Refractive Index, Critical Angle & Total Internal Reflection

Refractive Index Guide

Snell's Law and Refractive Index

Snell's law: n₁ sin θ₁ = n₂ sin θ₂. Refractive index of medium 2 relative to medium 1: n = sin θ_i / sin θ_r (when going from medium 1 to medium 2, with medium 1 being air/vacuum). The refractive index of a medium: n = c / v where c = speed of light in vacuum (3×10⁸ m/s) and v = speed of light in the medium. Air: n ≈ 1.0003 (effectively 1). Water: n ≈ 1.33. Glass: n ≈ 1.5. Diamond: n ≈ 2.42. Higher n means light slows more and bends more on entering the medium.

Critical Angle and Total Internal Reflection

When light travels from a denser medium to a less dense medium, it bends away from the normal. At the critical angle θ_c, the refracted ray travels along the boundary (90° to normal). Above θ_c: total internal reflection — all light is reflected back inside the denser medium. sin(θ_c) = n₂/n₁ = 1/n for glass-to-air. Diamond (n=2.42): θ_c = arcsin(1/2.42) = 24.4°. Very small critical angle means most light is totally internally reflected — this is why cut diamonds sparkle. Optical fibres: light i

Apparent Depth

Objects in water appear shallower than they really are due to refraction. Apparent depth = real depth / n. Water (n=1.33): a fish at 2m real depth appears at 2/1.33 = 1.50m. This apparent depth effect is why swimming pools look shallower than they are — a 2m deep pool appears to be approximately 1.5m deep from the surface. The effect explains the apparent bending of a straw in a glass of water — the submerged part appears displaced because light from it refracts at the water-air interface.

Applications of TIR

Optical fibres: core glass has higher n than cladding glass. Light entering within the acceptance cone is totally internally reflected repeatedly along the core. Monomode fibres (core 8-10μm): carry a single light mode, used for long-distance telecoms. Multimode fibres (core 50-62.5μm): multiple modes, used for shorter distances (local area networks). Endoscopes: flexible bundles of optical fibres transmit images from inside the body. Retroreflectors: cat's eyes in road markings and road signs u