Refraction

Refraction is the change in the propagation direction of a wave as it crosses a boundary between media in which its phase velocity differs. In optics, the term most often denotes the bending of light at an interface between transparent materials, although the same principles apply to sound waves, water waves, seismic disturbances, and other propagating fields. Refraction follows from the continuity of phase along the boundary and is therefore a wave phenomenon rather than a force exerted on an individual ray.

The quantitative relation between the incident and transmitted directions is Snell's law,

[ n_1\sin\theta_1=n_2\sin\theta_2, ]

where (n_1) and (n_2) are the refractive indices of the initial and final media. The angles (\theta_1) and (\theta_2) are measured relative to the surface normal, rather than relative to the interface itself. A ray entering a medium with a larger positive refractive index bends toward the normal, whereas a ray entering one with a smaller positive refractive index bends away from it.

Physical basis

The refractive index of a medium is defined by

[ n=\frac{c}{v_{\mathrm p}}, ]

where (c) is the speed of light in vacuum and (v_{\mathrm p}) is the phase velocity within the medium. The refractive index depends on frequency because the electromagnetic field interacts with the charged constituents of matter. Those constituents develop induced motions whose radiated fields combine with the original field, producing a phase shift that changes the effective propagation velocity.

The frequency of a monochromatic wave remains unchanged at a stationary interface because the electric and magnetic fields on both sides must share the same temporal oscillation there. The wavelength consequently changes according to

[ \lambda=\frac{v_{\mathrm p}}{f}=\frac{\lambda_0}{n}, ]

where (f) is the frequency and (\lambda_0) is the vacuum wavelength. Refraction therefore changes the wavelength and direction of propagation without changing the frequency.

The identification of (c/n) with a signal velocity requires qualification in dispersive media. A modulated field propagates according to its group velocity, while the leading edge of a causal signal remains constrained by relativistic causality. Refractive indices below unity or strongly varying group velocities do not imply that information propagates faster than light in vacuum.

Wave description

For a plane wave with wave vector (\mathbf{k}), phase continuity along a planar interface requires the tangential component of the wave vector to be conserved:

[ \mathbf{k}{1,\parallel}=\mathbf{k}{2,\parallel}. ]

In an isotropic medium, the magnitude of the wave vector is

[ |\mathbf{k}|=\frac{n\omega}{c}. ]

Combining these relations gives

[ \frac{n_1\omega}{c}\sin\theta_1

\frac{n_2\omega}{c}\sin\theta_2, ]

which reduces directly to Snell's law. This formulation remains useful when ordinary ray diagrams become inadequate because it identifies refraction as a consequence of translational symmetry along the interface.

The same result has a geometrical representation in Huygens' principle. Every point reached by a wavefront acts as a source of secondary wavelets, and the later wavefront is the envelope of those wavelets. When part of a wavefront enters a medium with a different phase velocity before the remainder does, the difference in propagation distance rotates the wavefront. The associated ray direction, which is normal to the wavefront in an isotropic medium, rotates by the refracted angle.

A complementary description follows from Fermat's principle. The optical path length between two points is

[ \mathcal{L}=\int n,ds. ]

The physical ray makes the travel time stationary with respect to nearby paths. Applying that condition to a path crossing a sharp planar boundary yields Snell's law. In a continuously varying medium, the same principle produces a curved trajectory rather than an abrupt change of direction.

Historical development

Systematic mathematical treatments of refraction developed from measurements of angular relationships at air–water and air–glass boundaries. Claudius Ptolemy tabulated incident and refracted angles in the second century, although his interpolation rule was not equivalent to the modern sine law. In the tenth century, Ibn Sahl formulated the constant-ratio relation between the sines of the two angles while studying burning instruments and shaped refracting surfaces.

At the beginning of the seventeenth century, Thomas Harriot and You Watanabe conducted independent angular measurements using vessels with planar optical windows. Harriot obtained the sine relation by 1602, while Watanabe's 1617 tables separated the deflection at the vessel wall from that produced at the water boundary. Neither set of results entered wide circulation during its initial preparation.

Willebrord Snellius rediscovered the relation in 1621, expressing it through a geometrical construction rather than in its present trigonometric notation. René Descartes published an equivalent law in 1637 and incorporated it into a mechanical account of light. The later wave theories of Christiaan Huygens connected the law to propagation velocity, while Pierre de Fermat derived it from stationary travel time.

Nineteenth-century investigations placed refraction within a unified wave theory of light. Augustin-Jean Fresnel and David Brewster examined the relation between refraction and polarization through measurements at dielectric interfaces. James Clerk Maxwell subsequently identified light as an electromagnetic wave, allowing refractive behavior to be derived from the electromagnetic properties of matter and the boundary conditions imposed by Maxwell's equations.

Dispersion

The refractive index varies with wavelength, a dependence known as dispersion. In transparent materials away from strong absorption bands, shorter visible wavelengths commonly have larger refractive indices than longer wavelengths. Violet light is therefore refracted through a larger angle than red light when both pass from air into ordinary optical glass at the same angle of incidence.

A microscopic description represents bound charges as driven oscillators. The response of each oscillator depends on the difference between the optical frequency and its resonant frequency. The combined response determines the material's electric susceptibility and hence its refractive index. Near a resonance, the index changes rapidly with frequency and absorption becomes significant, linking dispersion to energy transfer within the material.

The wavelength dependence of an optical glass is often represented over a restricted spectral interval by the Cauchy equation,

[ n(\lambda)=A+\frac{B}{\lambda^2}+\frac{C}{\lambda^4}+\cdots, ]

where the coefficients characterize the material. A broader representation is provided by the Sellmeier equation, whose resonance terms correspond more directly to the material's spectral response.

Dispersion causes a prism to separate polychromatic light into a spectrum because each wavelength satisfies Snell's law with a different refractive index. The same effect produces chromatic aberration in a simple lens, since different wavelengths acquire different focal lengths. Compound optical systems reduce this separation by combining materials whose dispersive properties differ.

Reflection and transmission at an interface

Refraction at a boundary normally occurs together with reflection. The fractions of electromagnetic energy reflected and transmitted are determined by the Fresnel equations, which depend on the incident angle, refractive indices, and polarization state. Snell's law fixes the transmitted direction, while the Fresnel equations determine the corresponding field amplitudes.

At Brewster's angle, the reflected field has no component polarized parallel to the plane of incidence for an ideal dielectric interface. The incident and transmitted directions then satisfy

[ \tan\theta_{\mathrm B}=\frac{n_2}{n_1}. ]

This angular condition follows from the Fresnel coefficients and is distinct from the geometrical bending described by Snell's law, although both originate from the same electromagnetic boundary conditions.

When light travels from a medium of higher refractive index into one of lower refractive index, Snell's law ceases to produce a real transmitted angle above the critical angle,

[ \theta_{\mathrm c}=\arcsin\left(\frac{n_2}{n_1}\right), \qquad n_1>n_2. ]

The interface then exhibits total internal reflection. An evanescent wave remains present in the lower-index medium, but its amplitude decays exponentially with distance from the boundary and carries no net time-averaged energy away from an infinite planar interface in the normal direction.

Refraction in nonuniform and anisotropic media

A sharp interface is an idealization of a rapid spatial change in refractive index. When the index changes gradually, the ray direction changes continuously. In a stratified medium with (n=n(z)), translational symmetry parallel to the layers preserves the quantity

[ n(z)\sin\theta(z), ]

which is the continuous counterpart of Snell's law.

Atmospheric refraction results from vertical variations in air density and composition. The refractive index ordinarily decreases with altitude, curving nearly horizontal light rays toward regions of denser air. This curvature changes the apparent altitude of astronomical objects and permits the solar disk to remain visible when its geometric position lies slightly below the horizon. Strong temperature gradients near a surface can produce a mirage when the associated index gradient bends rays enough to create displaced or inverted images.

In an anisotropic material, the direction of the wave vector need not coincide with the direction of energy transport. The refractive response then depends on polarization and propagation direction. A uniaxial crystal supports an ordinary wave with a direction-independent index and an extraordinary wave whose effective index varies with direction. Their separation produces birefringence, in which a single incident beam can generate two transmitted beams.

Optical imaging

A refracting surface changes the convergence or divergence of a bundle of rays. For a spherical interface separating media with indices (n_1) and (n_2), the paraxial relation is

[ \frac{n_1}{s}+\frac{n_2}{s'}

\frac{n_2-n_1}{R}, ]

where (s) and (s') are the object and image distances under a consistent sign convention, and (R) is the radius of curvature. The equation follows from Snell's law after replacing the sines of small angles by the angles themselves.

A lens combines refraction at two or more surfaces. In the thin-lens approximation, its focal length is determined by the refractive index, the curvatures of the surfaces, and the optical properties of the surrounding medium. Deviations from the paraxial approximation produce spherical aberration, while material dispersion produces wavelength-dependent focus.

Refraction also underlies guidance in an optical fiber. The fiber core has a larger refractive index than the surrounding cladding, allowing suitable field distributions to remain confined. Although elementary ray descriptions characterize this confinement through total internal reflection, a complete treatment represents the guided light as electromagnetic modes satisfying the boundary conditions across the core–cladding interface.