J JournalO of theU EuropeanR Optical SocietyN - RapidA PublicationsL2, 07012 O (2007) F www.jeos.org T H E E U R O P E A N Computing Zernike polynomials of arbitrary degree O PusingT theI discreteC A FourierL transformS O C I E T Y

Augustus J. E. M. Janssen Philips Research Europe, HTC 36 R a.j.e.m.janssen@philips.comA P I D P U B L I C A T I O N S Peter Dirksen Philips Research Europe, HTC 4 [email protected]

m The conventional representation of Zernike polynomials Rn (ρ) gives unacceptable numerical results for large values of the degree n. We present an algorithm for the computation of Zernike polynomials of arbitrary degree n. The algorithm has the form of a discrete Fourier (cosine) transform which comes with advantages over other methods in terms of computation time, accuracy and ease of implementation. As an application we consider the effect of NA-scaling on the lower-order aberrations of an optical system in the presence of a very high order aberration. [DOI: 10.2971/jeos.2007.07012]

Keywords: Zernike polynomial, aberration, high-order, discrete Fourier transform, NA-scaling

1 INTRODUCTION

m The (radial part of the) Zernike polynomials Rn (ρ) are widely and order. For instance, when the pupil function has a central used in the representation of the aberrations of optical sys- obstruction (0 and 1 on two concentric sets 0 ≤ ρ < a and 0 tems and in the computation of the diffraction integral defin- a ≤ ρ ≤ 1), the coefficient of Rn(ρ) in the Zernike expansion ing the point-spread function of these systems [1]-[5].When of the pupil function can be shown to decay only like n−1/2. we are dealing with smooth exit pupil functions, it is, in gen- Then Eq. (1) becomes cumbersome because of the high-order m eral, sufficient to consider the Rn for modest values of the de- factorials that are required. Also, for m = 0, it can be shown n−2s gree n and azimuthal order m, say, n + m ≤ 12. For such pupil from the Stirling’s formula that the largest coefficient√ of ρ functions, the conventional polynomial representation [1] occurring in the series in Eq. (1) behaves like (1 + 2)n. Ac- cordingly, when computing with d digits, Eq. (1) produces er- m √ Rn (ρ) = rors of the order of unity or larger from n = d/log(1 + 2) ( − ) n m /2 (n − s)! (−1)s onwards. Hence, for the commonly used 15 digits precision, = n−2s ≤ ≤ ∑  n − m   n + m  ρ , 0 ρ 1 , one has serious problems from n = 40 onwards, as shown in s=0 − s ! − s ! s! 2 2 Figure 1, top, right. An alternative to compute Zernike poly- (1) nomials is to use recursions for them such as those found in Ref. [6]. These recursion schemes are, however, computation- can be used to calculate the Zernike polynomials. Some low ally more expensive and less direct than a formula like Eq. (1) order Zernike polynomials are shown in Table 1. and their accuracy due to error propagation is also an issue.

In this paper, we present a new computation scheme in which m( ) Degree n m Rn ρ one can allow degrees as large as 105 without problems. This 0 0 1 new algorithm is of the discrete-cosine transform (DCT) type, 1 1 ρ and is direct and transparent. Furthermore, the computation 2 2 0 2ρ − 1 can be done using the FFT-algorithm which comes with the 2 2 2 ρ following advantages [7, 8]: 4 0 6ρ4 − 6ρ2 + 1 3 1 3ρ3 − 2ρ 3 3 ρ3 • Very favorable and well-established accuracy . . • Simultaneous computation of all Zernike polynomials of the same degree n in as few as O(n logn) operations TABLE 1 Low order Zernike polynomials

In the case that the exit pupil function contains discontinu- As an application we consider the effect of NA-scaling on the ities, or is roughly behaved in a more general sense, it is neces- lower-order aberrations of an optical system in the presence of sary to consider Zernike polynomials of much higher degree a very high order aberration. For this we use a recently found

Received March 16, 2007; published April 12, 2007 ISSN 1990-2573 Journal of the European Optical Society - Rapid Publications 2, 07012 (2007) A. J.E.M. Janssen, et. al.

n = 39, m=17 n = 50, m=0 formula [9], entirely in terms of Zernike polynomials, for the 1 1 Zernike coefficients of scaled pupils. Eq.(1) Eq.(1) Eq.(2) Eq.(2) 0.5 0.5

2 DCTFORMULAFORZERNIKE 0 0

POLYNOMIALS −0.5 −0.5

−1 −1 In Appendix A we show that 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 ρ ρ 1 N−1  2πk  2πmk m( ) = ≤ ≤ n = 10000, m=0 Rn ρ ∑ Un ρ cos cos , 0 ρ 1 , 1 N k=0 N N

(2) 0.5 where N is any integer > n + m. In Eq. (2) we have integer n, m ≥ 0 with n − m even and ≥ 0 (as usual), and Un is the 0 Chebyshev polynomial of the second kind and of degree n. No −0.5 matter how large n is, the evaluation of Un(x) is no problem

since we have −1 −5 −4.5 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0 sin(n + 1) v ρ − 1 −6 U (x) = , x = cos v. (3) x 10 n sin v m FIG. 1 Top: Rn (ρ) as a function of ρ, 0 ≤ ρ ≤ 1, computed according to Eq. (1) Eq. (2) is a consequence of the formula (A.10) that represents and Eq. (2), using 16 digits, for m = 17, n = 39 and for m = 0 , n = 50. Bottom: m m Rn (ρ) as an integral of a trigonometric polynomial of degree Rn (ρ), computed according to Eq. (2), with m = 0 and n = 10000 and ρ very close n + m over a periodicity interval. Such an integral can be com- to 1 puted error-free as a series when the number of equidistant sample points N exceeds the degree n + m. It also follows from Scaling to a pupil with relative size ε = NA/NAmax ≤ 1 this that the right-hand side of Eq. (2) (and that of Eq. (A.10)) m requires computation of the Zernike coefficients αn (ε) of the vanish when m > n or when n and m have different parity m scaled phase function Φ(ερ, ϑ). For m = 0, 1, ... the αn (ε) are (with again N > n + m). m given in terms of the αn as m( ) th m( ) = m [ n ( ) − n+2( )] = + Eq. (2) gives Rn ρ as the m component of the DCT of the se- αn ε ∑ αn0 Rn0 ε Rn0 ε , n m, m 2, ... , (6) m n0 quence (Un(ρ cos 2πk/N))k=0,1,...,N−1, hence we get all Rn (ρ), ≥ = − ( ) with m 0 and m n, n 2, ... , using O N log N operations. where the summation is over n0 = n, n + 2, ... (Rn+2 ≡ 0). ≤ > n Since m n, it is sufficient to take N any integer 2n. In case of a non-smooth phase function Φ, one should expect m 0 significant values of α 0 for very high degrees n . Also, scal- In Figure 1, top, we show Rm(ρ) as a function of ρ, 0 ≤ ρ ≤ 1, n n ing is normally done using values of ε close to its maximum 1, computed according to Eq. (1) and Eq. (2), using 16 decimal where Eq. (1) produces the largest numerical error. Thus, for- places, for m = 17, n = 39 and for m = 0 , n = 50. We see that mula (6) is not practicable in these cases when Eq. (1) is used Eq. (1) gives unacceptable results for the case m = 0, n = 50 n n+2 to evaluate R 0 (ε) − R 0 (ε), but becomes so when Eq. (2) is from ρ = 0.8 onwards. n n used instead.

m In Figure 1, bottom, we show Rn (ρ), computed according to ε = 0.50 ε = 0.98 0.6 0.6 Eq. (2), with m = 0 and n = 10000 and ρ very close to 1. We m 0.4 0.4 see that Rn (ρ = 1) = 1 which is in agreement with the theory [1]. 0.2 0.2

0 0 (0.50) (0.98) 0 n 0 n α α 3 ANAPPLICATION:HIGH-ORDER −0.2 −0.2 ABERRATIONSANDSCALING −0.4 −0.4

0 20 40 60 80 100 0 20 40 60 80 100 In lithographic imaging systems, the numerical aperture (NA) n − value n − value is varied intentionally below its maximum value so as to op- 0 timize the performance for the particular object to be imaged. FIG. 2 The disturbance αn(ε) of the aberration of order n = 0, 2, ··· , 100, due to the In Ref. [9] the effect of NA-scaling on the Zernike coefficients presence of an aberration of amplitude 1 and of the order n0 = 100 when the system describing the optical system has been concisely expressed in is scaled to relative size ε = 0.50 (left) and ε = 0.98 (right). terms of Zernike polynomials. Thus we consider a pupil func- tion P(ρ, ϑ) = exp {i Φ(ρ, ϑ)} , 0 ≤ ρ ≤ 1 , 0 ≤ ϑ ≤ 2π , (4) As an example, we consider the effect of a single high order m m aberration term αn0 on the totality of αn with n = m, m + in polar coordinates with real phase Φ, and we assume that Φ 0 m 2, ..., n while scaling to relative size ε. We take αn = 0 for is expanded as a Zernike series according to 0 m n 6= n and αn0 = 1, and get from Eq. (6) m m Φ(ρ, ϑ) = ∑ αn Rn (ρ) cos mϑ . (5) m n n+2 0 n,m αn (ε) = [Rn0 (ε) − Rn0 (ε)], n = m, m + 2, ..., n , (7)

07012- 2 Journal of the European Optical Society - Rapid Publications 2, 07012 (2007) A. J.E.M. Janssen, et. al.

m 0 n n+2 m0 ( ) = > ( ) − ( ) ( ) = 0 ( ) while αn ε 0 when n n . The numbers Rn0 ε Rn0 ε Un ν ∑ εm zn ν, µ , (A.8) required in Eq. (7), with n0 fixed and n = m, m + 2, ..., n0 , can m0 O(n0 n0) be computed simultaneously using log operations by i.e., that employing Eq. (2) in its DCT-mode. Figure 2 shows the result for m = 0 and n0 = 100, n = 0, 2, ··· , 100, α0 (ε) with ε = 0.50 m0 0 n Un(ρ cos ϑ) = ∑ εm0 Rn (ρ) cos m ϑ . (A.9) and ε = 0.98. m0

By orthogonality of the cos m0ϑ, ϑ ∈ [0, 2π], it follows that

A PROOFOFTHEMAINRESULT 2π Z m 1 R (ρ) = Un(ρ cos ϑ) cos mϑ dϑ . (A.10) We write for integer n, m ≥ 0 with n − m even and ≥ 0 n 2π 0 zm(ν, µ) = Zm(ρ, ϑ) = Rm(ρ) cos mϑ , (A.1) n n n Finally, Un(ρ cos ϑ) cos mϑ is a trigonometric polynomial of degree n + m. Therefore, the integral in Eq. (A.10) can be eval- in which the Cartesian coordinates ν, µ and polar coordinates uated using the sample values of the integrand at the points ρ, ϑ are related according to ν = ρ cos ϑ, µ = ρ sin ϑ and 0 ≤ 2πk/N, k = 0, 1, ..., N − 1 when N > n + m. This yields Eq. (2). ρ ≤ 1, 0 ≤ ϑ ≤ 2π. Furthermore, we let √ 1−ν2 1 Z f m(ν) = zm(ν, µ) dµ , − 1 ≤ ν ≤ 1 . Note. Eq. (A.10) can also be used to get accurate stationary n 2 1/2 n m 2(1 − ν ) √ phase approximations to Rn (ρ) when n gets large. Accord- − 1−ν2 m m ingly, Rn (ρ) is vanishing small when 0 ≤ ρ ≤ n − ε, and is os- (A.2) −1/2 m m cillatory and of amplitude O(n ) when n + ε ≤ ρ ≤ 1 − ε According to the formula for the Radon transform of Zn we (ε > 0 fixed, n → ∞). This can be used to explain some of the have, see Ref. [10], Eq. (8.13.17), phenomena that one observes in Figure 2. 1 f m(ν) = U (ν) , − 1 ≤ ν ≤ 1 , (A.3) n n + 1 n ACKNOWLEDGEMENTS with the Chebyshev polynomial Un given in Eq. (3). We con- m sider next the Zernike expansion of fn (ν), The authors thank the various referees of this paper for their

0 0 valuable and constructive comments. m( ) = m,m m ( ) fn ν ∑ βn,n0 zn0 ν, µ (A.4) n0,m0

in which the β’s are given, due to the orthogonality1 of the z’s, References by [1] M. Born and E. Wolf, Principles of Optics (7th rev. ed., Ch. 9 and 0 ZZ App. VII, Cambridge University Press, Cambridge, MA, 2001). m,m0 (n + 1)εm0 m m0 β = f (ν) z 0 (ν, µ) dν dµ . (A.5) n,n0 π n n [2] D. Malacara, Optical Shop Testing (2nd ed., Wiley, 1992). ν2+µ2≤1 [3] V. N. Mahajan, “Zernike circle polynomials and optical aberrations In Eq. (A.5) we have that m, m0, n, n0 all have the same parity of systems with circular pupils” Appl. Optics. 33, 8121–8124 (1994). 0 0 0 0 and n ≥ m , and εm0 = 1 for m = 0 and εm0 = 2 for m = [4] R. J. Noll, “Zernike polynomials and atmospheric turbulence” J. 1, 2, ... (Neumann’s symbol). According to Eq. (A.3) we have Opt. Soc. Am. 66, 207–211 (1976).  √  [5] See http://www.nijboerzernike.nl . 1 1−ν2 0 Z Z m,m0 (n + 1) εm0 m0 = U ( )  z 0 ( ) d  d [6] A. Prata and W. V. T. Rusch, “Algorithm for computation of βn,n0 n ν  n ν, µ µ ν . (n + 1) π √ Zernike polynomials expansion coefficients” Appl. Optics 28, 749– −1 2 − 1−ν 754 (1989). (A.6) Then using Eqs. (A.2), (A.3) with n0, m0 instead of n, m, we [7] D. Calvetti, “A stochastic roundoff error analysis for the Fast find Fourier Transform” Math. Comput. 56, 755–774 (1991).

1 [8] See http://en.wikipedia.org/wiki/Fast˙Fourier˙transform . Z m,m0 2εm0 2 1/2 εm0 β 0 = U (ν) U 0 (ν)(1 − ν ) dν = δ 0 , [9] A. J. E. M. Janssen and P. Dirksen, “Concise formula for the Zernike n,n π(n + 1) n n n + 1 n,n −1 coefficients of scaled pupils”, J. Microlith. Microfab. Microsyst. 5, (A.7) 030501 (2006). where δ denotes Kronecker’s delta, and where we have used [10] S. R. Deans, The Radon transform and some of its applications the orthogonality of the U’s, see Ref. [11], 22.2.5 on p. 774. (Wiley, New York, 1983, Sec. 7.6, Eq. 6.11). Handbook of Mathematical Func- We conclude from Eqs. (A.3), (A.4), (A.7) that [11] M. Abramowitz and I. A. Stegun, tions (Dover, New York, 1970).

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