Coherence is a property of waves that measures the ability of the waves to interfere with each other. Two waves that are coherent can be combined to produce an unmoving distribution of constructive and destructive interference (a visible interference pattern) depending on the relative phase of the waves at their meeting point. Waves that are incoherent, when combined, produce rapidly moving areas of constructive and destructive interference and therefore do not produce a visible interference pattern.
A wave can also be coherent with itself, a property known as temporal coherence. If a wave is combined with a delayed copy of itself (as in a Michelson interferometer), the duration of the delay over which it produces visible interference is known as the coherence time of the wave, Δtc. From this, a corresponding coherence length can be calculated:
c is the speed of the wave.
The temporal coherence of a wave is related to the spectral bandwidth of the source. A truly monochromatic (single frequency) wave would have an infinite coherence time and length. In practice, no wave is truly monochromatic (since this requires a wavetrain of infinite duration), but in general, the coherence time of the source is inversely proportional to its bandwidth.
Waves also have the related property of spatial coherence; this is the ability of any one spatial position of the wavefront to interfer with any other spatial position. Young's double-slit experiment relies on spatial coherence of the beam illuminating the two slits; if the beam was spatially incoherent, i.e. if the sunlight was not first passed through a single slit, then no interference pattern would be seen.
Spatial coherence is high for sphere waves and plane waves, and therefore is related to the size of the light source. A point source of zero diameter emits spatially coherent light, while the light from a collection of point-sources (or from a source of finite diameter) would have lower coherence. Spatial coherence can be increased with a spatial filter; a very small pinhole preceded by a condenser lens. The spatial coherence of light will increase as it travels away from the source and becomes more like a sphere or plane wave. Light from distant stars, though far from monochromatic, has extremely high spatial coherence. The science of stellar interferometry relies on the coherence of starlight.
Light waves produced by a laser often have high temporal and spatial coherence (though the degree of coherence depends strongly on the exact properties of the laser). For example, a stabilised helium-neon laser can produce light with coherence lengths in excess of 5 m. Light from common sources (such as light bulbs) is not monochromatic and has a very short coherence length (~1 μm), and can be considered totally temporally incoherent for most purposes. Spatial coherence of laser beams also manifests itself as speckle patterns and diffraction fringes seen at the edges of shadow.
Holography requires temporally and spatially coherent light. Its inventor, Dennis Gabor, produced successful holograms more than ten years before lasers were invented. To produce coherent light he passed the highly monochromatic light from a gas-discharge lamp through a pinhole.