1. Introduction

Artificial material with a magnetic response at terahertz (THz) frequencies that were recently demonstrated in a planar structure [1,2] are now of great interest because of their potential to produce a double-negative metamaterial (ε<0 and µ<0) simultaneously in the THz or potentially optical regimes. Negative-index metamaterials, whose permittivity and permeability are simultaneously negative and are known not to exist in a natural form in materials, were made by composing an array of metallic wires and split-ring resonators (SRRs) [3,4]. The negative refraction of electromagnetic waves was observed in a microwave transmission experiment as had been predicted theoretically by Veselago in his 1968 pioneering paper [5]. In addition to the negative refraction, some interesting phenomena such as reversed Doppler shift, reversed Cerenkov radiation, reversed radiation pressure, and imaging by a slab of a negative index metamaterial are also presented, all of which are direct results of the group velocity inversion of electromagnetic waves that propagate in such media. On the other hand, the artificially structured materials formed by an array of nonmagnetic conduction, SRRs exhibit negative permeability. However, when the artificially structured materials are combined with plasmonic wires that exhibit negative permittivity, the SRRs and wires cannot be patterned on the same planar substrate, The main reason is that the negative magnetic response of an artificial planar structure, in which both the SRRs and the wires are assumed to be formed on the same layer, can be achieved only for magnetic fields with a varying flux that is normal to the SRR plane. The square array of SRRs has a relative permeability of µ(ω)=1-(Fω ²)/(ω ²-${\omega }_0^2$+iΓω), where the strength (F), resonance frequency (ω ₀), and lifetime (Γ) of the magnetic dipole resonance are defined mainly by the structure parameters of SRRs, and the wire grids placed between the SRRs also have the relative permittivity of ε(ω)=1-${\omega }_p^2$ /ω ² with the plasma frequency (ω _p ) given by the width and radius of the wires [6,7]. To open a frequency range in which the ε and µ are negative simultaneously, the planar structure must satisfy the necessary condition of ω_p /ω ₀>1. After further consideration under the assumption that the period of the SRRs is 50µm for a typical THz response, we can conclude that the gap size between the inner and the outer rings in the SRR must be of the order of nanometers [8]. This requirement of a few nanometers pattern in the artificial planar structures makes it necessary for us to choose another structure, such as a layered planar structure with alternating layers having a SRR array for µ<0 and wire grids for ε<0. It has also been reported that periodic assembly of two alternating layers, one with negative permittivity and the other with negative permeability, can support backpropagating modes with an effectively negative index of refraction [9]. The backpropagating modes in the layered structure are Bloch waves for which propagation directions are limited to a surface normal. Therefore, such an artificial planar structure with SRRs might not be applicable for the system of alternating slabs because the magnetic fields of Bloch waves do not vary their flux normal to the SRR faces.

We show that the existence of backpropagating modes of surface polaritons that propagate along the single-boundary surface of two layers have µ<0 and ε<0, respectively. The boundary surface or cross-negative interface can be formed in a layered structure composed of artificial magnetic (SRRs) and electrical (plasmonic wires) layers since the surface polaritons that propagate on the cross-negative interface can have magnetic fields that oscillate normal to the surface of the SRR layer. We first find relative conditions for the material parameters of the two semi-infinite layers that support the backpropagating surface polaritons by considering dispersion relation and energy flow density transported by the surface polaritons. We then confirm our findings numerically by simulating backpropagation of the surface polaritons that are resonantly excited by attenuated total reflection (ATR) of a Gaussian beam incident on the cross-negative interface.

The surface polaritons defined by energy quanta of surface localized oscillations of electric or magnetic dipoles are substantially supported by negative values of the material parameters [10,11]. The negative permittivity enables resonant excitation of the p-polarized surface electric polaritons (SEPs), which are surface localized longitudinal oscillations of the electric dipoles, whereas the negative permeability makes possible the s-polarized surface magnetic polaritons (SMPs), which are surface localized longitudinal oscillations of the magnetic dipoles. The surface polaritons propagate along the boundary surface with their electric and magnetic fields localized and evanesce into both adjoined materials. Excitation of a single normal mode of the surface polaritons is influenced simultaneously by the material parameters of both media adjoined at the interface. Therefore, one can expect backpropagating modes of surface polaritons that have negative group velocity not only in double-negative media, but also at the cross-negative interface of two media in which only ε<0 is in one medium and only µ<0 is in the other. It is also known that, in thin slab geometry such as plasma or an ionic crystal [12,13] and a metal-film optical waveguide[14], surface polaritons with negative group velocity can exist even when only the permittivity is negative (and the permeability is equal to one everywhere). In these cases only the SEP modes can be excited on both sides of the slab, and they must be coupled to each other to concentrate more energy into the slab. If the thickness of the slab increases, the coupled SEP modes with negative group velocities would no longer exist. In contrast, single-interface modes that are excited at single boundaries of two semi-infinite media with double negativity or cross negativity, would be distinguishable from the coupled modes in thin slabs for the following two reasons: SMP modes as well as SEP modes are allowed for single-interface modes with negative group velocities; and a frequency range for the backpropagating SMP modes can be tuned flexibly by means of, for example, varying dimensions of the artificially structured materials [1–5]. Therefore, the single-interface modes play an important role for left-handed responses of composite media at THz and even higher frequencies.

2. Generalized conditions of material parameters for surface polariton excitation

First we analyze all the possible combinations of the four material parameters that support either SEPs or SMPs of two media that constitute a planar boundary. Consider a surface electromagnetic wave, φ(x,z)=A exp (iβx-γ_s |z|), localized near the interface (z=0 plane) of two media for which relative material parameters are given by (ε_s ,µ_s ), where medium index s has a value of 1 for z<0 or 2 for z>0. β is the propagation constant in the direction of the x axis and γ_s is the decay constant in each medium. Wave amplitude A represents the y component of the magnetic field or the electric field when the surface-localized electromagnetic wave is coupled to a SEP or a SMP, respectively. By requiring that all the field components satisfy Maxwell equations for all the positions including the interface, we can obtain a relationship between the propagation and the decay constants of the SMP-coupled waves (SMP modes) and the SEP-coupled waves (SEP modes) in terms of material parameters (ε ₁,µ ₁) and (ε ₂,µ ₂). For SMP modes, the relationship is

where k ₀=ω/c [15]. The relative conditions of the four material parameters that support excitation of the SMP modes can be derived from Eqs. (1) to (3). For an additional procedure, assume that the all-material parameters are complex valued but dominated by their real parts such that ε_s =ε′_s (1+ie_s ), |e_s |□ 1 and µ_s =µ′_s (1+im_s ), |m_s |□ 1, where all the symbols on the right-hand sides are real. When the propagation and decay constants are expanded by powers of e_s and m_s and terms up to their first orders are taken, they can be rewritten as

where the imaginary parts relative to the real parts are given by

It can be seen that b_SMP and g_SMP have values of the same orders of magnitude as e_s and m_s , because the terms that include the real parts of the material parameters in Eqs. (6) and (7) have numerators with multiplication factors of e_s and m_s , which are the same combinations of ε′_s and µ′_s as their denominators, respectively, and this results in overall factors of e_s and m_s to be of the order of 1. Therefore, the terms that include b_SMP and g_SMP in Eqs. (4) and (5) are negligible for the propagation and decay constants according to the previous assumption for the imaginary parts of the material parameters. If β′_SMP is a real value, the propagation constant βSMP has a dominant real part and the corresponding wave function expresses the propagation state with a relatively slow decaying profile along the interface. Otherwise, if β′_SMP is an imaginary value, the wave hardly propagates and decays rapidly. Thus it is reasonable to regard real-valued β′_SMP as a necessary condition, namely, in-plane propagation condition, for SMPs. To apply a similar manner to decay constant γ_SMP,s , a surface localization condition for SMPs could be obtained as γ′_SMP,s must be a real value. To complete the surface localization condition, it is necessary for γ′_SMP,s to be positive for both media. This condition requires µ′ ₁ µ′ ₂<0, when we approximate Eq. (3) to be γ′_SMP ,₁/µ′ ₁=-γ′_{SMP, 2}/µ′ ₂. In summary, the in-plane propagation condition reduces to µ′ ₁/ε′ ₁<µ′ ₂/ε′ ₂ when -1<µ′ ₂/µ′ ₁<0, or µ′ ₁/ε′ ₁>µ′ ₂/ε′ ₂ when µ′ ₂/µ′ ₁<-1. And the surface localization condition for a SMP reduces to ε′ ₁ µ′ ₁<ε′ ₂ µ′ ₂ when -1<µ′ ₂/µ′ ₁<0, orε′ ₁ µ′ ₁>ε′ ₂ µ′ ₂ when µ′ ₂/µ′ ₁<-1. For SEP modes the relative conditions for in-plane propagation and surface localization can be obtained simply by interchanging the relative permittivity and permeability positions with each other. Figure 1 shows diagrams for visualization of these relative conditions for SMP modes in the blue region and SEP modes in the red region. To represent four possible sign combinations of (a) (ε′ ₁>0,µ′ ₁>0), (b) (ε′ ₁/<0,µ′ ₁>0), (c) (ε′ ₁>0,µ′ ₁<0), and (d) (ε′ ₁<0,µ′ ₁<0). the normalized values of (ε′ ₁/|ε′ ₁|,µ′ ₁/|µ′ ₁|) are marked by black dots and their complementary values by white dots. The dashed curves in Figs. 1(a) and 1(d) represent the critical boundaries of ε′ ₂ µ′ ₂=ε′ ₁ µ′ ₁ for the surface localization conditions; the dashed lines in Figs. 1(b) and 1(c) reveal ε′ ₂/µ′ ₂=ε′ ₁/µ′ ₁ for the in-plane propagation conditions. The color densities in the figure indicate the normalized total energy flow densities of the SMP modes (η_SMP ) and the SEP modes (η_SEP ) that are defined in Eq. (3).

 

Fig. 1. Density plots of the normalized energy flow densities of the SEP (in red) and SMP (in blue) modes for four possible combinations of ε′ ₁ and µ′ ₁. The filled and open circles in (a)–(d) indicate the positions of (ε′ ₁/|ε′ ₁|,µ′ ₁/µ′ ₁) and (-ε′ ₁/|ε′ ₁|,-µ′ ₁/|µ′ ₁|), respectively; the white dashed boundaries represent the relations of ε′ ₁ µ′ ₁=ε′ ₂ µ′ ₂ in (a) and (d) and ε′ ₁/µ′ ₁=ε′ ₂/µ′ ₂ in (b) and (c). The arrows in (b) and (c) indicate the cross-negative areas where the surface polaritons have a negative group velocity.

Download Full Size | PDF

Some interesting properties of the surface polaritons can be observed in the Fig. 1 diagrams. Because the SEP and the SMP modes do not overlap, there are no possible combinations among the four material parameters that allow simultaneous excitation of the SEP and SMP modes. Even if we introduce absorption losses of the media by adding imaginary parts in the material parameters, some overlapped areas might appear only near the critical boundaries, but the SEP and SMP modes excited at the overlapped areas must be weakly localized, damping modes. Each of the regions with SEP or SMP modes consists of two parts separated by a white dot, indicating the (-ε′ ₁/|ε′ ₁|,-µ′ ₁/|µ′ ₁|). Physical implication of the two distinguished parts becomes obvious after evaluation of normalized total energy flow densities transported by the SEP and SMP modes:

where subscript a represents SEP or SMP, ξ′_SEP,s ≡ε′_s , and ξ′_SMP,s ≡µ′_s . p⃑_a,s represent the energy flow density that is derived by integrating time-averaged Poynting vector S⃑_a (z) over the respective surface normal distance through medium s:

From the localization condition of ξ′_a,1ξ′_a,2<0 the propagation directions of p⃑ _a,1 and p⃑ _a,2 are always opposite each other [16]. When ξ′_a,1<0 and |p⃑ _a,1|>|p⃑ _a,2|, for example, the normalized total energy flow density η_a is negative. This can be thought of as the mode with negative group velocity with respect to phase velocity because the Poynting vector direction is always equal to the group velocity for linear waves that propagate in a homogeneous medium with arbitrary spatial and temporal dispersion [17,18]. The calculation results of the normalized total energy flow densities are depicted by color density in Fig. 1. Note that the colored scale bars indicate the normalized ranges of the 1≥η_SEP ≥-1 in red and the 1≥η_SMP ≥-1 in blue. The surface polaritons are excited at the regions in which 0>η_a ≥-1 (1≥η_a >0) have a negative (positive) group velocity. These density plots obviously show that the regions that allow negative group velocity do not consist of only the double-negative areas in which either of the two media has ε′_s <0 and µ′_s <0 simultaneously, but also the cross-negative areas in which ε′ ₂>0 and µ′ ₂<0 while ε′ ₁<0 and µ′ ₁>0 or vice versa, such as is indicated by the arrows in Figs. 1(b) and 1(c).

Another interesting property of the surface polaritons is that the propagation and decay constants defined by Eqs. (4) and (5) are invariant when the signs of the real parts are simultaneously inversed, such as from (ε′ ₁,µ′ ₁,ε′ ₂,µ′ ₂) to (-ε′ ₁,-µ′ ₁,-ε′ ₂,-µ′ ₂). The excitation conditions of surface polaritons were derived from kinetic consideration of the propagation and decay constants, therefore it can be concluded that, if a SEP (SMP) can propagate on a boundary with material parameters of (ε′ ₁,µ′ ₁,ε′ ₂,µ′ ₂), the inverted material parameters of (-ε′ ₁,-µ′ ₁,-ε′ ₂,-µ′ ₂) also support a SEP (SMP) with the same propagation and decay constants as the original ones. For this reason, Figs. 1(a) and 1(d) and 1(b) and 1(c) show centrosymmetry with each other. However, there are differences in the field distributions between the two SEP (SMP) modes with (ε′ ₁,µ′ ₁,ε′ ₂,µ′ ₂) and (-ε′ ₁,-µ′ ₁,-ε′ ₂,-µ′ ₂). For example, the magnetic field of a SEP mode can be expressed by H=H ₀ exp(iβx-γ|z|)e _y . According to the Maxwell equations the corresponding electric field is given by E=(β e _x ±iγ e _z)×H/ωε_i where the +(-) sign is taken for z>0 (z<0). Inverting the signs of material parameters does not alter the magnetic field, whereas the electric field is changed by a π shift in its phase because of ε_i . As a consequence, only the direction of the Poynting vector is reversed. This is also confirmed by comparison of the upper-left region with the lower-right region in Figs. 1(a) and 1(d) or the upper-right region with the lower-left region in Figs. 1(b) and 1(c), where distribution of the normalized total energy flow density is centrosymmetric with different propagation directions. This dependence of surface polariton modes on simultaneous sign inversion of the real parts of the material parameters can be explained more generally by conjugation symmetry of the frequency domain Maxwell equations [19]. In a source-free and isotropic system, the frequency domain Maxwell equations are invariant under transformation such that

If E(r,ω) and H(r,ω) are solutions of the Maxwell equations in system Σ with the spatial distributions of material parameters ε(r,ω) and µ(r,ω), then Ē(r,ω)=E*(r,ω) and H̄(r,ω)=H*(r,ω) are also exact solutions for conjugate system $\overline{\Sigma }$, whose relative permittivity and permeability are -ε*(r,ω) and -µ*(r,ω), respectively. Note that $\overline{\Sigma }$ is given by the sign inversion of the real parts of the material parameters for Σ. It is also noted that the time-averaged Poynting vector given by Re[E(r,ω)×H*(r,ω)] is invariant under such transformation, and. as a consequence, the group velocity direction of the conjugate modes between Σ and $\overline{\Sigma }$ is equal for each mode. But complex conjugate operation on the spatial field amplitudes changes the signs of the wave vectors of plane-wave components of which E(r,ω) and H(r,ω) are composed, resulting in inversion of the phase velocities.

3. Backpropagating modes of surface magnetic and surface electric polaritons on crossnegative interfaces

We now consider characteristics of the surface polaritons excited at the cross-negative areas in Fig. 1(b). [Those in Fig. 1(c) are of symmetrical cases only.] Suppose that the two adjoined media are composed of a metal with (ε′ ₁<0, µ ₁=1) and a metamaterial with (ε ₂=1,µ′ ₂<0) over a frequency range lower neighbor of Ω₀. The material parameters of ε ₁ and µ ₂ can be expressed in the form of

Ω is a normalized frequency defined by ω/ω_p where ω_p is the plasma frequency of the metal. ε ₁(Ω) is a plasmonic form of a Drude model and µ ₂(Ω) is from an array of planar SRRs with a resonance frequency of Ω₀ [7]. F is the area fraction of the internal opening in the SRR and set to be 0.5, hereafter. To show schematically the frequency dependence of excitation of the SMP and SEP modes, we assume that Γ₁=0 and Γ₂=0 under the approximation of small damping losses in the media. Figure 2(a) shows a parametric plot of (ε′ ₂/|ε′ ₁|,µ′ ₂/|µ′ ₁|) as a function of Ω for three different Ω₀ of 0.8(>Ω _c ), Ω _c , and 0.4(<Ω _c ), where ${\Omega }_c=1⁄\sqrt{2+F}(=0.6325)$. Only the frequency range of 0<Ω<1 is considered since ε′ ₁<0, and the positions marked by A’s and B’s represent the boundary frequencies of Ω(A1)=0.4, Ω(A2)=0.4472, Ω(A3)=0.4714, Ω(A4)=1/√2, Ω(B1)=1/√2, Ω(B2)=0.8, Ω(B3)=0.8528, andΩ(B4)=0.8944. At the critical resonance frequency of Ω₀=Ω _c (dash-dot curves), there is no surface polariton mode in the cross-negative areas (ε′ ₂/|ε′ ₁|>0 and µ′ ₂/|µ′ ₁|<0). When Ω₀=0.8 (curves with open circles) as an example of Ω₀>Ω _c , a SEP(-) band that supports the SEP modes with negative group velocity is located in the frequency range of Ω(B2)<Ω<Ω(B3), whereas a SEP(+) band with positive group velocity is in 0<Ω<Ω(B1) and a SMP(+) band is in Ω(B3)<Ω<Ω(B4). In contrast, when Ω₀=0.4 (curves with solid circles), a single SMP(-) band appears in Ω(A2)<Ω<Ω(A3), and two SEP(+) bands appear in 0<Ω<Ω(A1) and (A3) Ω <Ω<Ω(A4), respectively. After further analysis, we can finally conclude that, for the cross-negative areas shown in Fig. 1(b), the SEP(-) modes appear only for the case of $1⁄\sqrt{2+F}<{\Omega }_0<1$, and at the frequency bands of $1⁄\sqrt{2}<\Omega <\sqrt{\left(1+F\right)⁄\left(1+F{{\Omega }_0}^2\right)}{\Omega }_0$ if $1⁄\sqrt{2+F}<{\Omega }_0<1⁄\sqrt{2}$ and ${\Omega }_0<\Omega <\sqrt{\left(1+F\right)⁄\left(1+F{{\Omega }_0}^2\right)}{\Omega }_0$ if 1/√2<Ω₀<1 ; the SMP(-) modes only for ${\Omega }_0<1⁄\sqrt{2+F}$, and at the frequency band of ${\Omega }_0<\Omega <\sqrt{\left(1+F\right)⁄\left(1+F{{\Omega }_0}^2\right)}{\Omega }_0$.

 

Fig. 2. (a) Parametric plot of (ε′ ₂/|ε′ ₁|,µ′ ₂/|µ′ ₁|) as a function of normalized frequency Ω. The positions marked by A’s and B’s represent the cutoff frequencies of Ω(A1)=0.4, Ω(A2)=0.4472, Ω(A3)=0.4714, Ω(A4)=1/√2, Ω(B1)=1/√2, Ω(B2)=0.8, Ω(B3)=0.8528, and Ω(B4)=0.8944. (b), (c), Dispersion curves of the SEP and SMP modes when (b) Ω₀=0.4 and (c) Ω₀=0.8, where the dashed curves represent dispersion relations of the bulk propagating mode in the metamaterial. The points marked by M1, M2, E1 and E2 are used in Fig. 3.

Download Full Size | PDF

The dispersion relations of the SMP modes in Eq. (1) are depicted by the open circles in Figs. 2(b) and 2(c), corresponding to the resonance frequencies of Ω₀=0.4 and Ω₀=0.8, respectively, and B≡cβ′/ω_p . For comparison, that of the SEP modes is also presented by solid circles and that of bulk modes that propagate in the metamaterial is represented by dashed curves. The negative slopes between Ω(A2) and Ω(A3) in Fig. 2(b) and between Ω(B2) and Ω(B3) in Fig. 2(c) clearly show the negative group velocities of the SMP(-) and SEP(-) modes, respectively.

4. Numerical demonstration of backpropagating modes on cross-negative interfaces

To demonstrate the backpropagation behavior of two modes with opposite group velocities we chose four points in the dispersion curves in Figs. 2(b) and 2(c), two typical SMP modes at M1 (Ω=0.87,B=0.53) for SMP(-) and M2 (Ω=0.87,B=0.53) for SMP(+), and two SEP modes at E1 (Ω=0.50,B=0.35) for SEP(+) and E2 (Ω=0.84,B=0.49) for SEP(-). With an ATR configuration composed of four stacked layers and a semi-infinite dielectric, metal, metamaterial, and semi-infinite air as indicated in Fig. 3(a), we evaluated propagation characteristics of two of the SMP and SEP modes on a cross-negative interface between the metal (ε′<0,µ′>0) and the metamaterial (ε′>0,µ′<0) by using the plane-wave expansion method. A Gaussian beam that is incident from the dielectric with a finite waist is assumed to be TE and TM polarized for SMP and SEP modes, respectively. The calculation results are shown in Fig. 3. In the calculation it is assumed that small damping constants of Γ₁=10^-3 and Γ₂=10^-3Ω₀ and that the dielectric (ε=2.25) and air (ε=1.0) are semi-infinite. The thickness of the metamaterial is 10×ƛ_p for all cases, where ƛ_p(≡c/ω_p ) is the plasma wavelength. The thicknesses of the metal layers are given differently by 2.7×ƛ _p , 4×ƛ _p , 3×ƛ _p and 3.25×ƛ _p for Figs. 3(a), 3(b), 3(c), and 3(d), respectively, to guarantee high coupling efficiency from the incident beam to the surface modes. It is apparent that the excited modes are not coupled modes but single-interface modes at the cross-negative (metal–metamaterial) interface, because no field enhancement is seen at the dielectric–metal or the metamaterial–air interfaces for all the cases. Figure 3(a) clearly shows the leftward propagation of the electric fields (E_y ) of the SMP(-) mode, as depicted by a dotted arrow near the cross-negative interface. The Gaussian beam incident from the dielectric is not only reflected from the dielectric–metal boundary, but it is coupled resonantly to the SMP(-) mode near the cross-negative interface. The electric field of the SMP(-) mode is more concentrated on the metamaterial layer with a group velocity antiparallel to its phase velocity. Evidence of the negative group velocity can also be found intuitively by observing the reemitted fields that radiate back into the dielectric medium. The reemitted fields that are shown just under the SMP(-) propagation region reveal wave fronts parallel to the reflected beam. As a consequence we can confirm that the phase velocity of the SMP(-) mode is positive in the x direction, but the group velocity is negative. The SMP(+) mode at point M2, on the other hand, is stretched toward the metal layer as shown in Fig. 3(b). It has a group velocity parallel to its phase velocity, which can also be checked by the wave fronts of the reemitted fields parallel to the reflected wave fronts. For the SEP modes depicted in Figs. 3(c) and 3(d), the SEP(+) mode has its magnetic field concentrated more in the metamaterial layer than in the metal layer, similar to the SMP(-) mode in Fig. 3(a), but it has rightward or forward propagation. The reverse is shown for the SEP(-) mode in Fig. 3(d) with the SMP(+) mode in Fig. 3(b). If we recall that the energy flow density is directly proportional to |E_y |² (SMP modes) or |H_y |² (SEP modes) as described in Eq. (10), these differences in field concentration can easily be understood by the fact that backpropagating SMP or SEP modes should have more energy in a diamagnetic or a metal layer, respectively.

 

Fig. 3. Simulation results of the ATR coupling from an incident Gaussian beam to (a) the SMP(-) mode of point M1 (Ω=0.46, B=0.41), (b) the SMP(+) mode at point M2 (Ω=0.87, B=0.53), (c) the SEP(+) mode of point E1 (Ω=0.50, B=0.35), (d) the SEP(-) mode of point E2 (Ω=0.84, B=0.49). Color density represents the y component of the electric field for (a) and (b) and the magnetic field for (c) and (d). Note that the y axis is normal to the page surface.

Download Full Size | PDF

One effect of material absorption to propagation loss is that backpropagating modes undergo the same losses as forward propagating modes. The propagation length of the mode decreases directly proportional to the imaginary part of the material parameters of both media, and this can be determined by the relative imaginary part of the propagation constant b_SMP in Eq. (6). We introduced Q_SMP ≡1/b_SMP as the effective number of spatial oscillations in the same manner as the definition of the resonance quality factor for temporal oscillations. For the SMP(-) mode in Fig. 3 (a), the relative damping constants of Γ₁=10^-3 and Γ₂=10^-3Ω₀ reveal Q_SMP =45.71. If we take into consideration ten times larger damping constants of Γ₁=10^-2 and Γ₂=10^-2Ω₀, Q_SMP =4.56, which is a ten times smaller value as expected from the dependence of b_SMP on the imaginary parts of the material parameters. For a more practical case, we consider an SMP(-) mode excited at the interface between a gold and a two-dimensional magnetic metamaterial as reported in Ref. 20. When ω_p =1.367×10¹⁶Hz, γ ₁=4.08×10¹³Hz for gold [21], and F=0.2, ω ₀=1.489×10¹⁴Hz, γ ₂=2.4×10¹²Hz for metamaterial, excitation of the SMP(-) modes can occur in the frequency range from 1.56×10¹⁴Hz to 1.63×10¹⁴Hz. At 1.5836×10¹⁴Hz, for example, (ε′ ₂/|ε′ ₁|,µ′ ₂/|µ′ ₁|)=(10^-4,-0.5) and the corresponding SMP(-) mode has Q_SMP =3.06, which means that the mode undergoes a significant amount of propagation loss.

5. Conclusion

We have derived the excitation conditions and propagation characteristics of backpropagating modes of surface polaritons by means of material parameters. We found four general properties of surface polaritons. First, simultaneous excitation of SEP and SMP modes is inhibited in general. Second, backpropagating surface modes with negative group velocity can be observed not only at the boundary of double-negative media but also at the cross- negative interface of two media where only ε<0 in one medium and only µ<0 in the other. Backpropagating modes of surface magnetic polaritons at cross-negative interfaces have been confirmed in detail by evaluation of their dispersion relations and ATR coupling behavior based on the plane-wave expansion method. Third, the two propagation directions, parallel and antiparallel to the phase velocity, are inherently determined by the values of the material parameters, regardless of their frequency dispersive characteristics. In particular, antiparallel propagation is also possible even when no double-negative medium is involved, such as the cross-negative media composed of two nontransparent media: one with negative permittivity only and the other with negative permeability only. Fourth, if a set of material parameters supports a parallel (antiparallel) propagating SEP (SMP), the sign inverted set supports antiparallel (parallel) propagating SEP (SMP) without changing the propagation and decay constants.

It is worth noting that the magnetic fields (B_x ) of the SMP(-) mode always have a varying flux normal to the metamaterial surface, which proves that efficient production of a negative magnetic response (µ<0) can be achieved in an artificial planar structure. Therefore, it is possible that such a cross-negative interface with a single-negative requirement imposed on the material parameters of two adjoined media can be implemented by stacking two different types of planar structure: one with a negative permeability, such as a SRR; the other with a negative permittivity, such as a metallic grid. The scalability of these separated planar structures could enable us to realize surface polaritonic devices with lefthanded behavior in THz and potentially optical frequencies.