Question

If the electric field radiated by the ions in a laser gain medium is written as e(t) Ege (el@ot +e iwpt where wo is the centre of the lineshape function and y is associated with the relaxation rate, show that the normalised lineshape as a function of frequency v is given by: I(v) I(vo)

Answer 1

E(w) = integral^infinity_0 e(t) dt e^i omega t (Laplace transform) = E_0 integral^infinity_0 e^- (r - i omega_0)t e^i omega t dt + E_0 integral^infinity_0 e^- (r + i omega_0)t e^i omega t dt = E_0 integral^infinity_0 e^- (r - i omega_0 + i omega)t dt + E_0 integral^infinity_0 e^- (r + i omega_0 - i omega)t dt = E_0 1/r - i omega_0 + i omega + E_0/r + i omega_0 - i omega because integral^infinity_0 x^n - 1 e^- lambda x dx = squareroot n/lambda^n = E_0[r + i omega_0 - i omega + r - i omega_0 + i omega/{r + i(omega - omega_0)}{r - i(omega - omega_0)}] = E_0 2r/r^2 - i^2(omega - omega_0)^2 E(omega) = E_0 2r/r^2 + (omega - omega_0)^2 omega = 2 pi gamma omega_0 = 2 pi gamma_0 so E(v) = E_0 2 r/[2 pi (v - v_gamma)^2 + r^2 E(v)/(2E_0/r) = r^2/[2 pi (v - v_0)]^2