Orthonormal aberration polynomials for optical systems with circular and annular sector pupils José Antonio Díaz1 and Virendra N. Mahajan2,* 1
Departamento de Óptica, Universidad de Granada, Granada 18071, Spain
2
The Aerospace Corporation, 2310 El Segundo Boulevard, El Segundo, California 90245, USA *Corresponding author:
[email protected] Received 28 November 2012; accepted 26 December 2012; posted 8 January 2013 (Doc. ID 180765); published 11 February 2013
Using the Zernike circle polynomials as the basis functions, we obtain the orthonormal polynomials for optical systems with circular and annular sector pupils by the Gram–Schmidt orthogonalization process. These polynomials represent balanced aberrations yielding minimum variance of the classical aberrations of rotationally symmetric systems. Use of the polynomials obtained is illustrated with numerical examples. © 2013 Optical Society of America OCIS codes: 110.0110, 010.7350, 220.1010, 120.3180, 220.0220.
1. Introduction
The interferogram of a cube-corner retroreflector consists of a fringe pattern inside a circle. It consists of six equal segments, where each segment has the form of a circular sector with an angular subtense of π∕3 [1]. The interferogram can also be hexagonal with six triangular segments [2]. The imaging properties of and phase retrieval on sector and annular sector pupils have also been discussed in the literature [3–5]. Two recent papers discuss the effects of the thermal gradient on, as well as the polarization and far-field diffraction patterns of cube corners [6,7]. In this paper, we discuss the orthonormal aberration polynomials for circular and annular sector pupils. In Section 2, we obtain the first four orthonormal polynomials for these pupils by orthogonalizing the Zernike circle polynomials [8,9] by the Gram– Schmidt orthogonalization process [10]. Due to the low symmetry of the sector pupils, the polynomials are sufficiently complex that the closed-form analytical expressions for only the first four polynomials 1559-128X/13/061136-12$15.00/0 © 2013 Optical Society of America 1136
APPLIED OPTICS / Vol. 52, No. 6 / 20 February 2013
are given. The details of the Gram–Schmidt process are illustrated only for the circular sector. However, the polynomials can be obtained numerically for any order and any obscuration value. The first 11 polynomials for a circular sector and an annular sector with an obscuration ratio of 0.5 with a semi-angular subtense of π∕3 are given as numerical examples. In these examples, the sector is assumed to be symmetrical about the x axis. We also outline the procedure for obtaining the orthonormal polynomials for a sector with an arbitrary orientation in an xy plane. Such a procedure is useful for obtaining the polynomials for the six circular sector segments of the interferogram of a cube-corner retroreflector. It is illustrated by applying it to obtain the polynomials for a circular sector that is symmetrical about the y axis. We point out that the defocus polynomial given in [1] for the annular and thereby the circular sector pupils is incorrect. The expansion of an aberration function for a certain pupil in terms of the polynomials that are orthonormal over it, and how to obtain the orthonormal expansion coefficients, are considered in Section 3. We illustrate the use of the sector polynomials in Section 4 by considering an aberration function that consists of
spherical aberration combined with defocus and tilt, and determine the aberration coefficients for circular and annular sector pupils that are symmetric about the x axis. To see how the coefficients change with the orientation of a sector, we also consider a circular sector pupil that is symmetric about the y axis. The coefficients for a pupil that is semi-circular, circular, or annular but aberrated by the same aberration function are also obtained. The interferograms of the aberration function for the various pupils are shown for the starting aberration function as well as when the first four polynomial terms representing the interferometer errors of piston, tip, tilt, and defocus are removed. A brief summary of the main results, and conclusions are given in Section 5. 2. Gram–Schmidt Orthogonalization of Zernike Circle Polynomials over a Sector Pupil A.
Circular Sector Pupil
Consider a circular sector, as illustrated in Fig. 1(a). The sector is symmetrical about the x axis and subtends a semi-angle α at the center of the circle. Let the orthonormal Zernike circle polynomials be represented by Zj ρ; θ. These polynomials may be written [8,9] p m ≠ 0; Zeven j ρ; θ 2n 1Rm n ρ cos mθ; (1a) p Zodd j ρ; θ 2n 1Rm n ρ sin mθ;
m ≠ 0; (1b)
Zj ρ; θ
p n 1R0n ρ;
m 0;
(1c)
where Rm n ρ are the radial polynomials given by Rm n ρ
n−m∕2 X s0
−1s n − s! nm n−m ρn−2s ; s! 2 − s ! 2 − s !
(2)
n and m are positive integers (including zero), and n − m ≥ 0 and even. The index n represents the radial degree or the order of the polynomial since it represents the highest power of ρ in the polynomial, and m is called the azimuthal frequency. The index j is a polynomial-ordering number and is a function of n and m. The first 11 orthonormal polynomials and the relationship among the indices j, n, and m are given in Table 1. They are ordered such that an even j corresponds to a symmetric polynomial varying as cos mθ, while an odd j corresponds to an antisymmetric polynomial varying as sin mθ. For a given value of n, a polynomial with a lower value of m is ordered first. The polynomials are orthonormal over a unit circular pupil according to Z 1 Z 2π Z 1 Z 2π Zj ρ; θZj0 ρ; θρdρdθ∕ ρdρdθ δjj0 : 0
0
0
0
(3) The circular sector polynomials Sρ; θ; α can be obtained by Gram–Schmidt orthogonalizing [10] the circle polynomials over the circular sector pupil according to "
# j X hZj1 Sk iSk ; Zj1 −
Sj1 N j1
(4)
k1
where S1 1;
(5)
the angular brackets represent an average value over the pupil, and N j1 is a normalization constant so that the S-polynomials are orthonormal, i.e., hSj Sj0 i
1 α
Z 1Z 0
α −α
Sj ρ; θ; αSj0 ρ; θ; αρdρdθ δjj0 : (6)
Here δjj0 is a Kronecker delta. Letting j0 1, we find that the mean value hSj i of a polynomial is zero (except when j 1). Similarly, letting j j0 , its mean square value hS2j i is unity (except when j 1). Letting j 1 and substituting Z2 2ρ cos θ into Eq. (4), we obtain S2 as follows: S2 N 2 Z2 − hZ2 i; Fig. 1. Sector pupil of unit radius and semi-angular subtense α symmetrical about the x axis. (a) Circular. (b) Annular with obscuration ratio ϵ.
1 hZ2 i α
Z 1Z 0
α −α
2ρ cos θρdρdθ
4 sin α ; 3 α
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Table 1.
Orthonormal Zernike Circle Polynomials Z j ρ;θ
j
n
m
Zj ρ; θ
Aberration
1 2 3 4 5 6 7 8 9 10 11
0 1 1 2 2 2 3 3 3 3 4
0 1 1 0 2 2 1 1 3 3 0
1 2ρ cos θ θ p2ρ sin 2 3 2ρ − 1 p 2 6ρ sin 2θ p 2 p 6ρ3 cos 2θ 83ρ − 2ρ sin θ p 3 8p 3ρ 3− 2ρ cos θ 8ρ sin 3θ p 3 p 8ρ4 cos23θ 56ρ − 6ρ 1
Piston x tilt y tilt Defocus 45° primary astigmatism 0° primary astigmatism Primary y coma Primary x coma
hZ22 i
4 α
Z 1Z 0
α
Since hS22 i 1, 1 4 sin α 2 −1∕2 2α sin 2α − 2α 3 α
S2 ρ; θ; α h
2ρ cos θ − 1 2α 2α
sin 2α −
i
4 sin α 2 1∕2 3 α
:
N 4 Z4 − hZ4 S2 iS2 − S3 ;
(7)
Letting j 2 and substituting Z3 2ρ sin θ into Eq. (4), we obtain S3 as follows:
since hZ4 i and hZ4 S3 i are both equal to zero. It can be shown that ( 1 p 2 S4 ρ; θ; α 32ρ − 1 N4 ) p 12 6 sin α2ρ cos θ − 4 sin α∕3α − p ; 5∕ 2α 9α2α sin 2α − 32 sin2 α
S3 N 3 Z3 − hZ3 i − hZ3 S2 iS2 : Since the integral of sin mθ between symmetric limits ∓α is zero, hZ3 i 0;
hZ3 S2 i 0;
hS3 i 0;
S3 N 3 Z3 :
Hence, hS23 i N 23 hZ23 i and hZ23 i
4 α
Z 1Z 0
α −α
ρ2 sin2 θρdρdθ 2α − sin 2α∕2α.
Since hS23 i 1, N 3 2α − sin 2α∕2α−1∕2 and 1138
(8)
S4 N 4 Z4 − hZ4 i − hZ4 S2 iS2 − hZ4 S3 iS3
and 4 sin α 3 α
2ρ sin θ : 2α − sin 2α∕2α1∕2
We see that even the expressions for the orthonormal tilt polynomials S2 and S3 are relatively complex. We also note that the polynomial S2 contains a piston term, since the mean value of the wavefront tilt ρ cos θ over the circular sector is not zero. p Letting j 3 and substituting Z4 32ρ2 − 1 into Eq. (4), the polynomial S4 is given by
hS22 i N 22 hZ2 − hZ2 i2 i N 22 hZ22 i − hZ2 i2 1 4 sin α 2 2α sin 2α − N 22 : 2α 3 α
N2
S3 ρ; θ; α
ρ2 cos2 θρdρdθ 2α sin 2α∕2α;
−α
Primary spherical
APPLIED OPTICS / Vol. 52, No. 6 / 20 February 2013
(9) where N4
1∕2 1 96 sin2 α 25 − : 5 9α2α sin 2α − 32 sin2 α
(10)
It is evident that the complexity of the polynomials increases considerably as we try to obtain the higherorder polynomials. It can be shown further that the polynomial S4 represents defocus aberration ρ2 balanced with an amount Bt of wavefront tilt aberration ρ cos θ in the form W ρ2 Bt ρ cos θ
(11)
such that it yields minimum variance of the balanced aberration. The orthonormal aberration is then given by S4
ρ2 Bt ρ cos θ − hWi ; σ
(12)
where Bt −
15α
1 4
sin α
; 2α sin8α2α − 4 sin 2 9α
(13)
and
hWi
1 16 sin2 α − 2 59α2α sin 2α − 32 sin2 α
(14)
sector. The annular sector reduces to a circular sector as ϵ → 0. The annular sector polynomials Sρ; θ; ϵ; α can be obtained by Gram–Schmidt orthogonalizing the circle polynomials over the annular sector pupil, i.e., by replacing the lower limit of radial integrations from zero in Section 2.A by ϵ. The first four polynomials thus obtained that are orthonormal over the annular sector pupil according to Z 1Z α 1 Sj ρ; θ; ϵ; αSj0 ρ; θ; ϵ; αρdρdθ hSj Sj0 i α1 − ϵ2 ϵ −α (17)
δjj0
is the mean value of the aberration and
are given by σ hW 2 i − hWi2 1∕2 1∕2 1 25 32 sin2 α − 10 3 9α2α sin 2α − 32 sin2 α
S1 1; (15)
2
S2 ρ;θ;ϵ;α h
is its standard deviation over the circular sector. The units of Bt, hWi, and σ are the same as those of the starting defocus aberration. The number of polynomials up to and including a certain degree n in ρ is given by the same number as in the case of Zernike circle polynomials, namely,
(18)
sin α 2ρ cos θ − 43 1ϵϵ 1ϵ α i ; 1ϵ2 4 1ϵϵ2 sin α 2 1∕2 2α sin 2α − 3 1ϵ α 2α
(19) S3 ρ; θ; ϵ; α
2ρ sin θ ; 1 ϵ2 2α − sin 2α∕2α1∕2
(20)
and p 3 161 − ϵ2 1 ϵ3 ϵ sin α−3α1 ϵρ cos θ 21 ϵ ϵ2 sin α 2 2 2ρ − 1 − ϵ ; S4 ρ; θ; ϵ; α N4 321 ϵ ϵ2 2 sin2 α − 9α1 ϵ2 1 ϵ2 2α sin 2α
(21)
where
1−ϵ 1287ϵ6 28ϵ5 50ϵ4 55ϵ3 50ϵ2 28ϵ 7sin2 α − 225α1 ϵ4 1 ϵ2 2α sin 2α 1∕2 × : N4 5 321 ϵ ϵ2 2 sin2 α − 9α1 ϵ2 1 ϵ2 2α sin 2α (22)
Nn
n 1n 2 : 2
(16)
In fact, if we let α π, the circular sector polynomials obtained above reduce to the corresponding Zernike circle polynomials. Similarly, if we let α π∕2, we obtain the first four orthonormal polynomials for a semi-circular pupil. B.
Annular Sector Pupil
Consider an annular sector pupil with inner and outer radii of ϵ and unity, and thus an obscuration ratio of ϵ, as illustrated in Fig. 1(b). The pupil is symmetrical about the x axis and subtends a semi-angle α at the center of the circles formed by the arcs of the
We see that the complexity of the polynomials increases because of the obscuration. As in the case of a circular sector, the orthonormal tilt polynomial S2 contains a piston term besides the wavefront tilt so that its mean value over the annular sector is zero. As in the case of a circular pupil, the orthonormal polynomial S4 ρ; θ; ϵ; α represents a defocus aberration ρ2 balanced with tilt aberration ρ cos θ in the form of Eq. (11) such that the variance of the balanced aberration over the annular sector pupil is minimum, the mean value of the polynomials is zero, and its mean square value is unity. It can be shown that, similar to the case of a circular sector pupil [see Eqs. (11)–(15)], the balancing tilt aberration Bt ϵ, the mean value hWϵi, and the standard deviation hσϵi of the balanced aberration are given by 20 February 2013 / Vol. 52, No. 6 / APPLIED OPTICS
1139
12α1 − ϵ2 1 ϵ1 ϵ3 ϵ sin α ; 59∕2α1 ϵ3 2α sin 2α − 161 ϵ ϵ2 sin2 α
(23a)
315α1 ϵ ϵ2 ϵ3 2 2α sin 2α − 641 ϵ ϵ2 1 ϵ ϵ2 ϵ3 ϵ4 sin2 α ; 209∕2α1 ϵ3 2α sin 2α − 161 ϵ ϵ2 sin2 α
(23b)
Bt ϵ −
hWϵi and
hσϵi
1 − ϵ2 2 81 − ϵ4 1 ϵ3 ϵsin2 α − 2 12 259α1 ϵ 1 ϵ2 2α sin 2α − 321 ϵ ϵ2 sin2 α
As α → π, the annular sector polynomials approach the annular polynomials that are orthonormal over an annular pupil with an obscuration ratio of ϵ [9,11,12]. C. Sector Pupil Symmetrical About an Arbitrary Orientation
The orthonormal polynomials for a circular sector pupil with an arbitrary orientation such that its sides make angles α1 and α2 with the x axis, as in Fig. 2(a), or an annular sector pupil with an obscuration ratio ϵ, as in Fig. 2(b), can be obtained in a manner similar to that in Section 2.A or 2.B, respectively. The angular integrations now will be from α1 to α2 . For example, the orthonormality of the polynomials for the circular and annular sector pupils will be described by Z 1Z α 2 1 S ρ; θ; α1 ; α2 hSj Sj0 i α2 − α1 0 α1 j × Sj0 ρ; θ; α1 ; α2 ρdρdθ δjj0 and 1 hSj S i α2 − α1 1 − ϵ2 j0
Z 1Z ϵ
α2 α1
:
(23c)
hWi −3.84, and σ 2 −1.40, while the correct numbers, as obtained from our Eqs. (13)–(15), are Bt −1.24, hWi −0.29, and σ 2 0.005. Similarly, when ϵ 0.8235 and α π∕6, they yield Bt −132.99, hWi −115.31, and σ 2 −65.57, while the correct numbers, as obtained from our Eqs. (23a)–(23c), are Bt −1.22, hWi −0.22, and σ 2 0.003. Hence, the Swantner and Chow equations referred to above are incorrect. 3. Expansion of an Aberration Function in Terms of Orthonormal Polynomials
The wave aberration function Wρ; θ of a sector pupil can be expanded in terms of the orthonormal sector polynomials Sj ρ; θ in the form
(24)
Sj ρ; θ; ϵ; α1 ; α2
× Sj0 ρ; θ; ϵ; α1 ; α2 ρdρdθ δjj0 ;
(25)
respectively. However, the closed-form expressions thus obtained are too complex to be of practical value. It is better to obtain the results for each specific case. As an example, the polynomials for circular and annular sector pupils that are symmetrical about the y axis, as illustrated in Figs. 3(a) and 3(b), can be obtained by letting α1 π∕2 − α and α2 π∕2 α. The first four polynomials thus obtained are similar to those for the corresponding pupil symmetrical about the x axis, except that S2 and S3 exchange with each other and cos θ is replaced by sin θ and vice versa. We have checked Eqs. (13)–(15) given by Swantner and Chow [1] numerically, and found that they yield a negative value of sigma. For example, when ϵ 0 and α π∕6, they yield (approximately) Bt −5.36, 1140
1∕2
APPLIED OPTICS / Vol. 52, No. 6 / 20 February 2013
Fig. 2. Sector pupil of unit radius with its sides making angles of α1 and α2 with the x axis. (a) Circular. (b) Annular with obscuration ratio ϵ.
hWρ; θi
∞ X
aj hSj ρ; θi a1 ;
(28)
j1
as may be seen from the orthonormality equation such as Eq. (6) with j0 1. The mean square value of the aberration function is given by hW 2 ρ; θi
1 A
Z
∞ X pupil j1
∞ X j1
aj hSj ρ; θi
∞ X j0 1
aj0 S0j ρ; θρdρdθ
a2j ;
(29)
where we have utilized the orthonormality of the polynomials. The variance σ 2 of the aberration function is accordingly given by σ 2 hW 2 ρ; θi − hWρ; θi2
∞ X j2
Fig. 3. Sector pupil of unit radius with its sides making angles of α1 π∕2 − α and α2 π∕2 α with the y axis. (a) Circular. (b) Annular with obscuration ratio ϵ.
Wρ; θ
∞ X j1
aj Sj ρ; θ;
(26)
where aj is an expansion or the aberration coefficient of the polynomial Sj ρ; θ. Multiplying both sides of Eq. (26) by S0j ρ; θ, integrating over the sector pupil of area A and utilizing the orthonormality of the polynomials, the aberration coefficients are given by 1 A
Z pupil
Wρ; θS0j ρ; θρdρdθ
∞ 1X a A j1 j
Z pupil
Sj ρ; θS0j ρ; θρdρdθ aj0 ;
or aj
1 A
Z pupil
Wρ; θSj ρ; θρdρdθ:
(27)
It is evident that the value of an expansion coefficient is independent of the number of polynomials used in the expansion. Accordingly, one or more terms can be added to or subtracted from the aberration function without affecting the other coefficients. It is a consequence of the orthogonality of the polynomials. The mean value of the aberration function is given by
a2j ;
(30)
where σ is the standard deviation or the sigma value of the aberration function. Since the mean value of a polynomial (except piston) is zero, each expansion coefficient aj represents the standard deviation of the corresponding polynomial term. The variance of the aberration function is simply the sum of the variances of these polynomial terms. It provides a measure of the quality of the image by way of the Strehl ratio, which for small aberrations is approximately given by exp−σ 2Φ , where σ Φ is the standard deviation of the phase aberration. The overall image quality may be estimated by averaging the variance over the six sectors of a cube-corner retroreflector. 4. Numerical Examples
In this section, we consider an aberration function Wρ; θ 4ρ4 − 5ρ2 10ρ cos θ
(31)
as measured by an interferometer, and determine the orthonormal aberration coefficients for a circular sector pupil and an annular sector pupil with obscuration ratio ϵ 0.5, each with an angular subtense of π∕3 (or α π∕6) and each symmetrical about the x axis. We also consider a circular sector symmetrical about the y axis to show how the orthonormal coefficients change. Finally, we consider the limiting cases of circular and annular [11,12] pupils with the same aberration function and determine the coefficients of these radially symmetric or full pupils. All the polynomials were determined by programming the nonrecursive matrix approach [13] using Mathematica software [14]. The first 11 orthonormal polynomials for a circular sector pupil of angular subtense π∕3 are given in Table 2 when it is symmetrical about the x axis, and in Table 3 when it is symmetrical about the y axis. The polynomials for a circular sector of angular subtense π∕2 symmetrical about the y axis are given in Table 4. We note that a polynomial consists of more and more terms as its order increases. The polynomials for a semi-circular pupil are given in Table 5. 20 February 2013 / Vol. 52, No. 6 / APPLIED OPTICS
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Orthonormal Polynomials for a Circular Sector Pupil with Angular Subtense of π∕3 Symmetrical about the x Axis, as in Fig. 1(a)
Table 2.
S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11
1 4.4081ρ cos θ − 2.8063 4.8084ρ sin θ 14.7738ρ2 − 18.2756ρ cos θ 4.2477 15.7199ρ2 sin 2θ − 23.1380ρ sin θ −2.3267ρ2 13.4384ρ2 cos 2θ − 11.5019ρ cos θ 2.9289 87.0864ρ3 sin θ − 65.2393ρ2 sin 2θ 37.9679ρ sin θ 72.1271ρ3 cos θ − 88.0240ρ2 − 35.9271ρ2 cos 2θ 61.3806ρ cos θ − 7.7589 7.5982ρ3 sin θ 42.8343ρ3 sin 3θ − 87.1692ρ2 sin 2θ 54.9874ρ sin θ −23.0378ρ3 cos θ 49.0241ρ3 cos 3θ 51.5225ρ2 − 83.7513ρ2 cos 2θ 10.8200ρ cos θ − 1.7027 237.8242ρ4 − 578.3556ρ3 cos θ 41.5354ρ3 cos 3θ 312.4650ρ2 95.9653ρ2 cos 2θ − 116.2166ρ cos θ 9.1348
Orthonormal Polynomials for a Circular Sector Pupil with Angular Subtense of π∕3 Symmetrical about the y Axis, as in Fig. 3(a)
Table 3.
S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11
1 4.8084ρ cos θ 4.4081ρ sin θ − 2.8063 14.7738ρ2 − 18.2756ρ sin θ 4.2477 15.7199ρ2 sin 2θ − 23.1380ρ cos θ 2.3267ρ2 13.4384ρ2 cos 2θ 11.5019ρ sin θ − 2.9289 72.1271ρ3 sin θ − 88.0240ρ2 35.9271ρ2 cos 2θ 61.3806ρ sin θ − 7.7589 87.0864ρ3 cos θ − 65.2393ρ2 sin 2θ 37.9679ρ cos θ 23.0378ρ3 sin θ 49.0240ρ3 sin 3θ − 51.5225ρ2 − 83.7513ρ2 cos 2θ − 10.8200ρ sin θ 1.7027 −7.5981ρ3 cos θ 42.8343ρ3 cos 3θ 87.1692ρ2 sin 2θ − 54.9874ρ cos θ 237.8243ρ4 − 578.3546ρ3 sin θ − 41.5354ρ3 sin 3θ 312.4651ρ2 − 95.9653ρ2 cos 2θ − 116.2156ρ sin θ 9.1348
Table 4.
S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11
Orthonormal Polynomials for a Circular Sector Pupil with Angular Subtense of π∕2 Symmetrical about the y Axis, as in Fig. 4
1 3.3178ρ cos θ 4.5221ρ sin θ − 2.7142 10.1720ρ2 − 12.4849ρ sin θ 2.4076 11.1500ρ2 sin 2θ − 14.7336ρ cos θ −7.0559ρ2 9.5665ρ2 cos 2θ 18.2521ρ sin θ − 4.3820 69.5749ρ3 sin θ − 82.2871ρ2 36.6255ρ2 cos 2θ 57.1668ρ sin θ − 6.5661 31.8696ρ3 cos θ − 22.6486ρ2 sin 2θ 8.6814ρ cos θ −17.9479ρ3 sin θ 25.2706ρ3 sin 3θ 11.0762ρ2 − 52.6627ρ2 cos 2θ − 28.0196ρ sin θ 4.0136 −34.0646ρ3 cos θ 27.8620ρ3 cos 3θ 73.1727ρ2 sin 2θ − 41.4386ρ cos θ 93.7045ρ4 − 223.2860ρ3 sin θ − 11.1546ρ3 sin 3θ 106.3560ρ2 − 51.7028ρ2 cos 2θ − 41.1621ρ sin θ 2.9078
Table 5.
S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 1142
Orthonormal Polynomials for a Semi-circular Pupil Symmetrical about the x Axis, as in Fig. 5(a)
1 3.7831ρ cos θ − 1.6056 2ρ sin θ 4.1683ρ2 − 2.5319ρ cos θ − 1.0096 4.4114ρ2 sin 2θ − 2.9956ρ sin θ 6.7981ρ2 7.5887ρ2 cos 2θ − 13.3480ρ cos θ 2.2660 8.9027ρ3 sin θ − 1.4006ρ2 sin 2θ − 4.9840ρ sin θ 20.5600ρ3 cos θ − 13.6275ρ2 − 7.6440ρ2 cos 2θ 0.3233ρ cos θ 1.4414 8.4228ρ3 sin θ 9.2844ρ3 sin 3θ − 14.4709ρ2 sin 2θ 4.2114ρ sin θ 40.7949ρ3 cos θ 15.2277ρ3 cos 3θ − 39.5924ρ2 − 41.0149ρ2 cos 2θ 31.8150ρ cos θ − 2.8023 18.2324ρ4 − 21.1998ρ3 cos θ − 2.1906ρ3 cos 3θ − 6.1076ρ2 7.8677ρ2 cos 2θ 2.5110ρ cos θ 1.1232
APPLIED OPTICS / Vol. 52, No. 6 / 20 February 2013
Table 6.
S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11
Orthonormal Polynomials for an Annular Sector Pupil with Obscuration Ratio ϵ 0.5 and Angular Subtense of π∕3 Symmetrical about the x Axis, as in Fig. 1(b)
1 7.1986ρ cos θ − 5.3465 4.3007ρ sin θ −28.7951ρ cos θ 19.0444ρ2 9.4841 17.5981ρ2 sin 2θ − 26.7660ρ sin θ 27.0338ρ2 cos 2θ − 74.4974ρ cos θ 24.0159ρ2 26.3481 83.4824ρ3 sin θ − 67.1102ρ2 sin 2θ 43.6343ρ sin θ 180.6545ρ3 cos θ − 267.7044ρ2 − 126.9429ρ2 cos 2θ 273.1875ρ cos θ − 59.1057 63.9617ρ3 sin θ 66.6774ρ3 sin 3θ − 191.8725ρ2 sin 2θ 135.5028ρ sin θ 257.4016ρ3 cos θ 103.2660ρ3 cos 3θ − 371.1898ρ2 − 416.5508ρ2 cos 2θ 561.8510ρ cos θ − 130.9662 351.1153ρ4 − 1032.8704ρ3 cos θ 30.8480ρ3 cos 3θ 705.9722ρ2 320.4290ρ2 cos 2θ − 440.8119ρ cos θ 66.3866
Table 7.
S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11
Orthonormal Polynomials for a Semi-annular Pupil with Obscuration Ratio ϵ 0.5 Symmetrical about the x Axis, as in Fig. 5(b)
1 3.8539ρ cos θ − 1.9083 1.7889ρ sin θ 4.9234ρ2 − 1.4225ρ cos θ − 2.3728 3.9259ρ2 sin 2θ − 2.7548ρ sin θ 6.0925ρ2 7.9347ρ2 cos 2θ − 14.6815ρ cos θ 3.4617 9.0714ρ3 sin θ − 0.9678ρ2 sin 2θ − 5.6709ρ sin θ 22.9120ρ3 cos θ − 14.6184ρ2 − 5.3639ρ2 cos 2θ − 6.0598ρ cos θ 4.6007 7.0047ρ3 sin θ 8.2895ρ3 sin 3θ − 13.0446ρ2 sin 2θ 4.2501ρ sin θ 39.3643ρ3 cos θ 16.1588ρ3 cos 3θ − 41.7012ρ2 − 44.7775ρ2 cos 2θ 39.2625ρ cos θ − 4.5536 25.8811ρ4 − 14.6339ρ3 cos θ − 2.5180ρ3 cos 3θ − 21.2183ρ2 8.0154ρ2 cos 2θ − 1.9705ρ cos θ 7.4515
Table 8.
Annular Polynomials Aj ρ;θ; ϵ 0.5 for an Annular Pupil with Obscuration Ratio ϵ 0.5
j
n
m
Aj ρ; θ; ϵ 0.5
Aberration
1 2 3 4 5 6 7 8 9 10 11
0 1 1 2 2 2 3 3 3 3 4
0 1 1 0 2 2 1 1 3 3 0
1 1.7889ρ cos θ 1.7889ρ sin θ 2.30942ρ2 − 1.25 2.1381ρ2 sin 2θ 2.1381ρ2 cos 2θ 2.34873.75ρ3 − 2.625ρ sin θ 2.34873.75ρ3 − 2.625ρ cos θ 2.4543ρ3 sin 3θ 2.4543ρ3 cos 3θ 3.97526ρ4 − 7.5ρ2 2.0625
Piston x tilt y tilt Defocus 45° primary astigmatism 0° primary astigmatism Primary y coma Primary x coma
For example, in Table 3, the S5 polynomial consists of Zernike 45° astigmatism balanced by tilt, and S6 consists of Zernike astigmatism balanced by not only tilt but additional defocus as well. The spherical aberration ρ4 in S11 is balanced not only by defocus but several other lower-order terms as well. Moreover, the balancing defocus has the same sign as the spherical aberration, instead of the opposite sign as in the corresponding Zernike circle polynomial. All of this is a consequence of the lower symmetry of the sector pupil. The orthonormal polynomials for an annular sector pupil, a semi-annular pupil, and an annular pupil of an obscuration ratio ϵ 0.5 are shown in Tables 6, 7, and 8, respectively. Of course, the lower
Spherical aberration
symmetry of the annular sector pupil results in similar balancing of an aberration as for a circular sector pupil. The balancing defocus for spherical aberration in semi-circular and semi-annular pupils does have opposite signs as for the circular and the annular pupils. Using Eq. (27), we obtain the orthonormal coefficients of the aberration function. Thus we may write the aberration function of Eq. (31) in terms of the orthonormal polynomials for the various pupils. They are given below along with their peak-to-valley and sigma values. Circular sector pupil of angular subtense π∕3 symmetrical about the x axis, as in Fig. 1(a): 20 February 2013 / Vol. 52, No. 6 / APPLIED OPTICS
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Wρ; θ; −π∕6; π∕6 5.1995S1 1.9345S2 0.1539S4 0.1395S6 0.1303S8 − 0.0143S10 0.0168S11 ; P−V 9
and
σ 1.9501:
(32a)
(32b)
Circular sector pupil of angular subtense π∕3 symmetrical about the y axis, as in Fig. 3(a): Wρ; θ; π∕3; 2π∕3 −1.6667S1 2.0797S2 − 0.3341S3 − 0.1539S4 − 0.1395S6 0.1303S7 − 0.0143S9 0.0168S11 ; (33a) P − V 10.0135
and
σ 1.9501:
(33b)
Circular sector pupil of angular subtense π∕2 symmetrical about the y axis, as in Fig. 4: Wρ; θ; π∕4; 3π∕4 −1.1667S1 3.0141S2
Fig. 5. Sector pupil of unit radius symmetrical about the x axis. (a) Semi-circular. (b) Semi-annular with obscuration ratio ϵ 0.5.
− 0.3231S3 0.0444S4 − 0.2087S6 0.1419S7 0.0188S9 0.0427S11 ; (34a) P − V 14.1421
and σ 3.2585:
and
σ 2.4798:
(35b)
(34b) Circular pupil (not shown):
Semi-circular pupil symmetrical about the x axis, as in Fig. 5(a):
Wρ; θ; 0; 2π −1.6667Z1 5.0000Z2 − 0.2887Z4 0.2981Z11 ;
Wρ; θ; −π∕2; π∕2 3.0775S1 2.4522S2 − 0.2194S4 0.1079S6
P − V 20
0.1636S8 − 0.0316S10 0.2194S11 ;
P − V 10.5625
(35a)
and σ 5.0172.
(36a) (36b)
Annular sector pupil of angular subtense π∕3 symmetrical about the x axis, as in Fig. 2(a) Wρ; θ; ϵ 0.5; −π∕6; π∕6 6.0522S1 1.3755S2 0.0546S4 0.1412S6 0.0700S8 − 0.0034S10
P − V 5.6699 Fig. 4. Circular sector pupil of unit radius and semi-angular subtense π∕2 with its sides making angles of α1 π∕4 and α2 3π∕4 with the x axis. 1144
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0.0114S11 ;
(37a)
and σ 1.3857:
(37b)
Semi-annular pupil symmetrical about the x axis, as in Fig. 5(b)
Wρ; θ; ϵ 0.5; −π∕2; π∕2 3.5765S1 2.5909S2 0.0018S4 0.0185S6 0.0573S8 − 0.0241S10 0.1546S11 ; P − V 10.5625
and σ 2.5953:
(38a) (38b)
aberration ρ4 balanced by appropriate amounts of defocus ρ2 and y tilt ρ sin θ to minimize its variance, as may be seen by dropping the first four polynomials in Eq. (34a): W R ρ; θ −0.2087S6 0.1419S7 0.0188S9 0.0427S11 4ρ4 − 1.3630ρ2 0.5040ρ sin θ
Annular pupil (not shown):
0.0458.
Wρ; θ; ϵ 0.5; 0; 2π −1.3750A1 5.5902A2 0.1677A11 ; P − V 20
and σ 5.5927:
(39a) (39b)
In Section 2.C, we showed how the orthonormal polynomials change as the orientation of the sector pupil changes from the x to the y axis. Tables 2 and 3 illustrate this fact over a larger number of polynomials. Equations (32a) and (33a) illustrate it with a numerical example. The aberration functions of Eqs. (33a) and (34a) for sector pupils symmetrical about the y axis contain both tilt polynomials S2 and S3 . Note that the defocus polynomial term A4 is missing in Eq. (39a), because the defocus term in the aberration function of Eq. (31) exactly balances its spherical aberration term for an annular pupil of obscuration ratio ϵ 0.5, as may be seen from the polynomial A11 in Table 8. Swantner and Chow also discussed a circular sector pupil of angular subtense π∕2 symmetrical about the y axis and aberrated by primary spherical aberration. It can be shown that the orthonormal polynomials obtained by orthonormalizing 1, ρ cos θ, ρ sin θ, ρ2 , and ρ4 over such a sector are 1;
40a
3.3178ρ cos θ;
(40b)
4.5221ρ sin θ − 2.7142;
(40c)
10.1720ρ2 − 12.4849ρ sin θ 2.4076;
(42)
The factor of 4 is simply a result of the 4 in 4ρ4 in the starting aberration function of Eq. (31), compared to only ρ4 in Eq. (41). It is not surprising that the residual aberration has the same form as the balanced spherical aberration of Eq. (41). Since the starting aberration function consists of spherical
(40d)
15.5885ρ4 − 21.2467ρ2 7.8559ρ sin θ 0.7120: (40e) If we divide the last polynomial by 15.5885, we obtain (approximately) the orthogonal spherical aberration ρ4 − 1.3630ρ2 0.5040ρ sin θ 0.0458;
(41)
considered by Swantner and Chow and plotted in their Fig. 5 [1]. This polynomial represents spherical
Fig. 6. Interferograms of the aberration function of Eq. (31), as described by Eqs. (32)–(36) for the various circular sectors. The left-side interferograms are for the aberration function without removal of the first four aberration polynomial terms of the piston, x and y tilts, and defocus, while the right side is for the residual aberration function after removing the first four terms. 20 February 2013 / Vol. 52, No. 6 / APPLIED OPTICS
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P-V = 20 σ = 5.5927
P-V = 0.5625 σ = 0.1677
P-V = 10.5625 σ = 2.5923
P-V = 0.5765 σ = 0.1678
P-V =5.6699 σ = 1.3857
P-V = 0.7003 σ = 0.1580
Fig. 7. Interferograms of the aberration function of Eq. (31), as described by Eqs. (37)–(39) for the various annular sectors with an obscuration ratio ϵ 0.5. The left-side interferograms are for the aberration function without removal of the first four aberration polynomial terms of the piston, x and y tilts, and defocus, while the right side is for the residual aberration function after removing the first four terms.
aberration combined with defocus and tilt, the residual aberration function has to be spherical aberration balanced with the amount of defocus and tilt that yields minimum variance. However, the sigma value p ofspherical aberration ρ4 over a π∕2 sector is 2∕3 5. When it is balanced by defocus and tilt only, as in Eq. (41), its sigma value decreases by a factor of 4.65. But, if it is balanced in the form of S11, as in Table 4, the sigma value decreases by a factor of 27.94, i.e., a reduction by an additional factor of 6. Figures 6 and 7 show the interferograms of the aberration function of Eq. (31) for the various pupils. In Fig. 6 they are given for the circular sectors, and in Fig. 7 for the annular sectors. The corresponding interferograms of the residual aberration function, when the first four polynomial terms, representing the interferometer setting errors of piston, tip, tilt, and defocus, are removed, are also shown side by side. The peak-to-valley and the sigma values of the aberration function are given in each case below the corresponding interferogram. 5. Discussion and Conclusions
We have considered the problem of a sector pupil, such as those that are formed in an interferogram of a cube corner. In Section 2.A, we obtained the first four orthonormal aberration polynomials by orthogonalizing the Zernike circle polynomials recursively over a sector pupil, as an illustration of the recursive 1146
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process of the Gram–Schmidt orthogonalization. The fourth polynomial represents defocus balanced with tilt. It is more convenient to obtain the orthonormal polynomials nonrecursively using a matrix approach [13]. Because of the lower symmetry of a sector pupil, the closed-form expressions of the polynomials for the general case are quite complex and lengthy, even with the computer algebra programs such as Mathematica, which was used for the calculations in this work [14]. However, it is straightforward to obtain the polynomial expressions for any specific pupil, as illustrated in Section 4 by considering circular and annular sector pupils with an angular subtense of π∕3 (as would be encountered in an interferogram of a cube-corner retroreflector), semi-circular and semi-annular pupils, and circular and annular pupils. The obscuration ratio of each annulus is 0.5. We have considered an aberration function consisting of primary spherical aberration combined with defocus and tilt, and obtained the orthonormal coefficients for the various pupils just mentioned. For the circular sector pupil, we also showed, as an example, how the orthonormal polynomials and the coefficients change when the symmetry axis of a pupil changes from the x to the y axis. We also illustrate a significant advantage in balancing spherical aberration with aberration terms as in the polynomial S11 compared to balancing with just defocus and tilt. The interferograms of the aberration function for the various pupils are shown for the starting aberration function as well as when the first four polynomial terms representing the interferometer errors of piston, x and y tilts, and defocus are removed. In practice, as in the more familiar case of a circular pupil, the aberration data will generally be available at a square array of points and the integrations will be carried out over the sector pupils in the x and y coordinates. The problem of obtaining the sector polynomials was suggested by William H. Swantner. One of the authors (VNM) gratefully acknowledges helpful discussions with him. References 1. W. Swantner and W. W. Chow, “Gram–Schmidt orthogonalization of Zernike polynomials for general aperture shapes,” Appl. Opt. 33, 1832–1837 (1994). 2. D. A. Thomas and J. C. Wyant, “Determination of the dihedral angle errors of a corner cube from its Twyman–Green interferogram,” J. Opt. Soc. Am. 67, 467–472 (1977). 3. R. A. Lessard and S. C. Som, “Imaging properties of sectorshaped apertures,” Appl. Opt. 11, 811–817 (1972). 4. G. Urcid and A. Padilla, “Far-field diffraction patterns of circular sectors and related apertures,” Appl. Opt. 44, 7677–7696 (2005). 5. S. Huang, F. Xi, C. Liu, and Z. Jiang, “Phase retrieval on annular and annular sector pupils by using the eigenfunctions method to solve the transport intensity equation,” J. Opt. Soc. Am. A 29, 513–520 (2012). 6. S. D. Goodrow and T. W. Murphy, Jr., “Effects of thermal gradients in total internal reflection corner cubes,” Appl. Opt. 51, 8793–8799 (2012). 7. T. W. Murphy and S. D. Goodrow, “Polarization and far-field diffraction patterns of total internal reflection corner cubes,” Appl. Opt. 52, 117–126 (2013).
8. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976). 9. V. N. Mahajan, Optical Imaging and Aberrations, Part II: Wave Diffraction Optics, 2nd ed. (SPIE, 2011). 10. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, 1968). 11. V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils,” J. Opt. Soc. Am. 71, 75–85 (1981).
12. V. N. Mahajan, “Zernike annular polynomials and optical aberrations of systems with annular pupils,” Appl. Opt. 33, 8125–8127 (1994). 13. G.-M. Dai and V. N. Mahajan, “Nonrecursive orthonormal polynomials with matrix formulation,” Opt. Lett. 32, 74–76 (2007). 14. Wolfram Research, Inc., Mathematica, Version 8.0, Champaign, Illinois (2010).
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