Influence of Nonlocality on Transmittance and Reflectance of Hyperbolic Metamaterials

Abstract

In this paper we investigate transmittance and reflectance spectra of multilayer hyperbolic metamaterials in the presence of strong spatial dispersion. Our analysis revealed a number of intriguing optical phenomena, which cannot be predicted with the local response approximation, such as total reflectance for small angles of incidence or multiple transmittance peaks of resonant character (instead of the respective local counterparts, where almost complete transparency is predicted for small angles of incidence and the broad-angle transparency can be observed within a range of larger angles of incidence). We believe that the observed effects may serve as a working principle in a number of new potential applications, such as spatial filtering, biosensing, or beam steering.

1. Introduction

Over the last two decades, a great deal of attention has been devoted to optical metamaterials providing new means for controlling wave propagation which are not achievable with conventional media [1,2,3]. A special class of uniaxially anisotropic metamaterials, called hyperbolic metamaterials (HMMs), have emerged as a particularly prospective media, due to their relatively high technological feasibility as well as wide applicability, including diffractionless lensing [4], biosensing [5,6,7,8], optical signal buffering/storing [9,10], efficient spectral [11], and spatial filtering [12], as well as many others [13,14,15,16,17,18,19,20,21].

The most common approach to predict the electromagnetic response of an HMM structure is the effective medium theory (EMT) which encompasses solely temporal dispersion, i.e., frequency-dependent effective electric permittivity tensor [13,22,23]. However, it has been shown that a correct description of realistic structures of finite dimensions often requires consideration of spatial dispersion, i.e., dependency of effective electric permittivity tensor components on wavevector [24,25,26]. Until now, a number of methods suitable for prediction of nonlocal response have been developed, including approaches based on Mie scattering theory [27] or Drude hydrodynamic model [28], as well as formalisms originating from the transfer matrix method [29,30].

The first studies devoted to spatial dispersion in hyperbolic media revealed that nonlocality substantially alters the intended performance of the structure by e.g., occurrence of an additional waveguide mode [31]. Further scientific efforts concerning nonlocality in HMMs were focused on investigating electromagnetic phenomena that are not present when spatial dispersion is negligible, such as direction controllable inverse transition radiation [32], spectral shift of intramolecular charge transfer emission [33], or the existence of Cherenkov radiation of controllable threshold [34]. Another branch of studies in this field was dedicated to possibility of controlling spatial dispersion at will. It has been shown that by proper layer arrangement in 1D HMM, e.g., Thue–Morse [35], or embedding nonuniform nanorods in 2D HMMs, e.g., cone-shaped [36], the nonlocality in HMMs can be greatly enhanced and employed, e.g., to obtain a stronger nonlinear response [36]. More recently, it has been also demonstrated that nonlocality can be efficiently controlled in spatially uniform hyperbolic metamaterials and may serve as a new mechanism for tailoring effective dispersion [37].

In this paper, we investigate the influence of nonlocality on reflectance and transmittance (RT) of an HMM structure. To unveil effects arising in the presence of strong spatial dispersion observed in an appropriately designed multilayer nanostructure [37], we perform numerical calculations based on a modified transfer matrix method (TMM), suitable for uniaxially and biaxially anisotropic media, for an HMM structure described with local and nonlocal EMT, respectively. By example of a multilayer structure based on graphene and silicon nitride, we investigate optical effects which arise in the presence of strong nonlocality, with a special emphasis on sensitivity of transmittance and reflectance to the angle of incidence. It is worth to underline that, since we do not consider spatial dispersion of constituent materials, all the observed nonlocal effects arise from appropriately chosen dimensions of the unit cell (in this case, thicknesses of the respective layers) [37]. Our analysis reveals a number of interesting nonlocality-induced effects, such as total reflectance for small angles of incidence or multiple transmittance peaks of resonant character (instead of the respective local counterparts, where almost complete transparency is predicted for small angles of incidence and the broad-angle transparency can be observed within a range of larger angles of incidence). We believe that controlling spatial dispersion in HMM structures opens new venues in various applications, e.g., requiring highly sensitive response to the angle of incidence.

2. Theoretical Model

In this section, we present theoretical background for our analysis. Firstly, we shortly discuss numerical models of the employed materials. Then, we present the local and nonlocal EMT formalisms, which are later used to describe the HMM structure. In the last subsection, we demonstrate a modified transfer matrix method (TMM), suitable for modelling propagation in planar biaxially anisotropic media, which is necessary when spatial dispersion is to be considered.

2.1. Dielectric Functions of Constituent Materials

We have selected graphene and silicon nitride (SiN) as nonmagnetic materials (relative magnetic permeability μ_r = 1) constituting the HMM structure, see Figure 1a. Complex relative electric permittivity of graphene is calculated based on the well-known Kubo formula [38], while the dielectric function of SiN is modelled based on the data available in the literature [39], which, within the considered spectral range, can be approximated with the help of Sellmeier formula [39], see Figure 1b.

It is worth to underline that of graphene and SiN are considered only in terms of their local response and have been chosen only as an example, which helps to demonstrate more general effects arising from the effective nonlocality in an appropriately designed periodic multilayer structure [37]. Technological feasibility of such structures is not impaired with significant difficulties and similar multilayer stacks have already been successfully manufactured [40,41]. Moreover, the chosen materials can be fabricated by means of well-established chemical vapor deposition [42,43] or reactive magnetron sputtering techniques [44,45].

2.2. Effective Medium Theory

With the help of local effective medium theory, a structure consisting of periodically arranged isotropic subwavelength layers (see Figure 1a) can be described as a uniform uniaxially anisotropic medium with effective diagonal permittivity tensor $\overline{\overline{{\epsilon }_{loc}}}(\omega )=diag\left(\begin{array}{ccc}{\epsilon }_{xx}^{loc},& {\epsilon }_{yy}^{loc},& {\epsilon }_{zz}^{loc}\end{array}\right)$, with components of the following form [23]

$$\begin{array}{cc}{\epsilon }_{xx}^{loc}={\epsilon }_{yy}^{loc}={\epsilon }_{\parallel }^{loc}=\frac{t_1{\epsilon }_1(\omega )+t_2{\epsilon }_2(\omega )}{t_1+t_2},& {\epsilon }_{zz}^{loc}\end{array}=\frac{{\epsilon }_1(\omega ){\epsilon }_2(\omega )\left(t_1+t_2\right)}{t_1{\epsilon }_2(\omega )+t_2{\epsilon }_1(\omega )},$$

(1)

Where ε_{1,2} and t_{1,2} are relative electric permittivities and layer thicknesses of constituent materials. However, this approximation is not correct when spatial dispersion becomes substantial, e.g., when condition t/λ→0 is not fulfilled, where t = t₁ + t₂ is the dimension of the structure’s unit cell (which, in this case, is the characteristic dimension of the structure) [46]. Thus, to predict the influence of nonlocality on the optical properties of an HMM structure, we need to employ a more comprehensive method, which allows for anticipating the impact of the wavevector on permittivity tensor components, apart from their frequency-dependent behavior.

In our analysis we use the formalism proposed by Chern [30], in which a two-constituent multilayer structure (see Figure 1a) can be effectively described as a biaxial anisotropic medium $\overline{\overline{\epsilon }}(\omega ,\overrightarrow{k})=diag\left(\begin{array}{ccc}{\epsilon }_{xx}^{nloc},& {\epsilon }_{yy}^{nloc},& {\epsilon }_{zz}^{nloc}\end{array}\right)$ having permittivity tensor components of the following form:

$${\epsilon }_{xx}^{nloc}=\frac{{\epsilon }_{\parallel }^{loc}-\frac{\alpha }{12}{k}_0^2{t}^2}{1-\frac{1}{12}{k}_z^2{t}^2},$$

(2a)

$${\epsilon }_{yy}^{nloc}={\epsilon }_{\parallel }^{loc}\left(1+\frac{1}{6}{k}_x^2{t}^2\right)+\frac{t^2}{12k_0^2}\left(k_z^4-k_x^4\right)-\frac{\alpha }{12}{k}_0^2{t}^2,$$

(2b)

$${\epsilon }_{zz}^{nloc}=\frac{{\epsilon }_{zz}^{loc}-\frac{\alpha }{12}{k}_0^2{t}^2}{1+\frac{{\epsilon }_{zz}^{loc}}{{\epsilon }_{\parallel }^{loc}}\left(\frac{\beta }{12}{k}_x^2{t}^2-\frac{\gamma }{6}{k}_0^2{t}^2\right)},$$

(2c)

Where k_x, k_z are components of the wavevector $\overrightarrow{k}$ and k_y = 0 (due to the fact, that x-z plane is the plane of incidence; see Figure 1a, and α, β, and γ are given as:

$$\alpha =\left[f_1^2{\epsilon }_1(\omega )+\left(1-f_1^2\right){\epsilon }_2(\omega )\right]\left[\left(1-f_2^2\right){\epsilon }_1(\omega )+f_2^2{\epsilon }_2(\omega )\right],$$

(3a)

$$\beta =\frac{1}{{\epsilon }_1(\omega ){\epsilon }_2(\omega )}\left[\left(1-2f_1{f}_2\right){\epsilon }_1(\omega )+2f_1{f}_2{\epsilon }_2(\omega )\right]\left[2f_1{f}_2{\epsilon }_1(\omega )+\left(1-2f_1{f}_2\right){\epsilon }_2(\omega )\right],$$

(3b)

$$\begin{array}{c}\gamma =\frac{1}{{\epsilon }_1(\omega ){\epsilon }_2(\omega )}[f_1^3{f}_2{\epsilon }_1^3(\omega )+f_1\left(1-2f_1^2{f}_2+f_2^3\right){\epsilon }_1^2(\omega ){\epsilon }_2(\omega )\\ +f_2\left(1-f_1{f}_2^2+f_1^3\right){\epsilon }_1(\omega ){\epsilon }_2^2(\omega )+f_1{f}_2^3{\epsilon }_2^3(\omega )],\end{array}$$

(3c)

And

$$f_i=t_i/t$$

(4)

is the so-called filling factor, where $i\in \left\{1,2\right\}$. It is worth to underline that such nonlocal EMT formalism is suitable for correct description of multilayered nanostructures consisted of at least 2 unit cells [24].

The introduced EMT formalisms will be further employed to provide local and nonlocal electric permittivity tensors, which are required to calculate transmittance and reflectance by means of the transfer matrix method (as discussed in the next section).

2.3. Transfer Matrix Method

In the principal coordinate system, permittivity tensor of a biaxial optical medium is of the diagonal form, i.e., $\overline{\overline{\epsilon }}=diag\left(\begin{array}{ccc}{\epsilon }_x,& {\epsilon }_y,& {\epsilon }_z\end{array}\right)$. In such a medium, two plane waves of orthogonal polarizations may exist, i.e., a transverse electric (TE) wave with field components E_y and H_x, and a transverse magnetic (TM) wave with field components H_y and E_x. In our case, we assume that time dependence of the waves is exp(-jωt) and the considered structure is nonuniform only in the z direction; thus, we have $\begin{array}{cc}\frac{\partial }{\partial x}=j{k}_x,& \frac{\partial }{\partial y}=j{k}_y\end{array}$. Because the plane of incidence is in the x-z plane, $\frac{\partial }{\partial y}=j{k}_y=0$. According to [47], starting from Maxwell equations, we may relate the field components E_y, H_x, H_y, and E_x as follows:

$$\frac{\partial E_x}{\partial z’}=\left({\mu }_r-\frac{{\overline{k_x}}^2}{{\epsilon }_z}\right)\overline{H_y},$$

(5a)

$$\frac{\partial E_y}{\partial z’}={\mu }_r\overline{H_x},$$

(5b)

$$\frac{\partial \overline{H_x}}{\partial z’}=\left({\epsilon }_y-\frac{{\overline{k_x}}^2}{{\mu }_r}\right)E_y,$$

(5c)

$$\frac{\partial \overline{H_y}}{\partial z’}={\epsilon }_x{E}_x,$$

(5d)

Where $\begin{array}{cccc}\overrightarrow{\overline{H}}=-j\sqrt{\frac{{\mu }_0}{{\epsilon }_0}}\overrightarrow{H},& k_0=\omega \sqrt{{\epsilon }_0{\mu }_0},& z’=k_{0}z,& \overline{k_x}=\frac{k_x}{k_0}\end{array}$ and μ_r denotes relative magnetic permeability (here, we consider nonmagnetic medium; thus, we have μ_r = 1).

The Equations (5a–d) can be reformulated as a single matrix equation:

$$\frac{\partial \psi }{\partial z’}-\Omega \psi =0,$$

(6)

Where

$$\begin{array}{cc}\Omega =\left[\begin{array}{cccc}0& 0& 0& 1-\frac{{\overline{k_x}}^2}{{\epsilon }_z}\\ 0& 0& 1& 0\\ 0& {\epsilon }_y-{\overline{k_x}}^2& 0& 0\\ -{\epsilon }_x& 0& 0& 0\end{array}\right],& \psi =\left[\begin{array}{c}{E}_x(z’)\\ E_y(z’)\\ \overline{H_x}(z’)\\ \overline{H_y}(z’)\end{array}\right]\end{array}.$$

(7)

Solution to the Equation (7) can be written in the following form:

$$\psi (z’)=e^{\Omega z’}\psi (0).$$

(8)

In order to solve this equation, we need to formulate an eigenvalue problem:

$$\psi (z’)=W{e}^{\lambda z’}c,$$

(9)

where c = W⁻¹ψ(0) is a vector of field amplitudes, while W and λ are eigen-vector and eigen-value matrices of Ω, respectively. Now, by employment of appropriate boundary conditions, i.e., ${\psi }_{in}={\psi }_{layer}(0)$ and ${\psi }_{layer}(k_{0}L)={\psi }_{out}$ (see Figure 2) and Equation (9), we can formulate a matrix equation describing the relationship between amplitudes in input (c_in) and output (c_out) media (see Figure 2):

$$c_{out}=W_{out}^{-1}{W}_{layer}{e}^{{\lambda }_{layer}{k}_{0}L}{W}_{layer}^{-1}{W}_{in}\cdot c_{in},$$

(10)

Where L is the thickness of an anisotropic film, $c_{\{in,out\}}=\left[\begin{array}{cccc}{c}_{\{in,out\}}^{TE+}& c_{\{in,out\}}^{TE-}& c_{\{in,out\}}^{TM+}& c_{\{in,out\}}^{TM-}\end{array}\right]$, while W_{{in,out,layer}} are eigen-vector matrices of Ω_{{in,out,layer}} of the input/output medium and the anisotropic layer, respectively (see Figure 2). Thus, the transfer matrix for an anisotropic layer (see Figure 2) takes the following form

$$T=\left[\begin{array}{cccc}{t}_{11}& t_{12}& t_{13}& t_{14}\\ t_{21}& t_{22}& t_{23}& t_{24}\\ t_{31}& t_{32}& t_{33}& t_{34}\\ t_{41}& t_{42}& t_{43}& t_{44}\end{array}\right]=W_{out}^{-1}{W}_{layer}{e}^{{\lambda }_{layer}{k}_{0}L}{W}_{layer}{W}_{in}.$$

(11)

Thus, the reflectance coefficients for TE and TM polarization can be straightforwardly derived from matrix multiplication [48]:

$$\left[\begin{array}{c}{c}_{in}^{TE+}\\ c_{in}^{TE-}\\ c_{in}^{TM+}\\ c_{in}^{TM-}\end{array}\right]={\left[\begin{array}{cccc}{t}_{11}& t_{12}& t_{13}& t_{14}\\ t_{21}& t_{22}& t_{23}& t_{24}\\ t_{31}& t_{32}& t_{33}& t_{34}\\ t_{41}& t_{42}& t_{43}& t_{44}\end{array}\right]}^{-1}\cdot \left[\begin{array}{c}{c}_{out}^{TE+}\\ 0\\ c_{out}^{TM+}\\ 0\end{array}\right],$$

(12)

resulting in:

$$R_{TE}\underset{c_{in}^{TM+}=0}{=}{\left|\frac{c_{in}^{TE-}}{c_{in}^{TE+}}\right|}^2={\left|\frac{t_{21}{t}_{33}-t_{23}{t}_{31}}{t_{11}{t}_{33}-t_{13}{t}_{31}}\right|}^2$$

(13)

and

$$R_{TM}\underset{c_{in}^{TE+}=0}{=}{\left|\frac{c_{in}^{TM-}}{c_{in}^{TM+}}\right|}^2={\left|\frac{t_{21}{t}_{43}-t_{41}{t}_{13}}{t_{11}{t}_{33}-t_{13}{t}_{31}}\right|}^2.$$

(14)

Having noted that the losses of the considered materials are low (see Figure 1b), we can estimate transmittance of the anisotropic film as follows:

$$T_{TE}\approx 1-R_{TE},$$

(15)

$$T_{TM}\approx 1-R_{TM}.$$

(16)

The Equations (13–16) will be further employed for the analysis of RT characteristics of the HMM structure depicted in the Figure 1a.

3. Results and Discussion

Here, we employed local and nonlocal EMT approaches to investigate influence of spatial dispersion on optical parameters of a hyperbolic metamaterial composed of periodically arranged layers of silicon nitride (SiN) and graphene, i.e., ε₁ = ε_SiN(ω) and ε₂ = ε_graphene(ω) (see Figure 1a). It is worth to reiterate that the obtained nonlocal response did not originate from the properties of constituent materials, but is a consequence of an appropriately designed multilayer arrangement, which led to a strong violation of the locality condition, i.e., t/λ→0 [37]. In this case, to observe a strong nonlocal response (according to [37]), we chose sufficiently thick layers constituting the unit cell, i.e., t₁ = t_SiN = 150 nm and t₂ = t_graphene = 0.35 nm (thickness of a graphene monolayer [49]). The total thickness of the HMM structure was set as t_total = 1.5 μm, which corresponds to 10 graphene/SiN bilayers—a number of unit cells ensuring validity of the EMT approach [20,40]. To determine the effects arising in the presence of nonlocality, the local and nonlocal permittivity tensor components of the considered HMM structure as well as respective RT characteristics plotted vs. wavelength were collated in the Figure 3a–d.

In general, effective permittivity tensor components obtained with local and nonlocal EMT become convergent for sufficiently long wavelengths (since t/λ→0) [37]. In the case discussed, due to strong nonlocal response provided by the proper design of the unit cell, the convergence within the considered spectral range occurs only for the permittivity tensor components ${\epsilon }_{\parallel }^{loc}$ and ${\epsilon }_{xx}^{nloc}$, ${\epsilon }_{yy}^{nloc}$ (${\epsilon }_{xx}^{nloc}\to {\epsilon }_{\parallel }^{loc}$ and ${\epsilon }_{yy}^{nloc}\to {\epsilon }_{\parallel }^{loc}$) (see Figure 3a,b). However, the local and nonlocal permittivity tensor components ${\epsilon }_{zz}^{loc}$ and ${\epsilon }_{zz}^{nloc}$ still differ considerably, causing inconsistency between respective RT characteristics over the wide spectral range (see Figure 3c,d). A particularly substantial difference can be observed within the range of shorter wavelengths (λ < 1μm), where a polarization-dependent RT as well as a total reflection, i.e., R≈1, occur, when nonlocality is considered. Since the nonlocal EMT inexplicitly considers plasmonic resonances, this effect may be related to the coupling between SPPs propagating at adjacent graphene/SiN interfaces [37,50,51].

For shorter wavelengths (λ < 1 μm), the influence of nonlocality on RT is more evident; thus, we focused our further analysis on this spectral range. For this purpose, reflectance and transmittance (RT) coefficients within λ = 0.5–0.7 μm and different angles of incidence, i.e., θ_inc= 0° (Figure 4a–d), θ_inc = 45° (Figure 4e–g), θ_inc = 85° (Figure 4h–l) were calculated for a HMM structure described with local (Figure 4a,c,e,g–i,k) and nonlocal EMT model (Figure 4b,d,f,h,j,l).

Reflectance and transmittance spectra predicted with the help of nonlocal EMT substantially differed from the respective characteristics determined with the local approach, see Figure 4a–l. It can be observed that, in contrast to local approach, nonlocal response, even for normal incidence, was polarization-dependent and highly dispersive, i.e., small changes of wavelength caused a significant modification of both reflectance and transmittance (see Figure 4a–d). What is more, the presence of nonlocality led to the occurrence of a rejection band, i.e., the spectral range where R ≈ 1 (see Figure 4a–f). It is also noteworthy that, in comparison to local RT, changing the angle incidence led to a much stronger alteration of nonlocal RT, which was caused by dependency of nonlocal permittivity tensor components on arrangement of wavevector of the incident radiation (see Equation (2a–c)). Thus, a responsible design should incorporate nonlocality to correctly predict spatial and spectral behavior of a multilayer HMM structure.

Let us now consider influence of the angle of incidence on optical properties of the considered structure. For this aim, permittivity tensor components along with transmittance and reflectance of the structure described with local (Figure 5a,c,e) and nonlocal EMT (Figure 5b,d,f) were plotted versus angle of incidence at a fixed wavelength of λ₀ = 0.55 μm. For the purpose of analysis, we considered free space (n_in = 1) as the incident medium.

The properties of the structure observed in the angular domain were still substantially influenced by the presence of nonlocality, compare Figure 5a,c,e and Figure 5b,d,f. It can be observed that, within the range of small angles, the structure reveals complete reflection for TM polarization, i.e., R ≈ 1, which was drastically different from the response predicted with the help of local approach, i.e., almost complete transparency T ≈ 1, compare Figure 5c,e and Figure 5d,f. What is more, the nonlocal reflectance and transmittance for TE polarization were also significantly different from the respective local characteristics. Furthermore, for larger angles of incidence, instead of a single broad minimum of reflectance predicted with the local approach, we obtained multiple minima of narrow width for both light polarizations, which may be useful in terms of potential applications, e.g., highly sensitive spatial filter.

To fully explore the influence of spatial dispersion, we calculated local and nonlocal transmittance as a function of angle of incidence for incident media characterized with various refractive indices n_in = 1.5 (Figure 6a,b), 2 (Figure 6c,d), and 3 (Figure 6e,f). The influence of nonlocality was particularly significant for TM polarization, for which we can observe multiple narrow peaks of transmittance that cannot be predicted with local EMT approach, compare Figure 6a,c,e and Figure 6b,d,f. On the other hand, we can observe an opposite effect for nonlocal TE polarization transmittance, for which the angular sensitivity was diminished with respect to its local counterpart, i.e., nonlocal transmittance is less sensitive to the angle of incidence.

In general, increasing the refractive index of the incident medium substantially alters transmittance of an arbitrary optical system. However, in our case, a clear-cut tendency in alteration of nonlocal transmittance cannot be observed, since the change of the refractive index, and thus the magnitude of the incident wavevector, modifies nonlocal permittivity tensor components of the medium (see Equation (2a–c)). Additionally, due to different material properties perceived by the incident wave, we can also observe a wider acceptance cone, i.e., higher value of the critical angle for total internal refraction, compare Figure 6e,f. It is worth noting that nonlocal transmittance of TM polarization preserves a discrete character, i.e., multiple narrow peaks, for an arbitrary medium, which may serve as a mechanism for experimental verification of the presence of nonlocality. Thus, the consideration of nonlocal effects allows us not only to predict a possible deviation from the intended performance, but also to exploit additional effects arising from nonlocality for the purpose of new potential applications requiring, e.g., highly efficient spatial discrimination. We believe that the nonlocality in HMMs can be viewed as an advantage and may help to deploy new exciting applications requiring tailored spatial and spectral response.

4. Conclusions

Within this paper, we have investigated local and nonlocal optical response of an HMM structure. By example of a multilayer structure based on graphene and SiN, represented by respective local dielectric functions, we demonstrated a substantial difference between local and nonlocal response by revealing effects arising in the presence of strong spatial dispersion arising in an appropriately designed HMM structure, such as a rejection band within the range of shorter wavelengths or highly selective angular transmittance characteristics. We believe that these distinctive features of nonlocal response that cannot be predicted with the help of local approximation, may serve as a working principle for a number of new potential applications, such as spatial filtering, biosensing, or beam steering.