Abstract

Wavefront expansion basis functions are important in representing ocular aberrations and phase perturbations due to atmospheric turbulence. A general discussion is presented for the conversions of the coefficients between any two sets of basis functions. Several popular sets of basis functions, namely, Zernike polynomials, Fourier series, and Taylor monomials, are discussed and the conversion matrices between any two of these basis functions are derived. Some analytical and numerical examples are given to demonstrate conversion of coefficients of different basis function sets.

© 2006 Optical Society of America

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  1. J. W. Hardy, J. E. Leferbre, and C. L. Koliopoulos, 'Real time atmospheric compensation,' J. Opt. Soc. Am. 67, 360-369 (1977).
  2. R. Foy and A. Labeyrie, 'Feasibility of adaptive optics telescope with laser probe,' Astron. Astrophys. 152, L29-L31 (1985).
  3. F. Roddier, 'Curvature sensing and compensation: a new concept in adaptive optics,' Appl. Opt. 27, 1223-1225 (1988).
  4. L. A. Thompson and C. S. Gardner, 'Experiments on laser guide stars at Mauna Kea Observatory for adaptive imaging in astronomy,' Nature 328, 229-231 (1987).
  5. G. Rousset, J.-C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Léna, C. Boyer, P. Jagourel, J.-P. Gaffard, and F. Merkle, 'First diffraction-limited astronomical images with adaptive optics,' Astron. Astrophys. 230, 29-32 (1990).
  6. J. Liang, B. Grimm, S. Goelz, and J. Bille, 'Objective measurement of the wave aberrations of the human eye with the use of a Hartmann-Shack wavefront sensor,' J. Opt. Soc. Am. A 11, 1949-1957 (1994).
  7. J. Liang and D. R. Williams, 'Aberrations and retinal image quality of the normal human eye,' J. Opt. Soc. Am. A 14, 2873-2883 (1997).
  8. J. Liang, D. R. Williams, and D. T. Miller, 'Supernormal vision and high-resolution retinal imaging through adaptive optics,' J. Opt. Soc. Am. A 14, 2884-2892 (1997).
  9. A. Roorda and D. R. Williams, 'The arrangement of the three cone classes in the living human eye,' Nature 397, 520-522 (1999).
  10. H. W. Babcock, 'The possibility of compensating astronomical seeing,' Publ. Astron. Soc. Pac. 65, 229-236 (1953).
  11. W. H. Southwell, 'Wavefront estimation from wavefront slope measurements,' J. Opt. Soc. Am. 70, 998-1006 (1980).
  12. E. P. Wallner, 'Optimal wavefront correction using slope measurement,' J. Opt. Soc. Am. 73, 1771-1776 (1983).
  13. B. M. Welsh and C. S. Gardner, 'Performance analysis of adaptive-optics systems using laser guide stars and slope sensors,' J. Opt. Soc. Am. A 6, 1913-1923 (1989).
  14. G.-m. Dai, 'Modal wavefront reconstruction with Zernike polynomials and Karhunen-Loève functions,' J. Opt. Soc. Am. A 13, 1218-1225 (1996).
  15. A. J. E. M. Janssen, 'Extended Nijboer-Zernike approach for the computation of optical point-spread functions,' J. Opt. Soc. Am. A 19, 849-857 (2002).
  16. J. Braat, P. Dirksen, and A. J. E. M. Janssen, 'Assessment of an extended Nijboer-Zernike approach for the computation of optical point-spread functions,' J. Opt. Soc. Am. A 19, 858-870 (2002).
  17. W. J. Donnelly III and A. Roorda, 'The optimal pupil size in the eye for axial resolution,' J. Opt. Soc. Am. A 20, 2010-2015 (2003).
  18. N. Roddier, 'Atmospheric wavefront simulation using Zernike polynomials,' Opt. Eng. 29, 1174-1180 (1990).
  19. G.-m. Dai, 'Wavefront simulation for atmospheric turbulence,' in Image Reconstruction and Restoration, T.J.Schulz and D.-L.Snyder, eds., Proc. SPIE 2302, 62-72 (1994).
  20. R. C. Cannon, 'Optimal bases for wavefront simulation and reconstruction on annular apertures,' J. Opt. Soc. Am. A 13, 862-867 (1996).
  21. H. Jakobsson, 'Simulations of time series of atmospherically distorted wave fronts,' Appl. Opt. 35, 1561-1565 (1996).
  22. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1965), Chap 9.
  23. R. J. Noll, 'Zernike polynomials and atmospheric turbulence,' J. Opt. Soc. Am. 66, 207-211 (1976).
  24. D. Malacara, Optical Shop Testing (Wiley, 1992), Chap. 13.
  25. G.-m. Dai, 'Modal compensation of atmospheric turbulence with the use of Zernike polynomials and Karhunen-Loève functions,' J. Opt. Soc. Am. A 12, 2182-2193 (1995).
  26. V. N. Mahajan, Optical Imaging and Aberrations, Part II: Wave Diffractive Optics (SPIE, 2002), Chap 5.
  27. H. C. Howland and B. Howland, 'A subjective method for the measurement of monochromatic aberrations of the eye,' J. Opt. Soc. Am. 67, 1508-1518 (1977).
  28. G. Walsh, W. N. Charman, and H. C. Howland, 'Objective technique for the determination of monochromatic aberrations of the human eye,' J. Opt. Soc. Am. A 1, 987-992 (1984).
  29. C. Cui and V. Laskshminarayanan, 'Choice of reference axis in ocular wavefront aberration measurement,' J. Opt. Soc. Am. A 15, 2488-2496 (1998).
  30. K. R. Freischlad and C. L. Koliopoulos, 'Modal estimation of wave front difference measurements using the discrete Fourier transform,' J. Opt. Soc. Am. A 3, 1852-1861 (1986).
  31. F. Roddier and C. Roddier, 'Wavefront reconstruction using iterative Fourier transform,' Appl. Opt. 30, 1325-1327 (1991).
  32. L. A. Poyneer, D. T. Gavel, and J. M. Brase, 'Fast wavefront reconstruction in large adaptive optics systems with use of the Fourier transform,' J. Opt. Soc. Am. A 19, 2100-2111 (2002).
  33. L. A. Poyneer and J.-P. Véran, 'Optimal modal Fourier-transform wavefront control,' J. Opt. Soc. Am. A 22, 1515-1526 (2005).
  34. G.-m. Dai, 'Zernike aberration coefficients transformed to and from Fourier series coefficients for wavefront representation,' Opt. Lett. 31, 501-503 (2006).
  35. L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, 'Standards for reporting the optical aberrations of eyes,' Vision Science and Its Applications, Vol. 35 of Trends in Optics and Photonics Series (Optical Society of America, 2000), pp. 232-244.
  36. R. K. Tyson, 'Zernike aberration coefficients from Seidel and higher-order power series aberration coefficients,' Opt. Lett. 7, 262-264 (1982).
  37. G. Conforti, 'Zernike aberration coefficients from Seidel and higher-order power series coefficients,' Opt. Lett. 8, 390-391 (1983).
  38. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Formulas, Graphs, and Mathematical Tables (Dover, 1972).
  39. P. R. Riera, G. S. Pankretz, and D. M. Topa, 'Efficient computation with special functions like the circle polynomials of Zernike,' in Optical Design and Analysis Software II, R.C.Juergens, ed., Proc. SPIE 4769, 130-144 (2002).
  40. J. Schwiegerling, 'Scaling Zernike expansion coefficients to different pupil sizes,' J. Opt. Soc. Am. A 19, 1937-1945 (2002).
  41. 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, 209-217 (2003).
  42. G.-m. Dai, 'Scaling Zernike expansion coefficients to smaller pupil sizes: a simpler formula,' J. Opt. Soc. Am. A 23, 539-543 (2006).
  43. Equation of Ref. ; the notation in Ref. is slightly different from that used in this paper, but the two formulas are equivalent.

2006

2005

2003

2002

1999

A. Roorda and D. R. Williams, 'The arrangement of the three cone classes in the living human eye,' Nature 397, 520-522 (1999).

1998

1997

1996

1995

1994

1991

1990

N. Roddier, 'Atmospheric wavefront simulation using Zernike polynomials,' Opt. Eng. 29, 1174-1180 (1990).

G. Rousset, J.-C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Léna, C. Boyer, P. Jagourel, J.-P. Gaffard, and F. Merkle, 'First diffraction-limited astronomical images with adaptive optics,' Astron. Astrophys. 230, 29-32 (1990).

1989

1988

1987

L. A. Thompson and C. S. Gardner, 'Experiments on laser guide stars at Mauna Kea Observatory for adaptive imaging in astronomy,' Nature 328, 229-231 (1987).

1986

1985

R. Foy and A. Labeyrie, 'Feasibility of adaptive optics telescope with laser probe,' Astron. Astrophys. 152, L29-L31 (1985).

1984

1983

G. Conforti, 'Zernike aberration coefficients from Seidel and higher-order power series coefficients,' Opt. Lett. 8, 390-391 (1983).

E. P. Wallner, 'Optimal wavefront correction using slope measurement,' J. Opt. Soc. Am. 73, 1771-1776 (1983).

1982

1980

1977

1976

1953

H. W. Babcock, 'The possibility of compensating astronomical seeing,' Publ. Astron. Soc. Pac. 65, 229-236 (1953).

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Formulas, Graphs, and Mathematical Tables (Dover, 1972).

Applegate, R. A.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, 'Standards for reporting the optical aberrations of eyes,' Vision Science and Its Applications, Vol. 35 of Trends in Optics and Photonics Series (Optical Society of America, 2000), pp. 232-244.

Babcock, H. W.

H. W. Babcock, 'The possibility of compensating astronomical seeing,' Publ. Astron. Soc. Pac. 65, 229-236 (1953).

Bille, J.

Born, M.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1965), Chap 9.

Boyer, C.

G. Rousset, J.-C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Léna, C. Boyer, P. Jagourel, J.-P. Gaffard, and F. Merkle, 'First diffraction-limited astronomical images with adaptive optics,' Astron. Astrophys. 230, 29-32 (1990).

Braat, J.

Brase, J. M.

Campbell, C. E.

Cannon, R. C.

Charman, W. N.

Conforti, G.

G. Conforti, 'Zernike aberration coefficients from Seidel and higher-order power series coefficients,' Opt. Lett. 8, 390-391 (1983).

Cui, C.

Dai, G.-m.

Dirksen, P.

Donnelly, W. J.

Fontanella, J.-C.

G. Rousset, J.-C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Léna, C. Boyer, P. Jagourel, J.-P. Gaffard, and F. Merkle, 'First diffraction-limited astronomical images with adaptive optics,' Astron. Astrophys. 230, 29-32 (1990).

Foy, R.

R. Foy and A. Labeyrie, 'Feasibility of adaptive optics telescope with laser probe,' Astron. Astrophys. 152, L29-L31 (1985).

Freischlad, K. R.

Gaffard, J.-P.

G. Rousset, J.-C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Léna, C. Boyer, P. Jagourel, J.-P. Gaffard, and F. Merkle, 'First diffraction-limited astronomical images with adaptive optics,' Astron. Astrophys. 230, 29-32 (1990).

Gardner, C. S.

B. M. Welsh and C. S. Gardner, 'Performance analysis of adaptive-optics systems using laser guide stars and slope sensors,' J. Opt. Soc. Am. A 6, 1913-1923 (1989).

L. A. Thompson and C. S. Gardner, 'Experiments on laser guide stars at Mauna Kea Observatory for adaptive imaging in astronomy,' Nature 328, 229-231 (1987).

Gavel, D. T.

Gigan, P.

G. Rousset, J.-C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Léna, C. Boyer, P. Jagourel, J.-P. Gaffard, and F. Merkle, 'First diffraction-limited astronomical images with adaptive optics,' Astron. Astrophys. 230, 29-32 (1990).

Goelz, S.

Grimm, B.

Hardy, J. W.

Howland, B.

Howland, H. C.

Jagourel, P.

G. Rousset, J.-C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Léna, C. Boyer, P. Jagourel, J.-P. Gaffard, and F. Merkle, 'First diffraction-limited astronomical images with adaptive optics,' Astron. Astrophys. 230, 29-32 (1990).

Jakobsson, H.

Janssen, A. J. E. M.

Kern, P.

G. Rousset, J.-C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Léna, C. Boyer, P. Jagourel, J.-P. Gaffard, and F. Merkle, 'First diffraction-limited astronomical images with adaptive optics,' Astron. Astrophys. 230, 29-32 (1990).

Koliopoulos, C. L.

Labeyrie, A.

R. Foy and A. Labeyrie, 'Feasibility of adaptive optics telescope with laser probe,' Astron. Astrophys. 152, L29-L31 (1985).

Laskshminarayanan, V.

Leferbre, J. E.

Léna, P.

G. Rousset, J.-C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Léna, C. Boyer, P. Jagourel, J.-P. Gaffard, and F. Merkle, 'First diffraction-limited astronomical images with adaptive optics,' Astron. Astrophys. 230, 29-32 (1990).

Liang, J.

Mahajan, V. N.

V. N. Mahajan, Optical Imaging and Aberrations, Part II: Wave Diffractive Optics (SPIE, 2002), Chap 5.

Malacara, D.

D. Malacara, Optical Shop Testing (Wiley, 1992), Chap. 13.

Merkle, F.

G. Rousset, J.-C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Léna, C. Boyer, P. Jagourel, J.-P. Gaffard, and F. Merkle, 'First diffraction-limited astronomical images with adaptive optics,' Astron. Astrophys. 230, 29-32 (1990).

Miller, D. T.

Noll, R. J.

Pankretz, G. S.

P. R. Riera, G. S. Pankretz, and D. M. Topa, 'Efficient computation with special functions like the circle polynomials of Zernike,' in Optical Design and Analysis Software II, R.C.Juergens, ed., Proc. SPIE 4769, 130-144 (2002).

Poyneer, L. A.

Riera, P. R.

P. R. Riera, G. S. Pankretz, and D. M. Topa, 'Efficient computation with special functions like the circle polynomials of Zernike,' in Optical Design and Analysis Software II, R.C.Juergens, ed., Proc. SPIE 4769, 130-144 (2002).

Rigaut, F.

G. Rousset, J.-C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Léna, C. Boyer, P. Jagourel, J.-P. Gaffard, and F. Merkle, 'First diffraction-limited astronomical images with adaptive optics,' Astron. Astrophys. 230, 29-32 (1990).

Roddier, C.

Roddier, F.

Roddier, N.

N. Roddier, 'Atmospheric wavefront simulation using Zernike polynomials,' Opt. Eng. 29, 1174-1180 (1990).

Roorda, A.

W. J. Donnelly III and A. Roorda, 'The optimal pupil size in the eye for axial resolution,' J. Opt. Soc. Am. A 20, 2010-2015 (2003).

A. Roorda and D. R. Williams, 'The arrangement of the three cone classes in the living human eye,' Nature 397, 520-522 (1999).

Rousset, G.

G. Rousset, J.-C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Léna, C. Boyer, P. Jagourel, J.-P. Gaffard, and F. Merkle, 'First diffraction-limited astronomical images with adaptive optics,' Astron. Astrophys. 230, 29-32 (1990).

Schwiegerling, J.

Schwiegerling, J. T.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, 'Standards for reporting the optical aberrations of eyes,' Vision Science and Its Applications, Vol. 35 of Trends in Optics and Photonics Series (Optical Society of America, 2000), pp. 232-244.

Southwell, W. H.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Formulas, Graphs, and Mathematical Tables (Dover, 1972).

Thibos, L. N.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, 'Standards for reporting the optical aberrations of eyes,' Vision Science and Its Applications, Vol. 35 of Trends in Optics and Photonics Series (Optical Society of America, 2000), pp. 232-244.

Thompson, L. A.

L. A. Thompson and C. S. Gardner, 'Experiments on laser guide stars at Mauna Kea Observatory for adaptive imaging in astronomy,' Nature 328, 229-231 (1987).

Topa, D. M.

P. R. Riera, G. S. Pankretz, and D. M. Topa, 'Efficient computation with special functions like the circle polynomials of Zernike,' in Optical Design and Analysis Software II, R.C.Juergens, ed., Proc. SPIE 4769, 130-144 (2002).

Tyson, R. K.

Véran, J.-P.

Wallner, E. P.

Walsh, G.

Webb, R.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, 'Standards for reporting the optical aberrations of eyes,' Vision Science and Its Applications, Vol. 35 of Trends in Optics and Photonics Series (Optical Society of America, 2000), pp. 232-244.

Welsh, B. M.

Williams, D. R.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1965), Chap 9.

Appl. Opt.

Astron. Astrophys.

G. Rousset, J.-C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Léna, C. Boyer, P. Jagourel, J.-P. Gaffard, and F. Merkle, 'First diffraction-limited astronomical images with adaptive optics,' Astron. Astrophys. 230, 29-32 (1990).

R. Foy and A. Labeyrie, 'Feasibility of adaptive optics telescope with laser probe,' Astron. Astrophys. 152, L29-L31 (1985).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

G.-m. Dai, 'Modal compensation of atmospheric turbulence with the use of Zernike polynomials and Karhunen-Loève functions,' J. Opt. Soc. Am. A 12, 2182-2193 (1995).

A. J. E. M. Janssen, 'Extended Nijboer-Zernike approach for the computation of optical point-spread functions,' J. Opt. Soc. Am. A 19, 849-857 (2002).

J. Braat, P. Dirksen, and A. J. E. M. Janssen, 'Assessment of an extended Nijboer-Zernike approach for the computation of optical point-spread functions,' J. Opt. Soc. Am. A 19, 858-870 (2002).

J. Schwiegerling, 'Scaling Zernike expansion coefficients to different pupil sizes,' J. Opt. Soc. Am. A 19, 1937-1945 (2002).

L. A. Poyneer, D. T. Gavel, and J. M. Brase, 'Fast wavefront reconstruction in large adaptive optics systems with use of the Fourier transform,' J. Opt. Soc. Am. A 19, 2100-2111 (2002).

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, 209-217 (2003).

W. J. Donnelly III and A. Roorda, 'The optimal pupil size in the eye for axial resolution,' J. Opt. Soc. Am. A 20, 2010-2015 (2003).

L. A. Poyneer and J.-P. Véran, 'Optimal modal Fourier-transform wavefront control,' J. Opt. Soc. Am. A 22, 1515-1526 (2005).

R. C. Cannon, 'Optimal bases for wavefront simulation and reconstruction on annular apertures,' J. Opt. Soc. Am. A 13, 862-867 (1996).

G.-m. Dai, 'Modal wavefront reconstruction with Zernike polynomials and Karhunen-Loève functions,' J. Opt. Soc. Am. A 13, 1218-1225 (1996).

G.-m. Dai, 'Scaling Zernike expansion coefficients to smaller pupil sizes: a simpler formula,' J. Opt. Soc. Am. A 23, 539-543 (2006).

J. Liang, B. Grimm, S. Goelz, and J. Bille, 'Objective measurement of the wave aberrations of the human eye with the use of a Hartmann-Shack wavefront sensor,' J. Opt. Soc. Am. A 11, 1949-1957 (1994).

G. Walsh, W. N. Charman, and H. C. Howland, 'Objective technique for the determination of monochromatic aberrations of the human eye,' J. Opt. Soc. Am. A 1, 987-992 (1984).

C. Cui and V. Laskshminarayanan, 'Choice of reference axis in ocular wavefront aberration measurement,' J. Opt. Soc. Am. A 15, 2488-2496 (1998).

J. Liang and D. R. Williams, 'Aberrations and retinal image quality of the normal human eye,' J. Opt. Soc. Am. A 14, 2873-2883 (1997).

J. Liang, D. R. Williams, and D. T. Miller, 'Supernormal vision and high-resolution retinal imaging through adaptive optics,' J. Opt. Soc. Am. A 14, 2884-2892 (1997).

K. R. Freischlad and C. L. Koliopoulos, 'Modal estimation of wave front difference measurements using the discrete Fourier transform,' J. Opt. Soc. Am. A 3, 1852-1861 (1986).

B. M. Welsh and C. S. Gardner, 'Performance analysis of adaptive-optics systems using laser guide stars and slope sensors,' J. Opt. Soc. Am. A 6, 1913-1923 (1989).

Nature

A. Roorda and D. R. Williams, 'The arrangement of the three cone classes in the living human eye,' Nature 397, 520-522 (1999).

L. A. Thompson and C. S. Gardner, 'Experiments on laser guide stars at Mauna Kea Observatory for adaptive imaging in astronomy,' Nature 328, 229-231 (1987).

Opt. Eng.

N. Roddier, 'Atmospheric wavefront simulation using Zernike polynomials,' Opt. Eng. 29, 1174-1180 (1990).

Opt. Lett.

Publ. Astron. Soc. Pac.

H. W. Babcock, 'The possibility of compensating astronomical seeing,' Publ. Astron. Soc. Pac. 65, 229-236 (1953).

Other

G.-m. Dai, 'Wavefront simulation for atmospheric turbulence,' in Image Reconstruction and Restoration, T.J.Schulz and D.-L.Snyder, eds., Proc. SPIE 2302, 62-72 (1994).

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1965), Chap 9.

D. Malacara, Optical Shop Testing (Wiley, 1992), Chap. 13.

V. N. Mahajan, Optical Imaging and Aberrations, Part II: Wave Diffractive Optics (SPIE, 2002), Chap 5.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, 'Standards for reporting the optical aberrations of eyes,' Vision Science and Its Applications, Vol. 35 of Trends in Optics and Photonics Series (Optical Society of America, 2000), pp. 232-244.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Formulas, Graphs, and Mathematical Tables (Dover, 1972).

P. R. Riera, G. S. Pankretz, and D. M. Topa, 'Efficient computation with special functions like the circle polynomials of Zernike,' in Optical Design and Analysis Software II, R.C.Juergens, ed., Proc. SPIE 4769, 130-144 (2002).

Equation of Ref. ; the notation in Ref. is slightly different from that used in this paper, but the two formulas are equivalent.

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