Critical Angle for Total Internal Reflection

When light travels from a denser medium (higher n) to a less dense medium (lower n), total internal reflection occurs if the angle of incidence exceeds the critical angle:

θ_c = arcsin(n₂/n₁) where n₁ > n₂

At angles greater than θ_c, ALL light is reflected back and none escapes. This is the basis of optical fiber communication.

Snell’s Law at the critical angle: n₁ sin(θ_c) = n₂ sin(90°) = n₂ → sin(θ_c) = n₂/n₁

Requirements:

• Light must travel from the denser medium (higher n) to the less dense one
• n₁ must be > n₂
• If n₁ < n₂, there is no critical angle, so total internal reflection cannot occur in this direction

Common material refractive indices:

• Air/vacuum: n = 1.000
• Water: n = 1.333
• Crown glass: n = 1.52
• Flint glass: n = 1.62
• Diamond: n = 2.417
• Cubic zirconia: n = 2.17

Applications:

• Optical fiber: Light travels in a glass or plastic core (n ≈ 1.5) surrounded by cladding (n ≈ 1.46). Critical angle ≈ 77°. Light bounces along the fiber with essentially zero loss.
• Diamonds: the high refractive index (2.417) gives a critical angle of only 24.4°, so light entering the stone is totally internally reflected many times before it escapes. That is where the famous “fire” and brilliance come from, and it is why a cutter who gets the pavilion angles wrong produces a stone that leaks light out of the bottom and looks dull.
• Prisms in binoculars: Roof prisms use total internal reflection to fold the optical path compactly.
• Swimming pools: Viewed from underwater beyond the critical angle, the surface appears like a mirror.

The escape cone, and why LEDs are hard to get light out of

Turn the critical angle around and it stops being about rays and starts being about budgets. A source buried inside the dense material radiates in every direction, but only the rays landing inside a cone of half-angle θ_c can leave. That cone’s share of the full sphere works out to (1 − cos θ_c) / 2 per surface, which the calculator reports alongside the angle.

Run gallium nitride at n = 2.4 and the number is brutal: about 4.5% of the light escapes through a given face on the first attempt. The rest bounces around inside and is slowly absorbed. That single figure is why LED makers roughen the surface, add a dome of encapsulant, or shape the chip, and why an LED that is 90% efficient at making photons can still be a mediocre lamp. The same arithmetic explains a diamond’s brilliance from the other direction, since light that struggles to get out has to bounce a lot on the way.