Abstract

Orthogonal polynomials are routinely used to represent complex surfaces over a specified domain. In optics, Zernike polynomials have found wide application in optical testing, wavefront sensing, and aberration theory. This set is orthogonal over the continuous unit circle matching the typical shape of optical components and pupils. A variety of techniques has been developed to scale Zernike expansion coefficients to concentric circular subregions to mimic, for example, stopping down the aperture size of an optical system. Here, similar techniques are used to rescale the expansion coefficients to new pupil sizes for a related orthogonal set: the pseudo-Zernike polynomials.

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References

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  1. F. Zernike, Physica 1, 689 (1934).
    [CrossRef]
  2. B. R. A. Nijboer, Physica 13, 605 (1947).
    [CrossRef]
  3. D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, 1992).
  4. R. J. Noll, J. Opt. Soc. Am. 66, 207 (1976).
    [CrossRef]
  5. M. Born and E. Wolf, Principles of Optics, 6th ed.(Pergamon, 1980).
  6. R. Mukundan and K. R. Ramakrishnan, Moment Functions in Image Analysis—Theory and Applications (World Scientific, 1998).
    [CrossRef]
  7. A. B. Bhatia and E. Wolf, Math. Proc. Camb. Phil. Soc. 50, 40 (1954).
    [CrossRef]
  8. D. R. Iskander, M. R. Morelande, M. J. Collins, and B. Davis, IEEE Trans. Biomed. Eng. 49, 320 (2002).
    [CrossRef] [PubMed]
  9. “Method for reporting aberrations of eyes,” Tech. Rep. ANSI Z80.28-2004 (American National Standards Institute, 2004).
  10. “Ophthalmic optics and instruments—reporting aberrations of the human eye,” Tech. Rep. ISO 24157:2008 (International Organization for Standardization, 2008).
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    [CrossRef]
  12. C. E. Campbell, J. Opt. Soc. Am. A 20, 209 (2003).
    [CrossRef]
  13. G.-M. Dai, J. Opt. Soc. Am. A 23, 539 (2006).
    [CrossRef]
  14. A. J. E. M. Janssen and P. Dirksen, J. Microlith. Microfab. Microsyst. 5, 030501 (2006).
    [CrossRef]

2008 (1)

“Ophthalmic optics and instruments—reporting aberrations of the human eye,” Tech. Rep. ISO 24157:2008 (International Organization for Standardization, 2008).

2006 (2)

G.-M. Dai, J. Opt. Soc. Am. A 23, 539 (2006).
[CrossRef]

A. J. E. M. Janssen and P. Dirksen, J. Microlith. Microfab. Microsyst. 5, 030501 (2006).
[CrossRef]

2004 (1)

“Method for reporting aberrations of eyes,” Tech. Rep. ANSI Z80.28-2004 (American National Standards Institute, 2004).

2003 (1)

2002 (2)

J. Schwiegerling, J. Opt. Soc. Am. A 19, 1937 (2002).
[CrossRef]

D. R. Iskander, M. R. Morelande, M. J. Collins, and B. Davis, IEEE Trans. Biomed. Eng. 49, 320 (2002).
[CrossRef] [PubMed]

1998 (1)

R. Mukundan and K. R. Ramakrishnan, Moment Functions in Image Analysis—Theory and Applications (World Scientific, 1998).
[CrossRef]

1992 (1)

D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, 1992).

1980 (1)

M. Born and E. Wolf, Principles of Optics, 6th ed.(Pergamon, 1980).

1976 (1)

1954 (1)

A. B. Bhatia and E. Wolf, Math. Proc. Camb. Phil. Soc. 50, 40 (1954).
[CrossRef]

1947 (1)

B. R. A. Nijboer, Physica 13, 605 (1947).
[CrossRef]

1934 (1)

F. Zernike, Physica 1, 689 (1934).
[CrossRef]

Bhatia, A. B.

A. B. Bhatia and E. Wolf, Math. Proc. Camb. Phil. Soc. 50, 40 (1954).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed.(Pergamon, 1980).

Campbell, C. E.

Collins, M. J.

D. R. Iskander, M. R. Morelande, M. J. Collins, and B. Davis, IEEE Trans. Biomed. Eng. 49, 320 (2002).
[CrossRef] [PubMed]

Dai, G.-M.

Davis, B.

D. R. Iskander, M. R. Morelande, M. J. Collins, and B. Davis, IEEE Trans. Biomed. Eng. 49, 320 (2002).
[CrossRef] [PubMed]

Dirksen, P.

A. J. E. M. Janssen and P. Dirksen, J. Microlith. Microfab. Microsyst. 5, 030501 (2006).
[CrossRef]

Iskander, D. R.

D. R. Iskander, M. R. Morelande, M. J. Collins, and B. Davis, IEEE Trans. Biomed. Eng. 49, 320 (2002).
[CrossRef] [PubMed]

Janssen, A. J. E. M.

A. J. E. M. Janssen and P. Dirksen, J. Microlith. Microfab. Microsyst. 5, 030501 (2006).
[CrossRef]

Malacara, D.

D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, 1992).

Morelande, M. R.

D. R. Iskander, M. R. Morelande, M. J. Collins, and B. Davis, IEEE Trans. Biomed. Eng. 49, 320 (2002).
[CrossRef] [PubMed]

Mukundan, R.

R. Mukundan and K. R. Ramakrishnan, Moment Functions in Image Analysis—Theory and Applications (World Scientific, 1998).
[CrossRef]

Nijboer, B. R. A.

B. R. A. Nijboer, Physica 13, 605 (1947).
[CrossRef]

Noll, R. J.

Ramakrishnan, K. R.

R. Mukundan and K. R. Ramakrishnan, Moment Functions in Image Analysis—Theory and Applications (World Scientific, 1998).
[CrossRef]

Schwiegerling, J.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 6th ed.(Pergamon, 1980).

A. B. Bhatia and E. Wolf, Math. Proc. Camb. Phil. Soc. 50, 40 (1954).
[CrossRef]

Zernike, F.

F. Zernike, Physica 1, 689 (1934).
[CrossRef]

IEEE Trans. Biomed. Eng. (1)

D. R. Iskander, M. R. Morelande, M. J. Collins, and B. Davis, IEEE Trans. Biomed. Eng. 49, 320 (2002).
[CrossRef] [PubMed]

J. Microlith. Microfab. Microsyst. (1)

A. J. E. M. Janssen and P. Dirksen, J. Microlith. Microfab. Microsyst. 5, 030501 (2006).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Math. Proc. Camb. Phil. Soc. (1)

A. B. Bhatia and E. Wolf, Math. Proc. Camb. Phil. Soc. 50, 40 (1954).
[CrossRef]

Physica (2)

F. Zernike, Physica 1, 689 (1934).
[CrossRef]

B. R. A. Nijboer, Physica 13, 605 (1947).
[CrossRef]

Other (5)

D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, 1992).

M. Born and E. Wolf, Principles of Optics, 6th ed.(Pergamon, 1980).

R. Mukundan and K. R. Ramakrishnan, Moment Functions in Image Analysis—Theory and Applications (World Scientific, 1998).
[CrossRef]

“Method for reporting aberrations of eyes,” Tech. Rep. ANSI Z80.28-2004 (American National Standards Institute, 2004).

“Ophthalmic optics and instruments—reporting aberrations of the human eye,” Tech. Rep. ISO 24157:2008 (International Organization for Standardization, 2008).

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Equations (23)

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Z n m ( ρ , θ ) = N n m R n m ( ρ ) Θ ( m θ ) .
N n m = 2 ( n + 1 ) 1 + δ m 0 ,
R n m ( ρ ) = s = 0 ( n | m | ) / 2 ( 1 ) s ( n s ) ! s ! [ 0.5 ( n + | m | ) s ] ! [ 0.5 ( n | m | ) s ] ! ρ n 2 s .
Θ ( m θ ) = { cos ( m θ ) ; for     m 0 sin ( m θ ) ; for     m < 0 .
Z ˜ n m ( ρ , θ ) = N n m R ˜ n m ( ρ ) Θ ( m θ ) .
R ˜ n m ( ρ ) = s = 0 n | m | ( 1 ) s ( 2 n + 1 s ) ! s ! [ n | m | s ] ! [ n + | m | + 1 s ] ! ρ n s .
0 1 R n m ( ρ ) R n m ( ρ ) ρ d ρ = 0 1 R ˜ n m ( ρ ) R ˜ n m ( ρ ) ρ d ρ = δ n n 2 ( n + 1 ) .
W ( ρ , θ ) = n , m a n m Z n m ( ρ , θ ) = n , m a n m N n m R n m ( ρ ) Θ ( m θ ) ,
W ( ε ρ , θ ) = n , m b n m N n m R n m ( ρ ) Θ ( m θ ) .
W ( ε ρ , θ ) = n , m a n m N n m R n m ( ε ρ ) Θ ( m θ ) .
n , m b n m N n m R n m ( ρ ) Θ ( m θ ) = n , m a n m N n m R n m ( ε ρ ) Θ ( m θ ) .
n , m b n m N n m [ 0 1 R n m ( ρ ) R n m ( ρ ) ρ d ρ ] Θ ( m θ ) = n , m a n m N n m [ 0 1 R n m ( ε ρ ) R n m ( ρ ) ρ d ρ ] Θ ( m θ ) .
b n m = 2 ( n + 1 ) N n m n a n m N n m M n n m ( ε ) ,
M n n m ( ε ) = 0 1 R n m ( ρ ) R n m ( ε ρ ) ρ d ρ .
M n n m ( ε ) = 1 2 ( n + 1 ) [ R n n ( ε ) R n n + 2 ( ε ) ] ,
b n m = 1 N n m n a n m N n m [ R n n ( ε ) R n n + 2 ( ε ) ] ,
W ( ρ , θ ) = n , m a ˜ n m Z ˜ n m ( ρ , θ ) = n , m a ˜ n m N n m R ˜ n m ( ρ ) Θ ( m θ ) ,
W ( ε ρ , θ ) = n , m b ˜ n m N n m R ˜ n m ( ρ ) Θ ( m θ ) .
b ˜ n m = 2 ( n + 1 ) N n m n a ˜ n m N n m [ 0 1 R ˜ n m ( ε ρ ) R ˜ n m ( ρ ) ρ d ρ ] .
ρ R ˜ n m ( ρ 2 ) = R 2 n + 1 2 m + 1 ( ρ ) .
b ˜ n m = 4 ( n + 1 ) ε N n m n a ˜ n m N n m [ 0 1 R 2 n + 1 2 m + 1 ( ε ρ ) R 2 n + 1 2 m + 1 ( ρ ) ρ d ρ ] = 4 ( n + 1 ) ε N n m n a ˜ n m N n m M 2 n + 1 , 2 n + 1 2 m + 1 ( ε ) .
b ˜ n m = 1 ε N n m n a ˜ n m N n m [ R 2 n + 1 2 n + 1 ( ε ) R 2 n + 1 2 n + 3 ( ε ) ]
b ˜ n m = 1 N n m n a ˜ n m N n m [ R ˜ n n ( ε ) R ˜ n n + 1 ( ε ) ] ,

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