## Robust and fast computation for the polynomials of optics

Optics Express, Vol. 18, Issue 13, pp. 13851-13862 (2010)

http://dx.doi.org/10.1364/OE.18.013851

Acrobat PDF (827 KB)

### 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.

© 2010 OSA

## 1. Introduction

*m*, is defined so thatwhere

*ϕ*is

*any*polynomial of order less than

*m*. It follows that

*q*,

*r*, and

*s*, are constants and

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]

## 2. Linear combinations of orthogonal polynomials and their derivatives

*j*’th derivative is written as

## 3. Change of basis

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]

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

## 4. Applications for Zernike polynomials

*x*in the second basis. The end result is then evidently just

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]

## 5. Applications for asphere shape specification

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]

*c*denotes the axial curvature and

*κ*the conic constant, the surface’s sag is written aswhere

*u*is just the normalized radial coordinate, i.e.

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]

## 6. Concluding remarks

## Appendix A: Directly evaluating linear combinations

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.

*n*in Eq. (A.3) with

*S*can then be determined simply with Eq. (A.7).

*β*’s from Eqs. (A.5) and (A.7): Since Eq. (A.5) establishes that

*P*’s, this is not effective for the optical applications considered here. It is important to note, however, that [9] gives a useful treatment of applications involving trigonometric and Bessel functions as well as a sketch of stability analysis for recurrence-based processes. For many cases of interest, including those considered in Sections 4 and 5, it turns out that

## Appendix B: Directly evaluating derivatives of linear combinations

*m*in

*x*and each

*x*, Smith [11

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]

*x*. With primes denoting derivatives, we now seek(Note that

*x*, differentiating Eq. (A.8) now leads directly toThis can be initialized with

*S*was evaluated beforehand—

*x*, it is straightforward to use the derivative of Eq. (A.8) to get a slight generalization of Eq. (B.3). For the most common case laid out above, it is also clear that higher derivatives can be determined in much the same way: if a superscript in parentheses is used to denote higher derivatives, it follows trivially from Eq. (A.8) thatThis is initialized by

## Appendix C: Directly changing to a new basis

*α*’s and

*β*’s are now just constant coefficients. By using Eq. (3.3a), the first term on the right-hand side of Eq. (C.1) can be re-expressed as

*n*replaced by

*k*except

*β*’s in Appendix A, Eq. (C.5) makes it easy to eliminate

*k*. With this as a given, Eq. (C.6) can be used, for fixed

*n*, by running over

## References and links

1. | M. Born, and E. Wolf, |

2. | A. E. Siegman, |

3. | M. Abramowitz, and I. Stegun, |

4. | A. B. Bhatia, E. Wolf, and M. Born, “On the circle polynomials of Zernike and related orthogonal sets,” Proc. Camb. Philos. Soc. |

5. | D. R. Myrick, “A generalization of the radial polynomials of F. Zernike,” J. Soc. Ind. Appl. Math. |

6. | E. C. Kintner, “On the mathematical properties of the Zernike polynomials,” Opt. Acta (Lond.) |

7. | R. Barakat, “Optimum balanced wave-front aberrations for radially symmetric amplitude distributions: generalizations of Zernike polynomials,” J. Opt. Soc. Am. |

8. | C.-W. Chong, P. Raveendran, and R. Mukundan, “A comparative analysis of algorithms for fast computation of Zernike moments,” Pattern Recognit. |

9. | W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, |

10. | C. W. Clenshaw, “A Note on the Summation of Chebyshev Series,” Math. Tables Other Aids Comput. |

11. | F. J. Smith, “An Algorithm for Summing Orthogonal Polynomial Series and their Derivatives with Applications to Curve-Fitting and Interpolation,” Math. Comput. |

12. | H. E. Salzer, “A Recurrence Scheme for Converting from One Orthogonal Expansion into Another,” Commun. ACM |

13. | E. H. Doha, “On the coefficients of differentiated expansions and derivatives of Jacobi polynomials,” J. Phys. Math. Gen. |

14. | R. Barrio and J. M. Peña, “Numerical evaluation of the p’th derivative of Jacobi series,” Appl. Numer. Math. |

15. | B. Y. Ting and Y. L. Luke, “Conversion of Polynomials between Different Polynomial Bases,” IMA J. Numer. Anal. |

16. | K. A. Goldberg and K. Geary, “Wave-front measurement errors from restricted concentric subdomains,” J. Opt. Soc. Am. A |

17. | J. Schwiegerling, “Scaling Zernike expansion coefficients to different pupil sizes,” J. Opt. Soc. Am. A |

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 |

19. | G. M. Dai, “Scaling Zernike expansion coefficients to smaller pupil sizes: a simpler formula,” J. Opt. Soc. Am. A |

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 |

21. | A. J. E. M. Janssen and P. Dirksen, “Concise formula for the Zernike coefficients of scaled pupils,” J. Microlith. Microfab Microsyst. |

22. | G. W. Forbes, “Shape specification for axially symmetric optical surfaces,” Opt. Express |

**OCIS Codes**

(000.3860) General : Mathematical methods in physics

(220.0220) Optical design and fabrication : Optical design and fabrication

(220.1250) Optical design and fabrication : Aspherics

(260.1960) Physical optics : Diffraction theory

**ToC Category:**

Optical Design and Fabrication

**History**

Original Manuscript: May 12, 2010

Revised Manuscript: June 2, 2010

Manuscript Accepted: June 3, 2010

Published: June 11, 2010

**Citation**

G. W. Forbes, "Robust and fast computation for the polynomials of optics," Opt. Express **18**, 13851-13862 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-13-13851

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### References

- 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.
- 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]
- D. R. Myrick, “A generalization of the radial polynomials of F. Zernike,” J. Soc. Ind. Appl. Math. 14(3), 476–492 (1966). [CrossRef]
- E. C. Kintner, “On the mathematical properties of the Zernike polynomials,” Opt. Acta (Lond.) 23, 679–680 (1976). [CrossRef]
- 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]
- 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]
- W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes: The Art of Scientific Computing (Cambridge University Press, 1992) Section 5.5.
- 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 .
- 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]
- H. E. Salzer, “A Recurrence Scheme for Converting from One Orthogonal Expansion into Another,” Commun. ACM 16(11), 705–707 (1973). [CrossRef]
- 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]
- B. Y. Ting and Y. L. Luke, “Conversion of Polynomials between Different Polynomial Bases,” IMA J. Numer. Anal. 1(2), 229–234 (1981). [CrossRef]
- 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]
- J. Schwiegerling, “Scaling Zernike expansion coefficients to different pupil sizes,” J. Opt. Soc. Am. A 19(10), 1937–1945 (2002). [CrossRef]
- 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]
- 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]
- 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]
- 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]
- 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]

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