# How to estimate change in refractive index due to acoustic wave travelling in quartz?

I'm studying acousto-optic modulators and the basic principle of operation is that an acoustic/sound wave is made to travel along one direction of a crystal (say quartz). This creates a pressure wave along the propagation direction of the sound, and thus a modulation in the refractive index in the material. Thus, light diffracts when passing through it.

I'm trying to do a very rough estimation of the relative change in refractive index, $$\Delta$$n/n. I know it should be a really small ratio, probably ppb or even less, but I want to do a back of the envelope calculation.

I found in the literature a reported change in refractive index for quartz of 1 ppb / bar, so if I calculate the relative change in pressure, I can estimate $$\Delta$$n/n. I know for example that I apply an RF power of about 30 dB (1W) to the AOM. That eventually gets actuated into a pressure wave propagating in the quartz crystal through a piezo. Say the overall dimensions of the quartz crystal are 10 cm x 5 cm x 2 cm. How would you estimate the pressure change?

I tried doing a really rough calculation saying the actuator is 100% efficient, so naively thinking E ~ P*V, I estimate an "overall" pressure E/V of about 0.1 bars, so the upper limit for the change in refractive index should be 0.1 ppb. Inefficiencies in the coupling between the actuator and the crystal, and the propagation of the mechanical energy through the crystal would only make the ratio even smaller. Would you say this argument is sound?

• Not sure enough to do a formal answer, but there is formula $E=\rho c^2$, so for constant density and assuming pressure changes c in the same way that E would, then $dP = \rho 2c dc$ and since $dn$ is related to $dc$ then $\frac{dn}{dP} = \frac{1}{2 \rho c}$using 5800m/s and density 2650 gives $\frac{dn}{dP} =0.0325$ppm, just an idea Commented Oct 3, 2021 at 21:19

This seems like a reasonable back-of-the-envelope method. Another way to approach this is as follows. Let's say the sound wave driving the AOM sets up some stress in the material that leads to a maximum strain $$s_0$$. For a photoelastic material, the change in refractive index will be given by $$\Delta n = -\frac{1}{2} p n_0^3 s_0$$, where $$p$$ is the photoelastic response coefficient, $$n_0$$ is the unperturbed index of refraction of the material, and $$s_0$$ is the maximum strain. If you look up the coefficients for your material of interest you could then estimate the change in index of refraction.
• I have seen the formula, and I have both p (~0.1-0.2) and n, but in this case $s_0$ is related to the amplitude of the acoustic wave driving the crystal. That's where the power I'm putting in will come in. How would you estimate the $s_0$? It cannot just be a lookup value, otherwise $\Delta n$ would be independent of the power I'm driving with. Commented Oct 3, 2021 at 22:38