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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 8 — Apr. 9, 2012
  • pp: 9151–9160
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Timing jitter from the optical spectrum in semiconductor passively mode locked lasers

Ricardo Rosales, Kamel Merghem, Anthony Martinez, Francois Lelarge, Alain Accard, and Abderrahim Ramdane  »View Author Affiliations


Optics Express, Vol. 20, Issue 8, pp. 9151-9160 (2012)
http://dx.doi.org/10.1364/OE.20.009151


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Abstract

An analysis of the passively mode locked regime in semiconductor lasers is presented, leading to an explicit expression relating the timing jitter diffusion constant to the optical linewidths in these devices. Experimental results for single section quantum-dash based lasers validating the theoretical analysis are presented for the first time. Timing jitter of mode locked lasers at rates of up to 130 GHz has been experimentally estimated from the optical spectra without requiring fast photodetection.

© 2012 OSA

1. Introduction

Semiconductor passively mode locked lasers (MLLs) are capable of emitting stable optical pulse trains in the absence of an external reference clock signal. Their compact size, ease of fabrication and low power consumption, make them interesting for a variety of applications including high bit rate optical time division multiplexing, clock recovery and millimeter wave generation. For all these applications low timing jitter is necessary in order to fulfill bit error rate and phase noise requirements. Understanding and quantifying this parameter is therefore of primary importance. Characterization of timing jitter is commonly performed by means of an electrical spectrum analyzer (ESA) after photodetection of the laser pulse train as described in [1

1. D. Von der Linde, “Characterization of noise in continuously operating mode-locked lasers,” Appl. Phys., B Photophys. Laser Chem. 39(4), 201–217 (1986). [CrossRef]

]. This method is however restricted to relatively low repetition frequencies due to the limited bandwidth of currently available photodiodes. Besides, the model in [1

1. D. Von der Linde, “Characterization of noise in continuously operating mode-locked lasers,” Appl. Phys., B Photophys. Laser Chem. 39(4), 201–217 (1986). [CrossRef]

] relies on the assumption of stationary timing fluctuations which is not the case in passively MLLs [2

2. D. Eliyahu, R. A. Salvatore, and A. Yariv, “Effect of noise on the power spectrum of passively mode-locked lasers,” J. Opt. Soc. Am. B 14(1), 167–174 (1997). [CrossRef]

], where the timing jitter exhibits a diffusion-like behavior characterized by a diffusion constant. Timing jitter can also be measured by the cross correlation technique presented in [3

3. L. A. Jiang, S. T. Wong, M. E. Grein, E. P. Ippen, and H. A. Haus, “Measuring timing jitter with optical cross correlations,” IEEE J. Quantum Electron. 38(8), 1047–1052 (2002). [CrossRef]

], however the high power levels required by the second harmonic generation nonlinear crystal limit its applicability.

2. Theory

Considering only the effects of phase fluctuations induced by quantum noise, the complex electric field in a semiconductor passively MLL can be written as:
E(t)=[n=1NEnej[ωnt+θn(t)+ϕn]]+c.c
(1)
where En, ωn, θn(t) and ϕn are respectively the real amplitude, angular frequency, phase noise and static phase, of each of theNlongitudinal modes. Amplitude noise has been neglected since phase noise dominates close to ωn [8

8. R. Paschotta, A. Schlatter, S. C. Zeller, H. R. Telle, and U. Keller, “Optical phase noise and carrier-envelope offset noise of mode-locked lasers,” Appl. Phys. B 82(2), 265–273 (2006). [CrossRef]

]. Under ML regime:
ddt[(2ωn+1ωnωn+2)t+2θn+1(t)θn(t)θn+2(t)+2ϕn+1ϕnϕn+2]=0
(2)
for n=1 to N2 and N3, which results in all the modes being phase correlated and equally separated, i.e.:
θn+1(t)θn(t)=Δθ(t)
(3)
ωn+1ωn=ωr
(4)
withωrthe angular repetition frequency such that the pulse train period T=2π/ωr. Equation (3) can also be written as:
θn(t)=θc(t)+(nnc)Δθ(t)
(5)
for n=1 to N, and with nca given mode number with corresponding phase noise θc(t). The complex electric field then becomes:
E(t)=[n=1NEnej[ωnt+θc(t)+(nnc)Δθ(t)+ϕn]]+c.c
(6)
and the laser intensity:
I(t)|E(t)|2=m=1N1l=1Nm2Em+lElcos[mωrt+mΔθ(t)+(ϕm+lϕl)]+m=1NEm2=m=1N1l=1Nm2Em+lElcos[mωr(t+Δtr(t))+(ϕm+lϕl)]+m=1NEm2
(7)
with Δtr(t)=Δθ(t)/ωr being the timing fluctuations in the pulse position, commonly referred to as timing jitter. It has been experimentally shown in two section quantum well [3

3. L. A. Jiang, S. T. Wong, M. E. Grein, E. P. Ippen, and H. A. Haus, “Measuring timing jitter with optical cross correlations,” IEEE J. Quantum Electron. 38(8), 1047–1052 (2002). [CrossRef]

] and single section QDash [15

15. J. P. Tourrenc, A. Akrout, K. Merghem, A. Martinez, F. Lelarge, A. Shen, G. H. Duan, and A. Ramdane, “Experimental investigation of the timing jitter in self-pulsating quantum-dash lasers operating at 1.55 µm,” Opt. Express 16(22), 17706–17713 (2008). [CrossRef] [PubMed]

] passively MLLs, through cross correlation measurements, that the variance of Δtr(t)grows linearly with time which implies that timing jitter undergoes a random walk. By the central limit theorem, Δtr(t)is normally distributed at large t, therefore it can be described by a Gaussian random walk with zero mean and variance|Δtr(t)|2=Dt, with D a diffusion coefficient. Substituting Δθ(t)=ωrΔtr(t) in Eq. (6) yields:

E(t)=[n=1NEnej[ωnt+θc(t)+ωr(nnc)Δtr(t)+ϕn]]+c.c
(8)

The phase fluctuations in each mode, and thereforeθc(t), are affected by amplified spontaneous emission noise and hence undergo a random walk process [11

11. H. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron. 29(3), 983–996 (1993). [CrossRef]

], which again for large twill be normally distributed. From Eq. (8), the optical spectrum |E(ω)|2 can be calculated by taking the electric field autocorrelation function R(τ)and Fourier transforming the result:
|E(ω)|2=R(τ)ejωτdτ=n=1NEn2ejωnτe12[|θc(τ)|2+ωr2(nnc)2D+2ωr(nnc)θc(τ)Δtr(τ)]ejωτdτn=1N{[12Δωθc+12ωr2(nnc)2D+ωr(nnc)γθc,Δtr]2+(ωωn)2}1
(9)
with γθc,Δtrthe slope at large tof the correlation term θc(τ)Δtr(τ). Consequently, the optical spectrum consists of NLorentzian lines centered atωnwith full width at half maximum (FWHM) linewidths:
Δωn=Δωmin+ωr2(nnmin)2D
(10)
withnmin=ncγθc,Δtr/ωrD, and Δωmin=Δωθcγθc,Δtr2/D. Equation (10) shows that the diffusion coefficient Daffects the linewidth of the modes; hence an estimation of timing jitter is readily available from the optical spectrum by measuring the mode linewidths aroundnmin. It can be noted that correlation of θc(t)andΔtr(t)affects the first and zero order terms in Eq. (10), resulting in a frequency shift of the mode with minimum linewidth relative to the case in which both random processes are statistically independent (γθc,Δtr=0), in agreement with [9

9. J. K. Wahlstrand, J. T. Willits, C. R. Menyuk, and S. T. Cundiff, “The quantum-limited comb lineshape of a mode-locked laser: Fundamental limits on frequency uncertainty,” Opt. Express 16(23), 18624–18630 (2008). [CrossRef] [PubMed]

]. As the value ofDis unambiguously determined from the second order term in Eq. (10), it can be extracted without requiring any knowledge of the correlation between θc(t)andΔtr(t). It should be noted, however, that these random processes are more likely to be uncorrelated if only quantum noise is considered as the temporal position of a pulse should not affect its optical phase changes [8

8. R. Paschotta, A. Schlatter, S. C. Zeller, H. R. Telle, and U. Keller, “Optical phase noise and carrier-envelope offset noise of mode-locked lasers,” Appl. Phys. B 82(2), 265–273 (2006). [CrossRef]

].

The value of D also determines the spectrum of the laser intensity |I(ω)|2, or more commonly called the RF (Radio Frequency) spectrum, which can be calculated straightforwardly from Eq. (7) in the same way as the optical one. Skipping the details, it is found to consist of a sum of Nm Lorentzian lines centered at mωr for m=1 toN1. At any given value ofm, each line will have identical FWHM linewidths given by:
ΔωRFm=m2ωr2D=m2ΔωRF1
(11)
withΔωRF1=ωr2D. Each line corresponds to the beating between a pair of modes separated by mmodes in the optical spectrum. We notice that the diffusion coefficient D could be directly derived fromΔωRF1, this is however not possible in practice at relatively high repetition frequencies due to the limited bandwidth of the photodiode. A relation between both the mode linewidths and those of their beatings can be established by combining (10) and (11) to yield:
Δωn=Δωmin+ΔωRFmm2(nnmin)2
(12)
or equivalently:
Δωn=Δωmin+ΔωRF1(nnmin)2
(13)
which was also obtained and experimentally observed in [5

5. T. Habruseva, S. O’Donoghue, N. Rebrova, F. Kéfélian, S. P. Hegarty, and G. Huyet, “Optical linewidth of a passively mode-locked semiconductor laser,” Opt. Lett. 34(21), 3307–3309 (2009). [CrossRef] [PubMed]

] for a two section passively MLLs emitting at 1.3 μm.

In order to be able to quantify the timing jitter from measurements of the mode linewidths, various basic aspects are first reviewed, some of which are also addressed in e.g [16

16. R. Paschotta, “Noise of mode-locked lasers (Part I): numerical model,” Appl. Phys. B 79(2), 153–162 (2004). [CrossRef]

,17

17. F. Kéfélian, S. O’Donoghue, M. T. Todaro, J. G. McInerney, and G. Huyet, “RF linewidth in monolithic passively mode-locked semiconductor laser,” IEEE Photon. Technol. Lett. 20(16), 1405–1407 (2008). [CrossRef]

]. Timing jitter is quantified as the standard deviation σofΔtr(t), hence, in a passively MLL, it will be given by:

σ=|Δtr(t)|2=Dt
(14)

Despite the lack of convergence ast, the timing jitter from pulse i at time ti to pulse i+Nat time ti+NTrcan be defined. This yields:

σ(N)=DNTr
(15)

For N=1this gives the pulse to pulse timing jitter σppthat is sometimes used in practice. Timing jitter can also be quantified in the frequency domain as:

σ=|Δtr(t)|2=+SΔtr(ω)dω2π=+Dω2dω2π
(16)

Clearly, the PSD of Δtr(t)does not converge asω0, which is expected from the lack of convergence in the time domain astand which is a consequence of the absence of a reference clock signal. The timing jitter can however be quantified between two given frequencies ω1, ω2>0 to yield the pulse to clock timing jitter, that is, the jitter relative to a perfect RF oscillator, also called r.m.s. integrated timing jitter:

σpc=ω1ω2Dω2dω2π=ω1ω2ΔωRF1ωr2ω2dω2π
(17)

3. Device

4. Experimental results

Figure 2(a)
Fig. 2 (a) Optical spectrum and (b) RF spectrum of the 29 consecutive mode beatings (m = 1) of the 1000 μm long laser under ML operation when biased at 150 mA.
shows the optical spectrum of the 1000 μm long laser when biased at 150 mA, from which a total of N=30 modes have been numbered. In order to verify the mode phase correlation and hence ML operation, the RF spectrum of all N1=29 mode beatings between consecutive pairs of modes (m=1) were measured. Figure 2(b) is a plot of the superposition of all these beatings. All beating spectra are very well fitted by lorentzian functions. The average beating linewidth is ΔωRF1/2π=89kHz with a standard deviation of 5.3 kHz, while the average center frequency is ωr/2π=39.6GHz with a standard deviation of 18.2 kHz. These small deviations demonstrate the mode phase correlation between all Nmodes and thus validate Eq. (3) and Eq. (4); consequently the laser is under ML regime. The lorentzian shape of all the lines in the RF spectrum for m=1 also confirms that Δtr(t) is in fact a Gaussian random walk process.

The linewidth of each of the N modes of the optical spectrum in Fig. 2(a) was then measured by the self-heterodyne technique [19

19. T. Okoshi, K. Kikuchi, and A. Nakayama, “Novel method for high resolution measurement of laser output spectrum,” Electron. Lett. 16(16), 630 (1980). [CrossRef]

]. Each longitudinal mode was individually filtered and split into two paths. The mode frequency in one path is offset using an acousto-optic modulator. On the second path, a fiber is inserted to introduce a delay exceeding the coherence time of the mode light so that the combining beams interfere as if they were independent. All the resulting self-mode beating spectra fitted very well lorentzian functions, as can be seen in Fig. 3(a)
Fig. 3 (a) Self-mode beating spectra with lorentzian fits for modes 11, 16, 21 and 26. (b) Measured mode linewidths as a function of mode number and parabolic fit for the 1000 μm long laser under ML operation when biased at 150 mA.
for several modes, confirming the lorentzian shape of the modes as expected. In Fig. 3(b) a plot of the measured linewidths as a function of mode number is shown. The solid line is a parabolic fit Δωn/2π=6.5+0.092(n12)2MHz, from which ΔωRF1/2π=92kHz is extracted with a standard error of ~3 kHz, which very much agrees with the measured linewidth of each beating in Fig. 2(b), verifying Eq. (13). We can also see from the fit thatnmin=12 which is the number of the less noisy mode with linewidth Δωθc/2π=6.5MHz, and this is what the linewidth of all the modes should be in the absence of timing jitter. The diffusion coefficient is also extracted from the parabolic fit to giveD=0.0092fs. By using Eq. (15) and Eq. (17), the pulse to pulse and the pulse to clock timing jitter from, e.g.16 MHz to 320 MHz [20

20. “The control of jitter and wander within digital networks which are based on the synchronous digital hierarchy (SDH),” Recommendation G.825, ITU-T, (2001).

] were then estimated at σpp=15.16fs and σpc=166fsrespectively.

In order to evaluate the limits in repetition rate for the applicability of the proposed method, estimations of the RF linewidth as a function of measured number of modes have been performed, from which corresponding relative errors have been calculated. Figure 7
Fig. 7 Relative error as a function of number of measured modes for the 1000 μm long laser.
illustrates this for the laser with repetition rate of 40 GHz, with errors being determined relative to the RF linewidth value directly measured from the photocurrent spectrum. This figure suggests that it is not required to perform the measurements of the entire number of available modes in order to obtain similar relative error values, which can be kept smaller than ~10% by measuring at least six of the available modes. This gives an idea on the maximum repetition frequency that the method could be applied to. In this example, six available modes is equivalent to performing the measurements every five modes, corresponding to a separation in frequency of ~200 GHz, for which an estimation in RF linewidth with a relative error of about 11% was obtained. Hence, application of the method might be possible at repetition frequencies up to 200 GHz yielding similar relative errors provided six modes are available for linewidth measurements. Timing jitter at even higher repetition rates could however be estimated if more modes are available. A lower limit in repetition frequency for the applicability of the method is also imposed since precise linewidth measurements on multiple lines of the optical spectrum becomes more difficult as the lines become closer to each other. If the self-heterodyne technique is used, this limit will then be determined by the minimum bandwidth of commercially available optical filters of ~0.1 nm in order to individually separate the optical modes, setting a repetition frequency limit down to ~10 GHz. Other linewidth measurement techniques, such as the heterodyne detection with narrow linewidth reference lasers, may however be implemented, allowing a lower limit in repetition frequency depending on the specific instrumentation used.

5. Summary

An analysis of the optical and intensity spectra of semiconductor lasers under passive mode locking was presented. This has allowed simple relations to be established between the mode linewidths, RF linewidths, L(f) and timing jitter. The use of these relations becomes particularly interesting at high repetion frequencies where direct measurement of the RF linewidths, L(f) and timing jitter is limited by the photodiode and the electronics instrumentation bandwidth. The analytical results have been experimentally validated for single section QDash passively MLLs at repetition frequencies of up to 130 GHz. As the presented theory is based on a phenomenological approach, the results can be applied regardless the MLL configuration.

Acknowledgments

This work was supported in part by the French National Research Agency through the project TELDOT.

References and links

1.

D. Von der Linde, “Characterization of noise in continuously operating mode-locked lasers,” Appl. Phys., B Photophys. Laser Chem. 39(4), 201–217 (1986). [CrossRef]

2.

D. Eliyahu, R. A. Salvatore, and A. Yariv, “Effect of noise on the power spectrum of passively mode-locked lasers,” J. Opt. Soc. Am. B 14(1), 167–174 (1997). [CrossRef]

3.

L. A. Jiang, S. T. Wong, M. E. Grein, E. P. Ippen, and H. A. Haus, “Measuring timing jitter with optical cross correlations,” IEEE J. Quantum Electron. 38(8), 1047–1052 (2002). [CrossRef]

4.

F. X. Kartner, U. Morgner, T. Schibli, R. Ell, H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Few-cycle pulses directly from a laser,” Top. Appl. Phys. 95, 73 (2004).

5.

T. Habruseva, S. O’Donoghue, N. Rebrova, F. Kéfélian, S. P. Hegarty, and G. Huyet, “Optical linewidth of a passively mode-locked semiconductor laser,” Opt. Lett. 34(21), 3307–3309 (2009). [CrossRef] [PubMed]

6.

M. J. Ablowitz, B. Ilan, and S. T. Cundiff, “Noise-induced linewidth in frequency combs,” Opt. Lett. 31(12), 1875–1877 (2006). [CrossRef] [PubMed]

7.

R. Paschotta, “Noise of mode-locked lasers (Part II): timing jitter and other fluctuations,” Appl. Phys. B 79(2), 163–173 (2004). [CrossRef]

8.

R. Paschotta, A. Schlatter, S. C. Zeller, H. R. Telle, and U. Keller, “Optical phase noise and carrier-envelope offset noise of mode-locked lasers,” Appl. Phys. B 82(2), 265–273 (2006). [CrossRef]

9.

J. K. Wahlstrand, J. T. Willits, C. R. Menyuk, and S. T. Cundiff, “The quantum-limited comb lineshape of a mode-locked laser: Fundamental limits on frequency uncertainty,” Opt. Express 16(23), 18624–18630 (2008). [CrossRef] [PubMed]

10.

Y. Takushima, H. Sotobayashi, M. E. Grein, E. P. Ippen, and H. A. Haus, “Linewidth of mode combs of passively and actively mode-locked semiconductor laser diodes,” Proc. SPIE 5595, 213–227 (2004). [CrossRef]

11.

H. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron. 29(3), 983–996 (1993). [CrossRef]

12.

R. Rosales, K. Merghem, A. Martinez, A. Akrout, J. P. Tourrenc, A. Accard, F. Lelarge, and A. Ramdane, “InAs/InP quantum-dot passively mode locked lasers for 1.55 μm applications,” IEEE J. Sel. Top. Quantum Electron. 17(5), 1292–1301 (2011). [CrossRef]

13.

F. Lelarge, B. Dagens, J. Renaudier, R. Brenot, A. Accard, F. van Dijk, D. Make, O. Le Gouezigou, J. G. Provost, F. Poingt, J. Landreau, O. Drisse, E. Derouin, B. Rousseau, F. Pommereau, and G. H. Duan, “Recent advances on InAs/InP quantum dash based semiconductor lasers and optical amplifiers operating at 1.55 μm,” IEEE J. Sel. Top. Quantum Electron. 13(1), 111–124 (2007). [CrossRef]

14.

K. Merghem, A. Akrout, A. Martinez, G. Aubin, A. Ramdane, F. Lelarge, and G. Duan, “Pulse generation at 346 GHz using a passively mode locked quantum-dash-based laser at 1.55 μm,” Appl. Phys. Lett. 94(2), 021107 (2009). [CrossRef]

15.

J. P. Tourrenc, A. Akrout, K. Merghem, A. Martinez, F. Lelarge, A. Shen, G. H. Duan, and A. Ramdane, “Experimental investigation of the timing jitter in self-pulsating quantum-dash lasers operating at 1.55 µm,” Opt. Express 16(22), 17706–17713 (2008). [CrossRef] [PubMed]

16.

R. Paschotta, “Noise of mode-locked lasers (Part I): numerical model,” Appl. Phys. B 79(2), 153–162 (2004). [CrossRef]

17.

F. Kéfélian, S. O’Donoghue, M. T. Todaro, J. G. McInerney, and G. Huyet, “RF linewidth in monolithic passively mode-locked semiconductor laser,” IEEE Photon. Technol. Lett. 20(16), 1405–1407 (2008). [CrossRef]

18.

A. Demir, A. Mehrotra, and J. Roychowdhury, “Phase noise in oscillators: a unifying theory and numerical methods for characterization,” IEEE Trans. Circ. Syst. I 47, 655–674 (2000).

19.

T. Okoshi, K. Kikuchi, and A. Nakayama, “Novel method for high resolution measurement of laser output spectrum,” Electron. Lett. 16(16), 630 (1980). [CrossRef]

20.

“The control of jitter and wander within digital networks which are based on the synchronous digital hierarchy (SDH),” Recommendation G.825, ITU-T, (2001).

OCIS Codes
(140.0140) Lasers and laser optics : Lasers and laser optics
(140.4050) Lasers and laser optics : Mode-locked lasers
(140.5960) Lasers and laser optics : Semiconductor lasers
(250.0250) Optoelectronics : Optoelectronics

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: August 31, 2011
Revised Manuscript: October 15, 2011
Manuscript Accepted: December 11, 2011
Published: April 5, 2012

Citation
Ricardo Rosales, Kamel Merghem, Anthony Martinez, Francois Lelarge, Alain Accard, and Abderrahim Ramdane, "Timing jitter from the optical spectrum in semiconductor passively mode locked lasers," Opt. Express 20, 9151-9160 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-8-9151


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References

  1. D. Von der Linde, “Characterization of noise in continuously operating mode-locked lasers,” Appl. Phys., B Photophys. Laser Chem.39(4), 201–217 (1986). [CrossRef]
  2. D. Eliyahu, R. A. Salvatore, and A. Yariv, “Effect of noise on the power spectrum of passively mode-locked lasers,” J. Opt. Soc. Am. B14(1), 167–174 (1997). [CrossRef]
  3. L. A. Jiang, S. T. Wong, M. E. Grein, E. P. Ippen, and H. A. Haus, “Measuring timing jitter with optical cross correlations,” IEEE J. Quantum Electron.38(8), 1047–1052 (2002). [CrossRef]
  4. F. X. Kartner, U. Morgner, T. Schibli, R. Ell, H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Few-cycle pulses directly from a laser,” Top. Appl. Phys.95, 73 (2004).
  5. T. Habruseva, S. O’Donoghue, N. Rebrova, F. Kéfélian, S. P. Hegarty, and G. Huyet, “Optical linewidth of a passively mode-locked semiconductor laser,” Opt. Lett.34(21), 3307–3309 (2009). [CrossRef] [PubMed]
  6. M. J. Ablowitz, B. Ilan, and S. T. Cundiff, “Noise-induced linewidth in frequency combs,” Opt. Lett.31(12), 1875–1877 (2006). [CrossRef] [PubMed]
  7. R. Paschotta, “Noise of mode-locked lasers (Part II): timing jitter and other fluctuations,” Appl. Phys. B79(2), 163–173 (2004). [CrossRef]
  8. R. Paschotta, A. Schlatter, S. C. Zeller, H. R. Telle, and U. Keller, “Optical phase noise and carrier-envelope offset noise of mode-locked lasers,” Appl. Phys. B82(2), 265–273 (2006). [CrossRef]
  9. J. K. Wahlstrand, J. T. Willits, C. R. Menyuk, and S. T. Cundiff, “The quantum-limited comb lineshape of a mode-locked laser: Fundamental limits on frequency uncertainty,” Opt. Express16(23), 18624–18630 (2008). [CrossRef] [PubMed]
  10. Y. Takushima, H. Sotobayashi, M. E. Grein, E. P. Ippen, and H. A. Haus, “Linewidth of mode combs of passively and actively mode-locked semiconductor laser diodes,” Proc. SPIE5595, 213–227 (2004). [CrossRef]
  11. H. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron.29(3), 983–996 (1993). [CrossRef]
  12. R. Rosales, K. Merghem, A. Martinez, A. Akrout, J. P. Tourrenc, A. Accard, F. Lelarge, and A. Ramdane, “InAs/InP quantum-dot passively mode locked lasers for 1.55 μm applications,” IEEE J. Sel. Top. Quantum Electron.17(5), 1292–1301 (2011). [CrossRef]
  13. F. Lelarge, B. Dagens, J. Renaudier, R. Brenot, A. Accard, F. van Dijk, D. Make, O. Le Gouezigou, J. G. Provost, F. Poingt, J. Landreau, O. Drisse, E. Derouin, B. Rousseau, F. Pommereau, and G. H. Duan, “Recent advances on InAs/InP quantum dash based semiconductor lasers and optical amplifiers operating at 1.55 μm,” IEEE J. Sel. Top. Quantum Electron.13(1), 111–124 (2007). [CrossRef]
  14. K. Merghem, A. Akrout, A. Martinez, G. Aubin, A. Ramdane, F. Lelarge, and G. Duan, “Pulse generation at 346 GHz using a passively mode locked quantum-dash-based laser at 1.55 μm,” Appl. Phys. Lett.94(2), 021107 (2009). [CrossRef]
  15. J. P. Tourrenc, A. Akrout, K. Merghem, A. Martinez, F. Lelarge, A. Shen, G. H. Duan, and A. Ramdane, “Experimental investigation of the timing jitter in self-pulsating quantum-dash lasers operating at 1.55 µm,” Opt. Express16(22), 17706–17713 (2008). [CrossRef] [PubMed]
  16. R. Paschotta, “Noise of mode-locked lasers (Part I): numerical model,” Appl. Phys. B79(2), 153–162 (2004). [CrossRef]
  17. F. Kéfélian, S. O’Donoghue, M. T. Todaro, J. G. McInerney, and G. Huyet, “RF linewidth in monolithic passively mode-locked semiconductor laser,” IEEE Photon. Technol. Lett.20(16), 1405–1407 (2008). [CrossRef]
  18. A. Demir, A. Mehrotra, and J. Roychowdhury, “Phase noise in oscillators: a unifying theory and numerical methods for characterization,” IEEE Trans. Circ. Syst. I 47, 655–674 (2000).
  19. T. Okoshi, K. Kikuchi, and A. Nakayama, “Novel method for high resolution measurement of laser output spectrum,” Electron. Lett.16(16), 630 (1980). [CrossRef]
  20. “The control of jitter and wander within digital networks which are based on the synchronous digital hierarchy (SDH),” Recommendation G.825, ITU-T, (2001).

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