Abstract

The power spectral density of the intensity of coherent Gaussian pulse trains suffering timing jitter after a dispersive line with arbitrary first- (β2) and second-order (β3) dispersion is computed in the small-signal approximation. Due to timing jitter noise, the initial radio-frequency spectrum shows noise bands whose bandwidth and position depend, respectively, on the jitter’s standard deviation and on the jitter’s pulse-to-pulse correlation. After setting the accumulated first-order dispersion to Talbot conditions, it is shown that the influence on the noise spectrum is a multiplicative factor with a multiple-bandpass structure. This factor depends on both the dispersive characteristics of the line and the pulse parameters, but not on the timing jitter’s correlation properties, and represents the filtering mechanism responsible for Talbot repetition-rate multiplication. It is shown that the integer or fractional temporal Talbot effect does not worsen the timing properties of the initial train. In addition, and depending on the type of jitter correlation, the pulse width, and the total dispersion, it is shown that the temporal Talbot effect may lead to significant jitter reduction. The theory is exemplified by use of simulations. The applicability of the model to practical situations is also analyzed.

© 2008 Optical Society of America

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    [CrossRef] [PubMed]

2008

D. Pudo, C. R. Fernández-Pousa, and L. R. Chen, “Timing jitter transfer function in the temporal Talbot effect,” IEEE Photon. Technol. Lett. 20, 496-498 (2008).
[CrossRef]

2007

2006

L. Chantada, C. R. Fernández-Pousa, and C. Gómez-Reino, “Spectral analysis of the temporal self-imaging phenomenon in fiber dispersive lines,” J. Lightwave Technol. 24, 2015-2025 (2006).
[CrossRef]

L. Chantada, C. R. Fernández-Pousa, and C. Gómez-Reino, “Theory of the partially coherent temporal Talbot effect,” Opt. Commun. 266, 393-398 (2006).
[CrossRef]

D. Pudo and L. R. Chen, “Estimating intensity fluctuations in high repetition rate pulse trains generated using the temporal Talbot effect,” IEEE Photon. Technol. Lett. 18, 658-660 (2006).
[CrossRef]

2005

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. II. Intensity spectrum,” J. Opt. Soc. Am. B 22, 753-763 (2005).
[CrossRef]

J. Lancis, J. Caraquitena, P. Andrés, and M. A. Muriel, “Temporal self-imaging effect for chirped laser pulse sequences: repetition rate and duty cycle tunability,” Opt. Commun. 253, 156-163 (2005).
[CrossRef]

2004

2003

2002

2001

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]

2000

1998

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]

S. Arahira, S. Kutsuzawa, Y. Matsui, D. Kunimatsu, and Y. Ogawa, “Generation of synchronized subterahertz optical pulse trains by repetition-frequency multiplication of subharmonic synchronous mode-locked semiconductor laser diode using fiber dispersion,” IEEE Photon. Technol. Lett. 10, 209-211 (1998).
[CrossRef]

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]

1993

1989

K. Patorski, “The self-imaging phenomenon and its application,” in Progress in Optics, E. Wolf, ed. (Elsevier Science, 1989), Vol. XXVII, pp. 1-108.
[CrossRef]

1986

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

1984

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

1981

1975

W. Gardner and L. Franks, “Characterization of cyclostationary random signal processes,” IEEE Trans. Inf. Theory 21, 4-14 (1975).
[CrossRef]

Abeles, J. H.

T. Yilmaz, C. M. DePriest, A. Braun, J. H. Abeles, and P. J. Delfyett, Jr., “Noise in fundamental and harmonic modelocked semiconductor lasers: experiments and simulations,” IEEE J. Quantum Electron. 39, 838-849 (2003).
[CrossRef]

T. Yilmaz, C. M. DePriest, P. J. Delfyett, Jr., 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).
[CrossRef] [PubMed]

Agogliati, B.

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).

Andrekson, P. A.

Andrés, P.

J. Lancis, J. Caraquitena, P. Andrés, and M. A. Muriel, “Temporal self-imaging effect for chirped laser pulse sequences: repetition rate and duty cycle tunability,” Opt. Commun. 253, 156-163 (2005).
[CrossRef]

Arahira, S.

S. Arahira, S. Kutsuzawa, Y. Matsui, D. Kunimatsu, and Y. Ogawa, “Generation of synchronized subterahertz optical pulse trains by repetition-frequency multiplication of subharmonic synchronous mode-locked semiconductor laser diode using fiber dispersion,” IEEE Photon. Technol. Lett. 10, 209-211 (1998).
[CrossRef]

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]

Arcangeli, L.

Azaña, J.

Bao, C.

Belmonte, M.

Braun, A.

T. Yilmaz, C. M. DePriest, P. J. Delfyett, Jr., 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).
[CrossRef] [PubMed]

T. Yilmaz, C. M. DePriest, A. Braun, J. H. Abeles, and P. J. Delfyett, Jr., “Noise in fundamental and harmonic modelocked semiconductor lasers: experiments and simulations,” IEEE J. Quantum Electron. 39, 838-849 (2003).
[CrossRef]

Caraquitena, J.

J. Lancis, J. Caraquitena, P. Andrés, and M. A. Muriel, “Temporal self-imaging effect for chirped laser pulse sequences: repetition rate and duty cycle tunability,” Opt. Commun. 253, 156-163 (2005).
[CrossRef]

Chantada, L.

Chen, L. R.

Delfyett, P. J.

Depa, M.

DePriest, C. M.

T. Yilmaz, C. M. DePriest, P. J. Delfyett, Jr., 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).
[CrossRef] [PubMed]

T. Yilmaz, C. M. DePriest, A. Braun, J. H. Abeles, and P. J. Delfyett, Jr., “Noise in fundamental and harmonic modelocked semiconductor lasers: experiments and simulations,” IEEE J. Quantum Electron. 39, 838-849 (2003).
[CrossRef]

Duchesne, D.

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.

Flores-Arias, M. T.

Franks, L.

W. Gardner and L. Franks, “Characterization of cyclostationary random signal processes,” IEEE Trans. Inf. Theory 21, 4-14 (1975).
[CrossRef]

Friberg, A. T.

Gardner, W.

W. Gardner and L. Franks, “Characterization of cyclostationary random signal processes,” IEEE Trans. Inf. Theory 21, 4-14 (1975).
[CrossRef]

Gee, S.

Gómez-Reino, C.

Grein, M. E.

Harvey, G. T.

Haus, H. A.

Ibsen, M.

Ippen, E. P.

Jannson, J.

Jannson, T.

Jiang, L. A.

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]

Kunimatsu, D.

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]

S. Arahira, S. Kutsuzawa, Y. Matsui, D. Kunimatsu, and Y. Ogawa, “Generation of synchronized subterahertz optical pulse trains by repetition-frequency multiplication of subharmonic synchronous mode-locked semiconductor laser diode using fiber dispersion,” IEEE Photon. Technol. Lett. 10, 209-211 (1998).
[CrossRef]

Kutsuzawa, S.

S. Arahira, S. Kutsuzawa, Y. Matsui, D. Kunimatsu, and Y. Ogawa, “Generation of synchronized subterahertz optical pulse trains by repetition-frequency multiplication of subharmonic synchronous mode-locked semiconductor laser diode using fiber dispersion,” IEEE Photon. Technol. Lett. 10, 209-211 (1998).
[CrossRef]

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]

Lajunen, H.

Lancis, J.

J. Lancis, J. Caraquitena, P. Andrés, and M. A. Muriel, “Temporal self-imaging effect for chirped laser pulse sequences: repetition rate and duty cycle tunability,” Opt. Commun. 253, 156-163 (2005).
[CrossRef]

Laporta, P.

Lee, H. L. T.

Longhi, S.

Marano, M.

Mateos, F.

Matsui, Y.

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]

S. Arahira, S. Kutsuzawa, Y. Matsui, D. Kunimatsu, and Y. Ogawa, “Generation of synchronized subterahertz optical pulse trains by repetition-frequency multiplication of subharmonic synchronous mode-locked semiconductor laser diode using fiber dispersion,” IEEE Photon. Technol. Lett. 10, 209-211 (1998).
[CrossRef]

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]

Mollenauer, L. F.

Morandotti, R.

Muriel, M. A.

J. Lancis, J. Caraquitena, P. Andrés, and M. A. Muriel, “Temporal self-imaging effect for chirped laser pulse sequences: repetition rate and duty cycle tunability,” Opt. Commun. 253, 156-163 (2005).
[CrossRef]

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]

Ogawa, Y.

S. Arahira, S. Kutsuzawa, Y. Matsui, D. Kunimatsu, and Y. Ogawa, “Generation of synchronized subterahertz optical pulse trains by repetition-frequency multiplication of subharmonic synchronous mode-locked semiconductor laser diode using fiber dispersion,” IEEE Photon. Technol. Lett. 10, 209-211 (1998).
[CrossRef]

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]

Ozharar, S.

Papoulis, A.

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

Patorski, K.

K. Patorski, “The self-imaging phenomenon and its application,” in Progress in Optics, E. Wolf, ed. (Elsevier Science, 1989), Vol. XXVII, pp. 1-108.
[CrossRef]

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.

Pudo, D.

D. Pudo, C. R. Fernández-Pousa, and L. R. Chen, “Timing jitter transfer function in the temporal Talbot effect,” IEEE Photon. Technol. Lett. 20, 496-498 (2008).
[CrossRef]

D. Pudo and L. R. Chen, “Simple estimation of the pulse amplitude noise and timing jitter evolution through the temporal Talbot effect,” Opt. Express 15, 6351-6357 (2007).
[CrossRef] [PubMed]

D. Pudo, M. Depa, and L. R. Chen, “Single and mutiwavelength all-optical clock recovery in single-mode fiber using the temporal Talbot effect,” J. Lightwave Technol. 25, 2898-2903 (2007).
[CrossRef]

D. Pudo and L. R. Chen, “Estimating intensity fluctuations in high repetition rate pulse trains generated using the temporal Talbot effect,” IEEE Photon. Technol. Lett. 18, 658-660 (2006).
[CrossRef]

Quinlan, F.

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]

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]

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]

Torres-Company, V.

von der Linde, D.

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

Yilmaz, T.

T. Yilmaz, C. M. DePriest, A. Braun, J. H. Abeles, and P. J. Delfyett, Jr., “Noise in fundamental and harmonic modelocked semiconductor lasers: experiments and simulations,” IEEE J. Quantum Electron. 39, 838-849 (2003).
[CrossRef]

T. Yilmaz, C. M. DePriest, P. J. Delfyett, Jr., 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).
[CrossRef] [PubMed]

Zervas, M. N.

Appl. Phys. B

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

Electron. Lett.

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]

J. Azaña, “Pulse repetition rate multiplication using phase-only filtering,” Electron. Lett. 40, 449-451 (2004).
[CrossRef]

IEEE J. Quantum Electron.

T. Yilmaz, C. M. DePriest, A. Braun, J. H. Abeles, and P. J. Delfyett, Jr., “Noise in fundamental and harmonic modelocked semiconductor lasers: experiments and simulations,” IEEE J. Quantum Electron. 39, 838-849 (2003).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

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]

IEEE Photon. Technol. Lett.

S. Arahira, S. Kutsuzawa, Y. Matsui, D. Kunimatsu, and Y. Ogawa, “Generation of synchronized subterahertz optical pulse trains by repetition-frequency multiplication of subharmonic synchronous mode-locked semiconductor laser diode using fiber dispersion,” IEEE Photon. Technol. Lett. 10, 209-211 (1998).
[CrossRef]

D. Pudo and L. R. Chen, “Estimating intensity fluctuations in high repetition rate pulse trains generated using the temporal Talbot effect,” IEEE Photon. Technol. Lett. 18, 658-660 (2006).
[CrossRef]

D. Pudo, C. R. Fernández-Pousa, and L. R. Chen, “Timing jitter transfer function in the temporal Talbot effect,” IEEE Photon. Technol. Lett. 20, 496-498 (2008).
[CrossRef]

IEEE Trans. Inf. Theory

W. Gardner and L. Franks, “Characterization of cyclostationary random signal processes,” IEEE Trans. Inf. Theory 21, 4-14 (1975).
[CrossRef]

J. Lightwave Technol.

J. Mod. Opt.

L. Chantada, C. R. Fernández-Pousa, and C. Gómez-Reino, “Matrix theory and entropy of the partially coherent temporal Talbot effect,” J. Mod. Opt. 54, 501-514 (2007).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. B

S. Gee, F. Quinlan, S. Ozharar, and P. J. Delfyett, “Correlation of supermode noise of harmonically mode-locked lasers,” J. Opt. Soc. Am. B 24, 1490-1497 (2007).
[CrossRef]

V. Torres-Company, H. Lajunen, and A. T. Friberg, “Coherence theory of noise in ultrashort pulse trains,” J. Opt. Soc. Am. B 24, 1441-1450 (2007).
[CrossRef]

D. Duchesne, R. Morandotti, and J. Azaña, “Temporal self-imaging phenomena in high-order dispersive media,” J. Opt. Soc. Am. B 20, 113-125 (2003).
[CrossRef]

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

J. Azaña and L. R. Chen, “General temporal self-imaging phenomena,” J. Opt. Soc. Am. B 20, 1447-1458 (2003).
[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]

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. II. Intensity spectrum,” J. Opt. Soc. Am. B 22, 753-763 (2005).
[CrossRef]

F. Rana, H. L. T. Lee, R. J. Ram, M. E. Grein, L. A. Jiang, E. P. Ippen, and H. A. Haus, “Characterization of the noise and correlations in harmonically modelocked-lasers,” J. Opt. Soc. Am. B 19, 2609-2621 (2002).
[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]

Opt. Commun.

J. Lancis, J. Caraquitena, P. Andrés, and M. A. Muriel, “Temporal self-imaging effect for chirped laser pulse sequences: repetition rate and duty cycle tunability,” Opt. Commun. 253, 156-163 (2005).
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J. Fatome, S. Pitois, and G. Millot, “Influence of third-order dispersion on the temporal Talbot effect,” Opt. Commun. 234, 29-34 (2004).
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L. Chantada, C. R. Fernández-Pousa, and C. Gómez-Reino, “Theory of the partially coherent temporal Talbot effect,” Opt. Commun. 266, 393-398 (2006).
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[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Intensity power spectral density (PSD) of a 10 GHz jittery train of chirp-free pulses ( σ j = 100 fs , η = 0.95 ) before and after a SMF-based temporal Talbot device with index γ / α = 1 . Pulse width t p = 8 ps . Top, numerical power spectral densities for the input and output trains (solid curves), and numerical PSD of the mean input signal (dotted curve). Bottom, analytical noise power spectral densities for the input (dashed curve) and output trains (solid curve).

Fig. 2
Fig. 2

Intensity PSD of a 10 GHz jittery train of chirp-free pulses ( σ j = 100 fs , η = 0.95 ) before and after a SMF-based temporal Talbot device with index γ / α = 1 / 2 . Pulse width t p = 8 ps .

Fig. 3
Fig. 3

Intensity PSD of a 10 GHz jittery train of chirp-free pulses ( σ j = 100 fs , η = 0.95 ) before and after a temporal Talbot device with index γ / α = 2 , and based on SMF with increased β 3 dispersion Pulse width t p = 3 ps .

Fig. 4
Fig. 4

Intensity PSD of a 10 GHz jittery train of chirp-free pulses ( σ j = 100 fs , η = 0.95 , and N = 2 neighbor correlation) before and after a SMF-based temporal Talbot device with index γ / α = 1 . Pulse width t p = 8 ps .

Fig. 5
Fig. 5

Intensity PSD of a 10 GHz jittery train of pulses ( σ j = 100 fs , η = 0.95 , and N = 2 neighbor correlation) before and after a SMF-based temporal Talbot device with index α / γ = 1 / 2 . Pulse width t p = 8 ps .

Equations (17)

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e z + i β 2 2 2 e t 2 β 3 6 3 e t 3 = 0 .
e ( t , 0 ) = k = + exp [ ( t k t 0 a k ) 2 ( 1 + i C ) / 2 t p 2 ] ,
E ( ω , L ) = ( 2 π ρ ) 1 / 2 exp ( ρ ω 2 / 2 ) exp ( i β 2 L ω 2 / 2 + i β 3 L ω 3 / 6 ) k = + exp ( i ω k t 0 + i ω a k ) ,
| β 2 | L = t 0 2 2 π γ α ,
S ( ω , z ) = + d t exp ( i ω t ) [ lim T T + T d t 2 T I ( t + t , z ) I ( t , z ) ] .
t 0 S ( N ) ( ω , L ) J ( ω ) | ζ ( ω ) | 3 Φ ( ω ) n = + G n ( ω ) M n ( ω ) ,
J ( ω ) = π t p 2 exp ( ω 2 t p 2 / 2 ) ,
ζ ( ω ) = 1 i β 3 L ω 2 t p 2 ,
Φ ( ω ) = k = + R ( k ) exp ( i ω k t 0 ) ,
G n ( ω ) = exp ( ω β 2 L n t 0 ) 2 / ( 2 t p 2 | ζ ( ω ) | 2 ) ,
M n ( ω ) = | A n exp ( i ω n t 0 / 2 ) + B n exp ( i ω n t 0 / 2 ) | 2 ,
A n = 1 2 t p 2 ζ ( ω ) | ζ ( ω ) | ( t p 2 ω i β 2 L ω i β 3 L ω 2 / 2 + i n t 0 ) ω ,
B n = A n ω .
a m = η a m 1 + ε m .
Φ ( ω ) = σ j 2 1 η 2 1 2 η cos ( ω t 0 ) + η 2 .
a m + N = η a m + ε m .
Φ ( ω ) = σ j 2 1 η 2 1 2 η cos ( ω N t 0 ) + η 2 .

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