Abstract

We introduce a numerical method that enables efficient modeling of light scattering by large, disordered ensembles of non-spherical particles incorporated in stratified media, including when the particles are in close vicinity to each other, to planar interfaces, and/or to localized light sources. The method consists of finding a small set of fictitious polarizable elements—or numerical dipoles—that quantitatively reproduces the field scattered by an individual particle for any excitation and at an arbitrary distance from the particle surface. The set of numerical dipoles is described by a global polarizability matrix that is determined numerically by solving an inverse problem relying on fullwave simulations. The latter are classical and may be performed with any Maxwell’s equations solver. Spatial non-locality is an important feature of the numerical dipoles set, providing additional degrees of freedom compared to classical coupled dipoles to reconstruct complex scattered fields. Once the polarizability matrix describing scattering by an individual particle is determined, the multiple scattering problem by ensembles of such particles in stratified media can be solved using a Green tensor formalism and only a few numerical dipoles, thereby with a low physical memory usage, even for dense systems in close vicinity to interfaces. The performance of the method is studied with the example of large high-aspect-ratio high-index dielectric cylinders. The method is easy to implement and may offer new possibilities for the study of complex nanostructured surfaces, which are becoming widespread in emerging photonic technologies.

© 2019 Optical Society of America

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References

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2017 (3)

M. Fruhnert, I. Fernandez-Corbaton, V. Yannopapas, and C. Rockstuhl, “Computing the T-matrix of a scattering object with multiple plane wave illuminations,” Beilstein J. Nanotechnol. 8, 614–626 (2017).
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A. Egel, Y. Eremin, T. Wriedt, D. Theobald, U. Lemmer, and G. Gomard, “Extending the applicability of the T-matrix method to light scattering by flat particles on a substrate via truncation of Sommerfeld integrals,” J. Quant. Spectrosc. Radiat. Transfer 202, 279–285 (2017).
[Crossref]

D. Theobald, A. Egel, G. Gomard, and U. Lemmer, “Plane-wave coupling formalism for T-matrix simulations of light scattering by nonspherical particles,” Phys. Rev. A 96, 033822 (2017).
[Crossref]

2016 (7)

J. Yang, J.-P. Hugonin, and P. Lalanne, “Near-to-far field transformations for radiative and guided waves,” ACS Photon. 3, 395–402 (2016).
[Crossref]

G. Gomard, J. B. Preinfalk, A. Egel, and U. Lemmer, “Photon management in solution-processed organic light-emitting diodes: a review of light outcoupling micro-and nanostructures,” J. Photon. Energy 6, 030901 (2016).
[Crossref]

A. Jouanin, J. P. Hugonin, and P. Lalanne, “Designer colloidal layers of disordered plasmonic nanoparticles for light extraction,” Adv. Funct. Mater. 26, 6215–6223 (2016).
[Crossref]

A. Baron, A. Aradian, V. Ponsinet, and P. Barois, “Self-assembled optical metamaterials,” Opt. Laser Technol. 82, 94–100 (2016).
[Crossref]

A. Egel, S. W. Kettlitz, and U. Lemmer, “Efficient evaluation of Sommerfeld integrals for the optical simulation of many scattering particles in planarly layered media,” J. Opt. Soc. Am. A 33, 698–706 (2016).
[Crossref]

O. Leseur, R. Pierrat, and R. Carminati, “High-density hyperuniform materials can be transparent,” Optica 3, 763–767 (2016).
[Crossref]

A. Egel, D. Theobald, Y. Donie, U. Lemmer, and G. Gomard, “Light scattering by oblate particles near planar interfaces: on the validity of the T-matrix approach,” Opt. Express 24, 25154–25168 (2016).
[Crossref]

2015 (4)

B. Gallinet, J. Butet, and O. J. Martin, “Numerical methods for nanophotonics: standard problems and future challenges,” Laser Photon. Rev. 9, 577–603 (2015).
[Crossref]

A. E. Miroshnichenko, A. B. Evlyukhin, Y. S. Kivshar, and B. N. Chichkov, “Substrate-induced resonant magnetoelectric effects for dielectric nanoparticles,” ACS Photon. 2, 1423–1428 (2015).
[Crossref]

G. M. Akselrod, J. Huang, T. B. Hoang, P. T. Bowen, L. Su, D. R. Smith, and M. H. Mikkelsen, “Large-area metasurface perfect absorbers from visible to near-infrared,” Adv. Mater. 27, 8028–8034 (2015).
[Crossref]

R. Faggiani, J. Yang, and P. Lalanne, “Quenching, plasmonic, and radiative decays in nanogap emitting devices,” ACS Photon. 2, 1739–1744 (2015).
[Crossref]

2014 (3)

G. Chardon, A. Cohen, and L. Daudet, “Sampling and reconstruction of solutions to the Helmholtz equation,” Sampl. Theory Signal Image Process. 13, 67–90 (2014).

D. M. Solis, J. M. Taboada, F. Obelleiro, L. M. Liz-Marzan, and F. J. Garcia de Abajo, “Toward ultimate nanoplasmonics modeling,” ACS Nano 8, 7559–7570 (2014).
[Crossref]

M. Langlais, J.-P. Hugonin, M. Besbes, and P. Ben-Abdallah, “Cooperative electromagnetic interactions between nanoparticles for solar energy harvesting,” Opt. Express 22, A577–A588 (2014).
[Crossref]

2013 (1)

Y. H. Fu, A. I. Kuznetsov, A. E. Miroshnichenko, Y. F. Yu, and B. Luk’yanchuk, “Directional visible light scattering by silicon nanoparticles,” Nat. Commun. 4, 1527 (2013).
[Crossref]

2012 (1)

E. Castanié, R. Vincent, R. Pierrat, and R. Carminati, “Absorption by an optical dipole antenna in a structured environment,” Int. J. Opt. 2012, 452047 (2012).
[Crossref]

2011 (3)

C. Forestiere, G. Iadarola, L. Dal Negro, and G. Miano, “Near-field calculation based on the T-matrix method with discrete sources,” J. Quant. Spectrosc. Radiat. Transfer 112, 2384–2394 (2011).
[Crossref]

T. Chung, S.-Y. Lee, E. Y. Song, H. Chun, and B. Lee, “Plasmonic nanostructures for nano-scale bio-sensing,” Sensors 11, 10907–10929 (2011).
[Crossref]

S. Mühlig, C. Rockstuhl, V. Yannopapas, T. Bürgi, N. Shalkevich, and F. Lederer, “Optical properties of a fabricated self-assembled bottom-up bulk metamaterial,” Opt. Express 19, 9607–9616 (2011).
[Crossref]

2010 (2)

A. Doicu and T. Wriedt, “Near-field computation using the null-field method,” J. Quant. Spectrosc. Radiat. Transfer 111, 466–473 (2010).
[Crossref]

T. K. Sau and A. L. Rogach, “Nonspherical noble metal nanoparticles: colloid-chemical synthesis and morphology control,” Adv. Mater. 22, 1781–1804 (2010).
[Crossref]

2009 (3)

N. A. Mirin and N. J. Halas, “Light-bending nanoparticles,” Nano Lett. 9, 1255–1259 (2009).
[Crossref]

L. Beghou, B. Liu, L. Pichon, and F. Costa, “Synthesis of equivalent 3-d models from near field measurements—application to the EMC of power printed circuit boards,” IEEE Trans. Magn. 45, 1650–1653 (2009).
[Crossref]

A. M. Kern and O. J. Martin, “Surface integral formulation for 3d simulations of plasmonic and high permittivity nanostructures,” J. Opt. Soc. Am. A 26, 732–740 (2009).
[Crossref]

2008 (1)

2007 (1)

Y. A. Eremin and A. G. Sveshnikov, “Mathematical models in nanooptics and biophotonics based on the discrete sources method,” Comput. Math. Math. Phys. 47, 262–279 (2007).
[Crossref]

2005 (1)

Y. Xia and N. J. Halas, “Shape-controlled synthesis and surface plasmonic properties of metallic nanostructures,” MRS Bull. 30(5), 338–348 (2005).
[Crossref]

2002 (2)

2000 (1)

M. Paulus, P. Gay-Balmaz, and O. J. Martin, “Accurate and efficient computation of the Green’s tensor for stratified media,” Phys. Rev. E 62, 5797–5807 (2000).
[Crossref]

1999 (1)

1998 (1)

S. Oldenburg, R. Averitt, S. Westcott, and N. Halas, “Nanoengineering of optical resonances,” Chem. Phys. Lett. 288, 243–247 (1998).
[Crossref]

1996 (3)

F. Zolla and R. Petit, “Method of fictitious sources as applied to the electromagnetic diffraction of a plane wave by a grating in conical diffraction mounts,” J. Opt. Soc. Am. A 13, 796–802 (1996).
[Crossref]

M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transfer 55, 535–575 (1996).
[Crossref]

Y.-L. Xu, “Calculation of the addition coefficients in electromagnetic multisphere-scattering theory,” J. Comput. Phys. 127, 285–298 (1996).
[Crossref]

1995 (1)

M. A. Green and M. J. Keevers, “Optical properties of intrinsic silicon at 300 k,” Prog. Photovolt. Res. Appl. 3, 189–192 (1995).
[Crossref]

1994 (2)

1987 (1)

Y. Leviatan and A. Boag, “Analysis of electromagnetic scattering from dielectric cylinders using a multifilament current model,” IEEE Trans. Anntenas Propag. 35, 1119–1127 (1987).
[Crossref]

1980 (1)

G. Kristensson, “Electromagnetic scattering from buried inhomogeneities—a general three-dimensional formalism,” J. Appl. Phys. 51, 3486–3500 (1980).
[Crossref]

1966 (1)

V. K. Ivanov, “The approximate solution of operator equations of the first kind,” USSR Comput. Math. Math. Phys. 6, 197–205 (1966).
[Crossref]

1965 (1)

P. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53, 805–812 (1965).
[Crossref]

1963 (1)

A. N. Tikhonov, “On the solution of ill-posed problems and the method of regularization,” Dokl. Akad. Nauk SSSR 151, 501–504 (1963).

Agio, M.

M. Agio and A. Alù, Optical Antennas (Cambridge University, 2013).

Akselrod, G. M.

G. M. Akselrod, J. Huang, T. B. Hoang, P. T. Bowen, L. Su, D. R. Smith, and M. H. Mikkelsen, “Large-area metasurface perfect absorbers from visible to near-infrared,” Adv. Mater. 27, 8028–8034 (2015).
[Crossref]

Alù, A.

M. Agio and A. Alù, Optical Antennas (Cambridge University, 2013).

Aradian, A.

A. Baron, A. Aradian, V. Ponsinet, and P. Barois, “Self-assembled optical metamaterials,” Opt. Laser Technol. 82, 94–100 (2016).
[Crossref]

Auger, J.-C.

Averitt, R.

S. Oldenburg, R. Averitt, S. Westcott, and N. Halas, “Nanoengineering of optical resonances,” Chem. Phys. Lett. 288, 243–247 (1998).
[Crossref]

Balanis, C. A.

C. A. Balanis, Antenna Theory: Analysis and Design (Wiley, 2016).

Barois, P.

A. Baron, A. Aradian, V. Ponsinet, and P. Barois, “Self-assembled optical metamaterials,” Opt. Laser Technol. 82, 94–100 (2016).
[Crossref]

Baron, A.

A. Baron, A. Aradian, V. Ponsinet, and P. Barois, “Self-assembled optical metamaterials,” Opt. Laser Technol. 82, 94–100 (2016).
[Crossref]

Beghou, L.

L. Beghou, B. Liu, L. Pichon, and F. Costa, “Synthesis of equivalent 3-d models from near field measurements—application to the EMC of power printed circuit boards,” IEEE Trans. Magn. 45, 1650–1653 (2009).
[Crossref]

Ben-Abdallah, P.

M. Langlais, J.-P. Hugonin, M. Besbes, and P. Ben-Abdallah, “Cooperative electromagnetic interactions between nanoparticles for solar energy harvesting,” Opt. Express 22, A577–A588 (2014).
[Crossref]

J.-P. Hugonin, M. Besbes, and P. Ben-Abdallah, “Photovoltaics: light energy harvesting with plasmonic nanoparticle networks,” in Nanotechnology for Energy Sustainability (2017), pp. 83–100.

Besbes, M.

M. Langlais, J.-P. Hugonin, M. Besbes, and P. Ben-Abdallah, “Cooperative electromagnetic interactions between nanoparticles for solar energy harvesting,” Opt. Express 22, A577–A588 (2014).
[Crossref]

J.-P. Hugonin, M. Besbes, and P. Ben-Abdallah, “Photovoltaics: light energy harvesting with plasmonic nanoparticle networks,” in Nanotechnology for Energy Sustainability (2017), pp. 83–100.

Boag, A.

Y. Leviatan and A. Boag, “Analysis of electromagnetic scattering from dielectric cylinders using a multifilament current model,” IEEE Trans. Anntenas Propag. 35, 1119–1127 (1987).
[Crossref]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 2008).

Bowen, P. T.

G. M. Akselrod, J. Huang, T. B. Hoang, P. T. Bowen, L. Su, D. R. Smith, and M. H. Mikkelsen, “Large-area metasurface perfect absorbers from visible to near-infrared,” Adv. Mater. 27, 8028–8034 (2015).
[Crossref]

Bringi, V.

V. Bringi and T. Seliga, “Surface currents and near zone fields,” in Acoustic, Electromagnetic and Elastic Wave Scattering-Focus on the T-Matrix Approach (1980), pp. 79–90.

Buffa, A.

A. Buffa and R. Hiptmair, “Galerkin boundary element methods for electromagnetic scattering,” in Topics in Computational Wave Propagation (Springer, 2003), pp. 83–124.

Bürgi, T.

Butet, J.

B. Gallinet, J. Butet, and O. J. Martin, “Numerical methods for nanophotonics: standard problems and future challenges,” Laser Photon. Rev. 9, 577–603 (2015).
[Crossref]

Cadilhac, M.

Carminati, R.

O. Leseur, R. Pierrat, and R. Carminati, “High-density hyperuniform materials can be transparent,” Optica 3, 763–767 (2016).
[Crossref]

E. Castanié, R. Vincent, R. Pierrat, and R. Carminati, “Absorption by an optical dipole antenna in a structured environment,” Int. J. Opt. 2012, 452047 (2012).
[Crossref]

Castanié, E.

E. Castanié, R. Vincent, R. Pierrat, and R. Carminati, “Absorption by an optical dipole antenna in a structured environment,” Int. J. Opt. 2012, 452047 (2012).
[Crossref]

Chardon, G.

G. Chardon, A. Cohen, and L. Daudet, “Sampling and reconstruction of solutions to the Helmholtz equation,” Sampl. Theory Signal Image Process. 13, 67–90 (2014).

S. Koyama, G. Chardon, and L. Daudet, “Joint source and sensor placement for sound field control based on empirical interpolation method,” in IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) (IEEE, 2018), pp. 501–505.

Chichkov, B. N.

A. E. Miroshnichenko, A. B. Evlyukhin, Y. S. Kivshar, and B. N. Chichkov, “Substrate-induced resonant magnetoelectric effects for dielectric nanoparticles,” ACS Photon. 2, 1423–1428 (2015).
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Figures (11)

Fig. 1.
Fig. 1. Illustration of the method. Light scattering by an arbitrary particle (left) is modeled by a small set of polarizable elements (right) capable of reproducing accurately the field $ {{\boldsymbol \Psi }_{\rm s }} = [ {{\textbf{E}_{\rm s }};{\textbf{H}_{\rm s }}} ] $ scattered by the particle everywhere around it, including in the near-field region, for an arbitrary background field $ {{\boldsymbol \Psi }_{\rm b}} = [ {{\textbf{E}_{\rm b}};{\textbf{H}_{\rm b}}} ] $. The numerical dipoles are described by a global polarizability tensor that is spatially non-local because every induced moment depends not only on the incident field at the dipole position but also on the incident field at the positions of the other dipoles. The spatial non-locality is sketched here as dotted blue lines and is formally described by the off-diagonal elements of the matrix, $ {{\boldsymbol {\cal A}}_{i \to j}} $, with $ i \ne j $. The scattering problem formulation of Eq. (1) is repeated here for $ {N_{\rm d}} = 3 $.
Fig. 2.
Fig. 2. Global polarizability matrix (GPM). (a) We consider light scattering from a long dielectric cylinder. Thirty localized sources (electric and magnetic point dipoles with polarizations along $ x $, $ y $, and $ z $ directions) are randomly distributed at 40 nm from the particle boundary (red dots). These sources are successively used to excite the particle, resulting in $ 6 \times 30 = 180 $ independent numerical fullwave simulations. The scattered field is evaluated at $ {N_{\rm l }} $ learning points on a closed surface lying at a distance $ {d_{\rm l }} $ (here, $ {d_{\rm l }} = 20\,\,{\rm nm} $) from the particle boundary (semi-transparent blue surface). $ {N_{\rm d}} $ numerical dipoles (white dots; here, $ {N_{\rm d}} = 10 $, regularly arranged on the cylinder axis) are used. (b) Polarizability matrix $ \textbf{A} $, computed for $ {N_{\rm d}} = 10 $, $ {d_{\rm l }} = 20\,\,{\rm nm} $, and $ {N_{\rm l }} = 338 $. Each $ 6 \times 6 $ block ($ i $, $ j $) describes the contribution of numerical dipole $ i $ to the electric and magnetic moments induced in numerical dipole $ j $. The insets show zooms of an off-diagonal block (here, $ i = 9 $ and $ j = 4 $) and an on-diagonal block (here, $ i = j = 10 $, short notation used for simplicity).
Fig. 3.
Fig. 3. Convergence of the scattered fields on the learning surface. The relative error on the electromagnetic energy density computed with the numerical dipoles compared to the learning set is averaged over all points of the surface and all learning sources. A fast convergence of the error with the number $ {N_{\rm d}} $ of numerical dipoles is obtained for the configuration of Fig. 2. The accuracy of the reconstruction also improves when increasing the distance $ {d_{\rm l }} $ between the learning surface and the particle boundary.
Fig. 4.
Fig. 4. Reconstruction of the total field around the particle for one of the learning sources. The cylinder is illuminated by one of the 180 learning sources, a $ z $-polarized magnetic point dipole at position $ (x,y,z) = ( - 24, - 219,90)\,\,{\rm nm} $. (a) $ yz $ map of the $ z $ component of the total magnetic field, normalized to its maximum amplitude outside the cylinder. The cylinder surface is indicated by a solid white line and the learning surface by the dashed white line. Ten numerical dipoles (white dots) are regularly distributed on the cylinder axis. (b) Log of the difference between the two field maps with 1% and 10% error contours. Errors superior to 10% can be observed only in the particle volume. Errors superior to 1% can be found near the source and at the ends of the cylinder but stay mainly confined within the volume bounded by the learning surface, thereby proving good agreement on the near field.
Fig. 5.
Fig. 5. Generalization capability for unknown dipole sources. (a) A point dipole source is placed at varying heights $ h $ above the particle surface (lateral position at 1/4 of the cylinder length). (b) Scattered power normalized by the power emitted by the source in the uniform background, for varying distances $ h $ and two different dipole sources $ {p_y} $ and $ {m_z} $, as calculated from the GPM method (solid lines) and with the reference (COMSOL, white markers). A quantitative agreement is achieved for source positions $ h \gtrsim 30\,\,{\rm nm} $.
Fig. 6.
Fig. 6. Generalization to planewave excitations. (a) The cylinder is illuminated by a planewave at an angle $ {\theta _{\rm i }} $. (b) Scattering cross sections for TE and TM polarizations. Predictions from numerical dipoles (solid lines) are in very good agreement with reference data (COMSOL, white markers). (c) Maps of the scattered electric field ($ x $ component) for TE polarization at $ {\theta _{\rm i } = 50^ \circ } $, normalized to its maximum amplitude outside the cylinder. The cylinder and numerical dipoles are the same as in the previous figures. (d) Difference between the field maps in $ {\log }_{10} $-scale with 1% and 10% error contours.
Fig. 7.
Fig. 7. Implementation of multiple scattering in stratified media with the GPM method. In the general case, particles may have different shapes and be incorporated in different layers. The excitation field of a specific numerical dipole (white disk) is due to the background field in the stratified medium (orange arrows) and to the field radiated by the other numerical dipoles (black disks). Numerical dipoles belonging to different particles interact via the total Green tensor $ {\boldsymbol {\cal G}} $ in the stratified medium (black arrows), while numerical dipoles belonging to the same particle interact only via the variation of the Green tensor $ \boldsymbol{\Delta {\cal G}} $ in the stratified medium (red arrows). Some arrows have been removed for clarity.
Fig. 8.
Fig. 8. Modeling of two particles interacting in their near field. (a) Two parallel cylinders, shifted by 100 nm along $ y $ and separated by a distance $ dz $, are illuminated by a planewave with incident angle $ {\theta _{\rm i }} $. (b) Scattering cross sections of the cylinder pair with $ dz = 40\,\,{\rm nm} $, for TM polarization with varying $ {\theta _{\rm i }} $, as predicted from the GPM method with numerical dipoles (solid line), the T-matrix method with VSH decomposition (dashed line) and COMSOL (white markers). While the T-matrix method expectedly fails to predict the response of the system for such strongly interacting particles, the GPM method is in very good agreement with exact simulations. (c) Scattered electric field maps, normalized to the maximum amplitude outside the cylinder, for a TM planewave at $ {\theta_{\rm i } = 80^ \circ } $ for the system with $ dz = 40\,\,{\rm nm} $. (d) Difference between the field maps from (c) with 1% and 10% error contours, shown in $ {\log }_{10} $-scale. Errors superior to 1% hardly go past the learning surface, proving the good near-field prediction.
Fig. 9.
Fig. 9. Relative error in the scattering cross section computed for the configuration in Fig. 8 (TM polarization, $ {\theta _{\rm i }}{ = 80^ \circ } $) as a function of distance $ dz $ between the two cylinders. The relative errors with respect to COMSOL results increase with decreasing distances $ dz $. The GPM method with its numerical dipoles is better than the T-matrix method with VSH decomposition (here, up to the 11th multipole order) by an order of magnitude. Note that the T-matrix method is expected to be valid only at distances $ dz \gtrsim 410\,\,{\rm nm} $.
Fig. 10.
Fig. 10. Modeling of one particle in the vicinity of a metallic interface. (a) A cylinder, placed at a distance $ dz $ above an interface, is illuminated by a planewave with incident angle $ {\theta _{\rm i }} $. (b) Scattering cross sections of the cylinder with $ dz = 20\,\,{\rm nm} $, for TE and TM polarizations with varying $ {\theta _{\rm i }} $, as predicted from the GPM method (solid line) and as computed with COMSOL (white markers). A good agreement is found. (c) Scattered electric field maps, normalized to the maximum amplitude outside the cylinder, for a TM planewave at $ \theta_{\rm i } { = 20^ \circ } $. (d) Difference between the field maps from (c) with 1% and 10% error contours, shown in $ {\log }_{10} $-scale. Errors superior to 1% are found mostly near the interface and in the direction of specular reflection. Nevertheless, the results are very good considering the extreme case considered here.
Fig. 11.
Fig. 11. Modeling of a dense ensemble of particles in a stratified geometry. Sixteen silicon cylinders are placed randomly in a 700 nm thick glass layer deposited on a gold substrate. The system is excited by an $ x $-polarized electric dipole placed in its center (red dot). (a) Lateral view ($ xz $) of the system. (b) Top view ($ xy $) of the system. (c) Free-space radiation diagram obtained with 10 numerical dipoles. (d) 2D cut of the free-space radiation diagram as computed with the GPM method (dashed line) and COMSOL (solid line). (e) Radiation diagrams in the guided modes, composed of four TE modes (left) and four TM modes (right), as computed with the GPM method and COMSOL. The agreement for both the free-space and the guided-mode radiation diagrams is very good considering the complexity of the present example. The COMSOL calculations required 78 Gb of physical memory to be completed, compared to 14.4 Mb for the GPM method implemented in MATLAB 2018a.

Tables (1)

Tables Icon

Table 1. Comparison of Computational Accuracy, User Time, and Physical Memory Usage between the GPM Method and a Numerical Method Based on Physical (Coupled) Dipoles for the Scattering Cross Section of the Silicon Cylinder upon Planewave Excitation at $ {\theta _{\rm i } = 90^ \circ } $

Equations (13)

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Ψ s ( r ) = i , j = 1 N d G b ( r , r j ) A ( r j , r i ) Ψ b ( r i ) .
G b = [ G b e e G b e m G b m e G b m m ] ,
I 1 = min X P s t r u e G b X 2 ,
I 2 = min A X H P b H A H 2 ,
P s = G b A P b .
Ψ s ( r ) = m = 1 N p i , j = 1 N m G ( r , r j m ) A m ( r j m , r i m ) Ψ e x c i m ( r i m ) f o r r Ω ,
Ψ e x c k n ( r k n ) = Ψ b ( r k n ) + m = 1 N p i , j = 1 N m [ G ( r k n , r j m ) δ m n G b , m ( r k n , r j m ) ] A m ( r j m , r i m ) Ψ e x c i m ( r i m ) ,
× × E s ( r ) k 0 2 ϵ b E s ( r ) = k 0 2 Δ ϵ ( r ) E ( r ) .
E s ( r ) = k 0 2 G b ( r , r ) Δ ϵ ( r ) E ( r ) d r ,
E ( r ) = E b ( r ) + k 0 2 G b ( r , r ) Δ ϵ ( r ) E ( r ) d r .
E ( r ) = E b ( r ) + k 0 2 G b ( r , r ) Δ ϵ ( r ) E b ( r ) d + k 0 4 G b ( r , r ) Δ ϵ ( ) G b ( r , r ) Δ ϵ ( r ) × E b ( r ) d r d r + .
T ( r , r ) = k 0 2 Δ ϵ [ δ ( r r ) I + G b ( r , r ) T ( r , ) d r ] ,
E s ( r ) = G b ( r , r ) T ( r , r ) E b ( r ) d r d r .

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