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Homework Help: Significance of the Saturation Photon-flux Density

  1. Jun 5, 2008 #1
    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?
  2. jcsd
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