Gustavsson, Magnus

Kristensson, Gerhard

Wellander, Niklas

English

Lund University Publications

LUND UNIVERSITYPO Box 117221 00 Lund+46 46-222 00 00The coherent electromagnetic field and the effect of the pair distribution functionGustavsson, Magnus; Kristensson, Gerhard; Wellander, Niklas2021Document Version:Publisher's PDF, also known as Version of recordLink to publicationCitation for published version (APA):Gustavsson, M., Kristensson, G., & Wellander, N. (2021). The coherent electromagnetic field and the effect ofthe pair distribution function. Abstract from Electromagnetic and Light Scattering Conference.Total number of authors:3Creative Commons License:UnspecifiedGeneral rightsUnless other specific re-use rights are stated the following general rights apply:Copyright and moral rights for the publications made accessible in the public portal are retained by the authorsand/or other copyright owners and it is a condition of accessing publications that users recognise and abide by thelegal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private studyor research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portalRead more about Creative commons licenses: https://creativecommons.org/licenses/Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will removeaccess to the work immediately and investigate your claim.Download date: 08. Oct. 2022The coherent electromagnetic field and the effect of thepair distribution functionMagnus Gustavsson1, Gerhard Kristensson2,*, and Niklas Wellander11Swedish Defence Research Agency, FOI, SE-581 11 Linköping, Sweden2Department of Electrical and Information Technology, Lund University,P.O. Box 118, SE-221 00 Lund, Sweden*corresponding author’s e-mail: gerhard.kristensson@eit.lth.se1 Theory and resultsThe coherent (ensemble average) transmitted and reflected fields from a particulate slab are most commonly computedby the effective wavenumber approach, see e.g. [5]. The effective wave number is obtained from the roots of adeterminant relation. An alternative method, presented in [1], solves the coherent transmitted and reflected fieldsfrom a particulate slab by solving a system of integral equations in the depth variable. We let a plane wave impingeat normal incidence on the slab, z ∈ [0, d], containing spherical dielectric particles of radius a. The scattered fields(ensemble average) on either side of the slab, [z1, z2] = [a, d− a], areE±s (r) =3f2(ka)3∑ni−l+τ−1An(±ẑ)k∫ z2z1e±ikz′fn(z′) dz′e±ikz,{z > dz < 0The summation is over the multi-index n = {τ, σ,m, l}, τ = 1, 2, σ = e, o, m = 0, 1, 2, . . . , l, and l = 1, 2, 3 . . ., andAn(k̂i) are the vector spherical harmonics, see [3] for more details. The volume fraction of the spheres is denoted f .The coefficients fn(z) satisfies a system of linear, one-dimensional integral equations in z [1], viz.,fn(z) = eikz∑n′Tnn′an′ + k∫ z2z1∑n′Knn′(z − z′)fn′(z′) dz′, z ∈ [z1, z2]The entries of the kernel in this set of integral equations consist of rapidly oscillating integrals. Fortunately, for thehole correction (HC), these integrals have a closed form solution in terms of a series of spherical waves [2]. Withoutthis analytic solution of the integrals, the integral equation approach offers challenging numerical integration. Theparticles are completely characterized by the transition matrix Tnn′ , which for a spherical particle is diagonal inits (pairwise) indices. The expansion coefficients of the plane wave in terms of regular spherical vector waves aredenoted an, see [3].The hole correction — an adequate approximation for gases and other tenuous media – gives less accurate resultsfor e.g., liquids or other amorphous materials. In this paper, we analyse the effect of the Percus-Yevick (P-Y)approximation of the pair distribution function on the transmitted and reflected fields [4]. This P-Y approximationenlarges the scope of the integral equation approach considerably, and we compare the effect of the P-Y approximationon reflection and transmission from a particulate slab of finite thickness. The kernel entries now include integrals withrapidly oscillating integrands, but, due to the form of the P-Y approximation, the integration interval is accuratelyapproximated by a finite interval, which makes numerical integration feasible.References[1] G. Kristensson. Coherent scattering by a collection of randomly located obstacles — an alternative integralequation formulation. J. Quant. Spectrosc. Radiat. Transfer, 164, 97–108, 2015.[2] G. Kristensson. Evaluation of some integrals relevant to multiple scattering by randomly distributed obstacles.Journal of Mathematical Analysis and Applications, 432(1), 324–337, 2015.[3] G. Kristensson. Scattering of Electromagnetic Waves by Obstacles. Mario Boella Series on Electromagnetism inInformation and Communication. SciTech Publishing, Edison, NJ, USA, 2016.[4] J. K. Percus and G. J. Yevick. Analysis of classical statistical mechanics by means of collective coordinates. Phys.Rev., 110(1), 1, 1958.[5] L. Tsang and J. A. Kong. Scattering of Electromagnetic Waves: Advanced Topics. John Wiley & Sons, NewYork, NY, 2001.1

The coherent electromagnetic field and the effect of the pair distribution function

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The coherent electromagnetic field and the effect of the pair distribution function

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