Abstract

The operational characteristics of a time-to-space processor based on three-wave mixing for instantaneous imaging of ultrafast waveforms are investigated. We assess the effects of various system parameters on the processor’s important attributes: time window of operation and signal conversion efficiency. Both linear and nonlinear operation regimes are considered, with use of a Gaussian pulse profile and a Gaussian spatial mode model. This model enables us to define a resolution measure for the processor, which is found to be an important characteristic. When the processor is operated in the linear interaction regime, we find that the conversion efficiency of a temporal signal to a spatial image is inversely proportional to the resolution measure. In the nonlinear interaction regime, nonuniform signal conversion due to fundamental wave depletion gives rise to a phenomenon that can be used to enhanced the imaging operation. We experimentally verify this nonlinear operation.

© 2001 Optical Society of America

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References

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  1. P. C. Sun, Y. T. Mazurenko, Y. Fainman, “Femtosecond pulse imaging: ultrafast optical oscilloscope,” J. Opt. Soc. Am. A 14, 1159–1170 (1997).
    [CrossRef]
  2. J. Janszky, G. Corradi, R. N. Gyuzalian, “On a possibility of analysing the temporal characteristics of short light pulses,” Opt. Commun. 23, 293–298 (1977).
    [CrossRef]
  3. F. Salin, P. Georges, G. Roger, A. Brun, “Single-shot measurement of a 52-fs pulse,” Appl. Opt. 26, 4528–4531 (1987).
    [CrossRef] [PubMed]
  4. A. M. Kan’an, A. M. Weiner, “Efficient time-to-space conversion of femtosecond optical pulses,” J. Opt. Soc. Am. B 15, 1242–1245 (1998).
    [CrossRef]
  5. P. C. Sun, Y. T. Mazurenko, Y. Fainman, “Real-time one-dimensional coherent imaging through single-mode fibers by space–time conversion processors,” Opt. Lett. 22, 1861–1863 (1997).
    [CrossRef]
  6. D. M. Marom, P.-C. Sun, Y. Fainman, “Analysis of spatial–temporal converters for all-optical communication links,” Appl. Opt. 37, 2858–2868 (1998).
    [CrossRef]
  7. B. Javidi, D. Painchaud, “Distortion-invariant pattern recognition with Fourier-plane nonlinear filters,” Appl. Opt. 35, 318–331 (1996).
    [CrossRef] [PubMed]
  8. R. N. Thurston, J. P. Heritage, A. M. Weiner, W. J. Tomlinson, “Analysis of picosecond pulse shape synthesis by spectral masking in a grating pulse compressor,” IEEE J. Quantum Electron. QE-22, 682–696 (1986).
    [CrossRef]
  9. A. M. Weiner, J. P. Heritage, E. M. Kirschner, “High-resolution femtosecond pulse shaping,” J. Opt. Soc. Am. B 5, 1563–1572 (1988).
    [CrossRef]
  10. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).
  11. Y. T. Mazurenko, “Holography of wave packets,” Appl. Phys. B 50, 101–114 (1990).
    [CrossRef]
  12. P. N. Butcher, D. Cotter, The Elements of Nonlinear Optics (Cambridge U. Press, New York, 1990).
  13. D. Eimerl, “High average power harmonic generation,” IEEE J. Quantum Electron. QE-23, 575–592 (1987).
    [CrossRef]
  14. C. Sang-Kyung, L. Ruo-Ding, K. Chonghoon, K. Prem, “Traveling-wave optical parametric amplifier: investigation of its phase-sensitive and phase-insensitive gain response,” J. Opt. Soc. Am. B 14, 1564–1575 (1997).
    [CrossRef]
  15. K. Chonghoon, L. Ruo-Ding, K. Prem, “Deamplification response of a traveling-wave phase-sensitive optical parametric amplifier,” Opt. Lett. 19, 132–134 (1994).
    [CrossRef]
  16. G. Imeshev, M. Proctor, M. M. Fejer, “Lateral patterning of nonlinear frequency conversion with transversely varying quasi-phase-matching gratings,” Opt. Lett. 23, 673–675 (1998).
    [CrossRef]

1998 (3)

1997 (3)

1996 (1)

1994 (1)

1990 (1)

Y. T. Mazurenko, “Holography of wave packets,” Appl. Phys. B 50, 101–114 (1990).
[CrossRef]

1988 (1)

1987 (2)

F. Salin, P. Georges, G. Roger, A. Brun, “Single-shot measurement of a 52-fs pulse,” Appl. Opt. 26, 4528–4531 (1987).
[CrossRef] [PubMed]

D. Eimerl, “High average power harmonic generation,” IEEE J. Quantum Electron. QE-23, 575–592 (1987).
[CrossRef]

1986 (1)

R. N. Thurston, J. P. Heritage, A. M. Weiner, W. J. Tomlinson, “Analysis of picosecond pulse shape synthesis by spectral masking in a grating pulse compressor,” IEEE J. Quantum Electron. QE-22, 682–696 (1986).
[CrossRef]

1977 (1)

J. Janszky, G. Corradi, R. N. Gyuzalian, “On a possibility of analysing the temporal characteristics of short light pulses,” Opt. Commun. 23, 293–298 (1977).
[CrossRef]

Brun, A.

Butcher, P. N.

P. N. Butcher, D. Cotter, The Elements of Nonlinear Optics (Cambridge U. Press, New York, 1990).

Chonghoon, K.

Corradi, G.

J. Janszky, G. Corradi, R. N. Gyuzalian, “On a possibility of analysing the temporal characteristics of short light pulses,” Opt. Commun. 23, 293–298 (1977).
[CrossRef]

Cotter, D.

P. N. Butcher, D. Cotter, The Elements of Nonlinear Optics (Cambridge U. Press, New York, 1990).

Eimerl, D.

D. Eimerl, “High average power harmonic generation,” IEEE J. Quantum Electron. QE-23, 575–592 (1987).
[CrossRef]

Fainman, Y.

Fejer, M. M.

Georges, P.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

Gyuzalian, R. N.

J. Janszky, G. Corradi, R. N. Gyuzalian, “On a possibility of analysing the temporal characteristics of short light pulses,” Opt. Commun. 23, 293–298 (1977).
[CrossRef]

Heritage, J. P.

A. M. Weiner, J. P. Heritage, E. M. Kirschner, “High-resolution femtosecond pulse shaping,” J. Opt. Soc. Am. B 5, 1563–1572 (1988).
[CrossRef]

R. N. Thurston, J. P. Heritage, A. M. Weiner, W. J. Tomlinson, “Analysis of picosecond pulse shape synthesis by spectral masking in a grating pulse compressor,” IEEE J. Quantum Electron. QE-22, 682–696 (1986).
[CrossRef]

Imeshev, G.

Janszky, J.

J. Janszky, G. Corradi, R. N. Gyuzalian, “On a possibility of analysing the temporal characteristics of short light pulses,” Opt. Commun. 23, 293–298 (1977).
[CrossRef]

Javidi, B.

Kan’an, A. M.

Kirschner, E. M.

Marom, D. M.

Mazurenko, Y. T.

Painchaud, D.

Prem, K.

Proctor, M.

Roger, G.

Ruo-Ding, L.

Salin, F.

Sang-Kyung, C.

Sun, P. C.

Sun, P.-C.

Thurston, R. N.

R. N. Thurston, J. P. Heritage, A. M. Weiner, W. J. Tomlinson, “Analysis of picosecond pulse shape synthesis by spectral masking in a grating pulse compressor,” IEEE J. Quantum Electron. QE-22, 682–696 (1986).
[CrossRef]

Tomlinson, W. J.

R. N. Thurston, J. P. Heritage, A. M. Weiner, W. J. Tomlinson, “Analysis of picosecond pulse shape synthesis by spectral masking in a grating pulse compressor,” IEEE J. Quantum Electron. QE-22, 682–696 (1986).
[CrossRef]

Weiner, A. M.

A. M. Kan’an, A. M. Weiner, “Efficient time-to-space conversion of femtosecond optical pulses,” J. Opt. Soc. Am. B 15, 1242–1245 (1998).
[CrossRef]

A. M. Weiner, J. P. Heritage, E. M. Kirschner, “High-resolution femtosecond pulse shaping,” J. Opt. Soc. Am. B 5, 1563–1572 (1988).
[CrossRef]

R. N. Thurston, J. P. Heritage, A. M. Weiner, W. J. Tomlinson, “Analysis of picosecond pulse shape synthesis by spectral masking in a grating pulse compressor,” IEEE J. Quantum Electron. QE-22, 682–696 (1986).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. B (1)

Y. T. Mazurenko, “Holography of wave packets,” Appl. Phys. B 50, 101–114 (1990).
[CrossRef]

IEEE J. Quantum Electron. (2)

D. Eimerl, “High average power harmonic generation,” IEEE J. Quantum Electron. QE-23, 575–592 (1987).
[CrossRef]

R. N. Thurston, J. P. Heritage, A. M. Weiner, W. J. Tomlinson, “Analysis of picosecond pulse shape synthesis by spectral masking in a grating pulse compressor,” IEEE J. Quantum Electron. QE-22, 682–696 (1986).
[CrossRef]

J. Opt. Soc. Am. A (1)

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

Opt. Commun. (1)

J. Janszky, G. Corradi, R. N. Gyuzalian, “On a possibility of analysing the temporal characteristics of short light pulses,” Opt. Commun. 23, 293–298 (1977).
[CrossRef]

Opt. Lett. (3)

Other (2)

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

P. N. Butcher, D. Cotter, The Elements of Nonlinear Optics (Cambridge U. Press, New York, 1990).

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Figures (8)

Fig. 1
Fig. 1

Time-to-space conversion by using wave mixing of inverted spectrally decomposed waves from a signal and a reference pulse: F1, F2, focal length.

Fig. 2
Fig. 2

Gaussian beam mode and Gaussian pulse envelope incident on the diffraction grating. The ratio of their spatial widths determines the resolution of the system.

Fig. 3
Fig. 3

Field distribution of the converted wave in the Fourier plane (left column) and the intensity distribution in the output plane after a spatial Fourier transform (right column) as a function of the reference pulse power (parameterized by K).

Fig. 4
Fig. 4

Enhanced-resolution imaging: comparison of the output image for linear (dashed curve) and nonlinear (solid curve for K=2.5) conversions.

Fig. 5
Fig. 5

Resolution improvement of the output signal as a function of reference pulse power K. Dashed curve, FWHM criteria; solid curve, standard deviation measure.

Fig. 6
Fig. 6

Conversion efficiency as a function of reference pulse power K. The linear conversion is shown to provide acceptable results for K<0.5.

Fig. 7
Fig. 7

Experimental results: (a) intensity distribution in the Fourier plane, (b) output pulse image, as functions of the reference pulse attenuation. A high ND setting corresponds to a small K value. Broadening in the Fourier plane and narrowing in the corresponding pulse images are exhibited.

Fig. 8
Fig. 8

Measured conversion efficiency as a function of the square root of the reference pulse power. Conversion follows the linear slope of 20 dB/dec.

Equations (47)

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Uin(x, y; t)=EswxLxwyLypt-t0-αx/cτ×exp(jω0t),
p(t)=exp-t22 TFT P(ω)=2πexp-ω22,
w(x)=exp(-x2) SFT W(fx)=πexp(-π2fx2),
Uin(x, y; t)=Esexp-x2Lx2exp-y2Ly2×exp-12τ2t-t0-αxc2exp(jω0t).
USDW(u, ν; t)=EsLxLyπλ0F11+N2exp-(t-t0)22τ2(1+N2)×exp-Lxω0u2cF1211+N2×exp-Lyω0ν2cF12×exp-j 2πλ0cuαF1N21+N2 (t-t0)×exp(jω0t).
U˜SDW(u, ν; ω)=EsLxLyπτ2πλ0F1exp-τ22 (1+N2)×ω-ω0+uω0αF1N21+N22×exp-Lω0u2cF1211+N2×exp-Lω0ν2cF12exp[-j(ω-ω0)t0].
PNL(u, ν, z; ω)
=20deff-U˜SDWS(u, ν; ϖ)U˜SDWR(u, ν; ω-ϖ)×exp[j[kS(ϖ)+kR(ω-ϖ)]z]dϖ,
kS(ϖ)=k(ω0)+(ϖ-ω0)k(Ω)ΩΩ=ω0+,
kR(ω-ϖ)=k(ω0)+(ω-ϖ-ω0)k(Ω)ΩΩ=ω0+ .
PNL(u, ν, z; ω)
=20deffexpj2k(ω0)+k(ω-2ω0)k(Ω)ΩΩ=ω0zG(u, ν; ω),
G(u, ν; ω)=-U˜SDWS(u, ν; ϖ)U˜SDWR(u, ν; ω-ϖ)dϖ
z U˜sum(u, ν, z; ω)=j2ω2μ0k(ω) PNL(u, ν, z; ω)×exp[-jksum(ω)z],
z U˜sum(u, ν, z; ω)
=j 2ω0cn2ω0 deffG(u, ν; ω)×exp[-j[k(2ω0)-2k(ω0)]z]exp{-j(ω-2ω0)×[(k/Ω)|Ω=2ω0-(k/Ω)|Ω=ω0]z}.
U¯sum(u, ν, Lc; ω)
=j 2ω0cn2ω0 EsErn2ω0LxLyπτ2πnω0λ0F12×πτ1+N2 deff Lcexp-2Lyω0ν2cF12×exp-21+N2Lxω0u2cF12exp-t024τ2(1+N2)×expjω0t0uαF1N21+N2A(ω)
A(ω)=exp-(ω-2ω0)2τ221+N22×sinc(ω-2ω0) βLc2×exp-j(ω-2ω0) βLc+t02,
a(t)=exp(j2ω0t)πτβLc1+N2exp-(t-t0/2)2τ2(1+N2)  recttβLc-12.
Uout(x, y; t)
=j deffn2ω0nω0 23π7/2EsErLxLyτλ02F2 Lca(t)×exp-t024τ2(1+N2)exp-2F1F2yLy2×exp-2F1F221+N2Lx2x-ct02αF2F1N21+N22.
Iout(x, y; t)=120μ |Uout(x, y; t)|2=deff2n2ω0nω0210c29π4λ04F22EsErLc2|a(t)|2×exp-t022τ2(1+N2)×exp-4F1F2yLy2×exp-4F1F221+N2L2×x-ct02αF2F1N21+N22.
Eout=---Iout(x, y; t)dxdydt=deff2n2ω0nω02 27π5EsEr10cLxLyLc2λ04F121+N2×exp-t022τ2(1+N2)-|a(t)|2dt,
-|a(t)|2dt=12π-|A(ω)|2dω=12π-exp-Ω2τ22 (1+N2)×sinc2Ω βLc2dΩ,
Eout=deff2n2ω0nω02 213/2π9/2EsEr10cLc2λ04F12LxLyτN2×exp-t022τ2N2.
ηN=deff2n2ω0ηω02 27π9/2ErLc2λ04F12Ly0αexp-t022τ2N2.
ηN=deff2n2ω0 29π5/2Er0F12αLy3λ02exp-t022τ2N2.
USDWS(u, ν; t)
=EsLycταλ0F1 Wω0Lyν2πcF1wc(t-t0)αLx×P-uτω0αF1exp-j ω0uαF (t-t0)exp(jω0t),
USDWS(u, ν; ω)
=EsLxLyτλ0F1 Wω0Lyν2πcFWαLxω0cωω0-1+uαF1×P-uτω0αF1exp[-j(ω-ω0)t0].
ωsig=ω0-uω0αF.
ωref=ω0+uω0αF.
Δk=ksum(ωsig+δωs+ωref+δωr)-kS(ωsig+δωs)-kR(ωref+δωr)=k(2ω0)+dkdω2ω0(ωsig+ωref-2ω0+δωs+δωr)-k(ω0)+dkdωω0(ωsig-ω0+δωs)-k(ω0)+dkdωω0(ωref-ω0+δωr)=[k(2ω0)-2k(ω0)]+β(ωsig+ωref-2ω0+δωs+δωr)=β(δωs+δωr)β|Δω|,
z USDWS(u, ν, z; t)
=j ω0cnω0 deffUsum(u, ν, z; t)USDWR*(u, ν; t),
z Usum(u, ν, z; t)
=j 2ω0cn2ω0 deffUSDWS(u, ν, z; t)USDWR(u, ν; t),
Usum(u, ν, z; t)
=j2nω0n2ω0USDWR(u, ν; t)|USDWR(u, ν; t)| USDWS(u, ν, z=0; t)×sin2ω02c2nω0n2ω0 deff |USDWR(u, ν; t)|z.
Isum(u, ν, Lc; t)
=2ISDWS(u, ν, 0; t)sin22π deffLcλ04ISDWR(u, ν; t)0cn2ω0nω021/2
-x2Iout(x)dx/-Iout(x)dx.
Iin(x, y; t)=120μ |Uin(x, y; t)|2=120μ Es2exp-2x2Lx2×exp-2y2Ly2exp-1τ2t-t0-αxc2.
---Iin(x, y; t)dxdydtEs,
Es=EsLxLyτ4π3/2μ01/2.

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