In this paper, we numerically study both bandpass and bandstop plasmonic filters based on MetalInsulatorMetal (MIM) waveguides and circular ring resonators. The bandpass filter consists of two MIM waveguides coupled to each other by a circular ring resonator. The bandstop filter is made up of an MIM waveguide coupled laterally to a circular ring resonator. The propagating modes of Surface Plasmon Polaritons (SPPs) are studied in these structures. By substituting a portion of the ring core with air, while the outer dimensions of the ring resonator are kept constant, we illustrate the possibility of redshift in resonant wavelengths in order to tune the resonance modes of the proposed filters. This feature is useful for integrated circuits in which we have limitations on the outer dimensions of the filter structure and it is not possible to enlarge the dimension of the ring resonator to reach to longer resonant wavelengths.The results are obtained by a 2D finitedifference timedomain (FDTD) method. The introduced structures have potential applications in plasmonic integrated circuits and can be simply fabricated.
I. INTRODUCTION
During recent years, plasmonics has been presented as a future technology in integrated circuits incorporating the compactness of electronics and wide bandwidth created by current optical networks
[1]
. In the region of integrated optics, plasmonic waveguides act as components to guide optical signals to different parts of the circuits. The capability of confining light beyond the diffraction limit and the ability to fabricate devices with dimensions below 100 nm has promised an evolution in optoelectronic circuits
[2]
. To guide plasmonic waves for various applications, different geometries have been proposed such as dielectricloaded surface plasmon polariton waveguides
[3]
, nanostructured plasmonic substrates for enhanced optical biosensing
[4]
, tunable coupledring reflector laser diodes
[5]
, dielectric gratings on flat metal surfaces
[6
,
7]
, MetalInsulatorMetal waveguides
[8
,
9]
nanowires
[10
,
11]
, chains of nanoparticles
[12

14]
, grooves and wedges
[15
,
16]
, etc. Although some of these nanoguides are inconvenient to implement in optical circuits, MIM waveguides have attracted considerable attention due to simplicity in fabrication and strong field confinement
[17]
.
During recent years, some simple structures have been suggested for plasmonic filters including channel drop filters with disk resonators
[18]
, rectangular geometry resonators
[19
,
20]
, and ring resonators
[18
,
21]
. In comparison with complex structures of Bragg reflectors, the aforementioned structures can be fabricated much more easily. Two typical types of plasmonic filters in MIM waveguides are bandpass and bandstop filters. In bandstop filters, majority of the input light spectrum is allowed to pass through the filter and only several wavelengths are prohibited from propagation in the structure. But, the bandpass filters let just some specific wavelengths pass through them and reject the majority of the input light spectrum.
One of the most interesting features of filters is to have tunable resonance wavelengths. In previous works, by varying the outer dimensions of the structure, i.e. cavity length or radius, tunable filters were achieved
[21

23]
. But in this paper, we want to tune the resonant wavelengths of circular ring resonator plasmonic filters, without changing the outer size of the structure. Here, by replacing a portion of the ring core with air, we show that tuning characteristics can be attained. As we know, according to some significant limitations on circuit components’ dimensions, it may not be possible to enlarge the size of the ring resonator to reach to longer resonant wavelengths. Therefore the new method is beneficial for miniaturizing the filters in the case of integrated circuits. The transmittance characteristics of the proposed filter are presented by 2D FDTD method.
This paper is organized as follows. In Section 2, the fundamental propagation mode of the MIM waveguide is reviewed. In Section 3, firstly a simple bandpass waveguide filter with a circular ring resonator is examined and then, to reach a tunable bandpass filter, a hollowcore ring resonator is introduced. The impact of hollowing out the core of the ring resonator is studied for a bandstop plasmonic filter in Section 4. Finally, Section 5 concludes the paper.
II. DISPERSION RELATION AND EFFECTIVE REFRACTIVE INDEX OF MIM WAVEGUIDE
Consider an MIM waveguide structure shown in figure 1. Each of the metaldielectric interfaces of the waveguide supports a Transverse Magnetic (TM) SPPs mode which propagates along the
x
direction. It would be anticipated that when the space between the two interfaces is comparable to the decay lengths of SPPs in the dielectric, the SPPs modes become coupled to each other
[24]
. The field components inside the dielectric layer can be calculated using Maxwell relations
Schematic of an MIM structure with two semiinfinitemetal slabs of permittivity ε_{2} surrounding a dielectric layer ofthickness h and permittivity ε_{1}.
where
β
, which has a complex value, is the propagation constant of a MIM waveguide and
is the wavevector perpendicular to the propagation direction (
k_{0}
is the propagation constant of freespace). For
h
<<
λ
, among different modes which can be supported by MIM, the first odd mode is the only mode that can propagate (
Ex(z)
is odd function,
Hy(z)
and
E_{z}(z)
are even, see figure 1). The corresponding propagation constant of MIM waveguide
β
, can be attained from the following dispersion relation
[25
,
26]
Indices 1 and 2 are pointing to dielectric layer and metal slabs, respectively. In order to characterize the dielectric constant of the metal (silver in this study) the sevenpole DrudeLorentz model is employed. The fitting model is described as
[27
,
28]
where Г
_{0}
= 11.5907 THz is the damping constant and ω
_{p}
= 2002.6 THz is the plasma frequency. The quantities of resonant frequencies ω
_{n}
, damping constants Г
_{n}
and oscillator strength
f_{n}
are given in
Table 1
[27
,
28]
.
Values of the Drude?Lorentz model parameters(metal is assumed to be silver)[2728]
Values of the Drude?Lorentz model parameters(metal is assumed to be silver)[27 28]
(a) Real part of effective refraction index as a functionof wavelength for four different widths of the air layer in the AgairAg waveguide. (b) The corresponding propagation length of SPPs as a function of wavelength for the samewidths.
We have analytically calculated the effective refraction index (
n_{eff}
=
β/k_{0}
) for different widths of MIM waveguides,using an IMSL Fortran subroutine by solving Eq. (2). Figure 2 shows the calculated Re[n
_{eff}
] and L
_{SPP}
(propagation length of SPPs defined as
L_{SPP}
=
1/(2Im[β])
), as a function of wavelength.
III. BANDPASS PLASMONIC FILTER WITH CIRCULAR RING RESONATOR
 3.1. Simple Bandpass Waveguide Filter with Circular Ring Resonator
Figure 3
shows the structure of a simple bandpass plasmonic filter which consists of two MIM waveguides coupled to each other by a circular ring resonator, which was first described in Ref
[21]
. The filter parameters h, Δ, R
_{core}
, R
_{Av}
, and R
_{out}
are the width of MIM waveguide, the gap distance between input/output MIM waveguides and circular ring resonator, the radius of core, the average radius of the ring and the outer radius of the ring resonator, respectively. We set
h
= 50
nm
which is much smaller than the operating spectrum.
Two power monitors are set at points P
_{1}
and P
_{2}
to detect the input power A
_{1}
(without the circular ring resonator) and the transmitted power A
_{2}
(with the circular ring resonator). So, the power transmittance is T = A
_{2}
/A
_{1}
[21
,
27]
. The transmittance of the filter is shownin
figure 4
(a), assuming
R_{Av}
=125
nm
and
Δ
=10
nm
. One can see in the figure that there are two resonance peaks at λ?1145 nm and λ?583.5 nm.
Figures 4
(b),
4
(c) depict the field patterns of 'H
_{y}
' for the first and second resonances, respectively. Here, we should note that the resonance wavelengths of the transmission spectrum cannot be achieved simply by λ = L[Re(n
_{eff}
)/N], where N is the mode number and L = 2πR
_{Av}
is the length of the ring
[21]
. The resonant wavelengths of a circular ring resonator are obtained by
[21
,
29]
Schematic of a simple bandpass plasmonic filterconsisting of two MIM waveguides coupled to each other bya circular ring resonator. h is set to 50 nm.
where
k
=
ω(ε_{0}ε_{r}μ_{0})^{1/2}
and ε
_{r}
=(n
_{eff}
)
^{2}
/
μ_{0}
is the relative permittivity. J
_{n}
is a first kind Bessel function of order n, and N
_{n}
is a second kind Bessel function of order n. So J′
_{n}
and N′
_{n}
are derivations of the Bessel functions with respect to the argument (
kR
). Eq. (4) must be numerically solved to determine the resonant wavelengths of the structure. A complete discussion of results obtained by this equation can be found in Ref
[21]
.
(a) Transmittance spectrum of the simple bandpass plasmonic filter with circular ring resonator (R_{Av}=125 nm Δ =10 nm). (b) The 'H_{y}' field profile of simple bandpass filterat the first resonance wavelength of λ=1145 nm. (c) The 'H_{y}' field pattern of simple bandpass filter at the second resonance wavelength of λ=583.5 nm.
 3.2. Bandpass Plasmonic Filter with Hollowcore Ring Resonator
In this section, we investigate the effect of hollowing out the core of the ring resonator on resonance wavelengths of the bandpass filter while keeping its outer dimensions constant (i.e. R
_{Av}
, h and Δ are kept constant). The proposed structure is depicted in
figure 5
. From the figure one can see that the core of the ring resonator is emptied with the radius of R
_{H}
.
Now, we look into the effect of varying R
_{H}
on shifting the
Schematic of a bandpass plasmonic filter withcircular hollowcore ring resonator. (h=50 nm R_{Av}=125 nm Δ= 10 nm). R_{H} is variable.
(a) Transmission spectrum of the bandpass plasmonic filter with circular hollowcore ring resonator shown in figure 5 fordifferent values of R_{H}. (b) Relationship between resonance wavelengths and hollow radius. (c) The 'H_{y}' field profile of bandpassplasmonic filter with circular hollow core ring resonator at the first resonance wavelength of λ= 1239 nm for R_{H}=85 nm. (d) The 'H_{y}' field pattern of the filter at the second resonance wavelength of λ=636 nm for R_{H}= 85 nm.
resonant wavelengths of the bandpass filter. The transmittance spectra of the proposed structure is calculated by 2DFDTD and illustrated in
figure 6
(a). One can see that by increasing RH from 70 nm to 85 nm, the entire transmission spectrum experiences a redshift.
Figure 6
(b) shows that the wavelengthshifts of modes 1 and 2 behave approximately linearly with respect to variation of RH. This result is in complete accordance with the physical fact that by increasing the resonance volume, resonant wavelength will be increased. According to the transmission curves illustrated in
figure 6
(a), it is clear that the bandpass filter can be easily tuned by adjusting RH.
 3.3. Enhancing the Transmittance Peak
It is seen from
figure 6
(a) that by increasing RH, the maximum value of the transmittance peak declines. To enhance the transmittance, we decrease the width of ring MIM waveguide in the coupling region. The proposed structure is depicted in
figure 7
. The procedure of transmittance increment versus W is illustrated in
figure 8
(a) for first and second resonance modes of the structure. It is obvious from the figure that the first resonance wavelength has experienced no shifting by varying W, but the second mode is slightly blueshifted (Δλ≈7 nm) with respect to decreasing W. The relation between decreasing the W and increment of transmittance for first and second modes is shown in
figure 8
(b). As can be
Schematic of a bandpass plasmonic filter withcircular hollowcore ring resonator and reduced width of ring MIM waveguide at the coupling region. (h=50 nm R_{Av}=125 nm R_{H}=85 nm Δ = 10 nm). W is variable.
(a) Transmission spectra of the structure shown in figure 7 for (h=50 nm R_{Av}=125 nm R_{H}=85 nm Δ =10 nm) and different values of W. (b) Relationship between transmittance of resonance wavelengths and W.
seen from this figure by decreasing W, the coupling area between input/output MIM and ring MIM waveguides increases and therefore more power will be transmitted.
Schematic of a simple bandstop plasmonic filter consisting of an MIM waveguide coupled to a circular ringresonator. h is set 50 nm.
IV. BANDSTOP PLASMONIC FILTER WITH CIRCULAR RING RESONATOR
 4.1. Simple Bandstop Waveguide Filter with Circular Ring Resonator
Another essential and useful component for photonic circuits is a bandstop filter.
Figure 9
depicts the geometry of a simple bandstop plasmonic filter which consists of an MIM waveguide laterally coupled to a circular ring resonator. The filter parameters h, Δ, R
_{core}
, R
_{Av}
, and R
_{out}
are the same as for the bandpass filter.
Two power monitors are set at points P
_{1}
and P
_{2}
to observe the incident power A
_{1}
and the transferred power A
_{2}
. The transmittance of the filter with
R_{Av}
=125
nm
and
Δ
=10
nm
is illustrated in
figure 10
(a). From
figure 10
(a), it is clear that there are two resonance valleys at λ?1145 nm and λ?583.5 nm.
Figures 10
(b),
10
(c) show the field profile of 'H
_{y}
' for the first and second valleys, respectively. As the transmission curve shows, the second resonance mode has a stronger coupling to the ring in comparison with the first one; this is because the wavelength of the second mode is shorter than that of the first one, so that the power density (W/m
^{2}
) of the second resonance is higher than the first mode. Therefore, more power can be coupled into the ring at the coupling region for the second mode.
 4.2. Bandstop Plasmonic Filter with Hollowcore Ring Resonator
In this part, by keeping constant the outer dimensions of the bandstop filter (i.e. R
_{Av}
, h and Δ) and hollowing out the core of the ring resonator (see
Fig. 11
); we check how the resonant wavelengths of the filter will be affected.
As for the bandpass filter case, we anticipate that by increasing R
_{H}
, the resonant wavelengths of the bandstop filter experience a redshift. This case is illustrated in
figure 12
(a). One can see that by varying R
_{H}
from 70 nm to 85 nm,the entire transmission spectrum has undergone a redshift.
Figure 12
(b) shows that the wavelengthshifts of the modes 1 and 2 have approximately lineal relations with respect to
(a) Transmittance spectrum of the simple bandstop plasmonic filter with circular ring resonator (R_{Av}=125nm Δ =10nm). (b) The 'H_{y}' field pattern of simple bandstop filter atthe first resonance wavelength of λ?1145 nm. (c) The 'H_{y}' field profile of the filter at the second resonance wavelength of λ?583.5 nm.
Schematic of a bandstop plasmonic filter with circular hollowcore ring resonator. (h=50 nm R_{Av}=125 nm Δ=10 nm). R_{H} is variable.
(a) Transmission spectrum of the bandstop plasmonic filter with circular hollowcore ring resonator shown in figure 11 for different values of R_{H}. (b) Relationship between resonance wavelengths and hollow radius. (c) The 'H_{y}' field pattern of bandstop filter with circular hollowcorering resonator at the first resonance wavelength of λ?1239nm for R_{H}=85nm. (d) The 'H_{y}' field pattern of the filter at the second resonance wavelength of λ?636 nm for R_{H}=85nm.
the variation of hollow radius. This effect is in complete agreement with the case for bandpass filters, i.e. by increasing the resonance volume, the resonant wavelength is increased.So, from the transmittance curves, it is explicit that by altering the R
_{H}
, the bandstop filter can be simply adjusted.
V. CONCLUSION
In this paper, firstly the propagation characteristics of a bandpass filter composed of two MIM waveguides coupled to each other by a circular ring resonator were studied. To make the proposed filter tunable, an easy way of emptying the core of the ring was introduced and investigated numerically. Then, a bandstop filter consisting of an MIM waveguide laterally coupled to a circular ring resonator was studied and the impact of hollowing out the core of this ring resonator was investigated. It was seen that for both cases, by increasing the radius of the hollow, the resonant wavelengths of the structures experienced a redshift. A significant advantage of the proposed method of making filters tunable is ease of fabrication. The resonant modes of the structures were calculated by FDTD and it was found that we can easily manipulate the central wavelengths of the resonance transmission by tuning the radius of the hollow while the outer dimensions of the ring are kept constant. The introduced structures can decrease plasmonic filter dimensions and so are potentially a choice for designing of alloptical integrated circuits for optical communication and optoelectronic circuits.
Acknowledgements
The authors acknowledge the financial support from the Education & Research Institute for ICT, Iran (Grant No.500/3653).
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