Significance of the Saturation Photon-flux Density

[tuanle007]

Homework Statement

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?

Homework Equations

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?

 

[Jhero]

Sure, I'd be happy to help with finding N2 and g. First, let's start with the expression for N2 in steady state:

N2 = (σ I / ε g2) / (1 + I / Isat)

Where I is the light intensity and Isat is the saturation photon-flux density. Now, let's substitute this into the expression for g:

g = 2(g2 N2 - g1 N1) σ

= 2(g2 (σ I / ε g2) / (1 + I / Isat) - g1 (N - (σ I / ε g2) / (1 + I / Isat))) σ

= 2(σ I / ε - g1 N) σ / (1 + I / Isat)

Now, let's solve for N by setting dN2/dt = 0:

dN2/dt = 0 = -(N2 - N2(T)) / τ + (σ I / ε) / (g1 N1 - g2 N2)

= -(N2 - N2(T)) / τ + (σ I / ε) / (g1 (N - N2) - g2 N2)

= -(N2 - N2(T)) / τ + (σ I / ε) / (g1 N - g1 N2 - g2 N2)

= -(N2 - N2(T)) / τ + (σ I / ε) / (g1 N - (g1 + g2) N2)

= (N2(T) / τ) - (σ I / ε) / (g1 N - (g1 + g2) N2)

= (N2(T) / τ) - (σ I / ε) / ((g1 + g2) (N / (g1 + g2)) - (g1 + g2) (N2 / (g1 + g2)))

= (N2(T) / τ) - (σ I / ε) / (N - (g1 + g2) (N2 / (g1 + g2)))

= (N2(T) / τ) - (σ I / ε) / (N - N2)

= 0

Therefore, N2 = N2(T) = N / (g1 + g2).

Now, substituting this into the