Long-range surface plasmons in dielectric-metal-dielectric structure with highly anisotropic substrates Page: 3
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LONG-RANGE SURFACE PLASMONS IN DIELECTRIC-...
t)p
s- l
1 + 6~(4)
C m 1m (2 - ,x6z)32(68m1 + )1/2
L(t) - 2 6
C," Em ,Z (Ea + exEz + 2Ez 1Em )(7)
at k- oc. Since (s is symmetric with respect to ex and ez, the
equivalent isotropic dielectric constant may be defined as
eis= E . The dispersion curve for the structure with
equivalent isotropic dielectrics (dotted line) lies in between
the solid and the dashed lines in Fig. 1. The dispersion
curves are plotted for silver film of thickness of d= 50 nm.
The numerical values of the principal dielectric constants are
taken to be 2 and 7.5. These are the homogenized values
obtained in the low-frequency limit for photonic crystal of Si
cylinders arranged in a square lattice with Si filling of 57%.15
The larger value 7.5 corresponds to the dielectric constant
along the cylinders. In the plane of periodicity, the photonic
crystal is isotropic with dielectric constant equals 2.
The asymmetry of the dispersion Eq. (3) with respect to
the interchange ex - Ez gives rise to different propagation
lengths for two different orientations. Since only the case of
weak dissipation is of practical interest, we consider the re-
gion of frequencies where the imaginary part of the dielectric
function of the metal film em()= E'(Wo)+ie"(wO) is small,
i.e., e" e'. For a typical metal this condition is valid in the
infrared and visible regions. In weakly dissipative medium,
the wave vector acquires small imaginary part, k=k' +ik".
Expanding the dispersion Eq. (3) over e" and k" and keeping
the linear terms, the following result for the propagation
length L(w)= 1/2k" is obtained:
L() 1 x - + d K- -MeK2
28 K2 z K1 Ex K1 Im ,
2 2 d m K Ex -1
X 1 2K2 + K2
m2c2K 4c2K2 Ex K2
(5)
Apart from the factor 1 / e", all the quantities on the right-
hand side (rhs) are calculated for a lossless metal, e"=0. The
propagation length L(w) -0 when w-+ s. As it was men-
tioned in the introduction, this effect is due to surface plas-
mon dispersion (vanishing of the group velocity) but not due
to increase in dissipation. The dispersion equation Eq. (3)
and the propagation length Eq. (5) can be simplified in the
limiting case of thick film.
The limit d--+ oc is readily obtained from Eqs. (3) and (5).
In this limit the surface plasmons propagating along two
metal-dielectric interfaces do not interact with each other,
therefore the dispersion relation is reduced to the result ob-
tained in Ref. 3 for semi-infinite metal
k- 6 6m12 ). (6)
C \mm- x6/z
Now, using Eq. (6), we obtain from Eq. (5) the following
result for the propagation length:Of course, in the case of isotropic dielectrics, ex= Ez, Eqs.
(3)-(8) are reduced to the well-known results.3,16-21
The calculation of propagation length in the limit when
d oc and in the limit when w - ws can both be considered
equivalent because at frequencies close to resonant fre-
quency os, the decay length K21 -0 (since the surface plas-
mon wave vector k- oc). For the evanescent field of surface
plasmon, this has the same effect as having a semi-infinite
metal film. Consequently, the propagation length
L(w) -- (w- )3/2 for a metal film of any given thickness
close to resonant frequency, as one obtains from Eq. (7).
In the opposite limit of thin film, d--+ 0, Eq. (5) is simpli-
fied in the long-wavelength regime, i.e., when surface plas-
mon behaves like a transverse light wave2 (c)3 E 3
L(w) - 2 3/2
S d2 Ex83/2(I m zm +(8)
In this approximation the dispersion relation w = kc/ 6 does
not contain characteristics of the metal but the propagation
length does. It follows from Eq. (8) that for fixed w, the
propagation length scales as d-2, which is a signature of the
long-range surface plasmon. Analysis of the long-wavelength
limit shows that the favorable orientation in the limit of thin
film (d 0) is not the same as the one in the case of a
semi-infinite film (d- oc). In the limit of thin film (for a
given frequency), [L(w)]-lo ex3/2, while in the case of a
semi-infinite film, [L(w)]- ex8 2. There is a certain critical
frequency Wcr (for a given thickness d of metal film) at
which the favorable orientation reverses (see Fig. 2). The
dependence of critical frequency on the thickness d of the
metal film is discussed in more detail later in this paper.
Further examination of asymptotes in the linear dispersion
regime (w= kc/ 6V) reveals some interesting details. The as-
ymptote in the limit of thin film (kd C 1) is given by Eq. (8),
and in the limit of thick film (kd > 1), we obtain from Eq. (5)(9)
L(w) ~ (m-6 kd > 1.
exmk (6mI + 2z)The linear dispersion regime in the limit of very thin metal
film spans over a broad range of frequencies, while in the
case of very thick metal film, the linear dispersion regime is
limited only to low frequencies (see Fig. 3). Although very
thin film favors a much longer propagation length (see Figs.
2 and 4), it is relatively more difficult to reach the resonance
frequency w (see Fig. 3). In addition to this, the surface
plasmon density of states dk/dw for very thin films is rela-
tively very low (due to the greater slope dOw/dk of the dis-
persion curve), leading to the decrease in probability of light
emission. We note that in the limit of very thick metal, the
form of Eq. (9) is different from Eq. (7), because, linear
dispersion does not span over a broad frequency range as
compared to linear dispersion in the limit of very thin metal.
However, at very low frequencies (in the long wavelength
regime), both Eqs. (7) and (9) are reduced to085426-3
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/m1/3/: accessed December 14, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT College of Arts and Sciences.