Hyperbolic surface waves on anisotropic materials without hyperbolic dispersion

Taavi Repän; Taavi Repän; Osamu Takayama; Andrei Lavrinenko

doi:10.1364/OE.398808

1. Introduction

Photonic surface waves, or simply surface waves, are propagating electromagnetic modes localized at the interface of two dissimilar media, whose electromagnetic fields decay exponentially away from the interface [1,2]. Various types of surface waves are known to date including surface plasmon polaritions supported at the interfaces with metals [3], Dyakonov surface waves at the interfaces with anisotropic dielectrics [4–10], Bloch surface waves on dielectric photonic crystals [11–13], and Tamm states on anisotropic materials [14,15]. Surface waves have been used for sensing [13], waveguiding [16], steering of signals [10,17,18]. Moreover, recently surface waves on anisotropic 2D materials [19,20] have been extensively investigated.

Hyperbolic metamaterials (HMMs) are distinguished by their extreme uniaxial anisotropy, having both negative and positive permittivity tensor components characterized by $\varepsilon _o$ and $\varepsilon _e$ [21–24]. Propagation of waves, bulk plasmons, in such media is characterized by hyperbolic dispersion. The unbounded nature of the hyperbolic dispersion relation corresponds to existence of modes with high spatial frequencies in a real case. These special properties lead to a wide range of applications, including far-field subwavelength imaging, engineering density of states for quantum emitters and sensing [25–27].

Propagating surface plasmons on anisotropic structures (Dyakonov plasmons) are also governed by hyperbolic-like dispersion in the plane of the interface [28,29]. In the literature so far it has been assumed that for hyperbolic surface waves (HSWs) it is required either (1) an interface between a metal and an anisotropic dielectric ($\varepsilon _o >$ 0 and $\varepsilon _e >$ 0) or (2) an interface between a HMM ($\varepsilon _o\varepsilon _e <$ 0) and an isotropic dielectric [28]. The former has been theoretically analysed by Li et al. [30], but so far most of attention has been devoted to the latter case (based on HMMs) [31], with a number of experimental demonstrations [32–35].

Losses are an important concern in engineering surface-wave-based devices. While material losses affect both bulk and surface waves, for the latter case additional challenges arise from the parasitic scattering from surface structures to free space modes [36–38]. Notably, surface waves on HMM based structures have an additional source of losses apart from absorption loss [39], via scattering to high-k modes present in the HMM (high photonic density of states). Thus enabling engineering of HSWs while avoiding HMMs allows to block a significant source of scattering losses.

Here we show that HSWs can exist under general conditions which do not require a HMM—dielectric interface. We explore the interface formed by an isotropic dielectric and an anisotropic medium described by negative permittivity tensor components ($\varepsilon _o <$ 0 and $\varepsilon _e <$ 0). We reveal that depending on the permittivity tensor components such anisotropic interfaces support two types of HSWs, which is similar to the ones on HMM-based interfaces but avoiding drawbacks of scattering loss on HMM-based interfaces. HSWs on anisotropic interface may enable engineering of surface wave propagation.

2. Theory

We consider a planar interface ($z=0$) between an anisotropic medium [with permittivity tensor $\hat {\varepsilon }=\mathrm {diag}\left (\varepsilon _{e},\varepsilon _{o},\varepsilon _{o}\right )$] and cover isotropic dielectric medium ($\varepsilon _{c}>0$), as illustrated in Fig. 1. Note that the anisotropic media can be either deep subwavelength metal-dielectric trenches (grating) or metal nanowires in dielectric host. In the dielectric medium there are typical TE and TM modes

(1)$$\begin{aligned}\boldsymbol{E}_{\mathrm{TE}} & =E_{0}^{\left(\mathrm{TE}\right)}\exp\left(\mathrm{i}\boldsymbol{k}_{c}\boldsymbol{r}\right)\left(k_{y}\hat{\boldsymbol{x}}-k_{x}\boldsymbol{\hat{y}}\right)/k_c,\end{aligned}$$

(2)$$\begin{aligned}\boldsymbol{H}_{\mathrm{TM}} & =H_{0}^{\left(\mathrm{TM}\right)}\exp\left(\mathrm{i}\boldsymbol{k}_{c}\boldsymbol{r}\right)\left(k_{y}\hat{\boldsymbol{x}}-k_{x}\boldsymbol{\hat{y}}\right)/k_c,\end{aligned}$$

while in the anisotropic media there are ordinary and extraordinary modes

(3)$$\begin{aligned}\boldsymbol{E}_{o} & = N_{o} \exp\left(\mathrm{i}\boldsymbol{k}_{o}\boldsymbol{r}\right)\left(k_{z}^{o}\hat{\boldsymbol{y}}-k_{y}\hat{\boldsymbol{z}}\right),\end{aligned}$$

(4)$$\begin{aligned}\boldsymbol{E}_{e} & = N_{e} \exp\left(\mathrm{i}\boldsymbol{k}_{e}\boldsymbol{r}\right)\left[-\left(k_{0}^{2}\varepsilon_{o}-k_{x}^{2}\right)\hat{\boldsymbol{x}}+k_{x}k_{y}\hat{\boldsymbol{y}}+k_{z}^{e}k_{x}\hat{\boldsymbol{z}}\right], \end{aligned}$$

where $N_{o}$ and $N_{e}$ are normalization coefficients [40]. The wavevectors are defined as $\boldsymbol {k}_{i}=k_{x}\hat {\boldsymbol {x}}+k_{y}\hat {\boldsymbol {y}}+k_{z}^{i}\hat {\boldsymbol {z}}$ with $i=c,o,e$. They have to satisfy dispersion relations [41,42]

(5)$$\begin{aligned}k_{x}^{2}+k_{y}^{2}+\left(k_{z}^{c}\right)^{2} & =\varepsilon_{c}k_{0}^{2},\end{aligned}$$

(6)$$\begin{aligned}k_{x}^{2}+k_{y}^{2}+\left(k_{z}^{o}\right)^{2} & =\varepsilon_{o}k_{0}^{2},\end{aligned}$$

(7)$$\begin{aligned}k_{x}^{2}/\varepsilon_{o}+k_{y}^{2}/\varepsilon_{e}+\left(k_{z}^{e}\right)^{2}/\varepsilon_{e} & =k_{0}^{2}\,. \end{aligned}$$

Fig. 1. Geometry of the anisotropic interfaces with optical axis (OA) of the uniaxial medium aligned along the $x$-axis with effective ordinary and extraordinary permittivities, $\varepsilon _o$ and $\varepsilon _e$, respectively and isotropic cover layer with permittivity $\varepsilon _c$. (a) Metal-dielectric multilayer with permittivity of metal and dielectric layers $\varepsilon _m$ and $\varepsilon _d$, respectively. (b) metal nanowire in dielectric host with permittivity of metal nanowire and dielectric host, $\varepsilon _i$ and $\varepsilon _h$, respectively.

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First we establish the limiting cases of propagation along either the $x$ or $y$ direction. Enforcing electromagnetic interface conditions and requiring fields to be evanescent on both sides of the interface leads to solutions with the propagation constants (for the ordinary and the extraordinary case, respectively)

(8)$$\begin{aligned} \beta_{o} & =\sqrt{\frac{\varepsilon_{o}\varepsilon_{c}}{\varepsilon_{o}+\varepsilon_{c}}}k_{0}\:,\end{aligned}$$

(9)$$\begin{aligned} \beta_{e} & =\sqrt{\frac{\varepsilon_{e}\varepsilon_{c}-\varepsilon_{c}^{2}}{\varepsilon_{e}-\varepsilon_{c}^{2}/\varepsilon_{o}}}k_{0}\,.\end{aligned}$$

From these equations we can identify the condition that must be fulfilled for the interface to support a propagating mode along the $y$ direction

(10)$$\varepsilon_{o}+\varepsilon_{c}<0$$

and the similar condition for the mode propagating along the $x$ axis

(11)$$\varepsilon_{e}-\varepsilon_{c}^{2}/\varepsilon_{o}<0\,.$$

In deriving condition (11), we assume that $\varepsilon _{e}<\varepsilon _{c}$, because for the other case $\beta _{e}<\sqrt {\varepsilon _{c}}k_{0}$ and thus the solution would not represent a surface wave. If the material parameters are such that only one of conditions (10) and (11) is fulfilled, then the anisotropic interface exhibits a transition between propagating and evanescent surface waves, leading to hyperbolic-like dispersion of surface waves. Here, we define two types of HSWs as HSW-1 that satisfies condition (10) and possesses hyperbolic dispersion in wavevector space (k-space) around y-axis perpendicular to the optical axis (OA, x-axis) while HSW-2 satisfies condition (11) and has hyperbolic dispersion in k-space axis is around the optical axis (x-axis). Note that these HSWs have hyperbolic dispersion in $k_{x,y}$ space, which distinguishes HMMs with hyperbolic dispersion of bulk plasmons in $k_{x,y,z}$ space.

In Fig. 2(a) we map the conditions of $\varepsilon _o$, $\varepsilon _e$, and $\varepsilon _c$, specifying two different regimes for HSWs and classify these regimes with HSW-1 and HSW-2. Note that HSW-1 also includes surface waves on type-II HMM regime ($\varepsilon _o <$ 0 and $\varepsilon _e >$ 0) and we point out that earlier works have considered subset of HSW-1 by limiting discussion to an interface between a HMM and dielectric [28]. We emphasize that an anisotropic plasmonic metamaterial ($\varepsilon _o <$ 0 and $\varepsilon _e <$ 0) is sufficient to support HSWs and we focus on this case as the novelty of this article, such that avoiding HMMs removes the problem of unwanted scattering into the high-k modes.

Fig. 2. (a) Conditions (10) (dash-dotted line) and (11) (dashed line) indicating two different types of hyperbolic surface waves. Striped region indicates the type-II HMM regime ($\varepsilon _{o}<0<\varepsilon _{e}$) inside the HSW-1 area. Inset shows typical spatial dispersion in $k_{x,y}$ space for HSW-1 and HSW-2 indicated by the blue and orange lines, respectively. Note that $\varepsilon _{c}$ = 2, corresponds to the vertical dashed line and dot-dashed line corresponds to $\varepsilon _{e}$ = $\varepsilon _{c}^{2}/\varepsilon _{o}$. (b) Operating wavelengths and metal filling fractions of Ag/TiO$_2$ trench, Ag/TiO$_2$ nanowire, and TiO$_ 2$/Ag nanowire systems, yielding effective medium parameters suitable for HSWs ($\varepsilon _{o} <$ 0 and $\varepsilon _{e} <$ 0) and types of HSWs, calculated without the loss of materials, imaginary part of permittivities. HSW-1 exhibits hyperbolic dispersion around y-axis perpendicular to the optical axis (OA, x-axis) while HSW-2 has that around the optical axis. Note that Ag/TiO$_2$ nanowires are Ag nanowires in TiO$_2$ host and TiO$_2$/Ag nanowires are TiO$_2$ nanowires in Ag host. The lower wavelength limit at 415 nm for both HSW-1 and -2 is where $\varepsilon _{m} + \varepsilon _d$ = 0. The upper wavelength limit of HSW-1 at 495 nm is at the crossing between $\varepsilon _{e} - \varepsilon _c^{2}/\varepsilon _o$ = 0 (upper boundary) and $\varepsilon _{e}$ = 0 (lower boundary), and that of HSW-2 is at the crossing between $\varepsilon _{o} + \varepsilon _c$ = 0 and 1/$\varepsilon _{e}$ = 0, respectively.

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Anisotropic metals and properties required for HSWs can be realized by periodic metamaterial structures where the period is significantly smaller than the wavelength. In such cases the structure can be approximated as a homogeneous anisotropic medium. From the fabrication standpoint the most straightforward structures exhibiting surface anisotropy are metal-dielectric trenches as illustrated in Fig. 1(a) [34]. The effective medium theory [43–45] for the trench structures (metal-dielectric multilayers) gives the effective material parameters as

(12)$$\begin{aligned}\varepsilon_{o} & =f_m\varepsilon_{m}+\left(1-f_m\right)\varepsilon_{d},\end{aligned}$$

(13)$$\begin{aligned}\varepsilon_{e}^{-1} & =f_m\varepsilon_{m}^{-1}+\left(1-f_m\right)\varepsilon_{d}^{-1},\end{aligned}$$

where $f_m$ is the volume filling fraction of metallic component with $\varepsilon _{m}$, permittivity of dielectric layers, $\varepsilon _{d}$. Assuming alternating Ag/TiO$2$ layers with $\varepsilon _d = 2.3^{2}$ – 2.4$^{2}$ for atomic layer deposited TiO$2$ film measured by ellipsometer and $\varepsilon _m$ of Ag from Ref [46], the multilayer structure behave as highly anisotropic material with different combination of effective permittivities. The structure behaves as anisotropic metal ($\varepsilon _o <$ 0 and $\varepsilon _e <$ 0) for certain volume fraction of metal and by checking the resulting effective medium parameters against conditions for HSWs [conditions (10) and (11)], they support HSW-2 for $\lambda$ = 415 nm – 495 nm as plotted in Fig. 2(b). Note that trench structures only allow obtaining HSW-2, but for achieving HSW-1 they will not suffice, as $\varepsilon _{o}>\varepsilon _{e}$ always holds whenever $\varepsilon _{e}<0$.

Instead, nanowire structures could be employed with effective anisotropic parameters calculated by the following equations

(14)$$\begin{aligned}\varepsilon_{o} & =\varepsilon_{h}\frac{\left(1+f_i\right)\varepsilon_{i}+\left(1-f_i\right)\varepsilon_{h}}{\left(1+f_i\right)\varepsilon_{h}+\left(1-f_i\right)\varepsilon_{i}},\end{aligned}$$

(15)$$\begin{aligned}\varepsilon_{e} & =f\varepsilon_{i}+\left(1-f\right)\varepsilon_{h},\end{aligned}$$

where $f_i$ is the volume filling fraction of the metallic inclusions (permittivity $\varepsilon _{i}$) in the host dielectric medium with permittivity $\varepsilon _{h}$, as illustrated in Fig. 1(b) [47]. It is possible to fulfill conditions for HSWs by aligning nanowires parallel to the interface, so that the optical axis of the effective medium is parallel to the interface. We consider Ag nanowires in the TiO$2$ host and show the domain where HSW-1 are supported for $\lambda$ = 415 nm – 495 nm [the red shaded region in Fig. 2(b)]. To fulfill HSW conditions at longer wavelengths TiO$2$ nanowires in the Ag host could also be considered.

Exact dispersion relation of surface waves on an anisotropic interface has been derived by Dyakonov [4]

(16)$$\left(k_{y}+k_{y}^{e}\right)\left(k_{y}+k_{y}^{o}\right)\left(\varepsilon_{c}k_{y}^{o}+\varepsilon_{o}k_{y}^{e}\right)=-\left(\varepsilon_{e}-\varepsilon_{c}\right)\left(\varepsilon_{c}-\varepsilon_{o}\right)k_{y}^{o}k_{0}^{2},$$

where $k_y$, $k_y^{o}$, and $k_y^{e}$ satisfy

(17)$$N^{2} - k_{y}^{2} = \varepsilon_c$$

(18)$$\frac{N^{2}sin^{2}\theta - (k_y^{e})^{2}}{\varepsilon_e} + \frac{N^{2}cos^{2}\theta}{\varepsilon_o} = 1$$

(19)$$N^{2} - (k_{y}^{o})^{2} = \varepsilon_o.$$

Here, $N$ is the normalized wavevector and $\theta$ is an angle from the optical axis in x-y plane. Although Eq. (16) was used in the context of anisotropic dielectric media in elliptic regime ($\varepsilon _o >$ 0 and $\varepsilon _e >$ 0) [4–6,9,10] and HMM ($\varepsilon _o <$ 0 and $\varepsilon _e >$ 0) – dielectric interfaces [28,31,34,35,48–50], this equation also covers HSW-1 and HSW-2 on anisotropic metals ($\varepsilon _o <$ 0 and $\varepsilon _e <$ 0) as introduced above. Equation (16) is in the implicit form, so obtaining dispersion of the HSWs (allowed $k_{x}$ and $k_{y}$) is not straightforward. However, after plotting the solution for Eq. (16) (Fig. 3) we see that it resembles hyperbolic dispersion of bulk modes in HMM given by

(20)$$\frac{k_{x}^{2}}{\varepsilon_{y}}+\frac{k_{y}^{2}}{\varepsilon_{x}}=k_{0}^{2},$$

where $\varepsilon _{x},\,\varepsilon _{y}$ are components of a corresponding HMM permittivity tensor. In contrast to Eq. (16), this explicit equation is easy to solve for $k_{x}$ and $k_{y}$, thus giving a useful approximation for HSW behavior. To show approximate equivalence we consider two limiting cases: $k_{x}=0$ and $k_{x}\to \infty$. In case of Eq. (20) we obtain

(21)$$\begin{aligned} \left.k_{y}\right|_{k_{x}=0} & =\sqrt{\varepsilon_{x}}k_{0},\end{aligned}$$

(22)$$\begin{aligned} \lim_{k_{x}\to\infty}k_{y} & =k_{x}\sqrt{-\frac{\varepsilon_{x}}{\varepsilon_{y}}}\,.\end{aligned}$$

Similarly, with Eq. (16) we get

(23)$$\begin{aligned} \left.k_{y}\right|_{k_{x}=0} & =\sqrt{\frac{\varepsilon_{c}\varepsilon_{o}}{\varepsilon_{c}+\varepsilon_{o}}}k_{0},\end{aligned}$$

(24)$$\begin{aligned} \lim_{k_{x}\to\infty}k_{y} & =\sqrt{-\frac{\varepsilon_{c}^{2}-\varepsilon_{e}\varepsilon_{o}}{\varepsilon_{c}^{2}-\varepsilon_{o}^{2}}}k_{x}\,.\end{aligned}$$

Now we can establish equivalence between the exact HSW dispersion Eq. (16) and the (approximate) hyperbolic dispersion Eq. (20). Equating Eq. (21) with Eq. (24) yields solution for equivalent parameters

(25)$$\begin{aligned}\varepsilon_{x}= & \frac{\varepsilon_{d}\varepsilon_{o}}{\varepsilon_{d}+\varepsilon_{o}},\end{aligned}$$

(26)$$\begin{aligned}\varepsilon_{y}= & \frac{\left(\varepsilon_{d}-\varepsilon_{o}\right)\varepsilon_{d}\varepsilon_{o}}{\varepsilon_{d}^{2}-\varepsilon_{e}\varepsilon_{o}}\,.\end{aligned}$$

Fig. 3. Spatial dispersion of HSW-1 with various permittivity parameters accordingly to the exact Dyakonov equation Eq. (16) and hyperbolic approximation Eq. (20), shown with solid and dashed lines, respectively.

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3. Result and discussion

Knowing that type-I ($\varepsilon _{o}>0$ and $\varepsilon _{e}<0$) and type-II ($\varepsilon _{o}<0$ and $\varepsilon _{e}>0$) HMMs [25] exhibit opposite phase propagation properties [51], we might expect the same for HSW-1 and HSW-2. This can be seen in Fig. 4, where fields and also phase propagation directions for the two types of HSWs are plotted. Comparing HSW-1 [Fig. 4(a)] and HSW-2 [Fig. 4(b)] indicates that phase propagation is flipped with respect to each other in the two cases. Considering propagation along the y-direction, the phase propagation is reversed for HSW-2. Backwards phase propagation has previously been reported for surface waves as well [29], although there only HMM-based HSWs were considered (allowing for only HSW-1) and thus magnetic properties were required for backwards phase propagation. In general, for HSW-1 on non-magnetic media phase propagation is always positive [29,48].

Fig. 4. Electric fields of a dipole source placed in the origin. The dipole is oriented perpendicular to the interface. A (a) HSW-1 and (b) HSW-2 are seen with $\varepsilon _{o}=-2.48+0.05\mathrm {i}$, $\varepsilon _{e}=-0.75+0.05\mathrm {i}$ and $\varepsilon _{o}=-1+0.05\mathrm {i}$, $\varepsilon _{e}=-7+0.05\mathrm {i}$, respectively. For both cases $\varepsilon _{c}=2$ is used. Arrows indicate direction of phase propagation.

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Backwards phase propagation of HSWs can arise, because the Poynting vector can be opposite in the metal and dielectric layers due to the electromagnetic interface conditions. Thus the total energy flux $\intop _{-\infty }^{\infty }P_{y}\mathrm {d}z$ (integrated Poynting vector) can be either parallel or antiparallel to the phase velocity. For isotropic case it is easy to demonstrate that the bigger part of the energy is contained in the dielectric and so the integrated Poynting vector coincides with the phase propagation direction. To show that this is not always the case for anisotropic media, we assume negligible contribution from TE mode in the dielectric to reach an approximate condition for backwards propagating HSWs (in the large $k_{x}$ limit)

(27)$$\frac{\varepsilon_{c}}{\varepsilon_{o}}-\frac{\varepsilon_{o}}{\varepsilon_{c}}<0\,.$$

Comparing this against Fig. 2(a) it is evident that HSW-1 and -2 are distinguished by phase propagation direction: HSW-1 (HSW-2) have always forwards (backwards) phase propagation, as seen in Fig. 4. Note that here we are interested in phase propagation in y direction, which is reversed and negative.

4. Conclusion

In conclusion, we show that HSWs exist for more general conditions and are not limited to the interface between an HMM and dielectric. We also show that there are two kinds of HSWs, differentiated by the phase propagation direction. Importantly the HMM–dielectric interface case only allows for HSW-1, so the general case allows for more flexibility in engineering the surface wave propagation, allowing to facilitate other designs or fabrication constraints. These surface waves are directional, and their propagation directions are sensitive to permittivities of the media at the interface. Hence, their propagation direction can be effectively controlled by changing a wavelength or material parameters for various nanophotonics applications from sensing to manipulation of light at the nanoscale.

Funding

Det Frie Forskningsråd (8022-00387B); Sihtasutus Archimedes (Kristjan Jaak scholarship); Villum Fonden (11116).

Disclosures

The authors declare no conflicts of interest.

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Hyperbolic surface waves on anisotropic materials without hyperbolic dispersion

Abstract

1. Introduction

2. Theory

3. Result and discussion

4. Conclusion

Funding

Disclosures

References

Cited By

Figures (4)

Equations (27)

Optics Express