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Abstract
In this study, we investigate an effect of spatial dispersion in anisotropic metamaterials of regular periodic geometry. We indicate conditions under which a local and nonlocal approach are convergent, as well as the areas of particularly strong nonlocality. Our analysis also reveals that new resonance transitions altering the topology of an iso-frequency surface arise in the presence of spatial dispersion. For the first time, we demonstrate that nonlocality can serve as a new mechanism for tailoring effective dispersion of an anisotropic metamaterial, which opens new venues for novel applications requiring strong direction discrimination of the incident radiation.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Designing conventional optical devices and systems usually requires knowledge of spectral responses of constituent materials or elements, which results from their temporal dispersion. However, due to ever-shrinking characteristic dimensions, modern nanophotonic systems demand a more detailed analysis, encompassing an effect of spatial dispersion, i.e., nonlocality, which, by means of inverse Fourier transformation, can be translated to the optical domain as a dependence of electric permittivity on the wavevector.
One of the first studies concerning nonlocality in photonic nanostructures has revealed that wire media exhibit strong nonlocality at any frequency, leading to substantial inconsistency with the local model description [1]. Another work considering this class of structures has demonstrated that spatial dispersion also leads to excitation of an additional TM mode which can be viewed either as an effect distorting propagation properties of the medium or as an additional information channel in the system [2]. It has been also shown that nonlocality plays an important role in other types of nanostructures, including metallic nano-particle arrays [3] and negative-index metamaterials composed of split ring resonators [4]. Moreover, practical development of hyperlenses [5] and scatter-free detectors [6] based on metamaterials also requires wavevector dependency to be considered.
On one hand, it has been demonstrated that nonlocality may result in deterioration of intended performance [4,7,8], and thus adequate schemes limiting its influence, such as reduction of the unit cell’s size [4,7] or increasing the capacitance and the inductance of the system [8], have been proposed. On the other hand, there has been an increasing amount of studies focused on unveiling new effects arising from the presence of spatial dispersion. Until now, it has been proven that nonlocality in metamaterials is responsible for enhancement of spontaneous emission [9] and strengthening nonlinear optical response [10], enabling new applications in quantum information processing. Spatial dispersion in this class of structures also opens new avenues in the field of transformation optics, where efforts are focused on overcoming degenerate-band-edge engineering effects [11,12]. What is more, it has been demonstrated both theoretically and experimentally that, in the presence of nonlocality, anisotropic metamaterials possessing hyperbolic dispersion (HMMs) reveal unusual features, such as nonlocal quantum gain of plasmons [13], inverse transition radiation of controllable direction [14], large enhancement of decay rate of an emitter located inside a hyperbolic metamaterial [15] or blueshift of intramolecular charge transfer emission [16]. Furthermore, a possibility of enhancing the nonlocal effect in HMMs has been obtained by applying Thue-Morse arrangement in 1D HMM multilayer structure [17,18] or by embedding cone-shaped rods into a dielectric matrix (2D HMM) [19], providing a stronger nonlinear response, which is suitable for low-intensity nonlinear optical applications [18,19].
So far, a variety of methods for description of spatial dispersion in nanorod media [20,21] and layered (planar) nanostructures [22–27] has been developed. Among them, Mie scattering theory has been recognized as an approach suitable for spatially-dispersive nanorod media [20,21], while in the case of multilayered structures, nonlocal models are based on either hydrodynamic Drude model, applicable only for metal-dielectric structures [22,23], or transfer matrix method (TMM), applicable for arbitrary planar media [24–27].
What is relevant for the TMM-based approach, spatial dispersion has been described by means of second-order corrections for permittivity tensor elements, applicable only for low magnitudes of the wavevector, i.e., |k|<6*|k0|, where k0 is the free-space wavevector [22–26]. A more accurate nonlocal model has been proposed by Chern [27], in which expansion of the dispersion relation to the fourth order terms with respect to wavevector components is applied. This approach, in contrast to [22–26], predicts additional effects from the presence of spatial dispersion, such as induced biaxiality, double eigenwave propagation, parabolic-like dispersion, as well as unusual negative refraction and backward wave for all positive tensor permittivity components. Moreover, it is applicable over a large range of wavevector magnitudes, up to ∼100*|k0| [27,28].
In this paper, we performed a comprehensive analysis of spatial dispersion and its influence on the optical properties of regular planar anisotropic metamaterials (further referred as regular AMM) by means of the nonlocal effective medium theory formalism developed by Chern [27]. In our analysis, we demonstrated both the previously confirmed phenomena arising from nonlocality, such as induced biaxiality [27,28], as well as a number of new effects that have not been yet investigated, including additional resonance transitions of permittivity tensor components altering the topology of the iso-frequency surface (further referred as topological points). Moreover, we indicated that, due to the spatial dispersion, a regular AMM may reveal completely different type of dispersion than predicted with a local approximation. Our analysis also showed that, for a given wavelength, a single nonlocal regular AMM structure can reveal various types of dispersion, i.e., dielectric, elliptic, metallic, epsilon-near-zero, type I and II hyperbolic, depending on the angle and polarization of the incident radiation, enabling a new mechanism for tuning effective properties of an AMM medium. Additionally, we showed that spatial dispersion can be either suppressed or enhanced by proper engineering of the unit cell of a periodic AMM structure, without disturbing its periodicity (in contrast to [17–19]). What is worth noting, all the aforementioned effects may occur even within low wavevector magnitudes range, which until now has been commonly recognized as an attribute of the local approximation.
Our paper is organized in the following manner: first section presents numerical tools and methods employed for the analysis, including homogenization formalism suitable for regular AMM, i.e., nonlocal EMT model developed by Chern [27], along with frequency-dependent dielectric functions of materials chosen for the analysis, i.e., graphene and hafnium dioxide. The following section is dedicated to analysis and discussion of the obtained results. The third section encompasses discussion of technological feasibility of the considered structure, including fabrication of the constituent materials and the complete multilayer structure. The final section summarizes the obtained results and a short discussion of possible applications requiring non-mechanical beam steering is given.
2. Theory
In this section we present our approach to the analysis of nonlocality in AMM structures. The first subsection covers the employed nonlocal EMT model along with a discussion of the scope of its applicability, whereas the second subsection demonstrates numerical models of constituent materials chosen for analysis, i.e., graphene and hafnium dioxide (HfO2).
2.1 Local and nonlocal EMT
According to the local effective medium theory (local EMT), any periodic multilayer medium composed of subwavelength layers of isotropic materials can be effectively treated as a homogenous uniaxial anisotropic medium described by a diagonal permittivity tensor [27,29]:
Without loss of generality, the analysis is focused on a periodic multilayer medium composed of multiple unit cells of two different materials with relative electric permittivities ɛ1, ɛ2 and layer thicknesses a1, a2, respectively, see Fig. 1. The plane of incidence is in the xz-plane, which means that the y component of the impinging wavevector equals zero. Thus, we have $\overrightarrow {{k_{inc}}} = |{{k_{inc}}} |\left[ {\begin{array}{ccc} {\sin {\theta_{inc}}}&0&{\cos {\theta_{inc}}} \end{array}} \right]$, where |kinc| is the wavevector magnitude of the incident wave and θinc is the angle of incidence.
Fig. 1. Scheme of the structure under consideration. The structure is composed of N unit cells of the materials described by the relative electric permittivities ɛ1, ɛ2 and thicknesses a1, a2.
Such two-constituent multilayer structure can be described with the help of a local EMT as a uniaxial medium of the effective permittivity tensor components of the following forms [29]:
This approximation is only valid for wavelengths much longer than the dimensions of the unit cell, i.e., λ0>>a, and when spatial dispersion is negligible, i.e., when a/λ→0. However, the local EMT approach often fails to correctly describe the response of a nanostructure, even though characteristic dimensions of the structure are sufficiently small in comparison to the wavelength considered [32]. Moreover, this approach is not suitable for larger wavevector magnitudes of impinging radiation, which can be encountered, e.g., when prism/grating coupling is applied.
Thus, predicting behavior of a device performance requires more accurate methods [27]. For this purpose, formalisms incorporating a nonlocal correction are typically applied [22–26]. Such approach, however, is valid only within a relatively small range of wavevector magnitudes, i.e., |k|≤6|k0|, and also does not predict biaxiality induced by nonlocality. Thereby, we employ a more general formalism proposed by Chern [27], which allows to describe the nonlocal response more precisely, within a larger range of wavevector magnitudes, and also anticipate additional optical axis arising from spatial dispersion. According to [27], dispersion relations for transverse electric (TE) and transverse magnetic (TM) polarization can be written down in the following form:
2.2. Numerical models of constituent materials
Without loss of generality, graphene and hafnium dioxide (HfO2) have been chosen as constituent materials of the AMM structure. Thus, we can substitute variables in Eqs. (2) and 6 with appropriate material parameters, i.e., ɛ1=ɛgraphene, ɛ2=ɛHfO2 and a1= agraphene, a2= aHfO2 [33,34].
For the purposes of our analysis, constituent materials, i.e., HfO2 and graphene, have been described with the help of their isotropic local complex dielectric functions ${\varepsilon _{HfO2}},{\varepsilon _{graphene}} = f(\omega )$, valid within the considered spectral range, i.e., 0.5-2µm [33,34]. Permittivities of graphene and HfO2 have been illustrated in Fig. 2. At this point, it is worth to underline that the dielectric functions of graphene and HfO2 do not take into account spatial dispersion of the constituent materials, which is valid as long as we do not consider geometries smaller than characteristic dimensions of the constituent materials [35], i.e., the mean free path of electrons in graphene (∼3µm [36]) and interatomic distance in HfO2 (in the order of tens of angstroms [37]). Thus, it is justified to assume that, within the considered spectral range and the dimensions of the structure, spatial dispersion of constituent materials can be neglected and observed effects arise solely from the effective nonlocality related to the multilayer arrangement [36–39].
Fig. 2. Complex permittivities of graphene (dotted lines) and HfO2 (solid lines) as functions of wavelength.
3. Results and discussion
We start our analysis with determination of local (Fig. 3(a)) and nonlocal effective permittivity tensor components (Fig. 3(b)) for a single AMM structure with the basic cell composed of 150 nm HfO2 layer and a single sheet of graphene (0.35 nm). To consider the influence of wavevector arrangement on the nonlocal permittivity tensor, the angle of incidence has been set to a nonzero value, i.e., θinc = 45°, which in turn gives nonzero values of the wavevector components.
Fig. 3. Components of the effective permittivity tensor determined with a local (a) and a nonlocal (b) EMT model. Radiation impinges from air (|kinc|=|k0|) at θinc=45°.
For a better insight, our analysis is focused on the real part of the effective permittivity tensor components, since it has a dominant influence on the topology of the iso-frequency surface of the structure [30,40]. As we can see, there is a substantial change between local and nonlocal response of the structure, compare Figs. 3(a) and 3(b). In particular, we can observe a resonance of the longitudinal component of the nonlocal permittivity tensor, i.e., $\varepsilon _{zz}^{nloc}$, occurring within the visible spectral range. According to [27], such a drastic change of electromagnetic response can be connected to the fact that nonlocal EMT takes into account additional effects arising from the multilayer arrangement of the structure, such as strongly varying field at graphene/HfO2 interfaces, which also causes substantially different high-k mode distribution. For shorter wavelengths, parallel components of the nonlocal permittivity tensor, i.e., $\varepsilon _{xx}^{nloc},\varepsilon _{yy}^{nloc}$, reach values below zero, which is significantly different from the behavior of the respective local tensor components i.e., $\varepsilon _{xx}^{loc},\varepsilon _{yy}^{loc}$ . What is intriguing, nonlocal parallel permittivity tensor components vary from each other in terms of magnitude, which means that nonlocality induces an effective biaxiality of the structure, even though the structure is spatially uniform in these directions, i.e., in the xy-plane. It is worth noting that, within the considered spectral range, the condition a/λ→0 is not fulfilled, thus spatial dispersion is no longer negligible, even though we consider a wavevector of a low magnitude, i.e., incidence from air, i.e., |kinc|=|k0|, which has been commonly recognized as valid and distinctive for the local approximation [35]. Thus, even under conditions that are commonly regarded as distinctive for local approximation, the design process should include nonlocal effects to accurately anticipate electromagnetic response.
Validity of a local or nonlocal approach can also be regarded in terms of the geometry of the structure i.e., thicknesses of layers constituting the unit cell. Since nonlocal EMT models predicts nonlinear influence of thicknesses of the layers, the commonly used filling factor parameter [41,42] is no longer unambiguous. Parallel, i.e., $\varepsilon _{xy}^{loc},\varepsilon _{xx}^{nloc},\varepsilon _{yy}^{nloc}$ (Fig. 4(a)), and longitudinal, i.e., $\varepsilon _{zz}^{loc},\varepsilon _{zz}^{nloc}$ (see Fig. 4(b)), components of local and nonlocal tensor were plotted against the dielectric thickness aHfO2. It can be observed that the local description is accurate for thin dielectric layers (aHfO2<25nm), since local and nonlocal components are convergent. However, increasing the layer thickness leads, for a given wavelength, to violation of local approximation condition, i.e., $a/\lambda \to 0$, and therefore to a stronger nonlocal response, which can be observed in a substantial change of nonlocal effective permittivity tensor components with respect to their local counterparts, see Figs. 4(a)–4(b). The influence of nonlocality is particularly apparent for the longitudinal permittivity tensor component, revealing strong resonant behavior at aHfO2≈90nm, see Fig. 4(b), related to a new topological transition not observable with the local EMT approach.
Fig. 4. Parallel (a) and longitudinal (b) components of local and nonlocal permittivity tensor, plotted versus dielectric thickness for θinc=45°, |kinc|=|k0|, λ=0.5µm
In order to investigate conditions for which nonlocality becomes particularly strong, colormaps illustrating dispersion type of a regular AMM were plotted against dielectric thickness aHfO2 and wavelength λ, see Figs. 5(a)–5(d). Based on respective dispersion relations, see Eqs. (3a)–(3b), as well as the existing terminology [30,31], we classified the types of dispersion as dielectric ($\varepsilon _{yy}^{nloc} > 1$), Epsilon-Near-Zero (ENZ; $0 < \varepsilon _{yy}^{nloc} < 1$) and metallic ($\varepsilon _{yy}^{nloc} < 0$) for TE polarization (Figs. 5(a) and 5(b)), as well as elliptic ($\varepsilon _{xx}^{nloc} > 1$ and $\varepsilon _{zz}^{nloc} > 1$), ENZ ($0 < \varepsilon _{xx}^{nloc} < 1$ and $0 < \varepsilon _{zz}^{nloc} < 1$), type I ($\varepsilon _{xx}^{nloc} > 0$ and $\varepsilon _{zz}^{nloc} < 0$) and type II HMM ($\varepsilon _{xx}^{nloc} < 0$ and $\varepsilon _{zz}^{nloc} > 0$) for TM polarization (Figs. 5(c) and 5(d)).
We can observe that, for longer wavelengths and thin dielectric layers, nonlocality does not alter the dispersion type, compare Figs. 5(a) and 5(c) and Figs. 5(b) and 5(d), which again confirms that the local and nonlocal EMT model become more convergent for low values of a/λ term. On the other hand, for shorter wavelengths (λ<0.8µm) and a sufficiently thick dielectric layer (aHfO2>85nm), very strong nonlocal response can be observed, leading to a counter-intuitive change of dispersion type, i.e., from dielectric to metallic for TE polarization and from elliptic to Type I/II hyperbolic for TM polarization. Thus, a structure designed with the help of a local EMT may reveal properties drastically different from those intended, even for incidence from air, i.e., |kinc|=|k0|, which is commonly recognized as a condition justifying use of a local approximation [35].
Fig. 5. Colormaps indicating occurrence of dispersion type as a function of dielectric thickness (with a fixed number of graphene sheets) and wavelength for a structure described with local (a,b) and nonlocal (c,d) model for TE (a,c) and TM modes (b,d).
Until now, our analysis has been performed for a wavevector of a constant arrangement, originating from a fixed angle of incidence (θinc=45°). However, to fully investigate nonlocality, dependency on the angle of incidence should be considered. For this purpose, local and nonlocal permittivity tensor components of a structure with the basic cell consisting of a single sheet of graphene and a 150 nm thick layer of HfO2 were plotted versus the angle of incidence, see Figs. 6(a)-(b). In this case, the magnitude of the wavevector and the wavelength have been set as |kinc|=3|k0| and λ0=0.5µm, respectively.
Fig. 6. Parallel (a) and longitudinal (b) components of local and nonlocal permittivity tensor plotted versus angle of incidence for |kinc|=3|k0|, aHfO2=150 nm, λ=500 nm.
In contrast to behavior predicted with the help of the local model, nonlocal permittivity tensor components reveal strong dependence on the angle of incidence, see Figs. 6(a)–6(b). Particularly strong angular-dependent behavior can be observed for the x component of the nonlocal permittivity tensor, i.e., $\varepsilon _{xx}^{nloc}$, for which a resonant transition occurs, giving arise to a topological point that cannot be anticipated by means of the local EMT. It can be observed that other permittivity tensor components, i.e., $\varepsilon _{yy}^{nloc},\varepsilon _{zz}^{nloc}$, are also influenced by the wavevector arrangement, which, in this case, leads to the increase of their magnitudes proportional to the angle of incidence.
Overall, the above analysis confirms a strong spatial dependence arising from nonlocality, which, in this class of structures, can lead to a drastic change of properties, i.e., the dispersion type, depending on the direction that the radiation is impinging from, i.e., angle of incidence.
To fully identify the conditions under which nonlocality occurs, the influence of the wavevector magnitude needs to be determined. For this purpose, colormaps of effective dispersion types for both polarizations, chosen wavelength λ=0.5µm and three different angles of incidence θ1=0°, θ2=45°, θ3=80° have been plotted against dielectric thickness and normalized wavevector magnitude |kinc/k0|, see Figs. 7(a)–7(f). The considered range of wavevector magnitudes may be obtain by means of coupling through dielectric/metallic grating, subwavelength aperture or prism coupling. The selection of angles of incidence aims to provide significantly varying wavevector arrangements.
Fig. 7. Colormaps indicating occurrence of dispersion type as a function of dielectric thickness (with fixed 0.35 nm thickness of graphene layer) and wavevector for a structure described with nonlocal EMT model for TE (a,c,e) and TM modes (b,d,f) and different angles of incidence: 0° (a,b), 45° (c,d) and 80° (e,f).
It can be observed that, nonlocality plays significant role for thicker dielectric layers (>90 nm) and/or for higher magnitudes of wavevector. Again, increasing dielectric thickness leads to drastic change in effective dispersion, which can be observed by metallic dispersion for TE polarization and type I/II hyperbolic dispersion TM polarization. This effect can be related to the fact that the employed nonlocal EMT model [27] takes into account influence of coupling between surface plasmons propagating at each metal/dielectric interface [43]. It is worth to underline that such increase of nonlocality strength is in line with the fact that the local approximation condition a/λ→0 is being violated, even though the size of the unit cell remains below subwavelength limit and nonlocal EMT is still valid [27]. Thus, decreasing the ratio a/λ by means of the structure’s dimensions or the wavelength considered can be viewed as a method to minimize effect of nonlocality in periodic AMMs.
On the other hand, strong wavevector dependence arising from nonlocality results in new topological points (indicated in Figs. 7(b), 7(d), 7(f)), again, which cannot be anticipated with the help of the local approximation. Hence, wavevector magnitude together with dielectric thickness and the angle of incidence can serve as new degrees of freedom in tailoring effective dispersion of an AMM structure. In particular, it can be observed that, for a given light polarization, dielectric thickness and wavevector magnitude, the type of dispersion can be almost freely chosen by the change of the incidence angle, compare Figs. 7(a)–7(f). Thus, a new mechanism for tuning dispersion properties of anisotropic metamaterials is obtained, which can be exploited in terms of new promising applications, such as all-optical beam steering or strongly discriminating directional coatings.
4. Discussion of feasibility and fabrication details
At this point, it is worth to underline that all considered nonlocal effects are not distinctive features of constituent materials and thus, under certain conditions, similar phenomena may be obtained for any AMM structure that is composed of metal and dielectric. However, it is still purposeful to shortly discuss feasibility of the considered exemplary structure.
Until now, similar multilayer stacks based on graphene have been successfully realized [44,45]. Both chosen materials, i.e., graphene and HfO2, may be fabricated by means of well-established deposition techniques [46–49]. Graphene layers may be obtained by either epitaxial growth or chemical vapor deposition (CVD), however CVD graphene is known to be more suitable for practical realization of multilayer structures [44]. The deposition of an HfO2 layer may also be performed by means of either a CVD process [46], or reactive magnetron sputtering [49]. However, manufacturing of the complete structure cannot be performed within a single CVD technological process. The fabrication of each basic cell of the structure requires transferring graphene sheets by, e.g., dry transfer procedure [50] or acetic acid method [51], onto previously deposited HfO2 layer. It is worth noting that preservation of graphene sheets from chemical or physical deterioration requires careful treatment of the interface throughout the technological processes [44,45]. The complete structure should consists of at least 5 unit cells to provide validity of effective medium approximation [44].
5. Summary
Within the scope of this work we have investigated nonlocality in regular periodic anisotropic metamaterials. We demonstrated schemes for diminishing and enhancing nonlocality, in contrast to previous works [17–19], in spatially uniform multilayer structures, and thus a possibility of tailoring spatial dispersion in this class of structures. Over the course of our analysis, we have identified conditions under which local approximation is correct, as well as areas that only ostensibly satisfy local approximation, and reveal strong spatial dispersion. Our study also confirmed the previously demonstrated nonlocal effects [27,28], such as induced biaxiality and additional resonance transition, arising for higher magnitudes of the wavevector. Moreover, we have shown that strong dependence on the angle of impinging radiation can serve as a new mechanism for tuning effective dispersion of AMMs, opening new avenues for applications requiring all-optical direction discrimination or beam steering, e.g., spatial filters, radars and lidars. Finally, we indicated that nonlocality in anisotropic metamaterials result in new topological points, that cannot be predicted with the local approximation, providing new possibilities for engineering effective dispersion.
Funding
Narodowe Centrum Nauki (UMO-2016/23/N/ST7/03573); Narodowe Centrum Badań i Rozwoju (TECHMATSTRATEG1/347012/3/NCBR/2017).
Acknowledgements
We would like to express gratitude to prof. Ruey-Lin Chern for insightful discussion concerning his nonlocal EMT model, as well as to Mr. Marcin Kieliszczyk for useful critiques of this research work.
Disclosures
The authors declare no conflicts of interest.
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