Abstract

The exact power spectrum of the detected optical envelope of a train of random pulses after the temporal Talbot effect is computed. The input train into a Talbot device consists of a sequence of chirped Gaussian pulses whose appearance in the train is probabilistic. Dispersion provides a Talbot replica of the original train. The resulting noise spectrum shows narrow spectral windows below a broadband output noise envelope. The noise-envelope width depends on the value of the chirp, coinciding with the single-pulse spectrum only if the pulses are unchirped. The locations and width of the spectral windows depend on the values of the chirp and the temporal width of the pulses in the train. For wide pulses, high output harmonics, and low dispersive devices, these windows are cosine-squared modulated. The properties of this modulation depend only on the statistics of the appearance of the pulses.

© 2004 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2003 (6)

N. K. Berger, B. Vodonos, S. Atkins, V. Smulakovsky, A. Bekker, and B. Fischer, “Compression of periodic light pulses using all-optical repetition rate multiplication,” Opt. Commun. 217, 343–349 (2003).
[CrossRef]

J. Azaña, P. Kockaert, R. Slavík, L. R. Chen, and S. LaRochelle, “Generation of a 100-GHz optical pulse train by pulse repetition-rate multiplication using superimposed fiber Bragg gratings,” IEEE Photon. Technol. Lett. 15, 413–415 (2003).
[CrossRef]

S. Atkins, B. Vodonos, A. Bekker, and B. Fischer, “Fractional dispersion modes in a pulsed fiber laser,” Opt. Commun. 222, 393–397 (2003).
[CrossRef]

S. Atkins and B. Fischer, “All-optical pulse rate multiplication using fractional Talbot effect and field-to-intensity conversion with cross-gain modulation,” IEEE Photon. Technol. Lett. 15, 132–134 (2003).
[CrossRef]

T. Yilmaz, C. M. Depriest, A. Braun, J. H. Abeles, and P. J. Delfyett, “Noise in fundamental and harmonic mode-locked semiconductor lasers: experiments and simulations,” IEEE J. Quantum Electron. 39, 838–848 (2003).
[CrossRef]

J. Azaña, “Temporal self-imaging effects for periodic optical pulse sequences of finite duration,” J. Opt. Soc. Am. B 20, 83–90 (2003).
[CrossRef]

2002 (2)

2001 (3)

M. Marano, S. Longhi, P. Laporta, M. Belmonte, and B. Agogliati, “All-optical square-pulse generation and multiplication at 1.5 μm by use a novel class of fiber Bragg gratings,” Opt. Lett. 26, 1615–1617 (2001).
[CrossRef]

R. P. Scott, C. Langrock, and B. H. Kolner, “High-dynamic-range laser amplitude and phase noise measurement techniques,” IEEE J. Sel. Top. Quantum Electron. 7, 641–655 (2001).
[CrossRef]

V. P. Minkovich, A. N. Starodumov, V. I. Borisov, V. I. Ledebev, and S. N. Perepechko, “Temporal interference of coherent laser pulses in optical fibers,” Opt. Commun. 192, 231–235 (2001).
[CrossRef]

2000 (2)

1999 (1)

1998 (2)

S. Arahira, S. Kutsuzawa, Y. Matsui, D. Kunimatsu, and Y. Ogawa, “Repetition-frequency multiplication of mode-locked pulses using fiber dispersion,” J. Lightwave Technol. 16, 405–410 (1998).
[CrossRef]

I. Shake, H. Tahara, S. Kawanishi, and M. Saruwatari, “High-repetition-rate optical pulse generation by using chirped optical pulses,” Electron. Lett. 34, 792–793 (1998).
[CrossRef]

1994 (1)

1989 (1)

K. Patorski, “The self-imaging phenomenon and its applications,” Prog. Opt. 27, 3–74 (1989).

1981 (1)

1978 (1)

A. Kalestynski and B. Smolinska, “Self-restoration of the autoidolon of defective periodic objects,” Opt. Acta 25, 125–134 (1978).
[CrossRef]

Abeles, J. H.

T. Yilmaz, C. M. Depriest, A. Braun, J. H. Abeles, and P. J. Delfyett, “Noise in fundamental and harmonic mode-locked semiconductor lasers: experiments and simulations,” IEEE J. Quantum Electron. 39, 838–848 (2003).
[CrossRef]

Agogliati, B.

Arahira, S.

Arcangeli, L.

Atkins, S.

S. Atkins and B. Fischer, “All-optical pulse rate multiplication using fractional Talbot effect and field-to-intensity conversion with cross-gain modulation,” IEEE Photon. Technol. Lett. 15, 132–134 (2003).
[CrossRef]

N. K. Berger, B. Vodonos, S. Atkins, V. Smulakovsky, A. Bekker, and B. Fischer, “Compression of periodic light pulses using all-optical repetition rate multiplication,” Opt. Commun. 217, 343–349 (2003).
[CrossRef]

S. Atkins, B. Vodonos, A. Bekker, and B. Fischer, “Fractional dispersion modes in a pulsed fiber laser,” Opt. Commun. 222, 393–397 (2003).
[CrossRef]

B. Fischer, B. Vodonos, S. Atkins, and A. Bekker, “Dispersion-mode pulsed laser,” Opt. Lett. 25, 728–730 (2000).
[CrossRef]

Azaña, J.

Bekker, A.

S. Atkins, B. Vodonos, A. Bekker, and B. Fischer, “Fractional dispersion modes in a pulsed fiber laser,” Opt. Commun. 222, 393–397 (2003).
[CrossRef]

N. K. Berger, B. Vodonos, S. Atkins, V. Smulakovsky, A. Bekker, and B. Fischer, “Compression of periodic light pulses using all-optical repetition rate multiplication,” Opt. Commun. 217, 343–349 (2003).
[CrossRef]

B. Fischer, B. Vodonos, S. Atkins, and A. Bekker, “Dispersion-mode pulsed laser,” Opt. Lett. 25, 728–730 (2000).
[CrossRef]

Belmonte, M.

Berger, N. K.

N. K. Berger, B. Vodonos, S. Atkins, V. Smulakovsky, A. Bekker, and B. Fischer, “Compression of periodic light pulses using all-optical repetition rate multiplication,” Opt. Commun. 217, 343–349 (2003).
[CrossRef]

Borisov, V. I.

V. P. Minkovich, A. N. Starodumov, V. I. Borisov, V. I. Ledebev, and S. N. Perepechko, “Temporal interference of coherent laser pulses in optical fibers,” Opt. Commun. 192, 231–235 (2001).
[CrossRef]

Braun, A.

T. Yilmaz, C. M. Depriest, A. Braun, J. H. Abeles, and P. J. Delfyett, “Noise in fundamental and harmonic mode-locked semiconductor lasers: experiments and simulations,” IEEE J. Quantum Electron. 39, 838–848 (2003).
[CrossRef]

Chen, L. R.

J. Azaña, P. Kockaert, R. Slavík, L. R. Chen, and S. LaRochelle, “Generation of a 100-GHz optical pulse train by pulse repetition-rate multiplication using superimposed fiber Bragg gratings,” IEEE Photon. Technol. Lett. 15, 413–415 (2003).
[CrossRef]

Chestnut, D. A.

de Matos, C. J. S.

Delfyett, P. J.

T. Yilmaz, C. M. Depriest, A. Braun, J. H. Abeles, and P. J. Delfyett, “Noise in fundamental and harmonic mode-locked semiconductor lasers: experiments and simulations,” IEEE J. Quantum Electron. 39, 838–848 (2003).
[CrossRef]

Depriest, C. M.

T. Yilmaz, C. M. Depriest, A. Braun, J. H. Abeles, and P. J. Delfyett, “Noise in fundamental and harmonic mode-locked semiconductor lasers: experiments and simulations,” IEEE J. Quantum Electron. 39, 838–848 (2003).
[CrossRef]

Fischer, B.

N. K. Berger, B. Vodonos, S. Atkins, V. Smulakovsky, A. Bekker, and B. Fischer, “Compression of periodic light pulses using all-optical repetition rate multiplication,” Opt. Commun. 217, 343–349 (2003).
[CrossRef]

S. Atkins and B. Fischer, “All-optical pulse rate multiplication using fractional Talbot effect and field-to-intensity conversion with cross-gain modulation,” IEEE Photon. Technol. Lett. 15, 132–134 (2003).
[CrossRef]

S. Atkins, B. Vodonos, A. Bekker, and B. Fischer, “Fractional dispersion modes in a pulsed fiber laser,” Opt. Commun. 222, 393–397 (2003).
[CrossRef]

B. Fischer, B. Vodonos, S. Atkins, and A. Bekker, “Dispersion-mode pulsed laser,” Opt. Lett. 25, 728–730 (2000).
[CrossRef]

Grein, M. E.

Haus, H. A.

Ibsen, M.

Ippen, E. P.

Jannson, J.

Jannson, T.

Jiang, L. A.

Kalestynski, A.

A. Kalestynski and B. Smolinska, “Self-restoration of the autoidolon of defective periodic objects,” Opt. Acta 25, 125–134 (1978).
[CrossRef]

Kawanishi, S.

I. Shake, H. Tahara, S. Kawanishi, and M. Saruwatari, “High-repetition-rate optical pulse generation by using chirped optical pulses,” Electron. Lett. 34, 792–793 (1998).
[CrossRef]

Kockaert, P.

J. Azaña, P. Kockaert, R. Slavík, L. R. Chen, and S. LaRochelle, “Generation of a 100-GHz optical pulse train by pulse repetition-rate multiplication using superimposed fiber Bragg gratings,” IEEE Photon. Technol. Lett. 15, 413–415 (2003).
[CrossRef]

Kolner, B. H.

R. P. Scott, C. Langrock, and B. H. Kolner, “High-dynamic-range laser amplitude and phase noise measurement techniques,” IEEE J. Sel. Top. Quantum Electron. 7, 641–655 (2001).
[CrossRef]

Kunimatsu, D.

Kutsuzawa, S.

Langrock, C.

R. P. Scott, C. Langrock, and B. H. Kolner, “High-dynamic-range laser amplitude and phase noise measurement techniques,” IEEE J. Sel. Top. Quantum Electron. 7, 641–655 (2001).
[CrossRef]

Laporta, P.

LaRochelle, S.

J. Azaña, P. Kockaert, R. Slavík, L. R. Chen, and S. LaRochelle, “Generation of a 100-GHz optical pulse train by pulse repetition-rate multiplication using superimposed fiber Bragg gratings,” IEEE Photon. Technol. Lett. 15, 413–415 (2003).
[CrossRef]

Ledebev, V. I.

V. P. Minkovich, A. N. Starodumov, V. I. Borisov, V. I. Ledebev, and S. N. Perepechko, “Temporal interference of coherent laser pulses in optical fibers,” Opt. Commun. 192, 231–235 (2001).
[CrossRef]

Lee, H. L. T.

Longhi, S.

Marano, M.

Matsui, Y.

Minkovich, V. P.

V. P. Minkovich, A. N. Starodumov, V. I. Borisov, V. I. Ledebev, and S. N. Perepechko, “Temporal interference of coherent laser pulses in optical fibers,” Opt. Commun. 192, 231–235 (2001).
[CrossRef]

Muriel, M. A.

Ogawa, Y.

Papoulis, A.

Patorski, K.

K. Patorski, “The self-imaging phenomenon and its applications,” Prog. Opt. 27, 3–74 (1989).

Perepechko, S. N.

V. P. Minkovich, A. N. Starodumov, V. I. Borisov, V. I. Ledebev, and S. N. Perepechko, “Temporal interference of coherent laser pulses in optical fibers,” Opt. Commun. 192, 231–235 (2001).
[CrossRef]

Prunei, V.

Ram, R. J.

Rana, F.

Saruwatari, M.

I. Shake, H. Tahara, S. Kawanishi, and M. Saruwatari, “High-repetition-rate optical pulse generation by using chirped optical pulses,” Electron. Lett. 34, 792–793 (1998).
[CrossRef]

Scott, R. P.

R. P. Scott, C. Langrock, and B. H. Kolner, “High-dynamic-range laser amplitude and phase noise measurement techniques,” IEEE J. Sel. Top. Quantum Electron. 7, 641–655 (2001).
[CrossRef]

Shake, I.

I. Shake, H. Tahara, S. Kawanishi, and M. Saruwatari, “High-repetition-rate optical pulse generation by using chirped optical pulses,” Electron. Lett. 34, 792–793 (1998).
[CrossRef]

Slavík, R.

J. Azaña, P. Kockaert, R. Slavík, L. R. Chen, and S. LaRochelle, “Generation of a 100-GHz optical pulse train by pulse repetition-rate multiplication using superimposed fiber Bragg gratings,” IEEE Photon. Technol. Lett. 15, 413–415 (2003).
[CrossRef]

Smolinska, B.

A. Kalestynski and B. Smolinska, “Self-restoration of the autoidolon of defective periodic objects,” Opt. Acta 25, 125–134 (1978).
[CrossRef]

Smulakovsky, V.

N. K. Berger, B. Vodonos, S. Atkins, V. Smulakovsky, A. Bekker, and B. Fischer, “Compression of periodic light pulses using all-optical repetition rate multiplication,” Opt. Commun. 217, 343–349 (2003).
[CrossRef]

Starodumov, A. N.

V. P. Minkovich, A. N. Starodumov, V. I. Borisov, V. I. Ledebev, and S. N. Perepechko, “Temporal interference of coherent laser pulses in optical fibers,” Opt. Commun. 192, 231–235 (2001).
[CrossRef]

Svelto, O.

Tahara, H.

I. Shake, H. Tahara, S. Kawanishi, and M. Saruwatari, “High-repetition-rate optical pulse generation by using chirped optical pulses,” Electron. Lett. 34, 792–793 (1998).
[CrossRef]

Taylor, J. R.

Vodonos, B.

N. K. Berger, B. Vodonos, S. Atkins, V. Smulakovsky, A. Bekker, and B. Fischer, “Compression of periodic light pulses using all-optical repetition rate multiplication,” Opt. Commun. 217, 343–349 (2003).
[CrossRef]

S. Atkins, B. Vodonos, A. Bekker, and B. Fischer, “Fractional dispersion modes in a pulsed fiber laser,” Opt. Commun. 222, 393–397 (2003).
[CrossRef]

B. Fischer, B. Vodonos, S. Atkins, and A. Bekker, “Dispersion-mode pulsed laser,” Opt. Lett. 25, 728–730 (2000).
[CrossRef]

Yilmaz, T.

T. Yilmaz, C. M. Depriest, A. Braun, J. H. Abeles, and P. J. Delfyett, “Noise in fundamental and harmonic mode-locked semiconductor lasers: experiments and simulations,” IEEE J. Quantum Electron. 39, 838–848 (2003).
[CrossRef]

Zervas, M. N.

Electron. Lett. (1)

I. Shake, H. Tahara, S. Kawanishi, and M. Saruwatari, “High-repetition-rate optical pulse generation by using chirped optical pulses,” Electron. Lett. 34, 792–793 (1998).
[CrossRef]

IEEE J. Quantum Electron. (1)

T. Yilmaz, C. M. Depriest, A. Braun, J. H. Abeles, and P. J. Delfyett, “Noise in fundamental and harmonic mode-locked semiconductor lasers: experiments and simulations,” IEEE J. Quantum Electron. 39, 838–848 (2003).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

R. P. Scott, C. Langrock, and B. H. Kolner, “High-dynamic-range laser amplitude and phase noise measurement techniques,” IEEE J. Sel. Top. Quantum Electron. 7, 641–655 (2001).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

J. Azaña, P. Kockaert, R. Slavík, L. R. Chen, and S. LaRochelle, “Generation of a 100-GHz optical pulse train by pulse repetition-rate multiplication using superimposed fiber Bragg gratings,” IEEE Photon. Technol. Lett. 15, 413–415 (2003).
[CrossRef]

S. Atkins and B. Fischer, “All-optical pulse rate multiplication using fractional Talbot effect and field-to-intensity conversion with cross-gain modulation,” IEEE Photon. Technol. Lett. 15, 132–134 (2003).
[CrossRef]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. (1)

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

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

Opt. Acta (1)

A. Kalestynski and B. Smolinska, “Self-restoration of the autoidolon of defective periodic objects,” Opt. Acta 25, 125–134 (1978).
[CrossRef]

Opt. Commun. (3)

S. Atkins, B. Vodonos, A. Bekker, and B. Fischer, “Fractional dispersion modes in a pulsed fiber laser,” Opt. Commun. 222, 393–397 (2003).
[CrossRef]

V. P. Minkovich, A. N. Starodumov, V. I. Borisov, V. I. Ledebev, and S. N. Perepechko, “Temporal interference of coherent laser pulses in optical fibers,” Opt. Commun. 192, 231–235 (2001).
[CrossRef]

N. K. Berger, B. Vodonos, S. Atkins, V. Smulakovsky, A. Bekker, and B. Fischer, “Compression of periodic light pulses using all-optical repetition rate multiplication,” Opt. Commun. 217, 343–349 (2003).
[CrossRef]

Opt. Lett. (5)

Prog. Opt. (1)

K. Patorski, “The self-imaging phenomenon and its applications,” Prog. Opt. 27, 3–74 (1989).

Other (3)

C. R. Fernández-Pousa, F. Mateos, L. Chantada, M. T. Flores-Arias, C. Bao, M. V. Pérez, and C. Gómez-Reino, “Random jitter smoothing by Talbot effect. I. Variance,” J. Opt. Soc. Am. B (to be published).

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, Boston, 2001).

A. Papoulis, Probability, Random Variables and Stochastic Processes, 2nd ed. (McGraw-Hill, New York, 1984).

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Figures (5)

Fig. 1
Fig. 1

Simulation of the form of the pulse envelope of 20-unit intervals of a 10-GHz, p=0.5 random train of unchirped, 12-ps rms Gaussian pulses before (above) and after (below) its pass through a Talbot device with γ/α=1.

Fig. 2
Fig. 2

Power spectrum of a detected 10-GHz, p=0.5 random train of unchirped pulses with 6.5-ps rms width after its pass through a Talbot filter with γ/α=3/2. The upper part of the plot shows (continuous curve) the power spectrum of the detected output train, (dashed curve) the power spectrum of the detected input train, and (dotted curve) the mean signal power spectrum of both input and output trains. The lower part shows analytical approximations to the noise power spectrum: (continuous curve) output noise; (dashed curve) input noise.

Fig. 3
Fig. 3

Same as in Fig. 2, but with chirped pulses with C=+1.

Fig. 4
Fig. 4

Power spectrum of a detected 10-GHz, p=0.9 random train of unchirped pulses with 10-ps rms width after its pass through a Talbot filter with γ/α=1. The upper part of the plot shows (continuous curve) the power spectrum of the detected output train, (dashed curve) the power spectrum of the detected input train, (dotted curve) the mean signal power spectrum of both input and output trains. The lower part shows analytical approximations to the noise power spectrum: (continuous curve) output noise; (dashed curve) input noise.

Fig. 5
Fig. 5

Same as in Fig. 2, but with chirped pulses with C=-1.

Equations (26)

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

xz=-iβ222xt2,
hξ(t-t)=(-2πiξ)-1/2 exp[-i(t-t)2/2ξ],
x0(t)=kbkf(t-kt0)=kbk exp[-(t-kt0)2(1+iC)/2tp2].
xξ(t)=tp(tp2+Cξ-iξ)-1/2kbk×exp[-(t-kt0)2(1+iC)/2(tp2+Cξ-iξ)],
Xˆξ(ω)=Hξ(ω)Xˆ0(ω)=tp2π1+iCexp(-ρω2/2)kbk exp(-iωkt0),
ρ=tp21+C2-iξ-iCtp21+C2.
Sξ(ω)=-+dτ exp(-iωτ)×limT12T-TTdtyξ(t+τ)yξ(t),
yξ(t)yξ(t)= dω12πdω22πexp[i(ω1t-ω2t)]×Yˆξ(ω1)Yˆξ(ω2)*.
Yˆξ(ω)=-+dt exp(-iωt)yξ(t)=-+dω2πXˆξ(ω+ω)Xˆξ(ω)*.
Yˆξ(ω)=Fˆ(ω)kBk(ω)exp(-iωkt0),
Bk(ω)=bksbk-shs(ω)exp(iωst0/2).
hs(ω)=expC2tp2ω24(1+C2)gs(ω)=expC2tp2ω24(1+C2)×exp-1+C24tp2ξω+Ctp2ω1+C2-st02,
Bk(ω1)Bm(ω2)*=Bk(ω1)Bm(ω2)*+δkmΩ(ω1, ω2)+Δk-m(ω1, ω2),
t0Sξ(N)(ω)=|Fˆ(ω)|2×Ω(ω, ω)+sΔs(ω, ω)exp(-iωst0).
bibjbkbl=p4+p2σb2×(δij+δik+δil+δjk+δjl+δkl)+σb2a(p)(δijδjk+δijδjl+δikδkl+δjkδkl)+σb4(δijδkl+δikδjl+δilδjk)+σb2b(p)δijδjkδkl,
Ω(ω, ω)=σb2p2|Z|2+2ah0(ω)Re(Z)+σb2shs2(ω)+bh02(ω),
sΔs(ω, ω)exp(-iωst0)=σb2p2(Z2+|Z|2+Z*2)+2ah0(ω)Re(Z)+σb2shs(ω)h-s(ω),
t0Sξ(N)(ω)=σb2|Fˆ(ω)|2h0(ω)+2ps>0[hs(ω)+h-s(ω)]cos(ωst0/2)2+σb2s>0[hs(ω)+h-s(ω)]2.
hs(ω)ξ=0=expCt0ωs2exp-(1+C2)s2t024tp2.
t0Sξ(N)(ω)σb2J(ω)g02(ω)+s0[4p2 cos2(ωst0/2)+σb2]gs2(ω),
ω=st0ξ+Ctp21+C2ωfsαγ1-sign(ξ)αγ2πC1+C2tp2t02=ωfsαγ+Δpeak(s)ω,
ΔWω=4.3tp1+C2ξ+Ctp21+C2-14.3ωfαγtpt011+C2,
C=-2.9 sign(ξ)ωfsΔpeak(s)ω(ΔWω)2.
ω=mωf/s,
ΔMω=ωf/s.
4.3tp>γαt0s.

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