Abstract

We employ the generalized Lorentz–Lorenz method to investigate how both magnetoelectric coupling and spatial dispersion influence the artificial magnetic capabilities at terahertz frequencies of the representative case of a metamaterial consisting of a three-dimensional (3D) lattice of TiO2 microspheres. The complex wavenumber dispersion relations pertaining to modes supported by the array, traveling along one of the principal axes of the array with electric or magnetic field polarized transversely and longitudinally (with respect to the mode traveling direction), are studied and thoroughly characterized. One mode with transverse polarization is dominant at any given frequency for the analyzed dimensions, proving that the 3D lattice can be treated as a homogeneous medium with defined electromagnetic material parameters. We show, however, that bianisotropy is a direct consequence of magnetoelectric coupling, and the dyadic expressions of both effective and equivalent material parameters are derived. In particular, we analyze the effect of spatial dispersion on the effective parameters relative to a composite material made by a 3D lattice of TiO2 microspheres with filling fraction around 30% and near the first Mie magnetic dipolar resonance. Finally, we homogenize the metamaterial in terms of equivalent index and impedance, and by comparison with full-wave simulations, we explain the presence of the unphysical antiresonance permittivity behavior observed in previous work.

© 2014 Optical Society of America

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    [CrossRef]
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  28. 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).
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  31. A. Alù and N. Engheta, “Polarizabilities and effective parameters for collections of spherical nanoparticles formed by pairs of concentric double-negative, single-negative, and/or double-positive metamaterial layers,” J. Appl. Phys. 97, 094310 (2005).
    [CrossRef]
  32. J. A. Kong, Electromagnetic Wave Theory (Wiley-Interscience, 1990).
  33. I. V. Lindell, A. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media (Artech House, 1994).
  34. M. G. Silveirinha, “Metamaterial homogenization approach with application to the characterization of microstructured composites with negative parameters,” Phys. Rev. B 75, 115104 (2007).
    [CrossRef]
  35. R. A. Shore and A. D. Yaghjian, “Complex waves on periodic arrays of lossy and lossless permeable spheres: 1. Theory,” Radio Sci. 47, RS2014 (2012).
  36. R. A. Shore and A. D. Yaghjian, “Complex waves on periodic arrays of lossy and lossless permeable spheres: 2. Numerical results,” Radio Sci. 47, RS2015 (2012).
  37. A. Alù, “Restoring the physical meaning of metamaterial constitutive parameters,” Phys. Rev. B 83, 081102 (2011).
    [CrossRef]
  38. P. Alitalo, A. Culhaoglu, C. Simovski, and S. Tretyakov, “Experimental study of anti-resonant behavior of material parameters in periodic and aperiodic composite materials,” J. Appl. Phys. 113, 224903 (2013).
    [CrossRef]
  39. R. Marqus, F. Medina, and R. Rafii-El-Idrissi, “Role of bianisotropy in negative permeability and left-handed metamaterials,” Phys. Rev. B 65, 144440 (2002).
    [CrossRef]
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  41. D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
    [CrossRef]
  42. T. Koschny, P. Markoš, D. R. Smith, and C. M. Soukoulis, “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. E 68, 065602 (2003).
    [CrossRef]
  43. D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E 71, 036617 (2005).
    [CrossRef]

2013 (3)

X. Liu and A. Alu, “Generalized retrieval method for metamaterial constitutive parameters based on a physically driven homogenization approach,” Phys. Rev. B 87, 235136 (2013).
[CrossRef]

S. Campione, M. Sinclair, and F. Capolino, “Effective medium representation and complex modes in 3d periodic metamaterials made of cubic resonators with large permittivity at mid-infrared frequencies,” Photon. Nanostr. Fundam. Appl. 11, 423–435 (2013).
[CrossRef]

P. Alitalo, A. Culhaoglu, C. Simovski, and S. Tretyakov, “Experimental study of anti-resonant behavior of material parameters in periodic and aperiodic composite materials,” J. Appl. Phys. 113, 224903 (2013).
[CrossRef]

2012 (4)

R. A. Shore and A. D. Yaghjian, “Complex waves on periodic arrays of lossy and lossless permeable spheres: 1. Theory,” Radio Sci. 47, RS2014 (2012).

R. A. Shore and A. D. Yaghjian, “Complex waves on periodic arrays of lossy and lossless permeable spheres: 2. Numerical results,” Radio Sci. 47, RS2015 (2012).

S. Campione, S. Lannebère, A. Aradian, M. Albani, and F. Capolino, “Complex modes and artificial magnetism in three-dimensional periodic arrays of titanium dioxide microspheres at millimeter waves,” J. Opt. Soc. Am. B 29, 1697–1706 (2012).
[CrossRef]

S. Campione and F. Capolino, “Ewald method for 3D periodic dyadic Green’s functions and complex modes in composite materials made of spherical particles under the dual dipole approximation,” Radio Sci. 47, RS0N06 (2012).
[CrossRef]

2011 (3)

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

A. Alù, “Restoring the physical meaning of metamaterial constitutive parameters,” Phys. Rev. B 83, 081102 (2011).
[CrossRef]

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]

2010 (1)

C. Fietz and G. Shvets, “Current-driven metamaterial homogenization,” Physica B 405, 2930–2934 (2010).
[CrossRef]

2009 (2)

C. R. Simovski, “Material parameters of metamaterials (a review),” Opt. Spectrosc. 107, 726–753 (2009).
[CrossRef]

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

2008 (2)

G. Lovat, P. Burghignoli, and R. Araneo, “Efficient evaluation of the 3-D periodic Green’s function through the Ewald method,” IEEE Trans. Microw. Theor. Tech. 56, 2069–2075 (2008).
[CrossRef]

N. Matsumoto, T. Hosokura, K. Kageyama, H. Takagi, Y. Sakabe, and M. Hangyo, “Analysis of dielectric response of TiO2 in terahertz frequency region by general harmonic oscillator model,” Jpn. J. Appl. Phys. 47, 7725–7728 (2008).
[CrossRef]

2007 (5)

M. G. Silveirinha, “Metamaterial homogenization approach with application to the characterization of microstructured composites with negative parameters,” Phys. Rev. B 75, 115104 (2007).
[CrossRef]

I. Stevanovic and J. Mosig, “Periodic Green’s function for skewed 3-D lattices using the Ewald transformation,” Microw. Opt. Technol. Lett. 49, 1353–1357 (2007).
[CrossRef]

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

C. R. Simovski and S. A. Tretyakov, “Local constitutive parameters of metamaterials from an effective-medium perspective,” Phys. Rev. B 75, 195111 (2007).

C. R. Simovski, “Bloch material parameters of magneto-dielectric metamaterials and the concept of bloch lattices,” Metamaterials 1, 62–80 (2007).
[CrossRef]

2006 (1)

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

2005 (5)

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, 3717–3734 (2005).
[CrossRef]

M. Wheeler, J. 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]

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

A. Alù and N. Engheta, “Polarizabilities and effective parameters for collections of spherical nanoparticles formed by pairs of concentric double-negative, single-negative, and/or double-positive metamaterial layers,” J. Appl. Phys. 97, 094310 (2005).
[CrossRef]

D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E 71, 036617 (2005).
[CrossRef]

2003 (1)

T. Koschny, P. Markoš, D. R. Smith, and C. M. Soukoulis, “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. E 68, 065602 (2003).
[CrossRef]

2002 (2)

R. Marqus, F. Medina, and R. Rafii-El-Idrissi, “Role of bianisotropy in negative permeability and left-handed metamaterials,” Phys. Rev. B 65, 144440 (2002).
[CrossRef]

D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[CrossRef]

2000 (1)

R. Ruppin, “Evaluation of extended Maxwell–Garnett theories,” Opt. Commun. 182, 273–279 (2000).
[CrossRef]

1994 (1)

G. Mulholland, C. Bohren, and K. Fuller, “Light scattering by agglomerates—coupled electric and magnetic dipole method,” Langmuir 10, 2533–2546 (1994).
[CrossRef]

1904 (1)

J. C. Maxwell-Garnett, “Colours in metal glasses and in metallic films,” Philos. Trans. R. Soc. London Ser. A 203, 385–420 (1904).

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1964).

Aitchison, J.

M. Wheeler, J. 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.

Alitalo, P.

P. Alitalo, A. Culhaoglu, C. Simovski, and S. Tretyakov, “Experimental study of anti-resonant behavior of material parameters in periodic and aperiodic composite materials,” J. Appl. Phys. 113, 224903 (2013).
[CrossRef]

Alu, A.

X. Liu and A. Alu, “Generalized retrieval method for metamaterial constitutive parameters based on a physically driven homogenization approach,” Phys. Rev. B 87, 235136 (2013).
[CrossRef]

Alù, A.

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

A. Alù, “Restoring the physical meaning of metamaterial constitutive parameters,” Phys. Rev. B 83, 081102 (2011).
[CrossRef]

A. Alù and N. Engheta, “Polarizabilities and effective parameters for collections of spherical nanoparticles formed by pairs of concentric double-negative, single-negative, and/or double-positive metamaterial layers,” J. Appl. Phys. 97, 094310 (2005).
[CrossRef]

Aradian, A.

Araneo, R.

G. Lovat, P. Burghignoli, and R. Araneo, “Efficient evaluation of the 3-D periodic Green’s function through the Ewald method,” IEEE Trans. Microw. Theor. Tech. 56, 2069–2075 (2008).
[CrossRef]

Berdel, K.

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

Bohren, C.

G. Mulholland, C. Bohren, and K. Fuller, “Light scattering by agglomerates—coupled electric and magnetic dipole method,” Langmuir 10, 2533–2546 (1994).
[CrossRef]

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

Bolivar, P.

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

Burghignoli, P.

G. Lovat, P. Burghignoli, and R. Araneo, “Efficient evaluation of the 3-D periodic Green’s function through the Ewald method,” IEEE Trans. Microw. Theor. Tech. 56, 2069–2075 (2008).
[CrossRef]

Campione, S.

S. Campione, M. Sinclair, and F. Capolino, “Effective medium representation and complex modes in 3d periodic metamaterials made of cubic resonators with large permittivity at mid-infrared frequencies,” Photon. Nanostr. Fundam. Appl. 11, 423–435 (2013).
[CrossRef]

S. Campione and F. Capolino, “Ewald method for 3D periodic dyadic Green’s functions and complex modes in composite materials made of spherical particles under the dual dipole approximation,” Radio Sci. 47, RS0N06 (2012).
[CrossRef]

S. Campione, S. Lannebère, A. Aradian, M. Albani, and F. Capolino, “Complex modes and artificial magnetism in three-dimensional periodic arrays of titanium dioxide microspheres at millimeter waves,” J. Opt. Soc. Am. B 29, 1697–1706 (2012).
[CrossRef]

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]

Capolino, F.

S. Campione, M. Sinclair, and F. Capolino, “Effective medium representation and complex modes in 3d periodic metamaterials made of cubic resonators with large permittivity at mid-infrared frequencies,” Photon. Nanostr. Fundam. Appl. 11, 423–435 (2013).
[CrossRef]

S. Campione and F. Capolino, “Ewald method for 3D periodic dyadic Green’s functions and complex modes in composite materials made of spherical particles under the dual dipole approximation,” Radio Sci. 47, RS0N06 (2012).
[CrossRef]

S. Campione, S. Lannebère, A. Aradian, M. Albani, and F. Capolino, “Complex modes and artificial magnetism in three-dimensional periodic arrays of titanium dioxide microspheres at millimeter waves,” J. Opt. Soc. Am. B 29, 1697–1706 (2012).
[CrossRef]

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]

S. Steshenko and F. Capolino, Metamaterials Handbook: Applications of Metamaterials (CRC, 2009), Chap. 8.

Collin, R. E.

R. E. Collin, Field Theory of Guided Waves (IEEE, 1991).

Culhaoglu, A.

P. Alitalo, A. Culhaoglu, C. Simovski, and S. Tretyakov, “Experimental study of anti-resonant behavior of material parameters in periodic and aperiodic composite materials,” J. Appl. Phys. 113, 224903 (2013).
[CrossRef]

de Maagt, P.

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

Engheta, N.

A. Alù and N. Engheta, “Polarizabilities and effective parameters for collections of spherical nanoparticles formed by pairs of concentric double-negative, single-negative, and/or double-positive metamaterial layers,” J. Appl. Phys. 97, 094310 (2005).
[CrossRef]

Fietz, C.

C. Fietz and G. Shvets, “Current-driven metamaterial homogenization,” Physica B 405, 2930–2934 (2010).
[CrossRef]

Fuller, K.

G. Mulholland, C. Bohren, and K. Fuller, “Light scattering by agglomerates—coupled electric and magnetic dipole method,” Langmuir 10, 2533–2546 (1994).
[CrossRef]

Gashinova, M.

O. Vendik and M. Gashinova, “Artificial double negative (DNG) media composed by two different dielectric sphere lattices embedded in a dielectric matrix,” in 34th European Microwave Conference, Vol. 3 (IEEE, 2005), pp. 1209–1212.

Hangyo, M.

N. Matsumoto, T. Hosokura, K. Kageyama, H. Takagi, Y. Sakabe, and M. Hangyo, “Analysis of dielectric response of TiO2 in terahertz frequency region by general harmonic oscillator model,” Jpn. J. Appl. Phys. 47, 7725–7728 (2008).
[CrossRef]

Hosokura, T.

N. Matsumoto, T. Hosokura, K. Kageyama, H. Takagi, Y. Sakabe, and M. Hangyo, “Analysis of dielectric response of TiO2 in terahertz frequency region by general harmonic oscillator model,” Jpn. J. Appl. Phys. 47, 7725–7728 (2008).
[CrossRef]

Huffman, D.

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

Jylhä, L.

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

Kageyama, K.

N. Matsumoto, T. Hosokura, K. Kageyama, H. Takagi, Y. Sakabe, and M. Hangyo, “Analysis of dielectric response of TiO2 in terahertz frequency region by general harmonic oscillator model,” Jpn. J. Appl. Phys. 47, 7725–7728 (2008).
[CrossRef]

Kolmakov, I.

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

Kong, J. A.

J. A. Kong, Electromagnetic Wave Theory (Wiley-Interscience, 1990).

Koschny, T.

D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E 71, 036617 (2005).
[CrossRef]

T. Koschny, P. Markoš, D. R. Smith, and C. M. Soukoulis, “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. E 68, 065602 (2003).
[CrossRef]

Kozlov, D.

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

Kurz, H.

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

Lannebère, S.

Lannebre, S.

S. Lannebre, “Étude théorique de métamatériaux formés de particules diélectriques résonantes dans la gamme submillimétrique: magnétisme artificiel et indice de réfraction négatif,” Ph.D. thesis (Université Bordeaux, 2011).

Lindell, I. V.

I. V. Lindell, A. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media (Artech House, 1994).

Liu, X.

X. Liu and A. Alu, “Generalized retrieval method for metamaterial constitutive parameters based on a physically driven homogenization approach,” Phys. Rev. B 87, 235136 (2013).
[CrossRef]

Lovat, G.

G. Lovat, P. Burghignoli, and R. Araneo, “Efficient evaluation of the 3-D periodic Green’s function through the Ewald method,” IEEE Trans. Microw. Theor. Tech. 56, 2069–2075 (2008).
[CrossRef]

Markoš, P.

T. Koschny, P. Markoš, D. R. Smith, and C. M. Soukoulis, “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. E 68, 065602 (2003).
[CrossRef]

D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[CrossRef]

Marqus, R.

R. Marqus, F. Medina, and R. Rafii-El-Idrissi, “Role of bianisotropy in negative permeability and left-handed metamaterials,” Phys. Rev. B 65, 144440 (2002).
[CrossRef]

Maslovski, S.

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

Matsumoto, N.

N. Matsumoto, T. Hosokura, K. Kageyama, H. Takagi, Y. Sakabe, and M. Hangyo, “Analysis of dielectric response of TiO2 in terahertz frequency region by general harmonic oscillator model,” Jpn. J. Appl. Phys. 47, 7725–7728 (2008).
[CrossRef]

Maxwell-Garnett, J. C.

J. C. Maxwell-Garnett, “Colours in metal glasses and in metallic films,” Philos. Trans. R. Soc. London Ser. A 203, 385–420 (1904).

Medina, F.

R. Marqus, F. Medina, and R. Rafii-El-Idrissi, “Role of bianisotropy in negative permeability and left-handed metamaterials,” Phys. Rev. B 65, 144440 (2002).
[CrossRef]

Mojahedi, M.

M. Wheeler, J. 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]

Moroz, A.

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, 3717–3734 (2005).
[CrossRef]

Mosig, J.

I. Stevanovic and J. Mosig, “Periodic Green’s function for skewed 3-D lattices using the Ewald transformation,” Microw. Opt. Technol. Lett. 49, 1353–1357 (2007).
[CrossRef]

Mulholland, G.

G. Mulholland, C. Bohren, and K. Fuller, “Light scattering by agglomerates—coupled electric and magnetic dipole method,” Langmuir 10, 2533–2546 (1994).
[CrossRef]

Odit, M.

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

Rafii-El-Idrissi, R.

R. Marqus, F. Medina, and R. Rafii-El-Idrissi, “Role of bianisotropy in negative permeability and left-handed metamaterials,” Phys. Rev. B 65, 144440 (2002).
[CrossRef]

Rivas, J.

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

Ruppin, R.

R. Ruppin, “Evaluation of extended Maxwell–Garnett theories,” Opt. Commun. 182, 273–279 (2000).
[CrossRef]

Sakabe, Y.

N. Matsumoto, T. Hosokura, K. Kageyama, H. Takagi, Y. Sakabe, and M. Hangyo, “Analysis of dielectric response of TiO2 in terahertz frequency region by general harmonic oscillator model,” Jpn. J. Appl. Phys. 47, 7725–7728 (2008).
[CrossRef]

Schultz, S.

D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[CrossRef]

Shore, R. A.

R. A. Shore and A. D. Yaghjian, “Complex waves on periodic arrays of lossy and lossless permeable spheres: 2. Numerical results,” Radio Sci. 47, RS2015 (2012).

R. A. Shore and A. D. Yaghjian, “Complex waves on periodic arrays of lossy and lossless permeable spheres: 1. Theory,” Radio Sci. 47, RS2014 (2012).

Shvets, G.

C. Fietz and G. Shvets, “Current-driven metamaterial homogenization,” Physica B 405, 2930–2934 (2010).
[CrossRef]

Sihvola, A.

I. V. Lindell, A. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media (Artech House, 1994).

Sihvola, A. H.

A. H. Sihvola, Electromagnetic Mixing Formulas and Applications (IET, 1999).

Silveirinha, M. G.

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

M. G. Silveirinha, “Metamaterial homogenization approach with application to the characterization of microstructured composites with negative parameters,” Phys. Rev. B 75, 115104 (2007).
[CrossRef]

Simovski, C.

P. Alitalo, A. Culhaoglu, C. Simovski, and S. Tretyakov, “Experimental study of anti-resonant behavior of material parameters in periodic and aperiodic composite materials,” J. Appl. Phys. 113, 224903 (2013).
[CrossRef]

Simovski, C. R.

C. R. Simovski, “Material parameters of metamaterials (a review),” Opt. Spectrosc. 107, 726–753 (2009).
[CrossRef]

C. R. Simovski and S. A. Tretyakov, “Local constitutive parameters of metamaterials from an effective-medium perspective,” Phys. Rev. B 75, 195111 (2007).

C. R. Simovski, “Bloch material parameters of magneto-dielectric metamaterials and the concept of bloch lattices,” Metamaterials 1, 62–80 (2007).
[CrossRef]

Sinclair, M.

S. Campione, M. Sinclair, and F. Capolino, “Effective medium representation and complex modes in 3d periodic metamaterials made of cubic resonators with large permittivity at mid-infrared frequencies,” Photon. Nanostr. Fundam. Appl. 11, 423–435 (2013).
[CrossRef]

Smith, D. R.

D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E 71, 036617 (2005).
[CrossRef]

T. Koschny, P. Markoš, D. R. Smith, and C. M. Soukoulis, “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. E 68, 065602 (2003).
[CrossRef]

D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[CrossRef]

Soukoulis, C. M.

D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E 71, 036617 (2005).
[CrossRef]

T. Koschny, P. Markoš, D. R. Smith, and C. M. Soukoulis, “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. E 68, 065602 (2003).
[CrossRef]

D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[CrossRef]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1964).

Steshenko, S.

Stevanovic, I.

I. Stevanovic and J. Mosig, “Periodic Green’s function for skewed 3-D lattices using the Ewald transformation,” Microw. Opt. Technol. Lett. 49, 1353–1357 (2007).
[CrossRef]

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

Takagi, H.

N. Matsumoto, T. Hosokura, K. Kageyama, H. Takagi, Y. Sakabe, and M. Hangyo, “Analysis of dielectric response of TiO2 in terahertz frequency region by general harmonic oscillator model,” Jpn. J. Appl. Phys. 47, 7725–7728 (2008).
[CrossRef]

Tretyakov, S.

P. Alitalo, A. Culhaoglu, C. Simovski, and S. Tretyakov, “Experimental study of anti-resonant behavior of material parameters in periodic and aperiodic composite materials,” J. Appl. Phys. 113, 224903 (2013).
[CrossRef]

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

S. Tretyakov, Analytical Modeling in Applied Electromagnetics (Artech House, 2003).

Tretyakov, S. A.

C. R. Simovski and S. A. Tretyakov, “Local constitutive parameters of metamaterials from an effective-medium perspective,” Phys. Rev. B 75, 195111 (2007).

I. V. Lindell, A. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media (Artech House, 1994).

Vendik, I.

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

Vendik, O.

O. Vendik and M. Gashinova, “Artificial double negative (DNG) media composed by two different dielectric sphere lattices embedded in a dielectric matrix,” in 34th European Microwave Conference, Vol. 3 (IEEE, 2005), pp. 1209–1212.

Vier, D. C.

D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E 71, 036617 (2005).
[CrossRef]

Viitanen, A. J.

I. V. Lindell, A. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media (Artech House, 1994).

Wheeler, M.

M. Wheeler, J. 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]

Yaghjian, A. D.

R. A. Shore and A. D. Yaghjian, “Complex waves on periodic arrays of lossy and lossless permeable spheres: 1. Theory,” Radio Sci. 47, RS2014 (2012).

R. A. Shore and A. D. Yaghjian, “Complex waves on periodic arrays of lossy and lossless permeable spheres: 2. Numerical results,” Radio Sci. 47, RS2015 (2012).

Yannopapas, V.

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, 3717–3734 (2005).
[CrossRef]

IEEE Trans. Microw. Theor. Tech. (2)

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

G. Lovat, P. Burghignoli, and R. Araneo, “Efficient evaluation of the 3-D periodic Green’s function through the Ewald method,” IEEE Trans. Microw. Theor. Tech. 56, 2069–2075 (2008).
[CrossRef]

J. Appl. Phys. (3)

A. Alù and N. Engheta, “Polarizabilities and effective parameters for collections of spherical nanoparticles formed by pairs of concentric double-negative, single-negative, and/or double-positive metamaterial layers,” J. Appl. Phys. 97, 094310 (2005).
[CrossRef]

P. Alitalo, A. Culhaoglu, C. Simovski, and S. Tretyakov, “Experimental study of anti-resonant behavior of material parameters in periodic and aperiodic composite materials,” J. Appl. Phys. 113, 224903 (2013).
[CrossRef]

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

J. Opt. Soc. Am. B (1)

J. Phys. Condens. Matter (1)

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, 3717–3734 (2005).
[CrossRef]

Jpn. J. Appl. Phys. (1)

N. Matsumoto, T. Hosokura, K. Kageyama, H. Takagi, Y. Sakabe, and M. Hangyo, “Analysis of dielectric response of TiO2 in terahertz frequency region by general harmonic oscillator model,” Jpn. J. Appl. Phys. 47, 7725–7728 (2008).
[CrossRef]

Langmuir (1)

G. Mulholland, C. Bohren, and K. Fuller, “Light scattering by agglomerates—coupled electric and magnetic dipole method,” Langmuir 10, 2533–2546 (1994).
[CrossRef]

Metamaterials (2)

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

C. R. Simovski, “Bloch material parameters of magneto-dielectric metamaterials and the concept of bloch lattices,” Metamaterials 1, 62–80 (2007).
[CrossRef]

Microw. Opt. Technol. Lett. (1)

I. Stevanovic and J. Mosig, “Periodic Green’s function for skewed 3-D lattices using the Ewald transformation,” Microw. Opt. Technol. Lett. 49, 1353–1357 (2007).
[CrossRef]

Opt. Commun. (1)

R. Ruppin, “Evaluation of extended Maxwell–Garnett theories,” Opt. Commun. 182, 273–279 (2000).
[CrossRef]

Opt. Express (1)

Opt. Spectrosc. (1)

C. R. Simovski, “Material parameters of metamaterials (a review),” Opt. Spectrosc. 107, 726–753 (2009).
[CrossRef]

Philos. Trans. R. Soc. London Ser. A (1)

J. C. Maxwell-Garnett, “Colours in metal glasses and in metallic films,” Philos. Trans. R. Soc. London Ser. A 203, 385–420 (1904).

Photon. Nanostr. Fundam. Appl. (1)

S. Campione, M. Sinclair, and F. Capolino, “Effective medium representation and complex modes in 3d periodic metamaterials made of cubic resonators with large permittivity at mid-infrared frequencies,” Photon. Nanostr. Fundam. Appl. 11, 423–435 (2013).
[CrossRef]

Phys. Rev. B (9)

X. Liu and A. Alu, “Generalized retrieval method for metamaterial constitutive parameters based on a physically driven homogenization approach,” Phys. Rev. B 87, 235136 (2013).
[CrossRef]

M. Wheeler, J. 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]

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

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

C. R. Simovski and S. A. Tretyakov, “Local constitutive parameters of metamaterials from an effective-medium perspective,” Phys. Rev. B 75, 195111 (2007).

A. Alù, “Restoring the physical meaning of metamaterial constitutive parameters,” Phys. Rev. B 83, 081102 (2011).
[CrossRef]

R. Marqus, F. Medina, and R. Rafii-El-Idrissi, “Role of bianisotropy in negative permeability and left-handed metamaterials,” Phys. Rev. B 65, 144440 (2002).
[CrossRef]

M. G. Silveirinha, “Metamaterial homogenization approach with application to the characterization of microstructured composites with negative parameters,” Phys. Rev. B 75, 115104 (2007).
[CrossRef]

D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[CrossRef]

Phys. Rev. E (2)

T. Koschny, P. Markoš, D. R. Smith, and C. M. Soukoulis, “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. E 68, 065602 (2003).
[CrossRef]

D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E 71, 036617 (2005).
[CrossRef]

Physica B (1)

C. Fietz and G. Shvets, “Current-driven metamaterial homogenization,” Physica B 405, 2930–2934 (2010).
[CrossRef]

Radio Sci. (3)

S. Campione and F. Capolino, “Ewald method for 3D periodic dyadic Green’s functions and complex modes in composite materials made of spherical particles under the dual dipole approximation,” Radio Sci. 47, RS0N06 (2012).
[CrossRef]

R. A. Shore and A. D. Yaghjian, “Complex waves on periodic arrays of lossy and lossless permeable spheres: 1. Theory,” Radio Sci. 47, RS2014 (2012).

R. A. Shore and A. D. Yaghjian, “Complex waves on periodic arrays of lossy and lossless permeable spheres: 2. Numerical results,” Radio Sci. 47, RS2015 (2012).

Other (11)

J. A. Kong, Electromagnetic Wave Theory (Wiley-Interscience, 1990).

I. V. Lindell, A. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media (Artech House, 1994).

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

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

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1964).

S. Steshenko and F. Capolino, Metamaterials Handbook: Applications of Metamaterials (CRC, 2009), Chap. 8.

S. Tretyakov, Analytical Modeling in Applied Electromagnetics (Artech House, 2003).

R. E. Collin, Field Theory of Guided Waves (IEEE, 1991).

A. H. Sihvola, Electromagnetic Mixing Formulas and Applications (IET, 1999).

S. Lannebre, “Étude théorique de métamatériaux formés de particules diélectriques résonantes dans la gamme submillimétrique: magnétisme artificiel et indice de réfraction négatif,” Ph.D. thesis (Université Bordeaux, 2011).

O. Vendik and M. Gashinova, “Artificial double negative (DNG) media composed by two different dielectric sphere lattices embedded in a dielectric matrix,” in 34th European Microwave Conference, Vol. 3 (IEEE, 2005), pp. 1209–1212.

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Figures (11)

Fig. 1.
Fig. 1.

TiO2 (rutile) microspheres with a diameter around 70 μm provided by Dr. Chung-Seu, CNRS, Univ. Bordeaux, ICMCB, UPR 9048, F-33600 Pessac, France.

Fig. 2.
Fig. 2.

On the left, schematic for a 3D periodic cubic array with lattice parameter a composed of TiO2 microspheres with radius R. On the right, frequency behavior of the magnitude of the dipolar Mie magnetic (b1) and electric (a1) coefficients given by Eq. (3), in free space for spheres of TiO2 with radius R=52μm.

Fig. 3.
Fig. 3.

Top row: trajectories in the complex plane of both longitudinal and transverse normalized wavenumbers computed with Eqs. (15) and (16) for a cubic array of TiO2 microspheres of 52 μm with a filling fraction of 29.44%. Bottom row: dispersion diagrams (real and imaginary parts) corresponding to the wavenumber of each mode.

Fig. 4.
Fig. 4.

yy, zz, and xx components of the relative effective permittivity in Eq. (12) computed for three different values of the wavevector k=0, π2ax^, πax^.

Fig. 5.
Fig. 5.

yy, zz, and xx components of the relative effective permeability in Eq. (12) computed for three different values of the wavevector k=0, π2ax^, πax^.

Fig. 6.
Fig. 6.

Main component of the relative out-of-diagonal tensors, normalized by ϵ0μ0=1/c, computed for three different values of the wavevector k=0, π2ax^, πax^.

Fig. 7.
Fig. 7.

On the left, representation of the five-layer slab with thickness d of TiO2 microspheres used for HFSS simulations. On the right, equivalent homogeneous slab of same thickness d.

Fig. 8.
Fig. 8.

Comparison between the magnitude and the phase of the S parameters obtained with HFSS and computed with the present method for a cubic array of TiO2 microspheres of 52 μm with a filling fraction of 29.44%.

Fig. 9.
Fig. 9.

Comparison between the yy component of the relative equivalent permittivity and permeability (18) and the full-wave relative permittivity and permeability retrieved with NRW method for a cubic array of TiO2 microspheres of 52 μm with a filling fraction of 29.44%.

Fig. 10.
Fig. 10.

Comparison between the equivalent refractive index obtained with HFSS (NRW), with the equivalent refractive index computed with the present method taking into account or not the C¯¯em dyadic, for a cubic array of TiO2 microspheres of 52 μm with a filling fraction of 29.44%.

Fig. 11.
Fig. 11.

Comparison between the equivalent relative impedance obtained with HFSS (NRW), with the yy component of the equivalent impedance computed with the present method, for a cubic array of TiO2 microspheres of 52 μm with a filling fraction of 29.44%.

Equations (21)

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

pe=εhαeeEloc,pm=αmmBloc,
αee=6πia11kh3,αmm=6πib11kh3.
a1=mψ1(mkhR)ψ1(khR)ψ1(khR)ψ1(mkhR)mψ1(mkhR)ξ1(khR)ξ1(khR)ψ1(mkhR),b1=ψ1(mkhR)ψ1(khR)mψ1(khR)ψ1(mkhR)ψ1(mkhR)ξ1(khR)mξ1(khR)ψ1(mkhR),
Je(r)=Je,aveik·r,
Eloc=Eav+C¯¯int(ω,k)·peεh+C¯¯em(ω,k)·pmεhμ0,Blocμ0=HavC¯¯em(ω,k)·peμ0εh+C¯¯int(ω,k)·pmμ0,
Hav=Bavμ0pmVcellμ0
(I¯¯αeeC¯¯int(ω,k))·peεhαeeC¯¯em(ω,k)·pmεhμ0=αeeEav,(I¯¯αmmC¯¯int(ω,k))·pmμ0+αmmC¯¯em(ω,k)·peμ0εh=αmmHav,
Pav=peVcell,Mav=pmVcellμ0,
[Pavμ0Mav]=[εhχ¯¯ee(ω,k)εhμ0χ¯¯em(ω,k)εhμ0χ¯¯me(ω,k)μ0χ¯¯mm(ω,k)]·[EavHav].
[χ¯¯ee(ω,k)χ¯¯em(ω,k)χ¯¯me(ω,k)χ¯¯mm(ω,k)]=1Vcell[I¯¯αeeC¯¯int(ω,k)αeeC¯¯em(ω,k)αmmC¯¯em(ω,k)I¯¯αmmC¯¯int(ω,k)]1·[αeeI¯¯0¯¯0¯¯αmmI¯¯].
Dav=ε¯¯eff(ω,k)·Eav+ξ¯¯eff(ω,k)·Hav,Bav=μ¯¯eff(ω,k)·Hav+ζ¯¯eff(ω,k)·Eav.
ε¯¯eff(ω,k)=εh(I¯¯+χ¯¯ee(ω,k)),ξ¯¯eff(ω,k)=εhμ0χ¯¯em(ω,k),ζ¯¯eff(ω,k)=εhμ0χ¯¯me(ω,k),μ¯¯eff(ω,k)=μ0(I¯¯+χ¯¯mm(ω,k)),
k×Eav=ωBav=ω(μ¯¯eff(ω,k)·Hav+ζ¯¯eff(ω,k)·Eav),k×Hav=ωDav=ω(ε¯¯eff(ω,k)·Eav+ξ¯¯eff(ω,k)·Hav).
Hav=μ¯¯eff1·(kω×I¯¯ζ¯¯eff)·Eav,Eav=ε¯¯eff1·(kω×I¯¯+ξ¯¯eff)·Hav.
[(kω×I¯¯+ξ¯¯eff(ω,k))·μ¯¯eff1(ω,k)·(kω×I¯¯ζ¯¯eff(ω,k))+ε¯¯eff(ω,k)]·Eav=0,
[(kω×I¯¯ζ¯¯eff(ω,k))·ε¯¯eff1(ω,k)·(kω×I¯¯+ξ¯¯eff(ω,k))+μ¯¯eff(ω,k)]·Hav=0.
Dav=ε¯¯eff·Eav+ξ¯¯eff·Hav=ε¯¯eq·Eav,Bav=μ¯¯eff·Hav+ζ¯¯eff·Eav=μ¯¯eq·Hav.
ε¯¯eq(ω,k)=ε¯¯eff+ξ¯¯eff·μ¯¯eff1·(kω×I¯¯ζ¯¯eff),μ¯¯eq(ω,k)=μ¯¯effζ¯¯eff·ε¯¯eff1·(kω×I¯¯+ξ¯¯eff).
ζ¯¯eff=[00000ζeffyz0ζeffzy0]ξ¯¯eff=[00000ξeffyz0ξeffzy0],
S11=Γ(1e2ikd)1Γ2·e2ikd,S21=4z(1+z)2·eikd1Γ2·e2ikd
z=ε0·μeqyyμ0·εeqyy,Γ=z1z+1,

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