# Long-range surface plasmons in dielectric-metal-dielectric structure with highly anisotropic substrates Page: 2

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PHYSICAL REVIEW B 81, 085426 (2010)

however, there are very limited options for variation in the

propagation length. One of them is to affect the level of

dissipation through the dielectric properties of the substrate.

The field in the substrate affects the field in the metal (and

thus the level of dissipation) through the boundary condi-

tions. Surface plasmons on anisotropic lossless substrate

were studied in Ref. 11. It was shown that the in-plane an-

isotropy gives rise to the splitting of surface plasmon by two

ordinarylike and extraordinarylike surface modes, if surface

plasmon propagates at an angle with respect to the optical

axis. The effects of anisotropy of the substrate were studied

for asymmetric plasmonic structures, i.e., for vacuum-metal-

dielectric structures. In Ref. 13, the proposed substrate was

two-dimensional (2D) photonic crystal (a periodic arrange-

ment of parallel cylinders) and in Ref. 12, the metallic film

was deposited on multilayered dielectric heterostructure

[one-dimensional (ID) photonic crystal]. In both cases an

essential increase in the propagation length has been re-

ported.

Here we consider a symmetric dielectric-metal-dielectric

plasmonic structure. The advantage of this configuration is

that it supports propagation of the long-range surface plas-

mon. The effects of anisotropy may lead to further increase

in the propagation length. Indeed, we report about 20-fold

increase in the propagation length of long-range surface plas-

mon in the symmetric configuration when compared to that

of the asymmetric (vacuum-metal-dielectric) one13 for the

same given parameters at X=1.57 lam. It is important to

mention at this point that an attempt to study the effects of

anisotropy on the propagation length was made in Ref. 14.

Unfortunately, the reported results are erroneous. The calcu-

lated propagation length turned out to be as short as

10-15 m, i.e., on the order of the classical electron radius.

Apart from this many-orders-of-magnitude error, the re-

ported tendency for the propagation length to grow toward

the resonant frequency is wrong. It is clear from the afore-

mentioned effect of vanishing of the total flux S in Eq. (1)

that even infinitesimally weak dissipation Q leads to zero

propagation length at w= ws. In addition, the authors of Ref.

14 report unreasonably strong sensitivity of the propagation

length to the level of anisotropy, taking into account rela-

tively weak anisotropy they used in their calculations.

II. CALCULATION OF THE PROPAGATION LENGTH

The symmetric configuration that we consider consists of

a metal film of thickness d clad with two identical semi-

infinite anisotropic dielectric substrates as shown in the inset

in Fig. 1. The metal film occupies the space between z= 0

and z=d (region 1). The dielectric on top of the metal film

occupies all space above z=d (region 2) and the dielectric

below the metal film occupies all space below z= 0 (region

3). The dielectrics are assumed to be isotropic in the x-y

plane, i.e., ex= y. The dielectric constant in the perpendicu-

lar to the interface direction (along z axis) is, however, dif-

ferent, ez E x.

The surface plasmon propagating along the metal-

dielectric interface is a p-polarized wave with the compo-nents of the electric field Ex and Ez and with the only com-

8 Ia Dielectric g

Setal (,,W) d

0.2 Dielectric

I Ix

0.0I

0 2 4 6 8 10 12 14

kc/m

FIG. 1. (Color online) Dispersion curves for different orienta-

tions of optical axis of the anisotropic dielectric crystals with metal

film of thickness d= 50 nm. The dotted curve represents dispersion

curve for equivalent isotropic dielectric with dielectric constant

8is= - z

ponent of the magnetic field Hy= H(x, z). The field inside the

metallic film is a superposition of two exponents, H (x, z)

=A exp(ikx+ Kz)+B exp(ikx-K1z), with Kl=k2_=Em2/c2

being the inverse skin depth, K11= . In the substrates, which

are two identical uniaxial dielectric crystals, the magnetic

field is obtained from the Helmholtz equation1 82H 1 82H (02

8X+ +- H= 0.

ez 8x2 8 xz2 C2(2)

The evanescent solutions of this equation are H2(x,z)

= C exp[ikx-K2(z-d)] and H3(x,z)= C exp(ikx+ K2z), where

K2= Ex( - j) is the inverse decay length of the surface

plasmon field in dielectric, and 8e(w)= 1- w2/ w2. From the

continuity of the magnetic field H(x, z) and electric field

Ex(x,z)=(c/ie6x)dH/dz at the interfaces z=0 and z=d, we

obtain the following dispersion equation for the surface plas-

mon:- tanh Kd

Kl1x 2(3)

This equation gives the dispersion for the mode with anti-

symmetric distribution of Ex with respect to the plane of

symmetry z=d/2, i.e., for the so-called long-range surface

plasmon. It is easy to see that interchanging the values of ex

and ez, two different dispersion equations are obtained. The

transformation ex Ez means 90 rotation of the optical axis

of the dielectric crystal with respect to the metal surface. In

Fig. 1 the solid and dashed curves are the dispersion Eq. (3)

for two different orientations. Both the curves approach the

same resonant frequency085426-2

NAGARAJ AND A. A. KROKHIN

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Nagaraj & Krokhin, Arkadii A. Long-range surface plasmons in dielectric-metal-dielectric structure with highly anisotropic substrates, article, February 22, 2010; [College Park, Maryland]. (digital.library.unt.edu/ark:/67531/metadc103273/m1/2/: accessed July 23, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT College of Arts and Sciences.