Consider a ruby crystal with two energy levels separated by an energy difference corresponding to a free-space wavelength lambda_0 = 694.3nm, with a Lorentzian lineshape of width Delta v = 330GHz. The spontaneous lifetime is t_sp = 3ms and the refractive index of ruby is n=1.76. What value should the population difference N be to achieve a gain coefficient gamma(v_0) = 0.5 cm^-1 ? How long should the crystal be to provide an overall gain of 4 at the central frequency when gamma(v_0) = 0.5 cm^-1 ? The gain coefficient at central wavelength for Lorentzian lineshape is given as: gamma (v_0) = N(lambda_0/2 pi n)^2 middot 1/t_sp Delta v