Dispersion relation of surface plasmon polaritons in non-local materials

Tatjana Gric

doi:10.1364/OE.27.013568

1. Introduction

The system is treated as non-local if its behavior at a given point is affected by its state at another spatially divided area. Quantum states of light and matter are inherently non-local. The former reflects the fundamental wave-particle duality [1]. Quantum entanglement is treated as one of the most amazing illustrations of non-localities in nature [2,3]. Inherently weak photon-photon interactions enable its efficient demonstration in optics with photons [4,5].

Here, we analyze theoretically a non-local environment employing a plasmonic nanowire metamaterial system. The former was employed aiming to allow for a topological transition between elliptic and hyperbolic dispersions. The work is systematized in the following way. Section 2 stands for to describe the effective medium formalism in case of local and non-local approaches of a composite metamaterial. Herein, an analytical approach providing sufficient explanation of electromagnetism in nanowire metamaterials considering nonlocal optical response taking its origin from the homogenization technique is presented. The presented formalism might be used to characterize optics of coaxial-cable-like media [6–8] and other uniaxial composites. Based on the presented theoretical approach, one may reconcile the local and nonlocal effective-medium approaches frequently employed to characterize the optics of nanowire media in different limits [9]. Moreover, the origin of optical nonlocality is related to collective (averaged over many nanorods) plasmonic excitation of nanowire media due to the presented formalism. Doing so, the methodology to fulfill additional boundary conditions in nanowire composites is also provided. The established approach is illustrated using the instance of plasmonic nanowire metamaterials. These are designed employing an assembly of systemized plasmonic nanowires [10–13] embedded in a dielectric host. To get a full picture, we vary the unit cell geometry of the composite. Moreover, we have considered metamaterials with different unit cell types, i.e., rectangular, hexagon. We make an assumption that the system operated in the effective-medium regime (its unit cell $S ≪ λ_{0}$ with λ₀ being the free-space wavelength).

2. Effective medium models of the metamaterial

It is worthwhile noting, that the optical response of nanowire composites resembles that of uniaxial media with optical axis parallel to the direction of the nanowires (z). Consequently, dielectric permittivity tensor characterizing features of modes propagating in the nanowire composite is diagonal with components $ε_{x} = ε_{y} = ε_{x, y}$ and $ε_{z}$ .

One may apply the effective permittivity aiming to describe the optical features of the metamaterial. The former approach can be applied to evaluate absorption spectra of the composite. The described formalism is valid if all the characteristic sizes of the composite, such as nanorod radius and their period are much smaller than light wavelengths. The optical features of a nanorod composite are similar to the properties of a homogeneous uniaxial anisotropic medium with optical axis parallel to the nanorods (Fig. 1) because of the symmetry. Macroscopically, a diagonal permittivity tensor $\hat{ε}$ with $ε_{x x} = ε_{y y} \equiv ε_{⊥} \neq ε_{z z}$ is used to describe these optical properties. Aiming to conduct analytical calculations, a metamaterial sheet was treated as a homogeneous one characterized by either local [14] or non-local [15] effective medium approaches.

Fig. 1 Schematic illustration and a unit cell of a nanowire composite in case of hexagonal unit cell (a), in case of the rectangular unit cell (b).

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3. Local EMT

Following conventional, local effective medium characterization, the permittivity of the metamaterial is presented by the Maxwell Garnett approach [16–19]:

ε_{⊥}^{m g} = ε_{d} \frac{(1 + p) ε_{A g} + (1 - p) ε_{d}}{(1 + p) ε_{d} + (1 - p) ε_{A g}}

ε_{z z}^{m g} = p ε_{A g} + (1 - p) ε_{d}

Here

ε_{d}

and

ε_{A g}

are the permittivities of the dielectric and silver [20], correspondingly. p is the fill coefficient of the metal inside the unit cell. It is worthwhile noting, that

p = \frac{π d^{2}}{2 \sqrt{3} S^{2}}

in case of the hexagon unit cell and

p = \frac{π d^{2}}{4 S^{2}}

in case of the rectangular unit cell. The spectral properties of the components of effective permittivity tensor of the metamaterial are depicted in Fig. 2.

Fig. 2 Local effective medium parameters of the metamaterials (Ag nanowires in dielectric ( $ε_{d} = 2.4$ )) calculated based on the geometry of the nanorod metamaterial: rectangular unit cell (a, b); hexagon unit cell (c, d). S = 80 nm in (a, c); d = 10 nm in (b, d).

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4. Non-local EMT

A more complicated, non-local effective medium response should be considered if optical losses in Ag nanorods are negligible. In this case a local effective medium approach of metamaterial becomes illegal. It is worthwhile mentioning that the components of the permittivity tensor perpendicular to the optical axis are still calculated by the Maxwell Garnett approach ( $ε_{x x} = ε_{y y} = ε_{⊥}^{M G}$ ) in case of non-local EMT. On the other hand, the component of the permittivity tensor along the optical axis becomes dramatically influenced by the wave vector as follows,

ε_{z z} (k_{z}) = ξ (k_{z}^{2} \frac{c^{2}}{ω^{2}} - {(n_{z}^{l})}^{2})

ξ = p \frac{ε_{A g} + ε_{h}}{ε_{h} - {(n_{\infty}^{l})}^{2}},

where

n_{z}^{l}

is the effective refractive coefficient of the cylindrical surface plasmons that propagate in a nanowire metamaterial with the nanorod permittivity

ε_{A g}

and

n_{\infty}^{l}

is the limit of

n_{z}^{l}

for perfectly conducting nanorods.

The extended considerations presented above can be applied for the case of propagation of waves at an angle to the optical axis. Herein, the case $k_{y} = 0$ , $k_{x} \neq 0$ is considered. It is comparatively straightforward to obtain a set of two uncoupled dispersion relations. In case of nanorod metamaterials, the first of these, $k_{x}^{2} + k_{z}^{2} = ε_{x, y}^{m g} ω^{2} / c^{2}$ defines the propagation of transverse-electric (TE) polarized waves. The second, $ε_{z} (k_{z}) (k_{z}^{2} - ε_{x, y}^{m g} \frac{ω^{2}}{c^{2}}) = - ε_{x, y}^{m g} k_{x}^{2}$ , defines the propagation of the transverse-magnetic (TM)-polarized waves. The latter relation can be rewritten as [21]:

(k_{z}^{2} - k_{z}^{l}^{2}) (k_{z}^{2} - ε_{x, y}^{m g} \frac{ω^{2}}{c^{2}}) = - \frac{ε_{x, y}^{m g}}{ξ} \frac{ω^{2}}{c^{2}} k_{x}^{2}

The former proves the fact that nanorod metamaterials support two TM-polarized waves propagating with different coefficients. The similar mode pattern takes place in case of other nonlocal materials [22].

It is clearly demonstrated by Eq. (6) that off-angle ( $k_{x} \neq 0$ ) propagation of the two TM-polarized waves in anisotropic wire media can be mapped to a microscopic model of optical features of a nanowire metamaterial. Following presented formalism, the two TM modes are characterized by the components of the effective permittivities taking their origin from (i) transverse (electron oscillations perpendicular to the nanowire axes) and (ii) longitudinal (electron oscillation and the wavevector parallel to the nanowire axes) parts of the cylindrical plasmons of the wires. The off-angle wavevector stands for as the coupling constant. This nonlocality is considered only in the effective medium model because of the homogenization technique. It is worthwhile noting, that in the microscopic model of the nanorod metamaterial, all the quantities are local.

We have used three different numerical techniques providing full-wave solutions of Maxwell equations aiming to analyze the problem under the consideration. First, the finite-difference time-domain method incorporating the approximate dispersion model, given by Eq. (4) was employed. Second, we have used a more accurate description of nonlocal EMT (Eq. (6)) aiming to solve Maxwell’s equations in the frequency domain with nonlocal transfer matrix formalism. Finally, we have employed the finite-element method (FEM) to assess the validities of the effective medium description of this complex media. Due to the utilization of the FEM calculations one may consider the full 3D structure of the composite without assuming any homogenization. From the implementation perspective, FEM results use signiﬁcantly more memory and time to make calculations than local or nonlocal EMTs. Understanding propagation of light through the interface between local and nonlocal layers requires one additional boundary condition (ABC). As shown in [21], continuity of E_z and D_x can be used to introduce such ABCs from the ﬁrst principles. Doing so, we extend the technique firstly introduced in [21]. Thus, ﬁelds of the two waves propagating in the wire media are represented as

E_{z} (y, z) = E_{z}^{m g} + (γ^{m g} + γ^{l}) E_{z}^{l} |_{(x = 0)}

E_{x} (y, z) = γ^{m g} E_{x}^{m g} + γ^{l} E_{z}^{l} |_{(x = 0)},

where

γ^{m g} = - \frac{ε_{⊥}^{m g} k_{z}}{ε_{x} k_{x}} \frac{ε_{x} - ε^{l}}{ε_{x}^{m g} - ε^{l}}

,

γ^{l} = - \frac{ε_{⊥}^{m g} k_{x}}{ε_{z} k_{z}} \frac{ε_{z} - ε_{z}^{e}}{ε^{l} - ε_{z}^{e}}

.

We increase diameter, d and the distance between nanowires, S aiming to figure out the origin of the SPP modes. The outputs of these investigations are depicted in Fig. 3. When S is increased, the energy band of the SPP modes gets closer and eventually converges to the light line, because the coupling between the nanorods becomes weaker. The modes in Figs. 3(a) and 3(c) are still distinguishable while modes in Figs. 3(b) and 3(d) almost overlap with each other.

Figure 3 The influence of the pore diameter (a, c) and distance S (b, d) on the modes pattern for different metamaterial geometry: rectangular unit cell (a, b); hexagon unit cell (c, d).

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5. Conclusion

To conclude, a formalism to treat optics of nonlocal nanowire metamaterials across the whole optical spectrum has been considered. The described theory can be applied to characterize the optics of other wire-like composites together with coated wire-structures, and coax-cable-based -systems [6–8]. We have investigated the SPP modes in the silver nanowire system by means of nonlocal EMT. The calculated dispersion relations of the SPP modes have been considered for different types of the metamaterial unit cell, i.e. either rectangular or hexagonal. Dispersion relations for the varied geometry of the metamaterial unit cell are presented.

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Dispersion relation of surface plasmon polaritons in non-local materials

Abstract

1. Introduction

2. Effective medium models of the metamaterial

3. Local EMT

4. Non-local EMT

5. Conclusion

References

Cited By

Figures (3)

Equations (7)

Optics Express