Van Cittert–Zernike theorem
Encyclopedia
The Van Cittert–Zernike theorem is a formula in coherence theory
that states that under certain conditions the Fourier transform
of the mutual coherence function of a distant, incoherent source is equal to its complex visibility. This implies that the wavefront
from an incoherent source will appear mostly coherent at large distances. Intuitively, this can be understood by considering the wavefronts created by two incoherent sources. If we measure the wavefront immediately in front of one of the sources, our measurement will be dominated by the nearby source. If we make the same measurement far from the sources, our measurement will no longer be dominated by a single source; both sources will contribute almost equally to the wavefront at large distances.
This reasoning can be easily visualized by dropping two stones in the center of a calm pond. Near the center of the pond, the disturbance created by the two stones will be very complicated. As the disturbance propagates towards the edge of pond, however, the waves will smooth out and will appear to be nearly circular.
The van Cittert–Zernike theorem has important implications for radio astronomy
. With the exception of pulsars and masers, all astronomical sources are spatially incoherent. Nevertheless, because they are observed at distances large enough to satisfy the van Cittert–Zernike theorem, these objects exhibit a non-zero degree of coherence at different points in the imaging plane. By measuring the degree of coherence at different points in the imaging plane (the so-called "visibility
function") of an astronomical object, a radio astronomer can thereby reconstruct the source's brightness distribution and make a two-dimensional map of the source's appearance.
is the mutual coherence function between two points on a plane perpendicular to the line of sight, then
where
and 
are the direction cosines of a point on a distant source and 
is the intensity of the source. This theorem was first derived by P. H. van Cittert in 1934 with a simpler proof provided by F. Zernike in 1938.
measured at two points in a plane of observation (call them 1 and 2), is defined to be
where
is the time offset between the measurement of 
at point 1 and point 2 and 
and 
give the number of wavelength
s between points 1 and 2 along the x- and y-axes of the observation plane, respectively. A special case of the mutual coherence function when
is called the visibility function.
The mutual coherence between two points may be thought of as the time-averaged cross-correlation between the electric fields at the two points over a time
. Thus, if we are observing two incoherent sources we should expect the mutual coherence function to be relatively small between the two random points in the observation plane since the sources will interfere destructively as well as constructively. From far away, however, we should expect the mutual coherence function to be relatively large since the sources will mostly interfere constructively.
Normalization of the mutual coherence function to the product of the square roots of the intensities of the two electric fields yields the complex degree of second-order coherence:
and 
, in the observation plane. The position of a point in the source may be referred to by its direction cosines 
. (Since the source is distant, its direction should be the same at 
as at 
.) The electric field measured at 
can then be written using phasor
s:
where
is the distance from the source to 
, 
is the angular frequency
of the light
, and
is the complex amplitude of the electric field. Similarly, the electric field measured at 
can be written as
Let us now calculate the time-averaged cross-correlation between the electric field at
and 
:
Because the quantity in the angle brackets is time-averaged an arbitrary offset to the temporal term of the amplitudes may be added as long as the same offset is added to both. Let us now add
to the temporal term of both amplitudes. The time-averaged cross-correlation of the electric field at the two points therefore simplifies to
But if the source is in the far field then the difference between
and 
will be small compared to the distance light travels in time 
. (
is on the same order as the inverse bandwidth
.) This small correction can therefore be neglected, further simplifying our expression for the cross-correlation of the electric field at
and 
to
Now,
is simply the intensity of the source at a particular point, 
. So our expression for the cross-correlation simplifies further to
To calculate the mutual coherence function from this expression, simply integrate over the entire source.
Note that cross terms of the form
are not included due to the assumption that the source is incoherent. The time-averaged correlation between two different points from the source will therefore be zero.
Next rewrite the
term using 
and 
. To do this, let 
and 
. This gives
where
is the distance between the center of the plane of observation and the center of the source. The difference between 
and 
thus becomes
But because
and 
are all much less than 
, the square roots may be Taylor expanded
, yielding, to first order,
which, after some algebraic manipulation, simplifies to
Now,
is the midpoint along the 
-axis between 
and 
, so 
gives us 
, one of the direction cosines to the sources. Similarly, 
. Moreover, recall that 
was defined to be the number of wavelengths along the 
-axis between 
and 
. So
Similarly,
is the number of wavelengths between 
and 
along the 
-axis, so
Hence
Because
and 
are all much less than 
, 
. The differential area element, 
, may then be written as a differential element of solid angle
of
. Our expression for the mutual coherence function becomes
Which reduces to
But the limits of these two integrals can be extended to cover the entire plane of the source as long as the source's intensity function is set to be zero over these regions. Hence,
which is the two-dimensional Fourier transform of the intensity function. This completes the proof.
and 
. Let us calculate the mutual coherence function between 
and 
in the plane of observation. From the principle of superposition, the electric field at 
is
and at
is
so the mutual coherence function is
Which becomes
If points
and 
are coherent then the cross terms in the above equation do not vanish. In this case, when we calculate the mutual coherence function for an extended coherent source, we would not be able to simply integrate over the intensity function of the source; the presence of non-zero cross terms would give the mutual coherence function no simple form.
This assumption holds for most astronomical sources. Pulsars and masers are the only astronomical sources which exhibit coherence.
and 
. That is, we assume that the distance to the source is much greater than the size of the observation area. More precisely, the van Cittert–Zernike theorem requires that we observe the source in the so-called far field. Hence if 
is the characteristic size of the observation area (e.g. in the case of a two-dish radio telescope
, the length of the baseline between the two telescopes) then
Using a reasonable baseline of 20 km for the Very Large Array
at a wavelength of 1 cm, the far field distance is of order
m. Hence any astronomical object farther away than a parsec is in the far field. Objects in the Solar System
are not necessarily in the far field, however, and so the van Cittert–Zernike theorem does not apply to them.
and 
as 
and 
. There is, however, a third direction cosine which is neglected since 
and 
; under these assumptions it is very close to unity. But if the source has a large angular extent, we cannot neglect this third direction cosine and the van Cittert–Zernike theorem no longer holds.
Because most astronomical sources subtend very small angles on the sky (typically much less than a degree), this assumption of the theorem is easily fulfilled in the domain of radio astronomy.
, with mean frequency 
, then it should satisfy
Moreover, the bandwidth must be narrow enough that
where
is again the direction cosine indicating the size of the source and 
is the number of wavelengths between one end of the aperture and the other. Without this assumption, we cannot neglect 
compared to 
This requirement implies that a radio astronomer must restrict signals through a bandpass filter. Because radio telescopes almost always pass the signal through a relatively narrow bandpass filter, this assumption is typically satisfied in practice.
relative to other regions of the source due to the difference in light travel time through the medium. In the case of a heterogeneous medium one must use a generalization of the van Cittert–Zernike theorem, called Hopkins's formula.
Because the wavefront does not pass through a perfectly uniform medium as it travels through the interstellar
(and possibly intergalactic) medium and into the Earth's atmosphere
, the van Cittert–Zernike theorem does not hold exactly true for astronomical sources. In practice, however, variations in the refractive index
of the interstellar and intergalactic media and Earth's atmosphere are small enough that the theorem is approximately true to within any reasonable experimental error. Such variations in the refractive index of the medium result only in slight perturbations from the case of a wavefront traveling through a homogeneous medium.
. Following a similar derivation as before, we find that
If we define
then the mutual coherence function becomes
which is Hopkins's generalization of the van Cittert–Zernike theorem. In the special case of a homogeneous medium, the transmission function becomes
in which case the mutual coherence function reduces to the Fourier transform of the brightness distribution of the source. The primary advantage of Hopkins's formula is that one may calculate the mutual coherence function of a source indirectly by measuring its brightness distribution.
or synthesis imaging.
In practice, radio astronomers rarely recover the brightness distribution of a source by directly taking the inverse Fourier transform of a measured visibility function. Such a process would require a sufficient number of samples to satisfy the Nyquist sampling theorem; this is many more observations than are needed to approximately reconstruct the brightness distribution of the source. Astronomers therefore take advantage of physical constraints on the brightness distribution of astronomical sources to reduce the number of observations which must be made. Because the brightness distribution must be real and positive everywhere, the visibility function cannot take on arbitrary values in unsampled regions. Thus, a non-linear deconvolution algorithm like CLEAN
or Maximum Entropy may be used to approximately reconstruct the brightness distribution of the source from a limited number of observations.
system. In an adaptive optics (AO) system, a distorted wavefront is provided and must be transformed to a distortion-free wavefront. An AO system must make a number of different corrections to remove the distortions from the wavefront. One such correction involves splitting the wavefront into two identical wavefronts and shifting one by some physical distance
in the plane of the wavefront. The two wavefronts are then superimposed, creating a fringe pattern. By measuring the size and separation of the fringes, the AO system can determine phase differences along the wavefront. This technique is known as "shearing."
The sensitivity of this technique is limited by the van Cittert–Zernike theorem. If an extended source is imaged, the contrast between the fringes will be reduced by a factor proportional to the Fourier transform of the brightness distribution of the source. The van Cittert–Zernike theorem implies that the mutual coherence of an extended source imaged by an AO system will be the Fourier transform of its brightness distribution. An extended source will therefore change the mutual coherence of the fringes, reducing their contrast.
Coherence theory
In physics, coherence theory is the study of optical effects arising from partially coherent light and radio sources. Partially coherent sources are sources where the coherence time or coherence length are limited by bandwidth, by thermal noise, or by other effect...
that states that under certain conditions the Fourier transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
of the mutual coherence function of a distant, incoherent source is equal to its complex visibility. This implies that the wavefront
Wavefront
In physics, a wavefront is the locus of points having the same phase. Since infrared, optical, x-ray and gamma-ray frequencies are so high, the temporal component of electromagnetic waves is usually ignored at these wavelengths, and it is only the phase of the spatial oscillation that is described...
from an incoherent source will appear mostly coherent at large distances. Intuitively, this can be understood by considering the wavefronts created by two incoherent sources. If we measure the wavefront immediately in front of one of the sources, our measurement will be dominated by the nearby source. If we make the same measurement far from the sources, our measurement will no longer be dominated by a single source; both sources will contribute almost equally to the wavefront at large distances.
This reasoning can be easily visualized by dropping two stones in the center of a calm pond. Near the center of the pond, the disturbance created by the two stones will be very complicated. As the disturbance propagates towards the edge of pond, however, the waves will smooth out and will appear to be nearly circular.
The van Cittert–Zernike theorem has important implications for radio astronomy
Radio astronomy
Radio astronomy is a subfield of astronomy that studies celestial objects at radio frequencies. The initial detection of radio waves from an astronomical object was made in the 1930s, when Karl Jansky observed radiation coming from the Milky Way. Subsequent observations have identified a number of...
. With the exception of pulsars and masers, all astronomical sources are spatially incoherent. Nevertheless, because they are observed at distances large enough to satisfy the van Cittert–Zernike theorem, these objects exhibit a non-zero degree of coherence at different points in the imaging plane. By measuring the degree of coherence at different points in the imaging plane (the so-called "visibility
Interferometric visibility
The interferometric visibility quantifies the contrast of interference in any system which has wave-like properties, such as optics, quantum mechanics, water waves, or electrical signals. Generally, two or more waves are combined and as the phase between them is changed The interferometric...
function") of an astronomical object, a radio astronomer can thereby reconstruct the source's brightness distribution and make a two-dimensional map of the source's appearance.
Statement of the theorem
If is the mutual coherence function between two points on a plane perpendicular to the line of sight, thenwhere
and are the direction cosines of a point on a distant source and is the intensity of the source. This theorem was first derived by P. H. van Cittert in 1934 with a simpler proof provided by F. Zernike in 1938.The mutual coherence function
The mutual coherence function for some electric fieldElectric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...
measured at two points in a plane of observation (call them 1 and 2), is defined to bewhere
is the time offset between the measurement of at point 1 and point 2 and and give the number of wavelengthWavelength
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...
s between points 1 and 2 along the x- and y-axes of the observation plane, respectively. A special case of the mutual coherence function when
is called the visibility function.The mutual coherence between two points may be thought of as the time-averaged cross-correlation between the electric fields at the two points over a time
. Thus, if we are observing two incoherent sources we should expect the mutual coherence function to be relatively small between the two random points in the observation plane since the sources will interfere destructively as well as constructively. From far away, however, we should expect the mutual coherence function to be relatively large since the sources will mostly interfere constructively.Normalization of the mutual coherence function to the product of the square roots of the intensities of the two electric fields yields the complex degree of second-order coherence:
Proof of the theorem
Consider a distant, incoherent, extended source located in a plane which is defined by two axes called the X- and Y-axes. This source is observed in a parallel plane defined by two axes which shall call the x- and y-axes. Suppose the electric field due to some point from this source is measured at two points, and , in the observation plane. The position of a point in the source may be referred to by its direction cosines . (Since the source is distant, its direction should be the same at as at .) The electric field measured at can then be written using phasorPhasor
Phasor is a phase vector representing a sine wave.Phasor may also be:* Phasor , a stereo music, sound and speech synthesizer for the Apple II computer* Phasor measurement unit, a device that measures phasors on an electricity grid...
s:
where
is the distance from the source to , is the angular frequencyAngular frequency
In physics, angular frequency ω is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity...
of the light
Light
Light or visible light is electromagnetic radiation that is visible to the human eye, and is responsible for the sense of sight. Visible light has wavelength in a range from about 380 nanometres to about 740 nm, with a frequency range of about 405 THz to 790 THz...
, and
is the complex amplitude of the electric field. Similarly, the electric field measured at can be written asLet us now calculate the time-averaged cross-correlation between the electric field at
and :Because the quantity in the angle brackets is time-averaged an arbitrary offset to the temporal term of the amplitudes may be added as long as the same offset is added to both. Let us now add
to the temporal term of both amplitudes. The time-averaged cross-correlation of the electric field at the two points therefore simplifies toBut if the source is in the far field then the difference between
and will be small compared to the distance light travels in time . ( is on the same order as the inverse bandwidthBand matrix
In mathematics, particularly matrix theory, a band matrix is a sparse matrix whose non-zero entries are confined to a diagonal band, comprising the main diagonal and zero or more diagonals on either side.-Matrix bandwidth:...
.) This small correction can therefore be neglected, further simplifying our expression for the cross-correlation of the electric field at
and toNow,
is simply the intensity of the source at a particular point, . So our expression for the cross-correlation simplifies further toTo calculate the mutual coherence function from this expression, simply integrate over the entire source.
Note that cross terms of the form
are not included due to the assumption that the source is incoherent. The time-averaged correlation between two different points from the source will therefore be zero.Next rewrite the
term using and . To do this, let and . This giveswhere
is the distance between the center of the plane of observation and the center of the source. The difference between and thus becomesBut because
and are all much less than , the square roots may be Taylor expandedTaylor series
In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....
, yielding, to first order,
which, after some algebraic manipulation, simplifies to
Now,
is the midpoint along the -axis between and , so gives us , one of the direction cosines to the sources. Similarly, . Moreover, recall that was defined to be the number of wavelengths along the -axis between and . SoSimilarly,
is the number of wavelengths between and along the -axis, soHence
Because
and are all much less than , . The differential area element, , may then be written as a differential element of solid angleSolid angle
The solid angle, Ω, is the two-dimensional angle in three-dimensional space that an object subtends at a point. It is a measure of how large that object appears to an observer looking from that point...
of
. Our expression for the mutual coherence function becomesWhich reduces to
But the limits of these two integrals can be extended to cover the entire plane of the source as long as the source's intensity function is set to be zero over these regions. Hence,
which is the two-dimensional Fourier transform of the intensity function. This completes the proof.
Assumptions of the theorem
The van Cittert–Zernike theorem rests on a number of assumptions, all of which are approximately true for nearly all astronomical sources. The most important assumptions of the theorem and their relevance to astronomical sources are discussed here.Incoherence of the source
A spatially coherent source does not obey the van Cittert–Zernike theorem. To see why this is, suppose we observe a source consisting of two points, and . Let us calculate the mutual coherence function between and in the plane of observation. From the principle of superposition, the electric field at isand at
isso the mutual coherence function is
Which becomes
If points
and are coherent then the cross terms in the above equation do not vanish. In this case, when we calculate the mutual coherence function for an extended coherent source, we would not be able to simply integrate over the intensity function of the source; the presence of non-zero cross terms would give the mutual coherence function no simple form.This assumption holds for most astronomical sources. Pulsars and masers are the only astronomical sources which exhibit coherence.
Distance to the source
In the proof of the theorem we assume that and . That is, we assume that the distance to the source is much greater than the size of the observation area. More precisely, the van Cittert–Zernike theorem requires that we observe the source in the so-called far field. Hence if is the characteristic size of the observation area (e.g. in the case of a two-dish radio telescopeRadio telescope
A radio telescope is a form of directional radio antenna used in radio astronomy. The same types of antennas are also used in tracking and collecting data from satellites and space probes...
, the length of the baseline between the two telescopes) then
Using a reasonable baseline of 20 km for the Very Large Array
Very Large Array
The Very Large Array is a radio astronomy observatory located on the Plains of San Agustin, between the towns of Magdalena and Datil, some fifty miles west of Socorro, New Mexico, USA...
at a wavelength of 1 cm, the far field distance is of order
m. Hence any astronomical object farther away than a parsec is in the far field. Objects in the Solar SystemSolar System
The Solar System consists of the Sun and the astronomical objects gravitationally bound in orbit around it, all of which formed from the collapse of a giant molecular cloud approximately 4.6 billion years ago. The vast majority of the system's mass is in the Sun...
are not necessarily in the far field, however, and so the van Cittert–Zernike theorem does not apply to them.
Angular size of the source
In the derivation of the van Cittert–Zernike theorem we write the direction cosines and as and . There is, however, a third direction cosine which is neglected since and ; under these assumptions it is very close to unity. But if the source has a large angular extent, we cannot neglect this third direction cosine and the van Cittert–Zernike theorem no longer holds.Because most astronomical sources subtend very small angles on the sky (typically much less than a degree), this assumption of the theorem is easily fulfilled in the domain of radio astronomy.
Quasi-monochromatic waves
The van Cittert–Zernike theorem assumes that the source is quasi-monochromatic. That is, if the source emits light over a range of frequencies,, with mean frequency , then it should satisfyMoreover, the bandwidth must be narrow enough that
where
is again the direction cosine indicating the size of the source and is the number of wavelengths between one end of the aperture and the other. Without this assumption, we cannot neglect compared to This requirement implies that a radio astronomer must restrict signals through a bandpass filter. Because radio telescopes almost always pass the signal through a relatively narrow bandpass filter, this assumption is typically satisfied in practice.
Two-dimensional source
We assume that our source lies in a two-dimensional plane. In reality, astronomical sources are three-dimensional. However, because they are in the far field, their angular distribution does not change with distance. Therefore when we measure an astronomical source, its three dimensional structure becomes projected upon a two-dimensional plane. This means that the van Cittert–Zernike theorem may be applied to measurements of astronomical sources, but we cannot determine structure along the line of sight with such measurements.Homogeneity of the medium
The van Cittert–Zernike theorem assumes that the medium between the source and the imaging plane is homogeneous. If the medium is not homogeneous then light from one region of the source will be differentially refractedRefraction
Refraction is the change in direction of a wave due to a change in its speed. It is essentially a surface phenomenon . The phenomenon is mainly in governance to the law of conservation of energy. The proper explanation would be that due to change of medium, the phase velocity of the wave is changed...
relative to other regions of the source due to the difference in light travel time through the medium. In the case of a heterogeneous medium one must use a generalization of the van Cittert–Zernike theorem, called Hopkins's formula.
Because the wavefront does not pass through a perfectly uniform medium as it travels through the interstellar
Interstellar medium
In astronomy, the interstellar medium is the matter that exists in the space between the star systems in a galaxy. This matter includes gas in ionic, atomic, and molecular form, dust, and cosmic rays. It fills interstellar space and blends smoothly into the surrounding intergalactic space...
(and possibly intergalactic) medium and into the Earth's atmosphere
Earth's atmosphere
The atmosphere of Earth is a layer of gases surrounding the planet Earth that is retained by Earth's gravity. The atmosphere protects life on Earth by absorbing ultraviolet solar radiation, warming the surface through heat retention , and reducing temperature extremes between day and night...
, the van Cittert–Zernike theorem does not hold exactly true for astronomical sources. In practice, however, variations in the refractive index
Refractive index
In optics the refractive index or index of refraction of a substance or medium is a measure of the speed of light in that medium. It is expressed as a ratio of the speed of light in vacuum relative to that in the considered medium....
of the interstellar and intergalactic media and Earth's atmosphere are small enough that the theorem is approximately true to within any reasonable experimental error. Such variations in the refractive index of the medium result only in slight perturbations from the case of a wavefront traveling through a homogeneous medium.
Hopkins' formula
Suppose we have a situation identical to that considered when the van Cittert–Zernike theorem was derived, except that the medium is now heterogeneous. We therefore introduce the transmission function of the medium,. Following a similar derivation as before, we find thatIf we define
then the mutual coherence function becomes
which is Hopkins's generalization of the van Cittert–Zernike theorem. In the special case of a homogeneous medium, the transmission function becomes
in which case the mutual coherence function reduces to the Fourier transform of the brightness distribution of the source. The primary advantage of Hopkins's formula is that one may calculate the mutual coherence function of a source indirectly by measuring its brightness distribution.
Aperture synthesis
The van Cittert–Zernike theorem is crucial to the measurement of the brightness distribution of a source. With two telescopes, a radio astronomer (or an infrared or submillimeter astronomer) can measure the correlation between the electric field at the two dishes due to some point from the source. By measuring this correlation for many points on the source, the astronomer can reconstruct the visibility function of the source. By applying the van Cittert–Zernike theorem, the astronomer can then take the inverse Fourier transform of the visibility function to discover the brightness distribution of the source. This technique is known as aperture synthesisAperture synthesis
Aperture synthesis or synthesis imaging is a type of interferometry that mixes signals from a collection of telescopes to produce images having the same angular resolution as an instrument the size of the entire collection...
or synthesis imaging.
In practice, radio astronomers rarely recover the brightness distribution of a source by directly taking the inverse Fourier transform of a measured visibility function. Such a process would require a sufficient number of samples to satisfy the Nyquist sampling theorem; this is many more observations than are needed to approximately reconstruct the brightness distribution of the source. Astronomers therefore take advantage of physical constraints on the brightness distribution of astronomical sources to reduce the number of observations which must be made. Because the brightness distribution must be real and positive everywhere, the visibility function cannot take on arbitrary values in unsampled regions. Thus, a non-linear deconvolution algorithm like CLEAN
CLEAN (algorithm)
The CLEAN algorithm is a computational algorithm to perform a deconvolution on images created in radio astronomy. It was published by Jan Högbom in 1974 and several variations have been proposed since then ....
or Maximum Entropy may be used to approximately reconstruct the brightness distribution of the source from a limited number of observations.
Adaptive optics
The van Cittert–Zernike theorem also places constraints on the sensitivity of an adaptive opticsAdaptive optics
Adaptive optics is a technology used to improve the performance of optical systems by reducing the effect of wavefront distortions. It is used in astronomical telescopes and laser communication systems to remove the effects of atmospheric distortion, and in retinal imaging systems to reduce the...
system. In an adaptive optics (AO) system, a distorted wavefront is provided and must be transformed to a distortion-free wavefront. An AO system must make a number of different corrections to remove the distortions from the wavefront. One such correction involves splitting the wavefront into two identical wavefronts and shifting one by some physical distance
in the plane of the wavefront. The two wavefronts are then superimposed, creating a fringe pattern. By measuring the size and separation of the fringes, the AO system can determine phase differences along the wavefront. This technique is known as "shearing."The sensitivity of this technique is limited by the van Cittert–Zernike theorem. If an extended source is imaged, the contrast between the fringes will be reduced by a factor proportional to the Fourier transform of the brightness distribution of the source. The van Cittert–Zernike theorem implies that the mutual coherence of an extended source imaged by an AO system will be the Fourier transform of its brightness distribution. An extended source will therefore change the mutual coherence of the fringes, reducing their contrast.
See also
- Degree of coherence
- Coherence theoryCoherence theoryIn physics, coherence theory is the study of optical effects arising from partially coherent light and radio sources. Partially coherent sources are sources where the coherence time or coherence length are limited by bandwidth, by thermal noise, or by other effect...
- VisibilityVisibilityIn meteorology, visibility is a measure of the distance at which an object or light can be clearly discerned. It is reported within surface weather observations and METAR code either in meters or statute miles, depending upon the country. Visibility affects all forms of traffic: roads, sailing...
- Hanbury Brown and Twiss effect
- Bose-Einstein correlationsBose-Einstein correlationsIn physics, Bose–Einstein correlations are correlations between identical bosons. They have important applications in astronomy, optics, particle and nuclear physics.- From intensity interferometry to Bose–Einstein correlations :...