Coherence (physics)

Coherence is the property of wave-like states that enable them to exhibit interference. It is also the parameter that quantifies the quality of the interference (also known as the degree of coherence). It was originally introduced in connection with Young’s double-slit experiment in optics but is now used in any field that involves waves, such as acoustics, electrical engineering, and quantum physics. In interference, at least two wave-like states are combined and, depending on the relative phase between them, they can add constructively or subtract destructively. The degree of coherence is equal to the interference visibility, a measure of how perfectly the waves can cancel due to destructive interference.

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Coherence and correlation

The coherence of two wave-like states follows from how well correlated the waves are as quantified by the cross-correlation function. Essentially, the cross-correlation quantifies the ability to predict the value of the second wave by knowing the value of the first. As an example, consider two waves perfectly correlated for all times. It follows that they are perfectly coherent since they can exhibit complete destructive interference at all times. As will be discussed below, the second wave need not be a separate entity. It could be the first wave at a different time or position. In this case, sometimes called self-coherence, the measure of correlation is the autocorrelation function.

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Examples of wave-like states

These states are unified by the fact that their behavior is described by a wave equation or some generalization thereof.

In most of these systems, one can measure the wave directly. Consequently, its correlation with another wave can simply be calculated. However, in optics one can not measure the electric field directly as it oscillates much faster than any detector’s time resolution. Instead, we measure 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.


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Temporal coherence

Figure 1: The amplitude of a single frequency wave as a function of time t (red) and a copy of the same wave delayed by τ(green). The coherence time of the wave is infinite since it is perfectly correlated with itself for all delays τ.
Figure 1: The amplitude of a single frequency wave as a function of time t (red) and a copy of the same wave delayed by τ(green). The coherence time of the wave is infinite since it is perfectly correlated with itself for all delays τ.
Figure 2: The amplitude of a wave whose phase drifts significantly in time τc as a function of time t (red) and a copy of the same wave delayed by 2τc(green). At any particular time t the wave can interfere perfectly with its delayed copy. But, since half the time the red and green waves are in phase and half the time out of phase, when averaged over t any interference disappears at this delay.
Figure 2: The amplitude of a wave whose phase drifts significantly in time τc as a function of time t (red) and a copy of the same wave delayed by 2τc(green). At any particular time t the wave can interfere perfectly with its delayed copy. But, since half the time the red and green waves are in phase and half the time out of phase, when averaged over t any interference disappears at this delay.
Figure 3: The amplitude of a wavepacket whose amplitude changes significantly in time τc (red) and a copy of the same wave delayed by 2τc(green) plotted as a function of time t. At any particular time the red and green waves are uncorrelated; one oscillates while the other is constant and so there will be no interference at this delay. Another way of looking at this is the wavepackets are not overlapped in time and so at any particular time there is only one nonzero field so no interference can occur.
Figure 3: The amplitude of a wavepacket whose amplitude changes significantly in time τc (red) and a copy of the same wave delayed by 2τc(green) plotted as a function of time t. At any particular time the red and green waves are uncorrelated; one oscillates while the other is constant and so there will be no interference at this delay. Another way of looking at this is the wavepackets are not overlapped in time and so at any particular time there is only one nonzero field so no interference can occur.

Temporal coherence is the measure of the average correlation between the value of a wave at every pair of times separated by delay τ. In other words, it characterizes how well to 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 τc. At τ=0 the degree of coherence is perfect whereas it drops significantly by delay τc. The coherence length Lc is defined as the distance the wave travels in time τc.

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.

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The relationship between coherence time and bandwidth

A change in phase or amplitude of a wave lengthens or shortens its period. Since period is the inverse of frequency, it follows that the faster a wave decorrelates (and hence the smaller τc is) the larger the range of frequencies Δf the wave contains. Thus there is a tradeoff:

<math>\tau_c \Delta f \approx 1</math>

This relation also follows from the convolution theorem in mathematics, which relates the fourier transform of a function to its autocorrelation.

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Examples of temporal coherence

We consider four examples of temporal coherence.

  • A wave containing only a single frequency (monochromatic) is perfectly coherent at all times according to 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.
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Measurement of temporal coherence

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 τ. A detector measures the time-averaged intensity of the light exiting the interferometer. The resulting interference visibility (e.g. see Figure 4) gives the temporal coherence at delay τ. 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 2. At a fixed delay, here 2τc an infinitely fast dector would measure an intensity that fluctuates significantly over a time t equal to τc. In this case, to find the temporal coherence at 2τc, one would manually time-average the intensity.

Figure 4: The time-averaged intensity (blue) detected at the output of an interferometer plotted as a function of delay τ for the example waves in Figures 2 and 3. As the delay is changed by half a period, the interference switches between constructive and destructive. The black lines indicate the interference envelope, which gives the degree of coherence
Figure 4: The time-averaged intensity (blue) detected at the output of an interferometer plotted as a function of delay τ for the example waves in Figures 2 and 3. As the delay is changed by half a period, the interference switches between constructive and destructive. The black lines indicate the interference envelope, which gives the degree of coherence


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Spatial coherence

Figure 5: A plane wave with an inifinite coherence length
Figure 5: A plane wave with an inifinite coherence length
Figure 6: A wave with a varying profile (wavefront) and inifinite coherence length
Figure 6: A wave with a varying profile (wavefront) and inifinite coherence length
Figure 7: A wave with a varying profile (wavefront) and finite coherence length
Figure 7: A wave with a varying profile (wavefront) and finite coherence length

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 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. The range of separation between the two points over which there is the significant interference is called the coherence area, Ac. This 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. 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.

The same definition can be applied to a source of light. A light-bulb filament is a spatially incoherent source because the waves emitted at points on opposite ends of the filament are incoherent. In contrast, a radio-antenna array, has large spatial coherence because antenna's at opposite ends of the array emit with a fixed phase-relationship.

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Examples of spatial coherence

  • Plane waves with an infinite coherence time have an infinite coherence area. See Figure 5.
  • A wave with distorted profile and with an infinite coherence time has an infinite coherence area. See Figure 6.
  • A wave with distorted profile and a finite coherence time has a finite coherence area. See Figure 7.
  • A wave with finite coherence area is incident on a pinhole (small aperture). The wave will diffract out of the pinholeFar from the pinhole the emerging spherical wavefronts are approximately flat. The coherence area is now infinite while the coherence length is unchanged. See Figure 8.
  • 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. See Figure 9.

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.

Figure 8: The wave with finite coherence length from Figure 7 is passed through a pinhole. The emerging wave has infinite coherence area. The coherence length (and coherence time) are unchanged by the pinhole.
Figure 8: The wave with finite coherence length from Figure 7 is passed through a pinhole. The emerging wave has infinite coherence area. The coherence length (and coherence time) are unchanged by the pinhole.
Figure 9: The wave with infinite coherence length from Figure 6 is combined with a spatially shifted copy of itself. For example a misaligned Mach-Zehnder interferometer will do this. A detector will will measure reduced visibility.
Figure 9: The wave with infinite coherence length from Figure 6 is combined with a spatially shifted copy of itself. For example a misaligned Mach-Zehnder interferometer will do this. A detector will will measure reduced visibility.


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Chromatic coherence

Waves of different freqencies (in light these are different colours) can interfere to form a pulse if they have a fixed relative phase (see Fourier transform). Conversely, if the the waves of different frequencies are not coherent then white light or white noise is created. This type of coherence is measured in an optical_autocorrelation.

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Polarization coherence

Light also has a polarization, which is the direction the electric field oscillates in. Unpolarized light is composed of two equally intense incoherent light waves with orthogonal polarizations. The electric field of the unpolarized light wanders in every direction and changes in phase over the coherence time of the two light waves. A polarizer rotated to any angle will always transmit half the incident intensity when averaged over time.

If the electric field wanders by a smaller amount the light will be partially polarized so that at some angle, the polarizer will transmit more than half the intensity. If a wave is combined with an orthogonally polarized copy of itself delayed by less than the coherence time, partially polarized light is created.

The polarization of a light beam is represented by a vector in the Poincare sphere. For polarized light the end of the vector lies on the surface of the sphere, whereas the vector has zero length for unpolarized light. The vector for partially polarized light lies within the sphere.

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Quantum coherence

In quantum mechanics, all objects have wave-like properties (see de Broglie waves). For instance, in Young's double-slit experiment electrons can be used in the place of light waves. Each electron can go through either slit and hence has two paths that it can take to a particular final position. In quantum mechanics these two paths interfere. If there is destructive interference, the electron never arrives at that particular position. This ability to interfere is called quantum coherence.

The quantum description of perfectly coherent paths is called a pure state, in which the two paths are combined in a superposition. The quantum description of imperfectly coherent paths is called a mixed state, described by a density matrix.

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References

  • Rolf G. Winter, Aephraim M. Steinberg, "Coherence", in AccessScience@McGraw-Hill, http://www.accessscience.com, DOI 10.1036/1097-8542.146900, last modified: October 24, 2001.
  • M. Born and E. Wolf, Principles of Optics, 7th ed., 1999
  • Loudon, Rodney, The Quantum Theory of Light (Oxford University Press, 2000), [ISBN 0198501773]
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