Abstract

In a previous paper [Opt. Commun. 225, 19–30 (2003) ] we presented a method to reconstruct two- dimensional complex amplitudes by using the ambiguity function of one-dimensional intensity scans, obtained from two optical setups involving cylindrical lenses. We demonstrate that the internal redundancy of the ambiguity function can be utilized to improve the efficiency of this method even further. We show that the phase reconstruction errors can be minimized with an appropriate algorithm, and we present experimental data that illustrate the efficient reconstruction of a two-dimensional phase element.

© 2008 Optical Society of America

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References

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  1. L. M. Foucault , “Memoire sur la construction des telescopes en verre argente,” Ann. Obs. Imp. Paris 5, 197-237 (1859).
    [CrossRef]
  2. R. V. Shack and B. C. Platt , “Production and use of a lenticular Hartmann screen,” J. Opt. Soc. Am. 61, 656 (1971).
  3. M. R. Teague , “Deterministic phase retrieval: a Green's function solution,” J. Opt. Soc. Am. A 73, 1434-1441 (1983).
  4. N. Streibl , “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6-10 (1984).
    [CrossRef]
  5. J. Ragazzoni , “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Opt. 43, 289-293 (1996).
    [CrossRef]
  6. R. W. Gerchberg and W. O. Saxton , “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Jena) 35, 237-246 (1972).
    [CrossRef]
  7. Z. Zalevsky , D. Mendlovic , and R. G. Dorsch , “Gerchberg-Saxton algorithm applied in the fractional Fourier or the Fresnel domain,” Opt. Lett. 21, 842-844 (1996).
  8. J. R. Fienup , “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27-29 (1978).
    [CrossRef] [PubMed]
  9. J. R. Fienup , “Space object imaging through the turbulent atmosphere,” Opt. Eng. 18, 529-534 (1979).
    [CrossRef] [PubMed]
  10. J. R. Fienup , “Iterative method applied to image reconstruction and to computer generated holograms,” Opt. Eng. 19, 297-305 (1980).
  11. G. Pedrini , W. Osten , and Y. Zhang , “Wave-front reconstruction from a sequence of interferograms recorded at different planes,” Opt. Lett. 30, 833-835 (2005).
  12. P. Almoro , G. Pedrini , and W. Osten , “Complete wavefront reconstruction using sequential intensity measurements of a volume speckle field,” Appl. Opt. 45, 8596-8605 (2006).
    [CrossRef] [PubMed]
  13. A. W. Lohmann , “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181-2186 (1993).
    [CrossRef] [PubMed]
  14. M. G. Raymer , M. Beck , and D. F. McAllister , “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137-1140 (1994).
    [CrossRef]
  15. D. F. McAlister , M. Beck , L. Clarke , A. Mayer , and M. G. Raymer , “Optical phase retrieval by phase-space tomography and fractional-order Fourier transformation,” Opt. Lett. 20, 1181-1183 (1995).
    [CrossRef] [PubMed]
  16. J. Tu and S. Tamura , “Wave field determination using tomography of the ambiguity function,” Phys. Rev. E 55, 1946-1949 (1997).
    [CrossRef] [PubMed]
  17. J. Tu and S. Tamura , “New technique for reconstruction of a complex wave field by means of measurement of three-dimensional intensity,” Proc. SPIE 3170, 108-115 (1997).
    [CrossRef]
  18. D. Dragoman , M. Dragoman , and K.-H. Brenner , “Amplitude and phase recovery of rotationally symmetric beams,” Appl. Opt. 41, 5512-5518 (2002).
    [CrossRef]
  19. D. Dragoman , M. Dragoman , and K.-H. Brenner , “Tomographic amplitude and phase recovery of vertical-cavity surface-emitting lasers by use of the ambiguity function,” Opt. Lett. 27, 1519-1521 (2002).
    [CrossRef] [PubMed]
  20. X. Liu and K.-H. Brenner , “Reconstruction of two-dimensional complex amplitudes from intensity measurements,” Opt. Commun. 225, 19-30 (2003).
    [CrossRef]
  21. D. Dragoman , “Redundancy of phase-space distribution functions in complex field recovery problems,” Appl. Opt. 42, 1932-1937 (2003).
    [CrossRef] [PubMed]
  22. D. Dragoman , “Reply to comment on 'redundancy of phase-space distribution functions in complex field recovery problems',” Appl. Opt. 44, 58-59 (2005).
  23. M. Testorf , “Comment on 'redundancy of phase-space distribution functions in complex field recovery problems',” Appl. Opt. 44, 55-57 (2005).
    [PubMed]
  24. A. Semichaevsky and M. Testorf , “Phase-space interpretation of deterministic phase retrieval,” J. Opt. Soc. Am. A 21, 2173-2179 (2004).
    [CrossRef]
  25. K.-H. Brenner , “Method for designing arbitrary two-dimensional continuous phase elements,” Opt. Lett. 25, 31-33(2000).
    [CrossRef]
  26. G. H. Golub and C. van Loan , Matrix Computations, 3rd ed. (Johns Hopkins U. Press, 1996).
  27. C. W. Groetsch , Inverse Problems in the Mathematical Sciences (Vieweg, 1993).
  28. W. H. Press , S. A. Teukolsky , W. T. Vetterling , and B. P. Flannery , Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1992).

2006 (1)

2005 (3)

2004 (1)

2003 (2)

X. Liu and K.-H. Brenner , “Reconstruction of two-dimensional complex amplitudes from intensity measurements,” Opt. Commun. 225, 19-30 (2003).
[CrossRef]

D. Dragoman , “Redundancy of phase-space distribution functions in complex field recovery problems,” Appl. Opt. 42, 1932-1937 (2003).
[CrossRef] [PubMed]

2002 (2)

2000 (1)

1997 (2)

J. Tu and S. Tamura , “Wave field determination using tomography of the ambiguity function,” Phys. Rev. E 55, 1946-1949 (1997).
[CrossRef] [PubMed]

J. Tu and S. Tamura , “New technique for reconstruction of a complex wave field by means of measurement of three-dimensional intensity,” Proc. SPIE 3170, 108-115 (1997).
[CrossRef]

1996 (3)

J. Ragazzoni , “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Opt. 43, 289-293 (1996).
[CrossRef]

Z. Zalevsky , D. Mendlovic , and R. G. Dorsch , “Gerchberg-Saxton algorithm applied in the fractional Fourier or the Fresnel domain,” Opt. Lett. 21, 842-844 (1996).

G. H. Golub and C. van Loan , Matrix Computations, 3rd ed. (Johns Hopkins U. Press, 1996).

1995 (1)

1994 (1)

M. G. Raymer , M. Beck , and D. F. McAllister , “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137-1140 (1994).
[CrossRef]

1993 (2)

1992 (1)

W. H. Press , S. A. Teukolsky , W. T. Vetterling , and B. P. Flannery , Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1992).

1984 (1)

N. Streibl , “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6-10 (1984).
[CrossRef]

1983 (1)

M. R. Teague , “Deterministic phase retrieval: a Green's function solution,” J. Opt. Soc. Am. A 73, 1434-1441 (1983).

1980 (1)

J. R. Fienup , “Iterative method applied to image reconstruction and to computer generated holograms,” Opt. Eng. 19, 297-305 (1980).

1979 (1)

J. R. Fienup , “Space object imaging through the turbulent atmosphere,” Opt. Eng. 18, 529-534 (1979).
[CrossRef] [PubMed]

1978 (1)

1972 (1)

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

1971 (1)

R. V. Shack and B. C. Platt , “Production and use of a lenticular Hartmann screen,” J. Opt. Soc. Am. 61, 656 (1971).

1859 (1)

L. M. Foucault , “Memoire sur la construction des telescopes en verre argente,” Ann. Obs. Imp. Paris 5, 197-237 (1859).
[CrossRef]

Almoro, P.

Beck, M.

Brenner, K.-H.

Clarke, L.

Dorsch, R. G.

Dragoman, D.

Dragoman, M.

Fienup, J. R.

J. R. Fienup , “Iterative method applied to image reconstruction and to computer generated holograms,” Opt. Eng. 19, 297-305 (1980).

J. R. Fienup , “Space object imaging through the turbulent atmosphere,” Opt. Eng. 18, 529-534 (1979).
[CrossRef] [PubMed]

J. R. Fienup , “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27-29 (1978).
[CrossRef] [PubMed]

Flannery, B. P.

W. H. Press , S. A. Teukolsky , W. T. Vetterling , and B. P. Flannery , Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1992).

Foucault, L. M.

L. M. Foucault , “Memoire sur la construction des telescopes en verre argente,” Ann. Obs. Imp. Paris 5, 197-237 (1859).
[CrossRef]

Gerchberg, R. W.

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

Golub, G. H.

G. H. Golub and C. van Loan , Matrix Computations, 3rd ed. (Johns Hopkins U. Press, 1996).

Groetsch, C. W.

C. W. Groetsch , Inverse Problems in the Mathematical Sciences (Vieweg, 1993).

Liu, X.

X. Liu and K.-H. Brenner , “Reconstruction of two-dimensional complex amplitudes from intensity measurements,” Opt. Commun. 225, 19-30 (2003).
[CrossRef]

Lohmann, A. W.

Mayer, A.

McAlister, D. F.

McAllister, D. F.

M. G. Raymer , M. Beck , and D. F. McAllister , “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137-1140 (1994).
[CrossRef]

Mendlovic, D.

Osten, W.

Pedrini, G.

Platt, B. C.

R. V. Shack and B. C. Platt , “Production and use of a lenticular Hartmann screen,” J. Opt. Soc. Am. 61, 656 (1971).

Press, W. H.

W. H. Press , S. A. Teukolsky , W. T. Vetterling , and B. P. Flannery , Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1992).

Ragazzoni, J.

J. Ragazzoni , “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Opt. 43, 289-293 (1996).
[CrossRef]

Raymer, M. G.

Saxton, W. O.

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

Semichaevsky, A.

Shack, R. V.

R. V. Shack and B. C. Platt , “Production and use of a lenticular Hartmann screen,” J. Opt. Soc. Am. 61, 656 (1971).

Streibl, N.

N. Streibl , “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6-10 (1984).
[CrossRef]

Tamura, S.

J. Tu and S. Tamura , “Wave field determination using tomography of the ambiguity function,” Phys. Rev. E 55, 1946-1949 (1997).
[CrossRef] [PubMed]

J. Tu and S. Tamura , “New technique for reconstruction of a complex wave field by means of measurement of three-dimensional intensity,” Proc. SPIE 3170, 108-115 (1997).
[CrossRef]

Teague, M. R.

M. R. Teague , “Deterministic phase retrieval: a Green's function solution,” J. Opt. Soc. Am. A 73, 1434-1441 (1983).

Testorf, M.

Teukolsky, S. A.

W. H. Press , S. A. Teukolsky , W. T. Vetterling , and B. P. Flannery , Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1992).

Tu, J.

J. Tu and S. Tamura , “New technique for reconstruction of a complex wave field by means of measurement of three-dimensional intensity,” Proc. SPIE 3170, 108-115 (1997).
[CrossRef]

J. Tu and S. Tamura , “Wave field determination using tomography of the ambiguity function,” Phys. Rev. E 55, 1946-1949 (1997).
[CrossRef] [PubMed]

van Loan, C.

G. H. Golub and C. van Loan , Matrix Computations, 3rd ed. (Johns Hopkins U. Press, 1996).

Vetterling, W. T.

W. H. Press , S. A. Teukolsky , W. T. Vetterling , and B. P. Flannery , Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1992).

Zalevsky, Z.

Zhang, Y.

Ann. Obs. Imp. Paris (1)

L. M. Foucault , “Memoire sur la construction des telescopes en verre argente,” Ann. Obs. Imp. Paris 5, 197-237 (1859).
[CrossRef]

Appl. Opt. (5)

J. Mod. Opt. (1)

J. Ragazzoni , “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Opt. 43, 289-293 (1996).
[CrossRef]

J. Opt. Soc. Am. (1)

R. V. Shack and B. C. Platt , “Production and use of a lenticular Hartmann screen,” J. Opt. Soc. Am. 61, 656 (1971).

J. Opt. Soc. Am. A (3)

Opt. Commun. (2)

X. Liu and K.-H. Brenner , “Reconstruction of two-dimensional complex amplitudes from intensity measurements,” Opt. Commun. 225, 19-30 (2003).
[CrossRef]

N. Streibl , “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6-10 (1984).
[CrossRef]

Opt. Eng. (2)

J. R. Fienup , “Space object imaging through the turbulent atmosphere,” Opt. Eng. 18, 529-534 (1979).
[CrossRef] [PubMed]

J. R. Fienup , “Iterative method applied to image reconstruction and to computer generated holograms,” Opt. Eng. 19, 297-305 (1980).

Opt. Lett. (6)

Optik (Jena) (1)

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

Phys. Rev. E (1)

J. Tu and S. Tamura , “Wave field determination using tomography of the ambiguity function,” Phys. Rev. E 55, 1946-1949 (1997).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

M. G. Raymer , M. Beck , and D. F. McAllister , “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137-1140 (1994).
[CrossRef]

Proc. SPIE (1)

J. Tu and S. Tamura , “New technique for reconstruction of a complex wave field by means of measurement of three-dimensional intensity,” Proc. SPIE 3170, 108-115 (1997).
[CrossRef]

Other (3)

G. H. Golub and C. van Loan , Matrix Computations, 3rd ed. (Johns Hopkins U. Press, 1996).

C. W. Groetsch , Inverse Problems in the Mathematical Sciences (Vieweg, 1993).

W. H. Press , S. A. Teukolsky , W. T. Vetterling , and B. P. Flannery , Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1992).

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Figures (4)

Fig. 1
Fig. 1

(a) 1D propagation operator with respect to the y axis. The x axis is imaged. (b) Camera picture.

Fig. 2
Fig. 2

Different domains of the AF: horizontal, ν axis; vertical, x axis. The dark regions are accessible by a series of finite propagation distances.

Fig. 3
Fig. 3

2D phase function of the lens array (in radians), measured by an interferometer.

Fig. 4
Fig. 4

Comparison of experimental reconstruction results with and without residue minimization (in radians). (a) Reconstructed 2D phase function. (b) Phase difference of (a) from the interferometric measurement. (c) 2D phase function reconstructed by the residue minimization algorithm. (d) Phase difference of (c) from the interferometric measurement.

Equations (28)

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A ( ν , x ; u ) = + u ( x + x 2 ) u * ( x x 2 ) e 2 π i x ν d x = + u ˜ ( ν + ν 2 ) u ˜ * ( ν ν 2 ) e 2 π i x ν d ν .
u ˜ ( ν ) = + u ( x ) e 2 π i x ν d x ,
u ( x ) = + u ˜ ( ν ) e 2 π i x ν d ν .
+ u z ( x ) u z * ( x ) e 2 π i x ν d x = A ( ν , λ z ν ; u 0 )
u ˜ z ( ν ) = u ˜ 0 ( ν ) e π i λ z ν 2 .
u ^ ( x ) = u ( x ) u * ( 0 ) = + A ( ν , x ; u ) e π i x ν d ν .
u ˜ z ( ν ; x i ) = u ˜ 0 ( ν ; x i ) e π i λ z ν 2
u ˜ ( ν ; x i ) = + u ( y ; x i ) e 2 π i y ν d ν .
u ^ c ( y ; x i ) = u * ( x i , 0 ) u ( x i , y ) = + A c ( μ , y ; u ( x i , y ) ) e i π y μ d μ .
u ^ r ( x ; y j ) = u * ( 0 , y j ) u ( x , y j ) = + A r ( ν , x ; u ( x , y j ) ) e i π x ν d ν .
u ^ ( x i , y j ) = u ^ c ( y j ; x i ) u ^ r ( x i ; y 0 ) u ^ c ( y 0 ; x i ) = u * ( x i , 0 ) u ( x i , y j ) u * ( 0 , y 0 ) u ( x i , y 0 ) u * ( x i , 0 ) u ( x i , y 0 ) = u ( x i , y j ) u * ( 0 , y 0 ) .
A ( ν , x ; u ) = + u ( x ) u * ( x x ) e 2 π i x ν d x e π i x ν .
F ( x , x ; u ) = u ( x ) u * ( x x ) = + A ( ν , x ; u ) e π i x ν e 2 π i x ν d ν .
F ( x , x ; u ) = F * ( x + x , x ; u ) ,
F ( x , n δ x ; u ) = k = 1 n 1 F ( x k δ x , δ x ; u ) F ( x k δ x , 0 ; u ) F ( x , δ x ; u ) .
F ( x , δ x ; u ) = F ^ ( x , δ x ; u ) + A ( 0 , δ x ; u ) N ,
u ( x ) = 0 for     | x | > W / 2 ,
F ( x , δ x ; u ) = u ( x ) u * ( x δ x ) = 0 for     | x | > W / 2.
F ^ ( x H , δ x ; u ) = A ( 0 , δ x ; u ) N for     x H { x ; | x | > W 2 } .
A ( 0 , δ x ; u ) = N F ^ ( x H , δ x ; u ) .
z m = δ x m λ δ ν , m = N / 2 N / 2 1 , m 0.
Γ ϕ d r = A ( × ϕ ) d A = 0 .
( ϕ j + 1 , k ϕ j , k ) + ( ϕ j + 1 , k + 1 ϕ j + 1 , k ) ( ϕ j + 1 , k + 1 ϕ j , k + 1 ) ( ϕ j , k + 1 ϕ j , k ) = 0 .
( ϕ ^ j + 1 , k r ϕ ^ j , k r ) + ( ϕ ^ j + 1 , k + 1 c ϕ ^ j + 1 , k c ) ( ϕ ^ j + 1 , k + 1 r ϕ ^ j , k + 1 r ) ( ϕ ^ j , k + 1 c ϕ ^ j , k c ) = r j , k .
δ ϕ ^ j , k r + δ ϕ ^ j + 1 , k c δ ϕ ^ j , k + 1 r δ ϕ ^ j , k c = r j , k .
( ( N 1 ) N   elements 1 0 0 1 0 0 N ( N 1 )   elements 1 1 0 0 ( N 1 ) N   elements 0 1 0 0 1 0 N ( N 1 )   elements 1 1 0 0 ( N 1 ) N   elements 0 0 1 0 0 1 N ( N 1 )   elements 1 1 0 0 ( N 1 ) N   elements 1 0 0 1 0 0 N ( N 1 )   elements 0 1 1 0 ) m atrix M ( δ ϕ ^ 0 , 0 r δ ϕ ^ 1 , 0 r δ ϕ ^ N 2 , 0 r δ ϕ ^ 0 , 1 r δ ϕ ^ N 2 , N 1 r δ ϕ ^ 0 , 0 c δ ϕ ^ 1 , 0 c δ ϕ ^ N 1 , 0 c δ ϕ ^ 1 , 2 c δ ϕ ^ N 1 , N 2 c ) v ector c = ( r 0 , 0 r 1 , 0 r N 2 , 0 r 0 , 1 r N 2 , N 2 ) v ector b .
[ M λ I ] c = [ b 0 ] ,
min c { [ M λ I ] c [ b 0 ] 2 2 } = min c { M c b 2 2 + λ 2 c 2 2 } .

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