why does anti-reflecting coating not affect critical angle? I did an experiment to find the refractive index of a plano-convex lens by measuring it's critical angle. I got a correct answer of 1.521 but the lens was coated with an anti-reflective coating and im not sure why that did't affect the critical angle. 
 A: The anti-reflective coating didn't affect the result because the coating has an even thickness, is thin compared to the dimensions of the lens, and has a refractive index greater than the refractive index of air.
Let $n_l$, $n_c$ and $n_a$ be the refractive index of the lens material, the coating, and air, respectively, and let $\theta_l$, $\theta_c$ and $\theta_a$ be the corresponding angles a ray of light makes with the surface normal as it passes through the three media. Because the coating has an even thickness, it's possible to talk about the surface normal, instead of having to deal with different surface normal vectors for the lens/coating interface and the coating/air interface.
Snell's law applied to the coating/air interface gives
$$\frac{\sin \theta_a}{\sin \theta_c}=\frac{n_c}{n_a}\ \ .$$
Snell's law applied to the lens/coating interface gives
$$\frac{\sin \theta_c}{\sin \theta_l}=\frac{n_l}{n_c}\ \ .$$
The above two equations can be combined as
$$\frac{\sin \theta_a}{\sin \theta_l}=\frac{\sin\theta_a}{\sin\theta_c}\frac{\sin\theta_c}{\sin\theta_l}=\frac{n_c}{n_a}\frac{n_l}{n_c}=\frac{n_l}{n_a}\ \ .$$
But
$$\frac{\sin \theta_a}{\sin \theta_l}=\frac{n_l}{n_a}$$
is exactly what Snell's law would give if the coating wasn't even there, i.e., the coating can just be ignored. There's a bit of a displacement orthogonal to the surface normal between where a ray goes through the air/coating interface and the where the ray goes through the coating/lens interface, but that small displacement can be neglected because the coating is so thin compared to the main lens material.
The above result is only valid as long as $\theta_c < \pi/2$. But in the critical angle experiment, we always have $\theta_a < \pi/2$, so the validity condition will hold as long as $\theta_c < \theta_a$. And since $\sin\theta$ is a monotonic increasing function in the range $0<\theta<\pi/2$, the validity condition will hold as long as $\sin\theta_c<\sin\theta_a$, which via Snell's law will hold as long as $n_c>n_a$. And the refractive index of pretty much any translucent solid in general is greater than the refractive index of air, so it doesn't really matter what the coating is made out of.
