wo is written as:

e =‖diag (Wo −ΛWo)‖

‖diag (ΛWo)‖· 100%. (3.87)

Finally, the output energy E found using the approximation (3.86) reads (see (3.76)):

E ≈N∑k=1

⎛⎜⎜⎜⎝

S∑s=1

(φ(k)s

)24ζkω

3k

(x20(2k−1) + x20(2k)

)⎞⎟⎟⎟⎠. (3.88)

This last relation indicates how each mode contributes to the observability of the sys-

tem.

3.3.1 Constrained beam case: observability Gramian

Consider the translational constrained beam example of figure 3.1 and let us determine

the observability Gramians. This example discusses only the measurements on displace-

ment, not on velocity. Therefore, the output matrix Cs is defined by equation (3.16).

To visualize observability of the DoF the bode diagram of figure 3.5 is presented. In

this bode diagram, an input u is chosen which is able to perturb all eigenmodes (3.31),

i.e. has full controllability over the system. The input vector (in this case scalar) is

chosen as:

u = pθ1 , (3.89)

for which in the next section full controllability will be shown. Although the observ-

ability Gramian does not take into account the input, it regards the system as fully

59


Magnitude(dB)

Phase(deg) θ1/pθ1

y2/pθ1θ3/pθ1θ4/pθ1

102 103-180

0

180

360

540

-150

-100

-50

0

Figure 3.5: Bode plots for all output DoF with input u = pθ1

controlled. As can be seen in the figure, the outputs θ1 and θ3 fully observe all 4 modes

of the system. Output y2 experiences only the first and third modes, which is also

seen in the mode visualization of figure 3.2. Output θ2 can only observe the second

and fourth mode. The observability Gramian will therefore be full rank for the output

cases θ1 and θ3 but not for y2 and θ2. Consider the case when displacement of y2 is

measured, i.e. y is defined by equation (3.41). The variable φs therefore reads:

φs =[φ(1)2 φ

(2)2 φ

(3)2 φ

(4)2

]=[1.29 0 −1.17 0

](3.90)

Substitution of the eigenfrequencies (3.30), modal damping (3.34) and eigenmodes

(3.31) in equation (3.85) results in the observability Gramian:

Wo =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

4.55 · 10−5 3.54 · 10−9 0 0 −3.30 · 10−15 4.70 · 10−12 0 0

3.54 · 10−9 4.55 · 10−5 0 0 −5.23 · 10−11 −3.63 · 10−14 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

−3.30 · 10−15 −5.23 · 10−11 0 0 3.06 · 10−9 2.11 · 10−12 0 0

4.70 · 10−12 −3.63 · 10−14 0 0 2.11 · 10−12 3.06 · 10−9 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

.

(3.91)

From the observability Gramian it can clearly be seen that the second and last modes

are not observable for output y2. The first mode has the highest observability since y2

60


has a larger contribution in mode 1 than in mode 3. The eigenvalues ΛWo of Wo read:

ΛWo = diag

([4.55 · 10−5 4.55 · 10−5 0 0 3.06 · 10−9 3.07 · 10−9 0 0

]T).

(3.92)

Because the damping is assumed low in this example, the approximation for the

observability Gramian (3.86) can be used and yields:

Wo ≈ diag

([4.55 · 10−5 4.55 · 10−5 0 0 3.06 · 10−9 3.06 · 10−9 0 0

]T)(3.93)

The relative error between the approximation (3.93) and the eigenvalues of Wo can

than be calculated in the two-norm format:

e =‖Wo −ΛWo‖

‖ΛWo‖100%

= 7.80 · 10−3%, (3.94)

showing that the approximation of the observability Gramian introduces a relative

error of less than 1%. It can therefore be concluded that the approximation (3.86) is

sufficiently accurate in this example. Finally it can be concluded that the first mode

has the highest energy contribution in outputs measured at y2, followed by the third

mode. Modes 2 and 4 have no contribution for the case that deformations are measured

at y2.

A second case is the observability Gramian defined for outputs measured at y2 and

θ2. Since the output θ2 only is sensitive to the second and last mode, the combination

of y2 and θ2 will result in full observability of the system. The variable φs reads:

φs =

[φ(1)2 φ

(2)2 φ

(3)2 φ

(4)2

φ(1)3 φ

(2)3 φ

(3)3 φ

(4)3

]

=

[1.29 0 −1.17 0

0 −9.94 0 26.29

]. (3.95)

As for the previous output case, it can also be shown for this case that the approxima-

tion of Wo (3.86) is sufficiently accurate. Equation (3.86) therefore yields:

Wo ≈ diag

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

4.55 · 10−5

4.55 · 10−5

8.74 · 10−6

8.74 · 10−6

3.06 · 10−9

3.06 · 10−9

1.41 · 10−7

1.41 · 10−7

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

, (3.96)

61


showing that all modes are observable. The first mode still remains the largest contrib-

utor in the output energy of the system.

The next section defines the controllability Gramian, which is used to derive a

measure for controllability of modes from predefined input DoF.

3.4 Controllability Gramian

Consider the state space system description given in (3.9). For the solution of control

problems, it is important to know if the system can be steered from an any given state

into any other given state. This property is described by the concept of controllability,

which is defined as [26]:

Definition 3.4.1. The state space system (3.9) is said to be fully controllable if the

states of the system can be transferred by the input u (t) from any initial state x (t0) at

any initial time t0 to any state x (T ) within a finite time T − t0.

Using the definition, the following theorem holds for the time solution of the system,

equation (3.48), [30]:

Theorem 3.4.1. The system (3.9) is controllable on [t0, t] if and only if the 2N rows

of the matrix function eAs(t−τ)Bs are linearly independent on the interval [t0, t].

The theorem becomes more clear when one considers the particular solution of the time

response of the system:

xp (t) =

t∫0

eAs(t−τ)Bsu (τ)dτ (3.97)

Forces acting on a DoF are only able to bring the system to a desired state if they

are able to move the DoF in an arbitrary direction. Subsequently, to have influence

on the system’s deformation, the particular solution of the system xp (t) cannot be 0.

Due to the integral action it can therefore be stated that a DoF is controllable if and

only if xp �= 0 for any nonzero force u (τ) in the timespan τ ∈ [t0, T ]. The system is

called fully controllable if the particular solution xp (t) never becomes 0 for all DoF

in case of force interactions. Therefore, for controllability to hold, every desired state

xp inside the spanned solution space must be achievable by the input u (τ). The next

step is to show that this is only possible when eAs(t−τ )Bs consists of linear independent

equations (i.e. is full rank).

An arbitrary vector z which lies in the solution space spanned by the states can be

implemented into (3.97) such that:

zT

t∫0

eAs(t−τ )Bsu (τ)dτ = 0 ∀u (τ) ∈ u ([t0, T ]) . (3.98)

62

3.4. CONTROLLABILITY GRAMIAN

When the system is full rank z = 0 because full controllability ensures that all attain-

able vectors xp to all possible choices of u are spanned by the state space. Consider

the following theorem for the inner product, or Gramian matrix, of a set of vectors:

Theorem 3.4.2. A set of vectors v1,v2, . . . ,v2N with a Gramian matrix G are linearly

independent if

det (G) �= 0. (3.99)

When (3.98) holds, the Gramian matrix of eAs(t−τ)Bs has a nonnegative determi-

nant, proving the linear independency of eAs(t−τ )Bs. Consider a second case where the

input is chosen as

u (τ) = BTs e

ATs (t−τ)z, (3.100)

with z �= 0, equation (3.98) becomes,

t∫0

zT eAs(t−τ )Bsu (τ)dτ =

t∫0

zT eAs(t−τ)BsBTs e

ATs (t−τ )zdτ ,

=

t∫0

zTGzdτ, (3.101)

with G the Gramian matrix of eAs(t−τ )Bs. To prove linear independence and thus

det (G) �= 0 the following theorem is used:

Theorem 3.4.3. An 2N × 2N real symmetric matrix G is positive definite if and only

if for any nonzero vector z ∈ R2N :

zTGz > 0 (3.102)

When (3.102) holds, then all eigenvalues of G are positive and

det (G) > 0. (3.103)

The proof of symmetry of G is relatively straightforward because:[BT eA

T (t−τ)]T

= eA(t−τ)B. (3.104)

Positive definiteness is found when further evaluating (3.101):

t∫0

zTGzdτ =

t∫0

u (τ)Tu (τ)dτ

=

t∫0

‖u (τ)‖2 dτ > 0, (3.105)

63


proving that G is positive definite and therefore has a positive determinant. From the

Gramian theory it then follows that eAsτBs has 2N linear equations and thus is fully

controllable.

There is a possibility to extract a simpler method from the above to determine

whether a system is fully controllable. Let us begin with considering the Taylor expan-

sion of eAs(t−τ) implemented in the particular solution:

xp =

t∫0

∞∑k=0

Aks (t− τ)k

k!Bsudτ. (3.106)

Because the state matrix and input matrix are time independent, they can be placed

outside the integral:

xp =∞∑k=0

AksBs

t∫0

(t− τ)k

k!udτ (3.107)

Equation (3.107) shows that xp is a linear combination of the columns[Bs AsBs . . . A∞

s Bs

](3.108)

It is now easily seen that the rows of (3.108) need to be linear independent for theorem

3.4.1 to hold. In other words, the row space of (3.108) for a system with 2N states is

2N for a controllable system. Since (3.108) holds powers of As up to ∞, it is not easy

to find a solution in practice. Therefore the Cayley-Hamilton theorem 3.3.2 is used to

find that the row space of (3.108) is equivalent to the row space of[Bs AsBs . . . A2N−1

s Bs

]. (3.109)

Therefore the system is fully controllable if (3.109) has 2N independent rows, i.e. is

full rank, and can therefore be used to check the controllability of a system. Equation

(3.109) is also known as the controllability matrix .

To check whether a system is fully controllable, also the controllability Gramian

can be used. The Gramian is based on the amount of energy brought into the system

by u in equation (3.9). For control purposes it is desirable to minimize this energy such

that the system can be brought to a desired state x (T ) = xT using a minimal amount

of effort. The quadratic cost function used to derive the minimum energy reads [30]:

J =

T∫0

uT (t)u (t) dt (3.110)

Minimization of the cost function subject to time response of the system (3.48) with

boundary conditions x0 and xT results in the optimal solution4 [21, 29]:

u0 (t) = −BTs e

ATs (T−t)Wc

−1 (t = T )(eAsTx0 − xT

), (3.111)

4By use of Riemann sums.

64

3.4. CONTROLLABILITY GRAMIAN

where Wc (t) is the controllability Gramian, defined as:

Wc (t) �t∫

0

eAs(t−τ)BBT eATs (t−τ)dτ . (3.112)

Using equations (3.101) and (3.105) it is found that the controllability Gramian is

positive definite and full rank in case of full controllability.

If the timespan of the obtained input energy goes to infinity (T → ∞), the Lya-

punov equation can be obtained in the same fashion as the observability Gramian, i.e.

equations (3.78)-(3.82) and C:

AsWc (t = ∞) +Wc (t = ∞)ATs +BsB

Ts = 0, (3.113)

which can also be solved in close form using the block diagonal matrix As:

Wc,(m−1:m)(j−1:j) =βkldkl

[2ωkωl (ζkωl + ζlωk) ωl

(ω2l − ω2

k

)−ωk

(ω2l − ω2

k

)2ωkωl (ζkωk + ζlωl)

](3.114)

βkl =

R∑r=1

φ(k)r φ

(l)r

μkμl


l

)2,

with the indices for each block element inside Wc:

m and j = 2, 4, 6, . . . , 2N

k =m

2

l =j

2.

As for the observability Gramian, the controllability Gramian can be simplified in case

of small damping terms (0 < ζk � 1). The off-diagonal terms of Wc (m �= j) become

negligible because of the large denominator dkl , resulting in a diagonally dominant

matrix:

Wc,(m−1:m)(m−1:m) ≈ diag

⎛⎜⎜⎜⎝

R∑r=1

(φ(k)r

)24ζkωkμk

,

R∑r=1

(φ(k)r

)24ζkωkμk

⎞⎟⎟⎟⎠ (3.115)

In case of small damping terms, the eigenvalues of the controllability Gramian are

equal to equation (3.115). These eigenvalues can be used to identify the error in the

approximation in a similar fashion to that of the observability Gramian, i.e.:

e =‖diag (Wc −ΛWc)‖

‖diag (ΛWc)‖· 100% (3.116)

65


3.4.1 Constrained beam case: controllability Gramian

The constrained beam example in previous section assumed that the input u = pθ1 is

fully controllable. The validity of this assumption will be proved using the controlla-

bility Gramian. The input matrix for this specific input reads:

Bs =[

φ(1)1μ1

0φ(2)1μ2

0φ(3)1μ3

0φ(4)1μ4

0

]T=[4.06 0 9.94 0 21.39 0 26.29 0

]T(3.117)

The variable φr in (3.114) therefore becomes:

φr =[

φ(1)1μ1

φ(2)1μ2

φ(3)1μ3

φ(4)1μ4

]=[4.06 9.94 21.39 26.29

](3.118)

Substitution of the eigenfrequencies (3.30), modal damping (3.34) and eigenmodes(3.31) in equation (3.114) results in the following controllability Gramian:

Wc =⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

171.21 0 5.22 · 10−7 0.02 1.62 · 10−7 0.01 5.91 · 10−8 0.01

0 171.21 −3.52 · 10−3 1.09 · 10−6 −1.15 · 10−3 8.04 · 10−7 −4.22 · 10−4 5.32 · 10−7

5.22 · 10−7 −3.52 · 10−3 65.26 0 9.57 · 10−6 0.04 2.76 · 10−6 0.02

0.02 1.09 · 10−6 0 65.26 −0.01 1.39 · 10−5 −4.78 · 10−3 6.59 · 10−6

1.62 · 10−7 −1.15 · 10−3 9.57 · 10−6 −0.01 48.40 0 6.93 · 10−5 0.06

0.01 8.04 · 10−7 0.04 1.39 · 10−5 0 48.40 −0.04 8.22 · 10−5

5.91 · 10−8 −4.22 · 10−4 2.76 · 10−6 −4.78 · 10−3 6.93 · 10−5 −0.04 22.03 0

0.01 5.32 · 10−7 0.02 6.59 · 10−6 0.06 8.22 · 10−5 0 22.03

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

.

(3.119)

Because of the nonzero and dominant elements on the diagonal, it is clear that all modes

are controlled by input u = pθ1 , resulting in a full rank controllability Gramian. The

first mode experiences the highest energy contribution from the input. The eigenvalues

ΛWc of Wc read:

ΛWc = diag

([171.21 171.21 65.26 65.26 48.40 48.40 22.02 22.02

]T).

(3.120)

Due to the small damping terms, the approximation (3.115) should hold. Computing

the approximation yields:

Wc ≈ diag

([171.21 171.21 65.26 65.26 48.40 48.40 22.03 22.03

]T)(3.121)

66

3.5. FIELD CURVATURE MODE SELECTION

The relative error between the approximation and the eigenvalues of Wc are next

calculated in the two-norm format:

e =‖diag (Wc −ΛWc)‖

‖(ΛWc)‖100%

= 9.14 · 10−5%, (3.122)

showing that the approximation of the controllability Gramian introduces a relative

error less than 1%. Therefore it can be concluded that the approximation (3.115) is

sufficiently accurate in this example.

The observability and controllability Gramians for a state space system description

were derived in the previous sections. The next section uses this theory to develop a

method in determining which modes of the system have the most influence on a specific

output deformation, namely curvature of a field.

3.5 Field curvature mode selection

Consider a structure described by N generalized coordinates qn and with low structural

damping. As explained in chapter 2, two sets of DoF q can be defined. The interface

DoF qf which experience external and connection forces and the internal DoF qi. The

equations of motion for such a system yield:[Mff Mfi

Mif Mii

][qf

qi

]+

[Cff Cfi

Cif Cii

] [qf

qi

]+

[Kff Kfi

Kif Kii

][qf

qi

]=

[p

0

]+

[g

0

],

(3.123)

where p and g are respectively the external and interface force vectors. The indices for

the internal and interface DoF read:

qf =[qf1 qf2 · · · qfF

]T(3.124)

qi =[qi1 qi2 · · · qiI

]T(3.125)

Suppose the system is reduced using the Craig-Bampton method. From section

2.2.3 it is known that the reduction basis is equal to:[qf

qi

]=

[I 0

S Φi

][qf

ηi

]

= T q, (2.30)

where T represents the reduction matrix. The matrix S holds the static modes used to

condense the internal DoF into the interface DoF and Φi is the truncated set of mass

normalized interface fixed vibration modes. The solution accuracy of the dynamic

67


analysis of the reduced system is determined by the number of interface fixed vibration

modes spanning the reduction basis. Usually, a selection of modes up to a certain

frequency band are used in the basis, causing the solution error to be determined by

the mode spillover. This section introduces a procedure for selecting interface fixed

modes, where output specifications on a system determine an additional selection of

modes to the common procedure of modes contributing in the reduction basis.

3.5.1 Specific output realization

Let the focus of the system lie on a specific deformation output, or shape, with de-

pendency on the DoF. By taking the internal modes of the system up to a certain

frequency band for the Craig-Bampton reduction basis it is not directly satisfied that

the information regarding the specific deformation of the system is preserved. The

observability Gramian information can be used to identify which internal modes have a

high energy contribution in the specific deformation of the structure. This creates the

possibility to select a set of internal modes for the extension of the common reduction

basis, optimizing the reduction scheme for the output specification. Given (3.123), the

EoM of the internal DoF qi read:

Miiqi +Ciiqi +Kiiqi = 0 (3.126)

Calculation of the systems observability Gramians gives insight into which internal

modes are observable at the DoF and which are not. Since the main goal of the model

is to extract specific shape during dynamic analysis, one would like to know which

modes are observable in terms of the shape rather than DoF. This can be obtained

by projecting the output of the system onto a basis Υ [I ×M ], with I the amount of

internal DoF and an arbitrary M < I. This basis holds the relations between the DoF

y and the specific deformation output y:

y ≈ Υy. (3.127)

Because M < I, the subspace spanned by y is not fully covering the subspace of y.

There are I −M orthogonal directions to Υ which lie in the subspace spanned by y.

This therefore means that in mapping y to y information could get lost in the nullspace

of Υ. For energy calculations this than results in finding only a fraction of the energy

found at y. To maintain all energy contributions, Υ can be extended with its nullspace.

This nullspace spans the subspace of y that is orthogonal to the subspace spanned by

y [33, 34].

Let Υ be the [I × I] projection matrix built from Υ and its nullspace:

Υ =[Υ null

(Υ) ]

(3.128)

68


Energy calculations now require that energy is preserved in mapping y onto y. This

is obtained in orthonormalization of the Υ such that no scaling is implemented in the

mapping. If Υ is orthogonal, a clear distinction can be made between the constants

in y. This is required because one is interested in the specific output determined from

the first M columns of Υ. Therefore, in order to determine energy contributions of

the independent elements of y, orthogonality of Υ is required. Orthonormalizing Υ is

allowed since the focus lies on the output of a shape and not on the amplitude of this

shape and can be found by (for instance) performing Gram-schmidt orthogonalization

on Υ [35] (see appendix D). Caution is however required in performing the orthonor-

malization because the process is dependent on the layout of the columns of Υ. The

column prior to the column that is made orthonormal determines the direction of the

vector, so the process could combine shapes from different columns. An example of

this difficulty will be presented in section 3.5.3.

Note that augmentation of the nullspace to the projection basis Υ serves merely as a

check to determine whether the projected output has preserved the total amount of

output energy found from the original output y. If one is only interested in the modal

energy contribution of one or more of the first M components defined in y, nullspace

augmentation to the projection basis is not required. In the remainder of the chapter,

the nullspace will be augmented to the projection basis, for the sake of completeness.

3.5.2 Observability Gramian

Given the state space representation of the system (3.126), the output is equal to (see

(3.10)):

y = Csxi. (3.129)

When the output y is required it can be found from (3.127) that:

y = ΥTCsxi, (3.130)

with Υ the orthonormal projection matrix. Because y is mapped onto y using a

projection matrix which spans the whole solution space of y, the observability Gramian

of y is identical to the observability Gramian of y. If however one chooses to calculate

the Gramian for one (or several) of the components of y, then the Gramian represents

the energy contributions of the modes to the particular shape related to the component.

Let yv =[y1 . . . yV

]T(V < I) be the shapes of interest. The output yv can be

found by inserting an extra boolean input matrix Cv [V × I] such that:

yv = CvΥTCsxi (3.131)

69


Conform section 3.3 the observability Gramian for (3.131) where Cs is based on dis-

placement measurements (3.16) is found as:


[2ωikωilekl ωik

(4ζilωilekl −

(ω2il− ω2

ik

))ωil

(4ζikωikekl +

(ω2il− ω2

ik

))2(ζilω

3il+ ζikω

3ik

)+ 8ζikζilωikωilekl

]

(3.132)

dkl = 4ωikωil (ζikωik + ζilωil) (ζikωil + ζilωik) +(ω2ik− ω2

il

)2ckl =

1

ωikωil

V∑s1=1

(I∑

s2=1

Cvs1s2

I∑s3=1

Υs3s2φ(k)is3

)(I∑

s2=1

Cvs1s2

I∑s3=1

Υs3s2φ(l)is3

)

ekl = ζikωik + ζilωil ,

with

m and j = 2, 4, 6 . . . 2I

k =m

2

l =j

2.

Comparing equation (3.132) to (3.85) it shows that only the term ckl is altered. There-

fore the diagonal dominance of Wo in case of small damping terms and well spaced

eigenfrequencies is also present in output converted Gramian calculation. The approx-

imation of (3.132) thus becomes:

Wo,(m−1:m)(m−1:m)

≈ diag

⎛⎜⎜⎜⎜⎜⎝

V∑s1=1

(I∑

s2=1Cvs1s2

I∑s3=1

Υs3s2φ(k)is3

)2

4ζikω3ik

,

V∑s1=1

(I∑

s2=1Cvs1s2

I∑s3=1

Υs3s2φ(k)is3

)2

4ζikω3ik

⎞⎟⎟⎟⎟⎟⎠

(3.133)

An approximation error can be calculated using the eigenvalues of (3.132), as presented

in section (3.3).

This thesis focuses on dynamic system analysis of field curvature correction perfor-

mance in lithography systems. As explained in the introduction, reduced FE models

are used to perform this analysis by reduction techniques of chapter 2. In the reduc-

tion of the photomask there is a specific deformation output which must be maintained

in the reduction in order to successfully perform a system analysis on field curvature

correction performance. The theory of the section will therefore be applied on this

benchmark.

70


3.5.3 Field curvature output based Gramians

Considering the EoM of the internal DoF (3.126), let y = [qi1 , qi2 , . . . , qiW ]T (W < I)

be defined as displacements in z-direction and let their initial position be 0. In order

to map y onto a basis which contains curvature information, the anticlastic curvature

augmented fitting function defined in section 1.2.1 gives the possibility to relate curva-

ture to the displacements in z-direction. In matrix notation the fitting function for the

internal DoF yields:

y = Υy

=

⎡⎢⎢⎢⎢⎣

1 −xi1 yi112

(νy2i1 − x2i1

)1 −xi2 yi2

12

(νy2i2 − x2i2

)...

......

...

1 −xiW yiW12

(νy2iW − x2iW

)

⎤⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎣

o

Ry

Rx

κxx

⎤⎥⎥⎥⎥⎦ (3.134)

where xi and yi are respectively the x and y positions of the nodes belonging to qi, κxx

is the first order mechanical curvature of the deformation, Rx and Ry are the rotation

over the x and y axis respectively and o is an offset term. Given that the nodes are

positioned symmetric with respect to the x and y plane, the columns related to rotation

are orthogonal in the spanned space. In other words, the offset and curvature terms

in the first and last columns have linear dependency and therefore are not orthogonal.

Since orthonormality is required, Gram-Schmidt orthonormalization is able to produce

an orthonormal basis for Υ. The shapes of the columns are only preserved if the column

related to offset is prior to the column related to curvature. This can easily be seen if

one considers a simple 2-D fit (conform section 1.2.2) without rotation:

y = Υy

=

⎡⎢⎢⎢⎢⎣

1 −12x

2i1

1 −12x

2i2

......

1 −12x

2iW

⎤⎥⎥⎥⎥⎦[

o

κxx

], (3.135)

where, for instance, xi =[−1 −0.8 . . . 0.8 1

]. By construction the first column

of Υ represents the shape related to offset and the second curvature. Performing the

Gram-Schmidt orthonormalization on the projection matrix yields:

y =

⎡⎢⎢⎢⎢⎢⎢⎢⎣

0.30 −0.51

0.30 −0.20...

...

0.30 −0.20

0.30 −0.51

⎤⎥⎥⎥⎥⎥⎥⎥⎦[

o

κxx

]. (3.136)

71


The cases y =[1 0

]Tand

[0 1

]Tare plotted in figure 3.6. It is clear to see that

a

x

yb

x

y

-1 -0.5 0 0.5 1-1 -0.5 0 0.5 1-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

-1

-0.5

0

0.5

1

1.5

Figure 3.6: y plot for (a) y = [1, 0]T and (b)y = [0, 1]T

the first column still represents the shape related to offset and the second to curvature.

If the columns of Υ are swapped, followed by Gram-Schmidt orthonormalization the

relation would become:

y =

⎡⎢⎢⎢⎢⎢⎢⎢⎣

−0.57 −0.18

−0.36 0.05...

...

−0.36 0.05

−0.57 −0.18

⎤⎥⎥⎥⎥⎥⎥⎥⎦[

κxx

o

]. (3.137)

The cases y =[1 0

]Tand

[0 1

]Tfor (3.137) are plotted in figure 3.7. Now the

first column still represents the shape related to curvature but the second also shows

curvature, while offset is required. It therefore can be concluded that the columns of

Υ need to be ordered as (3.136) for the orthonormalization to preserve the shapes of

the columns.

Once the orthonormal Υ is obtained a total of W − 4 orthogonal directions to the

space spanned by y exist in the solution space spanned by y . Energy can subsequently

disappear in these directions. To avoid this, Υ is built from Υ by nullspace augmenta-

tion. The nullspace is per definition orthogonal to Υ, but does not per definition have

orthogonal columns. To ensure orthonormality, the nullspace is made orthonormal

using the Gram-Schmidt scheme.

The output y was defined as:

y =[qi1 qi2 · · · qiW

]T, (3.138)

72


a

x

y

b

x

y

-1 -0.5 0 0.5 1-1 -0.5 0 0.5 1-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

Figure 3.7: y plot for (a) y = [1, 0]T and (b)y = [0, 1]T

which is equal to

y =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

0φ(1)i1ωi1

· · · 0φ(I)i1ωiI

0φ(1)i2ωi1

· · · 0φ(I)i2ωiI

......

. . ....

...

0φ(1)iWωi1

· · · 0φ(I)iWωiI

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

ηi1ωi1ηi1ηi2

ωi2ηi2...

ηiIωiIηiI

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(3.139)

= Csxi (3.140)

The conversion to output y is next found by using equation (3.131). To obtain the

desired scalar output for curvature the output matrix Cv is equal to:

Cv =[0 0 0 1 0 · · · 0

]. (3.141)

Calculating the observability Gramian for (3.141) conform section 3.3, Wo only differs

from equation (3.132) in the term ckl5:

ckl =1

ωikωil

W∑s2=1

Υs2s1φ(k)is2

·W∑

s2=1

Υs2s1φ(l)is2

(3.142)

s1 = 4.

5Equation (3.142) gives a good example why nullspace augmentation of the projection basis is

not always required. Because only curvature is specified as output, the observability Gramian is only

dependent on the fourth column of Υ.

73


The approximation of (3.142) becomes:


⎛⎜⎜⎜⎜⎜⎝

(W∑

s2=1Υs2s1φ

(k)is2

)2

4ζikω3ik

,

(W∑

s2=1Υs2s1φ

(k)is2

)2

4ζikω3ik

⎞⎟⎟⎟⎟⎟⎠ (3.143)

An approximation error can be calculated using the eigenvalues of Wo, as presented in

section (3.3).

The observability Gramian based on the output y now represents the energy con-

tributions of the internal modes of the system to deformation with curvature shape.

When this calculation is performed, information is gained about which modes have high

contribution to curvature.

Because in this thesis a model is created to perform dynamic analysis on curvature

deformation, it is wise to make sure the selected internal modes are present in the

reduction basis T for the Craig Bampton reduction, constructed at equation (2.30).

Note that a selection of only internal modes with high curvature influence is not ad-

visable since dynamic properties of the system also rely on other deformations than

curvature. Also, small deviations on curvature are important in the field curvature cor-

rection analysis, because they represent information about disturbance effects in the

dynamic analysis. Therefore a well defined reduction basis contains a combination of

interface fixed modes up to a certain frequency band and a selection of modes with

high curvature influence, increasing the reduction accuracy for curvature deformation.

3.5.4 Constrained beam case: curvature mode selection

It is possible to determine the curvature dependency of the modes of the constrained

beam example, but this requires an additional mapping step. Since the DoF of the

beam consist of rotations and only one translation, there is little information relating

translations to curvature. Additional information about the translations in the elements

of the beam can be gained by use of their shape functions. For a single beam element

the shape functions related to the DoF

qe =[y1 θ1 y2 θ2

]T(3.144)

yield [12]:

fe =

⎡⎢⎢⎢⎢⎣

1− 3ξ2 + 2ξ3

lξ (1− ξ)2

ξ2 (3− 2ξ)

lξ2 (ξ − 1)

⎤⎥⎥⎥⎥⎦

T

, (3.145)

74


where ξ represents the non dimensional variable xl , running from 0 to 1. The deflection

at any point ξ in the beam can now be calculated as:

y (x) = feqe. (3.146)

In the case of the constrained beam example, displayed in figure 3.1, the DoF can

be mapped onto a new vector holding the deflections of the beam at different x posi-

tions. For any even set of W equal spaced x positions symmetric about the y-axis, the

deflection of the beam can be calculated as:

y = Fq⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

y1...

yW2

yW2 +1

...

yW

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

fe2 (ξ (x1 + l)) fe3 (ξ (x1 + l)) fe4 (ξ (x1 + l)) 0...

......

...

fe2

(ξ(xW

2+ l))

fe3

(ξ(xW

2+ l))

fe4

(ξ(xW

2+ l))

0

0 fe1

(ξ(xW

2 +1

))fe2

(ξ(xW

2 +1

))fe4

(ξ(xW

2 +1

))...

......

...

0 fe1 (ξ (xW )) fe2 (ξ (xW )) fe4 (ξ (xW ))

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎣

θ1

y2

θ2

θ3

⎤⎥⎥⎥⎦ ,

(3.147)

providing a new set of DoF with W elements from which curvature can be distracted.

The matrix F is formulated from the shape functions of each element. The state space

output mapped onto y for the system now becomes:

y = FCsx, (3.148)

with

Cs =

⎡⎢⎢⎢⎣

0φ(1)1ω1

· · · 0φ(I)1ωI

......

. . ....

...

0φ(1)Iω1

· · · 0φ(I)IωI

⎤⎥⎥⎥⎦ (3.149)

with I = 4 and constructed for position measurements, as defined in equation (3.16).

The observability Gramian is for this output is calculated in the same fashion as

(3.132) although only the term ckl is altered:

ckl =1

ωikωil

W∑s1=1

(I∑

s2=1

Fs1s2φ(k)is2

·I∑

s2=1

Fs1s2φ(l)is2

)(3.150)

I = 4.

In the previous examples it became clear that the approximation for Wo is accurate

75


within 1%. The approximation in this setting yields:


⎛⎜⎜⎜⎜⎜⎝

W∑s1=1

(I∑

s2=1Fs1s2φ

(k)is2

)2

4ζikω3ik

,

(W∑

s1=1

I∑s2=1

Fs1s2φ(k)is2

)2

4ζikω3ik

⎞⎟⎟⎟⎟⎟⎠ ,

(3.151)

and will be used in the remainder of this example. Let us calculate the observability

Gramian approximation for W = 10. Substitution of the eigenfrequencies (3.30), modal

damping (3.34), eigenmodes (3.31) and projection matrix F into equation (3.151) re-

sults in:

Wo ≈ diag

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

2.01 · 10−4

2.01 · 10−4

6.55 · 10−7

6.55 · 10−7

1.64 · 10−8

1.64 · 10−8

1.50 · 10−9

1.50 · 10−9

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(3.152)

Note that Wo is full rank, which also is the case for the observability Gramian of

the full output y (section 3.3). Wo is now related to all the elements in y, showing

which modes are observable at its DoF. Therefore the energy for an initial state x0,

calculated using (3.76), sums up all energy measured at the DoF y, which is not equal

to calculating the energy for y.

Assume the initial state vector is unitary, the energy found for y is calculated as

(using (3.76)):

E = xT0 Wox0

≈ 4.04 · 10−4[m2]

(3.153)

To find an orthonormal basis containing curvature deformation on which y can be

76


projected, the starting point is the fitting function (3.134):

y Υy

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 0.5 −0.13

1 0.39 −7.6 · 10−2

1 0.28 −3.9 · 10−2

1 0.17 −1.4 · 10−2

1 5.6 · 10−2 −1.5 · 10−3

1 −5.6 · 10−2 −1.5 · 10−3

1 −0.17 −1.4 · 10−2

1 −0.28 −3.9 · 10−2

1 −0.39 −7.6 · 10−2

1 −0.5 −0.13

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎣

o

Ry

κxx

⎤⎥⎦ , (3.154)

where Rx = 0 since the example is 2-D. Orthonormalization of Υ yields:

Υ =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0.32 0.5 −0.52

0.32 0.39 −0.17

0.32 0.28 8.7 · 10−2

0.32 0.17 0.26

0.32 5.5 · 10−2 0.35

0.32 −5.5 · 10−2 0.35

0.32 −0.17 0.26

0.32 −0.28 8.7 · 10−2

0.32 −0.39 −0.17

0.32 −0.5 −0.52

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

, (3.155)

where the orthogonal columns preserve the shapes defined by y. Calculating the or-thonormal nullspace of Υ and augmenting it to Υ results in Υ:

Υ =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0.32 0.5 −0.52 −9.6 · 10−3 0.4 0.21 −7.4 · 10−2 0.18 −0.34 −0.17

0.32 0.39 −0.17 −0.18 −7.8 · 10−2 −0.17 0.2 −0.19 0.76 6.9 · 10−2

0.32 0.28 8.7 · 10−2 −0.13 −0.63 −5.8 · 10−2 0.16 −0.3 −0.5 0.2

0.32 0.17 0.26 0.3 −2.8 · 10−2 −0.43 −0.7 0.19 4.4 · 10−2 8.3 · 10−2

0.32 5.5 · 10−2 0.35 0.36 −0.13 0.64 0.18 0.37 0.17 0.12

0.32 −5.5 · 10−2 0.35 0.24 0.39 −0.42 0.55 −6.5 · 10−2 −0.16 −0.25

0.32 −0.17 0.26 −0.23 0.2 0.38 −0.33 −0.59 3.2 · 10−2 −0.33

0.32 −0.28 8.7 · 10−2 −0.68 −0.1 −9.6 · 10−2 8.0 · 10−3 0.55 −1.4 · 10−2 −0.19

0.32 −0.39 −0.17 −6.1 · 10−2 0.33 −1.2 · 10−3 9.5 · 10−3 −0.1 −5.6 · 10−2 0.77

0.32 −0.5 −0.52 0.39 −0.34 −5.3 · 10−2 −2.0 · 10−3 −4.4 · 10−2 6.5 · 10−2 −0.32

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(3.156)

creating a orthonormal projection matrix to map y onto y. The state space description

for the desired output yv based on curvature now becomes using (3.156), (3.148) and

(3.129) to (3.131) now becomes

yv = CvΥTFCsxi. (3.157)

77


The observability Gramian for displacement measurements is in the same fashion as

(3.132) but altered for the implementation of F . The only term changed is ckl:

ckl =1

ωikωil

V∑s1=1

(W∑

s2=1

Cvs1s2

W∑s3=1

Υs3s2

I∑s4=1

Fs3s4φ(k)is4

)(W∑

s2=1

Cvs1s2

W∑s3=1

Υs3s2

I∑s4=1

Fs3s4φ(l)is4

).

(3.158)

The approximation of Wo yields:

Wo,(m−1:m)(m−1:m) ≈

diag

⎛⎜⎜⎜⎜⎜⎝

V∑s1=1

(W∑

s2=1Cvs1s2

W∑s3=1

Υs3s2

I∑s4=1

Fs3s4φ(k)is4

)2

4ζikω3ik

,

V∑s1=1

(W∑

s2=1Cvs1s2

W∑s3=1

Υs3s2

I∑s4=1

Fs3s4φ(k)is4

)2

4ζikω3ik

⎞⎟⎟⎟⎟⎟⎠

(3.159)

When the complete output y is observed (i.e. Cv = I [W ×W ]) equation (3.159) is

found to be equal to (3.152) and the energy considering an initial unit state vector is

equal to (3.153). This shows that the transformation matrix Υ has not applied any

scaling of the output but only changed the basis of the output.

If one only wants to calculate which modes have energy contributions to the defor-

mations related to curvature one now simple changes Cv into:

Cv =[0 0 1 0 0 0 0 0 0 0

](3.160)

The approximation of the observability Gramian (3.159) then becomes:

Wo ≈ diag

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

5.75 · 10−5

5.75 · 10−5

0

0

2.34 · 10−9

2.34 · 10−9

0

0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

, (3.161)

showing that only the first and third modes have influence on curvature deformations.

To check whether this is true, a simple calculation can be performed on the modes

converted to match the output vector y using the shape functions:

Φ = FΦ. (3.162)

78

3.6. SUMMARY

Constant φ(1)

φ(2)

φ(3)

φ(4)

o 1.25 0 -0.13 0

Ry 0 1.74 0 0.73

κxx 10.25 0 -7.2 0

Table 3.1: Anticlastic curvature augmented fitting function constants for the separate

modes.

By extraction of the coefficients for the anticlastic curvature augmented fitting function

(section 1.2.2) one can easily determine which shapes can be found in the modes. Table

3.1 shows the coefficients extracted for each mode. From the table it is visible that only

the first and third mode have curvature dependency. This is equal to the findings using

the observability Gramian.

3.6 Summary

In this chapter a method was derived for selecting a Craig-Bampton reduction basis

based on the specific output requirements of a system. The analysis in this thesis fo-

cuses on curvature and using the observability Gramian one can find the internal fixed

modes which contribute to curvature deformation. These modes then can be used as

extension to the reduction basis, providing additional conservation of curvature de-

formation after reduction of the model. Before using this theory, some remarks are

required:

Controllability of the internal modes.

After finding the modes influencing curvature, the question remains if these modes

are controllable by the interface forces. Usually, this is easily obtained by calculating

the controllability Gramian, but in this setting that is not possible. The system is

fixed on its interfaces and the DoF on which forces are exerted (qf ) are not present.

Therefore no controllability Gramian can be obtained. A possible check is calculating

the controllability matrix or Gramian for the complete system (3.123) conform section

3.4. The modes of the complete system differ from the interface fixed system, therefore

the internal modes are controllable if and only if the controllability matrix or Gramian

is full rank. Indeed, this is true because every deformation in the solution space of the

complete system can be described by a linear combination of its modes (provided that

damping is low). Therefore, if fully controllable, the modes of the internal fixed system

can also be described and are controllable as well. However, some complications in

computation of the controllability could be encountered:

79


• Presence of rigid body modes require additional computation steps, see [20, 36].

• Large systems require large computational power in finding the controllability

Gramian.

Optimality of the reduction.

It was explained in chapter 2 that Craig-Bampton describes the internal DoF of a

system as:

qi = Sqf +Φiηi, (2.29)

where S represents the static modes and Φi the interface fixed modes of the system.

For field curvature output based Gramian calculation the first part of the internal DoF

description, Sqf , is not taken into consideration. Because the static modes, found

by Guyan’s reduction, also contribute to the response of the internal DoF and their

contribution is not by definition orthogonal to shapes of the interface fixed modes,

the reduction optimality is not ensured. It could be that a rather high amount of

contribution to curvature is already described by the static modes in the reduction

basis. It is therefore recommended that in future research the contribution of the static

modes to curvature deformation is investigated.

80

Part II

Photomask curvature analysis

methodology

81

Chapter 4

Curvature analysis procedure

In the introduction it was set out that state of the art wafer nonconformity corrections

in lithographic systems require improvement in order to meet future imaging require-

ments due to the continuous drive to manufacture smaller semiconductor products. In

lithographic systems leveling strategies position the substrate in the optimal focal plane

of the lens, creating an optimal projection of the photomask image onto the substrate.

Conventional leveling uses translational and rotational setpoints in order to determine

the optimal position. This thesis discuses an extension to this conventional method

by augmenting a curvature correction to the leveling by bending of the photomask,

creating the possibility to improve focus in the lithographic system. As presented in

the introduction, the following research topics are discussed:

• Specification derivation: Analysis of wafer flatness and identification of the

required amount of curvature correction in state of the art wafers.

• performance evaluation: Analysis on focus budget improvements using a me-

chanical model of the photomask bending actuator.

The analysis procedure of the specification derivation is presented in section 4.1, fol-

lowed by the performance evaluation procedure in section 4.2. This chapter can there-

fore be considered as the guide on how to achieve the thesis objectives.

4.1 Specification derivation

In order to analyze focus improvement potential of curvature augmented leveling, spec-

ifications must first be derived. For the specification derivation, an ideal photomask

bending actuator and wafer stage are assumed. Furthermore, state of the art substrate

flatness measurements are used for the analysis. The implementation of the curvature

augmented leveling can be subdivided in two approaches:

83

CHAPTER 4. CURVATURE ANALYSIS PROCEDURE Company-secret

• Focus improvement on dynamic shape adaptation of the photomask during a

scanning motion of an exposed substrate area (die). This process is also referred

to as intradie leveling.

• Focus improvement on static shape adaptation of the photomask during the scan-

ning motion of a die. In this procedure the photomask has a static shape during

scanning, based on the average curvature of the exposed die. This process is also

called interdie leveling.

Section 4.1.1 discusses the identification procedure of intradie curvature augmented

leveling focus improvement, followed by the identification procedure of the interdie

focus improvement in section 4.1.2.

Besides calculating the focus improvement for both curvature correction procedures,

the specification derivation process also provides the following information for the me-

chanical performance evaluation:

• Bending actuator resonance specifications: Due to the scanning nature of

the exposure process, the curvature setpoints for the photomask vary in time.

These setpoints can not be tracked if mechanical resonances are present in the

relevant frequency range of the curvature setpoints.

• Curvature setpoints: The derived curvature values are used as curvature set-

points for the mechanical model. From the model, realized photomask curvature

is derived.

The mechanical resonance specification theory is covered in section 4.1.3. The im-

plementation of curvature setpoints as actuator command signals for the mechanical

model is discussed in section 4.2.

4.1.1 Intradie leveling

This section covers the focus improvement potential analysis of intradie curvature aug-

mented leveling with respect to the present leveling strategy. As explained in the

introduction (figure 2), the projection of the photomask pattern onto the substrate

is performed using a light slit. To expose a whole die area, the reticle stage moves

through the projection beam whilst the wafer stage moves in the opposite direction.

To achieve optimal focus in the machine during scanning, the leveling setpoints are

continuously adopted to the changing wafer topology. For each position of the expo-

sure beam during scanning (also referred to as slit position) the leveling setpoints are

obtained from the measured height of the exposed substrate area using the anticlastic

curvature augmented fitting function which was derived in section 1.2. The curvature

augmented leveling (CAL) setpoints are:

84

4.1. SPECIFICATION DERIVATION

• Offset in z-direction, oz.

• Rotation around the y-axis, Ry.

• Rotation around the x-axis, Rx.

• Curvature in x-direction, κxx.

For a topology of m points the setpoints are calculated in a least squares sense using

equations (1.71) and (1.73):

α =(XTX

)−1XTh, (4.1)

with

α =[oz Ry Rx κxx

]T(4.2)

X =

⎡⎢⎢⎢⎢⎢⎢⎢⎣

1 −x1 y112

(νy21 − x21

)1 −x2 y2

12

(νy22 − x22

)1 −x3 y3

12

(νy23 − x23

)...

......

...

1 −xm ym12

(νy2m − x2m

)

⎤⎥⎥⎥⎥⎥⎥⎥⎦, (4.3)

and h the topology of each measured point on the substrate. Because in the actual

leveling process the curvature is adapted by the photomask, the Poisson ratio in (4.3) is

set to the material properties of the photomask. Because of the overdetermined nature

of the problem, the result of the fitted topology hfit is equal to:

hfit = Xα. (4.4)

The focus error for each point found between the fit and the actual topology of the

exposed area is next calculated using (1.69):

e (x, y) =

(oz (xs, ys)−Ry (xs, ys) qx +Rx (xs, ys) qy +

1

2κxx (xs, ys)

(νq2y − q2x

))−h (x, y) .

(1.69)

Once the setpoints and focus errors for every slit position are calculated, a focus

error ef is calculated for each measured point on the wafer. Due to the continuous

adaptation of the leveling setpoints during scanning, all datapoints are used multiple

times to calculate the setpoints of the slit positions. Therefore there are multiple focus

errors available for every measured datapoint on the wafer. To express a single focus

error ef for each datapoint, the following statistical error computation adopted:

ef (x, y) = e (x, y) + 3σe(x,y), (4.5)

85


where e is a vector holding all focus errors for a single data point. Equation (4.5) can

also be written as:

ef (x, y) = MA (x, y) +MSD (x, y) , (4.6)

with MA the Moving Average and MSD the Moving Standard Deviation. In this thesis,

only the MA error will be treated, since it represent the average error of each datapoint

on the wafer. Subsequently a single scalar focus error ew for the whole wafer is obtained

using the average and variational properties of the focus errors ef :

ew = |μMA|+ 3σMA, (4.7)

with |μMA| the absolute average of the MA focus error of all datapoints on the sub-

strate. Note that if not one but multiple substrates are considered for focus error

calculation, equation (4.7) can still be used. The solution then represents the sum of

the absolute mean and three times the standard deviation of all processed datapoints

from the set of substrates.

To make a comparison with the current leveling strategies, the same procedure

(equations (4.1)-(4.5)) is performed without the implementation of the fitting variable

κxx. The obtained setpoints oz, Rx and Ry for every slit position are used in the

conventional leveling strategy and are referred to as CL setpoints. Comparing the ew

of the CL to the ew of the CAL gives an indication of the focus improvement of the

curvature augmentation in leveling strategies.

The process used to identify the intradie leveling focus improvement is represented

by the flow chart of figure 4.1. Note that all variables concerning the CL strategy have

an appended tilde. It is also important to note that the leveling setpoints have become

vectors, because the setpoints are calculated for every slit position. The topology re-

construction hfit has become a matrix holding the reconstruction of every slit position.

The flowchart is used in section 5.2.1 to calculate the focus improvement of intradie

curvature augmented leveling for state of the art wafers. The next section treats the

procedure to identify the focus improvement of interdie curvature augmented leveling.

4.1.2 Interdie leveling

Next to intradie leveling also interdie leveling can be applied to reduce focus dependency

of substrate flatness. Interdie curvature augmented leveling applies a static curvature

shape to the photomask during the scan of a die. This shape follows from the av-

erage curvature setpoints calculated during the intradie curvature augmented leveling

strategy. The downside of interdie leveling is reduced focus improvement. For the im-

plementation of a bending actuator in lithographic systems however, interdie leveling

has advantages over intradie correction which are:

86


e

h

e

ew

substrate topology

CAL setpoints CL setpoints

oz,Ry,Rx,κxx oz, Ry, Rx

topology reconstruction

hfit hfit

ew

defocus error

++ −−

focus comparison

Figure 4.1: Intradie focus improvement identification flowchart.

87


• Reduced computational effort: Because a constant curvature is used during

scanning, less computational effort is needed in comparison to real time dynamic

adaptation of the photomask curvature.

• Reduced dynamic disturbances: Intradie leveling continuously introduces

disturbance forces in the system due to the dynamic adaptation of the photomask

shape. Interdie leveling only introduces disturbance forces when the static shape

of the photomask is adopted before scanning a specific die.

• Reduction in power: The photomask bending actuator requires electrical

power. This power results in heat dissipation on the reticle stage, resulting in

thermal deformation of the stages. When only a static shape of the photomaks

is set during scanning, a limited amount of power is consumed by the actuators,

resulting in less disturbance effects from heat loading.

From a design point of view it is interesting to investigate whether interdie leveling

is a suitable option for implementation inside lithographic systems in contradiction

to intradie leveling. When interdie leveling has a minimal loss of focus improvement

with respect to intradie leveling strategies, it is more efficient to implement interdie

curvature augmented leveling strategy into lithographic systems because of the above

mentioned advantages.

The interdie curvature augmented leveling process is based on the intradie leveling

but has the following additional process steps:

• Once the CAL setpoints are calculated for every slit position of the exposed area,

the offset and rotational setpoints oz, Ry and Rx respectively are discarded. The

curvature setpoints κxx are used to calculate an average (scalar) setpoint κxx for

the whole exposed area.

• Using the average curvature setpoint, offset and rotational setpoints are again

fitted for every slit position using the anticlastic curvature augmented fitting

function. They are referred to as oz, Ry and Rx respectively and are found in a

least squares sense by solving:⎡⎢⎢⎢⎢⎣

1 −x1 y1

1 −x2 y2...

......

1 −xm ym

⎤⎥⎥⎥⎥⎦⎡⎢⎣

oz

Ry

Rx

⎤⎥⎦ = h−

⎡⎢⎢⎢⎢⎣

12

(νy21 − x21

)12

(νy22 − x22

)...

12

(νy2m − x2m

)

⎤⎥⎥⎥⎥⎦ κxx

Xα = h, (4.8)

which leads to

α =(XT X

)−1XT h, (4.9)

88


for every slit position.

• The realized aerial image projection hfit for every slit position is found by:

hfit = X

[α

κxx

](4.10)

The flow chart of the interdie CAL focus improvement identification is presented in

figure 4.2. The flowchart is used in section 5.2.2 for calculating the focus improvement

e

h

e

ew

substrate topology

CAL setpointsCL setpointsκxx

oz, Ry , Rx


hfit hfit

ew

oz, Ry,Rx

average curvature

κxx

static CAL setpointsoz, Ry, Rx, κxx

defocus error

++ −−

focus comparison

Figure 4.2: Interdie focus improvement identification flowchart.

of interdie CAL using state of the art wafers. Comparing these improvements to the

intradie leveling focus improvements gives a clear indication of the relevance of the

dynamic adaptation of the photomask or if the static adaptation will suffice. The next

section covers the additional use of the specification derivation with respect to the

mechanical resonance specification for the photomask bending actuator.

89


4.1.3 Mechanical resonance specification

Intradie curvature augmented leveling strategies require the dynamic adaptation of

the shape of the photomask during a scan of an exposed die. Therefore the curvature

setpoints sent to the photomask bending actuator depend on time and inhabit frequency

content. Due to the mechanical actuation principle, the focus improvement of intradie

leveling is limited by the mechanical resonances of the bending design. These dynamic

properties prevent some frequency contents of the curvature setpoints to be tracked

by the bending actuator. To determine the influence of this performance limitation, a

study is performed on the loss of curvature augmented leveling focus improvement due

to mechanical limitations.

The bandwidth limiting effect of the mechanical design is modeled by filtering the

intradie curvature setpoints before calculating the photomask projection on the sub-

strate (as discussed in section 4.1.1). The offset and rotation setpoints are not filtered

because they are inputs for the wafer stage, which is assumed ideal. The photomask

projection following from the filtered curvature setpoints�κxx and unfiltered rotation

and offset setpoints then represents the leveling capability of a photomask bending

actuator with mechanical bandwidth limitations at the filtering frequency. The result-

ing focus error with respect to the intradie focus error from an ideal bending actuator

represents the loss of focus error due to mechanical limitations of the actuator. The

flowchart of the identification of the CAL focus improvement with the implementation

of a bandwidth limiter is presented in figure 4.3. The bandwidth limiter represents a

zero-phase IIR filtering sequence. The theory on signal filtering is discussed in [37].

When performing a loop with different filter cutoff frequencies a clear representation

can be made of the focus loss due to mechanical bandwidth limitation with respect

to an ideal mechanical bending actuator. A minimal attainable resonance frequency

requirement for the mechanical design can be extracted in order to balance the loss of

focus versus the design realization. Results of this topic are discussed in section 5.2.3.

The flowchart is used in section 5.2.3 for calculating the focus improvement of

bandwidth limited intradie curvature augmented leveling. From the results a mechan-

ical resonance specification is extracted which serves as a design specification for the

reticle bending actuator.

4.2 Performance evaluation

The analysis procedure of the previous section is the stepping stone for the analysis

procedure of the mechanical design performance investigation. To fulfill the assign-

ment definition an additional study is required in which the field curvature correction

90

4.2. PERFORMANCE EVALUATION

�e

h

e

�ef

substrate topology

CAL setpointsCL setpointsκxx

oz, Ry , Rx


�

hfit hfit

ef

oz, Ry,Rx

bandwidth limiter

�κxx

defocus error

++ −−

focus comparison

ˆ

Figure 4.3: Bandwidth limited intradie leveling focus improvement identification

flowchart.

91


performance of the mechanical plant (see P in figure 5) is investigated. In real life the

mechanical plant is the photomask actuator. In this case however the plant consists

of a model of the actuator which is able to apply bending moments to the photomask.

During exposure these moments generate a deformation of the photomask which cre-

ates an aerial image suited for correcting field curvature of the exposed substrate area.

The design of the mechanical plant is presented in figure 4.4. As seen in the figure,

Photomask

Bending actuator

Bending actuator

x

zy

Figure 4.4: Photomask-actuator assembly.

two piezoelectric actuator arrays capable of introducing a torque into the photomask

are positioned at the two opposite edges [38]. Figure 4.5 shows the deformation of the

reticle when a voltage is applied to the piezoelectric actuators. Chapter 6 will provide

Figure 4.5: Deformed photomask caused by the actuator torques.

92


more detailed information about the photomask-actuator assembly.

Field curvature correction induced by mechanically deforming the photomask re-

duces focus improvement compared to an ideal curvature correction, because of the

following effects:

• Non-uniform torque application onto the photomask by the bending actuator.

Because of the discrete actuator locations, the introduction of torque into the

photomask is not ideal.

• Mechanical resonances in the photomask-actuator assembly. Mechanical reso-

nances of the system can limit curvature tracking performance.

• Additional constraints on the photomask. Inside lithographic systems the pho-

tomask is overconstrained on three positions. These constraints are referred to as

z-supports. The z-supports restrain the photomask from deforming into a perfect

curved shape.

• Parasitic stiffness effects of the actuator design. Parasitic stiffnesses in the design

cause parasitic force introductions into the photomask where a pure moment is

desired.

The goal of the performance evaluation is to find out how much focus improvement

can actually be obtained during the scanning process of an exposed substrate area h

using a curvature correction achieved by the photomask and actuator configuration.

This identification requires a number of process steps which define the structure of the

remainder of this thesis:

1. Numerical modeling and validation of the plant.

Chapter 6, sections 6.1 and 6.2.

To predict the curvature tracking capability of the of the photomask-actuator assem-

bly, a numerical model is required. Because the complexity of the design complicates

the use of standard lumped mass modeling, the components of the photomask-actuator

assembly are modeled in FE. The similarity between the models and the actual design

is investigated by an experimental dynamic analysis of the components.

2. Reduction and assembly of the photomask-actuator parts.

Chapter 6, section 6.3.

The complexity of the different photomask-actuator assembly FE models require the

use of component reduction methods in order to keep the numerical models within

acceptable computational limits. Therefore the components are reduced and assembled

93


using the theory set out in chapter 2. The reduction method has influence on the

accuracy and efficiency of the model. To keep computational efforts to a minimum,

two reduction methods are investigated:

• Guyan reduction, see section 2.2.1.

• Craig-Bampton reduction, see section 2.2.3.

During the Craig-Bampton reduction of the photomask, special attention is given to

the definition of an optimal reduction basis which preserves the curvature output of

the mechanical plant. The reduction basis is created using the theory on field curva-

ture mode selection, discussed in section 3.5. After reduction, the mechanical plant is

modeled by assembly of the components.

3. Performance tracking.

Chapter 7.

After realization of the (reduced) numerical model of the mechanical plant the per-

formance tracking of the plant is studied. In performance tracking, the curvature

correction performance of the mechanical plant during scanning is investigated. This

requires the following steps:

• Curvature setpoints for the numerical model.

Using the anticlastic curvature augmented fitting function, for every slit position

during scan the required curvature κxx is determined from the exposed substrate

area. These values are then used to calculate setpoints uκ for the bending actu-

ators in the numerical model. From the anticlastic curvature augmented fitting

function, also the translation and rotation setpoints for the wafer stage are ob-

tained. During this analysis the wafer stage tracking performance is assumed

ideal.

• Time integration.

Because the curvature setpoints belong to specific slit positions during scan-

ning, they are dependent on the scanning time. The dynamic response of the

photomask-actuator assembly during the scan then can be obtained from a time

integration of the model during the input of the curvature setpoints.

• Curvature tracking.

Having obtained time response of the model, for every slit position the curvature

can be calculated from the deformation data of the photomask. In chapter 1 it was

discussed that the photomask is considered to be a plate like structure, therefore

the curvature of the photomask deformation is calculated with the anticlastic

94


curvature augmented fitting function. The presence of parasitic stiffnesses and

constraints in the photomask-actuator assembly introduce additional offset and

rotational outputs of the photomask. These outputs are assumed to be neutralized

by augmentation of the wafer stage setpoints.

• Defocus error calculation.

Once the curvature output of the numerical model is known, the realized pho-

tomask projection during the scan can be calculated. Comparing the projection

to the actual exposed substrate topology results in the MA focus error of the

scan.

• Focus improvement.

In this step the curvature augmented leveling MA focus error is compared to

the MA focus error of the conventional leveling strategy (determined in section

4.1). This then gives the focus improvement of the curvature augmented leveling

strategy using the mechanical plant.

Similar to the specification derivation, the effectiveness of the curvature augmented

leveling is considered in three cases:

• Intradie leveling.

• Filtered intradie leveling. Curvature setpoints filtered to a certain bandwidth to

reduce disturbance effects of the plants dynamic properties.

• Interdie leveling.

In chapter 8 conclusions are drawn for the photomask curvature analysis. Based on

the process steps in this section the following conclusion can be obtained:

• Overall focus improvement of the curvature augmented leveling using the photomask-

actuator.

• Best reduction method. The best reduction method is found to be the method

which has the largest reduction without loss of model accuracy.

• Best setpoint specification. From the focus improvements derivations conclusions

can be drawn between the use of intradie, filtered intradie and interdie leveling.

Figure 4.6 represents the flowchart of the performance evaluation for the curvature

analysis. This flowchart is used throughout the remainder of this thesis and guidance

towards the thesis objective.

95


−−

−

++

+

ˆ

V

V

Mechanical plant

Modeling & validation

Guyan reduction

oz, Rx, Ry,see fig 4.1

CB reduction

Assembly

Time integration Setpoint calculationκxx

κxx,oz ,Rx,Ry, see fig 4.1

κxx, oz , Rx, Ry, see fig 4.2

�κxx,

�oz,

�

Rx,�

Ry, see fig 4.3

oz,Rx,Ryo(P )z ,R

(P )x ,R

(P )yCurvature tracking

κ(P )xx

Topology reconstruction

h(P )fit

exposed topology

h

e(P ) e

Defocus error

Focus comparison

hfit

e(P )w ew

uκ (t)

Interface fixed modes

Figure 4.6: Performance evaluation flowchart.

96

Chapter 5

Specification derivation

procedure

This chapter discusses the analysis of wafer unflatness and the identified curvature in

state of the art wafers. Section 5.1 covers the measurement procedures of substrate

topologies in lithographic systems. Using these measurements, specifications are de-

duced in section 5.2 for the range of curvature setpoints, that will be used as input for

the mechanical plant. Once these specifications are known, the focus improvement of

lithographic systems using ideal CAL strategies can be identified. The CAL strategies

are divided into intradie (section 5.2.1) and interdie leveling (section 5.2.2). Finally, a

mechanical resonance specification for the mechanical plant is obtained in section 5.2.3.

5.1 Wafer topology measurements

In lithographic systems the topology of the substrates is measured before the wafer

undergoes exposure. The unflatness information is used to position the wafer stage in

the optimal focal plane of the lens, reducing wafer flatness influence during exposure.

The wafer topology measurement procedure can be divided into two steps:

• Raw data acquisition. In this step the topology of the whole substrate is

measured using a triangulation measurement principle.

• Data postprocessing. Because the exposure process is performed for individual

dies, the raw data of the topology map is sectioned into the die area’s which are

to be exposed.

The raw data extraction is discussed in section 5.1.1, followed by the data postprocess-

ing in section 5.1.2.

97

CHAPTER 5. SPECIFICATION DERIVATION PROCEDURE Company-secret

5.1.1 Raw data acquisition

The topology measurement of a substrate is performed using an array optical level

sensors. During the measurement the wafer stage, on which the substrate is clamped,

moves past the sensors, creating measurement strokes along the y-axis of the substrate.

The measurement procedure is visualized in figure 5.1. The level sensor array consists

start end

x

y

Figure 5.1: Measurement strokes of the wafer topology measurement.

of 9 sensors which are placed at intervals of 3.4 [mm] in x-direction. During scanning in

y-direction, discrete datapoints are obtained with intervals of 0.51 [mm]. The scanning

action on a stroke is visualized in figure 5.2.

x

y

z

0.51 [mm]

3.4 [mm]

Figure 5.2: Discrete level sensor measurements on a stroke of the substrate.

98

5.1. WAFER TOPOLOGY MEASUREMENTS

The obtained measurement data of the substrate is referred to as raw topology data.

Once the whole wafer is measured a topology map is obtained such as in figure 5.3.

The large deviations at the edges of the wafer are height measurements of the level

z[μm]

y [mm] x [mm]−200−100

0100

200

3

3.5

4

4.5

5

5.5

6

6.5

7

7.5

8

−200−1000

100200

2

4

6

8

10

Figure 5.3: Raw topology data of a measured substrate.

sensors which are not on the substrate, therefore they are called off wafer datapoints.

The off wafer datapoints are of no use for the leveling process and are discarded in the

postprocessing steps.

State of the art lithographic systems also perform an accuracy enhancing mea-

surement step called Agile correction. Using an air gage sensor (which has a higher

accuracy in comparison to the level sensors), several die topologies are remeasured on

the substrate. Thereafter, topologies measured by the level sensors and air gage sensor

are compared and a correction on the measured level sensor data is performed.

5.1.2 Data postprocessing

Lithographic systems are able to project all kinds of patterns onto substrates which

enable fabrication of customer specific IC designs. The dimensions of dies on the

substrate can therefore vary for each different lithographic step. Because the positioning

of the wafer stage in the optimal focal plane of the lens is dependent on die size, raw

topology data of the level sensor measurements must be divided into specific die area’s

before further postprocessing.

By construction, all dies lie within the stroke areas of the raw topology measure-

ment. In the previous section it was shown that the width of a stroke consists of 9 data

points in x-direction. The specific dimensions of a die however do not always cover the

full width of a stroke and therefore not all 9 topology datapoints are inside a die area.

99


The different die sizes can be divided into two classes, as presented in figure 5.4:

• Full field: The die width covers 7 of the 9 datapoints on each measured stroke.

• Non full field: The die width covers 5 of the 9 datapoints on each measured

stroke.

datapoint

y

xnon full field die width

full field die width

stroke width

Figure 5.4: Die classes displayed on a stroke of raw topology data.

The above classes identify which raw topology datapoints in x-direction lie within a die

area. Note however that the actual exposed die width for each specific die design can

still vary within the boundaries of the specific class. An example of a wafer divided

into full and non full field dies is presented in figures 5.5(a) and 5.5(b) respectively.

Because datapoints are discarded for both die classes at the edges of the stroke,

there is an extra postprocessing step available called Field Width Optimized Leveling

or FWOL. FWOL is an algorithm which uses all 9 datapoints of the stroke width

(see figure 5.4) and interpolates a new set of datapoints that are all within the actual

exposed die area. The properties of the FWOL algorithm are:

• Full Field: using all datapoints on the stroke width, 21 datapoints are generated

over the full die width.

• Non full field: 18 datapoints are generated over the non-full die width.

Because FWOL uses the outer raw topology datapoints on a stroke which would oth-

erwise be discarded, the algorithm provides a more accurate representation of the die

100

5.1. WAFER TOPOLOGY MEASUREMENTS

Figure 5.5: Raw topology data (in μm) sectioned into dies for: (a) full field dies, (b)

non full field dies.

topology. The FWOL postprocessed topology data of the wafers in figures 5.4(a) and

5.4(b) is presented in respectively figures 5.6(a) and 5.6(b).

Figure 5.6: FWOL topology data (in [μm]) for: (a) full field dies, (b) non full field dies.

The remainder of this thesis will only make use of substrates with full field class

and the absence of Agile correction to represent the most widely used wafers. For more

information on the non full field class and Agile correction, see [37]. The next section

discusses the theoretical study on required amount of curvature for the mechanical

plant and the focus improvements of lithographic systems using ideal CAL strategies.

101


5.2 Field curvature extraction

Once the topology of a substrate is measured by the level sensors and divided into dies

during the postprocessing step, a study is performed on the present curvature values

of wafer exposure area’s. These are used as input for the curvature actuator design.

Furthermore the focus improvement of ideal CAL strategies can be obtained for intradie

and interdie leveling, using procedures that were presented in sections 4.1.1 and 4.1.2.

Let us consider the postprocessed (FWOL is optional) topology of an arbitrary die.

In the introduction it was discussed that the exposure process in lithographic systems

is obtained by moving the reticle stage beneath a light slit, creating a projection of the

photomask image onto the substrate. At wafer level, the slit exposing the substrate to

the image of the photomask has the following properties:

• Dimensions: 26× 5.5 [mm].

• Moving speed: 1.6[ms

].

Recalling that the topology data points have intervals of 0.51 [mm] in y-direction, the

slit covers 11 datapoints. For the curvature extraction procedure, a slit with uniform

intensity is assumed, see figure 5.7. Furthermore only 7 datapoints in y direction are

taken into account for curvature calculation because the light intensity at the edges of

the slit neglectably low. To relate the position of the slit to the coordinates of the die,

y

y

slit coverage

slit window

topology data point

5.5 [mm]

3.25 [mm]

z

z

1

1

slit position

Figure 5.7: Applied window to account for light slit intensity.

the slit position is defined at the center datapoint of the slit.

102

5.2. FIELD CURVATURE EXTRACTION

Throughout the remainder of this chapter, the raw topology data of the wafer

displayed in figure 5.3 and its postprocessed topology data displayed in figure 5.5(a)

is used for explanation of the theory. Let us start with the extraction of curvature

setpoints from the measured topology. Consider the postprocessed data of figure 5.5(a).

As example, the topology of the die with its center position at (102.42, 70.45) [mm] is

plotted in figure 5.8. The black edged area is the window sized slitposition considered

for curvature setpoint calculation. Using the anticlastic curvature augmented fitting

z[μm]

y [mm]x [mm]

90

100

110

120

6.98

7

7.02

7.04

7.06

7.08

7.1

7.12

5060

7080

90

6.95

7

7.05

7.1

7.15

7.2

Figure 5.8: Topology of the die (102.42, 70.45) [mm] with slitposition in black.

function from section 1.2 the CAL setpoints are calculated and displayed in table 5.1.

Setpoint Value

oz [μm] 7.1

Ry

[μmm

]-3.8

Rx

[μmm

]-0.74

κxx[1m

]17.9 · 10−4

Table 5.1: CAL setpoints for the slit in figure 5.8.

Calculating the CAL setpoints for every slit position and plotting the required cur-

vature results in figure 5.9. To obtain an indication of the required curvature setpoints

for the mechanical plant the above procedure is performed on the postprocessed topol-

ogy data of a total of 6 state of the art wafers consisting of a total of 622 full field dies.

Figure 5.10(a) displays a histogram of the obtained curvature setpoints for the normal

postprocessing procedure and figure 5.10(b) the required curvatures for FWOL postpro-

103


κxx[ 1 m

]

Slitposition, y [mm]

55 60 65 70 75 80 85

×10−4

0.6

0.8

1

1.2

1.4

1.6

1.8

Figure 5.9: Curvature setpoints obtained from anticlastic curvature augmented fitting

function.

cessed topology data. The histograms show the percentage of occurrence for curvature

setpoints. Table 5.2 shows the statistical properties of the curvature setpoints in terms

of the mean value μ and variation 3σ. Considering a normal distribution, note that the

3σ indication covers 99.97% of all variations. From the table it can be concluded that

Property Full field Full field FWOL

Mean[1m

] −3.3 · 10−4 −2.7 · 10−4

3σ[1m

]57.0 · 10−4 40.3 · 10−4

Table 5.2: Mean and 3σ properties of the curvature setpoints.

the required curvature range of the mechanical plant is[−60.3 · 10−4, 53.7 · 10−4

] [1m

]in the case of normal postprocessed substrate topology data. For FWOL topology data

the required range yields[−43.0 · 10−4, 37.6 · 10−4

] [1m

].

5.2.1 Intradie curvature correction

In this section the focus improvement of intradie CAL strategies is investigated. The

flowchart of figure 4.1 is used to perform the investigation. In the previous section the

CAL setpoints of table 5.1 were derived for the slit position in figure 5.8. The fitted

topology of the slit position is next calculated using equation (4.4). A reconstruction

of this fit is presented in figure 5.11(b). Conform section 4.1.1 the same procedure is

performed for the CL setpoints. The reconstruction of the CL topology is displayed in

104


Field curvature[

1m

]

Percentage[%

]

Field curvature[

1m

]

Percentage[%

]

×10−3×10−3−1 −0.5 0 0.5 1−2 −1 0 1 20

2

4

6

8

10

12

0

1

2

3

4

5

6

7

8

Figure 5.10: Histogram of obtained curvature setpoints for full field dies with a) normal

postprocessing, b) FWOL postprocessing.

z[μm]

y[mm] x[mm]90100

110120

7.065

7.07

7.075

7.08

7.085

7.09

7.095

6869

7071

727.06

7.07

7.08

7.09

7.1

7.11

(a)

z[μm]

y[mm]x[mm]90

100110

120

7.065

7.07

7.075

7.08

7.085

7.09

6869

7071

727.06

7.07

7.08

7.09

7.1

7.11

(b)

Figure 5.11: Topology of the a) actual slit and b) CAL reconstruction.

105


figure 5.12(b).

z[μm]

y[mm] x[mm]90100

110120

7.065

7.07

7.075

7.08

7.085

7.09

7.095

6869

7071

727.06

7.07

7.08

7.09

7.1

7.11

(a)

z[μm]

y[mm]x[mm]

90100

110120

7.07

7.075

7.08

7.085

7.09

7.095

6869

7071

727.06

7.07

7.08

7.09

7.1

7.11

(b)

Figure 5.12: Topology of the a) actual slit and b) CL reconstruction.

When the above procedure is performed for the whole wafer (figure 5.5(a)) a MA

focus error between topology and fit is calculated for each point on the substrate. This

is both done for the CAL strategy as the CL strategy. The MA defocus error of the CAL

strategy is presented in figure 5.13. The next step is to express the MA focus error of the

topology datapoints in one scalar value ew. This is done by taking the absolute average

error |μ| plus the 3σ variation of the MA focus errors of all datapoints. The wafer focus

properties are displayed in table 5.3 for both CAL and CL strategies 1. From the table

Property CAL CL

|μ| [nm] 5.6 · 10−3 5.5 · 10−3

σ [nm] 3.8 4.4

ew, ew [nm] 11.5 13.2

Table 5.3: Absolute mean and 3σ properties of the wafer focus error for intradie CAL

and CL strategies.

the CAL focus improvement with respect to the CL strategy is calculated using:

Improvement =|ew − ew|

|ew| 100%

= 12.73% (5.1)

To investigate the intradie focus improvement of the CAL strategy with respect to

CL, the same set of substrates as discussed in the previous section is taken into account.

A distinction is made between normal postproccessing and FWOL postprocessing. Af-

ter calculating the MA focus errors for all wafers a scalar focus error is generated as is

1Note that, conform section 4.1.1, ew refers to CAL focus error and ew to CL focus error

106


Figure 5.13: MA focus error between intradie leveling and wafer topology (in [nm]).

done for the example wafer in table 5.3. The results are presented in table 5.4. From

Topology data ew [nm] ew [nm] Focus improvement [%]

Full field 14.9 19.0 21.5

Full field FWOL 11.7 16.3 28.1

Table 5.4: Focus error for intradie CAL and CL strategies and focus improvements for

the complete set of substrates.

table 5.4 it can be concluded that the intradie CAL results in a focus improvement of 21

% and 28 % for normal postprocessed and FWOL postprocessed substrate topologies.

5.2.2 Interdie curvature correction

The interdie focus improvement of CAL strategies is obtained in the same fashion as

the previous section, only the curvature of each slit position is taken as the average of

the curvature setpoints for the whole die, found using the anticlastic curvature aug-

mented fitting function. The flowchart used to obtain the interdie curvature correction

focus improvement is presented in section 4.1.2 by figure 4.2. As example the interdie

curvature correction focus improvement for the wafer of figure 5.5(a) is calculated.

Let us first consider the topology of the die presented by figure 5.8. Conform the

flowchart, the CAL setpoints are found using the anticlastic curvature augmented fitting

107


function (as displayed in figure 5.9). The average curvature for the whole die is found

to be:

κxx = 13.19 · 10−4

[1

m

]. (5.2)

The static CAL setpoints are found using equations (4.8) and (4.9). The calculated

values for slit position of figure 5.8 are displayed in table 5.5. In comparison to the

CAL setpoints of table 5.1, the offset setpoints is changed. If all static setpoints are

Setpoint Value

oz [μm] 7.8055

Ry

[μmm

]-3.8026

Rx

[μmm

]-0.7366

Table 5.5: Static CAL setpoints for the slit in figure 5.8.

calculated for the whole example wafer, the MA focus error is calculated for each point

and displayed in figure 5.14. The interdie CAL focus error ew for the whole substrate

Figure 5.14: MA focus error between interdie leveling and wafer topology (in [nm]).

and that of CL focus error is presented in table 5.6. Subsequently the interdie focus

108


Property CAL CL

|μ| [nm] 5.6 · 10−3 5.5 · 10−3

σ [nm] 4.1 4.4

ew, ew [nm] 12.4 13.2

Table 5.6: Absolute mean and 3σ properties of the wafer focus error for interdie CAL

and CL strategies.

improvement with respect to CL strategies yields:

Improvement =|ew − ew|

|ew| 100%

= 6.02% (5.3)

The interdie focus error based on the complete set of wafers is presented in table

5.7, with a distinction between normal and FWOL postprocessed wafer topology. The

last column represents the focus improvement of the interdie CAL with respect to CL

strategies. From the table it can be concluded that interdie CAL does not meet the

Topology data ew [nm] ew [nm] Focus improvement [%]

Full field 16.4 19.0 13.5

Full field FWOL 13.3 16.3 18.4

Table 5.7: Focus error for interdie CAL and CL strategies and focus improvements for

the complete set of substrates.

focus improvements of intradie CAL but still gains 14% and 18% with respect to CL

strategies for normal and FWOL postprocessed topology measurements of state of the

art substrates.

5.2.3 Mechanical resonance specification

This section discusses the influence of bandwidth limitations on curvature setpoints

of intradie CAL strategies. The bandwidth limiting effect of the mechanical plant is

modeled by implementing an IIR filter which filters the intradie curvature setpoints

up to a desired frequency before reconstructing the topology. The specification of the

implemented filter are (for more information about filtering theory see [37, 39, 40, 41]):

• Type: IIR Butterworth zero-phase filter.

• Filter order: 16.

• Implementation: Lowpass.

109


The flowchart of figure 4.3 is used for this analysis.

Let us start with the example die of figure 5.8. The curvature setpoints are found

using the anticlastic curvature augmented fitting function are displayed in figure 5.9.

To model the tracking ability of the mechanical plant, this profile is run through the

bandwidth limiter, which applies the lowpass filtering routine. The bandwidth limiter

is modeled for the cutoff frequencies Fc = 50 [Hz] to Fc = 500 [Hz] with intervals of

50 [Hz]. For each case, the curvature setpoints are filtered accordingly. An example of

a number of filtered curvature setpoints is presented in figure 5.15. The MA focus error

Fc = 50 [Hz]Fc = 100 [Hz]Fc = 250 [Hz]Fc = 400 [Hz]Original

κxx,� κ

xx

[ 1 m

]

Slitposition [mm]

55 60 65 70 75 80 85

×10−4

0.6

0.8

1

1.2

1.4

1.6

1.8

Figure 5.15: Curvature setpoints filtered with several cutoff frequencies.

is subsequently calculated for the different cutoff frequencies to get a clear indication

of focus losses due to bandwidth limitations.

Once the whole wafer is processed for all bandlimiter cases, the scalar wafer errors

are identified. Figure 5.16 shows the results for the example wafer topology of figure

5.5. It can be seen from the figure that cutoff frequencies lower than 200 [Hz] cause the

filtered intradie CAL strategy to gain focus error with respect to normal intradie CAL.

This phenomenon can be seen as the lack of ability of the mechanical plant to track the

original curvature profile. It can therefore be concluded that a mechanical plant with

resonance properties in the vicinity of 200 [Hz] (open loop) would have negative effects

on focus improvement of the intradie CAL strategy. From the figure it can also be

seen that curvature setpoints filtered with Fc = 50 [Hz] results in focus errors which

are larger than the CL strategy. In this case such a large frequency content of the

curvature profile is discarded by the bandwidth limiter topology that reconstruction

using the CAL setpoints does not represent an optimal topology description anymore.

110

5.3. SUMMARY

ew

ew

�ew

Wafer

error[nm]

Fc [Hz]

500 450 400 350 300 250 200 150 100 5011

11.5

12

12.5

13

13.5

14

14.5

15

Figure 5.16: Example wafer, scalar wafer focus errors for intradie filtered curvature

setpoints.

To specify a mechanical resonance requirement for the mechanical plant, the same

analysis is performed for the complete set of wafers. The results for normal and FWOL

postprocessed substrate topology are shown in figures 5.17 and 5.18 respectively.

From the figures it becomes clear that cutoff frequencies lower than 150 [Hz] result in

larger focus error for the intradie CAL strategy. Therefore it can be concluded that

the gross of frequency content available in the topology data lies within the frequency

range [0, 150] [Hz]. From a design point of view, the mechanical plant must be able

to track the frequency contents in this range in order to acquire the highest focus

improvement. To track these frequency contents, the first natural eigenfrequency of

the plant cannot lie in this frequency range. Subsequently, to obtain optimal results,

the design specification for the first natural eigenfrequency is set five times higher than

150 [Hz], ensuring no influence of mechanical resonances during intradie CAL.

5.3 Summary

In this chapter the specification derivation of ideal photomask curvature focus improve-

ment was investigated. The chapter started with the methodology of postprocessing

wafer measurement data into die topology, which can be used to derive CAL and CL set-

points. Using the postprocessed topology data of 6 substrates, the intradie and interdie

focus improvement of CAL strategies was investigated when using an ideal curvature

actuation system. Finally the frequency content of curvature values of substrates was

111


ew

ew

�ew

Wafer

error[nm]

Fc [Hz]

500 450 400 350 300 250 200 150 100 5014

14.5

15

15.5

16

16.5

17

17.5

18

18.5

19

Figure 5.17: Complete set of wafers, scalar wafer focus errors for intradie filtered CAL

strategy, normal postprocessed topology data.

ew

ew

�ew

Wafer

error[nm]

Fc [Hz]

500 450 400 350 300 250 200 150 100 5011

12

13

14

15

16

17

Figure 5.18: Complete set of wafers, scalar wafer focus errors for intradie filtered CAL

strategy, FWOL postprocessed topology data.

112

5.3. SUMMARY

identified by determining CAL focus improvement when using filtered curvature set-

points. From the obtained results a mechanical resonance specification was defined for

the design of the mechanical plant.

113


114

Part III

Results

115

Chapter 6

Numerical modeling and

validation

This chapter discusses the numerical modeling and validation of the photomask-actuator

assembly. The model will be used in the focus improvement analysis of chapter 7.

The chapter subsequently covers the ’Modeling & Validation’ block of the performance

evaluation flowchart of figure 4.6. Section 6.1 discusses the numerical modeling of the

different components in the bending-actuator assembly, followed by the experimental

validation of the photomask and bending clamp models in section 6.2. Finally, section

6.3 discusses the model reduction of the FE components and presents the reduced order

coupled model of the bending-actuator assembly.

6.1 Numerical modeling

In the introduction it was set out that the complexity of the bending-actuator design

demands the use of FE modeling. There are several parts in the bending-actuator

design which are separately modeled and discussed in this section. The design also

contains parts which are not very complex and will therefore be modeled using lumped

mass models. With the complete assembly introduced by figure 4.4, a cross sectional

schematic overview of the bending actuator design with the specific parts that are

discussed in the following subsections is presented in figure 6.1. For each part, the

following properties are covered:

• material properties

• mesh properties

• interfaces with other parts.

The next section starts with the modeling of the photomask.

117

CHAPTER 6. NUMERICAL MODELING AND VALIDATION Company-secret

photomaskrods

bending clamppiezo actuatorpreload spring

x

z

z-support

Figure 6.1: Schematic cross section of half the bending actuator assembly.

6.1.1 Photomask

The photomask is made of fused silica which has the following material properties [42]:

• Material model: isotropic.

• Modulus of elasticity: E = 72.6 [GPa].

• Poisson ratio: ν = 0.16 [−].

• Density: ρ = 2205.7[

kgm3

].

The dimensions of the photomask are (152.4 × 152.4 × 6.35) [mm]. The edges are cham-

fered with a radius of 3 [mm]. The meshed model is plotted in figure 6.2. The mesh

Figure 6.2: Meshed model of the photomask.

has the following characteristics:

• Model type: Solid 3D.

118

6.1. NUMERICAL MODELING

• Degrees of Freedom: 158502.

• Element type: triangular.

• Shape function: Lagrange quadratic.

The mesh consists of triangular elements, but the middle elements form a rectangular

domain which is also known as the Q-grid . The bottom of this domain is covered with

a chrome pattern which is projected onto the substrate during scanning. Furthermore,

this grid is also defined as a qualification grid to identify overlay and focus errors.

The area is therefore used to identify the realized curvature for the CAL analysis. In

specific, the deformation of the area is used by the anticlastic curvature augmented

fitting function to determine the curvature of the photomask after bending moment

application of the actuators. The Q-grid has the dimensions (101.76 × 129.74) [mm]

and consist of a grid of [49× 73] nodes of which the DoF in z-direction will be used

to identify curvature. The Q-grid (red), as well as the interfaces of other parts are

displayed in figure 6.3. The blue interfaces indicate the connections to the rods which

introduce the forces for the bending moments into the photomask. The z-support

locations are indicated with green dots.

Figure 6.3: Reticle with Q-grid, rod and z-support interfaces in respectively red, blue

and green.

6.1.2 Bending clamps

The bending clamps hold the piezo electric actuators and are designed to convert the

elongation of the actuators into a bending moment which is applied on the photomask

through the rods. The total bending configuration holds two bending clamps, but both

119


clamps have roughly equal dynamic properties (only the gaps for the z-supports differ).

The left bending clamp is displayed in figure 6.4. The triangular parts of the clamps are

interface to rodvacuum area

hole for z-supportinterface to reticle stageinterface to piezo assembly

leaf spring

Figure 6.4: Left bending clamp.

leaf springs. Beneath the leaf springs piezoelectric actuator assembly’s are positioned

(see also figure 4.4). As will be explained in section 6.1.6, each piezoelectric assembly

consists of a piezoelectric actuator, prestressed by a preload spring. In the photomask

assembly, the bending clamps are positioned beneath the photomask. There are a total

of 14 rods per clamp glued into the vacuum area. They realize the connection between

photomask and bending clamp and create a gap of 10 [μm] between photomask and

vacuum clamp. By creating the gap between photomask and bending clamp, the rods

provide a decoupling in x- and y-direction. Finally, a prestress force generated by a

vacuum holds the vacuum area in position with respect to the photomask.

The bending clamp is made of aluminum, which has the following material proper-

ties [43]:


• Modulus of elasticity: E = 70 [GPa].


• Density: ρ = 2700[

kgm3

].

The meshed model of the left bending clamp is plotted in figure 6.5. The mesh prop-

erties for both bending clamps are:

120


Figure 6.5: Meshed model of left bending clamp.

• Model type: Solid 3D.

• DoF left clamp: 234684.

• DoF right clamp: 240324.



The bending clamps have multiple interfaces to other parts, which are indicated

in figure 6.4. For each bending clamp there are 14 interfaces to the rods and an

equal amount for the piezo assemblies. Finally, there is one interface with the reticle

stage which is considered as the static boundary constraint of the photomask-actuator

assembly.

6.1.3 Rods

The rods are glued into the bending clamps and provide the connection between pho-

tomask and bending clamps. The have the following design specifications [43]:

• Length: 2 [mm].

• Diameter: 0.25 [mm]

121


• Spherical top radius connected to the photomask: 3.18 [mm].

• Axial stiffness: ka = 2.8 · 106 [Nm].• Radial stiffness: kr = 1.43 · 104 [Nm].

Because the rods cover only 8 · 10−4% of the total assembly mass and have a fairly

simple design, they are modeled as massless springs. Due to low radial stiffness the

rotational stiffnesses of the rods are neglected. For each rod, the vector qrod holding

the DoF reads:

qrod =[x1 y1 z1 x2 y2 z2

]T, (6.1)

where the index 1 and 2 refer to respectively the bottom and top nodes of the spring.

Node 1 is the interface with the photomask (see figure 6.3) and node 2 the interface to

the bending clamp. While the mass matrix of the rod equals zero, the stiffness matrix

reads:

Krod =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

kr 0 0 −kr 0 0

0 kr 0 0 −kr 0

0 0 ka 0 0 −ka

−kr 0 0 kr 0 0

0 −kr 0 0 kr 0

0 0 −ka 0 0 ka

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(6.2)

6.1.4 Z-supports

The z-supports are considered part of the surroundings of the photomask-actuator

assembly. They attach the reticle stage to the photomask, therefore they are regarded

as boundary conditions on the photomask. Their stiffness properties are [43]:

• Axial stiffness: ka = 2.5 · 106 [Nm].• Radial stiffness: kr = 1.76 · 104 [Nm].

To model the influence of the z-supports on the photomask, they are regarded as

massless springs. Because of low radial stiffness the rotational stiffnesses are neglected.

The properties of the massless springs are:

qzsup =[qx qy qz

]T(6.3)

Kzsup =

⎡⎢⎣

kr 0 0

0 kr 0

0 0 ka

⎤⎥⎦ (6.4)

The DoF qzsup are the interfaces with the photomask (see figure 6.3).

122


6.1.5 Preload springs

Piezo electric actuators are usually stacks of several actuators bonded with adhesive

material. Because they are not able to produce pulling forces, a prestress is applied on

the actuator using a preload spring. In the case of the photomask-actuator assembly,

the piezo actuator is glued in the spring of figure 6.6. The preload spring is made of

interface to bending clamp

interface to piezo actuator

Figure 6.6: CAD model of the preload spring.

STAVAX� and has the following properties [43]:


• Modulus of elasticity: E = 200 [GPa].


• Density: ρ = 7800[kgm3

].

The meshed model is plotted in figure 6.7 The properties of the mesh are:

Figure 6.7: Meshed model of the preload spring.

123


• Mesh type: Solid.

• DoF: 14088.



A total of four interfaces present in the preload spring (see figure 6.6). The inner

two area’s are connected to the piezo actuator and the outer area’s are connected to

the bending clamp (see also figure 4.4).

6.1.6 Piezoelectric actuators

The piezoelectric actuators are placed inside the preload springs as indicated in figure

6.6. In order to describe properties of the piezoelectric actuators, a model is required

which couples the electrical and mechanical properties of the piezoelectric actuators.

For that purpose a simplified model is extracted in [44] using the constitutive equations

for piezo material [45]. The actuator is modeled as a three dimensional beam without

shear deflection. The voltage which causes the piezo to elongate is modeled as forces

on the beam. These forces are dependent on the following actuator characteristics:

• Maximum displacement um: The displacement of the actuator in uncon-

strained configuration when applying a maximum voltage Um.

• Blocking force Fb: The force that is required to counteract the elongation of

the actuator when applying a maximum voltage Um.

The (simplified) linear relation between force Fp and voltage U representing the elon-

gation of the actuator is defined as:

Fp =Fb

UmU

= dpU. (6.5)

Having defined the force relating the elongation of the actuator to the applied

voltage, the actuator is modeled as in figure 6.8. Two nodes represent the interfaces to

the preload spring. The equations of motion for the beam read

Mq +Kq = dpU, (6.6)

with

q =[x1 y1 z1 θx1 θy1 θz1 x2 y2 z2 θx2 θy2 θz2

]T(6.7)

dp =[−dp 0 0 0 0 0 dp 0 0 0 0 0

]T. (6.8)

124


z1

θz1

θy1 y1

x1θx1

z2

θz2

θy2 y2

θx2x2

x

y

z

l

hb

E, ν,m, dp

Figure 6.8: Piezoelectric actuator model with its DoF and properties.

Using finite element theory for beam elements, the stiffness matrix yields [12]:

K =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

EAl

0 0 0 0 0 −EAl

0 0 0 0 0

0 12EIzl3

0 0 0 6EIzl2

0 −12EIzl3

0 0 0 6EIzl2

0 012EIy

l30

−6EIyl2

0 0 012EIy

l30

−6EIyl2

0

0 0 0 GJxl

0 0 0 0 0 −GJxl

0 0

0 0−6EIy

l20

4EIyl

0 0 06EIyl2

02EIy

l0

0 6EIzl2

0 0 0 4EIzl

0 −6EIzl2

0 0 0 2EIzl−EA

l0 0 0 0 0 EA

l0 0 0 0 0

0 −12EIzl3

0 0 0 −6EIzl2

0 12EIzl3

0 0 0 −6EIzl2

0 012EIy

l30

6EIyl2

0 0 012EIy

l30

6EIyl2

0

0 0 0 −GJxl

0 0 0 0 0 GJxl

0 0

0 0−6EIy

l20

2EIyl

0 0 06EIyl2

04EIy

l0

0 6EIzl2

0 0 0 2EIzl

0 −6EIzl2

0 0 0 4EIzl

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

,

(6.9)

with

G =E

2 (1 + ν)(6.10)

Iy =bh3

12(6.11)

Iz =hb3

12(6.12)

Jx = Iy + Iz (6.13)

A = bh (6.14)

125


The mass matrix yields (per unit of length):

M = ml

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

13

0 0 0 0 0 16

0 0 0 0 0

0 1335

0 0 0 11l210

0 790

0 0 0 − 13L420

0 0 1335

0 − 11l210

0 0 0 790

0 13L420

0

0 0 0 Jx3A

0 0 0 0 0 Jx6A

0 0

0 0 − 11l210

0 l2

1050 0 0 − 13L

4200 − L2

1400

0 11l210

0 0 0 l2

1050 13L

4200 0 0 − L2

14016

0 0 0 0 0 13

0 0 0 0 0

0 790

0 0 0 13L420

0 1335

0 0 0 − 11l210

0 0 790

0 − 13L420

0 0 0 1335

0 11l210

0

0 0 0 Jx6A

0 0 0 0 0 Jx3A

0 0

0 0 13L420

0 − L2

1400 0 0 11l

2100 l2

1050

0 − 13L420

0 0 0 − L2

1400 − 11l

2100 0 0 l2

105

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(6.15)

6.2 Experimental validation

The photomask and bending clamps are the main parts of the photomask-actuator

assembly. To validate the models with the manufactured subcomponents, experimental

modal analysis (EMA) is used. Section 6.2.1 first describes the correlation measure

between modeled and measured subcomponents, followed by section 6.2.2 and 6.2.3

which respectively discuss the experimental validation of the photomask and bending

clamps.

6.2.1 Correlation measures

To identify the correlation between measured and modeled parts, the Modal Assurance

Criterion or MAC is us as criterion. The MAC between two arbitrary modes φa and

φb is calculated as:

MAC =

∣∣φTaφb

∣∣2(φTaφa

) (φTb φb

) (6.16)

Mathematically the MAC describes how well two vectors are in the same direction

inside the space they span, without dependency on their amplitude. If the MAC value

is 1 for the modes in equation (6.16), the modes are exactly the same. If however

the MAC yields 0, the modes are orthogonal to each other and have no correlation.

From literature [27], modal correlation can be categorized as presented in table 6.1. In

0.9 > MAC � 1 Correlated modes

0.7 > MAC � 0.9 Doubtful correlation

0.1 > MAC � 0.7 Uncorrelated modes

MAC � 0.1 Modes nearly orthogonal

Table 6.1: Categorization of mode correlation for MAC analysis.

126

6.2. EXPERIMENTAL VALIDATION

experimental validation of a model, a correlation of 70 % is considered poor.

Next to the modal correlation, an eigenfrequency check will be performed where the

identified eigenfrequencies are compared to the eigenfrequencies of the model. By rule

of thumb is assumed that an eigenfrequency error within 10% is acceptable, therefore

modes with a larger eigenfrequency error will not be validated if their MAC show good

correlation [46].

6.2.2 Photomask validation

In this section an EMA is performed to validate the FE model of the photomask. During

the measurement the photomask was suspended vertically by weak elastic bands while

a Laser Doppler Vibrometer (LDV) was used to obtain the out of plane vibrations

of the object. Using the vibrometer, a grid of 13 × 13 discrete points was measured.

The excitation of the photomask was realized using different settings because several

complications had to be overcome when performing the analysis:

• Weight of the photomask: The photomask weight is 325.2 [g]. Excitation

using a shaker results in additional mass loading due to the use of a force sensor,

weighing 19.2 [g].

• Double modes: The photomask is a symmetric structure and inhabits symmet-

ric modes which can hardly be distinguished from the measurements because they

have the same resonance frequency.

Mass loading can also be advantageous. The mass effect of the added force sensor allows

to distinguish the double modes. Indeed, if the sensor is placed at a non-symmetrical

position, the properties of the system change and the double modes can be distinguished

The force sensor must be added in the FEM however to improve the correlation between

measurement and model. The downside of this approach is that not only the modes

change but also the eigenfrequencies. Indeed, the eigenfrequencies can be correlated

using the FE model with added mass of the force sensor, but additionally an extra

measurement can be performed to validate the eigenfrequencies of the model without

mass loading effects. The following measurements are proposed to validate the model:

1. EMA 1: A shaker is attached to the photomask at the bottom-right corner. This

analysis is discussed in section 6.2.2.1.

2. EMA 2: The shaker is attached at the bottom and in the center of the pho-

tomask. The results are provided in section 6.2.2.2.

3. Acoustic excitation: A speaker is used to excite the system. This eliminates

mass loading from force sensor placement. The downside of this measurement

127


is that no exact eigenmodes can be identified because the input energy is not

measured1. This measurement is therefore only used for eigenfrequency valida-

tion. The analysis is discussed in section 6.2.2.3. Details on the measurement

equipment can be found in appendix E.

A schematic setup for the measurements is presented in figure 6.9.

weak suspension

photomask

output: LDV measurement point

EMA 1: input, shaker & force sensor

EMA 2: input, shaker & force sensor

Acoustic excitation: input, speaker

Figure 6.9: Measurement setup for the two EMAs and acoustic excitation proposed for

the validation of the photomask model.

6.2.2.1 EMA 1

The schematic setup for EMA 1 is presented in figure 6.9 and considering the EMA 1

force input location. The excitation is obtained using a shaker with a pseudo random

noise signal ranging from 0−10 [kHz], which is attached to the back of the photomask.

The shaker signal is transferred to the photomask using a stinger and impedance head

(force sensor). The LDV measures the out of plane velocity of the discrete points

in figure 6.9. A picture of the measurement setup is presented in figure 6.10. The

photomask is coated with a thin white layer of commercially available paint to enhance

the reflection of the LDV. The spectral parameter settings for the pseudo-random

excitation are displayed in table 6.2.

Figure 6.11 shows some representative results of EMA1. Due to limitations of the

measurement software, the frequency domain was measured in two steps, 0 to 5 [kHz]

and 5 to 10 [kHz]. Unfortunately the driving point was not exactly measured, therefore

1From speaker measurements, it is possible to guess eigenmodes if the measurement peaks are sharp

and well separated.

128


weak suspension

shakerLDV focal pointphotomask

Figure 6.10: Measurement setup 1, shaker measurement picture.

Parameter Property

Measurement mode FFT complex

No. of spectral lines [−] 6400

Frequency resolution [Hz] 0.78

Window type Rectangle

Averaging type Linear

No. of averages [−] 10

Table 6.2: Spectral parameters for the experimental modal analysis of the photomask

using pseudo random shaker excitation.

129


figure 6.11 shows the obtained results of an arbitrary measurement point. In the figure

it can be seen that the coherence at some ranges is bad. After obtaining these results,

a new set of measurements was performed, zooming in on specific eigenfrequencies,

improving the coherence of the measurement. After performing the measurements, the

measured modes were identified using the Pole/Residue model [47]. Subsequently the

MAC was calculated to determine the correlation between the experimentally obtained

modes and the modes of the FE model. To overcome the problem of mass loading, the

force sensor was added in the FE model. The MAC results are plotted in figure 6.12.

In the MAC plot, the first 8 eigenmodes show good correlation and orthogonality

to the other modes. Mode 9 and 10 have some cross correlation to each other, this

is due to the fact that the eigenfrequencies are close to each other, indicating double

modes. Modes 10 to 16 have bad correlation. Mode 12 does show a high MAC but

has low orthogonality to mode 11. In figure 6.13 the obtained eigenfrequencies are

plotted against relative errors between the experimentally obtained eigenfrequencies

and the eigenfrequencies obtained from the FE model. The figure shows that the

relative error between the measured and FE eigenfrequencies is below 4.5 %. For the

first 10 eigenfrequencies the relative error is smaller than 3.5 %. Figure 6.14 shows the

identified damping ratios, they all remain below 0.8 %.

6.2.2.2 EMA 2

The schematic setup for EMA 2 is presented in figure 6.9 by input case EMA 2. The

measurement procedure is exactly the same as for EMA 1, only the sensor is at a

different position as indicated in the figure. The FE model is also altered to the new

measurement configuration. Figure 6.15 shows some results of the EMA 2 measurement.

Due to bad coherence, new measurements were performed when zoomed in on fre-

quency ranges around identified eigenfrequencies. After performing EMA 2 the MAC

results were calculated and plotted in figure 6.16. From the MAC results it can be

seen that there is a good mode correlation between the EMA and FEM modes 1 to 5

and 8 to 10. These modes also show good orthogonality properties. Mode 6 shows bad

correlation which is due to the cross-relation with mode 7. Modes 11 to 16 show bad

correlation properties.

The relative errors between the experimentally obtained eigenfrequencies and the

eigenfrequencies obtained from the FE model are plotted in figure 6.17 against the

identified eigenfrequencies. From the figure it is visible that the relative errors between

EMA and FEM eigenfrequencies are smaller than 5%, except for the first mode which

has an error about 6%.

In figure 6.18 the damping ratio’s obtained from the Pole/Residue estimation are

plotted against the eigenfrequencies of EMA 2. The figure shows that the identified

130


Coherence

Frequency [kHz]

Imaginairy part of the FRF

imag[m

/s/

N]

Frequency [kHz]

Real part of the FRF

real[m

/s/

N]

Frequency [kHz]

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

×10−3

×10−3

0

0.5

1

−5

0

5

−1

0

1

(a)

Coherence

Frequency [kHz]


imag[m

/s/

N]

Frequency [kHz]


real[m

/s/

N]

Frequency [kHz]

5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10

5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10

5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10

×10−4

×10−4

0

0.5

1

−5

0

5

−2

0

2

(b)

Figure 6.11: Selection of the measurement results from EMA 1 of the photomask, a) 0

to 5 [kHz] and b) 5 to 10 [kHz].

131


Iden

tified

modes

FE modes1 2 3 4 5 6 7 8 9 10111213141516

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1123456789

10111213141516

Figure 6.12: Results of the EMA 1 MAC analysis.

Frequen

cydifferen

ce[%

]

Frequency [kHz]0 1 2 3 4 5 6 7 8 9 10

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Figure 6.13: EMA 1 eigenfrequencies and relative frequency error between EMA and

FEM eigenfrequencies.

132


Dampingratio[%

]

Frequency [kHz]

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Figure 6.14: EMA 1 damping ratios plotted against the measured eigenfrequencies.

damping ratio’s for modes 1 and 2 are below 3%, which is much higher than the results

found in EMA 1. This could be due to the attachment of the shaker, which is at a

different location than for EMA 1. It is therefore possible that modes 1 and 2 were

more difficult to measure because less energy of the shaker excited the modes. The

remainder of the modes has a damping ratio of less then 0.8 %.

6.2.2.3 Acoustic excitation

The elimination of mass loading effects is realized by removing the shaker as input

source and replacing it by a speaker, as schematically depicted in figure 6.9. The signal

used to excite the system is a pseudo random noise, ranging from 0 to 10 [kHz]. The

spectral parameters of the measurement setup remain the same as for EMA 1 and

EMA 2, as presented in table 6.2. Figure 6.19 shows the measurement setup where

a speaker is positioned at the back of the weakly suspended photomask. To have a

clear view on the resonance frequencies of the system, the sum of all responses of the

measurement grid is presented in figure 6.20. The error between measurement and

FEM eigenfrequencies plotted against the obtained resonances is presented in figure

6.21. The figure highlights that the error between eigenfrequencies remain below 1%

up to 7 [kHz]. After 7 [kHz] the error remains to be low, but two eigenfrequencies were

not measured, as indicated by the red dashed lines. Because of the low eigenfrequency

error when comparing the measured resonance frequencies to the eigenfrequencies of

the model it can be concluded that the acoustic measurement produced the best results.

133


Coherence

Frequency [kHz]


imag[m

/s/

N]

Frequency [kHz]


real[m

/s/

N]

Frequency [kHz]

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

×10−4

×10−4

0

0.5

1

−5

0

5

−2

0

2

(a)

Coherence

Frequency [kHz]


imag[m

/s/

N]

Frequency [kHz]


real[m

/s/

N]

Frequency [kHz]

5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10

5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10

5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10

×10−4

×10−4

0

0.5

1

−2

0

2

−5

0

5

(b)

Figure 6.15: Selection of the measurement results from EMA 2 of the photomask, a) 0

to 5 [kHz] and b) 5 to 10 [kHz].

134


Iden

tified

modes

FE modes1 2 3 4 5 6 7 8 9 10111213141516

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1123456789

10111213141516

Figure 6.16: Results of the EMA 2 MAC analysis.

Frequen

cydifferen

ce[%

]

Frequency [kHz]0 1 2 3 4 5 6 7 8 9 10

0

1

2

3

4

5

6

Figure 6.17: EMA 2 eigenfrequencies and relative frequency error between EMA and

FEM eigenfrequencies.

135


Dampingratio[%

]

Frequency [kHz]

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3

Figure 6.18: EMA 2 damping ratios plotted against the measured eigenfrequencies.

weak suspension

speaker

LDV focal point

photomask

Figure 6.19: Measurement setup 3, speaker measurement picture.

136


Response (sum)

Velocity

[m/s]

Frequency [kHz]0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

×10−4

0

1

2

3

4

5

(a)

Response (sum)

Velocity

[m/s]

Frequency [kHz]5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10

×10−4

0

1

2

3

4

5

(b)

Figure 6.20: Sum of all measured responses, a) 0 to 5 [kHz] and b) 5 to 10 [kHz].

Missing eigenfrequencies

Eigenfrequency error

Frequen

cydifferen

ce[%

]

Frequency [kHz]1 2 3 4 5 6 7 8 9 10

0

0.2

0.4

0.6

0.8

1

Figure 6.21: Measured resonance frequencies and relative error between measurement

and FEM eigenfrequencies.

137


6.2.2.4 Summary

Using the results from the photomask measurements it can be concluded that the

photomask model is validated up to 7 [kHz]. The first 8 modes were validated during

EMA 1 and modes 1 to 4 and 8 to 10 where validated during EMA 2. All modes up

to 7 [kHz] except number 5 had a MAC value over 0.9. Mode 5 showed a MAC of 0.87

% which is sufficient for validation. Using the results of the acoustic measurement all

eigenfrequencies up to 7 [kHz] were validated with an error of less than 1% between

measurement and FEM eigenfrequencies. For future research it is recommended that

the photomask is validated without the effect of mass loading. It is also recommended

that the identified modal damping ratios of this measurement are implemented in the

FE model of the photomask.

6.2.3 Bending clamp validation

In this section, an EMA is performed to validate the FE model of the bending clamp.

Because the two designed bending clamps are practically equal, only one clamp was

validated. Suspended by weak elastic bands, the LDV was used to measure the out of

plane responses (see figures 6.22 and 6.23). The validation of the clamp was difficult

because of its size and weight. Because the weight of the clamp is only 13.6 [g], the

following problems were experienced:

• Mass loading: Attachment of the shaker introduces mass loading of the impedance

head.

• Environment disturbances: Due to the low weight of the clamp, it is very

sensitive to disturbances from the environment, like sound vibrations originating

from operational machinery.

From a variety of measurement setups, the setup using the shaker proved to generate the

best results. A secondary measurement was performed to validate the eigenfrequencies

of the FE model using a speaker as excitation source, but this measurement generated

no results of value. Therefore only the EMA using a shaker is discussed in this section.

Figure 6.22 shows the schematic setup of the measurement.

The excitation of the shaker is generated by a pseudo-random signal ranging from

0 to 5 [kHz]. The shaker is attached to the back of the clamp via a stinger and force

sensor. During the measurement, the LDV measured the clamps velocity at the discrete

measurement points which are displayed in figure 6.22. The spectral parameters of the

measurement equipment are the same as the EMA of the photomask and are presented

in table 6.2. A picture of the measurement setup is provided in figure 6.23. Due to

limitations of the measurement equipment and complexity of the measured system, the

138


weak suspension

bending clamp

output: LDV measurement point

input: shaker & force sensor (back)

Figure 6.22: Measurement setup for the EMA proposed for the validation of the bending

clamp.

shaker

weak suspension

bending clamp

LDV focal point

force sensor

Figure 6.23: Measurement setup for the bending clamp EMA.

139


overall range of 0 to 5 [kHz] was measured in blocks of 1 [kHz]. Figure 6.24 shows the

driving point measurement of the range 0 to 1 [kHz]. After obtaining the measurements

Coherence

Frequency [kHz]


imag[m

/s/N]

Frequency [kHz]


real[m

/s/N]

Frequency [kHz]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.70.80.9

−50

0

50

−50

0

50

Figure 6.24: Driving point measurement for 0 to 1 [kHz] of the bending clamp EMA.

for the whole frequency range, a new set of measurements was performed while zoomed

in on the eigenfrequencies to improve the coherences.

After identification of the modes the MAC was calculated with respect to the FEM

modes. To obtain the MAC, the FE model was augmented with the force sensor to

account for the mass loading effect. To validate the model, it proved that the radial

stiffness of the stinger also had to be incorporated into the FEM model because it had

constraining effects on the force sensor. The MAC results are plotted in figure 6.25.

The figure shows that modes 13, 17 and 23 show bad correlation between identified and

FEM modes. Mode 13 could not be validated because it has a very local deformation

shape which could not be measured properly. Modes 17 and 23 could not be validated

because they mainly show in plane deformations in the model, therefore they could not

be measured in the experimental setup. In figure 6.26 the relative error between FE

model and measured eigenfrequencies are plotted against the obtained eigenfrequencies.

From the figure it becomes clear that the FE eigenfrequencies 2, 4, 12 and 21 have a

correlation with the resonances of the FE model of less than 10%. The remaining

eigenfrequencies show an error of less than 5%.

The obtained damping ratios of the modes are plotted in figure 6.27 against the

obtained eigenfrequencies. The figure shows that the modes 1, 2 and 8 have a damping

140


Iden

tified

modes

FE modes1 2 3 4 5 6 7 8 9 1011121314151617181920212223

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1123456789

1011121314151617181920212223

Figure 6.25: Results of the EMA MAC analysis for the bending clamp.

Frequen

cydifferen

ce[%

]

Frequency [kHz]0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

0

1

2

3

4

5

6

7

8

9

Figure 6.26: EMA eigenfrequencies and relative frequency error between EMA and

FEM eigenfrequencies for the bending clamp.

141


Dampingratio[%

]

Frequency [kHz]0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0

0.5

1

1.5

2

2.5

3

3.5

4

Figure 6.27: Damping ratio’s plotted against the measured eigenfrequencies.

ratio of less then 4%, The remaining damping ratios are below 1.5%.

From the measurements it can be concluded that the model of the bending clamp

is validated up to 2.5 [kHz] because the 13th mode could not be validated. The MAC

values up to 2.5 [kHz] were all above 86% except for modes 1 and 3, which had a MAC

value above 83%. From the EMA it was found that these modes experienced high

influences of the radial stiffness of the stinger and therefore were difficult to validate.

All eigenfrequencies below 2.5 [kHz] had a relative error of less than 9%. Note that the

modes and eigenfrequencies of the system are found using a model and measurement

where the force sensor is attached to the bending clamp and the radial stiffness of the

stinger is augmented in the FE model. It is recommended that in future research the

eigenmodes and eigenfrequencies are validated without the effect of mass loading. It

is then also recommended to implement the obtained modal damping ratios in the FE

model of the bending clamp.

6.3 Reduced coupled model

After modeling all parts of the photomask-actuator assembly and performing the vali-

dation measurements on the photomask and bending clamp, the complete assembly is

constructed and used for the performance evaluation. Table 6.3 gives a summary of the

142

6.3. REDUCED COUPLED MODEL

number of parts and their amount of DoF in the total assembly. It highlights that the

large amount of DoF will demand a large number of numerical computations for CAL

performance evaluation simulations. To keep the model of the mechanical plant within

Part Amount DoF

Photomask 1 158502

Bending clamp left 1 234684

Bending clamp right 1 240324

Rods 28 6

Z-supports 3 3

Preload springs 14 14088

Piezo-actuators 14 12

Total assembly - 831087

Table 6.3: A summary of the photomask assembly parts and the total number of DoF.

acceptable computational limits, the nodal DoF of the FE components are reduced

using reduction techniques discussed in chapter 2.

In order to correctly perform the analysis on photomask curvature focus improve-

ment, reduction of components must not influence the properties of the system. This

effect would lead to additional lack of tracking performance due to the disability to

correctly describe the mechanical plant behavior. The goal of reducing the FE compo-

nents therefore is to be able to accurately describe the properties of the system with the

least amount of DoF. To identify the most suitable reduction scheme, i.e. one that does

not influence the curvature tracking abilities of the system but has the least amount of

DoF, three different reduced models will be used during the performance evaluation.

From the subsequent tracking performance results it can then be determined which re-

duction scheme is best suited for describing the dynamic properties of the mechanical

plant. The three reduced models are:

• Model 1: The FE components of the assembly are reduced using the Guyan

reduction (see 2.2.1). This reduction provides the largest reduction but also the

highest accuracy loss.

• Model 2: The FE components of the assembly are reduced using Craig-Bampton

mode component synthesis (see 2.2.3). The reduction basis is built using the first

10 interface fixed vibration modes. The reduced model has more DoF than model

1 but is able to describe higher dynamic responses of the system than model 1.

• Model 3: The FE components of the assembly are reduced using Craig-Bampton

mode component synthesis with a reduction basis containing 10 internally fixed

143


vibration modes. The reduction basis is built using the first 6 internal vibration

modes and 4 additional modes which are selected to have the highest energy

contribution to curvature deformation (see also 3.5).

The next section discusses the reduction of the FE parts and coupling procedure of the

reduced model.

6.3.1 Model coupling procedure

The reduction of the FE parts is performed conform the three methods discussed in the

previous section. Irrespective of the reduction method, the following reduction steps

are performed for each FE component:

• Photomask: The internal DoF of the photomask are condensed into the interface

DoF. The interfaces are the DoF connecting the photomask to the rods and the

z-supports. Because these interfaces are very small, they are modeled as point

interfaces. The Q-grid is also condensed into the interface DoF, which means

that obtaining deformations of the Q-grid after a dynamic analysis requires the

expansion of DoF from the solution.

• Bending clamps: The bending clamps are attached to the reticle stage. The

latter is assumed as the fixed world. The DoF on the interface of the reticle stage

are therefore constrained and left out of the model during dynamic analysis. Prior

to the reduction of the model, the interfaces to the piezo actuator assemblies

are individually reduced. The flexible interface responses are assumed negligible

because these interfaces are very small with respect to the rest of the structure.

The DoF on the interfaces are therefore reduced using rigid interface projection,

which is discussed in section 2.2.4. After creating the rigid interfaces the complete

component is reduced where the internal DoF are condensed into the interface

DoF.

• Preload springs: All interfaces on the preload springs are assumed to be rigid,

therefore they are reduced using the rigid interface projection. After reduction of

the interfaces the internal DoF are condensed into the interface DoF.

Table 6.4 shows the size of the individual reduced components for each reduction

method. Note that models 2 and 3 are reduced using the same method but have

different reduction bases. Therefore the sizes of the models are equal.

After the reduction, the photomask-actuator FE components are coupled in the

nodal domain using the theory of section 2.3. Conform equation (2.40) the assembled

block diagonal mass and stiffness matrices M and K are constructed using the mass

144


Component Model 1 Model 2/3

Photomask 93 103

Bending clamp 126 136

Rods 6 6

Z-supports 3 3

Preload spring 24 34

Piezoelectric actuators 12 12

Total 1026 1196

Table 6.4: Amount of DoF of the reduced photomask-actuator components.

and stiffness properties of the different substructures. As example, a spy plot of the

assembled stiffness matrix K of model 1 is presented in figure 6.28. In the figure the

component stiffness matrices K are indicated.

K(photomask)

K(clampright)

K(clampleft)

[K(rod1) . . . K(rod28)

][K(z−sup1) . . . K(z−sup3)

][K(spring1) . . . K(spring14)

]

[K(piezo1) . . . K(piezo14)

]

Figure 6.28: Spy plot of the assembled stiffness matrix K for model 1.

Coupling of the substructures is performed using primal assembly techniques which

are discussed in section 2.3.1. The advantage of primal assembly is that the assembled

model has less DoF than a model assembled using dual assembly. This is the effect of

the elimination of interface forces by constructing a unique set of DoF. The downside

is that the interface forces are no longer directly available after dynamic analyses. The

sizes of the complete reduced photomask-actuator models after assembly are displayed

in table 6.5. The table highlights that the reduction methods introduce a significant

reduction in model size and subsequently relax the required amount of computational

145


Model # DoF

Model 1 513

Model 2/3 683

Table 6.5: Amount of DoF of the reduced photomask-actuator components.

power for dynamic analysis of the mechanical plant. The reduced order numerical

model therefore facilitates the investigation of the wafer flatness correction ability of the

photomask bending actuator. The influence of reduction on dynamic properties of the

mechanical plant will be investigated by considering the three proposed models during

photomask curvature focus improvement analysis. Also the dynamic properties of the

reduced components are compared to the full FE models. The next section discusses

the model characteristic of the reduced components by performing MAC analyses and

eigenfrequency checks.

6.3.2 Reduced model accuracy

Reduction of the photomask-actuator components causes the models to describe an ap-

proximate dynamic behavior of the original. To determine the approximation accuracy

of the reduced components, modal analyses of the reduced components are performed

and the obtained modes and eigenfrequencies are compared to the reference model. The

reticle was validated up to 7 [kHz], therefore the modes and according eigenfrequencies

up to 7 [kHz] are used for the analysis. For the bending clamps, which were validated

up to 2.5 [kHz], the modes and according eigenfrequencies up to 2.5 [kHz] are used for

the analysis. Finally, the preload springs, which where not validated, are analyzed for

the eigenmodes and eigenfrequencies up to 5 [kHz] in order to get a clear indication of

the reduction accuracy.

The reduced model accuracies are discussed for the photomask, bending clamps

and preload springs in sections 6.3.2.1 to 6.3.2.3 respectively. For every component the

reduced model accuracies of the different reduction techniques, discussed at section 6.3,

are treated.

6.3.2.1 Photomask

In model 1 and 2 the photomask FE model is reduced using respectively Guyan and

Craig-Bampton reduction. The MAC and eigenfrequency checks with respect to their

reference models are plotted in figures 6.29 and 6.30. Note that rigid body modes are

left out of the figures. From figure 6.29 it is seen that Guyan reduction results in a

model which is only able to produce accurate results for quasi static analyses. The

eigenmodes 1 and 3 to 5 are correct but the eigenfrequencies of modes 3 to 5 differ

146


Referen

cemodel

Reduced model1 2 3 4 5 6 7 8 9 10

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

11

2

3

4

5

6

7

8

9

10

(a)

Frequen

cydifferen

ce[%

]

Frequency [kHz]0 1 2 3 4 5 6 70

10

20

30

40

50

60

70

80

(b)

Figure 6.29: Guyan reduction of the photomask, a) MAC analysis and b) eigenfrequency

error plotted against the eigenfrequencies of the reference system.

heavily with the reference model. Only the first mode is accurately described by the

reduced model. Figure 6.29 shows that the reduced systems eigenmodes up to 7 [kHz]

Referen

cemodel

Reduced model1 2 3 4 5 6 7 8 9 10

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

11

2

3

4

5

6

7

8

9

10

(a)

Frequen

cydifferen

ce[%

]

Frequency [kHz]

0 1 2 3 4 5 6 70

0.5

1

1.5

2

2.5

(b)

Figure 6.30: Craig-Bampton reduction of the photomask (reduction basis with 10 in-

terface fixed modes), a) MAC analysis and b) eigenfrequency error plotted against the

eigenfrequencies of the reference system.

are validated with MAC values higher than 98% and eigenfrequency errors below 2.5%.

The dynamic analysis of model 3 uses a Craig Bampton reduced order photomask-

actuator assembly. In specific, the photomask is reduced using a reduction basis which

contains modes of the interface fixed system that describe curvature deformation. The

selected modes are found using the Gramian theory, which is discussed in chapter 3.

As set out in chapter 3, the calculation of an observability Gramian for the interface

fixed system requires well spaced eigenfrequencies and low damping. During the EMA of

147


the photomask (section 6.2.2) it was shown that the damping ratio’s of the free system

lie below 0.8 [%] (based on the best measured results). Assuming that the damping

ratio’s for the interface fixed system are not higher than this value, the damping ratio

for all modes is set to the average of the damping ratio’s found during the EMA.

Figure 6.31 shows the eigenfrequencies of the interface fixed system up to 20 [kHz].

On the y-axis the frequency spacing to the next eigenfrequency is plotted. The plot

Spacing[kHz]

Frequency [kHz]

0 2 4 6 8 10 12 14 16 18 200

0.5

1

1.5

2

2.5

Figure 6.31: Frequency spacing of the eigenfrequencies of the interface fixed photomask

model.

shows that the eigenfrequencies are well spaced with at least 80 [Hz] in between two

eigenfrequencies.

To determine observability Gramians, the state space of the modal model was built

using equation (3.9). The output of the system (y) was chosen to be the DoF in

z-direction of the Q-grid. These DoF are used for the field curvature output based

Gramian calculation because the curvature shape of the Q-grid determines the focus

improvement of CAL strategies. The observability Gramian for field curvature based

output was next calculated using equation (3.132) where ckl was replaced by equation

(3.142). By inspection, the approximation for the field curvature based observability

Gramian, equation (3.143), is valid because it returned a relative error of 0.78 [%] with

respect to the full Gramian. Figure 6.32 shows the obtained field curvature based

observability Gramian approximation of equation (3.143). The results are plotted in

terms of energy contribution to curvature deformation for the modes of the interface

fixed photomask model. The figure shows that modes 1, 3, 8, 11, 13 and 26 contribute

to the deformations of the photomask inhabiting curvature shapes. It was assumed

148


Energy[ m

2]

Mode number

0 5 10 15 20 25 3010−12

10−10

10−8

10−6

10−4

10−2

100

Figure 6.32: Energy contribution per mode to curvature deformation for unitary initial

positions of the photomask model.

that these modes are also controllable by the interface forces and are present in the

responses of the internal DoF to the interface forces. This is indeed to be expected

because a controllability check of the modal truncated system (up to the 100th mode)

returned a full rank Gramian matrix.

The Craig-Bampton reduction basis for the photomask was next built using the

first 6 interface fixed eigenmodes, appended with modes 8, 11, 13 and 26 to ensure

preservation of curvature deformation properties after reduction of the system. The

MAC analysis and eigenfrequency check of the reduced system is plotted in figure 6.33.

The reduced system shows good correlation to the reference model, only mode 9 is not

properly described by the reduced system. The eigenfrequency errors all remain below

8%. It can be concluded that this reduced model of the photomask is able to accurately

describe the photomask dynamics up to 5 [kHz].

6.3.2.2 Bending clamps

The bending clamps are reduced using the Guyan method for model 1 and Craig-

Bampton method for models 2 and 3. The MAC analyses and eigenfrequency compar-

isons of the flexible modes for Guyan and Craig-Bampton reduction are displayed in

figures 6.34 and 6.35 respectively. Only one bending clamp is discussed in this section

because the two bending clamps show equal behavior. Figure 6.34 shows that Guyan re-

149


Referen

cemodel

Reduced model1 2 3 4 5 6 7 8 9 10

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

11

2

3

4

5

6

7

8

9

10

(a)Frequen

cydifferen

ce[%

]Frequency [kHz]

0 1 2 3 4 5 6 70

1

2

3

4

5

6

7

8

(b)

Figure 6.33: Craig-Bampton reduction of the photomask with Gramian selected reduc-

tion basis, a) MAC analysis and b) eigenfrequency error plotted against the eigenfre-

quencies of the reference system.

Referen

cemodel

Reduced model1 2 3 4 5 6 7 8 9 10 11

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

11

2

3

4

5

6

7

8

9

10

11

(a)

Frequen

cydifferen

ce[%

]

Frequency [kHz]0 0.5 1 1.5 2 2.50

1

2

3

4

5

6

7

8

9

(b)

Figure 6.34: Guyan reduction of the bending clamp, a) MAC analysis and b) eigenfre-

quency error plotted against the eigenfrequencies of the reference system.

150


duction results in a system which has dynamic properties approximating the properties

of the reference system up to about 2.2 [kHz]. Mode 12 has erroneous mode correlation

between the reduced and full model. The Craig-Bampton reduction produces a system

Referen

cemodel

Reduced model1 2 3 4 5 6 7 8 9 10 11

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

11

2

3

4

5

6

7

8

9

10

11

(a)

Frequen

cydifferen

ce[%

]

Frequency [kHz]0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

(b)

Figure 6.35: Craig-Bampton reduction of the bending clamp, a) MAC analysis and b)

eigenfrequency error plotted against the eigenfrequencies of the reference system.

which is able to approximate the dynamic properties of the reference system for the

complete range up to 2.5 [kHz]. All eigenfrequency errors remain below 1.8% and the

MAC analysis shows high correlations.

Prior to the Guyan and Craig-Bampton reduction, the interfaces with the piezo-

assemblies are reduced using rigid interface projection. The theory on this reduction

methodology is presented in section 2.2.4. To verify that the interfaces can indeed

be approximated by a rigid interface, a rigidness check of the interface responses is

performed. The rigidness formula was defined by equation (2.39) and is calculated for

all frequency response functions (FRF’s) in the frequency range of 0 to 2.5 [kHz]. The

results are displayed in figure 6.36. Because the clamp consists of multiple interfaces

which are rigified, the figure shows the average rigidness of all similar interfaces present

in the bending clamp. The figure shows that the front piezo-assembly interfaces behave

rigidly up to a frequency of 2.5 [kHz] because the rigidness remains above 90 %. After

2.5 [kHz] the rigidness drops down to 84 % which indicates that the interfaces show

flexible behavior and cannot be approximated with a rigid interface anymore. The back

piezo-assembly interfaces can be assumed to be rigid up to 1.5 [kHz].

6.3.2.3 Preload springs

The preload springs are reduced using the Guyan method for model 1 and Craig-

Bampton method for models 2 and 3. The MAC analyses and eigenfrequency com-

parisons of the flexible modes for Guyan and Craig-Bampton reduction are displayed

151


(a)

Front piezo interfaceBack piezo interface

Rigidness[%

]

Frequency [kHz]0 0.5 1 1.5 2 2.5

70

75

80

85

90

95

100

(b)

Figure 6.36: Rigidness of the rigid interface projected interfaces, a) indication of pro-

jected faces b) rigidness spectrum of the interfaces.

in figures 6.37 and 6.38 respectively. From figure 6.37 it is concluded that only the

Fullmodel

Reduced model1 2 3 4 5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1

2

3

4

5

(a)

Frequen

cydifferen

ce[%

]

Frequency [kHz]0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

20

40

60

80

100

120

140

(b)

Figure 6.37: Guyan reduction of the preload spring, a) MAC analysis and b) eigenfre-

quency error plotted against the eigenfrequencies of the reference system.

first mode of the reference system is validated. Because the eigenfrequency of this

mode is rather high, 1.4 [kHz], the model is valid up to the first eigenfrequency. The

Craig-Bampton reduction creates a model of the preload spring which is also valid up

to the first eigenfrequency. The first mode has high correlations to the reference system

but modes 2 to 5 are not accurately describing the dynamic properties of the reference

system.

Prior to the Guyan and Craig-Bampton reduction the interfaces of the preload

springs are rigified using rigid interface projection. To verify that the interface responses

can be approximated by a rigid interface the rigidness spectrum of the preload springs is

calculated using equation (2.39). The results are presented in figure 6.39. The rigidness

152


Fullmodel

Reduced model1 2 3 4 5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1

2

3

4

5

(a)

Frequen

cydifferen

ce[%

]

Frequency [kHz]0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

(b)

Figure 6.38: Craig-Bampton reduction of the preload spring, a) MAC analysis and b)

eigenfrequency error plotted against the eigenfrequencies of the reference system.

(a)

Back piezo interfaceFront piezo interfaceBack clamp interfaceFront clamp interface

Rigidness[%

]

Frequency [kHz]

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 560

65

70

75

80

85

90

95

100

(b)

Figure 6.39: Rigidness of the rigid interface projected interfaces, a) indication of pro-

jected faces b) rigidness spectrum of the interfaces.

153


spectrum shows that the interfaces can be assumed rigid up to 3.25 [kHz]. After this

frequency the interfaces show flexible behavior which can no longer be described by

rigid interfaces.

6.3.3 Summary

The photomask-actuator assembly is reduced and coupled using three methods: Guyan

reduction, Craig Bampton reduction and Craig Bampton reduction with a reduction

basis holding internally fixed eigenmodes which inhabit properties of curvature defor-

mation of the photomask.

After reduction of the FE components, the reduction accuracy with respect to the

reference systems was determined by MAC analysis and frequency error calculation.

From the obtained results it can be concluded that the assembled model using Guyan

reduced components will lead to a model which is able to describe quasi static behavior

of the photomask-actuator assembly. The Craig-Bampton reduced components (both

for model 2 and 3) will lead to an assembled model which is able describe the reference

system dynamics up to 3 [kHz].

During the specification derivation procedure (chapter 5) it was found that the fre-

quency content of the input specification for the photomask-actuator assembly holds

values up to 150 [Hz], therefore the systems responses to the input specification are

expected to be quasi static. This means that the Craig-Bampton reduced system is

well able to describe responses of the mechanical plant for the desired inputs. In the

next chapter it will therefore be shown which reduced photomask-actuator model is sat-

isfactory accurate for describing the mechanical plant during the photomask curvature

focus improvement analysis.

154

Chapter 7

Photomask curvature focus

improvement

This chapter discusses the photomask curvature focus improvement analysis of the

mechanical plant. The flowchart representing the analysis process is represented by

figure 4.6 and will be applied throughout this chapter.

To determine the behavior of the mechanical plant during an exposure process with

CAL application, the plant is described by the reduced photomask-actuator assembly

models derived in chapter 6. During the analysis the performance tracking ability of

the photomask-actuator assembly is identified with respect to ideal photomask CAL,

discussed in chapter 5. Three CAL strategies are adopted for evaluation: intradie,

filtered intradie and interdie.

Intradie and interdie CAL strategies were already discussed in chapter 4, though in

real life, the mechanical plant will not be subjected directly to piezo-actuator setpoints

obtained from the curvature values. To reduce high frequency noise on the actuator

signals and disturbance effects of resonances of the plant, the setpoints will be passed

through a low pass filter before presenting them to the actuators. To simulate the effect

of filtering the actuator setpoints, the mechanical resonance specification for the plant

design is considered (see 5.2.3). This specification was adopted using a criteria based

on the frequency content found in the curvature setpoints. It was determined that

the highest amount of frequency content was present in the range of 0 to 150 [kHz].

Therefore the filtered intradie CAL analysis in this chapter investigates the performance

tracking abilities of the mechanical plant when the actuator setpoints are run through

a lowpass filter with a cutoff frequency of 150 [kHz].

The proposed CAL strategies are evaluated for the three reduced order mechanical

models, derived in the previous section. With the analysis it is possible to identify

which reduction method is best suited for the specific purpose of this analysis. These

155

CHAPTER 7. PHOTOMASK CURVATURE FOCUS IMPROVEMENTCompany-secret

models are (see also 6.3):

• Model 1: Guyan reduced components.

• Model 2: Craig-Bampton reduced components using a reduction basis with 10

interface fixed eigenmodes.

• Model 3: Craig-Bampton reduced components using a reduction basis with 6

interface fixed eigenmodes and 4 eigenmodes selected to have high influence on

curvature deformation.

The substrates which are evaluated for CAL focus improvement analysis are iden-

tical to the substrates used in chapter 5. Therefore, the curvature setpoints for the

photomask-actuator assembly are already known from the specification derivation. Sec-

tion 7.1 discusses the analysis procedure and the settings of the numerical algorithm.

The following sections, 7.2, 7.3 and 7.4 discuss the performance tracking results of

intradie, filtered intradie and interdie CAL respectively.

7.1 Analysis procedure and settings

Consider the flowchart for the performance evaluation, which was presented in figure

4.6. Compared to ideal CAL strategies, which were applied in chapter 5 for the spec-

ification derivation, the analysis procedure is largely the same. The difference lies in

the fact that in this chapter the ideal mechanical plant is replaced by a numerical

model of the mechanical plant. To calculate focus improvements using the mechanical

plant model, several key procedures must be performed before calculating the focus

improvement:

• Model setpoint calculation: Consider a state of the art substrate topology.

The desired curvature values for CAL are calculated using the anticlastic cur-

vature augmented fitting function. To subsequently acquire an equal amount of

curvature from the photomask, the voltage applied to the piezo actuators must

be related to the resulting curvature deformation of the photomask. Using this

relation, the setpoints for the numerical model are constructed.

• Time integration: During scanning of a die, the mechanical plant is dynami-

cally or statically subjected to the loads on the piezo actuators. To obtain the

tracking performance, the photomask curvature over time, induced by the chang-

ing voltages of the piezoelectric actuators, is calculated using a Newmark time

integration scheme (see F).

156

7.1. ANALYSIS PROCEDURE AND SETTINGS

• Curvature tracking: Having obtained the time responses of the Q-grid during

scanning, the realized time dependent curvature values are calculated per slit

position, using the curvature augmented fitting function, resulting in a curvature

correction at wafer level. After the exposure to the light slit, the focus error

is calculated by subtraction of the corrections from the original wafer topology.

Additional offset and rotation of the Q-grid is neutralized by augmenting the

setpoints for the wafer stage.

The implementation of the above enumerations are further discussed in sections 7.1.1

to 7.1.3. The methodology is further clarified by the use of the topology of the die

displayed in figure 5.8.

7.1.1 Model setpoint implementation

The desired curvature values for the mechanical plant are derived from the substrate

topology using the anticlastic curvature augmented fitting functions. To subsequently

derive the setpoints for the numerical model, the applied piezo voltages are related to

curvature deformation by linear static calibration. First a unitary voltage of 1 [V ] is

supplied to the piezo actuators, followed by calculation of the equilibrium position of the

numerical model. Note that all three reduced models used for CAL focus improvement

calculation are producing exact solutions in static problems, the calibration thus holds

for all three models.

After obtaining the static solution to the unitary input, the curvature of the Q-

grid is calculated. Due to parasitic stiffnesses in the system, curvature calculation of

the whole Q-grid area would lead to an approximate curvature value. Therefore, to

get better insight in the curvature deformation, 2D line curvature κxx is calculated

for each slit position inside the Q-grid. The obtained curvature profile of the Q-grid

is presented in figure 7.1. The figure highlights that the central slit position has the

highest curvature. Because the overall difference in curvature is small, the calibration

value for the curvature to voltage ratio is adopted as the average of all curvature values:

κxxU

= 8.83 · 10−6

[1

m · V]

U =1

8.83 · 10−6κxx (7.1)

= 1.13 · 105κxx (7.2)

Subsequently, the setpoints for the numerical model are calculated as:

uκ (t) = ccalκxx (t)

= 1.13 · 105κxx (t) , (7.3)

157


average κxx

κxx

κxx

[ 1 m

]

y [mm]

−80 −60 −40 −20 0 20 40 60 80

×10−6

8.79

8.8

8.81

8.82

8.83

8.84

8.85

8.86

8.87

Figure 7.1: 2D curvature per slit position of the Q-grid to unitary actuator voltage

input.

with ccal the linear calibration constant. Note that the time dependency of the setpoints

originates from the scanning action of the reticle stage. Indeed, the average calibration

value suffices because the voltage difference calculated for the maximum and minimum

calibration values related to the highest found curvature setpoint in chapter 5 results

in (see page 104):

dU = κxx,max (ccal,min − ccal,max)

= 6.03 · 10−4

(1

8.80 · 10−6− 1

8.86 · 10−6

)

= 0.46 [V ] , (7.4)

which is 0.34 % difference considering the total voltage range of the piezo actuators:

U = ccal,min · ±κxx,max

= 1.13 · 105 · ±6.03 · 10−4

= ±68.3 [V ] . (7.5)

7.1.2 Newmark time integration

The time response of the numerical model to the command signals uκ (t) is calculated

using an integration method from the Newmark family. The Newmark method is a com-

monly used single step integration algorithm which can be tuned to be unconditionally

stable. The applied Newmark integration scheme in this setting is the unconditionally

stable Average Constant Acceleration [12]. A flowchart for the integration scheme is

presented in appendix F.

Before calculating the time response, boundary conditions must be formulated.

These boundaries are initial position and speed of the DoF. It is assumed that the

158

7.1. ANALYSIS PROCEDURE AND SETTINGS

photomask is positioned in the right deformation during the transfer from one die to

the next. The initial position is therefore adopted as the static equilibrium of the

model with respect to the actuator forces at t = t0. The initial speed of the DoF is also

assumed to be adopted during the transfer. The initial speed is therefore approximated

as the Euler forward differentiation [48] of the static equilibria with respect to the

actuator forces at t = t0 and t = t0 + dt. When applying these assumptions, the initial

boundary conditions read:

qp,0 = K−1p pp (t = t0) (7.6)

˙qp,0 =K−1

p pp (t = t0 + dt)− K−1p pp (t = t0)

dt, (7.7)

where qp is the DoF vector, Kp the primal assembled stiffness matrix and pp the force

vector holding the setpoints for the piezo actuators (see also 2.52 and 2.53).

Time integration is applied to the three reduced numerical models presented in

chapter 6. For each model the responses to intradie, filtered intradie and interdie

setpoints are calculated. For the filtered intradie setpoints, a lowpass zero phase filter,

identical to the filter used in section 5.2.3, with a bandwidth of 150 [Hz] is implemented

before construction of the force vector pp.

7.1.3 Performance tracking

The time response of the DoF inside the Q-grid are expanded from the time response of

the reduced systems by premultiplying the vector that represents the reduced DoF of

the photomask with transformation matrix T (s) of the photomask. Once these values

are known, the curvature tracking values for each slit position on the Q-grid are calcu-

lated using the anticlastic curvature augmented fitting function. The methodology of

obtaining curvature values from slit positions during scanning was described in section

4.1.

As example, figure 7.2 shows the curvature tracking values for the example die of

figure 5.8. The tracking values are derived from the responses of model 1, which was

built using Guyan reduced FE components. The black, red, green and blue lines repre-

sent respectively the desired curvature (from which the actuator setpoints are derived),

intradie curvature output, filtered intradie curvature output and interdie curvature out-

put obtained from the Q-grid responses. Note that the slit positions are dependent on

time due to the scanning speed of the reticle stage. The latter was assumed to be equal

to 1.6[ms

].

Figure 7.2 seems to indicate that the filtered intradie CAL disposes a lot of cur-

vature content from the setpoint signal, note however that the maximum deviation

between the original setpoint signal and filtered signal is 2.4 ·10−5[1m

]. This is only 2%

159


κ(P )xx output, interdie

κ(P )xx output, filtered intradie

κ(P)xx output, intradie

κxx setpoints

κ(P

)xx

[ 1 m

]

slit position y (t) [mm]

55 60 65 70 75 80 85

×10−4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Figure 7.2: Curvature tracking results of model 1 on the example die.

with respect to the range of curvature setpoints found in the specification derivation

(see also 5).

Using the curvature tracking results in addition to the offset and rotational setpoints

of the wafer stage, the topology is reconstructed using the theory set out in section 4.1.

Next the focus improvement is analyzed by calculating the focus error. The next section

discusses the intradie CAL focus improvement results obtained for the complete set of

substrate topologies.

7.2 Intradie performance tracking

When the intradie CAL setpoints are obtained for the slit positions on the Q-grid, the

focus error for each measured topology datapoint is found by calculating the difference

between actual topology and reconstructed topology, i.e. applying equation (1.69) on

the following values:

• offset found from the specification derivation, oz, adjusted for additional offset of

the photomask, o(P )z .

• Rotational setpoints from the specification derivation, Rx and Ry, adjusted for

additional rotations of the photomask, R(P )x and R

(P )y respectively.

• Curvature values derived from the deformation of the Q-grid during scanning,

κxx.

160

7.2. INTRADIE PERFORMANCE TRACKING

As explained in section 4.1.1, each topology datapoint is reconstructed multiple times

during this process. Therefore a MA error is calculated which represents the average

focus error of each individual datapoint on the substrate. As example, the resulting

MA focus errors of the substrate displayed in figure 5.5(a) are presented in figure 7.3.

y[m

m]

x [mm]

−100 −50 0 50 100

−25

−20

−15

−10

−5

0

5

10

15

−100

−50

0

50

100

Figure 7.3: MA focus error between intradie CAL and wafer topology (in [nm]).

Once a whole wafer is processed and MA focus errors of all datapoints on the

substrate are known, a single scalar focus error e(P )w for the whole wafer is calculated

(see also section 4.1.1):

e(P )w = |μMA|+ 3σMA. (7.8)

The wafer focus error of model 1, applied on the example wafer, is presented in table

7.1 together with the focus error of CL strategies (ew). From the table the intradie

Property CAL CL

|μ| [nm] 2.3 · 10−3 5.5 · 10−3

σ [nm] 3.8 4.4

e(P )w , ew [nm] 11.5 13.2

Table 7.1: Example of absolute mean and 3σ properties of the wafer focus error for

intradie CAL and CL strategies using reduced model 1.

CAL focus improvement of model 1 with respect to the CL strategy is subsequently

161


calculated as:

Improvement =

∣∣∣e(P )w − ew

∣∣∣|ew| 100%

= 12.78%. (7.9)

To investigate the intradie photomask curvature focus improvement, intradie CAL

focus error calculation for a set of 6 substrates is used for evaluation. The same sub-

strates as presented in chapter 5 are used for this analysis. The die topologies of a

substrate are available in normal postprocessed data and FWOL postprocessed data

formats, i.e. full field dies and full field FWOL dies respectively (see section 5.1.2).

These two formats are evaluated for comparison. Furthermore, the focus errors are

computed for the three numerical models (proposed in section 6.3) to identify the in-

fluence of reduction techniques. In table 7.2 the obtained intradie CAL focus errors of

the models are compared to CL focus errors for substrates with respectively full field

dies and full field FWOL dies. The last columns represent the obtained CAL focus im-

provement. The table shows that all three models produce the same focus improvement

Full Field

e(P )w [nm] ew [nm] Focus improvement [%]

Model 1 14.9

19.0

21.5

Model 2 14.9 21.5

Model 3 14.9 21.5

Full Field FWOL


Model 1 11.7

16.3

28.1

Model 2 11.7 28.1

Model 3 11.7 28.1

Table 7.2: Mechanically modeled intradie CAL focus improvement for full field and full

field FWOL topology substrates.

at nanometer level, which is equal to the ideal focus improvement calculated in section

5.2.1. Differences in focus error between plant model and ideal plant (which assumes

exact curvature output, as discussed in chapter 5), as well as between the different

numerical models was found on picometer level, which is neglected because this only

influences the focus improvement by 10−2 [%].

162

7.3. FILTERED INTRADIE PERFORMANCE TRACKING

7.3 Filtered intradie performance tracking

The procedure for the filtered intradie performance tracking is identical to the previ-

ous section, only the actuator setpoints are filtered (lowpass 150 [Hz]) before the time

integration. As example the filtered intradie CAL results for the example wafer using

mechanical plant model 1 is calculated. The MA focus errors with respect to the topol-

ogy data are not displayed in this section because there would be no visual difference

to figure 7.3. The wafer focus error characteristics are displayed in table 7.3. From the

Property CAL CL

|μ| [nm] 6.32 · 10−4 5.5 · 10−3

σ [nm] 3.9 4.4

e(P )w , ew [nm] 11.6 13.2

Table 7.3: Example of absolute mean |μ| and 3σ properties of the wafer focus error for

filtered intradie CAL and CL strategies.

table the filtered intradie CAL focus improvement of model 1 with respect to the CL

strategy is calculated as:

Improvement =

∣∣∣e(P )w − ew

∣∣∣|ew| 100%

= 12.21%, (7.10)

which is only 0.57 % lower than intradie CAL focus improvements. This indicates that

by filtering the signal up to 150 [Hz] most of the frequency content is present in the

command signals that are applied to the mechanical plant.

To investigate the filtered intradie photomask curvature focus improvement, the

intradie CAL focus error for the set of 6 substrates are used for evaluation. In table

7.4 the obtained filtered intradie CAL focus errors of the mechanical plant models are

compared to CL focus errors for substrates with respectively full field dies and full

field FWOL dies. The last columns represent the obtained CAL focus improvement.

The results show that filtered intradie CAL strategies introduce less than 1% loss on

focus improvement with respect to intradie leveling. This indicates that filtering up to

150 [Hz] only discards a small amount of frequency content in the setpoint signals for

the piezo actuators before correcting substrate unflatness. Therefore the mechanical

plant remains able to apply dynamic curvature correction of substrate unflatness. From

the results it is also seen that all three models produce the same focus improvement

at nanometer level. Differences in obtained focus error of the numerical models were

found on picometer level.

163


Full Field


Model 1 15.0

19.0

20.8

Model 2 15.0 20.8

Model 3 15.0 20.8

Full Field FWOL


Model 1 11.8

16.3

27.6

Model 2 11.8 27.6

Model 3 11.8 27.6

Table 7.4: Mechanically modeled filtered intradie CAL focus improvement for full field

and full field FWOL topology substrates.

7.4 Interdie performance tracking

For interdie CAL performance analysis, tracking results are identified from the re-

sponses of the mechanical plant with static load application on the piezoelectric ac-

tuators. Because only the average curvature is corrected using this procedure, less

focus improvement will be gained with respect to intradie and filtered intradie CAL

strategies.

As an example, the interdie CAL results for the example wafer using mechanical

plant model 1 is calculated. The resulting MA focus errors with respect to the topology

data are again not displayed because there would be no visual difference to figure 7.3.

The wafer focus error characteristics are displayed in table 7.5. The resulting interdie

Property CAL CL

|μ| [nm] 4.8 · 10−3 5.5 · 10−3

σ [nm] 4.1 4.4

e(P )w /ew [nm] 12.4 13.2

Table 7.5: Example of absolute mean |μ| and 3σ properties of the wafer focus error for

interdie CAL and CL strategies.

CAL focus improvement of model 1 with respect to the CL strategy yields:

Improvement =

∣∣∣e(P )w − ew

∣∣∣|ew| 100%

= 6.02%, (7.11)

164

7.5. SUMMARY

which is less than half the focus improvement found for intradie and filtered intradie

focus improvement.

The focus improvement of interdie CAL strategies using the mechanical plant mod-

els is next investigated for the set of substrates of the previous sections. In table 7.6

the obtained interdie CAL focus errors of the mechanical plant models are compared to

CL focus errors for substrates with respectively full field dies and full field FWOL dies.

The last columns represent the obtained CAL focus improvement. Interdie CAL using

Full Field


Model 1 16.4

19.0

13.5

Model 2 16.4 13.5

Model 3 16.4 13.5

Full Field FWOL


Model 1 13.3

16.3

18.4

Model 2 13.3 18.4

Model 3 13.3 18.4

Table 7.6: Mechanically modeled filtered interdie CAL focus improvement for full field

and full field FWOL topology substrates.

the mechanical plant produces the same results as the ideal plant focus improvements,

(see section 5.2.1). Only small deviations are found on picometer level. Compared to

intradie CAL there is 8% loss on focus improvement for full field dies and 10% loss for

full field FWOL dies.

7.5 Summary

This chapter discussed the photomask curvature focus improvement of intradie, filtered

intradie and interdie CAL. The flowchart for the analysis procedure was presented in

figure 4.6. To model the properties of the mechanical plant during the exposure process,

numerical reduced order models were introduced for performance tracking. During the

CAL procedure, the time responses of the model to the piezo actuator setpoints were

used to derive the subsequent focus error between adapted aerial image and substrate

topology. Finally, using the obtained focus errors of CAL the focus improvement of

photomask curvature adaptation was derived with respect to CL strategies.

To determine the influence of numerical reduction on the accuracy of the focus

improvement results, three different reduced models of the mechanical plant were im-

165


plemented. The first model was created using Guyan reduced components. The second

and third models were built using Craig-Bampton reduced components with a reduc-

tion basis using 10 interface fixed modes. While the second model contained the first

10 interface fixed modes, the third model used the first six interface fixed modes and

4 modes which have large influences on curvature deformation of the interface fixed

system.

After performing the analyses for the three reduced models, it was found that all

models produce the same focus improvement results for both intradie and filtered in-

tradie CAL, as well as interdie CAL. It was found that intradie CAL leveling introduces

a focus improvement of 21.5% and 28.1% with respect to CL strategies for substrates

with respectively full field and full field FWOL topology data. Filtered intradie CAL

resulted in focus improvements of respectively 20.8% and 27.6%. Finally, interdie CAL

resulted in focus improvements of respectively 13.5% and 18.4%. From these results

conclusions will be drawn in chapter 8, followed by recommendations for future research.

166

Chapter 8

Conclusions and

recommendations

In this thesis, field curvature correction performance in lithographic systems was in-

vestigated. In specific, a dynamic system analysis of field curvature optical element

manipulation was conducted. Field curvature correction makes it possible to reduce

the dependency of the lithographic focus budget on wafer flatness and subsequently

meet the imaging requirements for state of the art lithographic systems. Field curva-

ture correction is achieved by bending the photomask during its exposure, resulting in

a curved aerial image projection on the substrate.

To support the photomask curvature manipulation, a number of topics required

investigation. Assuming the manipulator design to be given, this thesis concentrated

on the following:

• Specification derivation: Analysis of state of the art substrate flatness and

identification of the required amount of curvature correction.

• Performance evaluation: Focus budget improvement analysis of curvature cor-

rection using the photomask bending actuator design.

To fulfill the thesis objectives, the research was divided into three parts. Part I

discussed the general theory on photomask curvature modeling and model reduction

techniques. Model reduction techniques were used to obtain a sufficiently accurate

photomask and manipulator model which require a minimal amount of computational

effort. Part II covered the photomask curvature analysis methodology and specification

derivation. Finally, part III covered the performance evaluation results of field curvature

correction using a numerical model of the photomask bending actuator design and state

of the art substrate topology information.

167

CHAPTER 8. CONCLUSIONS AND RECOMMENDATIONS Company-secret

Conclusions will be drawn for the performed research in section 8.1, followed by

recommendations for future work in section 8.2.

8.1 Conclusions

Chapter 1 introduced the anticlastic curvature augmented fitting function, derived

from plate theory using Kirchoff’s assumptions and Green strain expressions. It was

concluded that the photomask can be considered as a plate structure because the

ratio of its in-plane dimension with respect to the thickness places the photomask

in the category of thin plates. The fitting function is used to obtain offset, rotational

and curvature setpoints for CAL strategies from a set of discrete substrate topology

datapoints in a least squares sense.

In chapter 2 an introduction to model reduction techniques was given. These meth-

ods make it possible to obtain reduced order numerical models for the photomask bend-

ing actuator assembly. Chapter 3 concluded the discussion on model reduction tech-

niques by proposing a recipe for interface fixed mode selection for the Craig-Bampton

reduction method. It was defined that model reduction of the photomask required

the preservation of model properties regarding curvature deformation in order to suc-

cessfully evaluate the CAL performance of the photomask bending actuator design.

To identify the interface fixed modes which contribute to curvature deformation, the

theory on observability and controllability Gramians was set out. Thereafter, the ob-

servability Gramian theory was used to obtain a methodology on identifying modal

energy contribution for systems with field curvature based output specification. For

this chapter the following conclusions can be drawn:

• The observability Gramian is able to identify interface fixed modes which con-

tribute to curvature deformation output of the interface fixed system, provided

that eigenfrequencies are well spaced and damping is low.

• The controllability Gramian can be used to determine whether interface fixed

modes are controllable by the forces on the interfaces. Because the interfaces are

not present in the interface fixed system, the controllability of the free system has

to be full rank in order to conclude if the interface fixed modes are controllable.

In chapter 4 the analysis procedure for the specification derivation and performance

evaluation was set out. In order to determine the focus budget improvement potential of

ideal CAL (i.e. exact curvature correction) and bending actuator model induced CAL,

the focus error calculation procedure was discussed. The focus error of each point on

the substrate is obtained by the average of the focus errors that the point encounters

168

8.1. CONCLUSIONS

during exposure. Full wafer focus errors were thereafter determined by the avarage |μ|plus 3σ values for all parts on the wafer. This was done for two types of curvature

augmented leveling (CAL) strategies:

• intradie CAL

• interdie CAL

Furthermore, the amount of field curvature frequency content was derived from state

of the art substrate topologies. Using these properties, a mechanical resonance specifi-

cation for the bending actuator design was derived. For the photomask bending model

performance evaluation, the analysis procedure was set out for the CAL leveling strate-

gies using a reduced order model of the mechanical plant.

Chapter 5 discussed the specification derivation. Using a set of state of the art

wafer topologies, the intradie and interdie CAL focus improvements for ideal curvature

setpoint tracking were determined. To represent the most common used substrate

topology data in lithographic systems, substrates with normal postprocessed full field

die topology and FWOL postprocessed full field die topology were used during the

analysis. From the analysis procedure also a range of required curvature setpoints for

the photomask bending actuator was deduced. It was found that the curvature ranges

of state of the art substrates with full field dies are:

• Full field: κxx ⊆ [−60.3 · 10−4, 53.7 · 10−4] [

1m

].

• Full field FWOL: κxx ⊆ [−43.0 · 10−4, 37.6 · 10−4] [

1m

].

After obtaining the intradie ideal CAL focus improvement it was found that for full

field die topology data there is a focus improvement of 22% with respect to current

leveling (CL). For full field FWOL die topology data a focus improvement of 28% was

found. For interdie leveling the focus improvement of ideal CAL strategies with respect

to CL was found to be 14% and 18% for respectively full field and full field FWOL die

topology data. From the results the following conclusions can be drawn:

• Both intradie and interdie CAL reduce influences of image to wafer plane non-

conformities on the focus budgets in lithographic systems.

• Ideal intradie CAL has the highest curvature correction potential. The highest

focus improvements were found for full field FWOL die topology data, followed

by full field die topology data.

169


• Ideal interdie CAL does introduce a focus improvement with respect to CL but

is not as effective as intradie CAL. With respect to intradie CAL, the obtained

focus improvement results are about 7% less for full field die topologies and 10%

less for full field FWOL topologies.

After obtaining the ideal focus improvement results of intradie and interdie CAL strate-

gies, a study was performed on the amount of frequency content in the curvature set-

points. It was found that the largest amount of frequency content lies in the range

0 to 150 [Hz]. It therefore can be concluded that the photomask bending actuator

must be able to track frequency contents up to 150 [Hz] in order to successfully correct

field curvature. To ensure field curvature tracking a design specification for the first

mechanical resonance frequency of the bending actuator was set to 5 times the highest

frequency content.

Chapter 6 discussed the numerical modeling of the photomask bending actuator in

addition to the validation of the photomask and bending clamp model. The photomask,

bending clamps and preload springs were modeled using Finite Elements and the z-

supports and rods by massless springs, assuming no rotational stiffness. Finally, the

piezoelectric actuators were modeled as beam elements and assumed to be linearly

independent on voltage input.

From the results of the photomask validation measurement it can be concluded

that the model is validated up to 7 [kHz]. Up to this frequency, all identified modes

have a MAC value of 87% and higher with respect to the FE model. During the

acoustic measurement the eigenfrequencies up to 7 [kHz] were validated with less than

1% deviation between measurement results and FEM. The bending clamp was validated

up to 2.5 [kHz], with mode correlations higher than 84% and eigenfrequency errors

remaining below 9%.

The model reduction of the photomask FE model showed that only quasi-static

analyses can be performed with the Guyan reduced model. The Craig-Bampton reduc-

tion however proved to preserve the dynamic properties of the model up to 7 [kHz].

The Craig-Bamption reduced system with the reduction basis built using field curvature

mode selection is able to describe the dynamic properties of the system up to 5 [kHz].

The Guyan reduced model of the bending clamp showed that this model is able to

accurately describe dynamic behavior up to 1500 [H]. For the Craig-Bampton reduced

model, dynamic properties were preserved up to 2.5 [kHz]. The reduction of the preload

springs showed that for both the Guyan as well as Craig-Bampton reduction the models

are able to only accurately describe the first mode, therefore dynamic model accuracy

up to the first eigenfrequency is preserved.

For both the bending clamps and the preload springs, several interfaces were re-

170

8.2. RECOMMENDATIONS

duced using rigid interface projection. From the results it can be concluded that the

up to 1500 [kHz] all interfaces of the bending clamps can be assumed rigid. For the

preload spring interfaces rigidity is allowed up to 3.25 [kHz].

After constructing three different models of the photomask actuator assembly using

three different reduction techniques, focus improvement of CAL strategies and the me-

chanical plant models was analyzed for intradie, filtered intradie and interdie leveling.

Also both full field and full field FWOL topology substrates were considered. Regard-

ing the reduction methodology, it can be concluded that Guyan reduction proves to

be the best reduction method with the highest level of decrease in DoF. Because the

frequency content in curvature setpoints is found up to 150 [Hz] all reduced models

were able to track the setpoints during CAL and returned the same focus improvement

results. Regarding the leveling strategies, it can be concluded that for both intradie

CAL and interdie CAL the focus improvements are identical up to 0.1% with respect

to ideal CAL strategies. This indicates that the model of the mechanical plant is well

able to track the required curvature setpoints for intradie and interdie leveling. From

the results the following conclusions can be drawn:

• Both intradie and filtered intradie CAL, as well as interdie CAL reduce influences

of image to wafer plane nonconformities on the focus budgets in lithographic

systems.

• Intradie CAL has the highest curvature correction potential. For full field FWOL

die topology data the highest focus improvements were found (28%), followed by

full field die topology data (22%).

• Filtered intradie CAL has almost the same amount of curvature correction po-

tential as intradie CAL, with differences less than 1%. Therefore it can be also

concluded that filtered intradie CAL only discards a small amount of frequency

content in the curvature setpoints.

• Interdie CAL has focus improvement potential but results in lower focus improve-

ment than intradie CAL, about 7% less for full field die topologies and 10% less

for full field FWOL topologies.

8.2 Recommendations

In the previous sections, a number of aspects were mentioned which can provide impor-

tant contribution to the research performed in this thesis. The following recommenda-

tions are therefore given for future research:

171


• Gain more insight into the curvature deformation preservation properties of the

static modes in the Craig-Bampton description of the internal DoF. After Craig-

Bampton reduction, the internal DoF are also described by static modes, therefore

optimality of the reduction basis regarding curvature deformation output preser-

vation is not guaranteed. It could be possible that a large amount of curvature

output information is already preserved in the reduction basis described by the

static modes.

• Implement rotational stiffness of the rods in the photomask bending actuator

model. Up to now the rods are assumed to have negligible rotational stiffnesses.

In order to identify the influences of the parasitic stiffness to the CAL focus

improvement abilities of the mechanical plant, the rod models have to be extended

with rotational stiffness properties.

• Extend the piezoelectric actuator models with non-linear properties. In this the-

sis, the piezoelectric actuators were assumed linear, but in real life they are not

due to their hysteresis properties. It is therefore recommended to implement the

non-linear behavior and validate the resulting models with the actual actuators.

• Validate the modes and eigenfrequencies of the photomask and bending clamps

without the effect of mass loading. Because the modes of the photomask and

bending clamps could only be validated by augmenting the FE model with the

force sensor, it is recommended that an experimental modal analysis is performed

without the effects of mass loading. The obtained modal damping ratios are then

recommended to be implemented in the FE models.

• Use Guyan reduction to condense the internal DoF of the FE substructures of

the photomask actuator assembly. It was concluded that the Guyan reduction

provided the same analysis accuracy as the Craig-Bampton reduction method-

ologies but resulted in substructures with less DoF. Therefore Guyan reduction

is the best possible reduction method to be applied in modeling of the focus im-

provement potential of the photomask bending actuator. Note however that if

key properties of the system are changed, such as scanning speed or frequency

content in the curvature setpoints, dynamic modeling of the photomask bending

actuator could require more accurate reduction methodologies.

• Apply intradie CAL strategies in lithographic systems. Intradie CAL strategies

proved to acquire the highest focus improvements with respect to CL strategies.

If, due to the design of the plant, curvature setpoints are run through a lowpass

filter before presentation to the piezoelectric actuators, it is recommended that

172

8.2. RECOMMENDATIONS

frequency content up to 150 [Hz] is preserved in order to obtain focus improve-

ment results in the range of to intradie CAL focus improvement.

• Investigate the focus improvement potential of CAL strategies on substrates with

non-full field die topology data to complement the study on field curvature cor-

rection performance of the photomask bending actuator design.

• Investigate the focus improvement potential of CAL strategies on substrate topolo-

gies measured using Agile correction to complement the study on field curvature

correction performance of the photomask bending actuator design.

• Implement the flexible interface responses of the reticle stage. In this thesis, the

interfaces between bending clamps and reticle stage were assumed to be rigid.

Because the reticle stage inhabits dynamic properties, it is recommended to aug-

ment the photomask bending model with the dynamic properties of the reticle

stage.

Besides recommendations for future research, there are also a number of recommen-

dations to be made for the future usage of the reduced order numerical model of the

photomask bending actuator:

• Use the plant model for feedback controller tuning. Using the numerical model,

estimations can be calculated for realized curvature output and strain gauge sen-

sor output with respect to input curvature setpoints. Therefore the model can be

used for controller tuning of the piezo actuators.

• Use the plant model in formulating the feedforward controller. By approximating

stiffness effects of the mechanical plant with the numerical model, a stiffness

feedforward controller can be implemented into the control loop of the mechanical

plant.

• Perform overlay analyses of the photomask during dynamic field curvature adap-

tation. This thesis covered the analysis on focus budget improvement for CAL

but it is also possible to calculate overlay errors of the photomask Q-grid with

respect to the exposed substrate area.

173

CHAPTER 8. CONCLUSIONS AND RECOMMENDATIONS

174

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surface photomask substrates. 118

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Delft. 124

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Differentiaalvergelijkingen. VSSD, 2006. 159

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Engineering. John Wiley & Sons, LTD, 2000. 184

178

Appendix A

ASML Twinscan system

architecture

A lithographic system module, also called waferstepper, produced by ASML contains

various mechanical and electronic parts. The system architecture consists of several

parts (see also figure A.1):

Lens assembly

Reticle stage

Illuminator

Electronics cabinetMain body

Wafer stage

Reticle heating moduleWafer handling module

Figure A.1: Waferstepper system architecture (Twinscan platform)

• Main body

Frame on which all different parts are mounted, also known as the base frame.

• Reticle stage

Stage on which the reticle is loaded and used for projection. The reticle is held

by the short stroke that is moving on the long stroke. The long stroke is moving

on the base frame.

179

APPENDIX A. ASML TWINSCAN SYSTEM ARCHITECTURE

• Wafer stage

The stage on which the wafer is held when undergoing step & scan. The wafer is

held on a chuck, which is moving on a large stone. The wafer stage is capable of

holding two wafers, one chuck to measure the position and topology of the wafer

and one chuck on which the lithographic process is performed.

• Illuminator

Light source providing light for the lithographic process.

• Lens assembly

A system of lenses providing the imaging of the reticle image onto the wafer.

The lens assembly contains various lenses which are capable of altering the aerial

image in order to gain better focus and overlay accuracies. The lens assembly

creates a 4 times in plane and a 16 times out of plane image demagnification.

• Wafer handling module

Module which contains multiple wafers in order to handle multiple wafers auto-

matically. Furthermore, the module is used to load and unload wafers to and

from the wafer stage.

• Reticle handling module

A module which contains multiple reticles in order to automatically perform var-

ious illuminations steps. Furthermore, the module is used to load and unload

reticles to and from the reticle stage.

• Electronic cabinet

Cabinet containing various electronic components of the waferstepper.

The waferstepper is finally enclosed with an outside cover for protective and visual

purposes. When in operation the system works with dangerous laser sources and high

accelerations, so the cover provides the necessary safety. A representation of the Twin-

scan platform with a view on laser paths, wafer and reticle positions is represented in

figure A.2.

180

Reticle

Illuminator

Projection lens

Silicon wafers(images)

Figure A.2: Twinscan platform

181

APPENDIX A. ASML TWINSCAN SYSTEM ARCHITECTURE

182

Appendix B

Linear least squares methodology

Assume an overdetermined set of equations, ie. a problem with more equations than

unknowns:

Ax = b (B.1)

Due to the overdetermination A is not square but A ∈ Rm×n with m > n and b ∈

Rm. In practice, an overdetermined set of equations does not have a general solution,

therefore most suiting solution is to be found, minimizing the error between solution

and equation space. A possible method to find the best approximating solution is linear

least squares fitting , using the linear system (B.1).

The linear least squares method finds the solution of x (equation (B.1)) such that

the Euclidean norm ‖Ax− b‖2 is minimized. The Euclidean norm is defined as:

‖Ax− b‖2 =m∑j=1

ε2j (x) (B.2)

where εj is the jth component of Ax− b. The minimization of the Euclidean norm can

now be written as:

minx∈R

‖Ax− b‖2 (B.3)

The linear least squares solution is then found by rewriting equation (B.3) into (REF-

ERENCE):

minx∈R

‖Ax− b‖2 = minx∈R

[(Ax− b)T (Ax− b)

]= min

x∈R[(xTAT − bT

)(Ax− b)

]= min

x∈R

[xTATAx− xTATb− bTAx+ bTb

]= min

x∈R[xTATAx− 2xTATb+ bTb

].

(B.4)

183

APPENDIX B. LINEAR LEAST SQUARES METHODOLOGY

Note that xTATb and bTAx are equal scalars and can be added.

The second order nature of equation (B.4) makes it possible to find the solution

by setting the differentiated set to zero:

d

dx

(xTATAx− 2xTATh+ hTh

)= 0

2ATAx− 2ATh = 0

ATAx = ATh

x =(ATA

)−1ATh.

(B.5)

Note that equation (B.5) is requires the inverse of ATA, obligating this product

to be positive definite. A matrix Q is per definition positive definite when [49]:

xTQx > 0, (B.6)

for all nonzero vectors x. Replacing Q with ATA results in:

xTATAx > 0

(Ax)T Ax > 0.(B.7)

Equation (B.7) demands for a nonsingular matrixA. A matrixQ is per definition

nonsingular when

detQ �= 0. (B.8)

When the matrix A does not fulfill the requirements (B.7) and (B.8) no inverse

exists for the product ATA.

184

Appendix C

Derivation of the observability

Gramian

In this appendix, a more detailed description for the derivation of the observabil-

ity Gramian is presented. The derivation of the controllability Gramian can be

performed in the same fashion.

Consider the modal state space description of an arbitrary system with 2 DoF.

Using the theory set out in section 3.1, the state vector is defined as:

x =[η1 ω1η1 η2 ω2η2

]T, (C.1)

where η represents a DoF from the normal equations and ω an eigenfrequency of

the system. The state matrix is built as (see equation 3.13):

As =

⎡⎢⎢⎢⎣

−2ζ1ω1 −ω1 0 0

ω1 0 0 0

0 0 −2ζ2ω2 −ω2

0 0 ω2 0

⎤⎥⎥⎥⎦ . (C.2)

Next, the output is defined as the displacements of all DoF in the nodal system.

From equation 3.16 the output matrix then yields:

Cs =

⎡⎣ 0

φ(1)1

ω10

φ(2)1

ω2

0φ(1)2

ω10

φ(2)2

ω2

⎤⎦ , (C.3)

with φ(∗) represent the modes of the system.

The observability Gramian is calculated from the Lyapunov equation, which

was presented in equation (3.83):

ATs Wo (t = ∞) +Wo (t = ∞)As +CTC = 0. (3.83)

185

APPENDIX C. DERIVATION OF THE OBSERVABILITY GRAMIAN

Implementing the state matrix (C.2) and output matrix (C.3) into (3.83) andrearranging yields for the Lyapunov equation (the use of the block matrix partshighlighted in the equation will be explained hereafter):

⎡⎢⎢⎢⎣

−2ζ1ω1Wo,11 + ω1Wo,21 −2ζ1ω1Wo,12 + ω1Wo,22

−ω1Wo,11 −ω1Wo,12

[−2ζ1ω1Wo,13 + ω1Wo,23 −2ζ1ω1Wo,14 + ω1Wo,24

−ω1Wo,13 −ω1Wo,14

]

−2ζ2ω2Wo,31 + ω2Wo,41 −2ζ2ω2Wo,32 + ω2Wo,42

−ω2Wo,31 −ω2Wo,32

−2ζ2ω2Wo,33 + ω2Wo,43 −2ζ2ω2Wo,34 + ω2Wo,44

−ω2Wo,33 −ω2Wo,34

⎤⎥⎥⎥⎦+

⎡⎢⎢⎢⎣

−2ζ1ω1Wo,11 + ω1Wo,12 −ω1Wo,11

−2ζ1ω1Wo,21 + ω1Wo,22 −ω1Wo,21

[−2ζ2ω2Wo,13 + ω2Wo,14 −ω2Wo,13

−2ζ2ω2Wo,23 + ω2Wo,24 −ω2Wo,23

]

−2ζ1ω1Wo,31 + ω1Wo,32 −ω1Wo,31

−2ζ1ω1Wo,41 + ω1Wo,42 −ω1Wo,141

−2ζ2ω2Wo,33 + ω2Wo,34 −ω2Wo,33

−2ζ2ω2Wo,43 + ω2Wo,44 −ω2Wo,43

⎤⎥⎥⎥⎦ =

⎡⎢⎢⎢⎢⎢⎢⎢⎣

0 0

0 −2∑

s=1

φ(1)s φ

(1)s

ω1ω1

⎡⎢⎣ 0 0

0 −2∑

s=1

φ(1)s φ

(2)s

ω1ω2

⎤⎥⎦

0 0

0 −2∑

s=1

φ(2)s φ

(1)s

ω2ω1

0 0

0 −2∑

s=1

φ(2)s φ

(2)s

ω2ω2

⎤⎥⎥⎥⎥⎥⎥⎥⎦

(C.4)

Equation (C.4) shows that the Lyapunov equation has matrices which have horizontal

and vertical symmetric properties, only the indices of the matrix components differ. The

equation can be solved partially for Wo, when regarding the symmetric individual block

matrices of size [2× 2]. For instance, the solutions for Wo,13, Wo,14, Wo,23 and Wo,24

can be found by solving the equations in the upper right [2× 2] blocks of the matrices

in equation (C.4), which are highlighted in the equation. Indeed these unknowns are

only active in these block matrices due to the block diagonal nature of the state matrix.

In order to find a general solution to the observability Gramian, let us now derive

the solutions for the upper right [2× 2] block part of the Gramian matrix. From the

Lyapunov equation, the following four equations can be extracted from the highlighted

block parts:

−2ζ1ω1Wo,13 + ω1Wo,23 − 2ζ2ω2Wo,13 + ω2Wo,14 = 0 (C.5)

−2ζ1ω1Wo,14 + ω1Wo,24 − ω2Wo,13 = 0 (C.6)

−ω1Wo,13 − 2ζ2ω2Wo,23 + ω2Wo,24 = 0 (C.7)

−ω1Wo,14 − ω2Wo,23 = −2∑

s=1

φ(1)s φ

(2)s

ω1ω2(C.8)

Reorganizing (C.7) and (C.8) results in:

Wo,14 =1

ω1

2∑s=1

φ(1)s φ

(2)s

ω1ω2− ω2

ω1Wo,23 (C.9)

Wo,13 =ω2

ω1Wo,24 − 2ζ2ω2

ω1Wo,23 (C.10)

186

Reorganizing (C.6) and inserting (C.9) and (C.10) yields

Wo,24 = 2ζ1Wo,14 +ω2

ω1Wo,13 (C.11)

=2ζ1ω1

2∑s=1

φ(1)s φ

(2)s

ω1ω2− 2ζ1ω2

ω1Wo,23 +

ω22

ω21

Wo,24 − 2ζ2ω22

ω21

Wo,23 (C.12)

=2ζ1ω1

ω21 − ω2

2

2∑s=1

φ(1)s φ

(2)s

ω1ω2−Wo,23

(2ζ1ω1ω2

ω21 − ω2

2

+2ζ2ω

22

ω21 − ω2

2

)(C.13)

Let us now write

c =

2∑s=1

φ(1)s φ

(2)s

ω1ω2(C.14)

and insert (C.9), (C.10) and (C.13) into (C.5). This results into:

−(2ζ1ω2 +

2ζ2ω22

ω1

)(2ζ1ω1

ω21 − ω2

2

c−Wo,23

(2ζ1ω1ω2

ω21 − ω2

2

+2ζ2ω

22

ω21 − ω2

2

))+(

4ζ1ζ2ω2 + ω1 +4ζ22ω

22

ω1− ω2

2

ω1

)Wo,23 +

ω2

ω1= 0. (C.15)

By rearranging and further development (C.15) becomes:

Wo,23

⎛⎜⎜⎜⎝

4ζ21ω1ω22

ω21 − ω2

2

+4ζ1ζ2ω

32

ω21 − ω2

2

+ 4ζ1ζ2ω2 + ω1 +4ζ1ζ2ω1ω

32

ω1

(ω21 − ω2

2

)+

4ζ22ω42

ω1

(ω21 − ω2

2

) + 4ζ22ω22

ω1− ω2

2

ω1

⎞⎟⎟⎟⎠

= c

(4ζ21ω1ω2

ω21 − ω2

2

+4ζ1ζ2ω1ω

22

ω1

(ω21 − ω2

2

) − ω2

ω1

)(C.16)

Wo,23

ω1

(ω21 − ω2

2

)(4ζ21ω

21ω

22 + 4ζ1ζ2ω1ω

32 + 4ζ1ζ2ω1ω2

(ω21 − ω2

2

)+ ω2

1

(ω21 − ω2

2

)+4ζ1ζ2ω1ω

32 + 4ζ22ω

42 + 4ζ22ω

22

(ω21 − ω2

2

)− ω22

(ω21 − ω2

2

))

=c

ω1

(ω21 − ω2

2

) (4ζ21ω21ω2 + 4ζ1ζ2ω1ω

22 − ω2

(ω21 − ω2

2

))(C.17)

Wo,23 =c(4ζ21ω

21ω2 + 4ζ1ζ2ω1ω

22 − ω2

(ω21 − ω2

2

))(4ζ21ω

21ω

22 + 4ζ1ζ2ω1ω

32 + 4ζ1ζ2ω1ω2

(ω21 − ω2

2

)+ ω2

1

(ω21 − ω2

2

)+4ζ1ζ2ω1ω

32 + 4ζ22ω

42 + 4ζ22ω

22

(ω21 − ω2

2

)− ω22

(ω21 − ω2

2

)) (C.18)

Equation (C.18) can finally be written as:

Wo,23 =c

d

(ω2

(4ζ1ω1e+

(ω22 − ω2

1

))), (C.19)

with

d = 4ω1ω2 (ζ1ω1 + ζ2ω2) (ζ1ω2 + ζ2ω1) +(ω21 − ω2

2

)2(C.20)

e = ζ1ω1 + ζ2ω2. (C.21)

187


Inserting equations (C.19) to (C.21) into (C.9) then yields for Wo,14:

Wo,14 =c

ω1− ω2

2c

ω1d

(4ζ1ω1e+

(ω22 − ω2

1

))=

c

d

(4ζ1ζ2ω

21ω2 + 4ζ22ω1ω

22 + ω2

1 − ω1ω22

)=

c

d

(ω1

(4ζ2ω2e+

(ω22 − ω2

1

))). (C.22)

By inserting (C.19) to (C.21) into equation (C.13) Wo,24 is found:

Wo,24 =2ζ1ω1

ω21 − ω2

2

c− c

d

(4ζ1ω1e+

(ω21 − ω2

2

))(2ζ1ω1ω22

ω21 − ω2

2

+2ζ2ω

32

ω21 − ω2

2

)

=c

d

(2ζ2ω

32 + 2ζ1ω

31 + 8ζ21ζ2ω

21ω2 + 8ζ1ζ

22ω1ω

22

)=

c

d

(2(ζ2ω

32 + ζ1ω

31

)+ 8ζ1ζ2ω1ω2e

). (C.23)

Finally, Wo,13 is found by inserting (C.19) to (C.21) and (C.23) into equation (C.10):

Wo,13 =c

d

(2

(ζ2ω

42

ω1+ ζ1ω

21ω2

)+ 8ζ1ζ2ω

22e−

2ζ2ω22

ω1

(4ζ1ω1e+

(ω22 − ω2

1

)))

=2c

d

(ζ1ω

21ω2 + 2ζ2ω1ω

22

)=

c

d(ω1ω2e) . (C.24)

Combining the results found in equations (C.19), (C.22), (C.23) and (C.24) into matrix

form, the upper right block matrix of the observability Gramian is written as:[Wo,13 Wo,14

Wo,23 Wo,24

]=

c

d

[ω1ω2e ω1

(4ζ2ω2e+

(ω22 − ω2

1

))ω2

(4ζ1ω1e+

(ω22 − ω2

1

))2(ζ2ω

32 + ζ1ω

31

)+ 8ζ1ζ2ω1ω2e

].

(C.25)

Earlier, it was stated that the Lyapunov equation holds matrices which have hori-

zontal and vertical symmetric properties, only differences are found between indices of

the components. Therefore, if the other block matrix parts of the observability Gramian

are computed using the procedure set out in this appendix, the solution is found as:

Wo =

⎡⎢⎢⎢⎢⎣

c11d11

[ω1ω1e11 ω1

(4ζ1ω1e11 +

(ω21 − ω2

1

))ω1

(4ζ1ω1e11 +

(ω21 − ω2

1

))2(ζ1ω

31 + ζ1ω

31

)+ 8ζ1ζ1ω1ω1e11

]· · ·

c21d21

[ω2ω1e21 ω2

(4ζ1ω1e21 +

(ω21 − ω2

2

))ω1

(4ζ2ω2e21 +

(ω21 − ω2

2

))2(ζ1ω

31 + ζ2ω

32

)+ 8ζ2ζ1ω2ω1e21

]· · ·

c12d12

[ω1ω2e12 ω1

(4ζ2ω2e12 +

(ω22 − ω2

1

))ω2

(4ζ1ω1e12 +

(ω22 − ω2

1

))2(ζ2ω

32 + ζ1ω

31

)+ 8ζ1ζ2ω1ω2e12

]

c22d22

[ω2ω2e22 ω2

(4ζ2ω2e22 +

(ω22 − ω2

2

))ω2

(4ζ2ω2e22 +

(ω22 − ω2

2

))2(ζ2ω

32 + ζ2ω

32

)+ 8ζ2ζ2ω2ω2e22

]⎤⎥⎥⎥⎥⎦ (C.26)

188

with

c11 =2∑

s=1

φ(1)s φ

(1)s

ω1ω1c12 =

2∑s=1

φ(1)s φ

(2)s

ω1ω2

c21 =2∑

s=1

φ(2)s φ

(1)s

ω2ω1c22 =

2∑s=1

φ(2)s φ

(2)s

ω2ω2

(C.27)

e11 = 2ζ1ω1 e12 = ζ1ω1 + ζ2ω2

e21 = ζ1ω1 + ζ2ω2 e22 = 2ζ2ω2

(C.28)

and

d11 = 4ω1ω1 (ζ1ω1 + ζ1ω1) (ζ1ω1 + ζ1ω1) +(ω21 − ω2

1

)2d12 = 4ω1ω2 (ζ1ω1 + ζ2ω2) (ζ1ω2 + ζ2ω1) +

(ω21 − ω2

2

)2d21 = 4ω2ω1 (ζ2ω2 + ζ1ω1) (ζ2ω1 + ζ1ω2) +

(ω22 − ω2

1

)2(C.29)

d22 = 4ω2ω2 (ζ2ω2 + ζ2ω2) (ζ2ω2 + ζ2ω2) +(ω22 − ω2

2

)2Equations (C.26) to (C.29) show that for each block component in the solution only

different indices for the eigenfrequencies, damping ratios and eigenmodes are found. It

is therefore possible to construct a general solution for the observability Gramian by

generalizing the indices in equation (C.25):


[2ωkωlekl ωk

(4ζlωlekl −

(ω2l − ω2

k

))ωl

(4ζkωkekl +

(ω2l − ω2

k

))2(ζlω

3l + ζkω

3k

)+ 8ζkζlωkωlekl

]

(C.30)


l

)2ckl =

2∑s=1

φ(k)s φ

(l)s

ωkωl

ekl = ζkωk + ζlωl

with

m and j = 2, 4

k =m

2

l =j

2.

Equation (C.30) finalizes this appendix and will be used throughout this thesis as

general solution for the observability Gramian.

189


190

Appendix D

Gram-Schmidt

orthonormalization

The Gram-Schmidt orthonormalization is a methodology for orthonormalizing a set of

vectors in an inner product space Rn. Let Q =

[a1 a2 . . . aK

](K < n) be a

linearly independent set of vectors for a subspace of Rn. Using Q, an orthonormal

set of vectors Q =[b1 b2 . . . bK

]that spans the same subspace of Rn can be

extracted with the Gram-Schmidt methodology [34].

When performing a Gram-Schmidt orthonormalization, let the projection operator

be defined by:

proj[bk] (ak) =〈ak, bk〉〈bk, bk〉 bk k = 1, . . . ,K (D.1)

with 〈ak, bk〉 the inner product of the vectors ak and bk. The orthonormalized set of

vectors, Q, is next obtained by the following procedure:

b1 = a1 (D.2)

b1 =b1∥∥∥b1∥∥∥ (D.3)

b2 = a2 − proj[b1] (a2) (D.4)

b2 =b2∥∥∥b2∥∥∥ (D.5)

191

APPENDIX D. GRAM-SCHMIDT ORTHONORMALIZATION

b3 = a3 − proj[b1] (a3)− proj[b2] (a3) (D.6)

b3 =b3∥∥∥b3∥∥∥ (D.7)

...

bK = aK − proj[b1] (aK)− proj[b2] (aK)− · · · − proj[bK ] (aK) (D.8)

bK =bK∥∥∥bK∥∥∥ (D.9)

Q =[b1 b2 . . . bK

](D.10)

Note that the set[b1 b2 . . . bK

]is an orthogonal set of vectors. The calculation

of this set is also referred to as the Gram-Schmidt orthogonalization.

192

Appendix E

Specifications of measurement

equipment

This appendix will discuss several details of the measurement equipment used for the

validation measurements of the photomask and bending clamp.

E.1 Hardware

TIRA Vibration Exciter

Model, type TV 51110

Force rating 100 [N ]

Frequency range 0.002 − 7 [kHz]

First axial resonance > 6.5 [kHz]

Maximum bare table accelerations 440[ms2

]

193

APPENDIX E. SPECIFICATIONS OF MEASUREMENT EQUIPMENT

PCB ICP Impedance Head

Model, type 288M25, no. 1894

Force range (peak) ±445 [N ]

Sensitivity (acc.) 102[mVg

]Sensitivity (force) 11.32

[mVN

]Resonance frequency � 36 [kHz]

Transverse sensitivity 2,8%

Weight 19.3 [g]

Dimensions 11/16 [in]× 20.83 [mm]

PSV Scanning Vibrometer

Model, type PSV-I-400

Scanning frequency 40 [kHz]

Scanning range ±20◦ Rx and Ry

Angular resolution < 0.002◦

Angular stability < 0.01◦[

1hr

]Working distance > 0.4 [m]

Weight 7.5 [kg]

Dimensions 365× 160 × 190 [mm]

194

E.2. SOFTWARE

PSV Vibrometer Controller

Model, type OFV-5000

Velocity ranges 2, 10, 50, 100, 1000 [mm/s/V ]

Bandwidth 1.5 [mHz]

Analog low pass filters 2, 20, 100, 1500 [kHz]

E.2 Software

The software package used for the validation measurements of the photomask and

bending clamps is Polytec PSV software, version 8.3. The following acquisition settings

were applied:

Velocity acquisition V D − 0610 [mm/s/V/LP ]

Tracking filter off

Low pass filter 1.5 [mHz]

High pass filter off

195

APPENDIX E. SPECIFICATIONS OF MEASUREMENT EQUIPMENT

196

Appendix F

Newmark time integration

This appendix discusses the Newmark integration scheme for linear systems. The the-

ory covered in this section originates from [12].

Consider the equations of motion for a linear discrete system:

Mq (t) +Cq (t) +Kq (t) = p (t) , (F.1)

where the mass, damping and stiffness matrices (M , C and K respectively) are in-

dependent of the DoF q (t). The external force vector is represented by p (t). When

calculating the time response of the system with a Newmark time integration scheme,

the state vector of the system at time tn+1 = tn + h (with h serving as a time incre-

ment), is deduced from the already known state vector at time tn through a Taylor

series expansion of the displacements and velocities. From the Taylor series expansion,

an approximation of the velocities and displacements of the system at tn+1 can be com-

puted. Figure F.1 shows the flowchart of the Newmark integration scheme for linear

systems.

The constants γ and β are parameters associated with the integration scheme. The

choice of these parameters determines the stability limit, amplitude error and periodic-

ity error of the applied Newmark integration. Table F.1 shows a variety of possible pa-

rameter choices with the corresponding properties of the Newmark integration scheme.

In the table, ω refers to the highest frequency to be solved for, ρ is the amplitude of the

response, T the period of a free oscillation and α is an additional parameter associated

to the integration scheme.

197

APPENDIX F. NEWMARK TIME INTEGRATION

M ,C,K, q0 , q0

Compute q0q0 = M−1 (p0 −Cq0 −Kq0)

Time incrementationtn+1 = tn + h

Predictionq∗n+1 = qn + (1− γ)hqn

q∗n+1 = qn + hqn + (0.5− β) h2qn

Acceleration computing

S = M + hγC + h2βKSqn+1 = Pn+1 −Cq∗n+1 −Kq∗n+1

Correctionqn+1 = q∗n+1 + hγqn+1

qn+1 = q∗n+1 + h2βqn+1

Evaluation of energies(optional)

Figure F.1: Newmark integration scheme for linear systems.

198

Algorithm γ β

Stability Amplitude Periodicity

limit error error

ωh ρ− 1 ΔTT

Purely explicit 0 0 0 ω2h2

4 -

Central difference 12 0 2 0 −ω2h2

24

Fox & Goodwin 12

112 2.45 0 O

(h3)

Linear acceleration 12

16 3.46 0 ω2h2

24

Average constant acceleration 12

14 ∞ 0 ω2h2

12

Average constant acceleration 12 + α (1+α)2

4∞ −αω2h2

2ω2h2

12(modified)

Table F.1: Integration scheme of the Newmark family.

199

APPENDIX F. NEWMARK TIME INTEGRATION

200

Index

Agile correction, 99

anticlastic curvature, 19

anticlastic curvature augmented fitting func-

tion, 23

Cayley-Hamilton theorem, 56, 64

column space, 56

compatibility condition, 38

controllability Gramian, 41, 65

controllability matrix, 64

Craig Bampton, 33

critical dimension, 4

critical dimension uniformity, 4

curvature correction, 83

die, 3, 84, 99

dual assembly, 38

duhamel integral, 51

dynamic substructuring, 30, 37

eigenfrequency, 29

eigenmode, 28

eigenshape, see eigenmode

eigenspectrum, 33

equilibrium condition, 38

etching, 2

Euler, 53

exposure, 167

FEM, 7, 27

field curvature, 167

fitting function, 13

focus, 83

focus budget, 5, 167

focus error, 25, 85

full field die, 100

fused silica, 14

FWOL, 100

Gram-schmidt orthogonalization, 69

Gramian matrix, 42, 63

Gramian theory, 8

Green strain, 13

Guyan reduction, 31

IIR filter, 90

imaging, 167

integrating factor, 50

interdie, 84, 97

intradie, 84, 97

K-orthogonal, 29

Kirchoff assumptions, 13

Kirchoff plate theory, 14

Lagrange multipliers, 39

Laplace operator, 21

LDV, 127

least squares, 25

least squares fitting, 26, 183

level sensor, 98

leveling, 5

leveling strategies, 83

lithography, 1

201

INDEX

LTI system, 42

Lyapunov equation, 58, 65

M-orthogonal, 29

MAC, 126

modal mass, 33

modal stiffness, 33

modal superposition, 29, 32

modal truncation, 33

Moving Average (MA), 86

Moving Standard Deviation (MSD), 86

Newmark time integration, 156

non full field die, 100

normal equations, 33

nullspace, 68

observability Gramian, 41

observability matrix, 57

overlay, 3

parasitic stiffness, 93

particular solution, 28

photomask, 2

piezoelectric actuator, 92

plate theory, 13

Poisson ratio, 17, 19, 21

primal assemble, 38

Q-grid, 119

raw topology data, 99

reticle stage, 2

rigidness, 37

row space, 64

scanner, 2

slit position, 84

state space, 42, 43

stroke, 98, 99

Taylor expansion, 64

throughput, 3

topology, 85

wafer stage, 2, 98

yield, 4

z-supports, 93

202

kirchoff plate theory

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