Abstract

We analyze a new regime in the interaction between an optical pulse and a time lens (spectral Fraunhofer regime), where the input pulse amplitude is mapped from the time domain into the frequency domain (time-to-frequency conversion). Here we derive in detail the conditions for achieving time-to-frequency conversion with a single time lens (i.e., for entering the spectral Fraunhofer regime) as well as the expressions governing this operation. Our theoretical findings are demonstrated both numerically and experimentally. A comparative study between the proposed single-time-lens configuration and the conventional dispersion + time-lens configuration for time-to-frequency conversion is also conducted. Time-to-frequency conversion with a single time lens can be used for applications similar to those previously proposed for the conventional time-to-frequency converters, e.g., high-resolution measurement of fast optical temporal waveforms. Moreover, our results also indicate that the spectral Fraunhofer regime provides additional capabilities for controlling and processing optical pulses.

© 2004 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef]
  3. C. V. Bennett, B. H. Kolner, “Upconversion time microscope demonstrating 103× magnification of femtosecond waveforms,” Opt. Lett. 24, 783–785 (1999).
    [CrossRef]
  4. M. T. Kauffman, W. C. Banyai, A. A. Godil, D. M. Bloom, “Time-to-frequency converter for measuring picosecond optical pulses,” Appl. Phys. Lett. 64, 270–272 (1994).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  7. L. K. H. Mouradian, F. Louradour, V. Messager, A. Barthelemy, C. Froehly, “Spectro-temporal imaging of femtosecond events,” IEEE J. Quantum Electron. 36, 795–801 (2000).
    [CrossRef]
  8. J. Azaña, M. A. Muriel, “Real-time optical spectrum analysis based on the time-space duality in chirped fiber gratings,” IEEE J. Quantum Electron. 36, 517–526 (2000).
    [CrossRef]
  9. C. V. Bennett, B. H. Kolner, “Aberrations in temporal imaging,” IEEE J. Quantum Electron. 37, 20–32 (2001).
    [CrossRef]
  10. L. Cohen, “Time-frequency distributions—a review,” Proc. IEEE 77, 941–981 (1989).
    [CrossRef]
  11. S. Cheung, J. S. Lim, “Combined multiresolution (wide-band/narrow-band) spectrogram,” IEEE Trans. Signal Process. 40, 975–977 (1992).
    [CrossRef]

2001 (1)

C. V. Bennett, B. H. Kolner, “Aberrations in temporal imaging,” IEEE J. Quantum Electron. 37, 20–32 (2001).
[CrossRef]

2000 (3)

N. K. Berger, B. Levit, S. Atkins, B. Fischer, “Time-lens-based spectral analysis of optical pulses by electrooptic phase modulation,” Electron. Lett. 36, 1644–1646 (2000).
[CrossRef]

L. K. H. Mouradian, F. Louradour, V. Messager, A. Barthelemy, C. Froehly, “Spectro-temporal imaging of femtosecond events,” IEEE J. Quantum Electron. 36, 795–801 (2000).
[CrossRef]

J. Azaña, M. A. Muriel, “Real-time optical spectrum analysis based on the time-space duality in chirped fiber gratings,” IEEE J. Quantum Electron. 36, 517–526 (2000).
[CrossRef]

1999 (1)

1997 (1)

1994 (2)

M. T. Kauffman, W. C. Banyai, A. A. Godil, D. M. Bloom, “Time-to-frequency converter for measuring picosecond optical pulses,” Appl. Phys. Lett. 64, 270–272 (1994).
[CrossRef]

B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30, 1951–1963 (1994).
[CrossRef]

1992 (1)

S. Cheung, J. S. Lim, “Combined multiresolution (wide-band/narrow-band) spectrogram,” IEEE Trans. Signal Process. 40, 975–977 (1992).
[CrossRef]

1989 (1)

L. Cohen, “Time-frequency distributions—a review,” Proc. IEEE 77, 941–981 (1989).
[CrossRef]

1983 (1)

Arons, E.

Atkins, S.

N. K. Berger, B. Levit, S. Atkins, B. Fischer, “Time-lens-based spectral analysis of optical pulses by electrooptic phase modulation,” Electron. Lett. 36, 1644–1646 (2000).
[CrossRef]

Azaña, J.

J. Azaña, M. A. Muriel, “Real-time optical spectrum analysis based on the time-space duality in chirped fiber gratings,” IEEE J. Quantum Electron. 36, 517–526 (2000).
[CrossRef]

Banyai, W. C.

M. T. Kauffman, W. C. Banyai, A. A. Godil, D. M. Bloom, “Time-to-frequency converter for measuring picosecond optical pulses,” Appl. Phys. Lett. 64, 270–272 (1994).
[CrossRef]

Barthelemy, A.

L. K. H. Mouradian, F. Louradour, V. Messager, A. Barthelemy, C. Froehly, “Spectro-temporal imaging of femtosecond events,” IEEE J. Quantum Electron. 36, 795–801 (2000).
[CrossRef]

Bennett, C. V.

Berger, N. K.

N. K. Berger, B. Levit, S. Atkins, B. Fischer, “Time-lens-based spectral analysis of optical pulses by electrooptic phase modulation,” Electron. Lett. 36, 1644–1646 (2000).
[CrossRef]

Bloom, D. M.

M. T. Kauffman, W. C. Banyai, A. A. Godil, D. M. Bloom, “Time-to-frequency converter for measuring picosecond optical pulses,” Appl. Phys. Lett. 64, 270–272 (1994).
[CrossRef]

Cheung, S.

S. Cheung, J. S. Lim, “Combined multiresolution (wide-band/narrow-band) spectrogram,” IEEE Trans. Signal Process. 40, 975–977 (1992).
[CrossRef]

Cohen, L.

L. Cohen, “Time-frequency distributions—a review,” Proc. IEEE 77, 941–981 (1989).
[CrossRef]

Fischer, B.

N. K. Berger, B. Levit, S. Atkins, B. Fischer, “Time-lens-based spectral analysis of optical pulses by electrooptic phase modulation,” Electron. Lett. 36, 1644–1646 (2000).
[CrossRef]

Froehly, C.

L. K. H. Mouradian, F. Louradour, V. Messager, A. Barthelemy, C. Froehly, “Spectro-temporal imaging of femtosecond events,” IEEE J. Quantum Electron. 36, 795–801 (2000).
[CrossRef]

Godil, A. A.

M. T. Kauffman, W. C. Banyai, A. A. Godil, D. M. Bloom, “Time-to-frequency converter for measuring picosecond optical pulses,” Appl. Phys. Lett. 64, 270–272 (1994).
[CrossRef]

Jannson, T.

Kauffman, M. T.

M. T. Kauffman, W. C. Banyai, A. A. Godil, D. M. Bloom, “Time-to-frequency converter for measuring picosecond optical pulses,” Appl. Phys. Lett. 64, 270–272 (1994).
[CrossRef]

Kolner, B. H.

C. V. Bennett, B. H. Kolner, “Aberrations in temporal imaging,” IEEE J. Quantum Electron. 37, 20–32 (2001).
[CrossRef]

C. V. Bennett, B. H. Kolner, “Upconversion time microscope demonstrating 103× magnification of femtosecond waveforms,” Opt. Lett. 24, 783–785 (1999).
[CrossRef]

B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30, 1951–1963 (1994).
[CrossRef]

Leith, E. N.

Levit, B.

N. K. Berger, B. Levit, S. Atkins, B. Fischer, “Time-lens-based spectral analysis of optical pulses by electrooptic phase modulation,” Electron. Lett. 36, 1644–1646 (2000).
[CrossRef]

Lim, J. S.

S. Cheung, J. S. Lim, “Combined multiresolution (wide-band/narrow-band) spectrogram,” IEEE Trans. Signal Process. 40, 975–977 (1992).
[CrossRef]

Louradour, F.

L. K. H. Mouradian, F. Louradour, V. Messager, A. Barthelemy, C. Froehly, “Spectro-temporal imaging of femtosecond events,” IEEE J. Quantum Electron. 36, 795–801 (2000).
[CrossRef]

Messager, V.

L. K. H. Mouradian, F. Louradour, V. Messager, A. Barthelemy, C. Froehly, “Spectro-temporal imaging of femtosecond events,” IEEE J. Quantum Electron. 36, 795–801 (2000).
[CrossRef]

Mouradian, L. K. H.

L. K. H. Mouradian, F. Louradour, V. Messager, A. Barthelemy, C. Froehly, “Spectro-temporal imaging of femtosecond events,” IEEE J. Quantum Electron. 36, 795–801 (2000).
[CrossRef]

Muriel, M. A.

J. Azaña, M. A. Muriel, “Real-time optical spectrum analysis based on the time-space duality in chirped fiber gratings,” IEEE J. Quantum Electron. 36, 517–526 (2000).
[CrossRef]

Tien, A.-C.

Wagner, R.

Appl. Opt. (1)

Appl. Phys. Lett. (1)

M. T. Kauffman, W. C. Banyai, A. A. Godil, D. M. Bloom, “Time-to-frequency converter for measuring picosecond optical pulses,” Appl. Phys. Lett. 64, 270–272 (1994).
[CrossRef]

Electron. Lett. (1)

N. K. Berger, B. Levit, S. Atkins, B. Fischer, “Time-lens-based spectral analysis of optical pulses by electrooptic phase modulation,” Electron. Lett. 36, 1644–1646 (2000).
[CrossRef]

IEEE J. Quantum Electron. (4)

L. K. H. Mouradian, F. Louradour, V. Messager, A. Barthelemy, C. Froehly, “Spectro-temporal imaging of femtosecond events,” IEEE J. Quantum Electron. 36, 795–801 (2000).
[CrossRef]

J. Azaña, M. A. Muriel, “Real-time optical spectrum analysis based on the time-space duality in chirped fiber gratings,” IEEE J. Quantum Electron. 36, 517–526 (2000).
[CrossRef]

C. V. Bennett, B. H. Kolner, “Aberrations in temporal imaging,” IEEE J. Quantum Electron. 37, 20–32 (2001).
[CrossRef]

B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30, 1951–1963 (1994).
[CrossRef]

IEEE Trans. Signal Process. (1)

S. Cheung, J. S. Lim, “Combined multiresolution (wide-band/narrow-band) spectrogram,” IEEE Trans. Signal Process. 40, 975–977 (1992).
[CrossRef]

Opt. Lett. (2)

Proc. IEEE (1)

L. Cohen, “Time-frequency distributions—a review,” Proc. IEEE 77, 941–981 (1989).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Dual Fraunhofer regimes: (a) frequency-to-time conversion using dispersion, (b) time-to-frequency conversion using a single time lens.

Fig. 2
Fig. 2

Schematic of the dispersion-time-lens configuration for time-to-frequency conversion.

Fig. 3
Fig. 3

Simulation results for pulse propagation through a time lens operating within the spectral Fraunhofer regime. Plots at the bottom show the signal in the temporal domain, plots at the left show the signal in the frequency domain, and the larger 2D images show the joint time-frequency representation of the signal. (a) Input pulse to the time lens; (b) output pulse from the time lens.

Fig. 4
Fig. 4

Simulation results for pulse propagation through a time lens, assuming the same input pulse as in Fig. 3(a) but operating outside the spectral Fraunhofer regime. The representation shows the output pulse from the time lens, with the same definitions as for Fig. 3(a).

Fig. 5
Fig. 5

Simulation results for pulse propagation through a conventional time-to-frequency converter (dispersion-time-lens system) configured to implement time-to-frequency conversion (as shown in Fig. 2), assuming the same input pulse as in Fig. 3(a). (a) Output pulse from the dispersive medium, with the same definitions as for Fig. 3(a). (b) Output pulse from the time lens (system), with the same definitions as for Fig. 3(a).

Fig. 6
Fig. 6

Schematic of the experimental setup. Solid (dotted) lines are used for optical (electrical) signals. FBG, fiber Bragg grating; EDFA, erbium-doped fiber amplifier; C, fiber coupler; PC, polarization controller; PD, photodetector; Synch, synchronization.

Fig. 7
Fig. 7

Experimental results: (a) measured spectrum of the input pulse to the time lens; (b) measured spectrum of the output pulse from the time lens (solid curve, top scale) and measured temporal waveform of the input pulse to (output pulse from) the time lens (dotted curve, bottom scale).

Equations (9)

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

|Φ¨ω|Δt2/8π,
mt=expjϕt  expjϕ¨t/2t2,
Dω=Cω * Mω  Δω CΩexp-j1/2ϕ¨tω-Ω2dΩ  MωΔω CΩexp-j1/2ϕ¨tΩ2×expj1/ϕ¨tωΩdΩ,
|ϕ¨t|Δω2/8π,
Dω  MωΔω CΩexpj1/ϕ¨tωΩdΩ=Mωct=ω/ϕ¨t,
δt>5π/|ϕ¨t|4/|ϕ¨t|.
δt1/|ϕ¨t|.
Φ¨ωϕ¨t=1,
δt>2πτa |Φ¨ω|2πτa1|ϕ¨t|.

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