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Abstract
We investigated ghost surface phonon polaritons localized in a naturally uniaxial hyperbolic material, where the longitudinal axis (optical axis) is at any angle with respect to the surface plane. Two ghost surface phonon polaritons (GSPs) were found in the two Reststrahlen frequency bands, respectively. They are two TM surface waves, and their electromagnetic fields not only attenuate but also oscillate with the distance away from the surface. Unlike conventional surface phonon polaritons, GSPs are virtual surface phonon-polariton modes without electrostatic limit. The numerical simulations of the GSP dispersion and polarization properties were based on the hexagonal boron nitride, and the numerical simulation of the attenuated total reflection (ATR) experiment demonstrates the existence of the GSPs and prove their excitability and observability.
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1. INTRODUCTION
In recent years, the presence of hyperbolic materials brought about a new and attractive field [1–4]. These materials have received much attention in optics and technology due to their fascinating optical characteristics and applications, such as nano-imaging [5–7], sensing [8,9], guiding-waves [10,11], thermal conductivity and emission [12–14], and surface phonon polaritons [15–20], as well as guiding and confining of electromagnetic waves [10,21]. In particular, the surface phonon polaritons supported by naturally hyperbolic crystals can propagate for a long distance since very small optical-phonon damping [15,22], unlike surface plasmon polaritons in metals or metal metamaterials [23–25]. As typically hyperbolic materials, which have lower optical loss and whose physical parameters have been definite [2,21,26], naturally hyperbolic crystals are very good examples for theoretical investigations. The most typical example is the hexagonal boron nitride (hBN) with uniaxial anisotropy [21,27,28]. It exhibits two separated Reststrahlen frequency bands (RBs), or RB-I corresponding to the longitudinal principal-value of the hBN permittivity and RB-II related to the transverse principal values. In the RB-I or RB-II, the hBN is the Type-I or Type-II hyperbolic materials. It is of an ellipsoid shape outside both the RB-I and RB-II.
Conventional surface polaritons (CSPs) can be put into two categories. CSPs in the first category propagate along the relevant material surface and monotonously attenuate along the surface normally, meanwhile they are either TM surface waves whose magnetic-field is normal to the propagation plane or TE surface waves whose electric field is vertical to the propagation plane [29,30]. For example, surface magnon polaritons supported by magnetic crystals in the Voigt geometry are TE surface waves [29,31–33], but surface plasmon polaritons at the metal surface and surface phonon polaritons supported by ionic crystals are TM surface waves [30,34,35]. CSPs in the second category are of a hybrid polarization nature or are a mixture of TE and TM surface waves. Hybrid-polarization surface polaritons generally consist of two branch waves in the supporting materials. The Dyakonov-like surface polaritons [17,36–39] are typical hybrid-polarization surface polaritons, whose two branch waves are TM and TE waves in the supporting crystals or materials, respectively. One also can find that many metamaterials support hybrid-exciton surface polaritons [40,41].
In addition to the CSPs mentioned above, another kind of surface polaritons was predicted [42] in an ionic-crystal/dielectric metamaterial and was recently experimentally observed [43] at the surface of bulk calcite recently, where the used anisotropic materials have only one RB. They are called ghost surface phonon polaritons (GSPs) and are TM hyperbolic surface waves. Unlike CSPs that are purely evanescent away from the surface, GSPs are both propagating and evanescent in the supporting materials. The anisotropy and existence of the Reststrahlen frequency band are their necessary conditions. More recently, another different GSP was predicted at the surface of antiferromagnet and metamaterials in an external magnetic field [40,44], which is a hybrid-polarization surface wave and whose two branch waves in the materials are coherent.
Recently, the CSPs at the surface of naturally hyperbolic materials have been researched [16,18,22,27,28] in the different geometries, where the optical axis (or the longitudinal axis) is normal to the surface or lies below the surface. In this paper, we will investigate ghost surface polaritons localized at the surface of naturally hyperbolic material in a specific geometry, where the optical axis is not aligned along the surface and is at any angle with respect to the surface plane and lies in the propagation plane.
2. THEORETICAL MECHANISM
The geometry and coordinate systems are shown in Fig. 1, where we applied the uniaxial hBN crystal as the example that occupies the space of $x \gt 0$, and the space of $x \lt 0$ is filled with air or vacuum. The optical axis lies in the plane of propagation (the $x \text{-} y$ plane) and is at the angle $\phi$ with respect to the surface normal. In the $xyz$ coordinate system, the original permittivity of the hBN is changed into a nondiagonal matrix expressed by
Fig. 1. Geometries and coordinate system, where the dashed line represents the orientation of the optical axis (OA) of hBN that lies in the $x \text{-} y$ plane. (a) The geometry and system for the theoretical and numerical calculations. (b) The Otto geometry for the ATR calculation with the thickness of air layer ${d}$, where ${I}$ and ${ R}$ indicate the incident and reflected lights.
Table 1. Optical Parameters of the hBN Crystal
We assume that there exists a surface polariton propagating along the $y$-axis and decaying along the $x$-axis, and its electric field is expressed by
According to ${-}i\omega {\boldsymbol D} = \nabla \times {\boldsymbol H}$, we can directly express the electric field as a function of the magnetic field above the hBN, and we can further find that the electric field satisfies equations ${\varepsilon _{\textit{xx}}}{E_x} + {\varepsilon _{\textit{xy}}}{E_y} = - \delta k{H_z}/f$ and ${\varepsilon _{\textit{xy}}}{E_x} + {\varepsilon _{\textit{yy}}}{E_y} = i\delta \Gamma {H_z}/f$ in the hBN, where $\delta = \sqrt {{\mu _0}/{\varepsilon _0}} \cong 120\pi$. Therefore, the electric field of the surface polariton can be expressed with
This clearly shows that the real and imaginary parts of the formal attenuation constant are equal to $\alpha = - {\varepsilon _t}{\varepsilon _l}{\Gamma ^\prime}/{\varepsilon _{\textit{xx}}}$ and ${-}\beta = ik{\varepsilon _{\textit{xy}}}/{\varepsilon _{\textit{xx}}}$, where $\alpha$ and $\beta$ are called the attenuation and oscillatory constants in the hBN, respectively. Equation (5) is a complex equation, but substituting (5) into (3), we find a simple real dispersion equation of the surface polariton as follows:
which is the final dispersion relation of the surface polariton under the condition of ${\Gamma ^\prime} \gt 0$ and $\alpha \gt 0$. We see that although the geometry is of low symmetry, the dispersion equation is reciprocal $f(k) = f({- k})$. Now the magnetic field in the hBN can be explicitly expressed by ${H_z} = H\exp [{2\pi ({i\beta - \alpha})x}]$, and the electric field is determined with Eq. (4b). In the numerical calculations below, we will see two surface polaritons in the two Reststrahlen frequency bands, respectively. The two polaritons are both ghost surface polaritons, or GSP-I in the RB-I and GSP-II in the RB-II, whose electromagnetic fields attenuate and simultaneously oscillate with the distance away from the surface. The two GSPs also can be called ghost surface hyperbolic polaritons, as the hBN is hyperbolic in the GSPs’ plane of propagation. The numerical results of the dispersion relation (6) in the next section will show that either GSP present in the frequency interval, where ${\varepsilon _t}{\varepsilon _l} \lt 0$ and ${\varepsilon _{\textit{xx}}} \gt 0$. From the above expressions of the electromagnetic fields, the Poynting vector of the GSP can be obtained according to ${\boldsymbol S} = 0.5{\rm Re}({{{\boldsymbol E}^*} \times {\boldsymbol H}})$ and ${\Gamma} = \alpha + i \beta$ with $\alpha = - {\varepsilon _t}{\varepsilon _l}{\Gamma ^\prime}/{\varepsilon _{\textit{xx}}}$ and ${-}\beta = ik{\varepsilon _{\textit{xy}}}/{\varepsilon _{\textit{xx}}}$. We easily realize that the Poynting vector of either GSP is along the $y$-axis (the direction of propagation). After some analytical calculation, we obtain an explicit and simple expression of ${\textbf S}$ to beThe ratio of energy flux (EF) outside and inside the hBN is directly obtained with the integration of Eq. (7) with respect to $x$,
3. NUMERICAL RESULTS AND DISCUSSION
We first discuss the phase diagram of the polaritons propagating in the $x \text{-} y$ plane in a specific, as an example illustrated in Fig. 2. We clearly see two GSPs, which are situated in the bulk-polariton stop bands and are localized inside the two RBs, respectively. Either GSP start from the vacuum light line with ${\Gamma ^\prime} = 0$ and terminates at the right boundary of the relevant RB, where $\alpha = 0$ (${\varepsilon _t} = 0$ or ${\varepsilon _l} = 0$), so it is a virtual surface mode [29,30].
Fig. 2. Polariton phase-diagram in the wavenumber-frequency space for $\phi = {45^\circ}$, where the three areas surrounded by the dotted lines are the bulk-polariton continua and the dashed line represents the vacuum light line.
Dispersion properties of the GSPs are determined with dispersion Eq. (6). We illustrate dispersion curves of GSP-I and GSP-II for various orientations of the optical axis in Figs. 3(a) and 3(b). Dispersion curves of either GSP all are finite segments, whose starting points are on the vacuum light line, and terminal points are situated at the right boundary of the relevant RB. This directly proves that the GSPs are virtual surface modes without electrostatic limits. In the case of $\phi = {0}$ or 90°, the dotted-dashed dispersion curve represents a CSP since the oscillatory constant vanishes ($\beta = 0$). The dependence of GSP-I on the orientation of the optical axis is different from that of GSP-II. The two figures show that GSP-I exists in the angle range of ${23^\circ} \lt \phi \lt {90^\circ}$, but GSP-II presents in the angle range of ${0^\circ} \lt \phi \lt {62^\circ}$. For the orientation dependence, GSP-I looks opposite to GSP-II.
Fig. 3. Dispersion curves of the two GSPs for various orientations of the optical axis. (a) Dispersion curves of GSP-I and (b) those of GSP-II. The black dashed-dotted curve in (a) or (b) represents only a CSP since $\phi = 0$ or 90° does not correspond to any GSP. The dotted line serves as the scanning line of the ATR calculation.
The attenuation and oscillatory constants are illustrated in Fig. 4, which are two important characteristic parameters of the GSPs. The former reflects the localization of the GSPs at the surface, and the larger it is, the higher the localization is. The latter implies the oscillatory behavior of the GSPs in the direction normal to the surface, and the larger it is, the more remarkable the oscillatory behavior is. Generally, $\alpha$ has a peak value on the left side of each curve and is equal to 0 at the right terminal point of each dispersion curve in Fig. 3. $\beta$ of GSP-I is a positive value, but that of GSP-II is a negative value, which means the phases of the two GSPs are opposite in the oscillatory picture. It is easily found from the expression of $\beta$ that the values of $\beta$ are opposite in sign for GSP-I and GSP-II.
Fig. 4. Attenuation and oscillatory constants of GSPs for different orientations of the optical axis. (a) Those of GSP-I and (b) those of GSP-II, where the solid curves attached to the left axis represent the attenuation constant, and the dotted curves attached to the right axis indicate the oscillatory constant.
The electromagnetic fields satisfying Eq. (4) are the important ingredients of GSP solution. They exhibit the electromagnetic field distribution and the polarization direction of either GSP. From Eqs. (4b) and (5), we find that ${E_x}/H$ is complex, but ${E_y}/H$ is imaginary and opposite in sign to ${H_z}/H$. In the numerical calculation of the electromagnetic-field distributions along the $x$-axis, the oscillatory factor ${\rm exp}({2\pi \beta x})$ should be taken as ${\cos}({2\pi \beta x})$ to satisfy the continuity of ${H_z}$ at the surface. Figure 5 illustrates the electromagnetic field distribution in the specific case of $\phi = {45^\circ}$. In the hBN, the imaginary part of ${E_x}$ is almost equal to ${E_y}/i$, while the electric field is opposite in sign to ${H_z}$ for GSP-I, but the real part of ${E_x}$ is approximately the same as ${E_y}/i$ for GSP-II, while the imaginary part has the same sign as ${H_z}$. Figure 5 also obviously reflects the localization at the surface and the oscillatory behavior in the hBN.
Fig. 5. $x$-distributions of GSP electromagnetic fields for $\phi = {45^\circ}$. (a) Those of GSP-I for $k/{f_t} = 1.5$ and (b) those of GSP-II for $k/{f_t} = 2.5$, where the vertical dotted lines indicate the position of the surface.
The Poynting vector of either GSP reflects the energy propagation and the GSP intensity in the coordinate space. According to Eq. (7), it decays with $x$ and with attenuation constants $2{\Gamma}^\prime$ and ${2}\alpha$ on the two sides of the surface, respectively. Therefore, it is monotonously attenuating with the distance from the surface. We have obtained Eq. (8), which indicates the ratio of total energy fluxes above and below the surface. It is obvious that the ratio has similar behaviors for GSP-I and GSP-II. Figure 6 shows the ratio for GSP-I. This shows that the energy flux is mainly situated outside the hBN for the GSP on the left part of a dispersion curve but is mainly localized inside the hBN on the right part of the dispersion curve. In addition, the energy is always transferred along the GSP wave-vector, whether inside or outside. It is opposite to the CSPs supported by isotropic ionic crystals, where the energy flux is opposite in direction to the CSP wavevector.
Fig. 7. ATR spectra with ${p}$-incidence (the TM incident light), and fixed incident angle and air-layer thickness. (a) ATR spectra in the vicinity of the RB-I and (b) those in the vicinity of the RB-II.
Finally, we discuss the excitation and observation of the GSPs. One can numerically simulate the attenuated total reflection (ATR) experiment in theory to prove the excitability and observability of surface polaritons. A classical geometry generally used is the Otto geometry, as illustrated in Fig. 1(b), where a prism with dielectric constant ${\varepsilon _p}$ and a substrate supporting surface polaritons are separated by an air layer of proper thickness. In this paper, ${\varepsilon _p} = 2.3$, and the substrate is the hBN in Fig. 1. For the ATR simulation, we take the damping constant in the hBN permittivity to be $\tau = 2.0\;{\rm cm^{- 1}}$. An exciting light is incident on the interface between the prism and air layer and then is totally reflected if polaritons in the hBN are not excited but is partly reflected if the polaritons are excited. In general, the surface-polariton response exhibits a sharp dip on the reflective spectrum, and the bulk polariton response exhibits absorption bands in the reflective spectrum since bulk polaritons form continua in the space of the frequency-wavevector. Because the GSPs are surface waves with TM polarization, we take a $p$-polarized light as the incident light (the exciting light). Using the transfer matrix method [44,46], we find the reflective coefficient to be expressed with
4. CONCLUSION
We have investigated ghost surface polaritons (GSPs) at the surface of hexagonal boron nitride (hBN). A special geometry was used where the optical axis of the hBN is out of surface and is at any angle to the surface normal, while it lies in the plane of propagation (the $x \text{-} y$ plane). Two GSPs (GSP-I and GSP-II) were found and were situated in the RB-I and RB-II, respectively. These GSPs are TM surface waves whose magnetic field is pointed vertical to the plane of propagation (or along the $z$-axis), and the electric field is situated in this plane. Their electromagnetic fields attenuate and oscillate with the distance away from the surface, while their energy-flux density monotonously decays with the distance in the hBN. Unlike the CSP supported by an ordinary ionic crystal [29,30], the energy flux of either GSP is along its wavevector. Although this geometry is of low symmetry, the dispersion of both GPs is reciprocal, i.e., $f(k) = f({- k})$. The dependences of the two GSPs on the orientation of the optical axis are different, or the GSP-I exists in the range of a larger $\phi$, but the GSP-II appears in the region of a smaller $\phi$. The ATR spectra were numerically simulated. The results demonstrate that GSPs can be excited in the Otto geometry and can be exactly observed by the ATR method. GSPs are a new kind of surface polaritons and can exist in hyperbolic metamaterials, naturally hyperbolic crystals, anisotropic ionic crystals, and antiferromagnets. They open a new window for surface polaritons and surface-polariton applications.
Funding
Natural Science Foundation of Heilongjiang Province (ZD2009103).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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