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Realization of single-mode plasmonic bandpass filters using improved nanodisk resonators
Optics Communications 420 (2018) 147–156
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Realization of single-mode plasmonic bandpass filters using improved nanodisk resonators Shiva Khani, Mohammad Danaie *, Pejman Rezaei Electrical and computer Engineering Faculty, Semnan University, Semnan, Iran
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Keywords: Plasmonic filter Bandpass filter Optical resonator Drude–Lorentz Metal–insulator–metal (MIM) waveguide
ABSTRACT In this paper, symmetric and asymmetric plasmonic bandpass filter (BPF) topologies based on the metal– insulator–metal (MIM) configuration are proposed. These filters are numerically investigated using finite difference time domain (FDTD) method. The metal and dielectric used for the realization of the filters are silver and air, respectively. The real and imaginary parts of silver’s permittivity used in numerical simulation are based on the Drude–Lorentz and Palik models. Both structures are composed of two waveguides, a nanodisk, and parenthesis-shaped adjunctions. The inclusion of symmetrical and nonsymmetrical adjunctions results in single mode filters with higher transmission peaks compared to the original nanodisk-based filter. To provide better physical insight, various structural parameters of the filter are changed and their effects on filter’s response are presented. It is observed that the resonance mode of proposed BPFs can be tuned by changing the nanodisk resonator radius. Such structures can be employed in various plasmonic devices such as multiplexers and demultiplexers for optical communication purposes.
1. Introduction Surface plasmon polaritons (SPPs) are the electromagnetic excitations that propagate at the interface between metals and dielectric materials. Due to the remarkable capability to manipulate light in a nanoscale domain, SPPs are considered as an efficient basis for the realization of highly integrated optical circuits [1–3]. Growing demands for such elements motivates researchers to design numerous integrated nanoscale optical circuits and devices, such as optical filters, sensors [4,5], demultiplexers [6,7], switches [8,9], Mach–Zehnder modulators [10], splitters [11,12], couplers [13] and so on using the plasmonic technique. Also, at the plasmon resonance, the scattering and absorption cross sections of nanoparticles are enhanced. It can be used to increase the efficiency of solar cells [14]. Among the most important optical devices that have found wide application for wavelength selection are plasmonic filters. Such devices have been the center of attention in recent years. Among the plasmonic filters based on MIM structures, bandpass filters (BPFs) and bandstop filters (BSFs) are of high importance. In order to design an optical BPF the easiest method is to design an optical cavity which has a resonance frequency equal to the central frequency of the BPF. Such cavities are then laterally coupled to the input and output waveguides. If the resonance profile of the cavity matches the profile of the mode which propagates through the waveguides, a BPF can be conceived. To design BSFs, usually the cavity *
is side-coupled to a waveguide. The most common type of resonator used for realization of MIM plasmonic filters is a circular nanodisk. Such a disk is easy to implement and the resonance frequency can be tuned by variation of its radius. Over the last few years, many plasmonic waveguide filters based on ring-shaped or circular resonator configurations have been proposed. For example using ring resonators, different BPFs have been designed in [15,16]. Nanodisk resonators were used to design BPFs in [17,18]. To investigate the effects of silver slabs in nanodisk resonator, a plasmonic BPF is proposed in [19] that leads to reduction of the filter dimensions. To achieve a higher Q factor, a filter composed of three cascaded diskshaped cavities is presented in [20]. Drude model is used for simulation of metal behavior in [20]. Also, nanodisk resonators have been presented in a variety of different versions for filter applications which include hollow-core circular ring resonator [21], a nanodisk resonator coupled with stub resonator [22], L-shaped filter based on nanodisk resonator [23], and stub waveguide with nanodisk and fabry–perot resonator [24]. Other approaches such as plasmonic branch-shaped MIM waveguide with a triangular-annular channel [25], T-shaped plasmonic resonator [26], Archimedes’ spiral nanostructure [27], and isosceles trapezoid cavities [28] have also been presented. The metal–insulator–metal configuration is a relatively old technique for creation of microwave micro-strip filters [29–31]. In an
Corresponding author. E-mail address: [email protected] (M. Danaie).
https://doi.org/10.1016/j.optcom.2018.03.047 Received 23 December 2017; Received in revised form 17 March 2018; Accepted 20 March 2018 0030-4018/© 2018 Elsevier B.V. All rights reserved.
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Optics Communications 420 (2018) 147–156
integrated circuit the dielectric substrate and metal layers are already accessible. The same metal layers that convey the electric current between different transistors can be used to transport optical signals. MIM structures can be integrated with other electrical or optical components on a single chip. For example the photo-detectors, or laser sources needed for such structures can be implemented on-chip. An analog to digital convertor and a processor may also be integrated together to process the photodetector data. Although high quality optical filters and demultiplexers can be designed and implemented using photonic crystals [32,33], plasmonic devices can occupy far less area than similar photonic crystal devices [34–36], due to their ability to overcome the diffraction limit. Most importantly light amplification is possible in Schottky junction-based plasmonic waveguides. It creates unique capabilities [37,38]. It provides the ability to design a new genre of devices based on MIM topology. In case a Kerr-type nonlinear material is used as the dielectric, the resonance frequency of such filters can be tuned optically. Although Zheng et al. have proposed this technique for tuning the resonance filter the idea can also be employed for realization of all-optical switches [39]. Ring shaped structures can also be employed for design of power splitters [40]. Due to the relatively recent attention and interest to plasmonic filters, a huge amount of the results reported in the literature are still from the simulations only. Lots of such publications use plasma or Drude model for silver or gold in their finite difference time domain (FDTD) simulations. Such models provide nearly accurate results when simulating waveguides. However, we will show that they do not provide accurate outcome when the plasmonic structure contains a resonator. Here Drude, Drude–Lorentz and Palik models are used for numerical characterization of silver and the results are compared together. These methods will be briefly introduced in Section 2. It is observed that Palik and Drude–Lorentz model provide very similar results while the Drude model, which is very commonly used in many papers, is totally unacceptable for simulation of resonator based filters. There are some deficiencies associated with the waveguide-coupled disk resonator filter topologies reported in the literature. The first one is that for the wavelength range of 600 nm–2000 nm, which is the range reported in most of the papers, the central cavity is dual mode. Having a single-mode resonator is much more helpful when designing more complex structures such as optical demultiplexers. The second one is when such structures only provide high transmittance when the inaccurate Drude model (also known as plasma model) is used. We will show that when more accurate models are used the transmittance will be decreased to half of its original value. In this paper, parenthesisshaped adjunctions are added to the original resonator structure. Two types of filters are proposed based on using symmetric or asymmetric parentheses. These filters eliminate the lower and higher wavelength resonance modes respectively and provide a single mode behavior for the 600 nm–2000 nm range. Furthermore, the inclusion of the parenthesis shaped structure improves the transmittance of the filter compared to the original nanodisk filters. The proposed structures can thus be used for design of much more complex structures such as sensors and demultiplexers. Based on the direction of the light propagation, the plasmonic structures can be categorized into two main groups. The light can either propagate along the interface of metal or dielectric layers (in-plane propagation) or be perpendicular to a slab comprising an array of metal and dielectric shapes. In plasmonic crystals which use an array of metallic nanoparticles deposited on a dielectric layer or equally an array of holes etched in a metal surface, the incident light is usually perpendicular to the surface. For the first case, usually a plasmonic waveguide is created using a dielectric layer which is sandwiched between two metal layers. The MIM waveguides seem more appealing for those who seek to have all-optical integrated devices. They can be used for implementing much more complex structures such as demultiplexers [41], antennas [42,43], ring resonators [44,45], memristors [46], circulators [47], switches [48] etc. They can also be integrated with other electronic devices. The fact
Fig. 1. Schematic configuration of the initial plasmonic filter.
that such structures can be used to create Schottky plasmonic based amplifiers seems very promising [37,38]. Plasmonic crystals can act as an absorbers based on the localized surface plasmon resonance [49]. They are widely fabricated and used for bio-sensing applications [50,51]. Since the incident light is perpendicular to the surface, no coupling stage is needed for such structure. Only a light source and a photo detector are enough. There are lots of experimental papers published in this area. Some of them are reviewed in [52]. Although such structures are mostly used for sensors, they can also be used to improve solar cells [53,54], create nanolenses [55–57] etc. They are easy to fabricate. The fabrication methods and setups for such crystals are reviewed in [58]. To be able to use MIM waveguides, the optical fiber has to be coupled to it. Unlike the first method which involves metal layer deposition and lithography, it is much more expensive to fabricate and measure the results in these types of plasmonic devices. There has been numerous methods proposed to address the problem of coupling the light to the MIM waveguides. Experimentally tested solutions can be found in [59–62]. They usually involve an input grating coupler. In [63] experimental data and FDTD results for an MIM filter composed of SiO2 and Ag have been compared. Good agreement is observed between the simulation and measurement results in [63]. A same setup can be used for our proposed structure for experimental measurements. The rest of this paper is as follows: In Section 2, the original resonator-based filter is discussed; Drude, Drude Lorentz and Palik models are briefly reviewed and compared together. In Section 3 the proposed structure is presented. Section 4 discusses the results and the last section is devoted to conclusions. 2. Disk resonator-based filter Fig. 1 shows the topology of the initial plasmonic BPF consisting of two slits (two semi-infinite separated waveguides) and a nanodisk between them. The parameters of the structure are as follows: the radius of the nanodisk (𝑟 = 310 nm), the widths of the waveguides ( 𝑤 = 50 nm), and the coupling distances between the waveguides and the nanodisk (𝑠 = 16 nm). The insulator material in the white areas is set to be air with relative permittivity of 𝜀𝑑 = 1. The material of the gray areas is assumed silver, which is characterized by Drude, Drude–Lorentz, and Palik models that will be discussed shortly after. Silver due to its low Ohmic resistance, compared to other metals, is one of the most widely used metals in plasmonic. 2.1. Drude model The Drude model for the permittivity of a metal, which is the most simplified model, can be expressed as follows [16,23]: 𝜀𝑚 (𝜔) = 𝜀∞ − 148
𝜔2𝑝 𝜔(𝜔 − 𝑗𝛾)
(1)
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Fig. 3. Transmission spectrum with Palik model.
Fig. 2. Transmission spectrum obtained using the Drude model.
Table 1 Parameters of the Drude–Lorentz model for silver.
Where 𝜀∞ is the dielectric permittivity at infinite frequency, 𝜔𝑝 is the bulk plasma frequency of metal, 𝛾 is the electron collision frequency, 𝜔 is the√angular frequency of incident light, and 𝑗 is imaginary number (𝑗 = −1). The parameters of the Drude model for silver are assumed to be 𝜀∞ = 3.7, 𝜔𝑝 = 9.1 ev, and 𝛾 = 0.081 ev, respectively. The transmission characteristics of the filter shown in Fig. 1, are investigated using FDTD. Fig. 2 shows the filter transmission spectrum which is comprised of two resonance modes (mode 1 and mode 2) at the 1174 and 732 nm wavelengths, respectively. Because there are internal losses in the nanodisk and waveguides in the metal slits, the transmission peaks do not reach unity and the maximum transmissions of the two modes are 37% and 70%, respectively. The Drude model is accurate for a limited range of wavelengths [64,65]. Accordingly, for more precise description of the optical properties of metals through the electromagnetism spectrum, it is better that experimental refractive indices or more accurate models be used.
Another model that can be used for description of the permittivity of silver over a wider wavelength range is the Palik model which is introduced in [65]. In [65], different sample preparation methods and data from silver samples exposed to air are utilized by four research groups. There is however small inconsistencies in measuring representative properties of pristine silver due to the tarnishing caused by air exposure [66,67]. The transmission spectrum of the initial structure using Palik model is shown in Fig. 3. As seen, the transmission peak values are 13% and 32.6% at wavelengths of 1174 and 738 nm for mode 1 and mode 2, respectively. These values are much less than what was expected from the Drude model.
𝜔 (𝜔 − 𝑗𝛾)
5 ∑ 2 𝑛=1 𝜔𝑛
𝑓𝑛 𝜔2𝑛 − 𝜔2 + 𝑗𝜔𝛾𝑛
𝑓𝑛
939.62 109.29 15.71 221.49 584.91
7.9247 0.5013 0.0133 0.8266 1.1133
In the previous section the conventional filter structure was presented. One of the main problems of such a structure is its poor transmittance after the Drude–Lorentz model is used for numerical simulation. In this section to improve the initial structure and to design the proposed filter, these goals are pursued: First of all more accurate simulation with other models such as Palik and Drude–Lorentz. Secondly converting the two-mode filter to two one-mode filters, where each of these filters passes one of the modes and finally increasing the transmission peaks.
Here, the dielectric constant of the silver is characterized by the seven-pole Drude–Lorentz model, known to be reasonably accurate in the wavelength range of 200 to 2000 nm [21]. The permittivity model is presented as: +
𝛾𝑛 (THz)
197.3 1083.5 1979.1 4392.5 9812.1
3. The proposed structure and simulation results
2.3. Drude–Lorentz model
𝜔2𝑝
𝜔n (THz)
1 2 3 4 5
Fig. 4(a) illustrates the simulated transmittance using Drude–Lorentz model. It is superimposed on the results of Drude and Palik models in Fig. 4(b). As can be seen when the Drude–Lorentz or Palik models are used the transmission peaks are significantly reduced. Transmission peak values of two modes are 17% and 29%, respectively, which are too low for a filter. Therefore, the goal of this paper is to improve the transmittance. Fig. 5(a–c) shows the field profile of magnitude of 𝐻𝑍 for the initial structure at wavelengths of 1190 and 742 nm (resonance wavelengths of mode 1 and mode 2 in Drude–Lorentz model), and 1000 nm (nonresonance wavelength). As seen, in terms of 𝐻𝑍 , two resonance modes of the structure corresponding to wavelengths of 𝜆1 = 1190 and 𝜆2 = 742 nm (Fig. 5(a–b)) have appeared in the nanodisk and are transmitted to the output port of the filter. Also, Fig. 5(c) shows the field profile for 𝜆 = 1000 nm, which cannot be transmitted to the output port.
2.2. Palik model
𝜀𝑚 (𝜔) = 1 −
𝑛
3.1. The first proposed filter
(2)
To achieve better filters, parenthesis shaped adjunctions are added to the initial structure. The schematic illustration of the first proposed structure which passes the second mode is shown in Fig. 6. When the incident optical waves propagate forward along a waveguide, part of the energy will be reflected and the other part will be coupled into the
Where, 𝜔𝑝 = 2002.6 THz is the bulk plasma frequency of silver and 𝛾 = 11.61 THz is the damping constant. Other parameters include quantities of resonant frequencies 𝜔𝑛 , damping constants 𝛾𝑛 , and weights 𝑓𝑛 are listed in Table 1. It should be noted these values require to be changed to the Radian unit (multiplied by 2𝜋) before substitution in (2). 149
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Fig. 4. (a) Transmission spectrum with Drude–Lorentz model. (b) Comparison of transmission spectrum of three models.
resonator. The nanodisk resonance condition is given by [17,68,69]: ′
𝐾𝑑
𝐻𝑛(1) (𝐾𝑚 𝑟)
𝐻𝑛(1) (𝐾𝑚 𝑟) ( )1∕2 𝐾𝑑 = 𝐾 𝜀𝑑
= 𝐾𝑚
( )1∕2 𝐾 𝑚 = 𝐾 𝜀𝑚
𝐽𝑛′ (𝐾𝑑 𝑟) 𝐽𝑛 (𝐾𝑑 𝑟)
(3) (4) (5)
Where 𝐾𝑑 and 𝐾𝑚 are the wave vectors in the metal and dielectric nanodisk, respectively and K is the wave number. Also, 𝜀𝑚 and 𝜀𝑑 denote the relative dielectric constant of the metal and dielectric, respectively, ′ r is the radius of the nanodisk, 𝐻𝑛(1) and 𝐻𝑛(1) represent the first kind Hankel function of the order n and its derivation, 𝐽𝑛 and 𝐽𝑛′ are the first kind Bessel function of the order n and its derivation, respectively. The resonance wavelengths of a nanodisk cavity (with a radius equal to 𝑟) can be derived by solving Eq. (3). In other words, the resonance 1∕2 wavelengths depend on 𝑟 and 𝜀𝑑 . The parameters of the proposed structure are the inner radius of the resonator (𝑟𝑖𝑛 = 326 nm), the outer radius of the resonator (𝑟𝑜𝑢𝑡 = 376 nm), the gaps angle of the resonator (𝜃 = 75 degrees). Other parameters including 𝑤 and 𝑟 have been already explained. According to Fig. 7, the proposed structure generates one resonance mode at 742 nm wavelength with the maximum transmissions peak of 56%. It can be seen from Fig. 7 that the transmission peak value of mode 2 has increased about 27% and mode 1 is removed. We also intend to provide a view of the effect of different parameters on filter’s response. For this reason, the effect of increasing the disk
Fig. 5. Field profile of the magnitude of 𝐻𝑍 for wavelengths of (a) 𝜆1 = 1190 nm. (b) 𝜆2 = 742 nm. (c) 𝜆3 = 1000 nm.
radius on the transmission spectrum is investigated and presented in Fig. 8. Fig. 8(a) shows the transmission spectrum of the first proposed filter for different radii of the nanodisk. It is obvious that, by increasing 𝑟 from 270 to 350 nm, the transmission peak gradually increases, which is clearly shown in Fig. 8(b). Fig. 8(c) illustrates the relationship between the radius of the nanodisk and the resonance wavelength. It is easy to see that there is a nearly linear relationship between them. The FDTD simulation result is therefore in accordance with Eq. (3). As observed, by changing the radius of the nanodisk, the resonance mode can be tuned. Another structural parameter 𝜃 is a main factor which affects the transmission spectrum near the resonance mode. So, the transmission spectrums for various values of 𝜃 are investigated (Fig. 9(a)). It can be seen from Fig. 9(a) that by varying 𝜃 from 60 to 90 degrees, the peak 150
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Fig. 6. Schematic view of the first proposed plasmonic filter with a symmetric resonator topology.
Fig. 7. Comparison of transmission spectrum of the proposed filter and initial filter (a) using Drude–Lorentz model (b) using Drude model.
of the transmission spectrum is increased. Also, the relation between increasing the 𝜃 and increment of transmission peak for resonance mode is shown in Fig. 9(b). In Fig. 9(c), it is clear that by increasing the value of 𝜃, the resonance wavelength approximately remains unchanged. The profile of the magnitude of 𝐻𝑍 for wavelengths of 742 (resonance wavelength) and 1190 nm (non-resonance wavelength) are presented in Fig. 9. As it can be observed, the proposed filter transmits
Fig. 8. (a) Transmission spectrum for different values of ‘𝑟’ in first proposed filter. (b) Relationship between transmission peak and different values of ‘𝑟’. (c) Relationship between resonance wavelength and different values of ‘𝑟’.
151
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Fig. 10. Field profile of the magnitude of 𝐻𝑍 for wavelengths of (a) 𝜆2 = 742 nm. (b) 𝜆1 = 1190 nm.
3.2. The second proposed filter Based on the structure in previous section, another filter is proposed in Fig. 11. The only difference of this structure with the first proposed structure is 𝛼 degrees rotation of the two gaps of the resonator. The 𝛼 parameter value is set to be 22 degrees and other parameters have their previous values. According to Fig. 12, this structure has a resonance mode at 𝜆 = 1213 nm (almost equal to the first resonance wavelength of the initial structure) with the maximum transmissions peak of 35.2%. It is worth mentioning that this structure not only eliminates the second resonance mode of the initial structure but also enhances the maximum transmission peak of mode 1 as much as 18.2% (see Fig. 12). It should be noted that the different values of the disk radii can affect the transmission spectrum of this structure, too. So, the transmittance versus different values of r is shown in Fig. 13. When the nanodisk radius is increased, the obtained results show that the transmission peak is almost invariable and the resonance wavelength is linearly increasing. Fig. 13(b–c) show these relationships clearly. Accordingly, it can be said the resonance wavelength can be tuned by varying the value of 𝑟 in this case too. Finally, the magnetic profile of 𝐻𝑍 for wavelengths of 1213 nm (resonance wavelength) and 742 nm (non-resonance wavelength) is presented to clarify the operating mechanism of the second proposed filter. It should be noted that in Fig. 14 the resonance wavelength at 𝜆 = 1213 nm can be transmitted to the output port and the wavelength of 𝜆 = 742 nm cannot directed to the output port.
Fig. 9. (a) Transmission spectrum for different values of ‘𝜃’ in first proposed filter. (b) Relationship between transmission peak and different values of ‘𝜃’. (c) Relationship between resonance wavelength and different values of ‘𝜃’.
the second resonance mode of the initial structure (𝜆2 = 742 nm in Fig. 10(a)) and does not transmit the first resonance mode of the initial structure (𝜆1 = 1190 nm in Fig. 10(b)). 152
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Table 2 Comparisons between the proposed filters and other works. Ref.
𝑤 (nm)
𝑟 (nm)
𝑠 (nm)
Simulation model
Number of modes
𝜆 (nm)
Maximum transmittance (%)
𝑄-factor
[17]
50
200
20
Drude
2
520 & 816
82.3 & 60
32.5 & 54.4
[18]
50
410
8
Drude
2
956 & 1550
80.7 & 72.3
25.83 & 28.18
[19]
50
200
20
Drude
4
449 & 540 & 712 & 114
71.6 & 61 & 69 & 41
44.9 & 38.5 & 44.5 & 8.1
[20]
50
135 & 260
–
Drude
3
595 & 880 & 1550
11.85 & 55.45 & 88.66
9.15 & 11.89 & 25
[21]
50
150
10
[22]
50
305
5
[23] [68]
50 50
400 200
The first proposed filter
50
The first proposed filter
50
The second proposed filter The second proposed filter
Drude–Lorentz
2
583.5 & 1145
051 & 39
11.67 & 14.31
–
2
1110 & 1210
64.8 & 70.8
11.32 & 14
5 30
Drude Drude
1 2
940 517.3 & 803.4
89.8 74.3 & 45.7
23.5 64.66 & 114.7
310
16
Drude–Lorentz
1
742
56
22.48
310
16
Drude
1
732.5
89.9
26.2
50
310
16
Drude–Lorentz
1
1213
35.2
26.36
50
310
16
Drude
1
1197
56.4
36.27
Fig. 11. Schematic view of the second proposed plasmonic filter.
4. Results and comparison In order to be able to have a better prospect of obtained results, these results are compared with other works in this section. For this purpose, Table 2 summarizes the specifications of the nanodisk-based plasmonic filters reported in the literature with the current filters. These parameters include the waveguide widths (𝑤), nanodisk radiuses (𝑟), spacing between the waveguides and the nanodisk (𝑠), simulation model, number of modes (𝑛), resonance wavelengths (𝜆), maximum transmissions (𝑇max ), and quality factor (Q-factor). As previously mentioned, to have more accurate results, it is necessary to use the Drude– Lorentz model. Therefore the reported values which use the Drude in Table 2 such as 𝑇max and Q-factor are not valid. Based on this table, only the structure reported in [21] uses Drude–Lorentz model. However, it has a lower transmittance compared to our first proposed structure and it is not single-mode. To be able to compare the proposed filter to those reported in the literature, desirable filter characteristics have to be briefly reviewed. The figures of merit which are usually associated with filters are as follows: First, there is the filter transmittance. An ideal filter has to have a 0 dB transmittance for the resonance wavelength. It is equal to a 100% transmission ratio and requires the filter to have zero loss and input reflection. If the Drude model is used to simulate the proposed filter an 89.9% maximum transmittance is observed. It the highest among the reported structures in the literature which have used the Drude model. Second the filters should have a Lorentzian (bell-shaped) frequency spectrum. It means that the filter should have a perfect 100% transmission ratio for the resonance wavelength while maintaining zero
Fig. 12. Comparison of transmission spectrum of the proposed filter and initial filter (a) using Drude–Lorentz model (b) using Drude model.
transmission for undesired wavelengths. For many of the filter structures reported in the literature, high transmission ratios are obtained for the resonance frequency, however the frequency spectrums are not bell shaped, i.e. a residual transmittance remains for the undesired 153
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Fig. 14. Field profile of the magnitude of 𝐻𝑍 for wavelengths of (a) 𝜆1 = 742 nm. (b) 𝜆2 = 1213 nm.
than 7% for such wavelengths. In optical communication applications, such a transmittance can cause leakage from adjacent optical channels and be disastrous. The structures proposed here all have bell-shaped transmission spectrums. Third is the Q-factor of the filter. The quality factor required for a filter depends on its application. Although for some applications high Q-factors are needed but having a higher Q-factor is not always synonymous to better filter. The quality factors of the filters reported in this paper is nearly on a same order of magnitude with those listed in Table 2. Fourth is being single or multi-mode. A single mode filter is easier to cope with in circuits with higher complexity. For the topologies reported in this paper the main goal has been to obtain single-mode filters. Finally, there is the controversial issue of sensitivity of a filter’s frequency response to the variation of its parameters. For sensor applications, if a filter has a higher sensitivity to filter parameters, for example the refractive index of the dielectric which is employed in the plasmonic structure, higher resolutions are expected. However, for many other applications, lower sensitivity to filter parameter variations, such as those caused by lithography and fabrication errors leads to more robust and reliable devices. As seen in Figs. 9 and 13, for the proposed filters the parameters of the proposed filters have been swept and the results are superimposed to provide a better insight to filter’s sensitivity. Based on the mentioned points the proposed filter is single-mode, it has bell-shaped spectrum, it has a very high transmittance. Furthermore, the resonance wavelength can be easily tuned by changing the radius of the nano-disk. There is a perfectly linear relationship between the disk
Fig. 13. (a) Transmission spectrum for different values of ‘𝑟’ in the second proposed filter. (b) Relationship between maximum transmission and different values of ‘𝑟’. (c) Relationship between resonance wavelength and different values of ‘𝑟’.
wavelengths. As an example for the filter reported in [23] which has the highest transmittance reported in the literature, the bell-shaped transmission curve becomes zero at the right side (higher wavelengths) but it fails at the left side. It still shows a transmission ratio more 154
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radius and the resonance wavelength which makes the design process straightforward (Fig. 8(c) and Fig. 13(c)). Finally the filter topology is simple and it does not require extra stubs and does not occupy a large area.
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5. Conclusion Two single mode plasmonic filters were proposed in this paper. The structures were based on MIM waveguides laterally coupled to a disk resonator. Parenthesis shaped adjunctions were added to the resonator topology to increase the transmittance and provide a single mode behavior. The structures were numerically simulated using the FDTD method and a Drude–Lorentz model for silver. Although the first and second proposed filters are designed for the 742 nm and 1213 nm wavelengths but they can be easily tuned for other desired wavelengths as well. It makes them a good candidate for realization of plasmonic optical demultiplexers. Acknowledgments The authors would also like to thank the reviewers for their valuable comments. References [1] W.L. Barnes, A. Dereux, T.W. Ebbesen, Surface plasmon subwavelength optics, Nature 424 (6950) (2003) 824–830. [2] S.I. Bozhevolnyi, V.S. Volkov, E. Devaux, J.Y. Laluet, T.W. Ebbesen, Channel plasmon subwavelength waveguide components including interferometers and ring resonators, Nature 440 (7083) (2006) 508–511. [3] D.K. Gramotnev, S.I. Bozhevolnyi, Plasmonics beyond the diffraction limit, Nat. Photon. 4 (2) (2010) 83–91. [4] S. Luo, B. Li, D. Xiong, D. Zuo, X. Wang, A high performance plasmonic sensor based on metal–insulator–metal waveguide coupled with a double-cavity structure, Plasmonics 12 (2) (2017) 223–227. [5] Y. Xie, Y. Huang, W. Xu, W. Zhao, C. He, A plasmonic temperature-sensing structure based on dual laterally side-coupled hexagonal cavities, Sensors 16 (5) (2016) 706. [6] M.T.H. Azar, M. Zavvari, A. Arashmehr, Y. Zehforoosh, P. Mohammadi, Design of a high-performance metal–insulator–metal plasmonic demultiplexer, J. Nanophoton. 11 (2) (2017) 026002–026002. [7] Y.Y. Xie, C. He, J.C. Li, T.T. Song, Z.D. Zhang, Q.R. Mao, Theoretical investigation of a plasmonic demultiplexer in mim waveguide crossing with multiple side-coupled hexagonal resonators, IEEE Photon. J. 8 (5) (2016) 1–12. [8] H. Lu, X. Liu, L. Wang, Y. Gong, D. Mao, Ultrafast all-optical switching in nanoplasmonic waveguide with kerr nonlinear resonator, Opt. Express 19 (4) (2011) 2910– 2915. [9] A.N. Taheri, H. Kaatuzian, Numerical investigation of a nano-scale electro-plasmonic switch based on metal–insulator–metal stub filter, Opt. Quantum Electron. 47 (2) (2015) 159–168. [10] C. Haffner, W. Heni, Y. Fedoryshyn, J. Niegemann, A. Melikyan, D.L. Elder, et al., Allplasmonic Mach–Zehnder modulator enabling optical high-speed communication at the microscale, Nat. Photon. 9 (8) (2015) 525–528. [11] C.H. Chen, K.S. Liao, 1xN plasmonic power splitters based on metal–insulator–metal waveguides, Opt. Express 21 (4) (2013) 4036–4043. [12] D. Xiang, W. Li, MIM plasmonic waveguide splitter with tooth-shaped structures, J. Modern Opt. 61 (3) (2014) 222–226. [13] G. Magno, M. Grande, V. Petruzzelli, A. D’Orazio, Numerical analysis of the coupling mechanism in long-range plasmonic couplers at 1.55 μm, Opt. Pap. 38 (1) (2013) 46–48. [14] W. Zhang, G.G. Zheng, X.Y. Li, Design of light trapping structures for light-absorption enhancement in thin film solar cells, Optik 124 (16) (2013) 2531–2534. [15] G. Zheng, L. Xu, Y. Liu, Tunable plasmonic filter with circular metal–insulator–metal ring resonator containing double narrow gaps, Pramana 86 (5) (2016) 1091–1097. [16] B. Liu, Y.F. Liu, S.J. Li, X.D. He, Rotation and conversion of transmission mode based on a rotatable elliptical core ring resonator, Opt. Commun. 369 (2016) 44–49. [17] H. Lu, X. Liu, D. Mao, L. Wang, Y. Gong, Tunable band-pass plasmonic waveguide filters with nanodisk resonators, Opt. Express 18 (17) (2010) 17922–17927. [18] G. Zhan, R. Liang, H. Liang, J. Luo, R. Zhao, Asymmetric band-pass plasmonic nanodisk filter with mode inhibition and spectrally splitting capabilities, Opt. Express 22 (8) (2014) 9912–9919. [19] A. Ghorbanian, A.M. Kashani, A.M. Javan, The effects of silver slabs in nanodisk resonator of plasmonic tunable band-pass filter, Optik 127 (4) (2016) 1884–1888. [20] Z. Chen, J. Chen, Y. Li, J. Qian, J. Qi, J. Xu, Q. Sun, Highly efficient narrow-band plasmonic waveguide filter based on cascaded slot cavities, Chin. Opt. Pap. 11 (11) (2013) 112401.
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Realization of single-mode plasmonic bandpass filters using improved nanodisk resonators Shiva Khani, Mohammad Danaie *, Pejman Rezaei Electrical and computer Engineering Faculty, Semnan University, Semnan, Iran
ARTICLE
INFO
Keywords: Plasmonic filter Bandpass filter Optical resonator Drude–Lorentz Metal–insulator–metal (MIM) waveguide
ABSTRACT In this paper, symmetric and asymmetric plasmonic bandpass filter (BPF) topologies based on the metal– insulator–metal (MIM) configuration are proposed. These filters are numerically investigated using finite difference time domain (FDTD) method. The metal and dielectric used for the realization of the filters are silver and air, respectively. The real and imaginary parts of silver’s permittivity used in numerical simulation are based on the Drude–Lorentz and Palik models. Both structures are composed of two waveguides, a nanodisk, and parenthesis-shaped adjunctions. The inclusion of symmetrical and nonsymmetrical adjunctions results in single mode filters with higher transmission peaks compared to the original nanodisk-based filter. To provide better physical insight, various structural parameters of the filter are changed and their effects on filter’s response are presented. It is observed that the resonance mode of proposed BPFs can be tuned by changing the nanodisk resonator radius. Such structures can be employed in various plasmonic devices such as multiplexers and demultiplexers for optical communication purposes.
1. Introduction Surface plasmon polaritons (SPPs) are the electromagnetic excitations that propagate at the interface between metals and dielectric materials. Due to the remarkable capability to manipulate light in a nanoscale domain, SPPs are considered as an efficient basis for the realization of highly integrated optical circuits [1–3]. Growing demands for such elements motivates researchers to design numerous integrated nanoscale optical circuits and devices, such as optical filters, sensors [4,5], demultiplexers [6,7], switches [8,9], Mach–Zehnder modulators [10], splitters [11,12], couplers [13] and so on using the plasmonic technique. Also, at the plasmon resonance, the scattering and absorption cross sections of nanoparticles are enhanced. It can be used to increase the efficiency of solar cells [14]. Among the most important optical devices that have found wide application for wavelength selection are plasmonic filters. Such devices have been the center of attention in recent years. Among the plasmonic filters based on MIM structures, bandpass filters (BPFs) and bandstop filters (BSFs) are of high importance. In order to design an optical BPF the easiest method is to design an optical cavity which has a resonance frequency equal to the central frequency of the BPF. Such cavities are then laterally coupled to the input and output waveguides. If the resonance profile of the cavity matches the profile of the mode which propagates through the waveguides, a BPF can be conceived. To design BSFs, usually the cavity *
is side-coupled to a waveguide. The most common type of resonator used for realization of MIM plasmonic filters is a circular nanodisk. Such a disk is easy to implement and the resonance frequency can be tuned by variation of its radius. Over the last few years, many plasmonic waveguide filters based on ring-shaped or circular resonator configurations have been proposed. For example using ring resonators, different BPFs have been designed in [15,16]. Nanodisk resonators were used to design BPFs in [17,18]. To investigate the effects of silver slabs in nanodisk resonator, a plasmonic BPF is proposed in [19] that leads to reduction of the filter dimensions. To achieve a higher Q factor, a filter composed of three cascaded diskshaped cavities is presented in [20]. Drude model is used for simulation of metal behavior in [20]. Also, nanodisk resonators have been presented in a variety of different versions for filter applications which include hollow-core circular ring resonator [21], a nanodisk resonator coupled with stub resonator [22], L-shaped filter based on nanodisk resonator [23], and stub waveguide with nanodisk and fabry–perot resonator [24]. Other approaches such as plasmonic branch-shaped MIM waveguide with a triangular-annular channel [25], T-shaped plasmonic resonator [26], Archimedes’ spiral nanostructure [27], and isosceles trapezoid cavities [28] have also been presented. The metal–insulator–metal configuration is a relatively old technique for creation of microwave micro-strip filters [29–31]. In an
Corresponding author. E-mail address: [email protected] (M. Danaie).
https://doi.org/10.1016/j.optcom.2018.03.047 Received 23 December 2017; Received in revised form 17 March 2018; Accepted 20 March 2018 0030-4018/© 2018 Elsevier B.V. All rights reserved.
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integrated circuit the dielectric substrate and metal layers are already accessible. The same metal layers that convey the electric current between different transistors can be used to transport optical signals. MIM structures can be integrated with other electrical or optical components on a single chip. For example the photo-detectors, or laser sources needed for such structures can be implemented on-chip. An analog to digital convertor and a processor may also be integrated together to process the photodetector data. Although high quality optical filters and demultiplexers can be designed and implemented using photonic crystals [32,33], plasmonic devices can occupy far less area than similar photonic crystal devices [34–36], due to their ability to overcome the diffraction limit. Most importantly light amplification is possible in Schottky junction-based plasmonic waveguides. It creates unique capabilities [37,38]. It provides the ability to design a new genre of devices based on MIM topology. In case a Kerr-type nonlinear material is used as the dielectric, the resonance frequency of such filters can be tuned optically. Although Zheng et al. have proposed this technique for tuning the resonance filter the idea can also be employed for realization of all-optical switches [39]. Ring shaped structures can also be employed for design of power splitters [40]. Due to the relatively recent attention and interest to plasmonic filters, a huge amount of the results reported in the literature are still from the simulations only. Lots of such publications use plasma or Drude model for silver or gold in their finite difference time domain (FDTD) simulations. Such models provide nearly accurate results when simulating waveguides. However, we will show that they do not provide accurate outcome when the plasmonic structure contains a resonator. Here Drude, Drude–Lorentz and Palik models are used for numerical characterization of silver and the results are compared together. These methods will be briefly introduced in Section 2. It is observed that Palik and Drude–Lorentz model provide very similar results while the Drude model, which is very commonly used in many papers, is totally unacceptable for simulation of resonator based filters. There are some deficiencies associated with the waveguide-coupled disk resonator filter topologies reported in the literature. The first one is that for the wavelength range of 600 nm–2000 nm, which is the range reported in most of the papers, the central cavity is dual mode. Having a single-mode resonator is much more helpful when designing more complex structures such as optical demultiplexers. The second one is when such structures only provide high transmittance when the inaccurate Drude model (also known as plasma model) is used. We will show that when more accurate models are used the transmittance will be decreased to half of its original value. In this paper, parenthesisshaped adjunctions are added to the original resonator structure. Two types of filters are proposed based on using symmetric or asymmetric parentheses. These filters eliminate the lower and higher wavelength resonance modes respectively and provide a single mode behavior for the 600 nm–2000 nm range. Furthermore, the inclusion of the parenthesis shaped structure improves the transmittance of the filter compared to the original nanodisk filters. The proposed structures can thus be used for design of much more complex structures such as sensors and demultiplexers. Based on the direction of the light propagation, the plasmonic structures can be categorized into two main groups. The light can either propagate along the interface of metal or dielectric layers (in-plane propagation) or be perpendicular to a slab comprising an array of metal and dielectric shapes. In plasmonic crystals which use an array of metallic nanoparticles deposited on a dielectric layer or equally an array of holes etched in a metal surface, the incident light is usually perpendicular to the surface. For the first case, usually a plasmonic waveguide is created using a dielectric layer which is sandwiched between two metal layers. The MIM waveguides seem more appealing for those who seek to have all-optical integrated devices. They can be used for implementing much more complex structures such as demultiplexers [41], antennas [42,43], ring resonators [44,45], memristors [46], circulators [47], switches [48] etc. They can also be integrated with other electronic devices. The fact
Fig. 1. Schematic configuration of the initial plasmonic filter.
that such structures can be used to create Schottky plasmonic based amplifiers seems very promising [37,38]. Plasmonic crystals can act as an absorbers based on the localized surface plasmon resonance [49]. They are widely fabricated and used for bio-sensing applications [50,51]. Since the incident light is perpendicular to the surface, no coupling stage is needed for such structure. Only a light source and a photo detector are enough. There are lots of experimental papers published in this area. Some of them are reviewed in [52]. Although such structures are mostly used for sensors, they can also be used to improve solar cells [53,54], create nanolenses [55–57] etc. They are easy to fabricate. The fabrication methods and setups for such crystals are reviewed in [58]. To be able to use MIM waveguides, the optical fiber has to be coupled to it. Unlike the first method which involves metal layer deposition and lithography, it is much more expensive to fabricate and measure the results in these types of plasmonic devices. There has been numerous methods proposed to address the problem of coupling the light to the MIM waveguides. Experimentally tested solutions can be found in [59–62]. They usually involve an input grating coupler. In [63] experimental data and FDTD results for an MIM filter composed of SiO2 and Ag have been compared. Good agreement is observed between the simulation and measurement results in [63]. A same setup can be used for our proposed structure for experimental measurements. The rest of this paper is as follows: In Section 2, the original resonator-based filter is discussed; Drude, Drude Lorentz and Palik models are briefly reviewed and compared together. In Section 3 the proposed structure is presented. Section 4 discusses the results and the last section is devoted to conclusions. 2. Disk resonator-based filter Fig. 1 shows the topology of the initial plasmonic BPF consisting of two slits (two semi-infinite separated waveguides) and a nanodisk between them. The parameters of the structure are as follows: the radius of the nanodisk (𝑟 = 310 nm), the widths of the waveguides ( 𝑤 = 50 nm), and the coupling distances between the waveguides and the nanodisk (𝑠 = 16 nm). The insulator material in the white areas is set to be air with relative permittivity of 𝜀𝑑 = 1. The material of the gray areas is assumed silver, which is characterized by Drude, Drude–Lorentz, and Palik models that will be discussed shortly after. Silver due to its low Ohmic resistance, compared to other metals, is one of the most widely used metals in plasmonic. 2.1. Drude model The Drude model for the permittivity of a metal, which is the most simplified model, can be expressed as follows [16,23]: 𝜀𝑚 (𝜔) = 𝜀∞ − 148
𝜔2𝑝 𝜔(𝜔 − 𝑗𝛾)
(1)
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Fig. 3. Transmission spectrum with Palik model.
Fig. 2. Transmission spectrum obtained using the Drude model.
Table 1 Parameters of the Drude–Lorentz model for silver.
Where 𝜀∞ is the dielectric permittivity at infinite frequency, 𝜔𝑝 is the bulk plasma frequency of metal, 𝛾 is the electron collision frequency, 𝜔 is the√angular frequency of incident light, and 𝑗 is imaginary number (𝑗 = −1). The parameters of the Drude model for silver are assumed to be 𝜀∞ = 3.7, 𝜔𝑝 = 9.1 ev, and 𝛾 = 0.081 ev, respectively. The transmission characteristics of the filter shown in Fig. 1, are investigated using FDTD. Fig. 2 shows the filter transmission spectrum which is comprised of two resonance modes (mode 1 and mode 2) at the 1174 and 732 nm wavelengths, respectively. Because there are internal losses in the nanodisk and waveguides in the metal slits, the transmission peaks do not reach unity and the maximum transmissions of the two modes are 37% and 70%, respectively. The Drude model is accurate for a limited range of wavelengths [64,65]. Accordingly, for more precise description of the optical properties of metals through the electromagnetism spectrum, it is better that experimental refractive indices or more accurate models be used.
Another model that can be used for description of the permittivity of silver over a wider wavelength range is the Palik model which is introduced in [65]. In [65], different sample preparation methods and data from silver samples exposed to air are utilized by four research groups. There is however small inconsistencies in measuring representative properties of pristine silver due to the tarnishing caused by air exposure [66,67]. The transmission spectrum of the initial structure using Palik model is shown in Fig. 3. As seen, the transmission peak values are 13% and 32.6% at wavelengths of 1174 and 738 nm for mode 1 and mode 2, respectively. These values are much less than what was expected from the Drude model.
𝜔 (𝜔 − 𝑗𝛾)
5 ∑ 2 𝑛=1 𝜔𝑛
𝑓𝑛 𝜔2𝑛 − 𝜔2 + 𝑗𝜔𝛾𝑛
𝑓𝑛
939.62 109.29 15.71 221.49 584.91
7.9247 0.5013 0.0133 0.8266 1.1133
In the previous section the conventional filter structure was presented. One of the main problems of such a structure is its poor transmittance after the Drude–Lorentz model is used for numerical simulation. In this section to improve the initial structure and to design the proposed filter, these goals are pursued: First of all more accurate simulation with other models such as Palik and Drude–Lorentz. Secondly converting the two-mode filter to two one-mode filters, where each of these filters passes one of the modes and finally increasing the transmission peaks.
Here, the dielectric constant of the silver is characterized by the seven-pole Drude–Lorentz model, known to be reasonably accurate in the wavelength range of 200 to 2000 nm [21]. The permittivity model is presented as: +
𝛾𝑛 (THz)
197.3 1083.5 1979.1 4392.5 9812.1
3. The proposed structure and simulation results
2.3. Drude–Lorentz model
𝜔2𝑝
𝜔n (THz)
1 2 3 4 5
Fig. 4(a) illustrates the simulated transmittance using Drude–Lorentz model. It is superimposed on the results of Drude and Palik models in Fig. 4(b). As can be seen when the Drude–Lorentz or Palik models are used the transmission peaks are significantly reduced. Transmission peak values of two modes are 17% and 29%, respectively, which are too low for a filter. Therefore, the goal of this paper is to improve the transmittance. Fig. 5(a–c) shows the field profile of magnitude of 𝐻𝑍 for the initial structure at wavelengths of 1190 and 742 nm (resonance wavelengths of mode 1 and mode 2 in Drude–Lorentz model), and 1000 nm (nonresonance wavelength). As seen, in terms of 𝐻𝑍 , two resonance modes of the structure corresponding to wavelengths of 𝜆1 = 1190 and 𝜆2 = 742 nm (Fig. 5(a–b)) have appeared in the nanodisk and are transmitted to the output port of the filter. Also, Fig. 5(c) shows the field profile for 𝜆 = 1000 nm, which cannot be transmitted to the output port.
2.2. Palik model
𝜀𝑚 (𝜔) = 1 −
𝑛
3.1. The first proposed filter
(2)
To achieve better filters, parenthesis shaped adjunctions are added to the initial structure. The schematic illustration of the first proposed structure which passes the second mode is shown in Fig. 6. When the incident optical waves propagate forward along a waveguide, part of the energy will be reflected and the other part will be coupled into the
Where, 𝜔𝑝 = 2002.6 THz is the bulk plasma frequency of silver and 𝛾 = 11.61 THz is the damping constant. Other parameters include quantities of resonant frequencies 𝜔𝑛 , damping constants 𝛾𝑛 , and weights 𝑓𝑛 are listed in Table 1. It should be noted these values require to be changed to the Radian unit (multiplied by 2𝜋) before substitution in (2). 149
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Fig. 4. (a) Transmission spectrum with Drude–Lorentz model. (b) Comparison of transmission spectrum of three models.
resonator. The nanodisk resonance condition is given by [17,68,69]: ′
𝐾𝑑
𝐻𝑛(1) (𝐾𝑚 𝑟)
𝐻𝑛(1) (𝐾𝑚 𝑟) ( )1∕2 𝐾𝑑 = 𝐾 𝜀𝑑
= 𝐾𝑚
( )1∕2 𝐾 𝑚 = 𝐾 𝜀𝑚
𝐽𝑛′ (𝐾𝑑 𝑟) 𝐽𝑛 (𝐾𝑑 𝑟)
(3) (4) (5)
Where 𝐾𝑑 and 𝐾𝑚 are the wave vectors in the metal and dielectric nanodisk, respectively and K is the wave number. Also, 𝜀𝑚 and 𝜀𝑑 denote the relative dielectric constant of the metal and dielectric, respectively, ′ r is the radius of the nanodisk, 𝐻𝑛(1) and 𝐻𝑛(1) represent the first kind Hankel function of the order n and its derivation, 𝐽𝑛 and 𝐽𝑛′ are the first kind Bessel function of the order n and its derivation, respectively. The resonance wavelengths of a nanodisk cavity (with a radius equal to 𝑟) can be derived by solving Eq. (3). In other words, the resonance 1∕2 wavelengths depend on 𝑟 and 𝜀𝑑 . The parameters of the proposed structure are the inner radius of the resonator (𝑟𝑖𝑛 = 326 nm), the outer radius of the resonator (𝑟𝑜𝑢𝑡 = 376 nm), the gaps angle of the resonator (𝜃 = 75 degrees). Other parameters including 𝑤 and 𝑟 have been already explained. According to Fig. 7, the proposed structure generates one resonance mode at 742 nm wavelength with the maximum transmissions peak of 56%. It can be seen from Fig. 7 that the transmission peak value of mode 2 has increased about 27% and mode 1 is removed. We also intend to provide a view of the effect of different parameters on filter’s response. For this reason, the effect of increasing the disk
Fig. 5. Field profile of the magnitude of 𝐻𝑍 for wavelengths of (a) 𝜆1 = 1190 nm. (b) 𝜆2 = 742 nm. (c) 𝜆3 = 1000 nm.
radius on the transmission spectrum is investigated and presented in Fig. 8. Fig. 8(a) shows the transmission spectrum of the first proposed filter for different radii of the nanodisk. It is obvious that, by increasing 𝑟 from 270 to 350 nm, the transmission peak gradually increases, which is clearly shown in Fig. 8(b). Fig. 8(c) illustrates the relationship between the radius of the nanodisk and the resonance wavelength. It is easy to see that there is a nearly linear relationship between them. The FDTD simulation result is therefore in accordance with Eq. (3). As observed, by changing the radius of the nanodisk, the resonance mode can be tuned. Another structural parameter 𝜃 is a main factor which affects the transmission spectrum near the resonance mode. So, the transmission spectrums for various values of 𝜃 are investigated (Fig. 9(a)). It can be seen from Fig. 9(a) that by varying 𝜃 from 60 to 90 degrees, the peak 150
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Fig. 6. Schematic view of the first proposed plasmonic filter with a symmetric resonator topology.
Fig. 7. Comparison of transmission spectrum of the proposed filter and initial filter (a) using Drude–Lorentz model (b) using Drude model.
of the transmission spectrum is increased. Also, the relation between increasing the 𝜃 and increment of transmission peak for resonance mode is shown in Fig. 9(b). In Fig. 9(c), it is clear that by increasing the value of 𝜃, the resonance wavelength approximately remains unchanged. The profile of the magnitude of 𝐻𝑍 for wavelengths of 742 (resonance wavelength) and 1190 nm (non-resonance wavelength) are presented in Fig. 9. As it can be observed, the proposed filter transmits
Fig. 8. (a) Transmission spectrum for different values of ‘𝑟’ in first proposed filter. (b) Relationship between transmission peak and different values of ‘𝑟’. (c) Relationship between resonance wavelength and different values of ‘𝑟’.
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Fig. 10. Field profile of the magnitude of 𝐻𝑍 for wavelengths of (a) 𝜆2 = 742 nm. (b) 𝜆1 = 1190 nm.
3.2. The second proposed filter Based on the structure in previous section, another filter is proposed in Fig. 11. The only difference of this structure with the first proposed structure is 𝛼 degrees rotation of the two gaps of the resonator. The 𝛼 parameter value is set to be 22 degrees and other parameters have their previous values. According to Fig. 12, this structure has a resonance mode at 𝜆 = 1213 nm (almost equal to the first resonance wavelength of the initial structure) with the maximum transmissions peak of 35.2%. It is worth mentioning that this structure not only eliminates the second resonance mode of the initial structure but also enhances the maximum transmission peak of mode 1 as much as 18.2% (see Fig. 12). It should be noted that the different values of the disk radii can affect the transmission spectrum of this structure, too. So, the transmittance versus different values of r is shown in Fig. 13. When the nanodisk radius is increased, the obtained results show that the transmission peak is almost invariable and the resonance wavelength is linearly increasing. Fig. 13(b–c) show these relationships clearly. Accordingly, it can be said the resonance wavelength can be tuned by varying the value of 𝑟 in this case too. Finally, the magnetic profile of 𝐻𝑍 for wavelengths of 1213 nm (resonance wavelength) and 742 nm (non-resonance wavelength) is presented to clarify the operating mechanism of the second proposed filter. It should be noted that in Fig. 14 the resonance wavelength at 𝜆 = 1213 nm can be transmitted to the output port and the wavelength of 𝜆 = 742 nm cannot directed to the output port.
Fig. 9. (a) Transmission spectrum for different values of ‘𝜃’ in first proposed filter. (b) Relationship between transmission peak and different values of ‘𝜃’. (c) Relationship between resonance wavelength and different values of ‘𝜃’.
the second resonance mode of the initial structure (𝜆2 = 742 nm in Fig. 10(a)) and does not transmit the first resonance mode of the initial structure (𝜆1 = 1190 nm in Fig. 10(b)). 152
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Table 2 Comparisons between the proposed filters and other works. Ref.
𝑤 (nm)
𝑟 (nm)
𝑠 (nm)
Simulation model
Number of modes
𝜆 (nm)
Maximum transmittance (%)
𝑄-factor
[17]
50
200
20
Drude
2
520 & 816
82.3 & 60
32.5 & 54.4
[18]
50
410
8
Drude
2
956 & 1550
80.7 & 72.3
25.83 & 28.18
[19]
50
200
20
Drude
4
449 & 540 & 712 & 114
71.6 & 61 & 69 & 41
44.9 & 38.5 & 44.5 & 8.1
[20]
50
135 & 260
–
Drude
3
595 & 880 & 1550
11.85 & 55.45 & 88.66
9.15 & 11.89 & 25
[21]
50
150
10
[22]
50
305
5
[23] [68]
50 50
400 200
The first proposed filter
50
The first proposed filter
50
The second proposed filter The second proposed filter
Drude–Lorentz
2
583.5 & 1145
051 & 39
11.67 & 14.31
–
2
1110 & 1210
64.8 & 70.8
11.32 & 14
5 30
Drude Drude
1 2
940 517.3 & 803.4
89.8 74.3 & 45.7
23.5 64.66 & 114.7
310
16
Drude–Lorentz
1
742
56
22.48
310
16
Drude
1
732.5
89.9
26.2
50
310
16
Drude–Lorentz
1
1213
35.2
26.36
50
310
16
Drude
1
1197
56.4
36.27
Fig. 11. Schematic view of the second proposed plasmonic filter.
4. Results and comparison In order to be able to have a better prospect of obtained results, these results are compared with other works in this section. For this purpose, Table 2 summarizes the specifications of the nanodisk-based plasmonic filters reported in the literature with the current filters. These parameters include the waveguide widths (𝑤), nanodisk radiuses (𝑟), spacing between the waveguides and the nanodisk (𝑠), simulation model, number of modes (𝑛), resonance wavelengths (𝜆), maximum transmissions (𝑇max ), and quality factor (Q-factor). As previously mentioned, to have more accurate results, it is necessary to use the Drude– Lorentz model. Therefore the reported values which use the Drude in Table 2 such as 𝑇max and Q-factor are not valid. Based on this table, only the structure reported in [21] uses Drude–Lorentz model. However, it has a lower transmittance compared to our first proposed structure and it is not single-mode. To be able to compare the proposed filter to those reported in the literature, desirable filter characteristics have to be briefly reviewed. The figures of merit which are usually associated with filters are as follows: First, there is the filter transmittance. An ideal filter has to have a 0 dB transmittance for the resonance wavelength. It is equal to a 100% transmission ratio and requires the filter to have zero loss and input reflection. If the Drude model is used to simulate the proposed filter an 89.9% maximum transmittance is observed. It the highest among the reported structures in the literature which have used the Drude model. Second the filters should have a Lorentzian (bell-shaped) frequency spectrum. It means that the filter should have a perfect 100% transmission ratio for the resonance wavelength while maintaining zero
Fig. 12. Comparison of transmission spectrum of the proposed filter and initial filter (a) using Drude–Lorentz model (b) using Drude model.
transmission for undesired wavelengths. For many of the filter structures reported in the literature, high transmission ratios are obtained for the resonance frequency, however the frequency spectrums are not bell shaped, i.e. a residual transmittance remains for the undesired 153
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Fig. 14. Field profile of the magnitude of 𝐻𝑍 for wavelengths of (a) 𝜆1 = 742 nm. (b) 𝜆2 = 1213 nm.
than 7% for such wavelengths. In optical communication applications, such a transmittance can cause leakage from adjacent optical channels and be disastrous. The structures proposed here all have bell-shaped transmission spectrums. Third is the Q-factor of the filter. The quality factor required for a filter depends on its application. Although for some applications high Q-factors are needed but having a higher Q-factor is not always synonymous to better filter. The quality factors of the filters reported in this paper is nearly on a same order of magnitude with those listed in Table 2. Fourth is being single or multi-mode. A single mode filter is easier to cope with in circuits with higher complexity. For the topologies reported in this paper the main goal has been to obtain single-mode filters. Finally, there is the controversial issue of sensitivity of a filter’s frequency response to the variation of its parameters. For sensor applications, if a filter has a higher sensitivity to filter parameters, for example the refractive index of the dielectric which is employed in the plasmonic structure, higher resolutions are expected. However, for many other applications, lower sensitivity to filter parameter variations, such as those caused by lithography and fabrication errors leads to more robust and reliable devices. As seen in Figs. 9 and 13, for the proposed filters the parameters of the proposed filters have been swept and the results are superimposed to provide a better insight to filter’s sensitivity. Based on the mentioned points the proposed filter is single-mode, it has bell-shaped spectrum, it has a very high transmittance. Furthermore, the resonance wavelength can be easily tuned by changing the radius of the nano-disk. There is a perfectly linear relationship between the disk
Fig. 13. (a) Transmission spectrum for different values of ‘𝑟’ in the second proposed filter. (b) Relationship between maximum transmission and different values of ‘𝑟’. (c) Relationship between resonance wavelength and different values of ‘𝑟’.
wavelengths. As an example for the filter reported in [23] which has the highest transmittance reported in the literature, the bell-shaped transmission curve becomes zero at the right side (higher wavelengths) but it fails at the left side. It still shows a transmission ratio more 154
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radius and the resonance wavelength which makes the design process straightforward (Fig. 8(c) and Fig. 13(c)). Finally the filter topology is simple and it does not require extra stubs and does not occupy a large area.
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