Snell's law

Snell's law states that when light crosses the boundary between two media of different refractive index, the angles of incidence and refraction, both measured from the surface normal, obey

$$n_1\sin {\theta }_1=n_2\sin {\theta }_2\mathrm{.}$$

Light bends toward the normal on entering a higher-index medium and away from it on entering a lower-index one. A ray passing from air into BK7 glass ($n=1.5168$ at 587.6 nm) at 30° incidence travels at 19.2° inside the glass; the reverse path, glass to air at 30°, exits at 49.3°. The law fixes only directions. How much power is reflected and transmitted at each angle is given separately by the Fresnel equations.

Where the law comes from

Three independent arguments give the same result.

1. Huygens' principle. Each point on a wavefront radiates secondary wavelets; requiring the wavefronts to remain continuous across the interface, while the wavelets travel at $c\mathrm{/}{n}_1$ and $c\mathrm{/}{n}_2$, produces Snell's law.
2. Fermat's principle. Light follows a path of stationary optical path length. Minimizing $n_1{L}_1+n_2{L}_2$ between two fixed points on either side of the surface gives the same relation.
3. Electromagnetic boundary conditions. The tangential components of $\mathbf{E}$ and $\mathbf{H}$ must be continuous at the interface for all positions along it, which forces the tangential wavevector component to be the same on both sides:

$$k_{1\parallel }=k_{2\parallel },k_i=n_i\omega \mathrm{/}c\mathrm{.}$$

The third form is the most general. It holds when the transmitted wave cannot propagate at all, where it yields an evanescent wave, and it carries over directly to gratings and phase-matched couplers, where a structure adds a fixed amount to $k_{\parallel }$.

Vector form. Ray-tracing programs use a coordinate-free version. With unit surface normal $\widehat{\mathbf{n}}$ and unit ray directions ${\widehat{\mathbf{k}}}_1$, ${\widehat{\mathbf{k}}}_2$,

$$n_2(\widehat{\mathbf{n}}\times {\widehat{\mathbf{k}}}_2)=n_1(\widehat{\mathbf{n}}\times {\widehat{\mathbf{k}}}_1),$$

and ${\widehat{\mathbf{k}}}_2$ is found by adding the normal component needed to make $\mathrm{\mid }{\widehat{\mathbf{k}}}_2\mathrm{\mid }=1$. The incident ray, the refracted ray and the normal lie in one plane.

Special cases

• Normal incidence (${\theta }_1=0$): ${\theta }_2=0$ and the beam is undeviated, although a fraction is still reflected, about 4% at an uncoated glass–air surface.
• Critical angle: when $n_2<n_1$, ${\theta }_2$ reaches 90° at $\sin {\theta }_c=n_2\mathrm{/}{n}_1$. For BK7 to air ${\theta }_c=41.2\mathrm{°}$; for water ($n=1.333$) to air, 48.6°. Beyond it, total internal reflection occurs.
• Grazing incidence (${\theta }_1\to 90\mathrm{°}$ from the low-index side): the refracted angle approaches the critical angle. At X-ray wavelengths most materials have $n$ slightly below 1, so the vacuum side is the denser one and X-ray mirrors reflect totally only at grazing angles of a fraction of a degree.
• Brewster angle: at $\tan {\theta }_B=n_2\mathrm{/}{n}_1$ the p-polarized reflection vanishes and the reflected light is purely s-polarized.

Dispersion and worked numbers

Snell's law applies at each frequency separately. Because $n$ depends on wavelength (chromatic dispersion), a prism or lens sends different colors in different directions. For BK7:

The F–C spread at a single surface is 0.160°. Focused by a 100 mm lens, that angular spread corresponds to a lateral separation of about 280 μm, which is the scale of lateral color an uncorrected element can produce.

Plate displacement. A plane-parallel plate does not change a ray's direction but shifts it sideways. For a 5 mm BK7 plate at 45° the lateral shift is

$$d=t\sin {\theta }_1(1-\frac{\cos {\theta }_1}{\sqrt{n^2-{\sin }^2{\theta }_1}}),$$

which evaluates to 1.67 mm. Tilted plates of this kind are used deliberately as beam-steering elements and appear unintentionally as offsets when windows or beamsplitters are inserted into an aligned setup.

Measurement and practice

The law is checked, or $n$ is measured, by recording the deviation of a laser beam through a prism on a rotation stage; the minimum-deviation method gives $n$ to about ${10}^{-5}$ with a good goniometer. Abbe refractometers instead find the critical angle between a liquid sample and a high-index prism.

• Lens design: every refracting surface in a ray trace applies Snell's law exactly, without the small-angle approximation, and the departure of real rays from paraxial ones is what appears as aberrations.
• Fiber coupling: the acceptance angle of a fiber follows from applying the law at the end face and at the core–cladding boundary. Standard single-mode fiber with NA 0.14 accepts light within 8.0° of the axis in air.
• Prism spectrometers: angular dispersion $d{\theta }_2\mathrm{/}d\lambda$ follows from $dn\mathrm{/}d\lambda$.
• Grating couplers: the coupling angle is set by conservation of $k_{\parallel }$ with the grating vector added, the same boundary condition that gives Snell's law.

The Snell's law calculator gives the refracted angle, the critical and Brewster angles and the reflected power for any pair of indices, with a ray diagram.

Common questions

Does Snell's law apply to anisotropic crystals?

For the ordinary wave, yes. For the extraordinary wave in a birefringent crystal, $k_{\parallel }$ conservation still holds, but the index depends on direction, so the wavevector angle must be solved together with the index ellipsoid, and the ray (energy) direction can differ from the wavevector direction.

Why do the angles use the normal instead of the surface?

The boundary condition concerns the component of the wavevector along the surface, and $k\sin \theta$ is that component when $\theta$ is measured from the normal.

Who discovered it?

Ibn Sahl described the law in 984, Thomas Harriot found it in 1602, Willebrord Snellius in 1621, and Descartes published it in 1637; in France it is called the Snell–Descartes law.

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., 2017), Ch. 4.