Abstract

A general theoretical approach has been developed for the determination of orthonormal polynomials over any integrable domain, such as a hexagon. This approach is better than the classical Gram–Schmidt orthogonalization process because it is nonrecursvie and can be performed rapidly with matrix transformations. The determination of the orthonormal hexagonal polynomials is demonstrated as an example of the matrix approach.

© 2006 Optical Society of America

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References

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  1. Å. Björck, Numerical Methods for Least Squares Problems (Society for Industrial and Applied Mathematics, 1996).
    [CrossRef]
  2. G. H. Golub and C. F. van Loan, Matrix Computations, 3rd ed. (Johns Hopkins University, 1996).
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  4. M. H. Lee, Phys. Rev. Lett. 49, 1072 (1982).
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  5. B. Tatian, J. Opt. Soc. Am. 64, 1083 (1974).
    [CrossRef]
  6. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge, 1999), Chap. 9.
  7. V. N. Mahajan, Optical Imaging and Aberrations, Part II: Wave Diffraction Optics, 2nd printing (SPIE Press, 2004), Chap. 5.
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    [CrossRef] [PubMed]
  9. R. J. Noll, J. Opt. Soc. Am. 66, 207 (1976).
    [CrossRef]

2006 (1)

1982 (1)

M. H. Lee, Phys. Rev. Lett. 49, 1072 (1982).
[CrossRef]

1976 (1)

1974 (1)

1967 (1)

Å. Björck, BIT 7, 1 (1967).
[CrossRef]

Björck, Å.

Å. Björck, BIT 7, 1 (1967).
[CrossRef]

Å. Björck, Numerical Methods for Least Squares Problems (Society for Industrial and Applied Mathematics, 1996).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge, 1999), Chap. 9.

Dai, G.-m.

Golub, G. H.

G. H. Golub and C. F. van Loan, Matrix Computations, 3rd ed. (Johns Hopkins University, 1996).

Lee, M. H.

M. H. Lee, Phys. Rev. Lett. 49, 1072 (1982).
[CrossRef]

Mahajan, V. N.

V. N. Mahajan and G.-m. Dai, Opt. Lett. 31, 2462 (2006).
[CrossRef] [PubMed]

V. N. Mahajan, Optical Imaging and Aberrations, Part II: Wave Diffraction Optics, 2nd printing (SPIE Press, 2004), Chap. 5.

Noll, R. J.

Tatian, B.

van Loan, C. F.

G. H. Golub and C. F. van Loan, Matrix Computations, 3rd ed. (Johns Hopkins University, 1996).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge, 1999), Chap. 9.

BIT (1)

Å. Björck, BIT 7, 1 (1967).
[CrossRef]

J. Opt. Soc. Am. (2)

Opt. Lett. (1)

Phys. Rev. Lett. (1)

M. H. Lee, Phys. Rev. Lett. 49, 1072 (1982).
[CrossRef]

Other (4)

Å. Björck, Numerical Methods for Least Squares Problems (Society for Industrial and Applied Mathematics, 1996).
[CrossRef]

G. H. Golub and C. F. van Loan, Matrix Computations, 3rd ed. (Johns Hopkins University, 1996).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge, 1999), Chap. 9.

V. N. Mahajan, Optical Imaging and Aberrations, Part II: Wave Diffraction Optics, 2nd printing (SPIE Press, 2004), Chap. 5.

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Figures (1)

Fig. 1
Fig. 1

Coordinate system for a unit hexagon inscribed inside a unit circle.

Tables (1)

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Table 1 Nonzero Elements of the Conversion Matrix M through the Sixth Order

Equations (18)

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F i = j = 1 J M i j Z j ,
G i H j = 1 A Σ G i H j d S ,
F i F j = δ i j ,
F i Z k = j = 1 J M i j Z j Z k ,
Z k F i = j = 1 J [ M i j Z j Z k ] T = j = 1 J Z k Z j [ M i j ] T ,
C Z F = C Z Z M T ,
F i F k = j = 1 J M i j Z j F k = δ i k .
M C Z F = 1 .
M C Z Z M T = 1 .
M = ( Q T ) 1 ,
Q T Q = C Z Z .
Φ ( ρ , θ ) = j = 1 J b j Z j ( ρ , θ ) ,
Φ 2 = j = 1 J j = 1 J b j Z j Z j b j = b C Z Z b T > 0 .
c 11 = 1 , c 22 = c 33 = 5 6 , c 44 = 4 5 , c 14 = c 41 = 1 2 3 .
q 11 = 1 , q 22 = q 33 = 5 6 , q 44 = 1 2 43 15 , q 14 = 1 2 3 .
m 11 = 1 , m 22 = m 33 = 6 5 , m 44 = 2 15 43 , m 41 = 5 43 .
H 1 = Z 1 , H 2 = 6 5 Z 2 ,
H 3 = 6 5 Z 3 , H 4 = 2 15 43 Z 4 + 5 43 Z 1 ,

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