Scattering from a Point Source in Plane Clouds
JOSA, Vol. 46, Issue 11, pp. 927-932 (1956)
http://dx.doi.org/10.1364/JOSA.46.000927
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Abstract
A previously proposed modified form of diffusion theory [Phys. Rev. 100, 517 (1955)] is applied to problems involving point sources and plane clouds over an absorbing ground. A general but formal solution is first presented. Some analytic results are then obtained for a point source at the edge of a semi-infinite (half-space) cloud; specific formulas are given for the photon density at the edge of the cloud and for the net photon current there; asymptotic results are obtained for large distances inside the cloud along the source axis. Finally, numerical results are presented for the photon density at the mid-plane of an infinite plane cloud with thickness equal to two (transport) mean free paths and a point source at the center. An appendix summarizes analytic results of the modified theory for: a point source in an infinite medium, a point source at the center of a finite sphere, the albedo and transmission of an infinite plane slab with parallel (";solar";) illumination at an arbitrary angle, the albedo of the combination of a plane cloud over a ground plane with arbitrary albedo, and a rough analysis of ambient light levels in the ocean.
Citation
PAUL I. RICHARDS, "Scattering from a Point Source in Plane Clouds," J. Opt. Soc. Am. 46, 927-932 (1956)
http://www.opticsinfobase.org/josa/abstract.cfm?URI=josa-46-11-927
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References
- Paul I. Richards, Phys. Rev. 100, 517 (1955).
- G. N. Watson, Theory of Bessel Functions (Cambridge University Press, Cambridge, 1944), second edition.
- A. Erdelyi, editor, Tables of Integral Transforms (McGraw-Hill Book Company, Inc., New York, 1954), Vols. I and II.
- W. Magnus and F. Oberhettinger, Functions of Mathematical Physics (Chelsea Publishing Company, New York, 1954).
- D. V. Widder, Advanced Calculus (Prentice-Hall, Inc., New York, 1947).
- It is understood on the right-hand side that ƒ[(t^{2}-y^{2})^{½}]=0 for t<y.
- In this and subsequent manipulations the following identity is frequently useful:(a^{2}+b^{2})^{½}-a=b^{2}[(a^{2}+b^{2})^{½}+a]^{-1}.
- The notation O(g) denotes a quantity whose absolute values remains less than some constant times the absolute value of g.
- Diffusion theory applied to the same problem gives ρ=0 unless the "extrapolated end point" is used, in which case the result corresponding to expression (31) is ρ~3/2r^{3}, using the standard extrapolation distance (0.7104λ).
- G. Doetsch, Theorie und Anwendung der Laplacetransformation (Dover Publications, New York, 1945), p. 231.
- Diffusion theory, using the 0.7104λ extrapolated end point, gives 4.26/z^{2} in near agreement with Eq. (33).
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