Analytical Solution of Rate Equations Including Frequency Chirp of Modulated Quantum-Well Laser with Carrier Transport Processes

When used as light sources in modern fiber communication systems, the modulation bandwidth and chirp are crucial characteristics of high-speed quantum well (QW) lasers. These parameters are primarily constrained by two factors; namely, the transport of charge carriers in the separate confinement heterojunction (SCH) layer and their escape processes in the QW. To analyze the frequency chirp theoretically, a fourth rate equation is added to the existing system of three coupled rate equations, which describe the photon number in the QW and carrier numbers in both the QW and SCH layers. This study employs small-signal analysis to linearize these coupled equations and derives analytical expressions for both the intensity modulation (IM) response and its associated frequency chirp. The chirp is quantified using two metrics, first the chirp per modulated current (CCR), and second the chirp per modulated power (CPR). These analytical expressions are presented in a generalized form, making them applicable to any nonlinear gain mathematical formulation found in the literature. Through numerical calculations applied to high-speed QW lasers, we investigate the individual effects of transport and escape times on the frequency chirp. Our findings demonstrate that CCR reaches its minimum under two specific conditions: when the transport process is relaxed with a relatively long transport time, and when carrier escape in the QW occurs rapidly with a very short escape time. Notably, we found that CPR remains independent of the transport processes.