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The existence of surface waves at the boundary of a hyperbolic-gyromagnetic metamaterial is studied. The surface waves, which are analytically formulated in terms of the eigenfields, appear in the spatial gap between two elliptically polarized bulk modes of the metamaterial. The surface waves are chiral in the sense that they propagate unidirectionally along the edge and reverse the propagation direction upon changing sign of the gyrotropic parameter. The topological feature of the chiral surface waves can be characterized by the Berry phases of the bulk modes, showing the bulk-edge correspondence for the underlying medium. The unidirectionality of the chiral surface waves and their immunity to disorder are further demonstrated by the propagation of electromagnetic waves around sharp corners.
© 2017 Optical Society of America
1. Introduction
Electromagnetic surface waves have been the subject of intensive study over the past decades [1, 2]. The most widely studied surface waves, known as surface plasmon polaritons [3–6], normally occur at the interface between a metal and a dielectric. Surface waves may also exist at the boundary of an anisotropic dielectric medium as a new type of electromagnetic waves propagating at an interface, the so-called D’yakonov waves [7, 8], which arise from different symmetry between the media on two sides of the interface [7]. These surface waves are further supported by photonic crystals [9], chiral materials [10], and layered hyperbolic metamaterials [11–13].
Surface waves can be excited unidirectionally on a metasurface with interfacial phase discontinuity [14]. In biaxial hyperbolic metamaterials, surface waves are elliptically polarized with the helicity dependent on the propagation direction [15]. Inspired by the discovery of topological insulators in recent years [16, 17], surface waves may be further associated with the topological phase transition in photonic systems [18–32]. They are optical analogues of the chiral (helical) edge states in the quantum (spin) Hall systems [33–35], which propagate in one direction only without backscattering and are immune to disorder.
In photonic structures, the chiral or helical surface waves usually appear as gapless surface or edge states in the frequency (energy) gap between two bulk modes. These surface states also exist in the wave vector (momentum) gap, for example, in spiral waveguide arrays [22], where the propagation coordinate (say z) plays the role of ’time’. This feature may occur as well in a continuous medium characterized by the effective material parameters. For instance, the effective permittivity components in the hyperbolic medium have opposite signs in the ordinary and extraordinary directions, giving rise to two decoupled modes that touch at a single point. If a certain symmetry breaking, such as the chirality that breaks the mirror symmetry, is introduced in the hyperbolic medium, the two bulk modes are coupled together and a gap between the modes is opened, which is shown to be topologically nontrivial. A pair of helical edge states emerges inside the gap, corresponding to the topological phase transition in the photonic system treated as an effective medium [28]. The topological phase transition in the wave vector space is also identified in the surface states supported by the magnetized plasma [36].
In the present work, we study the existence of chiral surface waves on the hyperbolic-gyromagnetic metamaterials characterized by the hyperbolic permittivity and gyrotropic permeability tensors. The underlying medium is considered a variant of the hyperbolic metamaterial usually synthesized by metal-dielectric multilayers or nanowire arrays [37] with the inclusion of gyromagnetic material [38]. The surface waves, which are analytically formulated based on the eigenfields at the interface, exist in the spatial gap (along the extraordinary axis) between two elliptically polarized bulk modes of the metamaterial. The surface waves are chiral in the sense that they propagate unidirectionally on the edge (along the transverse direction) when the system is restricted to two dimensions. As the sign of gyrotropic parameter changes, the propagation direction of the surface wave is reversed and the polarization handedness is switched.
The topological feature of the chiral surface waves can be characterized by the Berry phases associated with the bulk eigenmodes, showing the bulk-edge correspondence for the hyperbolic-gyromagnetic metamaterials. This property is manifest on the calculation of Berry curvatures on the equifrequency surface based on the eigenfields of the metamaterial. The corresponding surface waves therefore represent an instance of topological phase transition in the effective medium. As in the quantum Hall systems, the chiral surface waves are unidirectional and immune to disorder. These features are confirmed by the propagation of surface waves around sharp corners at the boundary of the underlying medium.
2. Basic equations
2.1. Dispersion relations
Consider an effective medium characterized by the uniaxially anisotropic permittivity and gyrotropic permeability tensors as
where μεt > 0 and μεz < 0. This medium is termed as Type I or dielectric hyperbolic medium [39, 40] and is considered a variant of the hyperbolic metamaterial [37] with the inclusion of gyromagnetic material [38], where κ is responsible for the gyromagnetic effect [19]. The corresponding medium can be implemented by alternating layers made of dielectric and gyromagnetic materials [41, 42], with the effective constitutive parameters obtained by the effective medium theory.The time-harmonic wave propagation (with the convention e−iωt) in the hyperbolic-gyromagnetic medium can be described by the wave equation of the magnetic field H as
where is the identity tensor, k is the wave vector, and k0 = ω/c is the wave number in vacuum. The above equation is written in component form as The nontrivial solutions of H exist when the determinant of the square matrix is zero, which gives rise to the characteristic equation as where and . This is a bi-quadratic equation that represents a high-order dispersion. Equation (4) can be solved to yield the dispersion relation as where ε±= εt ± εz. There are in general two kz for each kt, as a result of the symmetry breaking implied in the constitutive relation [Eq. (1)]. The equifrequency surfaces thus consist of two parts: one with a larger |kz| and the other with a smaller |kz|. If the gyrotropic parameter κ is zero, Eq. (4) reduces to a product of two quadratic equations as which is the dispersion relation for two independent (mutually perpendicular) polarizations. In this situation, the equifrequency surfaces are the combination of a sphere and a hyperboloid, touching at two points on the ±kz axis. As κ increases from zero, the equifrequency surfaces for the two polarizations are no longer decoupled, giving rise to a mixed dispersion. Meanwhile, a spatial gap (along the kz axis) is opened between the two dispersion surfaces. A similar feature exists in the chiral hyperbolic metamaterials [28], where the chirality is responsible for the symmetry breaking. The coupled dispersion and the resulted spatial gap can be found in a more general bianisotropic medium [43].The eigen-magnetic field is determined by the nullspace of the square matrix in Eq. (3), given in component form as
Note that there are two solutions of kz for each kt [cf. Eq. (5)] and therefore two eigenwaves exist in the hyperbolic-gyromagnetic medium. The corresponding eigen-electric field is determined by as where . The eigenwaves of the hyperbolic-gyromagnetic medium are in general elliptically polarized and will be used in the formulation of surface waves stated below.2.2. Surface waves
Let the xz plane (y = 0) be the interface between vacuum (y > 0) and the hyperbolic-gyromagnetic medium (y < 0) characterized by εt, εz, μ, and κ. On the vacuum side, the eigenfields are given as
where the superscripts 1 and 2 refer to two linear (and independent) polarizations. The off-plane wave vector component ky is related to the in-plane components kx and kz as On the hyperbolic-gyromagnetic medium side, the eigenfields are denoted by where the superscripts + and − refer to two elliptical polarizations. Here, H and E are the vectors with the components defined in Eqs. (7)–(9) and Eqs. (10)–(12), respectively, in which ky should be replaced by with α±= μ2ε± ∓ κ2εz and The above equation comes from the characteristic equation [Eq. (4)] and is equivalent to the dispersion relation [Eq. (5)]. Note that the eigenfields in Eqs. (13)–(15) share the common tangential wave vector components kx and kz across the interface, as a direct consequence of the phase matching of the electromagnetic fields.The surface waves propagating at the interface (y = 0) are formulated according to Maxwell’s boundary conditions (the continuity of tangential electric and magnetic field components) as
where C1, C2, C+, and C− are constants. Using Eqs. (10) and (12) for Ex and Ez, respectively, the existence of a nontrivial solution of these constants requires that For the surface waves to be valid on the vacuum side, the normal (to interface) wave vector component ky should be purely imaginary . On the hyperbolic-gyromagnetic medium side, the normal components are in general complex numbers [cf. the conjugate expression of in Eq. (16)] and their imaginary parts should be negative ( with a > 0 and b > 0) to ensure that the eigenfields decay exponentially away from the interface.The surface waves on the hyperbolic-gyromagnetic medium are similar to the D’yakonov surface waves [7] in two aspects. First, they are surface waves in the anisotropic medium, which contain two different normal (to interface) wave vector components . Second, the surface wave propagation is only possible in limited ranges of direction. This feature is understood when ky and in Eq. (19) have been substituted [using and Eq. (16)], leading to an exact yet lengthy equation of kx and kz:
which is regarded as the characteristic equation of surface waves on the hyperbolic-gyromagnetic medium.3. Results and discussion
3.1. Dispersion surfaces
Figure 1(a) shows the equifrequency surfaces based on Eq. (5) for the hyperbolic-gyromagnetic metamaterial with εt = 2, εz = −1, μ = 1, and dispersion |κ| = 0.8. This plot is an illustration of the dispersion relation in the wave vector domain, with each wave vector component normalized by k0. The equifrequency surfaces consist of an ellipsoid-like surface at the center and a two-sheeted hyperboloid-like surface on top and bottom, both having rotational symmetry about the kz axis and reflection symmetry with respect to the kx–ky plane. The composite feature of the equifrequency surfaces appear when the hyperbolicity (in the permittivity) and the gyrotropy (in the permeability) exist simultaneously. This feature is also indicated in the hybrid character of the bi-quadratic equation [Eq. (4)]. Note that the dispersion surfaces remain the same when the sign of κ changes. The sign of κ, however, is relevant to the chiral nature of surface waves discussed later.
Fig. 1 Equifrequency surfaces of the hyperbolic-gyromagnetic metamaterial with (a) εt = 2, εz = −1, μ = 1, and |κ| = 0.8 and (b) εt = −2, εz = 1, μ = −1, and |κ| = 1.2. Thick black curves are equifrequency contours at ky = 0.
As |κ| increases, the ellipsoid-like surface gradually deforms by decreasing the thickness at the center. The thickness is reduced to zero when |κ| = |μ|, where Eq. (5) allows for the solution of kz = 0 at kt = 0. This is considered a transition point, across which the dispersion surface changes its character. For |κ|> |μ|, the ellipsoid-like surface transforms to a toroid, as shown in Fig. 1(b) for εt = −2, εz = 1, μ = −1, and |κ| = 1.2. The central part of the equifrequency surface has a different genus, a global mathematical property that usually interpreted as the number of holes or handles on the surface. Note that this condition is usually valid when the frequency exceeds the resonance frequency of the gyromagnetic medium [41,44]. In this situation, the diagonal element μ of the permeability tensor becomes negative. The signs of εt and εz are changed accordingly such that μεt > 0 and μεz < 0 as in Fig. 1(a) and a similar dispersion character of the equifrequency surfaces is preserved.
3.2. Chiral surface waves
Figure 2(a) shows the dispersion curves of the surface waves based on Eq. (20) for the hyperbolic-gyromagnetic metamaterial with the same constitutive parameters in Fig. 1(a), where |κ| < |μ|. At ky = 0, the dispersion of the bulk modes is represented by an ellipse-like curve at the center and a two-sheeted hyperbola-like curve on top and bottom [cf. thick black curves in Fig. 1(a)]. The surface modes are located inside the spatial gaps between the two bulk modes. The surface mode at kz > 0 is a mirror reflection (about the kx axis) of the mode at kz < 0. In particular, the surface waves are nonreciprocal (lacking symmetry) along the transverse direction (the kx axis in the present configuration). For κ > 0, the surface modes (in blue color) are to be connected to the hyperbola-like curve at the left end (with a negative kx) and to the ellipse-like curve at the right end (with a positive kx). For κ < 0, the surface modes (in red color) are mirror reflections (about the kz axis) of the modes for κ > 0. This feature is consistent with the asymmetry implied in the characteristic equation of surface waves [Eq. (20)], that is, F (kx, kz) ≠ F (−kx, kz).
Fig. 2 Dispersion curves of the surface waves for the hyperbolic-gyromagnetic metamaterial with (a) εt = 2, εz = −1, μ = 1, and |κ| = 0.8 and (b) εt = −2, εz = 1, μ = −1, and |κ| = 1.2. Black curves stand for bulk waves at ky = 0. Blue and red curves stand for surface waves for positive and negative values of κ, respectively. Gray regions correspond to spatial gaps between bulk modes. Arrows indicate the propagation directions of surface waves.
An immediate consequence of the nonreciprocity stated above is the unidirectional propagation of surface waves when the system is restricted to two dimensions. For a fixed kz (either positive or negative), the surface waves propagate toward the +kx (−kx) direction for κ > 0 (κ < 0), as indicated by the blue (red) arrows in Fig. 2(a). The surface waves are chiral in the sense that they propagate in one direction only along the edge and reverse the directions upon changing sign of the gyrotropic parameter. This feature is characteristic of the chiral edge states that occur in the quantum Hall systems [45]. In the wave vector space, the propagation direction of the surface wave can be determined by the average energy flux, given by the integral of the Poynting vector over a sufficient distance from the interface: , where is the time-averaged Poynting vector as a function of y. As the electromagnetic fields of the surface waves decay exponentially from the interface, the length d needs not to be large for accurately evaluating the integral. The average energy flux vector Savg is shown to be normal to the surface dispersion curve, as indicated by the arrows in the figure.
Note that the nonreciprocal dispersion curves of the surface waves have reflection symmetry with respect to the kx axis, that is, F(kx, kz) = F(kx, −kz). This is in contrast to the surface waves on the chiral hyperbolic metamaterials [28], which have point symmetry about the origin, that is, F(kx, kz) = F(−kx, −kz). The surface waves of the latter occur in pair with opposite helicity (spin) and counterpropagate at a given edge, which is characteristic of the helical edge states that occur in the quantum spin Hall systems [46]. The difference between the chiral and helical surface waves is also consistent with the breaking or preserving of the time reversal symmetry associated with the constitutive parameters [16, 17].
For |κ| > |μ|, the central part of the bulk dispersion curve is split into a pair of closed loops, as shown in Fig. 2(b) with the same constitutive parameters in Fig. 1(b). Here, the vacuum side is changed to the negative index medium with ε/ε0 = −1 and μ/μ0 = −1 (the complementary medium of vacuum [47, 48]) so that the surface waves have a similar dispersion as in the case where μ > 0. In this situation, E1 and E2 in Eqs. (13) and (14), respectively, are to be multiplied by a minus sign. The surface waves are located inside the gap between the hyperbola-like curve and the closed loop curve. The unidirectional feature of the surface waves is similar to that in Fig. 2(a), except that the propagation direction associated with the sign of κ is reversed.
The propagation directions of the chiral surface waves are further related to those of the bulk waves, which are determined by the time-averaged Poynting vectors of the eigenfields, without the need of integration over the distance as treated in the surface waves. The Poynting vectors associated with the parabola-like curves are directed toward the opening, and those with the ellipse-like or closed loop curves are pointed outward. The propagation directions of the surface waves are therefore conformed to those of the bulk waves.
In another aspect, the surface wave in the hyperbolic-gyromagnetic medium is composed of two eigenwaves, each with a complex normal (to interface) wave number component ( or ). The surface wave amplitude thus decays away from the interface in an oscillatory manner. Figure 3(a) shows the tangential electric field profile of the surface wave in Fig. 2(a) at kz/k0 = 1.2. Since the real parts of are in general nonzero, the two eigenwaves interfere each other in the hyperbolic-gyromagnetic medium. The time-averaged Poynting vector S (y) of the surface wave may change the orientation with y. In Fig. 3(b), the directions of Poynting vectors on two sides of the interface for the same surface wave in Fig. 2(b) are plotted as a trajectory by varying the distance from the interface. Note that the Poynting vectors sway around on the medium side (y < 0) as the distance increases, in contrast to the fixed orientation on the other side (y > 0). The direction of the average Poynting vector Savg over the distance (on both sides) from the interface, which makes an angle of 51.3° with respect to the x axis, is indicated by the dashed line. This angle is consistent with the direction normal to the dispersion curve of the surface wave [cf. the red arrow on the red dot in Fig. 2(b)].
Fig. 3 (a) Profile of the normalized tangential electric fields along the distance from the interface for the surface wave in Fig. 2(a) at kz/k0 = 1.2 (marked by blue dot) and (b) Trajectory of the directions of Poynting vectors by varying the distance from the interface for the surface wave in Fig. 2(b) at kz/k0 = 1.2 (marked by red dot). Dashed line indicates the direction of average Poynting vector.
The chiral nature of the surface waves is also manifest on the polarization handedness. Figure 4 shows the handedness in color for the surface as well as the bulk waves in Fig. 2. For κ > 0, the surface wave at kz > 0 is left-handed elliptically polarized (LEP), while the surface wave at kz < 0 is right-handed elliptically polarized (REP) [Fig. 4(a)]. The handedness of the surface waves is switched when the sign of κ is changed to negative [Fig. 4(b)]. Here, the handedness is evaluated by first calculating the electric (or magnetic) field components in the new coordinate system (x′, y′, z′), obtained by rotating the system (x, y, z) about the y axis such that the z′ axis is oriented to the time-averaged Poynting vector on the xz plane. Denoting θ the angle from the x (z) axis to the x′ (z′) axis, the electric field components in the new coordinate system are given by Ex′ = Ex cos θ + Ez sin θ, Ey′ = Ey, and Ez′ = −Ex sin θ + Ez cos θ. The polarization handedness is then determined by the phase difference between Ex′ and Ey′: δ = arctan Im[Ey′]/Re[Ey′] − arctan (Im[Ex′]/Re[Ex′]). The wave is REP (LEP) if δ = π/2 (−π/2), that is, the phase of Ey′ is delayed (advanced) by 90° relative to that of Ex′ (under the time-harmonic convention e−iωt).
Fig. 4 Polarization handedness of the surface and bulk waves for the hyperbolic-gyromagnetic metamaterial with (a) εt = 2, εz = −1, μ = 1, and κ = 0.8 and (b) εt = −2, εz = 1, μ = −1, and κ = −1.2. Blue and red curves denote right- and left-handed elliptical polarizations, respectively. Gray dashed circles are dispersion curves of (a) vacuum and (b) negative index medium.
In Fig. 4(a), the bulk wave with the hyperbola-like curve is shown to be LEP for the upper sheet and REP for the lower, which is consistent with the handedness of the corresponding surface wave (for κ > 0). On the other hand, the bulk wave with the ellipse-like curve has a mixed handedness. The handedness of the upper portion (kz > 0) is opposite to that of the lower (kz < 0). As in the surface waves, the handedness of the bulk waves is switched when the sign of κ is changed [Fig. 4(b)]. The above features of surface waves remain the same when the refractive index of the vacuum side (or negative index medium) is increased (or decreased) by a certain amount, except that the dispersion region of surface waves may be modified because of the change in radius of the dispersion circle on this side.
The topological feature intrinsic to the chiral surface waves on the hyperbolic-gyromagnetic medium can be characterized by the Berry phase associated with the bulk eigenmode as [49]
where |n〉 is the normalized eigenstate and 〈f |g〉 stands for the inner product of two states |f〉 and |g〉. Based on the eigenfields in Eq. (7)–(12), the Berry curvature i∇ × 〈n|∇n〉 can be calculated at every point (kx, ky, kz) on the equifrequency surface S, as shown in Fig. 5 for a cross section at ky = 0. Because of the azimuthal symmetry of S about the kz axis, the surface integral in Eq. (21) is simplified to an integral on the kt –kz plane, where [cf Eq. (4)]. The integral can be efficiently computed through a U(1) link variable [50], which correctly gives the quantized invariant of γn/(2π) even on a coarsely discretized zone. In the present configuration, the wave vector space is an unbounded region. The topological invariants are shown to be integers as in lattice structures, which is consistent with the feature of Chern numbers in continuous media [51].Fig. 5 Berry curvatures and Berry phases of the bulk eigenmodes for the hyperbolic-gyromagnetic metamaterial with (a) εt = 2, εz = −1, μ = 1, and κ = 0.8 and (b) εt = −2, εz = 1, μ = −1, and κ = −1.2. Blue and red arrows denote the outward and inward Berry curvatures, respectively.
The calculated results show that γn = ±2π × sgn(μκ) for the hyperboloid-like surface, which remains an invariant under continuous change of the dispersion surface by varying the material parameters. For the ellipsoid-like or toroid surface, the calculated results give γn = 0 with the same invariant nature. Note that the magnitude of Berry curvature on the hyperboloid-like surface decreases rapidly away from the center and a convergent and consistent Berry phase can be obtained with a computed range up to |kx|/k0 ≈ 20. On the other hand, the Berry curvature of the ellipsoid-like or toroid surface has opposite orientations for kz > 0 and kz < 0, which exactly cancels in the integral to give a zero Berry phase. Note also that the Berry curvatures reverse the orientations when the sign of κ is changed. The corresponding Berry phase changes sign accordingly. The difference of Berry phase between two bulk modes (separated by a gap) indicates the existence of chiral edge states as in the quantum Hall systems, showing the bulk-edge correspondence [16] for the underlying medium. The chiral surface waves therefore represent an instance of the topological phase transition in the hyperbolic-gyromagnetic metamaterial.
Finally, the topological feature of the chiral surface waves is demonstrated by numerical simulation of the electromagnetic wave propagating at the boundary (y = 0) between vacuum (or negative index medium) and the hyperbolic-gyromagnetic metamaterial, as shown in Fig. 6. Here, a dipole source is placed at the interface (marked by the asterisk symbol) to excite the surface wave as in Fig. 2, where kz/k0 = 1.2 (marked by the blue or red dot) is located inside the gap so that the fields decay evanescently in the metamaterial as well as in vacuum (or negative index medium). Note that the excited surface wave is outside the dispersion circle of vacuum (or negative index medium), that is, (cf. gray dashed circles in Fig. 4). It is shown that the surface waves propagate unidirectionally toward the right and are immune to disorder. In particular, the surface waves are able to propagate around sharp corners without backscattering, which is the typical feature of topological edge states in a topologically nontrivial system.
Fig. 6 Numerical simulation of the chiral surface wave at kz/k0 = 1.2 excited at the interface (marked by asterisk symbol) between (a) vacuum and the hyperbolic-gyromagnetic metamaterial with εt = 2, εz = −1, μ = 1, and κ = 0.8 ( Visualization 1) and (b) negative index medium and the metamaterial with εt = −2, εz = 1, μ = −1, and κ = −1.2 ( Visualization 2). The coordinates are scaled by λ0 = 2π/k0.
4. Concluding remarks
In conclusion, we have studied the existence of chiral surface waves on the hyperbolic-gyromagnetic metamaterials. The surface waves, which exist in the spatial gap between two elliptically polarized bulk modes, exhibit the nonreciprocal dispersion character in the wave vector space. In particular, the surface waves propagate unidirectionally along the edge when the system is restricted to two dimensions, which is characteristic of the chiral edge states that occur in the quantum Hall systems. The chiral surface waves reverse the propagation direction and switch the polarization handedness upon changing sign of the gyrotropic parameter. The topological feature of the chiral surface waves is further manifest on the Berry phases associated with the bulk modes and demonstrated by the propagation of electromagnetic waves around sharp corners.
Acknowledgments
The authors thank Dr. Ruei-Cheng Shiu for valuable discussion on the topological features. This work was supported in part by Ministry of Science and Technology of Republic of China under Contract No. MOST 105-2221-E-002-161-MY3.
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