Abstract

An orthonormal hexagonal Zernike basis set is generated from circular Zernike polynomials apodized by a hexagonal mask by use of the Gram–Schmidt orthogonalization technique. Results for the first 15 hexagonal Zernike polynomials are shown. The Gram–Schmidt orthogonalization technique presented can be extended to both apertures of arbitrary shape and other basis functions.

© 2004 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1993), pp. 464–468.
  2. D. Malacara and S. L. DeVore, in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), pp. 480–484.
  3. W. Swantner and W. Chow, Appl. Opt. 33, 1832 (1994).
    [CrossRef] [PubMed]
  4. V. N. Mahajan, J. Opt. Soc. Am. 71, 75 (1981).
  5. V. N. Mahajan, Optical Imaging and Aberrations Part II. Wave Diffraction Optics (SPIE Press, Bellingham, Wash., 2001), Sec. 3.4.
    [CrossRef]
  6. G. Strang, Linear Algebra and Its Applications (Harcourt Brace Jovanovitch, San Diego, Calif., 1988), pp. 166–174.
  7. D. G. Dudley, Mathematical Foundations for Electromagnetic Theory (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1994), pp. 16–18.
  8. R. J. Noll, J. Opt. Soc. Am. 66, 207 (1976).
  9. The Zemax optical design program is developed by the Zemax development corporation, www.zemax.com .
  10. The CODE V optical design program is developed by Optical Research Associates, www.opticalres.com .

1994 (1)

1981 (1)

1976 (1)

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1993), pp. 464–468.

Chow, W.

DeVore, S. L.

D. Malacara and S. L. DeVore, in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), pp. 480–484.

Dudley, D. G.

D. G. Dudley, Mathematical Foundations for Electromagnetic Theory (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1994), pp. 16–18.

Mahajan, V. N.

V. N. Mahajan, J. Opt. Soc. Am. 71, 75 (1981).

V. N. Mahajan, Optical Imaging and Aberrations Part II. Wave Diffraction Optics (SPIE Press, Bellingham, Wash., 2001), Sec. 3.4.
[CrossRef]

Malacara, D.

D. Malacara and S. L. DeVore, in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), pp. 480–484.

Noll, R. J.

Strang, G.

G. Strang, Linear Algebra and Its Applications (Harcourt Brace Jovanovitch, San Diego, Calif., 1988), pp. 166–174.

Swantner, W.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1993), pp. 464–468.

Appl. Opt. (1)

J. Opt. Soc. Am. (2)

Other (7)

The Zemax optical design program is developed by the Zemax development corporation, www.zemax.com .

The CODE V optical design program is developed by Optical Research Associates, www.opticalres.com .

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1993), pp. 464–468.

D. Malacara and S. L. DeVore, in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), pp. 480–484.

V. N. Mahajan, Optical Imaging and Aberrations Part II. Wave Diffraction Optics (SPIE Press, Bellingham, Wash., 2001), Sec. 3.4.
[CrossRef]

G. Strang, Linear Algebra and Its Applications (Harcourt Brace Jovanovitch, San Diego, Calif., 1988), pp. 166–174.

D. G. Dudley, Mathematical Foundations for Electromagnetic Theory (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1994), pp. 16–18.

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Figures (4)

Fig. 1
Fig. 1

Wave-front maps U1U3, V1V3, and U1-V1,U2-V2,U3-V3 truncated by the hexagonal aperture. The scale of the data is ±6 waves.

Fig. 2
Fig. 2

Wave-front maps U4U6, V4V6, and U4-V4,U5-V5,U6-V6 truncated by the hexagonal aperture. The scale of the data is ±6 waves.

Fig. 3
Fig. 3

Wave-front maps U7U9, V7V9, and U7-V7,U8-V8,U9-V9 truncated by the hexagonal aperture. The scale of the data is ±6 waves.

Fig. 4
Fig. 4

Plot of the rms residual arising from taking the difference between Un and Vn for a given Zernike number n.

Tables (1)

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Table 1 C Matrix Containing the Transformation from U1U9 to V1V9

Equations (9)

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f,g=drfrgrHrdrHr,
Vn=Un+m=1n-1DnmVm.
Vn=VnVn.
0=Vn,Vm=Un,Vm+l=1n-1DnlVl,Vm, Dnm=-Un,Vm
Vi=m=1nCimUm.
Uj=l=1nBjlVl=l=1nUj,VlVl,
Vi=m=1nCiml=1MBmlVl=l=1Nm=1NCimBmlVl.
V9=m=19C9mUm,
V9=0.0049U1+0.0204U3+0.0069U4+0.0112U5+0.0214U7+1.138U9.

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