Abstract

The phase of a signal at a plane is reconstructed from the intensity profiles at two close parallel screens connected by a small abcd canonical transform; this applies to propagation along harmonic and repulsive fibers and in free media. We analyze the relationship between the local spatial frequency (the signal phase derivative) and the derivative of the squared modulus of the signal under a one-parameter canonical transform with respect to the parameter. We thus generalize to all linear systems the results that have been obtained separately for Fresnel and fractional Fourier transforms.

© 2003 Optical Society of America

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References

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  1. Z. Zalevsky, D. Mendlovic, R. G. Dorsch, “Gerchberg–Saxton algorithm applied in the fractional Fourier or the Fresnel domain,” Opt. Lett. 21, 842–844 (1996).
    [CrossRef] [PubMed]
  2. H. M. Ozaktas, Z. Zalevsky, M. A. Kutay, The Fractional Fourier Transform (Wiley, Chichester, UK, 2001).
  3. W. X. Cong, N. X. Chen, B. Y. Gu, “Recursive algorithm for phase retrieval in the fractional Fourier-transform domain,” Appl. Opt. 37, 6906–6910 (1998).
    [CrossRef]
  4. W. X. Cong, N. X. Chen, B. Y. Gu, “Phase retrieval in the Fresnel transform system:  a recursive algorithm,” J. Opt. Soc. Am. A 16, 1827–1830 (1999).
    [CrossRef]
  5. K. A. Nugent, D. Paganin, “Matter-wave phase measurement:  a noninterferometric approach,” Phys. Rev. A 61, 063614 (2000).
    [CrossRef]
  6. D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. (Oxford) 206, 33–40 (2002).
    [CrossRef]
  7. D. Paganin, K. A. Nugent, “Phase measurement of waves that obey nonlinear equations,” Opt. Lett. 27, 622–624 (2002).
    [CrossRef]
  8. M. R. Teague, “Deterministic phase retrieval:  a Green function solution,” J. Opt. Soc. Am. 73, 1434–1441 (1983).
    [CrossRef]
  9. N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1984).
    [CrossRef]
  10. K. Ichikawa, A. W. Lohmann, M. Takeda, “Phase retrieval based on the Fourier transport method:  experiments,” Appl. Opt. 27, 3433–3436 (1988).
    [CrossRef] [PubMed]
  11. T. E. Gureev, A. Roberts, K. A. Nugent, “Partially coherent fields, the transport-of-intensity equation, and phase uniqueness,” J. Opt. Soc. Am. A 12, 1942–1946 (1995).
    [CrossRef]
  12. T. Alieva, M. J. Bastiaans, “On fractional Fourier transform moments,” IEEE Signal Process. Lett. 7, 320–323 (2000).
    [CrossRef]
  13. T. Alieva, M. J. Bastiaans, L. J. Stanković, “Signal reconstruction from two close fractional Fourier power spectra,” IEEE Trans. Signal Process. 51, 112–123 (2003).
    [CrossRef]
  14. M. Moshinsky, C. Quesne, “Oscillator systems,” in Proceedings of the 15th Solvay Conference in Physics (1970) (Gordon and Breach, New York, 1974).
  15. M. Moshinsky, C. Quesne, “Linear canonical transformations and their unitary representation,” J. Math. Phys. 12, 1772–1780 (1971).
    [CrossRef]
  16. C. Quesne, M. Moshinsky, “Canonical transformations and matrix elements,” J. Math. Phys. 12, 1780–1783 (1971).
    [CrossRef]
  17. K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979), Chap. 9.

2003 (1)

T. Alieva, M. J. Bastiaans, L. J. Stanković, “Signal reconstruction from two close fractional Fourier power spectra,” IEEE Trans. Signal Process. 51, 112–123 (2003).
[CrossRef]

2002 (2)

D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. (Oxford) 206, 33–40 (2002).
[CrossRef]

D. Paganin, K. A. Nugent, “Phase measurement of waves that obey nonlinear equations,” Opt. Lett. 27, 622–624 (2002).
[CrossRef]

2000 (2)

K. A. Nugent, D. Paganin, “Matter-wave phase measurement:  a noninterferometric approach,” Phys. Rev. A 61, 063614 (2000).
[CrossRef]

T. Alieva, M. J. Bastiaans, “On fractional Fourier transform moments,” IEEE Signal Process. Lett. 7, 320–323 (2000).
[CrossRef]

1999 (1)

1998 (1)

1996 (1)

1995 (1)

1988 (1)

1984 (1)

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

1983 (1)

1971 (2)

M. Moshinsky, C. Quesne, “Linear canonical transformations and their unitary representation,” J. Math. Phys. 12, 1772–1780 (1971).
[CrossRef]

C. Quesne, M. Moshinsky, “Canonical transformations and matrix elements,” J. Math. Phys. 12, 1780–1783 (1971).
[CrossRef]

Alieva, T.

T. Alieva, M. J. Bastiaans, L. J. Stanković, “Signal reconstruction from two close fractional Fourier power spectra,” IEEE Trans. Signal Process. 51, 112–123 (2003).
[CrossRef]

T. Alieva, M. J. Bastiaans, “On fractional Fourier transform moments,” IEEE Signal Process. Lett. 7, 320–323 (2000).
[CrossRef]

Bastiaans, M. J.

T. Alieva, M. J. Bastiaans, L. J. Stanković, “Signal reconstruction from two close fractional Fourier power spectra,” IEEE Trans. Signal Process. 51, 112–123 (2003).
[CrossRef]

T. Alieva, M. J. Bastiaans, “On fractional Fourier transform moments,” IEEE Signal Process. Lett. 7, 320–323 (2000).
[CrossRef]

Chen, N. X.

Cong, W. X.

Dorsch, R. G.

Gu, B. Y.

Gureev, T. E.

Gureyev, T. E.

D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. (Oxford) 206, 33–40 (2002).
[CrossRef]

Ichikawa, K.

Kutay, M. A.

H. M. Ozaktas, Z. Zalevsky, M. A. Kutay, The Fractional Fourier Transform (Wiley, Chichester, UK, 2001).

Lohmann, A. W.

Mayo, S. C.

D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. (Oxford) 206, 33–40 (2002).
[CrossRef]

Mendlovic, D.

Miller, P. R.

D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. (Oxford) 206, 33–40 (2002).
[CrossRef]

Moshinsky, M.

C. Quesne, M. Moshinsky, “Canonical transformations and matrix elements,” J. Math. Phys. 12, 1780–1783 (1971).
[CrossRef]

M. Moshinsky, C. Quesne, “Linear canonical transformations and their unitary representation,” J. Math. Phys. 12, 1772–1780 (1971).
[CrossRef]

M. Moshinsky, C. Quesne, “Oscillator systems,” in Proceedings of the 15th Solvay Conference in Physics (1970) (Gordon and Breach, New York, 1974).

Nugent, K. A.

Ozaktas, H. M.

H. M. Ozaktas, Z. Zalevsky, M. A. Kutay, The Fractional Fourier Transform (Wiley, Chichester, UK, 2001).

Paganin, D.

D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. (Oxford) 206, 33–40 (2002).
[CrossRef]

D. Paganin, K. A. Nugent, “Phase measurement of waves that obey nonlinear equations,” Opt. Lett. 27, 622–624 (2002).
[CrossRef]

K. A. Nugent, D. Paganin, “Matter-wave phase measurement:  a noninterferometric approach,” Phys. Rev. A 61, 063614 (2000).
[CrossRef]

Quesne, C.

C. Quesne, M. Moshinsky, “Canonical transformations and matrix elements,” J. Math. Phys. 12, 1780–1783 (1971).
[CrossRef]

M. Moshinsky, C. Quesne, “Linear canonical transformations and their unitary representation,” J. Math. Phys. 12, 1772–1780 (1971).
[CrossRef]

M. Moshinsky, C. Quesne, “Oscillator systems,” in Proceedings of the 15th Solvay Conference in Physics (1970) (Gordon and Breach, New York, 1974).

Roberts, A.

Stankovic, L. J.

T. Alieva, M. J. Bastiaans, L. J. Stanković, “Signal reconstruction from two close fractional Fourier power spectra,” IEEE Trans. Signal Process. 51, 112–123 (2003).
[CrossRef]

Streibl, N.

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

Takeda, M.

Teague, M. R.

Wilkins, S. W.

D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. (Oxford) 206, 33–40 (2002).
[CrossRef]

Wolf, K. B.

K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979), Chap. 9.

Zalevsky, Z.

Appl. Opt. (2)

IEEE Signal Process. Lett. (1)

T. Alieva, M. J. Bastiaans, “On fractional Fourier transform moments,” IEEE Signal Process. Lett. 7, 320–323 (2000).
[CrossRef]

IEEE Trans. Signal Process. (1)

T. Alieva, M. J. Bastiaans, L. J. Stanković, “Signal reconstruction from two close fractional Fourier power spectra,” IEEE Trans. Signal Process. 51, 112–123 (2003).
[CrossRef]

J. Math. Phys. (2)

M. Moshinsky, C. Quesne, “Linear canonical transformations and their unitary representation,” J. Math. Phys. 12, 1772–1780 (1971).
[CrossRef]

C. Quesne, M. Moshinsky, “Canonical transformations and matrix elements,” J. Math. Phys. 12, 1780–1783 (1971).
[CrossRef]

J. Microsc. (Oxford) (1)

D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. (Oxford) 206, 33–40 (2002).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

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

Opt. Lett. (2)

Phys. Rev. A (1)

K. A. Nugent, D. Paganin, “Matter-wave phase measurement:  a noninterferometric approach,” Phys. Rev. A 61, 063614 (2000).
[CrossRef]

Other (3)

H. M. Ozaktas, Z. Zalevsky, M. A. Kutay, The Fractional Fourier Transform (Wiley, Chichester, UK, 2001).

K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979), Chap. 9.

M. Moshinsky, C. Quesne, “Oscillator systems,” in Proceedings of the 15th Solvay Conference in Physics (1970) (Gordon and Breach, New York, 1974).

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Equations (23)

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p¯(x)  ϕ(x)  dϕ(x)dx,ψ(x)= |ψ(x)|exp[iϕ(x)].
p¯(x)=dϕ(x)dx=Imdln ψ(x)dx=Imψ(x)ψ(x)=12 iψ*(x)ψ*(x)-ψ(x)ψ(x)=12 i ψ*(x)ψ(x)-ψ*(x)ψ(x)ψ*(x)ψ(x).
p¯(x)|ψ(x)|2 =12 i[ψ*(x)(ψ)(x)-ψ*(x)ψ(x)].
h(x, k)=12no k2+νx2.
H  12 AP2+12B(QP+PQ)+12CQ2=H,
P=-i x,Q=x,P 2=-2x2,
QP+PQ=-2ix x+12 ,Q2=x2,
C(α)  exp(iαH)=C(-α),αRe,
C(α)ψ(x)  ψα(x)=-C(α; x, x)ψ(x)dx
Hψ(x)=-i ddα-C(α; x, x)ψ(x)dx|α=0 =-i dψα(x)dα α=0.
Ce(α; x, x)=1(2iπ sin α)1/2×exp-i x2cos α-2xx+x2cos α2 sin α,
Cp(α; x, x)=1(2iπα)1/2exp-i x2-2xx+x22α.
Ch(α; x, x)=1(2iπ sinh α)1/2
×exp-i x2cosh α-2xx+x2cosh α2 sinh α.
 
|ψα(x)|2α=ψα*ψαα=ψα*α ψα+ψα*ψαα=(iHψα)*ψα+ψα*(iHψα)=-i[-12Aψα*+iB(xψα*+12ψα*)+12Cx2ψα*]ψα+iψα*[-12Aψα-iB(xψα+12ψα)+12Cx2ψα]=12iA(ψα*ψα-ψα*ψα)+Bx(ψα*ψα+ψα*ψα)+Bψα*ψα,
|ψα(x)|2α=12 iA ddx [ψα*(x)ψα(x)-ψα*(x)ψα(x)]+B ddx [xψα*(x)ψα(x)].
d[Ap¯(x)+Bx]|ψ(x)|2dx=|ψα(x)|2α α=0,
-|ψα(x)|2α α=0dx=α-|ψα(x)|2dxα=0=α-|ψ(x)|2dxα=0=0,
[Ap¯(x)+Bx]|ψ(x)|2
=-x|ψα(x)|2α α=0dx=-|ψα(x)|2α α=0u(x-x)dx=12-|ψα(x)|2α α=0sgn(x-x)dx,
|ψ(x)|212(|ψ+(x)|2+|ψ-(x)|2),1.
p¯(x)-BA x+12|ψ(x)|2×-|ψ+(x)|2-|ψ-(x)|22sgn(x-x)dx.

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