Abstract

Zernike annular polynomials that represent orthogonal and balanced aberrations suitable for systems with annular pupils are described. Their numbering scheme is the same as for Zernike circle polynomials. Expressions for standard deviation of primary and balanced primary aberrations are given.

© 1994 Optical Society of America

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References

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  1. V. N. Mahajan, “Zernike circle polynomials and optical aberrations of systems with circular pupils.” Eng. Lab. Notes in Opt. & Phot. News 5, (1994).
  2. V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils,” J. Opt. Soc. Am. 71, 75–85, 1408 (1981),J. Opt. Soc. Am.A1, 685 (1984).
    [CrossRef]
  3. V. N. Mahajan, “Uniform versus Gaussian beams: A comparison of the effects of diffraction, obscuration, and aberrations,” J. Opt. Soc. Am. A3, 470–485 (1986).
    [CrossRef]
  4. W. H. Steel, “Etude des effets combines des aberrations et d'une obturation centrale de la pupille sur le contraste des images optiques,” Rev. Opt. (Paris) 32, 143–178 (1953).
  5. B. Tatian, “Aberration balancing in rotationally symmetric lenses,” J. Opt. Soc. Am. 64, 1083–1091 (1974).
    [CrossRef]
  6. S. Szapiel, “Aberration balancing techniques for radially symmetric amplitude distributions: a generalization of the Maréchal approach,” J. Opt. Soc. Am. 2, 947–956 (1985).
  7. V. N. Mahajan, Aberration Theory Made Simple, SPIE Press, 1991, Section 9.2.3. Note that the expressions for standard deviation of spherical aberration and astigmatism given in Table 9-2 of the first and second printing of this reference have some errors. Correct expressions appear in the third printing.
    [CrossRef]

1994 (1)

V. N. Mahajan, “Zernike circle polynomials and optical aberrations of systems with circular pupils.” Eng. Lab. Notes in Opt. & Phot. News 5, (1994).

1986 (1)

V. N. Mahajan, “Uniform versus Gaussian beams: A comparison of the effects of diffraction, obscuration, and aberrations,” J. Opt. Soc. Am. A3, 470–485 (1986).
[CrossRef]

1985 (1)

S. Szapiel, “Aberration balancing techniques for radially symmetric amplitude distributions: a generalization of the Maréchal approach,” J. Opt. Soc. Am. 2, 947–956 (1985).

1981 (1)

V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils,” J. Opt. Soc. Am. 71, 75–85, 1408 (1981),J. Opt. Soc. Am.A1, 685 (1984).
[CrossRef]

1974 (1)

1953 (1)

W. H. Steel, “Etude des effets combines des aberrations et d'une obturation centrale de la pupille sur le contraste des images optiques,” Rev. Opt. (Paris) 32, 143–178 (1953).

Mahajan, V. N.

V. N. Mahajan, “Zernike circle polynomials and optical aberrations of systems with circular pupils.” Eng. Lab. Notes in Opt. & Phot. News 5, (1994).

V. N. Mahajan, “Uniform versus Gaussian beams: A comparison of the effects of diffraction, obscuration, and aberrations,” J. Opt. Soc. Am. A3, 470–485 (1986).
[CrossRef]

V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils,” J. Opt. Soc. Am. 71, 75–85, 1408 (1981),J. Opt. Soc. Am.A1, 685 (1984).
[CrossRef]

V. N. Mahajan, Aberration Theory Made Simple, SPIE Press, 1991, Section 9.2.3. Note that the expressions for standard deviation of spherical aberration and astigmatism given in Table 9-2 of the first and second printing of this reference have some errors. Correct expressions appear in the third printing.
[CrossRef]

Steel, W. H.

W. H. Steel, “Etude des effets combines des aberrations et d'une obturation centrale de la pupille sur le contraste des images optiques,” Rev. Opt. (Paris) 32, 143–178 (1953).

Szapiel, S.

S. Szapiel, “Aberration balancing techniques for radially symmetric amplitude distributions: a generalization of the Maréchal approach,” J. Opt. Soc. Am. 2, 947–956 (1985).

Tatian, B.

Eng. Lab. Notes in Opt. & Phot. News (1)

V. N. Mahajan, “Zernike circle polynomials and optical aberrations of systems with circular pupils.” Eng. Lab. Notes in Opt. & Phot. News 5, (1994).

J. Opt. Soc. Am. (4)

V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils,” J. Opt. Soc. Am. 71, 75–85, 1408 (1981),J. Opt. Soc. Am.A1, 685 (1984).
[CrossRef]

V. N. Mahajan, “Uniform versus Gaussian beams: A comparison of the effects of diffraction, obscuration, and aberrations,” J. Opt. Soc. Am. A3, 470–485 (1986).
[CrossRef]

B. Tatian, “Aberration balancing in rotationally symmetric lenses,” J. Opt. Soc. Am. 64, 1083–1091 (1974).
[CrossRef]

S. Szapiel, “Aberration balancing techniques for radially symmetric amplitude distributions: a generalization of the Maréchal approach,” J. Opt. Soc. Am. 2, 947–956 (1985).

Rev. Opt. (Paris) (1)

W. H. Steel, “Etude des effets combines des aberrations et d'une obturation centrale de la pupille sur le contraste des images optiques,” Rev. Opt. (Paris) 32, 143–178 (1953).

Other (1)

V. N. Mahajan, Aberration Theory Made Simple, SPIE Press, 1991, Section 9.2.3. Note that the expressions for standard deviation of spherical aberration and astigmatism given in Table 9-2 of the first and second printing of this reference have some errors. Correct expressions appear in the third printing.
[CrossRef]

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Tables (2)

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Table 1 Zernike radial annular polynomials

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Table 2 Primary aberrations and their standard deviation for optical systems with annular pupils

Equations (32)

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W ( ρ ; θ ; ) = n = 0 m = 0 n [ 2 ( n + 1 ) / ( 1 + δ m 0 ) ] 1 / 2 R n m ( ρ ; ) ( c n m cos m θ + s n m sin m θ ) ,
R n m ( ρ ; ) = N n m [ R n m ( ρ ) i 1 ( n m ) / 2 ( n 2 i + 1 ) < R n m ( ρ ) R n 2 i m ( ρ ; ) > R n 2 i m ( ρ ; ) ] ,
< R n m ( ρ ) R n m ( ρ ; ) > = 2 1 2 1 R n m ( ρ ) R n m ( ρ ; ) ρ d ρ
1 R n m ( ρ ; ) R n m ( ρ ; ) ρ d ρ = 1 2 2 ( n + 1 ) δ n n .
R 2 n 0 ( ρ ; ) = P n [ 2 ( ρ 2 2 ) 1 2 1 ] .
R 2 n 0 ( ρ ; ) = R 2 n 0 [ ( ρ 2 2 1 2 ) 1 / 2 ] .
R n n ( ρ ; ) = ρ n / ( i = 0 n 2 i ) 1 / 2
= ρ n [ ( 1 2 ) / ( 1 2 ( n + 1 ) ) ] 1 / 2
R n n 2 ( ρ ; ) = n ρ n ( n 1 ) [ ( 1 ) 2 n / ( 1 2 ( n 1 ) ) ] ρ n 2 { ( 1 2 ) 1 [ n 2 ( 1 2 ( n + 1 ) ) ( n 2 1 ) ( 1 2 n ) 2 / ( 1 2 ( n 1 ) ) ] } 1 / 2
R n m ( 1 ; ) = 1 , m = 0 1 , m 0 .
W ( ρ , θ ; ) = j = 1 a j Z j ( ρ , θ ; ) ,
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 .
1 0 2 π Z j ( ρ , θ ; ) Z j ( ρ , θ ; ) ρ d ρ d θ / 1 0 2 π ρ d ρ d θ = δ j j .
( c n m , s n m ) = [ π ( 1 2 ) ] 1 [ 2 ( n + 1 ) / ( 1 + δ m 0 ) ] 1 / 2 1 0 2 π W ( ρ , θ ; ) R n m ( ρ ; ) ( cos m θ , sin m θ ) ρ d ρ d θ
a j = [ π ( 1 2 ) ] 1 1 0 2 π W ( ρ , θ ; ) Z j ( ρ , θ ; ) ρ d ρ d θ .
σ w 2 = < W 2 ( ρ , θ ; ) > < W ( ρ , θ ; ) > 2
= n = 1 m = 0 n ( c n m 2 + s n m 2 )
= j = 2 a j 2 ,
< W k ( ρ , θ ; ) > = 1 0 2 π W k ( ρ , θ ; ) ρ d ρ d θ / 1 0 2 π ρ d ρ d θ , k = 1 , 2 .
W ( ρ , ) = A s ρ 4 + A d ρ 2
A d = ( 1 + 2 ) A s .
R n m ( ρ ; )
( 4 2 6 4 6 + 4 8 ) 1 / 2 A s / 3 5
( 1 2 ) 2 A s / 6 5
( 1 + 2 + 4 + 6 ) 1 / 2 A c / 8
( 1 2 ) ( 1 + 4 2 + 4 ) 1 / 2 A c / 6 2 ( 1 + 2 ) 1 / 2
( 1 + 4 ) 1 / 2 A a / 4
( 1 + 2 + 4 ) 1 / 2 A a / 2 6
( 1 2 ) A d / 2 3
( 1 + 2 ) 1 / 2 A t / 2

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