Ray Optics and Optical Instruments

Snell’s law governs refraction at any boundary between two media of refractive index $n_1$ and $n_2$:

Going from a denser medium ($n_1$, e.g. glass) toward a rarer one ($n_2$, e.g. air), so $n_1 > n_2$, solving for $\sin\theta_2$ gives:

Since $n_1/n_2 > 1$, the right side grows faster than $\sin\theta_1$ itself. The critical angle $\theta_c$ is defined as the one angle of incidence at which the refracted ray grazes exactly along the boundary, $\theta_2 = 90^\circ$:

For any $\theta_1 > \theta_c$, the equation $\sin\theta_2 = (n_1/n_2)\sin\theta_1$ would demand $\sin\theta_2 > 1$, which is impossible. No refracted ray can exist, so all the light reflects back into the denser medium: total internal reflection.

Most students picture refraction “weakening” as the angle of incidence grows, until it eventually stops. That picture is wrong, and the simulation above is built to make the wrongness visible: refraction does not weaken, it is simply present, right up to $\theta_1 = \theta_c$, and then completely absent for every single angle beyond it.

At $\theta_1$ just below $\theta_c$, a real refracted ray exists, grazing almost flat along the surface at nearly 90°. At $\theta_1 = \theta_c$ exactly, that ray grazes at precisely $90^\circ$. One hundredth of a degree further, and Snell’s law has no real solution at all, there is no angle whose sine exceeds 1. The refracted ray does not get fainter or more grazing still, it simply ceases to exist, and 100% of the light reflects back internally instead. This is a genuine discontinuity in the physics, not a smooth transition, and exam questions are written specifically to catch students who assume otherwise.

This one discontinuity is the entire physical basis of optical fibres, where light launched at a steep angle keeps re-striking the core-cladding boundary beyond θc at every bounce, so it is reflected, never refracted out, carrying a signal the length of the fibre with almost no loss. It is also why diamonds sparkle: a diamond’s very high refractive index gives it an unusually small critical angle (≈24.4°), so light entering it keeps exceeding that small θc at its internal facets, bouncing around via total internal reflection several times before it finally escapes.