Tunable Double Negative Band Structure from Non-Magnetic Coated Rods

A system of periodic poly-disperse coated nano-rods is considered. Both the coated nano-rods and host material are non-magnetic. The exterior nano-coating has a frequency dependent dielectric constant and the rod has a high dielectric constant. A negative effective magnetic permeability is generated near the Mie resonances of the rods while the coating generates a negative permittivity through a field resonance controlled by the plasma frequency of the coating and the geometry of the crystal. The explicit band structure for the system is calculated in the sub-wavelength limit. Tunable pass bands exhibiting negative group velocity are generated and correspond to simultaneously negative effective dielectric permittivity and magnetic permeability. These can be explicitly controlled by adjusting the distance between rods, the coating thickness, and rod diameters.


Introduction
Metamaterials are new class of man made materials that impart unconventional electromagnetic properties derived from sub-wavelength configurations of different conventional materials [1]. The first such materials were seen to exhibit behavior associated with negative bulk dielectric constant [2] and were constructed from a cubic lattice of metal wires. Subsequently negative effective magnetic permeability at microwave frequencies [1] were derived from periodic arrays of non-magnetic metallic split ring resonators [1]. Double negative or left handed metamaterials with simultaneous negative bulk permeability and permittivity at microwave frequencies have been developed using periodic arrays of metallic posts and split ring resonators [3]. Subsequent work has delivered several new designs using different configurations of metallic resonators for double negative behavior [4,5,6,7,8,9,10].
For higher frequencies in the infrared and optical range an alternate strategy for constructing negative effective permeability from non-magnetic components relies on Mie resonances associated with small rods or particles made from high permittivity materials [11]. A double negative metamaterial may be achieved by coating the high permittivity material with a frequency dependent plasmonic or Drude type coating [12] as well as coatings with more general frequency dependence [13].
In this article we focus on the second approach and propose periodic assemblages of aligned non-magnetic dielectric rods each clad with a non-magnetic plasmonic coating. For this case we are able to explicitly calculate the propagation band structure in the sub-wavelength limit. The pass bands and stop bands are given by formulas that depend explicitly on the rod diameters and coating thickness. We show how to construct tunable pass bands associated with double negative effective properties. The physical origin of the negative effective permeability is due to the excitations of Mie resonances inside the rods. On the other hand the negative effective permittivity is caused by the extreme dielectric properties near the plasma resonance of the coating. We provide the explicit relationship linking Mie resonances and the frequency dependent effective permittivity to the spacing of the rods, the rod radii, and the coating thickness for the coated rod assemblage proposed here.
These relationships are found not by appealing to effective medium theory based on the Clausius-Mossotti formula, but instead we show that it is profitable to take a different approach and characterize wave propagation using explicit multiscale expansions for waves inside the photonic crystal. The wave number associated with a Bloch wave inside the d-periodic crystal is denoted by k = 2π/λ where λ is the wavelength. The approach taken here provides an explicit power series expansion for the fields in the parameter η = dk. We outline a systematic framework in which the homogenized dispersion relation is recovered directly from the expansion in the subwavelength d << λ limit. We introduce photonic crystals made from coated rod assemblages and obtain explicit homogenized dispersion relations for these geometries. The dispersion relations provide explicit conditions on the the distance between neighboring rods, rod radii, and coating thickness necessary for generating pass bands associated with double negative behavior. Two examples are provided showing how the band structure can be manipulated to generate double negative pass bands with negative group velocity.
We conclude referring to related work addressing wave propagation inside high contrast media. When the crystal period and wavelength have the same length scales, the ability to open band gaps for photonic crystals is established and developed in the set of papers [14], [15], and [16]. The power series strategy presented here for subwavelength analysis has been utilized earlier and developed in [17] for characterizing the dynamic dispersion relations for Bloch waves inside plasmonic crystals and for understanding the influence of effective negative permeability on the propagation of Bloch waves inside high contrast dielectrics [18]. Earlier work in two scale homogenization theory for high contrast dielectrics delivers a frequency dependent effective magnetic permiability [19], [20] identical to the generic effective permeability tensor recovered here, see also the work of [21] for homogenization in high contrast media. The connection between high contrast interfaces, homogenization and the associated generation of negative effective magnetic permeability is made in [22]. More recently two-scale homoginization theory has been developed for three dimensional split ring structures that deliver negative effective magnetic permeability [23] and for metal fibers delivering negative effective dielectric constant [24]. A method for creating metamaterials with prescribed effective dielectric permittivity and effective magnetic permeability at a fixed frequency is developed in [25].
2 Electromagnetic fields inside photonic crystals made from coated rods and the subwavelength limit The meta material is a two-dimensional photonic crystal made of parallel coated rods, see Figure 1. There can be one or more coated rods inside the crystal period. The time harmonic field is ppolarized and the magnetic field inside the crystal is H = H(x) exp (−iωt)e 3 where x = (x 1 , x 2 ) in the x 1 x 2 -plane. The period for the crystal is d and the dielectric coefficient ǫ d (x) takes the values in the host material. (2.1) The coating is a cylindrical shell of plasmonic material with dielectric constant ǫ p (ω 2 /c 2 ) = 1− Here ω p is the plasma frequency associated with the coating material and c is the speed of light in vacuum. The dielectric constant of the rod is chosen according to ǫ R = ǫr d 2 where ǫ r has units of area. The idea is to choose the dielectric permittivity of the rod to be large so that the corresponding Mie resonances are excited in the sub-wavelength limit. The dielectric constant of the host material is given by ǫ h = 1.
The time harmonic magnetic field for the d-periodic crystal is a Bloch wave where h is d-periodic in x, the wave number is given by k, and the direction of propagation in the x 1 x 2 -plane is described by the unit vectorκ = (κ 1 ,κ 2 ). H satisfies the Helmholtz equation with propagation bands described by dispersion relations involving ω and kκ. We examine the situation in the sub-wavelength limit when d tends to 0. Instead of developing an effective medium theory based on the Clausius-Mossotti formula we show that it is profitable to take a different approach and expand the Bloch wave in a multiscale power series in η = kd. Here we develop a multiscale power series expansion for h to recover "homogenized sub-wavelength" dispersion relations for plane waves inside a magnetically active double negative effective medium. We introduce the variable y = x/d that takes the period cell of size d into the unit period cell Y . Here Y is split into the subdomains given by the union of rod cross sections R, the rod coatings P and the host H. Figure 2 illustrates a unit cell Y containing one coated rod cross section. The expansion parameter is given by η = kd = 2πd λ . The sub-wavelength limit is described by η → 0 for λ fixed. In what follows it is useful to introduce ξ defined by ξ 2 = ω 2 c 2 k 2 . The propagation equations satisfied by h for y in the unit period Y are and the transmission conditions Here the interface separating the host phase and coating phase is denoted by H-C and the interface between the coating phase and the rod is denoted by R-C. Evaluation of quantities on the host side of the interface is denoted by the subscript H, on the coating side by C, and by R on the rod side.
The unit normal vector n on the host -coating interface points from the coating into the host and the unit normal vector on the rod -coating interface points from the rod into the coating. We expand h and ξ 2 as h(d y) = h 0 (y) + ηh 1 (y) + η 2 h 2 (y) + · · · , (2.9) where each term in the expansion is continuous in y and periodic on Y . Substitution of the expansion into (2.4), (2.7), and (2.8) equating like powers of η gives the following equations used to determine h 0 : and transmission conditions, Noting further that h 0 is continuous across these interfaces we apply (2.11), (2.12), (2.14), and (2.15) to discover that h 0 is a constant function h 0 = h outside the rods. From linearity it follows that the h 0 field inside the rod can be written as and m = 1 on the boundary of the rod. Define the function g(y) to be 1 for y outside the rod and g(y) to be m(y) for y inside the rod then h 0 (y) is defined up to a multiplicative constant h and is given by h 0 (y) = hg(y). Next we determine h 1 exterior to the rods. On equating like powers of η we find that − ∆h 1 = 0, (2.18) outside the rods, and the corresponding transmission conditions for h 1 are given by The equation ( where e 1 = (1, 0) and e 2 = (0, 1). We define the region inside the period cell exterior to the rods by D and introduce the position dependent dielectric constant ǫ(y) taking the value 1 inside the host and ǫ P (k 2 ξ 2 0 ) inside the coating. For future reference we introduce the tensor ǫ −1 ef f (ξ 2 0 k 2 ) defined by where I is the 2 × 2 identity tensor. Now we identify the effective magnetic permeability. Following [1], [11] the average component of the B field parallel to the rods is given by Here We recover the explicit form of the effective permeability µ ef f (ξ 2 0 k 2 ) for a distribution of circular rods inside the unit period cell. To do so recall the following Dirichlet eigenvalues E and eigenfunctions ψ for each rod cross section given by − ∆ψ = Eψ for y inside rod and ψ = 0 for y on rod boundary .
For rods with circular cross section of radius 0 < R r < 1 and polar coordinates 0 < ρ < R r , 0 ≤ θ < 2π, let χ nm be the mth zero of the nth Bessel function, then the Dirichlet eigenvalues are E nm = (χ nm /R r ) 2 , and the corresponding eigenfunctions are {J n ( χnm Rr ρ)e inθ , J n ( χnm Rr ρ)e −inθ }. The associated normalized eigenfunctions are denoted by ψ ± nm . Notice that the eigenvalue E 0m has only one associated normalized eigenfunction ψ 0m = J0(χ0m(ρ/Rr )) √ πR|J ′ 0 (χ0m)| and ψ 0m 2 = 4πR 2 χ 2 0m when n = 0. Here · denotes the area integral over a rod cross section. It is obvious that ψ ± nm 2 = 0 for n = 0. Expanding m(y) in the basis ψ ± nm noting that m − 1 = 0 outside the rods gives for each rod For a distribution of rods with radii R r j , j = 1, . . . , ℓ, we have where N (R rj ) is the number of rods with radius R rj and k 2 jm = Last we find the homogenized sub-wavelength dispersion relation. Recall that ξ 2 k 2 = ω 2 c 2 and from (2.10) The sub-wavelength dispersion relation identified as the dispersion relation between ω 0 and kκ given by We now recover this dispersion relation by deriving the formula for ξ 2 0 . Equating like powers of η gives the following equations for h 2 exterior to the rods.
− ∆h 2 + 2iηκ · ∇h 1 − η 2 h = ξ 2 0 h, in the host, (2.35) − ∆h 2 + 2iηκ · ∇h 1 − η 2 h = ξ 2 0 ǫ P (ξ 2 0 k 2 )h, in the coating, (2.36) and the transmission conditions , H-C interface, (2.37) Writing ω 2 0 /c 2 = ξ 2 0 k 2 and substitution into (2.40) delivers the homogenized subwavelength dispersion relation where the effective index of diffraction n 2 ef f is given by (2.42) Applying (2.9) and collecting results shows that to lowest order the Bloch waves H = h(x)e i(kκ·x−tω) inside the photonic crystal are given by the subwavelength expansion In the subwavelength limit the spatial averages of H correspond to spatial averages of plane waves associated with a magnetically active effective medium. To see this take any planar averaging domain S and pass to the d → 0 limit in the average to get In a similar way, taking averages over any interval a < x 3 < b on any line parallel to the x 3 axis not intersecting the coated rods gives (2.45) Thus the appropriate averages of the plane waves H hom = h exp i(kκ · x − tω 0 ) and B hom = µ ef f h exp i(kκ · x − tω 0 ) provide approximations to the average field seen in the d periodic photonic crystal for 0 < d << k.

Electromagnetic fields inside coated rod assemblages
Here we introduce a special class of photonic crystals made from coated rod assemblages and derive an explicit formula for the subwavelength dispersion relation (2.41). The formula shows how the band structure depends explicitly on the distribution of rod radii, coating thickness and distance between neighboring rods. The assemblage is described as follows. Consider first the unit period cell Y filled with disks with radii ranging down to the infinitesimal. Inside each disk we place a centered rod cross section then a concentric coating of plasmonic material and finally a concentric coating of host material, see Figure 3. The cross sections of the coated rods together with the connected host material is depicted in Figure 4. The ratio of the radii of the rod cross section, plasmonic coating, and host coating are the same for all the disks and we denote the area fractions of the host, coating, and core phases by θ H , θ P , and θ R respectively, see Figure 3. The radii of the rods used in the assemblage is denoted by R r1 > R r2 > R r3 · · · . The number of rods having radius R rj are denoted by N (R rj ). For rods of radii R rj the outer radii of the core is denoted by R pj and the distance to the nearest neighboring rod is R hj . for the coated rod assemblage it is clear that R hj is determined by R rj together with θ R and θ P . This type of configuration is known in the composites literature and is referred to as a doubly coated cylinder assemblage [26], [27], [28], [29]. The explicit formula for µ ef f follows from (2.32) and is given by (3.1) The poles of µ ef f (ω 2 0 /c 2 ) are given by k 2 jm , see Figure 5. The explicit formula for the effective dielectric permittivity follows from a standard calculation using (2.21), (2.21), (2.22) and (2.23) (see, for example [30]) and is given by On writing ǫ p (ω 2 0 /c 2 ) = 1 − ω 2 p /c 2 ω 2 0 /c 2 and ǫ H = 1 we obtain the formula for the frequency dependent effective dielectric tensor given by The dependence of µ ef f and ǫ ef f on the rod diameter and coating thickness and host area fraction is explicitly given by (3.1) and (3.4). This explicit dependence allows for the tuning of the dispersion relation where For each coated rod with radius R rj , outer coating radius R pj and nearest neighbor distance R hj the h 0 field inside the rod is given by (2.31) and the h 1 field outside the rod is where (r, γ) are local polar coordinates inside the coated rod and F is given by It is clear that h 0 diverges inside the rods with radii R rj as ω 2 0 /c 2 approaches k 2 jm while h 1 diverges in the neighborhoods of all the coated rods, R pj < r < R hj as ω 2 0 /c 2 approaches the zero of ǫ ef f ω 2 0 c 2 .

Tunable double negative behavior
When ǫ ef f and µ ef f have the same sign it is clear that n 2 ef f > 0 and wave propagation occurrs. In this section, we show that it is possible to tune the assemblage through the choice of rod radii, coating thickness and host volume fraction to exhibit single negative, double negative, and double positive behavior depending on the signs of ǫ ef f and µ ef f . For coated cylinder assemblages it is evident that the distance between neighboring rods and the coating thickness is explicitly determined by the area fraction of the host phase, the area fraction of the rods and the rod radii R rj . The dielectric function (3.4) has only one pole and one zero and these depend explicitly on the geometry of the coated rod assemblage and are given by and Figure 6: the relation between ǫ ef f and (ω 0 /c) 2 Observe that p < p * and ǫ ef f is negative when p < (ω 0 /c) 2 < p * . The relation between ǫ ef f and (ω 0 /c) 2 is displayed in Figure 6. From Figure 5 and Figure 6,it is clear that both ǫ ef f and µ ef f are negative for p < (ω 0 /c) 2 < p * provided that (ω 0 /c) 2 is simultaneously greater than but close to the poles of µ ef f . Therefore for ǫ ef f and µ ef f to be simultaneously negative it is required that p < k 2 jm < p * , i.e., the assemblage must contain rods with radii R rj satisfying the condition Now we fix the area fractions θ R = 0.8, θ P = 0.1, θ H = 0.1 , and set ω p = 25.8 THz and ǫ r = 200m 2 so that p = 0.349 × 10 10 m −2 and p * = 0.737 × 10 10 m −2 . The dispersion relation Figure 7 for an assemblage with the two largest rod radii chosen to be R r0 = 0.2 and R r1 = 0.17. These radii satisfy the requirement given by (4.3) while the radii of all other rods in the assemblage are chosen to lie below these two values and do not satisfy (4.3). It is clear from the figure that, ǫ ef f < 0 when p < (ω 0 /c) 2 < p * and otherwise ǫ ef f ≥ 0 . In this example, p is below k 2 00 and p * lies between k 2 00 and k 2 10 . Notice in Figure 7 that the slope is negative for the dispersion curve lying between k 2 00 and p * , indicating a negative group velocity.  In Figure 8, we choose the largest two radii to be given by R r0 = 0.2 and R r1 = 0.18 and all other rods have radii below these values. For this case, p is below k 2 00 but p * is between k 2 10 and k 2 01 .

Acknowledgements
This research is supported by NSF grant DMS-0807265 and AFOSR grant FA9550-05-0008.