In Chapter II, the freespace propagation of the coherence properties of partially coherent optical fields was discussed.
Initially, the propagation of the mutual intensity function,
ri 2(0),
was shown to be governed by a generalised form of the Huygens-Fresnelintegral, derived by both Zernike [1] and Hopkins [2]. When normalised by the local intensity, the mutual intensity function gives the degree of spatial coherence, Y 12- Then, in the general formulation of
coherence theory due to Wolf [3], this same law was presented in an extended form to include path length differences, thus applying to the propagation of the mutual coherence function,
r i 2(x).
Finally, the two wave equations derived by Wolf [3] that rigorously govern thepropagation of the mutual coherence function were presented, with an outline of their derivation. It was also shown that in the appropriate quasimonochromatic approximation, when Av |t | << 1, the mutual coherence function reduces to the mutual intensity function.
In Chapter III, it was noted that the wave equations were used by Parrent and Skinner [4] to investigate the far-field intensity
distribution in the diffraction pattern of a slit illuminated by
partially coherent light. Other analyses of the intensity distributions in the diffraction patterns of apertures illuminated by partially
coherent fields were also described. Most of these analyses have a basic similarity to the Huygens-Fresnel integral, and so could be used
to examine the coherence properties of the far-field, as was demonstrated in the study by Som and Biswas [5].
All of these analyses are for propagation in an unbounded and homogeneous medium- Hopkins' investigations of image formation in
coherent light (e.g. [6]) took account of finite apertures of an optical system, and in [21, the light was allowed to have traversed regions of various refractive indices in the analysis leading to the definition of the coherence function. However, specific effects of an optical system, or of some boundary or discontinuity in the medium, on the propgation of the coherence of the optical field were not examined.
The first example of an analysis that set out to find the effect of finite apertures in an optical system was that of Wolf in 1963
[7]. In this analysis he showed that the very high spatial coherence of laser light was not due to the stimulated emission of the light, but to the effective repetition of the diffraction of the light at the cavity mirrors and of the propagation of the field between them. An
interpretation of the result was that sufficient such repetitions singled out one mode of the optical system. In the case studied, the system was a sequence of equidistant plane apertures, and the initial field was the partially coherent light incident on the "first" mirror. The conclusion was reached that sufficient transmissions through any periodic structure would render even an incoherent, quasimonochromatic
field spatially totally coherent. The stimulated emission was seen as a mechanism that made up for losses during the transmissions of the field. As the investigation was of spatial coherence only, a quasimonochromatic field was considered, but it was thought that a more general analysis based on the mutual coherence function would give essentially the same
results.
A later study by Allen, Gatehouse and Jones [8] involving amplified spontaneous emission (which occurs in the superluminescent diode, cf. Chapter VI) showed again that a boundary in the medium through which the light passes can affect the coherence of the light.
In their analysis, they directed spatially incoherent, but essentially monochromatic, light through tubes of differing internal reflectivities and bores. The results showed that a small bore tube with a high
reflectance could markedly increase the spatial coherence of the emerging light. Lower reflectivities, larger bores, or short lengths caused smaller improvements over the freespace value. The conclusion
7.2
15 3reached was that images of the source, in the tube walls, increased the effective source size, thus increasing the degree of coherence of the
field. This was confirmed using the van Cittert-Zernike theorem, including cross terms, in amplitude, to account for the fact that virtual images were not independent incoherent sources.
The propagation of coherence along optical fibres has been examined (see e.g. [9-15]), but with most stress upon the temporal
coherence. This has a direct bearing on the pulse distortion introduced by mode dispersion, and is discussed at the end of the next section. A
general study of the propagation of partially coherent light through Selfoc fibres [11] gave the rather misleading result that the original coherence (or rather, incoherence) of the source is repeated
periodically along the fibre. This was a consequence of using an infinite parabolic graded index medium to represent the fibre. Hence, all radiation from the source into the "fibre" eventually returned to the axial region, exactly imaging the source. The two wave equations for the propagation of the mutual coherence function [3] and a modal expansion based on the infinite medium were used.
In the next section, the propagation of spatial coherence along an optical fibre is investigated, and the phenomenon of coherence enhancement, mentioned briefly in the study of temporal coherence
effects in [9], is clearly brought out.