Focusing of Partially Coherent Light
The focusing of fully coherent waves has been studied extensively over a period of many years.1 However, much less work has been done on the focusing of partially coherent fields. In fact, no satisfactory theory covering this subject exists. In the present paper we introduce a simple model for the analysis of problems of this kind.
Since fluctuating fields do not have well-defined phase fronts, the focusing of such wave-fields is best formulated in terms of the second-order coherence theory, which is concerned with correlations of the field variable at two space-time points, rather than with the field itself. We generalize the classic Debye theory2 of focusing of monochromatic waves to focusing of fluctuating fields and express the second-order coherence function of the field as a superposition of partially correlated pairs of plane waves. The correlation of the plane-wave modes is characterized by the so-called angular correlation function,3 whose precise form depends on the second-order coherence properties of the field. As an example, we consider two-dimensional focusing through a random phase screen, with Gaussian statistics.
We show that in systems of large Fresnel numbers the partially coherent field has simple symmetry properties with respect to the geometrical focus. Graphs are presented which illustrate the behavior of the light intensity in the focal region and we elucidate the changes in the focal depth and the focal spot size as the phase-screen parameters are varied. A fuller account of the analysis and the results, with illustrations, will be given in a forthcoming paper.
- 1.For a review of this subject see, for example, J.J. Stamnes, “Waves in Focal Regions,” Adam Hilger, Bristol (1986).Google Scholar
- 2.P. Debye, Ann. d. Physik 30:755(1909). An account of Debye’s theory in English is given in A. Sommerfeld, “Optics,” Academic Press, New York (1954), Sec. 45.Google Scholar
- 3.E.W. Marchand and E. Wolf, J. Opt. Soc. Am. 62: 379 (1972).Google Scholar