% Figures corresponding to this LaTeX file are fig1.ps...fig9.ps; % all this files are in the directory % http://formentor.uib.es/~marga/public_html/papers/chirp \documentstyle[aps,preprint] {revtex} \def\baselinestretch{1.2} \textheight 8.5in \topmargin .0in \textwidth 6in \oddsidemargin .25in \evensidemargin 0in \parskip .25in \pagestyle{myheadings} \begin{document} \baselineskip 24pt \title{Statistical properties of the spectrum of light pulses in fast pseudo random word modulation of a single-mode semiconductor laser} \author{S. Balle, M. Homar and M. San Miguel\\ Departament de F\'{\i}sica, Universitat de les Illes Balears\\ E--07071 Palma de Mallorca, Spain\\ Phone: 34--71--173234, Fax: 34-71--438028} \date{~} \maketitle \newpage \begin{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 turn-on 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. \end{abstract} \section{Introduction} The optimum performance of high capacity, long-haul fiber-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 \cite{Choy1}$-$\cite{APLBordoni} and chirp noise \cite{Andersson,Yamamoto}. In particular, differences in the spreading of optical pulses observed in fiber 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 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 different modulation schemes and different operating points of the laser are discussed for different 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 codification 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 \cite{Choy1}$-$\cite{APLBordoni}. In particular, the dependence of the mean turn-on time and timing jitter on the bias current \cite{Shen,Mecozzi,Claudio,APLBordoni} and on the modulation frequency \cite{Bottcher,Claudio,APLBordoni} 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 \cite{Claudio,Pere}. 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 fluctuations in fast PRWM. The usefulness of this proposal has been demonstrated for modulation of fiber lasers \cite{glorioso}. Chirp noise is another important factor in fiber-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 \cite{Ag-Dut}. This dynamic spectral broadening (laser frequency chirping) constitutes a span limitation for long-haul systems due to chromatic dispersion in the fiber and it can also cause a sensitivity penalty in repeaters arising from inter-symbol interference \cite{Yamamoto,Agrawal-fib}. For the SML under PRWM, the chirp range during each pulse fluctuates randomly (chirp noise) which gives a bit-error-rate (BER) floor \cite{Andersson}. The work in \cite{Andersson} 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 \cite{Salvador-Pere}, where spontaneous emission noise is the only source of fluctuations. It was also invoked \cite{ptl} to explain fluctuations 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 \cite{cleo}. In particular, the proposal in \cite{Claudio,Pere} of setting the bias value slightly below threshold to avoid the effect 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 \cite{Yamamoto}. 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 different 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 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 offers better pulse waveform reproducibility and pattern effects 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 specified worst case limit, then NRZ minimizes the error rate, and a bias current slightly below threshold does not offer any significant 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. \section{Model} Our analysis is based on the numerical integration of the SML rate equations including random terms that describe spontaneous emission noise and random non-radiative decay of the carriers \cite{Balle,peter}, \begin{eqnarray} \dot E &=& \frac{ 1 + i\alpha}{2}( G - \gamma)E + \sqrt{2\beta N}\xi_{E}(t) \label{eqnarray:camp} \\ \dot N &=& C(t) - \gamma_{e} N - G |E|^{2} - \sqrt{2 \beta N} \left( E^* \xi_{E}(t) + E \xi^*_{E}(t) \right) + \sqrt{2\gamma_{e}N}\xi_{N}(t) \label{eqnarray:port} \end{eqnarray} where $G = g(N - N_{0})(1 + s|E|^{2})^{-1}$, and units have been chosen so that the laser intensity $I=\vert E\vert^2$ 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 different parameters appearing in (1)-(2) are listed in Table~I. $\xi_{E}(t)$ is a complex Langevin noise term accounting for the stochastic nature of spontaneous emission and $\xi_{N}(t)$ describes random non-radiative carrier recombination due to thermal fluctuations; they are Gaussian noise terms of zero mean and correlations $\langle \xi_{i}(t) \xi_{j}(t') \rangle = \delta_{ij} \delta(t - t')$, where $i,\ j$ denote $Re(\xi_E)$, $Im(\xi_E)$ and $\xi_N$. The threshold current (expressed as number of injected carriers per unit time) is $C_{th} = \gamma_{e} ( N_{0} + \gamma/g )$ which corresponds to $C_{th}=3.76\times 10^{16} s^{-1}$ ($\approx$ 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 $T_{off} = T - T_{on}$, $T$ 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 different. 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 $T_{off}$, respectively. In order to study the dependence of the laser response on the bias current, we fix $C_{on}=14 \times 10^{16}\ s^{-1} \ (\approx 3.7\ C_{th})$, and to examine the dependence on the modulation frequency in the RZ case, we take $T_{on} = 90$~ps \cite{Claudio}. We have computed the field power spectrum and the chirp range, as well as the turn-on time if the bit is ``1'', during each modulation period. The field power spectrum is calculated via a Fast Fourier Transform of the field 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 (fixed by the minimum frequency during the relaxation oscillations in NRZ, and by the reference intensity in RZ \cite{Balle}). \section{Results} \subsection{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 field power spectra over ``1'' bits) and its relative power fluctuations (RPF) \cite{Balle}. 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 effects for the RZ scheme in \cite{Claudio,Pere}. The value $C_b/C_{th}=0.983$ corresponds to an optimum choice to avoid pattern effects 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 $C_b = 1.33 C_{th}$. However, the RPF is quite different for the different 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 central plateau of the average spectrum, but the RPF inside the plateau are smaller for $C_b = 0.983 C_{th}$. For $C_b = 1.1 C_{th}$, the RPF inside the plateau are enhanced, reflecting 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 field 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 defined 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 different 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 flat background corresponding to spontaneous emission noise with a very sharp dip at zero frequency characteristic of steady state operation. However, for the ``1'' bit in ``01'' sequences, we find 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, $f = 3.05 GHz$ and $f = 1.53GHz$ in Fig.3 and Fig.4. It is clear from these figures that the higher the modulation frequency the higher the bias level required to significantly reduce the chirp range. Based on this spectral analysis, it might seem that NRZ PRWM is better suited for fiber-based communication systems because of the narrower {\it average} spectrum. However, in fast PRWM the system performance is generally set by a ``worst case'' pulse \cite{Andersson} which, as we next discuss, yields rather different conclusions. The previous analysis reveals that the ``worst case'' will correspond to sequences of a ``1'' bit preceeded by a ``0'' bit. \subsection{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 different bias currents and modulation frequencies. It is seen that for both RZ and NRZ PRWM pattern effects 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 \cite{Claudio,Pere}. 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 effects; 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 $C_b = 1.1 C_{th}$ and $C_b = 0.983 C_{th}$, 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 $C_b = 1.33 C_{th}$, but in this case the increased intensity level of the ``0'' bits together with the relaxation oscillations yield a poorer ``on/off'' 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/off 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 \cite{Balle}, (the minimum frequency is nearly the same 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 defined as the time at which the laser intensity first 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 difference between the steady-state 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 definition 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 effects are evident as a clustering of the maximum frecuency of the optical pulses according to the bit sequence considered. These effects are especially clear for bias currents above threshold (cases c) and d) in Figs.7 and 8), where there is little overlapping between the different clusters. For bias currents below threshold (case a) in Figs. 7 and 8), pattern effects 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 \cite{Salvador-Pere}. For the bias value corresponding to Figs.7.b and 8.b, pattern effects 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 \cite{Pere}. As a consequence, in spite of a nominal bias current above threshold, the {\it effective} 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 fluctuations in these quantities. For the ``01'' sequences, the turn-on time and the maximum frequency during the ``1'' bit ---for fixed 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 difference 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 turn-on time and no chirp. This is simply due to the different current pulseform, but it implies that dispersion will affect NRZ pulses in rather different 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 different pulses and bit sequences. Another important point to be characterized is the dependence of the SML response on the bias current for a fixed 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 different 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 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 $\alpha$-factor gives a reduced value of the chirp range, as is expected, without modifying the turn-on time statistics. Results for $\alpha=2$ 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 $\alpha$-factors. The chirp noise shows a lesser reduction due to memory effects. In order to estimate the probability of error in the bit emission from the SML, we have calculated the probability of error as \cite{Marshall} \begin{eqnarray*} P_e(T)=1/2 \left[1-erf(SNR(T)/2) \right] \end{eqnarray*} 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, defined 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 different 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 different, 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 fulfilled by realistic optical pulses emitted by the SML wathever the modulation scheme, except 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 define the BER in the same fashion, but considering the {\it integrated} photon number for the bit instead of the instantaneous photon number. These two alternative definitions 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 definitions 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. \section{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 effects 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 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 suffer a similar distortion due to fiber 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 eye-diagrams and quite narrow power spectra, though some pattern effects are noticeable. For higher modulation frequencies, strong pattern effects 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 effects 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 offset 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 different pulses is less than in NRZ, hence the pulses are more similar at the fiber output for RZ than for NRZ. If the design of the decision circuit is adjusted after the average pulses as in \cite{Andersson}, RZ PRWM with a bias current slightly below threshold offers 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 specified worst-case limit, then NRZ PRWM is advantageous. 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Marshall, {\sl Principles of Digital Communications}, (McGraw-Hill, Maidenhead (UK), 1980) \end{thebibliography} \begin{table} \caption{Meaning and values of the different parameters appearing in the model} \begin{tabular}{|c|c|c|c|} Parameter & Meaning & Value & Units \\ \hline \hline g & Gain parameter & 5.6 $\times 10^4$ & $s^{-1}$ \\ $\gamma_{0}$ & Inverse photon lifetime & 4 $\times 10^{11}$ & $s^{-1}$ \\ $\gamma_e$ & Inverse carrier lifetime & 5 $\times 10^{8}$ & $s^{-1}$ \\ $n_0$ & Carrier number at transparency & 6.8 $\times 10^7$ & ---\\ s & Gain saturation factor & 6 $\times 10^{-7}$ & --- \\ $\alpha$ & Enhancement linewidth factor & 6 & ---\\ $\beta$ & Spontaneous emission rate & 1.1$\times 10^{4}$ & $s^{-1}$ \\ $\omega_{t}$ & Tranparency frequency & 4 $\pi\times 10^{5}$ rad & $ns^{-1}$ \\ J & Injection current & 7$\times 10^{16}$ & $s^{-1}$ \\ \end{tabular} \end{table} \begin{table} \caption {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. $\alpha = 6$.} \begin{tabular} {|c|c|c|c|c|} $C_b/C_{th}$ & Turn-on time (ps) & Timing jitter (ps) & Chirp (GHz) & Chirp noise (GHz)\\ \hline \hline 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 \\ \end{tabular} \end{table} \begin{table} \caption {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. $\alpha = 6$.} \begin{tabular} {|c|c|c|c|c|} $C_b/C_{th}$ & Turn-on time (ps) & Timing jitter (ps)& Chirp (GHz) & Chirp noise (GHz) \\ \hline \hline 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 \\ \end{tabular} \end{table} \begin{table} \caption {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. $\alpha = 6$.} \begin{tabular} {|c|c|c|c|c|} $C_b/C_{th}$ & Turn-on time (ps) & Timing jitter (ps) & Chirp (GHz) & Chirp noise (GHz)\\ \hline \hline 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 \\ \end{tabular} \end{table} \begin{table} \caption {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. $\alpha = 6$.} \begin{tabular} {|c|c|c|c|c|} $C_b/C_{th}$ & Turn-on time (ps) & Timing jitter (ps)& Chirp (GHz) & Chirp noise (GHz) \\ \hline \hline 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 \\ \end{tabular} \end{table} \begin{table} \caption {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. $\alpha = 2$.} \begin{tabular} {|c|c|c|c|c|} $C_b/C_{th}$ & Turn-on time (ps) & Timing jitter (ps) & Chirp (GHz) & Chirp noise (GHz)\\ \hline \hline 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 \\ \end{tabular} \end{table} \begin{table} \caption {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. $\alpha = 2$. } \begin{tabular} {|c|c|c|c|c|} $C_b/C_{th}$ & Turn-on time (ps) & Timing jitter (ps)& Chirp (GHz) & Chirp noise (GHz) \\ \hline \hline 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 \\ \end{tabular} \end{table} \newpage \begin{figure} \caption{: Mean power spectrum (left column) and relative power fluctuations (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).} \end{figure} \begin{figure} \caption{:Mean power spectrum (left column) and relative power fluctuations (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).} \end{figure} \begin{figure} \caption{: Mean power spectrum (left column) and relative power fluctuations (right column) at modulation frequency of $f = 3.05$ GHz for RZ modulation (a, c), and NRZ (b, d).} \end{figure} \begin{figure} \caption{: Mean power spectrum (left column) and relative power fluctuations (right column) at modulation frequency of $f = 1.53$ GHz for RZ modulation (a, c), and NRZ (b, d).} \end{figure} \begin{figure} \caption{: Eye diagram for a random stream of 1000 bits for RZ modulation scheme. The modulation frequency associated with different $T_{off}$ is indicated at the top and the different bias current values at the left. $T_{on}$ is fixed at the value of $T_{on}=90$ ps.} \end{figure} \begin{figure} \caption{: Same as in Fig.5 but for NRZ modulation scheme.} \end{figure} \begin{figure} \caption{a: Chirp range vs pulse emission time for $f = 6.13$ GHz and the same bias currents than in fig. 1 for RZ modulation scheme. The different symbols correspond to different bit sequences preceding the considered "1" bit: stars, 1111; diamonds, 0111; $\cdot$ 1011; +, 0011; squares, 0101; triangles, 1101; x, 1001; solid line, at least three "0" bits before the "1" bit.} \end{figure} \begin{figure} \caption{: Same as in Fig.7 for NRZ.} \end{figure} \end{document}