Abstract

Mathematical methods that are poorly known in the field of optics are adapted and shown to have striking significance. Orthogonal polynomials are common tools in physics and optics, but problems are encountered when they are used to higher orders. Applications to arbitrarily high orders are shown to be enabled by remarkably simple and robust algorithms that are derived from well known recurrence relations. Such methods are demonstrated for a couple of familiar optical applications where, just as in other areas, there is a clear trend to higher orders.

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References

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  1. M. Born, and E. Wolf, Principles of Optics (Cambridge, 1999), see Sec. 9.2 and Appendix VII.
  2. A. E. Siegman, Lasers (University Science Books, 1986), Chaps. 16–17.
  3. M. Abramowitz, and I. Stegun, Handbook of Mathematical Functions (Dover, 1978), Chap. 22.
  4. A. B. Bhatia, E. Wolf, and M. Born, “On the circle polynomials of Zernike and related orthogonal sets,” Proc. Camb. Philos. Soc. 50(01), 40–48 (1954).
    [CrossRef]
  5. D. R. Myrick, “A generalization of the radial polynomials of F. Zernike,” J. Soc. Ind. Appl. Math. 14(3), 476–492 (1966).
    [CrossRef]
  6. E. C. Kintner, “On the mathematical properties of the Zernike polynomials,” Opt. Acta (Lond.) 23, 679–680 (1976).
    [CrossRef]
  7. R. Barakat, “Optimum balanced wave-front aberrations for radially symmetric amplitude distributions: generalizations of Zernike polynomials,” J. Opt. Soc. Am. 70(6), 739–742 (1980).
    [CrossRef]
  8. C.-W. Chong, P. Raveendran, and R. Mukundan, “A comparative analysis of algorithms for fast computation of Zernike moments,” Pattern Recognit. 36(3), 731–742 (2003).
    [CrossRef]
  9. W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes: The Art of Scientific Computing (Cambridge University Press, 1992) Section 5.5.
  10. C. W. Clenshaw, “A Note on the Summation of Chebyshev Series,” Math. Tables Other Aids Comput. 9, 118–120 (1955), http://www.jstor.org/stable/2002068 .
  11. F. J. Smith, “An Algorithm for Summing Orthogonal Polynomial Series and their Derivatives with Applications to Curve-Fitting and Interpolation,” Math. Comput. 19(89), 33–36 (1965).
    [CrossRef]
  12. H. E. Salzer, “A Recurrence Scheme for Converting from One Orthogonal Expansion into Another,” Commun. ACM 16(11), 705–707 (1973).
    [CrossRef]
  13. E. H. Doha, “On the coefficients of differentiated expansions and derivatives of Jacobi polynomials,” J. Phys. Math. Gen. 35(15), 3467–3478 (2002).
    [CrossRef]
  14. R. Barrio and J. M. Peña, “Numerical evaluation of the p’th derivative of Jacobi series,” Appl. Numer. Math. 43(4), 335–357 (2002).
    [CrossRef]
  15. B. Y. Ting and Y. L. Luke, “Conversion of Polynomials between Different Polynomial Bases,” IMA J. Numer. Anal. 1(2), 229–234 (1981).
    [CrossRef]
  16. K. A. Goldberg and K. Geary, “Wave-front measurement errors from restricted concentric subdomains,” J. Opt. Soc. Am. A 18(9), 2146–2152 (2001).
    [CrossRef]
  17. J. Schwiegerling, “Scaling Zernike expansion coefficients to different pupil sizes,” J. Opt. Soc. Am. A 19(10), 1937–1945 (2002).
    [CrossRef]
  18. 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(2), 209–217 (2003).
    [CrossRef]
  19. G. M. Dai, “Scaling Zernike expansion coefficients to smaller pupil sizes: a simpler formula,” J. Opt. Soc. Am. A 23(3), 539–543 (2006).
    [CrossRef]
  20. 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(8), 1960–1966 (2006).
    [CrossRef]
  21. A. J. E. M. Janssen and P. Dirksen, “Concise formula for the Zernike coefficients of scaled pupils,” J. Microlith. Microfab Microsyst. 5(3), 030501–3 (2006).
    [CrossRef]
  22. G. W. Forbes, “Shape specification for axially symmetric optical surfaces,” Opt. Express 15(8), 5218–5226 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-8-5218 .
    [CrossRef] [PubMed]

2007 (1)

2006 (3)

2003 (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(2), 209–217 (2003).
[CrossRef]

C.-W. Chong, P. Raveendran, and R. Mukundan, “A comparative analysis of algorithms for fast computation of Zernike moments,” Pattern Recognit. 36(3), 731–742 (2003).
[CrossRef]

2002 (3)

E. H. Doha, “On the coefficients of differentiated expansions and derivatives of Jacobi polynomials,” J. Phys. Math. Gen. 35(15), 3467–3478 (2002).
[CrossRef]

R. Barrio and J. M. Peña, “Numerical evaluation of the p’th derivative of Jacobi series,” Appl. Numer. Math. 43(4), 335–357 (2002).
[CrossRef]

J. Schwiegerling, “Scaling Zernike expansion coefficients to different pupil sizes,” J. Opt. Soc. Am. A 19(10), 1937–1945 (2002).
[CrossRef]

2001 (1)

1981 (1)

B. Y. Ting and Y. L. Luke, “Conversion of Polynomials between Different Polynomial Bases,” IMA J. Numer. Anal. 1(2), 229–234 (1981).
[CrossRef]

1980 (1)

1976 (1)

E. C. Kintner, “On the mathematical properties of the Zernike polynomials,” Opt. Acta (Lond.) 23, 679–680 (1976).
[CrossRef]

1973 (1)

H. E. Salzer, “A Recurrence Scheme for Converting from One Orthogonal Expansion into Another,” Commun. ACM 16(11), 705–707 (1973).
[CrossRef]

1966 (1)

D. R. Myrick, “A generalization of the radial polynomials of F. Zernike,” J. Soc. Ind. Appl. Math. 14(3), 476–492 (1966).
[CrossRef]

1965 (1)

F. J. Smith, “An Algorithm for Summing Orthogonal Polynomial Series and their Derivatives with Applications to Curve-Fitting and Interpolation,” Math. Comput. 19(89), 33–36 (1965).
[CrossRef]

1955 (1)

C. W. Clenshaw, “A Note on the Summation of Chebyshev Series,” Math. Tables Other Aids Comput. 9, 118–120 (1955), http://www.jstor.org/stable/2002068 .

1954 (1)

A. B. Bhatia, E. Wolf, and M. Born, “On the circle polynomials of Zernike and related orthogonal sets,” Proc. Camb. Philos. Soc. 50(01), 40–48 (1954).
[CrossRef]

Barakat, R.

Barrio, R.

R. Barrio and J. M. Peña, “Numerical evaluation of the p’th derivative of Jacobi series,” Appl. Numer. Math. 43(4), 335–357 (2002).
[CrossRef]

Bhatia, A. B.

A. B. Bhatia, E. Wolf, and M. Born, “On the circle polynomials of Zernike and related orthogonal sets,” Proc. Camb. Philos. Soc. 50(01), 40–48 (1954).
[CrossRef]

Born, M.

A. B. Bhatia, E. Wolf, and M. Born, “On the circle polynomials of Zernike and related orthogonal sets,” Proc. Camb. Philos. Soc. 50(01), 40–48 (1954).
[CrossRef]

Campbell, C. E.

Chong, C.-W.

C.-W. Chong, P. Raveendran, and R. Mukundan, “A comparative analysis of algorithms for fast computation of Zernike moments,” Pattern Recognit. 36(3), 731–742 (2003).
[CrossRef]

Clenshaw, C. W.

C. W. Clenshaw, “A Note on the Summation of Chebyshev Series,” Math. Tables Other Aids Comput. 9, 118–120 (1955), http://www.jstor.org/stable/2002068 .

Coatrieux, J.-L.

Dai, G. M.

Dirksen, P.

A. J. E. M. Janssen and P. Dirksen, “Concise formula for the Zernike coefficients of scaled pupils,” J. Microlith. Microfab Microsyst. 5(3), 030501–3 (2006).
[CrossRef]

Doha, E. H.

E. H. Doha, “On the coefficients of differentiated expansions and derivatives of Jacobi polynomials,” J. Phys. Math. Gen. 35(15), 3467–3478 (2002).
[CrossRef]

Forbes, G. W.

Geary, K.

Goldberg, K. A.

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(3), 030501–3 (2006).
[CrossRef]

Kintner, E. C.

E. C. Kintner, “On the mathematical properties of the Zernike polynomials,” Opt. Acta (Lond.) 23, 679–680 (1976).
[CrossRef]

Luke, Y. L.

B. Y. Ting and Y. L. Luke, “Conversion of Polynomials between Different Polynomial Bases,” IMA J. Numer. Anal. 1(2), 229–234 (1981).
[CrossRef]

Luo, L.

Mukundan, R.

C.-W. Chong, P. Raveendran, and R. Mukundan, “A comparative analysis of algorithms for fast computation of Zernike moments,” Pattern Recognit. 36(3), 731–742 (2003).
[CrossRef]

Myrick, D. R.

D. R. Myrick, “A generalization of the radial polynomials of F. Zernike,” J. Soc. Ind. Appl. Math. 14(3), 476–492 (1966).
[CrossRef]

Peña, J. M.

R. Barrio and J. M. Peña, “Numerical evaluation of the p’th derivative of Jacobi series,” Appl. Numer. Math. 43(4), 335–357 (2002).
[CrossRef]

Raveendran, P.

C.-W. Chong, P. Raveendran, and R. Mukundan, “A comparative analysis of algorithms for fast computation of Zernike moments,” Pattern Recognit. 36(3), 731–742 (2003).
[CrossRef]

Salzer, H. E.

H. E. Salzer, “A Recurrence Scheme for Converting from One Orthogonal Expansion into Another,” Commun. ACM 16(11), 705–707 (1973).
[CrossRef]

Schwiegerling, J.

Shu, H.

Smith, F. J.

F. J. Smith, “An Algorithm for Summing Orthogonal Polynomial Series and their Derivatives with Applications to Curve-Fitting and Interpolation,” Math. Comput. 19(89), 33–36 (1965).
[CrossRef]

Ting, B. Y.

B. Y. Ting and Y. L. Luke, “Conversion of Polynomials between Different Polynomial Bases,” IMA J. Numer. Anal. 1(2), 229–234 (1981).
[CrossRef]

Wolf, E.

A. B. Bhatia, E. Wolf, and M. Born, “On the circle polynomials of Zernike and related orthogonal sets,” Proc. Camb. Philos. Soc. 50(01), 40–48 (1954).
[CrossRef]

Appl. Numer. Math. (1)

R. Barrio and J. M. Peña, “Numerical evaluation of the p’th derivative of Jacobi series,” Appl. Numer. Math. 43(4), 335–357 (2002).
[CrossRef]

Commun. ACM (1)

H. E. Salzer, “A Recurrence Scheme for Converting from One Orthogonal Expansion into Another,” Commun. ACM 16(11), 705–707 (1973).
[CrossRef]

IMA J. Numer. Anal. (1)

B. Y. Ting and Y. L. Luke, “Conversion of Polynomials between Different Polynomial Bases,” IMA J. Numer. Anal. 1(2), 229–234 (1981).
[CrossRef]

J. Microlith. Microfab Microsyst. (1)

A. J. E. M. Janssen and P. Dirksen, “Concise formula for the Zernike coefficients of scaled pupils,” J. Microlith. Microfab Microsyst. 5(3), 030501–3 (2006).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

J. Phys. Math. Gen. (1)

E. H. Doha, “On the coefficients of differentiated expansions and derivatives of Jacobi polynomials,” J. Phys. Math. Gen. 35(15), 3467–3478 (2002).
[CrossRef]

J. Soc. Ind. Appl. Math. (1)

D. R. Myrick, “A generalization of the radial polynomials of F. Zernike,” J. Soc. Ind. Appl. Math. 14(3), 476–492 (1966).
[CrossRef]

Math. Comput. (1)

F. J. Smith, “An Algorithm for Summing Orthogonal Polynomial Series and their Derivatives with Applications to Curve-Fitting and Interpolation,” Math. Comput. 19(89), 33–36 (1965).
[CrossRef]

Math. Tables Other Aids Comput. (1)

C. W. Clenshaw, “A Note on the Summation of Chebyshev Series,” Math. Tables Other Aids Comput. 9, 118–120 (1955), http://www.jstor.org/stable/2002068 .

Opt. Acta (Lond.) (1)

E. C. Kintner, “On the mathematical properties of the Zernike polynomials,” Opt. Acta (Lond.) 23, 679–680 (1976).
[CrossRef]

Opt. Express (1)

Pattern Recognit. (1)

C.-W. Chong, P. Raveendran, and R. Mukundan, “A comparative analysis of algorithms for fast computation of Zernike moments,” Pattern Recognit. 36(3), 731–742 (2003).
[CrossRef]

Proc. Camb. Philos. Soc. (1)

A. B. Bhatia, E. Wolf, and M. Born, “On the circle polynomials of Zernike and related orthogonal sets,” Proc. Camb. Philos. Soc. 50(01), 40–48 (1954).
[CrossRef]

Other (4)

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes: The Art of Scientific Computing (Cambridge University Press, 1992) Section 5.5.

M. Born, and E. Wolf, Principles of Optics (Cambridge, 1999), see Sec. 9.2 and Appendix VII.

A. E. Siegman, Lasers (University Science Books, 1986), Chaps. 16–17.

M. Abramowitz, and I. Stegun, Handbook of Mathematical Functions (Dover, 1978), Chap. 22.

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Figures (2)

Fig. 1
Fig. 1

The red curve at left is a plot of a high-order rotationally symmetric Zernike over just 0.95 < x < 1; the green curve is 1014 times the error when this function is evaluated by using Eq. (1.6), so there are typically 14 significant digits in this double-precision result. The curve on the right is the catastrophic error when Eq. (1.7) is used at double precision. Although exact to within round-off, the explicit polynomial gives values with the wrong order of magnitude and chaotic sign. This failure grows exponentially with the polynomial’s order.

Fig. 2
Fig. 2

A sample of the members of the basis used in Eq. (5.1).

Equations (52)

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P m , P n     =     0 ,     when   m n ,
S ( x )     =     m s m P m ( x )
s m     =     S , P m .
S , S     =     m s m     2 .
x P n ( x )     =     q P n + 1 ( x ) + r P n ( x ) + s P n 1 ( x ) ,
P n + 1 ( x )     =     ( a n + b n x ) P n ( x ) c n P n 1 ( x ) .
Z n m ( x )     =     j = 0 n ( 1 ) n j ( n j ) ( m + n + j n ) x j     =     j = 0 n ( 1 ) n j ( m + n + j ) ! j ! ( m + j ) ! ( n j ) ! x j .
Z 10 0 ( x )     =     1 110 x + 2 , 970 x 2 34 , 320 x 3 + 210 , 210 x 4 756 , 756 x 5         + 1 , 681 , 680 x 6 2 , 333 , 760 x 7 + 1 , 969 , 110 x 8 923 , 780 x 9 + 184 , 756 x 10 .
S ( x )     : =     m = 0 M s m P m ( x ) .
α n     =     s n + ( a n + b n x ) α n + 1 c n + 1 α n + 2 ,
α n ( j )     =     j b n α n + 1 ( j 1 ) + ( a n + b n x ) α n + 1 ( j ) c n + 1 α n + 2 ( j )
S     =     m = 0 M s m I P m I ,
S     =     m = 0 M s m II P m II .
P n + 1 I     =     ( a n I + b n I x ) P n I c n I P n 1 I ,       P n + 1 II     =     ( a n II + b n II x ) P n II c n II P n 1 II .
f k n : =     b n I / b k 1 II ,     g k n : =     a n I b n I a k II / b k II ,     h k n : =     b n I c k + 1 II / b k + 1 II .
α 0 n     =     s n I + g 0 n α 0 n + 1 + h 0 n α 1 n + 1 c n + 1 I α 0 n + 2 .
α k n     =     f k n α k 1 n + 1 + g k n α k n + 1 + h k n α k + 1 n + 1 c n + 1 I α k n + 2 .
α M n 1 n     =     f M n 1 n α M n 2 n + 1 + g M n 1 n α M n 1 n + 1 ,
α M n n     =     f M n n α M n 1 n + 1 .
a n = ( s + 1 ) [ ( s n ) 2 + n 2 + s ] ( n + 1 ) ( s n + 1 ) s ,     b n = ( s + 2 ) ( s + 1 ) ( n + 1 ) ( s + n 1 ) ,     c n = ( s + 2 ) ( s n ) n ( n + 1 ) ( s n + 1 ) s .
n = 0 M s n P n I ( x )         n = 0 M ( t n / ε m ) P n II ( x / ε 2 ) ,
z ( ρ )     =     c ρ 2 / [ 1 + 1 ( 1 + κ ) c 2 ρ 2 ]     + u 4 m = 0 M s m Q m con ( u 2 ) ,
a n = ( 2 n + 5 ) ( n 2 + 5 n + 10 ) ( n + 1 ) ( n + 2 ) ( n + 5 ) , b n = 2 ( n + 3 ) ( 2 n + 5 ) ( n + 1 ) ( n + 5 ) , c n = ( n + 3 ) ( n + 4 ) n ( n + 1 ) ( n + 2 ) ( n + 5 ) .
z ( ρ )     =     c ρ 2 / ( 1 + ϕ )     + u 4 S ( u 2 ) ,
z ( ρ )     =     c ρ / ϕ     + 2 u 3 ρ max [ 2 S ( u 2 ) + u 2 S ( u 2 ) ] ,
z ( ρ )     =     c / ϕ 3     + 2 u 2 ρ max 2 [ 6 S ( u 2 ) + 9 u 2 S ( u 2 ) + 2 u 4 S ( u 2 ) ] .
u 4 m = 0 M s m Q m con ( u 2 )     =     m = 0 M A 2 m + 4 ρ 2 m + 4     =     ( u ρ max ) 4 m = 0 M A 2 m + 4 ( u ρ max ) 2 m .
m = 0 M s m Q m con ( x )     =     m = 0 M t m x m ,
a n = 0 ,     b n = 1 ,     c n = 0 .
P n + 1     =     u n P n v n P n 1 ,
S     : =     m = 0 M s m P m .
S     =     α n + 1 P n + 1 + β n + 1 P n + m = 0 n 1 s m P m ,
S     =     α n + 1 ( u n P n v n P n 1 ) + β n + 1 P n + m = 0 n 1 s m P m =     ( u n α n + 1 + β n + 1 ) P n + ( s n 1 v n α n + 1 ) P n 1 + m = 0 n 2 s m P m .
α n     =     u n α n + 1 + β n + 1 ,
β n     =     s n 1 v n α n + 1 .
S     =     α 1 P 1 + β 1 P 0 .
α n     =     s n + u n α n + 1 v n + 1 α n + 2 ,
S     =     ( α 0 u 0 α 1 ) P 0 + α 1 P 1 .
S     =     α 0 .
S     =     m = 1 M s m P m .
u n     =     a n + b n x ,
α n     =     b n α n + 1 + u n α n + 1 v n + 1 α n + 2 .
S     =     α 0 .
α n ( j )     =     j b n α n + 1 ( j 1 ) + u n α n + 1 ( j ) v n + 1 α n + 2 ( j ) .
S ( j )     =     m = j M s m P m ( j )     =     α 0 ( j ) .
S     =     ( k = 0 M n 1 α k n + 1 P k II ) P n + 1 I + ( k = 0 M n β k n + 1 P k II ) P n I + m = 0 n 1 s m I P m I ,
( k = 0 M n 1 α k n + 1 P k II ) P n + 1 I     =     ( k = 0 M n 1 α k n + 1 P k II ) [ ( a n I + b n I x ) P n I c n I P n 1 I ]     =     ( k = 0 M n 1 α k n + 1 x P k II ) b n I P n I + ( k = 0 M n 1 α k n + 1 P k II ) [ a n I P n I c n I P n 1 I ] .
x P k II     =     [ P k + 1 II a k II P k II + c k II P k 1 II ] / b k II .
S     =     ( k = 0 M n 1 α k n + 1 [ P k + 1 II a k II P k II + c k II P k 1 II ] / b k II ) b n I P n I     + ( k = 0 M n 1 α k n + 1 P k II ) [ a n I P n I c n I P n 1 I ] + ( k = 0 M n β k n + 1 P k II ) P n I + m = 0 n 1 s m I P m I .
β k n     =     s n 1 I δ k 0 c n I α k n + 1 ,
α k n     =     s n I δ k 0 + ( b n I b k 1 II ) α k 1 n + 1 + ( a n I b n I a k II b k II ) α k n + 1 + ( b n I c k + 1 II b k + 1 II ) α k + 1 n + 1 c n + 1 I α k n + 2 .
S     =     k = 0 M α k 0 P k II .

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