关于速率方程和增益饱和的习题.doc

习题1(a) 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.i. Write down the rate of change of the photon density, resulting from absorption, spontaneous emission and stimulated emission.ii. Why can we neglect the spontaneous emission in the derivation of the gain coefficient in a laser?iii. Given Einsteins relations (g1B12 = g2B21 and A21 / B21 = 8phn3/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.(b) 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.(c) What is the population of the lower lasing level in an ideal 4-level laser? Derive an expression for the saturation intensity.Solution:(a) The energy of level 1 is E1, the energy of level 2 is E2, and the photon energy is hn=E1-E2. The radiation energy density is r(n) and A21, B12 and B21 are the usual Einstein coefficients (or, constants of proportionality defined by the rate equations).(i) 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 f is the number of photons per unit volume, where the photons are created / destroyed by the above processes. Hence .(ii) 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 initiating the laser oscillation)(iii) First, realize that you are being asked to find dI/dz, where I is the radiation intensity and z is the distance of propagation in the material (given in the question).Second, recall that the light intensity (which depends on the light frequency in the general case) is related to the energy density by I(n)=cr(n). Use the units to help you remember: W/m2=(m/s) (J/m3) Third, realise that you are going to use the previous calculation of df/dt, so you need to relate r and f: r=fhn.Fourth, make the change of variable from time to distance:.Hence ,using expression from part (ii), where the small spontaneous term has been neglected.Fifth, make the substitutions using Einsteins equations (given).Henceor, where g(n) is the gain coefficient(b)Advantages of a 4-level laser system: pumping transition (level 0 to level 3) may involve large number of states (hence high transition rates) without affecting the lasing transition (level 2 to 1). Hence high pumping efficiency. Depopulation of lower lasing level is fast, so that N1 is very small, and population inversion is easily achieved. Re-population of upper lasing level (2) is fast, leading to large N2 and hence high gain In an optimal / ideal laser, the following approximations are valid: no pumping into lower lasing state: W31=0 rapid depopulation of lower lasing level: t1=1/W10=0 no non-radiative depopulation of upper lasing level: W20=0(c) Firstly, recognise that the existence of a saturation intensity implies saturation in the optical gain g(n) - and therefore population inversion - with increasing optical intensity I(n). So we need to calculate the population inversion N2-N1 as a function of optical intensity I(n) (this was done for the more general case in the lectures).Secondly, consider all the processes contributing to the population/depopulation of the lasing levels 1 and 2, as shown on the previous diagram, with the approximations listed for the ideal 4-level system.Finally, write down the rate equations for the case of illumination intensity I(n):For level 2: For level 1:and at equilibrium Level 1 equation gives: which is proportional to t1 which tends to zero for the ideal system. Therefore there is no population in the lower lasing level, irrespective of the optical density. (or by inspection)Level 2 equation (with N1=0) givesAt zero optical density, this reduces to . Hence the population inversion and optical gain reduce with optical intensity r:Using the given Einstein relation and relation between r(n) and I(n) used earlier:where is the saturation intensity. For a wavelength of 1.06mm, the saturation intensity isor 13.2 Watts for a 2 mm diameter beam.习题 2 Gaussian beam relay: A Ga