Statistical properties of the spectrum of light pulses in fast. pseudo random word modulation of a single-mode. semiconductor laser

Statistical properties of the spectrum of light pulses in fast pseudo random word modulation of a single-mode semiconductor laser S. Balle, M. Homar and M. San Miguel Departament de Fsica, Universitat de les Illes Balears E{07071 Palma de Mallorca, Spain Phone: 34{71{173234, Fax: 34-71{438028 () 1

2 Abstract The spectrum of Single Mode Laser pulses generated by fast Pseudo Random Word Modulation is studied numerically for Return-to-Zero and Non-Return-to-Zero control signals. We analyze both statistics and the worst cases for the frequency chirp during each optical pulse, and we study the connection between these frequency chirps and the turnon times. We show that patterns in the modulation signal sequences contribute to chirp noise. The worst case values of the turn-on time and the chirp range are very similar in the two modulation schemes, hence, the optimum choice depends mainly on the characteristics of the decision circuit and on the driver and detector bandwidths. I. INTRODUCTION The optimum performance of high capacity, long-haul ber-based, communication systems is obtained with monochromatic light sources that remain single-mode even under fast, large-signal pseudo random word modulation (PRWM). The light sources used in such systems are tipically single-mode semiconductor lasers (SMLs) either DFB or DBR lasers which have narrow CW linewidths and which remain in stable single-mode operation under fast modulation. For such laser sources two main factors that limit the system performance are timing jitter [1]? [7] and chirp noise [8,9]. In particular, dierences in the spreading of optical pulses observed in ber transmission are due to chirp noise. In this paper we present a statistical analysis of the spectral properties of the optical pulses emitted by a SML under fast PRWM. We characterize the relation between timing jitter and chirp noise for a repetitively modulated laser

3 and demonstrate that patterns in the modulation sequence and the resulting patterns in the timing jitter are the main cause of chirp noise. The relative advantages of dierent modulation schemes and dierent operating points of the laser are discussed for dierent detector bandwidths and digital decision circuitry. Previous work on PRWM focused on either timing jitter or chirp noise as separate issues. The time required by the SML to emit an optical pulse after an electrical pulse is applied (turn-on time) determines the maximum transmission rate for the system. But, for a SML under large-signal PRWM the turn-on time becomes a random quantity, which may cause errors in the optical codication stage. The statistical properties of the turn-on time (mean turn-on time and its standard deviation, or timing jitter) have been extensively studied both experimentally and theoretically [1]? [7]. In particular, the dependence of the mean turn-on time and timing jitter on the bias current [3,5{7] and on the modulation frequency [2,6,7] have been analyzed, showing that signal patterns and spontaneous emission noise contribute in rather different ways to the mean turn-on time and timing jitter [6,10]. Based on these analysis the use of a bias current slightly below threshold was proposed to minimize the sensitivity to pattern sequences, by reducing the timing uctuations in fast PRWM. The usefulness of this proposal has been demonstrated for modulation of ber lasers [11]. Chirp noise is another important factor in ber-based single frequency communication systems. It arises from the dependence of the laser frequency on the index of refraction which varies with the evolving carrier density as the SML is modulated [12]. This dynamic spectral broadening (laser frequency chirping) constitutes a span limitation for long-haul systems due to chromatic dispersion in the ber and it can also cause a sensitivity penalty in repeaters arising from inter-symbol interference [9,13].

4 For the SML under PRWM, the chirp range during each pulse uctuates randomly (chirp noise) which gives a bit-error-rate (BER) oor [8]. The work in [8] concluded that chirp noise originates from spontaneous emission noise which is largest when the laser is operated at or below threshold, but the dependences of the chirp noise on the modulation scheme or on the modulation rate were not analyzed. Since both timing jitter and chirp noise depend on spontaneous emission noise, some connection between them is expected. This relationship was analyzed for a gain-switched SML [14], where spontaneous emission noise is the only source of uctuations. It was also invoked [15] to explain uctuations observed in transmission of solitons generated by periodically gain switched lasers. For a SML under fast PRWM it is important to assess in which way patterns in the past modulation sequence contribute to the frequency chirp during the successive pulse [16]. In particular, the proposal in [6,10] of setting the bias value slightly below threshold to avoid the eect of patterns under fast PRWM might be detrimental from an spectral point of view since across-threshold modulation is expected to induce larger frequency chirping during the optical pulses, thereby increasing the chirp power penalty [9]. Also, the dependences of the turn-on time and the frequency chirp on the modulation scheme have to be analyzed in order to devise strategies that allow for a simultaneous minimization of the dierent BER sources. We consider in this paper the return-to-zero (RZ) and non-return-to-zero (NRZ) schemes for fast PRWM of a SML. We perform in both cases a statistical analysis as well as a worst case analysis of the frequency chirp during each optical pulse. Our analysis reveals that for both RZ and NRZ, the worst-case turn-on-time and chirp range occur for \1" bits preceeded by short sequences of \0" bits. The worst

5 case values of both turn-on-time and chirp range are similar for the two modulation schemes, for the same bias current and modulation frequency. Hence, the choice of one modulation scheme or the other depends mainly on the characteristics of the decision circuit and on the driver and detector bandwidths. If the the decision circuit is designed after the average pulses with some tolerance range, RZ oers better pulse waveform reproducibility and pattern eects can be suppressed by biasing the SML slightly below threshold. By contrast, if the decision circuit is designed to accept pulse waveforms up to a specied worst case limit, then NRZ minimizes the error rate, and a bias current slightly below threshold does not oer any signicant advantage. The paper is organized as follows: in Section II we present the model used to describe the system. In Section III we present our results for the spectral response of the laser, and we relate them to the temporal response. This section also contains a BER analysis for the two modulation schemes and previously unreported results for the temporal response of NRZ-PRWM of SML. Section IV contains a summary of the results and conclusions. II. MODEL Our analysis is based on the numerical integration of the SML rate equations including random terms that describe spontaneous emission noise and random nonradiative decay of the carriers [17,18], _E = 1 + q i (G? )E + 2N E (t) (1) 2 q q _N = C(t)? e N? GjEj 2? 2N (E E (t) + EE(t)) + 2 e N N (t) (2)

6 where G = g(n? N 0 )(1 + sjej 2 )?1, and units have been chosen so that the laser intensity I = jej 2 and N correspond to the number of photons and the number of minority carriers within the active layer, respectively. The meaning of the symbols and the values of the dierent parameters appearing in (1)-(2) are listed in Table I. E (t) is a complex Langevin noise term accounting for the stochastic nature of spontaneous emission and N (t) describes random non-radiative carrier recombination due to thermal uctuations; they are Gaussian noise terms of zero mean and correlations h i (t) j (t 0 )i = ij (t? t 0 ), where i; j denote Re( E ), Im( E ) and N. The threshold current (expressed as number of injected carriers per unit time) is C th = e (N 0 + =g) which corresponds to C th = 3:76 10 16 s?1 ( 6 ma) for the parameters considered. In our numerical simulations of (1) and (2), the current applied to the SML, C(t), is a random sequence of one thousand bits, either \0" or \1", with equal probability for the two symbols. The bit sequence is the same for all values of bias current and modulation frequency. The form of C(t) depends on the modulation scheme chosen. In RZ PRWM and for a \0" bit, the current stays constant at the bias level, C b. For the \1" bits, the current takes its \on" value, C on, during T on, and then drops to the bias level, where it stays during T of f = T? T on, T being the period of modulation. In NRZ, for a \0" bit the current stays at C b, and for a \1" bit the current stays at its \on" value C on, except for the case in which the preceeding bit is dierent. In all cases we take 5 ps for the rise- and fall-times for the current pulse, which for RZ are included in T on and T of f, respectively. In order to study the dependence of the laser response on the bias current, we x C on = 14 10 16 s?1 ( 3:7 C th ), and to examine the dependence on the modulation frequency in the RZ case, we take T on = 90 ps [6]. We have computed the eld power spectrum and the chirp range, as well as the

7 turn-on time if the bit is \1", during each modulation period. The eld power spectrum is calculated via a Fast Fourier Transform of the eld sampled at a rate of T =1024, and we consider the chirp range to be accurately described by the maximum frequency reached during the pulse, since the minimum frequency is almost constant for the \1" bits (xed by the minimum frequency during the relaxation oscillations in NRZ, and by the reference intensity in RZ [17]). III. RESULTS A. Statistical analysis of the spectral response A statistical characterization of the spectra is given in Fig. 1 in terms of the mean power spectrum (average of the eld power spectra over \1" bits) and its relative power uctuations (RPF) [17]. These results correspond to a modulation frequency f = 6:13 GHz and to bias levels C b =C th = 0:9, 0:983, 1:1 and 1:33. Zero frequency corresponds to the CW lasing frequency of the SML in the \on" state. The choice of bias values is suggested by the analysis of timing jitter and pattern eects for the RZ scheme in [6,10]. The value C b =C th = 0:983 corresponds to an optimum choice to avoid pattern eects at large modulation frequencies. In the RZ case (Figs.1.a, 1.c), the mean spectral width is quite similar for C b =C th = 0:9, 0:983 and 1:1, being markedly narrower only for C b = 1:33C th. However, the RPF is quite dierent for the dierent bias levels. The RPF exhibits in all cases a dip whose width roughly corresponds to the spectral width at -10 db, together with a peak on the high frequency wing of the spectrum. The RPF are almost the same outside the

8 central plateau of the average spectrum, but the RPF inside the plateau are smaller for C b = 0:983C th. For C b = 1:1C th, the RPF inside the plateau are enhanced, reecting the larger pulse-to-pulse variations of the intensity due to patterns in the data sequences, as is also the case for C b =C th = 1:33. In Figs. 1.b and 1.d we plot the average spectrum and RPF for the NRZ. It can be observed that the spectral width is narrower than in the RZ modulation scheme, as expected from the current pulses of larger duration. However, the results for NRZ deserve some detailed analysis. In Fig.2, we plot separately the eld power spectrum averaged over the \1" bits preceeded by a \1" (Fig.2.a) and over the \1" bits preceeded by a \0" bit (Fig.2.b). Fig.1.b can be understood as the weighted average of Fig.2.a and Fig.2.b and also Fig.1.d corresponds to the weighted average of Fig.2.c and Fig.2.d. These results show that the existence of a better dened central frequency in the NRZ scheme, as compared with the RZ scheme, is due to the fact that, on average, half of the \1" bits are preceeded by another \1" bit, so that half of the pulses have no chirp, which yields a narrower spectrum in NRZ than in RZ. In addition is worth noting that the \1" bits preceeded by \0" bits display an average spectrum very similar to the one obtained in the RZ case. However, the NRZ scheme allows the optical pulses to almost reach a constant power level, which yields a more peaked spectrum at low frequencies together with secondary peaks which are the signature of the transient relaxation oscillations. As a consequence, the characteristic plateaux in RZ spectra are strongly reduced in NRZ. Moreover the RPF are very dierent for the two kinds of \1" bits (see Figs. 2.c and 2.d). For the second \1" bit in \11" sequences, the RPF display a at background corresponding to spontaneous emission noise with a very sharp dip at zero frequency

9 characteristic of steady state operation. However, for the \1" bit in \01" sequences, we nd that RPF inside the plateau are strongly enhanced with respect to the RZ case. The origin of this larger RPF is that, for NRZ modulation there is a long time interval in each \1" bit during which the laser frequency is almost the stationary one, hence spontaneous emission noise contributes to the low frequency spectrum. Oppositely, for fast RZ modulation the laser frequency during the \1" bits is continuously evolving, so that frequency chirping avoids the contribution of spontaneous emission noise to low frequencies. In addition, it should be noted that for NRZ, the minimum RPF inside the "plateau" now corresponds to C b = 1:33 C th, while for RZ it corresponds to C b = 0:983 C th. We show the results obtained for two lower modulation frequencies, f = 3:05GHz and f = 1:53GHz in Fig.3 and Fig.4. It is clear from these gures that the higher the modulation frequency the higher the bias level required to signicantly reduce the chirp range. Based on this spectral analysis, it might seem that NRZ PRWM is better suited for ber-based communication systems because of the narrower average spectrum. However, in fast PRWM the system performance is generally set by a \worst case" pulse [8] which, as we next discuss, yields rather dierent conclusions. The previous analysis reveals that the \worst case" will correspond to sequences of a \1" bit preceeded by a \0" bit.

10 B. Worst case analysis of the SML response Eye diagrams for a random stream of 1000 bits are shown in Fig.5 (RZ) and Fig.6 (NRZ) for dierent bias currents and modulation frequencies. It is seen that for both RZ and NRZ PRWM pattern eects appear for all values of C b except for C b = 0:983 C th. For this bias current, the temporal spread of the pulses is minimized, in agreement with [6,10]. Two contributions to the temporal spread of the optical pulses can be distinguished: on one hand, the clustering of the optical pulses according to their starting conditions that yields the pattern eects; on the other hand, within each cluster some spread is observed due to the jitter induced by spontaneous emission noise. It is worth mentioning that, for bias currents C b = 1:1C th and C b = 0:983C th, the worst case pulses have very similar turn-on times for both RZ and NRZ PRWM. The worst-case turn-on time, and hence the resulting temporal eye-closure, can be reduced by choosing a bias current well above threshold, like C b = 1:33C th, but in this case the increased intensity level of the \0" bits together with the relaxation oscillations yield a poorer \on/o" ratio as well as a vertical eye-closure which place stronger constraints on the selection of the decision level in a direct detection scheme. Therefore, the main advantage of NRZ as compared to RZ PRWM is that it relaxes the requirements for the sampling time, though a RZ modulation yields better on/o ratios together with a higher uniformity of the optical pulseforms. The eye diagrams yield only qualitative information on the spectral characteristics of the optical pulses. In order to perform a worst case analysis from the spectral point of view, we characterize each optical pulse by its turn-on time. We also note that the amount of frequency chirp during a pulse is correctly described by the maximum frequency reached during this pulse [17], (the minimum frequency is nearly the same

11 for each pulse). In Figs. 7 and 8 we plot the maximum frequency during the pulse for the \1" bits vs the pulse turn-on delay for the RZ and NRZ modulation schemes. The turn-on time is dened as the time at which the laser intensity rst surpasses a given reference value. We take the reference intensity to be 5% of the steady-state intensity corresponding to C on for bias currents C b < C th ; if C b > C th, we take the reference value for the laser intensity to be 50% of the dierence between the steadystate intensities corresponding to C on and C b, since in this way we avoid errors due to spurious turn-on during \0" bits induced by relaxation oscillations. With this denition for the turn-on time, it is obvious that the second \1" bit in a NRZ \11" sequence has a null turn-on time. Pattern eects are evident as a clustering of the maximum frecuency of the optical pulses according to the bit sequence considered. These eects are especially clear for bias currents above threshold (cases c) and d) in Figs.7 and 8), where there is little overlapping between the dierent clusters. For bias currents below threshold (case a) in Figs. 7 and 8), pattern eects in the chirp range also exist, but they are masked by the contribution of spontaneous emission noise which yields an almost linear relation between the maximum frequency during each pulse and its turn-on time, as in the case of a gain-switched SML [14]. For the bias value corresponding to Figs.7.b and 8.b, pattern eects are suppressed and the maximum frequency is a linear function of the turn-on-time independent of the bit sequence. The worst case pulses always correspond to \1" bits preceeded by either one or two \0" bits whatever the modulation scheme. The reason is that, for fast PRWM, after an optical pulse is emitted, the carrier density drops to a value below threshold and due to its long relaxation time, it is unable to recover before the next electrical pulse is applied unless a long enough sequence of \0" bits follows [10]. As a consequence,

12 in spite of a nominal bias current above threshold, the eective initial conditions for the next optical pulse in general correspond to biasing below threshold. Hence, for these bit sequences, the build-up of the optical pulse is dominated by spontaneous emission which yields long turn-on times and large chirp ranges, together with strong uctuations in these quantities. For the \01" sequences, the turn-on time and the maximum frequency during the \1" bit for xed bias current are almost equal for RZ and NRZ, though they are slightly larger for NRZ due to the fact that for RZ the carrier density has some extra time to recover as compared to NRZ. The major dierence in the response of the SML between RZ and NRZ is related to the second bit in \11" sequences, which in a NRZ scheme has vanishing turnon time and no chirp. This is simply due to the dierent current pulseform, but it implies that dispersion will aect NRZ pulses in rather dierent way according to the preceeding bit sequence, since in NRZ, pulses with zero turn-on time and null chirp coexist with pulses with long turn-on times and large chirp ranges. Instead, for RZ, turn-on times and chirp ranges are more uniform among the dierent pulses and bit sequences. Another important point to be characterized is the dependence of the SML response on the bias current for a xed modulation scheme which is summarized in table II and table III. For bias currents close to threshold, the worst case turn-on time and chirp range are slightly larger for C b =C th = 0:983 than for C b =C th = 1:1. However, the pulses are more uniform for bias currents slightly below threshold, and the corresponding variation in turn-on time and chirp range among the dierent pulses are lesser for C b =C th = 0:983 than C b =C th = 1:1. The preceeding discussion applies to the case of fast PRWM (f > 5 GHz). Bias currents above threshold yield better SML

13 responses as the modulation frequency is lowered, as it is seen comparing the results in tables IV and V with those of tables II and III. We also note that a reduced - factor gives a reduced value of the chirp range, as is expected, without modifying the turn-on time statistics. Results for = 2 and high modulation frequency are shown in tables VI and VII. The chirp range is reduced, for any bias current, in a factor roughly equal to the ratio of the -factors. The chirp noise shows a lesser reduction due to memory eects. In order to estimate the probability of error in the bit emission from the SML, we have calculated the probability of error as [20] P e (T ) = 1=2 [1? erf(snr(t )=2)] where erf(x) is the error function and T is the sampling time measured respect to the start of the electrical pulse. SNR(T ) is the signal to noise ratio at the sampling time, dened as the quotient of the mean square value of the signal divided by the mean square value of the noise. In Fig. 9, we plot P e (T ) for both NRZ and RZ and dierent bias currents. These results show that in the NRZ modulation scheme, the lowest P e is obtained for a sampling time window placed where the steady state for bits \1" and \0" has been reached. There are other time windows, between 1=3 and 1=2 of the modulation period depending on the bias level, where P e presents a secondary minimum, the lowest value of this minimum corresponding to the special bias current. The behavior for RZ is essentially dierent, with the lowest value at the minimum corresponding to the highest bias current, C b =C th = 1:4. However, it must be emphasized that the above standard calculation of the BER is not reliable since it is based on the assuption of Gaussian noise. This assumption is not fullled by realistic optical pulses emitted by the SML wathever the modulation scheme, except

14 perhaps at the very end of the time slot for each bit in NRZ. Due to this limitation of the above standard error calculation, other possibilities have been considered. If almost instantaneous sampling is permitted, then an erroneous bit will be sent whenever we sample a \1" at a time when it has a photon number lower than a certain decission level, or when a \0"bit is sampled at a time when it has too a high photon number. If instantaneous sampling is not possible, one can yet dene the BER in the same fashion, but considering the integrated photon number for the bit instead of the instantaneous photon number. These two alternative denitions of BER are related to the consideration of wort-case bits: a \1" will have a low photon number (integrated or not) only if its turn-on time is very late. However, a reliable calculation of BER from any of these two worst-case denitions requires either the consideration of a extremely high number of bits, or a detailed knowledge of the tails of the turn-on time probability density function, both of which are well beyond the scope of this paper. IV. SUMMARY AND CONCLUSIONS We have characterized the temporal and spectral response of a SML to PRWM as a function of the bias current, modulation frequency and the modulation scheme from both a statistical and a worst-case point of view. We have shown that pattern eects contribute to interpulse chirp noise in the same way that they contribute to timing jitter of the pulses. Though from a statistical analysis NRZ may seem to yield better SML response than RZ, the worst case turn-on time and chirp range are very similar for the two

15 kinds of PRWM at high speeds. The worst case bits are the \1" bits preceeded by short sequences of \0" bits (one or two zeros, typically), which will suer a similar distortion due to ber dispersion whatever the modulation scheme. For low modulation frequencies (f < 2 GHz), biasing 10% above threshold yields a good response both in time and frequency, with almost complete opening of the eyediagrams and quite narrow power spectra, though some pattern eects are noticeable. For higher modulation frequencies, strong pattern eects appear at this bias level which degrade both the temporal and spectral response of the SML because of an increase in the timing jitter and the interpulse chirp noise. These memory eects can be reduced in NRZ PRWM (or even completely suppressed in RZ PRWM) by setting the bias current slightly below threshold with only a minor increase in both the worst case turn-on time and chirp range. From the previous analysis, the main advantage of a NRZ scheme is that it requires a lower bandwidth both in the driver (due to the longer current pulseforms) and in the detector (due to the longer sampling times available). This advantage is partly oset in a RZ modulation scheme by the higher uniformity of the optical pulses. For RZ, the spread of turn-on times and chirp ranges among the dierent pulses is less than in NRZ, hence the pulses are more similar at the ber output for RZ than for NRZ. If the design of the decision circuit is adjusted after the average pulses as in [8], RZ PRWM with a bias current slightly below threshold oers a better pulseform reproducibility at the cost of an increased detector and driver bandwidth. However, if the decision circuit is designed to accept all pulseforms up to a specied worst-case limit, then NRZ PRWM is advantageous. In this case, a bias current slightly below threshold does not oer any advantage over a bias current above threshold since the

16 worst case turn-on time and chirp range are larger; it simply reduces the variability of the \01" sequences, but not the overall range of variations. V. ACKNOWLEDGMENT Financial support from Comision Interministerial de Ciencia y Tecnologia (CI- CYT) Project TIC93-0744 is acknowledged. We thank Neal Abraham and Pere Colet for useful comments.

17 REFERENCES [1] M. M. Choy, P. L. Liu, P. W. Shumate, T. P. Lee and S. Tsuji, "Measurement of Dynamic Photon Fluctuations in a Directly Modulated 1.5 m InGaAsP Distributed Feedback Laser", Appl. Phys. Lett. 47, 448-449 (1985). [2] E. H. Bottcher, K. Ketterer and D. Bimberg, "Turn-on Delay Time Fluctuations in Gain-Switched AlGaAs/GaAs Multiple-Quantum-Well Lasers", J. Appl. Phys. 63, 2469-2471, (1988). [3] T. M. Shen, "Timing Jitter In Semiconductor Lasers under Pseudorandom Word Modulation", J. Lightwave Technol. 7, 1394-1399 (1989). [4] C. R. Mirasso, A. Valle, L. Pesquera and P. Colet, "Simple method for estimating the memory diagram in single mode semiconductor lasers", IEE Proc.- Optoelectron., 141, 109-113, (1994). [5] A. Mecozzi, P. Spano, A. D'Ottavi and S. Piazzolla, "Analysis of Transients in Pulse Modulated Semiconductor Lasers Biased Near Threshold", Appl. Phys. Lett. 55, 769-771 (1989). [6] C. Mirasso, P. Colet and M. San Miguel, "Pulse Statistics in Single-mode Semiconductor Lasers Modulated at Gigahertz Rates", Optics Lett. 16, 1753-1755 (1991);, "Dependence of Timing Jitter on Bias Level for Single-Mode Semiconductor Lasers under High Speed Operation", IEEE J. Quantum Electron. 29, 23-32 (1993). [7] A. Sapia, P. Spano, C. R. Mirasso, P. Colet and M. San Miguel, " Pattern Eects in Time Jitter of Semiconductor Lasers", Appl. Phys. Lett. 61, 1748-1750 (1991).

18 [8] P. O. Andersson and K. Akermak, "Generation of BER oors from Laser Diode Chirp Noise", Electron. Lett. 28, 472-474 (1992). [9] S. Yamamoto, M. Kuwazuru, H. Wakabayashi and Y. Iwamoto, "Analysis of Chirp Power Penalty in 1.55 m DFB-LD High-speed Optical Fiber Transmission Systems", J. Lightwave Technol. LT-5, 1518-1524 (1987). [10] P. Colet, C. Mirasso and M. San Miguel, "Memory Diagram of Single-Mode Semiconductor Lasers", IEEE J. Quantum Electron. 29, 1624-1630 (1993). [11] J. C. Garreau, P. Y. Wang and P. Glorieux, "Bit Correlation and Memory Eects in High Speed Pump Modulation of a Fiber Laser", IEEE J. Quantum Electron. 30, 1058-1075 (1994). [12] G. P. Agrawal and N. K. Dutta, Long-Wavelength Semiconductor Lasers, (van Nostrand-Reinhold, New York 1986). [13] G. P. Agrawal, Nonlinear Fiber Optics, (Academic Press, Inc., San Diego, 1989). [14] S. Balle, P. Colet and M. San Miguel, "Statistics of Transient Response of a Single-Mode Semiconductor Laser Gain-Switching", Phys. Rev. A 43, 498-506 (1991). [15] Claudio R. Mirasso, Luis Pesquera and Antonio Mecozzi, "Pulse-to-Pulse Frequency Jitter in Diode Lasers and Soliton Transmission", IEEE Photonics Technology Letters,5, 1455-1458 (1993). [16] S. Balle, M. Homar and M. San Miguel, "Minimization of chirp noise and timing jitter of light pulses from a single-mode semiconductor laser under fast RZ-PRW modulation", Paper CWJ39, OSA/CLEO'93, May 2-7 1993, Baltimore, MD, USA;, "Statistical properties of the spectrum of light pulses in PRW NRZ

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20 TABLES TABLE I. Meaning and values of the dierent parameters appearing in the model Parameter Meaning Value Units g Gain parameter 5.6 10 4 s?1 0 Inverse photon lifetime 4 10 11 s?1 e Inverse carrier lifetime 5 10 8 s?1 n 0 Carrier number at transparency 6.8 10 7 s Gain saturation factor 6 10?7 Enhancement linewidth factor 6 Spontaneous emission rate 1.110 4 s?1! t Tranparency frequency 4 10 5 rad ns?1 J Injection current 710 16 s?1 TABLE II. Average turn-on time delay, timing jitter, average chirp and chirp noise at a f = 6:13 GHz as a function of the bias current for RZ PWRM. = 6. C b =C th Turn-on time (ps) Timing jitter (ps) Chirp (GHz) Chirp noise (GHz) 0.905 61.30 10.36 90.56 13.04 0.983 54.54 6.43 88.89 12.48 1.1 50.24 8.05 69.35 15.28 1.33 34.87 8.74 40.99 8.96 TABLE III. Average turn-on time delay, timing jitter, average chirp and chirp noise at a 6:13 GHz as a fuction of the bias current for NRZ PRWM. = 6. C b =C th Turn-on time (ps) Timing jitter (ps) Chirp (GHz) Chirp noise (GHz) 0.905 32.45 33.04 50.32 48.58 0.983 28.65 29.15 50.52 49.49 1.1 25.63 26.78 40.03 40.86 1.33 16.83 17.15 22.79 23.49

21 TABLE IV. Average turn-on time delay, timing jitter, average chirp and chirp noise at a f = 3:05 GHz as a function of the bias current for RZ PWRM. = 6. C b =C th Turn-on time (ps) Timing jitter (ps) Chirp (GHz) Chirp noise (GHz) 0.905 72.78 9.53 98.84 9.20 0.983 59.98 5.07 99.27 9.89 1.1 44.79 12.93 67.43 25.03 1.33 26.25 4.43 31.33 3.26 TABLE V. Average turn-on time delay, timing jitter, average chirp and chirp noise at a 3:05 GHz as a fuction of the bias current for NRZ PRWM. = 6. C b =C th Turn-on time (ps) Timing jitter (ps) Chirp (GHz) Chirp noise (GHz) 0.905 36.25 37.16 50.21 49.68 0.983 28.42 28.86 49.78 49.31 1.1 21.53 22.66 33.28 35.73 1.33 15.91 16.09 17.83 16.66 TABLE VI. Average turn-on time delay, timing jitter, average chirp and chirp noise at a f = 6:13 GHz as a function of the bias current for RZ PWRM. = 2. C b =C th Turn-on time (ps) Timing jitter (ps) Chirp (GHz) Chirp noise (GHz) 0.905 62.00 11.03 36.55 5.23 0.983 54.78 6.69 35.67 5.19 1.1 52.10 8.02 28.93 5.82 1.33 34.92 8.79 16.50 3.61 TABLE VII. Average turn-on time delay, timing jitter, average chirp and chirp noise at a 6:13 GHz as a fuction of the bias current for NRZ PRWM. = 2. C b =C th Turn-on time (ps) Timing jitter (ps) Chirp (GHz) Chirp noise (GHz) 0.905 33.05 33.54 20.66 19.67 0.983 28.75 29.01 20.40 19.59 1.1 25.96 27.03 16.41 16.41 1.33 16.90 17.09 9.38 9.23

22 FIGURES FIG. 1. : Mean power spectrum (left column) and relative power uctuations (right column) at modulation frequency of f = 6:13 GHz for RZ modulation (top row), and NRZ (bottom row). In all cases, C b =C th = 0.9 (solid), 0.983 (dotted), 1.1 (dashed), and 1.33 (dot-dashed). FIG. 2. :Mean power spectrum (left column) and relative power uctuations (right column) at modulation frequency of f = 6:13 GHz for NRZ modulation for: "1" bits preceeded by a "1" (a, c); "1" bits preceeded by a "0" (b, d). FIG. 3. : Mean power spectrum (left column) and relative power uctuations (right column) at modulation frequency of f = 3:05 GHz for RZ modulation (a, c), and NRZ (b, d). FIG. 4. : Mean power spectrum (left column) and relative power uctuations (right column) at modulation frequency of f = 1:53 GHz for RZ modulation (a, c), and NRZ (b, d). FIG. 5. : Eye diagram for a random stream of 1000 bits for RZ modulation scheme. The modulation frequency associated with dierent T of f is indicated at the top and the dierent bias current values at the left. T on is xed at the value of T on = 90 ps. FIG. 6. : Same as in Fig.5 but for NRZ modulation scheme. FIG. 7. a: Chirp range vs pulse emission time for f = 6:13 GHz and the same bias currents than in g. 1 for RZ modulation scheme. The dierent symbols correspond to dierent bit sequences preceding the considered "1" bit: stars, 1111; diamonds, 0111; 1011; +, 0011; squares, 0101; triangles, 1101; x, 1001; solid line, at least three "0" bits before the "1" bit. FIG. 8. : Same as in Fig.7 for NRZ.