Abstract

An analytical formulation that relates the modal expansion coefficients of a given wave front to its local transverse phase derivatives is proposed. The modal coefficients are calculated as a weighted integral over the wave-front slopes. The weighting functions for each mode are the components of a two-dimensional vector whose divergence equals the corresponding mode function. This approach is useful for analytical phase reconstruction from the input data provided by shearing interferometers or Hartmann–Shack wave-front sensors. Numerical results for a simulated experiment in terms of a set of Zernike polynomials are given.

© 1995 Optical Society of America

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References

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  1. F. Merkle, in International Trends in Optics, J. W. Goodman, ed. (Academic, London, 1991), pp. 375–390.
  2. J. Primot, G. Rousset, J. C. Fontanella, J. Opt. Soc. Am. A 7, 1598 (1990).
    [CrossRef]
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    [PubMed]
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    [CrossRef] [PubMed]
  5. W. H. Southwell, J. Opt. Soc. Am. 70, 988 (1980).
    [CrossRef]
  6. D. L. Fried, J. Opt. Soc. Am. 67, 370 (1977).
    [CrossRef]
  7. R. Cubalchini, J. Opt. Soc. Am. 69, 972 (1979).
    [CrossRef]
  8. V. P. Aksenov, Yu. N. Isaev, Opt. Lett. 17, 1180 (1992).
    [CrossRef] [PubMed]
  9. J. D. Logan, Applied Mathematics: A Contemporary Approach (Wiley, New York, 1987), Chap. 8, p. 534.
  10. D. Malacara, ed., Optical Shop Testing (Wiley Inter-science, New York, 1978), App. 2, p. 490.
  11. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Chap. 9, p. 465.
  12. S. S. Kuo, Computer Applications of Numerical Methods (Addison-Wesley, Reading, Mass., 1972), Chap. 12, pp. 299–304.
  13. N. Bakhvalov, Métodos Numéricos (Paraninfo, Madrid, 1980), Chap. 3, pp. 135–136.
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    [CrossRef] [PubMed]

1992 (2)

1990 (1)

1980 (2)

1979 (1)

1977 (1)

1975 (1)

Aksenov, V. P.

Bakhvalov, N.

N. Bakhvalov, Métodos Numéricos (Paraninfo, Madrid, 1980), Chap. 3, pp. 135–136.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Chap. 9, p. 465.

Cubalchini, R.

Fontanella, J. C.

Fried, D. L.

Isaev, Yu. N.

Kuo, S. S.

S. S. Kuo, Computer Applications of Numerical Methods (Addison-Wesley, Reading, Mass., 1972), Chap. 12, pp. 299–304.

Lane, R. G.

Logan, J. D.

J. D. Logan, Applied Mathematics: A Contemporary Approach (Wiley, New York, 1987), Chap. 8, p. 534.

Merkle, F.

F. Merkle, in International Trends in Optics, J. W. Goodman, ed. (Academic, London, 1991), pp. 375–390.

Primot, J.

Rimmer, M. P.

Rousset, G.

Silva, D. E.

Southwell, W. H.

W. H. Southwell, J. Opt. Soc. Am. 70, 988 (1980).
[CrossRef]

Tallon, M.

Wang, J. Y.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Chap. 9, p. 465.

Wyant, J. C.

Appl. Opt. (3)

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (1)

Opt. Lett. (1)

Other (6)

J. D. Logan, Applied Mathematics: A Contemporary Approach (Wiley, New York, 1987), Chap. 8, p. 534.

D. Malacara, ed., Optical Shop Testing (Wiley Inter-science, New York, 1978), App. 2, p. 490.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Chap. 9, p. 465.

S. S. Kuo, Computer Applications of Numerical Methods (Addison-Wesley, Reading, Mass., 1972), Chap. 12, pp. 299–304.

N. Bakhvalov, Métodos Numéricos (Paraninfo, Madrid, 1980), Chap. 3, pp. 135–136.

F. Merkle, in International Trends in Optics, J. W. Goodman, ed. (Academic, London, 1991), pp. 375–390.

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Figures (2)

Fig. 1
Fig. 1

Definition domain of the wave-front function.

Fig. 2
Fig. 2

Phase plot of the simulated wave front.

Tables (1)

Tables Icon

Table 1 radial and angular components for the weighting functions of nl zernike Modes, Mode Coefficients for the Simulated Wave Front (anl), and Reconstructed Values ( a n l )

Equations (12)

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S ( r ) = k a k ψ k ( r )
a n = ψ n , S / ψ n , ψ n ,
f , g = σ f * ( r ) g ( r ) d σ ,
· F k ( r ) = ψ k * ( r ) ,
a n = c n σ · ( S F n ) d σ - c n σ ( S ) · F n d σ ,
( · F n ) S = · ( S F n ) - ( S ) · F n .
a n = c n C S F n · d c - c n σ ( S ) · F n d σ ,
a n = - c n σ ( S ) · F n d σ ,
ψ n l ( ρ , θ ) = R n l ( ρ ) sin ( l θ )             for     l > 0 , ψ n l ( ρ , θ ) = R n l ( ρ ) cos ( l θ )             for l 0 ,
( 1 / ρ ) ( ρ F n l , ρ ) / ρ + ( 1 / ρ ) F n l , θ / θ = ψ n l ( ρ , θ ) ,
F n l , θ ( ρ , θ ) = - ( ρ / l ) R n l ( ρ ) cos ( l θ ) + u ( ρ )             for     l > 0 , F n l , θ ( ρ , θ ) = ( ρ / l ) R n l ( ρ ) sin ( l θ ) + w ( ρ )             for     l < 0 ,
F n 0 , ρ ( ρ , θ ) = ( 1 / ρ ) 0 ρ R n 0 ( ρ ) ρ d ρ .

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