intensities) for the ith mode, N is the carrier density, Nom is the carrier density for transparency, k is the gain constant in c =c ±g(N-Nomd cm /unit inversion), Gh is the unsaturated gain in cm B +terms of higher orders in B half of the fraction of spontaneous emission entering the sing mode, z is the distance along the active medium with The gain factor g, differs only slightly (parts in 10)from one z=0 at the center of the laser and g is the gain factor for the mode to another and therefore the distributions xt's are ith mode, which is commonly approximated by a lorentzian almost identical, except for the proportionality factor C with its maximum at the ioth mode which determines the power in the ith mode. The small dif- ferences in the g, s, however, plays a crucial role in determi- g=1/[1+a(i-i2] (ld) nating the Ci's when applying boundary conditions(2) The mode io where the gain maximum occurs varies with Given that the photon distribution of each mode has carrier density and is approximately given by nearly identical shapes, one can derive the following io=[0.095KN]+an arbitrary integer A, constant independent of i Typical values areB=10,a=5X10+, and KNom=200 ∑g[x(z)+x(2]=S[xdz)+x(2],(6 the boundary conditions where SeEP/Po is the ratio of the total power to that in an rbitrarily chosen Oth mode. The photon number in each x;(L/2)=R2,*(L/2);x * (-L/2)=Rx(-L/2), mode is then computed as follows: given a value for S, the n coupled equations(1) can be reduced to two equations for the where L is the length of the laser, typically 250 um; R and Oth mode R2 are the reflectivities of the end mirrors. This boundary value problem involving n coupled nonlinear differential =g01N一Nmxd+BN equations does not lend itself to even easy numerical solu dz tions. Major features can, however be extracted with some KN-Nom)=G, /[1+S(xo*+xo) (8 manipulations and a minimal amount of computations This can then be solved with boundary condition (2)and the Equations (la)and (1b)are averaged over the entire cav- resulting photon and carrier distributions can be used to ty, and the forward and backward photons are summed to compute from Eq (3c)the factors B for each mode. The give the total photon density in the ith mode ratio of the photon number in the ith mode to that in the Oth [A,-B K(N-Nom)]P,=BxN (3a) mode is derived from Eq (3a) where Pi=x *+x;and P 4=1PL2)I(R,R,y12+111-(RR2)21 F-B+(B。-B:M1-Nn/NP 1+R1 B-2∫N=Nm ind P;=(1/L)SP dz= average photon density in the cav- intra-cavity ity, N=(1/L)Sn dz= average carrier density. The factor A, involves the photon density at the mirror facet and is related to the rate of photon loss from the end mirrors. B can be regarded as the overall efficiency of stimulated emis on in the cavity, which is considerably smaller than 1 be cause of the nonperfect overlap of the carriers and photons the photon density is highest near the end facet where the 8 carrier density is lowest because of local gain saturation In the case where both mirrors are sufficiently reflective >20%), the carrier distribution is almost uniform along the length of the cavity and B, approaches g Equation(3a)can be physically interpreted as that the spontaneous emission KBN makes up for the slightly excessive cavity loss A, P,over the stimulated gain KB, N-Nom)P The factors A; and B, for mode i depends on the shape of °00200300a400500 the photon and carrier distributions. Now, the fact is that the Gh +KNo (cm) shape of these distributions depends strongly on the level of pumping and the mirror reflectivities, but only weakly FIG. 1. Output optica id lines) from the exit facet of (a]a commor the mode number This conclusion can be drawn from a di laser with R,=R2=0,3 and(b) an Ar coated laser with R,=0.01 R,=0.3, plotted as a function of G+ +KN.m, which is proportional to the rect integration of Eqs.(la)and(1b), which giv pump current. Appl. Phys. Lett., Vol 40, No 9, 1 May 1982 K. Y Lau and A. Yariv Downloaded24May2006to131.215.240.9.RedistributionsubjecttoAlplicenseorcopyrightseehttp:/laplaip.org/apl/copyright.jspDownloaded 24 May 2006 to 131.215.240.9. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp