A novel triangularshaped plasmonic metalinsulatormetal (MIM) Bragg grating waveguide is introduced, whose bandgap is narrower than that of the conventional step type and wider than that of the sawtoothshaped one. Moreover apodized triangularshaped MIM Bragg grating structures are proposed in order to reduce the side lobes of the transmission spectrum, because the Bragg reflector with a sawtooth profile has a smoother transmission spectrum than that of a triangularshaped one. The performance of the proposed structures is simulated by using the finite difference time domain method.
I. INTRODUCTION
During the past few decades, due to the diffraction limit of light
[1]
, design, implementation, and application of subwavelength photonic devices, such as waveguides, couplers etc. have faced basic problems and difficulties. In recent years, researchers in the field of plasmonics, investigating the properties and applications of the surface plasmon polaritons (SPP) and localized plasmons [
2

4
], have introduced a new class of highly miniaturized optical devices to the world [
5
,
6
]. These days SPPs have an important role in the fabrication of devices, with a greater confinement of light at nanoscales without limitations due to lightwave diffraction
[7]
.
Surface plasmon polaritons are the coupling of the electromagnetic fields to coherent charge oscillations of conduction electrons [
8
,
9
], at the interface of the metallic and dielectric materials, that are excited in visible and near infrared wavelengths. Exciting the SPPs, part of the light energy is transferred to the surface plasmons, so that the reflected wave has lower energy compared to the incident light.
From the application point of view, some novel photonic devices based on SPPs, such as waveguides, directional couplers, reflectors, absorption switches and generalized sensors have been proposed in recent years[
10

16
].
Two specific kinds of multilayer plasmonic structures are insulatormetalinsulator (IMI) and metalinsulatormetal (MIM)
[8]
, but the MIM heterostructures are appropriate geometries in order to obtain better light confinement. The propagation loss of the MIM structures is higher than that of the IMI ones, but in the nanoscale devices it is negligible [
17
,
18
].
Among the bandgap structures, some kinds of plasmonic Bragg reflectors have recently been introduced. A lowloss indexmodulated (InM) Bragg reflector was investigated by Hosseini
et al
.
[14]
. A thicknessmodulated (ThM)
[19]
and one with both thickness and index modulated profile
[13]
have been proposed in order to achieve a wider bandgap. Most recently Shibayama
et al.
have introduced an apodized ThM Bragg reflector
[20]
to reduce the side lobes of the transmission spectrum. Liu
et al.
have introduced a sawtooth profile Bragg reflector
[21]
which has a narrower bandgap and slightly reduced rippling in the transmission spectrum, compared to the step Bragg reflector. But the main problem with such a profile is that we cannot directly apply the Bragg condition to the structure in order to calculate the lengths of the different layers because the sawtooth profile is repeated completely through the structure and its length equals the sum of lengths of different parts in conventional Bragg structures.
In this paper, we have introduced two new thicknessmodulated (ThM) and indexthicknessmodulated (InThM) triangularshaped grating MIM Bragg reflectors, whose bandgaps are narrower than those of the same ThM and InThM steptype structures respectively, and wider than those of the similar sawtooth profile structures. In comparison between the sawtooth and triangularshaped structures, it is revealed that a smoother transmission spectrum could be achieved by the sawtooth profile. Thus, in order to compensate the smaller reduction of rippling, the apodization operation is performed on the new triangular Bragg gratings and a smoother transmission spectrum could be achieved compared to the conventional Bragg reflectors.
II. ANALYSIS METHOD
The dispersion relation of a simple MIM waveguide with the assumption of infinite structure and the form of exp[
i
(
β
x
?
ω
t
)] for the field components propagating in the xdirection for the fundamental TM mode, with
E
_{x}
,
E
_{y}
and
H
_{z}
field components, is given by the following equations
[8
,
22]
where
k
_{0}
is the free space wavenumber and
ε
_{d}
,
ε
_{m}
, and
t
are the dielectric constants of dielectric and metal and the dielectric thickness, respectively, as shown in
FIG. 1
. The effective refractive index of the waveguide can be determined by:
In this paper, the propagation of SPPs in these structures is simulated, using a twodimensional FDTD method, accomplished with an auxiliary differential equation (ADE) approach
[23]
. All the software has been prepared in C++ and Matlab language and environment. The absorbing boundary conditions for all the boundaries of the computational window are convolutional perfectly matched layer (CPML) with the absorption
A simple MIM waveguide with dielectric thickness of t.
loss of about 90 dB
[24]
. The grid sizes in
x
and
y
directions are
Δ
x
=
Δ
y
=4 nm, and considering the Courant limit, the time step is
where
c
is the speed of light in free space. A modulated Gaussian point source is located at the middle of the feeding waveguide.The number of time steps in our simulation is 60000.
In our simulations, the metallic cladding layers are assumed to be silver, characterized by the Drude dispersion model
[25]
:
where the materialdependant constants
ω
_{p}
and
γ
_{p}
are the bulk plasma and damping frequency, respectively.
ε
_{∞}
is the dielectric constant at the infinite frequency and the parameters of the Drude model are chosen to be
ε
_{∞}
=3.7,
γ
p
=2.73×10
^{13}
Hz and
ω
p
= 1.38×10
^{16}
Hz at λ=1550 nm
[25]
.
The input and output sampling planes are displayed in
FIG. 2.
(a), and the number of periods for all the structures is N=19. The normalized transmission can be defined as │
H
_{y}
_{,}
_{out}
(λ )/
H
_{y}
_{,}
_{in}
(λ)│, where
H
_{y}
_{,}
_{out}
(λ) and
H
_{y}
_{,}
_{in}
(λ) are the Fourier transform of the
H
_{y}
component at the output and
(a) The whole structure of a step InM MIM displaying the input and output sampling planes locations, (b) InM Bragg grating waveguide for two different set of insulators, first: air and silica with d1=245 nm , d2=167 nm , and second: porous silica and silica with d1=169 nm , d2=167 nm, and t=30 nm, and (c) comparison of transmission spectra of these two structures.
Effective refractive indices of different MIMs
Effective refractive indices of different MIMs
input planes, respectively.
Solving the dispersion relation equations (12) in mathematical software, we obtain the effective refractive indices of MIM structures with various dielectrics and insulator widths as given in
TABLE 1.
According to the Bragg condition d1
Re
{
n
_{eff}
_{,}
_{1}
}+d2
Re
{
n
_{eff ,2}
}=nλ
_{b}
/2 where λ
_{b}
is the Bragg wavelength, which is assumed to be 1550 nm here, the thicknesses are chosen as d1 and d2 in order to realize the Bragg condition.
III. RESULTS AND DISCUSSIONS
Two different InM MIM Bragg gratings, with different insulators, illustrated in
FIG. 2
have been simulated. In the first one air and silica with
ε
_{d}
=2.13, and in the second structure porous silica with
ε
_{d}
=1.51 and silica have been chosen as the insulators. As depicted in
FIG. 2
. (c), the bandgap of the second Bragg grating is narrower than that of the first one, due to the lower contrast of the effective refractive indices.
FIG. 3
represents the InM, ThM, and InThM MIM structures, which have been analyzed in
[13
,
14]
and
[19]
,respectively by a transfer matrix method. For the Bragg reflector shown in
FIG. 3.
(a), the dielectric width is w1=30 nm,
ε
_{d1}
=2.13 (SiO
_{2}
) and
ε
_{d2}
=1 (air), the effective indices are Re{
n
_{eff,1}
}=2.3, and Re{
n
_{eff,2}
}=1.5 and their corresponding lengths are d1=168 nm and d2=244 nm. In
FIG. 3.
(b),the widths of dielectric slits for two alternating stacked MIM waveguides are w1=30 nm and w2=100 nm, which are filled with air, and their corresponding lengths are d1=244 nm and d2=324 nm. In
FIG. 3.
(c), the slit widths are w1=30 nm and w2=100 nm which are filled with SiO
_{2}
and air, respectively and d1=168 nm and d2=324 nm.
Considering the approximated formula for calculating the bandgap width of a 1D photonic crystal
[26]
, and also the equations (1) and (2), for a structure that consists of two alternately stacked MIM waveguides, the parameters that affect the
n
_{eff}
and thereupon
Δ
λ
_{g}
, the bandgap width,are mostly permittivity and thickness of the dielectric region. Here we have shown that with different configurations made by changing these principle factors different widths of bandgaps can be obtained.
As demonstrated in
FIG. 3.
(d), the bandgap of a ThM Bragg reflector compared to that of the InM one, and also
Three different MIM structures (a) a part of an InMMIM with w1=30 nm, ε d1=1 (air), and ε d2=2.13 (SiO2), (b) apart of a ThM MIM with w1=30 nm and w2=100 nm whichare filled with air, (c) a part of an InThM MIM with w1=30nm which is filled with SiO2 and w2=100 nm, filled with air,(d) the normalized transmission spectra of b, c and dstructures.
the bandgap of an InThM structure compared to that of the ThM one are wider. In the InThM structure because of utilizing different dielectrics and thicknesses in these MIM waveguides, the contrast between two effective refractive indices of MIM waveguides increases and a wider bandgap is expected.
A new triangularshaped Bragg grating structure shown in
FIG. 4.
(a), is proposed here. This proposed structure,compared to a simple ThM MIM with the same w1, w2,d1, and d2, has a narrower bandgap. If we define an effective insulator width as
w
_{eff}
=
A
_{s}
/d2 in the MIM structures,where
A
_{s}
is the area of the corrugated part, for triangularshaped grating waveguides this parameter is smaller than that of a simple ThM structure. Due to Eqs. (1) and (2),and also according to the parameter values of
TABLE 1
,the structures with smaller insulator width, have larger
n
_{eff}
.
(a) A triangular shaped ThM MIM with w1=30 nmand w2=100 nm filled with air and (b) its transmissionspectrum, compared to those of similar InM and ThM MIMones.
So, the contrast between the effective refractive indices of the triangularshaped grating waveguide parts is less than that of a simple ThM structure, and a narrower photonic bandgap is expected from the triangularshaped one, as shown in
FIG. 4.
(b).
With the same reasoning, the bandgap of a triangularshaped grating InThM MIM will be narrower than that of the InM MIM and wider than that of the ThM MIM,as illustrated in
FIG. 5.
(b).
Liu
et al.
have introduced a kind of grating with sawtooth profile which has a narrower bandgap than a similar conventional step Bragg reflector
[21]
. Moreover they have claimed that with the sawtooth profile the ripples in the passband of the transmission spectrum would be reduced. We have simulated similar structures of conventional step shape, sawtooth and triangular shaped Bragg grating profiles and compared the results.
FIG. 6
represents the transmission spectra of these three different gratings simulated in the same conditions.The bandgap width of the triangularshaped structure (800 nm) is 26% narrower than that of the step one and 23%wider than that of the sawtoothshaped one. The principle parameters of the Bragg structures, such as the widths and the lengths of the layers in both the sawtooth and triangularshaped reflectors are the same as those of the step structure,but according to different shapes, the bandgaps with different widths are expected.
With smaller w2 the bandgaps become narrower. The variation of the bandgap width versus w2 is depicted in
(a) A triangularshaped grating InThM MIM withw1=30 nm filled with SiO2 and w2=100 nm filled with air and(b) the transmission spectrum of the displayed structure,compared to those of ThM and InThM MIMs. Its bandgapis wider than that of the ThM and is narrower than that of theInThM one.
FIG. 7
, in which the bandgap is defined as the band where the transmission coefficient becomes under 30dB
[20]
. In
FIG. 7
, as w2 increases, the bandgap becomes wider, but as illustrated in
FIG. 7
for all values of parameter w2, in both of the triangularshaped structures the bandgaps are narrower than the similar simple gratings.
In addition, the depths of the ripples in the triangular and sawtooth shaped reflectors are respectively (18%) and (50%)less than that of the conventional step one. In order to compensate this smaller reduction of the ripples in our proposed structure compared to the sawtoothshaped reflector, we have utilized the apodization operation.
Shibayama
et al.
have investigated the effect of apodization on a simple ThM step structure
[20]
and have shown that with this operation on the 5 or 9 periods of the grating at both input and output ports, the side lobes in the transmission spectrum are well suppressed
[27]
. Again, to verify our code, we have first simulated the same structure as that of
[20]
as depicted in
FIG. 8.
(a). The same results have been obtained, as shown in
FIG. 8.
(b).
Now we show that if a similar procedure is performed on the triangularshaped grating MIMs, the resultant bandgaps would approximately remain as wide as those of the simple gratings, but the side lobes would be noticeably reduced.
FIG. 8
illustrates the apodized triangularshaped grating,ThM and InThM gratings and also their transmission
A (a) step, (b) sawtooth and (c) triangularshapedBragg reflector. (d) The transmission spectra of these threekinds of Bragg gratings.
Variation of the bandgap width versus the height ofthe triangle in two new proposed structures.
Side lobes are well suppressed in apodized structures,(a) an apodized ThM MIM, (b) comparison of simple andapodized ThM MIMs transmission spectra, (c) an apodizedtriangularshaped grating ThM MIM, (d) comparison of thesimple and apodized triangularshaped grating ThM MIMstransmission spectra, (e) an apodized triangularshaped gratingInThM MIM, (f) comparison of the simple and apodizedtriangularshaped grating InThM MIMs transmission spectra.
IV. CONCLUSION
In this paper, we have proposed a new triangularshaped MIM Bragg grating structure, whose bandgap is narrower than that of the simple step and wider than that of the sawtoothshaped Bragg grating. Also, an apodization procedure has been carried out on these new MIM waveguides, in order to cause the side lobes of the transmission coefficient to be considerably suppressed compared to the conventional simple step Bragg reflectors.
Acknowledgements
The authors would like to thank The Education and Research Institute for ICT (ERICT) (formerly, The Iran Telecommunication Research Center (ITRC)) for the financial support of this project.
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