Geometric Optics: Reflection and Refraction
Geometric optics treats light as rays traveling in straight lines, bending only at boundaries between materials — a simplification that works beautifully as long as whatever the light interacts with is much larger than its wavelength (true for mirrors, lenses, and prisms, not true for the diffraction and interference in Units 305-306, where the wave nature comes back to the foreground). Snell's law, n₁sinθ₁=n₂sinθ₂, is the single equation this entire unit is built around.
What you'll learn
- Apply the law of reflection to find the direction of a reflected ray.
- Apply Snell's law to find the angle of refraction at a boundary between two media.
- Calculate the speed and wavelength of light inside a medium of known refractive index.
- Calculate the critical angle for total internal reflection at a given boundary.
- Determine whether a given ray undergoes refraction or total internal reflection.
- Explain why refractive index depends on wavelength (dispersion) at a qualitative level.
1. Reflection and Snell's Law
The law of reflection — angle of incidence equals angle of reflection, both measured from the surface's normal — holds for any reflecting surface, flat or curved. Refraction at a boundary between two transparent media follows Snell's law: n₁sinθ₁=n₂sinθ₂, where refractive index n=c/v measures how much slower light travels in a medium compared to vacuum.
A higher refractive index means a *slower* speed, not faster — a common point of confusion worth deliberately unlearning if it comes up.
2. Total Internal Reflection
When light tries to go from a higher-index medium into a lower-index one, Snell's law eventually runs out of solutions as the angle of incidence increases — beyond the critical angle θc (where sinθc=n₂/n₁), no refracted ray can exist at all, and 100% of the light reflects back into the original medium. This total internal reflection, not ordinary mirror reflection, is what keeps light traveling down an optical fiber.
Key equations
- θᵢ = θᵣ — The law of reflection — always true, for any reflecting surface.
- n = c/v — Definition of refractive index — always ≥1, since light never travels faster in a medium than in vacuum.
- n₁ sinθ₁ = n₂ sinθ₂ — Snell's law — relates the angles of incidence and refraction at a boundary between two media of different refractive index.
- λ_medium = λ_vacuum/n — Wavelength shortens inside a medium with n>1 (frequency stays the same, since it's set by the source; speed decreases, so wavelength must decrease too, since v=fλ).
- sinθc = n₂/n₁ (for n₁>n₂) — Total internal reflection occurs for any angle of incidence beyond θc, when light tries to go from a higher-index medium into a lower-index one — beyond this angle, no refracted ray exists at all, and 100% of the light reflects.