## Orthonormal polynomials in wavefront analysis: analytical solution

JOSA A, Vol. 24, Issue 9, pp. 2994-3016 (2007)

http://dx.doi.org/10.1364/JOSAA.24.002994

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### Abstract

Zernike circle polynomials are in widespread use for wavefront analysis because of their orthogonality over a circular pupil and their representation of balanced classical aberrations. In recent papers, we derived closed-form polynomials that are orthonormal over a hexagonal pupil, such as the hexagonal segments of a large mirror. We extend our work to elliptical, rectangular, and square pupils. Using the circle polynomials as the basis functions for their orthogonalization over such pupils, we derive closed-form polynomials that are orthonormal over them. These polynomials are unique in that they are not only orthogonal across such pupils, but also represent balanced classical aberrations, just as the Zernike circle polynomials are unique in these respects for circular pupils. The polynomials are given in terms of the circle polynomials as well as in polar and Cartesian coordinates. Relationships between the orthonormal coefficients and the corresponding Zernike coefficients for a given pupil are also obtained. The orthonormal polynomials for a one-dimensional slit pupil are obtained as a limiting case of a rectangular pupil.

© 2007 Optical Society of America

**OCIS Codes**

(010.1080) Atmospheric and oceanic optics : Active or adaptive optics

(010.7350) Atmospheric and oceanic optics : Wave-front sensing

(120.3180) Instrumentation, measurement, and metrology : Interferometry

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

(220.1010) Optical design and fabrication : Aberrations (global)

**ToC Category:**

Atmospheric and oceanic optics

**History**

Original Manuscript: March 8, 2007

Manuscript Accepted: April 18, 2007

Published: August 30, 2007

**Virtual Issues**

Vol. 2, Iss. 10 *Virtual Journal for Biomedical Optics*

**Citation**

Virendra N. Mahajan and Guang-ming Dai, "Orthonormal polynomials in wavefront analysis: analytical solution," J. Opt. Soc. Am. A **24**, 2994-3016 (2007)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=josaa-24-9-2994

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### References

- F. Zernike, "Diffraction theory of knife-edge test and its improved form, the phase contrast method," Mon. Not. R. Astron. Soc. 94, 377-384 (1934).
- R. J. Noll, "Zernike polynomials and atmospheric turbulence," J. Opt. Soc. Am. 66, 207-211 (1976).
- M. Born and E. Wolf, Principles of Optics, 7th ed. (Oxford, 1999).
- V. N. Mahajan, Optical Imaging and Aberrations, Part II: Wave Diffraction Optics (SPIE, 2004).
- V. N. Mahajan, "Zernike polynomials and aberration balancing," Proc. SPIE 5173, 1-17 (2003).
- V. N. Mahajan, "Zernike polynomials and wavefront fitting," in Optical Shop Testing, 3rd ed., D.Malacara, ed. (Wiley, 2007) pp. 498-546.
- V. N. Mahajan, "Zernike annular polynomials for imaging systems with annular pupils," J. Opt. Soc. Am. 71, 75-85 (1981).
- V. N. Mahajan, "Zernike annular polynomials for imaging systems with annular pupils: errata," J. Opt. Soc. Am. 71, 1408 (1981).
- V. N. Mahajan, "Zernike annular polynomials for imaging systems with annular pupils," J. Opt. Soc. Am. 1, 685 (1984).
- V. N. Mahajan, "Zernike annular polynomials and optical aberrations of systems with annular pupils," Appl. Opt. 33, 8125-8127 (1994).
- V. N. Mahajan, "Uniform versus Gaussian beams: a comparison of the effects of diffraction, obscuration, and aberrations," J. Opt. Soc. Am. A 3, 470-485 (1986).
- V. N. Mahajan, "Zernike-Gauss polynomials and optical aberrations of systems with Gaussian pupils," Appl. Opt. 34, 8057-8059 (1995).
- S. Szapiel, "Aberration balancing techniques for radially symmetric amplitude distributions; a generalization of the Maréchal approach," J. Opt. Soc. Am. 72, 947-956 (1982).
- G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, 1968).
- http://scikits.com/KFacts.html.
- W. B. King, "The approximation of vignetted pupil shape by an ellipse," Appl. Opt. 7, 197-201 (1968).
- G. Harbers, P. J. Kunst, and G. W. R. Leibbrandt, "Analysis of lateral shearing interferograms by use of Zernike polynomials," Appl. Opt. 35, 6162-6172 (1996).
- H. Sumita, "Orthogonal expansion of the aberration difference function and its application to image evaluation," Jpn. J. Appl. Phys. 8, 1027-1036 (1969).
- K. N. LaFortune, R. L. Hurd, S. N. Fochs, M. D. Rotter, P. H. Pax, R. L. Combs, S. S. Olivier, J. M. Brase, and R. M. Yamamoto, "Technical challenges for the future of high energy lasers," Proc. SPIE 6454, 1-11 (2007).
- V. N. Mahajan and G.-m. Dai, "Orthonormal polynomials for hexagonal pupils," Opt. Lett. 31, 2462-2465 (2006). [CrossRef]
- G.-m. Dai and V. N. Mahajan, "Nonrecursive orthonormal polynomials with matrix formulation," Opt. Lett. 32, 74-76 (2007).
- R. Barakat and L. Riseberg, "Diffraction theory of the aberrations of a slit aperture," J. Opt. Soc. Am. 55, 878-881 (1965). There is an error in their polynomial S2, which should read as x2−1/3.
- M. Bray, "Orthogonal polynomials: a set for square areas," Proc. SPIE 5252, 314-320 (2004).
- J. L. Rayces, "Least-squares fitting of orthogonal polynomials to the wave-aberration function," Appl. Opt. 31, 2223-2228 (1992).

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