Surface plasmon polaritons in enantiomeric ... - OSA Publishing

Surface plasmon polaritons in enantiomeric chiroplasmonic structures due to bianisotropy Roland H. Tarkhanyan* Institute of Radiophysics & Electronics of NAS, 378410 Ashtarack-2, Armenia *rolandtarkhanyan@yahoo.com

Abstract: A new class of surface plasmon polaritons supported in identical enantiomeric chiral plasmonic structures is presented. The waves are caused by bianisotropy and are absent in the case of bi-isotropic media as well as in anisotropic structures without a magnetoelectric coupling. The existence of two distinct modes of the surface plasmon polaritons with unusual dispersion and polarization properties is predicted. The role of losses is investigated and the propagation length of the surface waves is determined. © 2011 Optical Society of America OCIS Codes: (240.5420) Polaritons; (1601190) Anisotropic optical materials; (240.6690) Surface waves; (160.1585) Chiral media.

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Received 7 Apr 2011; revised 21 Jun 2011; accepted 7 Jul 2011; published 29 Jul 2011

1 August 2011 / Vol. 1, No. 4 / OPTICAL MATERIALS EXPRESS 742

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, z0). 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

 ξ⊥  ξL =  0 0  ∧

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0

ξ⊥ 0

0  0 . ξ|| 

(1)



In the case of reciprocal media both

ξ⊥

and ξ|| are assumed to be real, positive and

nondimensional constants smaller than 1. The corresponding tensor for the R-material is given ∧

∧

by ξ R = − ξ L . Both media are described by the same relative permittivity tensor

ε⊥ 0 ∧  ε =  0 ε⊥ 0 

0

0  0 , ε || 

(2)

where ε ⊥ ,|| = ε ⊥' ,|| + iε ⊥'' ,|| . Using Drude free electron gas model and neglecting contributions from the bound charges, we can set [15]

ε ⊥' ,|| = 1 −

ω p2 ⊥,||τ ⊥2 ,|| '' ω p2 ⊥ ,||τ ⊥,|| , , = ε ,|| ⊥ 1 + ω 2τ ⊥2 ,|| ω (1 + ω 2τ ⊥2 ,|| )

(3)

where ω p2 ⊥,|| = Ne 2 / m⊥,|| , e is the charge and N is the density of the free charge carriers,

m||, ⊥ and τ ||, ⊥ are the effective mass and relaxation time along and across the optical axis, respectively. 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 k = k0 {n, 0, iq}, where

k0 = ω / c is the free space wave number at angular frequency ω. Assuming that the relative magnetic permeability of the structure is equal to unity and time enters as a factor of the form exp(−iωt ), one can find evanescent solutions of the Maxwell equations rot H = −iω D, rot E = iω B (4a) with constitutive relations for a bianisotropic medium [16,17] ∧ i ∧ D = ε0 ε E − ξ H , c

(4b)

i ∧ B = µ0 H + ξ E , c

(4c)

decaying with increasing distance from the interface plane z = 0. In (4b), ε 0 and µ0 are permittivity and permeability of free space, c = (ε 0 µ0 ) −1/ 2 . At a given value of the tangential wave vector component k x = k0 n, in each medium can exist two evanescent partial waves with different complex values of the normal component (k z ) ± = ik0 q± , where

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Fig. 1. Geometry of the problem. The region z>0 is occupied by L-material and the region z0 (the common term exp[i (k0 nx − ωt )] is emitted) and

R R E R = E + exp(k0 q+ z ) + E − exp(k0 q− z )

(8)

in the region z> (ωτ ||, ⊥ ) −1 .

(39b)

Then in the first nonvanishing approximation over 1/ ε ||," ⊥ one can obtain from Eq. (37)

L(ω ) = n1 (ω ) / k0 [ε ||" + ζ Q0 (α||' − α ⊥' ) −2 ],

(40)

where n1 (ω ) is given by Eq. (15a). However, Eq. (40) is not applicable for frequencies close to the resonance and cutoff frequencies. Assuming that | δ | /ζ