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T-matrix method

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The Transition Matrix Method (T-matrix method, TMM) is a computational technique of <a href="Light_scattering" class="mw-redirect" title="Light scattering">light scattering</a> by nonspherical particles originally formulated by Peter C. Waterman (1928–2012) in 1965.<a href="#cite_note-Waterman1965-1">[1]</a><a href="#cite_note-2">[2]</a>

The technique is also known as null field method and extended boundary condition method (EBCM).<a href="#cite_note-Mishchenko1996-3">[3]</a> In the method, matrix elements are obtained by <a href="Impedance_matching" title="Impedance matching">matching</a> boundary conditions for solutions of <a href="Maxwell_equations" class="mw-redirect" title="Maxwell equations">Maxwell equations</a>. It has been greatly extended to incorporate diverse types of linear media occupying the region enclosing the scatterer.<a href="#cite_note-4">[4]</a>

T-matrix method proves to be highly efficient and has been widely used in computing electromagnetic scattering of single and compound particles.<a href="#cite_note-5">[5]</a>

<div class="mw-heading mw-heading2"><h2 id="Definition_of_the_T-matrix">Definition of the T-matrix</h2></div>

The incident and scattered <a href="Electric_field" title="Electric field">electric field</a> are expanded into spherical vector wave functions (SVWF), which are also encountered in <a href="Mie_scattering" title="Mie scattering">Mie scattering</a>. They are the <a href="Fundamental_solution" title="Fundamental solution">fundamental solutions</a> of the vector <a href="Helmholtz_equation" title="Helmholtz equation">Helmholtz equation</a> and can be generated from the scalar fundamental solutions in <a href="Spherical_coordinates" class="mw-redirect" title="Spherical coordinates">spherical coordinates</a>, the spherical <a href="Bessel_functions" class="mw-redirect" title="Bessel functions">Bessel functions</a> of the first kind and the spherical Hankel functions. Accordingly, there are two linearly independent sets of solutions denoted as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {M} ^{1},\mathbf {N} ^{1}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {M} ^{1},\mathbf {N} ^{1}}</annotation>

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</math><img src="./cb4d8a07600e665a6c526c67467cb38b8d1771b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.771ex; height:3.009ex;" alt="{\displaystyle \mathbf {M} ^{1},\mathbf {N} ^{1}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {M} ^{3},\mathbf {N} ^{3}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {M} ^{3},\mathbf {N} ^{3}}</annotation>

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</math><img src="./837a070c89bea07fe7a4e676c632cfa38ed20979.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.771ex; height:3.009ex;" alt="{\displaystyle \mathbf {M} ^{3},\mathbf {N} ^{3}}" loading="lazy">, respectively. They are also called regular and outgoing SVWFs, respectively. With this, we can write the incident field as

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {E} _{inc}=\sum _{n=1}^{\infty }\sum _{m=-n}^{n}\left(a_{mn}\mathbf {M} _{mn}^{1}+b_{mn}\mathbf {N} _{mn}^{1}\right).}</annotation>

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</math><img src="./aa1747b97cb838375ac8b9a508ec32234530d32e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:39.974ex; height:6.843ex;" alt="{\displaystyle \mathbf {E} _{inc}=\sum _{n=1}^{\infty }\sum _{m=-n}^{n}\left(a_{mn}\mathbf {M} _{mn}^{1}+b_{mn}\mathbf {N} _{mn}^{1}\right).}" loading="lazy"></dd></dl>

The scattered field is expanded into radiating SVWFs:

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {E} _{scat}=\sum _{n=1}^{\infty }\sum _{m=-n}^{n}\left(f_{mn}\mathbf {M} _{mn}^{3}+g_{mn}\mathbf {N} _{mn}^{3}\right).}</annotation>

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</math><img src="./3a2c32b55e51a573dd61552881f1bf46ccdb48a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:40.676ex; height:6.843ex;" alt="{\displaystyle \mathbf {E} _{scat}=\sum _{n=1}^{\infty }\sum _{m=-n}^{n}\left(f_{mn}\mathbf {M} _{mn}^{3}+g_{mn}\mathbf {N} _{mn}^{3}\right).}" loading="lazy"></dd></dl>

The T-matrix relates the expansion coefficients of the incident field to those of the scattered field.

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<annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}f_{mn}\\g_{mn}\end{pmatrix}}=T{\begin{pmatrix}a_{mn}\\b_{mn}\end{pmatrix}}}</annotation>

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</math><img src="./1cb1b913781ca48d7bdbd0c2170fa175745b3df1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.772ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}f_{mn}\\g_{mn}\end{pmatrix}}=T{\begin{pmatrix}a_{mn}\\b_{mn}\end{pmatrix}}}" loading="lazy"></dd></dl>

The T-matrix is determined by the scatterer shape and material and for a given incident field allows one to calculate the scattered field.

<div class="mw-heading mw-heading2"><h2 id="Calculation_of_the_T-matrix">Calculation of the T-matrix</h2></div>

The standard way to calculate the T-matrix is the null-field method, which relies on the Stratton–Chu equations.<a href="#cite_note-6">[6]</a> They basically state that the electromagnetic fields outside a given volume can be expressed as integrals over the surface enclosing the volume involving only the tangential components of the fields on the surface. If the observation point is located inside this volume, the integrals vanish.

By making use of the <a href="Boundary_conditions" class="mw-redirect" title="Boundary conditions">boundary conditions</a> for the tangential field components on the scatterer surface,

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {n} \times (\mathbf {E} _{scat}+\mathbf {E} _{inc})=\mathbf {n} \times \mathbf {E} _{int}}</annotation>

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</math><img src="./5295dadd560ad5a53510b7ba90d1014d5c77d75d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.727ex; height:2.843ex;" alt="{\displaystyle \mathbf {n} \times (\mathbf {E} _{scat}+\mathbf {E} _{inc})=\mathbf {n} \times \mathbf {E} _{int}}" loading="lazy"></dd></dl>

and

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {n} \times (\mathbf {H} _{scat}+\mathbf {H} _{inc})=\mathbf {n} \times \mathbf {H} _{int}}</annotation>

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</math><img src="./922c4bfbff7dd942aab2dbdedbbe5bd426a8884d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.73ex; height:2.843ex;" alt="{\displaystyle \mathbf {n} \times (\mathbf {H} _{scat}+\mathbf {H} _{inc})=\mathbf {n} \times \mathbf {H} _{int}}" loading="lazy">,</dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {n} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {n} }</annotation>

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</math><img src="./4a720c341f39f52fd96028dab83edd34d400be46.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:1.676ex;" alt="{\displaystyle \mathbf {n} }" loading="lazy"> is the <a href="Normal_vector" class="mw-redirect" title="Normal vector">normal vector</a> to the scatterer surface, one can derive an integral representation of the scattered field in terms of the tangential components of the internal fields on the scatterer surface. A similar representation can be derived for the incident field.

By expanding the internal field in terms of SVWFs and exploiting their orthogonality on spherical surfaces, one arrives at an expression for the T-matrix. The T-matrix can also be computed from far field data.<a href="#cite_note-7">[7]</a> This approach avoids <a href="Numerical_stability" title="Numerical stability">numerical stability</a> issues associated with the null-field method.<a href="#cite_note-8">[8]</a>

Several numerical codes for the evaluation of the T-matrix can be found online <a rel="nofollow" class="external autonumber" href="http://www.scattport.org/index.php/light-scattering-software/t-matrix-codes/list">[1]</a> <a rel="nofollow" class="external autonumber" href="https://www.ugr.es/~aquiran/codigos.htm">[2]</a> <a rel="nofollow" class="external autonumber" href="https://www.giss.nasa.gov/staff/mmishchenko/t_matrix.html">[3]</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20221216165719/https://www.giss.nasa.gov/staff/mmishchenko/t_matrix.html">Archived</a> 2022-12-16 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a>.

The T matrix can be found with methods other than null field method and extended boundary condition method (EBCM); therefore, the term "T-matrix method" is infelicitous.

Improvement of traditional T-matrix includes Invariant-imbedding T-matrix Method (IITM) by B. R. Johnson.<a href="#cite_note-9">[9]</a> The numerical code of IITM is developed by Lei Bi, based on Mishchenko's EBCM code.<a href="#cite_note-Mishchenko1996-3">[3]</a><a href="#cite_note-10">[10]</a> It is more powerful than EBCM as it is more efficient and increases the upper limit of particle size during the computation.

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFWaterman1965" class="citation journal cs1">Waterman, P.C. (1965). "Matrix formulation of electromagnetic scattering". Proceedings of the IEEE. 53 (8). Institute of Electrical and Electronics Engineers (IEEE): 805–812. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2Fproc.1965.4058">10.1109/proc.1965.4058</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0018-9219">0018-9219</a>.</cite>

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<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFWaterman1971" class="citation journal cs1">Waterman, Peter C. (1971). "Symmetry, unitarity, and geometry in electromagnetic scattering". Physical Review D. 3 (4): 825–839. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1971PhRvD...3..825W">1971PhRvD...3..825W</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevD.3.825">10.1103/PhysRevD.3.825</a>.</cite>

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<li id="cite_note-Mishchenko1996-3">^ <a href="#cite_ref-Mishchenko1996_3-0">a</a> <a href="#cite_ref-Mishchenko1996_3-1">b</a> <cite id="CITEREFMishchenkoTravisMackowski1996" class="citation journal cs1">Mishchenko, Michael I.; Travis, Larry D.; Mackowski, Daniel W. (1996). "T-matrix computations of light scattering by nonspherical particles: A review". Journal of Quantitative Spectroscopy and Radiative Transfer. 55 (5). Elsevier BV: 535–575. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0022-4073%2896%2900002-7">10.1016/0022-4073(96)00002-7</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0022-4073">0022-4073</a>.</cite>

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<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite id="CITEREFLakhtakia2018" class="citation book cs1">Lakhtakia, Akhlesh (2018). The Ewald–Oseen Extinction Theorem and the Extended Boundary Condition Method, in: The World of Applied Electromagnetics. Cham, Switzerland: Springer. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-319-58403-4_19">10.1007/978-3-319-58403-4_19</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-319-58403-4</bdi>.</cite>

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<li id="cite_note-5"><a href="#cite_ref-5">^</a> <cite id="CITEREFMishchenkoTravisLacis2002" class="citation book cs1">Mishchenko, Michael I.; Travis, Larry D.; Lacis, Andrew A. (2002). Scattering, Absorption, and Emission of Light by Small Particles. Cambridge, UK: Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780521782524</bdi>.</cite>

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<li id="cite_note-6"><a href="#cite_ref-6">^</a> <cite id="CITEREFStrattonChu1939" class="citation journal cs1">Stratton, J. A.; Chu, L. J. (1939-07-01). "Diffraction Theory of Electromagnetic Waves". Physical Review. 56 (1). American Physical Society (APS): 99–107. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1939PhRv...56...99S">1939PhRv...56...99S</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2Fphysrev.56.99">10.1103/physrev.56.99</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0031-899X">0031-899X</a>.</cite>

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<li id="cite_note-7"><a href="#cite_ref-7">^</a> <cite id="CITEREFGaneshHawkins2010" class="citation journal cs1">Ganesh, M.; Hawkins, Stuart C. (2010). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.cam.2009.08.018">"Three dimensional electromagnetic scattering T-matrix computations"</a>. Journal of Computational and Applied Mathematics. 234 (6): 1702–1709. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.cam.2009.08.018">10.1016/j.cam.2009.08.018</a>.</cite>

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<li id="cite_note-8"><a href="#cite_ref-8">^</a> <cite id="CITEREFGaneshHawkins2017" class="citation journal cs1">Ganesh, M.; Hawkins, Stuart C. (2017). "Algorithm 975: TMATROM - A T-matrix Reduced Order Model Software". ACM Transactions on Mathematical Software. 44: 9:1–9:18. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F3054945">10.1145/3054945</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:24838138">24838138</a>.</cite>

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<li id="cite_note-9"><a href="#cite_ref-9">^</a> <cite id="CITEREFJohnson1988" class="citation journal cs1">Johnson, B. R. (1988-12-01). <a rel="nofollow" class="external text" href="https://opg.optica.org/ao/abstract.cfm?uri=ao-27-23-4861">"Invariant imbedding T matrix approach to electromagnetic scattering"</a>. Applied Optics. 27 (23): 4861–4873. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1364%2FAO.27.004861">10.1364/AO.27.004861</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2155-3165">2155-3165</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/20539668">20539668</a>.</cite>

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<li id="cite_note-10"><a href="#cite_ref-10">^</a> <cite id="CITEREFBiYangKattawarMishchenko2013" class="citation journal cs1">Bi, Lei; Yang, Ping; Kattawar, George W.; Mishchenko, Michael I. (2013-02-01). <a rel="nofollow" class="external text" href="https://www.sciencedirect.com/science/article/pii/S0022407312005201">"Efficient implementation of the invariant imbedding T-matrix method and the separation of variables method applied to large nonspherical inhomogeneous particles"</a>. Journal of Quantitative Spectroscopy and Radiative Transfer. 116: 169–183. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jqsrt.2012.11.014">10.1016/j.jqsrt.2012.11.014</a>. <a href="Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/2060%2F20140010884">2060/20140010884</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0022-4073">0022-4073</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:11722624">11722624</a>.</cite>

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