Information about Coherence (physics)
Coherence is the property of wave-like states that enables 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, neuroscience, and quantum physics. In interference, at least two wave-like entities 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. The property of coherence is the basis for commercial applications such as holography, the Sagnac gyroscope, radio antenna arrays, optical coherence tomography and telescope interferometers (astronomical optical interferometers and radio telescopes).
Temporal coherence is the measure of the average correlation between the value of a wave at any pair of times, separated by delay τ. 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 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.
One should be careful not to confuse the coherence time with the time duration of the signal, nor the coherence length with the coherence area (see below).
In terms of wavelength (
) this relationship becomes,
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.
Holography requires light with a long coherence time. In contrast, Optical coherence tomography uses light with a short 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 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 3. At a fixed delay, here 2τc, an infinitely fast detector 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.
. Since for a white-light source such as a light-bulb 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.
Waves of different frequencies (in light these are different colours) can interfere to form a pulse if they have a fixed relative phase-relationship (see Fourier transform). Conversely, if waves of different frequencies are not coherent, then, when combined, they create a wave that is continuous in time (e.g. white light or white noise). The temporal duration of the pulse
is limited by the spectral bandwidth of the light 
according to:
which follows from the properties of the Fourier transform (for quantum particles it also follows from the Heisenberg uncertainty principle).
If the phase depends linearly on the frequency (i.e.
) then the pulse will have the minimum time duration for its bandwidth (a transform-limited pulse), otherwise it is chirped (see dispersion).
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.
The quantum description of perfectly coherent paths is called a pure state, in which the two paths are combined in a superposition. The correlation between the two particles exceeds what would be predicted for classical correlation alone (see Bell's inequalities). If this two-particle system is decohered (which would occur in a measurement via Einselection), then there is no longer any phase relationship between the two states. The quantum description of imperfectly coherent paths is called a mixed state, described by a density matrix and is entirely analogous to a classical system of mixed probabilities (the correlations are classical).
Large-scale (macroscopic) quantum coherence leads to very amazing phenomena. For instance, the laser, superconductivity, and superfluidity are examples of highly coherent quantum systems. One example that shows the amazing possibilities of macroscopic quantum coherence is the Schrödinger's cat thought experiment. Another example of quantum coherence is in a Bose-Einstein condensate. Here, all the atoms that make up the condensate are in-phase. They are thus all described by a single quantum wavefunction. Their behavior is communal and inseparable until the coherence is destroyed.
Coherence and correlation
The coherence of two waves follows from how well correlated the waves are as quantified by the cross-correlation function. 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. At any time, if the first wave changes, the second will change in the same way. If combined they can exhibit complete constructive interference at all times. It follows that they are perfectly coherent. 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.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.- Waves in a rope (up and down) or slinky (compression and expansion)
- Surface waves in a liquid
- Electric signals (fields) in transmission cables
- Sound
- Radio and Microwaves
- Light (optics)
- Electrons, atoms, and any other object (as described by quantum physics)
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 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.
Temporal coherence is the measure of the average correlation between the value of a wave at any pair of times, separated by delay τ. 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 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.
One should be careful not to confuse the coherence time with the time duration of the signal, nor the coherence length with the coherence area (see below).
The relationship between coherence time and bandwidth
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:
.
In terms of wavelength (
) this relationship becomes,
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.
Examples of temporal coherence
We consider four examples of temporal coherence.- A wave containing only a single frequency (monochromatic) is perfectly correlated 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.
Holography requires light with a long coherence time. In contrast, Optical coherence tomography uses light with a short coherence time.
Measurement of temporal coherence
)
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 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. 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 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 3. At a fixed delay, here 2τc, an infinitely fast detector 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.
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 points in space, 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 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.Examples of spatial coherence
)
Figure 5: A plane wave with an infinite coherence length.
- 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 pinhole. Far 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.
. Since for a white-light source such as a light-bulb 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.
)
Figure 6: A wave with a varying profile (wavefront) and infinite coherence length.
)
Figure 7: A wave with a varying profile (wavefront) and finite coherence length.
)
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 (or 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 measure reduced visibility.
Spectral coherence
)
Figure 10: Waves of different frequencies (i.e. colors) interfere to form a pulse if they are coherent.
)
Figure 11: Spectrally incoherent light interferes to form continuous light with a randomly varying phase and amplitude
Waves of different frequencies (in light these are different colours) can interfere to form a pulse if they have a fixed relative phase-relationship (see Fourier transform). Conversely, if waves of different frequencies are not coherent, then, when combined, they create a wave that is continuous in time (e.g. white light or white noise). The temporal duration of the pulse
is limited by the spectral bandwidth of the light according to:
,
which follows from the properties of the Fourier transform (for quantum particles it also follows from the Heisenberg uncertainty principle).
If the phase depends linearly on the frequency (i.e.
) then the pulse will have the minimum time duration for its bandwidth (a transform-limited pulse), otherwise it is chirped (see dispersion).
Measurement of spectral coherence
Measurement of the spectral coherence of light requires a nonlinear optical interferometer, such as an intensity optical correlator, frequency-resolved optical gating (FROG), or Spectral phase interferometry for direct electric-field reconstruction (SPIDER).Polarization coherence
Light also has a polarization, which is the direction in which the electric field oscillates. 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.
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 correlation between the two particles exceeds what would be predicted for classical correlation alone (see Bell's inequalities). If this two-particle system is decohered (which would occur in a measurement via Einselection), then there is no longer any phase relationship between the two states. The quantum description of imperfectly coherent paths is called a mixed state, described by a density matrix and is entirely analogous to a classical system of mixed probabilities (the correlations are classical).
Large-scale (macroscopic) quantum coherence leads to very amazing phenomena. For instance, the laser, superconductivity, and superfluidity are examples of highly coherent quantum systems. One example that shows the amazing possibilities of macroscopic quantum coherence is the Schrödinger's cat thought experiment. Another example of quantum coherence is in a Bose-Einstein condensate. Here, all the atoms that make up the condensate are in-phase. They are thus all described by a single quantum wavefunction. Their behavior is communal and inseparable until the coherence is destroyed.
See also
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 0-19-850177-3]
- Leonard Mandel and Emil Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995), [ISBN 0521417112]
- Arvind Marathay, Elements of Optical Coherence Theory (John Wiley & Sons Inc, 1982), [ISBN 0471567892]
Interference is the addition (superposition) of two or more waves that results in a new wave pattern.
As most commonly used, the term interference usually refers to the interaction of waves which are correlated or coherent with each other, either because they
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As most commonly used, the term interference usually refers to the interaction of waves which are correlated or coherent with each other, either because they
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degree of coherence is the normalized correlation of electric fields. In its simplest form, termed , it is useful for quantifying the coherence between two electric fields, as measured in an Michelson or other linear optical interferometer.
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π due to reflection at the interface of a denser medium)
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Quantum version of experiment
By the 1920s, various other experiments (such as the photoelectric effect) had demonstrated that light interacts with matter only in discrete, "quantum"-sized packets called photons...... Click the link for more information.
Optics (ὀπτική appearance or look in Ancient Greek) is a branch of physics that describes the behavior and properties of light and the interaction of light with matter.
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Acoustics is the branch of physics concerned with the study of sound (mechanical waves in gases, liquids, and solids). A scientist who works in the field of acoustics is an acoustician. The application of acoustics in technology is called acoustical engineering.
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Electrical engineering (sometimes referred to as electrical and electronic engineering) is an engineering field that deals with the study and/or application of electricity, electronics and electromagnetism.
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Neuroscience is a field that is devoted to the scientific study of the nervous system. Such studies may include the structure, function, evolutionary history, development, genetics, biochemistry, physiology, pharmacology, and pathology of the nervous system.
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quantum mechanics is the study of the relationship between energy quanta (radiation) and matter, in particular that between valence shell electrons and photons. Quantum mechanics is a fundamental branch of physics with wide applications in both experimental and theoretical physics.
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phase can be readily understood in terms of simple harmonic motion. The same concept applies to wave motion, viewed either at a point in space over an interval of time or across an interval of space at a moment in time.
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The interferometric visibility (also known as "interference visibility" or "fringe visibility" or just "visibility") quantifies the contrast of interference in any system which has wave-like properties, such as optics, quantum mechanics, water waves, or electrical signals.
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Holography (from the Greek, όλος-hòlòs whole + γραφή-grafè write) is the science of producing holograms
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The Sagnac effect (also called Sagnac Interference), named after French physicist Georges Sagnac, is a phenomenon encountered in interferometry that is elicited by rotation. The Sagnac effect manifests itself in a setup called ring interferometry.
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A gyroscope is a device for measuring or maintaining orientation, based on the principle of conservation of angular momentum. The device is a spinning wheel whose axle is free to take any orientation.
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Radio is the wireless transmission of signals, by modulation of electromagnetic waves with frequencies below those of visible light. Electromagnetic radiation travels by means of oscillating electromagnetic fields that pass through the air and the vacuum of space.
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Antenna array may refer to:
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- an interferometric array of Radio telescopes used in radio astronomy.
- a Phased array, an electronically steerable directional antenna typically used in RADAR.
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Optical coherence tomography (OCT) is an interferometric, non-invasive optical tomographic imaging technique offering millimeter penetration (approximately 2-3 mm in tissue) with micrometer-scale axial and lateral resolution.
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radio telescope is a form of directional radio antenna used in radio astronomy and in tracking and collecting data from satellites and space probes. In their astronomical role they differ from optical telescopes in that they operate in the radio frequency portion of the
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In statistics, the term cross-correlation is sometimes used to refer to the covariance cov(X, Y) between two random vectors X and Y, in order to distinguish that concept from the "covariance" of a random vector X
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Autocorrelation is a mathematical tool used frequently in signal processing for analysing functions or series of values, such as time domain signals. Informally, it is a measure of how well a signal matches a time-shifted version of itself, as a function of the amount of time shift.
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The wave equation is an important second-order linear partial differential equation that describes the propagation of a variety of waves, such as sound waves, light waves and water waves. It arises in fields such as acoustics, electromagnetics, and fluid dynamics.
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Slinky, or "Lazy-Spring," is a coil-shaped toy invented by mechanical engineer Richard James in Philadelphia, Pennsylvania. [1]
Slinkys come in various sizes. The shape is a simple helix, or coil design, of a ribbon of material, originally metal.
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Slinkys come in various sizes. The shape is a simple helix, or coil design, of a ribbon of material, originally metal.
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In physics, surface wave can refer to a mechanical wave that propagates along the interface between differing media, usually two fluids with different densities. A surface wave can also be an electromagnetic wave guided by a refractive index gradient.
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Sound is a disturbance of mechanical energy that propagates through matter as a wave (through fluids as a compression wave, and through solids as both compression and shear waves).
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Radio is the wireless transmission of signals, by modulation of electromagnetic waves with frequencies below those of visible light. Electromagnetic radiation travels by means of oscillating electromagnetic fields that pass through the air and the vacuum of space.
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Microwaves are electromagnetic waves with wavelengths shorter than one meter and longer than one millimeter, or frequencies between 300 megahertz and 300 gigahertz.
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Light is electromagnetic radiation of a wavelength that is visible to the eye (visible light). In a scientific context, the word "light" is sometimes used to refer to the entire electromagnetic spectrum.
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Optics (ὀπτική appearance or look in Ancient Greek) is a branch of physics that describes the behavior and properties of light and the interaction of light with matter.
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quantum mechanics is the study of the relationship between energy quanta (radiation) and matter, in particular that between valence shell electrons and photons. Quantum mechanics is a fundamental branch of physics with wide applications in both experimental and theoretical physics.
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electric field. This electric field exerts a force on other electrically charged objects. The concept of electric field was introduced by Michael Faraday.
The electric field is a vector field with SI units of newtons per coulomb (N C−1
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The electric field is a vector field with SI units of newtons per coulomb (N C−1
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In physics, intensity is a measure of the time-averaged energy flux. To find the intensity, take the energy density (that is, the energy per unit volume) and multiply it by the velocity at which the energy is moving. The resulting vector has the units of power divided by area (i.e.
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