# Long-range surface plasmons in dielectric-metal-dielectric structure with highly anisotropic substrates Page: 7

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LONG-RANGE SURFACE PLASMONS IN DIELECTRIC-...

0 2 4 6 8 10

at/c

FIG. 9. The first four allowed bands for a plasmonic crystal of

Si semispace and periodic substrate with Eo=3.

and q diverge. As one can see from Eq. (17) the Bloch vector

k also formally diverges. The frequency s= op/ is the

limiting frequency in the dispersion Eq. (17). It is the reso-

nant frequency of the plasmonic crystal. The resonant fre-

quency is independent on the details of the substrate since in

the Kronig-Penney model the filling fraction of the dielectric

component is exactly zero. Near the resonance the wave-

length 2wrr/k vanishes, therefore the field of the surface plas-

mon is insensitive to the presence of the dielectric sheets,

which are separated by a finite distance a, i.e., by infinite

number of wavelength. In the case of nonzero filling fraction

of the dielectric component the resonant frequency depends

on the properties of the substrate.24

The structure of the low-frequency bands of Eq. (17) is

similar to that of any periodic system. Few first bands are

shown in Fig. 9. Unlike standard Kronig-Penney model, the

width of the allowed bands decreases with frequency. Near

the resonant frequency the spectrum becomes pointlike.

Zero-width frequency bands are accumulated toward wp/2.

Here the spectrum becomes dense and asymptotically it co-

incides with the spectrum of surface plasmon near its reso-

nant frequency. This tendency is clearly seen in Fig. 10.

Long wavelength limit. In the limit ka,qa C 1 Eq. (17) is

simplifiedk2- 1 o .

(22)

8

Dispersion of surface plasmon with anisotropic homoge-

neous substrate is given by Eq. (6). Substituting in Eq. (6)

the effective dielectric constants of the substrate given by

Eqs. (12) and (13) the following dispersion equation is ob-

tained:2 (1 + o)m()I[1+ E m()I]

k =

c2 27 0m (1 )(23)

a 3-

. 2

0 1............. ............. .........

01

CN

0

ca -2

o -3

0

0 5 10 15 20 25

awlc

FIG. 10. The rhs of Eq. (17) vs frequency. The band edges are

the intersections of the curve with two horizontal lines at the level

+1.

11. In fact, the dispersion curves in Fig. 11 are very close to

each other not only in the low-frequency limit but within a

wide range of frequencies. This means that the substrate can

be considered as an effective homogeneous medium once the

conditions qa, ka C 1 are satisfied. Two dispersion curves ex-

hibit essential difference only near the resonant frequencies

wp/2 and ws=wp/1 +0=wp/2. Here q,k--oo and the dis-

persion Eq. (22) is not valid since the conditions qa, ka ~ 1

cannot be satisfied.

Contribution of higher harmonics. The contribution of

higher harmonics is evaluated from the Fourier expansion of

magnetic field

h(x) = exp(ikx)fk(x) = exp(ikx)1 fk(n)exp(2rrinx/a).

n

(24)

The distribution of the field h(x) within a period

-a/2<x<a/2 is obtained from Eq. (14), where the ratiokc/c

sIn the limit w 0 Eqs. (22) and (23) have the same linear

asymptotic, k=(wi/c) 1l+eo. This means that in the low-

frequency limit the Kronig-Penney plasmonic crystal ho-

mogenizes and the effective dielectric constants coincide

with those obtained for 1D photonic crystal. Two dispersion

curves corresponding to Eqs. (22) and (23) are shown in Fig.FIG. 11. (Color online) Dispersion curves for plasmonic crystal

with so=3 in the limit ka,qa 1 [Eq. (22), solid line] and for

plasmonic structure with a homogeneous dielectric substrate with

ex= 1 and z= 1 +s0 [Eq. (23), dashed line]. The inset shows the

region of low frequencies. The frequency is normalized to

w =wl l +0= %p/2.085426-7

PHYSICAL REVIEW B 81, 085426 (2010)

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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/m1/7/: accessed July 26, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT College of Arts and Sciences.