Degree of coherence

In quantum optics, correlation functions are used to characterize the statistical and coherence properties of an electromagnetic field. The 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 a Michelson or other linear optical interferometer. The correlation between pairs of fields, , typically is used to find the statistical character of intensity fluctuations. First order correlation is actually the amplitude-amplitude correlation and the second order correlation is the intensity-intensity correlation. It is also used to differentiate between states of light that require a quantum mechanical description and those for which classical fields are sufficient. Analogous considerations apply to any Bose field in subatomic physics, in particular to mesons (cf. Bose–Einstein correlations). The normalized first order correlation function is written as: where denotes a (statistical) ensemble average. For non-stationary states, such as pulses, the ensemble is made up of many pulses. When one deals with stationary states, where the statistical properties do not change with time, one can replace the ensemble average with a time average. If we restrict ourselves to plane parallel to each other waves then . In this case, the result for stationary states will not depend on , but on the time delay (or if ). This allows us to write a simplified form where we have now averaged over t. In optical interferometers such as the Michelson interferometer, Mach–Zehnder interferometer, or Sagnac interferometer, one splits an electric field into two components, introduces a time delay to one of the components, and then recombines them. The intensity of resulting field is measured as a function of the time delay. In this specific case involving two equal input intensities, the visibility of the resulting interference pattern is given by: where the second expression involves combining two space-time points from a field.