Physics

Refraction at a boundary is governed by Snell's law, which relates the angles of incidence and refraction to the refractive indices—but the key insight is that the component of the wave vector parallel to the interface is conserved. Since the $x$-$z$ plane is the boundary, the $\hat{i}$ and $\hat{k}$ components of the direction vectors don't "bend," only the perpendicular ($\hat{j}$) component does. This means you can set up a relationship between the parallel components using $\mu_1 \sin\theta_1 = \mu_2 \sin\theta_2$, where the sines come directly from the $\hat{i}$ (or $\hat{k}$) parts of your unit vectors.

Refraction at a boundary is governed by Snell's law, which relates the angles of incidence and refraction to the refractive indices—but the key insight is that the component of the wave vector parallel to the interface is conserved. Since the $x$-$z$ plane is the boundary, the $\hat{i}$ and $\hat{k}$ components of the direction vectors don't "bend," only the perpendicular ($\hat{j}$) component does. This means you can set up a relationship between the parallel components using $\mu_1 \sin\theta_1 = \mu_2 \sin\theta_2$, where the sines come directly from the $\hat{i}$ (or $\hat{k}$) parts of your unit vectors.