Coherence (physics)
Coherence ( from Latin: cohaerere = related ) called in physics the property of waves to follow the dynamic development of a common set rule. The coherence is also defined as the set of all correlation properties between shaft sizes. A particularly strong form of consistency is when stationary interference phenomena are visible in the superposition of waves. Not available a desired form of coherence, it is called in the context of incoherence. Often the presence of coherence suggests a common genesis of the waves.
- 3.1 Coherence and contrast of an interferogram
- 3.2 Temporal coherence 3.2.1 Wiener- Khinchin theorem
- 3.2.2 Intuitive Explanation of temporal coherence by finite wave trains
- 3.3.1 Van - Cittert - Zernike theorem
- 3.5.1 Temporal coherence
- 3.5.2 Spatial coherence
Detailed description
All physical waves such as light waves, radar waves, sound waves or water waves may be to other shafts in a certain way coherent, or it can be coherent between corresponding half-shafts. Cause of coherence may be a common genesis of the waves. For example, if the wave generation, the same causal mechanism underlying lay, constant vibration patterns can arise in the wave train, which can be made visible in a comparison of partial waves later. If the wave amplitudes of two waves is directly correlated with each other, so this is shown in the superposition of the waves in the occurrence of stationary ( spatial and time invariant ) interference phenomena. In other cases, in part, a technically more effort or a more complicated mathematical consideration of the waveform is necessary to demonstrate a consistency in the waves.
In simple cases, such as periodic waves, two partial waves are coherent if a fixed phase relationship to each other. In optics, this phase relationship often means a consistent difference between the phases of the oscillation period. Partial waves, which overlap at a fixed location to a specific ( time-averaged ) intensity (for example, on a viewing screen ), can then be dependent on the phase relationship either enhance or extinguish (full coherence), increase a little or weaken (partial coherence) or to an average intensity offset ( incoherence ). Incoherence is here especially at different frequencies, when all of the phase differences equal to occur frequently and thus no constructive or destructive interference is possible.
On the other hand, can also waves with different frequencies have a coherence to each other. Technically, this type of consistency, plays a role in the frequency comb, or in the radar art. This coherence is generated by mode coupling or frequency doubling or multiplication.
In wave fields, one can distinguish the cases of a temporal and a spatial coherence, although using both forms of coherence must be present. Temporal coherence is present when (often visually identified with the spatial axis parallel to the propagation direction), there is a fixed phase difference along the time axis. Spatial coherence is present when ( often reduced to the propagation direction of the spatial axes perpendicular ) is along a single axis, a fixed phase difference.
Mathematical representation
Coherence and correlation
The time required for the interference ability consistency in waves can be quantified using the correlation function. This function provides a measure of the similarity of the time course of two placed on the connecting shaft amplitudes.
The function
First defines the (complex) cross-correlation function between the time courses of two amplitudes considered one. The two amplitudes are compared to the location points A and B of the shaft and at a time difference of singled out and as a function of time.
The contrast function for spatio-temporal coherence, by
Is given, then directly delivers the strength of coherence as a value between 0 and 1, In general, there are three cases:
In the case of purely temporal coherence only correlations with A = B are considered. Here delivers the contrast function for temporal coherence
The strength of the temporal coherence function of the time interval. has the maximum value at 1 and falls depending on the consistency more or less rapidly to 0 off. The coherence time is defined as the time interval in which the contrast function is decreased to 1 / e. If the coherence between different waves are calculated, the cross-correlation function is
The shafts and used.
In the case of purely spatial coherence only correlations are considered with. Here delivers the contrast function for spatial coherence
The strength of the spatial coherence between the points A and B. A volume in which all the pairs of points A, B have a contrast, forms a so-called coherence volume within the spatial coherence is present. Most often understood transverse to the propagation direction of the wave, the term spatial coherence only the coherence, which would have to be more specific designated transverse spatial coherence. The spatial coherence along the propagation direction, ie, the longitudinal spatial coherence on the other hand is often equated with the temporal coherence, which is only approximately correct.
Multiple-beam interference
The mathematical definition of consistency shown only describes the correlation between two points of a wave. In many applications, however, the condition must be met that many partial waves can be overlapped to a common interference pattern. The pairwise coherence of the wave alone is not sufficient. The coherence of this term must be extended or combined with additional conditions.
In the example of a diffraction grating in the optical system in which the need to interfere with a very large number of partial waves, the pair-wise spatial coherence is not enough to be let to sharp diffraction spectra visible. In addition, a simultaneous correlation between the phases of all partial waves must be present for the pairwise interference-capable partial beams come in a common diffraction maximum on the screen to cover. This condition is particularly satisfied if a plane wave fronts meet at a plane diffraction grating. Two other applications where multi-beam interference plays a role are the Bragg reflection, and the Fabry-Perot interferometer.
Coherence in classical optics
In classical optics consistency is placed in direct connection with the ability of the interference light. The contrast of the interference pattern V ( engl. Visibility) is a measure of the coherence of the light. In particular, in optics, the two special cases of spatial and temporal coherence play a major role.
Coherence and contrast of an interferogram
Means in the optical coherent interference ability with respect to a particular experiment and can be connected to the contrast of the interference pattern, the maximum of 1 ( fully coherent light ) and a minimum of 0 (fully incoherent light ), associated. The interference pattern of two light sources depends on their complex mutual coherence function and the complex mutual degree of coherence, and the contrast
For two-beam interference of a wave with their spatially and temporally shifted copy of the well-known two-beam interference formula gives
Temporal coherence
Light is produced from discontinuous emission files that emit photon wave trains. These wave trains are each connected to a regularly oscillating field that randomly changes its phase. " This interval, in which the optical wave is a sine wave, is a measure of their temporal coherence. " The coherence time is therefore defined by the average time interval in which the light wave oscillates in a predictable manner. A higher coherence time corresponds to a higher temporal coherence of a light -emitting source.
Temporal coherence is necessary when the wave to a time-shifted copy of itself to be coherent. This is for example the case when in a Michelson interferometer, the path lengths of the object and reference arm have different lengths. The time after which have significantly changed the relative values of phase and / or amplitude ( so that the correlation decreases to a great extent ) is defined as the coherence time. When the consistency is perfect yet, but it has decreased significantly after time. The coherence length is defined as the distance traveled by the wave within the coherence time.
Wiener- Khinchin theorem
In a light source, the coherence time is determined by the spectral composition of light. Light from a monochromatic light source is temporally completely coherent. Light, which is composed of different wavelengths (eg due to Doppler broadening ) - depending on the type of composition - partially coherent or incoherent. This relationship is described by the Wiener- Khinchin theorem, which states that the degree of coherence corresponds (as autocorrelation function of the field strength) of the normalized Fourier transform of the light spectrum. The coherence length of the light is defined as the point at which the degree of coherence is dropped on.
The relationship between the spectrum of the light source and the temporal coherence can be illustrated using the example of the Michelson interferometer. If tilting the reference mirror of the path length difference of the two beams is linearly dependent on the tilt direction. Corresponding to the path length to an integer multiple of the wavelength, the rays interfere constructively, and the interference pattern has a maximum. When monochromatic light of a stripe pattern on the screen is visible.
The light has various wavelengths, the individual stripes are shifted from each other. The strips are all the wider the longer the wavelength is. The superposition of the stripe pattern on a viewing screen, the strips in some places cancel each other out or reinforce each other ( partial coherence ). The return of the contrast can not be explained in view of the finite-length wave trains.
If we calculate according to the Wiener- Khinchin theorem, the coherence function for the case of a laser with a Gaussian spectrum (bandwidth FWHM = centroid wavelength ), we obtain a Gaussian coherence function with the coherence length.
Depending on the shape of the spectrum ( in the above case of the Gaussian spectrum, for example, does not change, but, for example, for a beat, in which the autocorrelation function is periodic ) - - even for larger path length as again a high degree of coherence from the Fourier transform that directly follows can be achieved. This property of coherence can not be explained in the intuitive picture of the finite-length wave trains (see below).
Intuitive Explanation of temporal coherence by finite wave trains
"Natural " light is generated when an electron passes into an atom from an excited in a less excited state. The decay of the excited state oscillates in semiclassical idea the electron a certain time. During this time (= life) it will emit a photon ( damped oscillation ). Typical lifetimes of such atomic processes ( = coherence time ). This leads to wave packets with a length of ( = coherence length ) with a frequency of focus of about 100 MHz.
The resulting light is the sum of wave packets together, who were sent out by many different atoms and differ in phase and in frequency. Since the atoms are mostly in thermal motion, the light emitted from light atoms such Doppler broadening, with strong mutual interaction shows (eg shocks) of the atoms and pressure broadening. Both effects reduce the coherence time or length of the emitted light substantially.
The model of finite wave trains can not explain all aspects of the temporal coherence! It therefore serves only as an auxiliary notion in very simple cases.
Spatial coherence
If the wave interfere with a spatially shifted copy of their own, spatial coherence is necessary. This is the case for example in the young between double slit experiment: Here are the two column two points from the incident wave singled out and brought to interference. How far these two points may lie apart, describes the extent of the area of the spatial coherence.
Van Cittert - Zernike theorem
With an extended light source having a random phase distribution, i.e. applicable for LED light bulb and the gas discharge lamp but not for laser, the spatial coherence is determined by the extent and shape of the light source. It 's more about the angular extent than the actual expansion, so that the spatial coherence therefore increases with increasing distance. A point light source has a full spatial coherence even at close range. This relationship is determined by the Van Cittert - Zernike theorem - described, which states that the complex degree of coherence of the normalized Fourier transform of the intensity distribution of the light source corresponds to ( conditions - by Pieter Hendrik van Cittert ( 1889 to 1959 ) and Frits Zernike: small stretches of light source and the observation area, sufficiently large observation distance ). For a circular light source, the spatial coherence drops rapidly and reaches its minimum in dependence on the distance of the observation screen by the light source. After that, the coherence is not lost, but is eligible for greater distances ( in a very weak form) again.
The relationship between extent of the light source and spatial coherence can be illustrated by the example of the double-slit interference experiment. At the observation screen is created depending on the time differences of the two beams an interference pattern. For this purpose, a sufficiently high temporal coherence of the light beams is required. For the point of the observation screen, which lies between the two columns, the light rays have no runtime difference. Here, the interference pattern has the zeroth maximum. In an extended light source, the point is slightly shifted with time difference equal to zero for each point of the light source. The individual interference pattern blur each other depending on the size of the light source.
Generating coherent light
Coherence is not a property of a light source, but the light beams, because the interference ability of the light may change during its propagation.
If one sends spatially non-coherent light through a very narrow gap, the light behaves behind it, as if the gap a point light source ( in one dimension ), the elementary waves emits (see Huygens' principle). The size of the spatial coherence area is in the case of a simple gap inversely proportional to the gap size ( van Cittert - Zernike theorem, Verdet coherence condition ). With increasing distance from the light source, the angular extent of the light source decreases and thus, the spatial coherence.
The temporal coherence of the light can be increased by inserting a wavelength filter, which limits the spectrum of the light source.
The choice of light source is crucial for the coherence. Fluorescent, incandescent and gas discharge lamps are spatially extended light sources ( spatially incoherent ), ( temporally incoherent ) that produce white light from a large amount of different frequencies. Thus, by pinhole and wavelength filter from spatially and temporally coherent light are generated, while maintaining the remaining intensity of the light is greatly reduced, so that this method is not very practical.
Laser light, however, is considered the best producible monochromatic light at all and has the largest coherence length ( up to several kilometers ). A helium -neon laser, for example, produce light with coherence lengths in excess of 1 km. However, not all laser monochromatic ( eg titanium - sapphire laser Δλ ≈ 2 nm - 70 nm). LEDs are less monochromatic ( Δλ ≈ 30 nm) and therefore have shorter coherence times than most monochromatic lasers. Since a laser is generally of the same phase over its entire output aperture of time, the emitted laser light has also a very high spatial coherence.
Measurement of the coherence
Temporal coherence
One can determine the coherence time and coherence length of a light wave by splitting them into two sub-beams and unites them again later - as in a Michelson interferometer or Mach- Zehnder interferometer. Interference phenomena can be seen in such an arrangement, only when the delay time difference and the path difference between the partial waves is less than the coherence time and coherence length of the light emitted by the atoms of wave trains.
And from the measurement of the spectrum can be explained by Fourier transform to determine the temporal coherence. Conversely, the spectrum of a light source are determined by the interference contrast is measured in a Michelson interferometer, while the path length is varied (FTIR ) spectrometer.
Spatial coherence
Similarly to the case of the temporal coherence, the spatial coherence can be determined by measuring the contrast of an interference pattern, if an interferometer is used that is sensitive to the spatial coherence ( of the related double-slit structure ). Wherein the angular extent of Stellarinterferometrie stars is determined by measuring the contrast of the spatial coherence.
Coherent vs.. incoherent superposition in quantum mechanics and statistical physics
With coherent superposition of states ( superposition of the field amplitudes ) one has to do it also in quantum mechanics, although the relationship with the measured variables is complicated: A quantum mechanical state vector, interpreted as an ensemble of probability amplitudes ( more precisely: their densities ) defined by a complex-valued function of position be shown, can be superposed linearly in an arbitrary orthonormal basis with complex constants, although the measurement probabilities themselves depend quadratically on (eg applies to the probability in a small volume the following statement: ). The linear Superponierbarkeit states that it is also true, ie ( The index * denotes here the complex conjugate size.) The probability thus depends quadratically (more precisely, bilinear) of the from, although the states themselves linear ( i.e. coherent) superimposed be. The aspects discussed here are exploited in quantum computers.
In general, the quantum mechanical expectation value of a measured variable, which is represented by a self-adjoint operator, given by the following formula (wherein, the terms in brackets the quantum mechanical scalar product on which at this point can not be discussed ): Although this expression does not depend linearly on, is the coherent states Superponierbarkeit of the essentials: the off-diagonal elements give, in general, equal significant contributions to the result as the diagonal elements.
In contrast, in statistical physics, including quantum statistics, averaging a priori incoherent ( superposition of field intensities). Here it is assumed probabilities that the quantum mechanical state of the system is in the state. The statistical expectation values are accordingly
With and
There are so really not the states weighted, but the expectation values themselves (that is, not the " amplitudes ", but the " intensities " ), in contrast to the following paragraph does not occur off-diagonal processes. The associated entropy - an important physical and information theoretic size - this does not disappear. ( In contrast, quantum statistical states " pure" if the following holds: .. A, for all other (eg ) The entropy S vanishes, like this is the so-called Boltzmann constant. )
The breaking point between the coherent physics of quantum mechanics and the incoherent physics of quantum statistics lies in a subtle section of the advanced quantum mechanics exactly in the above- described " non-diagonal processes," namely, where the " time -dependent perturbation theory, " the transition rates in the lowest relevant approximation as proportional prove to the intensity squares of the time-dependent perturbation and the higher terms are neglected. This is the case of temporally incoherent (quasi- " random " ) disturbance appropriate, but for example in laser radiation is generally not useful.
The qualitative difference between final and initial state of the system, coherent radiation field versus random initial state is here so neglected precisely as "higher approximation ", as is done in the formulas of statistical physics, where no distinction is basically between initial and final states. The transition from the reversibility of quantum mechanics (or classical counterparts, the so-called Hamiltonian mechanics) irreversibility general operations of statistical physics is to locate exactly at this point ( with significant direction of the arrow, for example, with the reduced Planck constant, the Dirac function, the energies of the initial or final state. and the angular frequency of the assumed to be monochromatic interference ).
Statistical methods in which, for example, signal amplification is based on the end of an optical fiber, so detrimental to the consistency, which, inter alia, leads to the range limit of quantum cryptography, which can be currently performed only on distances up to about 100 km, while the methods of classical computer science are virtually unlimited in scope.