Abstract

We characterize the modes with real and complex wavenumbers for both longitudinal and transverse polarization states (with respect to the mode traveling direction) in three dimensional (3D) periodic arrays of titanium dioxide (TiO2) microspheres in the frequency range between 250 GHz and 350 GHz. Modal results are computed by using a single magnetic dipole approximation (SDA) and an SDA model with correction (SDA-WC) that assumes the array to be embedded in a host with an effective permittivity computed through Maxwell Garnett formulas. Moreover, for the transverse polarization case, modal wavenumbers are computed also by fitting the full-wave simulation magnetic field (one point per unit cell) in a finite thickness structure, and their agreement and disagreement are discussed. The longitudinal polarization is not affected by the artificial correction introduced in the SDA-WC; in the transverse polarization case, instead, the correction is needed to obtain results in better agreement with the full-wave data fit. In the observed frequency range, there are one longitudinal mode and two transverse modes, one forward and one backward, where the forward one is “dominant” (i.e., it contributes mostly to the field in the array). Therefore, in the case of transverse polarization, we describe the composite material in terms of homogenized refractive index and relative permeability, comparing results from (i) modal analysis (with and without correction), (ii) Maxwell Garnett formulas, and (iii) Nicolson–Ross–Weir retrieval method from scattering parameters of finite thickness structures. The agreement among the different methods justifies the performed homogenization procedure in the case of transverse polarization. We show that artificial magnetism is generated from a nonmagnetic composite material.

© 2012 Optical Society of America

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2012

H. Nemec, C. Kadlec, F. Kadlec, P. Kuzel, R. Yahiaoui, U. C. Chung, C. Elissalde, M. Maglione, and P. Mounaix, “Resonant magnetic response of TiO2 microspheres at terahertz frequencies,” Appl. Phys. Lett. 100, 061117 (2012).
[CrossRef]

2011

S. Campione, S. Steshenko, M. Albani, and F. Capolino, “Complex modes and effective refractive index in 3D periodic arrays of plasmonic nanospheres,” Opt. Express 19, 26027–26043 (2011).
[CrossRef]

A. Alù, “First-principles homogenization theory for periodic metamaterials,” Phys. Rev. B 84, 075153 (2011).
[CrossRef]

J. Liu and N. Bowler, “Analysis of double-negative (DNG) bandwidth for a metamaterial composed of magnetodielectric spheres embedded in a matrix,” IEEE Antennas Wireless Propagat. Lett. 10, 399–402 (2011).
[CrossRef]

G. Nehmetallah, R. Aylo, and P. P. Banerjee, “Binary and core-shell nanoparticle dispersed liquid crystal cells for metamaterial applications,” J. Nanophoton. 5, 051603 (2011).
[CrossRef]

E. F. Kuester, N. Memic, S. Shen, A. D. Scher, S. Kim, K. Kumley, and H. Loui, “A negative refractive index metamaterial based on a cubic array of layered nonmagnetic spherical particles,” Progr. Electromag. Res. B 33, 175–202 (2011).

A. Vallecchi, M. Albani, and F. Capolino, “Collective electric and magnetic plasmonic resonances in spherical nanoclusters,” Opt. Express 19, 2754–2772 (2011).
[CrossRef]

2010

A. Vallecchi, S. Campione, and F. Capolino, “Symmetric and antisymmetric resonances in a pair of metal-dielectric nanoshells: tunability and closed-form formulas,” J. Nanophoton. 4, 041577 (2010).
[CrossRef]

S. Campione, A. Vallecchi, and F. Capolino, “Closed form formulas and tunability of resonances in pairs of gold-dielectric nanoshells,” Proc. SPIE 7757, 775738 (2010).
[CrossRef]

2009

I. B. Vendik, M. A. Odit, and D. S. Kozlov, “3D isotropic metamaterial based on a regular array of resonant dielectric spherical inclusions,” Metamaterials 3, 140–147 (2009).

G. Donzelli, A. Vallecchi, F. Capolino, and A. Schuchinsky, “Metamaterial made of paired planar conductors: particle resonances, phenomena and properties,” Metamaterials 3, 10–27 (2009).

A. Vallecchi, F. Capolino, and A. G. Schuchinsky, “2-D isotropic effective negative refractive index metamaterial in planar technology,” IEEE Microw. Wireless Compon. Lett. 19, 269–271 (2009).
[CrossRef]

A. Vallecchi and F. Capolino, “Tightly coupled tripole conductor pairs as constituents for a planar 2D-isotropic negative refractive index metamaterial,” Opt. Express 17, 15216–15227 (2009).
[CrossRef]

C. R. Simovski and S. A. Tretyakov, “Model of isotropic resonant magnetism in the visible range based on core-shell clusters,” Phys. Rev. B 79, 045111 (2009).
[CrossRef]

I. Vendik, M. Odit, and D. Kozlov, “3D metamaterial based on a regular array of resonant dielectric inclusions,” Radioengineering 18, 111–116 (2009).

R. A. Shore and A. D. Yaghjian, “Traveling waves on three-dimensional periodic arrays of two different alternating magnetodielectric spheres,” IEEE Trans. Antennas Propag. 57, 3077–3091 (2009).
[CrossRef]

2008

C.-W. Qiu and L. Gao, “Resonant light scattering by small coated nonmagnetic spheres: magnetic resonances, negative refraction, and prediction,” J. Opt. Soc. Am. B 25, 1728–1737 (2008).
[CrossRef]

A. Alu and N. Engheta, “Dynamical theory of artificial optical magnetism produced by rings of plasmonic nanoparticles,” Phys. Rev. B 78, 085112 (2008).
[CrossRef]

2007

R. A. Shore and A. D. Yaghjian, “Traveling waves on two- and three-dimensional periodic arrays of lossless scatterers,” Radio Sci. 42, RS6S21 (2007).
[CrossRef]

A. Alu, and N. Engheta, “Three-dimensional nanotransmission lines at optical frequencies: a recipe for broadband negative-refraction optical metamaterials,” Phys. Rev. B 75, 024304 (2007).
[CrossRef]

H. K. Yuan, U. K. Chettiar, W. S. Cai, A. V. Kildishev, A. Boltasseva, V. P. Drachev, and V. M. Shalaev, “A negative permeability material at red light,” Opt. Express 15, 1076–1083 (2007).
[CrossRef]

W. Cai, U. K. Chettiar, H. K. Yuan, V. C. De Silva, A. V. Kildishev, V. P. Drachev, and V. M. Shalaev, “Metamagnetics with rainbow colors,” Opt. Express 15, 3333–3341 (2007).
[CrossRef]

I. V. Shadrivov, A. N. Reznik, and Y. S. Kivshar, “Magnetoinductive waves in arrays of split-ring resonators,” Physica B 394, 180–183 (2007).
[CrossRef]

M. G. Silveirinha, “Generalized Lorentz–Lorenz formulas for microstructured materials,” Phys. Rev. B 76, 245117 (2007).
[CrossRef]

2006

O. Ouchetto, Q. Cheng-Wei, S. Zouhdi, L. Le-Wei, and A. Razek, “Homogenization of 3-D periodic bianisotropic metamaterials,” IEEE Trans. Microwave Theory Tech. 54, 3893–3898 (2006).
[CrossRef]

M. W. Klein, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Single-slit split-ring resonators at optical frequencies: limits of size scaling,” Opt. Lett. 31, 1259–1261 (2006).
[CrossRef]

I. Vendik, O. Vendik, and M. Odit, “Isotropic artificial media with simultaneously negative permittivity and permeability,” Microw. Opt. Technol. Lett. 48, 2553–2556 (2006).
[CrossRef]

T. G. Mackay and A. Lakhtakia, “Correlation length and negative phase velocity in isotropic dielectric-magnetic materials,” J. Appl. Phys. 100, 063533–063535 (2006).
[CrossRef]

L. Jylha, I. Kolmakov, S. Maslovski, and S. Tretyakov, “Modeling of isotropic backward-wave materials composed of resonant spheres,” J. Appl. Phys. 99, 043102–043107 (2006).
[CrossRef]

I. B. Vendik, O. G. Vendik, and M. S. Gashinova, “Artificial dielectric medium possessing simultaneously negative permittivity and magnetic permeability,” Tech. Phys. Lett. 32, 429–433 (2006).
[CrossRef]

I. Vendik, O. Vendik, I. Kolmakov, and M. Odit, “Modelling of isotropic double negative media for microwave applications,” Opto-Electron. Rev. 14, 179–186 (2006).
[CrossRef]

2005

M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Three-dimensional array of dielectric spheres with an isotropic negative permeability at infrared frequencies,” Phys. Rev. B 72, 193103 (2005).
[CrossRef]

C. L. Holloway, M. A. Mohamed, E. F. Kuester, and A. Dienstfrey, “Reflection and transmission properties of a metafilm: with an application to a controllable surface composed of resonant particles,” IEEE Trans. Electromagn. Compat. 47, 853–865 (2005).
[CrossRef]

V. Yannopapas and A. Moroz, “Negative refractive index metamaterials from inherently non-magnetic materials for deep infrared to terahertz frequency ranges,” J. Phys. Condens. Matter 17, 3717 (2005).
[CrossRef]

V. M. Shalaev, W. Cai, U. K. Chettiar, H.-K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. 30, 3356–3358 (2005).
[CrossRef]

S. Zhang, W. Fan, K. J. Malloy, S. R. Brueck, N. C. Panoiu, and R. M. Osgood, “Near-infrared double negative metamaterials,” Opt. Express 13, 4922–4930 (2005).
[CrossRef]

K. Berdel, J. G. Rivas, P. H. Bolivar, P. de Maagt, and H. Kurz, “Temperature dependence of the permittivity and loss tangent of high-permittivity materials at terahertz frequencies,” IEEE Trans. Microwave Theory Tech. 53, 1266–1271 (2005).
[CrossRef]

2004

V. A. Markel, V. N. Pustovit, S. V. Karpov, A. V. Obuschenko, V. S. Gerasimov, and I. L. Isaev, “Electromagnetic density of states and absorption of radiation by aggregates of nanospheres with multipole interactions,” Phys. Rev. B 70, 054202 (2004).
[CrossRef]

T. J. Yen, W. Padilla, N. Fang, D. Vier, D. Smith, J. Pendry, D. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science 303, 1494 (2004).
[CrossRef]

2003

C. L. Holloway, E. F. Kuester, J. Baker-Jarvis, and P. Kabos, “A double negative (DNG) composite medium composed of magnetodielectric spherical particles embedded in a matrix,” IEEE Trans. Antennas Propag. 51, 2596–2603 (2003).
[CrossRef]

2002

E. Shamonina, V. A. Kalinin, K. H. Ringhofer, and L. Solymar, “Magnetoinductive waves in one, two, and three dimensions,” J. Appl. Phys. 92, 6252–6261 (2002).
[CrossRef]

1999

J. Pendry, A. Holden, D. Robbins, and W. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
[CrossRef]

1997

A. H. Boughriet, C. Legrand, and A. Chapoton, “Noniterative stable transmission/reflection method for low-loss material complex permittivity determination,” IEEE Trans. Microwave Theory Tech. 45, 52–57 (1997).
[CrossRef]

1994

1974

W. B. Weir, “Automatic measurement of complex dielectric constant and permeability at microwave frequencies,” Proc. IEEE 62, 33–36 (1974).
[CrossRef]

1971

L. S. Benenson, “Dispersion equations of periodic structures,” Radio Eng. Electron. Phys. 16, 1280–1290 (1971).

1970

A. M. Nicolson and G. F. Ross, “Measurement of the intrinsic properties of materials by time-domain techniques,” IEEE Trans. Instrum. Meas. 19, 377–382 (1970).
[CrossRef]

1961

F. S. Ham and B. Segall, “Energy bands in periodic lattices-Green’s function method,” Phys. Rev. 124, 1786 (1961).
[CrossRef]

1921

P. P. Ewald, “The calculation of optical and electrostatic grid potential,” Ann. Phys. 64, 253–287 (1921).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1965).

Aitchison, J. S.

M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Three-dimensional array of dielectric spheres with an isotropic negative permeability at infrared frequencies,” Phys. Rev. B 72, 193103 (2005).
[CrossRef]

Albani, M.

Alu, A.

A. Alu and N. Engheta, “Dynamical theory of artificial optical magnetism produced by rings of plasmonic nanoparticles,” Phys. Rev. B 78, 085112 (2008).
[CrossRef]

A. Alu, and N. Engheta, “Three-dimensional nanotransmission lines at optical frequencies: a recipe for broadband negative-refraction optical metamaterials,” Phys. Rev. B 75, 024304 (2007).
[CrossRef]

Alù, A.

A. Alù, “First-principles homogenization theory for periodic metamaterials,” Phys. Rev. B 84, 075153 (2011).
[CrossRef]

Aylo, R.

G. Nehmetallah, R. Aylo, and P. P. Banerjee, “Binary and core-shell nanoparticle dispersed liquid crystal cells for metamaterial applications,” J. Nanophoton. 5, 051603 (2011).
[CrossRef]

Baker-Jarvis, J.

C. L. Holloway, E. F. Kuester, J. Baker-Jarvis, and P. Kabos, “A double negative (DNG) composite medium composed of magnetodielectric spherical particles embedded in a matrix,” IEEE Trans. Antennas Propag. 51, 2596–2603 (2003).
[CrossRef]

Banerjee, P. P.

G. Nehmetallah, R. Aylo, and P. P. Banerjee, “Binary and core-shell nanoparticle dispersed liquid crystal cells for metamaterial applications,” J. Nanophoton. 5, 051603 (2011).
[CrossRef]

Basov, D.

T. J. Yen, W. Padilla, N. Fang, D. Vier, D. Smith, J. Pendry, D. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science 303, 1494 (2004).
[CrossRef]

Benenson, L. S.

L. S. Benenson, “Dispersion equations of periodic structures,” Radio Eng. Electron. Phys. 16, 1280–1290 (1971).

Berdel, K.

K. Berdel, J. G. Rivas, P. H. Bolivar, P. de Maagt, and H. Kurz, “Temperature dependence of the permittivity and loss tangent of high-permittivity materials at terahertz frequencies,” IEEE Trans. Microwave Theory Tech. 53, 1266–1271 (2005).
[CrossRef]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Bolivar, P. H.

K. Berdel, J. G. Rivas, P. H. Bolivar, P. de Maagt, and H. Kurz, “Temperature dependence of the permittivity and loss tangent of high-permittivity materials at terahertz frequencies,” IEEE Trans. Microwave Theory Tech. 53, 1266–1271 (2005).
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F. S. Ham and B. Segall, “Energy bands in periodic lattices-Green’s function method,” Phys. Rev. 124, 1786 (1961).
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H. Nemec, C. Kadlec, F. Kadlec, P. Kuzel, R. Yahiaoui, U. C. Chung, C. Elissalde, M. Maglione, and P. Mounaix, “Resonant magnetic response of TiO2 microspheres at terahertz frequencies,” Appl. Phys. Lett. 100, 061117 (2012).
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V. A. Markel, V. N. Pustovit, S. V. Karpov, A. V. Obuschenko, V. S. Gerasimov, and I. L. Isaev, “Electromagnetic density of states and absorption of radiation by aggregates of nanospheres with multipole interactions,” Phys. Rev. B 70, 054202 (2004).
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L. Jylha, I. Kolmakov, S. Maslovski, and S. Tretyakov, “Modeling of isotropic backward-wave materials composed of resonant spheres,” J. Appl. Phys. 99, 043102–043107 (2006).
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E. F. Kuester, N. Memic, S. Shen, A. D. Scher, S. Kim, K. Kumley, and H. Loui, “A negative refractive index metamaterial based on a cubic array of layered nonmagnetic spherical particles,” Progr. Electromag. Res. B 33, 175–202 (2011).

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[CrossRef]

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O. Ouchetto, Q. Cheng-Wei, S. Zouhdi, L. Le-Wei, and A. Razek, “Homogenization of 3-D periodic bianisotropic metamaterials,” IEEE Trans. Microwave Theory Tech. 54, 3893–3898 (2006).
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Liu, J.

J. Liu and N. Bowler, “Analysis of double-negative (DNG) bandwidth for a metamaterial composed of magnetodielectric spheres embedded in a matrix,” IEEE Antennas Wireless Propagat. Lett. 10, 399–402 (2011).
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E. F. Kuester, N. Memic, S. Shen, A. D. Scher, S. Kim, K. Kumley, and H. Loui, “A negative refractive index metamaterial based on a cubic array of layered nonmagnetic spherical particles,” Progr. Electromag. Res. B 33, 175–202 (2011).

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E. F. Kuester, N. Memic, S. Shen, A. D. Scher, S. Kim, K. Kumley, and H. Loui, “A negative refractive index metamaterial based on a cubic array of layered nonmagnetic spherical particles,” Progr. Electromag. Res. B 33, 175–202 (2011).

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I. Vendik, M. Odit, and D. Kozlov, “3D metamaterial based on a regular array of resonant dielectric inclusions,” Radioengineering 18, 111–116 (2009).

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I. Vendik, O. Vendik, and M. Odit, “Isotropic artificial media with simultaneously negative permittivity and permeability,” Microw. Opt. Technol. Lett. 48, 2553–2556 (2006).
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I. B. Vendik, M. A. Odit, and D. S. Kozlov, “3D isotropic metamaterial based on a regular array of resonant dielectric spherical inclusions,” Metamaterials 3, 140–147 (2009).

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O. Ouchetto, Q. Cheng-Wei, S. Zouhdi, L. Le-Wei, and A. Razek, “Homogenization of 3-D periodic bianisotropic metamaterials,” IEEE Trans. Microwave Theory Tech. 54, 3893–3898 (2006).
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O. Ouchetto, Q. Cheng-Wei, S. Zouhdi, L. Le-Wei, and A. Razek, “Homogenization of 3-D periodic bianisotropic metamaterials,” IEEE Trans. Microwave Theory Tech. 54, 3893–3898 (2006).
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[CrossRef]

Other

I. Vendik, O. G. Vendik, and M. Odit, “Isotropic double-negative materials,” in Theory and Phenomena of Metamaterials, F. Capolino, ed. (CRC Press, 2009), p. 21.1.

I. Vendik, M. Odit, and D. Kozlov, “All-dielectric metamaterials based on spherical and cubic inclusions,” in Selected Topics in Metamaterials and Photonic Crystals, A. Andreone, A. Cusano, A. Cutolo, and V. Galdi, eds. (World Scientific, 2011).

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Figures (11)

Fig. 1.
Fig. 1.

(a) Schematic for a 3D periodic array of TiO 2 microspheres embedded in a homogeneous medium with permittivity ε h . r is the radius of each microsphere, and a , b , and c are the periodicities along x -, y -, and z -directions, respectively. (b) Frequency behavior of the magnitude of the electric ( a 1 ) and magnetic ( b 1 ) Mie coefficients in Eq. (2) in the case of TiO 2 microspheres in free space, with constant permittivity ε m = 94 + 2.35 i , and radius r = 52 μm .

Fig. 2.
Fig. 2.

Dispersion diagram for T -pol for structure II in Table 1. (a) Real part and (b) imaginary part of the wavenumber k z . Solid curves, SDA; dashed curves, SDA-WC; dotted-circled curves, fitting of the HFSS full-wave simulation fields as explained in Section 4.

Fig. 3.
Fig. 3.

Trajectories of modal wavenumbers in the complex k z plane for T -pol, with respect to (a) the periodicity a , and (b) the free space wavenumber k 0 . Notice that in (b), crossing the vertical black dash-dotted line at “ 1 ” and “1” means crossing the light line β z = k 0 . Arrows indicate direction of increasing frequency. Solid curves, SDA; dashed curves, SDA-WC.

Fig. 4.
Fig. 4.

Dispersion diagram for L -pol for structure II in Table 1. (a) Real part and (b) imaginary part of the wavenumber k z . Solid curves, SDA; dashed curves, SDA-WC.

Fig. 5.
Fig. 5.

Trajectories of modal wavenumbers in the complex k z plane for L -pol, with respect to (a) the periodicity a , and (b) the free space wavenumber k 0 . Notice that in (b), crossing the vertical black dash-dotted line at “ 1 ” and “1” means crossing the light line β z = k 0 . Arrows indicate direction of increasing frequency. Solid curves, SDA; dashed curves, SDA-WC.

Fig. 6.
Fig. 6.

Comparison between the HFSS full-wave field in 11 layers of microsphere arrays and the fitting result using Eq. (9) at 290 GHz. (a) Normalized magnitude and (b) phase of the extracted magnetic field.

Fig. 7.
Fig. 7.

(a) Real part and (b) imaginary part of the relative effective permittivity for the structures in Table 1 for T -pol retrieved through Maxwell Garnett formulas in Eq. (10).

Fig. 8.
Fig. 8.

Comparison of the permeability for structure II in Table 1 for T -pol retrieved through different methods.

Fig. 9.
Fig. 9.

Comparison of the refractive index for structure II in Table 1 for T -pol retrieved through different methods.

Fig. 10.
Fig. 10.

Comparison of the permeability for the three structures in Table 1 for T -pol according to NRW-HFSS retrieval method.

Fig. 11.
Fig. 11.

Comparison of the refractive index for the three structures in Table 1 for T -pol according to NRW-HFSS retrieval method.

Tables (1)

Tables Icon

Table 1. Array Parameters

Equations (15)

Equations on this page are rendered with MathJax. Learn more.

p = α e e E loc , m = α m m H loc ,
a 1 = m ψ 1 ( m k r ) ψ 1 ( k r ) ψ 1 ( k r ) ψ 1 ( m k r ) m ψ 1 ( m k r ) ξ 1 ( k r ) ξ 1 ( k r ) ψ 1 ( m k r ) , b 1 = ψ 1 ( m k r ) ψ 1 ( k r ) m ψ 1 ( k r ) ψ 1 ( m k r ) ψ 1 ( m k r ) ξ 1 ( k r ) m ξ 1 ( k r ) ψ 1 ( m k r ) ,
α e e = 6 π i ε 0 ε h k 3 a 1 , α m m = 6 π i k 3 b 1 ,
H loc ( r 0 , k B ) = H inc ( r 0 ) + Ğ ̲ ( r 0 , r 0 , k B ) · m 0 ,
G ̲ ( r , r 0 , k B ) = n G ̲ ( r , r 0 + d n ) e i k B · d n
G ̲ ( r m , r n ) = e i k r m n 4 π [ ( k 2 r m n + i k r m n 2 1 r m n 3 ) I ̲ ( k 2 r m n + 3 i k r m n 2 3 r m n 3 ) r ^ m n r ^ m n ] ,
m 0 = α m m [ H inc ( r 0 ) + Ğ ̲ ( r , r 0 , k B ) · m 0 ] ,
A ̲ ( k B ) · m 0 = α m m H inc ( r 0 ) , A ̲ ( k B ) = I ̲ α m m Ğ ̲ ( r , r 0 , k B ) .
H y ( n ) = A + e i k z , f ( n 1 2 ) a + A e i k z , f ( n 1 2 ) a + B + e i k z , b ( n 1 2 ) a + B e i k z , b ( n 1 2 ) a ,
ε eff = ε h + ε h N D 1 [ ε 0 ε h α e e 1 + i k 3 6 π ] 1 3 , μ eff = 1 + 1 N D 1 [ α m m 1 + i k 3 6 π ] 1 3 ,
n eff = ± cos 1 ( 1 R 2 + T 2 T ) k 0 t + 2 π q k 0 t , Z eff = ± μ 0 ϵ 0 ( 1 + R ) 2 T 2 ( 1 R ) 2 T 2 ,
Ğ ̲ ( r , r 0 , k B ) = [ k 2 I ̲ + ] Ğ ( r , r 0 , k B ) .
Ğ ̲ ( r , r 0 , k B ) = G ̲ spectral ( r , r 0 , k B ) + Ğ ̲ spatial ( r , r 0 , k B ) ,
G ̲ spectral ( r , r 0 , k B ) = [ k 2 I ̲ + ] G spectral ( r , r 0 , k B ) ,
Ğ ̲ spatial ( r , r 0 , k B ) = [ k 2 I ̲ + ] Ğ spatial ( r , r 0 , k B ) ,

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