Turn-on-time statistics of modulated lasers subjected to resonant weak optical feedback

We present analytical calculations of the turn-on-time probability distribution of intensity-modulated lasers under resonant weak optical feedback. Under resonant conditions, the external cavity round-trip time is taken to be equal to the modulation period. The probability distribution of the solitary laser results are modified to give reduced values of the mean turn-on-time and its variance. Numerical simulations have been carried out showing good agreement with the analytical results.


I. INTRODUCTION
Lasers subjected to external feedback have been exten- sively studied in the last years.Light fed back into the cavity greatly modifies the steady-state operation of the solitary la- ser, giving frequency shifts [1], multistability [2] and low- [3] and high- [4] frequency fluctuations, depending on the amount of light reinjected.Moderate to strong feedback lev- els drive the laser dynamics through different routes to chaos [5] to a regime of coherence collapse [6].Controlled in- phase weak optical feedback provides a well-known method for linewidth reduction and output stabilization in lasers [7].
Q switching of gas, dye, or solid-state lasers is a common setup for narrow, high peak power pulse generation [8,9].The losses are periodically modulated by means of tilted mirrors, saturable absorber, acousto-optics or electroabsorption modulators, and others.By a careful experimental ar- rangement, high repetition rates have been obtained [10].
Gain-switched semiconductor lasers provide power peaks of several mW at GHz modulation rates.The material gain is periodically driven lower to higher values by means of an increase of the current injection through the diode junction [11].Synchronization of the different pulses is of special importance in practical applications.Desynchronized pulses cause a degradation of the temporal resolution and act as a limiting factor in high-bit rate optical communication sys- tems, where semiconductor lasers are used as light sources [12].Some works considered the transient dynamics of the la- ser.Analytical calculations of turn-on-time statistics are available for lasers in which the polarization (class-B lasers) [13], or polarization and population inversion (class-A la- sers) [14] could be adiabatically eliminated.Transient dy- namics of lasers subjected to feedback have also been ana- *Present address: School of Electronic and Electrical Engineering, University of Bath, Claverton Down, Bath BA2 7AY, United King- dom.
Present address: Institute for Nonlinear Science, 0402, University of California, San Diego, La Jolla, CA 92093-0340.lyzed .Frequency selection during the switch-on [15], mean turn-on-time and its variance were calculated for gainand Q-switched conditions with short external cavities [16,17], where the theory is valid.Only numerical simulations of the equations describing time evolution have been done to study the transient behavior of the laser under periodic or pseudo- random word modulations [18,19].
In this paper we study the transient behavior of both class-A and -B lasers externally modulated, whether in losses or in gain, and subjected to weak optical feedback.We consider the special, but interesting situation, in which the period of the modulation coincides with the external cavity round-trip time, for any cavity length.Under this condition, we calculate the mean switch-on time and its variance.The most important assumption is that the field fed back into the laser cavity has the same statistical properties as the one that is generated at that time.This resonance condition is particu- larly interesting.The light pulse that is building up receives an incoming feedback from the previous light pulse, which we assume as statistically identical.The situation is very different to other feedback conditions analytically studied in previous works, where the light fed back was assumed to have their origin in the same light pulse [16,17,19].This assumption allows us to perform analytical calculations which, as will be shown, are in good agreement with the numerical simulation of noise driven rate equations.The pa- per is organized as follows.In Secs.II and III, analytical results for class-A and -B lasers, respectively, are presented.Numerical solutions and discussion of the results are given in Sec.IV.A summary and conclusions are given in Sec.V.

II. ANALYTICAL RESULTS: CLASS-A LASERS
We describe the dynamics of a class-A laser by means of the time evolution of the slowly varying complex electric field E(t).The material variables (polarization and popula- tion inversion) have been adiabatically eliminated.The evolution of the electric field of a class-A laser is described through the following stochastic differential equation, where the possibility for a cavity detuning 0 has been included [20,21,15 + Q~g(t), where g(t) is a white Gaussian process taking into account spontaneous emission with zero mean and a correlation (((t) (*(t ') ) = 2 6'(t t ')-.Under Q-switching conditions, the cavity losses are modulated [8].During the modulation period T=T,"+T, z, the losses parameter takes a value y= y& below threshold in T,~~, while during T," the losses are reduced to y= y2.
The delayed term Ke '"'E(t r) is a-dded to the equation in order to take into account the amount of light fed back into the laser.The external reflector couples the two electric fields through the feedback strength ~.The external cavity round time ~and the emission frequency of the solitary laser co combine to give the rejative phase between the two fields.The external round time can be set arbitrarily to any value.In the present work, this value is adjusted to be ~=T.Our model also applies to optical feedback coming from other emitted pulses giving r=nT, with n an integer.
The time is set to t=0 at one of the Q-switching times.
Gain saturation can be neglected up to a given intensity ref- erence I, at which the switch-on time, defined as the time the optical intensity takes to reach the prefixed value I, , is reached after the losses switch.As the time evolution of the electric field, or equivalently the optical intensity, is affected by spontaneous emission noise, the switch-on time is a sto- chastic quantity with a mean time and a variance.Both quantities will be analyzed.By neglecting gain saturation effects, instantaneous gain-switching turns-out to be formally equivalent to Q switching in class-A lasers.
In the case of the solitary laser (K=O), the net losses ap= yig and the noise strength e combine to give the mean intensity emitted in the off-state.Weak feedback is known to modify the expression to Ip= ao -Kcos (cow) '   (2) so that light emission is enhanced when the external field is reinjected in phase with the internal one.The Q switch in- creases suddenly the net gain to a = g -y2.In the usual ana- lytical approach, an amplification for the electrical field is assumed in the way E(t) = h(t) et'+' ~".h(t) results to be a stochastic quantity containing information about the seeding process, which gives rise to light amplification [14].
This relation is totally independent of both laser parameters and the initial conditions for the electrical field.We will also assume that this equality holds in the case of being C&, C2, C3 time-independent coefficients.In previous works where other situations were analyzed (solitary lasers [14], weak feedback [16],weak injected field [22], . . ., ) the variance oh(t) was shown .to relax to an asymptotic value o. ( h)txbefore reaching the intensity reference.The stochastic process h(t) was then treated as a stochastic variable, and the expressions for both the mean switch-on time and its variance were analytically solved to get simple expressions.
In the present case, however, ot, (t) diverges expo.nentially at long times even for very low feedback levels.A method to overcome this limitation was used in the case of strong light injection [23].A more general method [24] based on the exact probability distribution of switch-on times is needed here because of the lack of a deterministic drift.Our analysis considers the transient dynamics of the laser, from the initial below threshold emission up to the switch-on time, always before reaching continuous-wave operation.
Under this assumption [24], the electric field is a zero-mean complex circularly Gaussian distributed variable, so that the probability density distribution for the light intensity at a given time P(I;t) reads I P(I;t) = ( ( )) e (t(t)).(10) (I(t)) Q(t I ) (I t )2 e (t(t))l, .
The above distribution allows a simple way of evaluating all the moments of the switch-on time distribution From this expression we can find the probability density dis- tribution for the switch-on time at a given intensity reference I, as It was recently shown in [27] the intrinsic difference be- tween Q switching and gain switching in class-B lasers.Our attention concentrates on gain switching because this operating condition is the most usual one in semiconductor lasers.
On the other hand, the switching dynamics of Q-switched class-B lasers can be described in terms of that of class-A lasers, due to the fact that the losses switch is fast and the population inversion is clamped to a fixed value.
The presence of weak optical feedback [7] modifies the mean light intensity emitted in the off-state before the cur- rent switch to (t") = t"Q(t;I,)dt p (12) 4pCb Ip= yy,g(Cby,Np) -2y, t~cos(cpr) ' (15) The mean intensity functions (I(t)) needed in all these cal- culations can be readily evaluated in terms of ab(t) as (I(t)) = ob(t) e " to give a final result.
It is important to note what is the role of the external round-trip time and the modulation period.Because of the resonance, the results become quite independent of these two times.In fact, only the relative phase between the internal and the reinjected field given by (2) appears in the calcula- tions through the constants C;.All modulation periods equally spaced in time to give the same value of cos(cur) would also give the same mean switch-on time and variance.This is quite in contrast with other feedback situations, where any independent variation of these parameters (exter- nal round time or frequency modulation) is linked to signifi- cant changes in the switch-on time distribution [19,25].As in the solitary laser case, the detuning parameter 0 only affects phase dynamics, and is irrelevant in the calculation of switch-on time statistics [26].

III. ANALYTICAL RESULTS: CLASS-B LASERS
In class-B lasers, the slow dynamics of the population inversion does not allow an adiabatic elimination of this ma- terial variable.The simpler mathematical description of these lasers involves a set of two coupled stochastic nonlinear dif- ferential equations for both the slowly varying complex elec- tric field and the population inversion.We will use from now on the time evolution equations for a semiconductor laser subjected to feedback [1]: Under conditions of moderate or strong feedback, the above expression must be replaced to give the actual mean values for both light intensity and carrier number.The presence of a significant amount of photons in the cavity due to the optical feedback reduces the initial carrier number through the non- linear coupling in (14).
During the modulation period T=T,"+T, «, the pump parameter (current injection) takes a value C=Cb below threshold in T,«, while during T,"the pumping is increased to a value C = C,"above threshold.T,«must be large enough in order to guarantee the return to the same initial conditions for all the gain switches.The carrier lifetime y, is the intrinsic parameter controlling the relaxation time of the population inversion to the steady state.For typical values of y, ' in semiconductor lasers, T,ff has to be much longer than the time needed to trigger the switch-on [28].This limitation was not present for class-A lasers, where population inver- sion follows almost immediately the pump control.
Similar to the case of class-A lasers, our modulation scheme assumes ~= T. Gain compression effects are irrel- evant (s = 0) during the switch-on.When the nonlinear term in the carrier equation can be neglected because of low in- tensity, population inversion dynamics decouple from light evolution.Carrier evolution can then be readily solved as [26] C, , C." N(t) =e ~"+ (1 -e ~"), Ye Ye (16) where the time is set to t=0 when the population inversion crosses threshold after one current switch.The current value C, = y, (Np+ ylg) corresponds to the threshold value for la- ser emission.The time delay between the current switch and the threshold crossing is given by dN g(N Np) dt =c -y, N - 1+S~E~( 14) where g(t) is a white Gaussian process taking into account spontaneous emission, with zero mean and a correlation given by (g(t) g*(t')) = 2 6(t t').Feedback a-nd noise ad- ditive terms have the same meaning as in the previous section.The linewidth enhancement factor o. couples the inten- sity and phase dynamics, resulting in changes of the emission frequency with carrier variations [26].
As usual, the switch-on time is defined as the time delay between the current switch and light emission, and is there- fore given by t,h+ t, Light amplification is only possible for carrier populations above threshold, when t)0.We use the decomposition E(t)=h(t)e"t')' to solve (13).laser.The values correspond to the solitary laser (solid line) and to the feedback levels a=0.03 ps ' (dashed line) and a. =0.1 ps (dotted line).

FIG. 2.
Probability distribution of switch-on times Q(t;I") for the class-A laser.The values correspond to the solitary laser (solid line) and to the feedback levels a. =0.03 ps ' (dashed line) and &&=0.1 p, s ' (dotted line).

IV. NUMERICAL RESULTS AND DISCUSSION
We show in Fig. 1 the variance crh(t) for several feedback strengths (class-A dye laser).This function grows exponen-  tially for long times separating from the solitary laser behav- ior and giving an earlier switch-on.The probability distribu- tions of switch-on time are plotted in Fig. 2. The introduction of feedback induces a shift to shorter times in the probability distribution and causes a narrower width.The usual asym- metric shape with long tails at long times [29] persists also when the laser is subjected to feedback.The same behavior can be shown for class-B lasers as well.
The stochastic differential equations (1), (13), and (14)  have been numerically solved by means of a first-order Euler algorithm [30] taking into account the delayed additive terms.Statistics were performed after a significant number (100) of initial modulation periods.For the class-A laser, 10" turn-on events were considered in the averages to get the mean switch-on time and its variance.The integration time step was 0.01 p, s.In the case of a class-B laser, these values were 10 and 0.01 ps, respectively.The laser parameters and the driving conditions of each laser are summarized in Tables I and II.The results of these numerical simulations are plotted in Figs. 3 and 4 for class-A and class-B lasers, respectively.The theoretical results of Secs.II and III are also plotted in these figures.Although the external reflector dis- tance for the dye laser is unrealistically large (several kilometers) in order to meet resonance with the time constants involved, the analytical results are expected to apply to class-A lasers with arbitrary resonant modulation periods.
Figure 3 shows a good agreement between theory and simulation.The shift to short times in the probability distri-  3. Mean switch-on time and variance for the class-A laser at different feedback levels.The points correspond to the numerical simulation (asterisks) and to the theory developed (circles).The triangles show the result expected when the feedback comes from the same light pulse.FIG. 4. Mean switch-on time and variance for the class-B laser at different feedback levels.The points correspond to the numerical simulation (asterisks) and to the theory developed (circles).The squares show the improvement in the theory when using the actual initial conditions for each light pulse.
bution is now evident.The mean switch-on time and its vari- ance are significantly reduced when feedback is present.In our theory, feedback coming from previous light pulses has been considered.For completeness, we compare the results with those corresponding to feedback coming from the same light pulse [E(t r) =-E(t) instead of E(t r) =-E(t T)]. - The reduction of the mean switch-on time and its variance starts at lower ~values, and is always larger than the one present with feedback from previous pulses.The explanation of this behavior lies in the reduction of the effective laser threshold (lower cavity losses) and the elimination of extra noise (intrinsically identical light pulses).Our theoretical model, developed in the previous sections, is able to distin- guish the fine differences between these two feedback conditions.
The theoretical results for the class-B laser seem to be correct only at small feedback levels.In the calculations, the expression (15) was used to give account of the initial con- ditions for each switch-on.In fact, that expression is only valid for low ~values.We have checked in the numerical simulations that the mean value for the initial carrier number is significantly lower than Cbl y, and the mean intensity is much higher than the one given by (15).A lower carrier number increases the mean switch-on through t,h, while a high initial intensity improves the seeding process, accelerating the switch-on.We include in Fig. 4 the results of the theory developed in Sec.III using the actual initial condi- tions for both carrier number and light intensity"as given by numerical simulations.In class-B lasers, the switch-on time variance turns out to be much less sensitive to feedback than in class-A lasers, even at strong feedback levels.

V. SUMMARY AND CONCLUSIONS
We have analytically calculated the mean switch on time and its variance for both class-A and -B lasers under periodic modulation, whether in losses or in gain, and subjected to optical feedback.We considered the special situation in which the period of the modulation T is equal to the external cavity round-trip time 7. .For this situation our theory is valid at any cavity length.We observed that for weak feedback both the mean switch on time and its variance decrease with feedback strength.Our analytical results fit well with nu- merical simulation of noise driven rate equations.For class-A lasers the fit is especially good, while for class-B lasers the theory fails at feedback strengths for which the weak feedback condition is lost.From a practical point of view, we demonstrate an important jitter reduction when a reflector element is placed in front of a laser and the reso- nance with the modulation period is met.

14 FIG. 1 .
FIG. 1. Variance o"(t) as a .function of time for the class-A FIG.3.Mean switch-on time and variance for the class-A laser

TABLE I .
Parameters corresponding to the class-A laser.

TABLE II .
Parameters corresponding to the class-B laser.