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Keywords:

  • Ewald method;
  • dual dipole approximation;
  • dyadic Green's function;
  • homogenization;
  • metamaterials;
  • modal analysis

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Formulation Using Periodic Green's Functions
  5. 3. Ewald Representation for the Dyadic GFs for 3D Periodic Arrays
  6. 4. Modes With Real or Complex Wave Number and Description of Composite Materials Through Effective Refractive Index
  7. 5. Conclusion
  8. Appendix A:: Regularization of the Spatial Terms of the Dyadic Green's Functions
  9. Appendix B:: Supporting Mathematical Expressions
  10. Acknowledgments
  11. References

[1] We derive the Ewald representation for the dyadic periodic Green's functions to represent the electromagnetic field in a three dimensional (3D) periodic array of electric and magnetic dipoles. Then we use the developed theory to analyze the modes with real and complex wave number in a 3D periodic lattice of lead telluride (PbTe) microspheres at infrared frequencies and in a 3D periodic lattice of titanium dioxide (TiO2) microspheres at millimeter waves. Each microsphere is equivalently modeled with both an electric and a magnetic dipole, via a method here called the dual dipole approximation (DDA). The 3D lattices exhibit first a magnetic-induced then an electric-induced feature determined by microsphere magnetic and electric resonances. The DDA wave number results are compared to the ones computed with single electric or single magnetic dipole approximation and to the ones retrieved by using the Nicolson-Ross-Weir (NRW) retrieval method from reflection and transmission of finite thickness slabs computed by a full-wave simulation. It is shown that the DDA method is in very good agreement with NRW, in contrast to the previously reported single dipole approximation methods that fail to predict one of the two features (either electric or magnetic). A mode with transverse polarization is found to be dominant and able to propagate inside the lattice, and therefore the composite material can be treated as a homogeneous one with effective refractive index. This is obtained by adopting five different retrieval procedures for each lattice, and their agreement or disagreement is discussed.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Formulation Using Periodic Green's Functions
  5. 3. Ewald Representation for the Dyadic GFs for 3D Periodic Arrays
  6. 4. Modes With Real or Complex Wave Number and Description of Composite Materials Through Effective Refractive Index
  7. 5. Conclusion
  8. Appendix A:: Regularization of the Spatial Terms of the Dyadic Green's Functions
  9. Appendix B:: Supporting Mathematical Expressions
  10. Acknowledgments
  11. References

[2] Artificial composite materials (metamaterials) proved to be a feasible way to increase the degrees of freedom in the interaction of electromagnetic fields with matter from microwaves to optical frequencies.

[3] Collective resonances and wave propagation in composite materials can be characterized by modal analyses of arrays periodic in three dimensions (3D) [Alù and Engheta, 2007; Benenson, 1971; Campione et al., 2011a, 2011b, 2012; Ham and Segall, 1961; Shore and Yaghjian, 2007, 2010, 2012; Steshenko and Capolino, 2009; Wheeler et al., 2005b]. In particular, under certain circumstances of polarization and excitation, a 3D periodic array of particles with finite thickness could be described with good approximation as a homogeneous slab with effective parameters [Campione et al., 2011a, 2011b, 2012; Collin, 1960; Shore and Yaghjian, 2012], such as relative permittivity (εeff), relative permeability (μeff), and refractive index (neff).

[4] The aim of this paper is to characterize the modes with real and complex wave number in 3D periodic arrays made of particles with spherical shape modeled through the dual (electric and magnetic) dipole approximation (DDA). The DDA is a good approximation when the two dipolar terms (or any of them) dominate the scattered-field multipole expansion. For transverse polarization, the structure is treated as a homogeneous slab with effective refractive index. Here we are interested in analyzing composite materials made of large dielectric permittivity materials (e.g., lead telluride, titanium dioxide, etc.) which in general exhibit first a magnetic then an electric resonance. The DDA is more accurate than the single dipole approximation (SDA) approach, since it accounts for electric and magnetic couplings, and the method described here could also be used to characterize the artificial magnetic (i.e.,μeff ≠ 1) or electric (i.e., εeff ≠ 1) properties of the 3D lattice. Possible routes for obtaining artificial magnetism, such as the use of split ring resonators, binary mixtures, etc., have been recently summarized in the introduction of Campione et al. [2012]. The present work has been indeed partly motivated by the observations made in [Campione et al., 2012] about the fact that the results shown in that paper could be further improved by modeling the particle with the DDA model, instead of the two SDA models based only on magnetic dipoles shown therein.

[5] The approach described in the present paper allows for the tracking and especially for the characterization of the evolution of modes varying frequency. First, we develop the formulation with coupled electric and magnetic dipoles through DDA to model the particle behavior (as done, for example, in Mulholland et al. [1994] and Shore and Yaghjian [2010, 2012]) in order to analyze modes with real and complex wave number in 3D periodic arrays of spherical particles. The numerical procedure developed in this paper for evaluating the complex zeros of the dispersion relation uses the Ewald representation for the dyadic periodic Green's functions (GFs) to represent the field in 3D periodic arrays, and is partly based on previous scalar developments [Kustepeli and Martin, 2000; Lovat et al., 2008; Park et al., 1998; Stevanoviæ and Mosig, 2007] and dyadic developments [Campione et al., 2011b, 2012]. The Ewald representation, besides providing analytic continuation to the complex wave number space, results in two series with Gaussian convergence where only a handful of terms are needed [Ewald, 1921; Kustepeli and Martin, 2000; Lovat et al., 2008; Park et al., 1998; Stevanoviæ and Mosig, 2007]. In Shore and Yaghjian [2010, 2012] the authors treat similar problems of arrays of spherical particles where both electric and magnetic dipoles have been considered. They evaluate the modes with complex wave number by using a method based on polylogarithmic functions, which is different from the Ewald method employed here. Furthermore, in our formulation, the direction of electric and magnetic dipoles is arbitrary.

[6] The structure of the paper is as follows. We discuss in section 2 the rigorous representation of the field in 3D periodic arrays using the dyadic GFs including the case with coupled electric and magnetic dipoles. Then, in section 3, we derive the new expressions for the dyadic GFs using the Ewald method related to both electric and magnetic dipole excitations. The convergence properties of scalar and dyadic GFs are briefly inspected as well. Last, in section 4, we analyze the modal wave numbers for transverse polarization in two composite metamaterials: one made of an array of non-magnetic lead telluride (PbTe) microspheres, and the other made of non-magnetic titanium dioxide (TiO2) microspheres. Modal wave numbers are computed through various methods: DDA, SDA based on electric dipoles only (SDA-E), SDA based on magnetic dipoles only (SDA-M), and Nicolson-Ross-Weir (NRW) retrieval method from reflection and transmission of finite thickness slabs computed by full-wave simulations. The 3D lattices are then described in terms of effective refractive index, computed by using the four methods above and Maxwell Garnett (MG) formulas, and their agreement or disagreement is discussed. The procedure for the singularity regularization of the spatial series counterparts in the Ewald representations is discussed inAppendix A. Supporting mathematical expressions are reported in Appendix B.

2. Formulation Using Periodic Green's Functions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Formulation Using Periodic Green's Functions
  5. 3. Ewald Representation for the Dyadic GFs for 3D Periodic Arrays
  6. 4. Modes With Real or Complex Wave Number and Description of Composite Materials Through Effective Refractive Index
  7. 5. Conclusion
  8. Appendix A:: Regularization of the Spatial Terms of the Dyadic Green's Functions
  9. Appendix B:: Supporting Mathematical Expressions
  10. Acknowledgments
  11. References

[7] The monochromatic time harmonic convention exp(−iωt) is assumed here and throughout the paper, and is therefore suppressed hereafter. In the following equations, bold letters refer to vector quantities, a caret ‘^’ on top of a bold letter refers to unit vectors, and a bar under a bold letter refers to dyadic quantities.

2.1. Field in Periodic Arrays

[8] Consider a 3D periodic array of particles (Figure 1) described by dipole-like electric and magnetic polarizabilities. Each particle is characterized by its induced electric (p) and magnetic (m) dipole moments as described in section 2.4. The array is immersed in a homogeneous background, with relative permittivity εh. Particles are placed at positions rn = r0 + dn, where dn = n1a inline image + n2b inline image + n3c inline image, n = (n1, n2, n3) is a triple index with n1, n2, n3 = 0, ±1, ±2, …, r0 = x0 inline image + y0 inline image + z0 inline image denotes a reference particle location, and a, b and c are the periodicities along the x-,y- andz-directions, respectively [Campione et al., 2011b, 2012; Steshenko and Capolino, 2009].

image

Figure 1. Three-dimensional periodic array made of spherical particles with relative permittivityεm embedded in a homogeneous medium with relative permittivity εh. The radius of each sphere is denoted as r and a, b and c are the periodicities along the x-,y- andz-directions, respectively.

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[9] Suppose that the 3D array is either excited by an external incident field or supports a mode, with wave vector kB = kx inline image + ky inline image + kz inline image, such that the nth dipole (at rn) is related to the 0th one (at r0) by pn = p0 exp(ikB · dn) and mn = m0 exp(ikB · dn). Then, the electric and magnetic fields at a general position r are given by

  • display math

[10] The Green's function matrix in (1) accounts for all the coupling between electric and magnetic dipoles. The blocks inline imageEP and inline imageHM in the principal diagonal of (1) represent the periodic Green's functions that connect all pn to E and all mn to H, respectively. The blocks in the antidiagonal inline imageEM and inline imageHP represent the periodic Green's functions that connect all mn to E and all pn to H, respectively.

2.2. Principal Diagonal Blocks: inline imageEP and inline imageHM

[11] Both the principal diagonal blocks inline imageEP(r, r0, kB) = (ε0εh)−1 inline imagepd(r, r0, kB) [Campione et al., 2011b] and inline imageHM(r, r0, kB) = inline imagepd(r, r0, kB) [Campione et al., 2012] are defined in terms of a “principal diagonal” dyad inline imagepd for the periodically phased array of dipoles, defined as

  • display math

[12] Here inline imagepd(r, r0) denotes the “free space” dyadic Green's function in the homogeneous background, given by Campione et al. [2012, equation (6)]. The principal diagonal dyad inline imagepd can be also expressed as

  • display math

which will be used later in section 3.1, where the scalar term G is represented as a sum of “spatial” spherical waves as

  • display math

where G(r, r0) = eikR/(4πR), with R = |R| and R = rr0. Moreover, k = ω inline image /c0 = k0 inline image is the host wave number, where k0 denotes the free space wave number and c0 the speed of light in free space, εh is the relative permittivity of the host medium, and inline image is the unit dyad.

2.3. Antidiagonal Blocks: inline imageEM and inline imageHP

[13] Both antidiagonal blocks inline imageEM(r, r0, kB) = iμω inline imagead(r, r0, kB) and inline imageHP(r, r0, kB) = − inline imagead(r, r0, kB) are defined in terms of an “antidiagonal” dyad inline imagead for the periodic phased array of dipoles, computed as

  • display math

[14] Here inline imagead(r, r0) denotes the “free space” dyadic Green's function in the homogeneous background, defined as inline imagead(r, r0) = ∇G(r, r0) × inline image, and hence

  • display math

where R = rr0, R = |R| and inline image = R/R. The antidiagonal dyad inline imagead for the 3D periodic array can be also expressed as

  • display math

with G(r, r0, kB) the scalar GF in (4), and this will be used later in section 3.2.

2.4. Spherical Particle Modeling and Modal Analysis in 3D Arrays

[15] The induced electric (p0) and magnetic (m0) dipole moments for the 0th spherical particle at position r0 are given by [Alù and Engheta, 2006; Ishimaru et al., 2003]

  • display math

where inline image is a 6 × 6 matrix, inline imageee = αee inline image and inline imagemm = αmm inline image, with αee and αmm the isotropic electric and magnetic polarizabilities of the spherical particle. Moreover, here we assume inline imageem = inline imageme = inline image in the dipole approximation [Alù and Engheta, 2006], with inline image being the null dyad. Therefore in this paper we are limiting the nanoparticle interactions to only the dipole-dipole terms, however including both electric and magnetic ones, and this approximation is very satisfactory for a large class of problems involving collections of nanoparticles or microparticles [Capolino, 2009].

[16] According to Mie theory, the electric and magnetic polarizabilities of a sphere are [Bohren and Huffman, 1983; Steshenko and Capolino, 2009; Wheeler et al., 2005a]

  • display math

with

  • display math
  • display math

[17] In (9)(11), r is the sphere radius, a1 and b1 are the electric and magnetic Mie dipole scattering coefficients [Bohren and Huffman, 1983], respectively, ψ1(ρ) = ρj1(ρ) = sin ρ/ρ − cos ρ and ξ1(ρ) = ρh1(1)(ρ) = (−i/ρ − 1)eare the Riccati-Bessel functions [Abramowitz and Stegun, 1965], mr = inline image is the relative refractive index contrast, and εm is the relative permittivity of the spheres. Notice that a prime in (10) and (11) refers to the first derivative of the functions ψ1(ρ) and ξ1(ρ) with respect to their argument ρ.

[18] The local fields Eloc and Hloc in (8) acting on the 0th particle at position r0 are computed as [Campione et al., 2011b, 2012; Steshenko and Capolino, 2009]

  • display math

where inline image is a 6 × 6 matrix, since each block is a 3 × 3 matrix. In the definition of the local fields in (12) the contribution from the 0th particle itself has been removed and the value of the regularized periodic GF inline image(r0, r0, kB) (either in the principal or antidiagonal) is determined by the limit inline image(r0, r0, kB) = inline image[ inline image(r, r0, kB) − inline image(r, r0)] since both inline image and inline image are singular at r = r0. In particular, the principal diagonal GFs possess a singularity at the source location which is of one higher power in |r − r0| than the antidiagonal ones, as also stated in Yaghjian [1980].

[19] By combining (12) with (8), one obtains a linear system from which one could compute the modal complex wave numbers by solving

  • display math

for complex kB.

3. Ewald Representation for the Dyadic GFs for 3D Periodic Arrays

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Formulation Using Periodic Green's Functions
  5. 3. Ewald Representation for the Dyadic GFs for 3D Periodic Arrays
  6. 4. Modes With Real or Complex Wave Number and Description of Composite Materials Through Effective Refractive Index
  7. 5. Conclusion
  8. Appendix A:: Regularization of the Spatial Terms of the Dyadic Green's Functions
  9. Appendix B:: Supporting Mathematical Expressions
  10. Acknowledgments
  11. References

3.1. Expressions for the Regularized Principal Diagonal Dyad inline imagepd

[20] According to the Ewald representation [Ewald, 1921; Ham and Segall, 1961] the scalar GF in (4) for a 3D array of dipoles is split into the hybrid sum of spectral and spatial scalar terms

  • display math

as described also in Kustepeli and Martin [2000], Lovat et al. [2008], Park et al. [1998], and Stevanoviæ and Mosig [2007]. Accordingly,

  • display math
  • display math

where γn2 = |kB + kn|2k2, with kn = (2πn1/a) inline image + (2πn2/b) inline image + (2πn3/c) inline image, and

  • display math

[21] The function erfc(ν) denotes the complementary error function [Abramowitz and Stegun, 1965] of argument β± = RnE ± ik/(2E). The adopted Ewald parameter E is [Kustepeli and Martin, 2000]

  • display math

which is chosen based on optimizing the total number of necessary terms in both the scalar spatial and spectral series, since with this choice they both exhibit the same Gaussian convergence rate. Note that the spatial 1/R singularity is fully represented by the n = 0 term of the spatial sum.

[22] Similarly to what discussed after (12), the regularized scalar periodic GF evaluated at r0 is given by inline imagespatial(r0, r0, kB) = inline image[Gspatial(r, r0, kB) − G(r, r0)]. After performing the limit for r [RIGHTWARDS ARROW] r0 (see details in the Appendix A) one has

  • display math

[23] In (19) a prime (f′) denotes a derivative of f in (17) with respect to its argument Rn, whose expression is given in Appendix B.

[24] Then, we accordingly split the dyadic GF in (3) as

  • display math

where

  • display math

where “type” is either spectral or spatial. The first term of the dyad in (21) is proportional to the scalar GF which is given in (15) for the spectral and in (16) for the spatial type. The terms ∇∇Gspectral and ∇∇Gspatial are derived based on (15) and (16) leading to

  • display math

and

  • display math

where

  • display math

and inline imagen = Rn/Rn. Furthermore, f″ and f‴ are the second and third derivatives of f in (17) with respect to its argument Rn, respectively, whose expressions are given in Appendix B. Similarly to what discussed for the scalar GF, here ∇∇ inline imagespatial(r0, r0, kB) = inline image[∇∇Gspatial(r, r0, kB) − ∇∇G(r, r0)]. In its computation we need to evaluate the limit because both ∇∇Gspatial and ∇∇G have singular terms at r = r0 of the kind 1/R0p with p = 1, 2, 3. After performing the limit for r [RIGHTWARDS ARROW] r0 (see details in the Appendix A) one has

  • display math

3.2. Expressions for the Regularized Antidiagonal Dyad inline imagead

[25] Similarly to what done for the principal diagonal dyad in the previous section, also the dyadic GF in (7) is split into the hybrid sum of spectral and spatial dyadic terms

  • display math

through the Ewald method, where

  • display math

and “type” is either spectral or spatial. The terms ∇Gspectral and ∇Gspatial are derived based on (15) and (16) leading to

  • display math
  • display math

[26] Analogously to what discussed for the scalar GF, here ∇ inline imagespatial(r0, r0, kB) = inline image[∇Gspatial(r, r0, kB) − ∇G(r, r0)]. In its computation we need to evaluate the limit because both ∇Gspatial and ∇G have singular terms at r = r0 of the kind 1/R0 and 1/R02. After performing the limit for r [RIGHTWARDS ARROW] r0 (see details in Appendix A) one has

  • display math

3.3. Convergence of the Scalar GFs and of the Principal and Antidiagonal Blocks of the Dyadic GF

[27] As stated in section 3.1, the Ewald method allows for obtaining series with Gaussian convergence rate, so only a handful of summation terms are needed to achieve convergence. In order to verify this, we define the relative error as

  • display math

with Kspectral either Gspectral(r0, r0, kB), or ∇Gspectral,i(r0, r0, kB), or ∇∇Gspectral,ij(r0, r0, kB) and inline imagespatial either inline imagespatial(r0, r0, kB), or ∇ inline imagespatial,i(r0, r0, kB), or ∇∇ inline imagespatial,ij (r0, r0, kB), and ij = xyz indicating the vector and dyad components. In (31), Kspectralexact and inline imagespatialexact are evaluated with a sufficiently large number of terms to achieve high numerical accuracy, whereas KspectralN and inline imagespatialN are evaluated with n1 = n2 = n3 = 0, ±1, …, ±N terms only, as was done in Capolino et al. [2005, 2007]. As shown in section 2, the two principal blocks in (12) are proportional to one regularized principal diagonal dyad inline imagepd whereas the two antidiagonal blocks are proportional to one regularized antidiagonal dyad inline imagead. This fact implies that it is sufficient to check the convergence rate of inline imagepd and inline imagead for a given wave vector kB.

[28] Consider the 3D lattice of PbTe microspheres shown in Figure 1, and analyzed in section 4, and assume a pseudo-periodic wavefield with frequency of 25 THz. Assume first, for simplicity, a wave vector kB = kz inline image; in Figure 2 we show the relative error Err in (31) of the scalar spectral and spatial GFs in (15) and (19), and of the inline image (for ∇G terms) and the inline image (for the ∇∇G terms) components versus the total number of summation terms N computed for their evaluation. Note that for symmetry reasons, it follows that ∇Gx = ∇Gy = 0, ∇∇Gxx = ∇∇Gyy ≠ ∇∇Gzz, ∇∇Gxy = ∇∇Gyx = 0, and ∇∇Gxz = ∇∇Gyz = ∇∇Gzx = ∇∇Gzy = 0.

image

Figure 2. Relative error Err in (31) of the scalar and dyadic quantities in the legends for kB = kz inline image for the 3D lattice of PbTe microspheres analyzed in section 4 at 25 THz.

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[29] In general, however, under oblique field propagation also other vector and dyad components play a role. As an example, considering again the 3D lattice of PbTe microspheres analyzed in Figure 1, and in section 4, at 25 THz, and assuming kB = kB (sin θ inline image + cos θ inline image) with θ = 30°, we show in Figure 3 the relative error Err in (31) of the inline image and inline image (for ∇G terms) and the inline image and inline image (for the ∇∇G terms) components versus the total number of summation terms N computed for their evaluation.

image

Figure 3. Relative error Err in (31) of the dyadic quantities in the legends for kB = kB (sin θ inline image + cos θ inline image) with θ = 30° for the 3D lattice of PbTe microspheres analyzed in section 4 at 25 THz.

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[30] Note that, as expected, scalar spectral and spatial components in Figures 2 and 3 converge very rapidly, with about the same rate (i.e., same number of summation terms N). The same applies to the shown dyadic quantities. For example, in Figures 2 and 3, Err < 10−8 for all the series when using N = 2. Note that even N = 1 would already guarantee a relative error Err < 10−3.

[31] To provide enough evidence of the functionality of the proposed DDA method, consider the 3D lattice of TiO2 microspheres shown in Figure 1, and analyzed in section 4, and assume a field is present with frequency of 300 GHz. Assume a wave vector kB = kz inline image; in Figure 4 we show the relative error Err in (31) of the scalar spectral and spatial GFs in (15) and (19), and of the inline image (for ∇G terms) and the inline image (for the ∇∇G terms) components versus the total number of summation terms N computed for their evaluation. Similar conclusions to the results in Figure 2 apply to Figure 4.

image

Figure 4. Relative error Err in (31) of the scalar and dyadic quantities in the legends for kB = kz inline image for the 3D lattice of TiO2 microspheres analyzed in section 4 at 300 GHz.

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image

Figure 5. Frequency behavior of the magnitude of the electric (a1) and magnetic (b1) Mie dipole coefficients for (a) a PbTe microsphere with radius 1 μm and (b) a TiO2 microsphere with radius 52 μm.

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[32] Last, from a numerical point of view, one can notice that still in the provided numerical examples only a relatively small number of summation terms (N = 2) are needed to achieve convergence when using the Ewald method proposed here, with respect to the fully spatial (singular when computed at the dipole location, as in the present case) or the fully spectral (which needs an infinite number of terms to reconstruct the singularity when computed at the dipole location, as in the present case) counterparts.

4. Modes With Real or Complex Wave Number and Description of Composite Materials Through Effective Refractive Index

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Formulation Using Periodic Green's Functions
  5. 3. Ewald Representation for the Dyadic GFs for 3D Periodic Arrays
  6. 4. Modes With Real or Complex Wave Number and Description of Composite Materials Through Effective Refractive Index
  7. 5. Conclusion
  8. Appendix A:: Regularization of the Spatial Terms of the Dyadic Green's Functions
  9. Appendix B:: Supporting Mathematical Expressions
  10. Acknowledgments
  11. References

[33] We analyze some electromagnetic properties of two composite materials made of a 3D array of (i) PbTe microspheres at infrared frequencies and (ii) TiO2 microspheres at millimeter waves. Due to the high frequency values, one may assume that the chosen material composing the microspheres has μ = μ0 (though the effective permeability of the composite μeff may differ from unity). We compute and compare the modal wave numbers through mode analysis based on three different approximations: (i) single dipole approximation (SDA) using only electric dipoles [Campione et al., 2011b], (ii) only magnetic dipoles [Campione et al., 2012], and (iii) the DDA that uses both electric and magnetic dipoles discussed in sections 2 and 3 (we assumed N= 2 in the evaluation of the GF summations because it already guarantees convergence as explained in the previous section, in the shown frequency range), from which also the effective refractive index is retrieved. We want to stress that the DDA is a good approximation when the two dipole moments (or any of them) dominate the scattered-field multipole expansion. We also adopt Maxwell Garnett (MG) homogenization theory [Sihvola, 1999, 2009], and Nicolson-Ross-Weir (NRW) retrieval method [Boughriet et al., 1997; Nicolson and Ross, 1970; Simovski, 2009; Smith et al., 2002; Weir, 1974] from scattering parameters for finite thickness structures, computed here through a full-wave simulation software employing the finite element method in frequency domain (high frequency structure simulator, HFSS by Ansys Inc.). In general, the NRW solution should be proven to be consistent for varying the number of layers, as shown inCampione et al. [2011b] for example. For simplicity in Figures 69 we show only the result obtained with 4 layers, because results with other number of layers are found to be in good agreement.

image

Figure 6. Wave number kz = βz + zversus normalized frequency dispersion diagram for T-pol modes, for the considered 3D periodic array made of PbTe microspheres, retrieved with four different methods. The DDA is the one that better computes the mode wave number, when compared with NRW-HFSS.

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image

Figure 7. As in Figure 6, for TiO2 microspheres.

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image

Figure 8. Effective refractive index retrieved by different methods for the array of PbTe microspheres. The DDA method is the one that better computes the refractive index, when compared with the NRW-HFSS. MG is the worst approximation.

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image

Figure 9. As in Figure 8, for TiO2 microspheres.

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4.1. Microsphere Resonances

[34] We consider microspheres with radius r = 1 μm made of PbTe, with permittivity εm= 32.04 + 0.0524i [Basilio et al., 2011; Palik, 1985], and with r = 52 μm made of TiO2, with permittivity εm = εm + iεm, where εm = 3.33f + 92.34 and εm = 0.28f2 + 7.64f − 1.54, with f being the frequency in THz [Berdel et al., 2005; Lannebere, 2011]. PbTe microspheres resonate at infrared frequencies between 20 THz and 40 THz, and TiO2 microspheres resonate at millimeter waves between 200 GHz and 500 GHz. With these assumptions, in Figure 5we report the magnitude of the Mie dipole-like scattering coefficientsa1 and b1 (see (10) and (11)) versus frequency. It can be observed that the PbTe (TiO2) particles, for increasing frequency, exhibit first a magnetic resonance around 26 THz (300 GHz) (due to |b1| > |a1|) with residual electric contributions, and then an electric resonance around 36 THz (420 GHz) (due to |a1| > |b1|) with residual magnetic contributions. All these effects are accounted for in the DDA model proposed here, whereas the residual ones are not considered in the SDA models. This is why the DDA solution is expected to provide results in better agreement with the full-wave NRW result than simpler SDA models. Since these two electric and magnetic dipole-like terms are dominant in the scattering process, microsphere electromagnetic interactions can be based only on dipole-dipole mechanisms, including both electric and magnetic kinds.

4.2. Modes With Real or Complex Wave Number

[35] In this section, we first show the modes with real or complex wave number for traveling modes along the z-direction (i.e.,kB = kz inline image, with kz = βz + z) with dipole moments polarized along either the x- ory-direction (transversal polarization, “T-pol,” with respect to the mode traveling direction) in a 3D periodic array of PbTe microspheres embedded in free space (i.e.,k = k0, εh = 1) with cubic lattice (i.e., a = b = c), with c = 3 μm.

[36] The dispersion diagrams for the 3D periodic array described above are shown in Figure 6, computed with various methods: (i) the SDA method (including only magnetic effects Mode-SDA-M, obtained by imposingαee = 0 in (13) and neglecting the contribution from inline imageHP), (ii) the SDA with only electric effects Mode-SDA-E, obtained by imposingαmm = 0 in (13) and neglecting the contribution from inline imageEM), (iii) the DDA method in (13), and (iv) the NRW method. The latter method is applied as follows: first, reflection and transmission coefficients are retrieved through a HFSS simulation for 4 layers of microspheres. These coefficients are used to retrieve the effective refractive index neff (see the next section). Then, the modal wave number is retrieved as kz = neffk0. The plot in Figure 6 shows only the dominant transverse mode. Other modes with transverse polarization with large attenuation constant αz, symptom of a weak spatial dispersion, are present and much weaker than the fundamental one. Indeed, the effect of these highly attenuated modes can be neglected as discussed in Campione et al. [2011b, 2012]. In Figure 6one should note how the Mode-SDA-M fails to predict the feature in the range 0.6 <kc/π < 0.8 (i.e., frequency range between 30 and 40 THz), which is mainly related to the strong scattering by the electric dipoles in the same frequency range, shown in Figure 5a. Vice versa, the Mode-SDA-E fails to predict the feature in the range 0.4 <kc/π< 0.6 (i.e., frequency range between 20 and 30 THz), associated to the strong scattering by the magnetic-dipoles in the same frequency range, shown inFigure 5a. On the contrary, the DDA method allows getting results in good agreement with NRW in the entire analyzed frequency range. In this particular case however, the Mode-SDA-M is in good agreement with DDA and NRW in the range 0.45 <kc/π< 0.55 (i.e., frequency range between 22.5 and 27.5 THz, where the magnetic dipole-like scattering is dominant), and the same happens for the SDA-E in the range 0.65 <kc/π< 0.8 (i.e., frequency range between 32.5 and 40 THz, where the electric dipole-like scattering is dominant).

[37] Note however that in general there could be cases where the SDA approximations provide quite different results than the ones computed through NRW in the whole frequency range, as for example shown in Campione et al. [2012]. For this reason, we report in Figure 7 the modes with real or complex wave number for traveling modes along the z-direction (i.e.,kB = kz inline image, with kz = βz + z) with dipole moments polarized along either the x- ory-direction (transversal polarization, “T-pol,” with respect to the mode traveling direction) in a 3D periodic array of TiO2 microspheres embedded in free space (i.e., k = k0, εh = 1) with cubic lattice (i.e., a = b = c), with c = 106 μm. This case is quite extreme because the spheres are very packed in the lattice (i.e., multipolar terms may have some effect), however the DDA provides results still in good agreement with NRW since it accounts for both electric and magnetic dipolar interactions among microspheres. Note again that the Mode-SDA-M fails to predict the feature in the range 0.25 <kc/π < 0.35 (i.e., frequency range between 350 and 500 GHz), which is mainly related to the strong scattering by the electric dipoles in the same frequency range, shown in Figure 5b. Vice versa, the Mode-SDA-E fails to predict the feature in the range 0.15 <kc/π< 0.25 (i.e., frequency range between 200 and 350 GHz), associated to the strong scattering by the magnetic-dipoles in the same frequency range, shown inFigure 5b. In this case, however, the two SDA models provide curves with only salient propagation features. In contrast, the DDA model allows for a better estimation of the complex propagation constant kzwhen compared to the full-wave simulation result (NRW).

4.3. Effective Refractive Index

[38] As described in the previous section one mode only, with transverse polarization, is dominant and able to propagate inside the lattice, and therefore the composite material is now considered for refractive index homogenization. We compute the effective refractive index by using the four methods employed in the previous section, and also by using Maxwell Garnett method [Sihvola, 1999, 2009]. In particular, the refractive index can be easily retrieved through modal analysis (SDA or DDA) as neff = kz/k0, with kz being the complex wave number shown in Figures 6 and 7. With Maxwell Garnett, first εeff and μeff are computed independently [Sihvola, 1999, 2009], then neff = inline image. Through NRW, instead, neff is computed after retrieving reflection and transmission coefficients through a HFSS simulation with four layers of microspheres. The effective refractive index computed by the five different methods is shown in Figures 8 and 9. Notice how in the considered 3D arrays the DDA method provides a result in very good agreement with the NRW method. Maxwell Garnett is the one that disagrees the most because it does not include field retardation effects and electric-magnetic particle interactions accounted for in all the other methods. Also, in the MG method, electric and magnetic effects are considered separately, and not together as in the DDA or in the NRW methods.

5. Conclusion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Formulation Using Periodic Green's Functions
  5. 3. Ewald Representation for the Dyadic GFs for 3D Periodic Arrays
  6. 4. Modes With Real or Complex Wave Number and Description of Composite Materials Through Effective Refractive Index
  7. 5. Conclusion
  8. Appendix A:: Regularization of the Spatial Terms of the Dyadic Green's Functions
  9. Appendix B:: Supporting Mathematical Expressions
  10. Acknowledgments
  11. References

[39] A formulation for electromagnetic analysis of composite materials made of periodic arrays of electric and magnetic dipoles has been carried out by using new expressions for the dyadic GF for periodic arrays based on the Ewald method. Then we have used the developed theory to analyze two 3D periodic arrays made of (i) lead telluride (PbTe) and (ii) titanium dioxide (TiO2) microspheres, for which we have evaluated the modes with real or complex wave number. Each microsphere has been modeled according to the dual (electric and magnetic) dipole approximation (DDA). A mode with transverse polarization has been found to be dominant and able to propagate inside the lattice, thus the composite material has been described through an effective refractive index. Both the modal wave number and homogenized refractive index results have been compared with those obtained from the algorithm based on scattering parameters for finite thicknesses, and from single dipole approximation based on electric or magnetic dipoles only. The effective refractive index has also been calculated from Maxwell Garnett homogenization theory. The proposed DDA method has been found to be in very good agreement with full-wave simulations and showed better performance than the MG and SDA methods.

Appendix A:: Regularization of the Spatial Terms of the Dyadic Green's Functions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Formulation Using Periodic Green's Functions
  5. 3. Ewald Representation for the Dyadic GFs for 3D Periodic Arrays
  6. 4. Modes With Real or Complex Wave Number and Description of Composite Materials Through Effective Refractive Index
  7. 5. Conclusion
  8. Appendix A:: Regularization of the Spatial Terms of the Dyadic Green's Functions
  9. Appendix B:: Supporting Mathematical Expressions
  10. Acknowledgments
  11. References

[40] The regularized GF inline imagespatial(r0, r0, kB) reported in (19) is evaluated by subtracting the n = (0, 0, 0) spherical wave term of the purely spatial GF. This leads to

  • display math

[41] The same regularization is applied to ∇∇ inline imagespatial(r0, r0, kB) in (25), which leads to

  • display math

[42] The limit is evaluated as follows:

  • display math

[43] The same regularization is applied to ∇ inline imagespatial(r0, r0, kB) in (30), which leads to

  • display math

[44] The limit is evaluated as follows:

  • display math

Appendix B:: Supporting Mathematical Expressions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Formulation Using Periodic Green's Functions
  5. 3. Ewald Representation for the Dyadic GFs for 3D Periodic Arrays
  6. 4. Modes With Real or Complex Wave Number and Description of Composite Materials Through Effective Refractive Index
  7. 5. Conclusion
  8. Appendix A:: Regularization of the Spatial Terms of the Dyadic Green's Functions
  9. Appendix B:: Supporting Mathematical Expressions
  10. Acknowledgments
  11. References

[45] The expressions for the derivatives of the function f in (17) are here summarized. The first derivative f′ (with respect to the argument Rn) is given by

  • display math

where

  • display math

[46] The second derivative f″ is given by

  • display math

where

  • display math

[47] The third derivative f‴ is given by

  • display math

[48] Last, note that (17), (B1), (B3) and (B5), when computed for Rn = 0 as required in (A1), (A3) and (A5), assume the following values:

  • display math

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Formulation Using Periodic Green's Functions
  5. 3. Ewald Representation for the Dyadic GFs for 3D Periodic Arrays
  6. 4. Modes With Real or Complex Wave Number and Description of Composite Materials Through Effective Refractive Index
  7. 5. Conclusion
  8. Appendix A:: Regularization of the Spatial Terms of the Dyadic Green's Functions
  9. Appendix B:: Supporting Mathematical Expressions
  10. Acknowledgments
  11. References

[49] We acknowledge partial support from the National Science Foundation, award NSF-CMMI 1101074. The authors also thank Ansys Inc. for providing HFSS that was instrumental in this analysis. We also acknowledge useful discussions with R. Shore and A. Yaghjian, originally with Air Force Research Laboratory, Hanscom AFB, Massachusetts, and with M. Albani from the University of Siena, Italy.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Formulation Using Periodic Green's Functions
  5. 3. Ewald Representation for the Dyadic GFs for 3D Periodic Arrays
  6. 4. Modes With Real or Complex Wave Number and Description of Composite Materials Through Effective Refractive Index
  7. 5. Conclusion
  8. Appendix A:: Regularization of the Spatial Terms of the Dyadic Green's Functions
  9. Appendix B:: Supporting Mathematical Expressions
  10. Acknowledgments
  11. References
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