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The amount of coherence can readily be measured by the interference visibility, which looks at the size of the interference fringes relative to the input waves (as the phase offset is varied); a precise mathematical definition of the degree of coherence is given by means of correlation functions. More broadly, coherence describes the ...
Coherence functions, as introduced by Roy Glauber and others in the 1960s, capture the mathematics behind the intuition by defining correlation between the electric field components as coherence. [3] These correlations between electric field components can be measured to arbitrary orders, hence leading to the concept of different orders or ...
The coherence of a linear system therefore represents the fractional part of the output signal power that is produced by the input at that frequency. We can also view the quantity 1 − C x y {\displaystyle 1-C_{xy}} as an estimate of the fractional power of the output that is not contributed by the input at a particular frequency.
[12] [13] [clarification needed] After calculating the cross-correlation between the two signals, the maximum (or minimum if the signals are negatively correlated) of the cross-correlation function indicates the point in time where the signals are best aligned; i.e., the time delay between the two signals is determined by the argument of the ...
This is the approach on which quantum optics is based and it is only through this more general approach that quantum statistical coherence, lasers and condensates could be interpreted or discovered. Another more recent phenomenon discovered via this approach is the Bose–Einstein correlation between particles and antiparticles [citation needed].
The perfect coherent state has all n-orders of correlation equal to 1 (coherent). It is perfectly coherent to all orders. The second-order correlation coefficient () gives a direct measure of the degree of coherence of photon states in terms of the variance of the photon statistics in the beam under study. [13]
For example, an electrical utility may produce less power on a mild day based on the correlation between electricity demand and weather. In this example, there is a causal relationship, because extreme weather causes people to use more electricity for heating or cooling.
With any number of random variables in excess of 1, the variables can be stacked into a random vector whose i th element is the i th random variable. Then the variances and covariances can be placed in a covariance matrix, in which the (i, j) element is the covariance between the i th random variable and the j th one.