×

Light scattering as a Poisson process and first-passage probability. (English) Zbl 1456.60272

Summary: A particle entering a scattering and absorbing medium executes a random walk through a sequence of scattering events. The particle ultimately either achieves first-passage, leaving the medium, or it is absorbed. The Kubelka-Munk model describes a flux of such particles moving perpendicular to the surface of a plane-parallel medium with a scattering rate and an absorption rate. The particle path alternates between the positive direction into the medium and the negative direction back towards the surface. Backscattering events from the positive to the negative direction occur at local maxima or peaks, while backscattering from the negative to the positive direction occur at local minima or valleys. The probability of a particle avoiding absorption as it follows its path decreases exponentially with the path-length \(\lambda \). The reflectance of a semi-infinite slab is therefore the Laplace transform of the distribution of path-length that ends with a first-passage out of the medium. In the case of a constant scattering rate the random walk is a Poisson process. We verify our results with two iterative calculations, one using the properties of iterated convolution with a symmetric kernel and the other via direct calculation with an exponential step-length distribution. We present a novel demonstration, based on fluctuation theory of sums of random variables, that the first-passage probability as a function of the number of peaks \(n\) in the alternating path is a step-length distribution-free combinatoric expression involving Catalan numbers. Counting paths with backscattering on the real half-line results in the same Catalan number coefficients as Dyck paths on the whole numbers. Including a separate forward-scattering Poisson process results in a combinatoric expression related to counting Motzkin paths. We therefore connect walks on the real line to discrete path combinatorics.

MSC:

60K37 Processes in random environments
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
60G50 Sums of independent random variables; random walks

Software:

OEIS
PDFBibTeX XMLCite
Full Text: DOI arXiv

Online Encyclopedia of Integer Sequences:

Triangular array of Motzkin polynomial coefficients.

References:

[1] Kubelka P and Munk F 1931 An article on optics of paint layers Z. Tech. Phys.12 593-601
[2] Yule J A C and Nielsen W J 1951 The penetration of light into paper and its effect on halftone reproduction Proc. TAGAvol 3 65-76
[3] Rudnick J and George G 2004 Elements of the Random Walk: An Introduction for Advanced Students and Researchers (Cambridge: Cambridge University Press) · Zbl 1086.60003 · doi:10.1017/CBO9780511610912
[4] Redner S 2001 A Guide to First-Passage Processes (Cambridge: Cambridge University Press) · Zbl 0980.60006 · doi:10.1017/CBO9780511606014
[5] Spitzer F 1964 Principles of Random Walk(Graduate Texts in Mathematics vol 34) (New York: Springer) · Zbl 0119.34304 · doi:10.1007/978-1-4757-4229-9
[6] Philips-Invernizzi B, Dupont D and Caze C 2001 Bibliographical review for reflectance of diffusing media Opt. Eng.40 1082-92 · doi:10.1117/1.1370387
[7] Schwarzschild K 1906 On the equilibrium of the sun’s atmosphere Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen(Math.-Phys. Klasse vol 195) pp 41-53 · JFM 37.0970.02
[8] Chandrasekhar S 1950 Radiative Transfer (Oxford: Clarendon) · Zbl 0037.43201
[9] Chandrasekhar S 1943 Stochastic problems in physics and astronomy Rev. Mod. Phys.15 1 · Zbl 0061.46403 · doi:10.1103/revmodphys.15.1
[10] Bonner R F, Nossal R, Havlin S and Weiss G H 1987 Model for photon migration in turbid biological media J. Opt. Soc. Am. A 4 423-32 · doi:10.1364/josaa.4.000423
[11] Nossal R, Kiefer J, Weiss G H, Bonner R, Taitelbaum H and Havlin S 1988 Photon migration in layered media Appl. Opt.27 3382-91 · doi:10.1364/ao.27.003382
[12] Schuster A 1905 Radiation through a foggy atmosphere Astrophys. J.21 1 · doi:10.1086/141186
[13] Gate L F 1974 Comparison of the photon diffusion model and Kubelka-Munk equation with the exact solution of the radiative transport equation Appl. Opt.13 236-8 · doi:10.1364/ao.13.000236
[14] Sandoval C and Kim A D 2014 Deriving Kubelka-Munk theory from radiative transport J. Opt. Soc. Am. A 31 628 · doi:10.1364/josaa.31.000628
[15] Youngquist R C, Carr S and Davies D 1987 Optical coherence-domain reflectometry: a new optical evaluation technique Opt. Lett.12 158-60 · doi:10.1364/ol.12.000158
[16] Haney M M and van Wijk K 2007 Modified Kubelka-Munk equations for localized waves inside a layered medium Phys. Rev. E 75 036601 · doi:10.1103/physreve.75.036601
[17] Hébert M and Becker J-M 2008 Correspondence between continuous and discrete two-flux models for reflectance and transmittance of diffusing layers J. Opt. A: Pure Appl. Opt.10 035006 · doi:10.1088/1464-4258/10/3/035006
[18] Ballestra L V, Pacelli G and Radi D 2016 A very efficient approach to compute the first-passage probability density function in a time-changed Brownian model: applications in finance Physica A 463 330-44 · Zbl 1400.60104 · doi:10.1016/j.physa.2016.07.016
[19] Simon K and Trachsler B 2003 A random walk approach for light scattering in material Discrete Random Walks, DRW’03 pp 289-300 · Zbl 1034.60093
[20] Jacques S L 2010 Monte Carlo modeling of light transport in tissue (steady state and time of flight) Optical-Thermal Response of Laser-Irradiated Tissue (Berlin: Springer) pp 109-44 · doi:10.1007/978-90-481-8831-4_5
[21] Doering C R, Ray T S and Lawrence Glasser M 1992 Long transmission times for transport through a weakly scattering slab Phys. Rev. A 45 825-8 · doi:10.1103/physreva.45.825
[22] Antal T and Redner S 2006 Escape of a uniform random walk from an interval J. Stat. Phys.123 1129-44 · Zbl 1113.82025 · doi:10.1007/s10955-006-9139-2
[23] Wuttke J 2014 The zig-zag walk with scattering and absorption on the real half line and in a lattice model J. Phys. A: Math. Theor.47 215203 · Zbl 1295.82012 · doi:10.1088/1751-8113/47/21/215203
[24] Darwin C G 1922 XCII. The reflexion of x-rays from imperfect crystals London, Edinburgh, and Dublin Phil. Mag. J. Sci.43 800-29 · doi:10.1080/14786442208633940
[25] Hamilton W C 1957 The effect of crystal shape and setting on secondary extinction Acta Crystallogr.10 629-34 · doi:10.1107/s0365110x57002212
[26] Andersen E S 1962 The equivalence principle in the theory of fluctuations of sums of random variables Colloquium on Combinatorial Methods in Probability Theory (Aarhus, Denmark: Matematisk Institut Aarhus Universitet) pp 13-6 · Zbl 0158.34903
[27] Sparre Andersen E 1953 On the fluctuations of sums of random variables Math. Scand.1 263-85 · Zbl 0053.09701 · doi:10.7146/math.scand.a-10385
[28] Myrick M L, Simcock M N, Baranowski M, Brooke H, Morgan S L and McCutcheon J N 2011 The Kubelka-Munk diffuse reflectance formula revisited Appl. Spectrosc. Rev.46 140-65 · doi:10.1080/05704928.2010.537004
[29] Beer A 1852 Bestimmung der absorption des rothen lichts in farbigen flussigkeiten Ann. Phys.162 78-88 · doi:10.1002/andp.18521620505
[30] Lambert J H 1760 Photometria, sive de mensura et gradibus luminis. Colorum et Umbrae (Augsberg: Eberhard Klett)
[31] Sloane N J A et al 2019 The On-Line Encyclopedia of Integer Sequences (https://oeis.org/A055151)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.