Photonic band structure of silver nanorod waveguide
using embedding method
N A Giannakis1, J E Inglesfield2, P A Belov1, Y Zhao1, and Y Hao1
11
Department of Electronic Engineering, Queen Mary University of London, Mile End
Road, London, E1 4NS, United Kingdom,……
2
School of Physics and Astronomy, Cardiff University, The Parade, Cardiff, CF24 3AA
Abstract
In this work we calculate the photonic band structure of an infinite line array of long silver nanorods, using
the embedding method. The boundaries between the array of nanorods and free space is treated exactly by
the embedding method. The calculation of the dispersion diagram reveals branches of longitudinal and
transverse guided modes as well as flat plasmon bands. At lower frequencies a phase matching occurs
between the transverse modes and the light in free space. Flat bands are observed around the single rod
resonance frequency.
1. Introduction
A plasmonic waveguide consists of a chain of metallic particles (like Au, Ag), and the
energy of light is propagated along the array, due to the plasmon near-field interaction
between the particles. The lateral confinement of the propagating wave is below the
diffraction limit  / 2n (where n is the refractive index and  is the wavelength). This
makes it a good candidate for a sub-wavelength light guide. Theoretically this light
waveguide was first introduced by Quinten et al [1]. Maier and et al subsequently
fabricated and did experimental measurements on such a guide [3]. A number of authors
have calculated the dispersion diagram of a metallic chain of spheres using Mie theory
[1], a quasistatic approximation [2], as well as point dipole approximations [4]. Here we
will use the embedding method, which is a full wave solution, for an infinite onedimensional array of silver nanorods, with infinite length, for TE polarization (magnetic
field parallel to the rods).
2. Conceptual description of the model
The embedding method [5, 6] is a plane wave method, which calculates the region of
interest by applying an embedding potential along the boundary surface. Using this
method we solve the problem in region I (see figure 1), which is in between the inner
boundary of unit cell and the outer surface part of cylinder, by defining the embedding
potential operator along the vacuum and cylindrical boundaries.
vacuum
Periodic boundary
I
Periodic boundary
y
II
vacuum
x
Figure 1, Unit cell with periodic and vacuum boundaries.
In region that we solve the problem, region I, the dielectric constant is a constant value
(   1), and eventually we do not need to expand the dielectric constant, like in classic
plane wave methods, and this makes the embedding method more efficient.
By defining the Green function of region I [5], which is expanded in terms of electric field
basis functions of region I, we can calculate the (photonic) local density of states (LDOS)

 ( r , ) 
2

 
ImG I ( r , r ;  i )
(1)
and density of states (DOS)


nI ()   d r  ( r ,)
I
(2)
in region I. By definition the LDOS is the number of photon states per unit volume per
unit frequency and DOS is the number of photon states per unit frequency. We notice
that the LDOS is proportional to the electric field intensity [8].
In our calculations, we assume that the angular frequency and wavevector are
~
implemented in reduced units,   a / 2c , k y  k y a / 2 respectively. The lattice
~  a / 2 ) and the radius is   25nm ( ~  a / 3 ). The dielectric
constant is a  75nm ( a
constant of metallic rods is expressed by the Drude dielectric constant,
where 
 2p
,
 ( )  1 
(  i )
is the damping constant. The plasma angular frequency for silver is
 p  6.18eV ( ~ p  0.37383 ) and we set   0.005 which corresponds to an almost
lossless metal (for simplicity).
3. Results
The DOS converges for 140 plane waves and it is plotted for different wavectors
~
ky .
The frequency peak points of DOS (fig.2) construct the photonic band structure (3.1a). In
the photonic band structure the lower branch is the transverse guided modes, where at
lower frequencies occurs a phase matching between transverse modes and the light
200
180
160
140
DOS
120
100
80
60
40
20
0
0.00
0.05
0.10
0.15
0.20
0.25
0.30

Figure 2, DOS versus frequency for wavector
~
k y  0.33 . The first two peak values correspond
to transverse and longitudinal modes respectively. The next two peaks are plasmon modes.
line. At higher frequencies the group velocity changes smoothly and beyond 0.415 the
group velocity becomes zero. The next upper branch is a longitudinal guided mode. In
those modes the zero group velocity appears from frequency 0.458. The flat bands
regime are plasmon states, which is centered at the single rod plasmon frequency,
 p / 2 , in which the incident light is strongly absorbed [7,9].
0.30
0.25

0.20
0.15
0.10
0.05
0.00
0.0
0.1
0.2
0.3
0.4
0.5
ky
1)
2)
Figure 3, 1) Photonic band structure. The black lines are allowed modes of region I and the blue
line is the light line, 2) LDOS of region I for allowed frequencies of wavector
transverse, b) longitudinal, c), d) plasmon modes.
~
k y  0.381 . 2a)
The polarization of the mentioned modes can proved, by plotting the LDOS of electric
~
field at wavector k y  0.381 for different frequency states, shown in figure 3.2. Starting
from the lowest point, a strong electric field intensity is on the poles along the x direction,
corresponding to the transverse modes (fig. 3.2a). The second upper state is polarized
along the y-axis which is a longitudinal mode (fig. 3.2b). And the next two edge states of
flat bands are plasmon states and the LDOS pattern shows strong field localization on
the cylinder surface and higher order plasmon poles (fig. 3.2c,d).
4. Conclusion
In conclusion we have formalized the embedding method to silver nanorod chain. The
advantage of this method is that excludes difficult area of dispersive cylinder by applying
the embedding operator. The dispersive diagram was calculated by the DOS and the
LDOS of electric field was used to confirm the polarization of transverse, longitudinal and
plasmon modes.
References
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