These states are unified by the fact that their behavior is described by a
wave equation or some generalization thereof. • Waves in a rope (up and down) or
slinky (compression and expansion) •
Surface waves in a liquid •
Electromagnetic signals (fields) in
transmission lines •
Sound •
Radio waves and
microwaves •
Light waves (
optics) •
Matter waves associated with, for examples,
electrons and
atoms In system with macroscopic waves, one can measure the wave directly. Consequently, its correlation with another wave can simply be calculated. However, in optics one cannot measure the
electric field directly as it oscillates much faster than any detector's time resolution. Instead, one measures the
intensity of the light. Most of the concepts involving coherence which will be introduced below were developed in the field of optics and then used in other fields. Therefore, many of the standard measurements of coherence are indirect measurements, even in fields where the wave can be measured directly. ==Temporal coherence== Temporal coherence is the measure of the average correlation between the value of a wave and itself delayed by \tau, at any pair of times. Temporal coherence tells us how monochromatic a source is. In other words, it characterizes how well a wave can interfere with itself at a different time. The delay over which the phase or amplitude wanders by a significant amount (and hence the correlation decreases by significant amount) is defined as the
coherence time \tau_\mathrm{c}. At a delay of \tau=0 the degree of coherence is perfect, whereas it drops significantly as the delay passes \tau=\tau_\mathrm{c}. The
coherence length L_\mathrm{c} is defined as the distance the wave travels in time \tau_\mathrm{c}. The coherence time is not the time duration of the signal; the coherence length differs from the coherence area (see below).
The relationship between coherence time and bandwidth The larger the bandwidth – range of frequencies Δf a wave contains – the faster the wave decorrelates (and hence the smaller \tau_\mathrm{c} is): :\tau_c \Delta f \gtrsim 1. Formally, this follows from the
convolution theorem in mathematics, which relates the
Fourier transform of the power spectrum (the intensity of each frequency) to its autocorrelation. Narrow bandwidth
lasers have long coherence lengths (up to hundreds of meters). For example, a stabilized and monomode
helium–neon laser can easily produce light with coherence lengths of 300 m. Not all lasers have a high monochromaticity, however (e.g. for a mode-locked
Ti-sapphire laser, Δλ ≈ 2 nm – 70 nm). LEDs are characterized by Δλ ≈ 50 nm, and tungsten filament lights exhibit Δλ ≈ 600 nm, so these sources have shorter coherence times than the most monochromatic lasers.
Examples of temporal coherence Examples of temporal coherence include: • A wave containing only a single frequency (monochromatic) is perfectly correlated with itself at all time delays, in accordance with the above relation. (See Figure 1) • Conversely, a wave whose phase drifts quickly will have a short coherence time. (See Figure 2) • Similarly, pulses (
wave packets) of waves, which naturally have a broad range of frequencies, also have a short coherence time since the amplitude of the wave changes quickly. (See Figure 3) • Finally, white light, which has a very broad range of frequencies, is a wave which varies quickly in both amplitude and phase. Since it consequently has a very short coherence time (just 10 periods or so), it is often called incoherent.
Holography requires light with a long coherence time. In contrast,
optical coherence tomography, in its classical version, uses light with a short coherence time.
Measurement of temporal coherence . Although the waves in Figures 2 and 3 have different time durations, they have the same coherence time. In optics, temporal coherence is measured in an interferometer such as the
Michelson interferometer or
Mach–Zehnder interferometer. In these devices, a wave is combined with a copy of itself that is delayed by time \tau. A detector measures the time-averaged
intensity of the light exiting the interferometer. The resulting visibility of the interference pattern (e.g. see Figure 4) gives the temporal coherence at delay \tau. Since for most natural light sources, the coherence time is much shorter than the time resolution of any detector, the detector itself does the time averaging. Consider the example shown in Figure 3. At a fixed delay, here 2\tau, an infinitely fast detector would measure an intensity that fluctuates significantly over a time
t equal to \tau. In this case, to find the temporal coherence at 2\tau_\mathrm{c}, one would manually time-average the intensity. ==Spatial coherence== In some systems, such as water waves or optics, wave-like states can extend over one or two dimensions. Spatial coherence describes the ability for two spatial points
x1 and
x2 in the extent of a wave to interfere when averaged over time. More precisely, the spatial coherence is the cross-correlation between two points in a wave for all times. If a wave has only 1 value of amplitude over an infinite length, it is perfectly spatially coherent. The range of separation between the two points over which there is significant interference defines the diameter of the coherence area, A_\mathrm{c} (Coherence length l_\mathrm{c}, often a feature of a source, is usually an industrial term related to the coherence time of the source, not the coherence area in the medium). A_\mathrm{c} is the relevant type of coherence for the Young's double-slit interferometer. It is also used in optical imaging systems and particularly in various types of astronomy telescopes. A distance z away from an incoherent source with surface area A_\mathrm{s} , A_\mathrm{c}=\frac{\lambda^2 z^2}{A_\mathrm{s}} Sometimes people also use "spatial coherence" to refer to the visibility when a wave-like state is combined with a spatially shifted copy of itself.
Examples File:spatial coherence infinite ex1.svg|Figure 5: A plane wave with an infinite
coherence length. File:spatial coherence infinite ex2.png|Figure 6: A wave with a varying profile (wavefront) and infinite coherence length. File:spatial coherence finite.png|Figure 7: A wave with a varying profile (wavefront) and finite coherence length. File:spatial coherence pinhole.png|Figure 8: A wave with finite coherence area is incident on a pinhole (small aperture). The wave will
diffract out of the pinhole. Far from the pinhole the emerging spherical wavefronts are approximately flat. The coherence area is now infinite while the coherence length is unchanged. File:spatial coherence detector.png|Figure 9: A wave with infinite coherence area is combined with a spatially shifted copy of itself. Some sections in the wave interfere constructively and some will interfere destructively. Averaging over these sections, a detector with length D will measure reduced
interference visibility. For example, a misaligned
Mach–Zehnder interferometer will do this. Consider a tungsten light-bulb filament. Different points in the filament emit light independently and have no fixed phase-relationship. In detail, at any point in time the profile of the emitted light is going to be distorted. The profile will change randomly over the coherence time \tau_c. Since for a white-light source such as a light-bulb \tau_c is small, the filament is considered a spatially incoherent source. In contrast, a radio
antenna array, has large spatial coherence because antennas at opposite ends of the array emit with a fixed phase-relationship. 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). 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. In February 2011 it was reported that
helium atoms, cooled to near
absolute zero /
Bose–Einstein condensate state, can be made to flow and behave as a coherent beam as occurs in a laser. == Spectral coherence of short pulses ==