Long-range surface plasmons in dielectric-metal-dielectric structure with highly anisotropic substrates Page: 8
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PHYSICAL REVIEW B 81, 085426 (2010)
B eiqa - eika
A e-iqa - eika (25)
Normalizing the magnetic field by the condition h(0)= 1, we
obtain the following result:
h(x) = [eika sin qx + sin q(a - x)]/sin qa, - a/2 < x < a/2.
The Fourier harmonics are easily calculated from Eqs. (24)
(-1)n eia_ eiqa sin[a(q - k)/2]+
fk () - ia sin qa (eq- k- 2rn/a (eqa
eika sin[a(q + k)/2] 1 (27)
q + k + 2 rn/a J
In the long wavelength limit ka < 1 all the harmonics vanish
but the one with n= 0O
fk() (-1)n+la(q - k)2 n 0. (28)
f _ +, n + 0. (28)
Thus, the contribution of higher harmonics to the electro-
magnetic field of surface plasmon decays as a2(k-q)2. This
justifies the approximation of the effective medium theory in
calculation of the propagation length of surface plasmon Eq.
(7) propagating along a photonic crystal substrate.
We have studied the propagation of long-range surface
plasmons in a dielectric-metal-dielectric structure with
highly anisotropic substrates. We have derived the formula
for propagation length valid in a wide range of frequencies,
including the telecommunication region, and have shown
that the proper orientation of optical axis of uniaxial dielec-
tric substrates with respect to the metal surface enhances the
propagation length as well as the penetration depth of surface
plasmons. The frequency ocr [at which L(w) is independent
of the orientation of optical axis] shifts toward lower fre-
quencies for thinner metal films indicating that the parallel
orientation (ez < Ex) becomes increasingly favorable as thick-
ness d of the metal film is reduced. We demonstrated an
important general property of long-range surface plasmons
that the propagation length tends to zero close to resonance
frequency ws not because of increase in dissipation but due
to the compensation of electromagnetic energy fluxes in the
metal and in the dielectrics. This is a direct consequence of
dispersion resulting from the negative value of the dielectric
constant of metal. We have also shown that the propagation
length for symmetric configuration is one order of magnitude
greater than the propagation length for asymmetric configu-
ration. This enhancement in propagation length can cut the
limitation on the size of photonic chip or component of op-
tical circuit containing plasmonic structure. We proposed a
simple, analytically solvable Kronig-Penney model for plas-
monic crystal. The obtained dispersion equation has a band
structure. In the long-wavelength limit this equation is re-
duced to the equation obtained for surface plasmon propa-
gating along a homogeneous anisotropic substrate. This re-
sult justifies application of the homogenization procedure for
surface plasmon. We show that in the case of 1D periodic
substrate the contribution of higher harmonics to electromag-
netic fields vanishes in the long-wavelength limit. A specific
feature of the Kronig-Penney model for plasmonic crystal is
independence of the resonant frequency of surface plasmon
of dielectric properties of the substrate. This is due to zero
filling fraction of the dielectric component.
This work was supported by the U.S. Department of En-
ergy under Grant No. DE-FG02-06ER46312.
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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/m1/8/: accessed January 17, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT College of Arts and Sciences.