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Discrete dipole approximation

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr">The discrete dipole approximation (DDA), also known as the coupled dipole approximation, is a numerical method for computing the scattering and absorption of electromagnetic radiation by particles of arbitrary shape and composition. The method represents a continuum target as a finite array of small, polarizable dipoles, and solves for their interactions with the incident field and with each other. DDA can handle targets with inhomogeneous composition and anisotropic material properties, as well as periodic structures. It is widely applied in fields such as <a href="Nanophotonics" title="Nanophotonics">nanophotonics</a>, <a href="Radar" title="Radar">radar</a> scattering, <a href="Aerosol" title="Aerosol">aerosol</a> physics, <a href="Biomedical_optics" class="mw-redirect" title="Biomedical optics">biomedical optics</a>, and <a href="Astrophysics" title="Astrophysics">astrophysics</a>.

<div class="mw-heading mw-heading2"><h2 id="Basic_concepts">Basic concepts</h2></div>

The basic idea of the DDA was introduced in 1964 by DeVoe<a href="#cite_note-devoe1964-1">[1]</a> who applied it to study the optical properties of molecular aggregates; retardation effects were not included, so DeVoe's treatment was limited to aggregates that were small compared with the wavelength. The DDA, including retardation effects, was proposed in 1973 by <a href="Edward_Mills_Purcell" title="Edward Mills Purcell">Purcell</a> and Pennypacker<a href="#cite_note-purcell1973-2">[2]</a>

who used it to study interstellar dust grains. Simply stated, the DDA is an approximation of the continuum target by a finite array of polarizable points. The points acquire dipole moments in response to the local electric field. The dipoles interact with one another via their electric fields, so the DDA is also sometimes referred to as the coupled dipole approximation.<a href="#cite_note-singham1986-3">[3]</a><a href="#cite_note-singham1987-4">[4]</a>

Nature provides the physical inspiration for the DDA - in 1909 <a href="Hendrik_Lorentz" title="Hendrik Lorentz">Lorentz</a><a href="#cite_note-lorentz1909-5">[5]</a>

showed that the dielectric properties of a substance could be directly related to the polarizabilities of the individual atoms of which it was composed, with a particularly simple and exact relationship, the <a href="Clausius-Mossotti_relation" class="mw-redirect" title="Clausius-Mossotti relation">Clausius-Mossotti relation</a> (or Lorentz-Lorenz), when the atoms are located on a cubical lattice. We may expect that, just as a continuum representation of a solid is appropriate on length scales that are large compared with the interatomic spacing, an array of polarizable points can accurately approximate the response of a continuum target on length scales that are large compared with the interdipole separation.

For a finite array of point dipoles the scattering problem may be solved exactly, so the only approximation that is present in the DDA is the replacement of the continuum target by an array of N-point dipoles. The replacement requires specification of both the geometry (location of the dipoles) and the dipole polarizabilities. For monochromatic incident waves the self-consistent solution for the oscillating dipole moments may be found; from these the absorption and scattering cross sections are computed. If DDA solutions are obtained for two independent polarizations of the incident wave, then the complete amplitude scattering matrix can be determined.

Alternatively, the DDA can be derived from <a href="Electric-field_integral_equation" title="Electric-field integral equation">volume integral equation for the electric field</a>.<a href="#cite_note-Yurkin2007a-6">[6]</a> This highlights that the approximation of point dipoles is equivalent to that of discretizing the integral equation, and thus decreases with decreasing dipole size.

With the recognition that the polarizabilities may be tensors, the DDA can readily be applied to anisotropic materials. The extension of the DDA to treat materials with nonzero <a href="Magnetic_susceptibility" title="Magnetic susceptibility">magnetic susceptibility</a> is also straightforward, although for most applications magnetic effects are negligible.

There are several reviews of DDA method. <a href="#cite_note-draine1994-7">[7]</a><a href="#cite_note-Yurkin2007a-6">[6]</a><a href="#cite_note-Yurkin2023-8">[8]</a><a href="#cite_note-chaumet2022discrete-9">[9]</a>

The method was improved by <a href="Bruce_T._Draine" title="Bruce T. Draine">Draine</a>, Flatau, and Goodman, who applied the <a href="Fast_Fourier_transform" title="Fast Fourier transform">fast Fourier transform</a> to solve fast convolution problems arising in the discrete dipole approximation (DDA). This allowed for the calculation of scattering by large targets. They distributed an open-source code DDSCAT.<a href="#cite_note-draine1994-7">[7]</a><a href="#cite_note-draine2008-10">[10]</a>

There are now several DDA implementations,<a href="#cite_note-Yurkin2007a-6">[6]</a> extensions to periodic targets,<a href="#cite_note-chaumet2003-11">[11]</a> and particles placed on or near a plane substrate.<a href="#cite_note-schmehl1997-12">[12]</a><a href="#cite_note-yurkin2015-13">[13]</a> Comparisons with exact techniques have also been published.<a href="#cite_note-penttila2007-14">[14]</a>

Other aspects, such as the validity criteria of the discrete dipole approximation, were published.<a href="#cite_note-zubko2010-15">[15]</a> The DDA was also extended to employ rectangular or cuboid dipoles,<a href="#cite_note-smunev2015-16">[16]</a> which are more efficient for highly oblate or prolate particles.

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

In the discrete dipole approximation, a target object is represented as a finite array of N point dipoles located at positions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} _{j}}">

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

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</math><img src="./609bdd34e1060e0d89cb81aad617e36d6116b2a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.012ex; height:2.343ex;" alt="{\displaystyle \mathbf {r} _{j}}" loading="lazy"> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j=1,2,\dots ,N}">

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<annotation encoding="application/x-tex">{\displaystyle j=1,2,\dots ,N}</annotation>

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</math><img src="./6176bdcf4d1c207122117d9338a0f33bccebbbb4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:14.684ex; height:2.509ex;" alt="{\displaystyle j=1,2,\dots ,N}" loading="lazy">). The polarization vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} _{j}}">

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

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</math><img src="./478d5de6fcd4dd679fe271f13f9331e0d1114f6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.736ex; height:2.843ex;" alt="{\displaystyle \mathbf {P} _{j}}" loading="lazy"> of each dipole is related to the local electric field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} _{j}}">

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

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</math><img src="./b767571edcdc06dfbfa267959cb490bb937de10f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.667ex; height:2.843ex;" alt="{\displaystyle \mathbf {E} _{j}}" loading="lazy"> at that dipole by its polarizability tensor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\alpha }}_{j}}">

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<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\alpha }}_{j}}</annotation>

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</math><img src="./9f0838e683a340cd729f1b58ef1c7720784b5ae2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.678ex; height:2.343ex;" alt="{\displaystyle {\boldsymbol {\alpha }}_{j}}" loading="lazy">:

In anisotropic case (diagonal polarizability)

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {P} _{j}={\boldsymbol {\alpha }}_{j}\cdot \mathbf {E} _{j}}</annotation>

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</math><img src="./cb0c6e378c3bffff3e358cbfdfbcf43db17e9a73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.859ex; height:2.843ex;" alt="{\displaystyle \mathbf {P} _{j}={\boldsymbol {\alpha }}_{j}\cdot \mathbf {E} _{j}}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\alpha }}_{j}}">

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<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\alpha }}_{j}}</annotation>

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</math><img src="./9f0838e683a340cd729f1b58ef1c7720784b5ae2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.678ex; height:2.343ex;" alt="{\displaystyle {\boldsymbol {\alpha }}_{j}}" loading="lazy"> is diagonal

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<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\alpha }}_{j}={\begin{pmatrix}\alpha _{x,j}&0&0\\0&\alpha _{y,j}&0\\0&0&\alpha _{z,j}\end{pmatrix}}.}</annotation>

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</math><img src="./56dc45c041ebcde99d04a60688eb0a00bbcabc0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:26.978ex; height:9.843ex;" alt="{\displaystyle {\boldsymbol {\alpha }}_{j}={\begin{pmatrix}\alpha _{x,j}&0&0\\0&\alpha _{y,j}&0\\0&0&\alpha _{z,j}\end{pmatrix}}.}" loading="lazy"></dd></dl>

This leads to componentwise relations:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}P_{x,j}&=\alpha _{x,j}E_{x,j},\\P_{y,j}&=\alpha _{y,j}E_{y,j},\\P_{z,j}&=\alpha _{z,j}E_{z,j}.\end{aligned}}}</annotation>

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</math><img src="./bce3dc52eac82debeaf5fa56699e8999b2a75ede.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:16.114ex; height:9.509ex;" alt="{\displaystyle {\begin{aligned}P_{x,j}&=\alpha _{x,j}E_{x,j},\\P_{y,j}&=\alpha _{y,j}E_{y,j},\\P_{z,j}&=\alpha _{z,j}E_{z,j}.\end{aligned}}}" loading="lazy"></dd></dl>

For isotropic materials, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{x,j}=\alpha _{y,j}=\alpha _{z,j}=\alpha _{j}}">

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<annotation encoding="application/x-tex">{\displaystyle \alpha _{x,j}=\alpha _{y,j}=\alpha _{z,j}=\alpha _{j}}</annotation>

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</math><img src="./6a4a4b6eaf627bb1ae337357470804e8e990bdf9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.784ex; height:2.343ex;" alt="{\displaystyle \alpha _{x,j}=\alpha _{y,j}=\alpha _{z,j}=\alpha _{j}}" loading="lazy">, so

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {P} _{j}=\alpha _{j}\mathbf {E} _{j}}</annotation>

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</math><img src="./634ebbe0032501191267502df4d260c860337fa9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.899ex; height:2.843ex;" alt="{\displaystyle \mathbf {P} _{j}=\alpha _{j}\mathbf {E} _{j}}" loading="lazy">.</dd></dl>

The local electric field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} _{j}}">

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

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</math><img src="./b767571edcdc06dfbfa267959cb490bb937de10f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.667ex; height:2.843ex;" alt="{\displaystyle \mathbf {E} _{j}}" loading="lazy"> acting on the j‑th dipole is given by the sum of the incident field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} _{\mathrm {inc} }(\mathbf {r} _{j})}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {E} _{\mathrm {inc} }(\mathbf {r} _{j})}</annotation>

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</math><img src="./8efea9b2ececcb1d39ea518c8562ae1b20cb948d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.912ex; height:3.009ex;" alt="{\displaystyle \mathbf {E} _{\mathrm {inc} }(\mathbf {r} _{j})}" loading="lazy"> and the fields radiated by all other dipoles:

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {E} _{j}=\mathbf {E} _{\mathrm {inc} }(\mathbf {r} _{j})+\sum _{k\neq j}\mathbf {G} (\mathbf {r} _{j}-\mathbf {r} _{k})\cdot \mathbf {P} _{k}}</annotation>

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</math><img src="./f73ee5ebd0e04b66ce4a8188b06a798c60f988e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:35.807ex; height:6.009ex;" alt="{\displaystyle \mathbf {E} _{j}=\mathbf {E} _{\mathrm {inc} }(\mathbf {r} _{j})+\sum _{k\neq j}\mathbf {G} (\mathbf {r} _{j}-\mathbf {r} _{k})\cdot \mathbf {P} _{k}}" loading="lazy"></dd></dl>

Here, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {G} (\mathbf {r} )}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {G} (\mathbf {r} )}</annotation>

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</math><img src="./be2fb63903f0a53e8a1a9c6eb8236b4f298e2f05.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.012ex; height:2.843ex;" alt="{\displaystyle \mathbf {G} (\mathbf {r} )}" loading="lazy"> is the dyadic Green's function describing the field at position <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} }">

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

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</math><img src="./eca0f46511c4c986c48b254073732c0bd98ae0c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }" loading="lazy"> due to a unit dipole at the origin.

<div class="mw-heading mw-heading3"><h3 id="Dyadic_Green’s_function">Dyadic Green’s function</h3></div>

The free-space dyadic Green's function used in the discrete dipole approximation (DDA) can be expressed as the action of a differential operator on the scalar Green's function:

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {G} (\mathbf {r} )=\left[\nabla \nabla +k^{2}\mathbf {I} \right]{\frac {e^{ikr}}{r}},}</annotation>

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</math><img src="./892f434a36bc97fd19e8613e20efbac0e11dea51.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.393ex; height:5.676ex;" alt="{\displaystyle \mathbf {G} (\mathbf {r} )=\left[\nabla \nabla +k^{2}\mathbf {I} \right]{\frac {e^{ikr}}{r}},}" loading="lazy"></dd></dl>

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

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> is the wavenumber, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {I} }">

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

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</math><img src="./8a458c8aeb096ce732abf346ae8edf3e4f53a126.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.014ex; height:2.176ex;" alt="{\displaystyle \mathbf {I} }" loading="lazy"> is the identity matrix, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} }">

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</math><img src="./eca0f46511c4c986c48b254073732c0bd98ae0c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }" loading="lazy"> is the vector from the source dipole to the observation point. Evaluating the derivatives leads to the explicit form:

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {G} (\mathbf {r} )={\frac {e^{ikr}}{r^{3}}}\left[k^{2}r^{2}\left(\mathbf {I} -{\hat {\mathbf {r} }}{\hat {\mathbf {r} }}\right)+(1-ikr)\left(3{\hat {\mathbf {r} }}{\hat {\mathbf {r} }}-\mathbf {I} \right)\right],}</annotation>

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</math><img src="./4bc361189304a9f1da8351daed81062446c38420.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:49.785ex; height:6.009ex;" alt="{\displaystyle \mathbf {G} (\mathbf {r} )={\frac {e^{ikr}}{r^{3}}}\left[k^{2}r^{2}\left(\mathbf {I} -{\hat {\mathbf {r} }}{\hat {\mathbf {r} }}\right)+(1-ikr)\left(3{\hat {\mathbf {r} }}{\hat {\mathbf {r} }}-\mathbf {I} \right)\right],}" loading="lazy"></dd></dl>

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

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

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</math><img src="./930a88b12bae14cb42f2c390fcac932f8d346169.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.921ex; height:2.843ex;" alt="{\displaystyle {\hat {\mathbf {r} }}=\mathbf {r} /|\mathbf {r} |}" loading="lazy"> is the unit vector pointing from the source to the observation point.

This Green’s tensor describes the electric field generated by a dipole in a homogeneous medium. It is used to compute the off-diagonal blocks of the interaction matrix in DDA, that is, the interaction between distinct dipoles <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j\neq k}">

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<annotation encoding="application/x-tex">{\displaystyle j\neq k}</annotation>

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</math><img src="./a08a8f7e7c65621ea80f6989770e39aa591d0886.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.027ex; width:5.294ex; height:2.676ex;" alt="{\displaystyle j\neq k}" loading="lazy">. The singular self-term <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {G} (\mathbf {r} =0)}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {G} (\mathbf {r} =0)}</annotation>

</semantics>

</math><img src="./8efde63dbd34b60d103d0676650ad6e713ff92d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.273ex; height:2.843ex;" alt="{\displaystyle \mathbf {G} (\mathbf {r} =0)}" loading="lazy"> is excluded and replaced by a prescribed local term involving the inverse polarizability tensor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\alpha }}_{j}^{-1}}">

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\alpha }}_{j}^{-1}}</annotation>

</semantics>

</math><img src="./52958a218077006a14d08a9dff1ebc09b678679b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:4.101ex; height:3.676ex;" alt="{\displaystyle {\boldsymbol {\alpha }}_{j}^{-1}}" loading="lazy">.

Thus, the electric field at dipole <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy"> due to dipole <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> is given by

<msub>

</mrow>

</mrow>

</msub>

<mfrac>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mrow>

</mrow>

</msubsup>

</mfrac>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msubsup>

<mrow>

<mrow>

</mrow>

<msub>

<mover>

</mrow>

</mover>

</mrow>

</mrow>

</msub>

<msub>

<mover>

</mrow>

</mover>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<mrow>

<mrow>

<msub>

<mover>

</mrow>

</mover>

</mrow>

</mrow>

</msub>

<msub>

<mover>

</mrow>

</mover>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {G} _{jk}={\frac {e^{ikr_{jk}}}{r_{jk}^{3}}}\left[k^{2}r_{jk}^{2}\left(\mathbf {I} -{\hat {\mathbf {r} }}_{jk}{\hat {\mathbf {r} }}_{jk}\right)+\left(1-ikr_{jk}\right)\left(3{\hat {\mathbf {r} }}_{jk}{\hat {\mathbf {r} }}_{jk}-\mathbf {I} \right)\right],}</annotation>

</semantics>

</math><img src="./b9223aa8c1c7da397653e2726610212f751c86b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:59.848ex; height:7.176ex;" alt="{\displaystyle \mathbf {G} _{jk}={\frac {e^{ikr_{jk}}}{r_{jk}^{3}}}\left[k^{2}r_{jk}^{2}\left(\mathbf {I} -{\hat {\mathbf {r} }}_{jk}{\hat {\mathbf {r} }}_{jk}\right)+\left(1-ikr_{jk}\right)\left(3{\hat {\mathbf {r} }}_{jk}{\hat {\mathbf {r} }}_{jk}-\mathbf {I} \right)\right],}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} _{jk}=\mathbf {r} _{j}-\mathbf {r} _{k}}">

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {r} _{jk}=\mathbf {r} _{j}-\mathbf {r} _{k}}</annotation>

</semantics>

</math><img src="./61ea938271d3fedf74c17a3bb55d459449eba158.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.01ex; height:2.676ex;" alt="{\displaystyle \mathbf {r} _{jk}=\mathbf {r} _{j}-\mathbf {r} _{k}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{jk}=|\mathbf {r} _{jk}|}">

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r_{jk}=|\mathbf {r} _{jk}|}</annotation>

</semantics>

</math><img src="./19e039701167f6ee5ea30ad385014ffe004284f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.075ex; height:3.009ex;" alt="{\displaystyle r_{jk}=|\mathbf {r} _{jk}|}" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\mathbf {r} }}_{jk}=\mathbf {r} _{jk}/r_{jk}}">

<msub>

<mover>

</mrow>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {r} }}_{jk}=\mathbf {r} _{jk}/r_{jk}}</annotation>

</semantics>

</math><img src="./9aa7b83c528cad4e264af66dce977d22b9067ebf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.873ex; height:3.009ex;" alt="{\displaystyle {\hat {\mathbf {r} }}_{jk}=\mathbf {r} _{jk}/r_{jk}}" loading="lazy">. Here <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {I} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./8a458c8aeb096ce732abf346ae8edf3e4f53a126.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.014ex; height:2.176ex;" alt="{\displaystyle \mathbf {I} }" loading="lazy"> is the identity matrix and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=2\pi /\lambda }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k=2\pi /\lambda }</annotation>

</semantics>

</math><img src="./eecfda2366cb3a64ef6d4747e2460284155a1960.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.322ex; height:2.843ex;" alt="{\displaystyle k=2\pi /\lambda }" loading="lazy"> is the vacuum wavenumber.

Define

<msub>

</mrow>

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<msub>

</mrow>

</msub>

<mfrac>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mrow>

</mrow>

</msubsup>

</mfrac>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{1}(r_{jk})={\frac {e^{ikr_{jk}}}{r_{jk}^{3}}}\left(k^{2}r_{jk}^{2}+ikr_{jk}-1\right),}</annotation>

</semantics>

</math><img src="./a4c26598c1efeb93e2774b89146df1392abdad0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:37.114ex; height:7.176ex;" alt="{\displaystyle C_{1}(r_{jk})={\frac {e^{ikr_{jk}}}{r_{jk}^{3}}}\left(k^{2}r_{jk}^{2}+ikr_{jk}-1\right),}" loading="lazy"></dd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mrow>

</mrow>

</msubsup>

</mfrac>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{2}(r_{jk})={\frac {e^{ikr_{jk}}}{r_{jk}^{3}}}\left(3-3ikr_{jk}-k^{2}r_{jk}^{2}\right).}</annotation>

</semantics>

</math><img src="./57af636a2220b31eff967f2ea254435fcfe9a7e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:38.277ex; height:7.176ex;" alt="{\displaystyle C_{2}(r_{jk})={\frac {e^{ikr_{jk}}}{r_{jk}^{3}}}\left(3-3ikr_{jk}-k^{2}r_{jk}^{2}\right).}" loading="lazy"></dd></dl>

The dyadic Green’s function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {G} _{jk}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./41b6b8f17d51c88f1651739eb81fba4a39f2b3bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.867ex; height:2.843ex;" alt="{\displaystyle \mathbf {G} _{jk}}" loading="lazy"> is:

<msub>

</mrow>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mover>

</mover>

</mrow>

</mrow>

</mrow>

</msubsup>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mover>

</mover>

</mrow>

</mrow>

</mrow>

</msubsup>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mover>

</mover>

</mrow>

</mrow>

</mrow>

</msubsup>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {G} _{jk}={\begin{pmatrix}C_{1}+C_{2}{\hat {r}}_{jk,x}^{2}&C_{2}{\hat {r}}_{jk,x}{\hat {r}}_{jk,y}&C_{2}{\hat {r}}_{jk,x}{\hat {r}}_{jk,z}\\C_{2}{\hat {r}}_{jk,y}{\hat {r}}_{jk,x}&C_{1}+C_{2}{\hat {r}}_{jk,y}^{2}&C_{2}{\hat {r}}_{jk,y}{\hat {r}}_{jk,z}\\C_{2}{\hat {r}}_{jk,z}{\hat {r}}_{jk,x}&C_{2}{\hat {r}}_{jk,z}{\hat {r}}_{jk,y}&C_{1}+C_{2}{\hat {r}}_{jk,z}^{2}\end{pmatrix}},}</annotation>

</semantics>

</math><img src="./cb08dd58b756a9982aff94048a897b96e2a5a849.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:54.965ex; height:11.509ex;" alt="{\displaystyle \mathbf {G} _{jk}={\begin{pmatrix}C_{1}+C_{2}{\hat {r}}_{jk,x}^{2}&C_{2}{\hat {r}}_{jk,x}{\hat {r}}_{jk,y}&C_{2}{\hat {r}}_{jk,x}{\hat {r}}_{jk,z}\\C_{2}{\hat {r}}_{jk,y}{\hat {r}}_{jk,x}&C_{1}+C_{2}{\hat {r}}_{jk,y}^{2}&C_{2}{\hat {r}}_{jk,y}{\hat {r}}_{jk,z}\\C_{2}{\hat {r}}_{jk,z}{\hat {r}}_{jk,x}&C_{2}{\hat {r}}_{jk,z}{\hat {r}}_{jk,y}&C_{1}+C_{2}{\hat {r}}_{jk,z}^{2}\end{pmatrix}},}" loading="lazy"></dd></dl>

Notice that it is symmetric: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{jk}^{yx}=G_{jk}^{xy},\;G_{jk}^{zx}=G_{jk}^{xz},\;G_{jk}^{zy}=G_{jk}^{yz}}">

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G_{jk}^{yx}=G_{jk}^{xy},\;G_{jk}^{zx}=G_{jk}^{xz},\;G_{jk}^{zy}=G_{jk}^{yz}}</annotation>

</semantics>

</math><img src="./c47420ec419fa5eff2fb823c2a0a31c3e7f974fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:35.114ex; height:3.509ex;" alt="{\displaystyle G_{jk}^{yx}=G_{jk}^{xy},\;G_{jk}^{zx}=G_{jk}^{xz},\;G_{jk}^{zy}=G_{jk}^{yz}}" loading="lazy">.

Here <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} _{jk}=\mathbf {r} _{j}-\mathbf {r} _{k}}">

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {r} _{jk}=\mathbf {r} _{j}-\mathbf {r} _{k}}</annotation>

</semantics>

</math><img src="./61ea938271d3fedf74c17a3bb55d459449eba158.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.01ex; height:2.676ex;" alt="{\displaystyle \mathbf {r} _{jk}=\mathbf {r} _{j}-\mathbf {r} _{k}}" loading="lazy"> is the displacement vector from dipole <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> to dipole <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{jk}=|\mathbf {r} _{jk}|}">

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r_{jk}=|\mathbf {r} _{jk}|}</annotation>

</semantics>

</math><img src="./19e039701167f6ee5ea30ad385014ffe004284f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.075ex; height:3.009ex;" alt="{\displaystyle r_{jk}=|\mathbf {r} _{jk}|}" loading="lazy"> is the distance between them, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\mathbf {r} }}_{jk}=\mathbf {r} _{jk}/r_{jk}}">

<msub>

<mover>

</mrow>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {r} }}_{jk}=\mathbf {r} _{jk}/r_{jk}}</annotation>

</semantics>

</math><img src="./9aa7b83c528cad4e264af66dce977d22b9067ebf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.873ex; height:3.009ex;" alt="{\displaystyle {\hat {\mathbf {r} }}_{jk}=\mathbf {r} _{jk}/r_{jk}}" loading="lazy"> is the unit vector pointing from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy">. The components of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\mathbf {r} }}_{jk}}">

<msub>

<mover>

</mrow>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {r} }}_{jk}}</annotation>

</semantics>

</math><img src="./2e5e6683296c7d7f1921d8fb671afe1293e980d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.929ex; height:3.009ex;" alt="{\displaystyle {\hat {\mathbf {r} }}_{jk}}" loading="lazy"> are defined as:

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {r}}_{jk,x}={\frac {r_{j,x}-r_{k,x}}{r_{jk}}},\;{\hat {r}}_{jk,y}={\frac {r_{j,y}-r_{k,y}}{r_{jk}}},\;{\hat {r}}_{jk,z}={\frac {r_{j,z}-r_{k,z}}{r_{jk}}}.}</annotation>

</semantics>

</math><img src="./676d48ac8d63a9897c392f4b650baf565c5350b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:57.487ex; height:6.009ex;" alt="{\displaystyle {\hat {r}}_{jk,x}={\frac {r_{j,x}-r_{k,x}}{r_{jk}}},\;{\hat {r}}_{jk,y}={\frac {r_{j,y}-r_{k,y}}{r_{jk}}},\;{\hat {r}}_{jk,z}={\frac {r_{j,z}-r_{k,z}}{r_{jk}}}.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Polarizability">Polarizability</h3></div>

/* start https://en.wikipedia.org/ */

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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Polarizability" title="Polarizability">Polarizability</a></div>

In the discrete dipole approximation, the electromagnetic response of a target is modeled by replacing the continuous material with a finite array of point dipoles. Each dipole represents a small volume of the material and acts as a polarizable unit that interacts with both the incident field and the fields radiated by all other dipoles. The key parameter that describes how each dipole responds to the local electric field is its polarizability <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{j}}">

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<annotation encoding="application/x-tex">{\displaystyle \alpha _{j}}</annotation>

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</math><img src="./293a364991ab1ee55c25b0f60fd9e52af7b7dbde.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.397ex; height:2.343ex;" alt="{\displaystyle \alpha _{j}}" loading="lazy">. For a homogeneous material, the polarizability of a dipole is determined by the material’s complex dielectric function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon (\lambda )}">

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<annotation encoding="application/x-tex">{\displaystyle \varepsilon (\lambda )}</annotation>

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</math><img src="./1da6bc807274d14590766a7c6d9a8d63463a6737.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.248ex; height:2.843ex;" alt="{\displaystyle \varepsilon (\lambda )}" loading="lazy">, which depends on the wavelength <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

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

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</math><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"> of light in vacuum. The dielectric function is related to the complex refractive index <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=n'+in''}">

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

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</math><img src="./54fb4dd2edcf1637560262662e75fbcc901fa60e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.747ex; height:2.676ex;" alt="{\displaystyle n=n'+in''}" loading="lazy"> through <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon =n^{2}}">

<msup>

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</msup>

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<annotation encoding="application/x-tex">{\displaystyle \varepsilon =n^{2}}</annotation>

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</math><img src="./9db2c03479ff09889fe2dfd388066d2f31ad4c69.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.631ex; height:2.676ex;" alt="{\displaystyle \varepsilon =n^{2}}" loading="lazy">. The goal in DDA is to assign to each dipole a polarizability <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{j}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha _{j}}</annotation>

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</math><img src="./293a364991ab1ee55c25b0f60fd9e52af7b7dbde.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.397ex; height:2.343ex;" alt="{\displaystyle \alpha _{j}}" loading="lazy"> such that the array of dipoles reproduces, as accurately as possible, the scattering and absorption behavior of the original continuous medium. For isotropic materials, a common starting point is the <a href="Clausius%E2%80%93Mossotti_relation" title="Clausius–Mossotti relation">Clausius–Mossotti relation</a>, which connects the polarizability to the dielectric function:

In the discrete dipole approximation, the total volume of the target is divided into small cubic cells of volume <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\mathrm {dipole} }=d^{3}}">

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<annotation encoding="application/x-tex">{\displaystyle V_{\mathrm {dipole} }=d^{3}}</annotation>

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</math><img src="./67aa13d56c9e405bd9c8c9484db73c97a452776d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.253ex; height:3.343ex;" alt="{\displaystyle V_{\mathrm {dipole} }=d^{3}}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}">

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

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</math><img src="./e85ff03cbe0c7341af6b982e47e9f90d235c66ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" loading="lazy"> is the lattice spacing. The Clausius–Mossotti polarizability for each dipole is

<msub>

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<mfrac>

<mrow>

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<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle \alpha _{j}={\frac {3V_{\mathrm {dipole} }}{4\pi }}{\frac {\varepsilon _{j}-1}{\varepsilon _{j}+2}},}</annotation>

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</math><img src="./f944e8c14be2b7becbb8c3c0211d1f298f0997d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.856ex; height:6.176ex;" alt="{\displaystyle \alpha _{j}={\frac {3V_{\mathrm {dipole} }}{4\pi }}{\frac {\varepsilon _{j}-1}{\varepsilon _{j}+2}},}" loading="lazy"></dd></dl>

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

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \varepsilon _{j}}</annotation>

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</math><img src="./00f968845121a91c222089c0edebd9db9250fa2e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.993ex; height:2.343ex;" alt="{\displaystyle \varepsilon _{j}}" loading="lazy"> is the relative permittivity of the material at the dipole’s position. The dipole volume <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\mathrm {dipole} }}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle V_{\mathrm {dipole} }}</annotation>

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</math><img src="./fda9f0016eb392c67624e4a11c7c7c96cc1f5c67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.882ex; height:2.843ex;" alt="{\displaystyle V_{\mathrm {dipole} }}" loading="lazy"> is constant across all dipoles.

This formula assumes that each dipole occupies a volume embedded in an otherwise uniform dielectric medium. In most implementations of DDA the formulation is expressed in Gaussian units (CGS). In these units, the polarizability <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{j}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \alpha _{j}}</annotation>

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</math><img src="./293a364991ab1ee55c25b0f60fd9e52af7b7dbde.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.397ex; height:2.343ex;" alt="{\displaystyle \alpha _{j}}" loading="lazy"> has dimensions of volume (cm³). In the discrete dipole approximation, the total volume of the target is divided into <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

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

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</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> small cubic cells of volume <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\mathrm {dipole} }=d^{3}}">

<msub>

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</msub>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle V_{\mathrm {dipole} }=d^{3}}</annotation>

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</math><img src="./67aa13d56c9e405bd9c8c9484db73c97a452776d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.253ex; height:3.343ex;" alt="{\displaystyle V_{\mathrm {dipole} }=d^{3}}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}">

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

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</math><img src="./e85ff03cbe0c7341af6b982e47e9f90d235c66ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" loading="lazy"> is the lattice spacing and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

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

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</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> is the total number of dipoles. The total target volume is thus

<msub>

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<msup>

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</msup>

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<annotation encoding="application/x-tex">{\displaystyle V_{\mathrm {target} }=Nd^{3}.}</annotation>

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</math><img src="./4bd5287aa86f9c70a0074510b999bbb40d9b3e75.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.967ex; height:3.343ex;" alt="{\displaystyle V_{\mathrm {target} }=Nd^{3}.}" loading="lazy"></dd></dl>

To improve the accuracy of the method various corrections to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{j}}">

<msub>

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</msub>

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<annotation encoding="application/x-tex">{\displaystyle \alpha _{j}}</annotation>

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</math><img src="./293a364991ab1ee55c25b0f60fd9e52af7b7dbde.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.397ex; height:2.343ex;" alt="{\displaystyle \alpha _{j}}" loading="lazy"> are applied. These include: the lattice dispersion relation (LDR) polarizability (Draine & Goodman, 1993), which adjusts <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{j}}">

<msub>

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</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha _{j}}</annotation>

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</math><img src="./293a364991ab1ee55c25b0f60fd9e52af7b7dbde.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.397ex; height:2.343ex;" alt="{\displaystyle \alpha _{j}}" loading="lazy"> to ensure that the dispersion relation of an infinite lattice of dipoles matches that of the continuous material; the radiative reaction (RR) correction, which compensates for the fact that each dipole radiates energy and is influenced by its own radiation field.

<div class="mw-heading mw-heading3"><h3 id="Size_parameter">Size parameter</h3></div>

The size parameter is a dimensionless quantity used in scattering theory to characterize the size of a particle relative to the wavelength of the incident light. For a sphere, it is defined as:

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle x={\frac {2\pi r}{\lambda }}=kr}</annotation>

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</math><img src="./c5ae739267ca926228f1a296103b3633a4d55d98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.166ex; height:5.343ex;" alt="{\displaystyle x={\frac {2\pi r}{\lambda }}=kr}" loading="lazy"></dd></dl>

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

</mstyle>

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

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> is the size parameter (dimensionless), <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

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

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</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> is the radius of the sphere, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

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

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</math><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"> is the wavelength of light in vacuum,

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<annotation encoding="application/x-tex">{\displaystyle k={\frac {2\pi }{\lambda }}}</annotation>

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</math><img src="./f567225915e2b51c00573536e20eee76db99740b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.64ex; height:5.343ex;" alt="{\displaystyle k={\frac {2\pi }{\lambda }}}" loading="lazy"> is the wavenumber.</dd></dl>

In case of a sphere, the size parameter determines the scattering regime:

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<annotation encoding="application/x-tex">{\displaystyle x\ll 1}</annotation>

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</math><img src="./735b23270ff44ee7804e7412d368833a63a716fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.106ex; height:2.176ex;" alt="{\displaystyle x\ll 1}" loading="lazy">, <a href="Rayleigh_scattering" title="Rayleigh scattering">Rayleigh scattering</a> dominates.</li>

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<annotation encoding="application/x-tex">{\displaystyle x\sim 1}</annotation>

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</math><img src="./ee623d3d4e2e737df83255352223d37a930d9fe8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x\sim 1}" loading="lazy">, the scattering is in the regime of <a href="Mie_scattering" title="Mie scattering">Mie scattering</a>.</li>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle x\gg 1}</annotation>

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</math><img src="./03817e8d3d6b485b7304c4b481d7a352da1f4f0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.106ex; height:2.176ex;" alt="{\displaystyle x\gg 1}" loading="lazy">, the <a href="Geometrical_optics" title="Geometrical optics">geometric optics</a> approximation becomes valid.</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Effective_radius_and_dipole_discretization">Effective radius and dipole discretization</h3></div>

For nonspherical targets with the same volume as a sphere, the effective radius <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{\text{eff}}}">

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<annotation encoding="application/x-tex">{\displaystyle r_{\text{eff}}}</annotation>

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</math><img src="./c6cd0c1eae1324b5960491509843a273bea02e28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.018ex; height:2.009ex;" alt="{\displaystyle r_{\text{eff}}}" loading="lazy"> is often used in place of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

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

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</math><img src="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy">, with:

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle r_{\text{eff}}=\left({\frac {3V_{\text{tot}}}{4\pi }}\right)^{1/3}=\left({\frac {3Nd^{3}}{4\pi }}\right)^{1/3}}</annotation>

</semantics>

</math><img src="./af68fb700331d1885753012c5b9d2bf9f0f28269.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.475ex; height:6.843ex;" alt="{\displaystyle r_{\text{eff}}=\left({\frac {3V_{\text{tot}}}{4\pi }}\right)^{1/3}=\left({\frac {3Nd^{3}}{4\pi }}\right)^{1/3}}" loading="lazy"></dd></dl>

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

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> is the total number of dipoles, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e85ff03cbe0c7341af6b982e47e9f90d235c66ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" loading="lazy"> is the dipole spacing, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\text{tot}}=Nd^{3}}">

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle V_{\text{tot}}=Nd^{3}}</annotation>

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</math><img src="./3f70f1e76d55e403a3a255ede23733616c72c6fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.123ex; height:3.009ex;" alt="{\displaystyle V_{\text{tot}}=Nd^{3}}" loading="lazy"> is the total volume represented by the dipoles. This gives effective size parameter

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<annotation encoding="application/x-tex">{\displaystyle x_{\text{eff}}={\frac {2\pi r_{\text{eff}}}{\lambda }}}</annotation>

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</math><img src="./5fb2ad3676e4839be093b0e51b5fca228d28aab8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.745ex; height:5.343ex;" alt="{\displaystyle x_{\text{eff}}={\frac {2\pi r_{\text{eff}}}{\lambda }}}" loading="lazy"></dd></dl>

One convenient trick in certain DDA accuracy tests is to define wavelength as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi }">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation>

</semantics>

</math><img src="./73efd1f6493490b058097060a572606d2c550a06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }" loading="lazy">, in such a case effective radius is the same as effective size parameter.

<div class="mw-heading mw-heading3"><h3 id="Dipole-scale_size_parameter">Dipole-scale size parameter</h3></div>

Each polarizable point (dipole) occupies a cubic volume with side length <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e85ff03cbe0c7341af6b982e47e9f90d235c66ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" loading="lazy">. Analogous to the global size parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=2\pi r/\lambda }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=2\pi r/\lambda }</annotation>

</semantics>

</math><img src="./88849403ca91f06cc26bc228bc77c50ce1483279.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.489ex; height:2.843ex;" alt="{\displaystyle x=2\pi r/\lambda }" loading="lazy"> used for whole particles, one can define a local size parameter for each dipole:

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<mfrac>

<mrow>

</mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{d}=|m|kd={\frac {2\pi |m|d}{\lambda }}}</annotation>

</semantics>

</math><img src="./aa4891fd8ed1e680e7a673f983bd5cd3245fa42a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:22.26ex; height:5.843ex;" alt="{\displaystyle x_{d}=|m|kd={\frac {2\pi |m|d}{\lambda }}}" loading="lazy"></dd></dl>

This local parameter quantifies the ratio of the dipole size to the wavelength of light inside the material. For the DDA to be accurate, the field should vary slowly over the size of each dipole. This condition is satisfied when:

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{d}=|m|kd\lesssim 0.5}</annotation>

</semantics>

</math><img src="./b35bcdc6012cc9216bab8e43c8e8ad925d114af7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.352ex; height:2.843ex;" alt="{\displaystyle x_{d}=|m|kd\lesssim 0.5}" loading="lazy"></dd></dl>

This ensures that each dipole is optically small, fields vary slowly over the dipole and the polarizability formula used for each dipole is accurate. Notice that a similar parameter plays a crucial role in the <a href="Anomalous_diffraction_theory" title="Anomalous diffraction theory">anomalous diffraction theory</a> of van de Hulst, where the total phase shift experienced by light rays traveling through or around the particle is given by:

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta =2\pi (m-1){\frac {r}{\lambda }}}</annotation>

</semantics>

</math><img src="./f3d388ab9efda11395f69c9d6123d53b77cb8f6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:16.685ex; height:4.843ex;" alt="{\displaystyle \delta =2\pi (m-1){\frac {r}{\lambda }}}" loading="lazy"></dd></dl>

This describes the optical path difference introduced by the particle (or in the case of DDA by a dipole).

<div class="mw-heading mw-heading3"><h3 id="Explicit_Matrix_Form_of_the_DDA_System">Explicit Matrix Form of the DDA System</h3></div>

The Discrete Dipole Approximation (DDA) linear system is expressed as:

</mrow>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {A} \mathbf {P} =\mathbf {E} _{\mathrm {inc} }}</annotation>

</semantics>

</math><img src="./5a750e36fe200bf2f80d43c1b033fc745f140314.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.035ex; height:2.509ex;" alt="{\displaystyle \mathbf {A} \mathbf {P} =\mathbf {E} _{\mathrm {inc} }}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} \in \mathbb {C} ^{3N\times 3N}}">

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {A} \in \mathbb {C} ^{3N\times 3N}}</annotation>

</semantics>

</math><img src="./b654ea1b82d1b8ed0bd610a82521dd790733c07b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.611ex; height:2.676ex;" alt="{\displaystyle \mathbf {A} \in \mathbb {C} ^{3N\times 3N}}" loading="lazy"> is the system matrix, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} \in \mathbb {C} ^{3N}}">

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {P} \in \mathbb {C} ^{3N}}</annotation>

</semantics>

</math><img src="./e67e1b0da026ee1270074563456e3b9a8cece326.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.859ex; height:2.676ex;" alt="{\displaystyle \mathbf {P} \in \mathbb {C} ^{3N}}" loading="lazy"> is the unknown polarization vector, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} _{\mathrm {inc} }\in \mathbb {C} ^{3N}}">

<msub>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {E} _{\mathrm {inc} }\in \mathbb {C} ^{3N}}</annotation>

</semantics>

</math><img src="./32ebc9c3ea646ae27c48beff3110475ef6a3c8df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.123ex; height:3.009ex;" alt="{\displaystyle \mathbf {E} _{\mathrm {inc} }\in \mathbb {C} ^{3N}}" loading="lazy"> is the incident electric field vector.

We have

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {A} =-\mathbf {G} +\mathrm {diag} ({\boldsymbol {\alpha }}_{1}^{-1},\dots ,{\boldsymbol {\alpha }}_{N}^{-1})}</annotation>

</semantics>

</math><img src="./9bfbbb0dfc61aa647ab433a35f41d1c24e498538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.322ex; height:3.343ex;" alt="{\displaystyle \mathbf {A} =-\mathbf {G} +\mathrm {diag} ({\boldsymbol {\alpha }}_{1}^{-1},\dots ,{\boldsymbol {\alpha }}_{N}^{-1})}" loading="lazy">.</dd></dl>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f6d9c60d3cf462a9812e9a9d021d17c7bc272a5a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.101ex; height:2.176ex;" alt="{\displaystyle \mathbf {G} }" loading="lazy"> encodes interactions between dipoles via the Green’s tensor (non-local), and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {diag} ({\boldsymbol {\alpha }}_{1}^{-1},\dots ,{\boldsymbol {\alpha }}_{N}^{-1})}">

</mrow>

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {diag} ({\boldsymbol {\alpha }}_{1}^{-1},\dots ,{\boldsymbol {\alpha }}_{N}^{-1})}</annotation>

</semantics>

</math><img src="./3b10b2e1085e2fb7c80169e6637883dfa75985bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.455ex; height:3.343ex;" alt="{\displaystyle \mathrm {diag} ({\boldsymbol {\alpha }}_{1}^{-1},\dots ,{\boldsymbol {\alpha }}_{N}^{-1})}" loading="lazy"> is a block-diagonal matrix with each block <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\alpha }}_{j}^{-1}\in \mathbb {C} ^{3\times 3}}">

</mrow>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\alpha }}_{j}^{-1}\in \mathbb {C} ^{3\times 3}}</annotation>

</semantics>

</math><img src="./f8ba3e8ce5dd35747afbf067553a3e3ae06d236d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:11.775ex; height:3.676ex;" alt="{\displaystyle {\boldsymbol {\alpha }}_{j}^{-1}\in \mathbb {C} ^{3\times 3}}" loading="lazy">.

Let N be the number of dipoles. Each dipole has a polarization vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} _{j}\in \mathbb {C} ^{3}}">

<msub>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {P} _{j}\in \mathbb {C} ^{3}}</annotation>

</semantics>

</math><img src="./ba2e676506b89b05b17a67a5d9acdeaf64b9bf75.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.309ex; height:3.343ex;" alt="{\displaystyle \mathbf {P} _{j}\in \mathbb {C} ^{3}}" loading="lazy">. The total system is a matrix equation of size <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3N\times 3N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 3N\times 3N}</annotation>

</semantics>

</math><img src="./43c8eaa59b4e8cc2d17cc7f193f5d68a6b437024.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.293ex; height:2.176ex;" alt="{\displaystyle 3N\times 3N}" loading="lazy">:

<mrow>

<mtr>

<mtd>

</mrow>

</mrow>

</mrow>

</msubsup>

</mtd>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

</mrow>

</mrow>

</mrow>

</msubsup>

</mtd>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

</mrow>

</mrow>

</mrow>

</msubsup>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

</mrow>

</mrow>

</mrow>

</msubsup>

</mtd>

</mtr>

</mtable>

</mrow>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}{\boldsymbol {\alpha }}_{1}^{-1}&-\mathbf {G} _{12}&-\mathbf {G} _{13}&\cdots &-\mathbf {G} _{1N}\\-\mathbf {G} _{21}&{\boldsymbol {\alpha }}_{2}^{-1}&-\mathbf {G} _{23}&\cdots &-\mathbf {G} _{2N}\\-\mathbf {G} _{31}&-\mathbf {G} _{32}&{\boldsymbol {\alpha }}_{3}^{-1}&\cdots &-\mathbf {G} _{3N}\\\vdots &\vdots &\vdots &\ddots &\vdots \\-\mathbf {G} _{N1}&-\mathbf {G} _{N2}&-\mathbf {G} _{N3}&\cdots &{\boldsymbol {\alpha }}_{N}^{-1}\end{bmatrix}}{\begin{bmatrix}\mathbf {P} _{1}\\\mathbf {P} _{2}\\\vdots \\\mathbf {P} _{N}\end{bmatrix}}={\begin{bmatrix}\mathbf {E} _{\mathrm {inc} ,1}\\\mathbf {E} _{\mathrm {inc} ,2}\\\vdots \\\mathbf {E} _{\mathrm {inc} ,N}\end{bmatrix}}}</annotation>

</semantics>

</math><img src="./84fc0d550f25107e125542bfe2577da828d8ebd0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.005ex; width:62.139ex; height:19.176ex;" alt="{\displaystyle {\begin{bmatrix}{\boldsymbol {\alpha }}_{1}^{-1}&-\mathbf {G} _{12}&-\mathbf {G} _{13}&\cdots &-\mathbf {G} _{1N}\\-\mathbf {G} _{21}&{\boldsymbol {\alpha }}_{2}^{-1}&-\mathbf {G} _{23}&\cdots &-\mathbf {G} _{2N}\\-\mathbf {G} _{31}&-\mathbf {G} _{32}&{\boldsymbol {\alpha }}_{3}^{-1}&\cdots &-\mathbf {G} _{3N}\\\vdots &\vdots &\vdots &\ddots &\vdots \\-\mathbf {G} _{N1}&-\mathbf {G} _{N2}&-\mathbf {G} _{N3}&\cdots &{\boldsymbol {\alpha }}_{N}^{-1}\end{bmatrix}}{\begin{bmatrix}\mathbf {P} _{1}\\\mathbf {P} _{2}\\\vdots \\\mathbf {P} _{N}\end{bmatrix}}={\begin{bmatrix}\mathbf {E} _{\mathrm {inc} ,1}\\\mathbf {E} _{\mathrm {inc} ,2}\\\vdots \\\mathbf {E} _{\mathrm {inc} ,N}\end{bmatrix}}}" loading="lazy"></dd></dl>

Each block <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} _{jk}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a3c4a0872ed58170220c2b892862732e84d9f11e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.786ex; height:2.843ex;" alt="{\displaystyle \mathbf {A} _{jk}}" loading="lazy"> is a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3\times 3}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 3\times 3}</annotation>

</semantics>

</math><img src="./ddc0d4d6106875f8006be1d898512ca5843bad8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 3\times 3}" loading="lazy"> complex matrix, defined by:

<msub>

</mrow>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

</mrow>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</mrow>

</msup>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {A} _{jk}={\begin{cases}{\boldsymbol {\alpha }}_{j}^{-1}\in \mathbb {C} ^{3\times 3},&{\text{if }}j=k\\-\mathbf {G} _{jk}\in \mathbb {C} ^{3\times 3},&{\text{if }}j\neq k\end{cases}}}</annotation>

</semantics>

</math><img src="./f69233e0c8d8391db8e47dae888a30bb5c091269.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:33.034ex; height:7.509ex;" alt="{\displaystyle \mathbf {A} _{jk}={\begin{cases}{\boldsymbol {\alpha }}_{j}^{-1}\in \mathbb {C} ^{3\times 3},&{\text{if }}j=k\\-\mathbf {G} _{jk}\in \mathbb {C} ^{3\times 3},&{\text{if }}j\neq k\end{cases}}}" loading="lazy"></dd></dl>

So <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} \in \mathbb {C} ^{3N\times 3N}}">

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {A} \in \mathbb {C} ^{3N\times 3N}}</annotation>

</semantics>

</math><img src="./b654ea1b82d1b8ed0bd610a82521dd790733c07b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.611ex; height:2.676ex;" alt="{\displaystyle \mathbf {A} \in \mathbb {C} ^{3N\times 3N}}" loading="lazy"> is composed of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N\times N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N\times N}</annotation>

</semantics>

</math><img src="./99a86c5231bb3cbb863d9d428ebe9ac8db8d4ffb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.968ex; height:2.176ex;" alt="{\displaystyle N\times N}" loading="lazy"> blocks, each of size <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3\times 3}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 3\times 3}</annotation>

</semantics>

</math><img src="./ddc0d4d6106875f8006be1d898512ca5843bad8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 3\times 3}" loading="lazy">. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\alpha }}_{j}^{-1}\in \mathbb {C} ^{3\times 3}}">

</mrow>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\alpha }}_{j}^{-1}\in \mathbb {C} ^{3\times 3}}</annotation>

</semantics>

</math><img src="./f8ba3e8ce5dd35747afbf067553a3e3ae06d236d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:11.775ex; height:3.676ex;" alt="{\displaystyle {\boldsymbol {\alpha }}_{j}^{-1}\in \mathbb {C} ^{3\times 3}}" loading="lazy"> is the inverse polarizability tensor, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {G} _{jk}\in \mathbb {C} ^{3\times 3}}">

<msub>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {G} _{jk}\in \mathbb {C} ^{3\times 3}}</annotation>

</semantics>

</math><img src="./068258165887e877e8720c9632dbb87faf5f111c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.54ex; height:3.343ex;" alt="{\displaystyle \mathbf {G} _{jk}\in \mathbb {C} ^{3\times 3}}" loading="lazy"> is the dyadic Green’s tensor for interaction between dipoles <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} _{j},\mathbf {E} _{\mathrm {inc} ,j}\in \mathbb {C} ^{3}}">

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {P} _{j},\mathbf {E} _{\mathrm {inc} ,j}\in \mathbb {C} ^{3}}</annotation>

</semantics>

</math><img src="./c7169e7d349437d70219ecc9d595dc7cb8469f9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.569ex; height:3.343ex;" alt="{\displaystyle \mathbf {P} _{j},\mathbf {E} _{\mathrm {inc} ,j}\in \mathbb {C} ^{3}}" loading="lazy"> are the dipole polarization and incident electric field at dipole <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy">, respectively.

Typically dipoles are arranged on a regular grid. This implies translational invariance:

<msub>

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</mrow>

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</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

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</msub>

<msub>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {G} _{jk}=\mathbf {G} (\mathbf {r} _{j}-\mathbf {r} _{k})=\mathbf {G} _{|j-k|}}</annotation>

</semantics>

</math><img src="./92b5271e65d6fb2f7c30c03afbddef2ee9b433a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:27.077ex; height:3.176ex;" alt="{\displaystyle \mathbf {G} _{jk}=\mathbf {G} (\mathbf {r} _{j}-\mathbf {r} _{k})=\mathbf {G} _{|j-k|}}" loading="lazy"></dd></dl>

Because <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {G} _{|j-k|}=\mathbf {G} _{|k-j|}}">

<msub>

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</mrow>

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</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {G} _{|j-k|}=\mathbf {G} _{|k-j|}}</annotation>

</semantics>

</math><img src="./9e6f4c7ec82218eac7a1d2ad119e3882c4182519.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:15.219ex; height:3.009ex;" alt="{\displaystyle \mathbf {G} _{|j-k|}=\mathbf {G} _{|k-j|}}" loading="lazy">, the matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./0795cc96c75d81520a120482662b90f024c9a1a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }" loading="lazy"> is symmetric:

<msub>

</mrow>

</mrow>

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<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {A} _{jk}=\mathbf {A} _{kj}\quad {\text{for all }}j,k}</annotation>

</semantics>

</math><img src="./d693df2e796260582389c5608e6752eb98c4fb14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.599ex; height:2.843ex;" alt="{\displaystyle \mathbf {A} _{jk}=\mathbf {A} _{kj}\quad {\text{for all }}j,k}" loading="lazy"></dd></dl>

Each dipole has three vector components (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy">), so we can rearrange the unknown vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c0c250ef2a112c86b93c637dfa288c6d7f34ac3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle \mathbf {P} }" loading="lazy"> by grouping all x-components together, then y-components, then z-components:

</mrow>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

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</mrow>

</mrow>

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</mtd>

</mtr>

<mtr>

<mtd>

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</mrow>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

<mtext>where</mtext>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

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</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

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</mtd>

</mtr>

</mtable>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

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</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

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</mtable>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<mrow>

<mtr>

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</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

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</mrow>

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</mtd>

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<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

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</mtd>

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</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {P} ={\begin{bmatrix}\mathbf {P} _{x}\\\mathbf {P} _{y}\\\mathbf {P} _{z}\end{bmatrix}}\in \mathbb {C} ^{3N}\quad {\text{where}}\quad \mathbf {P} _{x}={\begin{bmatrix}P_{1x}\\P_{2x}\\\vdots \\P_{Nx}\end{bmatrix}},\quad \mathbf {P} _{y}={\begin{bmatrix}P_{1y}\\P_{2y}\\\vdots \\P_{Ny}\end{bmatrix}},\quad \mathbf {P} _{z}={\begin{bmatrix}P_{1z}\\P_{2z}\\\vdots \\P_{Nz}\end{bmatrix}}}</annotation>

</semantics>

</math><img src="./5d934752564c05035af543c708a381e9c6d8f10a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:77.747ex; height:14.509ex;" alt="{\displaystyle \mathbf {P} ={\begin{bmatrix}\mathbf {P} _{x}\\\mathbf {P} _{y}\\\mathbf {P} _{z}\end{bmatrix}}\in \mathbb {C} ^{3N}\quad {\text{where}}\quad \mathbf {P} _{x}={\begin{bmatrix}P_{1x}\\P_{2x}\\\vdots \\P_{Nx}\end{bmatrix}},\quad \mathbf {P} _{y}={\begin{bmatrix}P_{1y}\\P_{2y}\\\vdots \\P_{Ny}\end{bmatrix}},\quad \mathbf {P} _{z}={\begin{bmatrix}P_{1z}\\P_{2z}\\\vdots \\P_{Nz}\end{bmatrix}}}" loading="lazy"></dd></dl>

Similarly, the incident field can be grouped as:

<msub>

</mrow>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

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</mrow>

</mrow>

</msubsup>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mrow>

</mrow>

</msubsup>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mrow>

</mrow>

</msubsup>

</mtd>

</mtr>

</mtable>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {E} _{\mathrm {inc} }={\begin{bmatrix}\mathbf {E} _{x}^{\mathrm {inc} }\\\mathbf {E} _{y}^{\mathrm {inc} }\\\mathbf {E} _{z}^{\mathrm {inc} }\end{bmatrix}}\in \mathbb {C} ^{3N}}</annotation>

</semantics>

</math><img src="./f57819d5a854abb932730385cd967c21d9759f2e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.47ex; margin-bottom: -0.201ex; width:22.164ex; height:10.509ex;" alt="{\displaystyle \mathbf {E} _{\mathrm {inc} }={\begin{bmatrix}\mathbf {E} _{x}^{\mathrm {inc} }\\\mathbf {E} _{y}^{\mathrm {inc} }\\\mathbf {E} _{z}^{\mathrm {inc} }\end{bmatrix}}\in \mathbb {C} ^{3N}}" loading="lazy"></dd></dl>

Because the system is linear, we can equivalently rewrite it in block matrix form, that describe how the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./7ed48a5e36207156fb792fa79d29925d2f7901e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" loading="lazy">-component of polarization affects the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy">-component of the resulting field:

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</mrow>

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</mtd>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

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</mrow>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

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</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

<mrow>

<mtr>

<mtd>

</mrow>

</mrow>

</mrow>

</msubsup>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mrow>

</mrow>

</msubsup>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mrow>

</mrow>

</msubsup>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}\mathbf {A} _{xx}&\mathbf {A} _{xy}&\mathbf {A} _{xz}\\\mathbf {A} _{yx}&\mathbf {A} _{yy}&\mathbf {A} _{yz}\\\mathbf {A} _{zx}&\mathbf {A} _{zy}&\mathbf {A} _{zz}\end{bmatrix}}{\begin{bmatrix}\mathbf {P} _{x}\\\mathbf {P} _{y}\\\mathbf {P} _{z}\end{bmatrix}}={\begin{bmatrix}\mathbf {E} _{x}^{\mathrm {inc} }\\\mathbf {E} _{y}^{\mathrm {inc} }\\\mathbf {E} _{z}^{\mathrm {inc} }\end{bmatrix}}}</annotation>

</semantics>

</math><img src="./781ddbf876e83d0a28db7959a37b384843a8c332.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.47ex; margin-bottom: -0.201ex; width:38.493ex; height:10.509ex;" alt="{\displaystyle {\begin{bmatrix}\mathbf {A} _{xx}&\mathbf {A} _{xy}&\mathbf {A} _{xz}\\\mathbf {A} _{yx}&\mathbf {A} _{yy}&\mathbf {A} _{yz}\\\mathbf {A} _{zx}&\mathbf {A} _{zy}&\mathbf {A} _{zz}\end{bmatrix}}{\begin{bmatrix}\mathbf {P} _{x}\\\mathbf {P} _{y}\\\mathbf {P} _{z}\end{bmatrix}}={\begin{bmatrix}\mathbf {E} _{x}^{\mathrm {inc} }\\\mathbf {E} _{y}^{\mathrm {inc} }\\\mathbf {E} _{z}^{\mathrm {inc} }\end{bmatrix}}}" loading="lazy">

The expanded form of the equations is:

<mtr>

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<msub>

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<msub>

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<msub>

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<msup>

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</mrow>

</msup>

</mrow>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

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</mrow>

</mrow>

</msubsup>

</mtd>

</mtr>

<mtr>

<mtd>

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</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

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</msub>

<msub>

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</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mrow>

</mrow>

</mrow>

</msubsup>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mrow>

</mrow>

</mrow>

</msubsup>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}-\mathbf {G} _{xx}\mathbf {P} _{x}-\mathbf {G} _{xy}\mathbf {P} _{y}-\mathbf {G} _{xz}\mathbf {P} _{z}+({\boldsymbol {\alpha }}^{-1}\mathbf {P} )_{x}&=\mathbf {E} _{x}^{\mathrm {inc} }\\-\mathbf {G} _{yx}\mathbf {P} _{x}-\mathbf {G} _{yy}\mathbf {P} _{y}-\mathbf {G} _{yz}\mathbf {P} _{z}+({\boldsymbol {\alpha }}^{-1}\mathbf {P} )_{y}&=\mathbf {E} _{y}^{\mathrm {inc} }\\-\mathbf {G} _{zx}\mathbf {P} _{x}-\mathbf {G} _{zy}\mathbf {P} _{y}-\mathbf {G} _{zz}\mathbf {P} _{z}+({\boldsymbol {\alpha }}^{-1}\mathbf {P} )_{z}&=\mathbf {E} _{z}^{\mathrm {inc} }\end{aligned}}}</annotation>

</semantics>

</math><img src="./c1d969a53c1a78f9b8c7fbd490cce134c9ae5743.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.341ex; margin-bottom: -0.331ex; width:48.23ex; height:10.509ex;" alt="{\displaystyle {\begin{aligned}-\mathbf {G} _{xx}\mathbf {P} _{x}-\mathbf {G} _{xy}\mathbf {P} _{y}-\mathbf {G} _{xz}\mathbf {P} _{z}+({\boldsymbol {\alpha }}^{-1}\mathbf {P} )_{x}&=\mathbf {E} _{x}^{\mathrm {inc} }\\-\mathbf {G} _{yx}\mathbf {P} _{x}-\mathbf {G} _{yy}\mathbf {P} _{y}-\mathbf {G} _{yz}\mathbf {P} _{z}+({\boldsymbol {\alpha }}^{-1}\mathbf {P} )_{y}&=\mathbf {E} _{y}^{\mathrm {inc} }\\-\mathbf {G} _{zx}\mathbf {P} _{x}-\mathbf {G} _{zy}\mathbf {P} _{y}-\mathbf {G} _{zz}\mathbf {P} _{z}+({\boldsymbol {\alpha }}^{-1}\mathbf {P} )_{z}&=\mathbf {E} _{z}^{\mathrm {inc} }\end{aligned}}}" loading="lazy">

Each block <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {G} _{ij}\in \mathbb {C} ^{N\times N}}">

<msub>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {G} _{ij}\in \mathbb {C} ^{N\times N}}</annotation>

</semantics>

</math><img src="./d99d055ff6c7f6819ca98044de7f7cd29ada0a59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.526ex; height:3.343ex;" alt="{\displaystyle \mathbf {G} _{ij}\in \mathbb {C} ^{N\times N}}" loading="lazy"> and the total system size is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3N\times 3N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 3N\times 3N}</annotation>

</semantics>

</math><img src="./43c8eaa59b4e8cc2d17cc7f193f5d68a6b437024.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.293ex; height:2.176ex;" alt="{\displaystyle 3N\times 3N}" loading="lazy">. The interaction matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {G} \in \mathbb {C} ^{3N\times 3N}}">

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {G} \in \mathbb {C} ^{3N\times 3N}}</annotation>

</semantics>

</math><img src="./5bdc22599ae11111445e3d98047d8106b1da8235.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.693ex; height:2.676ex;" alt="{\displaystyle \mathbf {G} \in \mathbb {C} ^{3N\times 3N}}" loading="lazy"> is composed of 9 blocks:

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {G} _{xx},\mathbf {G} _{xy},\mathbf {G} _{xz},\mathbf {G} _{yx},\mathbf {G} _{yy},\mathbf {G} _{yz},\mathbf {G} _{zx},\mathbf {G} _{zy},\mathbf {G} _{zz}}</annotation>

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</math><img src="./4217f759eae82f92972f9dfc31aa96c0df877439.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:44.429ex; height:2.843ex;" alt="{\displaystyle \mathbf {G} _{xx},\mathbf {G} _{xy},\mathbf {G} _{xz},\mathbf {G} _{yx},\mathbf {G} _{yy},\mathbf {G} _{yz},\mathbf {G} _{zx},\mathbf {G} _{zy},\mathbf {G} _{zz}}" loading="lazy"> (only 6 of them need to be evaluated due to symmetry).

Each matrix-vector multiplication <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {G} _{\alpha \beta }\mathbf {P} _{\beta }}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {G} _{\alpha \beta }\mathbf {P} _{\beta }}</annotation>

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</math><img src="./a40c7a9e924220ec27035db5380ac273c3d14c03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.328ex; height:2.843ex;" alt="{\displaystyle \mathbf {G} _{\alpha \beta }\mathbf {P} _{\beta }}" loading="lazy"> can be computed as a convolution when the dipoles are arranged on a regular grid, allowing the use of Fast Fourier Transforms (FFTs) to accelerate the solution.

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\beta }}_{j}={\boldsymbol {\alpha }}_{j}^{-1}}">

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<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\beta }}_{j}={\boldsymbol {\alpha }}_{j}^{-1}}</annotation>

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</math><img src="./02af206050c42932c51561bb47ee13e4df4aa1fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:9.644ex; height:3.676ex;" alt="{\displaystyle {\boldsymbol {\beta }}_{j}={\boldsymbol {\alpha }}_{j}^{-1}}" loading="lazy"> denote the inverse polarizability tensor for dipole <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}">

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

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</math><img src="./2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy">. Each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\beta }}_{j}}">

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<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\beta }}_{j}}</annotation>

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</math><img src="./42a14eb92be02addb4d056166a60cc0bf3e471b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.444ex; height:3.009ex;" alt="{\displaystyle {\boldsymbol {\beta }}_{j}}" loading="lazy"> is a complex-valued <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3\times 3}">

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<annotation encoding="application/x-tex">{\displaystyle 3\times 3}</annotation>

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</math><img src="./ddc0d4d6106875f8006be1d898512ca5843bad8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 3\times 3}" loading="lazy"> matrix. This gives:

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<annotation encoding="application/x-tex">{\displaystyle ({\boldsymbol {\beta }}\mathbf {P} )_{x}=\mathrm {diag} (\beta _{1,xx},\dots ,\beta _{N,xx})\,\mathbf {P} _{x}+\mathrm {diag} (\beta _{1,xy},\dots ,\beta _{N,xy})\,\mathbf {P} _{y}+\mathrm {diag} (\beta _{1,xz},\dots ,\beta _{N,xz})\,\mathbf {P} _{z}}</annotation>

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</math><img src="./339f40753234884b9e829dbaceef98fd3144af8f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:88.313ex; height:3.009ex;" alt="{\displaystyle ({\boldsymbol {\beta }}\mathbf {P} )_{x}=\mathrm {diag} (\beta _{1,xx},\dots ,\beta _{N,xx})\,\mathbf {P} _{x}+\mathrm {diag} (\beta _{1,xy},\dots ,\beta _{N,xy})\,\mathbf {P} _{y}+\mathrm {diag} (\beta _{1,xz},\dots ,\beta _{N,xz})\,\mathbf {P} _{z}}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle ({\boldsymbol {\beta }}\mathbf {P} )_{y}=\mathrm {diag} (\beta _{1,yx},\dots ,\beta _{N,yx})\,\mathbf {P} _{x}+\mathrm {diag} (\beta _{1,yy},\dots ,\beta _{N,yy})\,\mathbf {P} _{y}+\mathrm {diag} (\beta _{1,yz},\dots ,\beta _{N,yz})\,\mathbf {P} _{z}}</annotation>

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</math><img src="./a11bc846480033ae338243169941261ab3f4db88.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:87.451ex; height:3.009ex;" alt="{\displaystyle ({\boldsymbol {\beta }}\mathbf {P} )_{y}=\mathrm {diag} (\beta _{1,yx},\dots ,\beta _{N,yx})\,\mathbf {P} _{x}+\mathrm {diag} (\beta _{1,yy},\dots ,\beta _{N,yy})\,\mathbf {P} _{y}+\mathrm {diag} (\beta _{1,yz},\dots ,\beta _{N,yz})\,\mathbf {P} _{z}}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle ({\boldsymbol {\beta }}\mathbf {P} )_{z}=\mathrm {diag} (\beta _{1,zx},\dots ,\beta _{N,zx})\,\mathbf {P} _{x}+\mathrm {diag} (\beta _{1,zy},\dots ,\beta _{N,zy})\,\mathbf {P} _{y}+\mathrm {diag} (\beta _{1,zz},\dots ,\beta _{N,zz})\,\mathbf {P} _{z}}</annotation>

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</math><img src="./52d5cb3081a88520b811b1622fdd3447b3c1531a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:87.117ex; height:3.009ex;" alt="{\displaystyle ({\boldsymbol {\beta }}\mathbf {P} )_{z}=\mathrm {diag} (\beta _{1,zx},\dots ,\beta _{N,zx})\,\mathbf {P} _{x}+\mathrm {diag} (\beta _{1,zy},\dots ,\beta _{N,zy})\,\mathbf {P} _{y}+\mathrm {diag} (\beta _{1,zz},\dots ,\beta _{N,zz})\,\mathbf {P} _{z}}" loading="lazy">

In the special case of an isotropic and homogeneous particle, the polarizabilities <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\alpha }}_{j}}">

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<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\alpha }}_{j}}</annotation>

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</math><img src="./9f0838e683a340cd729f1b58ef1c7720784b5ae2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.678ex; height:2.343ex;" alt="{\displaystyle {\boldsymbol {\alpha }}_{j}}" loading="lazy"> are identical for all dipoles and proportional to the identity matrix: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\alpha }}_{j}=\alpha \,\mathbf {I} }">

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<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\alpha }}_{j}=\alpha \,\mathbf {I} }</annotation>

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</math><img src="./560ec0af12670f9f8e1894742d77bab0d8d832e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.665ex; height:2.843ex;" alt="{\displaystyle {\boldsymbol {\alpha }}_{j}=\alpha \,\mathbf {I} }" loading="lazy">. Then, the inverse becomes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\beta }}_{j}=\alpha ^{-1}\,\mathbf {I} }">

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<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\beta }}_{j}=\alpha ^{-1}\,\mathbf {I} }</annotation>

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</math><img src="./542b4bf12ede43ffb11b40c5e39ef3012a6ddbb9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:10.764ex; height:3.509ex;" alt="{\displaystyle {\boldsymbol {\beta }}_{j}=\alpha ^{-1}\,\mathbf {I} }" loading="lazy">, all off-diagonal elements vanish, and the expressions reduce to a simple element-wise division:

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<annotation encoding="application/x-tex">{\displaystyle ({\boldsymbol {\beta }}\mathbf {P} )={\frac {1}{\alpha }}\,\mathbf {P} }</annotation>

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</math><img src="./2c8fab1497844e4c6788e8ef18638b5dd60a6068.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.806ex; height:5.176ex;" alt="{\displaystyle ({\boldsymbol {\beta }}\mathbf {P} )={\frac {1}{\alpha }}\,\mathbf {P} }" loading="lazy">

Note on practical implementation. In Fortran and MATLAB, arrays such as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} (n_{x},n_{y},n_{z},3)}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {P} (n_{x},n_{y},n_{z},3)}</annotation>

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</math><img src="./c4a947341ab8c5eb89365b874f44ba3ea140a4f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.308ex; height:3.009ex;" alt="{\displaystyle \mathbf {P} (n_{x},n_{y},n_{z},3)}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} ^{\mathrm {inc} }(n_{x},n_{y},n_{z},3)}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {E} ^{\mathrm {inc} }(n_{x},n_{y},n_{z},3)}</annotation>

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</math><img src="./6fcb4b7a1bd2a9433dd9741df5becaef7d39a7a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.572ex; height:3.343ex;" alt="{\displaystyle \mathbf {E} ^{\mathrm {inc} }(n_{x},n_{y},n_{z},3)}" loading="lazy"> are stored in column-major order, where the first index varies fastest in memory (anti-lexicographic). This means that all x-components <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} _{x}=\mathbf {P} (:,:,:,1)}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {P} _{x}=\mathbf {P} (:,:,:,1)}</annotation>

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</math><img src="./b123681dc82c9c8663010f459a36b39e15e11e0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.938ex; height:2.843ex;" alt="{\displaystyle \mathbf {P} _{x}=\mathbf {P} (:,:,:,1)}" loading="lazy"> are contiguous in memory, followed by all y-components <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} _{y}=\mathbf {P} (:,:,:,2)}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {P} _{y}=\mathbf {P} (:,:,:,2)}</annotation>

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</math><img src="./2f4a6cc45564b8873b880c23157d266320d89b82.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.815ex; height:3.009ex;" alt="{\displaystyle \mathbf {P} _{y}=\mathbf {P} (:,:,:,2)}" loading="lazy">, and then all z-components <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} _{z}=\mathbf {P} (:,:,:,3)}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {P} _{z}=\mathbf {P} (:,:,:,3)}</annotation>

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</math><img src="./9cde624330f05d65506e3e4da168c56f5a7ffdf5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.768ex; height:2.843ex;" alt="{\displaystyle \mathbf {P} _{z}=\mathbf {P} (:,:,:,3)}" loading="lazy">. In contrast, Python (NumPy) uses row-major order by default (lexicographic, last index varies fastest). To achieve the same contiguous layout of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} _{x}}">

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

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</math><img src="./a435b447b1c2895cca26c393aced83fbe8cff001.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.999ex; height:2.509ex;" alt="{\displaystyle \mathbf {P} _{x}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} _{y}}">

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

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</math><img src="./a66416826a5dc416c684b806049d990fd8e4368d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.876ex; height:2.843ex;" alt="{\displaystyle \mathbf {P} _{y}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} _{z}}">

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

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</math><img src="./1465633729955281bf9100a7ee3b496ab85f238b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.828ex; height:2.509ex;" alt="{\displaystyle \mathbf {P} _{z}}" loading="lazy"> in memory, the array should be defined in Python as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} (3,n_{x},n_{y},n_{z})}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {P} (3,n_{x},n_{y},n_{z})}</annotation>

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</math><img src="./eae2794f09a48a7ac58b05080c87e35e4b376367.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.308ex; height:3.009ex;" alt="{\displaystyle \mathbf {P} (3,n_{x},n_{y},n_{z})}" loading="lazy">, with the vector component index (x, y, z) first. This ensures that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} _{x}=\mathbf {P} [0,:,:,:]}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {P} _{x}=\mathbf {P} [0,:,:,:]}</annotation>

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</math><img src="./c7f7806f1aca8d20e87b1f68fe289e628c0ead91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.423ex; height:2.843ex;" alt="{\displaystyle \mathbf {P} _{x}=\mathbf {P} [0,:,:,:]}" loading="lazy"> is stored contiguously in memory, followed by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} _{y}}">

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</math><img src="./a66416826a5dc416c684b806049d990fd8e4368d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.876ex; height:2.843ex;" alt="{\displaystyle \mathbf {P} _{y}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} _{z}}">

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<div class="mw-heading mw-heading2"><h2 id="Conjugate_gradient_iteration_schemes_and_preconditioning">Conjugate gradient iteration schemes and preconditioning</h2></div>

The solution of the linear system <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} \cdot \mathbf {P} =\mathbf {E} ^{\mathrm {inc} }}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {A} \cdot \mathbf {P} =\mathbf {E} ^{\mathrm {inc} }}</annotation>

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</math><img src="./18fc1741ab23d7cdc96e88ef85450b1183ea1e8c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.714ex; height:2.676ex;" alt="{\displaystyle \mathbf {A} \cdot \mathbf {P} =\mathbf {E} ^{\mathrm {inc} }}" loading="lazy"> in the DDA is typically performed using iterative methods. These methods aim to minimize the residual vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} =\mathbf {E} ^{\mathrm {inc} }-\mathbf {A} \cdot \mathbf {P} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {r} =\mathbf {E} ^{\mathrm {inc} }-\mathbf {A} \cdot \mathbf {P} }</annotation>

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</math><img src="./5dcc19b8aa97ad7d398e59b150be497474aa09dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.657ex; height:2.843ex;" alt="{\displaystyle \mathbf {r} =\mathbf {E} ^{\mathrm {inc} }-\mathbf {A} \cdot \mathbf {P} }" loading="lazy"> through successive approximations of the polarization vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} }">

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

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</math><img src="./c0c250ef2a112c86b93c637dfa288c6d7f34ac3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle \mathbf {P} }" loading="lazy">. Among the earliest implementations were those based on direct matrix inversion,<a href="#cite_note-purcell1973-2">[2]</a> as well as the use of the conjugate gradient (CG) algorithm of Petravic and Kuo-Petravic.<a href="#cite_note-Petravic1979-17">[17]</a> Subsequently, various conjugate gradient methods have been explored and improved for DDA applications.<a href="#cite_note-chaumet2024-18">[18]</a> These methods are particularly well-suited for large systems because they require only the matrix-vector product <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} \cdot \mathbf {P} }">

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</mstyle>

</mrow>

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

</semantics>

</math><img src="./b7601f2b126e38738edadb95d3f196370a5640f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.525ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} \cdot \mathbf {P} }" loading="lazy"> and do not require storing the full matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} }">

</mrow>

</mstyle>

</mrow>

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

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</math><img src="./0795cc96c75d81520a120482662b90f024c9a1a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }" loading="lazy"> explicitly.

In practice, the dominant computational cost in DDA arises from the repeated evaluation of matrix-vector products during the iteration process. When the vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} }">

</mrow>

</mstyle>

</mrow>

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

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</math><img src="./c0c250ef2a112c86b93c637dfa288c6d7f34ac3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle \mathbf {P} }" loading="lazy"> is stored in component-block form (as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} _{x}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

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

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</math><img src="./a435b447b1c2895cca26c393aced83fbe8cff001.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.999ex; height:2.509ex;" alt="{\displaystyle \mathbf {P} _{x}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} _{y}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

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

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</math><img src="./a66416826a5dc416c684b806049d990fd8e4368d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.876ex; height:2.843ex;" alt="{\displaystyle \mathbf {P} _{y}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} _{z}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

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

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</math><img src="./1465633729955281bf9100a7ee3b496ab85f238b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.828ex; height:2.509ex;" alt="{\displaystyle \mathbf {P} _{z}}" loading="lazy">), the action of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./0795cc96c75d81520a120482662b90f024c9a1a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }" loading="lazy"> reduces to evaluating nine sub-products of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} _{\alpha \beta }\mathbf {P} _{\beta }}">

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {A} _{\alpha \beta }\mathbf {P} _{\beta }}</annotation>

</semantics>

</math><img src="./342e9247be9811604c3d34d22a0ebb266177fe9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.246ex; height:2.843ex;" alt="{\displaystyle \mathbf {A} _{\alpha \beta }\mathbf {P} _{\beta }}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ,\beta \in \{x,y,z\}}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \alpha ,\beta \in \{x,y,z\}}</annotation>

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</math><img src="./c575e7562bdabd6137dc0c4757e8747eae859116.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.66ex; height:2.843ex;" alt="{\displaystyle \alpha ,\beta \in \{x,y,z\}}" loading="lazy">. These operations can be computed efficiently using convolution and FFT-based techniques when the dipole geometry is grid-based.

<div class="mw-heading mw-heading2"><h2 id="Fast_Fourier_Transform_for_fast_convolution_calculations">Fast Fourier Transform for fast convolution calculations</h2></div>

The use of the Fast Fourier Transform (FFT) to accelerate convolution operations in the discrete dipole approximation (DDA) was introduced by Goodman, Draine, and Flatau in 1991<a href="#cite_note-Goodman1991-19">[19]</a>. Their approach utilized a 3D FFT algorithm (GPFA) developed by Clive Temperton<a href="#cite_note-Temperton1983-20">[20]</a>, and involved extending the interaction matrix to twice its original dimensions. This extension was accomplished by flipping and mirroring the Green’s function blocks to incorporate negative lags, allowing FFT-based convolution. The technique of sign-flipping and block extension became a foundational step in efficient implementations of DDA. A similar variant was adopted in the 2021 MATLAB implementation by Shabaninezhad and Ramakrishna<a href="#cite_note-matlab2021-21">[21]</a>.

Several variants have been proposed since then. In 2001, Barrowes, Teixeira, and Kong<a href="#cite_note-Barrowes2001-22">[22]</a> introduced a method based on block reordering, zero padding, and a reconstruction algorithm to minimize memory requirements. In 2009, McDonald, Golden, and Jennings<a href="#cite_note-mcdonald2009-23">[23]</a> proposed a different scheme utilizing sequences of 1D FFTs, extending the interaction matrix separately in the x, y, and z directions. They argued that their approach leads to reduced memory consumption.

More generally, advanced FFT-based convolution methods have been developed in the machine learning and numerical analysis communities, offering potential benefits for DDA solvers as well. These include FlashFFTConv<a href="#cite_note-fu2023flashfftconv-24">[24]</a> and frequency-domain low-rank techniques<a href="#cite_note-bowman2011efficient-25">[25]</a> that aim to reduce the computational burden of large-scale convolutions.

<div class="mw-heading mw-heading2"><h2 id="Thermal_discrete_dipole_approximation">Thermal discrete dipole approximation</h2></div>

Thermal discrete dipole approximation is an extension of the original DDA to simulations of near-field heat transfer between 3D arbitrarily-shaped objects.<a href="#cite_note-edalatpour2015-26">[26]</a><a href="#cite_note-Moncada-Villa2022-27">[27]</a>

<div class="mw-heading mw-heading2"><h2 id="Discrete_dipole_approximation_codes">Discrete dipole approximation codes</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Discrete_dipole_approximation_codes" title="Discrete dipole approximation codes">Discrete dipole approximation codes</a></div>

Most of the codes apply to arbitrary-shaped inhomogeneous nonmagnetic particles and particle systems in free space or homogeneous dielectric host medium. The calculated quantities typically include the <a href="Mueller_calculus#Mueller_matrices" title="Mueller calculus">Mueller matrices</a>, <a href="Cross_section_(physics)#Scattering_of_light" title="Cross section (physics)">integral cross-sections</a> (extinction, absorption, and scattering), internal fields and angle-resolved scattered fields (phase function). There are some published comparisons of existing DDA codes.<a href="#cite_note-penttila2007-14">[14]</a>

<div class="mw-heading mw-heading2"><h2 id="Gallery_of_shapes">Gallery of shapes</h2></div>

<div class="gallerytext">Scattering by periodic structures such as slabs, gratings, of periodic cubes placed on a surface, can be solved in the discrete dipole approximation.</div>

</li>

<div class="gallerytext">Scattering by infinite object (such as cylinder) can be solved in the discrete dipole approximation.</div>

</li>

</ul>

<ul><li><a href="Computational_electromagnetics" title="Computational electromagnetics">Computational electromagnetics</a></li>

<li><a href="Mie_theory" class="mw-redirect" title="Mie theory">Mie theory</a></li>

<li><a href="Finite-difference_time-domain_method" title="Finite-difference time-domain method">Finite-difference time-domain method</a></li>

<li><a href="Method_of_moments_(electromagnetics)" title="Method of moments (electromagnetics)">Method of moments (electromagnetics)</a></li></ul>

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

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