eindhoven university of technology master deflectometry on … · cft -philips tule 2....

Eindhoven University of Technology

MASTER

Deflectometry on aspherical surfaces

van der Beek, N.A.J.

Award date:2004

Link to publication

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https://research.tue.nl/en/studentthesis/deflectometry-on-aspherical-surfaces(ee10fbeb-6080-477e-a9ed-d8c22791dec1).html

Supervisors:

Deflectometry on Aspherical Surfaces

Niels van der Beek October 2003

AQT 03-06

Eindhoven University of Technology Prof.Dr. H.C.W. Beijennek Philips CFT Drs. W.D. van Amstel

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Abstract

This report describes last 1.5 years workof research on a new method for measuring free

form aspherics. Feasibility is shown for measuring and reconstructing free form aspherics by

using the deflectometer for aspherical surfaces. Feasibility has been shown by measuring and

reconstructing aspherical contact lens inserts with a precision of several micrometers over 15 mm

integration paths. Using a mathematica! misalignment model we have been able to extract and

correct for misalignments in the set-up. However absolute radius reconstruction has not been

achieved, to achleve this, more work on the misalignment model and fitting procedures must take

place. Also calibration procedures have been developed using flat mirror and sphere inserts.

Three different sensor types have been tested and compared of which the PBS-type en

PPSC type sensor yielded the best results, typical slope accuracies are in the order of 40 J..lfad and

typ i cal slope measuring range is +\- 4 o.

NAJ van der Beek: Deflectometry on aspheric Surfaces

CFT -Philips TUle

Table of contents

1. Introduetion ................................................................................................................. 4 1.1. General Introduetion .............................................................................................. 4

1.2. History .................................................................................................................... 5

1.3. Project goal. ............................................................................................................ 5

1.4. This report .............................................................................................................. 5

2. Deflectometry ............................................................................................................... 7

2 .1. Introduetion ............................................................................................................ 7

2.2. Aspherical surfaces ................................................................................................ 8

2. 3. Deflection angle ..................................................................................................... 9

2.4. Surface reconstruction .......................................................................................... 11

2.5. Misalignments ...................................................................................................... 13

2.6. Resolution ............................................................................................................. 16 2.6.1. Detection unit .............................................................................................................. 16 2.6.2. Spot size ....................................................................................................................... 17 2.6.3. Desired specs ............................................................................................................... 18

2. 7. Sensor types .......................................................................................................... 21

3. Sensor type 1: Prism (P)-type ................................................................................... 22

3 .1. Introduetion .......................................................................................................... 22

3 .2. Schematics of the P-type sensor ........................................................................... 22

3.3. Laser unit .............................................................................................................. 23

3.4. Spot characteristics ............................................................................................... 25

4. Sensor type 11: Prism with Polarization Sensitive Coating (PPSC)-type ............. 28

4 .1. Introduetion .......................................................................................................... 28

4.2. Schematics ofthe PPSC-type sensor .................................................................... 28

4. 3. Spot characteristics ............................................................................................... 29

5. Sensor type 111: Polarizing Beam Splitter (PBS)-type ............................................ 31

5. 1. Introduetion .......................................................................................................... 31

5.2. Schematics ofthe PBS-type sensor ...................................................................... 31

5.3. Spot characteristics ............................................................................................... 32

6. Tilted flat mirror experiments .................................................................................. 34

6.1. Introduetion .......................................................................................................... 34

6.2. Flat scan ................................................................................................................ 34

6.3. False reflections .................................................................................................... 38

6.4. Wobbie experiments ............................................................................................. 40

6. 5. Analysis of 'wobble '-experiments ....................................................................... 41

6.6. Surface reconstruction from 'wobble' experiments ............................................ .43

6. 7. Conclusions .......................................................................................................... 45

7. Sphere experiments ................................................................................................... 46


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7 .1. Introduetion .......................................................................................................... 46

7.2. Experimental set-up .............................................................................................. 46

7.3. Slope data processing from sphere measurement for rnisalignment extraction ... 46

7.4. Surface reconstruction of the sphere .................................................................... 48

7. 5. Conclusions .......................................................................................................... 51

8. Asphere experiments ................................................................................................. 52

8.1. Toric sample ......................................................................................................... 52

8.2. Bi-focal sample .................................................................................................... 54

9. Concluding remarks .................................................................................................. 57

10. Appendices .............................................................................................................. 59

10 .1. Appendix 1: PSD ................................................................................................ 59

1 0.2. Appendix 11: Misalignments .............................................................................. 61

10.3. Appendix 111: Transformation ofspec 2 into slope domain ............................... 64

1 0.4. Appendix N: laser characteristics ..................................................................... 65

10.5. Appendix V: Gaussian opties ............................................................................ 66

1 0. 6. Appendix N: Op ti cal system efficiency ........................................................... 69

11. References ................................................................................................................ 71


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1. Introduetion

1.1. General Introduetion

In metrology, the science of measuring the topology of surfaces, two different techniques

are commonly used. A frrst technique uses a mechanica! probe, which scans the surface much like

a blind person reads Braille. The second is interferometry, an optica! technique that measures the

path difference between a laser beam reflected at a reference surface and one at the surface under

investigation. With a known reference surface the investigated surface can he reconstructed.

About 2.5 year ago, a new technique was developed at the Philips CFT in Eindhoven,

capable of competing with the two methods mentioned above. This technique, called

deflectometry, is an optica! method. It uses a laser beam to obtain the local slope of the surface on

a small spot. This slope is determined by measuring the angle between the incident and reflected

laser beam. The slope of the surface is scanned point by point along different lines on the surface.

If these local slopes are stitched together, a slope map is created and with this map the original

surface can he reconstructed. This technique is generic in the sense that it can he used to measure

spherical and aspherical surfaces of a wide variety of radii of curvature (both concave and

convex) with the same set-up. This flexibility is an important asset ofthe technique that makes it

possible to implement this technique on a lathe in a later stadium.

Compared with the established techniques, deflectometry has two major advantages that

justify the research being done in this field. It would take an off-line measurement to use a

mechanica! probe to scan a surface with the desired resolution, because these measurements take

too long to do on-line. Non-destructive measuring is the main cause ofthis long duration. Ifwe

compare this to a measurement which uses deflectometry, the same surface could he scanned in

less then five minutes with the same resolution, resulting in measurements that can he done online

in the production process. This speed advantage has two causes, one being the fact that

deflectometry is non-destructive by nature, the other concerns the usage of analogue electronics,

which allow very fast data-acquisition.

The lateral range, i.e. the range in the plane perpendicular to the investigated surface, of

an interferometer measurement is in most cases a limiting factor. On an ordinary CCD-camera,

normally used in interferometers, 400 to 1000 pixels are used to distinguish between the different

fringes. Therefore, a distinction can he made over 20 to 50 fringes (wavelengths) of lateral range,

resulting in an ability to measure aspherical corrections up to 5 to 25 microns. Furthermore, the

NAJ van der Beek: Deflectometry on aspheric Surfaces 4

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necessity fora reference fonn close to the actual fonn ofthe surface under investigation lirnits the

flexibility of this technique. The lateral range of a deflectometry set-up can be up to 40 times

larger than is the case for an interferometer. Again the advantage can be found within the

analogue electronics, which provide the range but allow for enough details to be measured.

A possible application for the deflectometer would be in the ophthalrnic industry, which

produces spectacle-glasses and contact lenses. Here a trend towards complex designs for lenses is

seen. For example, one can buy spectacle-glasses that have two different regions. One region is

used for close reading and the other for distant looking. The design for such a lens is very

complex and has no symmetry. To check a surface with such a design the whole surface has to be

measured, due to the lack of symmetry. These free-fonn designs, as they are called, also need a

large lateral range. The development of a machine, which can measure these surfaces, is therefore

a hot topic of research within metrology. Deflectometry may prove a valuable technique for such

an application.

Concluding, it can be said that deflectometry combines the better of the two existing

worlds, i.e. speed and lateral range.

1.2. Bistory

The goal ofthe previous two projects [1,2] was to build a FuMo (Functional

Model) of the deflectometer that could actually measure aspherical surfaces and reconstruct them.

Due to circumstances, this goal was not achieved within the time that was available for it. Rather

than a FuMo, a 'Breadboard' model was achieved that was capable ofmeasuring surfaces, but the

software and models needed to reconstruct the surfaces were not finished.

1.3. Project goal

The first goal for this project was to build and test three different sensor types and to

compare them. The next goal was to develop calibration procedures and the final goal was to

measure aspheric surfaces and prove feasibility of the deflectometer to measure aspherical

surfaces.

1.4. This report

In this report, the results are described of the research done on the deflectometer for

aspherical surfaces that has been built within the Philips CFT -department. This research is done


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in the form of a masters project (9 months) for the physics department of the Eindhoven

University ofTechnology. We report the design and manufacturing ofthree different sensors, the

calibration procedures and finally some measurements on aspherical surfaces.


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2. Deflectometry

2.1. Introduetion

In this chapter, the concept of deflectometry is presented. Deflectometry or optical slope

sensing is an existing technique, which has found new life through the use of Position Sensitive

Diodes (PSD). Deflectometry can be used in topography and shape measurements although it

does nat measure height differences or distances directly. The main advantage of deflectometry is

that it rather measures the first derivative ofthe shape or the so-called local slope ofthe surface.

Measuring the slope yields better results when determining local curvature (important in lens

designs) compared to measuring in the height domain (figure 2.1).

g g dx dx ~

~ ~

~ fd x fd x -E-- -E--

a) height domain b) slope domain c) curvature domain

figure 2.1 (a): Surface shape (a) and its slope signa! (b) as a dejlectometer delects it and the corresponding curvature plot (c)

In order to compare the measurements with the specifications, the design (solid shape)

can be differentiated or the measured slopes can be integrated. An intrinsic advantage of

measuring slopes is the fact that smalllocal errors in the shape cause large signal variations in the

slope domain and can be measured therefore with greater accuracy. A second advantage is that it

has a larger height range that can be measured than for example interferometry.

Deflectometry measures the deflection angle (a".) between an incident laser beam and the

reflected laser beam. The reileetion law tells us the angle between the incident laser beam and the

normal (n) ofthe surface is equal to the angle between the normal n and the reflected beam (see

figure 2.2)


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n <Xm

figure 2.2: Rejlection law

Firstly we can state ~ = a,., and, if we know the

direction ofthe incident laser beam, the direction ofthe

surface normal can be calculated from the measured

reflection angle. The angle between the incident laser

beam and the normal a; = am/2. The situation described

in the tigure above is true for the plane - which is called

the 'plane of incidence' - containing the incident laser

beam, the surface normal and also true for the reflected

beam and for any (projection) plane containing the surface normaL

By scanning the surface, while the direction ofthe incident laser beam is equal to the

normal of an arbitrarily chosen reference plane, a map of the surface is obtained with the

difference between the normal ofthe surface and the reference plane. Thus, at every point at the

surface we know the vector deviation of the surface relative to the reference plane. If these local

slopes, which have two perpendicular components at each pixel, are combined, a slope map of the

surface is created from which the surface form can be reconstructed by integration.

2.2. Aspherical surfaces

Slope measurements are relative measurements, therefore, a reference plane has to be

defined perpendicular to the incident laser beam. For measuring aspherical surfaces we choose a

spherical surface as a reference plane. Thus a spherical surface under investigation will yield a

zerosignaland any non-zero signal will give us information about the artifact ofthe Surface

Under Test (SUT). We then map these deviations and this makes it possible to campare these

with the design specifications to give a measure of the shape deviations.

Using spherical symmetry is the obvious choice for the reference plane, because the

aspheric corrections are small compared to the base form, which is spherical. Thus, we can use

the main part of the dynamic range of the sensor for measuring the aspherical deviations. This

way we can design the sensor such that it can measure small details, while it still has the dynamic

range to measure different radii of curvature in a single design. For example, this is the case in a

lens design incorporating an astigmatic correction, a so-called 'toric' lens.

This choice of a sphere as a reference plane is converted into the concept for the set-up.

Two rotational axes are placed orthogonal toeach other. These two rotations are called q> and e, respectively and describe the reference sphere. This means that our sensor can travel over the

surface of an imaginary sphere, while continuously pointing to the center of that sphere. In this


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situation the measurement is independent of the radius of the reference sphere, we only need the

two rotational directions ( q>,8) to make the imaginary surface. A schematic layout of these two

orthogonal directionsis shown in figure 2.3:

SUT

I

~-Ws i~~ •. -· -- -----;8· 8-ax.is _

/' slope sensor

~-., __

figure 2.3: To measure aspherical surfaces, a system with two orthogonal rotational axes ( rp and B) is used with the Surface Under Test (SUT) on the rp-axis and the sensor on the 8-axis.

The q>-axis lies in the plane ofthe drawing (0::; q>::; 27t). The SUT is mounted onto this

rotation axis. The SUT can than be turned around 360 °. The 8-axis that stands perpendicular to

the plane ofthe drawing rotates the slope sensor around the imaginary center of curvature ofthe

reference surface (0::; 8::; 7t/2). Most ofthe symmetry in the aspheric surface can be used this

way.

2.3. Deflection angle

To detect the angle between the incident and the reflected beam we use a detection unit

that consists of a lens and a 2-Dimensional Position Sensitive Diode (2D-PSD, Appendix I). The

PSD measures the position at which a laser beam hits its surface and returns this position as an X

and Y -signal (ranging from -10 V to + 1 OV). It also returns a signal corresponding to the intensity

(or the total amount oflight) ofthe light hitting its surface.

In combination with a lens, the PSD can be used to measure pure angles instead of

position. If the PSD is placed in the focal plane of a lens, all incoming beams that make the same

angle with the op ti cal axis will be focused on the same point on the PSD, independent of their

point of origin (figure 2.4a). Soa parallel incoming beam will be focused on a single point. Fora

divergent beam the same holds, but with a slight difference. Now, the incoming beam will not be

focused on a single point, but will forma wider spot on the PSD (figure 2.4b), because ofthe

slightly different incoming angles. Since the PSD returns the position ofthe 'center ofweight' of

the spot this method can be used to determine the average incoming angle of a divergent beam.


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f f

(a) (b)

figure 2.4: Incident parallel (a) and divergent (b) beams with different angles with respect to the optica/ axis arefocused dif!erently on the PSD, but resulting in the same output. (center ofweight of spot is the same).

Since we use a 2D-PSD, we are able to measure the slope in two directions. At the PSD,

we call the component of the slope in the 9-direction, the 9-signal and the component orthogonal

at the 9-direction, the q>-component. In figure 2.3 the 9-direction lies in the plane ofthe drawing

and the q>-direction is perpendicular to the plane ofthe drawing. Forthese slope components the

following conversion factors, or slope scale factors (C) are used to convert the voltages ofthe

PSD into slope values.

1 L V: S = tana = _ _fH!_ __ e- =CV: (J (J 2 f L\.V: (J

PSD

1 L V S = tana =- PSD __ ffJ_=CV ffJ tp 2 f L\.V ffJ

PSD

in which

Thus:

V9,Vfj) is the measured voltage in the 9,q> direction

Lpso is the size ofthe PSD (4 mm).

ö V Pso is the voltage difference between the edges of the PSD (20V)

f is the distance between the lens and the PSD (30 mm).

a9,afj) is the angle ofthe surface normal in the 9,q> direction.

c =..!.. Lpsn _I_~ 3.333o.o-3 v-1 2 f L\.VPSD


(2.1)

(2.2)

(2.3)

10

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2.4. Surface reconstruction

figure 2.5: Integration routinefor reconstruction ofan aspherical surface.

lntegrating (2.5) over 8 results in:

Se= dRe = dRe ds R8dB

1 S8dB=-dR8

Re

TUle

Since we measure in the slope

domain, we have to integrate the

measured slopes for reconstructing the

actual shape of a SUT (figure 2.1).

For this integration we use the

following routine. In figure 2.5 we see

an aspherical surface with slope Se,

for which the following holds:

(2.4)

(2.5)

(} Ru

fS8dB= f~R =lnRi~ =In Re o RoR Ro (2.6)

So for the relative asphericity (Re!Ro) we get:

(2.7)

and for the absolute asphericity:

(2.8)

The integration in the <p-direction goes the same way, resulting in:

(2.9)


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ft gure 2. 6: Integration paths for surface reconstruction from therp-slope data. One line of B-slope data functions as the start line for the rp-slope data.

jigure 2.8: Integration paths for surface construction from the B-slope data. One point (or very small rp-circle) functions asthestart pointlline for the B-slope data.

TUle

ft gure 2. 7: Integration paths for surface reconstruction from theB-slope data. One circle of rp-slope data functions as the start line for the B-slope data

Now we can reconstruct the surface by

integrating the measured slope components.

Since we measure two slope components (i.e.

<p-and 8-component), we get two independent

sets of data that we can use to reconstruct the

surface. To reconstruct the surface weneed to

detine the integration paths along which to

integrate the slope data. We use different

integration paths ( one for each slope

component). These integration paths are

shown in tigure 2.6, tigure 2.7 and tigure 2.8. In tigure 2.6 we integrate one line of8-slope data

in the 8 direction, that functions as the starting line from where the <p-slope data integrations start.

In tigure 2. 7 and tigure 2.8 we use one circle of <p-slope data in the <p-direction that functions as

the start line for the 8-slope data integration. In formulas:


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Integration path <p-slope data:

M.( ffJ, (J) = 11, { exp 0 S,(ffJ, (J ~d (J} I} (2.10)

M,(ffJ,(J) = 11, sine{ exp(jS,(ffJ', (J)dffJ'] -i} (2.11)

lntegration path e-slope data:

M, ( ffJ, (J) = R, sin (J {expO S,('fJ ', (J)dffJ ']-I} (2.12)

MI,( ffJ, (J) = R, { exp ( j S, ( ffJ, (J~d(J ']-I} (2.13)

2.5. Misalignments

Before we can reconstruct a surface, we need to consider the possible misalignments that can

occur in the set-up, since these misalignments will influence the measurement data. In figure 2.9

the misalignments are shown in three different views. Since we use a spherical system, we

describe these misalignments on the basis of a misaligned sphere, see figure 2.9.

We consider four different misalignments. At first we consider the crossing error C; crossing

occurs when the 9- and q>-axis don't interseet in the samepoint (i.e. the center ofthe reference

sphere). Next we consider the height h misalignment, i.e. when the 9-plane lies above or below

the q>-axis. Otherwise stated, h is introduced when the laser beam is aimed above, or below the

center of the reference sphere. Thirdly we consider the eccentricity E, when the SUT is not

centered correctly on the q>-axis. And the last misalignment considered is the shift ~z of the insert

along the q>-axis. The misalignments crossing C and height h are fixed machine errors and typical

for the actual set up, since replacing a SUT on the set up does not change these two

misalignments. Eccentricity E and shift ~z are not fixed; they will change every time a new SUT

is mounted.


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(a) sideview

(c) top view

figure 2.9: Four different misalignments that can occur in the set-up in three different views: side view (a),front view (b) and top view (c). The shifted sphere (solid line) with radiusRis shown with the different misalignments: Where crossing C is the misalignment that occurs when the Band cp-axis don 't inters eet, eccentricity & is introduced when the SUT is not centered on the cpaxis. The height h misalignment is introduced when the B-plane lies above or below the cp-axis and L1z is the shift of the SUT along the cp-axis.


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R'

- ............ LlZ

h

figure 2.10: I/lustration ofthe effect ofthe misalignments on the measured slope angle in the ~ direction where inslead of the expected arp=O, a different angle ( arp:;;t(J) is measured.

In tigure 2.10 we illustrate the effect of the misalignments on the measured slope angle.

Here we look what the effect of the misalignments is on the measured slope angle in the <p- ·

direction, i.e. the side view (a) in tigure 2.9. When the sphere is perfectly aligned we expect to

measure ~=0°, but due to the misalignments we measure a<i>*Ü0• Now we can derive an

analytica! formula to describe the effects that the misalignments will have on the measured angle

in the <p-direction, this is done in appendix II. Fortheresult we tind:

2 . { h + & sin( rp - rp0 ) } arp = arcsm

~R2 - { C + &cos(rp- rp0 )} 2

(2.14)

For the 9-component the effect of the misalignments are shown in tigure 2.11, again we

derive the analytica! formula (appendix IJ) resulting in:

where:

[

r2

+ R'2 - { C + &cos(rp- rp0 )}2

+ (~)2 J ae = 2arccos

2rR'


(2.15)

15

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R' = ~R2 -{h + &sin(ÇJ- ÇJ0 )}2

r = (J1 C +&cos(q.>- q.>0 )}2 + (Liz)'lcos{ 11-arctan [ C +&c~q.>- q.>,) ]}

± [ { C + ecos(q.>- q.>0 ) )' + (Liz)2 J{ cos' (11- arctan[ C + sc~q.>- q.>,)} -I}+ R '2

The purpose of these derivations is to he able to detennine the misalignments from the set

up by the measurements on metal spheres. By using the misalignment model to detennine the

signature of the misalignment, the measurements can than he corrected for these misalignments

by subtracting the misalignment signature from the slope signals.

I I

laser in - T -/ I '-

/ I ' n / I '

I I ' I

cp

figure 2.11: Illustration of the effect of the misalignments on the measured slope angle in the Bdirection where insteadof the expected ae=O, a different angle (a~) is measured

2.6. Resolution

2.6.1. Deleetion unit

Now let us take a look at some ofthe features of deflectometry, namely the resolution of

the system and the slope range that can and needs to he measured. At first we need to know what

the resolution is for the detection unit. Looking at the detection unit we can calculate its Object

Angular Resolution at the surface under test (öa) with the following formula:


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oa =-arctan -1 (oxJ~. 2 f , (2.16)

with 8x the resolution ofthe PSD and fis the distance from the lens to the detection unit (figure

2.12).

f

Jens PSD

figure 2.12: Determination of detector resolution.

For this unit, we use a 4x4

mm 2D-PSD that can distinguish 104

'pixels' resulting in a resolution for

the PSD of 8x, 8)=0.4 )liD.

Calculating the object angular

resolution oa with formula (2.16)

results in an object angular resolution

8a=6.67 )lrad at the SUT.

This detection unit determines another feature ofthe sensor, namely the maximal slope

range that can be measured. In the case of a 4x4 mm PSD the slope that can be measured ranges

from -0.07 rad to +0.07 rad. Since the PSD has the tendency to have a non-linear response at the

edges, only about 80% of its surface can be used, limiting the range to roughly -0.06 to 0.06 rad.

A summary ofthese values is shown in table 2.1 for different PSD sizes.

table 2.1: Object Angular Resolution and slope range for different PSD 's.

PSD size 4x4mm 10x10 mm 8x,8y 0.4j.L_m 1J.lm Object Angular Resolution (öo:) 6.67 J.lrad 16.67 J.lrad Slope Range -0.07 rad through 0.07 rad -0.17 rad through 0.17 rad

2. 6.2. Spot size

Another feature that determines the resolution of the system is the chosen spot size on the

SUT. This willlimit the lateral resolution (the size of smallest features that can beseen on the

surface).

In our set up we use a laser beam that forms a spot with a Gaussian intensity distribution

at the measuring position as given by (in terms ofbeam diameter d):

Where do is the quantity that determines the diameter of the beam at the position where the

intensity is I0e-2, i.e.:


(2.17)

17

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(2.18)

This value do is the so-called waist diameter of a gaussian beam. From now on when we

talk about spot size we mean the waist diameter do.

By experience this lateral resolution (in case of a gaussian spot) is about the same as the

Full Width Half Maximum of a gaussian spot.

dFWHM = Û.58d0 (2.19)

Thus in case of a 100 J..lm spot the lateral resolution is ~60 J..lffi. Since we 're only interested in

surface topography and not in surface roughness, this should he sufficient at this point.

2.6.3. Desired specs

Since the main costurner for the deflectometer at this moment is a contact lens manufacturer,

we would like the set up to he capable of functioning withintheir specs. For the contact lens

inserts, the following specs are defined:

1) Radius of curvature in the OZ (Optica/ Zone, 8-mm diameter) has to be within 5 fJ11'l of

the nomina/ value. The typ i ca/ average value of the radius of curvature is 7. 5 mm.

2) Peak-to-val/ey deviationfrom the nomina/ value ofthe height must be less than 0.52

(0.22for spherical inserts).

Note that these specs are defined in two different domains: spec 1 is defmed in the

curvature domain and spec 2 is defined in the height domain. Since deflectometry measurements

take place in the slope domain we need to translate these specs into the slope domain.

First we look at spec 1. In figure 2.13 a visualization of this spec is shown. Note that the

figure is not drawn to scale. Here we see the basic spherical shape with radius R, and an

additional sphere with radius R-L~R is superimposed. We assume that L\R. is the maximum

deviation of the nomina! value of the radius of curvature, in our case 5 ).ll11 (spec 1 ).


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figure 2.13: Maximum slope change (Liço) that occurs when a deviation (LIR) ofthe radius of curvature occurs along a certain arc-length (la) i.e. a sphere with radius R:i:LIR is superimposed on the original shape.

Since we work in the slope domain, we need to define the are length 2a along which this

deviation occurs to transform the curvature/ radius spec into a slope spec. This are length is also

shown in tigure 2.13. To change the are length along which the deviation occurs, the center ofthe

superimposed sphere is moved along the vertical dashed line. Now we need to know what slope

change occurs when such a radius deviation occurs along a certain are length. These values are

shown in table 2.2 as a function of arelengthand are calculated as follows. Weneed todetermine

the angle ~<pas a function oflength a, that is halfthe are length. For ~<p, the following equation

holds:

l:!lrp =a- fJ = arcsin(;)-arcsin(R :M) (2.20)


CFT -Philips TUle

table 2.2: Slope ranges as a function of are lengths from spec 1

are length 2a ~<p [mm] [!!fad] 0.001 0.044

~<p:::;; 0.005 0.222 0.01 0.45

resolution

0.05 2.22 0.1 4.45 0.5 22.25 1 44.57 2 89.75 ~<p ~

4 184.59 resolution 6 291.17 8 420.66

From tahle 2.2, the condusion can he drawn that we should he ahle to measure within

spec 1, provided that the are length ofthe defect is larger than 0.5 mm when the are length

hecomes smaller, the resolution of the detector is not sufficient to measure the defects.

Next, we take a doser look at spec 2. Since this specis defmed in the height domain, we

also need totranslate this spec into the slope domain. In this case, a height ~H of0.2À (:::.100 nm)

is superimposed on the surface (for simplicity we assume this to have a spherical shape). This is

shown in tigure 2.14. Again wedefine the are length (2a) along which this deviation takes place,

to derive the neerled slope range (appendix lil).

' ' '

R=7.5 mm

' ' ' ' ' ' ' I I I

' I I

' ' ' I I I

' ' I I

' ' ' ' I '' ,,

* are length (2a)

ft gure 2.14 Maximum slope change ( LlqJ) that occurs when a height defect is superimposed on the surface along a certain are length (2a).

In tahle 2.3 the slope range as a ftmction ofthe arc-length is shown, with d=0.2À (:::.100

nm). From this tahle the condusion can he drawn that we should he ahle to measure within spec


CFT -Philips TUle

2, provided that the effects do not occur with an are length smaller than 0.01 mm. This limitation

is because ofthe limited slope range ofthe sensor. Note that these defects smaller than 0.1 mm

will also show an effect in the detected intensity at the PSD. This is due tothefact that most of

the reflected light will not reach the detector because it is reflected at a higher angle than the

detector is capab1e ofmeasuring. This will result in less light on the detector. Soit might be

possible to detect andreconstruct these defects via this signal. Ifthe are length ofthe defect

becomes larger then 8 mm, the resolution ofthe detector will also again become a limiting factor.

table 2. 3: Slope range as a function of arc-length for spec 2.

Are length (mm) Ll<p (J..Lrad) out ofslope

0.001 394788.55 0.005 79956.84

range

0.01 39994.39 0.05 7999.88 0.1 3999.92 0.5 799.77 in slope 1 399.55 2 199.10

range

4 98.19 6 63.88 8 46.14

2. 7. Sensor types

We use a <p-8 actuator mechanism to describe an imaginary sphere. Independenee from

the (local) radius of the object is needed to measure the local slope without an effect of the shape

of the surface. Independenee from the radius is achieved by scanning normal to the imaginary

sphere. This implies that the measured angles will be very small and so the reflected beam will

follow almost the same path as the incident beam. To be able to detect the reflected angle we

have to separate the reflected beam from the incident beam. We have chosen to test two different

methods and to compare them. In the first two sensor types we use a prism to separate the beams

and in the third type we use a Polarizing Beam Splitter (PBS). In the following chapters we will

look in these sensor types in more detail.


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3. Sensor type 1: Prism (P)-type

3.1. Introduetion

The first sensor type we build was the 'over the counter' prism type. In which we used

standard, uncoated components. As a laser souree a high power laser diode is used to increase the

amount of light reaching the PSD. The spot at the SUT is 150 !liD. In the following section we

describe the set up and its characteristics

3.2. Schematics of the P-type sensor

The P-type sensor is basedon a standard catalogue 45° prism, which refracts the

'probing' beam onto the SUT and reflects the 'measuring' beam conring from the SUT towards

the detector (see figure 3.1):

toPSD

from laser

figure 3.1: Light path in the prism

The reason for choosing an incidence

angle of ten degrees is basedon a

combination of low reflectance at the

prism/air interface and a high reflectivity

of the prism for the light conring from the

SUT.

The biggest disadvantage of using

a prism is the highlossof light (~90 %).

The major part ofthe lossis contributed by

the light refracted back towards the laser by the prism and the amount of light that is transmitted

back into the prism instead ofbeing reflected towards the detector. Another disadvantage ofthe

system with the prism is the asymmetry introduced in the measuring beam due to the asymmetry

of the prism, resulting in an elliptical spot instead of a circular one and astigmatism.

The complete set-up ofthe P-type sensor is shown in figure 3.2. With the 'insert' we

mean the SUT. The prism type slope sensor consists ofthree main parts: a laser unit, a prism and

the PSD unit. From the laser unit (figure 3.2) a laser beam is generated, which is modified by an

optica} system to create a circular spot of 150 !liD diameter at the SUT. The details of this laser

unit will be described in the next section.


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e

Insert Prism

figure 3.2: Slope Sensorbasedon the prism design

3.3. Laser unit

To create a spot size of 150 f..Lm at the SUT the following setup (figure 3.3) is built. The

details of the optica! scheme are given in table 3 .1.

Dl D2 D3

Fl F2 d .. F3

figure 3.3: Optica! scheme ofthe laser unit.

table 3.1: Dimensionsfor the optica! scheme shown infigure 3.3.

Diaphragm Diameter Lens es Focallen_g_th Waist Diameter Dl 3mm Fl 30mm do! 26J.lm D2 50 J.lm F2 80mm do2 150 J.lm

F3 lOmm

A diode laser creates a divergent beam that is collimated by lens Fl. Since we use a solid

state laser the beam is elliptical due to different divergence angles as can beseen in tigure 3.4. In

order to create the circular beam we need, a diaphragm (D 1) is used to create a more circular

beam.


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b

a

figure 3.4: Elliptical beam comingfrom solid-state laser, due to reetangu/ar shape of laser cavity. The ratio between the elliptical axes a:b=22:8.5.

figure 3.5: Airy pattern around beam due to diffraction on diaphragm 1.

0.8

0.6 0

~ 0.4

0.2

~ 4

ft gure 3. 6: Gaussian (solid fine) and Airy (dotted fine) intensity profile.

At this point, due to diffraction at the edges

of diaphragm D 1, an Airy pattem around the beam is

introduced. A typical example of this Airy pattem is

shown in tigure 3.5. For measuring purposes, we

would like to have a circular beam with a gaussian

intensity distri bution and to be free of astimgatism.

Note that the bright spot in the center

(zeroth order ofthe Airy distribution) already has a

nearly gaussian intensity distribution as can be seen

in tigure 3.6 where the Airy intensity distribution is

shown together with a gaussian intensity

distribution.

Now we need to remove the higher orders of

the Airy pattem from the beam. We do this by using

a second lens (F2) that focuses the beam onto a

small spot do1. At this point it is possible to insert a

pinhole at position D2 that has the correct diameter

to fall into the frrst dark ring of the Airy pattem, thus

removing the outer rings, see tigure 3. 7. The next

lens (F3) is than chosen to focusthespot do1 at the SUT.


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At frrst a laser souree has to be choseno To

ensure that enough light reaches the detector we use

a high output power Mitsubishi laser diode for

which the specifications are shown in appendix IV 0

At normal operation conditions the output power is

P0 (40mW)o

In order to determine what lenses,

diaphragms and pinholes to use we calculate from figure 30 7: Diameter ofthe pinhole to remave the Airy patterns from the beam

the desired spot at the SUT, back to the laser, using

Gaussian opties (appendix V). We would like to have a spot of 150 J.Ul1 at 60 mm from the last

lens (F3)o For this lens we use a FlO, (focallength is 10 mm)o Using the formulas from appendix

V this results in a spot do2 of size 26 J.l.m at position D2. Thus we use a diaphragm of 50 J.Ul1 for

D2, since weneed to be in the first darkring ofthe Airy spot, which is roughly 2*do. To make

this spot, a lens with focal distance 80 mm is used at position F2 and a beam diameter (Dl) of ~3

mmo To collimate the laser light a lens with focal distance 30 mm is used at Fl. For an estimation

ofthe efficiency ofthis system look in appendix VI.

3.4. Spot characteristics

To check whether the spot has the correct size and is at the correct position, the beam has

been measured withaMeiles Griot Laser Beam Profilero In figure 308 the width ofthe beam (do2)

is shown as a function of the distance from lens F3 0 From the figure we see that the size and the

place of the spot are correct.


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350~----------------------~

• 300

• 250 •

'iii' • s 200 • ... u

i 150 0 'C

100

50

••• • •

0+-------~--------~------~ 0 50 100 150

Distance from lens F3 [mm]

jigure 3.8: Beam width as ajunetion of distancefrom lens F3 without a prism present

TUle

We've also measured the actual beam profile at the waist position to see whether the

beam is gaussian, this measurement and the gaussian fit on the measurement can be seen in figure

3.9, concluding that the spot is almost (>99%) gaussian.

1000

800

.wo

200

08580 8640 8660 8680

Pooltion [,.m)

figure 3.9: Beam profile (solid fine) and gaussianfit (dotted fine) at waist position.

Next, the prism is inserted into the beam. Again measuring the light intensity after the

prism results in 3.0 mW. Finally, the intensity reaching the PSD is only 80 ).!W. Showing the

major problem with this system, namely the low Signal to Noise ratio on the PSD due to the small

amount of light reaching the PSD.

Due to the nature ofthe prism this insertion results in an asymmetrical deformation ofthe

beam as can beseen in figure 3.10. Note that this is measured in the horizontal direction ofthe


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beam. In the vertical direction the beam doesn't change. As can be seen, introducing the prism

results in a measuring spot (at 60 mm) that is 150 fliD in the vertical direction and 300 fliD in the

horizontal direction.

800~------------------------~

700

600 'iii' c 500

300 0

e u :§.

400

"C 200 • 100 ..

.. .. H .. .. .. .. ..

I' V .. . ... I •a•••••

•

0+--------r--------r-------~

0 50 100 150

Distance from lens F3 [mm)

figure 3.10: Beam width as a function of distance from lens F3 without a prism in vertical (circles) direction and with a prism in vertical (lines) and horizontal (triangles) direction.


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4. Sensor type 11: Prism with Polarization Sensitive

Coating (PPSC)-type

4.1. Introduetion

The second sensor type is the Prism with Polarization Sensitive Coatings (PPSC)-type

sensor in which we use a customized prism that has three different coatings on its sides, namely

an AR coating, an adsorption coating and a polarization sensitive coating to try to reduce the

occurring false reflections in the p-type sensor. Also the other components in the set up are

coated. We also tried a new approach with the laser source: we now use a Polarization

Maintaining Fibre (PMF) to 'wash' the beam to create a gaussian spot of 105 J..lffi at the SUT. In

the following section we describe the set-up and its characteristics.

4.2. Schematics of the PPSC-type sensor

The principle of the polarization sensitive coating at the prism is shown in figure 4.1.

surface Linearly P-polarized light coming from the laser

cnular polamat1on

S-polamatlon

P-polamation

ft gure 4.1: Principle for the prism with polarization sensitive coating.

enters the prism and is refracted towards the SUT via a

114 A wave plate, resulting in circularly polarized light

at the SUT. After reflection ofthe SUT the light

passes the 114 A wave plate again resulting in linearly S

polarized light. The S-polarization is rotated 90° with

respect to P-polarized light. The coating at the upper

face ofthe prism (towards the SUT) has a higher

transmittance for P-polarized light and has a higher

reflectance for S-polarized light. This results in a

lower amount of light that is coupled back into the

prism after being reflected from the SUT. This light is one ofthe major contributors to false

reflections and we like it to be minimized therefore. On the side towards the laser, the prism has

an AR coating to reduce reflections of its surface, and on the left si de there is an absorption

coating to absorb the light that might still enter the prism

For the PPSC set up (figure 4.2), we tested a new approach to 'wash' the laser beam, i.e.


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to create a circular spot with a gaussian intensity distribution. To achleve this we used a

Polarization Maintaining Fiber (PMF). The purpose ofthe fiber is to function as a new 'laser

source' with the desired parameters. The samehigh power laser souree as in the P-type approach

is used and the light from this laser enters the fiber via two f1 0 lenses. Now the fiber will function

as a new light souree with a circular beam and a gaussian intensity distribution and the same

polarization as the laser source. Coming from the fiber the light is than focused on spot of 105 1-lm

at the SUT by the four lenses (flO, f80, flO and f30).

flO flO

~-- --{}- -E~--------------- fiber/~------- -

'-la_s_er _____ __,, /SUT '~ /< _ =' =

I

(a) I

(b)

I

I

! J

--

-,--- r--

(f I---

" ~ r---~ -

(

-

figure 4.2: Side (a) and top view (b) ofthe PPSC-type set-up. The laser light is coupled into the system via the polarization maintainingfiber.


Again we measured the beam diameter (spot size) at the SUT and these results are shown

in tigure 4.3. !t's clear that due to the asymmetry ofthe prism the beam is distorted, resulting in

an elliptical shape, that is 105 1-lm in the vertical direction and 140 1-lffi in the horizontal direction.

This means that we have two different lateral resolutions as shown in table 4.1.


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table 4.1: Lateral resolutions for the horizontal and vertical direction

Direction beam diameter r llffi l lateral resolution r llffi 1 Vertical 105 63 Horizontal 140 84

We also measured the beam intensity profile at the waist position to verify its gaussian

distribution. These results are shown in tigure 4.4 and tigure 4.5, including the gaussian fit. Thus,

although the spot has a different diameter in the horizontal and vertical direction the energy

distribution is still almast gaussian.

150 Vi 145 r:: e 14o • !::! 135 .§. 130 .!i 125 Cl)

~ 120 :s 115 E 110 lll al 105

100 25

..

---...

27

~ ....,_ .....

- - :::.-

29 31 33 35

Posltion [mm]

figure 4. 3: Beam diameter measured at different positions from lens j3 0 in horizontal ( diamonds) and vertical (squares) direction.

1200..---~--~--~--~----..,

1000

; 800

~ ~600 ïii c Cll

~ 400

200

8~50 9150 9200 Position [11m]

figure 4.4: Beam profile (continuous fine) and gaussian fit ( dotted fine) at the waist position in the horizontal direction.


1DOD..---~-----~----,------,

BOD

~ 600

~ 11)

c 400

~ 200

9000 9050 Position [11m]

figure 4.5: Beam profile (continuous fine) and gaussianfit (dotted fine) at the waist position in the vertical direction

30

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5. Sensor type 111: Polarizing Beam Splitter (PBS)-type

5.1. Introduetion

The third sensor type is a Polarizing Beam Splitter (PBS) type, in which we use

polarization sensitive and anti reflection coated components to reduce possible false reflections.

We also expect to gain about 5 times more light at the PSD resulting in a better Signal To Noise

ratio. In the following section we describe the design of the set up and its characteristics.

5.2. Schematics of the PBS-type sensor

The PBS is a cubic beam splitter for which the reflection ( or transmission) is dependent

on the polarization ofthe laser beam. In figure 5.1 the principle ofthis PBS is shown.

to detector

S-polamat1on

P-polamat1on ....__ __ __"

figure 5.1: Principle ofthe PBS set up.

surrace

circular polamation

At the left, P-polarized light

from the laser, is transmitted through

the PBS. lt is than transmitted through

a 14 Ä wave plate, which changes the

polarization to circular. After reflecting

on the SUT, the beam is transmitted

back through the 14 Ä wave plate and

thus the polarization is changed into S-

polarization. Because of the S-polarization, it will now reflect on the diagonal surface of the PBS

onto the detector.

A possible drawback for the PBS set up is that due to the parallel surfaces ofthe PBS,

interference can occur, probably resulting in a less accurate measurement.

The setup ofthe PBS system is shown infigure 5.2. In this set-up we used the samefiber

system as in the PPSC-type sensor. Thus, coming from the fiber the light is focused on a spot of

105 fJ.m at the SUT by the four lenses: f1 0, f80, f1 0 and f30.


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laser beam fiber f!O

-------------- ----------- -----------

figure 5.2: Schematic ofthe PBS-system that is conneeled to the laser via a Polarization

Maintaining Fiber.

f80

TUle

The light from the fiber is P-polarized and will therefore pass through the PBS with

almost no reflection losses. After the PBS the light will pass through the Y.. /.. wave plate that

changes the P-polarized light into circular polarized light (see also tigure 2.9). The light retuming

from the SUT will again pass this Y.. /.. wave plate and this results into S-polarized light (rotated

90° with respect to P-polarized light). The diagonal face ofthe PBS is highly (>99%) reflective

for S-polarized light so the light will be reflected towards the PSD.


Again we measured the beam diameter (spot size) at the SUT and these results are shown

in tigure 5.3. We also measured the beam intensity profile at the waist position to verify its

gaussian distribution. These results are shown in tigure 5.4, including the gaussian fit.

Concluding from the figures the spot is almost gaussian, the correct size and at the correct

position.


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116

'iii' 114 c e 112 ... 1

110 ... J!l Cl>

! E 108 • ca '6 • - • • E 106 z ca ! -Cl> lil 104 •

102 20 25 30 35

Position [mm]

figure 5.3: Beam diameter as ajunetion of distancefrom lensj30 in the horizontal (lines) and vertical (triangles) direction ofthe beam.

TUle

1200

1000

800

::; ~ ~ 600 c !l s

400

200

8700 8750 Posltion[l.m]

figure 5.4: Beam profile (continuous fine) and gaussian fit (dotted fine) at waist position, d0= 105

J-Un.


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6. Tilted flat mirror experiments

6.1. Introduetion

The first experiments carried out with the different sensors are mainly for calibration

purposes. We start by using a known surface, i.e. a flat mirror, and test the sensors signatures

under different conditions. Later we can than subtract these signatures from the measurements to

achleve more accurate results. From these experiments we can also derive calibration results to be

used in later experiments when unknown surfaces are scanned.

6.2. Flat scan

First, we mount a flat mirror perpendicular at the <p-axis. Now we give this mirror a small

tilt angle 8, thus the surface normal of the mirror makes an angle 8 with the <p-axis, see figure 6.1.

We then rotate the sensor along its 9-axis, while measuring the different slope components. Next

we rotate the <p-axis by 30° and repeat the measurement. This is repeated until the <p-axis has been

rotated 360 °. The expected outcome of these measurements should then be as follows. We expect

the <p-component of the signal to be constant ( during 9-rotation) and the 9-component to be linear

increasing with the rotation around the 9-axis.

<p-axis

figure 6.1: Flat mirror mounted on the cp-axis with a smal! tilt angle t5.

has been chopped off.

In figure 6.2 a typical result for a flat scan

measurement is shown and we see the constant <p

component and the linear increasing 9-component. The

first part of the <P- and 9-components shows some

strange behavior because ofthe 'over scanning' ofthe

sensor i.e. the spot goes over the edge of the PSD

resulting in invalid signals. Note that the samehappens

at the other end of the graphs but in this figure that part


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10,-----------------,

5

~ ii c .!!' 0 .. 0 ~ ······ ....................................................................................... - ............ ,_ ........ .

-5

-1 ~3~~~--"'':------::-0 -~---::-2----.:3

e["]

figure 6.2: B-response (solid fine) and ~ response (dashed fine) ofthe deflectometer on a finearscan across afZat mirror. For 8<-1 ° the response of the dejl.ectometer becomes unreliable because of the spot going over the edge ofthe PSD the response shows the same behavior for B> 2 o but this part is not shown.

0.02

I 0.015'

0.01

0.005 ~

0

-0.005

-0.01

-0.01~1 -0.5 0 0.5 1.5 e["]

figure 6.3: Deviationfrom the expected Bresponse (sofid fine) and ~response (dotted fine) when scanning a flat mirror.

To analyze the linearity of the a-component we fit the straight part of the data with a

linear function, and we subtract this fit from the a-component. This residue will then show the

non-linear behavior ofthe a-component. For the cp-component we do a horizontal fit (i.e. average

the signals) and subtract this from the signal. In figure 6.3 these two residues are shown. The first

point to be noted is the slightly increasing <p-signal; this is mainly because of a slight rotation in

the position ofthe PSD. To overcome this problem wedetermine the rotation angle f3 ofthe PSD

from these measurements. In case ofthe measurement shown in figure 6.3 the rotation angle

f3=5.4 mrad. We now can correct the signals from future experiments for this PSD rotation by

using a coordinate transformation around the rotation angle f3, as follows.

(S 'Sr/)=(S Se)(c~sfJ -sinfJJ rp rp sm fJ cos fJ (6.1)

In order to determine the slope scale factor and to get a feeling for the accuracy of the

different sensor types, we do this experiment with each type and with different tilt angles 8. Due

to the experimental setup, it was not possible to use the same tilt angle in each measurement. In

figure 6.4 to figure 6. 7 we show the slope scale factors determined for the different experiments.

The tilt angles shown in the captions are calculated from the measurement itself and are not

calibrated. In figure 6.4 we see the results for the P-type sensor (uncoated prism) in combination

with a 2L4 (4*4 mm) PSD. In figure 6.5 and figure 6.6 we see the results for the PBS-type sensor,

in figure 6.5 with a 2L4 PSD and in figure 6.6 with a 2Ll 0 (1 0* 10 mm) PSD, in figure 6. 7 we see


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the results for the PPSC-type sensor with a 2L10 PSD. Note that the scales in the figures are

different!

3.6 ,--------------" 3.55 ,.__ __________ _,

~ 3.5 +:------------j "0

~ 3.45 -+-~"=---~-.-. --------1

0- 3.4 -~ ~ . • •····

3.35 :.. .. ··~·····~·-.

3.3 +-----,------,-------,-.._j

0 100 200

phl [deg]

300

figure 6.4: P-type sensor/2L4 PSD slope scale factor at different tilt angles:5=0.3432 (sofid fine) and o=l.7871. 0 (dotted fine).

3.66 , ......... -...... ,_,_,_ .................. -................. _ ................................. ,_,_", .... -........ ~,

3.64 -+--------~· -----;<1.!

~ 3.62 +--------------!·-: _,···------..;-j· : ~ . i ~OE 3.6+-~ .• ~ .• ~.-~ .• ~ .• ~·~--•. -_.~.-~.-..• -.~--~j

· .•. - ••. ...... i 3.58 ~ ...... =-.:-=-------'----:_ /~"".. --...-. .......... ---3.56 +----,------,------,-----'

0 100 200 300

ph i [deg]

figure 6.6: PBS-type sensor/2L4 PSD slope scale factor at different tilt angles: o=0.2036 (solid fine) and OF1.2298 °(dotted fine).

8.68 ,----------------,

8.675 +-----,/--->>r---------1

~ 8.67 +:----r------'.....-------1 "0 ~ 8.665 +---+--------"-~---1

ü 8.66 +-f-----------'r---1

8.655 -11---------------'~

8.65 +---.,.----.,.----,----'

0 100 200

ph i [deg]

300

figure 6.5: PBS-type sensor/2Ll 0 PSD slope scale factor at tilt angle: o=O.l311 o

9.26 , .... __________ . __ ..... _ ......... ,_ ..... - .. --·-··-................................. _"_/'

9.24 +---,.-------.i' .. , -----! ~ 9,22 I \

è 9.2 ++-' .....L\ ------,,.'---;,-----:i .§. 9.18 T--.. ~-';\-:-------;-----+-, - ..... --i .. -1 0 9.16 .. ~~~--'--""~---"""""~~~~~ · .•...•. -:·::.-..-.:-.... - • ...-1 .......... ·

9.14 +---=--=."..::...;=-------==---==--------! 9.12 +-----,-------,..-----,----'

0 100 200

phl[deg]

figure 6. 7: PPSC-type sensor/2LJO PSD slope scalefactor at different tilt angles:o=0.0859 (sofid fine), o= 0.353 (dotted fine) and o= 0.8399 °(dashed fine).

From these experiments we determine the slope scale factor C and its standard deviation

from linearity, these results are shown in table 6.1.


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table 6.1: Slope scale factor Cexp' its standard deviation, percentage deviation from the expected theoretica! values for Che and the resulting accuracy in (he slope signals.

Cexp cr( Cexp) o-(Cexp) Cthe Cexp -Cthe (j<p cre

cexp ct he

[mrad/V] [mrad!V] [%] [mrad/V] [%] [J..Lrad] fJ..Lrad] I (Prism old,

3.40 0.04 1.25 3.333 2.0 39 58 2L4 PSD) 11 (PBS, 2L4

3.59 0.02 0.54 3.333 7.7 21 41 PSD 11 (PBS,

8.675 0.007 0.08 8.333 4.1 29 84 2Ll0 PSD) lil (Prism new, 2L10 9.16 0.02 0.21 8.333 9.9 33 45 PSD)

As can be seen, there are differences between the slope scale factors from the different

sensors. From formula 2.3 it is clear that the slope scale factor is mainly determined by the size of

the PSD and the distance between the PSD and the lens. Theoretically we expect the slope scale

factor to be: 3.333 mrad/V in case of a 2L4 PSD and 8.333 mrad/V in case of a 2L10 PSD. For

the measured deviations we can give some possible explanations. A major contributor to the

deviations is probably the (mechanica!) distance between the lens and the PSD, it is possible that

this distance is not exactly 30 mm, resulting in a different value for the slope scale factor. When

for example a 2L4 PSD is shifted 0.5 mm this results in a 1.8% deviation for the slope scale

factor. In case of a 2L 10 PSD this is 1. 7%. Another possible explanation for the deviations lies in

the so-called 'false reflections' resulting in ghost spots on the PSD and therefore affecting the

measurements. Thirdly there are the optical components that are not perfect. Another possible

contributor is the presence of dust particles and contaminations in the setup, since we work in a

normal room. These will have their influence especially at the places in the optica! path where the

beam is very small ( -10-50 J..Lm). This is for the P-type sensor at the 50 J..Lffi pinhole and for the

other two types at the edges of the PMF where the beam diameter is 10 J..Lm.

Next we take a look at the deviations from the ideal case, as shown in figure 6.2, to

derive the accuracy ofthe different sensor types ifthe linear fit is used. Wedetermine the

standard deviation ofthe measurement signals and translate this by the slope scale factor into

angular accuracies. The results for the different sensor types are also shown in table 6.1. Here we

see that the accuracy in the 8-component is worse than in the <p-component, this can be explained

by the fact that the false reflections have their strongest effect in this component. We expect that

the accuracy ofthe PBS and PPSC-sensor types is better than the P-type, since we use better


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optical parts. It is clear that the PBS with a 2L4 PSD has a better accuracy, but the PBS and PPSC

type with a 2Ll0 PSD are not significantly better. The possible explanation for this is the fact that

the 2L10 PSD has a glass covertoproteet its surface. Since this is a non-coated surface it will

introduce more false reflection and therefore possibly reducing the sensors accuracy.

6.3. False reflections

One of the lirnitations for the accuracy of the sensor comes from the occurrence of false

reflections on the PSD, therefore resulting in a less accurate measurement. Todetermine whether

or not these false reflections are of significant influence we have mounted a CCD camera at the

position ofthe PSD to take pictures ofwhat the PSD actually sees. In figure 6.8 and figure 6.9 we

see a compilation of these images while scanning a flat rnirror, as intheflat 8-scan from chapter

6.2. It is clear that the false reflections still are a contribution, although measures have been taken

to reduce them. To get a feeling for the influence of the false reflections we make an estimation

of the introduced error as a function of the amount of light that is in the false reflections. In table

6.2 this estimation is done in the case where the measuring spot is at 5 mm on the PSD and the

false reflections are in the center.

table 6.2: Estimation ofthe error in slope signa/ when the spot is at 5 mm at the PSD as a function of the percentage of light in false rejlections at the center of the PSD.

Percentage light in false Shift in center of weight Shift in slope signa! reflections spot [%] [rnm] [JJiad] 2 0.1 333 1 0.05 167 0.5 0.025 83.3 0.05 0.0025 8.33


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false reflections ->

spot->

fr.gure 6. 8: CCD images at the position of P SD as the dejlectometer scans over a flat mirror, using the new prism dejlectometef. Clearly visible are the undesired fa/se rejlections injluencing the measurement.

spot->

false reflections ->

fr.gure 6.9: CCD images at the position ofthe PSD as the dejlectometer scans over a flat mirror, using the PBS dejlectometer. Clearly visible are the undesiredfalse rejlections injluencing the measurement.


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6.4. Wobbie experiments

The next experiments carried out with the different sensor types are the so-called

'wobble' -experiments, in which again a flat mirror (/./4) is mounted onto the <p-axis at a small

angle (8). But contrarily to the fi.rst experiments where we did rotate the 8-axis while keeping the

<p-axis at a certain angle, we now rotate the <p-axis while keeping the 8-angle constant. Then the

8-angle is changed and the experiment is repeated until the complete 8-range is done. We now

expect, according to the laws of reflection, the reflected laser beam to project a (more or less)

circular trajectory on the PSD, i.e. the surface normal ofthe mirror will precess around the <p-axis

(tigure 6.10). The location of these trajectories on the PSD then depends on the angle 8 between

the <p-axis and the incident laser beam. In tigure 6.11 two different important orientations of the

<p-axis are shown in top and side view, all for a 8=0, i.e. the incident laser beam lies in the

extension of the <p-axis. At the <p=O the mirror is rotated in such a way that the tilt angle lies in the

8-plane, as can beseen best in the side view. This position results in a zero <p-signal and

maximum 8-signal. Now the <p-axis rotates to rr./2 and this results in a zero 8-signal and maximum

<p-signal.

.ft gure 6.10: Precession of surface normal of a rotating tilted mirror, resulting in circular trajectories of the spot at the PSD.

top-view si de-view

<p=O

e <p-axis

.ft gure 6.11: Two views of a flat mirror mounted at a smal/ angle ( w) on the phi axis; for the two different ~position resulting in a zero ~signa/ (cp=O) and a zero 8-signal (cp=m'2). The dotted fine is the actual laser beam and its rejlection.

In tigure 6.12 aresult of a typical 'wobble' experiment is shown. Note that ifthe axis

would have the same scale, the ellipses are circles. In tigure 6.13 and figure 6.14, theseparate <p


CFT -Philips

and 8-components are shown.

a ~ 0.5 -~ '(> ..

f\ I \ I: \

·----1--:---t--.!, I I: I

----l+·j---1 ' \: I: I --- -----·r\·:--;--

: ; : , I

.o.~:----7-& -~-4:-----7--:o:-----72 ---'4.

e-signal [V)

TU/e

1.5r--------r----,----,----,

~'·············V\ ---------- ------- -.~- ~----------- ~-----------

"0·5o:----:1-:-:oo,----~2o-=-o ----=-3o=-=-o-----:-:40o

+1"1

figure 6.12: Typical result of a 'wobble 'experiment for Jour different e angles.

figure 6.13: cp-component of the PSD signa/ at Jour different e angles.

4r---~--~----r----. ' ' .L.-------:---.....__ :

2 -~~::~:----------- ~-------- :·~:.::.:.:---

100 200

+1"1 300 400

figure 6.14: B-component ofthe PSD signa/ at Jour different B-ang/es.

6.5. Analysis of 'wobble' -experiments

To have a compact description ofthe measurement results we performa Fourier analysis

on the 8 and cp component (figure 6.13 and tigure 6.14) ofthe signals, with M=40 termsas given

by:

M

ff/J(x),/8 (x) = a0 + Lan cos(nx)+bn sin(n.x) (6.2) n=l

1 1r

a0 =- J f(x)dx 1C

-Ir

(6.3)


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1 1t

an =- f f(x)cos(nx)dx 2:r -1t

(6.4)

1 1t

bn =- J f(x)sin(nx)dx 2:r

(6.5) -1t

The first detail that can be extracted from these data is again the slope scale factor C from

chapter 2.3. To do this, we look at the 8-component ofthe signal (figure 6.13). For each

measurement angle 8, the average value ofthe 8-signal is taken, that is the Fourier term ao ofthe

8-component. This is supposed to be a straight line (figure 6.15) as insection 6.2. Intigure 6.15 it

is clear that the behavior ofthe sensor becomes non-linearat the larger angles. This probably bas

its origin in two different effects. At first it is a known fact that a PSD does not behave linear at

its edges. And the second effect is the fact that the spot goes over the edge of the PSD when its

follows its circular path at the edge ofthe PSD.

To avoid these non-linear parts ofthe data, we performa linear fit (y=Cx+b) on the

'linear' part ofthe average 8-slope. This 'best linear' part is shown in figure 6.16 between the

circles, where the difference between the measured 8-slope and the linear fit is shown.

0.04,-----r----.-----,------,--,------, 0.03,.----~-~-~--~-~----,

~ 0.02 ·········:········-~---·····-~---·····-:

! ' ' ' 0 ......... ; ......... ; ......... : ......... ; ......... ; ....... .

I I 0 I I

·2

' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' '

·1 0

e[~ 2 3

figure 6.15: Average B-slope to determine slope scale factor.

IC 0.02 ···· ··· ; ......... ; ......... ; ......... ~----····-:---······ ~ a:> 11)

g. 0.01 ·········! ·······+·······l········-:·-········!········· 'jij ' : : 11) : : Cl f ~ .E ' ' ' g ·0.01 ·········:········-~·-·······>·····-~·-·······, ....... .

Cll

·2 ·1 0 e[~

2

figure 6.16: Error in average 8-slope, i.e. difference between B-signal and the B-fit.

3

From the amplitude (ao, formula (6.2)) ofthe q>- and 8-component, the tilt-angle (8) ofthe

mirror can be calculated, namely:

(6.6)

in which Cis the slope scale factor from figure 6.15.

From the previous measurements we determine the average slope scale factors


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determined by the 'wobble' experiments and the flat 8-scan experiments for the different sensor

types, to be used in further experiments, see table 6.3. We see that the two different methods yield

almast identical results.

table 6.3: Average slope scalefactor C and its standard deviation determined by thejlat B-scan and "wobble" experiments.

flat scan Wobbie AVERAGE c cr c cr c cr [mrad!V] .Imrad/V] [mrad/V] [mrad!V] [mrad!V] [mrad/V]

I (Prism old, 2L4 PSD) 3.40 0.04 3.40 0.01 3.40 0.04 II (PBS, 2L4 PSD 3.59 0.02 3.62 0.06 3.61 0.06 II (PBS, 2Ll0 PSD) 8.675 0.007 lil (Prism new, 2Ll0 PSD) 9.16 0.02 9.16 0.020 9.16 0.03

6.6. Surface reconstruction from 'wobble' experiments

The next step is to reconstruct the surface of the flat mirror by the integration routine

described in section 2.4. Since we know this surface we can predict what the measured slopes

should look like ( tigure 6.17). We try todetermine the absolute asphericity (formula 2.8) ~Re,

bath experimentally and theoretically. By using simple geometry we can calculate the distance

~Re (tigure 6.17), that is:

M 8 =Ro(-1 -1) cose (6.7)

Todetermine ~Re from the measured slope we use formula (2.8) farm section 2.4:

(6.8)

note that S9=tan(8).

For this reconstruction, we use the 8 slope data at which the mirror is rotated such that

there is no tilt in the 8-direction, i.e. the mirror is tilted at a maximum angle in the <p-direction.

This means that we do a line scan along the mirror at which we determine the asphericity of the

mirror with respect to the reference sphere.

The results for a typical surface reconstruction (line scan) are shown in tigure 6.18

tagether with the theoretically expected values. In tigure 6.19 we show the difference between the

two graphs in tigure 6.18, i.e. the error in the absolute measured asphericity.


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mwror

ft gure 6.17:Asphericity&?.0of a flat surface with respect to the reference sphere.

5 ........ \: __ ...... _; ....... --~-- ·····-~-- ....... : ....... .

' ' ' ' ' I ' 0 I 0

4 ....... ··:· .... ·--~ ........ -~-- ...... ~- ······ . :····· .. .

E : ! ! i i ..=. 3 ......... : ........ ; ......... ~ ........ ; ........ ; ....... . er."' : : : : ' <I : : : ' '

2 ·······-~·-···· --~·-······-~·-·····-~-- ···-··:········ ' '

1 ......... : ......... ! ........ : ........ ·:·········i·-······

figure 6.18: Experimental (solid fine) and theoretica! (dashed fine) resultfor the absolute asphericity &?.0 of a flat mirror.

100 ... ·········!··············i·-···········

'E 5o ······· .... ; .............. ; .............. ···· ········ s : : er."' <I .5 ..

' ' ' ' ' '

~ -50 ·············j··············j·············· ·············

' ' ' '

-100 ··············!··············i············· ·············

- 15<!2'----~-1---~0---~----'2

9['1

figure 6.19: Difference between the experimental and theoretica! absolute asphericity of a flat mirror.

Again to determine the accuracy of the different sensor types we look at the deviations

between the measurements and the theoretica! results and the results are shown in table 6.4. To

compare these results with the expected errors we estimate the effect of the slope signa!

inaccuracies from table 6.1 in the absolute asphericity determined by formula (6.8). The results

forthese estimations are shown in table 6.4 and are compared with the (averaged) experimentally

determined values.


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table 6.4: Error estimation in absolute asphericity (&r) determined by using the inaccuracies in the slope signals from table 6.1 compared to the experimentally determined values for &(}

Sensortype Experimentally determined Inaccuracy in 8-component Estimated error in äRe from averaged error in ~ inaccuracy in slope signals.

[nm] [!Jiad] [nm]

P-type/214 PSD) 0.52 58 1.21

PBS-type/214 PSD 4.5 41 0.86

PPSC-type/211 0 PSD) 1.51 45 0.94

Looking at the values for the experimental error in Ll~ and the estimated error we see an

agreement (order of magnitude) between the values for the P and PPSC-sensor type. But for the

PBS sensor type there is a disagreement At this moment we have no explanation for this

disagreement

6.7. Conclusions

By using the flat rnirror experiments we have been able to deterrnine the accuracies of the

different sensor types, showing the PBS and PPSC type to be slightly better than the P-type ·

sensor, as expected. Still, the inaccuracies are mainly deterrnined by the occurring false

reflections. These false reflections have been made visible by CCD camera pictures. By

reconstructing the surface ofthe flat rnirror with a line scan we deterrnined the inaccuracies in the

surface reconstruction to be in agreement with the inaccuracies in the deterrnined slope signals.

Only the PBS-type sensor showed a slight disagreement, but no explanation has been found.

Another feature of the flat rnirror experiments was a method to deterrnine the rota ti on of

the PSD and to correct for this. Typical rotation angles for the PSD are in the order of 5 rnrad.

Also we used the flat rnirror experiments to calibrate the slope scale factor to be used in later

experiments.


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7. Sphere experiments

7.1. Introduetion

Following the flat mirror experiments we mounted a polished metal sphere (R=8 mm) as

a SUT. The purpose of these sphere measurements is mainly tolook at the misalignments ofthe

set up, since a perfectly aligned sphere would yield a zero slope signal (in both directions). Any

deviations from this zero signal can, in combination with the mathematica! model from section

2.4, tellus something about the misalignments in the set up.

7.2. Experimental set-up

To carry out the experiments we mount the sphere intentionally with some variabie

degree of misalignments, and these misalignment are then verified with mechanica! pro bes. The

sphere is measured by the deflectometer by starting the <p- and 8-axis at zero. The <p-axis is then

rotated 360° following by a rotation ofthe 8-axis of0.5°. This is repeated until the 8-axis is at 42"

i.e. 81 steps. During the 360° <p-rotation, 307 measurement samples are taken, so resulting in a

total of8lx307=24867 data points for each measurement. After measuring the sphere with the

deflectometer we fit the measuring data with the model to extract the misalignments from the

deflectometer measurement. Forthese experiments we used the PPSC-type sensor for the

deflectometer set up.

7.3. Slope data processing from sphere measurement for

misalignment extraction

After the measurement we start the processing of the slope data by performing the

Fourier analysis from section 6.4.2 on the slope data. We perform this analysis for each 8-angle

on both the slope signal components. At this point we're interested only in the signals that are

produced by the misalignments and not in the surface topography of the sphere. Therefore we use

only the zeroth (ao) and frrst (ai and bi) terms ofthe Fourier analysis, since the signals ofthe

misalignment will show a periodicity of 21t.

We now insert this frrst order Fourier analysis into the misalignment model to fit the

misalignment parameters on the measurement data and to extract these. Again this is done for


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each 8-angle. For this fit we use the 'Least Square Curve Fit' procedure from Matlab (REF!!).

20,-------------------------~

'ë' .=.

19.8

19.6

"' 19.4

10 20 30 40

er~

figure 7.1: Misalignment parameter &

determined by fitting the first order Fourier analysis ofthe qrcomponent ofthe slope data on the misalignment model.

100

50

-"'V ·tv ~"V 'ë' .=. 0 (.)

·50

·1000 10 20 30 40

9[~

figure 7.3: Misalignment parameter crossing determined by fitting the first order Fourier analysis ofthe qrcomponent ofthe slope data on the misalignment model.

·1.1 r-------------------------,

·1.15

·1.2

~ ·1.3

= ·1.35

·1.4

·1.45

' 1·5o:------:1'=-o -----=2~0 -----=3'=-o -----,4~0-

e[~

figure 7.2: Misalignment parameter height determined by fitting the first order F ourier analysis ofthe q>-component ofthe slope data on the misalignment model.

1.02r----------------------------,

0.98 c

C> .. 0.96

0.94

0.920 10 20 30 40

er~

figure 7.4: Misalignment parameter rp0

determined by fitting the q>-component of the first order Fourier analysis of the slope data on the misalignment model.

In tigure 7.1 to tigure 7.4 the results from the misalignment fit in the q>-direction are

shown. Here we see the misalignments E, h, C and the phase q>0 determined from the fit. To

campare these results we also measured these values with the mechanica! probes. In table 7.1 the

values determined by the deflectometer measurement and from the mechanica! probe

measurements are shown. Where it is clear that the values determined from the deflectometer

measurement are consistent with the mechanica! probe measurements, except for the values


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around 8=0°. The deviations at 8=0° occur because of the extreme small signals occurring at this

angle, therefore the fit procedure has difficulties determining the best-fit value. To overcome this

problem we use the part > 1 Û0 for the misalignment extraction.

table 7.1: Misalignment parameters determined with bath the dejlectometer and the mechanica/ probe.

fit parameter deflectometer measurement mechanica! probe measurement [J,tm] [J,tm]

E: 19.18 ± 0.02 19.3 ± 0.5 h -1.3±0.1 -1.5 ± 0.5 c 23 ±3 22±2 M, -- 1.0 ± 0.5

As mentioned before, this misalignment fit is done in the <p-direction because of the

relative 'simple' model in that direction (formula 2.15). We did not yet succeed in performing the

fit in the 8-direction. Therefore we do not have information about the misalignment ~z from the

deflectometer measurements. Since we are not able to determine the misalignment ~z from the

sphere measurements we try to minimize this value by using the mechanica} probes. Typical

value for this misalignment ~z that remains is then in the order of a micron.

7.4. Surface reconstruction of the sphere

Another test that we are interested in, of course, is the surface reconstruction of the

sphere. To reconstruct the surface we use the following procedure. First wedetermine the

misalignment parameters as in the previous section. We than subtract the theoretica! signals that a

sphere with the determined misalignments would produce from the measured signals by using the

misalignments model. Now the effect ofthe misalignments is removed from the signals and we're

left with the signals that describe the actual surface ofthe sphere relative to the reference sphere.

These signals are then integrated following the procedure of section 2.3 to reconstruct the surface.

Where, for the q>-component, we first should integrate a line ofthe 8-component data at <p=0° as a

starting point for our <p-integrations. But due to the fact that we cannot reconstruct the

misalignment ~z from the 8-component, we have no information on the misalignment ~z, which

weneed todetermine the absolute radius ofthe SUT. As a result, the starting line ofthe q>

integration is not known! This means that we cannot determine the absolute radius ofthe

reference sphere, but only the anomalies relative to the reference sphere can be determined.

To overcome this problem, wetook the following approach. We take the first order

Fourier analysis and subtract this from the 8-component ofthe signals, thus removing the


CFT-Philips TUle

misalignment signature. This leaves us with the surface details, but without knowledge about the

absolute radii of the reference sphere or misalignments. From this data we now make the starting

line for the integration of the <p-component.

In tigure 7.5 we show the result of this surface reconstruction. This is a height map of the

difference between the measured sphere and the reference sphere, looking on top of the sphere.

The next tigure (figure 7.6) shows the so-called closing error ofthe integration procedure,

because we use line integrals where the end point ( <p=360°) is the same as the starting point

( <p=0°) we expect the values at this point to be the same. The closing error is therefore the

difference between the height at the end point and that at the starting point of the integrated <p

circle. This closing error gives us an in di cation for the accuracy of the model and the

deflectometer itself. To reconstruct the surface in figure 7.5 we used the best misalignment fit for

each 8 line, i.e. the misalignment parameters were not averaged, resulting in a small closing error.

When doing the same surface reconstruction, but now using the average misalignment parameters

from table 7.1 we see that the result becomes different, as is shown in tigure 7. 7. Looking at

tigure 7.8 we see that this approach has increased significantly the closing error. Looking at the

shape ofthe closing error graph, it is clear that it resembles the misalignment parameter h (figure

7 .2). We assume that increasing the accuracy in determining this parameter should result in a

lower closing error. Apparently the fit procedure is very sensitive for this parameter.

To see whether the closing error is consistent with the inaccuracy in the slope

measurements determined in table 6.1 we make an estimation of the error that is produced in the

surface reconstruction by these inaccuracies. In table 7.2 we see the results for this estimation.

Concluding that an inaccuracy of 33 J..IIad in the detected <p-slope yields an error of 1.6 J...tm in the

surface reconstruction after integrating over the <p-circle, i.e. we expect the maximum closing

error to be in the order of 1.5 J...l.ID. In the results ofthe surface reconstruction we see a closing

error of ~.5 J...tm. This can be explained by the way we estimated the error; we assumed the

maximal inaccuracy to be occurring at each sample point and added these over the complete

integration line resulting in the maximal closing error. In the real case the inaccuracy will not be

maximal at all the sample points resulting in a smaller closing error, but, the order of magnitude

for the error complies.


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table 7.2: Error in surface reconstruction occurring because of slope measuring accuracy for the new prism design with a 2Ll 0 PSD.

accuracy <p- accuracy height error height error error after line error after line comp: 9-comp per sample per sample integration in integration in [J.Lrad] [J.Lrad] point in <p- point in 9- <p-direction 9-direction

direction direction [J.Lm] [JJ.m] [r..tm] [r..tm]

I Prismnew, 2Ll0 PSD 33 45 0.0052 0.0030 1.60 0.25

For showing feasibility ofthe deflectometer metbod for aspherical surfaces, an error of

0.42 J.Ull over an integration pathof R:15 mm is at this point acceptable. In order to increase the

accuracy of the surface reconstruction it is necessary to improve the quality of the measurements,

i.e. to reduce the false reflections and to work in cleaner conditions, since these are the main

probable contributors to the inaccuracies. Another point for impravement is the linear

approximation for the sensor, using a higher order approximation should also result in a higher

accuracy. Another way to imprave the surface reconstruction is to use the redundancy between

the <p, 9-component and the intensity signals. At this point, no use has been made of this

redundancy.

0.02 ,--~--~--~---------,

'ë' .§. l(

y[mm]

figure 7.5: Surface reconstructionfrom q;-slope data. Here the best misalignmentfitfor each 8-angle is used.


0.01

Ë :. .. e .. 111

0 OI c: iii 0 ü

.0.01 0 10 20 30 40 50

ef']

ft gure 7. 6: Closing error for the surface reconstruction infigure 7.5. This is the difference between the starting point and the ending point of the integration.

50

CFT -Philips

Ê .§. )(

figure 7. 7. Surface reconstruction from ~slope data. Here the averaged value of all the fit data is used, i.e. the bestfit misaligned sphere is subtracted

7 .5. Conclusions

TUle

0.5

0.4

0.3 Ê .:. 0.2 .. ê 0.1 .. &:I> c Ui 0 u

10 20 30 40 50 o["]

figure 7. 8: Closing error for the surface reconstruction in figure 7. 7. This is the difference between the starting point and the ending point of the integration.

By using the misalignment model and the measurements on a metal sphere we have been

able to extract the misalignments eccentricity, crossing and height from the experimental data for

the q> component. For the 8-component the misalignment extraction failed due to the complexity

of the misalignment model in this direction, making the fitting procedure very difficult. Therefore

no information on the misalignment shift dz was gained. Lack of information on shift dz leaves

us without information about the absolute radius ofthe SUT, therefore only a relative surface

construction with respect to the chosen reference sphere is possible. Using this relative surface

reconstruction in combination with the misalignment extraction for the q>-component showed us

the closing error to he a good indication of the accuracy of the measurement and reconstruction.

A typical closing error for a sphere measurement is ~o.5 J.Ul1 over an integration path of~ 15 mm,

enough for proving feasibility for the deflectometer at this point.


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8. Asphere experiments

8.1. Toric sample

Finally, to prove feasibility ofthe deflectometer for aspherical surfaces, we have

measured actual aspherical surfaces. First we measured aso-called 'toric' sample that has two

different radii on perpendicular lines as is shown in tigure 8.1. So on line A it has a radius

R1=7.505 mm and online Ba radius R2=8.005 mm.

figure 8.1: Top view of the toric sample with the two radii. Along fine A the sample has a radius R1 (7.505 mm) and along fineBaradius R2 (8.005 mm). Outside the area with the two radii, the shape is not specified.

In tigure 8.2 we see the result ofthe surface reconstruction from the measurement ofthis

sample. In tigure 8.3 we show the closing error for the surface reconstruction. Note that the

closing error in the part with the unknown shape is ~.02 ).LID!

y[mm]

figure 8.2 Surface reconstruction of a toric sample and the fines A and B with the radii R1=8.005 mm and R2=7.5005 mm.

).l.m


0

~ -1

~ -2 CD c ~ -3 u

-\~---1~0~--~270--~7.30~--~4~0----~50

efi figure 8.3: Closing error for the surface reconstruction ofthe toric sample infigure 8.2.

52

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·!s=-------70--- --'----:5 croaaectlon [mm)

10

0

"1'!s~-----:-o-------!.s

croaaec:tlon [mm)

TUle

figure 8.4 Cross section along fine A (RI) ofthe toric sample.

figure 8.5: Cross section along fine B (R2) ofthe toric sample.

In tigure 8.4 and tigure 8.5 we show the cross section along the two lines A and B ofthe

toric sample. Since we are not able todetermine the absolute value ofthe radius we cannot give

an absolute surface reconstruction. But we can look at the relative height difference between the

cross sections along the symmetry axis A and B for comparison with theoretica! values. We are

interested in the distance pin tigure 8.6 for which the following formula holds:

. (8.1)

p

jigure 8.6: Height difference p between the two radii ofthe toric sample at an o.ff-axis distance!.

In tigure 8.7 the difference between the experimentally CPcxp) and theoretically (Ptheor)

determined values for p is shown as a function of 1. Again we see that the deviation from the

theoretically determined values is ofthe sameorder of magnitude as the closing error. This

indicates again the closing error to he a good indication ofthe worst-case error ofthe

deflectometer when using the linear model and no data redundancies.


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4

0

-~JL__--~2---1~-~o-~--~2 _ __,3

I [mm]

figure 8. 7:Difference between the experimentally (pex~ and theoretically (ptheoJ determined values jor p.

8.2. Bi-focal sample

Another sample we measured is aso-called bi-focal sample. This is a lens that corrects

for vision at two different distances; therefore two different radii are incorporated into the design.

On this sample there are 6 areas that have two altemating radii. A height plot of the design data,

with respect to the base shape (figure 8.9) ofthis sample is shown in figure 8.8. Note that the base

shape ofthe bi-focal sample is nota sphere, as can be with respect to the plotted sphere of radius

R=7.505 mm. In figure 8.8 we can clearly see the different areas with the altemating radii as the

light and dark areas. In figure 8.10 the height map of the surface reconstruction is shown relative

to the reference sphere (R=7.505 mm) and in figure 8.11 the closing error is shown.

llm

figure 8.8: Design plot ofthe bi-focal sample (relative to the base sphere), where the two different radii-areas are visible as the dark and light areas.

7

x[mm]

figure 8.9: Base shape ojbi-focal sample (solid line) and the corresponding sphere shape (dashed /ine) with a radius R=7.505 mm.


CFT -Philips

y(mm]

figure 8.10: Surface reconstruction of the bifocal sample relative to the reference sphere, where the reconstruction is done on the cpcomponent of the slope data.

Ê .:. .,g Dl c ii .g

TUle

1 .5 ,---------~-~--

0.5

40 50

figure 8.11 Closing error for the surface reconstruction of the bi-focal sample.

Tolook in more detail into the surface reconstruction, we take a cross section ofthe

design data (figure 8.12) and the measurement result (figure 8.13). Due to different data

defmitions it is not easily possible to campare the two sets numerically and to plot them in the

same plot. Therefore, to campare the two cross sections, we look at the maximum and minimum

values ofthe cross sections and determine the numerical values forthese positions (table 8.1).

8,---------~-----~

-2

-4

-6

-~:5'-------~0 --------' 5

cross sectlon [mm]

figure 8.12: Cross section of design datafor bifocal sample.


8

6

4

2

Io N

-2

-4

-6

~5 0 crosseetion [mm]

ft gure 8.13 Cross section of surface reconstruction of the bi-focal sample.

5

55

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table 8.1: Numerical val u es for the maximum and minimum of the cross sections.

Surface reconstruction Surface reconstruction Design of q>-slope data [Jlm] of6-slope data [Jlm] data [Jlm]

max 6.56 6.28 6.15 min -6.33 -6.29 -6.15 t:, 12.89 12.57 12.3

We can see in table 8.1 that the difference between the design data and the surface

reconstruction is within a micron, again what we expected when looking at the closing error and

the error estimation from section 7.4. In this case, also an additional explanation for the deviation

can he given. For the surface reconstruction wetook the base shape to he spherical (R=7.505

mm), but since the base shape ofthis sample is not spherical (figure 8.9) we expect deviations

between the reconstructed surface and the design data.

For this bi-focal samplewetried to reduce all misalignments as much as possible (<1

J.Lm) to attempt to reconstruct the surface also with the 9-slope data. The result for this

reconstruction is shown in figure 8.14 and the numerical values for the cross-section (not shown)

are given in table 8.1. This shows us that we can reconstruct the surface using both slope

components, providing we reduce the misalignments as much as possible.

'ë' .s )(

y(mm]

figu.re 8.14: Surface reconstruction ofthe bi-focal sample by using the B-component ofthe slope data.


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9. Concluding remarks

We've shown feasibility for measuring aspherical surfaces with the deflectometer,

although absolute surface reconstruction has notbeen achieved yet. We've managed to

reconstruct the surface of aspherical shapes relatively to the base sphere with only errors of ~1

f.i.m over 15 mm integration path distance. But information ofthe base sphere is not achieved,

thus no absolute surface reconstruction was possible.

Designing, constructing and testing the three different sensor design resulted in

comparable accuracies for the designs, where the new prism and PBS design had a slightly better

performance. For the current set ups, i.e. the prism and PBS design; typical slope angle

measurement accuracies are in the order of 30-40 f.i.fad, although the theoretica! achievable

accuracies are a factor 10 higher. To increase accuracy for the deflectometer method several steps

can and should be taken. At first, even after the effort taken to reduce false reflections in the set

up, there still remain false reflections influencing the accuracy ofthe measurements. To reduce

the influence of these false reflections several steps can be taken. With the prism set up, the si de

of the prism that is not used should be 100% adsorptive to reduce the false reflections occurring

by light traveling around in the prism. Also removing the glass cover from the PSD should reduce

the false reflections. If not succeeding in removing the false reflections by adapting the optical

coatings another option could be to use a CCD camera insteadof a PSD. This would reduce the

speed ofthe system, but by using 'spot-tracking' the influence ofthe false reflections could be

removed. Thus improving the accuracy of the system. Another way of increasing the accuracy

would be by creating a look up table for the PSD signals, i.e. by using a calibrated angle stage one

could offer the deflectometer allangles (in its range) and store the response ofthe deflectometer

in the look up table. When measuring an insert one can then look up the slopes that belong to the

PSD signals. This should also remove a lot ofthe signatures ofthe set up, including false

reflections.

Using the flat mirror for the flat 9-scan and wobbie experiments we developed calibration

procedures by using a linear model to have a first order calibration. Also these experiments

gained us the information about the accuracies of the sensors.

By measuring the metal polisbed sphere we've succeeded in extracting some ofthe

misalignments from the set-up, making it possible to correct forthese misalignments, although

we succeeded only for this by using the q>-slope component. To use the 9-component ofthe slope

the fit procedure on the mathematica! model should be improved, ifthis succeeds then absolute


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surface reconstruction can be achieved. Relative surface reconstruction of the sphere succeeded

with errors <0.5 Jlffi over 15 mm integration paths using linear modeling ofthe deflectometer.


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10. Appendices

10.1. Appendix 1: PSD

A PSD is built up from a sandwich of three layers. The centrallayer is a semiconductor

material acting as a photodiode. This layer is stacked between two thin ohmic layers (see figure

8.1 ). In principle, when light ( or a laser beam) interacts with the surface the semiconductor frees

electrons. These electrons will travel through the ohmic layers towards the contacts at the border

of the PSD, where the current, produced by the freed electrons, is measured.

In a two-dimensional PSD, which is used in the detector, the contacts ofthe upper ohmic

layer are placed orthogonal to the contacts ofthe lower layer. This way, it can pinpoint the center

ofweight ofthe light (also called spot) interacting with the PSD in two directions. This is done by

comparing the currents of the upper layer with each other and the same for the lower layer. Thus,

the PSD can sense the position, where the light intersects its surface, in two dimensions (X- and

Y -signal). With analog electranies the currents are converted in two voltages each between -10

and 10 Volts. These voltages can be directly translated into coordinates on the PSD, where (0,0)

Volts is the center ofthe PSD. For example, a 4x4 mm PSD is used, then a displacement ofthe

light spot lmm in the X-direction will yield a signal (X,Y) of (5,0) V.

1 t 1 I

! p ~

~ ('

I

I ! !

N 12 h i I l x

I ltotaa i

1 ) i ! L L i i .. ! ... l ..

ft gure 8.1: Cross-section of a PSD

The accuracy ofthe measured position is, in principle, independent ofthe diameter and

intensity ofthe spot on the PSD. The intensity ofthe spot must be large enough to overpower

ambient and stray light such that small variations in these have little to no effect on the measured

position. Additionally, the spot diameter must not be too large, because changes in the intensity


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distribution, due to inhomogenities in the reflecting surface, may then cause a large apparent

position variation. The upper limit ofthis error is the spot size on the PSD, but in practice the

error will not he larger then 11100 or 1/1000 ofthe spot size. Thus, the smaller the spot on the

PSD the better, but as the spot gets smaller the energy density gets higher and saturation effects

may occur making the PSD slowerin its response.

When the PSD is used under i deal conditions, the bandwidth of the device with its

electranies is approximately 100 kHz. This frequency is a maximum that depends on the light

intensity incident on the PSD. The more light is present the more accurate the PSD will he at high

frequencies.


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10.2. Appendix 11: Misalignments

Following is the denvation for the analytica} formulas that describe the effects ofthe

misalignments on the measured slope signals.

(a) Esin(<p) (b)

<p R'

- ........... b.z h R'

V h

CL~ J I

/ _ _.....

figure 10.1: Illustration ofthe effect ofthe misalignments on the measured slope angle in the (pdirection (a) where instead ofthe expected a'P=O, a different angle (a'P:;r()) is measured. In (b) and a simplified schematic to delermine a'P is shown. Note that the misalignment L1z does not affect the measured slope angle in this direction.

First we start with the effect in the cp-component. In figure 10.1 the effect ofthe

misalignment on the cp component is shown. Due to the eccentricity E, the SUT will show an

effective radius R' at the position ofthe measuring beam:

(1 0.1)

resulting for the measured slope angle:

(10.2)


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- - T - .....,

\

R

(j)

\

I

B

TUle

A

figure 10.2: Illustration ofthe effect ofthe misalignments on the measured slope angle in the Bdirection where instead of the expected ae=O, a different angle ( ae:;t()) is measured

figure 10.3: Simplijled schematic to de termine aefromfigure 10.2.

For thee component the effect ofthe misalignments are shown in tigure 10.2. In tigure

10.3a simplitied schematic from tigure 10.2 is shown in order to show the mathematica!

derivation. Here the distance AB=R+~, BC=R, CD=e(q>-q>0)=C+Ecos(q>-q>0 ) and AD=~. The

different angles are called: LBAD=e, LBAC=n and LCAD=I3.

Now we can calculate r( q>,8) as follows. At tirst we calculate the angle 13( q>) as a function

of the distances CD=e( q>-q>0) and AD=L\z:

ft(cp) = arctan((cp- CfJo)) L\z (0.3)

From here it follows that when 8>13: fl=8-l3 and when 8<13: fl=l3-8, because n has to stay

positive. The next step is to calculate the distance AC=a by using Phytagoras:

Now the cosine-rule can be applied to the triangle ABC, from which follows:

R '2 = r 2 + a2- 2ra cos(Q) (0.5)

where:

From this we can extract r( q>,8), for which follows:

r=acos(Q)±~a2 (cos2 (0)-l)+R'2 ,

again implementing the eosine rule results in:


(10.6)

(0.7)

62

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(10.8)

(10.9)

concluding:

(10.10)

where:

R' = ~R2 - { h + esin(tp- tp0 )}2

r = \ J{ C +& cos(q.>- q.>0 )} 2 + (M )')cos { B- arctan [ C +& c~ q.>- I" u)]}

± [ { C + &cos(q.>- q.>0 J} 2 + (M)' J{ cos' (B- arctan[ C + EC~q.>- q.>,)} -I}+ R '1

(10.11)


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10.3. Appendix 111: Transformation of spec 2 into slope

domain

are length (2a)

ft gure 10.4 Maximum slope change ( Lltp) that occurs when a height defect is superimposed on the surface along a certain are length (2a).

To find Llq> we have to calculate the following formulas (according to figure 10.4):

D.cp =a- fJ = arcsin(;)- (;r- 2y)

r = arctan (-a ) c+d


(10.12)

(10.13)

(10.14)

64

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10.4. Appendix IV: laser characteristics

For the laser used in the sensors the following characteristics apply:

Mitsubishi Laser Diode: ML101J8 (Tc=25°C) Symbol Parameter Test Conditions Min. Typ. Max. Unit I tb Threshold current cw - 57 - inA IQII Operating Current CW,P0=40mW - 117 - inA Voo Operating Voltage CW,P0=40mW - 2.5 3.0 V

1'] Slope efficiency CW,P0=40mW - 0.67 - mW/inA

À" Peak wavelength CW,P0=40mW 655 660 666 nm ex Beam divergence angle (x) CW,P0 =40mW - 8.5 - deg

9_y_ Beam divergence angle (y) CW,P0=40mW - 22 - deg


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10.5. Appendix V: Gaussian opties

do B ZR

do do= do d = 4xA, d, ;tx~xz, 0 1rxB

B B= 4xA- B=B 0=14xÄ

1rxd0 1CXZR

ZR 1CXd 2 4xA ZR =zR z = 0 z =--R 4xA R 1CX gz

dz d =d I+( 4dxz )' d = 4xÄ I+( ncxO' xz )' d,=~ 4x~xz,++(:.J] z 0~ d 2 z 1rxB ~ 4xA, 1CX 0

ZR

e

z

Gaussian Beam parameters.


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z2 zl'

e 9'

x f f x'

Transformation of a Gaussian beam by a thin lens.

Formulas for the transformation of a Gaussian beam by a thin lens.

B'=~ a


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ZR

D e

z

f

dz ( J d ~ 4.< I+( mJ'z J J,~~ 4~R[J+(:.)'] 4Àz dz =do~ 1 + ;rdg z ;r{} ~ 4À

dz als z 0 zR z d =Bz J, ~ ..:__ ~4ÀZR d =d- z

z 0 D ZR

B=- ZR 7r

f

Another useful formula for dz


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10.6. Appendix IV: Optica! system efficiency

For estimating the efficiency ofthe system described insection 3.3, we calculate the

irradiance [W/m2] distribution in the beam. Ifwe would approximate this with a gaussian beam,

the irradiance distribution is:

(10.15)

where r0=do/2 is the distance from the center ofthe beam to the place where the intensity is 10/e2

and do is the so-called waist ofthe beam (figure 3.6).

Since the beam coming from the laser is elliptical we need to adjust for this. When

looking at the divergence angles of the laser beam, i.e. 8.5 ° and 22 °, the ratio R=22/8.5=2.5,

resulting for the irradiance distribution in the x and y-direction in:

I(x) ~ ! 0 exp( -2:,:) (10.16)

(10.17)

In table 10.1, the results ofthe calculations are shown as a function ofthe normalized

distance x/r0• and y/ro.

tab ie 10.1: Energy dis tribution (normalized on Jo)

Position (xlrO)

Position _{y/rOl 0.00 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.00 1.00 1.00 1.00 0.99 0.99 0.98 0.97 0.96 0.00

0.02 0.99 0.99 0.99 0.99 0.98 0.98 0.97 0.96 0.00

0.04 0.98 0.98 0.98 0.97 0.97 0.96 0.95 0.94 0.00

0.06 0.95 0.95 0.95 0.95 0.94 0.94 0.93 0.92 0.00

0.08 0.92 0.92 0.92 0.91 0.91 0.90 0.89 0.00 0.00

0.10 0.88 0.88 0.88 0.87 0.87 0.86 0.00 0.00 0.00

0.12 0.83 0.83 0.83 0.82 0.82 0.00 0.00 0.00 0.00

0.14 0.78 0.77 0.77 0.77 0.00 0.00 0.00 0.00 0.00

0.16 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.18 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Meao Irradiance [Wm"2

] 7.33 7.33 7.31 7.28 6.47 5.61 4.71 3.78 0.00 Subtotal ener-gy [Wl 0.0029 0.0029 0.0029 0.0029 0.0026 0.0022 0.0019 0.0015 0.0029 Total [W] 0.02

As can beseen in table 10.1 the light is cut off after x/r0 and y/r0 =0.14, because ofthe

diaphragm (Dl). The exact value ofthis cut offposition is 0.155. This results in a total energy


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coming out ofthis system of0.02 W. To calculate the coupling efficiency ofthis system, weneed

to know how much light goes through when there is no diaphragm present. This is carried out the

same way as in table 10.1, but this results in a very large table and is therefore not shown here.

The calculated total amount ofintensity in the casewithno diaphragm is 0.16 W. So we expect a

coupling efficiency of0.02/0.16x100%=12.5%. Thus when using the 40 mW laser, we expect ~s

mW corning out ofthe system in figure 3.3. The actual measurement ofthe light leaving the

system resulted in 3.5 mW. The fact that it is lower than predicted is not surprising since we

didn't include the transmission and reflection losses ofthe lenses in the system. Also the first lens

(Fl) is notlarge enough (in diameter) to 'capture' all the light corning from the laser, thus notall

the light from the laser is coupled into the system, resulting in a lower output than expected.


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11. References

[1] Asphericai surfaces: Deflectometry as a new tooi, ir. D. van Kaathoven, ISBN 90-

444-0189-0, 2002

[2] Deflectometry on Asphericai Surfaces: Improving the Bread.Board design. NAJ van

der Beek, AQT 02-04, 2002

[3] Optimization Tooibox User's Guide, The Mathworks, Inc, Isqcurvefit: 4-104 .. 4-113,

2000


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