Unique surface polaritons and their transitions in metamaterials

Hongyan Peng; Xuan-Zhang Wang; Xuan-Zhang Wang

doi:10.1364/OE.460832

1. Introduction

Some magnetic materials or magnetic metamaterials were demonstrated to be a kind of hyperbolic materials in specific situations [1–5]. As naturally hyperbolic crystals, insulative antiferromagnets (AFs) in the case of no external magnetic field, such as transition metal oxides, fluorides and sulfides (NiO and MnO, FeF₂ and MnF₂, and MnS and CrS) as well as KNiF₃ and RdMnF₃ [6–8], possess a unique magnetic permeability with opposite-sign principal values in the AF reststrahlen band. Due to their hyperbolism, they have recently been attracting the attention of physicists again [9–12]. These materials support surface magnons and magnon polaritons [13–16] in the far-infrared range. A simple AF crystal contains only two mutually-staggered magnetic sublattices. Either magnetic sublattice is composed of atoms with the same magnetic moment, whereas the strong exchange field leads to that the atomic magnetic moments of the two sublattices are opposite in direction. It is more interesting that antiferromagnetically anisotropic field holds the atomic magnetic moments of the two sublattices to be parallel and antiparallel to the easy-axis [7–8], respectively. The polar crystals (PCs) can support surface phonon polaritons in the infrared window [17–18]. For some IV-VI semiconductors, their response frequencies are situated in the far-infrared range [19], similar to those of the AFs. The surface polaritons found previously have a common feature [17–18,20–21], i.e., they propagate along the relevant material surface and their electromagnetic fields exponentially and monotonously attenuate with the distance away from the surface. We call them the conventional surface polaritons. In addition to them, there are surface hybridization polaritons. A surface hybrid-polarization polariton means that it is localized at the surface of relevant medium, and consists of surface waves with different polarizations (e.g., TE wave and TM wave) in the medium, namely it is the mixture of different-polarization surface waves. In general, it exists at the surface of anisotropic materials (including metamaterials). Another surface hybridization polariton is the mixture of two surface excitation polaritons, e.g., a phonon-plasmon polariton or phonon-magnon polariton.

Recently, a new style of surface polaritons was found [22–23], named the ghost surface polariton (GSP). Unlike the conventional ones, the electromagnetic fields of GSP not only exponentially attenuate but also oscillate with the distance away from the surface.

In this paper, we theoretically structure a metamaterial with alternately stacked AF layers and PC layers, where AF layers are anisotropic, but PC layers are isotropic. We suppose that the AF response frequency range is completely or partially covered by the PC response frequency range so that magnon polaritons in AF layers and phonon polaritons in PC layers sufficiently couple together in the metamaterial to form new magnon-phonon polaritons. We focus on surface polaritons in this metamaterial and discuss their features.

2. Dispersion and polarization of surface polaritons

The metamaterial used in this paper is composed of alternately AF layers and PC layers, whose surface is perpendicular to the layers and parallel to the y-z plane, as illustrated in Fig. 1. The permeability of AF layers in an external magnetic field along the AF easy axis (the z-axis) is a nondiagonal matrix [13–16]. Its nondiagonal elements result in the AF gyromagnetism and its expression is

(1)$${\mathop \mu \limits^ \leftrightarrow}_a = {\mu _0}\left( {\begin{array}{ccc} {{\mu_1}}&{i{\mu_2}}&0\\ { - i{\mu_2}}&{{\mu_1}}&0\\ 0&0&1 \end{array}} \right), $$

where ${\mu _1} - 1\, + \,{\psi _ + } + {\psi _ - }$ and ${\mu _2} = {\psi _ + } - {\psi _ - }$ with ${\psi _ \pm } = {\omega _m}{\omega _a}/[\omega _r^2 - {(\omega \pm \omega + i{\tau _a})^2}]$ including damping constant τ_a [7–8]. We should remind that the specific frequencies included in the formulae are determined by the physical parameters of AF layers, i.e., ${\omega _m} = 4\pi \gamma {M_0}$, ${\omega _a} = \gamma {H_a}$, ${\omega _0} = \gamma {H_0}$ and $\omega _r^2 = {\gamma ^2}{H_a}(2{H_e} + {H_a})$, where H_a, H_e and H₀ are the anisotropic, exchange and external fields, respectively. M₀ is the sublattice magnetization and meanwhile γ is the gyromagnetic ratio [7–8,12]. These parameters will be offered in subsequent numerical calculations. The permittivity of AF layers is considered as a constant ${\varepsilon _a}$. However, the permeability of PC layers is assumed to be equal to ${\mu _0}$ (the vacuum permeability) and their isotropic permittivity is a scalar quantity shown by [17–18]

(2)$${\varepsilon _p} = {\varepsilon _h} + \frac{{({\varepsilon _l} - {\varepsilon _h})\omega _T^2}}{{\omega _T^2 - {\omega ^2} - i\omega {\tau _p}}}, $$

where ${\omega _T}$ is the transverse optical-phonon frequency, and ${\varepsilon _h}$ and ${\varepsilon _l}$ are the dielectric constants in the high- and low-frequency limits, respectively. ${\tau _p}$ is the phonon damping constant. For the metamaterial composed of AF and PC layers, its effective permittivity and permeability can be determined with the effective-medium method when the two layer-thicknesses are much less than the wavelength of polariton. We assume the thicknesses of AF layers and polar-crystal layers are ${l_a}$ and ${l_p}$, respectively. The period of metamaterial is $P\, = \,{l_a} + {l_p}$ and then the AF ratio ${r_a}\, = \,{l_a}/P$ and the polar-crystal ratio ${r_p}\, = \,1 - {r_a}$. In the geometry and coordinate system shown in Fig. 1, the effective permittivity tensor of the metamaterial is a diagonal matrix with elements

(3)$${\varepsilon _{xx}} = {\varepsilon _{zz}} = {r_p}{\varepsilon _p} + {r_a}{\varepsilon _a},\,\,{\varepsilon _{yy}} = \frac{{{\varepsilon _a}{\varepsilon _p}}}{{{r_a}{\varepsilon _p} + {r_p}{\varepsilon _a}}},$$

The effective permeability tensor is a nondiagonal matrix to be,

(4)$$\mu = {\mu _0}\left( {\begin{array}{ccc} {{\mu_{xx}}}&{i{\mu_{xy}}}&0\\ { - i{\mu_{xy}}}&{{\mu_{yy}}}&0\\ 0&0&1 \end{array}} \right), $$

with nonzero elements

(5)$${\mu _{xx}} = {r_a}{\mu _1} + {r_p} - \frac{{{r_a}{r_p}\mu _2^2}}{{{r_p}{\mu _1} + {r_a}}},\,\,{\mu _{yy}} = \frac{{{\mu _1}}}{{{r_p}{\mu _1} + {r_a}}},\,\,{\mu _{xy}} = \frac{{{r_a}{\mu _2}}}{{{r_p}{\mu _1} + {r_a}}}, $$

Fig. 1. Metamaterial structure and coordinate system applied in the paper, where the metamaterial consists of alternately antiferromagnetic (AF) layers and polar-crystal (PC) layers and its surface is the y-z plane and normal to the layers, and the AF easy axis and external magnetic field both are parallel to the z-axis. The space above the surface occupied by air or vacuum. The surface polaritons propagate along the z-axis.

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We assume that a surface polariton (SP) propagates along the z-axis and its electromagnetic fields exponentially attenuate with the distance (x) away from the surface. Consequently, its magnetic field is written as $\textbf{H}^{\prime}\exp (2\pi \Gamma^{\prime}x + i2\pi kz - i\omega t)$ with ${\varGamma^{\prime}} = {({k^2} - {f^2})^{1/2}}$ above the surface, where $f = \omega /2\pi c$ is the reduced frequency, and meanwhile $\textbf{H}\exp ( - 2\pi \Gamma x + i2\pi kz - i\omega t)$ in the metamaterial with a formal attenuation constant ±β. The vector in either magnetic-field expression is the magnetic-field amplitude at the surface. Г´ and the real part of Г both are positive real quantities for the surface polariton. It should be noted that the imaginary part of Γ is an oscillating constant. In the metamaterial, the field amplitude satisfies the wave equations as follows

(6a)$$({k^2} - {\varepsilon _{yy}}{\mu _{xx}}{f^2}) \;{H_x} - i{\varepsilon _{yy}}{\mu _{xy}}{f^2}{H_y} - i\Gamma k{H_z} = 0, $$

(6b)$$- ik\Gamma {H_x} - ({\Gamma ^2} + {\varepsilon _{yy}}{f^2}){H_z} = 0, $$

(6c)$$i{\varepsilon _{xx}}{\mu _{xy}}{f^2}{H_x} + ({k^2} - {\Gamma ^2} - {\varepsilon _{xx}}{\mu _{yy}}{f^2}){H_y} = 0. $$

Equation system (6) leads to ${\varGamma ^4} + b{\varGamma ^2} + c = 0$ for any nonzero solution of H. It is evident that there are two evanescent-wave solutions (or two evanescent branches) in the metamaterial, which correspond to

(7)$$\Gamma_ \pm ^2 = \frac{1}{2}[ - b \pm {({b^2} - 4c)^{1/2}}], $$

with

(8a)$$b = ({\varepsilon _{yy}}{f^2} - {k^2}/{\mu _{xx}}) + ({\varepsilon _{xx}}{\mu _{yy}}{f^2} - {k^2}) - {\varepsilon _{xx}}\mu _{xy}^2{f^2}/{\mu _{xx}}, $$

(8b)$$c = ({\varepsilon _{yy}}{f^2} - {k^2}/{\mu _{xx}})({\varepsilon _{xx}}{\mu _{yy}}{f^2} - {k^2}) - {\varepsilon _{xx}}{\varepsilon _{yy}}\mu _{xy}^2{f^4}/{\mu _{xx}}. $$

For convenience of discussion, we ignore the damping terms in permeability (Eq. 1) and permittivity (Eq. 2) so that b and c are real in (Eq. 7). There are two cases for surface polaritons, which are the first case of ${b^2} - 4c > 0$ and the second case of ${b^2} - 4c < 0$. In the first case, one may find surface hybrid-polarization polaritons that are abbreviated to SHPs. In the second case, $\Gamma_ \pm ^2 = a^{\prime} \pm ib^{\prime}$ with real ${a^{\prime}}$ and ${b^{\prime}}$. It is evident that if we are able to find any solution in the second case, it will be a new surface polariton, a ghost surface polariton (GSP) since ${\varGamma _ \pm }$ includes an imaginary part, or the oscillatory constant. Because the real part of ${\varGamma _ \pm }$ must be positive for the GSP, we achieve

(9)$${\Gamma_ + } = \alpha + i\beta ,\,\,{\Gamma_ - } = \alpha - i\beta, $$

where

(10a)$$\alpha \, = \,{\lceil{{a^{\prime}}\, + \,{{({a^{{\prime}2}} + {b^{{\prime}2}})}^{1/2}}} \rceil ^{1/2}}/\sqrt 2 ,$$

(10b)$$\beta \, = \,{\lceil{ - {a^{\prime}}\, + \,{{({a^{{\prime}2}} + {b^{{\prime}2}})}^{1/2}}} \rceil ^{1/2}}/\sqrt 2 .$$

α is defined as the attenuating constant and β is defined as the oscillatory constant, which are determined by ${a^{\prime}} ={-} b/2$ and ${b^{\prime}} = {(4c - {b^2})^{1/2}}/2$. We realize from Eq. (10) that the two constants both are positive.

In both the cases, the surface polariton is a surface hybridized-polarization polariton whose magnetic field in the metamaterial should be the sum of two branch-fields related to Г₊ and Г_-, or

(11)$$\textbf{H} = [{\textbf{H}^\textrm{ + }}\exp ( - 2\pi {\Gamma_\textrm{ + }}x) + {\textbf{H}^ - }\exp ( - 2\pi {\Gamma_ - }x)], $$

where the common factor $\exp (i2\pi kz - i\omega t)$ is not put in the expression (hereinafter). The three magnetic-field components of either branch-wave couple together in the response-frequency range, so we can express the other components as functions of the x-component. According to Eqs. (6b) and (6c), we find the functions to be

(12a)$$H_y^ \pm{=} \frac{{ - i{\varepsilon _{xx}}{\mu _{xy}}{f^2}H_x^ \pm }}{{{k^2} - \Gamma_ \pm ^2 - {\varepsilon _{xx}}{\mu _{yy}}{f^2}}} = i{\lambda _ \pm }H_x^ \pm, $$

(12b)$$H_z^ \pm{=} \frac{{ - ik{\Gamma_ \pm }H_x^ \pm }}{{\Gamma_ \pm ^2 + {\varepsilon _{yy}}{f^2}}} = i{\gamma _ \pm }H_x^ \pm. $$

We also will use the electric-field tangential components to solve the dispersion relation of surface polariton. Applying the Maxwell equation $- i\omega {\boldsymbol D}\textrm{ = }\boldsymbol{\nabla } \times {\boldsymbol {\rm H}}$ to either branch wave in the metamaterial, we get

(13)$$E_y^ \pm{=} \frac{{ - 1}}{{{\varepsilon _0}{\varepsilon _{yy}}\omega }}(k + {\Gamma_ \pm }{\gamma _ \pm })H_x^ \pm ,\,\,E_z^ \pm{=} \frac{{{\Gamma_ \pm }{\lambda _ \pm }}}{{{\varepsilon _0}{\varepsilon _{zz}}\omega }}H_x^ \pm. $$

The tangential components of electromagnetic fields must be continual at the surface. As a result, it means that

(14a)$${H^{\prime}_y} = i{\lambda _ + }H_x^ +{+} i{\lambda _ - }H_x^ - ,\,\,{H^{\prime}_z} = i{\gamma _ + }H_x^ +{+} i{\gamma _ - }H_x^ -, $$

(14b)$${E^{\prime}_y} = \frac{{ - 1}}{{{\varepsilon _0}{\varepsilon _{yy}}\omega }}[(k + {\Gamma_ + }{\gamma _ + })H_x^ +{+} (k + {\Gamma_ - }{\gamma _ - }H_x^ - )],\,\,{E^{\prime}_z} = \frac{1}{{{\varepsilon _0}{\varepsilon _{zz}}\omega }}({\Gamma_ + }{\lambda _ + }H_x^ +{+} {\Gamma_ - }{\lambda _ - }H_x^ - ), $$

where E′ is the electric-field amplitude on the surface and is determined with ${E^{\prime}}_y = i{f^2}{H^{\prime}}_z/{\varepsilon _0}\omega {\Gamma^{\prime}}$ and ${E^{\prime}}_z = i{\Gamma^{\prime}}{H^{\prime}}_y/{\varepsilon _0}\omega$. It is obvious that Eq. (14) leads to a dispersion equation,

(15)$$\left|{\begin{array}{cc} {{\lambda_ + }({\varepsilon_{zz}}\Gamma^{\prime} + {\Gamma_ + })}&{{\lambda_ - }({\varepsilon_{zz}}\Gamma^{\prime} + {\Gamma_ - })}\\ {{\gamma_ + }({\varepsilon_{yy}}{f^2} - \Gamma^{\prime}{\Gamma_ + }) - k\Gamma^{\prime}}&{{\gamma_ - }({\varepsilon_{yy}}{f^2} - \Gamma^{\prime}{\Gamma_ - }) - k\Gamma^{\prime}} \end{array}} \right|= 0. $$

This dispersion equation is a real equation in the first case and will offer dispersion curves of SHP. It is a complex equation in the second case and will result in dispersion curves of GSP.

We subsequently discuss this complex dispersion equation in the second case. Owing to $\Gamma_ + ^\ast{=} {\Gamma_ - }$ in the second case, we find $\lambda _ + ^\ast{=} {\lambda _ - }$ and $\gamma _ + ^\ast{=} {\gamma _ - }$ from formulae (12), so Eq. (15) is simplified into a simpler real dispersion relation, i.e.,

(16)$${\mathop{\rm Im}\nolimits} \{ \lambda ({\varepsilon _{zz}}\Gamma^{\prime} + \Gamma)[{\gamma ^\ast }({\varepsilon _{yy}}{f^2} - \Gamma^{\prime}{\Gamma^\ast }) - k\Gamma^{\prime})]\} = 0, $$

where $\Gamma= {\Gamma_ + }$, $\lambda = {\lambda _ + }$ and $\gamma = {\gamma _ + }$. Equation (16) will offer dispersion curves of GSP. From the above equations, we conclude that dispersion Eqs. (15) and (16) are reciprocal, or $f(k) = f( - k)$ and $f({H_0}) = f( - {H_0})$. In addition, the external magnetic field and the hyperbolicity of the effective permittivity play key roles for the existence of GSP, which can be realized from Eqs. (7) and (8). We are going to put emphasis on the magnetic field of GSP in the metamaterial, where the direction of the magnetic-field amplitude is used to represent the GSP polarization. Assuming $H_x^ -{=} \wedge H_x^ +$, we find from Eq. (15) that

(17)$$\Lambda ={-} \frac{{{\lambda _ + }({\varepsilon _{zz}}\Gamma^{\prime} + {\Gamma_ + })}}{{{\lambda _ - }({\varepsilon _{zz}}\Gamma^{\prime} + {\Gamma_ - })}}, $$

which implies ${ \wedge ^{ - 1}} = { \wedge ^\ast }$. We easily prove that $|{H_x^ - } |= |{H_x^ + } |$, $|{H_y^ - } |= |{H_y^ + } |$ and $|{H_z^ - } |= |{H_z^ + } |$, and moreover also can find similar relations between the branch-wave electric fields. Therefore, the corresponding field components of the two branches have the same amplitude and different phases for the GSP. We obtain the magnetic-field components of the GSP in the metamaterial to be

(18a)$${H_x} ={-} 2i{H^{\prime}_y}{e^{ - 2\pi \alpha x}}[{R_x}\cos (2\pi \beta x) + {I_x}\sin (2\pi \beta x)], $$

where R_x and I_x are the real and imaginary parts of ${({\lambda _ + } + {\lambda _ - } \wedge )^{ - 1}}$, and

(18b)$${H_{y,z}} = 2{H^{\prime}_y}{e^{ - 2\pi \alpha x}}[{R_{y,z}}\cos (2\pi \beta x) + {I_{y,z}}\sin (2\pi \beta x)], $$

where ${R_y}$ and ${I_y}$ are the real and imaginary parts of ${\lambda _ + }/({\lambda _ + } + {\lambda _ - } \wedge )$, but ${R_z}$ and ${I_z}$ are the real and imaginary parts of ${\gamma _ + }/({\lambda _ + } + {\lambda _ - } \wedge )$. The three expressions of magnetic-field components reflect the interference of the two branches in the metamaterial and show that the magnetic field exponentially attenuates and sinusoidally oscillates with the distance away from the surface. The GSP and SHP both are surface magnon-phonon polaritons with hybridized-polarization.

3. Numerical results and discussions

Numerical calculations are based on the FeF₂/TlBr metamaterial [24]. Antiferromagnetic FeF₂-layers have dielectric constant ${\varepsilon _a} = 5.5$ and the gyromagnetic ratio $\gamma = 1.97 \times {10^{10}}$ (cm⁻¹.rad.s⁻¹/kG). The other physical parameters are the sublattice magnetization 4πM₀ = 7.04kG (${\omega _m} = 0.736c{m^{ - 1}}$), the exchange field H_e = 540.0kG (${\omega _e} = 56.44c{m^{ - 1}}$), anisotropic field H_a = 200.0kG (${\omega _a} = 20.9c{m^{ - 1}}$) [12,25]. The AF resonant frequency is ${\omega _r}\textrm{ = 52}\textrm{.877c}{\textrm{m}^{ - 1}}$ without external magnetic field. In the external magnetic field, there are two resonant frequencies, ${\omega _1} = {\omega _r} - {\omega _0}$ and ${\omega _2} = {\omega _r} - {\omega _0}$ For ${H_0} = 1kG$ (${\omega _0} = 0.1046c{m^{ - 1}}$), ${\omega _1} = 52.772$ and ${\omega _2} = 52.982$. The polar-crystal TlBr is nonmagnetic [26–27] and its permeability is equal to ${\mu _0}$. Its permittivity is a function of operating frequency, including the high- and low-frequency dielectric constants ${\varepsilon _h} = 5.34$ and ${\varepsilon _l} = 30.4$, as well as the transversely optical phonon frequency ${\omega _\tau } = 48c{m^{ - 1}}$. For the FeF₂/TIBr metamaterial, the vacuum wavelength is larger than 180µm for operating frequency $f = 53c{m^{ - 1}}$, and the polariton wavelength in the metamaterial is about 20µm for wavenumber k = 500cm⁻¹. As a result, the effective permeability and permittivity are absolutely available for the metamaterial of micron- or submicron-period.

The effective permeability and permittivity of the metamaterial depend on H₀ and ${r_a}$. As an example, Fig. 2 shows the effective permeability and permittivity as functions of operating frequency for H₀ = 1.0kG and ${r_a} = 0.7$. We will see that the surface polaritons are situated in the two grey areas in Fig. 2(a), where ${\mu _{xx}}$ and ${\mu _{yy}}$ have the same sign but they are opposite to ${\mu _{xy}}$. Equation (8b) implies that no GSP exists for ${H_0} = 0$ since ${\mu _{xy}} = 0$ and then ${b^2} - 4c > 0$ in this case. We also have not found any surface polariton for ${H_0} = 0$. In the same frequency window, the effective permittivity is illustrated in Fig. 2(b), where ${\varepsilon _{xx}} = {\varepsilon _{zz}}$ is negative but ${\varepsilon _{yy}}$ is positive. The two figures are helpful for discussion about dispersion properties of the surface polaritons.

Fig. 2. The effective permeability and permittivity of TlBr/FeF₂ metamaterial for ${r_a} = 0.7$ and ${H_0} = 1.0kG$. (a) illustrates the permeability. ${\mu _{xx}}{\mu _{yy}} > 0$ in the two grey areas where the SHP and GSP will be found. (b) shows the permittivity.

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We use the progressive scanning and dichotomy for solving transcendental equation (the dispersion equation) under the control conditions, edited with Fortran 90. We first discuss basic properties of GSP and SHP for external magnetic field ${H_0} = 1.0\textrm{kG}$. Figures 3(a) and 3(b) illustrate dispersion curves of GSP and SHP for various values of ${r_a}$. The surface polaritons occupy two small frequency ranges, e.g., for ${r_a} = 0.7$ the two small frequency ranges are shown by the grey areas in Fig. 2(a). Figure 3(a) shows the dispersion curves in the left grey region. Figure 3(b) presents the dispersion curves in the right grey region. ${\varepsilon _{xx}} > 0$ and ${\varepsilon _{yy}} < 0$, and meanwhile ${\mu _{xx}}{\mu _{yy}} > 0$ in the two grey regions. We first see that the GSP starts from the vacuum photon line with f = k and then undergoes a transition to the SHP as k is increased to a certain value. We also find the other picture in the right area of Fig. 3(a), where the SHP undergoes a transition to the GSP as k is increased to another certain value. In other words, the GSP can not only change into the SHP, but also the SHP can change into the GSP with the increase of k. Figure 3(a) also indicates that the surface polaritons in the left grey region have either positive group-velocity or negative group-velocity, dependent on wavenumber k or frequency f. In the right grey region, Fig. 3(b) shows that there is only the transition from the GSP to SHP with the increase of k and the group-velocity of GSP or SHP is positive. In numerical calculations, we find that no surface polariton exists without the external magnetic field in the used geometry. Therefore, the external field performs the necessary condition for the existence of surface polaritons. Except the grey regions in Fig. 2(a), we also search for surface polaritons in other frequency ranges, but no surface polariton is found for the specific external field. It should be noted that the change of external field will bring about the shift of the grey regions.

Fig. 3. Dispersion curves of SHP and GSP for a fixed external magnetic field and various values of ${r_a}$, where the surface polaritons appear in the two frequency regions. (a) The surface polaritons in the lower-frequency region and (b) those in the upper-frequency region.

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The SHP is composed of the two branches with different attenuation constants in the metamaterial, but the GSP consists of the two branches with the same attenuation constant $\alpha $ and with opposite-sign oscillatory constants ${\pm} \beta $. Attenuation constants characterize the localization of surface polaritons at the surface. The oscillatory constant determines the oscillating behavior of the GSP in the x-direction. We first discuss the attenuation constants of the SHP indicated by the green curves in Fig. 3, as shown in Fig. 4. For various values of ${r_a}$, we see interesting dispersion properties from Fig. 4(a). For ${r_a} = 0.7$, the two curves form an ellipse, the two connection points correspond to the two terminals of the green dispersion curve in Fig. 3(a), respectively. At the connection points, the two attenuation constants of the SHP are the same as the attenuation constant of the GSP (see Fig. 5(a)), which also can be realized from Eqs. (7) and (8). For ${r_a} = 0.5$, the SHP has the two separated dispersion curves. At the left endpoint of the left curve, the two attenuation constants are equal, but ${\Gamma_ - } = 0$ at its right endpoint. For the right dispersion curve, ${\Gamma_ - } = 0$ at its left endpoint and the two attenuation constants are equal at the right endpoint. Figure 4(b) is related to Fig. 3(b). We see that the SHP in the right region has shorter dispersion curves, whose two endpoints correspond to ${\Gamma_ + } = {\Gamma_ - }$ and ${\Gamma_ - } = 0$, respectively. Therefore, the dispersion curves of the SHP are finite curves, whose two terminals correspond to either ${\Gamma_ + } = {\Gamma_ - }$ or ${\Gamma_ - } = 0$.

Fig. 4. The two attenuation constants of SHP as functions of wavenumber for a fixed external magnetic field and various values of ${r_a}$. (a) illustrates both constants of the SHP described in Fig. 3(a). (b) shows those of the SHP illustrated in Fig. 3(b).

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Fig. 5. Attenuation constant (α) and oscillatory constant (β) of GSP for a fixed external magnetic field and various values of ${r_a}$. (a) The two constants related to the dispersion curves in Fig. 3(a), and (b) the two constants corresponding to the dispersion curves in Fig. 3(b).

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In the second case (${b^2} - 4c < 0$), ${\Gamma_ \pm } = \alpha \pm i\beta $ with attenuation constant α and oscillatory constants β. α and β are important to characterize the GSP. α shows the GSP localization at the surface and $\beta $ determines the oscillatory behavior. Figure 5(a) illustrates the two constants related to the GSP dispersion curves in Fig. 3(a). Combining Fig. 5(a) with 3(a), we find that the left dispersion curve of the GSP starts from the vacuum photon line and terminates at $\beta = 0$, but the two terminals of the right dispersion curve both are at $\beta = 0$. Figure 5(b) shows the two constants corresponding to the GSP dispersion curves in Fig. 3(b). Because every dispersion curve starts from the vacuum light line, the two constants both are finite at its left endpoint, but the curve terminates at $\beta = 0$. This figure also explores that the oscillatory constant is obviously larger than the attenuation constant, so the GSP in the left frequency region will exhibit evidently oscillatory behavior.

We see from Eqs. (7) and (8) that the external magnetic field plays a necessary condition for the existence of GSP or SHP. Due to very large effective dielectric-constants ${\varepsilon _{xx}}$ and ${\varepsilon _{yy}}$ in the FeF₂/TlBr metamaterial, we find that the GSP can still exist for a very small external magnetic field. We offer Fig. 6 to illustrate the field-dependence of surface-polariton frequency for fixed wavenumbers and an AF-ratio (${r_a}$). One finds the two dispersion curves of GSP for k = 70cm⁻¹, the lower one comes from Fig. 3(a) and the upper is related to Fig. 3(b). The upper linearly increases in frequency with ${H_0}$, and the lower first slightly increases and then linearly decreases in frequency with the increase of ${H_0}$. For k = 77 and 90cm⁻¹, it is very interesting that the GSP and SHP occupy different parts of one smooth curve, where there is the switch between SHP and GSP by changing ${H_0}$. In other words, one can use an external magnetic field to switch GSP to SHP, or switch SHP to GSP.

Fig. 6. The field-dependence of surface-polariton frequency for various wavenumbers and a fixed value of ${r_a}$. The red curves are attached to the GSP and the green curves correspond to the SHP. The switch between GSP and SHP can be actualized by changing external magnetic field.

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Due to the interference of the two wave-branches of the GSP in the metamaterial, Eq. (18) foretells the oscillatory behavior of GSP magnetic field or electric field in the x-direction. However, the relative amplitude of oscillatory constant β is key for whether this behavior is clearly visible. One cannot clearly see the oscillatory behavior if α is obviously larger than β. As demonstrated in Eq. (18), the GSP magnetic field oscillate and attenuate with the distance away from the surface. We apply Fig. 7 to intuitively illustrate the two distributions of GSP magnetic field for k = 60.1cm⁻¹, which correspond to the two specified points on the red curves in Figs. 3(a) and 3(b), respectively. Figure 7(a) comes from the point on red curve in Fig. 3(a) where the GSP is in the left frequency-region, and Fig. 7(b) is obtained from the point on the red curve in Fig. 3(b), where the GSP is situated in the right frequency-region. The components of electromagnetic fields exhibit their different feature due to their different oscillatory factors. For the magnetic field in the metamaterial, its x-component is not continual at the surface. We find that the polarization of GSP is different in the two regions. In the left frequency-region, the magnetic-field components synchronously oscillate and attenuate with the distance from the surface, as shown in Fig. 7(a). In the right frequency range, the components non-synchronously oscillate with the distance, where we see more oscillatory periods as β is evidently larger than α. The maximum of GSP magnetic-field amplitude is not situated at the surface, but is localized inside the metamaterial, which is completely different from the SHP and conventional surface polaritons. In addition, the figures obviously show the continuity of H_y and H_z at the surface.

Fig. 7. The distribution of GSP magnetic field in the x-direction normal to the surface. (a) and (b) show the distributions corresponding to the two specific points on the red curves in Figs. 3(a) and 3(b). The magnetic field is measured in $H_y^{\prime}$ and x = 0 is the position of the surface.

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4. Conclusions

We have predicted new surface polaritons in a metamaterial alternately composed of antiferromagnetic (AF) layers and polar-crystal (PC) layers in a specific geometry. In the used geometry, the layer-array direction is pointed along the y-axis and the AF easy axis and external magnetic field both are in the z-direction. The metamaterial surface is parallel to the y-z plane and surface polaritons move along the z-axis and decay in the x-direction. The external magnetic field is applied to produce the gyromagnetic permeability in AF layers. We find that this metamaterial simultaneously supports two kinds of new surface polaritons in this specific geometry, i.e., a normal surface hybrid-polariton polariton (SHP) and a ghost surface polariton (GSP). The electromagnetic fields of SHP monotonously and exponentially decay with the distance from the metamaterial surface, but those of GSP attenuate and oscillate with the distance. The GSP is different from the ghost surface polariton found previously [22–23]. This ghost surface polariton is composed of two branch-waves in the metamaterial and the two branch waves are coherent. The polariton dispersion can be linearly controlled with the external magnetic field. The external magnetic field and the hyperbolicity of the effective permittivity perform necessary conditions for the existence of GSP and SHP. The GSP and SHP occupy different segments of a smooth dispersion curve or a smooth field-dependence curve of surface-polariton frequency, so to change the external field or wavenumber can bring about the switch or transition between the SHP and GSP. These interesting properties may offer a new possibility of surface-polariton applications for the far-infrared or THz technology.

Funding

Natural Science Foundation of Hainan Province (YSPTZX202207); 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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Unique surface polaritons and their transitions in metamaterials

Abstract

1. Introduction

2. Dispersion and polarization of surface polaritons

3. Numerical results and discussions

4. Conclusions

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (7)

Equations (25)

Optics Express