Fast Track Communication

A single-image method of aberrationretrieval for imaging systems under partiallycoherent illumination

Shuang Xu1, Chuanwei Zhang2,3, Haiqing Wei2 and Shiyuan Liu1,2,3

1Wuhan National Laboratory for Optoelectronics, Huazhong University of Science and Technology,Wuhan 430074, People’s Republic of China2 State Key Laboratory of Digital Manufacturing Equipment and Technology, Huazhong University ofScience and Technology, Wuhan 430074, People’s Republic of China

E-mail: chuanweizhang@mail.hust.edu.cn and shyliu@mail.hust.edu.cn

Received 2 April 2014, revised 30 May 2014Accepted for publication 4 June 2014Published 25 June 2014

AbstractWe propose a method for retrieving small lens aberrations in optical imaging systems underpartially coherent illumination, which only requires to measure one single defocused image ofintensity. By deriving a linear theory of imaging systems, we obtain a generalized formulation ofaberration sensitivity in a matrix form, which provides a set of analytic kernels that relate themeasured intensity distribution directly to the unknown Zernike coefficients. Sensitivity analysisis performed and test patterns are optimized to ensure well-posedness of the inverse problem.Optical lithography simulations have validated the theoretical derivation and confirmed itssimplicity and superior performance in retrieving small lens aberrations.

Keywords: partial coherence in imaging, aberration retrieval, object pattern optimization, opticallithographyPACS numbers: 06.20.-f, 42.30.-d, 42.15.Fr

(Some figures may appear in colour only in the online journal)

1. Introduction

Information of lens aberration of an imaging system isimportant as it directly affects the intensity distribution in theimage plane [1, 2]. Zernike polynomials are commonly usedfor a mathematical description of lens aberrations [3]. Currentapproaches of aberration retrieval are broadly classified intotwo categories: one uses wavefront sensors and the other isbased on intensity measurement. A variety of techniquesusing wavefront sensors have been reported, such as Hart-mann–Shack wavefront sensors [4, 5], and point diffractioninterferometers [6, 7]. Though capable of fast and accurateaberration retrieval, these techniques need complex and costlyapparatus such as embedded interferometers and micro lens

arrays. Due to the advantage of lower cost and easierimplementation of tools, techniques based on intensity mea-surement have been widely used. A common way to retrieve awavefront aberration uses the intensity point spread function,as in the Extended Nijboer–Zernike approach [8], and thephase retrieval and phase diversity approach [9, 10]. How-ever, Being limited to optical imaging systems with eitherfully spatially coherent or completely incoherent sources,such methods are not well suited for systems under partiallycoherent illumination.

In practice, many imaging systems, such as lithographytools [11] and optical microscopes [12], are typically partiallycoherent that are characterized by an optimized source mapand a lens pupil function. The imaging process of such par-tially coherent systems is described by a bilinear model [13],which entails time-consuming calculations and does not lend

Journal of Optics

J. Opt. 16 (2014) 072001 (6pp) doi:10.1088/2040-8978/16/7/072001

3 Authors to whom any correspondence should be addressed

2040-8978/14/072001+06$33.00 © 2014 IOP Publishing Ltd Printed in the UK1

a simple and intuitive relationship between lens aberrationsand the resulted images. Previous methods for retrieving lensaberrations in such partially coherent systems involvethrough-focus image measurements [14, 15] and time-con-suming iterative algorithms [16]. In this work, we propose amethod for small aberration retrieval in optical imaging sys-tems under partially coherent illumination, which onlyrequires to measure one single defocused image of intensitydistribution. A linear formulation is derived in a matrix formthat directly relates the measured image to the unknownZernike coefficients. Consequently, an efficient non-iterativesolution is obtained.

2. Theory

According to the Hopkins’ formulation [17], an imageintensity I(x) for a system under partially coherent illumina-tion is characterized by a bilinear transform

∬= * ⋅ π− − ⋅( ) ( ) ( )I O O Tx f f f f f f( ) , e d d , (1)( )i f f x1 2 1 2

21 2

1 2

where x is a two-dimensional real vector representing thespatial coordinate, f is the normalized pupil coordinate, O(f) isthe diffraction spectrum of the object, * denotes complexconjugation, and T(f1, f2), the so-called transmission crosscoefficient (TCC) function, is defined as

∫= + * +( ) ( ) ( )T J H Hf f f f f f f f, ( ) d . (2)1 2 1 2

Here J(f) describes the source map in the pupil plane underKohler illumination, the objective pupil function H(f) repre-sents effects of lens aberration, which can be represented as

= ∑H Pf f( ) ( )e , (3)ik z R f( )n n n

where k = 2π/λ is the wave number, λ is the wavelength of themonochromatic light source, and n is the Zernike index, sothat zn is the nth Zernike coefficient and Rn denotes the nthZernike polynomial in normalized coordinate over the pupilplane. P(f) is the unit disc modulated by a defocus phasefactor:

= − −( )P f f( ) circ ( ) e . (4)ikh f1 NA 12 2

Here, h is the defocus (in nm) of the image plane, NA is theimage-side numerical aperture of the projection lens. Sub-stituting equation (3) into equation (2), the TCC function iswritten as

∫= + * +

× ∑ + −∑ +⎡⎣ ⎤⎦

( ) ( ) ( )T J P Pf f f f f f f

f

, ( )

e d . (5)( ) ( )ik z R z Rf f f f

1 2 1 2

n n n n n n1 2

Since the exponential term inside the integral in equation (5)is expanded as an infinite Taylor series, the TCC function can

be further represented as

∫∑=!

+ * +

× ∑ + − ∑ +=

∞

⎡⎣ ⎤⎦

( ) ( ) ( )

( ) ( )

Tik

mJ P P

z R z R

f f f f f f f

f f f f f

,( )

( )

d . (6)

m

m

n n n n n n

m

1 2

0

1 2

1 2

In the case that the unknown aberration is small, weapproximate the TCC function by truncating the Taylor seriesto the linear term:

= +( ) ( ) ( )T T Tf f f f f f, , , , (7)1 2 0 1 2 1 1 2

where T0(f1, f2) represents the TCC without aberration:

∫= + * +( ) ( ) ( )T J P Pf f f f f f f f, ( ) d , (8)0 1 2 1 2

and T1(f1, f2) represents the linearly aberrated TCC, which hasthe Zernike coefficients {zn} separated from the integrals overthe spatial frequency:

∑=( ) ( )T z Bf f f f, , , (9)n

n n1 1 2 1 2

∫= + * +

× + − +⎡⎣ ⎤⎦( ) ( ) ( )

( ) ( )

B ik J P P

R R

f f f f f f f

f f f f f

, ( )

d . (10)

n

n n

1 2 1 2

1 2

Consequently, the total image intensity in equation (1) isapproximated as

∑= +I I z Ax x x( ) ( ) ( ). (11)n

n n0

Here we define An(x) as the sensitivity function for the nthZernike order, and I0(x) and An(x) can both be directly cal-culated from T0(f1, f2) and Bn(f1, f2) respectively:

∬= * ⋅ π− − ⋅( ) ( ) ( )I O O Tx f f f f f f( ) , e d d , (12)( )i f f x0 1 2 0 1 2

21 2

1 2

∬= * ⋅ π− − ⋅( ) ( ) ( )A O O Bx f f f f f f( ) , e d d . (13)( )n n

i f f x1 2 1 2

21 2

1 2

Actually, both I0(x) and An(x) can be calculated inadvance and stored in memory for a given object at a givendefocus. This means that for the purpose of aberrationretrieval, we only need to measure one defocused image forsome specially designed object. In this way, the total imageintensity I(x) in equation (11) is treated as the observed data,and a linear relationship for aberration retrieval is formulatedas

∑Δ = − =I I I z Ax x x x( ) ( ) ( ) ( ). (14)n

n n

def

0

To suit numerical calculations, equation (14) is dis-cretized into matrix and vector operations as

Δ =I AZ, (15)

where ΔI is an M× 1 vector representing M discrete samples(pixels) from the image intensity ΔI(x), Z is an N× 1 vector ofthe unknown Zernike coefficients {zn}, and A is an M×Nmatrix of samples of the sensitivity function {An(x)}. Testpatterns on the object and sampling of the image need to beoptimized such that the linear system is over-determined,

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namely, the rank of A saturates its upper limit N, when Z canbe solved using the least-square method.

It is worth pointing out that the sensitivity matrix of theobject pattern is critical in the aberration retrieval, as eachaberration has a unique characteristics and interacts with testpatterns differently. Test patterns should be carefullydesigned and optimized to be most effective under a givenillumination condition. We use the condition number of thesensitivity matrix as the objective function for optimizing thetest patterns, and formulate the optimization problem as

=

= ‖ ‖ ⋅ ‖ ‖

+

+ + −

⎡⎣ ⎤⎦⎡⎣ ⎤⎦

( )

( )

min

min

O A A

A A A A

arg cond

arg , (16)

O

O

1

where Ο̂ represents the optimized object pattern parameters,containing width of each patterns and pitch of adjacent pat-terns, A is the sensitivity matrix of the given object pattern,which can be calculated efficiently from equation (13), A+ isthe Hermitian conjugate of A, and‖ ⋅ ‖ is the spectral norm ofmatrix. A small condition number guarantees fast con-vergence and a stable solution to the inverse problem. Manymulti-parameter optimization algorithms can be used to solvethe inverse problem. Here we adopt the Powell algorithm forthis purpose [18].

A flowchart of the proposed aberration retrieval methodis shown in figure 1, which highlights two key technicalaspects. An aberration retrieval model is first established,serving as the theoretical basis for efficiently calculating thesensitivity matrix. An object pattern optimization algorithmbased on sensitivity analysis is another important module fordesigning optimal test patterns whose images are sensitive toZernike aberrations.

3. Numerical results

We take a projection lithography tool as an example partiallycoherent system and apply the proposed method. The con-sidered tool has a partially coherent illumination with wave-length λ= 193 nm, projection lens with NA= 0.8, andoperates at defocus h = 180 nm. Figure 2 depicts a typicalimage intensity of a two-level 0/π phase-shift object patternwith the width of central contact 210 nm, the width of sur-rounding contact 80 nm, the pitch between central and sur-rounding contact 85 nm, and the simulation range [−500 nm,500 nm], under a quadrupole source σout/σin/θ= 0.8/0.4/45°and lens aberration shown below in figure 5(a). The nor-malized intensity error is on the order of 10−4, which justifiesthe use of linear approximation.

From equation (13), each element of the sensitivitymatrix A has a compact analytic expression as a function of xunder fixed test patterns O(f), which serves an analytic kernelsfor measuring the Zernike coefficients. A suitable amount ofdefocus is desired so that odd aberrations lead to lateral shiftsand even aberrations interact with defocus to distort images.Figure 3 depicts the first fifteen analytic kernels for retrievingaberrations from z2 to z16, as Ai(x) (i= 2, 3,…, 16) is the ithrow of the matrix A in equation (15). A quadrupole sourcewith σout/σin/θ= 0.8/0.4/45° was set, and an object patternshown in figure 2 was used. As expected, the analytic kernelshave the same symmetry properties as the correspondingZernike polynomials. In addition, with the help of defocus,the sensitivities to even aberrations (A4(x), A5(x), A6(x), A9(x),A12(x), A13(x) and A16(x) in figure 3) are in similar magni-tudes as those to the odd aberrations (A2(x), A3(x), A7(x),A8(x), A10(x), A11(x), A14(x) and A15(x) in figure 3), whichfacilitates reliable retrieval of both the even and odd aberra-tions with only one single intensity measurement.

It is customary to represent aberrations of lens pupils byZernike coefficients, which weight the corresponding Zernikepolynomials that form a complete and orthogonal basis overthe interior of the unit circle. There the Zernike representationaffords a linear relationship between the Zernike coefficientsand the point-wise aberration of pupil phase. Analogously,equations (13) and (14) establish a similar linear map betweenthe Zernike coefficients and the image intensity points,through our derived analytic kernels. Although the analytickernels are not necessarily orthogonal, they incorporate andcondense information of the illumination source and testpatterns, so to lend a simple and compact linear equation,

Figure 1. Flowchart of small aberration retrieval using one singleintensity measurement.

Figure 2. Image intensity calculation results for the object pattern with aberration shown in figure 5(a).

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which is amenable to an efficient and robust solution andsensitivity analysis.

As the test patterns should be carefully designed andoptimized to be most effective under a given illuminationcondition, figure 4 shows some of the results of our optimi-zation simulation. We used two kinds of source: one was setas a traditional one with σ= 0.8, and the other a quadrupole

one with σout/σin/θ= 0.8/0.4/45°. There are four kinds ofobject patterns shown in the figure, which fall into two types:binary objects shown as black-and-white diagrams, and 0/πphase-shift objects as gray-scale diagrams. In this figure,original object patterns are in the first row, object patternsafter optimization with a traditional source are in the secondrow, and with a quadrupole source in the third row. The

Figure 3. Analytic kernels for measuring aberrations.

Figure 4. Optimization results of some object patterns, with (a)–(d) as four kinds of object patterns.

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histogram in the bottom shows the condition numbers of thesensitivity matrices of object patterns above. In the legend,‘B’ is short for a binary object pattern, ‘P’ for a phase-shiftobject pattern, ‘T’ for a traditional source, ‘Q’ for a quadru-pole one, ‘O’ for original, and ‘A’ for after optimization.Therefore, ‘B-T-O’ means the condition number of the ori-ginal binary object pattern with a traditional source, etc. It isnoted from the histogram that the condition number of sen-sitivity matrix becomes much smaller after optimization, andthe results are different under different sources. We utilizedthe combination of phase-shift object and a quadrupole sourceshown in figure 2 as an example for the demonstration of ouraberration retrieval method. Theoretically lens aberrationshould be express as the sum of infinite orthogonal Zernikepolynomials, and this method is applicable up to any high-order Zernike term. However, for practical applications, theaberrations are expressed as the sum of finite Zernike termsup to zn plus a much smaller residual aberration which is thesum of infinite Zernike terms higher than zn, where n is thehighest concerned order. To guarantee the accuracy, theaberrations should be measured up to a sufficient high-orderZernike term [19], which is usually the 37th order for prac-tical lithography tools, where the 37th Zernike term is actuallythe 49th Zernike term [1, 3]. In these simulations, to guar-antee that equation (15) is over-determined and can be solvedby the least-square method accurately, for measuringunknown Zernike coefficients up to the 37th (49th) order(N = 36), the discretized number M is set to be 81 × 81.

Figure 5 depicts one of the simulation results with anoptimized object shown in figure 2. An unknown aberrationas shown in figure 5(a) is induced by setting the Zernikecoefficients z2 to z37 (z49) with a random value in the range of[−15 mλ, 15 mλ], which leads to the aberration in the range of[−60 mλ, 60 mλ]. Figure 5(b) depicts the wavefront of theretrieved Zernike coefficients up to the 37th (49th) order,while figure 5(c) shows the difference between retrieved andunknown wavefront. The absolute error in figure 5(c) is1.61 mλ, which is decreased by 1.1 mλ compared to the errorof retrieval using the original object, indicating a satisfactoryaccuracy for aberration retrieval.

To further evaluate the accuracy and effective range ofthe proposed method, we performed another series of simu-lations with unknown aberrations in different ranges. Figure 6shows the absolute error for different ranges of input

aberrations. It is clear that the proposed method achieves avery good accuracy on the order of mλ s when the inputaberrations are small (less than 60 mλ), and the retrieval errorincreases significantly as the magnitude of aberrationincreases. When the aberration range is smaller than 50 mλ,the absolute error is less than 1.5 mλ. However, if the aber-ration range surpasses 80 mλ, the absolute wavefront errorsharply increases to the order of 10 mλ s, which is unac-ceptable for the aberration retrieval application. This simu-lations demonstrate that the proposed linear formulationworks best under the condition of small aberrations.

4. Conclusion

In conclusion, a method is proposed for small aberrationretrieval in optical imaging systems under spatially partiallycoherent illumination using one single measurement ofdefocused image of an object containing optimized test pat-terns. Numerical results are presented to verify the theoreticalderivation and demonstrate the efficacy of test pattern opti-mization. The proposed method of aberration retrieval issimple to implement and yields superior performance inretrieving small lens aberrations.

In the proposed method, the effective source intensitydistribution and the NA are treated as known system para-meters for sensitivity analysis, test patterns optimization andaberration retrieval. In practical applications, however, thereare imaging systems that do not have all the system para-meters known or given. To ensure accurate aberrationretrieval, it is best to have such system parameters measuredor calibrated in advance. Inevitably, uncertainty in suchmeasured system parameters will propagate to the retrievedZernike coefficients. There are interesting questions and

Figure 5. Example of aberration retrieval. (a) Unknown aberration, (b) retrieved aberration, and (c) difference between retrieved andunknown wavefront.

Figure 6. Measurement accuracy for different ranges of inputaberrations.

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challenges for further investigations, regarding error oruncertainty analysis and optimizations of the measurementprocedures as well as the data processing algorithms tominimize errors in the final results.

Acknowledgements

This work was funded by the National Natural ScienceFoundation of China (grants 91023032 and 51005091), theSpecialized Research Fund for the Doctoral Program ofHigher Education of China (20120142110019), the NationalScience and Technology Major Project of China(2012ZX02701001), and the Program for Changjiang Scho-lars and Innovative Research Team in University of China.

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