Snell's Law and Refraction — The Complete Physics Guide
Snell's Law quantitatively describes how waves bend when crossing the boundary between two media with different wave speeds: n₁sinθ₁ = n₂sinθ₂. Named after Dutch mathematician Willebrord Snellius who derived it geometrically around 1621, the law explains why a straw looks bent in water, how lenses focus light, why diamonds sparkle, how optical fibres carry the internet's data, and how the human eye forms images on the retina.
The law was discovered and rediscovered multiple times — by Ibn Sahl in 984 CE (the earliest known derivation), by Thomas Harriot in 1602, by Snellius circa 1621, and independently by Descartes in 1637, who was the first to publish it. Despite the name "Snell's Law" in English-speaking countries, French scientists often call it "la loi de Descartes." Whatever its name, the underlying physics is the same: waves slow down and bend toward the normal in denser media, speed up and bend away from the normal in less dense media.
The Refractive Index
The refractive index n of a medium is defined as the ratio of the speed of light in vacuum to the speed of light in that medium: n = c/v. Since light slows in any material medium, n ≥ 1 always. Vacuum: n = 1.000 (exactly, by definition). Air: n ≈ 1.0003 (negligible difference from vacuum for most purposes). Water: n = 1.333. Crown glass: n ≈ 1.523. Diamond: n = 2.417.
Refractive index also varies with the wavelength of light — a phenomenon called dispersion. In most transparent media, blue light has a higher refractive index than red light. This means blue light bends more at interfaces. Prisms separate white light into rainbow spectra by exploiting this wavelength-dependence: each colour refracts by a slightly different angle, separating them spatially. Raindrops act as tiny prisms, separating sunlight into the colours of a rainbow.
The higher refractive index at shorter wavelengths also causes chromatic aberration in lenses — blue and red light focus at slightly different distances, producing colour fringing in images. Modern cameras use achromatic doublets (two lens elements of different glass types bonded together) to compensate for this, with the dispersions of the two glasses cancelling each other out.
Snell's Law Derived from Wave Principles
Snell's Law follows from the requirement that wave crests remain continuous across the interface — the wave must not break up at the boundary. Consider a plane wave approaching an interface at angle θ₁. Along the interface, the wave crests advance at speed v₁/sinθ₁ in the first medium and v₂/sinθ₂ in the second. Continuity requires these to be equal:
This derivation shows that Snell's Law is not specific to light — it holds for any wave that changes speed at an interface. Sound waves refract at water-air boundaries. Seismic waves refract at geological layer boundaries (this is how seismologists map Earth's interior). Radio waves refract in the ionosphere.
Total Internal Reflection
When light travels from a denser to a less dense medium (n₁ > n₂), Snell's Law predicts sinθ₂ = (n₁/n₂)sinθ₁. Since n₁/n₂ > 1, there is a maximum angle of incidence θ₁ = θ_c beyond which sinθ₂ would exceed 1 — geometrically impossible. At this critical angle and beyond, no refracted ray exists; all light is reflected back into the denser medium. This is total internal reflection (TIR).
Critical angle: sin(θ_c) = n₂/n₁. For glass (n = 1.5) to air (n = 1): sin(θ_c) = 1/1.5 = 0.667 → θ_c = 41.8°. Any ray in glass hitting an air interface at angles greater than 41.8° undergoes TIR and stays entirely within the glass.
TIR is the physical mechanism behind optical fibres. Light enters the core of the fibre (high n) and hits the cladding (lower n) at angles exceeding the critical angle, bouncing along the fibre with essentially no loss. Modern single-mode silica fibres can carry terabits of data per second over thousands of kilometres with signal losses as low as 0.2 dB/km — corresponding to only 5% power loss per 100 km. The entire modern internet backbone runs on this application of Snell's Law.
Worked Example 1 — Light Entering Glass
Problem: A ray of light in air strikes a glass surface (n = 1.52) at an angle of incidence of 35°. Find the angle of refraction.
n₁sinθ₁ = n₂sinθ₂ → 1 × sin(35°) = 1.52 × sinθ₂
sinθ₂ = sin(35°)/1.52 = 0.5736/1.52 = 0.3774
θ₂ = arcsin(0.3774) = 22.2° — the ray bends toward the normal (correct: entering denser medium)
Worked Example 2 — TIR in a Diamond
Problem: Calculate the critical angle for diamond (n = 2.417) in air, and explain why diamonds are cut with many facets.
sin(θ_c) = n_air/n_diamond = 1/2.417 = 0.4137 → θ_c = arcsin(0.4137) = 24.4°
This tiny critical angle means light entering from almost any direction will undergo multiple TIR reflections inside the diamond before finally exiting from the top. A brilliant-cut diamond has 58 facets specifically angled to maximise this multiple-reflection pathway — producing the characteristic sparkle and "fire." Glass (θ_c ≈ 42°) sparkles far less because more light leaks through the sides.
Applications of Snell's Law
Lenses and optics: Every lens works by applying Snell's Law at two curved glass-air interfaces. The lens equation (1/f = 1/u + 1/v) and the lensmaker's equation both derive from Snell's Law applied to curved surfaces. Camera lenses, microscope objectives, telescope mirrors, and spectacle lenses are all designed using Snell's Law as the fundamental constraint.
The human eye: The cornea (n ≈ 1.376) provides approximately 75% of the eye's focusing power — the most powerful refracting surface in the body. Light from air enters the cornea, refracts at the air-cornea interface (Snell's Law with n₁ = 1.0, n₂ = 1.376), and continues through the aqueous humour, crystalline lens, and vitreous to focus on the retina. Myopia (short-sightedness) occurs when this system focuses in front of the retina; corrective lenses pre-refract light to compensate.
LASIK eye surgery: Laser ablation reshapes the cornea's curvature, permanently altering the refraction at the air-cornea interface — applying Snell's Law permanently. Over 40 million LASIK procedures have been performed worldwide, restoring normal vision by changing the geometry of the first refracting surface.
Atmospheric refraction and mirages: Air density decreases with altitude, so n decreases with altitude. Light bends continuously as it travels through this gradient — the same physics as Snell's Law applied continuously through a medium with gradually varying n. This makes stars near the horizon appear slightly higher than they actually are, and allows the Sun to be visible for a few minutes after it has geometrically set below the horizon.