When light passes from one medium into another, its direction changes according to Snell’s law:
\[
n_1\sin\theta_i=n_2\sin\theta_t.
\]
Here \(n_1\) and \(n_2\) are the refractive indices of the two media, \(\theta_i\) is the angle of incidence, and \(\theta_t\) is the angle of transmission or refraction.
All of these angles are measured from the normal to the surface, not from the surface itself. This is one of the most important conventions in geometric optics, because confusing the normal and the surface angle leads to many wrong answers.
A very special situation occurs when light travels from a medium with higher refractive index into one with lower refractive index. In that case, the refracted ray bends away from the normal.
As the incident angle increases, the refracted ray is pushed closer and closer to the interface. There is one specific incident angle for which the refracted ray just grazes the boundary.
That threshold angle is called the critical angle, and it satisfies
\[
\theta_c=\sin^{-1}\left(\frac{n_2}{n_1}\right),
\]
but only when \(n_1>n_2\). If \(n_1\le n_2\), then this quantity is not physically relevant as a critical angle for that direction of travel, because the refracted ray never reaches the grazing limit in the same way.
The critical angle comes directly from Snell’s law by setting the transmitted angle to \(90^\circ\). At that point,
\[
\sin\theta_t=\sin 90^\circ = 1,
\]
so Snell’s law becomes
\[
n_1\sin\theta_c=n_2,
\]
and therefore
\[
\sin\theta_c=\frac{n_2}{n_1}.
\]
If the incident angle becomes larger than this threshold, Snell’s law would require \(\sin\theta_t>1\), which is impossible for a real propagating angle.
This means the refracted ray can no longer propagate into the second medium. Instead, the wave is reflected entirely back into the first medium.
This phenomenon is called total internal reflection, or TIR.
The reflected ray still obeys the law of reflection,
\[
\theta_r=\theta_i,
\]
so from a geometric-optics point of view the path looks like a clean mirror-like bounce at the boundary.
That is why TIR plays such an important role in optical fibers, prisms, and sparkle effects in high-index materials such as diamond.
However, the full electromagnetic picture is slightly richer than the simple ray picture. Even though no real transmitted ray propagates into the second medium during TIR,
the electric and magnetic fields do not drop abruptly to zero at the interface. Instead, a non-propagating field appears on the second-medium side and decays exponentially with distance from the boundary.
This is called an evanescent wave. Its amplitude is strongest at the interface and fades rapidly away from it.
In this calculator, the evanescent-wave graphic is shown only as a qualitative preview so students can connect the geometric picture with the underlying wave behavior.
For the sample case of water to air, with \(n_1=1.33\) and \(n_2=1.00\), the critical angle is
\[
\theta_c=\sin^{-1}\left(\frac{1.00}{1.33}\right)\approx 48.6^\circ.
\]
If the incident angle is \(50^\circ\), then \(50^\circ > 48.6^\circ\), so total internal reflection occurs.
The ray remains in the water, and no real propagating transmitted ray enters the air.
This idea explains many familiar optical effects. Swimming pools can appear brighter or dimmer depending on viewing angle.
Diamond sparkles strongly because its refractive index is high, so its critical angle is relatively small and many internal rays undergo repeated TIR before escaping.
Optical fibers guide light precisely because the core has a slightly higher refractive index than the cladding, allowing repeated total internal reflections that keep the light trapped over long distances.
At a more advanced university level, TIR connects naturally to waveguides, Fresnel coefficients, phase shifts on reflection, frustrated total internal reflection, and evanescent-field coupling.
But even at the introductory level, the key picture is already powerful: below the critical angle, light can refract; above it, the interface acts like a perfect internal reflector.
