Abstract

The method for phase retrieval from a set of intensity measurements is described and thoroughly investigated. This method is based on application and further minimization by gradient procedures of the functional of reconstruction error in all the planes of recording. The method is shown to result in the well-known Gerchberg–Saxton algorithm [ Optik (Stuttgart) 35, 237 ( 1972)] if two intensity measurements (in the pupil and the focal planes of a lens) are used. Numerical simulation revealed the following advantages of the method: the convergence of iterative procedures is improved, the range of reconstruction turns out to be wider, and the stability of procedures with respect to the additive noise in intensity measurements is enhanced. Experimental data confirming the conclusions of numerical simulation are presented. A nonlinear optical spatial filter is also described. The possibility of applying this filter to the problem of recovering phase from intensity measurements in the input and the output planes of the filter is shown.

© 1992 Optical Society of America

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References

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  1. H. P. Baltes, ed., Inverse Problems in Optics (Springer-Verlag, New York, 1978).
    [CrossRef]
  2. W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy (Academic, New York, 1978).
  3. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  4. M. V. Klibanov, “On the uniqueness of determination of a finite function by its Fourier transform modulus,” Dokl. Akad. Nauk SSSR 285, 278–280 (1985).
  5. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).
  6. D. L. Misell, “A new method for the solution of the phase problem in electron microscopy,”J. Phys. D 6, 16–19 (1973).
    [CrossRef]
  7. R. Rollestiow, N. George, “Image reconstruction from partial Fresnel zone information,” Appl. Opt. 25, 178–183 (1986).
    [CrossRef]
  8. S. Sali, A. P. Anderson, “New phase retrieval technique for microwave imaging diagnostics,” presented at the Colloquium on Inverse Methods of Image Processing; Colloq. Digest IEE 65 (Institution of Electrical Engineers, London, 1985), pp. 1–5.
  9. B. R. Frieden, ed., The Computer in Optical Research (Springer-Verlag, New York, 1980).
    [CrossRef]
  10. M. A. Vorontsov, V. I. Shmalhauzen, Principles of Adaptive Optics (Nauka, Moscow, 1988).
  11. W. H. Southwell, “Wave-front analyzer using a maximum likelihood algorithm,”J. Opt. Soc. Am. 67, 396–399 (1977).
    [CrossRef]
  12. D. A. Nahrstedt, W. H. Southwell, “Maximum likelihood phase-retrieval algorithm: applications,” Appl. Opt. 23, 4328–4331 (1984).
    [CrossRef] [PubMed]
  13. M. A. Vorontsov, A. N. Matveev, V. P. Sivokon, “Phase retrieval from registered intensity distributions,” Dokl. Akad. Nauk SSSR 296, 842–846 (1987).
  14. M. A. Vorontsov, A. V. Koriabin, V. I. Shmalhauzen, Controlled Optical Systems (Nauka, Moscow, 1988).
  15. F. P. Vasiliev, Methods of Solution of Extreme Problems (Nauka, Moscow, 1985).
  16. Yang Guozhen, Wang Li, Dong Bizhen, Gu Benyuan, “On the amplitude-phase retrieval problem in an optical system involved non-unitary transformation,” Optik (Stuttgart) 75, 68–74 (1987).
  17. E. A. Sziclas, A. E. Siegman, “Diffraction calculations using fast Fourier transform methods,” Proc. IEEE 62, 410–412 (1974).
    [CrossRef]
  18. M. A. Vorontsov, I. A. Kudriashov, V. I. Shmalhauzen, “Methods of adaptive optics in wave front reconstruction from intensity measurement,” Opt. Spectrosc. (USSR) 63, 842–846 (1987).
  19. M. A. Vorontsov, V. Y. Ivanov, A. N. Matveev, “Adaptive compensation of phase distortions by systems with optical feedback,” Vestn. Mosk. Univ. Fiz. 30, 28–31 (1989).
  20. H. M. Gibbs, Optical Bistability: Controlling Light with Light (Academic, New York, 1985).
  21. D. Malacara, ed., Optical Shop Testing (Wiley, Toronto, 1977), pp. 264–270.

1989 (1)

M. A. Vorontsov, V. Y. Ivanov, A. N. Matveev, “Adaptive compensation of phase distortions by systems with optical feedback,” Vestn. Mosk. Univ. Fiz. 30, 28–31 (1989).

1987 (3)

M. A. Vorontsov, I. A. Kudriashov, V. I. Shmalhauzen, “Methods of adaptive optics in wave front reconstruction from intensity measurement,” Opt. Spectrosc. (USSR) 63, 842–846 (1987).

M. A. Vorontsov, A. N. Matveev, V. P. Sivokon, “Phase retrieval from registered intensity distributions,” Dokl. Akad. Nauk SSSR 296, 842–846 (1987).

Yang Guozhen, Wang Li, Dong Bizhen, Gu Benyuan, “On the amplitude-phase retrieval problem in an optical system involved non-unitary transformation,” Optik (Stuttgart) 75, 68–74 (1987).

1986 (1)

1985 (1)

M. V. Klibanov, “On the uniqueness of determination of a finite function by its Fourier transform modulus,” Dokl. Akad. Nauk SSSR 285, 278–280 (1985).

1984 (1)

1982 (1)

1977 (1)

1974 (1)

E. A. Sziclas, A. E. Siegman, “Diffraction calculations using fast Fourier transform methods,” Proc. IEEE 62, 410–412 (1974).
[CrossRef]

1973 (1)

D. L. Misell, “A new method for the solution of the phase problem in electron microscopy,”J. Phys. D 6, 16–19 (1973).
[CrossRef]

1972 (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Anderson, A. P.

S. Sali, A. P. Anderson, “New phase retrieval technique for microwave imaging diagnostics,” presented at the Colloquium on Inverse Methods of Image Processing; Colloq. Digest IEE 65 (Institution of Electrical Engineers, London, 1985), pp. 1–5.

Benyuan, Gu

Yang Guozhen, Wang Li, Dong Bizhen, Gu Benyuan, “On the amplitude-phase retrieval problem in an optical system involved non-unitary transformation,” Optik (Stuttgart) 75, 68–74 (1987).

Bizhen, Dong

Yang Guozhen, Wang Li, Dong Bizhen, Gu Benyuan, “On the amplitude-phase retrieval problem in an optical system involved non-unitary transformation,” Optik (Stuttgart) 75, 68–74 (1987).

Fienup, J. R.

George, N.

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Gibbs, H. M.

H. M. Gibbs, Optical Bistability: Controlling Light with Light (Academic, New York, 1985).

Guozhen, Yang

Yang Guozhen, Wang Li, Dong Bizhen, Gu Benyuan, “On the amplitude-phase retrieval problem in an optical system involved non-unitary transformation,” Optik (Stuttgart) 75, 68–74 (1987).

Ivanov, V. Y.

M. A. Vorontsov, V. Y. Ivanov, A. N. Matveev, “Adaptive compensation of phase distortions by systems with optical feedback,” Vestn. Mosk. Univ. Fiz. 30, 28–31 (1989).

Klibanov, M. V.

M. V. Klibanov, “On the uniqueness of determination of a finite function by its Fourier transform modulus,” Dokl. Akad. Nauk SSSR 285, 278–280 (1985).

Koriabin, A. V.

M. A. Vorontsov, A. V. Koriabin, V. I. Shmalhauzen, Controlled Optical Systems (Nauka, Moscow, 1988).

Kudriashov, I. A.

M. A. Vorontsov, I. A. Kudriashov, V. I. Shmalhauzen, “Methods of adaptive optics in wave front reconstruction from intensity measurement,” Opt. Spectrosc. (USSR) 63, 842–846 (1987).

Li, Wang

Yang Guozhen, Wang Li, Dong Bizhen, Gu Benyuan, “On the amplitude-phase retrieval problem in an optical system involved non-unitary transformation,” Optik (Stuttgart) 75, 68–74 (1987).

Matveev, A. N.

M. A. Vorontsov, V. Y. Ivanov, A. N. Matveev, “Adaptive compensation of phase distortions by systems with optical feedback,” Vestn. Mosk. Univ. Fiz. 30, 28–31 (1989).

M. A. Vorontsov, A. N. Matveev, V. P. Sivokon, “Phase retrieval from registered intensity distributions,” Dokl. Akad. Nauk SSSR 296, 842–846 (1987).

Misell, D. L.

D. L. Misell, “A new method for the solution of the phase problem in electron microscopy,”J. Phys. D 6, 16–19 (1973).
[CrossRef]

Nahrstedt, D. A.

Rollestiow, R.

Sali, S.

S. Sali, A. P. Anderson, “New phase retrieval technique for microwave imaging diagnostics,” presented at the Colloquium on Inverse Methods of Image Processing; Colloq. Digest IEE 65 (Institution of Electrical Engineers, London, 1985), pp. 1–5.

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy (Academic, New York, 1978).

Shmalhauzen, V. I.

M. A. Vorontsov, I. A. Kudriashov, V. I. Shmalhauzen, “Methods of adaptive optics in wave front reconstruction from intensity measurement,” Opt. Spectrosc. (USSR) 63, 842–846 (1987).

M. A. Vorontsov, A. V. Koriabin, V. I. Shmalhauzen, Controlled Optical Systems (Nauka, Moscow, 1988).

M. A. Vorontsov, V. I. Shmalhauzen, Principles of Adaptive Optics (Nauka, Moscow, 1988).

Siegman, A. E.

E. A. Sziclas, A. E. Siegman, “Diffraction calculations using fast Fourier transform methods,” Proc. IEEE 62, 410–412 (1974).
[CrossRef]

Sivokon, V. P.

M. A. Vorontsov, A. N. Matveev, V. P. Sivokon, “Phase retrieval from registered intensity distributions,” Dokl. Akad. Nauk SSSR 296, 842–846 (1987).

Southwell, W. H.

Sziclas, E. A.

E. A. Sziclas, A. E. Siegman, “Diffraction calculations using fast Fourier transform methods,” Proc. IEEE 62, 410–412 (1974).
[CrossRef]

Vasiliev, F. P.

F. P. Vasiliev, Methods of Solution of Extreme Problems (Nauka, Moscow, 1985).

Vorontsov, M. A.

M. A. Vorontsov, V. Y. Ivanov, A. N. Matveev, “Adaptive compensation of phase distortions by systems with optical feedback,” Vestn. Mosk. Univ. Fiz. 30, 28–31 (1989).

M. A. Vorontsov, A. N. Matveev, V. P. Sivokon, “Phase retrieval from registered intensity distributions,” Dokl. Akad. Nauk SSSR 296, 842–846 (1987).

M. A. Vorontsov, I. A. Kudriashov, V. I. Shmalhauzen, “Methods of adaptive optics in wave front reconstruction from intensity measurement,” Opt. Spectrosc. (USSR) 63, 842–846 (1987).

M. A. Vorontsov, V. I. Shmalhauzen, Principles of Adaptive Optics (Nauka, Moscow, 1988).

M. A. Vorontsov, A. V. Koriabin, V. I. Shmalhauzen, Controlled Optical Systems (Nauka, Moscow, 1988).

Appl. Opt. (3)

Dokl. Akad. Nauk SSSR (2)

M. V. Klibanov, “On the uniqueness of determination of a finite function by its Fourier transform modulus,” Dokl. Akad. Nauk SSSR 285, 278–280 (1985).

M. A. Vorontsov, A. N. Matveev, V. P. Sivokon, “Phase retrieval from registered intensity distributions,” Dokl. Akad. Nauk SSSR 296, 842–846 (1987).

J. Opt. Soc. Am. (1)

J. Phys. D (1)

D. L. Misell, “A new method for the solution of the phase problem in electron microscopy,”J. Phys. D 6, 16–19 (1973).
[CrossRef]

Opt. Spectrosc. (USSR) (1)

M. A. Vorontsov, I. A. Kudriashov, V. I. Shmalhauzen, “Methods of adaptive optics in wave front reconstruction from intensity measurement,” Opt. Spectrosc. (USSR) 63, 842–846 (1987).

Optik (Stuttgart) (2)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Yang Guozhen, Wang Li, Dong Bizhen, Gu Benyuan, “On the amplitude-phase retrieval problem in an optical system involved non-unitary transformation,” Optik (Stuttgart) 75, 68–74 (1987).

Proc. IEEE (1)

E. A. Sziclas, A. E. Siegman, “Diffraction calculations using fast Fourier transform methods,” Proc. IEEE 62, 410–412 (1974).
[CrossRef]

Vestn. Mosk. Univ. Fiz. (1)

M. A. Vorontsov, V. Y. Ivanov, A. N. Matveev, “Adaptive compensation of phase distortions by systems with optical feedback,” Vestn. Mosk. Univ. Fiz. 30, 28–31 (1989).

Other (9)

H. M. Gibbs, Optical Bistability: Controlling Light with Light (Academic, New York, 1985).

D. Malacara, ed., Optical Shop Testing (Wiley, Toronto, 1977), pp. 264–270.

H. P. Baltes, ed., Inverse Problems in Optics (Springer-Verlag, New York, 1978).
[CrossRef]

W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy (Academic, New York, 1978).

M. A. Vorontsov, A. V. Koriabin, V. I. Shmalhauzen, Controlled Optical Systems (Nauka, Moscow, 1988).

F. P. Vasiliev, Methods of Solution of Extreme Problems (Nauka, Moscow, 1985).

S. Sali, A. P. Anderson, “New phase retrieval technique for microwave imaging diagnostics,” presented at the Colloquium on Inverse Methods of Image Processing; Colloq. Digest IEE 65 (Institution of Electrical Engineers, London, 1985), pp. 1–5.

B. R. Frieden, ed., The Computer in Optical Research (Springer-Verlag, New York, 1980).
[CrossRef]

M. A. Vorontsov, V. I. Shmalhauzen, Principles of Adaptive Optics (Nauka, Moscow, 1988).

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Figures (11)

Fig. 1
Fig. 1

Block diagram of the iterative procedure for phase retrieval (a) from two intensity measurements and (b) from a set of intensity measurements.

Fig. 2
Fig. 2

Block diagram of the iterative procedure for reconstruction of the polynomial expansion coefficients of the phase.

Fig. 3
Fig. 3

Dynamics of variation of the functional Ĵ in reconstructing the phase from a set of intensity measurements in the planes {Lk} (Ĵ0 is the initial value of the functional and n is the iteration number): 1, Lk = {0; ∞}; 2, Lk = {0; 0.1}; 3, Lk = {0; 0.1; ∞}; 4, Lk = {0; 0.1; 0.5; ∞}; 5, Lk = {0; 0.01; ∞}.

Fig. 4
Fig. 4

Efficiency of phase retrieval from (a) three and (b) four intensity measurements (c = 0.15λ).

Fig. 5
Fig. 5

Intensity distributions recorded in various output planes of the lens analyzer (c = 0.15λ, Δ = 3.2Δ) (a) in the focal plane (Lk = ∞) and in the plane with (b) Lk = 0.01 and (c) Lk = 0.1; (d) intensity distribution in the focal plane when phase aberrations are absent.

Fig. 6
Fig. 6

Dynamics of variation of the functional Ĵ in reconstructing the Zernike polynomial expansion coefficients of the phase (Lk = {0; 0.1; ∞}): 1, c = 0.1λ; 2, c = 0.2λ; 3, c = 0.25λ.

Fig. 7
Fig. 7

Schematic of the experimental setup for phase retrieval.

Fig. 8
Fig. 8

Intensity distributions in the iterative process of phase retrieval: (a) images obtained with a piece of window glass fixed in the optical channel of the system; (b) initial images (n = 0); (c) n = 7; (d) n = 12.

Fig. 9
Fig. 9

Nonlinear spatial filter. L1 and L2 are lenses, and NL is a thin layer of a nonlinear medium (z = 0 is the input plane; z = Z0 is the output plane).

Fig. 10
Fig. 10

Block diagram of the iterative procedure for phase retrieval from intensity measurements in the input and the output planes of the filter.

Fig. 11
Fig. 11

Dynamics of variation of the functional J in the phase retrieval procedure with a nonlinear filter (c = 0.15λ; J0 is the initial value of the functional corresponding to case 1): 1, phase initial approximation taken in the form of Eq. (17); 2, initial approximation taken in the form of Eq. (25) (α = 1).

Tables (1)

Tables Icon

Table 1 Comparison of Phase Retrieval Efficiencies

Equations (46)

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a ( r ) = a 0 ( r ) exp [ i u ( r ) ]
u ( r ) = m = 1 M c m S m ( r ) .
W = ( a , a ) = ( A , A ) = const . ,
G { α f + β g } = α G { f } + β G ( g ) ,
2 i k A ^ z = Δ A ^ ,             0 z Z 0 ,
I ( r ) = G { a 0 ( r ) exp [ i v ( r ) ] } 2 ,
J = [ A ( r ) - I 0 ( r ) ] 2 d r .
J ^ = k = 1 N J k = k = 1 N [ ( A k - I k 0 ) 2 d r ] ,
v n + 1 ( r ) = v n ( r ) + h J { v n } ,             n = 0 , 1 , 2 , .
J = - Im ( a Ψ * ) ,
Ψ = G - 1 { ψ } ,             ψ = I 0 A / A .
Ψ = k = 1 N Ψ k .
Ψ k = G k - 1 { ψ k }
ψ k = I k 0 A k / A k .
v n + 1 ( r ) = φ n ( r ) ,
c m n + 1 = c m n + h J { c m n } .
J m = J / c m = - Im ( a Ψ * ) S m d r .
v 0 ( r ) = c 3 S 3 ( r ) + c 4 S 4 ( r ) ,
L k = Z k F z d ( F - Z k ) = a d a F ,
J ^ = k = 1 2 ( I k 0 - A A * z = z k ) 2 d r .
A F ( - ) ( κ ) = F { a ( r ) } ,
t ( κ ) = exp [ i β I F ( κ ) ] .
A F ( + ) ( κ ) = t ( κ ) A F ( - ) ( κ ) .
I 0 ( r ) = A ( r ) 2 = F { A F ( + ) ( κ ) } 2 ,
I 0 ( - r ) γ u ( r ) + const . ,
v 0 ( r ) = α I 0 ( - r ) ,
ψ = I 0 A / A ,
ψ ( - ) = F - 1 { ψ } ,
ψ ( + ) = ψ ( - ) exp [ - i β A F ( - ) 2 ] - 2 β Im [ A F ( + ) ψ ( - ) * ] A F ( - ) ,
Ψ = F - 1 { ψ ( + ) } .
J = A 2 d r + I 0 d r - 2 I 0 A d r = 2 W - 2 I 0 A d r .
J 1 = I 0 A d r .
Δ J 1 = I 0 Δ A d r .
Δ f = Δ ( f f * ) = Δ ( f f * ) 2 f f * = 1 f Re ( f * Δ f ) ,
Δ J 1 = Re ( I 0 A * / A Δ A ) d r = Re ( I 0 A * / A G { Δ a } ) d r = Re ( G { Δ a } , I 0 A / A ) .
Δ J 1 = Re ( G { Δ a } , ψ ) = Re ( Δ a , G - 1 { ψ } ) = Re ( Δ a , Ψ ) = - Im ( a Δ u , Ψ ) = - Im ( a Ψ * ) Δ u d r .
J ^ = 2 N W - 2 J ^ 1 ,
J ^ 1 = k = 1 N I k 0 A k d r .
Δ J ^ 1 = k = 1 N Re ( G { Δ a } , I k 0 A k / A k ) = k = 1 N Re ( Δ a , G k - 1 { ψ k } ) = k = 1 N Re ( Δ a , Ψ k ) = Re ( Δ a , k = 1 N Ψ k ) = Re ( Δ a , Ψ ) ,
min v n + 1 Δ J = min v n + 1 { - 2 Re ( a 0 exp ( i v n + 1 ) , Ψ ) + 2 Re ( a 0 exp ( i v n ) , Ψ ) } = max v n + 1 [ 2 a 0 Ψ cos ( v n + 1 - φ n ) d r ] ,
Δ J 1 = Re ( I 0 A * / A Δ A ) d r = Re ( Δ A , ψ ) ,
Δ J 1 = Re ( F { Δ A F ( + ) } , ψ ) = Re ( Δ A F ( + ) , ψ ( - ) ) .
Δ A F ( + ) = t Δ A F ( - ) + 2 i β t A F ( - ) A F ( - ) Δ A F ( - ) .
Δ A F ( + ) = t Δ A F ( - ) + 2 i β A F ( + ) Re [ Δ A F ( - ) A F ( - ) * ] .
Δ J 1 = Re ( t Δ A F ( - ) , ψ ( - ) ) + Re ( 2 i β A F ( + ) Re [ Δ A F ( - ) A F ( - ) * ] , ψ ( - ) ) = Re ( Δ A F ( - ) , t * ψ ( - ) ) - Re ( Δ A F ( - ) , 2 β Im [ A F ( + ) ψ ( - ) * ] A F ( - ) ) = Re ( Δ A F ( - ) , ψ ( + ) ) ,
Δ J 1 = Re ( F { Δ a } , ψ ( + ) ) = Re ( Δ a , Ψ ) .

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