Abstract

We analyze the effects of noise on the power spectrum of pulse trains generated by a continuously operating passively mode-locked laser. The shape of the different harmonics of the power spectrum is calculated in the presence of correlated timing fluctuations between neighboring pulses and in the presence of amplitude fluctuations. The spectra at the different harmonics are influenced mainly by the nonstationary timing-jitter fluctuations; amplitude fluctuations slightly modify the spectral tails. Estimation of the coupling term between the longitudinal cavity modes or the effective saturable absorber coefficient is made from the timing-jitter correlation time. Experimental results from an external cavity two-section semiconductor laser are given. The results show timing-jitter fluctuations having a relaxation time much longer than the repetition period.

© 1997 Optical Society of America

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References

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  1. D. von der Linde, “Characterization of the noise in continuously operating mode-locked lasers,” Appl. Phys. B 39, 201–217 (1986).
    [CrossRef]
  2. D. Eliyahu, R. A. Salvatore, and A. Yariv, “Noise characterization of pulse train generated by actively mode-locked lasers,” J. Opt. Soc. Am. B 13, 1619–1626 (1996).
    [CrossRef]
  3. H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron. 29, 983–996 (1993).
    [CrossRef]
  4. A. Yariv, Optical Electronics, 4th ed. (Saunders, Philadelphia, Pa., 1991).
  5. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 4th ed. (Academic, New York, 1965).
  6. R. A. Salvatore, T. Scharns, and A. Yariv, “Pulse characteristics of passively mode-locked diode lasers,” Opt. Lett. 20, 737–739 (1995).
    [CrossRef] [PubMed]
  7. S. Sanders, A. Yariv, J. Paslaski, J. E. Ungar, and H. A. Zarem, “Passive mode locking of a two-section multiple quantum well laser at harmonics of the cavity round-trip frequency,” Appl. Phys. Lett. 58, 681–683 (1991).
    [CrossRef]

1996 (1)

1995 (1)

1993 (1)

H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron. 29, 983–996 (1993).
[CrossRef]

1991 (1)

S. Sanders, A. Yariv, J. Paslaski, J. E. Ungar, and H. A. Zarem, “Passive mode locking of a two-section multiple quantum well laser at harmonics of the cavity round-trip frequency,” Appl. Phys. Lett. 58, 681–683 (1991).
[CrossRef]

1986 (1)

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

Eliyahu, D.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 4th ed. (Academic, New York, 1965).

Haus, H. A.

H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron. 29, 983–996 (1993).
[CrossRef]

Mecozzi, A.

H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron. 29, 983–996 (1993).
[CrossRef]

Paslaski, J.

S. Sanders, A. Yariv, J. Paslaski, J. E. Ungar, and H. A. Zarem, “Passive mode locking of a two-section multiple quantum well laser at harmonics of the cavity round-trip frequency,” Appl. Phys. Lett. 58, 681–683 (1991).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 4th ed. (Academic, New York, 1965).

Salvatore, R. A.

Sanders, S.

S. Sanders, A. Yariv, J. Paslaski, J. E. Ungar, and H. A. Zarem, “Passive mode locking of a two-section multiple quantum well laser at harmonics of the cavity round-trip frequency,” Appl. Phys. Lett. 58, 681–683 (1991).
[CrossRef]

Scharns, T.

Ungar, J. E.

S. Sanders, A. Yariv, J. Paslaski, J. E. Ungar, and H. A. Zarem, “Passive mode locking of a two-section multiple quantum well laser at harmonics of the cavity round-trip frequency,” Appl. Phys. Lett. 58, 681–683 (1991).
[CrossRef]

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]

Yariv, A.

D. Eliyahu, R. A. Salvatore, and A. Yariv, “Noise characterization of pulse train generated by actively mode-locked lasers,” J. Opt. Soc. Am. B 13, 1619–1626 (1996).
[CrossRef]

R. A. Salvatore, T. Scharns, and A. Yariv, “Pulse characteristics of passively mode-locked diode lasers,” Opt. Lett. 20, 737–739 (1995).
[CrossRef] [PubMed]

S. Sanders, A. Yariv, J. Paslaski, J. E. Ungar, and H. A. Zarem, “Passive mode locking of a two-section multiple quantum well laser at harmonics of the cavity round-trip frequency,” Appl. Phys. Lett. 58, 681–683 (1991).
[CrossRef]

A. Yariv, Optical Electronics, 4th ed. (Saunders, Philadelphia, Pa., 1991).

Zarem, H. A.

S. Sanders, A. Yariv, J. Paslaski, J. E. Ungar, and H. A. Zarem, “Passive mode locking of a two-section multiple quantum well laser at harmonics of the cavity round-trip frequency,” Appl. Phys. Lett. 58, 681–683 (1991).
[CrossRef]

Appl. Phys. B (1)

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

Appl. Phys. Lett. (1)

S. Sanders, A. Yariv, J. Paslaski, J. E. Ungar, and H. A. Zarem, “Passive mode locking of a two-section multiple quantum well laser at harmonics of the cavity round-trip frequency,” Appl. Phys. Lett. 58, 681–683 (1991).
[CrossRef]

IEEE J. Quantum Electron. (1)

H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron. 29, 983–996 (1993).
[CrossRef]

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

Opt. Lett. (1)

Other (2)

A. Yariv, Optical Electronics, 4th ed. (Saunders, Philadelphia, Pa., 1991).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 4th ed. (Academic, New York, 1965).

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Figures (5)

Fig. 1
Fig. 1

Timing-jitter noise in the two opposed cases. (a) The stationary fluctuations in actively mode-locked lasers. For long time scale compared with the fluctuations, correlation time τt has a bounded (δTn-δTm)2 that reaches the constant value 2GT(0). (b) The nonstationary fluctuations in passively mode-locked lasers. For short time compared with the correlation time, τt has (δTn-δTm)2 that shows a time-squared (|n-m|2) dependence while for long time (τt) it is unbounded and reveals a linear dependence on time (|n-m|).

Fig. 2
Fig. 2

Numerical calculation, based on Eqs. (10) and (15), of the normalized spectra around different harmonic numbers versus the scaled offset frequency for ΔT21/2/T=2×10-4. (a) τt/T=100, (b) τt/T=500, (c) τt/T=2000. The innermost curve in each part represents the spectra at the first harmonic. As the harmonic number increases, its spectrum becomes wider and approaches the Gaussian shape.

Fig. 3
Fig. 3

The effect of amplitude fluctuations on the normalized spectra of long-range timing-jitter correlation time τt [the normalized outer parentheses of the summation in Eq. (28)] (a) Versus the scaled frequency offset for different values of scaled amplitude correlation time. The solid curve describes the normalized spectra for δA2=0. The dotted, dashed, and dotted–dashed curves stand for (T/τa)/(2ω2ΔT2)1/2 equal to 1, 2, and 10, respectively, where maximum amplitude fluctuation was taken into account (δA2/A2=1). (b) Versus the scaled amplitude correlation time for several values of scaled frequency offset. The solid curve describes the normalized spectra for ΔωnT/(2ω2ΔT2)1/2=2. The dotted, dashed, and dotted–dashed lines stand for ΔωnT/(2ω2ΔT2)1/2 equal to 4, 6, and 10, respectively.

Fig. 4
Fig. 4

External cavity configuration for the two-section semiconductor passively mode-locked laser used in the experiment.

Fig. 5
Fig. 5

Normalized intensity spectra of the first six harmonic numbers versus the deviation frequency divided by the harmonic number n. The solid curve describes a Gaussian with FWHM of 2.83×10-3.

Equations (30)

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IΔt(t)=n=-NNfn(t-Tn),
PI(ω)=limΔt 1ΔtIΔt(ω)IΔt*(ω),
IΔt(ω)=-Δt/2Δt/2IΔt(t)eiωtdt.
IΔt(ω)=n=-NNFn(ω)exp(iωTn),
Fn(ω)=-fn(t)exp(iωt)dt.
PI(ω)=limN 12N+1n,m=-NNFn(ω)Fm*(ω)×exp[iω(Tn-Tm)].
Tn=nT+δTn,
PI(ω)=limN 12N+1n,m=-NN exp[iωT(n-m)]×Fn(ω)Fm*(ω)exp[iω(δTn-δTm)].
(δTn-δTm)2=2[GT(0)-GT(|n-m|)],
PI(ω)=|F(ω)|2limN 12N+1n,m=-NN exp[iωT(n-m)]×exp[iω(δTn-δTm)],
δTn-δTm=sgn(n-m)i=min(n,m)+1max(n,m)ΔTi.
PI(ω)=|F(ω)|2limN 12N+1n,m=-NN exp[iωT(n-m)]×exp-ω22(δTn-δTm)2,
(δTn-δTm)2=i,j=min(n,m)+1max(n,m)ΔTiΔTj.
ΔTiΔTj=ΔT2exp[-|i-j|T/τt],
(δTn-δTm)2=2ΔT2 exp(-T/τt)[1-exp(-T/τt)]2×|n-m| 1-exp(-2T/τt)2 exp(-T/τt)-1+exp(-|n-m|T/τt).
ΔTiΔTjΔT2δi,j,
(δTn-δTm)2=ΔT2|n-m|.
PI(ω)=|F(ω)|2 sinh(ω2ΔT2/2)cosh(ω2ΔT2/2)-cos(ωT).
PI(ω)=|F(ω)|2n=0 ω2ΔT2(ω2ΔT2/2)2+(ΔωnT)2,
(δTn-δTm)2=2ΔT2τtT2|n-m| Tτt-1+exp(-|n-m|T/τt).
(δTn-δTm)2
=ΔT2|n-m|22|n-m|τtT|n-m|Tτt |n-m|Tτt.
PI(ω)=|F(ω)|22πω2ΔT21/2n=- exp-(ΔωnT)22ω2ΔT2.
TτtgΩg2τp2,
gΩg2τp2γAp2,
δAnδAm=δA2exp(-|n-m|T/τa),
PI(ω)=|S(ω)|2limN 12N+1n,m=-NN[A2+δA2×exp(-|n-m|T/τa)]exp[iωT(n-m)]×exp[-ω22(δTn-δTm)2].
PI(ω)=A2|S(ω)|2sinh(ω2ΔT2/2)cosh(ω2ΔT2/2)-cos(ωT)+δA2A2sinh(ω2ΔT2/2+T/τa)cosh(ω2ΔT2/2+T/τa)-cos(ωT).
PI(ω)=A2|S(ω)|22πω2ΔT21/2×n=-exp-(ΔωnT)22ω2ΔT2+δA2A2×Reexp(T/τa+iΔωnT)22ω2ΔT2×erfcT/τa+iΔωnT2ω2ΔT2,
2ω2ΔT2π1/2 T/τa(T/τa)2+(ΔωnT)2.

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