Abstract

A systematic treatment of modal estimation of a wave-front phase from its gradients is given. We introduce a gradient matrix and use it to describe cross coupling of aberrations (lack of orthogonality of its column vectors) and aliasing of aberrations (lack of linear independence of its column vectors).

© 1981 Optical Society of America

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References

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  1. R. Cubalchini, “Modal wave-front estimates from phase derivative measurements,” J. Opt. Soc. Am. 69, 972–977 (1979).
    [CrossRef]
  2. D. L. Fried, “Least-squares fitting a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. 67, 370–375 (1977).
    [CrossRef]
  3. R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am. 67, 374–378 (1977).
    [CrossRef]
  4. J. Herrmann, “Least-squares wave front errors of minimum norm,” J. Opt. Soc. Am. 70, 28–35 (1980).
    [CrossRef]
  5. M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1965).
  6. C. L. Lawson and R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, Englewood Cliffs, N.J., 1974).
  7. A. Ben-Israel and N. E. Greville, Generalized Inverses: Theory and Applications (Wiley, New York, 1974).
  8. J. W. Hardy, J. E. Lefebvre, and C. L. Koliopoulos, “Real-time atmospheric compensation,” J. Opt. Soc. Am. 67, 360–369 (1977).
    [CrossRef]
  9. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]

1980 (1)

1979 (1)

1977 (3)

1976 (1)

Ben-Israel, A.

A. Ben-Israel and N. E. Greville, Generalized Inverses: Theory and Applications (Wiley, New York, 1974).

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1965).

Cubalchini, R.

Fried, D. L.

Greville, N. E.

A. Ben-Israel and N. E. Greville, Generalized Inverses: Theory and Applications (Wiley, New York, 1974).

Hanson, R. J.

C. L. Lawson and R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, Englewood Cliffs, N.J., 1974).

Hardy, J. W.

Herrmann, J.

Hudgin, R. H.

R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am. 67, 374–378 (1977).
[CrossRef]

Koliopoulos, C. L.

Lawson, C. L.

C. L. Lawson and R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, Englewood Cliffs, N.J., 1974).

Lefebvre, J. E.

Noll, R. J.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1965).

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Figures (5)

Fig. 1
Fig. 1

Array for which a modal decomposition was performed. M = 42, N = 21.

Fig. 2
Fig. 2

Weighting array for a decomposition using an expansion up to focus only, and the first three rows of the cross-coupling matrix.

Fig. 3
Fig. 3

Weighting array for a decomposition using an expansion up to Zernike polynomial number 22.

Fig. 4
Fig. 4

Weighting array for a decomposition using an expansion up to Zernike polynomial number 22 with an average gradient. The cross-coupling matrix is for a measuring model using a gradient evaluated at one point for a measurement that actually performs an average.

Fig. 5
Fig. 5

Distinction between aberration cross coupling and aliasing. Gradient g = Dc. Column vectors, D = D2D3D4DN. Lack of orthogonality is aberration cross coupling; lack of linear independence is aberration aliasing.

Tables (3)

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Table 1 Zernike Polynomials ( U n m = R n m ( ρ ) cos m θ , U n - m = R n m ( ρ ) sin m θ)

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Table 2 Aberration Aliasing

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Table 3 Zernike Representation of Kolmogorov Spectrum of Turbulencea

Equations (10)

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ϕ = k = 1 N c k Z k ,
g = k = 2 N c k Z k .
g = D c ,
g = D c + n .
( D c - g ) T ( D c - g ) = min
ĉ = ( D T D ) - 1 D T g .
ĉ s = D s + g ,
g = D f c f
ĉ s = D s + D f c f = C s , f c f ,
c = D + g + ( I - D + D ) y