“…where k3=wic is the vacuum wave number, n(F,w0) = n'(i.w0) + in(F,w0)and ng(r.wo) = n'(r.w0) 1n(i,w(,) are the complex refractive phase and group indices evaluated at the nominal angular frequency wi), and the dimensionless frequency parameter i', defined as EQ) = f(z; r)ct(r) , (4) where for simplicity of notation we have dropped the argument o. The vertical part of the solutionf(z; r) is assumed to be normalized according to I -j-+ k0 n (r, z) if(z; r) = Veff(r)k0 n(r, Z)flg (r, z)f(z; r) , Under steady-state conditions, the imaginary part of i should satisfy the threshold condition Im(v) = 0, which provides information about the modal gain necessary for each mode to reach threshold.…”