We define photon absorption rate as R12, which means the rate of generating new electrons by reducing the number of photons. Stimulated emission rate R21 represents the rate of generating new photons by recombining EHPs. Rsp is the spontaneous emission rate related to the spontaneous emission. Rst is the rate of density of electrons recombined through stimulated recombination. Thus: Rst = R21 – R12; Although Rst represents the rate of carriers recombination, it does not represents the rate of photons generation in the cavity. Because in a cavity, the EHP recombination only occurs in the active region with the volume of V = d*w*La, while the area of the photons that can be observed in the cavity is Vp = deff*weff*L which is larger than the active region (figure 1). We define Vp as the cavity region and use Γ (confinement factor) to represents the ratio of these two volumes: Γ= V/ Vp. Normally, Γ<1. So the actual photon generation rate is Γ* Rst.
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Figure 1: Active Region and Photons-occupied Region
Firstly, we analyse equation 1a:
This equation is obtained after the current excesses the threshold current Itr. ε is the gain compression coefficient which we can assume to be 0 in the analytical calculation. Thus the last section of this equation (stimulated recombination of EHPs) is equivalent to vg*a(N-Ntr)*Np.
In this equation, ηiI/qV (rate of the injection carrier density) is the source of the carrier which comes from the injected current. ηi is the carrier injection efficiency: the ratio of the number of carriers that injected into the active region and the number of carriers that supplied by the current. The first loss of the carrier density is due to thermal loss, spontaneous emission and non-radiation recombination. We use N/τ to represents these losses, in which τ is the electron lifetime. The second loss is caused by electrons’ recombination: Rst.