Optics
Snell's Law
Light bends at an interface; the sines of incidence and refraction angles obey n₁ sin θ₁ = n₂ sin θ₂
Snell's law: when a wave (light, sound, water wave) passes from a medium of refractive index n₁ to one of n₂, the angles to the surface normal satisfy n₁ sin θ₁ = n₂ sin θ₂. Discovered by Persian scientist Ibn Sahl (984), independently by Thomas Harriot (1602), and named after Willebrord Snellius (1621); first published derivation by Descartes (1637). Equivalent to Fermat's principle of least time (light minimizes optical path). At critical angle θ_c = sin⁻¹(n₂/n₁) for n₁ > n₂, refraction transitions to total internal reflection. Examples: pencil-in-water bending, swimming pool depth illusion, fiber optic communication (TIR), prism color separation, lens design, mirage formation (refraction in heated air gradient).
- Statementn₁ sin θ₁ = n₂ sin θ₂
- FirstIbn Sahl 984
- NamedSnellius 1621
- EquivalentFermat's least time
- Critical anglesin⁻¹(n₂/n₁)
- Refractive indexWater 1.33, glass 1.5, diamond 2.42
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Why Snell's law matters
- Lens optics. Every camera, telescope, microscope, and corrective eyeglass shapes a glass interface so that Snell's law bends rays toward a focus. Aberrations are residual deviations from the ideal Snell-traced focus.
- Fiber optics. Internet backbone fibers trap pulses by total internal reflection — an extreme limit of Snell's law where the angle just exceeds θ_c. Tens of terabits per second per fiber pair.
- Atmospheric optics. Mirages, twilight sun-flattening, the green flash, and astronomical refraction (stars near horizon appear higher than they are) are all Snell's law in a graded-index atmosphere.
- Computer graphics ray tracing. Refraction in glass, water, gemstones, and skin subsurface scattering all use Snell's law per ray. Modern path tracers compute it billions of times per frame.
- Prism color separation. Spectrometers split incoming light by wavelength using Snell's law with n(λ) — the foundation of all optical spectroscopy from chemistry to astrophysics.
- Seismology. Earth's interior layers refract seismic P and S waves at boundaries (Mohorovičić, Gutenberg, inner-core). Snell's law applied to seismic velocity contrasts maps the Earth's structure.
- Sonar and underwater acoustics. Sound speed varies with temperature, salinity, and pressure. Snell's law in this varying-c profile creates the SOFAR channel — sound trapped at ~1 km depth that propagates thousands of kilometers.
Common misconceptions
- "Light always bends toward the normal." Only true going from low-n to high-n. From high-n to low-n (glass to air), light bends away from the normal — and beyond critical angle, doesn't refract at all.
- "Frequency changes when light enters glass." No — frequency is fixed by the source and conserved across boundaries. What changes is wavelength: λ_medium = λ_vacuum / n. Color (which the eye perceives as frequency) is preserved.
- "Snell's law is just for light." It applies to any wave at an interface where speed differs: ocean waves bending toward shore, sound waves at temperature gradients, seismic waves at rock boundaries, electron waves in solids.
- "Critical angle exists for any pair of media." Only when n₁ > n₂ — light traveling from the denser side. Air-to-glass has no critical angle; glass-to-air does.
- "Refractive index is constant for a material." n depends weakly on wavelength (dispersion), strongly on temperature and pressure for gases, and on stress (photoelasticity) in solids. Tabulated values are usually quoted at 589 nm (sodium D line).
- "Snell's law gives the intensity of refracted light." No — Snell only fixes the geometry. The Fresnel equations give what fraction of energy refracts vs reflects, depending on polarization and angle.
Frequently asked questions
Why does light bend at an interface?
Light slows down when entering a medium of higher refractive index because its phase velocity is c/n. To keep the wavefront continuous across the boundary, the wavelength shortens while frequency stays fixed. The propagation direction must rotate so that the unchanged tangential component of the wavevector still matches across the interface — Snell's law n₁ sin θ₁ = n₂ sin θ₂ is precisely the conservation of that tangential component. Mechanically, picture a marching band crossing from pavement to mud at an angle: rows on the slow side step shorter, the line bends.
What is Fermat's principle of least time?
Fermat's principle states that light traveling between two points takes the path that minimizes total time (more precisely, makes optical path length stationary). In a single medium that gives a straight line. Across an interface where speeds differ, the time-minimizing path is bent. Setting the time derivative to zero with respect to crossing point yields exactly n₁ sin θ₁ = n₂ sin θ₂. The principle generalizes to graded-index media (path becomes a curve) and to general relativity (light follows null geodesics).
When does total internal reflection happen?
When light travels from a denser medium (higher n₁) toward a less dense medium (lower n₂), Snell's law gives sin θ₂ = (n₁/n₂) sin θ₁ > sin θ₁. As θ₁ grows, sin θ₂ approaches 1; the critical angle θ_c = sin⁻¹(n₂/n₁) is where θ₂ = 90°. Beyond θ_c there is no real solution for the refracted ray, and 100% of the light reflects. For glass-to-air θ_c ≈ 41.8°; for water-to-air ≈ 48.6°. This is what makes optical fibers and binoculars-prisms work.
Why does white light split through a prism (chromatic dispersion)?
The refractive index n depends slightly on wavelength: n(λ). For typical glass (BK7), n decreases as λ grows — about 1.530 at 400 nm (violet) and 1.514 at 700 nm (red). Snell's law then bends each color by a different angle: violet bends most, red least. Two refractions (entering and leaving the prism) compound the spread. Newton (1666) used this to demonstrate white light is composed of a spectrum. The same effect causes chromatic aberration in simple lenses and the rainbow in raindrops.
How do mirages form?
Hot air near a sunbaked road has lower density and lower refractive index than cooler air above it. Light from the sky descending toward the road continuously refracts upward through this gradient, eventually curving back to your eye — making distant sky look like reflective water on the road. Above-sea inversion mirages (Fata Morgana) work the reverse: cold dense air below refracts ship-light downward, producing elevated, distorted, sometimes inverted images. Snell's law applied to a graded n(z) gives the curved ray path.
How is Snell's law used in fiber optics?
An optical fiber is a glass core (n_core ≈ 1.47) surrounded by a cladding of slightly lower index (n_clad ≈ 1.46). Light entering the core at small enough angle to the axis hits the core-cladding interface above the critical angle and totally internally reflects — bouncing along the fiber for kilometers with very low loss. The acceptance cone is described by the numerical aperture NA = √(n_core² − n_clad²), which sets how steeply you can launch light and still trap it. Single-mode fibers carry the entire transatlantic Internet.