Research Journal of Applied Sciences, Engineering and Technology 4(15): 2422-2426, 2012
ISSN: 2040-7467
© Maxwell Scientifc Organization, 2012
Submitted: February 13, 2012
Accepted: March 08, 2012
Published: August 01, 2012
Theoretical Investigation of Discrete Mode in Waveguiding Plasmonic Structures
P. Azimi Anaraki
Department of Physics, Takestan Branch, Islamic Azad University, Takestan-Iran
Abstract: In this study, we are interested in studying the refraction of a surface polariton by a vertical interface
between two different surface active dielectric media. In this study, the interaction volume is limited by placing
two shorting plasmons above and below the surface on which propagation take place. These planes are perfect
conductors that force the tangential components of the electric field to vanish on their surfaces. The resulting
structure is a closed waveguide that the electromagnetic modes of this waveguide are discrete.
Keywords: Attenuation modes, electromagnetic modes, plasmonic wave guide, propagation modes, surface
plasmons
INTRODUCTION
Study of surface plasmon polaritons became a rapidly
developing field of nanophotonics that is expected to
enhance functionalities of the key optical components for
information processing on a photonic chip, providing light
confinement at subwavelength scale and overcoming the
diffraction limit (Maire, 2007; Gramotnev and
Bozhevolnyi, 2010). Some plasmonic elements are metaldielectric waveguides that support surface plasmon
polaritons. Analysis of the modes in plasmonic
waveguides is one of the most important problem which
was addressed in many studies, including the modes in
circular (Chen et al., 2009; Novony and Hafner, 1994)
and planar waveguides (Kocabas et al., 2009; Dionne
et al., 2005; Satuby and Orenstein, 2007; Feng and Negro,
2007; Yang and Crozier, 2009). To date, the properties of
guided modes in plasmonic waveguides are studied in two
limiting cases:
C
C
Ideal structures without losses (Davoyan et al., 2010)
Structures with real losses in metals
In conventional approach (Ibanescu et al., 2004,
2005), which is usually employed in optics for the
analysis of dielectric waveguides (Snyder and Love,
1983; Zhang et al., 2010).
On the other hand, the study of such interactions is of
interest both for basic physics reasons and for possible
applications of the results in the design of all-optical,
integrated optical devices. Since the electromagnetic field
of a surface plasmon polariton is confined to the near
vicinity of the surface supporting it, it can be used as an
experimental probe of the properties of the transition layer
between vacuum and the bulk of the substrate or of thin
films deposited on the substrate (Jablonski, 1994; Sturman
et al., 2007).
The refraction of guided wave polaritons incident on
the interface between two waveguides or the end face of
a waveguide is also of interest in guided wave technology.
With the recent interest in bistability and similar alloptical operations in integrated optic structures it has been
realized that the guided wave end face reflectivity
determines the finesse of a guided wave cavity and hence
the critical power required for switching. A different
approach to the problem of the refraction of a surface or
guided wave polariton by material and/or geometrical
discontinuities in its path has been taken by the others
(Sterke et al., 2009; Feigenbaum et al., 2009).
In this study we will obtain discrete modes of the
closed structure consisting of a metal characterized by a
dielectric function g(T). This structure consists of two
closed waveguides of the type depicted in Fig. 1. By
calculating these modes we can show the time averaged
power flow is directed in the waveguide.
MODAL EXPANSIONS
In the study of the refraction of a surface plasmon
polariton by a vertical interface between two different
surface active dielectric media, characterized by isotropic,
frequency-dependent, dielectric functions g1(T) and g2(T),
which are both negative at the frequency T of the surface
polariton, the open structure depicted in Fig. 1a is
replaced by the closed waveguide structure depicted in
Fig. 1b. The latter structure consists of two closed
waveguides of the type depicted in Fig. 2 joined at the
plane x1 = 0.
The waveguide modes in the structure of Fig. 2 that
describe a p-polarized wave propagating in the x1 direction are:
2422
H2 x1 , x 3 ω =
cosh α 0 (d − x 3 )
cosh α 0 d
e iβ x1 , 0 ≤ x 3 ≤ d
(1)
Res. J. Appl. Sci. Eng. Technol., 4(15): 2422-2426, 2012
Fig. 1: (a) An open structure consisting of a metal characterized by a dielectric function g1, (T) in contact across a vertical
interface with a second metal characterized by a dielectric function g2, (T), when both metals have a common interface
with vacuum, (b) The closed waveguide structure that replaces theopen structure depicted in, (a)
Fig. 2: A closed waveguide of the kind used in creating the structure depicted in Fig. 1b
H 2 x1 , x 3 ω =
E1 x1 , x3 ω =
cosh α (d + x 3 )
cosh αd
e iβx1 , − d ≤ x 3 ≤ 0
icα 0 sinh α 0 (d − x 3 ) iβx1
e , 0 ≤ x3 ≤ d
ω
cosh α 0 d
icα sinh α (d + x 3 ) iβx1
E1 x1 , x 3 ω =
e , − d ≤ x3 ≤ 0
cosh αd
ωε (ω )
cβ cosh α 0 (d − x3 ) iβx1
e , 0 ≤ x3 ≤ d
ω
cosh α 0 d
E 3 x1 , x3 ω =
E 3 x1 , x 3 ω = −
cβ cosh α (d + x 3 ) iβx
e , − d ≤ x3 ≤ 0
cosh αd
ωε (ω )
1
(3)
(4)
(5)
(6)
⎛
⎝
ω2 ⎞
⎟
c ⎠
2
1/ 2
, Re α 0 > 0, Im α 0 < 0
(7)
ω2 ⎞
⎟
c ⎠
2
1/ 2
, Re α > 0, Im α < 0
(8)
For a given value of the frequency T the allowed
values of $ for which the modes (1)-(6) are defined are
the solutions of the dispersion relation:
ε (ω )
where:
α 0 (β , ω ) = ⎜ β 2 −
⎛
⎝
α ( β , ω ) = ⎜ β 2 − ε (ω )
(2)
α 0 ( β ,ω )
tanh α ( β ,ω )d
=−
α ( β ,ω )
tanh α 0 ( β ,ω )d
(9)
The solutions of this equation are discrete and are
labeled by an index m = 0, 1, 2, ….They are complex in
general, $ = $R + i$I , where $R > 0, $I >0 ,for a wave that
propagates in the +x1 -direction, or decays exponentially
with increasing +x1 .The modes are ordered according to
decreasing $R with $I = 0 (which corresponds to
propagating modes) until $R = 0. They are then ordered
according to increasing $I (which corresponds to
evanescent modes). The value of $ corresponding to the
surface polariton is real and is denoted by $0.
The remaining solutions of Eq. (9) describe modes of
several types. These include modes which, in the present
closed waveguide structure, are standing waves in the
coordinate x3 normal to the surface of propagation in the
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Res. J. Appl. Sci. Eng. Technol., 4(15): 2422-2426, 2012
region 0 < x3 < d of the waveguide filled with a dielectric
medium whose dielectric constant is positive. Hence in
these modes energy is being returned to the surface of
propagation from the region of the perfectly conducting
planes as well as flowing away from it. A similar
necessity of including incoming as well as outgoing
waves in obtaining the normal modes of open waveguides
has been noted (Davoyan et al. (2011). Finally, the
electromagnetic modes of our waveguide include some
that decay exponentially in either the +x1 -or –x1directions.
If it is assumed that the surface polariton is incident
on the interface x1 = 0 from the region x1 < 0, the
components of the electromagnetic field in each of the
regions x1 < 0 and x1 > 0 can be expanded in the
corresponding waveguide modes. Thus, in the region
x1 < 0 we have:
⎧ H (10) x , x ω + rH (10) x , x ω + ⎫
1
3
2
1
3
⎪ 2
⎪ −iωt
H2< ( x; t ) = ⎨
⎬e
(1m )
ω
R
H
x
,
x
∑
2
1
3
m
⎪
⎪
⎩ m ( > 0)
⎭
(12)
⎧
E1> ( x; t ) = ⎨ tE1( 20) x1 , x3 ω +
⎩
∑T E
m( >0)
⎧
E3> ( x; t ) = ⎨ tE3( 20) x1 , x3 ω +
⎩
⎫
H 2( 2 m) x1 , x3 ω ⎬ e − iωt
⎭
( 2 m)
1
m
∑T E
m( > 0)
m
⎫
x1 , x3 ω ⎬ e −iωt
⎭
( 2 m)
3
⎫
x1 , x3 ω ⎬ e −iωt
⎭
m
( 2 m)
2
H2(1m) 0, x3 ω
0, x3 ω , − d < x3 < d
(16)
∑R
m( >0)
m
E3(1m) 0, x3 ω
∑TE
m( > 0 )
m
( 2 m)
3
0, x3 ω , − d < x3 < d
(17)
RESULTS
While in the region x1 < 0 we have:
m
∑TH
m( > 0)
m
and
= tE3( 20) 0, x3 ω +
⎧ E (10) x , x ω + rE (10) x , x ω + ⎫
1
3
3
1
3
⎪ 3
⎪ − iω t
E 3< ( x; t ) = ⎨
⎬e
(1m)
ω
R
E
x
,
x
1
3
⎪∑ m 3
⎪
⎩ m( > 0)
⎭
∑T
= tH2( 20) 0, x3 ω +
∑R
m( >0)
(10)
(11)
m( > 0 )
H2(10) 0, x3 ω + rH2(10) 0, x3 ω +
E3(10) 0, x3 ω + rE3(10) 0, x3 ω +
⎧ E (10) x , x ω + rE (10) x , x ω + ⎫
1
3
1
1
3
⎪ 1
⎪ − iω t
E1< ( x; t ) = ⎨
⎬e
(1m)
R
E
x
,
x
ω
∑
m
1
1
3
⎪
⎪
⎩ m( > 0 )
⎭
⎧
H 2> ( x; t ) = ⎨ tH 2( 20) x1 , x 3 ω +
⎩
Eq. (13)-(15) represents the transmitted surface polariton,
while the second represents all the transmitted modes
other than the surface polariton.
The coefficients in these expansions are obtained
from the Maxwell boundary conditions at the interface
x1 = 0, which require the continuity of the tangential
components of the magnetic and electric field, viz. of
H2(x;t) and E3(x;t), across this interface. The equations
expressing these conditions are:
(13)
(14)
(15)
In these expansions the waveguide modes have been
labeled by a double index (im), where i = 1, 2, indicates
which dielectric function, g1(T) or g2(T) , respectively,
appears in the dispersion relation (9), while m labels the
solution for $ for a given g(T). In addition, modes for
which $ are replaced by - $, corresponding to reflected
waves, are denoted by a bar over the symbols of the
corresponding field components.
Thus, the first term on the right hand side of each of
Eq. (10)-(12) represents the incident surface polariton; the
second term represents the reflected surface polariton.
Similarly, the first term on the right hand side of each of
The mathematical necessity for including waveguide
modes other than those corresponding to the incident,
transmitted and reflected surface polaritons in solving the
problem of the refraction of a surface polariton by a
transverse discontinuity is due to the impossibility of
satisfying the boundary conditions at the transverse
boundaries of the refracting system by the use of the
electromagnetic fields of the surface polaritons on the two
sides of the boundaries alone. This is caused by the
different decay lengths of the surface polaritons in the
directions normal to the plane of propagation. An
admixture of the remaining types of waveguide modes is
needed to satisfy these boundary conditions. Physically,
the presence of volume modes in the expansion of the
electromagnetic field in each region of the refracting
system corresponds to the conversion of a portion of the
energy carried by the incident surface polariton into
volume modes when the surface polariton strikes a
transverse discontinuity.
The method generally used to solve a system of
equations of this type was the collocation method. In this
method these equations are required to be satisfied at a
discrete set of N equally spaced values of x3 in the
interval (!d, d) along the x3 -axis. This requirement leads
to a set of 2N equations in the case of Eq. (16)-(17). The
first N modes are used in the expansion of the field
components on each side of the interface. Thus, a set of
2N unknowns has to be solved for the coefficients, r, t, Rm
and Tm. The number of points N is increased until
convergence of the solution is achieved. The convergence
criterion used was that the energy in the system be
conserved, which can be formulated in the following way.
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Res. J. Appl. Sci. Eng. Technol., 4(15): 2422-2426, 2012
The time averaged power flow in the x1-direction per
unit width in the x2-direction in any waveguide mode is
given by:
ω /ωp (variable)
ωP /ω = 1.5
ωP /ω = 1.5
ωp / ω (variable)
1.0
d
c
Re ∫ dx 3 E 3 x1 , x 3 ω H 2* x1 , x3 ω
−d
8π
0.9
(18)
This quantity is nonzero only for those modes for
which the corresponding $ is real. This result can be used
to obtain the reflection and transmission coefficients for
the surface polariton. These are obtained by normalizing
the power flow per unit width in the reflected and
transmitted surface polaritons by that in the incident
surface polariton. In this way we obtain the results:
∫
∫
(
2
Rsp = r
d
−d
d
−d
∫
∫
d
(
2
Tsp = t
−d
d
−d
dx 3 E 3(10) x1 , x3 ω H 2(10) x1 , x 3 ω
*
dx 3 E 3(10) x1 , x3 ω H2(10) x1 , x3 ω
*
dx3 E3( 20) x1 , x3 ω H2( 20) x1 , x3 ω
dx3 E
(10 )
3
x1 , x3 ω H
(10 )
2
x1 , x3 ω
= r
(
2
Rm = Rm
∫
∫
−d
d
−d
∫
∫
d
(
2
Tm = Tm
−d
d
−d
dx3 E3(1m) x1 , x3 ω H2(1m) x1 , x3 ω
*
dx3 E3(10) x1 , x3 ω H2(10) x1 , x3 ω
*
dx3 E3( 2 m) x1 , x3 ω H2( 2 m) x1 , x3 ω
*
dx3 E3(10) x1 , x3 ω H2(10) x1 , x3 ω
*
m
(
m
0.4
Vacuum
mode
transmission
0.2
0.0
1.5
2.0
2.5
3.0
3.5
ωP /ω
4.5
5.0
2πc
ω
frequency Tp characterizing the two metals. Two
situations were considered. In the first, it is assumed that
the frequency T of the surface polariton incident on the
interface x1 = 0 is such that
.
4.0
polaritons for d = d0 = 1 in units of
(20)
ωp
= 15
.
ω
where Tp is the plasma
(21)
frequency of the metal in the region x1 = 0. The plasma
frequency of the metal in the region x1 > 0 is
assumed to be variable and to increase in such a way
(22)
that
nonzero only for those modes on either side of the
interface x1 = 0 for which the corresponding $m is real.
Since we are dealing with lossless dielectric media the
requirement of energy conservation is that all of the
energy in the incident surface polariton is converted into
the energy carried by the reflected and transmitted surface
polaritons and by the reflected and transmitted volume
modes. This condition can be expressed as:
∑T
Surface
polarion
transmission
0.5
Fig. 3: The energy transmission coefficient for surface plasmon
(
(
It should be kept in mind that Rm and Tm are
(
(
(
Rsp + Tsp + ∑ Rm +
0.6
0.3
(19)
*
*
0.7
0.1
2
The coefficients of converting the incident surface
polariton into reflected and transmitted volume modes are
defined in the same way:
d
Total
transmission
0.8
Transmission coefficient
P1 =
ωp
ω
increases from 1.5 to 5 while T is kept fixed. It is
seen from Fig. 3 that the transmission coefficient of the
ω
surface polariton decreases as the mismatch ∆ ⎡⎢ ω ⎤⎥ between
p
⎣
⎦
the surface polariton fields on both sides of the transverse
discontinuity increases. In the second case considered the
surface polariton of frequency T was incident on the
interface from the region x1 < 0, in which the plasma
frequency Tp was variable, so that the ratio
ωp
ω
increased
from 1.5 to 5. The plasma frequency of the metal in the
region x1 > 0 was kept constant in such a way that the
ratio ωω was fixed at a value of 1.5. In this case the
p
=1
(23)
transmission coefficient remains constant at nearly unity
m
This condition could be satisfied with an error of less
than 1% by making N sufficiently large, e.g., 50 to 80,
depending on the specific structure studied.
The surface plasmon polariton transmission and
reflection coefficients have been calculated for the
structure depicted in Fig. 1b and the transmission
coefficient is plotted in Fig. 3. The dielectric function
g(T) for each metal was assumed to have the simple
free electron form , ε (ω ) = 1 − ωω , with a different plasma
2
p
2
as the value of
ωp
ω
the region of incidence increases. The
most interesting point from the results depicted in Fig. 3
is the lack of reciprocity of the transmission process. That
is, given a fixed set of material parameters, the surface
polariton transmission coefficient depends on which side
of the transverse interface the surface polariton is incident
form. The reason for this non reciprocity is the generation
of volume fields and nonpropagating modes that
accompanies the transmission phenomenon.
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Res. J. Appl. Sci. Eng. Technol., 4(15): 2422-2426, 2012
CONCLUSION
We have revealed that discrete modes describing
propagation of surface plasmon polariton in a closed
waveguide appear a mixture of the originally propagating
and evanescent guided modes. We have described the
transformation of guided modes for a closed metaldielectric structures. We have shown the time averaged
power flow in the specific direction in any waveguide
mode and then we could obtain reflection and
transmission coefficients for the surface polariton. Finally
we have calculated the transmission and reflected
coefficients from Fig. 3 and then pointed out the
interesting results that are the lack of reciprocity of the
transmission process. Our results are not limited to
plasmonic structures and could be extended to pure
dielectric structures as well, where complex modes could
exist even losses are ignored. All of these make our
analysis quite universal and applicable to a wide variety
of light waveguiding structures.
ACKNOWLEDGMENT
The author likes to acknowledge the support given by
the office of vice president for research and technology of
Islamic Azad University, Takestan Branch.
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