Population inversion and the atomic

The propagation of light through an excited gas was considered in detail in Chapter 10. If spontaneous emission is neglected, the intensity of radiation of angular frequency ( ) emerging from a column of gas of length L, Ij (L), is related to the intensity of radiation incident at z=0 by 

The gain coefficient, a (u), given by equation (11.4) is known as the small signal or unsaturated gain coefficient. It determines the gain which exists before amplification of radiation or laser oscillation starts to draw energy from the system thus reducing the population inversion, 

By substituting in equation (11.4), the explicit expression for the gain coefficient in the limit of dominant Doppler 

It is important to realize that in this situation the standing electromagnetic wave in the cavity at the angular frequency to is able to interact only with those atoms whose axial velocities v cause them to be Doppler shifted into resonance, that is 

The electromagnetic field in the laser therefore draws energy from only a small fraction of the total excited state population and consequently this is termed an inhomogeneously broadened transition. 

---