关于速率方程和增益饱和的习题.doc

习题1 Consider an optical gain medium consisting of idealised atoms, with two non-broadened energy levels 1 and 2, having populations N1 and N2 and degeneracies g1 and g2 respectively. Write down the rate of change of the photon density, resulting from absorption, spontaneous emission and stimulated emission. Why can we neglect the spontaneous emission in the derivation of the gain coefficient in a laser? Given Einsteins relations (g1B12 = g2B21 and A21 / B21 = 8?h?3/c3), find the change in intensity of the radiation field with distance of propagation through the gain medium, and hence find an expression for the gain coefficient. Illustrate with the aid of an energy-level diagram the operating principle of a 4-level laser. What are the advantages compared to a 2-level laser? Write down approximate conditions pertaining in an ideal 4-level laser. What is the population of the lower lasing level in an ideal 4-level laser? Derive an expression for the saturation intensity. Solution: The energy of level 1 is E1, the energy of level 2 is E2, and the photon energy is h?=E1-E2. The radiation energy density is ?(?) and A21, B12 and B21 are the usual Einstein coefficients (or, constants of proportionality defined by the rate equations). For induced absorption, the transition rate from level 1 to 2 is W12 and is given by . For spontaneous emission, the transition rate from level 2 to 1 is Wsp and is given by . For stimulated emission, the transition rate from level 2 to 1 is W21 and is given by . The photon density ? is the number of photons per unit volume, where the photons are created / destroyed by the above processes. Hence . In a laser oscillator, the photon density is dominated by stimulated emission of photons into a few (or a single) cavity mode(s). Spontaneous emission is isotropic, and only a small fraction of emitted photons couple into cavity modes. Therefore spontaneous emission is negligible in the rate equations. (However, it is an essential process in init