Abstract

By expressing a scaled Zernike radial polynomial as a linear combination of the unscaled radial polynomials, we give a simple derivation for determining the Zernike coefficients of an aberration function of a scaled pupil in terms of their values for a corresponding unscaled pupil.

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References

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  1. J. Schwiegerling, “Scaling Zernike expansion coefficients to different pupil sizes,” J. Opt. Soc. Am. A 19, 1937–1945(2002).
    [CrossRef]
  2. C. E. Campbell, “Matrix method to find a new set of Zernike coefficients from an original set when the aperture radius is changed,” J. Opt. Soc. Am. A 20, 209–217 (2003).
    [CrossRef]
  3. G.-m. Dai, “Scaling Zernike expansion coefficients to smaller pupil sizes: a simpler formula,” J. Opt. Soc. Am. A 23, 539–547 (2006).
    [CrossRef]
  4. H. Shu, L. Luo, G. Han, and J.-L. Coatrieux, “General method to derive the relationship between two sets of Zernike coefficients corresponding to different aperture sizes,” J. Opt. Soc. Am. A 23, 1960–1966 (2006).
    [CrossRef]
  5. A. J. E. M. Janssen and P. Dirksen, “Concise formula for the Zernike coefficients of scaled pupils,” J. Microlith., Microfab., Microsyst. 5, 030501 (2006).
    [CrossRef]
  6. J. A. Diaz, J. Fernandez-Dorado, C. Pizarro, and J. Arasa, “Zernike coefficients for concentric, circular pupils: an equivalent expression,” J. Mod. Opt. 56, 131 (2009).
    [CrossRef]
  7. V. N. Mahajan, Optical Imaging and Aberrations, Part II: Wave Diffraction Optics (SPIE Press, 2004), second printing.
  8. G.-m. Dai and V. N. Mahajan, “Zernike annular polynomials and atmospheric turbulence,” J. Opt. Soc. Am. A 24, 139–155 (2007).
    [CrossRef]
  9. V. N. Mahajan, “Zernike polynomials and wavefront fitting,” in Optical Shop Testing, 3rd ed., D.Malacara, ed. (Wiley, 2007), pp. 498–546.
    [CrossRef]

2009

J. A. Diaz, J. Fernandez-Dorado, C. Pizarro, and J. Arasa, “Zernike coefficients for concentric, circular pupils: an equivalent expression,” J. Mod. Opt. 56, 131 (2009).
[CrossRef]

2007

2006

2003

2002

Arasa, J.

J. A. Diaz, J. Fernandez-Dorado, C. Pizarro, and J. Arasa, “Zernike coefficients for concentric, circular pupils: an equivalent expression,” J. Mod. Opt. 56, 131 (2009).
[CrossRef]

Campbell, C. E.

Coatrieux, J.-L.

Dai, G.-m.

Diaz, J. A.

J. A. Diaz, J. Fernandez-Dorado, C. Pizarro, and J. Arasa, “Zernike coefficients for concentric, circular pupils: an equivalent expression,” J. Mod. Opt. 56, 131 (2009).
[CrossRef]

Dirksen, P.

A. J. E. M. Janssen and P. Dirksen, “Concise formula for the Zernike coefficients of scaled pupils,” J. Microlith., Microfab., Microsyst. 5, 030501 (2006).
[CrossRef]

Fernandez-Dorado, J.

J. A. Diaz, J. Fernandez-Dorado, C. Pizarro, and J. Arasa, “Zernike coefficients for concentric, circular pupils: an equivalent expression,” J. Mod. Opt. 56, 131 (2009).
[CrossRef]

Han, G.

Janssen, A. J. E. M.

A. J. E. M. Janssen and P. Dirksen, “Concise formula for the Zernike coefficients of scaled pupils,” J. Microlith., Microfab., Microsyst. 5, 030501 (2006).
[CrossRef]

Luo, L.

Mahajan, V. N.

G.-m. Dai and V. N. Mahajan, “Zernike annular polynomials and atmospheric turbulence,” J. Opt. Soc. Am. A 24, 139–155 (2007).
[CrossRef]

V. N. Mahajan, Optical Imaging and Aberrations, Part II: Wave Diffraction Optics (SPIE Press, 2004), second printing.

V. N. Mahajan, “Zernike polynomials and wavefront fitting,” in Optical Shop Testing, 3rd ed., D.Malacara, ed. (Wiley, 2007), pp. 498–546.
[CrossRef]

Pizarro, C.

J. A. Diaz, J. Fernandez-Dorado, C. Pizarro, and J. Arasa, “Zernike coefficients for concentric, circular pupils: an equivalent expression,” J. Mod. Opt. 56, 131 (2009).
[CrossRef]

Schwiegerling, J.

Shu, H.

J. Microlith., Microfab., Microsyst.

A. J. E. M. Janssen and P. Dirksen, “Concise formula for the Zernike coefficients of scaled pupils,” J. Microlith., Microfab., Microsyst. 5, 030501 (2006).
[CrossRef]

J. Mod. Opt.

J. A. Diaz, J. Fernandez-Dorado, C. Pizarro, and J. Arasa, “Zernike coefficients for concentric, circular pupils: an equivalent expression,” J. Mod. Opt. 56, 131 (2009).
[CrossRef]

J. Opt. Soc. Am. A

Other

V. N. Mahajan, Optical Imaging and Aberrations, Part II: Wave Diffraction Optics (SPIE Press, 2004), second printing.

V. N. Mahajan, “Zernike polynomials and wavefront fitting,” in Optical Shop Testing, 3rd ed., D.Malacara, ed. (Wiley, 2007), pp. 498–546.
[CrossRef]

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Equations (38)

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W ( ρ , θ ) = j 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 π 0 1 0 2 π Z j ( ρ , θ ) Z j ( ρ , θ ) ρ d ρ d θ = δ j j ,
δ j j
a j = 1 π 0 1 0 2 π W ( ρ , θ ) Z j ( ρ , θ ) ρ d ρ d θ ,
W ( ϵ ρ , θ ) = j a j Z j ( ϵ ρ , θ ) .
W ϵ ( ρ , θ ) = j b j Z j ( ρ , θ ) ,
b j = 1 π 0 1 0 2 π W ϵ ( ρ , θ ) Z j ( ρ , θ ) ρ d ρ d θ ,
b j = 1 π 0 1 0 2 π W ( ϵ ρ , θ ) Z j ( ρ , θ ) ρ d ρ d θ .
b j = 1 π j 0 1 0 2 π a j Z j ( ϵ ρ , θ ) Z j ( ρ , θ ) ρ d ρ d θ .
b n m = 2 n + 1 n n + 1 a n m 0 1 R n m ( ϵ ρ ) R n m ( ρ ) ρ d ρ ,
R n m ( ϵ ρ ) = n = 0 n h n ( n ; ϵ ) R n m ( ρ ) ,
h n ( n ; ϵ ) = ( n + 1 ) s s ( 1 ) s ( n s ) ! ϵ n 2 s s ! s ! ( n + s + 1 ) ! ,
0 1 R n m ( ρ ) R n m ( ρ ) ρ d ρ = 1 2 ( n + 1 ) δ n n ,
b n m = n n + 1 n + 1 h n ( n ; ϵ ) a n m .
W ( ρ , θ ) = c 11 ρ cos θ + c 20 ρ 2 + c 22 ρ 2 cos 2 θ + c 31 ρ 3 cos θ + c 40 ρ 4 ,
W ( ρ , θ ) = a 00 Z 0 0 + a 11 Z 1 1 + a 20 Z 2 0 + a 22 Z 2 2 + a 31 Z 3 1 + a 40 Z 4 0 ,
a 00 = c 20 2 + c 22 4 + c 40 3 , a 11 = c 11 2 + c 31 3 , a 20 = c 20 2 3 + c 22 4 3 + c 40 2 3 ,
a 22 = c 22 2 6 , a 31 = c 31 6 2 , a 40 = c 40 6 5 .
σ 2 = a 11 2 + a 20 2 + a 22 2 + a 31 2 + a 40 2 .
h n ( n ; ϵ ) = ϵ n ,
h n 2 ( n ; ϵ ) = ( n 1 ) ( 1 ϵ 2 ) ϵ n 2 ,
h n 4 ( n ; ϵ ) = n 3 2 ( 1 ϵ 2 ) ( n 2 n ϵ 2 ) ϵ n 4 .
W ϵ ( ρ , θ ) = b 00 Z 0 0 + b 11 Z 1 1 + b 20 Z 2 0 + b 22 Z 2 2 + b 31 Z 3 1 + b 40 Z 4 0 ,
b 00 = a 00 h 0 ( 0 ; ϵ ) + 3 h 0 ( 2 ; ϵ ) a 20 + 5 h 0 ( 4 ; ϵ ) a 40 = a 00 3 ( 1 ϵ 2 ) a 20 + 5 ( 1 ϵ 2 ) ( 1 2 ϵ 2 ) a 40 ,
b 11 = h 1 ( 1 ; ϵ ) a 11 + 2 h 1 ( 3 ; ϵ ) a 31 = ϵ [ a 11 2 2 ( 1 ϵ 2 ) a 31 ] ,
b 20 = h 2 ( 2 ; ϵ ) a 20 + 5 / 3 h 2 ( 4 ; ϵ ) a 40 = ϵ 2 [ a 20 15 ( 1 ϵ 2 ) a 40 ] ,
b 22 = h 2 ( 2 ; ϵ ) a 22 = ϵ 2 a 22 ,
b 31 = h 3 ( 3 ; ϵ ) a 31 = ϵ 3 a 31 ,
b 40 = h 4 ( 4 ; ϵ ) a 40 = ϵ 4 a 40 .
σ ϵ 2 = b 11 2 + b 20 2 + b 22 2 + b 31 2 + b 40 2 .
W ( ϵ ρ , θ ) = c 11 ϵ ρ cos θ + c 20 ϵ 2 ρ 2 + c 22 ϵ 2 ρ 2 cos 2 θ + c 31 ϵ 3 ρ 3 cos θ + c 40 ϵ 4 ρ 4 .
W ϵ ( ρ , θ ) = c 11 ρ cos θ + c 20 ρ 2 + c 22 ρ 2 cos 2 θ + c 31 ρ 3 cos θ + c 40 ρ 4 ,
c 11 = c 11 ϵ , c 20 = c 20 ϵ 2 , c 22 = c 22 ϵ 2 , c 31 = c 31 ϵ 3 , c 40 = c 40 ϵ 4 .
a 00 = 13 / 12 , a 11 = 5 / 6 , a 20 = 5 / 4 3 , a 22 = 1 / 2 6 , a 31 = 1 / 6 2 , a 40 = 1 / 6 5 .
b 00 = 0.6165 , b 11 = 0.5707 , b 20 = 0.3954 , b 22 = 0.1306 , b 31 = 0.0603 , b 40 = 0.0305 .

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