2D Geometric Optics Simulation

A Gentle Introduction to Geometric Optics

This is an introductory educational demo that shows an interesting way to visualize optical phenomena: modeling light as a stream of photons with probabilistic behavior. While this approach trades physical rigor for simplicity and visual clarity, it can help build intuition about refraction, reflection, and dispersion.

Rendering Perspective:

At every interface between materials with different refractive indices, incident light is partially reflected and partially transmitted (refracted). The Fresnel equations determine the exact energy split between these two components. In path tracing and Monte Carlo rendering, we cannot afford to deterministically split rays at every interface—instead, we use stochastic sampling: each ray makes a probabilistic choice between reflection and refraction, weighted by the Fresnel reflectance coefficient R. With probability R, the ray reflects; with probability (1-R), it refracts. Over many samples, the aggregate distribution converges to the correct energy partition. This approach is unbiased and naturally handles complex multi-bounce scenarios without exponential ray splitting.

What This Demo Shows:

• White Light as a Statistical Mixture: We model "white" light as randomly sampled wavelengths across the visible spectrum (380-750 nm).
• Wavelength-Dependent Refraction: Each photon's refraction angle is determined by Snell's law with a wavelength-dependent refractive index (Cauchy's approximation: n(λ) ≈ n₀ + B/λ²). Shorter wavelengths (blue) bend more than longer ones (red).
• Fresnel Reflection: At each interface, we use the Fresnel equations to calculate reflection probability:
  R_s = ||(n₁cosθᵢ - n₂cosθₜ) / (n₁cosθᵢ + n₂cosθₜ)||²
  R_p = ||(n₁cosθₜ - n₂cosθᵢ) / (n₁cosθₜ + n₂cosθᵢ)||²
  For unpolarized light: R = (R_s + R_p) / 2
  We then randomly decide whether each photon reflects or refracts based on this probability.
• Total Internal Reflection: When sin²θₜ > 1 (beyond the critical angle), all light reflects. This creates the sharp cutoff in the TIR demo.
• Statistical Patterns: The sensor accumulates individual photon hits over time. Though each photon's path involves randomness, the ensemble average reveals familiar patterns like rainbow dispersion.

Important Limitations:

This simulation significantly simplifies real optics. We treat photons as rays (no wave effects like interference or diffraction), exaggerate dispersion for visibility, and use geometric approximations throughout. Real optical phenomena involve electromagnetic wave theory, polarization, coherence, and many other effects not captured here. This demo is purely for educational visualization—not for accurate optical design.

Metamerism: Different Spectra, Same Appearance

An important phenomenon this demo illustrates is metamerism: different spectral distributions can appear identical to the human eye. For example, the "Purple (Red+Blue)" mode mixes red (~650nm) and blue (~420nm) photons, creating the visual perception of purple—yet a "true" single-wavelength purple (~420nm) looks nearly the same to us. This is because human color vision is trichromatic: we have only three types of cone cells, so any combination of wavelengths that stimulates them in the same ratio appears identical.

Our simplified model maps wavelength directly to RGB, which approximates but oversimplifies real eye sensitivity curves (the actual spectral response functions are more complex bell curves). Despite this limitation, the demo shows why spectral analysis with prisms is valuable: two colors that look identical to the eye reveal their true composition when dispersed—a single-wavelength purple creates one narrow band, while red+blue creates two distinct separated bands. This principle is used in spectroscopy to identify materials.

Geometric Optics with Chromatic Effects:

One interesting aspect: we can demonstrate wavelength-dependent phenomena (dispersion, color) using only geometric ray tracing, without implementing full wave mechanics. The wavelength is just a parameter that affects the refractive index and perceived color. This shows the power (and limitations) of geometric optics as an approximation.