Nonlinear electric susceptibility and degenerate frequencies

[Yorre]

Hi there,

I'm having a little trouble understanding the "distinguishability" of frequencies in the nonlinear electric susceptibility tensor. As far as I understand, if we have a SHG process with two collinear beams of the same polarization and frequency ω, there is only one susceptibility component 2ω;ω,ω. But if these beams propagate to different directions, still with the same frequency ω, must we label the frequencies as ω1 and ω2 and end up with components 2ω;ω1,ω1 , 2ω;ω2,ω2 , 2ω;ω1,ω2 and 2ω;ω2,ω1? So can the propagation alone make the beams distinct?

Thanks in advance!

 

[blue_leaf77]

The second order susceptibility, as well as susceptibility in the other orders, are material specific and is not dependent on the external field. So, I believe propagation direction should not change the value of $\chi^{(2)}(2\omega;\omega,\omega)$. The quantities which will look different for different propagation alignments are, among others, the effective susceptibility and the phase matching condition.

 

[Yorre]

Thanks for the reply, blue_leaf. So when calculating the NL source polarization, the complex amplitudes of the two fields create four propagation terms (1,1), (1,2), (2,1), (2,2) and four phase matching conditions, while the effective susceptibility can be taken as the common factor when both fields have the same polarization?

 

[blue_leaf77]

Sorry, I forgot that the effective susceptibility $d_{eff}$ was defined when the two beams are propagating in the same direction, and it reads
$$
P(\omega_3) = d_{eff}E(\omega_1)E(\omega_2)
$$
where $P(\omega_3) = |\mathbf{P}(\omega_3)|$ and $E(\omega_i) = |\mathbf{E}(\omega_i)|$. For crossing beams, I am not sure if you can derive a similar expression which relates the magnitudes of the polarization and fields like that above. An example of the derivation of $d_{eff}$ can be found in "Applied Nonlinear Optics" by Zernike and Midwinter for some cases. May be you can derive $d_{eff}$ for general case of crossing beams.