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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Editor: Stephen A. Burns
  • Vol. 24, Iss. 8 — Aug. 1, 2007
  • pp: 2363–2371

Ray propagation in nonuniform random lattices. Part II

Anna Martini, Renzo Azaro, Massimo Franceschetti, and Andrea Massa  »View Author Affiliations

JOSA A, Vol. 24, Issue 8, pp. 2363-2371 (2007)

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In this paper and its companion [ J. Opt. Soc. Am. A. 23, 2251 (2006) ], the problem of ray propagation in nonuniform random half-plane lattices is considered. Cells can be independently occupied according to a density profile that depends on the lattice depth. An electromagnetic source external to the lattice radiates a monochromatic plane wave that undergoes specular reflections on the occupied sites. The probability of penetrating up to level k inside the lattice is analytically evaluated using two different approaches, the former applying the theory of Markov chains (Markov approach) and the latter using the theory of Martingale random processes (Martingale approach). The full theory concerned with the Martingale approach is presented here, along with an innovative modification that leads to some improved results. Numerical validation shows that it outperforms the Markov approach when dealing with ray propagation in dense lattices described by a slowly varying density profile.

© 2007 Optical Society of America

OCIS Codes
(000.3860) General : Mathematical methods in physics
(000.5490) General : Probability theory, stochastic processes, and statistics
(030.6600) Coherence and statistical optics : Statistical optics
(080.2710) Geometric optics : Inhomogeneous optical media
(350.5500) Other areas of optics : Propagation

Original Manuscript: February 2, 2007
Manuscript Accepted: March 18, 2007
Published: July 11, 2007

Anna Martini, Renzo Azaro, Massimo Franceschetti, and Andrea Massa, "Ray propagation in nonuniform random lattices. Part II," J. Opt. Soc. Am. A 24, 2363-2371 (2007)

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  1. G. Grimmett, Percolation (Springer-Verlag, 1989).
  2. D. Stauffer, Introduction to Percolation Theory (Taylor and Francis, 1985). [CrossRef]
  3. A. Martini, M. Franceschetti, and A. Massa, "Ray propagation in nonuniform random lattices," J. Opt. Soc. Am. A 23, 2251-2261, 2006. [CrossRef]
  4. J. R. Norris, Markov Chains (Cambridge U. Press, 1998).
  5. G. Franceschetti, S. Marano, and F. Palmieri, "Propagation without wave equation toward an urban area model," IEEE Trans. Antennas Propag. 47, 1393-1404, 1999. [CrossRef]
  6. R. M. Ross, Stochastic Processes (Wiley, 1983).

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