# Temporal and spatial coherence

The interference between two beams can exist only if the frequency of the two beams remains constant in a certain period of time, this is indicated by the term temporal coherence. This condition is usually satisfied only if the two waves are generated by the same electronic transition in a particular atom. The time interval in which the characteristics remain constant, the time of coherence tco, is related to the frequency range of the emitted light by the equation

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t x = co Af fA

The coherence length £co = ctco = X2/AX limits the difference in path lengths of the two beams for which interference can still be generated. The light originated from a light source with thermal spontaneous emission, passed through a filter with an interferometric bandwidth, has a coherence length of the order of mm; for a laser, the coherence length is of the order of 1m (£co = c Af = 2L).

It must also be ensured that the light emitted from two different points of a source of finite dimensions (width d) and a certain divergence angle & can still interfere (spatial coherence). The answer is provided by the classical Young experiment: the necessary condition for the spatial coherence between the two beams is dsin© << X/ 2.

The high degree of spatial coherence of laser light is shown by the extreme parallelism of the beam of the laser. Because of the laws of diffraction, no laser beam is perfectly parallel: the slightest difference for a given diameter is obtained with a laser beam with a Gaussian profile (mode TEM00) for which the opening angle is in the order of mrad. In a laser beam that has an initial diameter of 0.8 mm and an angle of divergence of 1.1 mrad, the beam diameter is about 110 mm at a distance of 100 m.

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