Total Internal Reflection and the Critical Angle

Total internal reflection (TIR) is one of the most dramatic phenomena in optics. You're shining light inside a dense medium (water, glass, diamond) toward a boundary with a less-dense medium (air). At small angles, some light escapes and some reflects — business as usual. But as you increase the angle of incidence, the refracted ray bends farther and farther away from the normal. At a certain angle — the critical angle $\theta_{c}$ — the refracted ray skims along the surface at exactly 90°. And beyond $\theta_{c}$, refraction becomes impossible: all the light bounces back inside the dense medium. 100% reflection. Zero transmission. It's as if the boundary becomes a perfect mirror.

This isn't gradual dimming followed by cutoff — it's a sharp transition. Below $\theta_{c}$, there's a refracted beam. At $\theta_{c}$, it grazes the surface. Above $\theta_{c}$, it vanishes completely. That dramatic cutoff is what makes TIR so useful in technology.

Finding the critical angle

The critical angle is the angle of incidence at which $\theta_{2} = 90°$. At this angle, $\sin\theta_{2} = 1$, so Snell's law gives:

$n_{1}\sin\theta_{c} = n_{2}\sin 90° = n_{2}$

$\sin\theta_{c} = \dfrac{n_{2}}{n_{1}}$

$\theta_{c} = \arcsin\!\left(\dfrac{n_{2}}{n_{1}}\right)$

For this to have a solution, we need $n_{2}/n_{1} < 1$, which means $n_{1} > n_{2}$: TIR only happens when light goes from a denser medium to a less-dense medium. You'll never see TIR when light goes from air into glass — only from glass into air.

For our problem: water → air, $n_{1} = 1.33$, $n_{2} = 1.00$.

$\theta_{c} = \arcsin\!\left(\dfrac{1.00}{1.33}\right) = \arcsin(0.7519) \approx 48.75°$

So: any ray hitting the water-air boundary at more than 48.75° from the normal will be totally internally reflected.

What happens at each angle range

$\theta_{1} < \theta_{c}$ (below critical angle): Some light refracts into the air (bending away from the normal), and some reflects back into the water. The refracted beam gets weaker and the reflected beam gets stronger as $\theta_{1}$ approaches $\theta_{c}$.

$\theta_{1} = \theta_{c}$ (at the critical angle): The refracted ray grazes along the surface at 90°. In practice, the refracted beam is infinitely weak at this angle — all the energy goes into reflection.

$\theta_{1} > \theta_{c}$ (above critical angle): No refracted ray exists. Snell's law gives $\sin\theta_{2} > 1$, which is impossible. 100% of the light reflects. The boundary acts as a perfect mirror with zero absorption loss.

Critical angles for common pairs

• Glass (1.50) → Air: $\theta_{c} = \arcsin(1/1.50) = 41.8°$
• Water (1.33) → Air: $\theta_{c} = 48.75°$ (our problem)
• Diamond (2.42) → Air: $\theta_{c} = \arcsin(1/2.42) = 24.4°$
• Glass (1.50) → Water (1.33): $\theta_{c} = \arcsin(1.33/1.50) = 62.5°$

Diamond's low critical angle (24.4°) is why diamonds sparkle so intensely. Light entering a diamond bounces around inside due to TIR at most internal surfaces, and only escapes through carefully angled facets. A gem cutter's job is to shape the facets so that the most dramatic internal reflections direct light back toward the viewer's eye.

Why TIR is a perfect mirror

An ordinary metal mirror absorbs a few percent of the light at each reflection. Over many bounces, the signal degrades. TIR, on the other hand, reflects 100% of the light — there's no absorption at the interface, no metal coating to degrade. That's why TIR is the operating principle behind:

• Optical fibers: A glass core ($n \approx 1.50$) surrounded by a glass cladding ($n \approx 1.48$). Light entering the core at a shallow angle repeatedly hits the core-cladding boundary at an angle greater than $\theta_{c}$ and bounces back, zigzagging down the fiber for kilometers with almost no loss. This is how the internet's backbone works — undersea cables carry data as light pulses via TIR.

• Binoculars and periscopes: Use glass prisms with internal surfaces at 45°. Since $\theta_{c}$ for glass-air is about 42°, the 45° angle exceeds it, creating a perfect reflecting surface without a metal coating. Lighter, more durable, and higher-quality than mirror-based designs.

• Diamonds: Cut so that most light entering from the top hits internal facets at angles exceeding 24.4° and reflects back out the top.

• Endoscopes and light pipes: Use TIR to channel light around curves for medical imaging.

The evanescent wave (advanced)

Even during TIR, there's a subtle effect: an evanescent wave penetrates a tiny distance (about one wavelength) into the less-dense medium. This wave doesn't carry energy away — it decays exponentially. But if you bring another piece of glass very close (within the evanescent wave's reach), the light can "tunnel" across the gap. This is called frustrated total internal reflection and is the optical analog of quantum tunneling. It's used in touchscreens, fingerprint sensors, and beam splitters.

Common mistakes

• Thinking TIR can happen going from less-dense to more-dense. It can't. Light going from air into glass always has a refracted ray (it bends toward the normal). TIR requires $n_{1} > n_{2}$.
• Using the wrong ratio. The critical angle formula is $\sin\theta_{c} = n_{2}/n_{1}$ (smaller $n$ over larger $n$). If you invert it, you get $\sin\theta_{c} > 1$, which should signal an error.
• Ignoring the reflected ray below $\theta_{c}$. Even below the critical angle, there's always a partially reflected ray. TIR doesn't create reflection — it makes it total. Partial reflection exists at all angles.

Try it in the visualization

Start at a small angle and slowly increase it. Watch the refracted ray in the upper medium bend further and further from the normal. At the critical angle, it lies flat along the surface — then poof, it vanishes, and the reflected beam suddenly brightens to full intensity. Switch between water→air, glass→air, and diamond→air to see how the critical angle changes. Diamond's tiny critical angle (24.4°) means TIR kicks in almost immediately.