Abstract

We describe a new method—chronocyclic tomography—for determining the amplitude and phase structure of a short optical pulse. The technique is based on measurements of the energy spectrum of the pulse after it has passed through a time–frequency-domain imaging system. Tomographic inversion of these measured spectra yields the time–frequency Wigner distribution of the pulse, which uniquely determines the amplitude and phase structure.

© 1993 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J.-C. Diels, J. J. Fontaine, I. C. McMichael, F. Simoni, Appl. Opt. 24, 1270 (1985); C. Yan, J.-C. Diels, J. Opt. Soc. Am. B 8, 1259 (1991).
    [CrossRef] [PubMed]
  2. K. Naganuma, K. Mogi, H. Yamada, IEEE J. Quantum Electron. 25, 1225 (1989); Appl. Phys. Lett. 54, 1201 (1989).
    [CrossRef]
  3. J. L. A. Chilla, O. E. Martinez, IEEE J. Quantum Electron. 27, 1228 (1991).
    [CrossRef]
  4. D. J. Kane, R. Trebino, Opt. Lett. 18, 823 (1993).
    [CrossRef] [PubMed]
  5. E. B. Treacy, IEEE J. Quantum Electron. QE-5, 454 (1969).
    [CrossRef]
  6. B. H. Kolner, M. Nazarathy, Opt. Lett. 14, 630 (1989).
    [CrossRef] [PubMed]
  7. M. T. Kauffman, A. A. Godil, B. A. Auld, W. C. Banyai, D. M. Bloom, Electron. Lett. 29, 268 (1993).
    [CrossRef]
  8. M. T. Kauffman, W. C. Banyai, D. M. Bloom, in Conference on Lasers and Electro-Optics, Vol. 11 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), paper CPD33.
  9. K.-H. Brenner, K. Wodkiewicz, Opt. Commun. 43, 103 (1982); J. Paye, IEEE J. Quantum Electron. 28, 2262 (1992).
    [CrossRef]
  10. O. E. Martinez, IEEE J. Quantum Electron. QE-23, 59 (1987).
    [CrossRef]
  11. All Fourier-transform pairs will be represented by f˜(ω)=(2π)−1/2∫−∞∞f(t)exp(iωt)dt,where ω denotes the frequency as measured from some average carrier frequencyω¯. All times t and frequencies ω will be dimensionless quantities (i.e., scaled by a characteristic time T).
  12. Our definition differs from that of Ref. 9 (see Ref. 13).
  13. D. T. Smithy, M. Beck, A. Faridani, M. G. Raymer, Phys. Rev. Lett. 70, 1244 (1993).
    [CrossRef]
  14. G. T. Herman, Image Reconstruction from Projections: The Fundamentals of Computerized Tomography (Academic, New York, 1980), Chaps. 6–8, p. 90.
  15. The probability distributions are normalized such that ∫−∞∞Pθ(ωθ)dωθ=1.

1993 (3)

M. T. Kauffman, A. A. Godil, B. A. Auld, W. C. Banyai, D. M. Bloom, Electron. Lett. 29, 268 (1993).
[CrossRef]

D. T. Smithy, M. Beck, A. Faridani, M. G. Raymer, Phys. Rev. Lett. 70, 1244 (1993).
[CrossRef]

D. J. Kane, R. Trebino, Opt. Lett. 18, 823 (1993).
[CrossRef] [PubMed]

1991 (1)

J. L. A. Chilla, O. E. Martinez, IEEE J. Quantum Electron. 27, 1228 (1991).
[CrossRef]

1989 (2)

B. H. Kolner, M. Nazarathy, Opt. Lett. 14, 630 (1989).
[CrossRef] [PubMed]

K. Naganuma, K. Mogi, H. Yamada, IEEE J. Quantum Electron. 25, 1225 (1989); Appl. Phys. Lett. 54, 1201 (1989).
[CrossRef]

1987 (1)

O. E. Martinez, IEEE J. Quantum Electron. QE-23, 59 (1987).
[CrossRef]

1985 (1)

1982 (1)

K.-H. Brenner, K. Wodkiewicz, Opt. Commun. 43, 103 (1982); J. Paye, IEEE J. Quantum Electron. 28, 2262 (1992).
[CrossRef]

1969 (1)

E. B. Treacy, IEEE J. Quantum Electron. QE-5, 454 (1969).
[CrossRef]

Auld, B. A.

M. T. Kauffman, A. A. Godil, B. A. Auld, W. C. Banyai, D. M. Bloom, Electron. Lett. 29, 268 (1993).
[CrossRef]

Banyai, W. C.

M. T. Kauffman, A. A. Godil, B. A. Auld, W. C. Banyai, D. M. Bloom, Electron. Lett. 29, 268 (1993).
[CrossRef]

M. T. Kauffman, W. C. Banyai, D. M. Bloom, in Conference on Lasers and Electro-Optics, Vol. 11 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), paper CPD33.

Beck, M.

D. T. Smithy, M. Beck, A. Faridani, M. G. Raymer, Phys. Rev. Lett. 70, 1244 (1993).
[CrossRef]

Bloom, D. M.

M. T. Kauffman, A. A. Godil, B. A. Auld, W. C. Banyai, D. M. Bloom, Electron. Lett. 29, 268 (1993).
[CrossRef]

M. T. Kauffman, W. C. Banyai, D. M. Bloom, in Conference on Lasers and Electro-Optics, Vol. 11 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), paper CPD33.

Brenner, K.-H.

K.-H. Brenner, K. Wodkiewicz, Opt. Commun. 43, 103 (1982); J. Paye, IEEE J. Quantum Electron. 28, 2262 (1992).
[CrossRef]

Chilla, J. L. A.

J. L. A. Chilla, O. E. Martinez, IEEE J. Quantum Electron. 27, 1228 (1991).
[CrossRef]

Diels, J.-C.

Faridani, A.

D. T. Smithy, M. Beck, A. Faridani, M. G. Raymer, Phys. Rev. Lett. 70, 1244 (1993).
[CrossRef]

Fontaine, J. J.

Godil, A. A.

M. T. Kauffman, A. A. Godil, B. A. Auld, W. C. Banyai, D. M. Bloom, Electron. Lett. 29, 268 (1993).
[CrossRef]

Herman, G. T.

G. T. Herman, Image Reconstruction from Projections: The Fundamentals of Computerized Tomography (Academic, New York, 1980), Chaps. 6–8, p. 90.

Kane, D. J.

Kauffman, M. T.

M. T. Kauffman, A. A. Godil, B. A. Auld, W. C. Banyai, D. M. Bloom, Electron. Lett. 29, 268 (1993).
[CrossRef]

M. T. Kauffman, W. C. Banyai, D. M. Bloom, in Conference on Lasers and Electro-Optics, Vol. 11 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), paper CPD33.

Kolner, B. H.

Martinez, O. E.

J. L. A. Chilla, O. E. Martinez, IEEE J. Quantum Electron. 27, 1228 (1991).
[CrossRef]

O. E. Martinez, IEEE J. Quantum Electron. QE-23, 59 (1987).
[CrossRef]

McMichael, I. C.

Mogi, K.

K. Naganuma, K. Mogi, H. Yamada, IEEE J. Quantum Electron. 25, 1225 (1989); Appl. Phys. Lett. 54, 1201 (1989).
[CrossRef]

Naganuma, K.

K. Naganuma, K. Mogi, H. Yamada, IEEE J. Quantum Electron. 25, 1225 (1989); Appl. Phys. Lett. 54, 1201 (1989).
[CrossRef]

Nazarathy, M.

Raymer, M. G.

D. T. Smithy, M. Beck, A. Faridani, M. G. Raymer, Phys. Rev. Lett. 70, 1244 (1993).
[CrossRef]

Simoni, F.

Smithy, D. T.

D. T. Smithy, M. Beck, A. Faridani, M. G. Raymer, Phys. Rev. Lett. 70, 1244 (1993).
[CrossRef]

Treacy, E. B.

E. B. Treacy, IEEE J. Quantum Electron. QE-5, 454 (1969).
[CrossRef]

Trebino, R.

Wodkiewicz, K.

K.-H. Brenner, K. Wodkiewicz, Opt. Commun. 43, 103 (1982); J. Paye, IEEE J. Quantum Electron. 28, 2262 (1992).
[CrossRef]

Yamada, H.

K. Naganuma, K. Mogi, H. Yamada, IEEE J. Quantum Electron. 25, 1225 (1989); Appl. Phys. Lett. 54, 1201 (1989).
[CrossRef]

Appl. Opt. (1)

Electron. Lett. (1)

M. T. Kauffman, A. A. Godil, B. A. Auld, W. C. Banyai, D. M. Bloom, Electron. Lett. 29, 268 (1993).
[CrossRef]

IEEE J. Quantum Electron. (4)

O. E. Martinez, IEEE J. Quantum Electron. QE-23, 59 (1987).
[CrossRef]

K. Naganuma, K. Mogi, H. Yamada, IEEE J. Quantum Electron. 25, 1225 (1989); Appl. Phys. Lett. 54, 1201 (1989).
[CrossRef]

J. L. A. Chilla, O. E. Martinez, IEEE J. Quantum Electron. 27, 1228 (1991).
[CrossRef]

E. B. Treacy, IEEE J. Quantum Electron. QE-5, 454 (1969).
[CrossRef]

Opt. Commun. (1)

K.-H. Brenner, K. Wodkiewicz, Opt. Commun. 43, 103 (1982); J. Paye, IEEE J. Quantum Electron. 28, 2262 (1992).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. Lett. (1)

D. T. Smithy, M. Beck, A. Faridani, M. G. Raymer, Phys. Rev. Lett. 70, 1244 (1993).
[CrossRef]

Other (5)

G. T. Herman, Image Reconstruction from Projections: The Fundamentals of Computerized Tomography (Academic, New York, 1980), Chaps. 6–8, p. 90.

The probability distributions are normalized such that ∫−∞∞Pθ(ωθ)dωθ=1.

All Fourier-transform pairs will be represented by f˜(ω)=(2π)−1/2∫−∞∞f(t)exp(iωt)dt,where ω denotes the frequency as measured from some average carrier frequencyω¯. All times t and frequencies ω will be dimensionless quantities (i.e., scaled by a characteristic time T).

Our definition differs from that of Ref. 9 (see Ref. 13).

M. T. Kauffman, W. C. Banyai, D. M. Bloom, in Conference on Lasers and Electro-Optics, Vol. 11 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), paper CPD33.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (2)

Fig. 1
Fig. 1

(a) Pulse intensity |E(t)|2 plotted versus time t for the input pulse of relation (6) (dashed curve) and the reconstructed pulse (circles). Also plotted is the phase of the input pulse (solid curve) and of the reconstructed pulse (×’s). (b) The pulse-energy spectrum |(ω)|2 plotted versus frequency ω for the input pulse of relation (6) (dashed curve) and the reconstructed pulse (circles).

Fig. 2
Fig. 2

The reconstructed Wigner function W(ω, t) of the pulse given by relation (6). The view is perpendicular to the frequency axis, with time increasing into the page.

Equations (8)

Equations on this page are rendered with MathJax. Learn more.

E ˜ out ( ω ) = ( i / 2 π Φ t ¯ ) 1 / 2 exp ( i ω 2 / 2 Φ t ¯ ) × exp { i [ ω ω / Φ t ¯ + 1 2 ( ϕ ω ¯ + 1 / Φ t ¯ ) ω 2 ] } E ˜ ( ω ) d ω .
W ( ω , t ) = 1 π E ( t + t ) E * ( t t ) exp ( i 2 ω t ) d t
= 1 π E ( ω + ω ) E * ( ω ω ) × exp ( i 2 ω t ) d ω .
P θ ( ω θ ) = W ( ω θ cos θ t θ sin θ , ω θ sin θ + t θ cos θ ) d t θ ,
ψ ˜ ( ω θ ) = ( 2 π | sin θ | ) 1 / 2 exp [ i ( ω θ ω csc θ 1 2 ω 2 cot θ ) ] E ˜ ( ω ) d ω .
cot θ = ( ϕ ω ¯ + 1 / Φ t ¯ ) ,
ω θ / sin θ = ω / Φ t ¯ ,
E ( t ) { exp [ ( t τ ) 2 / 2 ] + exp [ ( t τ ) 2 / 2 ] } exp ( i γ t 2 ) .

Metrics