Long-range surface plasmons in dielectric-metal-dielectric structure with highly anisotropic substrates Page: 6
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PHYSICAL REVIEW B 81, 085426 (2010)
E
, 1.
a
0 1.
0
o_
.0
2
0.
40_
Q)
a)0.18 0.19 0.20 0.21 0.22
0/0
S
FIG. 7. (Color online) Penetration depth of
different orientations of dielectric crystal witl
ness d=50 nm.6(x) = 1 + eoa, S(x- na), z > 0.
In the long-wavelength limit this substrate behaves as a
uniaxial optical crystal with effective dielectric constantsez = 6y - e(x)dx = 1 + o,
a -a/2
1 1 a/2 1
6xa ( - a dx = 1.
ex a -a/2 8(xThe magnetic component H(x, z) of surface plasmon field
decays exponentially away from the interface: in the metal
(z <0), H(x,z) =exp(Klz- iwt)h(x) and in the substrate
0.23 0.24 0.25 (z>0), H(x,z)=exp(-K2z-iw(t)h(x). The continuity of
H(x, z) at z=0 is satisfied and within the interval
-a/2<x<a/2 the function h(x) is a superposition of two
surface plasmons for plane wavesh metal film of thick-
h(x) = Aeiqx + Be-qx,
where
enization of metamaterials. Being technically different, how-
ever, they lead to the equivalent results for the effective di-
electric constants in the low-frequency region away from
internal resonances, see, e.g.22 It is not clear, however,
whether the "bulk" effective dielectric constants can be used
in the case of surface modes which are strongly inhomoge-
neous in the direction perpendicular to the interface. This
inhomogeneity may also enhance the contribution of higher
harmonics to the dissipation and, thus, reduce the propaga-
tion length in Eq. (5). The latter was calculated assuming that
the substrate is a homogeneous anisotropic dielectric, i.e., all
the spacial harmonics of the electromagnetic field of surface
plasmon have been ignored. To clarify the possibility of ho-
mogenization of plasmonic crystal and evaluate the contribu-
tion of higher harmonics, we consider here a simple Kronig-
Penney model. Usually, a periodic structure in a plasmonic
crystal is associated with a periodic dielectric pattern on the
surface of a metal film.23 Here we consider a periodic ar-
rangement of infinitesimally thin dielectric sheets parallel to
axis y and oriented perpendicular to semi-infinite metal, see
Fig. 8. The dielectric constant of this structure can be repre-
sented as a series of delta peaksq2 = K -I(/C)2m() = K + (W/C)2.
The values of h(x) on the whole axis x are obtained from the
Bloch theorem, h(x+a) =exp(ikx)h(x). h(x) is a continuous
function at x=0 but its derivative has a discontinuous jump2
h'(+ 0) - h' (- 0) = - boa-2-h(0).
c"The continuity of h(x) at x=0 together with Eq. (16) form a
set of homogeneous equations for A and B. Equating its de-
terminant to zero, the following dispersion equation is ob-
tained:
coa2 2 sin qa
cos ka = cos qa- 2 (17)
2 c2 qa
In this equation the wave vector q is a function of frequency
w. The dependence q(w) is obtained from the continuity of
the tangential electric field Ex=-(ic/ w6) 8H/l z at z=0. This
condition givesK1 = m() K2.
Equations (15) and (18) determine the frequency dependen-
cies K1(W), K2(Cw), and q(w)FIG. 8. Kronig-Penney model for plasmonic crystal.
2 2(0(2
2C (W -1'
C1 C2 1'
q 2
C2 Em(O) - 1(19)
(20)
(21)Since 18m((o) > 1, the following inequality holds,
K2 < q< K1. When w- p/ 2 all three parameters K1, K2,085426-6
(11)
(12)
(13)(14)
(15)(16)
(18)
NAGARAJ AND A. A. KROKHIN
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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/m1/6/: accessed October 21, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT College of Arts and Sciences.