Abstract

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

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References

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2013

2012

2007

2005

1994

1981

1977

1976

1972

Chow, W. W.

Dai, G.-M.

Goodrow, S. D.

Huang, S.

Jiang, Z.

Korn, A.

A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, 1968).

Korn, T. M.

A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, 1968).

Lessard, R. A.

Liu, C.

Mahajan, V. N.

Murphy, T. W.

Noll, R. J.

Padilla, A.

Som, S. C.

Swantner, W.

Thomas, D. A.

Urcid, G.

Wyant, J. C.

Xi, F.

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Lett.

Other

V. N. Mahajan, Optical Imaging and Aberrations, Part II: Wave Diffraction Optics, 2nd ed. (SPIE, 2011).

A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, 1968).

Wolfram Research, Inc., Mathematica, Version 8.0, Champaign, Illinois (2010).

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Figures (7)

Fig. 1.
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 .

Fig. 2.
Fig. 2.

Sector pupil of unit radius with its sides making angles of α 1 and α 2 with the x axis. (a) Circular. (b) Annular with obscuration ratio ϵ .

Fig. 3.
Fig. 3.

Sector pupil of unit radius with its sides making angles of α 1 = π / 2 α and α 2 = π / 2 + α with the y axis. (a) Circular. (b) Annular with obscuration ratio ϵ .

Fig. 4.
Fig. 4.

Circular sector pupil of unit radius and semi-angular subtense π / 2 with its sides making angles of α 1 = π / 4 and α 2 = 3 π / 4 with the x axis.

Fig. 5.
Fig. 5.

Sector pupil of unit radius symmetrical about the x axis. (a) Semi-circular. (b) Semi-annular with obscuration ratio ϵ = 0.5 .

Fig. 6.
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.

Fig. 7.
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.

Tables (8)

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Table 1. Orthonormal Zernike Circle Polynomials Z j ( ρ , θ )

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Table 2. Orthonormal Polynomials for a Circular Sector Pupil with Angular Subtense of π / 3 Symmetrical about the x Axis, as in Fig. 1(a)

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Table 3. Orthonormal Polynomials for a Circular Sector Pupil with Angular Subtense of π / 3 Symmetrical about the y Axis, as in Fig. 3(a)

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Table 4. Orthonormal Polynomials for a Circular Sector Pupil with Angular Subtense of π / 2 Symmetrical about the y Axis, as in Fig. 4

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Table 5. Orthonormal Polynomials for a Semi-circular Pupil Symmetrical about the x Axis, as in Fig. 5(a)

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Table 6. 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)

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Table 7. Orthonormal Polynomials for a Semi-annular Pupil with Obscuration Ratio ϵ = 0.5 Symmetrical about the x Axis, as in Fig. 5(b)

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Table 8. Annular Polynomials A j ( ρ , θ ; ϵ = 0.5 ) for an Annular Pupil with Obscuration Ratio ϵ = 0.5

Equations (70)

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Z even j ( ρ , θ ) = 2 ( n + 1 ) R n m ( ρ ) cos m θ , m 0 ,
Z odd j ( ρ , θ ) = 2 ( n + 1 ) R n m ( ρ ) sin m θ , m 0 ,
Z j ( ρ , θ ) = n + 1 R n 0 ( ρ ) , m = 0 ,
R n m ( ρ ) = s = 0 ( n m ) / 2 ( 1 ) s ( n s ) ! s ! ( n + m 2 s ) ! ( n m 2 s ) ! ρ n 2 s ,
0 1 0 2 π Z j ( ρ , θ ) Z j ( ρ , θ ) ρ d ρ d θ / 0 1 0 2 π ρ d ρ d θ = δ j j .
S j + 1 = N j + 1 [ Z j + 1 k = 1 j Z j + 1 S k S k ] ,
S 1 = 1 ,
S j S j = 1 α 0 1 α α S j ( ρ , θ ; α ) S j ( ρ , θ ; α ) ρ d ρ d θ = δ j j .
S 2 = N 2 ( Z 2 Z 2 ) ,
Z 2 = 1 α 0 1 α α 2 ρ cos θ ρ d ρ d θ = 4 3 sin α α ,
Z 2 2 = 4 α 0 1 α α ρ 2 cos 2 θ ρ d ρ d θ = ( 2 α + sin 2 α ) / 2 α ,
S 2 2 = N 2 2 ( Z 2 Z 2 ) 2 = N 2 2 [ Z 2 2 Z 2 2 ] = N 2 2 [ 1 2 α ( 2 α + sin 2 α ) ( 4 3 sin α α ) 2 ] .
N 2 = [ 1 2 α ( 2 α + sin 2 α ) ( 4 3 sin α α ) 2 ] 1 / 2
S 2 ( ρ , θ ; α ) = 2 ρ cos θ 4 3 sin α α [ 1 2 α ( 2 α + sin 2 α ) ( 4 3 sin α α ) 2 ] 1 / 2 .
S 3 = N 3 ( Z 3 Z 3 Z 3 S 2 S 2 ) .
Z 3 = 0 , Z 3 S 2 = 0 , S 3 = 0 , S 3 = N 3 Z 3 .
S 3 2 = N 3 2 Z 3 2
Z 3 2 = 4 α 0 1 α α ρ 2 sin 2 θ ρ d ρ d θ = ( 2 α sin 2 α ) / 2 α .
N 3 = [ ( 2 α sin 2 α ) / 2 α ] 1 / 2
S 3 ( ρ , θ ; α ) = 2 ρ sin θ [ ( 2 α sin 2 α ) / 2 α ] 1 / 2 .
S 4 = N 4 ( Z 4 Z 4 Z 4 S 2 S 2 Z 4 S 3 S 3 ) = N 4 ( Z 4 Z 4 S 2 S 2 S 3 ) ,
S 4 ( ρ , θ ; α ) = 1 N 4 { 3 ( 2 ρ 2 1 ) 12 6 sin α ( 2 ρ cos θ 4 sin α / 3 α ) ( 5 / 2 α ) [ 9 α ( 2 α + sin 2 α ) 32 sin 2 α ] } ,
N 4 = 1 5 [ 25 96 sin 2 α 9 α ( 2 α + sin 2 α ) 32 sin 2 α ] 1 / 2 .
W = ρ 2 + B t ρ cos θ
S 4 = ρ 2 + B t ρ cos θ W σ ,
B t = sin α 15 α ( 1 4 + sin 2 α 8 α 4 sin 2 α 9 α 2 ) ,
W = 1 2 16 sin 2 α 5 [ 9 α ( 2 α + sin 2 α ) 32 sin 2 α ]
σ = [ W 2 W 2 ] 1 / 2 = 1 10 [ 25 3 32 sin 2 α 9 α ( 2 α + sin 2 α ) 32 sin 2 α ] 1 / 2
N n = ( n + 1 ) ( n + 2 ) 2 .
S j S j = 1 α ( 1 ϵ 2 ) ϵ 1 α α S j ( ρ , θ ; ϵ ; α ) S j ( ρ , θ ; ϵ ; α ) ρ d ρ d θ = δ j j
S 1 = 1 ,
S 2 ( ρ , θ ; ϵ ; α ) = 2 ρ cos θ 4 3 1 + ϵ + ϵ 2 1 + ϵ sin α α [ ( 1 + ϵ 2 ) 2 α ( 2 α + sin 2 α ) ( 4 3 1 + ϵ + ϵ 2 1 + ϵ sin α α ) 2 ] 1 / 2 ,
S 3 ( ρ , θ ; ϵ ; α ) = 2 ρ sin θ [ ( 1 + ϵ 2 ) ( 2 α sin 2 α ) / 2 α ] 1 / 2 ,
S 4 ( ρ , θ ; ϵ ; α ) = 3 N 4 [ 2 ρ 2 1 ϵ 2 + 16 ( 1 ϵ 2 ) [ 1 + ϵ ( 3 + ϵ ) ] sin α [ 3 α ( 1 + ϵ ) ρ cos θ + 2 ( 1 + ϵ + ϵ 2 ) sin α ] 32 ( 1 + ϵ + ϵ 2 ) 2 sin 2 α 9 α ( 1 + ϵ ) 2 ( 1 + ϵ 2 ) ( 2 α + sin 2 α ) ] ,
N 4 = 1 ϵ 5 × { 128 [ ( 7 ϵ 6 + 28 ϵ 5 + 50 ϵ 4 + 55 ϵ 3 + 50 ϵ 2 + 28 ϵ + 7 ) sin 2 α 225 α ( 1 + ϵ ) 4 ( 1 + ϵ 2 ) ( 2 α + sin 2 α ) ] 32 ( 1 + ϵ + ϵ 2 ) 2 sin 2 α 9 α ( 1 + ϵ ) 2 ( 1 + ϵ 2 ) ( 2 α + sin 2 α ) } 1 / 2 .
B t ( ϵ ) = 12 α ( 1 ϵ 2 ) ( 1 + ϵ ) [ 1 + ϵ ( 3 + ϵ ) ] sin α 5 [ ( 9 / 2 ) α ( 1 + ϵ ) 3 ( 2 α + sin 2 α ) 16 ( 1 + ϵ + ϵ 2 ) sin 2 α ] ,
W ( ϵ ) = 3 [ 15 α ( 1 + ϵ + ϵ 2 + ϵ 3 ) 2 ( 2 α + sin 2 α ) 64 ( 1 + ϵ + ϵ 2 ) ( 1 + ϵ + ϵ 2 + ϵ 3 + ϵ 4 ) sin 2 α ] 20 [ ( 9 / 2 ) α ( 1 + ϵ ) 3 ( 2 α + sin 2 α ) 16 ( 1 + ϵ + ϵ 2 ) sin 2 α ] ,
σ ( ϵ ) = { ( 1 ϵ 2 ) 2 12 8 ( 1 ϵ ) 4 [ 1 + ϵ ( 3 + ϵ ) ] sin 2 α 25 [ 9 α ( 1 + ϵ ) 2 ( 1 + ϵ 2 ) ( 2 α + sin 2 α ) 32 ( 1 + ϵ + ϵ 2 ) sin 2 α ] } 1 / 2 .
S j S j = 1 α 2 α 1 0 1 α 1 α 2 S j ( ρ , θ ; α 1 ; α 2 ) S j ( ρ , θ ; α 1 ; α 2 ) ρ d ρ d θ = δ j j
S j S j = 1 ( α 2 α 1 ) ( 1 ϵ 2 ) ϵ 1 α 1 α 2 S j ( ρ , θ ; ϵ ; α 1 ; α 2 ) S j ( ρ , θ ; ϵ ; α 1 ; α 2 ) ρ d ρ d θ = δ j j ,
W ( ρ , θ ) = j = 1 a j S j ( ρ , θ ) ,
1 A pupil W ( ρ , θ ) S j ( ρ , θ ) ρ d ρ d θ = 1 A j = 1 a j pupil S j ( ρ , θ ) S j ( ρ , θ ) ρ d ρ d θ = a j ,
a j = 1 A pupil W ( ρ , θ ) S j ( ρ , θ ) ρ d ρ d θ .
W ( ρ , θ ) = j = 1 a j S j ( ρ , θ ) = a 1 ,
W 2 ( ρ , θ ) = 1 A pupil j = 1 a j S j ( ρ , θ ) j = 1 a j S j ( ρ , θ ) ρ d ρ d θ = j = 1 a j 2 ,
σ 2 = W 2 ( ρ , θ ) W ( ρ , θ ) 2 = j = 2 a j 2 ,
W ( ρ , θ ) = 4 ρ 4 5 ρ 2 + 10 ρ cos θ
W ( ρ , θ ; π / 6 , π / 6 ) = 5.1995 S 1 + 1.9345 S 2 + 0.1539 S 4 + 0.1395 S 6 + 0.1303 S 8 0.0143 S 10 + 0.0168 S 11 ,
P V = 9 and σ = 1.9501 .
W ( ρ , θ ; π / 3 , 2 π / 3 ) = 1.6667 S 1 + 2.0797 S 2 0.3341 S 3 0.1539 S 4 0.1395 S 6 + 0.1303 S 7 0.0143 S 9 + 0.0168 S 11 ,
P V = 10.0135 and σ = 1.9501 .
W ( ρ , θ ; π / 4 , 3 π / 4 ) = 1.1667 S 1 + 3.0141 S 2 0.3231 S 3 + 0.0444 S 4 0.2087 S 6 + 0.1419 S 7 + 0.0188 S 9 + 0.0427 S 11 ,
P V = 14.1421 and σ = 3.2585 .
W ( ρ , θ ; π / 2 , π / 2 ) = 3.0775 S 1 + 2.4522 S 2 0.2194 S 4 + 0.1079 S 6 + 0.1636 S 8 0.0316 S 10 + 0.2194 S 11 ,
P V = 10.5625 and σ = 2.4798 .
W ( ρ , θ ; 0 , 2 π ) = 1.6667 Z 1 + 5.0000 Z 2 0.2887 Z 4 + 0.2981 Z 11 ,
P V = 20 and σ = 5.0172.
W ( ρ , θ ; ϵ = 0.5 ; π / 6 , π / 6 ) = 6.0522 S 1 + 1.3755 S 2 + 0.0546 S 4 + 0.1412 S 6 + 0.0700 S 8 0.0034 S 10 + 0.0114 S 11 ,
P V = 5.6699 and σ = 1.3857 .
W ( ρ , θ ; ϵ = 0.5 ; π / 2 , π / 2 ) = 3.5765 S 1 + 2.5909 S 2 + 0.0018 S 4 + 0.0185 S 6 + 0.0573 S 8 0.0241 S 10 + 0.1546 S 11 ,
P V = 10.5625 and σ = 2.5953 .
W ( ρ , θ ; ϵ = 0.5 ; 0 , 2 π ) = 1.3750 A 1 + 5.5902 A 2 + 0.1677 A 11 ,
P V = 20 and σ = 5.5927 .
1 ,
3.3178 ρ cos θ ,
4.5221 ρ sin θ 2.7142 ,
10.1720 ρ 2 12.4849 ρ sin θ + 2.4076 ,
15.5885 ρ 4 21.2467 ρ 2 + 7.8559 ρ sin θ + 0.7120 .
ρ 4 1.3630 ρ 2 + 0.5040 ρ sin θ + 0.0458 ,
W R ( ρ , θ ) = 0.2087 S 6 + 0.1419 S 7 + 0.0188 S 9 + 0.0427 S 11 = 4 ( ρ 4 1.3630 ρ 2 + 0.5040 ρ sin θ + 0.0458 ) .

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