Define coherence and explain temporal and spatial coherence. Show that visibility is a maxim of coherence. Define the 'Q' factor for a spectral line. Explain coherence length and coherence time. Show that visibility is a measure of degree of coherence. 

An important consideration for the study of lasers is the interaction of two electromagnetic waves that have only slightly different frequencies of that originate from points of small separation. Spatially, e.g., two close located but separate laser beams of a single beam illuminating two closely positioned apertures. In these cases, the two beams will interfere and give some important effects like longitudinal modes, mode locking, phase matching and frequency multiplication etc. When complete is not achieved still some interference takes place, which is known as partial interference. Partial interference condition can be grouped into two categories. (1) Temporal coherence associated with frequency difference of two waves. It is also referred to as longitudinal coherence. (2) Spatial coherence associated with location of two waves. It is also referred to the transverse coherence.

Temporal coherence   It refers to the relative phase or coherence of the two waves at two separate locations along the propagation direction of the two beams. If we assume that the two beams ate exactly in phase at initial location then they will be at least partially in phase in some other location up to some length say which is called coherence length and may be found as where represents the wavelength difference and corresponds to their average wavelength.

Spatial Coherence   Spatial coherence is also referred to as transverse coherence and it describes, how far apart two sources or two portion of the same source can be located in a direction transverse to the direction of observation and still exhibit coherent properties over a range of observation points. This is sometimes called lateral coherence. In other words, it means that the distance 1 by which two points are separated in transverse direction and still interference effects are observed at Specific distance from the source. If two sources be separated by a distance S in transverse direction to the direction of observation and distance r from point of observation. Temporal coherence can be studied from the Michelson's interferometer by moving one of the    mirrors. In the case of an interferometer, the visibility V of the fringes is a measure of degree of coherence. In the case of Michelson interferometer, path difference is adjusted to be zero to get maximum visibility. From this position, one of the mirrors is moves to a position where no triangle are observed. Then distance between these positions gives us the maximum range over which fringes can be observed. The time is the time taken by the wave to travel a distance 2d and v₁, v₂ are the frequencies of the two waves. The distance for which the field remains sinusoidal is called coherence length and is given by where c is the velocity of light. Now if there will be a definite phase relationship between  two interfering beams and if 2d, no definite phase relationship between the beams will be observed, so starting with equal separation of mirrors from the silvered glass plate, if now the path difference is continuously increased, the dark and bright fringe contrast goes on reducing till the fringes disappear. The path difference at disappearance of fringes given us an estimate of coherence length.                                                    

Visibility as a Measure of Coherence   The measure of contrast of fringes is called the fringe visibility which serves as a useful measure of coherence according to Michelson, the visibility of fringes is defined as where is the relative energy of a bright fringe and the relative energy of a neighboring dark fringe. If the fringes are produced by coherent by coherent beams of equal amplitude, the visibility of fringes is unity, while that of the fringes produced by non-coherent beams is zero i.e. no fringes. Impractical the visibility of the fringes is less than unity, even with the waves of equal amplitude. The ability of light waves to produce interference is measured by the degree of coherence of light waves. Higher the degree of coherence higher is the probability that the waves produce a contrast interference pattern. Hence the degree of coherence, V is to be equal to the visibility of fringes when the path difference between the beams is small and the amplitudes are equal. In Young experiment the fringe visibility can be taken as a direct measure of the degree of coherence conventionally, if V>0.85, the two secondary sources are said to be highly coherent.