# Homework Help: Significance of the Saturation Photon-flux Density

1. Jun 5, 2008

### tuanle007

1. The problem statement, all variables and given/known data
In the general two-level atomic system of Fig. 14.2-3, τ2
Significance of the Saturation Photon-flux Density.
In the general two-level atomic system of Fig. 14.2-3, τ2 represents the lifetime of level 2 in the absence of stimulated emission. In the presence of stimulated emission, the rate of decay from level 2 increases and the effective lifetime decreases. Find the photon-flux density Φ at which the lifetime decreases to half its value. How is that photon-flux density related to the saturation photon-flux density Φs?

2. Relevant equations

3. The attempt at a solution

A simple rate equation for the upper level population N2 is

dN2/dt = -(N2-N2(T))/τ + (σ I/ε) / (g1N1-g2 N2),

where N1=lower level population N2(T) =thermal equilibrium population when light intensity I=0, ε=photon energy,σ=transition cross section, g1,2 =degeneracy factors. τ =level lifetime. Neglect N2(T). The answer to the first question is obtained by solving

1/τ = (σ I /ε) g2 for I. Thus I= ε/(τ σ g2) Call this I'.

To find Isat , let dN2/dt=0 N1=N-N2. Solve for N2, and write the denominator of the resulting expression in the form
1+I/Isat. Then Isat is found as

Isat= ε/[τ σ (g1+g2)] = g2 I' /(g1+g2)]

(For nondegenerate levels g1=g2=1)

If pumping is present, as for an amplifying medium, add a constant term R to the right hand side of the above rate equation.

Other questions, involving gain, can be solved using the following expression for the gain per unit length:

g = 2(g2 N2 -g1N1)σ

In steady state, dN2/dt=0, solve the rate equation for N2 and N1=N-N2, where N is the total concentration of active atoms, and substitute in the expression for g.

can someone help me with finding N2 and g?