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Wafer-based aberration metrology for lithographic systemsusing overlay measurements on targets imaged from phase-shift gratingsCitation for published version (APA):Haver, van, S., Coene, W. M. J. M., D'havé, K., Geypen, N., Adrichem, van, P., Winter, de, L., Janssen, A. J. E.M., & Cheng, S. (2014). Wafer-based aberration metrology for lithographic systems using overlay measurementson targets imaged from phase-shift gratings. Applied Optics, 53(12), 2562-2582.https://doi.org/10.1364/AO.53.002562

DOI:10.1364/AO.53.002562

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https://doi.org/10.1364/AO.53.002562

https://doi.org/10.1364/AO.53.002562

https://research.tue.nl/en/publications/f6845017-a0e5-457a-8f80-b5456ab2e47d

Wafer-based aberration metrology for lithographicsystems using overlay measurements on targets

imaged from phase-shift gratings

Sven van Haver,1,2,* Wim M. J. Coene,1 Koen D’havé,3 Niels Geypen,1,4

Paul van Adrichem,1 Laurens de Winter,1 Augustus J. E. M. Janssen,5

and Shaunee Cheng3

1ASML, Flight Forum 1900, 5657 EZ Eindhoven, The Netherlands2S[&]T Experts Pool (STEP), Olof Palmestraat 14, 2616 LR Delft, The Netherlands

3IMEC, Kapeldreef 75, B-3001 Leuven, Belgium4Sioux, LIME BV, Esp 405, 5633 AJ Eindhoven, The Netherlands

5Department of Mathematics and Computer Science, Eindhoven University of Technology,P.O. Box 513, 5600 MB Eindhoven, The Netherlands

*Corresponding author: [email protected]

Received 10 January 2014; revised 28 February 2014; accepted 1 March 2014;posted 10 March 2014 (Doc. ID 204555); published 15 April 2014

In this paper, a new methodology is presented to derive the aberration state of a lithographic projectionsystem from wafer metrology data. For this purpose, new types of phase-shift gratings (PSGs) are intro-duced, with special features that give rise to a simple linear relation between the PSG image displace-ment and the phase aberration function of the imaging system. By using the PSGs as the top grating in adiffraction-based overlay stack, their displacement can be measured as an overlay error using a standardwafer metrology tool. In this way, the overlay error can be used as a measurand based on which the phaseaberration function in the exit pupil of the lithographic system can be reconstructed. In practice, theoverlay error is measured for a set of different PSG targets, after which this information serves as inputto a least-squares optimization problem that, upon solving, provides estimates for the Zernike coefficientsdescribing the aberration state of the lithographic system. In addition to a detailed method description,this paper also deals with the additional complications that arise when the method is implemented ex-perimentally and this leads to a number of model refinements and a required calibration step. Finally, theoverall performance of the method is assessed through a number of experiments in which the aberrationstate of the lithographic system is intentionally detuned and subsequently estimated by the newmethod.These experiments show a remarkably good agreement, with an error smaller than 5 mλ, among therequested aberrations, the aberrations measured by the on-tool aberration sensor, and the results ofthe new wafer-based method. © 2014 Optical Society of AmericaOCIS codes: (120.0120) Instrumentation, measurement, and metrology; (220.3740) Lithography;

(100.5070) Phase retrieval; (220.1010) Aberrations (global); (050.0050) Diffraction and gratings.http://dx.doi.org/10.1364/AO.53.002562

1. Introduction

In recent years, the specifications and tolerances foraberration control in the lithographic industry havereached a level at which the contributions from

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http://dx.doi.org/10.1364/AO.53.002562

dynamic effects also form a significant fraction of theoverall aberration budget of a lithographic system. Inthis context, an aberration is considered dynamicwhenitscontributionmaychangeduringthecompleteexposure of a wafer. This includes, for example, lensheating, vibrational modes of lens elements due tothescanningmotionof thereticle stage,orevenrefrac-tive index variations of the immersion fluid caused byflow or bubbles. Although, modern day lithographicsystems are all equipped with on-tool sensors to mea-sure their aberrations, these sensors commonly oper-ate in between wafer exposures and consequently areunable to pick up the dynamics of the mentionedeffects. In the present work we aim at developing analternativewafer-basedaberrationmetrologymethodthat can pick-up and study these effects.

In the semiconductor industry, lithography is ap-plied to transfer a desired pattern from a reticle ontoa light-sensitive layer on a wafer. If aberrations arepresent, these give rise to deformations of the trans-ferred pattern, like lateral feature displacementsand blurring effects, which in turn may lead tomalfunctioning electronic circuits or chips. As aconsequence, the tolerances for these effects areextremely tight. For example, current and futurelithographic nodes have an overall feature displace-ment budget that is of the order of a few nanometers,and this includes, in addition to aberrations, contri-butions from other sources, such as wafer stagepositioning errors and reticle defects. Effectively thismeans that, for state of the art lithographic proc-esses, aberrations should be controlled down to thesubnanometer regime and this implies that, for alithographic system operating at a wavelength of193 nm, a phase measurement accuracy of the orderof a few milliwaves is required. In the current workwe therefore aim at a desired aberration measure-ment accuracy that is better than 2.5 mλ.

Since aberration control in the semiconductor in-dustry is mainly driven by the tight error tolerancesfor the features being printed, it is also true thataberrations are considered relevant only if theysignificantly alter the pattern being printed. Lookingat the problem from this perspective, it seems natu-ral to monitor the aberration state of a lithographicsystem based on information extracted from thewafer. However, quantitative wafer-based aberrationanalysis, down to the required accuracy, proves tobe very challenging. Not only are the aberration-induced changes very small, they are also stronglycorrelated with other feature shape changing effects,such as focusing and wafer processing. Because ofthese and other complications, wafer-based aberra-tion metrology methods have never become anindustry standard that is widely used. Nevertheless,we will show in this paper that the above-mentionedissues can be overcome and that a wafer-basedmethod can be devised that is an excellent candidatefor studying dynamic aberration effects.

The wafer-based aberrationmetrology method pre-sented here is inspired by a paper by Nomura

published in 2001 [1]. In this paper, Nomura intro-duced a special type of grating, based on which amethod could be devised to distinguish betweenodd and even aberrations simply by measuring theaberration-induced image shift for a few of these gra-tings. The targets used by Nomura are of the phase-shift grating (PSG) type and basically consist ofbinary line-space gratings for which part of the lighttransmitting area is phase shifted by means of anetch into the grating substrate of the mask. The cur-rent work builds upon Nomura’s pioneering ideasand further exploits the aberration informationcontained in the image position of a PSG.

By careful design of a PSG, and satisfying a num-ber of constraints, a simple linear relation betweenthe PSG image displacement and the phase aberra-tion function of the imaging system can be estab-lished. This will be explained in detail in Section 2and basically provides the possibility to obtain phaseinformation, for a single point in the exit pupil of theimaging optics, by measuring the PSG image dis-placement. However, in order to determine the aber-rations of an optical system, the phase distribution inthe entire pupil is required. In the proposed method,we therefore use many 1D- and 2D-PSGs, having dif-ferent grating pitches and rotation in the objectplane, to effectively probe the phase distribution atdifferent locations in the pupil. In this way, enoughphase information can be collected to reconstructthe aberration function by means of solving aleast-squares optimization problem, and the onlytask that remains is the accurate measurement ofthe lateral image displacements of all PSGs to serveas input.

Measuring relative displacements is a commontask in the semiconductor industry. An electronic cir-cuit is built up frommany different layers that shouldbe stacked on top of each other very accurately to as-sure a correctly functioning chip. Layer placement ismonitored by dedicated wafer overlay metrology sys-tems, which basically measure the relative displace-ment (known in the industry as the overlay error)between two functional layers on the wafer by com-paring dedicated targets included in both layers. Inthe current work we exploit this existing overlaymeasurement technology by replacing the target inone layer by a PSG-based target to directly measurethe aberration-induced PSG image shift relative tothe target in the other (reference) layer.

By the end of Section 2 we will have introduced allbasic concepts and measurement principles requiredto devise a PSG overlay (PSG-OVL) based aberrationmetrology method, and we will give a proof-of-principle in simulation to show its feasibility. Fromthere on, the remainder of the paper is organizedas follows. Section 3 deals with the additional com-plications that arise when the proposed method isimplemented experimentally. There it will be shownthat, although some of the distinguished experimen-tal issues have the potential to significantly degradethe performance of the method, they can all be

20 April 2014 / Vol. 53, No. 12 / APPLIED OPTICS 2563

accounted for, or compensated for, through a numberof model refinements and a calibration operation. InSection 4 the implementation of the method is dis-cussed and experimental results are given for a num-ber of aberration reconstruction examples. This willalso illustrate the impact of the various model refine-ments and the calibration operation introduced inSection 3. Section 4 is concluded with an experimentin which the relative aberrations measured by thenew wafer-based method and an on-tool aberrationsensor (based on shearing interferometry [2]) arecompared, showing both methods to agree to a levelbetter than 5 mλ. We will conclude with the overviewof our work in Section 5, where we shall summarizeand comment on the results obtained and where weshall give recommendations for further developmentof the method.

2. PSG-OVL Aberration Metrology

In this section, the basic concepts and measurementprinciples exploited in this paper are discussed.Based on a simplified image formation model andnew types of PSGs, it is shown that a straightforwardrelation can be realized between the displacement ofa PSG image and the aberrations of the imaging sys-tem. In addition, it is explained how standard wafermetrology tools can be exploited to measure thisaberration-induced displacement for a tailored setof different PSGs, in this way collecting enough infor-mation to allow estimation of the aberration functionof the optical system.

A. Aerial Image and Two-Beam Interference

The simplified image formation model applied in thissection is based on the following assumptions. Forthe illumination conditions in the lithographic sys-tem, the effective illumination is assumed to consistof a circular centered monopole with a radius σ thatis very small compared to the numerical aperture(NA) of the lithographic projection lens and thus ef-fectively generates a plane wave toward the reticle(Köhler illumination configuration). Upon transmis-sion of the incident plane wave through a periodicstructure on the reticle, a discrete number w ofdiffracted beams are generated that lie within theentrance pupil NA of the projection lens (see alsoFig. 1, where the special case of a PSG as object isillustrated). As a result, the aerial image of the dif-fraction grating can be considered as the superposi-tion of a number W of two-beam interferences, withW � �w2�. Each of these two-beam interferences givesrise to a cosine fringe in the aerial image, character-ized by two parameters that are its phase offset andits amplitude. The two beams themselves are charac-terized by their complex-valued diffraction ampli-tudes given by C1 exp iφ1 and C2 exp iφ2, and bytheir reciprocal space coordinates denoted by�h1; g1� and �h2; g2�, respectively. The interferenceof these two beams leads to a spatial frequencycomponent of the aerial image, at the coordinates�h � h1 − h2; g � g1 − g2�, given by

~Ih;g � C21 � C2

2 � 2C1C2 cos�2π�h1 − h2�x� 2π�g1 − g2�y� φ1 − φ2 � ΔΦ1;2�; (1)

where we will assume for now that φ1 � φ2 (the φ1 ≠φ2 case is discussed at the end of Subsection 3.D). Inthis case, the overall phase offset of the cosine fringe,denoted by ΔΦ1;2, is given by the difference of thewave-aberration function taken at the two spatialfrequencies or diffraction angles in the exit pupilof the lens; the amplitude of the cosine fringe is de-termined by the diffraction amplitudes of the gratingat mask level. The pattern shift of the aerial image isa complicated function of the phase shift and ampli-tudes for all the two-beam interferences that giverise to the aerial image. A directly interpretable

fj :j :

10

03

i2

1+i1

0th +1st-1st-2nd

-3rd

+2nd+3rd

Normal incidenceplanewave

Entrancepupil

PSG

ρ ρ≡1

Diffractionorders

P

Exitpupil

Image plane

Projection optics

(h2,g2) (h1,g1)

Fig. 1. PSG-based image formation. A schematic representationof image formation for a 1D PSG, having a unit cell with N � 4different taps and period P, is shown. The PSG at the reticle levelis illuminated by a normal incidence plane wave. Each tap in thePSG unit cell has an effective scattering factor, f j, withj � 0; � � � ;N − 1. The scattering factors shown correspond to thetap design values given in Eq. (11), where a complex valued f j in-dicates a relative phase shift for the light passing through thatparticular area of the unit cell. Interaction between the incidentlight and the PSG will generate a number of diffraction ordersin the entrance pupil of the projection optics. Due to the complexscattering factors, destructive interference can take place and anumber of diffraction orders are forbidden (shown in gray). We ba-sically end up with only two non-zero orders within the NA, thezeroth and −1st diffraction orders, and only these will propagatethrough the projection optics to contribute to the image. Note thatthe reciprocal space coordinates, �h1;2; g1;2�, as used in Eq. (1), aregiven by the coordinates of the two allowed diffraction orders in thepupil and that ρ is the distance between the first diffraction orderand the pupil center in normalized pupil coordinates.


and unambiguous relation between the pattern shiftand the aberrations of the projection lens can beachieved when the aerial image is generated by onlya single two-beam interference (see Figs. 1 and 2). Inthis case the observed pattern shift of the aerial im-age, ΔXAI, is simply given by

ΔXAI �ΔΦ1;2P

2π; (2)

with P the period or pitch of the generated fringepattern.

Given that, for the aimed aberrationmetrology, theentrance pupil consists of a single monopole centeredat zero, a two-beam interference can only be realizedby a diffraction grating at mask level that has atleast one diffraction order that is “forbidden,” i.e.has zero amplitude.

B. Gratings with Forbidden Diffraction Orders

The analysis of forbidden orders of a 1D diffractiongrating with N equidistant subdivisions or taps inits unit cell (see Fig. 1), and with for each jth tapa respective scattering factor denoted by f j, withj � 0; 1;…; N − 1, is typically carried out in termsof the unit cell’s structure factor, which is, for thekth diffraction order, given by, (e.g., Chapter 2,Eq. (46) in [3], with i2 � −1)

FN�k� �XN−1

j�0

f j exp�2πi

jkN

�: (3)

First, it should be noted that the tap-based scatter-ing factors f j have to be complex valued in order to

realize a forbidden order only at the �k diffractionorder position, while the −k diffraction order hasnon-zero amplitude. Indeed, for real-valued scatter-ing factors f j, the symmetry relation jFN�−k�j �jFN��k�j is always satisfied (also known as Friedel’slaw, a well-known property of the Fourier transformof a real-valued function). Next, the question of howmany taps N are to be distinguished within the unitcell should be addressed. For N � 2, only the trivialsolution for the set f j is obtained, equal to f1; 1g,which actually implies a pitch halving. For N � 3,only a real-valued solution for the set f j is obtainedgiven by f1; 1; 1g, which does not qualify. For N � 4,nontrivial complex-valued solutions are obtainedfor the set f j. Note that without loss of generality,f �j�0� can be set equal to 1. Possible sequences of dif-fraction orders in the reciprocal space unit cell k �0;…; N are given in Table 1. Cases 1 and 2 in Table 1will be shown to be the (independent) relevant cases.

Case 1 requires that jF�N�4��1�j � 0, which leads tothe condition

1� if 1 − f 2 − if 3 � 0: (4)

This single equation on ff 1; f 2; f 3g leaves 2 deg offreedom. By making the choice f 2 � 0 � f 3, the rela-tion reduces to 1� if 1 � 0, which yields the set f jgiven by f1; i; 0; 0g, which is identified as Nomura’sPSG [1].

Case 2 requires that both jF�N�4��1�j � 0 andjF�N�4��2�j � 0, which leads to the simultaneousconditions

1� if 1 − f 2 − if 3 � 0; (5)

1 − f 1 � f 2 − f 3 � 0: (6)

The latter equations still leave 1 deg of freedom onff 1; f 2; f 3g. By making the choice f 3 � 0, a non-trivialsolution for the set f j is found to be f1; 1� i; i; 0g.Clearly, by having two consecutive diffraction orders(k � 1 and k � 2) that are forbidden, a larger rangeof pitches can be accommodated before a higherdiffraction order enters the NA of the projection lens.In that sense, case 3 would hypothetically be even

beam1

beam2

x

z

(z=0)

Fig. 2. Two-beam interference image formation. Beams 1 and 2have equal wavelengths with dashed and solid lines representingtops and valleys of the plane wave, respectively. Where solid (ordashed) lines intersect, both beams are in phase and constructiveinterference takes place to form the image (interference fringes in-dicated by the thick black lines). A shift of the image plane in the zdirection (defocus) and/or a phase change of either beam will onlyshift the complete interference fringe in the image plane (z � 0) inthe lateral (x) direction.

Table 1. Seven Possible Sequences of Diffraction Orders forN � 4, with “O” and “X” Indicating Nonforbidden and ForbiddenDiffraction Orders, Respectively, and the Top Row Giving the

Indices k of the Diffraction Order

k 0a 1 2 3 4

1 O X O O O2 O X X O O3 O X X X O4 O X O X O5 O O O X O6 O O X X O7 O O X O O

aThe 0th order relates to the centered monopoleillumination.


more preferable, but its simultaneous conditions,given by

1� if 1 − f 2 − if 3 � 0; (7)

1 − f 1 � f 2 − f 3 � 0; (8)

1 − if 1 − f 2 � if 3 � 0; (9)

just allow for the trivial solution of the set f j given byf1; 1; 1; 1g, which corresponds to an effective pitchthat is a quarter of the original one. Similarly, case4 corresponds to a pitch halving. Cases 5 and 6 arefurther equivalent to cases 1 and 2, respectively,while case 7 obviously cannot lead to a situation withjust two non-zero diffraction orders within the NA ofthe exit pupil. Concluding, for N � 4, there are twosolutions for the set f j that yield a diffraction patternthat can be used to generate a two-beam interferenceprocess within the NA of the optical system, given by

f1; i; 0; 0g OXOOO; (10)

f1; 1� i; i; 0g OXXOO: (11)

Note that a complex scattering factor implies thatlight transmitted by the corresponding tap is phaseshifted with respect to other taps. This is why agrating involving complex-valued taps is commonlyreferred to as a PSG and we will use this nomencla-ture to address this type of grating throughout theremainder of this paper.

C. Sampling the Phase Aberration Function

Using the grating designs given in Eqs. (10) and (11),it is possible to generate a diffraction pattern havingonly two non-zero (allowed) diffraction orders withinthe NA of the optical system, while all other diffrac-tion orders are either zero (forbidden) or lie com-pletely outside the NA. In this case, the image ofthe grating will be generated from a single two-beaminterference and the simple relation given in Eq. (2)applies. Consequently, the phase difference of theaberration function evaluated at the position of thetwo allowed diffraction orders can be determinedby measuring the displacement of the aerial imageat the wafer level.

As the aberration function is probed at the locationof the allowed diffraction orders, we can control theprobing locations by varying the grating parameters.Assuming normally incident plane wave illumina-tion, the 0th diffraction order of the grating will begenerated at the center of the exit pupil, while thehigher orders can be generated anywhere in the pu-pil by changing the pitch and rotation of the gratingaccordingly (see Fig. 3). Consequently, the gratingsdefined in Eqs. (10) and (11) can be used to samplethe phase difference of the aberration function be-tween the center position and another position

elsewhere in the pupil, by controlling the 1st diffrac-tion order position via the pitch and orientation ofthe grating.

However, the grating designs given in Eqs. (10) and(11) cannot be used to sample the entire pupil. This isbecause Eq. (2) is valid only in the case of pure two-beam imaging. This means that, for larger pitches,when the diffraction pattern is compressed relativeto the size of the pupil, the condition should be sat-isfied that no non-zero higher diffraction orders enterthe pupil. As a result, the following constraintapplies on the maximum allowed grating pitch:

P ≤kminλ

NA; (12)

where kmin specifies the lowest diffraction orders thatshould be kept outside the NA to satisfy the two-beam requirement. Note that kmin is different forthe designs given in Eqs. (10) and (11); the first(Nomura) design has one allowed 2nd diffraction or-der implying kmin � 2, while the second design hasboth 2nd diffraction orders forbidden, thus allowingkmin � 3. As a result of the constraint given inEq. (12), the aberration function cannot be sampledclose to the center of the pupil (see also Fig. 5), withthe excluded area being larger for the design inEq. (10) than that in Eq. (11).

D. Aberration Information from Overlay Measurements

In the previous subsections it was explained that theimage displacement of a PSG directly providesthe phase difference between two discrete points ofthe aberration function. However, since the aberra-tions we intend to measure are expected to be verysmall, typically of the order of 5 mλ for well-corrected(lithographic) systems, it can prove quite challengingto measure the aberration-induced image displace-ment with a high enough accuracy to pick up theseeffects. Fortunately, the measurement of image dis-placements is a common task in the lithographicindustry, known as overlay metrology, and we canexploit existing commercially available tools to

xx

xx

1

xxxxθ

xxxx

ρ

Fig. 3. PSG-based diffraction pattern control. Dots and crossesrepresent allowed and forbidden diffraction orders, respectively.Variation of the PSG pitch scales the diffraction pattern andchanges the radial sampling position of the −1st diffraction order.A rotation, θ, of the PSG grating in the x–y plane gives rise to anidentical rotation of its diffraction pattern in pupil space.


measure our grating displacements accuratelyenough for the aimed aberration metrology describedin this paper.

In the lithographic industry several differentoverlay metrology methods, which can be eitherdiffraction-based or image-based, are available todetermine the shift between layers, known as theoverlay error. They all have in common that bothlayers for which the displacement should be com-pared should contain standard grating features fromwhich it is possible to deduce the relative layer place-ment error. In the remainder of this paper we shalluse an overlay measurement methodology known asdiffraction-based overlay (DBO) to measure the PSGimage displacements. We choose DBO as it is themethod that currently provides the highest measure-ment accuracy and reproducibility based on acommercially available system (ASML YieldStarS/T-200). Nonetheless, the basic ideas and principlesenabling the aberration metrology method presentedin this paper can also be used to devise a similaraberration metrology method based on another over-lay measurement technique.

In DBO, the displacement between two layers isassessed by including identical line-space (LS) gra-tings in each layer, which are then effectively printedon top of each other. What results is a stack with twogratings and possibly some other unstructuredlayers in between (see Fig. 4). The relative shift be-tween the two gratings in the stack is determined viaan asymmetry in the intensity of the �1st and −1stdiffraction orders; the stack is illuminated by a spotafter which the backreflected light, containing thefirst diffraction orders coming from the gratings, iscollected on a CCD camera. Combining pupil imagesfrom two such grating stacks, both with a differentyet known displacement between top and bottomgratings, it is then possible to accurately compute

the overlay error between both layers (for DBOdetails see [4,5]).

The above-described standard DBO measurementcan be adapted for our application as follows. We gen-erate the top resist grating in the stacks from a PSG.As a bottom grating we use a standard LS patternetched in silicon having the same pitch as the PSGand which we assume is manufactured and placedperfectly so that it may act as an absolute reference.In this case the overlay error measured between bothgratings in the stacks, directly gives us the PSG im-age shift due to aberrations present during the expo-sure of the PSG layer (potential issues originatingfrom a non-perfect bottom reference grating and/oradditional overlay contributions coming from othersources than aberrations are discussed in Section 3).

A restrictive consequence of using DBO is that itimposes a constraint on the minimum pitch of thePSG gratings that can be used. If the pitch of the gra-tings in the stack is too small, their first diffractionorders will not lie within the NA of the wafer metrol-ogy sensor and, consequently, no overlay error can bedetermined. This leads to the following constraint onthe minimum pitch:

P >λDBO

DNADBO; (13)

where NADBO and λDBO are the NA and radiationwavelength of the DBO tool and D is a parametercontrolling the amount of the first diffraction orderpresent within the NA of the DBO tool. For D ≤ 2,first diffraction order information is available; how-ever, to allow accurate displacement measurementsa typical value of D � 1.8 is required. Consequently,the minimum grating pitch allowed by the DBO over-lay measurement (with a YieldStar S/T-200;NA � 0.95, λ � 425 nm) is P ≥ 249 nm. In pupilspace, this constraint translates to a sampling areain the pupil bounded by the radius ρ ≤ 0.57, thus pre-venting us from sampling close to the edge of thepupil (see Fig. 5).

E. Sampling Close to the Pupil Edge: 2D-PSGs

The combination of the constraint on the minimumPSG pitch, which was given in Eq. (13), and the con-straint on the maximum PSG pitch in Eq. (12) im-plies that the resulting pupil area that can beaddressed (see Fig. 5) is clearly insufficient to collectenough phase information to successfully retrievethe aberration function. Also note that the pupil areathat can be addressed by the design in Eq. (10) iscompletely contained by the area accessible withthe design in Eq. (11), we will therefore exclusivelyuse the latter design in the remainder of this paperand shall refer to it as 1D-PSG. To enlarge the frac-tion of the pupil that can be sampled using PSG tar-gets in combination with DBO, we shall nowintroduce a new type of two-dimensional PSGs(2D-PSGs) on top of the 1D-PSGs introduced in Sub-section 2.B and will show how the additional design

OV < 0

I0 I+1I-1

OV < 0

I0 I+1I-1

OV = 0

I0 I+1I-1

OV = 0

I0 I+1I-1

OV > 0

I0 I+1I-1

OV > 0

I0 I+1I-1

Symmetrical Shift in + directionShift in - direction

Fig. 4. Basic principle of DBO. A focused spot, with an annularpupil distribution, is used to illuminate a stack with two gratings.The backreflected light, containing diffraction orders, is collectedand imaged on a CCD camera (top row images). In the case thatboth gratings are perfectly aligned, the image on the CCD will besymmetrical (center column), while if a displacement is present,the recorded CCD image is asymmetrical (left and right columns).Then using two such grating stacks, with known biases in overlay,the relative displacement between the top and bottom gratinglayers can be determined from the measured asymmetries inthe free first diffraction orders reflected by both stacks.


freedom for these 2D-PSGs enables one to sample theaberration function near the pupil edge.

A 2D-PSG is defined in this context as having atypically oblique unit cell that allows for two (truly)diffracted beams within the NA of the projection lens,on top of the zeroth-order beam. This implies thatthere are three independent two-beam interferencesthat make up the aerial image at the wafer level; twoof these three two-beam interferences comprise thezeroth-order beam, and the third two-beam interfer-ence occurs between the two truly diffracted beams[with non-zero reciprocal space coordinates �h1; k1�and �h2; k2�, respectively]. The parameters of the2D unit cell, i.e., lattice parameters a and b andthe angle ϕ between them, are chosen such thatthe DBO metrology tool is only sensitive to the lattertwo-beam interference. This will be explained inmore detail below.

The structure factor of the 2D unit cell, comprisingN taps per direction and M � N2 objects, is given by

FN�h; k� �XM−1

j�0

f j expf2πi�xjh� yjk�g; (14)

with �h; k� the indices of the diffraction order, and f jand 0 ≤ xj, yj < 1 the scattering factor and the (frac-tional) 2D coordinates of the jth object, respectively.A 2D-PSG must then satisfy the following conditions(indicating a number of so-called forbidden diffrac-tion orders):

FN�−1; 0� � 0;

FN�0; 1� � 0;

FN�−1;−1� � 0;

FN�1; 1� � 0: (15)

These conditions yield a diffraction pattern withforbidden orders as shown in Table 2.

Two structures that satisfy the conditions inEq. (15) are shown in Fig. 6, both having four non-trivial (non-zero) scattering objects in the unit cell.Note that outside of the scattering object, the unitcell has no transmission. It should further be notedthat some of the objects need to have a complex-valued scattering factor to enable the generation offorbidden diffraction orders, following a similar argu-mentation as in Subsection 2.B for the 1D-PSG case.The first type of unit cell has three different phases,0, �90 and −90 deg, while the second type has onlytwo different phases, 0 and −90 deg (parameters forboth types are given in Table 3). The second type of2D-PSG has the benefit that it uses only two phases,thus requiring fewer process steps in reticle manu-facturing, and has better diffraction efficiency dueto its larger scattering objects. In the remainder ofthis paper we shall exclusively use the second typeand we shall simply refer to it as 2D-PSG.

Next, the 2D unit cell can be deformed from asquare into an oblique one, thereby retaining the con-ditions of the forbidden orders. At this stage it provesmore convenient to use an alternative, yet equiva-lent, representation of the 2D-PSG unit cell as givenin the upper left of Fig. 7. The obliquity operation forthe original unit cell in Fig. 6 is equivalent to a ratiochange between the width, A, and height, B, of thealternative design (see second row of Fig. 7). Inaddition to this, we can also apply an obliquityoperation to the new unit cell definition, as shown

Fig. 5. Pupil accessibility of 1D-PSGs in combination with DBOmetrology. The concentric circle at pupil radius 0.57 represents themaximum radial sampling position allowed by the constraint inEq. (13) (for DBO with a YieldStar S/T-200; NA � 0.95,λ � 425 nm). The circles at pupil radii 0.45 and 0.36 representthe minimum radial pupil position allowed by the constraint inEq. (12) for the PSG designs given in Eqs. (10) and (11), respec-tively. Consequently, phase information from the hatched Pupilarea 1 can be obtained using the PSG design in Eq. (11) whilethe design in Eq. (10) can only be used to sample the smaller pupilsubset labeled Pupil area 2.

Table 2. Schematic Representation of the DiffractionPattern Defined by the Conditions in Eq. (15), with “O” and

“X” Indicating Nonforbidden and Forbidden Orders,Respectively

�h; k� −1 0 1

1 X X O0 O O X−1 O O X

Fig. 6. Two 2D-PSG unit cell designs satisfying the conditions inEq. (15). Both designs contain four scattering objects and their rel-ative scattering factors are color coded according to the legend onthe right; i and −i indicate relative phase offsets of �90 and−90 deg, respectively. Note that the correct positioning and rela-tive scattering factor of the objects in the unit cell are essentialin achieving destructive interference for certain diffraction orders;the actual shape of the scattering objects is, in this respect, ofminor importance.


in the bottom left of Fig. 7. The impact of these trans-formations on the gratings diffraction pattern isshown on the right of Fig. 7. Going from the firstto the second row, it can be observed that a ratiochange between A and B compresses the diffractionpattern in one direction, effectively bringing theallowed diffraction orders (−1, 0) and (0, 1) closer to-gether. On the other hand, the obliquity transforma-tion causes the two allowed orders to be generated atdifferent radial distances from the pupil center, andthis will prove to be an essential feature in measur-ing spherical aberration components later on. Fulldetails on the relation between the 2D-PSG unit cellparameters and the resulting position of its alloweddiffraction orders are provided in Appendix B.

Since the diffraction pattern of a 2D-PSG containsthree allowed orders within the pupil, its aerial im-age is comprised of a superposition of three two-beaminterference fringes [with their respective reciprocalspace coordinates �h; k�]:

�0; 0� − �1; 0� � �−1; 0�;�0; 0� − �0;−1� � �0; 1�;�1; 0� − �0;−1� � �1; 1�: (16)

Now the important observation to be made here isthat the first two fringes in Eq. (16), involving thezeroth order, will have a small period, as the distancebetween the orders is relatively large, and will there-fore not satisfy the constraint in Eq. (13). On theother hand, the third fringe in Eq. (16) stems frominterference between orders (0, −1) and (1, 0) and,since these orders are relatively close together, gen-erates a fringe �h � h1 − h2; k � k1 − k2� with a largeenough period to be observed by the DBO tool. As aresult, the complicated 2D resist grating generatedfrom a 2D-PSG will be observed by the DBO toolas a simple 1D pattern as if it was generated exclu-sively from interference between orders (0, −1) and(1, 0). Moreover, it turns out that even the simple re-lation given in Eq. (2) remains valid for the fringegenerated by the orders (0, −1) and (1, 0). Then, byapplying a resist grating imaged from the 2D-PSGas the top grating in an overlay stack, and using a

LS grating etched in silicon matched to the DBOobservable 1D component, the phase difference be-tween two points close to the pupil edge can againbe measured as an overlay value.

Similar as in the 1D case, a multitude of 2D-PSGgratings at different orientations at mask level canbe used so that the aberration function is sampledat different azimuths in the pupil. Moreover, somefreedom in the allowed unit cell parameters can beused to achieve additional measurement diversity(for more details, see Appendix B). Consequently,combining 1D-PSGs and 2D-PSGs, it is possible tocollect differential phase information on the aberra-tion function for the larger part of the pupil via DBOmeasurements (see Fig. 8). This, in principle, opensup the route toward a method capable of estimatingthe aberrational state of a lithographic system basedon a large number of overlay measurements.

Table 3. Parameters Defining the Left and Right Unit CellsShown in Fig. 6, Respectively, with the Four Scattering

Objects Listed for Each Case

j f j xj yj

0 1 18

18

1 �i 78

18

2 −i 18

78

3 1 78

78

j f j xj yj0 −i 1

214

1 −i 34

12

2 1 14

12

3 1 12

34

Fig. 7. 2D-PSG unit cells and their corresponding diffraction pat-terns (dots and crosses represent allowed and forbidden orders, re-spectively). The first row pertains to the nominal 2D-PSG design,which is equivalent to the minimum unit cell given on the right-hand side of Fig 6. In the second row, the unit cell is transformedsuch that A ≠ B, resulting in a compression of the diffraction pat-tern in one direction. In the third row, the obliquity transformationis applied, resulting in the generation of the orders (1, 0) and(0, −1) at different distances from the pupil center. Note thatthe diffraction orders (1, 0) and (0, −1) are the two truly diffractedbeams. The orders that are forbidden because of the 2D-PSG are(−1, 0), (0, 1), �−1;−1�, and (1, 1). Some higher orders, for example�−2;−2� and (2, 2), are not forbidden, but remain outside of the NAof the optical system. Note that the nomenclature used here, andthroughout this paper, to indicate a diffraction order pertains tothe minimum unit cell as defined in the right-hand side of Fig. 6.


F. PSG-OVL Aberration Model

In the previous subsections it was explained that thephase difference, measured as an overlay value, be-tween different points in the pupil can be obtained byvarying the type, unit cell parameters, and orienta-tion of the PSG targets used in the overlay stack. Inthis way, by using both 1D- and 2D-PSGs with variedparameters, it is in principle possible to collectenough information on the aberration function to re-trieve it. However, to estimate the aberration func-tion based on differential phase information weneed to define an appropriate model and aberrationrepresentation and for this purpose we proceed asfollows.

Let Φ�ρ; θ� be the aberration function definedon the pupil in normalized polar coordinates(0 ≤ ρ ≤ 1 and 0 ≤ θ ≤ 2π). The classical Zernike ex-pansion of the aberration function is then given by

Φ�ρ; θ� �Xn;m

αmn Zmn �ρ; θ�; (17)

where αmn denotes a Zernike coefficient for integer n,m such that n − jmj is even and non-negative andZmn �ρ; θ� denotes a classical Zernike circle polynomial

defined as

Zmn �ρ; θ� � Rjmj

n �ρ��cos�jmjθ� for m ≥ 0sin�jmjθ� for m < 0

: (18)

Note that we choose to use the double index Zernikeconvention here as it will prove mathematically moreconvenient to construct an appropriate model. On theother hand, we will use the single index Zernike con-vention in the text and figures to refer to specificaberration terms. A conversion rule, to go from thesingle index Zernikes (FRINGE convention) tothe double index, and vice versa, is provided inAppendix A.

Next, let �ρ1; θ1� and �ρ2; θ2� be the polar pupil co-ordinates of the two nonforbidden diffraction ordersof a given PSG and let ΔΦ1;2 be the phase differenceof the aberration function evaluated at thesepositions:

ΔΦ1;2 � Φ�ρ2; θ2� −Φ�ρ1; θ1�: (19)

Then, using Eqs. (2), (17), and (19), we can write theaberration induced overlay error, OVL, for thecurrent PSG as

OVL � ΔXAI �ΔΦ1;2P

2π

� P2π

Xn;m

αmn �Zmn �ρ2; θ2� − Zm

n �ρ1; θ1��; (20)

with the Zernike coefficients αmn describing the aber-ration state of the imaging system.

If the aberration state of a system is unknown, theexpression in Eq. (20) allows one to devise a linearsystem of equations relating the unknown Zernikecoefficients, αmn , to measured overlay values. Esti-mates for the αmn are then obtained by solving thestandard least-square minimization problem definedby [6]:

αmin � arg minα

‖Eα −OVLmeas‖2; (21)

with α ∈ RG the vector of unknown Zernike coeffi-cients, OVLmeas ∈ RH the vector of measured overlayvalues per PSG target, and E ∈ RH×G a 2D matrixcontaining the model predicted overlay per Zerniketerm per PSG as constructed using Eq. (20). Notethat H > G.

G. Proof-of-Principle in Simulation

In Subsection 2.F it was shown how to relate themeasured PSG overlay values to the aberrations ofthe lithographic system. Based on this model, a lin-ear system of equations can be constructed that,upon solving, provides estimates for the Zernike co-efficients representing the aberration function. Toevaluate the potential of our method we haveexecuted the following numerical experiment.

1. A set of 420 PSG targets (with a set of match-ing 1D LS reference gratings at the bottom of the gra-ting stack) is defined, aimed at sampling throughoutthe pupil while satisfying the constraints given inEqs. (12) and (13) and Appendix B.

2. Using a commercial lithographic simulator[Prolith (64-bit) Version 14.2.0.29], the aerial imageof all PSG targets is simulated under realistic expo-sure conditions, given an arbitrary aberration stateof the lithographic system.

3. The images of the reference gratings are simu-lated in the same way. However, since these gratingsare on a different layer, the aberration conditionsduring their exposure are possibly different, whichis accounted for in the simulation by using another(unknown) aberration state.

4. The aberration-induced overlay error per PSGtarget is then obtained as the shift between the aber-rated PSG and the reference grating aerial image.

5. Finally, the simulated overlay values serve asinput to the least-square minimization problemgiven in Eq. (21), which, upon solving, provides

Fig. 8. Pupil accessibility of 1D- and 2D-PSGs in combinationwith DBO.


estimates for the αmn describing the aberrationstate as defined during the simulation of the PSGimages.

The results of the above-described numerical ex-periment are presented in Fig. 9. In the top figure,the arbitrary set of 64 Zernikes used for this simula-tion is shown. In the bottom panel, the error per es-timated Zernike is displayed. These simulationresults clearly show that the Zernike coefficients de-scribing the aberrational state of a lithographic sys-tem can in principle be estimated based on measuredrelative image displacements. Note that an error isobserved in the estimated coefficients due to the factthat the various constraints do not allow the pupil tobe sampled optimally and because the simulation in-volved a reference grating that was produced underdifferent aberration conditions (random set with thesame order of magnitude as shown in the top ofFig. 9). Nevertheless, the presented simulationshows that the method set out in this paper iscapable of estimating the aberrational state of alithographic system under these conditions, withan accuracy better than 5 mλ.

3. Experimental Challenges and Refinements

In the previous section, the basic concepts and anumerical proof-of-principle were presented for ourwafer-based aberration metrology method. In thepresent section, we discuss the complications thatarise when the proposed method is implemented ex-perimentally, and we will evaluate the impact ofthese potential issues on the overall performanceof the method. Where necessary, refinements of thePSG-OVL model and measurement procedure areproposed in order to minimize their negative impact.

A. Finite Illumination Source

During the discussion of the basic measurementprinciples in Section 2 it was assumed that thePSG targets at the reticle are illuminated coherentlyby a plane wave at normal incidence. However, inreality all lithographic systems have a finite source,commonly combined with a Köhler illumination sys-tem. In such an illumination configuration, everysource point generates a plane wave at a slightly dif-ferent angle toward the reticle and, consequently, thereticle image can be considered as the sum of manycoherent contributions. In addition to this, the sourceis imaged in the pupil planes of the projection optics.This implies that light incident on a PSG, originatingfrom different points on the source, will generate dif-fraction orders at slightly different spatial frequen-cies in the pupil and consequently will accumulatea different relative phase on propagation throughthe projection optics if aberrations are present (seealso Fig. 10). Under these conditions, Eq. (2) is stillvalid for predicting the aberration-induced fringedisplacement for a single coherent contribution,but does not give a good prediction for the overall dis-placement of the total PSG image, which consists ofthe superposition of all differently displaced coherentcontributions. Consequently, the method presentedin Section 2 will also not provide optimal aberrationestimates for realistic lithographic systems due tothe impact of their finite sources.

A straightforward way to reduce the impact of thefinite source is to make it smaller so that it betterapproximates the coherent (single point) illumina-tion case. In modern day lithographic systems, thisis possible via a user-defined effective source inthe entrance pupil; for example a small centereddisk. However, theminimum area of such an effective

Fig. 9. Numerical proof-of-principle. A set of 60 Zernikes (Z5 toZ64) is defined with random values between −50 and �50 mλ(top row). For this Zernike set, the aberration- induced shift ofall 420 PSG targets is computed and the resulting simulated val-ues serve as input to the minimization problem defined in Eq. (21).The bottom row presents the error in the estimated Zernike coef-ficients, which is better than 5 mλ.

Fig. 10. Schematic representation of the projection optics exit pu-pil for the case of a 1D-PSG in combination with an on-axis finitemonopole effective source. The monopole source, the size of whichis defined by the radius σ relative to the NA of the projection sys-tem, is convoluted with the PSG diffraction orders at locations�ρ1; θ1� and �ρ2; θ2�. As the source is assumed spatially incoherent,each point on the resulting diffraction order disks can interfereonly with its corresponding point on the other disk and eachof these point pairs effectively generates a single coherentcontribution.


source is constrained, because making it too small ei-ther increases the exposure time too much or re-quires the energy density to be so high that theoptics can be damaged. Consequently, the source rep-resentation in the pupil cannot be small enough toallow for an accurate coherent treatment of the prob-lem and we need to accommodate for the finiteness ofthe effective source through the PSG-OVL model, forwhich we proceed as follows.

We assume the effective source in the pupil to be asmall disk (monopole), with radius σ, that is spatiallyincoherent and uniform. In addition, it is assumedthat the diffraction efficiency of the scattering objectsat the reticle do not vary with the angle of incidenceof the light. The total PSG image is then obtained asthe incoherent summation of all fringe intensitieswhile integrating over the source, and the effectiveaberration-induced shift simply becomes the averageof the individual fringe displacements. In this case,an expression similar to Eq. (2) can be obtained, re-lating the overall PSG-image shift, in the case of afinite source, to the average phase difference be-tween two areas in the pupil:

ΔXAI �P2π

�Φ̄σ�ρ2; θ2� − Φ̄σ�ρ1; θ1��; (22)

where Φ̄σ denotes the average of the aberration func-tion, taken over a disk with radius σ at the diffractionorder positions �ρ; θ� (see Fig. 10). Next, the phaseaverage over a disk-shaped area in the pupil can alsobe written in terms of Zernikes and we can finallywrite an equivalent expression for Eq. (20) in whichthe finiteness of the source is accounted for:

OVL � ΔXAI �P2π

�Φ̄σ�ρ2; θ2� − Φ̄σ�ρ1; θ1��

� P2π

Xn;m

αmn �Z̄mn;σ�ρ2; θ2� − Z̄m

n;σ�ρ1; θ1��; (23)

with Z̄mn;σ�ρ; θ� the average of a Zernike function taken

over a possibly truncated disk with radius σ at thepupil position �ρ; θ�. Accurate computation of Zernikefunction averages required in Eq. (23) is discussed inAppendix C. It should be noted that the 2D-PSGs in-troduced in Subsection 2.E require some additionalattention. As they are meant to sample the aberra-tion function near the pupil edge, we also have todeal with the case that, for either one or both ofthe relevant interfering diffraction orders, the aver-aging disk is partly outside the NA of the litho-graphic system. Since the effective source isconsidered incoherent, a point on the averaging diskwill contribute only if the corresponding points in theother diffraction order also lie within the NA of thesystem. Source points for which only one diffractionorder lies within the NA cannot interfere and, conse-quently, produce only a spatially uniform yet irrel-evant contribution to the image. In Fig. 11 thecommon area of both allowed diffraction orders is

indicated for a general case that both orders arecut differently by the NA limited pupil. The taskof computing the relative phase for the resultingstrangely shaped common area, which is indicatedin Fig. 11, is also dealt with in Appendix C.

In Fig. 12 the impact of a finite source (on-axis mo-nopole) on the effective phase of a diffraction orderarea is shown. On the left, the effective diffractionorder phase, for Zernike function Z36, is shown asa function of the source radius, σ, for four radial po-sitions of the averaging disk in the pupil. It can beseen that the effective diffraction order phase devi-ates substantially from the coherent case (σ � 0), al-ready for modest illumination σ, clearly showing thenecessity of the finite source correction. The right-hand figure compares the effective diffraction orderphase for σ � 0.0 (coherent case) and σ � 0.122

1 2

Pupil

Common Area

Pupil

Fig. 11. 2D-PSG diffraction orders partly outside the pupil. Apoint on one diffraction order disk can only interfere, and thus con-tribute to the overall image, if its corresponding point on the otherdisk also lies within the pupil. As a result, the effective diffractionorder phase is obtained by integration over the common area ofboth orders, which will be an irregular shaped area when one,or both, diffraction orders lie partly outside the pupil. Systemati-cally computing the average phase for such areas is dealt with inAppendix C.

Fig. 12. Finite source impact. Assume a phase distribution in thepupil, defined by Zernike function Z36. On the left, we plot Z36

averaged over a disk as a function of the averaging disk radius,σ, for four different radial positions in the pupil. On the right,the disk average is plotted as a function of its radial position toillustrate the impact of a finite monopole source (σ � 0.122) com-pared to the coherent case (σ � 0.0).


(realistic lower bound for the σ in a lithographicsystem). Especially for averaging areas near the pu-pil edge (ρ → 1), as encountered for the 2D-PSGs, thedifference between the coherent and finite sourcemodels is non-negligible.

B. Asymmetric Resist Gratings

Another experimental complication comes from thefact that the PSGs are imaged by a two-beam processinto a layer of resist with a finite thickness. As a re-sult of this, the resist grating formed from a 1D-PSGwill be asymmetric (see left-hand side of Fig. 13). Inthe 2D-PSG case, where the image is formed by mu-tual interference among three orders, the resultingresist grating looks even more exotic, with effectivegrating lines built up from tilted pillars (see right-hand side of Fig. 13). These irregular grating shapesintroduce an offset in the asymmetry signal of a DBOmeasurement and, consequently, also give rise to aprocess and PSG dependent, yet constant, offset inthe measured overlay values. When not accountedfor, these offsets introduce very large errors on thereconstructed spherical aberration terms (see alsoFig. 16). Fortunately, these artifacts are independentof the aberrations that we want to determine, thusmaking it possible to remove them via a calibrationstep. For this purpose one can execute the presentedaberration metrology method on a very well-corrected lithographic system and assume that theoverlay values measured on that system for all PSGsare exclusively due to the PSG grating asymmetries.Now by subtracting these measured overlay resid-uals, as a reference measurement, from any sub-sequent PSG-OVL measurement, one effectivelyremoves the Zernike reconstruction error comingfrom the resist grating asymmetry. The price thatis paid for this is that the reconstructed Zernikesare now no longer independent and describe theaberration state of the unknown system relative tothe reference system.

C. Other Overlay Contributions

In this paper, the aberrations are estimated based onmeasured overlay values. Therefore, it can be easilyunderstood that overlay contributions coming fromsources other than aberrations directly propagateinto the Zernike reconstruction error. Possible othersources contributing to the overlay are:

1. Mechanical overlay.The actual displacement between different layers

of the wafer, caused by wafer stage and reticle stagepositioning errors.

2. Reticle registration errors.The misplacement of features on the reticles with

respect to the reticle designs.3. Reticle and wafer deformations.Temperature gradients, (thermo-)mechanical

strain and/or processing can cause the reticle andwafer to deform, introducing position-dependent rel-ative displacements of the features they contain.

In its simplest form, mechanical overlay imposes arelative �x; y� position shift between features in twodifferent layers of the wafer; this shift is identicalthroughout our small-sized target area in the die.In our method, where we measure the grating dis-placement in the direction of the grating vector only,the mechanical overlay contribution is given by theprojection of the vector �x; y� on the grating vector.As a result, mechanical overlay gives a pitch-independent contribution to the PSG grating dis-placement, but does vary with the PSG orientation.It turns out that the mechanical overlay fingerprint,the fingerprint being the combined response of allPSG targets to a given overlay source, is identicalto that of a tilt aberration (linear combination ofZ2 and Z3), and we may conclude that mechanicaloverlay does not contribute to the relevant Zernikes(Zt for t > 3).

The second overlay contribution listed is comingfrom the reticle. If features on the reticle are notplaced exactly at the position defined in the reticledesign, this displacement contributes directly tothe observed overlay for the displaced target. Sincethe overlay is defined as the displacement betweentwo gratings in two different layers imaged fromtwo different reticles, the combined overlay contribu-tion of both reticles is in fact the target specific rel-ative displacement between its bottom and topgratings, also known as the registration error be-tween the two reticles. This registration error, whichis basically a manufacturing defect that remainsstatic during the lifetime of a reticle, is commonlymeasured during manufacturing of a reticle setand could therefore be compensated for in thePSG-OVLmodel. However, this is not even necessarywhen a calibration as described in Subsection 3.B isperformed, as this calibration step also automati-cally compensates for the static overlay contributioninduced by the reticle registration error. Other reticledefects also contribute to the Zernike reconstruction

250nm 250nm

Fig. 13. Cross-section scanning electron microscopy (SEM) im-ages. On the left a cross-section SEM image of a 1D-PSG waferstack is shown and one can observe a significant difference inside-wall angle between the sides of the grating lines. On the right,a cross-section SEM image of a 2D-PSG wafer stack is shown. It isclearly observed that, although the line appears to be built up fromtilted pillars, the 2D-PSG generates effectively a 1D grating with apitch matched to the bottom grating. The fine structure in the gra-ting line is subresolution for the DBO measurement, but can pos-sibly contribute to the asymmetry signal generated by the wholestack.


error of our method, but these are discussed in moredetail in the next subsection.

The final contribution listed above is trickier. Thementioned deformations introduce overlay errorsaccording to the samemechanism as the reticle regis-tration error, but in this case depend on environmen-tal and exposure conditions, implying that they arenot static and are, therefore, not automatically re-moved by the calibration. The overlay contributiongenerated by wafer and reticle deformations is posi-tion dependent and can in principle result in anarbitrary overlay fingerprint for a PSG-target set.However, temperature, strain, and processing defor-mations typically give rise to overlay contributionsthat vary slowly over the wafer. By placing allPSG targets required for our method close together,we canminimize the variation and, therefore, the im-pact of these deformations to an acceptably low level.If this is insufficient, one may resort to complex com-putational models to predict temperature-induceddeformations, but this has not been investigated fur-ther in this paper.

D. Reticle 3D Effects and Manufacturing Errors

A typical lithographic reticle consists of a quartz sub-strate on which a thin layer of chromium is depositedto make it opaque. By selectively removing chro-mium using electron beam lithography, a desiredbinary pattern can be transferred. In addition to this,our PSG designs require that certain transmissiveareas on the reticle are assigned a different phase,and this can be achieved by selectively etching thoseareas into the substrate. Combined, this allows us tomanufacture a reticle with PSG targets according tothe designs described in Section 2. However, sincethe resulting PSG feature sizes on the reticle areof the same order of magnitude as the illuminationwavelength and the grating cannot be considered op-tically thin, the diffraction efficiency of the PSGtargets is not predicted well by the Kirchhoff diffrac-tion model. Instead, a rigorous 3D-diffraction modelshould be applied to correctly account for the light–matter interactions taking place at the reticle. In thefirst two rows of Fig. 14, the simulated diffractionpattern is shown for these two cases, clearly illustrat-ing that the forbidden diffraction orders of the nomi-nal PSG design are no longer zero when the 3Dstructure of the reticle is accounted for. At the bottomrow of Fig. 14, however, it is shown that this issue canbe solved by optimizing the nominal PSG design sothat it compensates for the 3D-diffraction effects toagain achieve forbidden diffraction orders.

In addition to the 3D-diffraction effects discussedabove, reticle manufacturing errors can also contrib-ute to unwanted energy in the forbidden orders.These are, for example, corner rounding, under-etch,and nonvertical etch side walls, and they can allmodify the diffraction efficiency of the features onthe reticle (see also Fig. 15). As these imperfectionsare difficult to predict and to parameterize, it isimpractical to accurately estimate their impact via

computational means. Instead, one may choose tofurther optimize the PSG designs for these effects ex-perimentally. This would be done by making a testreticle containing PSGs for different combinationsof design and processing parameters. Next, theamount of energy in the forbidden orders is assessedby measuring the diffraction pattern, which can, forexample, be done using a Zeiss AIMS system. Thecombination of parameters for which the forbiddenorder strength is minimal is then used to manufac-ture the actual measurement reticle.

In reality, the forbidden diffraction orders willnever be completely zero. The first order effect of asmall to moderate non-zero forbidden order is thatit reduces the sensitivity of our proposed methodfor aberrations having an even azimuthal depend-ence (αmn with m is even), while not influencing thesensitivity of odd aberrations (m is odd). However,since we can, in principle, measure the relativeforbidden-order strengths of all PSGs with an AIMSmicroscope, this effect can also be accounted for inthe model by adjusting the sensitivity matrix [E inEq. (21)] accordingly. Nevertheless, it remains bestpractice to strive for the lowest possible amount ofenergy in the forbidden orders to avoid higher ordereffects and to have the highest possible sensitivity forall aberrations.

Fig. 14. Computed diffraction patterns for the nominal 1D-PSGdefined in Eq. (11). The top image shows the diffraction patterncomputed according to Kirchhoff, with, as expected, only twonon-zero orders within the NA (indicated by the white circle).The middle image shows the diffraction pattern for the same tar-get when correctly accounting for the reticle 3D effects. In thiscase, three non-zero orders are predicted within the NA, destroy-ing the pure two-beam process. In the bottom image, the rigorouslycomputed diffraction pattern is shown for an optimized 1D-PSGdesign, showing that the pure two-beam process is restored.


Even in the case of perfectly forbidden orders,there is one other reticle characteristic that can im-pact the overlay measured in our method and has notyet been discussed. This is the relative phase of theallowed diffraction orders, φ1 and φ2, as occurring inEq. (1). Looking at Eq. (1), it can be understood that ifφ1 − φ2 ≠ 0 this directly adds up to, and cannot bedistinguished from ΔΦ1;2, the aberration-inducedphase contribution and, consequently, contributesdirectly to the error of the Zernikes reconstructedby our method. Furthermore, the diffraction orderphase is not easily measured using existing semicon-ductor equipment, which implies that a model-basedcorrection of this artifact is not feasible. But again,this reticle effect can also be assumed static, assuringthat it is also effectively removed by the calibrationprocedure described in Subsection 3.B.

E. Stochastic Overlay Variations

The final topic we want to discuss in this section isthe uncertainty in the overlay measurement. Twodistinct sources generating stochastic variations inthe measured overlay signal are recognized. The firstis the uncertainty coming from the DBO measure-ment itself and is extremely small, of the order ofa few tenths of an angstrom. This very high measure-ment reproducibility was, in fact, the main reasonwhy DBO was selected as the overlay methodology

of choice for our application. It basically means thatno significant contribution to the method error is ex-pected from the DBO measurement. The second con-tribution is caused by small differences in the resistgratings used in the overlay measurement. Thesedifferences may, for example, originate from the com-plicated and random interaction between exposureradiation and light-sensitive molecules in the resistlayer or any other stochastic process during exposureor processing of the wafers. The uncertainty intro-duced by these mechanisms is much larger than thatof the DBOmeasurement and experiments have alsoshown that their magnitude may vary as a functionof the PSG parameters.

Under these conditions, the ordinary least-squaresminimization problem given in Eq. (21) is no longeroptimal and should be replaced by the weightedleast-squares minimization problem given by

~αmin � arg minα

‖Q�Eα − �OVLmeas −OVLref ��‖2; (24)

where OVLref denotes the calibration measurementon the reference system, which we assume to be in-dependent of OVLmeas, in which case K simply be-comes the sum of the experimentally determinedcovariance matrices

K � Kmeas �Kref ; (25)

and Q denotes the Cholesky decomposition of K−1

according to

K−1 � QQ�: (26)

The optimal generalized least-squares estimator,~αmin, for the Zernike coefficients describing the aber-ration state of the lithographic system, can then beexplicitly written as

~αmin � �E�K−1E�−1E�K−1�OVLmeas −OVLref �; (27)

where we have now correctly accounted for the PSG-target-specific overlay measurement uncertaintyand the additional uncertainty introduced by the cal-ibration measurement.

An additional uncertainty related problem that weare facing for the current measurement reticles isthat the 420 targets included in the measurementmodules do not sample the pupil optimally. As a re-sult, the conditioning of the resulting linear system isnot very good and this gives rise to a relatively largevariance for specific (spherical) Zernikes. In thiscase, the least-squares estimation can be further im-proved by exploiting a Tikhonov-like regularizationapproach, which introduces an additional term inthe cost function:

~αmin�ζ� � arg minα

‖Q�Eα − �OVLmeas −OVLref ��‖2

� ζ2‖D�α − αp�‖2: (28)

Fig. 15. Illustration and impact of reticle manufacturing defects.The top row shows SEM images of typical 1D- and 2D-PSGs on thereticle. The 1D SEM image was taken before the phase etch step,while the 2D SEM image was taken afterward. One can observesignificant corner rounding and additional deformations due tothe phase etch into the substrate. In the bottom row, typical dif-fraction patterns for 1D- and 2D-PSGs are shown as measuredby an AIMS. Due to the writing and etching defects, the intendedforbidden orders (indicated by dashed circles) are non-zero. Notethat the monopole source of the AIMS is relatively large, resultingin overlapping orders and higher diffraction orders showing up atthe pupil edge that are of no concern in the actual experiment inthe lithographic system where a much smaller source is used.


HereD is the regularization operator and αp the priormean, which are both set based on prior knowledge oftypical aberration states of lithographic systems. Theparameter ζ ≥ 0 determines the importance of theregularization term. Obtaining a good recipe for anoptimal choice of ζ is a non-trivial task. For a moredetailed description of the applied regularizationtechniques and the determination of the regulariza-tion parameter, the reader is referred to [7].

4. Experimental Verification of the Method

In this section we present the experimental valida-tion of our method. We shall start by discussingthe hardware used and we will describe the appliedexperimental procedures in detail, aimed at avoidingthe experimental complications discussed in the pre-vious section. Subsequently, aberration retrievalresults are presented for the case that a known aber-ration was intentionally introduced to the projectionlens of a lithographic system and, additionally, thesewafer-based reconstruction results are compared tothe aberrations measured by the on-tool aberrationsensors.

A. Reticle Design

The most important components required by thePSG-OVL aberration metrology method are thetwo matched PSG and REF reticles required to con-struct the PSG-OVL targets. For the reticle set usedin the experiments presented in this paper, the top-level layout is identical. They contain 7 × 7 measure-ment modules, where each module consists of 420PSG-based overlay targets. In turn, each overlay tar-get consists of several grating stacks that togetherallow one to determine a single overlay value (seealso Fig. 4). The grating stacks themselves are builtup from two stacked 40 μm× 40 μm gratings, wherethe top and bottom grating are constructed from thePSG and REF reticle, respectively. Basically, the onlydifference between the two reticle designs is the unitcell used to fill the 40 μm× 40 μm grating areas.[Note that dimensions are given at wafer level(1X) and that dimensions at reticle level are 4 timeslarger.] For the PSG reticle, the unit cell definitionsare given in Subsections 2.B and 2.E for the 1D- and2D-PSG cases, respectively. The gratings used in theREF reticle design are simple LS gratings, whichhave the same effective pitch, P, and grating orien-tation, θ, as the corresponding grating on the PSGreticle.

As mentioned above, the modules on the aberra-tion measurement reticles contain 420 PSG-OVLtargets. This set is composed from 28 differentPSG-OVL targets (three 1D-PSGs and 25 2D-PSGs),all included at 15 different orientations (rotation an-gles in the reticle plane). For the current reticles, wehave included three grating stacks per PSG-OVL tar-get, and this results in a minimum area required fora complete module that is a little bit larger than2 mm2. Note that on our test reticles the module sizeis substantially larger due to the inclusion of visually

observable labels and additional test and calibrationfeatures. Nevertheless, our test reticles already in-clude 7 × 7 modules, thus allowing one to also mea-sure the field dependence of the aberrations in asingle exposure.

B. Experiment Description

Experimental verification of the method is executedas follows. First, a set of reference wafers is created:using the complimentary binary REF reticle, coatedsilicon wafers are exposed and subsequently etched,stripped, and cleaned. The chosen illumination con-dition used to print the reference gratings was a wideconventional setting (0.94σ at 1.35 NA), this in orderto minimize the impact of possible aberrations on thereference features. The etch depth for the referencegratings is 30 nm.

The chosen resist process for all our experiments is105 nm JSR resist AIM5484 on top of 95 nm BrewerBARC ARC29SR; this should result in sufficient pla-narization of the etched reference gratings. All coat-ings and development are done inline on a SokudoDuo track fitted to an ASML NXT:1950Ai scanner.Before starting experiments, we determined thesmallest safe conventional illumination possible onthe scanner (0.122σ at 1.35 NA). We then determinedthe dose at which all required phase gratings wouldprint simultaneously to an acceptable pattern byqualitative SEM inspection. That exposure dosewas then used for all experiments.

Each experiment consists of a set of exposures onthe reference wafers using the above-mentionedresist process and tools. The use of so-called image-tuner subrecipes allowed us to dial in specificaberration offsets per wafer. The validity of each sub-recipe was first verified by measuring the wavefrontusing the scanner’s own integrated interferometerwhen applying the offsets requested in the subrecipe.For the exposures themselves, we used a specificscanner test that would also allow us to measurethe wavefront before and after each exposure withthe scanner’s interferometer while still working in-line with the track to bake and develop each waferimmediately after exposure.

Measurements were then performed on each waferon a YieldStar S200 and raw pupil data was gatheredthat was subsequently processed offline to determinethe overlay error for each PSG target. These mea-sured overlay values, together with uncertainty in-formation gathered from multiple measurements,then serve as input to the minimization problems de-fined in Eqs. (24) and (28), which, upon solving, pro-vide us with estimates of the aberration state of thelithographic scanner at the time of the PSG reticleexposure.

C. Experimental Results

In this subsection we give an overview of the exper-imental work done to validate the method. We beginwith an experiment in which a single non-zeroZernike (Z7 � 3nm) has been dialed in as an


aberration offset of the lithographic system. This in-tentionally detuned system is subsequently used toprint the PSG reticle on top of the reference featuresalready on the wafers. Next, the overlay is measuredfor all targets in a single module and, based on thesemeasured overlay values, we try to recover the mag-nitude of the dialed-in Zernike term. In Fig. 16 theZernike estimates obtained with the PSG-OVLmethod are shown. The bar plot on the top pertainsto retrieval based on the raw measured overlayvalues, while the finite source is correctly accountedfor according to Subsection 3.A. It may be observedthat, although a Z7 with a magnitude between 2 and3 nm is detected, the method erroneously recon-structs large values for several other Zernikes, withmost profound values for the spherical aberrationterms (Z16, Z25, and Z36). These large spherical termsare mainly due to the PSG asymmetry effect dis-cussed in Subsection 3.B. Therefore, by applyingthe calibration proposed in Subsection 3.B, a strongimprovement can be achieved, resulting in the aber-ration retrieval result shown in the middle row ofFig. 16. Note that, due to this calibration step, theZernikes shown in the middle row of Fig. 16, andin all subsequent figures, now represent the aberra-tions of the measured system relative to the refer-ence system on which the calibration was based.Even for the calibrated result, some reconstructedspherical Zernikes remain substantial. A systematicanalysis of this phenomenon has shown that it iscaused by the suboptimal sampling of the aberrationfunction achieved by the current set of measurementreticles. As a result, the conditioning of the linearsystem being solved is not optimal and, in this case,results in relatively large uncertainty, specifically forthe spherical Zernikes. A proven concept to improveestimates from poorly conditioned systems is the useof regularization, and doing so further improves theZernikes estimates obtained with the PSG-OVLmethod, as is shown in the bottom row of Fig. 16.In the remainder of this subsection, all presentedreconstruction results will be obtained with regulari-zation and are based on calibrated overlay data.

The results of a second experiment, in which acocktail of three non-zero Zernikes (Z8 � 2 nm,Z10 � −2 nm, and Z27 � 2 nm) was dialed in, isshown in Fig. 17. Also in this experiment, a verygood correlation is observed between the requestedand retrieved Zernike terms and all residualZernikes are smaller than 1 nm (1 nm ≈ 5 mλ for alithographic system operating at 193 nm). This givesa good indication that the PSG-OVL method can de-termine arbitrary aberration states and does not suf-fer from a strong cross correlation between differentaberration terms.

So far, reconstruction results are shown for onlyodd Zernike terms. As explained in Subsection 3.D,the aberration-induced overlay sensitivity of evenZernikes is reduced when the intended forbiddenorders of a PSG are non-zero. The impact of this phe-nomenon may be observed in the retrieval results

shown in Fig. 18, where a cocktail of even Zernikes(Z5 � 2 nm, Z13 � −2 nm, and Z17 � 2 nm) isdialed in during exposure. In the top row, the

Fig. 16. Zernikes estimated with the PSG-OVL method for thecase that a single non-zero coefficient (Z7 � 3 nm) was dialed induring exposure of the PSG layer onto the wafer. On the top, es-timates for the Zernikes (Z5 − Z36), obtained from the raw overlaymeasurements, are shown. In the middle, the correspondingretrieval result is shown for the case that the measured overlayvalues are first calibrated with respect to an aberration-free refer-ence system. The bottom row shows the calibrated result when ad-ditional regularization is applied in the fitting procedure.

Fig. 17. PSG-OVL reconstructed Zernikes for a requested cock-tail of three non-zero coefficients (Z8 � 2 nm, Z10 � −2 nm, andZ27 � 2 nm).


PSG-OVL-based estimates are shown when this ef-fect is neglected, indeed showing Zernike estimatesthat are too small. A significant improvement is ob-served when this effect is accounted for in the modelusing AIMS-measured forbidden-order information(see bottom row Fig. 18). Note that AIMS measure-ments were available only for a subset of all PSGsand that missing data was supplemented by extrapo-lated values. Therefore, the improvement achievedby this correction is expected to be even more drasticwhen forbidden-order information is available for allPSGs used in the reconstruction.

Finally, we present in Fig. 19 a comparisonbetween the aberration coefficients measured withthe on-tool aberration sensor of the exposure system

and those estimated by the PSG-OVL method. Alsohere, a very good agreement between both measure-ments is observed for the imposed non-zero Zernikecoefficients. A comparison of the residuals of bothmethods shows larger differences between the twomethods, where it should be pointed out that the re-siduals of the PSG-OVL method are slightly largerthan those of the on-tool sensor. Nevertheless, it isremarkable that such a good agreement, down tothe subnanometer regime, can be achieved betweena wafer-based method and a dedicated on-tool sensorthat is based on shearing interferometry.

5. Conclusions and Discussion

In this paper we have presented a new methodologyfor deriving the aberration state of a lithographicprojection system based on wafer metrology data.This method uses the aberration-induced image shiftof specially designed overlay targets as a measurandbased on which the phase aberration function in theexit pupil of the lithographic system can be recon-structed. In Section 2 it was explained how PSG tar-gets can be designed such that a very simple relationemerges between their relative image position andthe difference of the phase aberration functionevaluated at the location of their diffraction orders.Moreover, it was shown that, using a combinationof 1D- and 2D-PSGs, the phase difference betweendifferent points throughout the pupil can be probed,eventually providing enough information to accu-rately estimate the Zernikes representing the phaseaberration function. At the end of Section 2, thismethodology was proven to be feasible in simulationvia a numerical experiment based on a lithographicsimulator and showed an aberration reconstructionaccuracy of better than 5 mλ.

Before continuing with the experimental verifica-tion of the method, first a number of practical com-plications were addressed in Section 3. There it wasshown that, for all complications currently recog-nized, their impact can either be accounted for inthe PSG-OVL model or their negative effect can bestrongly reduced through a calibration step. Thiswas illustrated in Section 4 through a number of ex-periments, with and without the mentioned correc-tions, which showed the most drastic improvementwhen the reference scanner calibration was included.Altogether, using all model refinement and correc-tions discussed in Section 3, the accuracy of thewafer-based aberration estimation method intro-duced in this paper was shown to be remarkablygood. The retrieval error, with respect to thedialed-in aberration and independent measurementwith the on-tool aberration sensors, is below 1 nm,which corresponds to approximately 5 mλ at theexposure wavelength of the lithographic systembeing evaluated.

Nonetheless, we still observe residuals in the re-constructed Zernikes that are between 0.5 and1 nm. Although extremely small, aberrations ofthis order of magnitude are nowadays considered

Fig. 18. Zernikes estimated with the PSG-OVL method for thecase that a cocktail of three even Zernikes (Z5 � 2 nm,Z13 � −2 nm, and Z17 � 2 nm) is dialed in. The top axis showsthe Zernike estimates obtained using both calibration and regu-larization. The results in the bottom panel are obtained usingan additional correction based on the AIMS-measured residual for-bidden-order intensities generated by the PSG reticle.

Fig. 19. Comparison between the aberration coefficients obtainedwith the on-tool aberration sensors of the exposure tool and thePSG-OVL method (identical experimental settings as in Fig. 17).


significant by the lithographic industry. Therefore,the aim of the current research was to achieve anaberration estimation accuracy better than 0.5 nmand, in this respect, the current work still requiresadditional effort to reach the desired accuracy. Fortu-nately, we do see several opportunities to furtherimprove and enhance the presented method andwe shall discuss those here briefly.

For the experiments presented in this paper, wehave used a set of reticles that was designed basedon our initial understanding and ideas regardingwafer-based aberration metrology. Since that time,we have gained much more insight on the physicalprocesses and error sources relevant for our method.Basically, we now know that our reticles are, in manyways, not optimal for the aimed aberrationmetrologyapplication. For example, the current set of 420 PSGtargets does not sample the pupil in an optimal wayand this leads to poor conditioning and significantcross correlation in the linear system being solvedto estimate Zernikes. In fact, this is the main reasonwhy regularization does improve the aberrationreconstruction results obtained with the current re-ticles. Apart from selecting the optimal PSG targets,there is also still room to optimize the targets them-selves. It was discussed in Subsection 3.D that,although the PSG designs are optimized to compen-sate mask 3D effects, there will always remaindifferences between the design and realized features,leading to non-zero forbidden diffraction orders. Anexperimental optimization approach, in which PSGshaving slightly different design parameters areevaluated in terms of their measured forbidden-order strength, could be a way to compensatethese manufacturing defects, which are hard toparameterize.

Parallel to improving the PSG set and the individ-ual designs, also the analysis and corrections pre-sented in this paper can be further improved. Forexample, the forbidden-order correction mentionedin Subsection 3.D was based on AIMSmeasurementson just a subset off all PSGs, with the missing databeing filled in by extrapolated values. Also, deter-mining the target specific variance for the overlayvia a large number of experiments, which then servesas input to the generalized least-squares problem,will contribute to an improved stability of thePSG-OVL method.

Altogether, we are convinced that, with a set of newoptimized measurement reticles, complete reticlemetrology data and accurate knowledge of the sto-chastic overlay behavior of the PSGs, the aimed aber-ration reconstruction accuracy for the methodpresented in this paper is well within reach. Conse-quently, we consider this method an excellent candi-date to study dynamic aberration effects occurringduring exposure of a wafer.

Appendix A: Conversion between Z t and Zmn

In this appendix we provide a conversion rule to gofrom a Zernike function in single-index notation, Zt,

(FRINGE convention) to a Zernike function indouble-index notation, Zm

n , and vice versa.

a. Conversion: Z t → Zmn

For t � 1; 2; � � �, we let

q � ⌊��t − 1

p⌋; p � ⌊

t − q2 − 12

⌋; (A1)

and

n � q� p; m � q − p; (A2)

then a single-index Zernike function, Zt�ρ; θ� equals

Zt�ρ; θ� � Rmn �ρ�f�t − q2 − 2p� cos mθ

� �1 − �t − q2 − 2p�� sin mθg� Zm

n �ρ; θ�: (A3)

b. Conversion: Zmn → Z t

When n, m are positive integers such that n −m iseven and non-negative, then

Rmn �ρ� cos mθ � Z�n�m

2 �2�n−m�1; (A4)

Rmn �ρ� sin mθ � Z�n�m

2 �2�n−m�2; (A5)

with Eq. (A5) only valid for m ≠ 0.

Appendix B: 2D-PSG Unit Cell Definition

In this appendix we discuss the relation between the2D-PSG unit cell definition, its corresponding pupilpositions of its allowed diffraction orders, and theparameters we use to define them.

We use four parameters that completely define the2D-PSG targets used by our method: the pitch, P, ofthe resulting 1D pattern generated from the interfer-ence between the two orders close the pupil edge, thedistance ρ1 between the pupil center and the alloweddiffraction order closest to the pupil edge, the frac-tion ξ being the ratio between ρ3 and ρ0 that are de-fined in Fig. 20, and, finally, the grating orientationϕ. All distances defined in Fig. 20 are then given by

ρ0 � λ

NA P; (B1)

ρ1 � ρ1; (B2)

ρ2 ��1 − ξ�2ρ20 � ρ21 − ξ2ρ20

q; (B3)

ρ3 � ξρ0; (B4)

ρA � �1 − ξ�ρ0 (B5)


ρB ��

pρ21 − ξ2ρ20�; (B6)

where λ and NA are again the wavelength and NA ofthe lithographic system.

Next, we can also express the real-space parame-ters defined in Fig. 21 in terms of the 2D-PSG param-eters and the distances defined in Fig. 20:

ω � arccos�ρ21 � ρ22 − ρ20

2ρ1ρ2

�; (B7)

Ψ � arccos�ρ20 �

�2 sin

�ω2

�ρ2�2− �ρ1 − ρ2�2

4 sin�ω2

�ρ0ρ2

�; (B8)

Amin � λ

2NAρ1

��1

cos�ω2

�2�

�1

sin�ω2

�2s

; (B9)

Bmin � ρ1ρ2

Amin; (B10)

AC � ΞλNA ξρ0

; (B11)

BC � λ

NA��ρ21 − ξ2ρ20

q ; (B12)

where ξ is a rational number between �1∕2� ≤ ξ ≤ 1,and Ξ is the numerator of the smallest possiblerational representation of � ξ

1−ξ�. Note that the anglesω and Ψ, together with the minimum unit cell riblengths, are sufficient to define a 2D-PSG in a reticledesign. However, the super cell defined by AC and BCis also provided because lithographic simulatorscommonly require such a rectangular (Manhatten-type) unit cell to simulate the proposed 2D-PSGs.

For a given 2D-PSG design to be applicable, itshould satisfy requirements similar to those in the1D-PSG case; nonforbidden higher orders should re-main outside the pupil and the effective pitch of thegrating image should be observable by the DBO tool.For a 2D-PSG, this translates into the followingconstraints:

2λNA �1� σ� ≤ P ≤

λDBO

CNADBO; (B13)

��1� σ�2 − �4 − 4ξ�

�λ

NA P

2

s≤ ρ1 ≤ 1; (B14)

1 −

��1� σ�2 − ρ21��NA P�24λ2

≥ ξ ≥12: (B15)

Appendix C: Semianalytic Computation of PhaseDifference Averages

After linearizing the exponential comprising thephase difference, we must evaluate the average ofthe phase difference over a specific subset of thereference pupil. This subset consists of the inter-section region of three disks, namely, the pupil disk,a displaced copy of the pupil disk, and the supportingdisk of the scanner’s illumination monopole. For thesake of mathematical convenience we will use in thisappendix the double-index complex exponential rep-resentation of the Zernike functions, and we assumethat the disk pertaining to the scanner monopole hasradius 1 and center o, while the two pupil disks haveradius r and centers ν1 and ν2, respectively. SeeFig. 22 for the configuration, in which the hatchedregion S is the averaging region.

Fig. 20. Definition of the relevant distances in reciprocal (pupil)space.

Fig. 21. Definition of the real-space variables and relation be-tween the minimum unit cell (dotted box), computational unit cell(dashed box) and the unit cell as defined in Fig. 7.


The mathematical problem at hand thus consistsof solving the integrals

ZZSZm0n0

�ν − ν1;2

r

dν; (C1)

where Zm0n0 denotes the complex Zernike circle polyno-

mial vanishing outside the unit disk, given as

Zm0n0 �ν�≡ Zm0

n0 �ρeiθ� � Rjm0 jn0 �ρ�eim0θ;

0 ≤ ρ ≤ 1; 0 ≤ θ ≤ 2π; (C2)

for integer n0, m0 such that n0− jm0j is even and non-

negative. The evaluation of Eq. (C1) can be doneusing the approach in [8]. Accordingly, focusing onthe case ν1, we have

ZZSZm0n0

�ν − ν1r

dν

�ZZ

ν≤1Zm0n0

�ν − ν1r

Z00�ν�

�Z00

�ν − ν2r

Z00�ν�

�dν

�Xn;m

π

n� 1βmn;1�βmn;2��; (C3)

with βmn;1 and βmn;2 the Zernike coefficients with re-spect to the unit disk of Zm0

n0 �ν−ν1r �Z00�ν� and

Z00�ν−ν2r �Z0

0�ν�, respectively. These β can be cast intothe form of a correlation integral, as in [8], Eq. (8):

βmn;1 � n� 1π

ZZZm0n0

�ν − ν1r

�Zm

n �−ν��dν; (C4)

βmn;2 � n� 1π

ZZZ00

�ν − ν2r

�Zm

n �−ν��dν; (C5)

and these correlation integrals have been computedin [8], Eq. (9) and Theorem 2.1. Thus (choosing c0 � r,c � 1, �τ0; η0� � −ν1, �τ; η� � �o�),

βmn;1 �n�1π

Xn00

Cm0mn0n;n00Zm0

−mn00

�−

ν1r�1

; jν1j≤ r�1;

(C6)

and a similar formula for βmn;2. The summation inEq. (C6) is over all integers n00 such that n00

− �n0 � n�is even and non-negative. The C are given as

Cm0mn0n;n00 �

�r

r� 1

2 �−1�n�n00 � 1�π�n0 � 1��n� 1�

× �Sn00�1n0n − Sn00�1

n0�2;n − Sn00�1n0;n�2 � Sn00�1

n0�2;n�2�;(C7)

with

Sk�1ij �

12 �k� i� j�

�!12 �k − i − j�

�!

12 �k� i − j�

�!12 �k − i� j�

�!

×ri

�r� 1�i�j

�P�i;j�

12�k−i−j�

�1 − r1� r

2; (C8)

for integers i, j, k ≥ 0 such that k − i − j is non-negative and even and Sk�1

i;j � 0 otherwise. TheP�α;β�l �x� are Jacobi polynomials; see [9], Chapter 22.The series in Eqs. (C3) and (C6) are rather slowly

convergent, and so the computations must be doneefficiently. The Sk�1

ij can be expressed in terms ofthe generalized Zernike functions of [10] according to

ri

�r� 1�i�j

�P�i;j�

12�k−i−j�

�1 − r1� r

2� �1 − ρ2�− i

2Rj;ik−i�ρ�;

(C9)

with ρ � �1� r�−1∕2 ∈ �0; 1�. Finally, according to[10], Theorem 5.2 (correcting two minor typos), thegeneralized Zernike functions can be computed ina DFT format as follows. Let n and m be integerssuch that n − jmj is even and non-negative, and letq � �1∕2��n� jmj�. Furthermore, let α > −1 anddenote by Cα�1

n the Gegenbauer polynomial of [9],Chapter 22. Then for any integer N such thatN > n� jmj, we have�

q� α

q

�1 − ρ2�−αRjmj;α

n �ρ�

� 1N

XN−1

k�0

Cα�1n

�ρ cos

2πkN

e−2πi

kmN : (C10)

The prefactors

ak �

12 �k� i� j�

�!12 �k − i − j�

�!

12 �k� i − j�

�!12 �k − i� j�

�!

�C11�

in Eq. (C8) should be computed for all k such thatk − i − j is even and non-negative. We can do thatrecursively according to

Fig. 22. Integration range S in Eq. (C1) consisting of the intersec-tion of two pupil disks centered at ν1 and ν2 and a third disk cen-tered at o defined by the scanner’s illumination monopole, withchoice of origin and radii facilitating solving the mathematicalproblem.


ak�i�j ��i� ji

; (C12)

and, for k � i� j; i� j� 2; � � �,

ak�2 �

12 �k� i� j� � 1

�12 �k − i − j� � 1

�12 �k� i − j� � 1

�12 �k − i� j� � 1

�ak: (C13)

The authors would like to acknowledge Brid Con-nolly, Martin Sczyrba, and their team (from ToppanPhotomask, Inc. in Dresden) for their excellent sup-port and input during realization and optimization ofthe nonstandard reticles required by our method. Inaddition, we would like to thank our ASML co-workers Twan Schellekens, Joep de Vocht, FahongLi, Ton Kiers, Jan Baselmans, Nico Vanroose, MaximPisarenco, Patrick Tinnemans, and Jan Mulkens,who all contributed in their own way to this work.

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Baselmans, and M. Hemerik, “Full optical column characteri-zation of DUV lithographic projection tools,” Proc. SPIE 5377,1960–1970 (2004).

3. C. Kittel, Introduction to Solid State Physics, 5th ed. (Wiley,1976).

4. P. Vanoppen and T. Theeuwes, “Lithographic scanner stabilityimprovements through advanced metrology and control,”Proc. SPIE 7640, 764010 (2010).

5. M. Ebert, H. Cramer, W. Tel, M. Kubis, and H. Megens, “Com-bined overlay, focus and CD metrology for leading edge lithog-raphy,” Proc. SPIE 7973, 797311 (2011).

6. T. Hastie, R. Tibshirani, and J. J. H. Friedman, The Elementsof Statistical Learning (Springer, 2001), Vol. 1.

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8. A. J. E. M. Janssen, “Computation of Hopkins’ 3-circle inte-grals using Zernike expansions,” J. Eur. Opt. Soc. RapidPub. 6, 11059 (2011).

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