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
