PHYSICAL REVIEW B 81, 085426 (2010)

Long-range surface plasmons in dielectric-metal-dielectric structure

with highly anisotropic substrates

Nagaraj and A. A. Krokhin

Department of Physics, University of North Texas, 1155 Union Circle 311427, Denton, Texas 76203, USA

(Received 8 October 2009; revised manuscript received 23 December 2009; published 22 February 2010)

We present a theoretical study of long-range surface plasmons propagating in a thin metallic film clad

between two identical uniaxial anisotropic dielectric crystals. We show that the proper orientation of the optical

axis of the crystal with respect to the metal surface enhances the propagation length of surface plasmons. Since

the proper orientation depends on surface plasmon frequency, we give the results for the propagation length in

a wide range of frequencies, including the telecommunication region. To increase the role of anisotropy, we

consider artificial substrates from photonic crystals, which possess an order of magnitude stronger anisotropy

than the natural optical crystals. We propose Kronig-Penney model for plasmonic crystal where the substrate is

a periodic sequence of dielectric delta peaks. In this model the dispersion relation for surface plasmon has a

band structure where the band width tends to zero when the frequency approaches the resonant frequency.

DOI: 10.1103/PhysRevB.81.085426

I. INTRODUCTION

The efficiency of plasmonic devices is limited by finite

propagation length L(w) of surface plasmon polaritons. The

main source of attenuation of propagating surface plasmon is

Joule losses in the metal. The Joule losses Q= fj . EdV are

reduced if the electric field has a mode, enforced by symme-

try, somewhere inside the metal plate. Since in a bulk con-

ductor, the field decays at the skin depth S, the effect is

noticeable if the plate thickness d does not exceed S.1 Strong

reduction in dissipation occurs if the plasmonic structure is

symmetric, i.e., the dielectrics on both sides of the plate are

the same. Then, the electric field E vanishes exactly at the

plane of symmetry, minimizing the integral Q.2 This plas-

monic mode with antisymmetric distribution of parallel-to-

the-plate component of electric field is usually called long-

range surface plasmon.3 The propagation length of this mode

scales as L- 1/d2 and may be as long as few millimeters or

even centimeters in the near infrared4 for silver films of

thickness d< 50 nm. If surface plasmon propagates along a

metal strip instead of an infinite plane, some increase in

propagation length can be reached for special choice of cross

section of the strip.5 Short propagation length limits the size

of photonic chip or component of optical circuit containing

plasmonic structure. In order to reduce dissipation in plas-

monic waveguides, it was proposed to groove V-shaped

channels in metal.6 These channels support propagation of

long-range surface plasmons and allow experimental realiza-

tion of interference, splitting, and switching of surface

waves.7

The most interesting features of surface plasmon polariton

are manifested at frequencies close to the limiting frequency

Ws in the spectrum of surface plasmon w= cw(k). For the case

of isotro ic substrate, this frequency is given by as

= w/i +e, where op is plasma frequency of the metal and e

is the dielectric constant. Close to the resonant frequency, the

surface plasmon density of states dk/dcw grows infinitely,

leading to the enhancement of light emission from quantum

semiconductor structures.8 Also the subwavelength reso-

lution of plasmonic devices is strongly enhanced near the

PACS number(s): 42.70.Qs, 41.20.Jb, 42.25.Lc

resonant frequency since the penetration depth into the di-

electric vanishes at w= ws.9 Unfortunately, the propagation

length L(w) quickly tends to zero near ws. Because of this

property, any plasmonic device cannot operate in the fre-

quency region close to surface-plasmonic resonance. It is

worthwhile to discuss here the physical reasons for such

strong decay.

Propagation length, being the distance at which the energy

of the wave decays by a factor of e, can be expressed through

the rate of dissipation Q and flux of energy S(w) as follows:

S(w)

L(Q(w))

- 9 ) *

(1)

It is clear that the denominator Q, while grows smoothly

with cw, remains finite at any frequency. Fast decay of surface

plasmon is due to vanishing of the energy flux S at w= w .

The total flux S associated with propagating surface plasmon

is a sum of two terms, S= 2Sd+ Sm. Here Sd and Sm are Poyn-

ting vectors in one of the dielectrics and in the metal, respec-

tively. Since the dielectric constant of the metal film is nega-

tive, em(o)< 0, the energy in the metal and in the dielectrics

flows in opposite directions, i.e., Sd>O and Sm <0. While

the total flux S is positive [surface plasmon is a wave with

normal dispersion, dw/dk > 0, if the metal (silver) film

thickness is not less than 30 nm], the interior of the metal

gives negative contribution. It is easy to derive that

2Sd+ Sm a (() - E2. The resonant frequency is obtained

from the equation em(os)+ e= 0. Therefore, the net flux S(w)

vanishes linearly at w= ws. This simple calculation shows

that the main reason for strong decay of surface plasmon

near the resonant frequency is related to its dispersion but

not to dissipation. For frequencies near ws, the most efficient

way to increase the propagation length is detuning from the

resonance, which affects the numerator in Eq. (1). Thus, the

most attractive for applications region of frequencies is un-

reachable because of very short propagation length.

At low temperatures, the Joule losses are strongly reduced

due to the decrease in denominator in Eq. (1), leading to

increase in the propagation length.1'0 At room temperatures,

2010 The American Physical Society

1098-0121/2010/81(8)/085426(9)

085426-1