Total internal reflection

Refraction at a flat boundary obeys n₁ sin θ₁ = n₂ sin θ₂, both angles measured from the normal. Rearranged for the transmitted ray, sin θ₂ = (n₁/n₂) sin θ₁. When n₁ > n₂ that prefactor exceeds one, so θ₂ is always larger than θ₁ and the transmitted ray leans towards the surface. Increase θ₁ and θ₂ closes on 90°. The equation reaches its own limit when sin θ₂ = 1, which happens at n₁ sin θ_c = n₂, so sin θ_c = n₂/n₁ and θ_c = arcsin(n₂/n₁). For crown glass against air that is arcsin(1.00/1.52) = 41.1°; for water against air, 48.8°; for diamond, whose index is 2.42, only 24.4°. Beyond θ_c no real angle has the sine the equation is asking for, so the transmitted ray does not merely grow faint — it does not exist, and the surface returns the whole beam.

The ratio n₂/n₁ can only be a sine if it is at most 1, so a critical angle exists only when n₁ > n₂: the light must already be inside the denser medium. Run the same boundary the other way, air into crown glass, and sin θ₂ = sin θ₁/1.52 ≤ 0.658, so the refracted ray inside the glass never exceeds 41.1° from the normal however you aim the incoming beam. The same number appears, now as a ceiling on refraction rather than a threshold for reflection — the two statements are one fact read in both directions, which is ray reversibility. What matters is the pair of indices, not density in the everyday sense: glass at 1.52 against water at 1.33 still reflects totally, but only past 61.0°, and a fibre core barely denser than its cladding has a critical angle of 81.5°, within 8.5° of grazing.

A boundary already reflects some light below the critical angle; total reflection is the end of a trend, not the start of one. Head-on, the reflected fraction at a glass-air surface is R₀ = ((1.52 − 1)/(1.52 + 1))² = 0.043, about 4%. That fraction climbs slowly as θ₁ grows, rises steeply over the last few degrees before θ_c, and equals exactly 1 at θ_c and everywhere beyond. No coating does that. A good aluminium mirror returns about 90% of visible light and a protected silver one perhaps 97%, and every bounce multiplies the loss: a two-mirror periscope keeps 0.90² = 0.81 of what enters it, while the two total reflections inside a prism periscope cost nothing whatever. The prism is not free — it still gives up about 4% at each of its own entry and exit faces unless they are coated — but the bounces themselves are exact. A right-angle prism in air reflects totally at 45° provided sin 45° ≥ 1/n, that is n ≥ √2 = 1.414, which crown glass at 1.52 comfortably satisfies.

The electromagnetic boundary conditions do not let a field stop dead at a surface, so even above θ_c something exists in the rarer medium. It does not propagate away from the surface: its amplitude decays as e^(−z/d) with distance z from the boundary, where d = λ₀ / (2π √(n₁² sin²θ₁ − n₂²)). This evanescent wave runs along the interface and, averaged over a cycle, carries no net power across it, so energy conservation and the word total both survive. The depth is small and shrinks as you move past θ_c: for crown glass against air at λ₀ = 589 nm and θ₁ = 45.0°, d = 238 nm. Bring a second glass surface within a few d and the evanescent field reaches it, a genuine transmitted ray reappears on the far side, and the reflection is frustrated. That is the mechanism of beam-splitter cubes, optical fingerprint readers, and microscopes that illuminate only the first hundred nanometres of a cell.

A step-index fibre is a high-index core inside a slightly lower-index cladding, and any ray meeting the wall past θ_c bounces along it indefinitely. What limits the fibre is the cone it will accept at its end face. A ray entering at θₐ in air refracts to θᵣ inside the core and then strikes the wall at 90° − θᵣ, so guiding demands 90° − θᵣ ≥ θ_c; combining that with n₀ sin θₐ = n_core sin θᵣ gives n₀ sin θₐ ≤ √(n_core² − n_clad²), the numerical aperture. With a core of 1.4682 and a cladding of 1.4519, NA = √(2.15561 − 2.10801) = 0.2182, an acceptance half-angle of 12.6° in air. Light arriving steeper refracts into the cladding and is gone within centimetres. A wide cone has a price: the steepest guided ray travels 1/cos θᵣ as far as the axial one, 11.2 m extra per kilometre, arriving about 55 ns late and smearing the pulse.

Total internal reflection is a statement about a flat, clean, unbounded boundary, and real surfaces fall short in specific ways. A bend tilts the wall relative to the rays, and below a critical bend radius the rays at the outer wall arrive inside θ_c and escape, which is why a kinked fibre leaks. Anything touching the surface changes n₂: a fingerprint replaces air at 1.00 with oil near 1.47, pushing θ_c for crown glass from 41.1° to 75.3° and killing the reflection at 45°. The reflected beam is also displaced sideways by a fraction of a wavelength, the Goos-Hanchen shift, because the field spends time in the evanescent skin — invisible in a ray diagram, measurable on a bench. And because n depends on wavelength, so does θ_c: in crown glass violet sees 40.9° and red 41.3°, so a boundary set at 41.0° reflects violet totally while red still escapes.