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Photon conservation in Raman amplification

  1. Apr 8, 2013 #1
    Part 1:
    1. The problem statement, all variables and given/known data

    Use two coupled-wave equations for the
    Raman amplification process depicted to the right
    to show that for every photon at Stokes frequency
    omega_s created (destroyed) one photon in the laser
    pump field omega_L is destroyed (created).

    I have the coupled-wave equations, they're given in part 3. Basically, I have to show that the sum of intensities is constant, which means the sum of the derivatives is 0.

    2. Relevant equations

    See (3) below


    3. The attempt at a solution

    [itex] \text{For Manley-Rowe, must prove that sum of intensities is constant.}\\
    \text{That is, sum of derivatives is 0.}\\
    I=2 n_i \epsilon_0 c A_i A^*_i,\ \frac{dI_i}{dz}=2n_i \epsilon_0 c \left( A_i^* \frac{dA_i}{dz}+A_i \frac{dA^*_i}{dz} \right )\\
    \text{We have: }\left\{\begin{matrix}
    \frac{dA_s}{dz}=\alpha_s A_s,\ where\ \alpha_s=3i\frac{w_s^2}{n_s c}\chi_R^{(3)}(w_s)|A_L|^2\\
    \frac{dA_L}{dz}=\alpha_L A_L,\ where\ \alpha_L=3i\frac{w_L^2}{n_L c}\chi_R^{(3)}(w_L)|A_s|^2\\
    \chi_R(w_L)=\chi^*_R(w_s)
    \end{matrix}\right.\\ \\
    \frac{dI_s}{dz}+\frac{dI_L}{dz}=2n_i \epsilon_0 c \left( \left[ 3i\frac{w_s^2}{n_s c}|A_L|^2 |A_s|^2 \right ]2\chi_R(w_s)+\left[ 3i\frac{w_L^2}{n_L c}|A_L|^2 |A_s|^2 \right ]2\chi_R(w_L)\right )\\
    [/itex]

    The sum of derivatives is supposed to be 0, but I don't see how. Adding chi_R to its complex conjugate gives a real number that isn't 0:

    [itex]
    \epsilon_0 \left( \frac{N}{6m} \right)\left( \frac{\partial a}{\partial q} \right)^2_0 \left[ \frac{1}{w_v^2-(w_s-w_L)^2 +2i(w_s-w_L)\gamma}+ \frac{1}{w_v^2-(w_L-w_s)^2 +2i(w_L-w_s)\gamma} \right ]\\
    =\frac{(w_s+w_L^2-2w_sw_L)-2i(w_s-w_L)\gamma+(w_s+w_L^2-2w_sw_L)+2i(w_s-w_L)\gamma}{(w_s+w_L^2-2w_sw_L)^2+4(w_s-w_L)^2 \gamma^2}\\=\frac{2}{[(w_s+w_L^2-2w_sw_L)+4\gamma^2]}
    [/itex]

    Part 2:
    1. The problem statement, all variables and given/known data
    In part 2, values are given for chi_R(w_s), Stokes and laser frequencies, refractive indices, and intensities for both the laser wave and Stokes wave at z=0. I need to find the intensity of the laser wave at z=1 cm (that is, after propagating 1 cm in the Raman medium.


    2. Relevant equations
    Same as Part 1


    3. The attempt at a solution

    I think we can just plug into the intensity equation, where dz is 1 cm. It seems to me that we need numbers for amplitude values, but they aren't given.
    Thanks for reading all that.
     
  2. jcsd
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