## Abstract

We present general properties of surface modes in binary metal-dielectric metamaterials. We show mechanism for surface mode formation and analyze their existence conditions for semi-infinite metamaterials in the frame of couple mode theory.

©2010 Optical Society of America

## 1. Introduction

Understanding of localized states on the materials’ surfaces is crucial to understand a multitude of physical and chemical phenomena occurring on the surfaces. Owing to the pioneering works by Tamm and Shockley and following numerous efforts, the origin and nature of the surface states has been revealed [1,2]. Optical analogues of the surface states also have been studied in photonic lattice systems such as photonic crystals [3–6] and optical waveguide arrays [7,8]. The photonic lattices provide ideal and controlled environments for investigating surface modes, which have enabled new approaches to photon manipulation. Localized modes also can be supported on the surface of metamaterials made of metal-dielectric multilayers, leading to deep subwavelength mode profiles [9]. Such surface modes supported either in solids or in photonic lattices are formed by termination of the periodicity and can be categorized into two types, Tamm and Shockley states: while Tamm states appear due to strong perturbation above a certain threshold at the surface, Shockley states requires a different condition, break of the stronger bonding (coupling) at the surface in the chains of two alternating weak and strong bondings. In a band diagram, Shockley states show up in an inverted bandgap which is formed with crossing of adjacent bands.

The metal-dielectric metamaterials offer intriguing platforms for investigating surface modes. Weak and strong (binary) couplings for formation of Shockley-like modes can be realized not only with different strengths but also with combinations of signs for the two coupling coefficients, which have not been realized in the previous configurations [9]. In this paper, we present the origins of the surface mode formation arising from termination of the periodicity and identify their existence conditions in metal-dielectric metamaterials. We describe how internal interactions and surface perturbations affect the formation of surface modes in semi-infinite metamaterials through a simple model, which also allows clear physical insight into the localized modes in general photonic systems.

## 2. Internal interactions & band crossing in binary metamaterials

We begin by describing unique bulk properties of infinite binary metamaterials in Fig. 1(b)
and 1(c) via comparison with conventional metal-dielectric metamaterials in Fig. 1(a). The metal-dielectric multilayers in Fig. 1(a) have been studied as metamaterials for subwavelength imaging and focusing since we can tailor their dispersions at will by adjusting layer parameters [10–14]. For the conventional structure in Fig. 1(a), the effective medium dispersion curve ${k}_{x}{}^{2}/{\epsilon}_{\parallel}+{k}_{z}{}^{2}/{\epsilon}_{\perp}={k}_{0}^{2}$, where ${\epsilon}_{\perp}{}^{-1}={\displaystyle {\sum}_{i}{f}_{i}{\epsilon}_{i}{}^{-1}}$and ${\epsilon}_{\parallel}={\displaystyle {\sum}_{i}{f}_{i}{\epsilon}_{i}}$ for electric fields acting along the direction normal and parallel to the layer plane and ${f}_{i}$is a filling ratio of the *i*th layer, agrees well with the accurate dispersion calculated by the transfer matrix method [15] for small *κ* ( = ${k}_{x}\mathrm{\Lambda}$, *Λ*is the lattice period). However, this approach can break down for the structures that have multiple interactions between constituents in metamaterials. Since the effective medium parameters are obtained by macroscopically averaging out electromagnetic properties of each constituent, internal microscopic features are not fully taken into account. Figure 2(d)
–2(h) display the microscopic details for the coupling between surface plasmon polariton (SPP) modes in the layers [16]. The single gap SPP mode (TM polarization) in Fig. 2(d) couples to the adjacent identical mode through a thin metal layer as shown in Fig. 2(e), which causes mode splitting into symmetric and antisymmetric mode. For this case the propagation constant of the antisymmetric mode (${\beta}_{a}$) is bigger than that of the symmetric mode (${\beta}_{s}$) unlike conventional index-guided waveguides. Thus the corresponding coupling coefficient defined by $C=({\beta}_{s}-{\beta}_{a})/2$is negative (anomalous coupling) [17]. Now let us consider the structure in Fig. 1(b). The multilayer is similar to the one in Fig. 1(a) except for the alternating metal layer thicknesses, 40nm and 20nm, which leads to two different strengths of the couplings between adjacent SPP modes. As a result, two split dispersion curves are obtained. We can easily notice that the dispersion by effective medium approximation fails, even for small*κ*. Similar effect is observed with the structure in Fig. 1(c), whose unit cell consists of one metal layer (20nm) and three dielectric layers, consecutively n = 1.34 (50nm), n = 3.48(200nm), and n = 1.34(50nm) supporting single hybrid SPP mode as seen Fig. 2(f) [18]. Although the structure looks more complicated than the one in Fig. 1(b), there are also two types of couplings between SPP modes through the high index dielectric layer [normal coupling, $C>0$, Fig. 1(g)] and through the metal layer [anomalous coupling, $C<0$, Fig. 1(h)]. Thus, the resulting two dispersion curves with huge splitting cannot be approximated by the single effective medium curve in Fig. 1(c). The examples in Fig. 1(b) and 1(c) clearly demonstrate the effect of the multiple interactions in metamaterials, which also plays an essential role in supporting surface modes in metal-dielectric metamaterials.

For the binary metamaterials in Fig. 1(b) and 1(c), the band structure evolves with the relative strength of the couplings between SPP modes. We define asymmetry constant as the ratio of coupling coefficients, $\eta ={C}_{-}/{C}_{+}$. The asymmetry constant *η*can take both negative and positive values depending on the sign of ${C}_{+}$ and ${C}_{-}$. For example, *η*is positive for the structure in Fig. 1(b) and negative for the one in Fig. 1(c) [9]. The evolutions of bands are shown in Fig. 2(a)–2(c) for the structure in Fig. 1(c) with increasing magnitude of*η*. To control the magnitude (through${C}_{-}$), we adjust the thickness of the metal layer to 15nm, 8.4nm, and 5nm, respectively. The thickness of the other layers is fixed (see the figure caption). The thinner the metal layer, the stronger the coupling becomes, thus $\left|{C}_{-}\right|$ increases. The evolution of two-band structure as a function of*η* is simplified in Fig. 2(d) by coupled mode theory [2,9]. With$\eta =0$, we obtain only two discrete values corresponding to the symmetric and antisymmetric modes from the normal coupling (${C}_{+}$) with no band formation [two red circles in (d)]. As the magnitude of *η*increases, the bandwidth linearly increases until$\left|\eta \right|=1$. The bandgap with $\left|\eta \right|<1$ is a direct bandgap [the yellow region D in (d)]. Figure 2(a) corresponds to this case. The coupling through the metal layer (d_{Au} = 15nm) is not strong enough yet, so *η*is in the range of$-1<\eta <0$and the bandgap is direct. The eigenmodes at the bandedges B and T are shown in Fig. 2(e) and 2(f), respectively. With reduced metal thickness (8.4nm from 15nm), ${C}_{-}$ is increased and reaches the same magnitude as${C}_{+}$ ($\eta =-1$). The bandgap is closed [Fig. 2(b)] at$\kappa =0$and the eigenmodes in (e) and (f) become degenerate. The touching point becomes a singularity [green arrow in (b)] called diabolical point or Dirac point. It brings important consequences for the light propagation dynamics in the nano-scale as the intersection of energy bands (conical intersection) has great importance for molecular dynamics and electron transport [14]. As *η*increases more ($\left|\eta \right|>1$), the two bands cross and an inverted bandgap appears [grey region in (d)]. As a consequence of the band crossing, the eigenmodes at the bandedges are interchanged. Figure 2(c) shows a crossed band with$\eta <-1$when the metal layer thickness is more reduced to 5nm and $\left|{C}_{-}\right|$ further increases. The crossed band can be easily identified by the eigenmodes at the bandedges. We can observe the exchanged eigenmode symmetry at B’ and T’ in Fig. 2(g) and 2(h). The band crossing feature in the binary metamaterial is analogous to the electronic band crossing for the chain of atoms with s- and p- orbitals [2]. The band crossing phenomena is one of the essential features of binary metamaterials and causes a direct consequence on the formation of surface modes.

## 3. Formation of surface modes and their existence condition

When the periodicity is terminated, localized modes whose amplitudes decay away from the surface can be supported. To analyze the existence conditions of the surface modes, we describe their formations based on coupled mode theory. For a semi-infinite metamaterial with termination of ${C}_{-}$coupling on one side as seen at the top in Fig. 3(c) , the governing coupled differential equations for the bulk are given by Eq. (1):

*nth*odd-numbered and even-numbered SPP modes from the surface of binary metamaterials, respectively. Next we apply the boundary condition on the surface$(i\partial /\partial z+{\beta}^{\text{'}})\text{}{a}_{1}+{C}_{+}{b}_{1}=0$, where ${\beta}^{\text{'}}$is the propagation constant of the surface mode, and ${a}_{1}$ and ${b}_{1}$ are the amplitudes for the first and second SPP modes from the surface, respectively. For trial solutions, we use ${a}_{n}={p}_{n}{e}^{i\beta z}$ and ${b}_{n}={q}_{n}{e}^{i\beta z}$ where ${p}_{n}={p}_{0}{r}^{n-1}$ and ${q}_{n}={q}_{0}{r}^{n-1}$. For the localized modes decaying away from the surface, the condition $\left|r\right|<1$ must be met. By inserting the ansatz into Eq. (1), we obtain as below:

*η*case and extension to positive

*η*case is straightforward.

- (1).
*No surface perturbation*($Z=0$): We obtain$K=0$,${q}_{0}=0$, and$r=-1/\eta $ from the Eq. (2). Thus we have localized surface modes at the center of the bandgap ($\beta ={\beta}_{0}$) under the condition of$\left|\eta \right|>1$. This localized mode corresponds to the Shockley states in condensed matter. This mode appears only in the inverted bandgap, meaning that the stronger coupling (${C}_{-}$) is terminated at the surface. Only the odd- numbered SPP modes (${a}_{n}$) are non-zero. - (2).
*With surface perturbation*($Z\ne 0$): For this case, the following solution is obtained for*r*from Eq. (2).

*η*and

*Z*are identified in Fig. 3(a) and 3(b) from the two solutions in Eq. (3). The contours for constant

*r*indicate the degree of localization; the smaller values, the stronger localization. Depending on the combination of

*η*and

*Z*, the whole plane is divided into several regions. The existence and character of the surface modes is determined by which region the modes belong to. We describe the detailed properties of the modes in each region below.

- (i) $-1\le \eta \le 0$& $Z\le \left|\eta \right|$: In this region no surface mode exists. The bandgap is direct and the surface perturbation is too weak to induce surface modes.
- (ii) $-1\le \eta \le 0$& $Z>\left|\eta \right|$ (the region
**e**and**f**): Tamm-like surface modes are supported due to strong surface perturbations. Two different types of solutions are obtained in this region. One of these is located in the center bandgap. These Tamm-like modes move toward the center from the bandedges with increasing surface perturbations. The other type of solution is located out of the bands (not between) and move away from the band edges with increasing perturbation. Since those two types are located in different positions in the band diagram, their mode symmetries are also different. The mode profiles for those located in the center bandgap are shown in Fig. 3(d) for opposite signs of*Z*. The mode energy is more localized in the even-numbered SPP modes (${b}_{n}$). - (iii) $\eta \le -1$& $Z<\left|\eta \right|$ (the region
**a**and**b**): Although surface perturbation is weak, Shockley-like modes are supported due to the break of stronger coupling. The increase of surface perturbation $\left|Z\right|$ makes the surface modes more delocalized until it reaches the line of $Z=\pm \eta $ where the modes become extended, which is in sharp contrast to the region $-1<\eta <0$where the modes need strong surface perturbation ($\left|Z\right|>\left|\eta \right|$) to be localized. The shift of mode locations in Fig. 3(c) also confirms this. The modes move toward the bandedges from the bandgap center with increasing perturbation. The corresponding eigenmode profiles are also shown in Fig. 3(d). The mode energy is localized only in ${a}_{n}$SPP modes without surface perturbation and the field amplitude decays into the bulk without changing the sign. With increasing perturbation, the mode energy in ${b}_{n}$SPP modes starts to increase. - (iv) $\eta \le -1$& $Z>\left|\eta \right|$ (the region
**c**and**d**): Shockely-like modes do not exist in this strong perturbation region but Tamm-like modes can be supported. The modes are located out of the bands and move away from the bandedges with increasing perturbation. The corresponding eigenmodes shown in Fig. 3(d) change the sign of the field amplitude at every other SPP modes.

The mode symmetries for positive *η*are opposite compared to the case of negative *η*due to the flipped band curvatures. For finite metal-dielectric metamaterials with strong internal interactions, surfaces at each side can interact, thus can complicate the analysis [9,19]. Nevertheless the simple approach described in this paper sheds clear insight on the fundamental physics of the surface modes in metal-dielectric metamaterials.

## 5. Conclusion

We have analyzed how internal interactions and surface perturbations affect the formation of surface modes in binary metal-dielectric metamaterials. A simple model has been introduced to analyze the existence conditions and mode symmetries of the surface modes in semi-infinite metamaterials. The surface modes in metamaterials could be utilized for manipulating surface plasmon polaritons in the nano-scale.

## Acknowledgements

This work has been supported by U.S. Army Research Office (ARO) MURI program 50432 PH-MUR.

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