Does Snell's law still apply after total internal reflection has occurred?

Question

Snell's law states that:

\begin{align} n_1\sin\theta_1 &= n_2\sin\theta_2\, ,\\ \sin\theta_1 &= \frac{n_2}{n_1}\sin\theta_2\, ,\\ \sin\theta_1&\propto \sin\theta_2\, . \end{align}

The critical angle $\theta_1 = \theta_c$ is reached when $\sin\theta_2 = 1$, assuming that $n_2 > n_1$.

Total internal reflection occurs when $\theta_1 > \theta_c$, and when $\theta_1$ increases between 0 and 90 so does $\sin\theta_1$, and since $\sin\theta_1 \propto \sin\theta_2$, $\sin\theta_2$ should increase also, which should be impossible since $\sin\theta_2 = 1$.

Does the proportionality (and therefore Snell's law) no longer hold once the critical angle $\theta_c$ has been reached, or have I got my maths wrong?

Answer 1

First, let's calculate the critical angle. As you noted, at $\theta_1=\theta_c$, $\sin\theta_2=1$, and thus $\theta_c=\arcsin(n_2/n_1)$. Obviously, for a critical angle to exist, we must have $n_1>n_2$, so that $n_2/n_1<1$.

Now, from Snell's law, we know that \begin{align} \sin\theta_1=\frac{n_2}{n_1}\sin\theta_2\end{align} As $\theta_1$ increases to $\theta_c$, $\sin\theta_2$ goes to $90$ degrees, as you noted:

\begin{align} \sin \theta_c =\frac{n_2}{n_1}=\frac{n_2}{n_1}\sin\theta_2\rightarrow \sin\theta_2=1 \end{align}

For $\theta_1>\theta_c$, we find that

\begin{align} \sin \theta_1 \equiv C = \frac{n_2}{n_1}\sin\theta_2\rightarrow \sin\theta_2=\frac{C}{n_2/n_1} \end{align}

where $C$ is some number larger than $(n_2/n_1)$ but less than 1, implying that $\frac{C}{n_2/n_1}>1$. Thus, you are correct; Snell's law will not be able to solve for the refracted angle for an incident angle larger than $\theta_c$. Instead, the system in this limit obeys the law of reflection. For more about Snell's law and total internal reflection, see here for a good reference.

Answer 2

Snell's law is obtained by applying the electromagnetic boundary conditions to the problem; therefore it holds under all circumstances where Maxwell's equations hold.

In order to see what happens for angles of incidence greater than the critical angle, we derive Snell's law. The electromagnetic boundary conditions take the following form for both TE and TM polarizations of the incident wave:

$$A_ie^{j\mathbf{k_+ \cdot r}} + A_re^{j\mathbf{k_- \cdot r}} = A_te^{j\mathbf{k'\cdot r}}$$

Snell's law is derived by taking care of the exponential parts for $z=0$:

$$k_x=k_x' \rightarrow k\sin \theta = k'\sin \theta' \rightarrow \boxed{n\sin \theta =n'\sin \theta'}$$

Now we obtain the wavevectors in the $z$ direction, $k_z$ and $k_z'$:

$$k_z^2+k_x^2=k^2=n^2 k_0^2$$

$$k_z'^2+k_x'^2=k'^2=n'^2 k_0^2$$

Using the Snell's law ($k_x' = k_x$, so $k_x' = k \sin \theta = n k_0 \sin \theta $) we find, from the second equation:

$$k_z'^2 = k_0^2 \left( n'^2 - n^2 \sin^2 \theta \right) \Rightarrow k_z' = k_0\sqrt{n'^2-n^2\sin^2 \theta}$$

Using this equation we can see what happens for angles of incidence greater than the critical angle. If $n'<n\sin \theta$ the wavevector in the $z$ direction would become imaginary. $k_z' = j \alpha$, with

$$\alpha = k_0 \sqrt{n^2\sin^2 \theta - n'^2}$$

and the wave function in the right-hand medium would be

$$()e^{- \alpha z + j k_x' x}$$

Therefore, there exists a wave in the second medium even for $\theta >\theta_0$, but it decays exponentially with $z$ and doesn't carry any energy in the $z$ direction (an evanescent wave).