Abstract

We outline a method for the determination of the unknown wave-front aberration function of an optical system from noisy measurements of the corresponding point-spread function. The problem is cast as a nonlinear unconstrained minimization problem, and trust region techniques are employed for its solution in conjunction with analytic evaluations of the Jacobian and Hessian matrices governing slope and curvature information. Some illustrative numerical results are presented and discussed.

© 1992 Optical Society of America

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References

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  1. R. Gonsalves, “Phase retrieval from modulus data,”J. Opt. Soc. Am. 66, 961–964 (1976).
    [Crossref]
  2. R. Barakat, G. Newsam, “Numerically stable iterative method for the inversion of wave-front aberrations from measured point-spread-function data,”J. Opt. Soc. Am. 70, 1255–1263 (1980).
    [Crossref]
  3. R. Barakat, G. Newsam, “Algorithms for reconstruction of partially known, bandlimited Fourier transform pairs from noisy data. I. The prototypical linear problem; II. The nonlinear problem of phase retrieval,”J. Integr. Eqs. 9, 49–125 (1985).
  4. R. Barakat, G. Newsam, “Necessary conditions for a unique solution to two-dimensional phase retrieval,”J. Math. Phys. (NY) 23, 3190–3193 (1984).
    [Crossref]
  5. J. Fienup, “Reconstruction of an object from the modulus of its Fourier transforms,” Opt. Lett. 3, 27–29 (1978).
    [Crossref] [PubMed]
  6. J. Fienup, “Phase retrieval algorithms,” Appl. Opt. 21, 2758–2769 (1982). This paper contains references to many of his previous papers on the Fienup algorithm. For an application of the Fienup algorithm to the wave-front recovery problem, see his paper “Characterizing the Hubble space telescope using retrieval algorithms,” Opt. Photon. News 12, (12) 41–42 (1991).
    [Crossref] [PubMed]
  7. G. Newsam, R. Barakat, “Phase retrieval in two dimensions,” (Centre for Mathematical Analysis, Australian National University, Canberra, Australia, 1986).
  8. N. E. Hurt, Phase Retrieval and Zero Crossings (Kluwer, Boston, Mass., 1989).
    [Crossref]
  9. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, London, 1986).
  10. P. Davis, P. Rabinowitz, Methods of Numerical Integration (Academic, New York, 1975), Chap. 2.
  11. J. Dennis, R. Schnable, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Prentice-Hall, Englewood Cliffs, N.J., 1983).
  12. R. Fletcher, Practical Methods of Optimization, 2nd ed. (Wiley, New York, 1987).
  13. J. J. More, “Recent developments in algorithms and software for trust region methods,” in Mathematical Programming: the State of the Art, A. Bachem, M. Grotschel, B. Korte, eds. (Springer-Verlag, Berlin, 1983), pp. 258–287.
    [Crossref]
  14. A. N. Tikhonov, V. Arsenin, Solutions of Ill-Posed Problems (Wiley, New York, 1977).
  15. Y. A. Morozov, Methods of Solving Incorrectly Posed Problems (Springer-Verlag, New York, 1984), Chap. 2.
    [Crossref]
  16. W. Lawton, “Uniqueness results for the phase-retrieval problem for radial functions,”J. Opt. Soc. Am. 71, 1519–1522 (1981).
    [Crossref]

1985 (1)

R. Barakat, G. Newsam, “Algorithms for reconstruction of partially known, bandlimited Fourier transform pairs from noisy data. I. The prototypical linear problem; II. The nonlinear problem of phase retrieval,”J. Integr. Eqs. 9, 49–125 (1985).

1984 (1)

R. Barakat, G. Newsam, “Necessary conditions for a unique solution to two-dimensional phase retrieval,”J. Math. Phys. (NY) 23, 3190–3193 (1984).
[Crossref]

1982 (1)

1981 (1)

1980 (1)

1978 (1)

1976 (1)

Arsenin, V.

A. N. Tikhonov, V. Arsenin, Solutions of Ill-Posed Problems (Wiley, New York, 1977).

Barakat, R.

R. Barakat, G. Newsam, “Algorithms for reconstruction of partially known, bandlimited Fourier transform pairs from noisy data. I. The prototypical linear problem; II. The nonlinear problem of phase retrieval,”J. Integr. Eqs. 9, 49–125 (1985).

R. Barakat, G. Newsam, “Necessary conditions for a unique solution to two-dimensional phase retrieval,”J. Math. Phys. (NY) 23, 3190–3193 (1984).
[Crossref]

R. Barakat, G. Newsam, “Numerically stable iterative method for the inversion of wave-front aberrations from measured point-spread-function data,”J. Opt. Soc. Am. 70, 1255–1263 (1980).
[Crossref]

G. Newsam, R. Barakat, “Phase retrieval in two dimensions,” (Centre for Mathematical Analysis, Australian National University, Canberra, Australia, 1986).

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, London, 1986).

Davis, P.

P. Davis, P. Rabinowitz, Methods of Numerical Integration (Academic, New York, 1975), Chap. 2.

Dennis, J.

J. Dennis, R. Schnable, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Prentice-Hall, Englewood Cliffs, N.J., 1983).

Fienup, J.

Fletcher, R.

R. Fletcher, Practical Methods of Optimization, 2nd ed. (Wiley, New York, 1987).

Gonsalves, R.

Hurt, N. E.

N. E. Hurt, Phase Retrieval and Zero Crossings (Kluwer, Boston, Mass., 1989).
[Crossref]

Lawton, W.

More, J. J.

J. J. More, “Recent developments in algorithms and software for trust region methods,” in Mathematical Programming: the State of the Art, A. Bachem, M. Grotschel, B. Korte, eds. (Springer-Verlag, Berlin, 1983), pp. 258–287.
[Crossref]

Morozov, Y. A.

Y. A. Morozov, Methods of Solving Incorrectly Posed Problems (Springer-Verlag, New York, 1984), Chap. 2.
[Crossref]

Newsam, G.

R. Barakat, G. Newsam, “Algorithms for reconstruction of partially known, bandlimited Fourier transform pairs from noisy data. I. The prototypical linear problem; II. The nonlinear problem of phase retrieval,”J. Integr. Eqs. 9, 49–125 (1985).

R. Barakat, G. Newsam, “Necessary conditions for a unique solution to two-dimensional phase retrieval,”J. Math. Phys. (NY) 23, 3190–3193 (1984).
[Crossref]

R. Barakat, G. Newsam, “Numerically stable iterative method for the inversion of wave-front aberrations from measured point-spread-function data,”J. Opt. Soc. Am. 70, 1255–1263 (1980).
[Crossref]

G. Newsam, R. Barakat, “Phase retrieval in two dimensions,” (Centre for Mathematical Analysis, Australian National University, Canberra, Australia, 1986).

Rabinowitz, P.

P. Davis, P. Rabinowitz, Methods of Numerical Integration (Academic, New York, 1975), Chap. 2.

Schnable, R.

J. Dennis, R. Schnable, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Prentice-Hall, Englewood Cliffs, N.J., 1983).

Tikhonov, A. N.

A. N. Tikhonov, V. Arsenin, Solutions of Ill-Posed Problems (Wiley, New York, 1977).

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, London, 1986).

Appl. Opt. (1)

J. Integr. Eqs. (1)

R. Barakat, G. Newsam, “Algorithms for reconstruction of partially known, bandlimited Fourier transform pairs from noisy data. I. The prototypical linear problem; II. The nonlinear problem of phase retrieval,”J. Integr. Eqs. 9, 49–125 (1985).

J. Math. Phys. (NY) (1)

R. Barakat, G. Newsam, “Necessary conditions for a unique solution to two-dimensional phase retrieval,”J. Math. Phys. (NY) 23, 3190–3193 (1984).
[Crossref]

J. Opt. Soc. Am. (3)

Opt. Lett. (1)

Other (9)

G. Newsam, R. Barakat, “Phase retrieval in two dimensions,” (Centre for Mathematical Analysis, Australian National University, Canberra, Australia, 1986).

N. E. Hurt, Phase Retrieval and Zero Crossings (Kluwer, Boston, Mass., 1989).
[Crossref]

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, London, 1986).

P. Davis, P. Rabinowitz, Methods of Numerical Integration (Academic, New York, 1975), Chap. 2.

J. Dennis, R. Schnable, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Prentice-Hall, Englewood Cliffs, N.J., 1983).

R. Fletcher, Practical Methods of Optimization, 2nd ed. (Wiley, New York, 1987).

J. J. More, “Recent developments in algorithms and software for trust region methods,” in Mathematical Programming: the State of the Art, A. Bachem, M. Grotschel, B. Korte, eds. (Springer-Verlag, Berlin, 1983), pp. 258–287.
[Crossref]

A. N. Tikhonov, V. Arsenin, Solutions of Ill-Posed Problems (Wiley, New York, 1977).

Y. A. Morozov, Methods of Solving Incorrectly Posed Problems (Springer-Verlag, New York, 1984), Chap. 2.
[Crossref]

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Figures (7)

Fig. 1
Fig. 1

Locations of the 11-point Gauss quadrature points located within the circular aperture.

Fig. 2
Fig. 2

Locations of the 17-point Gauss quadrature points located within the circular aperture.

Fig. 3
Fig. 3

Theoretical values of the defocused wave front W2 = 0.5λ at 11 × 11 Gauss quadrature points inside the unit circle 0 ≤ p2 + q2 ≤ 1 of the exit pupil.

Fig. 4
Fig. 4

Numerical values of the estimated wave front for data in Fig. 3 for a sample realization from the measured point-spread function with 5% noise.

Fig. 5
Fig. 5

Theoretical value of the optionally balanced rotationally symmetric wave front W2 = −0.45λ, W4 = 0.45λ at 17 × 17 Gauss quadrature points inside the unit circle 0 ≤ p2 + q2 ≤ 1 of the exit pupil.

Fig. 6
Fig. 6

Numerical values of the estimated wave front for data in Fig. 5 for a sample realization from the measured point-spread function with 10% noise.

Fig. 7
Fig. 7

Numerical values of the estimated wave front for data in Fig. 5 for a sample realization from the measured point-spread function data with 10% noise.

Tables (1)

Tables Icon

Table 1 Values of Gauss Quadrature Points Truncated to Five Digits

Equations (35)

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u = ( k a f ) x ,             v = ( k a f ) y .
u m = β m ,             v n = β n ,             m , n = 0 , ± 1 , ± 2 , , ± M ,
τ ( u M , v M ) 0.
t ( x , y ) = | C 0 ξ 2 + η 2 a 2 A ( ξ , η ) exp [ i k f ( x ξ + y η ) ] d ξ d η | 2 ,
A ( ξ , η ) = B ( ξ , η ) exp [ i k W ( ξ , η ) ] ,
W ( 0 , 0 ) = 0.
p = ξ / a ,             q = η / a .
t ( u , v ) = | o p 2 + q 2 1 exp [ i k W ( p , q ) ] exp [ i ( u p + v q ) ] d p d q | 2 ,
τ = T ( W ) .
t ( u m , v n ) t m n = | k = 1 N l = 1 N α k α l exp [ i 2 π W ( p k , q l ) ] exp [ i ( u m p k + v n q l ) ] | 2 .
( W ) ( W * ) ,             W .
( W + Δ W ) ( W ) + G T ( W ) ( Δ W ) + ½ ( Δ W ) T H ( W ) ( Δ W ) ,
G T | W 1 ,             W 2 , , W n ˜ | ,
H | 2 W 1 2 2 W 1 W n ˜ 2 W n ˜ W 1 2 W n ˜ 2 | .
G ( W * ) = 0.
( Δ W ) T H ( W * ) ( Δ W ) > 0 ,
Δ W = - H ( W ) - 1 G ( W ) .
Δ ( k ) = { W : W - W ( k ) h ( k ) } ,
W ( k + 1 ) = W ( k ) + Δ ( k ) ,
minimize ( local model ) subject to constraint Δ 2 h ( k ) .
r k [ W ( k ) + Δ ( k ) ] - [ W ( k ) ] q [ W ( k ) + Δ ( k ) ] - q [ W ( k ) ] ,
( W ) = i = 1 m ˜ [ t j ( W ) - τ i ] 2 .
( W ) = i = 1 m ˜ t i ( W ) - τ i .
Φ ( W ) = T ( W ) - τ ,
( W ) = Φ T ( W ) Φ ( W ) .
J | ϕ 1 W 1 ϕ 1 W n ˜ ϕ m ˜ W 1 ϕ m ˜ W n ˜ | .
W j = 2 i = 1 m ˜ ϕ i ϕ i W j
G = 2 J T Φ .
2 W k W l = 2 i = 1 m ˜ ϕ i W k ϕ i W l + 2 i = 1 m ˜ ϕ i 2 ϕ i W k W l ,
H = 2 J T J + S .
τ ( u m , v n ) = [ 1 + δ μ ( u m , v n ) ] t ( u m , v n ) ,
f ( μ ) = 1 / 2 , μ < 1 , = 0 , μ > 1.
W ( p , q ) = W [ ( p 2 + q 2 ) 1 / 2 ] ,
W ( p , q ) = W 2 ( p 2 + q 2 ) ,
W ( p , q ) = W 4 ( p 2 + q 2 ) 2 + W 2 ( p 2 + q 2 )

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