The single-mode rate equations for semiconductor lasers with thermal effects

Mathematical Modelling of Electrical-Optical Effects in Semiconductor Laser Operation

“…The latter is precisely the optical time-scale identified in the isothermal lumped model of [7], the only other relevant time-scale there being an electrical one corresponding here to An, the time-scale for carrier variations. In the model discussed here we have thus identified a total of four physical time-scales: the time-scale for carrier variations ("" 1O-8 s), that for an electromagnetic wave to traverse the laser in the transverse direction ("" 1O- 13 S), that for traverse in the longitudinal direction ("" 1O-12 S) and the reciprocal of the frequency ("" 1O-15 S).…”

“…The generation of electron-hole pairs due to the mechanisms of impact ionization and absorption of spontaneous emission are negligible (see [13, p. 44] and [7]). We assume that the normal component of the electric field and the fluxes of electrons and holes are zero on the insulated boundaries, so that (see Figure 1 the silicon dioxide is thus treated as insulating.…”

“…Resonance occurs in isolated intervals of the electromagnetic spectrum, for example at visible frequencies at which gallium arsenide has an electronic transition. What we are aiming for here is a tractable constitutive law which is guided by the physically established facts that ([9, p. 242], [8]) the damping is relatively large in practice (that is o'/w = 0(1), where w is the relevant electronic resonant angular frequency of the semiconductor (given by w = 21rEg /h) and the positive constant 0' is the damping factor) and that the relevant mode of operation is single-mode with angular frequency 0 = w (with neighbouring modes having lower gain). We adopt a simple-minded approach which reproduces these features and is adequate for our purposes, the result (see (30)) being consistent with the corresponding constitutive assumptions adopted elsewhere (see [9]).…”

“…However, some partially lumped models for the active region have also been reported which comprise one ordinary differential equation and two first-order wave equations [4], [5]. A fully lumped model in terms of two ordinary differential equations (for electron and photon concentrations) is very well-established (see, for example, [6], [7]). In this paper, we attempt to address deficiencies in the modelling of the fully space-dependent case.…”

“…Strictly, longitudinal modes must satisfy the standing wave condition ).m = 2L/m, where m is a positive integer, ).m is the wavelength of the mth mode and L is the axial length of the cavity, but in general the photon energy of a number of these modes will be close to the band-gap energy and it is with one such mode that our modelling is concerned. Although several longitudinal modes will experience similar gain, it is reasonable to assume single-mode operation, because this corresponds to the only stable steady state [8].…”