Standing wave

A standing wave is the field produced when two waves of equal frequency travel in opposite directions through the same region, most often a wave and its reflection from a mirror. Their interference gives a pattern that does not move: nodes, where the field is always zero, and antinodes, where it oscillates with twice the amplitude of either wave. Adjacent nodes are half a wavelength in the medium apart, $\lambda/(2n)$: 532 nm for 1064 nm light in air and 292 nm inside an Nd:YAG crystal with $n$ = 1.82. Every linear laser cavity holds its light as standing waves, and the pattern matters wherever thin layers, atoms or gain sit at fixed positions in it.

Field and intensity

For a wave of amplitude $E_0$ reflected at normal incidence from a perfect metal mirror at $z = 0$, the incident and reflected waves add to

$$E(z,t) = 2E_0 \sin(kz)\,\sin(\omega t),$$

with $k = 2\pi n/\lambda$. The spatial and temporal factors are separate, so the field at each position oscillates in time with a fixed local amplitude $2E_0|\sin kz|$. The electric field has a node at the metal surface and further nodes every $\lambda/(2n)$; the time-averaged intensity varies as $\sin^2 kz$ between zero and four times the intensity of one wave. The magnetic field has its antinodes where the electric field has its nodes, a quarter wavelength away, and oscillates a quarter period out of phase. With equal amplitudes the time-averaged Poynting vector is zero: energy is exchanged locally between the electric and magnetic fields with no net flow.

Partial reflection

When the reflected wave is weaker, with power reflectance $R$, part of the field travels and part stands. The intensity is

$$I(z) \propto 1 + R + 2\sqrt{R}\cos(2kz + \phi),$$

which swings between $(1+\sqrt{R})^2$ and $(1-\sqrt{R})^2$. For the 4% reflection of an uncoated glass surface, $\sqrt{R}$ = 0.2, so the intensity runs from 1.44 to 0.64 times the incident intensity, a ratio of 2.25 and a fringe visibility of $2\sqrt{R}/(1+R)$ = 0.38. The field amplitude ratio $(1+\sqrt{R})/(1-\sqrt{R})$ = 1.5 is the standing-wave ratio of microwave engineering. A weak reflection therefore leaves a strongly modulated pattern, which is why stray back-reflections matter in thin-film and lithography processes.

Observation: Wiener's experiment and photoresist

Otto Wiener in 1890 placed a thin photographic emulsion at a small angle to a silvered mirror, so that it cut obliquely through the standing wave and recorded the pattern as widely spaced bands. The emulsion stayed unexposed at the mirror surface, where the electric field has a node, which showed that the photographic action follows the electric field. The same effect appears in photolithography: light reflected from the substrate forms a standing wave in the resist, and the developed sidewalls show ripples with period $\lambda/(2n)$, about 73 nm for 248 nm exposure in a resist with an index near 1.7. Bottom anti-reflection coatings are used to suppress it.

Standing waves in laser cavities

The modes of a two-mirror optical cavity are standing waves that fit an integer number of half-wavelengths between the mirrors, which sets the frequencies of the longitudinal modes. Because the gain medium sees the intensity pattern, the lasing mode saturates the gain at its antinodes and leaves it unused at its nodes. A neighboring mode with a slightly different period overlaps the unused gain and can also reach threshold; this spatial hole burning is why standing-wave Nd:YAG lasers tend to run on several longitudinal modes. A unidirectional ring cavity supports traveling waves only and removes the effect, and a twisted-mode cavity, with quarter-wave plates on either side of the gain medium, makes the intensity uniform along the crystal. In a VCSEL the opposite strategy is used: the quantum wells are placed at antinodes, which doubles their effective gain relative to a uniform distribution.

Other uses

The first fiber Bragg gratings, reported by Hill and colleagues in 1978, were written by the standing wave formed when blue argon-ion laser light launched into a germanosilicate fiber reflected from its far end; the pattern photoinduced an index grating of period $\lambda/(2n)$ that then reflected the writing wavelength. Modern fiber Bragg gratings are written with two crossing beams or a phase mask, but the principle is the same interference. In atomic physics, two counter-propagating laser beams form an optical lattice, a periodic potential with wells $\lambda/2$ apart in vacuum: 406.5 nm for the 813 nm lattice of strontium clocks. The same counter-propagating geometry is used in laser cooling.

Common questions

What is the difference between a standing wave and a traveling wave?

A traveling wave carries its crests along at the phase velocity and transports energy. A standing wave has fixed nodes and antinodes and, for equal counter-propagating amplitudes, no net energy flow.

Why are the nodes half a wavelength apart?

The two waves change their relative phase by $2kz$ over a distance $z$, twice as fast as either wave alone, so the pattern repeats when $2kz$ advances by $2\pi$, at $z = \lambda/(2n)$.

Is the light in a laser a standing wave?

Inside a linear cavity, yes: each longitudinal mode is a standing wave between the mirrors. The output beam that leaves through the partially transmitting mirror is a traveling wave.

References: E. Hecht, Optics, 5th ed. (Pearson, 2017); M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999); A. E. Siegman, Lasers (University Science Books, 1986); K. O. Hill, Y. Fujii, D. C. Johnson and B. S. Kawasaki, "Photosensitivity in optical fiber waveguides: application to reflection filter fabrication," Appl. Phys. Lett. 32, 647 (1978).