Geometric Optics — Where Rays Rule
When the wavelength of light is much smaller than the objects it encounters, the wave nature of light can be ignored and we treat it as a ray travelling in straight lines. Geometric optics gives us Snell's law, total internal reflection, and lens equations — and all of these are exact in the ray limit.
Snell's Law
Drag a ray across a glass boundary. The refracted angle updates live as you tune n₁ and n₂ — with the full vector derivation visible.
Total Internal Reflection
Increase the angle of incidence past the critical angle and watch the refracted ray vanish — evanescent field, Goos-Hänchen shift, and fibre-optic waveguide mode all shown.
Mirrors & Lenses
Converging and diverging lenses, concave and convex mirrors — drag the object and watch the image relocate via the thin-lens equation and ray-transfer matrix.
Optical Fibre
Multi-mode and single-mode fibres: numerical aperture, V-number, evanescent coupling, and mode confinement — all derived from the TIR condition.
Key Relations — Geometric Optics
Snell's law: n₁ sin θ₁ = n₂ sin θ₂
Critical angle: θ_c = arcsin(n₂ / n₁) (n₁ > n₂)
Thin-lens: 1/f = 1/d_o + 1/d_i
Numerical aperture: NA = n_core · sin(θ_max) = √(n_core² − n_clad²)
Fresnel reflectance (s-pol): r_s = (n₁cosθ_i − n₂cosθ_t) / (n₁cosθ_i + n₂cosθ_t)
Wave Optics — Where Interference Appears
When the wavelength of light is comparable to aperture sizes or slit separations, the wave nature dominates. Interference and diffraction are purely wave phenomena — the ray model cannot explain them.
Double-Slit Experiment
Young's experiment in full — slit separation d, wavelength λ, screen distance L → fringe spacing y = λL/d. Switch to single-photon mode to see the quantum weirdness.
Single-Slit Diffraction
Fraunhofer single-slit pattern: intensity I(θ) ∝ sinc²(πasinθ/λ). Tune aperture width a and observe central maximum spreading as a → λ.
Rayleigh Scattering
Elastic scattering cross-section ∝ λ⁻⁴ explains blue sky, red sunsets, and why overhead sun is white — adjustable atmospheric depth and sun angle.
Colours of Light
CIE 1931 XYZ colour matching, spectral to RGB, additive/subtractive mixing, metamerism — the full colour science behind human vision.
Natural Phenomena & Quantum Optics
Rainbow Formation
Descartes' raindrop — ray tracing inside a sphere: two refractions plus one internal reflection produce the primary bow at 42°, the secondary at 51°, and Alexander's dark band.
Water Caustics
Refracted light rays focusing through a water surface create bright caustic patterns on the pool floor — rendered in real time with a GPU-accelerated photon map.
Photoelectric Effect
Einstein's 1905 paper in simulation form: E_photon = hν, work function φ, stopping voltage V_s = (hν − φ)/e — tune frequency and metal type to see threshold behaviour.
Why does TIR enable optical fibres? When light travels from a denser medium (glass, n≈1.5) into a less dense one (air, n=1), total internal reflection occurs above the critical angle θ_c = arcsin(1/1.5) ≈ 41.8°. A fibre's core-cladding interface is engineered so that guided rays always exceed this angle — they bounce indefinitely without loss to radiation. The evanescent field that extends a few hundred nanometres into the cladding is exploited for coupling between fibres and in integrated photonic chips.
Algorithms & Methods
Suggested Learning Paths
📐 Geometric Optics First
- Snell's Law — build the refraction intuition
- Total Internal Reflection — understand critical angle
- Optical Fibre — apply TIR to waveguides
- Mirrors & Lenses — ray-transfer matrix
- Rainbow Formation — Descartes + dispersion
🌊 Wave Optics Track
- Colours of Light — CIE colour space foundation
- Rayleigh Scattering — λ-dependence of scattering
- Diffraction — single aperture, sinc² pattern
- Double Slit — Young's experiment + QM mode
- Photoelectric Effect — quantisation of light