Simulation of Bistable Laser Diodes with lnhomogeneous Excitation

A comprehensive time-dependent 1-D computer model is developed for the simulation of multi-section bistable laser diodes. Analysis of a Fabry-Perot (FP) cavity laser diode indicates that the height and shape of the optical input pulse as well as the wavelength play important roles in the set/reset operation of optical switching.


INTRODUCTION
In recent years, BiStable Laser Diodes (BSLDs) have attracted much attention because of their potential applications in optical switching and wavelength conversion. The simulation of BSLDs involves large-signal analysis, which currently existing LD simulators (such as CLADISS [1]) cannot perform. Most studies of BSLDs are conducted using a set of rate equations that are expressed as a function of time only [2]. Such an approach has two main limitations. First, an approximation of distributed loss is required. Second, the spatial dependence of the gain saturation inside the cavity is not taken into account. Most existing models use a single variable to represent the total photon density without considering the difference in wavelength.
Moreover, these models often do not include all pertinent physical processes which compete with each other to influence the dynamic characteristics of the device. Therefore a more sophisticated simulator is needed to assist the design as well as the understanding of the physics of BSLDs.
In this work, a comprehensive time-dependent one-dimensional computer model is developed for the simulation of multi-section bistable laser diodes. Longitudinal spatial hole-burning, nonlinear gain, carrier induced refractive index change (frequency chirp), and thermal effects are rigorously accounted for.

PHYSICAL MODEL
A typical longitudinal view of a multi-section laser diode is shown in Figure 1. The laser consists of two gain sections and a saturable absorption section in between.
The optical field E(z, t) can be separated into the forward ( + ) and backward (-) propagating parts, satisfying the traveling-wave rate equations, +0/7+ 10 (27r Oznr (1) vg at -+i-G +/-, where (A..(z)) r + r/x(A. (z)). and F is the confinement factor, n,r is the effective refractive index determined by waveguide structure, g is the gain function, OZac and cc represent the absorption coefficients in the active layer and cladding layers, respectively, and Anr is the carrier induced change of refractive index.
If we assume +(z, t) E(z, t)e ick+/-(z't), after substituting into equation (1) and separating the real and imaginary parts, we obtain OE + 10E + + Oz + Vg Ot 2G(A'n(z))E+' (2) Oc + O0 + 27r -'-----2 -g lot , nr A, n z (3)  For each longitudinal mode (Am) the amplitude and phase satisfy the following boundary conditions atz 0andz L: where m is an integer, rl and r2 are reflective coefficients of the front and rear facets, respectively. However, for an incident optical field (at z 0) of an arbitrary wavelength (Ain), since the incident optical field tIEex(t) interferes with the reflected field rE-(O, t) just inside the left (front) facet, the boundary condition at z 0 becomes and at z L, it is given by equations (6)  is the round-trip phase change. For the light in the m th mode, A( must be equal to 2mTr. Thus we can calculate wavelength change (frequency chirp) by solving this nonlinear equation. For the incident light, since its wavelength is determined by the external light source, Ate)in will change with the carrier density in the active region. Since the round-trip time (2L/vg lOps) is typically much shorter than the response time (>0.2ns) for devices under the study, we can still use the boundary condition (8) for the time-dependent problems, instead of solving the rate equations for the phases.
For the incident light (Pin, Qin), the rate equations remain the same, while the boundary conditions are different, given by 4Pin(0, t) cos(0ff 4R1Qin(0, t) cos(0ff +4T1Pex(t),(t), In our model, the linear gain is approximated by gA" (nno), gL(A, n) g2 n2 + gl n + go, g6 (n-no). n < nlo n20 > n > nl0 n > n20 where gA and g6 are the differential gain coefficients in the gain and loss region, no is the transparency carrier density. These parameters can be determined by theoretical calculation or experiment. It should be emphasized that different values of ga, ga, and no are used for each longitudinal mode and optical input.
The stimulated emission becomes nonlinear whenever large optical intensities exist in the active layer. To account for this effect a simplified gain model is used, i.e., 4Pin(0, t) sin(O-) 4R1Qin(0, t) sin(0ff), (18 where an is assumed to be a constant.

NUMERICAL SOLUTION METHODS
In this work, we have improved the spatial discretization scheme described in [3]. A first-order backward finite difference formula is applied to P(z, t) and a forward difference applied to Q(z, t).
The photon density is defined at the nodes while the carrier density is defined between two adjacent nodes. The change in the carrier density in a region now depends on the average number of photons surrounding that region. A trapezoidal rule/backward-differentiation-formula composite method [4] has been implemented for the time integration.
The system of discretized nonlinear equations is solved simultaneously using the full Newton method. The simulation result shown in Figure 4 can be obtained within 20 minutes (CPU time) on a DECstation 5000/200.

SIMULATION RESULTS
A two-electrode FP cavity InGaAs/InP laser diode [5] has been simulated. An example of set-reset operation (undershoot switching) using a single wavelength (Ain=1557.6nm) light is shown in Figure 2. The bias current is chosen to be I1--18.4 mA. The intensities of pulses for set and reset are 0.874 mW and 13.12 mW, respectively.
The mechanism of undershoot switching is believed to be dependent on different changing rates of carrier densities in the gain and absorption regions under a variation of the optical excitation. We have found that the shape of the pulses plays an important role in undershoot switching, as shown in Figures 3 and 4. From our simulation results, a subnanosecond switch-on (rise) time is observed, while the switchoff (fall) time of the device is about 3 ns. We also found that an input pulse with a higher intensity and faster fall time would reduce the switch-off time somewhat but still no less than 2 ns. The wavelength of the injected optical pulses is also crucial for the undershoot switching. The minimum set/reset powers will be modulated by

CONCLUSION
In this paper, a comprehensive transient 1-D computer model for the simulation of multi-section bistable laser diodes is presented. Traveling wave rate equations are solved for each longitudinal mode and optical input. All the important effects have been included in this model.
Analyses of two-electrode FP cavity InGaAsP/ InP laser diodes incorporating an absorber region are carried out to illustrate the capability of the simulator. Special attention is paid to the optical set-reset operation (undershoot switching). Both the height and shape (rise/fall time) as well as the wavelength of the optical input pulse are shown to play an important roles in undershoot switching.