Abstract

In the first two Notes of this series,1,2 we discussed Zernike circle and annular polynomials that represent optimally balanced classical aberrations of systems with uniform circular or annular pupils, respectively. Here we discuss Zernike-Gauss polynomials which are the corresponding polynomials for systems with Gaussian circular or annular pupils.35 Such pupils, called apodized pupils, are used in optical imaging to reduce the secondary rings of the point-spread functions of uniform pupils.6 Propagation of Gaussian laser beams also involves such pupils.

© 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 (1994).
  2. V. N. Mahajan, “Zernike annular polynomials and optical aberrations of systems with annular pupils.” Eng. Lab. Notes in Opt. & Phot. News 5 (11) (1994).
  3. V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils,” J. Opt. Soc. Am. 71, 75–85,1408 (1981), Al, 685 (1984).
    [CrossRef]
  4. 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]
  5. S. Szapiel, “Aberration balancing techniques for radially symmetric amplitude distributions: A generalization of the Maréchal approach,” J. Opt. Soc. Am. 72,947–956 (1982).
    [CrossRef]
  6. P. Jacquinot, B. Roisen-Dossier, “Apodization,” Prog. Opt. 3,29–186 (1964).
    [CrossRef]

1994 (2)

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

1982 (1)

1981 (1)

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

1964 (1)

P. Jacquinot, B. Roisen-Dossier, “Apodization,” Prog. Opt. 3,29–186 (1964).
[CrossRef]

Jacquinot, P.

P. Jacquinot, B. Roisen-Dossier, “Apodization,” Prog. Opt. 3,29–186 (1964).
[CrossRef]

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, “Zernike annular polynomials and optical aberrations of systems with annular pupils.” Eng. Lab. Notes in Opt. & Phot. News 5 (11) (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), Al, 685 (1984).
[CrossRef]

Roisen-Dossier, B.

P. Jacquinot, B. Roisen-Dossier, “Apodization,” Prog. Opt. 3,29–186 (1964).
[CrossRef]

Szapiel, S.

Eng. Lab. Notes in Opt. & Phot. News (2)

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, “Zernike annular polynomials and optical aberrations of systems with annular pupils.” Eng. Lab. Notes in Opt. & Phot. News 5 (11) (1994).

J. Opt. Soc. Am. (3)

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

S. Szapiel, “Aberration balancing techniques for radially symmetric amplitude distributions: A generalization of the Maréchal approach,” J. Opt. Soc. Am. 72,947–956 (1982).
[CrossRef]

Prog. Opt. (1)

P. Jacquinot, B. Roisen-Dossier, “Apodization,” Prog. Opt. 3,29–186 (1964).
[CrossRef]

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Tables (2)

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Table 1 Radial polynomials for orthogonal primary aberrations for uniform and Gaussian pupils

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Table 2 Coefficients of ρ n in radial polynomials R n m ( ρ ; γ ; ) for orthogonal primary aberrations for uniform (γ = 0) and Gaussian (γ = 1) pupils.1

Equations (13)

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A ( ρ ) = A 0 exp ( - γ ρ 2 )
γ = ( a / ω ) 2
W ( ρ , θ ; γ ) = n = 0 m = 0 n [ 2 ( n + 1 ) / ( 1 + δ m 0 ] 1 / 2             R n m ( ρ ; γ ) ( c n m cos m θ + s n m sin m θ ) ,
R n m ( ρ ; γ ) = a n m ρ n + b n m ρ n - 2 + + d n m ρ m ,
R n m ( ρ ; γ ) = M 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 ; γ ) > R n - 2 i ( ρ ; γ ) ] .
< R n m ( ρ ) R n - 2 i m ( ρ ; γ ) > = 0 1 R n m ( ρ ) R n - 2 i m ( ρ ; γ ) A ( ρ ) ρ d ρ / 0 1 A ( ρ ) ρ d ρ .
0 1 R n m ( ρ ; γ ) R n m ( ρ ; γ ) A ( ρ ) ρ d ρ / 0 1 A ( ρ ) ρ d ρ = 1 n + 1 δ n n .
R n n ( ρ ; γ ) = M n n R n n ( ρ ) .
( c n m , s n m ) = [ 2 ( n + 1 ) / ( 1 + δ m 0 ) ] 1 / 2 0 1 0 2 π W ( ρ , θ ; γ ) R n m ( ρ ; γ ) ( cos m θ , sin m θ ) A ( ρ ) ρ d ρ d θ / 0 1 0 2 π A ( ρ ) ρ d ρ d θ .
σ w 2 = < W 2 ( ρ , θ ; γ ) > - < W ( ρ , θ ; γ ) > 2 = n = 1 x m = 1 n ( c n m 2 + s n m 2 ) ,
< W k ( ρ , θ ; γ ) > = 0 1 0 2 π W k ( ρ , θ ; γ ) A ( ρ ) ρ d ρ d θ / 0 1 0 2 π A ( ρ ) ρ d ρ d θ ,             k = 1 , 2.
1 A ( ρ ) R n m ( ρ ; γ ; ) R n m ( ρ ; γ ; ) ρ d ρ / 1 A ( ρ ) ρ d ρ = 1 n + 1 δ n n .
1 0 2 π A ( P ) Z j ( ρ , θ ; ; γ ) Z j ( ρ , θ ; ; γ ) ρ d ρ d θ / 1 0 2 π A ( P ) ρ d ρ d θ = δ j j .

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