# Coherence (physics)

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Sine_waves_same_phase.png
Monochromatic waves of the same frequency are coherent with each other, and combine constructively if they are in phase with each other (example: several light beams originating from one laser, each with the same phase)
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Sine_waves_different_phase.png
Monochromatic waves which are not in phase with each other do not combine constructively. If the phase difference between the waves remains constant (i.e. they have the same frequency as in this example), the different waves are still coherent with each other, and the total light produced will depend on how the waves interfere with each other (example:several light beams from a single laser which has high temporal coherence, each beam delayed by a different amount from the others, and thus having a different phase)
Several monochromatic waves each having a different frequency (not all the same colour) combine incoherently to give light which is not monochromatic. (example:combining light from lasers with different colours, to produce an incoherent light source)

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. Truly monochromatic plane waves with exactly the same frequency are always coherent, but waves do not have to be monochromatic in order to be coherent with each other. Two non-monochromatic waves are fully coherent with each other if they both have exactly the same range of wavelengths and the same phase difference at each constituent wavelength. For waves to be coherent with each other, they generally must either both come from, or be phase-locked to, the same source (e.g. the sunlight passing through the two slits of Young's double-slit experiment), or be monochromatic with precisely the same frequency (e.g. two extremely stable oscillators -- two or more different sources can only be used to produce interference when there is a fixed phase relation between them (they are phase-locked), but in this case the interference generated is the same as with a single source; see Huygens's principle).

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:

[itex]\Delta x_c = c \Delta t_c \,\![itex]

where

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 (see second diagram on right which shows parts of truly monochromatic waves delayed by different amounts, which remain fully coherent with each other). 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. The most monochromatic sources are usually lasers, and thus have the longest coherence times. Not all lasers are monochromatic, however. LEDs are less monochromatic than the most-monochromatic lasers, and tungsten filament lights are even less monochromatic, and so these sources have shorter coherence times than the most-monochromatic lasers.

Waves also have the related property of spatial coherence; this is the ability of any one spatial position of the wavefront to interfere 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 spherical waves and plane waves as it is related to the size and coherence of the light source. A non-monochromatic point source of zero diameter emits spatially coherent light, while the light from a collection of incoherent point-sources which are resolved by the imaging system would have lower coherence (as the point sources are not coherent with each other, the resulting light has low spatial coherence). If multiple point sources are coherent with each other (e.g. they are the slits in a Young's slit experiment) then there is no loss of spatial coherence from having multiple point sources. An incoherent source of finite size can be thought of as an arrangement of many different incoherent point sources making up the finite size of the source (like many pixels making up an image of the source). Spatial coherence can be increased with a spatial filter; a very small pinhole preceded by a condenser lens which excludes all light apart from that coming from a central "pixel", thus converting a source of finite size into a point source. The spatial coherence of light will increase as it travels away from the source as the source appears smaller (more pointlike) and the waves become more like a sphere or plane wave. Light from pointlike stars, though far from monochromatic, have extremely high spatial coherence, while light from binary stars or stars with finite size have lower spatial coherence. The science of stellar interferometry measures the spatial coherence of starlight and uses it to determine which stars consist of two or more points (these are binaries or multiple stars) or to measure the diameter of stars. In some cases images of the surfaces of stars (http://www.mrao.cam.ac.uk/telescopes/coast/astronomy.html#supergiants02-04) can be made from these interferometric measurements.

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 monochromatic light from an emission line of a mercury-vapor lamp through a pinhole spatial filter.de:Kohärenz (Physik) fr:Cohérence (physique) ja:コヒーレンス sv:Koherent

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