As shown on previous pages, most available light sources emit a range wavelengths. For many photonic applications it is desirable to have a "monochromatic" (ideal) point light source. An ideal light source would emit only one exact frequency f₀ and its physical size would be infinitely small (an ideal point source). Such a source cannot exist in reality. Every real source of light has some emission uncertainty which appears as linewidth Df. Also, every real light source has some (non-zero) physical size.
 

Coherence  is a concept that that establishes the limits within which a real light source can be considered ideal.

There are two types of coherence:

(1) Temporal Coherence (related to the emitted linewidth)

(2) Spatial Coherence (related to the physical size of the source).
 

Most quasi-monochromatic sources of light have spectral intensity profile that can be approximated by a Gaussian curve. Temporal coherence is determined by the coherence length Lc which depends on the linewidth Dl (wavelength uncertainty within FWHM of relative irradiance) of central wavelength l0. It can also be expressed in terms of frequency bandwidth (uncertainty) Df of the central optical frequency f0.   (Note that when Df << f0 and  Dl << l0 , then  Df/f0= Dl /l0). In free space, coherence length  Lc = c/Df = l02 / Dl. Coherence length  Lc is the distance in the direction of wavefront propagation within which the amplitude and phase of the wave can be considered well defined, predictable, and therefore subject to possible wave interference. An ideal light source would have an infinite coherence length. On the other hand, thermal sources of light (such as the sun or a light bulb) cover a relatively broad range of wavelengths Dl (see Planck's radiant function) and therefore have very small coherence length. For that reason, under normal lighting conditions, we can observe interference only in very thin regions, such as the thickness of a soap bubble.

Note that in a dielectric medium with refractive index n, the speed of light, wavelength, and therefore also the coherence length, is decreased by the factor n
thus L_c= c/(nDf) = l² / (nDl).
 
 

Michelson interferometer is an instrument in which a collimated beam of light is split into two beams travelling separate paths (1) and (2), and then reassembled as (3). If the path difference DL = 2L = path(2) - path(1)  is smaller than the coherence length Lc, interference can be observed. If  DL > Lc  the interference pattern disappears (the phase relation between the reassembled beams no longer exists). We can thus determine the coherence length Lc = l02/Dl by increasing the distance (2) and finding the path difference DL = 2L at which the interference pattern disappears. From this measurement we can determine the linewidth Dl and the quality of the quasi-monochromatic light source l0.

With increasing distance L the interference pattern fades

Spatial Coherence

Spatial coherence is determined by the coherence width
W_c = k (l/D) R, where k is a constant dependent on the shape of the source (for a circular source k = 1.22), D is the approximate diameter of the source and R is the distance from the source. Coherence width is the distance along the wavefront (perpendicular to the direction of propagation), within which the amplitude and phase of the wave can be considered well defined and therefore predictable. An ideal light source would be a point (D = 0) generating ideal spherical wavefronts. An ideal point source would have infinite coherence width.

The degree of spatial coherence can be estimated by inspection of a shadow cast by an illuminated object. The sharper the shadow, the better spatial coherence of the source.

Idealized improvement of coherence from an incoherent light source

Click: Additional Info on the Coherence of Light