Photon conservation in Raman amplification

[snickersnee]

Part 1:

Homework Statement

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.

Homework Equations

See (3) below

The Attempt at a Solution

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

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:

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

Part 2:

Homework Statement

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.

Homework Equations

Same as Part 1

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.

 

[Nate810]

\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 )\\=0\\\therefore\ I_s(z=1cm)+I_L(z=1cm)=I_s(z=0)+I_L(z=0)=constant.