Prism

A prism is a block of transparent material with flat polished faces set at defined angles. Depending on its geometry it does one of four jobs: it deviates a beam by refraction, it disperses light into its component wavelengths, it folds or inverts a beam by internal reflection, or, when made of a birefringent crystal, it separates polarizations. A 60° equilateral prism of N-BK7 deviates yellow light by about 39°; a right-angle prism folds a beam through 90° with near-total reflection; a thin wedge of 1° steers a beam by about half a degree.

Deviation and minimum deviation

A ray passing through a prism of apex angle $A$ is bent twice, toward the base each time, and the total deviation $\delta$ depends on the angle of incidence. It is smallest when the ray passes symmetrically, parallel to the base inside an isosceles prism. At this minimum deviation

$$n = \frac{\sin\!\big[(A + \delta_{\min})/2\big]}{\sin(A/2)} .$$

For N-BK7 with $A = 60^\circ$, $\delta_{\min}$ is 38.65° at 587.6 nm ($n = 1.5168$), 39.14° at 486.1 nm ($n = 1.5224$) and 38.43° at 656.3 nm ($n = 1.5143$). The blue and red lines leave 0.71° apart, the angular dispersion that makes the prism a spectrometer element. Measuring $\delta_{\min}$ on a goniometer and inverting the formula is the classical way to determine the refractive index of glass, to parts in $10^6$ with care.

For a thin wedge the deviation is nearly independent of incidence angle,

$$\delta \approx (n - 1)\,A ,$$

so a 1° N-BK7 wedge deviates a beam by 0.52°. Wedge pairs rotated against each other (Risley prisms) steer a beam over a cone, and ophthalmic prescriptions express the same deviation in prism diopters, centimeters of displacement per meter.

Dispersing prisms compared with gratings

A prism's dispersion comes from the wavelength dependence of $n$, which the Abbe number summarizes in the visible. Dense flint glasses disperse three to four times more than crown glasses: SF11 has $n_F - n_C$ = 0.030 against 0.008 for N-BK7 (Abbe numbers near 26 and 64). Compared with a diffraction grating, a prism has lower and strongly nonlinear dispersion, larger in the blue than in the red, but sends all the light into a single spectrum with no overlapping orders and high throughput. Prisms remain in use for broadband work, for ultraviolet spectroscopy with fused silica or CaF₂, and wherever grating orders would be a nuisance.

Reflecting prisms

When the angle of incidence on an internal face exceeds the critical angle, the face reflects by total internal reflection with no coating. In a right-angle prism the light meets the hypotenuse at 45°, above the 41.2° critical angle of N-BK7, so the fold is lossless apart from the entrance and exit faces. Other common geometries:

• Penta prism. Deviates a beam by 90° regardless of small rotations of the prism; used in rangefinders and optical squares. Its two reflecting faces are metallized because the angles do not satisfy TIR.
• Porro and roof prisms. Invert and revert an image in binoculars.
• Dove prism. Rotates an image by twice the prism's rotation angle.
• Corner cube. Three mutually perpendicular faces return a beam parallel to itself; used as retroreflectors in interferometers and surveying.

Prisms in laser systems

An anamorphic prism pair expands a beam in one axis only and is the standard way to circularize the elliptical output of a laser diode. Brewster-angled prisms introduce almost no reflection loss for p-polarized light and appear inside laser cavities as tuning elements. A pair of prisms arranged so that different wavelengths travel different path lengths in glass produces adjustable negative group-velocity dispersion, the method introduced by Fork, Martinez and Gordon to compress femtosecond pulses. Crystal prisms of calcite or α-BBO, exploiting birefringence, form Glan-Taylor and Wollaston polarizers with extinction ratios of $10^5$ and better.

Pitfalls

A dispersing prism's deviation and dispersion change with angle of incidence, so a spectrometer must be set at or calibrated near minimum deviation. Reflecting prisms add glass path, which shifts focus and adds chromatic aberration in a converging beam. Any contamination in contact with a TIR face raises the outside index locally and lets light escape, so these faces must stay clean and uncemented.

Common questions

Why does a prism split white light into colors?

Glass has a slightly higher refractive index for shorter wavelengths, so blue light is bent more than red at each face. White light entering the prism leaves as a fan of colors, with violet deviated most.

What is the angle of minimum deviation?

The angle of incidence at which the total deviation through the prism is smallest, reached when the ray crosses the prism symmetrically. At that point small changes in incidence angle barely change the output direction, which makes it the preferred operating point for spectroscopy and index measurement.

What is the difference between a prism and a beamsplitter cube?

A cube beamsplitter is built from two right-angle prisms cemented along their hypotenuses with a partially reflecting or polarizing coating between them. The prisms serve as a housing for the coating; the splitting is done by the coating.

References: E. Hecht, Optics, 5th ed. (Pearson, 2017), Ch. 5; M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999), Ch. 4; W. J. Smith, Modern Optical Engineering, 4th ed. (McGraw-Hill, 2008); R. L. Fork, O. E. Martinez, J. P. Gordon, Opt. Lett. 9, 150 (1984).