Snell's law

Snell's law states that when light crosses the boundary between two media with different refractive indices, the angles of incidence and refraction (measured from the surface normal) obey:

$$n_1 \sin\theta_1 \;=\; n_2 \sin\theta_2,$$

where $n_1, n_2$ are the refractive indices and $\theta_1, \theta_2$ are the angles in each medium. Light bends toward the normal when entering a higher-index medium ($\theta_2 < \theta_1$) and bends away when entering a lower-index medium.

Derivation. Snell's law follows from any of three independent principles:

1. Huygens' principle: each wavefront acts as a source of spherical wavelets; the boundary condition that wavefronts remain continuous across the interface produces Snell's law
2. Fermat's principle: light travels along the path of stationary optical path length; minimizing $n_1 L_1 + n_2 L_2$ between fixed endpoints gives Snell's law
3. Maxwell's equations boundary conditions: matching the tangential components of $\mathbf{E}$ and $\mathbf{H}$ at the interface forces the parallel component of the wave vector $k_\parallel$ to be continuous, which is exactly Snell's law

The boundary-condition derivation is fundamental and explains why Snell's law holds even for waves where the geometric-optics picture breaks down (evanescent waves, near-field optics).

Vector form. A coordinate-free version useful for ray-tracing software:

$$\mathbf{n}_2 \times \hat{\mathbf{k}}_2 \;=\; \mathbf{n}_2 \times \hat{\mathbf{k}}_1 \, \frac{n_1}{n_2},$$

where $\hat{\mathbf{k}}_1$ and $\hat{\mathbf{k}}_2$ are the unit propagation directions and $\mathbf{n}_2$ is the surface normal. The transmitted direction is obtained by solving for $\hat{\mathbf{k}}_2$ subject to the constraint $|\hat{\mathbf{k}}_2| = 1$.

Special cases.

• Normal incidence ($\theta_1 = 0$): $\theta_2 = 0$, the beam passes straight through with no bending. Reflection still occurs at the interface with magnitude given by the Fresnel coefficients.
• Critical angle ($\theta_2 = 90°$): occurs only when $n_2 < n_1$; for $\theta_1 > \theta_c$, total internal reflection occurs.
• Grazing incidence ($\theta_1 \to 90°$): $\sin\theta_2 = (n_1/n_2)$, useful for X-ray mirrors operating at very shallow angles where reflectance can be high.
• Brewster's angle: at this specific incidence angle, the reflected wave is purely s-polarized; the p-polarized component is fully transmitted. See Brewster angle.

Frequency dependence. Snell's law applies at each frequency separately. Since refractive index depends on frequency (chromatic dispersion), different wavelengths refract at different angles — this is the origin of spectral dispersion in prisms and the chromatic-aberration problem in lenses.

For typical optical glass:

• $n_\text{BK7}$(486 nm) = 1.522
• $n_\text{BK7}$(589 nm) = 1.517
• $n_\text{BK7}$(656 nm) = 1.514

A ray incident at 45° refracts at $\theta_2 = 27.7°$ for 486 nm and $\theta_2 = 27.8°$ for 656 nm — a 0.1° angular spread that focuses to a visible chromatic blur on a 100 mm focal-length lens.

Practical applications in photonics.

• Lens design: ray tracing of all optical systems uses Snell's law at each refracting surface
• Fiber coupling: calculate acceptance angle of a fiber from $n_\text{core}$, $n_\text{clad}$, surrounding medium
• Prism spectrometers: angular dispersion comes from $\partial \theta_2 / \partial \lambda$ via $\partial n / \partial \lambda$
• Coupler design: angle of incidence at grating couplers is set by Snell-like phase matching between fiber and grating
• Refractometry: measure liquid refractive index by finding the critical angle on a glass prism

Historical note. Named for Willebrord Snellius (1580 – 1626), who formulated the law in 1621. Earlier formulations by Ibn Sahl (984) and Thomas Harriot (1602) predate Snellius, but his name was attached by Descartes who published the law in 1637.

References: Saleh & Teich, Fundamentals of Photonics (3rd ed., 2019), Ch. 1 (ray optics); Born & Wolf, Principles of Optics (7th ed., Cambridge University Press, 1999), Ch. 1; Hecht, Optics (5th ed.), Ch. 4.