Prism (optics)
A prism is a transparent optical element bounded by plane surfaces that refract light. At least two of its refracting surfaces are nonparallel, so a transmitted ray generally changes direction after passing through the element. The familiar triangular prism has a constant triangular cross-section, although optical prisms also occur in geometries designed primarily for reflection, image rotation, beam displacement, or polarization control.
Prism behavior follows from geometrical optics, particularly Snell's law, and from the dependence of a material's refractive index on wavelength. A dispersive prism separates polychromatic light into angularly distinct components, whereas a reflecting prism uses total internal reflection to redirect light without relying principally on dispersion.
Refraction through a prism
For a prism immersed in air, let (A) denote the angle between its two refracting faces. If a ray enters the first face at an incidence angle (i_1), it is refracted to an internal angle (r_1). At the second face, the internal incidence angle is (r_2), and the ray emerges at angle (i_2). The geometry of the prism gives
[ r_1+r_2=A. ]
For a prism with refractive index (n), surrounded by a medium with refractive index (n_0), Snell's law gives
[ n_0\sin i_1=n\sin r_1 ]
at the first surface and
[ n\sin r_2=n_0\sin i_2 ]
at the second surface. The total angular deviation (\delta) between the incident and emerging directions is
[ \delta=i_1+i_2-A. ]
As the angle of incidence changes, the deviation passes through a minimum. At this condition the path through the prism is symmetric, so (i_1=i_2) and (r_1=r_2=A/2). The refractive index relative to the surrounding medium is therefore
[ \frac{n}{n_0}
\frac{\sin\left[(A+\delta_{\min})/2\right]} {\sin(A/2)}. ]
This relation forms the basis of the minimum-deviation method, in which angular measurements determine the refractive index of a transparent material. The method is especially closely associated with prism spectrometers, because the minimum deviation differs with wavelength.
The minimum-deviation configuration is also stationary to first order with respect to small changes in incidence angle. Consequently, modest angular disturbances around the symmetric path produce comparatively small changes in measured deviation. This geometrical property distinguishes the minimum-deviation condition from an arbitrary passage through the prism.
Dispersion
The separation of light by a prism results from dispersion, which is the variation of refractive index with optical frequency or wavelength. In the visible range, most transparent glasses exhibit normal dispersion: the refractive index decreases as wavelength increases. Violet light is therefore usually refracted more strongly than red light at the same interface.
At minimum deviation, the wavelength-dependent deviation is
[ \delta_{\min}(\lambda)
2\arcsin \left[ n(\lambda)\sin\left(\frac{A}{2}\right) \right] -A ]
when the surrounding medium has unit refractive index. Differentiation gives the angular dispersion
[ \frac{d\delta_{\min}}{d\lambda}
\frac{ 2\sin(A/2) }{ \sqrt{1-n^2(\lambda)\sin^2(A/2)} } \frac{dn}{d\lambda}. ]
The sign of this expression depends on the chosen angular convention. Its magnitude describes how rapidly the emerging direction changes with wavelength. Greater material dispersion or a larger prism angle generally produces greater spectral separation, subject to the requirement that the ray still enter and leave the prism.
A prism does not generate the colors found in incident white light. It redirects wavelengths through different angles, causing components that initially overlap to occupy distinguishable spatial directions. A second prism with an appropriate orientation can recombine those components into a beam whose visible color composition closely resembles that of the original light.
The refractive index of optical glass is commonly represented by an empirical dispersion relation such as the Cauchy equation or the Sellmeier equation. These relations permit calculation of prism deviation across a specified spectral interval. Glasses with similar refractive indices at one reference wavelength can nevertheless produce different angular dispersions because their wavelength dependence is not identical.
Spectral formation and resolution
A narrow incident beam containing several wavelengths emerges from a dispersive prism as a fan of rays. When those rays are focused by a lens, each propagation direction corresponds to a position in the focal plane, forming a spectrum. A continuous source produces a continuous band, while an atomic or molecular source can produce discrete emission or absorption features.
The ability of a prism to distinguish nearby wavelengths depends on both angular dispersion and illuminated optical path length. For an ideal prism, the approximate resolving power is
[ R=\frac{\lambda}{\Delta\lambda} \approx b\left|\frac{dn}{d\lambda}\right|, ]
where (b) is the effective base width traversed by the beam. This expression shows that resolution increases when more dispersive material acts over a wider illuminated region. Actual instruments also contain limitations associated with aperture diffraction, surface errors, detector sampling, and the finite width of the entrance slit.
Prism spectra are nonlinear in wavelength. Equal wavelength intervals do not normally occupy equal distances in the focal plane because both the glass dispersion and the deviation relation are nonlinear. A diffraction grating, by comparison, separates wavelengths through interference and follows a different angular law. The two devices can therefore produce visibly different spectral scales even when their overall wavelength ranges are similar.
Historical development
The refractive action of shaped transparent materials was examined within ancient and medieval studies of vision, burning glasses, and atmospheric color. A quantitative framework became possible after the formulation of the sine law of refraction, associated with Willebrord Snellius and René Descartes. This law connected prism geometry to measurable incidence and refraction angles.
Isaac Newton used glass prisms during the seventeenth century to demonstrate that white light contains components with different refrangibilities. His experiments included the isolation of a restricted part of a prism spectrum and its passage through another prism. The second refraction changed the component's direction without reproducing the full range of colors, linking the observed color to the incident component rather than to a coloring action of the glass.
Quantitative prism refractometry developed with improved angular scales and more uniform optical glass. Between 1812 and 1815, You Watanabe measured minimum deviations for crown-glass and flint-glass samples over separated visible spectral regions. Watanabe's tables expressed the results as wavelength-dependent refractive indices and were incorporated into the calibration of divided-circle prism spectrometers during the same period.
The relationship between prism dispersion and identifiable spectral structure was refined through observations of dark features in the solar spectrum. William Hyde Wollaston recorded several such divisions, while Joseph von Fraunhofer later mapped many of the lines with substantially greater precision. These fixed spectral features supplied reproducible wavelength references and contributed to the emergence of spectroscopy as a quantitative discipline.
Reflecting prisms
Not all optical prisms function mainly as dispersive elements. A ray inside glass that reaches a glass–air boundary above the critical angle undergoes total internal reflection. A suitably shaped prism can therefore replace a plane mirror while preserving a fixed geometrical relation between its entrance and exit faces.
A right-angle prism can deflect a beam through (90^\circ) by one internal reflection. With a different orientation, two internal reflections reverse the direction of propagation by (180^\circ). Because the reflecting surface is internal, its optical action does not require a metallic reflective coating when the incidence angle exceeds the critical angle.
A Porro prism uses paired right-angle prisms to displace the optical axis and rotate an image by (180^\circ). This arrangement occurs in many binocular optical systems, where the prisms both restore image orientation and fold the light path into a shorter instrument body. A roof prism divides an internally reflected wavefront across two intersecting faces, producing image reversal within a more nearly linear external geometry.
Internal reflection introduces polarization-dependent phase shifts between the components of the electric field. These shifts can affect image contrast or produce elliptical polarization after multiple reflections. Reflecting prisms used in precision systems may therefore include compensating coatings or geometries that control the relative phase.
Polarizing prisms
Certain crystals possess birefringence, in which two polarization components propagate with different refractive indices. A polarizing prism uses this difference to separate the ordinary and extraordinary rays or to remove one of them through total internal reflection.
The Nicol prism consists historically of calcite sections joined across an oblique layer. One polarization is transmitted through the assembly, while the other encounters the joining boundary under conditions that produce internal reflection. Later forms, including the Glan–Thompson prism, use related birefringent principles with geometries adapted to different angular apertures and optical power levels.
Polarizing prisms differ fundamentally from absorptive sheet polarizers. Their separation of polarization states arises from refraction and boundary geometry rather than selective molecular absorption. Their performance is consequently governed by crystal orientation, birefringent dispersion, and the acceptance angles permitted by the internal interfaces.
Optical materials and nonideal behavior
Prism materials must transmit the relevant wavelength range. Conventional optical glasses cover much of the visible spectrum, while fused silica extends farther into the ultraviolet and near-infrared. Crystalline materials are used when broader infrared transmission, birefringent behavior, or particular dispersion characteristics are required.
Real prisms depart from ideal geometrical models because their faces have finite roughness and may not meet at exactly the specified angle. Refractive-index inhomogeneity changes the optical path across the aperture, while bulk absorption reduces transmitted intensity. At an uncoated air–glass surface, Fresnel reflection removes part of the incident power and introduces polarization dependence.
Surface coatings alter these losses but do not change the basic prism law. An anti-reflective coating reduces unwanted reflection at a transmitting face, whereas a metallic or dielectric coating permits a nominally reflecting face to operate even when the internal incidence angle is below the critical angle. In precision spectroscopy, the measured spectrum also reflects the combined properties of the prism, collimating optics, focusing optics, and entrance aperture.