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Derive the Manley-Rowe relation for the process of the second harmonics generation.

Manley-Rowe relation: ~ The Manley-Rowe relations are mathematical expressions developed originally for electrical engineers to predict the amount of energy in a wave that has multiple frequencies.

Second-harmonic generation: ~ Second harmonic generation (SHG); also called frequency doubling; is a nonlinear optical process, in which photons interacting with a nonlinear material are effectively “combined” to form new photons with twice the energy, and therefore twice the frequency and half the wavelength of the initial photons. It is example of nonlinear phenomena (I, 3). In the SHG process, the intense wave at the frequency $\omega$ propagates in the medium with second-order nonlinearity ( VI, 1).

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The Manley-Rowe relation arises from conservation of energy and momentum. For the case of SHG, the presence of the nonlinear optical (NLO) material eliminates the conservation of momentum (any momentum difference between the initial and final photons can be provided by the bulk material). So what's left is conservation of energy.

Let $N_\omega$ and $N_{2\omega}$ be the number of the fundamental frequency and the second harmonic. Usually NLO people care about how these sorts of things change with the distance that the wave moves through the material. So let the direction of propagation be $x$. Then, by energy conservation:
$$2\frac{dN_{\omega}(x)}{dx} + \frac{dN_{2\omega}(x)}{dx} = 0.$$ This follows from $E=\hbar\omega$. That is, it takes two of the $\omega$ photons to provide the energy in one $2\omega$ photon.

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