Abstract

We propose a tomographic method for determining the Wigner function of a short pulse, which may be used with a wide class of optical systems.

© 1995 Optical Society of America

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References

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  1. E. Wigner, “On the quantum corrections for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
    [CrossRef]
  2. A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 58, 1256–1259 (1968).
    [CrossRef]
  3. H. O. Bartelt, K.-H. Brenner, and A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
    [CrossRef]
  4. M. Beck, M. G. Raymer, I. A. Walmsley, and V. Wong, “Chronocyclic tomography for measuring the amplitude and phase structure of optical pulses,” Opt. Lett. 18, 2041–2043 (1993).
    [CrossRef] [PubMed]
  5. Z. Namias, “The fractional Fourier transform and its applications in quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
    [CrossRef]
  6. A. C. McBride and F. H. Kerr, “On Namias’s fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
    [CrossRef]
  7. D. Mendelovic and H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation: 1,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
    [CrossRef]
  8. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
    [CrossRef]
  9. G. S. Agarwal and R. Simon, “A simple realization of fractional Fourier transforms and the relation to the harmonic oscillator Green’s function,” Opt. Commun. 110, 23–26 (1994).
    [CrossRef]
  10. E. B. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. QE-5, 454–458 (1969).
    [CrossRef]
  11. B. H. Kolner and M. Nazarathy, “Temporal imaging with time lenses,” Opt. Lett. 14, 630–632 (1989).
    [CrossRef] [PubMed]
  12. S. P. Dijaili, A. Dienes, and J. S. Smith, “ABCD matrices for dispersive pulse propagation,” IEEE J. Quantum Electron. QE-26, 1158–1164 (1990).
    [CrossRef]
  13. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1989), Chap. 9.
  14. E. Merzbacher, Quantum Mechanics (Wiley, New York, 1961), Sec. 8.
  15. The general properties of linear transforms with kernels of the type given by Eq. (11) is discussed in detail in D. F. V. James and G. S. Agarwal, “Generalized Fresnel transform in optics,” to be submitted Opt. Commun.
  16. G. S. Agarwal, “Wigner-function description of quantum noise in interferometers,” J. Mod. Opt. 34, 909–921 (1987).
    [CrossRef]
  17. A. T. Friberg, “Propagation of a generalized radiance in paraxial optical systems,” Appl. Opt. 30, 2443–2446 (1991).
    [CrossRef] [PubMed]
  18. H. H. Barrett, “The Radon transform and its applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1984), Vol. XXI, pp. 217–286.
    [CrossRef]
  19. G. T. Herman, Image Reconstruction from Projections (Academic, New York, 1980), Chap. 6.
  20. A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, New York, 1988), Chap. 3.
  21. F. Gori, M. Santarsiero, and G. Guattari, “Coherence and the spatial distribution of intensity,” J. Opt. Soc. Am. A 10, 673–679 (1993).
    [CrossRef]
  22. M. W. Kowarz, “Noninterferometric reconstruction of optical-field correlations,” Phys. Rev. E 49, 890–893 (1994).
    [CrossRef]
  23. M. G. Raymer, M. Beck, and D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
    [CrossRef] [PubMed]

1994 (3)

G. S. Agarwal and R. Simon, “A simple realization of fractional Fourier transforms and the relation to the harmonic oscillator Green’s function,” Opt. Commun. 110, 23–26 (1994).
[CrossRef]

M. W. Kowarz, “Noninterferometric reconstruction of optical-field correlations,” Phys. Rev. E 49, 890–893 (1994).
[CrossRef]

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

1993 (4)

1991 (1)

1990 (1)

S. P. Dijaili, A. Dienes, and J. S. Smith, “ABCD matrices for dispersive pulse propagation,” IEEE J. Quantum Electron. QE-26, 1158–1164 (1990).
[CrossRef]

1989 (1)

1987 (2)

G. S. Agarwal, “Wigner-function description of quantum noise in interferometers,” J. Mod. Opt. 34, 909–921 (1987).
[CrossRef]

A. C. McBride and F. H. Kerr, “On Namias’s fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

1980 (2)

H. O. Bartelt, K.-H. Brenner, and A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

Z. Namias, “The fractional Fourier transform and its applications in quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
[CrossRef]

1969 (1)

E. B. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. QE-5, 454–458 (1969).
[CrossRef]

1968 (1)

1932 (1)

E. Wigner, “On the quantum corrections for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Agarwal, G. S.

G. S. Agarwal and R. Simon, “A simple realization of fractional Fourier transforms and the relation to the harmonic oscillator Green’s function,” Opt. Commun. 110, 23–26 (1994).
[CrossRef]

G. S. Agarwal, “Wigner-function description of quantum noise in interferometers,” J. Mod. Opt. 34, 909–921 (1987).
[CrossRef]

The general properties of linear transforms with kernels of the type given by Eq. (11) is discussed in detail in D. F. V. James and G. S. Agarwal, “Generalized Fresnel transform in optics,” to be submitted Opt. Commun.

Barrett, H. H.

H. H. Barrett, “The Radon transform and its applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1984), Vol. XXI, pp. 217–286.
[CrossRef]

Bartelt, H. O.

H. O. Bartelt, K.-H. Brenner, and A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

Beck, M.

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

M. Beck, M. G. Raymer, I. A. Walmsley, and V. Wong, “Chronocyclic tomography for measuring the amplitude and phase structure of optical pulses,” Opt. Lett. 18, 2041–2043 (1993).
[CrossRef] [PubMed]

Brenner, K.-H.

H. O. Bartelt, K.-H. Brenner, and A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

Dienes, A.

S. P. Dijaili, A. Dienes, and J. S. Smith, “ABCD matrices for dispersive pulse propagation,” IEEE J. Quantum Electron. QE-26, 1158–1164 (1990).
[CrossRef]

Dijaili, S. P.

S. P. Dijaili, A. Dienes, and J. S. Smith, “ABCD matrices for dispersive pulse propagation,” IEEE J. Quantum Electron. QE-26, 1158–1164 (1990).
[CrossRef]

Friberg, A. T.

Gori, F.

Guattari, G.

Herman, G. T.

G. T. Herman, Image Reconstruction from Projections (Academic, New York, 1980), Chap. 6.

James, D. F. V.

The general properties of linear transforms with kernels of the type given by Eq. (11) is discussed in detail in D. F. V. James and G. S. Agarwal, “Generalized Fresnel transform in optics,” to be submitted Opt. Commun.

Kak, A. C.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, New York, 1988), Chap. 3.

Kerr, F. H.

A. C. McBride and F. H. Kerr, “On Namias’s fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

Kolner, B. H.

Kowarz, M. W.

M. W. Kowarz, “Noninterferometric reconstruction of optical-field correlations,” Phys. Rev. E 49, 890–893 (1994).
[CrossRef]

Lohmann, A. W.

A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
[CrossRef]

H. O. Bartelt, K.-H. Brenner, and A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

McAlister, D. F.

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

McBride, A. C.

A. C. McBride and F. H. Kerr, “On Namias’s fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

Mendelovic, D.

Merzbacher, E.

E. Merzbacher, Quantum Mechanics (Wiley, New York, 1961), Sec. 8.

Namias, Z.

Z. Namias, “The fractional Fourier transform and its applications in quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
[CrossRef]

Nazarathy, M.

Ozaktas, H. M.

Raymer, M. G.

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

M. Beck, M. G. Raymer, I. A. Walmsley, and V. Wong, “Chronocyclic tomography for measuring the amplitude and phase structure of optical pulses,” Opt. Lett. 18, 2041–2043 (1993).
[CrossRef] [PubMed]

Santarsiero, M.

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1989), Chap. 9.

Simon, R.

G. S. Agarwal and R. Simon, “A simple realization of fractional Fourier transforms and the relation to the harmonic oscillator Green’s function,” Opt. Commun. 110, 23–26 (1994).
[CrossRef]

Slaney, M.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, New York, 1988), Chap. 3.

Smith, J. S.

S. P. Dijaili, A. Dienes, and J. S. Smith, “ABCD matrices for dispersive pulse propagation,” IEEE J. Quantum Electron. QE-26, 1158–1164 (1990).
[CrossRef]

Treacy, E. B.

E. B. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. QE-5, 454–458 (1969).
[CrossRef]

Walmsley, I. A.

Walther, A.

Wigner, E.

E. Wigner, “On the quantum corrections for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Wong, V.

Appl. Opt. (1)

IEEE J. Quantum Electron. (2)

S. P. Dijaili, A. Dienes, and J. S. Smith, “ABCD matrices for dispersive pulse propagation,” IEEE J. Quantum Electron. QE-26, 1158–1164 (1990).
[CrossRef]

E. B. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. QE-5, 454–458 (1969).
[CrossRef]

IMA J. Appl. Math. (1)

A. C. McBride and F. H. Kerr, “On Namias’s fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

J. Inst. Math. Appl. (1)

Z. Namias, “The fractional Fourier transform and its applications in quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
[CrossRef]

J. Mod. Opt. (1)

G. S. Agarwal, “Wigner-function description of quantum noise in interferometers,” J. Mod. Opt. 34, 909–921 (1987).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Commun. (2)

G. S. Agarwal and R. Simon, “A simple realization of fractional Fourier transforms and the relation to the harmonic oscillator Green’s function,” Opt. Commun. 110, 23–26 (1994).
[CrossRef]

H. O. Bartelt, K.-H. Brenner, and A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. (1)

E. Wigner, “On the quantum corrections for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Phys. Rev. E (1)

M. W. Kowarz, “Noninterferometric reconstruction of optical-field correlations,” Phys. Rev. E 49, 890–893 (1994).
[CrossRef]

Phys. Rev. Lett. (1)

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

Other (6)

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1989), Chap. 9.

E. Merzbacher, Quantum Mechanics (Wiley, New York, 1961), Sec. 8.

The general properties of linear transforms with kernels of the type given by Eq. (11) is discussed in detail in D. F. V. James and G. S. Agarwal, “Generalized Fresnel transform in optics,” to be submitted Opt. Commun.

H. H. Barrett, “The Radon transform and its applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1984), Vol. XXI, pp. 217–286.
[CrossRef]

G. T. Herman, Image Reconstruction from Projections (Academic, New York, 1980), Chap. 6.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, New York, 1988), Chap. 3.

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Figures (2)

Fig. 1
Fig. 1

Optical system used in the example of the generalized Radon transform discussed in the text. (a) The optical system for transmission of short pulses is shown symbolically with the thick lines representing lengths of a dispersive optical fiber and the chirp modulators (or time lenses) shown as blocks marked C.M. (b) The equivalent paraxial optical system for beam propagation is shown with a pair of lenses of focal length f that are separated from each other and from the input and the output planes by the distance z, as shown. The lengthening of the time width of the optical pulse as it propagates through the system in (a) is mathematically analogous to the broadening of an optical beam as it is diffracted through the system in (b).

Fig. 2
Fig. 2

Variation of the ratio B(f)/A(f), with the focal length f for the optical system shown in Fig. 1.

Equations (31)

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( z , ω ) = 1 2 π E ( z , t ) exp ( i ω t ) d t ,
[ 2 z 2 + k ( ω ) 2 ] ( z , ω ) = 0 .
( z , ω ) = A ( ω ) exp [ i k ( ω ) z ] + B ( ω ) exp [ i k ( ω ) z ] ,
E ( z , t ) = A ( ω ) exp [ i k ( ω ) z i ω t ] d ω .
k ( ω ) k 0 + k 0 ( ω ω 0 ) + k 0 2 ( ω ω 0 ) 2 .
E ( z , t ) = u ( z , t k 0 z ) exp [ i ( k 0 z ω 0 t ) ] .
u ( z , t ) = α ( ω ) exp ( i 2 ω 2 k 0 z i ω t ) d ω .
i u ( z , t ) z = k 0 2 2 u ( z , t ) t 2 .
u out ( t ) = u in ( t ) K 0 ( t , t ) d t ,
K 0 ( t , t ) = exp ( i π / 4 ) 2 π k 0 z exp [ i ( t t ) 2 2 k 0 z ] .
K ( t , t ) = exp ( i π / 4 ) 2 π B exp [ i ( A t 2 2 t t + D t 2 ) 2 B ] ,
[ A B C D ] = [ 1 k 0 z 0 1 ] .
u out ( t ) = u in ( t ) exp ( i t 2 / 2 k 0 f ) ,
u out ( t ) = u in ( t ) K ( t , t ) d t ,
K ( t , t ) = δ ( t t ) exp ( i t 2 / 2 k 0 f ) = lim σ 0 + exp ( i π / 4 ) 2 π σ exp [ i 2 σ ( t t ) 2 i t 2 2 k 0 f ]
[ A B C D ] = lim σ 0 + [ 1 σ 1 / k 0 f 1 σ / k 0 f ] = [ 1 0 1 / k 0 f 1 ] .
W in ( t , ω ) = 1 2 π u in * ( t + τ 2 ) u in ( t τ 2 ) exp ( i ω τ ) d τ .
I in ( t ) = 0 W in ( t , ω ) d ω .
W out ( t , ω ) = W in ( D t B ω , A ω C t ) .
I out ( t ) = 0 W in ( D t B ω , A ω C t ) d ω .
K θ FracFT ( t , t ) = exp ( i π / 4 ) 2 π τ 2 sin θ × exp { i 2 [ ( cos θ ) t 2 + ( cos θ ) t 2 2 t t ] τ 2 sin θ } ,
[ A B C D ] = [ cos θ τ 2 sin θ sin θ / τ 2 cos θ ] .
I out FracFT ( t ) Λ θ ( t ) = 0 W in [ t cos θ + ( ω τ 2 ) sin θ , ω cos θ + ( t / τ 2 ) sin θ ] d ω .
Λ ( t , f ) 0 W in [ D ( f ) t B ( f ) ω , A ( f ) ω C ( f ) t ] d ω .
1 ( 2 π ) 2 Λ ( t , f ) exp ( i ξ t ) d t = W in [ A ( f ) ξ , B ( f ) ξ ] ,
W in ( K 1 , K 2 ) 1 ( 2 π ) 2 W in ( t , ω ) × exp [ i ( K 1 t + K 2 ω ) ] d t d ω .
W in ( t , ω ) = W in [ A ( f ) ξ , B ( f ) ξ ] × exp { i ξ [ A ( f ) t + B ( f ) ω ] } d [ A ( f ) ξ ] d [ B ( f ) ξ ] .
W in ( t , ω ) = 1 ( 2 π ) 2 d t d ξ f 1 f 2 d f × | A ( f ) d B ( f ) d f B ( f ) d A ( f ) d f | × | ξ | Λ ( t , f ) exp { i ξ [ A ( f ) t + B ( f ) ω t ] } ,
[ A ( f ) B ( f ) C ( f ) D ( f ) ] = [ ( z / f ) 2 3 z / f + 1 k 0 z [ ( z / f ) 2 4 z / f + 3 ] ( z / f 2 ) / k 0 f ( z / f ) 2 3 z / f + 1 ] .
W in ( t , ω ) = k 0 ( 2 π ) 2 d t d ξ 0.382 z 2.618 z d f ( z f ) 2 × [ ( z f ) 2 4 ( z f ) + 5 ] × | ξ | Λ ( t , f ) exp { i ξ [ A ( f ) t + B ( f ) ω t ] } ,
f = ( k 0 Φ 0 ω ¯ 2 ) 1 ,

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