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
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
k2- 1 o .
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-
2 (1 + o)m()I[1+ E m()I]
c2 27 0m (1 )
0 1............. ............. .........
0 5 10 15 20 25
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
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
h(x) = exp(ikx)fk(x) = exp(ikx)1 fk(n)exp(2rrinx/a).
The distribution of the field h(x) within a period
-a/2<x<a/2 is obtained from Eq. (14), where the ratio
In 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.
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.