Population Inversion the Threshold Condition

It is clear that in order to get stimulated emission, a pumping process is required to excite the system into its high quantum energy level. Real materials can be pumped in many ways, as will be mentioned later. For laser action to occur, the pumping process must produce not merely excited atoms, but the condition of population inversion. 

However, a series of factors introduce losses in the system namely, the reflectivities of the mirrors (RiandR2) on the figure, which reflect only a fraction, Ri and R2, of the intensity. Additional losses can be produced by absorption in the windows of the cell that contains the active medium (if this is the case), diffraction by apertures, and scattering due to particles or imperfect surfaces. All of those losses can be included in a loss factor per trip, expressed as e. Thus, considering both amplification and intensity decrease per round trip, the intensity after a single round trip through a resonator of length d is 

The wave is amplified if the gain overcomes the losses per round trip that is, 

This implies a threshold condition (G = y) for the population difference zlA th = (N2 - A i)th = y/2aL. If the population inversion, AN, is larger than ZiiVth, a wave reflected back and forth in the resonator will be amplified and its intensity will increase, in spite of the losses. 

EXAMPLE 2.3 Suppose that a 7.5 cm length rod ofaNd Y3AI5O12 (Nd YAG) laser crystal is located in a linear cavity with two mirrors of transmittances Ti =0 and T2 = 0.5 at the laser wavelength (1.06 /am). If the cross section is a = 8.8 X 10- crn, determine the population inversion density at threshold. Assume that losses are only due to the output mirror transmittance. 

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