1. Introduction

During last five decades, the surface plasmon polaritons (SPP) propagating along an interface between a plasma-like medium (e.g., metal or semiconductor) and a dielectric, became an object of intensive theoretical and experimental studies (see, for example, Refs. [1–8] and citations therein). At present, SPP are widely used in several areas of science and technology, particularly in near-field optical spectroscopy and optoelectronic devices [9,10]. It has commonly been assumed, that SPP occur at a planar interface of different isotropic materials with different permittivities and only in the case when the permittivity of one of the media in contact is negative. In the case when at least one of the media is optically anisotropic, the existence of interface waves does not impose any conditions on the permittivity [11]. Such nondispersive, so called Dyakonov surface waves appear due to differences between the symmetry of the media and recently have been observed experimentally [12]. The boundary–value problem for Dyakonov-Tamm waves propagating along a planar interface of two structurally chiral nonmagnetic materials was formulated and numerically solved in Ref. [13], assuming that either of the materials is twisted arbitrarily with respect to the direction of propagation. The waves considered in [13] are result of anisotropy; they occur when at least one of the optical axes is parallel to the interface and exist even in the absence of the chirality. In our recent work [14], the existence of nondispersive surface electromagnetic waves guided by an interface between two transparent enantiomeric bianisotropic media has been predicted in the case when the optic axis is perpendicular to the interface. Note that in this case there are no Dyakonov-like surface waves which can only propagate in a certain range of angles with respect to the axis (see Ref. [11]).The waves predicted in [14] are caused by anisotropic magnetoelectric coupling in contacting materials which exhibit right- and left-handedness, and are absent in the case of enantiomeric bi-isotropic media.

The purpose of this paper is an investigation of SPP localized at the plane interface between two identical uniaxially bianisotropic plasmonic media, one of which is the mirror image of the second one. Such materials can be realized by arranging chiral objects (for example, metal helixes or omega-shaped particles) in a host medium with a preferable direction. The existence of a new class of SPP with unusual dispersion and polarization properties is predicted.

The paper is organized in six main sections. In section 2, the field structure of evanescent partial waves is studied and general dispersion relations for two distinct modes of SPP are obtained. Section 3 describes polarization of the waves. In section 4, the dispersion properties of SPP are studied rigorously in nondissipative case and the frequency regions of existence are determined analytically. The role of losses is examined in section 5. In section 6 the concluded remarks are highlighted.

2. SPPs at the Interface of Enantiomeric Media. Partial Waves

We assume that the z = 0 plane separates a semi-infinite plasma-like medium (by definition, R-material, z<0) from the identical material of opposite handedness (L-material, z>0). Optic axes in both materials are supposed to be perpendicular to the interface plane. For brevity we consider only one plasma component, for instance the electron component. The L-material is characterized by diagonal chirality admittance tensor

In the case of reciprocal media both ${\xi }_{\perp }$ and ${\xi }_{\left|\right|}$ are assumed to be real, positive and nondimensional constants smaller than 1. The corresponding tensor for the R-material is given by $\stackrel{\wedge }{{\xi }_R}=-\stackrel{\wedge }{{\xi }_L}.$ Both media are described by the same relative permittivity tensor

Consider surface waves which travel and are attenuated in the x-direction along the interface (see Fig. 1 ). Then the wave vector can be represented as $\overrightarrow{k}=k_0\{n,0,iq\},$ where $k_0=\omega /c$ is the free space wave number at angular frequency $\omega .$ Assuming that the relative magnetic permeability of the structure is equal to unity and time enters as a factor of the form $\exp (-i\omega t),$ one can find evanescent solutions of the Maxwell equations

 

Fig. 1 Geometry of the problem. The region z>0 is occupied by L-material and the region z<0 –by R-material which are the mirror image of each other. The surface wave is propagating along the positive x-axis.

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At a given value of the tangential wave vector component $k_x=k_{0}n,$ in each medium can exist two evanescent partial waves with different complex values of the normal component ${(k_z)}_{\pm }=i{k}_0{q}_{\pm },$ where

In each medium, the electric field vector is a linear combination of the field vectors of both partial waves:

In this paper we will restrict our consideration in the region of frequency-tangential wave vector plane, where both partial waves are evanescent nonradiative waves, that is, the imaginary parts of the normal wave vector components are positive: $q_{\pm }^{\text{'}}\equiv \mathrm{Re}{q}_{\pm }>0.$ In general case $q_{\pm }^{"}\equiv \mathrm{Im}{q}_{\pm }\ne 0,$ and then the wave fields decay in both half-spaces executing spatial oscillations. To this end, using standard boundary conditions for tangential field components, we find two distinct modes of SPP which are described by the following dispersion relations:

It is important to note that these modes of SPP exist only in the case of bianisotropic enantiomeric structure, for which ${\epsilon }_{\perp }\ne {\epsilon }_{\left|\right|}$ _and ${\xi }_{\perp }\ne {\xi }_{\left|\right|}.$ At a planar interface of two bi-isotropic enantiomeric media there are no surface waves [14].

3. Polarization Properties of the Waves

In SPP governed by Eq. (9a), tangential components of the magnetic field in both R- and L-materials coincide: ${\overrightarrow{H}}_{\pm t}^L={\overrightarrow{H}}_{\pm t}^R.$ As to the tangential components of the electric fields, they are parallel but $\left|{\overrightarrow{E}}_{\pm t}^L\right|\ne \left|{\overrightarrow{E}}_{\pm t}^R\right|.$ These components are polarized elliptically:

In SPP described by Eq. (9b), the properties of the electric and magnetic fields are interchanged: the tangential components of the electric field in R- and L-materials coincide: ${\overrightarrow{E}}_{\pm t}^L={\overrightarrow{E}}_{\pm t}^R$ while those of the magnetic fields are parallel, but $\left|{\overrightarrow{H}}_{\pm t}^L\right|\ne \left|{\overrightarrow{H}}_{\pm t}^R\right|.$These components are characterized by elliptical polarization:

Polarization of the field components parallel to the xz-plane are described by relations

4. SPP in Nondissipative Case. Frequency Regions of the Wave Existence

Before a rigorous examination of the SPP’s dispersion properties in general case when both permittivity tensor components are complex quantities, an investigation of the simple situation is required, when there is no wave dissipation. In this case SPP can be described, like in the case of well known Fano modes [18], by real frequencies and real tangential wave vectors. Except that, the polarization parameters given by Eqs. (10b) and (12b) are real too, that is, the tangential field components are polarized linearly. Note also that in the lossless case ImA = ImC = 0 [see Eqs. (5b) and (5c)] and therefore the propagation of true surface waves with no space oscillations of the fields is possible, if ._. From Eq. (5a) it is clear, that such a situation can only be realized if the following conditions are fulfilled simultaneously:

If $C<0,$ the partial wave with $q_{-}$ is a bulk wave while that with $q_{+}$ is an evanescent one, but it loses energy via the propagating wave (so-called pseudosurface wave). If $C>A^2,$ both $q_{+}$ and $q_{-}$ are complex quantities. In this section, we restrict our consideration to the case of surface waves described by unequalities (13).

Using the Drude expressions for the collisionless plasma

Furthermore, for the brevity we will restrict us to the consideration of the mode described by relations (15a) and (10a)–(11b). Properties of the second mode governed by Eqs. (16a) and (12a)–(12d) can be examined analogically.

It is clear from Eqs. (15a) and (15b) that the wave is cut off (k _x = 0) at $\omega ={\omega }_1$ and $\omega ={\omega }_2$ if $\gamma >1,$ and only at $\omega ={\omega }_1$ if $\gamma <1.$ Moreover, the wave has a resonance (k _x →∞) only if the expression on the right hand side of Eq. (15d) is positive. It means that surface plasmon modes with frequency $\omega ={\omega }_r$ in the absence of retardation occur only when $\gamma >1$ if ${\xi }_{\left|\right|}{}_{|}<{\xi }_{\perp },$ and vice versa: at $\gamma <1$ if ${\xi }_{\left|\right|}{}_{|}>{\xi }_{\perp }.$

Note that at the resonance frequency ${\epsilon }_{\parallel }({\omega }_r)-{\epsilon }_{\perp }({\omega }_r)={\xi }_{\left|\right|}^2-{\xi }_{\perp }^2$ while at cutoff frequencies _{${\epsilon }_{\left|\right|}({\omega }_1)={\xi }_{\left|\right|}^2$ and ${\epsilon }_{\left|\right|}({\omega }_2)-{\epsilon }_{\perp }({\omega }_2)=-\psi .$ Localization conditions (13) give}

Using now Eqs. (6), (14) and the left condition (18), one can conclude that SPPs exist in the frequency range

Note that

In Fig. 2 , the region in the plane $({\alpha }_{\left|\right|},{\alpha }_{\perp })$ is shown where conditions (18) are satisfied and both $q_{+}$ and $q_{-}$ are positive (shaded). It is easily to see that for a given value of ${\alpha }_{\left|\right|}$ in the interval $-{({\xi }_{\left|\right|}-{\xi }_{\perp })}^2/4<{\alpha }_{\left|\right|}<\psi ,$ that is, inside the frequency range

 

Fig. 2 The region in the plane $({\alpha }_{\left|\right|},{\alpha }_{\perp })$ where the conditions (18) are satisfied and both $q_{+}$ and $q_{-}$ are real positive quantities (shaded).

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The corresponding frequency forbidden gap is given by

With increasing ${\alpha }_{\left|\right|},$ the forbidden gap decreases and vanishes at ${\alpha }_{\left|\right|}=\psi .$It means that for ${\alpha }_{\left|\right|}\ge \psi ,$ SPPs exist at any negative value of ${\alpha }_{\perp }.$ Corresponding frequency range is given by

With decreasing ${\alpha }_{\left|\right|}$ in the region ${\alpha }_{\left|\right|}\le -{({\xi }_{\left|\right|}-{\xi }_{\perp })}^2/4,$ the forbidden gap is expanded and can be represented by unequality ${\alpha }_{-}({\alpha }_{\left|\right|})<{\alpha }_{\perp }<{\alpha }_{\left|\right|},$ where ${\alpha }_{-}({\alpha }_{\left|\right|})$ is given by Eq. (24b). Using now Eqs. (9a) and (18), it is easily to see that SPPs can only exist (n ²>0) if ${\alpha }_{\left|\right|}>0,$ that is, in the frequency range

The dispersion curves of SPP are shown schematically in Figs. 3 and 4 for different values of the ratio $\xi ={\xi }_{\left|\right|}/{\xi }_{\perp }$ and different intervals of the parameter $\gamma .$ In the case when $\xi <1$ and $1<\gamma <{\gamma }_0,$ there are two branches for which n² >0: low-frequency branch corresponding to the propagating (bulk) waves (dashed lines in Fig. 3(a)) and higher-frequency branch, only two sections of which correspond to the surface waves of interest (solid lines). In fact, the low frequency branch corresponds to the Brewster wave: both attenuation constants $q_{+}$ and $q_{-}$ given by Eq. (5a) are pure imaginary (A>C>0) and thus describe polaritons which are not bound to the interface. There is a forbidden bandgap given by Eq. (25a), so as SPP exists only in the following two frequency ranges:

 

Fig. 3 Dispersion curves of SPP (solid lines) in the case ${\xi }_{\left|\right|}<{\xi }_{\perp }.$ Dashed lines correspond to the bulk waves.

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Fig. 4 Dispersion curves of SPP (solid lines) in the case ${\xi }_{\left|\right|}>{\xi }_{\perp }.$ _a. ${\gamma }_1<\gamma \le {\gamma }_2.$ b. $\gamma \le {\gamma }_1.$

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In the case of ${\gamma }_1<\gamma <1,$ where

Thus, we conclude that the dispersion properties of the SPP are very sensitive to the ratio ${\xi }_{\left|\right|}/{\xi }_{\perp }$ as well as to the value of the anisotropy parameter $\gamma$ given by Eq. (17).

5. Role of the Wave Dissipation. Refraction Index and Propagation Length

Consider now briefly the main properties of SPP governed by Eq. (9a) taking damping into account. Substituting Eqs. (6) into (9a) and separating real and imaginary parts $(n=n^{\text{'}}+i{n}^{\text{'}\text{'}})$ we obtain

The relative attenuation of the wave on the wavelength distance is given by

Since the square of the refraction index defined by Eq. (32a) is always positive independly on the sign of P, it is clear that the SPP can propagate not only in the frequency regions defined in the previous section, but also at any other frequencies at which the dissipation is not very strong. Besides, the presence of losses leads to a finite value of the refraction index at resonance frequency ${\omega }_r,$ as well as to nonzero values at cutoff frequencies ${\omega }_{1,2}$ given by Eqs. (15b) and (15c). Indeed,

One of the most important characteristics of SPP in lossy structures is the propagation length defined as a distance along the propagation direction where the mode power decays by a factor of 1/e. This distance can be calculated using the expression

It is evident that in the absence of damping $L\to \infty$ while the absorption leads to its finite value depending on ω. Note that all the expressions given by Eqs. (32)–(35) and (37) are highly general and applicable for all frequencies at any level of dissipation. Assuming that the dissipation is weak $(n^{\text{'}\text{'}}/n^{\text{'}}<<1),$ in Eq. (37) one can put $n^{\text{'}}\approx \sqrt{\left|P\right|},$ at this the extinction coefficient $n^{"}\approx Q/2\sqrt{\left|P\right|}$. Note also that in the limiting case $\omega >>{\tau }_{\left|\right|,\perp }^{-1}$ (for example, at visible and shorter wavelengths), the imaginary parts of the permittivity tensor components can be approximated as

Then in the first nonvanishing approximation over $1/{\epsilon }_{\left|\right|,\perp }^{"}$ one can obtain from Eq. (37)

6. Conclusions

The existence of a novel class of surface plasmon polaritons at an interface between two enantiomeric uniaxial chiroplasmonic media has been shown. The waves are caused by bianisotropy and are absent in the case of bi-isotropic media. The field structure and polarization properties of the partial waves are studied. A possibility for a propagation of two distinct surface modes with different dispersion relations is predicted.

Frequency regions of existence, as well as cutoff and resonance frequencies of the waves are found, localization conditions and dispersion curves in the absence of dissipation are examined. The effect of losses on the wave’s refractive index is investigated analytically and the propagation length is determined for frequencies far enough from the plasma frequencies as well as in the vicinity of characteristical frequencies. It is shown that for visible and shorter wavelengths, the propagation length of SPP in the vicinity of cutoff frequencies becomes large enough for the observation of the surface polaritons in a possible experimental situation.