Abstract

The power spectral density of the intensity of jittery trains after an integer temporal Talbot dispersive line is computed in the small-signal approximation. The influence in the spectrum of the optical linewidth and chirp of the Gaussian pulses of the train and also of different pulse-to-pulse timing jitter correlations is addressed. Before entering the Talbot dispersive line, timing jitter produces noise sidebands around the harmonics of the train. The temporal Talbot effect adds a multiplicative factor to the noise spectral density that depends on the characteristics of both the pulses and the dispersive line but not on the pulse-to-pulse correlation or the value of the timing jitter’s standard deviation. The structure of this multiplicative term is peaked, resulting in narrowband noise patterns in specific locations of the spectrum and, in particular, around the harmonics of the train. Thus the temporal Talbot effect provides a dispersive mechanism for noise filtering. The bandwidth of the dispersion-induced noise peaks is ∼1 order of magnitude below the repetition-rate frequency.

© 2005 Optical Society of America

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2004 (5)

J. Fatome, S. Pitois, and G. Millot, "Influence of third-order dispersion on the temporal Talbot effect," Opt. Commun. 234, 29-34 (2004).
[CrossRef]

J. T. Mok and B. J. Eggleton, "Impact of group delay ripple on repetition-rate multiplication through Talbot self-imaging effect," Opt. Commun. 232, 167-178 (2004).
[CrossRef]

N. K. Berger, B. Levit, A. Bekker, and B. Fischer, "Compression of periodic optical pulses using temporal fractional Talbot effect," IEEE Photonics Technol. Lett. 16, 1855-1857 (2004).
[CrossRef]

C. R. Fernández-Pousa, F. Mateos, L. Chantada, M. T. Flores-Arias, C. Bao, M. V. Pérez, and C. Gómez-Reino, "Broadband noise filtering in random sequences of coherent pulses using the temporal Talbot effect," J. Opt. Soc. Am. B 21, 914-922 (2004).
[CrossRef]

C. R. Fernández-Pousa, F. Mateos, L. Chantada, M. T. Flores-Arias, C. Bao, M. V. Pérez, and C. Gómez-Reino, "Timing jitter smoothing by Talbot effect. I. Variance," J. Opt. Soc. Am. B 21, 1170-1177 (2004).
[CrossRef]

2003 (6)

T. Yilmaz, C. M. DePriest, P. J. Defyett, S. Etemad, A. Braun, and J. H. Abeles, "Supermode suppression to below −130 dBc/Hz in a 10 GHz harmonically mode-locked external sigma cavity semiconductor laser," Opt. Express 11, 1090-1095 (2003), http://www.opticsexpress.org.
[CrossRef] [PubMed]

J. Capmany, D. Pastor, S. Sales, and M. A. Muriel, "Pulse distortion in optical fibers and waveguides with arbitrary chromatic dispersion," J. Opt. Soc. Am. B 20, 2523-2533 (2003).
[CrossRef]

J. Azaña, P. Kockaert, R. Slavík, L. R. Chen, and S. LaRochelle, "Generation of a 100-GHz optical pulses train by pulse repetition-rate multiplication using superimposed fiber Bragg gratings," IEEE Photonics 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 Photonics 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 modelocked semiconductor lasers: experiments and simulations," IEEE J. Quantum Electron. 39, 838-848 (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]

2002 (1)

2001 (2)

J. Azaña and M. A. Muriel, "Temporal self-imaging effects: theory and application for multiplying pulse repetition rates," IEEE J. Sel. Top. Quantum Electron. 7, 728-744 (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]

2000 (1)

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]

1996 (1)

M. V. Berry and S. Klein, "Integer, fractional and fractal Talbot effects," J. Mod. Opt. 43, 2139-2164 (1996).
[CrossRef]

1990 (1)

A. S. Hou, R. S. Tucker, and G. Eisentein, "Pulse compression of an actively modelocked diode laser using linear dispersion in fiber," IEEE Photonics Technol. Lett. 2, 322-324 (1990).
[CrossRef]

1986 (1)

D. von der Linde, "Characterization of the noise in continuously operating mode-locked lasers," Appl. Phys. B: Photophys. Laser Chem. 39, 201-217 (1986).
[CrossRef]

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, P. J. Defyett, S. Etemad, A. Braun, and J. H. Abeles, "Supermode suppression to below −130 dBc/Hz in a 10 GHz harmonically mode-locked external sigma cavity semiconductor laser," Opt. Express 11, 1090-1095 (2003), http://www.opticsexpress.org.
[CrossRef] [PubMed]

T. Yilmaz, C. M. Depriest, A. Braun, J. H. Abeles, and P. J. Delfyett, "Noise in fundamental and harmonic modelocked 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 Photonics Technol. Lett. 15, 132-134 (2003).
[CrossRef]

Atkins, S.

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]

Azaña, J.

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

Azaña , J.

J. Azaña and M. A. Muriel, "Temporal self-imaging effects: theory and application for multiplying pulse repetition rates," IEEE J. Sel. Top. Quantum Electron. 7, 728-744 (2001).
[CrossRef]

J. Azaña and M. A. Muriel, "Technique for multiplying the repetition rates of periodic trains of pulses by means of a temporal self-imaging effect in chirped fiber gratings," Opt. Lett. 24, 1672-1674 (1999).
[CrossRef]

Bao, C.

Bekker, A.

N. K. Berger, B. Levit, A. Bekker, and B. Fischer, "Compression of periodic optical pulses using temporal fractional Talbot effect," IEEE Photonics Technol. Lett. 16, 1855-1857 (2004).
[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]

Belmonte, M.

Berger, N. K.

N. K. Berger, B. Levit, A. Bekker, and B. Fischer, "Compression of periodic optical pulses using temporal fractional Talbot effect," IEEE Photonics Technol. Lett. 16, 1855-1857 (2004).
[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]

Berry , M. V.

M. V. Berry and S. Klein, "Integer, fractional and fractal Talbot effects," J. Mod. Opt. 43, 2139-2164 (1996).
[CrossRef]

Braun, A.

T. Yilmaz, C. M. DePriest, P. J. Defyett, S. Etemad, A. Braun, and J. H. Abeles, "Supermode suppression to below −130 dBc/Hz in a 10 GHz harmonically mode-locked external sigma cavity semiconductor laser," Opt. Express 11, 1090-1095 (2003), http://www.opticsexpress.org.
[CrossRef] [PubMed]

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

Capmany, J.

Chantada, L.

Chen, L. R.

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

Defyett, P. J.

Delfyett, P. J.

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

T. Yilmaz, C. M. DePriest, P. J. Defyett, S. Etemad, A. Braun, and J. H. Abeles, "Supermode suppression to below −130 dBc/Hz in a 10 GHz harmonically mode-locked external sigma cavity semiconductor laser," Opt. Express 11, 1090-1095 (2003), http://www.opticsexpress.org.
[CrossRef] [PubMed]

Eggleton, B. J.

J. T. Mok and B. J. Eggleton, "Impact of group delay ripple on repetition-rate multiplication through Talbot self-imaging effect," Opt. Commun. 232, 167-178 (2004).
[CrossRef]

Eisentein, G.

A. S. Hou, R. S. Tucker, and G. Eisentein, "Pulse compression of an actively modelocked diode laser using linear dispersion in fiber," IEEE Photonics Technol. Lett. 2, 322-324 (1990).
[CrossRef]

Etemad, S.

Fatome, J.

J. Fatome, S. Pitois, and G. Millot, "Influence of third-order dispersion on the temporal Talbot effect," Opt. Commun. 234, 29-34 (2004).
[CrossRef]

Fernández-Pousa, C. R.

Fischer, B.

N. K. Berger, B. Levit, A. Bekker, and B. Fischer, "Compression of periodic optical pulses using temporal fractional Talbot effect," IEEE Photonics Technol. Lett. 16, 1855-1857 (2004).
[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 Photonics 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]

Flores-Arias, M. T.

Gómez-Reino, C.

Grein, M. E.

Haus, H. A.

Hou, A. S.

A. S. Hou, R. S. Tucker, and G. Eisentein, "Pulse compression of an actively modelocked diode laser using linear dispersion in fiber," IEEE Photonics Technol. Lett. 2, 322-324 (1990).
[CrossRef]

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]

Klein, S.

M. V. Berry and S. Klein, "Integer, fractional and fractal Talbot effects," J. Mod. Opt. 43, 2139-2164 (1996).
[CrossRef]

Kockaert, P.

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

Lee, H. L. T.

Levit, B.

N. K. Berger, B. Levit, A. Bekker, and B. Fischer, "Compression of periodic optical pulses using temporal fractional Talbot effect," IEEE Photonics Technol. Lett. 16, 1855-1857 (2004).
[CrossRef]

Longhi, S.

Marano, M.

Mateos, F.

Matsui, Y.

Millot, G.

J. Fatome, S. Pitois, and G. Millot, "Influence of third-order dispersion on the temporal Talbot effect," Opt. Commun. 234, 29-34 (2004).
[CrossRef]

Mok , J. T.

J. T. Mok and B. J. Eggleton, "Impact of group delay ripple on repetition-rate multiplication through Talbot self-imaging effect," Opt. Commun. 232, 167-178 (2004).
[CrossRef]

Muriel, M. A.

Ogawa, Y.

Pastor, D.

Pérez, M. V.

Pitois, S.

J. Fatome, S. Pitois, and G. Millot, "Influence of third-order dispersion on the temporal Talbot effect," Opt. Commun. 234, 29-34 (2004).
[CrossRef]

Prunei, V.

Ram, R. J.

Rana, F.

Sales, S.

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 pulses train by pulse repetition-rate multiplication using superimposed fiber Bragg gratings," IEEE Photonics 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]

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]

Tucker, R. S.

A. S. Hou, R. S. Tucker, and G. Eisentein, "Pulse compression of an actively modelocked diode laser using linear dispersion in fiber," IEEE Photonics Technol. Lett. 2, 322-324 (1990).
[CrossRef]

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]

von der Linde, D.

D. von der Linde, "Characterization of the noise in continuously operating mode-locked lasers," Appl. Phys. B: Photophys. Laser Chem. 39, 201-217 (1986).
[CrossRef]

Yilmaz, T.

T. Yilmaz, C. M. DePriest, P. J. Defyett, S. Etemad, A. Braun, and J. H. Abeles, "Supermode suppression to below −130 dBc/Hz in a 10 GHz harmonically mode-locked external sigma cavity semiconductor laser," Opt. Express 11, 1090-1095 (2003), http://www.opticsexpress.org.
[CrossRef] [PubMed]

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

Zervas, M. N.

Appl. Phys. B: Photophys. Laser Chem. (1)

D. von der Linde, "Characterization of the noise in continuously operating mode-locked lasers," Appl. Phys. B: Photophys. Laser Chem. 39, 201-217 (1986).
[CrossRef]

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 modelocked semiconductor lasers: experiments and simulations," IEEE J. Quantum Electron. 39, 838-848 (2003).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (2)

J. Azaña and M. A. Muriel, "Temporal self-imaging effects: theory and application for multiplying pulse repetition rates," IEEE J. Sel. Top. Quantum Electron. 7, 728-744 (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]

IEEE Photonics Technol. Lett. (4)

N. K. Berger, B. Levit, A. Bekker, and B. Fischer, "Compression of periodic optical pulses using temporal fractional Talbot effect," IEEE Photonics Technol. Lett. 16, 1855-1857 (2004).
[CrossRef]

J. Azaña, P. Kockaert, R. Slavík, L. R. Chen, and S. LaRochelle, "Generation of a 100-GHz optical pulses train by pulse repetition-rate multiplication using superimposed fiber Bragg gratings," IEEE Photonics 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 Photonics Technol. Lett. 15, 132-134 (2003).
[CrossRef]

A. S. Hou, R. S. Tucker, and G. Eisentein, "Pulse compression of an actively modelocked diode laser using linear dispersion in fiber," IEEE Photonics Technol. Lett. 2, 322-324 (1990).
[CrossRef]

J. Lightwave Technol. (1)

J. Mod. Opt. (1)

M. V. Berry and S. Klein, "Integer, fractional and fractal Talbot effects," J. Mod. Opt. 43, 2139-2164 (1996).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (4)

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)

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. Fatome, S. Pitois, and G. Millot, "Influence of third-order dispersion on the temporal Talbot effect," Opt. Commun. 234, 29-34 (2004).
[CrossRef]

J. T. Mok and B. J. Eggleton, "Impact of group delay ripple on repetition-rate multiplication through Talbot self-imaging effect," Opt. Commun. 232, 167-178 (2004).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Other (4)

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

J. G. Proakis and D. G. Manolakis, Digital Signal Processing , 3rd ed. (Prentice-Hall, Upper Saddle River, N.J., 1996).

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

K. Patorski, "The self-imaging phenomenon and its applications," in Progress in Optics , E. Wolf, ed. (Elsevier, Amsterdam, 1989), Vol. XXVII, pp. 1-108.

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Figures (11)

Fig. 1
Fig. 1

Intensity power spectral density (PSD) of a jittery train of pulses (σj=100 fs, η=0.7) before and after a temporal Talbot device with index γ=1. Pulse width, tp=10 ps; chirp, C=0; repetition rate, 10 GHz; Q=1.59. Above, numerical power spectral densities for the input and output trains (continuous curves) and numerical power spectral density of the mean signal (dotted curve). Below, analytical noise power spectral densities for the input (dashed curve) and output (continuous curve) trains.

Fig. 2
Fig. 2

Contributions to the right-hand offset noise spectrum of the train in Fig. 1 near the fifth harmonic (n=h=5). Left, dashed curve, jitter power spectrum Φ(ω); dotted curve, spectral window function Gn=5(ω); continuous curve, modulation function Mn=5(ω). Right, dashed curve, jitter power spectrum Φ(ω); dotted curve, spectral window function Gn=5(ω); continuous curve, product Φ(ω)Gn=5(ω)Mn=5(ω). PSD, power spectral density.

Fig. 3
Fig. 3

Frequency offsets corresponding to the location of the zeros of the modulation function for the fifth harmonic (h=5) and the first integer Talbot replica (γ=1). Chirp: C=0, continuous curves; C=1, dashed curves; C=-1, dotted curves.

Fig. 4
Fig. 4

Intensity power spectral density (PSD) of a jittery train of pulses (σj=100 fs, η=0.7) before and after a temporal Talbot device with index with γ=1. Pulse width, tp=10 ps, chirp, C=1; repetition rate, 10 GHz; Q=2.25. Above, numerical power spectral densities for the input and output trains (continuous curves) and numerical power spectral density of the mean signal (dotted curve). Below, analytical noise power spectral densities for the input (dashed curve) and output (continuous curve) trains.

Fig. 5
Fig. 5

Left and right offset noise power spectra of the train of Fig. 4 around the fifth harmonic (n=h=5). Dashed curves, analytical input noise power spectral density (PSD). Continuous curves, analytical output noise power spectral density. Numerical data extracted from Fig. 4 superimposed: dots, input noise; crosses, output noise.

Fig. 6
Fig. 6

Intensity power spectral density (PSD) of a jittery train of pulses (σj=100 fs, η=0.5) before and after a temporal Talbot device with index γ=2. Pulse width, tp=10 ps; chirp, C=1; repetition rate, 10 GHz; Q=3.18. Above, numerical power-spectral densities for the input and output trains (continuous curves), and numerical power spectral density of the mean signal (dotted curve). Below, analytical noise power spectral densities for the input (dashed curve) and output (continuous curve) trains.

Fig. 7
Fig. 7

Left and right offset noise power spectra of the train of Fig. 6 around the fifth harmonic (h=5, n=10). Dashed curves, analytical input noise power spectral density (PSD). Continuous curves, analytical output noise power spectral density. Numerical data extracted from Fig. 6 superimposed: dots, input noise; crosses, output noise.

Fig. 8
Fig. 8

Ratio ρ(h) for the fifth harmonic h=5 (fifth spectral window, n=5) and for the first integer Talbot replica γ=1 as a function of correlation parameter η for three values of dispersion parameter Q: continuous curves, Q=2; dashed curves, Q=5; dotted curves Q=7.

Fig. 9
Fig. 9

Ratio ρ(h) for the first integer Talbot replica γ=1 and for jittery trains with correlation parameter η=0.7 as a function of dispersion parameter Q for three harmonics h: continuous curves, h=1; dashed curves, h=5; dotted curves, h=7.

Fig. 10
Fig. 10

Ratio ρ(h) for the fifth harmonic h=5 (tenth spectral window, n=10) and for the second integer Talbot replica γ=2 as a function of correlation parameter η for three values of dispersion parameter Q: continuous curves, Q=2; dashed curves, Q=5; dotted curves, Q=7.

Fig. 11
Fig. 11

Ratio ρ(h) for the second integer Talbot replica γ=2 and for jittery trains with correlation parameter η=0.7 as a function of dispersion parameter Q for three harmonics h: continuous curves, h=1; dashed curves, h=5; dotted curves, h=7.

Equations (30)

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Ez=-i β22 2Et2,
E0(t)=kf(t-kt0-ak)=k exp-(t-kt0-ak)2(1+iC)/2tp2.
Eˆξ(ω)=Hξ(ω)Eˆ0(ω)=tp[2π/(1+iC)]1/2 exp(-ρω2/2)×k exp(iωkt0+iωak),
|ξ|=t022π γα,
tp=tp1+(C+Δξ/τ2)21+C21/2.
Sξ(ω)=-+dτ exp(iωτ)limT 12T -TTdtIξ(t+τ)Iξ(t).
t0Sξ(N)(ω)ω2J(ω)Φ(ω)nGn(ω)Mn(ω).
ω2J(ω)=πtp2ω2 exp(-ω2τ2/2).
Φ(ω)=kR(k)exp(iωkt0).
Gn(ω)=exp[-(ωξ+ωCτ2-nt0)2/(2τ2)].
Mn(ω)=[cos(ωnt0/2)-(ξ+Cτ2-nt0/ω)×sin(ωnt0/2)/τ2]2.
t0S0(N)(ω)ω2|F(ω)|2Φ(ω),
ω(n)=nt0|ξ+Cτ2|ωf nγ 1-sign(ξ) 2πCγ τ2t02.
ΔWω=22log10 e τ|ξ+Cτ2|4.3 τ|ξ|=4.3t0Q.
Q=γ2π t0τ.
cot(πhγx)=sign(ξ)C+2πQ2γ xx+h.
am=ηam-1+m.
Φ(ω)=σj2 1-η21-2η cos(ωt0)+η2.
(h-1/2)ωf(h+1/2)ωfdωS0(N)(ω)
t0-1ωf2h2|F(hωf)|2(h-1/2)ωf(h+1/2)ωfdωΦ(ω)=t0-1ωf3h2|F(hωf)|2σj2.
(h-1/2)ωf(h+1/2)ωfdωSξ(N)(ω)
t0-1ωf2h2J(hωf)×(h-1/2)ωf(h+1/2)ωfdωΦ(ω)Gn=hγ(ω)Mn=hγ(ω).
ρ(h)expn2C2/(2Q2)Ln=hγ(η, Q, C),
Ln=hγ(η, Q, C)=-1/21/2dxΦ¯(x)Gn=hγ(x)Mn=hγ(x).
Iξ(t)Iξ(t)= dω12π dω22π exp[-i(ω1t-ω2t)]×Iˆξ(ω1)Iˆξ(ω2)*,
Iˆξ(ω)=dt exp(iωt)Iξ(t)= dω2πEˆξ(ω+ω)Eˆξ(ω)*.
Iˆξ(ω1)Iˆξ(ω2)*=Iˆξ(ω1)Iˆξ(ω2)*+2πK(ω1, ω2)×sδ(ω1-ω2-2πs/t0).
Iˆξ(ω)=tpπ exp(-ρω2/2)pk exp(iωkt0+τ2μp-k2+iμp-k*ak+iμp-kap)×exp[-(ap-ak)2/(4τ2)].
Iˆξ(ω1)Iˆξ*(ω2)=Iˆξ(ω1)Iˆξ*(ω2)+πtp2 exp(-ρω12/2-ρ*ω22/2)k exp[i(ω1-ω2)kt0]×rsq exp[iω2qt0]×exp[τ2(μr2+μs*2)]×[μs*μrR(q-s+r)+μr*μs*R(q-s)+μsμr*R(q)+μrμsR(q+r)],
t0Sξ(N)(ω)J(ω)nGn(ω)q exp(iωqt0)[2|μn|2R(q)+μn*2R(q-n)+μn2R(q+n)],

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