《关于速率方程和增益饱和的习题》由会员分享，可在线阅读，更多相关《关于速率方程和增益饱和的习题（5页珍藏版）》请在金锄头文库上搜索。
1、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 stimul
2、ated 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 = 8h3/c3), find the change in intensity of the radiation field with distance of propagation through the gain medium, and henc
3、e 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 t
4、he 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 h=E1-E2. The radiation energy density is () and A21, B12 and B21 are the usual Einstein coefficients (or,
5、 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.121BNdtFor spontaneous emission, the transition rate from level 2 to 1 is Wsp and is given by.spAt212For stimulated emission, the transition rate from
6、 level 2 to 1 is W21 and is given by.212BNdtThe photon density is the number of photons per unit volume, where the photons are created / destroyed by the above processes. Hence .122121122 BNANdttdtNabssp (ii) In a laser oscillator, the photon density is dominated by stimulated emission of photons in
7、to 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, r
8、ealize 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()=c(
9、). 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 d/dt, so you need to relate and : =h.Fourth, make the change of variable from time to distance:.dtIctzdI1Hence ,122BNhthttIzusing expression from part (ii), where the small s
10、pontaneous term has been neglected.Fifth, make the substitutions using Einsteins equations (given).Hence IcANgchdzI 211238or, ,zIGdzIwhere is the gain coefficient21128cANg(b)Wpump Wpump 30W10W322-levl aser 4-levl aser21W12 W21Wsp W31 2112 W21WspW20Advantages of a 4-level laser system: pumping transi
11、tion (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-popu
12、lation 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: 1=1/W10=0 no non-radiative depopulation of upper lasing level: W2
13、0=0(c) Firstly, recognise that the existence of a saturation intensity implies saturation in the optical gain - and therefore population inversion - with increasing optical intensity I(). So we need to calculate the population inversion N2-N1 as a function of optical intensity I() (this was done for
14、 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
15、illumination intensity I:For level 2: 2123212132 BNgWBNNWdt spsp For level 1: 212112211 BNgNBNdtN spsp and at equilibrium 021dtLevel 1 equation gives: 1212BgNspspwhich is proportional to 1 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) gives spBWN213At zero optical density, this reduces to . sp02Hence the population inversion and optical gain reduce with optical intensity :spBNgG2102120, Using the given Einst
