Abstract

A numerically efficient algorithm for expanding a function in a series of Zernike polynomials is presented. The algorithm evaluates the expansion coefficients through the standard 2-D integration formula derived from the Zernike polynomials’ orthogonal properties. Quadratic approximations are used along with the function to be expanded to eliminate the computational problems associated with integrating the oscillatory behavior of the Zernike polynomials. This yields a procedure that is both fast and numerically accurate. Comparisons are made between the proposed scheme and a procedure using a nested 2-D Simpson’s integration rule. The results show that typically at least a fourfold improvement in computational speed can be expected in practical use.

© 1989 Optical Society of America

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References

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  1. F. Zernike, “Beugungstheorie des Schneidenverfahrens und Seiner Verbesserten Form, der Phasenkontrastmethode,” Physica I, 689 (1934).
    [CrossRef]
  2. A. B. Bhatia, E. Wolf, “On The Circle Polynomials of Zernike and Related Orthogonal Sets,” Proc. Cambr. Philos. Soc. 50, Part 1, 40 (1954).
    [CrossRef]
  3. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1983), Chap. 9.
  4. S. N. Bezdidko, “The Use of Zernike Polynomials in Optics,” Sov. J. Opt. Technol. 41, 425 (1974).
  5. S. N. Bezdidko, “Determination of the Zernike Polynomial Expansion Coefficients of the Wave Aberration,” Sov. J. Opt. Technol. 42, 426 (1975).
  6. J. Y. Wang, D. E. Silva, “Wave-front Interpretation with Zernike Polynomials,” Appl. Opt. 19, 1510 (1980).
    [CrossRef] [PubMed]
  7. M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (U.S. G.P.O., Washington, DC, 1972), Chap. 22.
  8. R. J. Noll, “Zernike Polynomials and Atmospheric Turbulence,” J. Opt. Soc. Am. 66, 207 (1976).
    [CrossRef]
  9. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U.P., London, 1986), Chap. 12.
  10. L. N. G. Filon, “On a Quadrature Formula for Trigonometric Integrals,” Proc. R. Soc. Edinburgh49, 38 (1928–1929).
  11. P. F. Davis, R. Rabinowitz, Methods of Numerical Integration2nd. edition (Academic, New York, 1984), Sec. 2.10.

1980 (1)

1976 (1)

1975 (1)

S. N. Bezdidko, “Determination of the Zernike Polynomial Expansion Coefficients of the Wave Aberration,” Sov. J. Opt. Technol. 42, 426 (1975).

1974 (1)

S. N. Bezdidko, “The Use of Zernike Polynomials in Optics,” Sov. J. Opt. Technol. 41, 425 (1974).

1954 (1)

A. B. Bhatia, E. Wolf, “On The Circle Polynomials of Zernike and Related Orthogonal Sets,” Proc. Cambr. Philos. Soc. 50, Part 1, 40 (1954).
[CrossRef]

1934 (1)

F. Zernike, “Beugungstheorie des Schneidenverfahrens und Seiner Verbesserten Form, der Phasenkontrastmethode,” Physica I, 689 (1934).
[CrossRef]

Bezdidko, S. N.

S. N. Bezdidko, “Determination of the Zernike Polynomial Expansion Coefficients of the Wave Aberration,” Sov. J. Opt. Technol. 42, 426 (1975).

S. N. Bezdidko, “The Use of Zernike Polynomials in Optics,” Sov. J. Opt. Technol. 41, 425 (1974).

Bhatia, A. B.

A. B. Bhatia, E. Wolf, “On The Circle Polynomials of Zernike and Related Orthogonal Sets,” Proc. Cambr. Philos. Soc. 50, Part 1, 40 (1954).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1983), Chap. 9.

Davis, P. F.

P. F. Davis, R. Rabinowitz, Methods of Numerical Integration2nd. edition (Academic, New York, 1984), Sec. 2.10.

Filon, L. N. G.

L. N. G. Filon, “On a Quadrature Formula for Trigonometric Integrals,” Proc. R. Soc. Edinburgh49, 38 (1928–1929).

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U.P., London, 1986), Chap. 12.

Noll, R. J.

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U.P., London, 1986), Chap. 12.

Rabinowitz, R.

P. F. Davis, R. Rabinowitz, Methods of Numerical Integration2nd. edition (Academic, New York, 1984), Sec. 2.10.

Silva, D. E.

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U.P., London, 1986), Chap. 12.

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U.P., London, 1986), Chap. 12.

Wang, J. Y.

Wolf, E.

A. B. Bhatia, E. Wolf, “On The Circle Polynomials of Zernike and Related Orthogonal Sets,” Proc. Cambr. Philos. Soc. 50, Part 1, 40 (1954).
[CrossRef]

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1983), Chap. 9.

Zernike, F.

F. Zernike, “Beugungstheorie des Schneidenverfahrens und Seiner Verbesserten Form, der Phasenkontrastmethode,” Physica I, 689 (1934).
[CrossRef]

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

Physica (1)

F. Zernike, “Beugungstheorie des Schneidenverfahrens und Seiner Verbesserten Form, der Phasenkontrastmethode,” Physica I, 689 (1934).
[CrossRef]

Proc. Cambr. Philos. Soc. (1)

A. B. Bhatia, E. Wolf, “On The Circle Polynomials of Zernike and Related Orthogonal Sets,” Proc. Cambr. Philos. Soc. 50, Part 1, 40 (1954).
[CrossRef]

Sov. J. Opt. Technol. (2)

S. N. Bezdidko, “The Use of Zernike Polynomials in Optics,” Sov. J. Opt. Technol. 41, 425 (1974).

S. N. Bezdidko, “Determination of the Zernike Polynomial Expansion Coefficients of the Wave Aberration,” Sov. J. Opt. Technol. 42, 426 (1975).

Other (5)

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1983), Chap. 9.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U.P., London, 1986), Chap. 12.

L. N. G. Filon, “On a Quadrature Formula for Trigonometric Integrals,” Proc. R. Soc. Edinburgh49, 38 (1928–1929).

P. F. Davis, R. Rabinowitz, Methods of Numerical Integration2nd. edition (Academic, New York, 1984), Sec. 2.10.

M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (U.S. G.P.O., Washington, DC, 1972), Chap. 22.

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Figures (7)

Fig. 1
Fig. 1

Radial functions R n m ( ρ ) with azimuthal index m = 0.

Fig. 2
Fig. 2

Radial functions R n m ( ρ ) with azimuthal index m = 1.

Fig. 3
Fig. 3

Radial functions R n m ( ρ ) with azimuthal index m = 2.

Fig. 4
Fig. 4

Piecewise approximation of the integrand by a second-degree polynomial.

Fig. 5
Fig. 5

Relative spurious power generated when expanding the function F(ρ,ϕ) = 6ρ4 − 6ρ2 + 1 (primary spherical aberration) in a series of Zernike polynomials (N = M = 10).

Fig. 6
Fig. 6

Relative spurious power generated when expanding the function F(ρ,ϕ) = (3ρ3 − 2ρ) cosϕ (primary coma) in a series of Zernike polynomials (N = M = 10).

Fig. 7
Fig. 7

Relative spurious power generated when expanding the function F(ρ,ϕ) = ρ2 cosϕ (primary astigmatism) in a series of Zernike polynomials (N = M = 10).

Equations (43)

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U o n m ( ρ , ϕ ) U e n m ( ρ , ϕ ) = R n m ( ρ ) sin cos m ϕ ,
R n m ( ρ ) = l = 0 ( n - m ) / 2 ( - 1 ) l ( n - l ) ! l ! [ ( n + m ) / 2 - l ] ! [ ( n - m ) / 2 - l ] ! ρ n - 2 l
R n m ( ρ ) = 0.
P n ( α , β ) ( x ) = ( - 1 ) n R n m ( ρ ) ρ α ,
ρ R n m ( ρ ) = 1 2 ( n + 1 ) [ ( n + m + 2 ) R n + 1 m + 1 ( ρ ) + ( n - m ) R n - 1 m + 1 ( ρ ) ] ,
R n + 2 m ( ρ ) = n + 2 ( n + 2 ) 2 - m 2 { [ 4 ( n + 1 ) ρ 2 - ( n + m ) 2 n - ( n - m + 2 ) 2 n + 2 ] R n m ( ρ ) - ( n 2 - m 2 ) n R n - 2 m ( ρ ) } .
S n m ( 0 , ρ ) = R n m ( ρ ) d ρ .
R n m ( ρ ) + R n m + 2 ( ρ ) = 1 n + 1 d [ R n + 1 m + 1 ( ρ ) - R n - 1 m + 1 ( ρ ) ] d ρ ,
S n m ( 0 , ρ ) = 1 n + 1 { [ R n + 1 m + 1 ( ρ ) - R n + 1 m + 3 ( ρ ) + ] - [ R n - 1 m + 1 ( ρ ) - R n - 1 m + 3 ( ρ ) + ] } ,
S n m ( s , ρ ) = ρ s R n m ( ρ ) d ρ
S n m ( 1 , ρ ) = 1 2 ( n + 1 ) [ ( n + m + 2 ) S n + 1 m + 1 ( 0 , ρ ) + ( n - m ) S n - 1 m + 1 ( 0 , ρ ) ] ,
S n m ( 2 , ρ ) = 1 2 ( n + 1 ) { ( n - m ) ( n - m - 2 ) 2 n S n - 2 m + 2 ( 0 , ρ ) + n + m + 2 n + 2 [ ( n - m ) ( n + 1 ) n S n m + 2 ( 0 , ρ ) + ( n + m + 4 ) 2 S n + 2 m + 2 ( 0 , ρ ) ] } ,
S n m ( 3 , ρ ) = 1 2 ( n + 1 ) { ( n - m ) ( n - m - 2 ) ( n - m - 4 ) 4 n ( n - 1 ) S n - 3 m + 3 ( 0 , ρ ) + n + m + 2 4 [ 3 ( n - m ) ( n - m - 2 ) ( n - 1 ) ( n + 2 ) S n - 1 m + 3 ( 0 , ρ ) + 3 ( n - m ) ( n + m + 4 ) n ( n + 3 ) S n + 1 m + 3 ( 0 , ρ ) + ( n + m + 4 ) ( n + m + 6 ) ( n + 2 ) ( n + 3 ) S n + 3 m + 3 ( 0 , ρ ) ] } .
F ( ρ , ϕ ) m = 0 M n = m N [ A n m U o n m ( ρ , ϕ ) + B n m U e n m ( ρ , ϕ ) ] ,
A n m B n m = m ( n + 1 ) π 0 1 0 2 π F ( ρ , ϕ ) U o n m ( ρ , ϕ ) U e n m ( ρ , ϕ ) ρ d ϕ d ρ ,
A n m B n m = m ( n + 1 ) π 0 1 G o ( ρ ) G e ( ρ ) R n m ( ρ ) ρ d ρ ,
G o ( ρ ) G e ( ρ ) = 0 2 π F ( ρ , ϕ ) sin cos m ϕ d ϕ .
G ( ρ ) C 2 ( k ) ρ 2 + C 1 ( k ) ρ + C 0 ( k ) .
C 2 ( k ) = G ( ρ i ) - 2 G ( ρ c ) + G ( ρ f ) 2 h ρ 2 ,
C 1 ( k ) = G ( ρ f ) - G ( ρ i ) 2 h ρ - 2 ρ c C 2 ( k ) ,
C 0 ( k ) = G ( ρ c ) - ρ c [ ρ c C 2 ( k ) + C 1 ( k ) ] ,
ρ i = 2 ( k - 1 ) h ρ ,
ρ c = ( 2 k - 1 ) h p ,
ρ f = 2 k h ρ .
A n m B n m m ( n + 1 ) π k = 1 N ρ / 2 × ρ i ρ f [ C o 2 ( k ) C e 2 ( k ) ρ 2 + C o 1 ( k ) C e 1 ( k ) ρ + C o 0 ( k ) C e 0 ( k ) ] R n m ( ρ ) ρ d ρ .
A n m B n m m ( n + 1 ) π k = 1 N ρ / 2 { C o 2 ( k ) C e 2 ( k ) [ S n m ( 3 , ρ f ) - S n m ( 3 , ρ i ) ] + C o 1 ( k ) C e 1 ( k ) [ S n m ( 2 , ρ f ) - S n m ( 2 , ρ i ) ] + C o 0 ( k ) C e 0 ( k ) [ S n m ( 1 , ρ f ) - S n m ( 1 , ρ i ) ] } .
G o ( ρ ) e G ( ρ ) 1 m k = 1 N ϕ / 2 { - + [ F ( ρ , ϕ f ) - F ( ρ , ϕ i ) ] γ i cos sin m ϕ c + { [ F ( ρ , ϕ f ) + F ( ρ , ϕ i ) ] γ 2 + F ( ρ , ϕ f ) - 2 F ( ρ , ϕ c ) + 2 F ( ρ , ϕ i ) ] γ 3 } sin cos m ϕ c } ,
ϕ i = 2 ( k - 1 ) h ϕ ,
ϕ c = ( 2 k - 1 ) h ϕ ,
ϕ f = 2 k h ϕ ,
γ 1 = cos m h ϕ - sin m h ϕ m h ϕ ,
γ 2 = sin m h ϕ ,
γ 3 = 2 γ 1 m h ϕ .
R m m ( ρ ) = ρ m ,
R m + 2 m ( ρ ) = R m m ( ρ ) [ ( m + 2 ) ρ 2 - m - 1 ] ,
N ϕ = 2 π ρ N ρ .
N F π N ρ 2 + 3 N ρ .
F 1 ( ρ , ϕ ) = 6 ρ 4 - 6 ρ 2 + 1 = U e 4 0 ( ρ , ϕ ) ,
F 2 ( ρ , ϕ ) = ( 3 ρ 3 - 2 ρ ) cos ϕ = U e 3 1 ( ρ , ϕ ) ,
F 3 ( ρ , ϕ ) = ρ 2 cos 2 ϕ = e U 2 2 ( ρ , ϕ ) .
P ¯ spur = | P exact - P expan P exact | ,
P exact = 0 1 0 2 π F ( ρ , ϕ ) * F ( ρ , ϕ ) ρ d ϕ d ρ ,
P expan = π m = 0 M n = m N 1 m ( n + 1 ) ( A n m * A n m + B n m * B n m ) .

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