polarization, and direction of propagation as the incident photons. The photons involved are thus mutually coherent. When a population inversion () is present, therefore, optical amplification of incident radiation will take place. Although energy generated by stimulated emission is always at the exact frequency of the field which has stimulated it, the above rate equation refers only to excitation at the particular optical frequency corresponding to the energy of the transition. At frequencies offset from the strength of stimulated (or spontaneous) emission will be decreased according to the so-called line shape. Considering only homogeneous broadening affecting an atomic or molecular resonance, the spectral line shape function is described as a Lorentzian distribution where is the full width at half maximum or FWHM bandwidth. The peak value of the Lorentzian line shape occurs at the line center, . A line shape function can be normalized so that its value at is unity; in the case of a Lorentzian we obtain Thus stimulated emission at frequencies away from is reduced by this factor. In practice there may also be broadening of the line shape due to inhomogeneous broadening, most notably due to the Doppler effect resulting from the distribution of velocities in a gas at a certain temperature. This has a Gaussian shape and reduces the peak strength of the line shape function. In a practical problem the full line shape function can be computed through a convolution of the individual line shape functions involved. Therefore, optical amplification will add power to an incident optical field at frequency at a rate given by Stimulated emission cross section The stimulated emission cross section is where A21 is the Einstein A coefficient, λ is the wavelength in vacuum, n is the refractive index of the medium (dimensionless), and g(ν) is the spectral line shape function. Optical amplification Stimulated emission can provide a physical mechanism for optical amplification. If an external source of energy stimulates more than 50% of the atoms in the ground state to transition into the excited state, then what is called a population inversion is created. When light of the appropriate frequency passes through the inverted medium, the photons are either absorbed by the atoms that remain in the ground state or the photons stimulate the excited atoms to emit additional photons of the same frequency, phase, and direction. Since more atoms are in the excited state than in the ground