Abstract

The use of Zernike polynomials to calculate the standard deviation of a primary aberration across a circular, annular, or a Gaussian pupil is described. The standard deviation of secondary aberrations is also discussed briefly.

© 1995 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, 8 (August1994).
  2. V. N. Mahajan, “Zernike annular polynomials and optical aberrations of systems with annular pupils,” Eng. & Lab Notes, in Opt. & Phot. News 5, 11 (November1994).
  3. V. N. Mahajan, “Zernike-Gauss polynomials and optical aberrations of systems with Gaussian pupils,” Eng. & Lab Notes, in Opt. & Phot. News 6, 2 (February1995).
  4. V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils,” J. Opt. Soc. Am. 71, 75–85, 1408 (1981), A1, 685 (1984).
    [CrossRef]
  5. 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]
  6. V. N. Mahajan, Aberration Theory Made Simple, SPIE Press, Bellingham, Wash., 76, (1991).
  7. V. N. Mahajan, Aberration Theory Made Simple, SPIE Press, Bellingham, Wash., 120, (1991).

1995 (1)

V. N. Mahajan, “Zernike-Gauss polynomials and optical aberrations of systems with Gaussian pupils,” Eng. & Lab Notes, in Opt. & Phot. News 6, 2 (February1995).

1994 (2)

V. N. Mahajan, “Zernike circle polynomials and optical aberrations of systems with circular pupils,” Eng. & Lab Notes, in Opt. & Phot. News 5, 8 (August1994).

V. N. Mahajan, “Zernike annular polynomials and optical aberrations of systems with annular pupils,” Eng. & Lab Notes, in Opt. & Phot. News 5, 11 (November1994).

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]

1981 (1)

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

Mahajan, V. N.

V. N. Mahajan, “Zernike-Gauss polynomials and optical aberrations of systems with Gaussian pupils,” Eng. & Lab Notes, in Opt. & Phot. News 6, 2 (February1995).

V. N. Mahajan, “Zernike circle polynomials and optical aberrations of systems with circular pupils,” Eng. & Lab Notes, in Opt. & Phot. News 5, 8 (August1994).

V. N. Mahajan, “Zernike annular polynomials and optical aberrations of systems with annular pupils,” Eng. & Lab Notes, in Opt. & Phot. News 5, 11 (November1994).

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), A1, 685 (1984).
[CrossRef]

V. N. Mahajan, Aberration Theory Made Simple, SPIE Press, Bellingham, Wash., 76, (1991).

V. N. Mahajan, Aberration Theory Made Simple, SPIE Press, Bellingham, Wash., 120, (1991).

Eng. & Lab Notes, in Opt. & Phot. News (3)

V. N. Mahajan, “Zernike circle polynomials and optical aberrations of systems with circular pupils,” Eng. & Lab Notes, in Opt. & Phot. News 5, 8 (August1994).

V. N. Mahajan, “Zernike annular polynomials and optical aberrations of systems with annular pupils,” Eng. & Lab Notes, in Opt. & Phot. News 5, 11 (November1994).

V. N. Mahajan, “Zernike-Gauss polynomials and optical aberrations of systems with Gaussian pupils,” Eng. & Lab Notes, in Opt. & Phot. News 6, 2 (February1995).

J. Opt. Soc. Am. (2)

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

Other (2)

V. N. Mahajan, Aberration Theory Made Simple, SPIE Press, Bellingham, Wash., 76, (1991).

V. N. Mahajan, Aberration Theory Made Simple, SPIE Press, Bellingham, Wash., 120, (1991).

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Tables (2)

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Table 1 Standard deviation of primary and balanced primary aberrations for uniform and Gaussian pupils.

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Table 2 Standard deviation of secondary and balanced secondary aberrations for uniform annular pupils.

Equations (20)

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W n m ( ρ , θ ; ; γ ) = c n m [ 2 ( n + 1 ) / ( 1 + δ m 0 ] ) 1 / 2 R n m ( ρ ; ; γ ) cos m θ ,
1 0 2 π W n m ( ρ , θ ; ; γ ) W n m ( ρ , θ ; ; γ ) A ( ρ ) ρ d ρ d θ / 1 0 2 π A ( P ) ρ d ρ d θ = c m m 2 δ n n δ m m ,
A ( ρ ) = A 0 exp ( - γ ρ 2 )
W s ( ρ ) = A s ρ 4 ,
W c ( ρ , θ ) = A c ρ 3 cos θ ,
W a ( ρ , θ ) = A a ρ 3 cos 2 θ ,
W s ( ρ ) = 5 c 40 R 4 0 + 3 c 20 R 2 0 + c 00 R 0 0 ,
c 40 = A s / 5 a 4 0 , c 20 = - A s b 4 0 3 a 4 0 a 2 0 , and c 00 = - ( A s / a 4 0 ) ( c 4 0 - b 4 0 b 2 0 / a 2 0 ) ,
< W s > = c 00
σ s = ( c 40 2 + c 20 2 ) 1 / 2 = A s a 4 0 [ 1 5 + 1 3 ( b 4 0 a 2 0 ) 2 ] 1 / 2 .
W c ( ρ , θ ) = 8 c 31 R 3 1 cos θ + 2 c 11 R 1 1 cos θ ,
c 31 = A c / 8 a 3 1 and c 11 = - 2 c 31 b 3 1 / a 1 1 .
σ c = ( c 31 2 + c 11 2 ) 1 / 2 = A c [ 1 2 + ( b 3 1 / a 1 1 ) 2 ] 1 / 2 2 a 3 1 .
W a ( ρ , θ ) = 6 c 22 R 2 2 cos 2 θ + 3 c 20 R 2 0 + c 00 ,
c 22 = A a / 2 6 a 2 2 , c 20 = A a / 2 3 a 2 0 , and c 00 = - A a b 2 0 / 2 a 2 0 .
σ a = ( c 22 2 + c 20 2 ) 1 / 2 = A a 2 3 [ 1 2 ( a 2 2 ) - 2 + ( a 2 0 ) - 2 ] 1 / 2 .
σ b s = A s / 5 a 4 0 ,
σ b c = A c / 2 2 a 3 1 ,
σ b a = A a / 2 6 a 2 2 .
S = exp ( - σ Φ 2 ) , S 0.3 ,

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