The rate equations may be written heuristically by considering various physical phenomena through which the number of photons, p, and the number of electrons, n, change with time inside the active region. for a single-mode laser, these equations take the form -gp+r dt sp dn i n g is the net rate of stimulated emission and rsp is the rate of spontaneous emission into the lasing mode. note that rsp is much smaller than the total spontaneous-emission rate. τ.is the lifetime of carrier and tp is the lifetime of photon. (a) the modulation response of semiconductor lasers is studied by solving the rate equations provide in the above with a time-dependent current of the form where ib is the bias current, im is the current amplitude, and ω represents the angular frequency of the modulating current. determine the response of the semiconductor laser and 3-db bandwidth. (b) the system of differential equations for a small perturbation δ n in the carrier and δ p in photons densities constitute a second order linear system much like the equations for the current and the voltage in an lcr circuit. solve the second order differential equation and determine the relaxation frequency. compare the result with that from (a)