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Abstract
In the present study, we have devised and conducted an investigation into a real-time tunable notched waveguide, employing a voltage-controllable plasmonic resonator. This plasmonic resonator is meticulously engineered from a ferroelectric substrate featuring a compound multilayer structure, thereby conferring it with the remarkable capability of flexible permittivity control. Furthermore, we have implemented two non-intersecting Archimedean spiral electrodes on the surface of the ferroelectric substrate, dedicated to applying the bias field onto the controllable plasmonic ferroelectric resonator (CPFR). Notably, our system affords the capability to finely tune both the magnetic and electric modes, achieving precise adjustments of 8.7% and 11%, respectively. The performance is complemented by minimal insertion loss, rapid response times, and a broad range of potential applications, positioning it as a candidate for a diverse array of notched waveguide scenarios.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The natural surface plasmons (SPs) can be divided into two categorizations, namely, the surface plasmon polaritons (SPPs) and the localized surface plasmons (LSPs). The former is a mixed resonance state of photons and electrons, which can propagate on metal surfaces with strong field binding ability on the interface and exponentially decaying property in the normal direction [1,2]. Meanwhile, the latter is limited near the interface of small metal particles and is highly sensitive to particle geometry and local dielectric environment, which possesses field enhancement characteristics and has been exploited in a broad range of technological areas, such us optical antenna [3,4], optoelectronic devices [5–10], surface-enhanced Raman scattering [11,12] and sensors [13–19]. However, since EM wave is difficult to penetrate the metal and resonate with free electrons at much lower frequency band, in THz or GHz band, due to PEC property of metal, the above two SPs do not be motivated [20,21]. In 2004, J. B. Pendry et al. firstly proposed spoof SPPs (SSPPs) conception, with the periodically structured engineering on metal surface, which can mimic all the characteristics of nature SPP in THz or GHz frequency band [22]. With the opening up of the artificial plasmonic investigation, spoof LSPs (SLSPs) structures, as an extension of the SSPPs structure, have also been proposed [23]. Especially, Tiejun Cui et al put forward the design of ultra-thin SLSPs structures, which not only provides great convenience for industry applications [23,24] but also allows for the first investigation of multi-level resonance, where the higher-order modes distributed along the radial direction of the SLSPs structure [25].
However, most of these technologies are static circuits and do not provide real time tunable devices, which is not suitable for complex and switchable communication scenarios. For example, various filters have been proposed due to their important applications in modern communications, such as suppressing interference signals and removing specific frequency components [26–30]. However, few of these waveguides can work in dynamically controllable mode, which brings a lot of inconvenience to the interference elimination of mobile communication. For instance, Wang et al. loaded a corrugated disk resonator to SSPPs transmission line to fabricate notched waveguide. The corrugated disk can introduce notched characteristics with multiple resonant modes, in which the Q value of the octupole mode can reach 368.3 in the experiment [31]. Zhang et al provided large bandwidth and wide dynamic range multi-band notched filters by loading interdigital capacitance loaded loop resonators into the SSPPs transmission lines, which can make the device more compact and flexibly adjusted bandwidth [32] than the traditional split ring resonators loaded notched filters. By transforming the connecting microstrip into a rectangular box and embedding an open-circuited stub with a quarter notch wavelength inside the box, extremely narrow notched bands in the passband with -10 dB fractional bandwidth of about 6.5% can be achieved an ultra-wideband bandpass filter [33]. Notching can also be introduced by loading a step impedance resonant ring on the original ultra-wideband filter [34]. However, all the functions of these notched filters are solidified once the structures are processed and cannot be modulated in a real time manner. Therefore, dynamically tunable notched filters need to be further developed.
Bearing this in mind, researchers have proposed several techniques for realizing tunability filters by combining them with active components, such as varactors [35–40], pin diodes [41,42], graphene [43] and phase-change materials [44,45]. For example, Hamed Saghaei et al have designed a band-stop SSPPs filter in the mid-infrared region with small overall laminated structure, large bandwidth, and high Q factors. Especially, with the employment of the graphene layer, the resonance wavelength, modulation depth, and bandwidth of the advanced filter can be dynamically adjusted in the range of 11.5 µm to 30 µm, −57 dB to −60 dB, and 2 µm to 4 µm respectively. Nevertheless, with the fast development of high power endurance SSPPs devices, a continuously tunable, compact size and flexible filtering design with active components is still a challenge.
To solve this problem, active tunable materials should be further developed. One of the most promising candidate materials is voltage controllable ferroelectrics, which possess high power endurance as well as real time tunable dielectric permittivity. They are capable of being synthesized to fabricate various kinds of real time tunable microwave resonators, filters as well as antennas. Against this background, we establish voltage controllable ferroelectric resonator in the present design. The ferroelectric resonator is composed of a home-made ferroelectric Ba0.85Ca0.15Zr0.1Ti0.9 (BCZT), whose permittivity can be adjusted by more than 60% under a DC biased voltage depending on the composition and the capacitance structure during processing [46,47]. With this high permittivity tunability, especially the high-power endurance, the voltage controllable ferroelectric resonator can be feasibly utilized to fabricate notched waveguides, which are widely investigated due to their advantages in eliminating noises in adjustable communication systems.
In this paper, we explore the possibility of a voltage tunable CPFR, which is based on BCZT ferroelectric substrate. The SLSPs metallic pattern on the surface of the ferroelectric substrate is designed in term of extinction cross section (ECS) engineering. An Archimedes electrode is proposed for the convenient voltage control on composite BCZT ferroelectric resonator. Full wave simulations show a well performed voltage tunable notched transmission in GHz frequencies. Also, the ECS properties can provide insight into the fundamental physics of EM field in this CPFR. Due to the merits of high permittivity and voltage controllable behaviors, the compact size can be achieved in the CPFR, also, the notched frequency of the SSPPs waveguide can be dynamically tuned by changing the voltage on the loaded CPFR resonator, which is beneficial for complex and switchable communication scenarios.
2. Design principle
The structure of CPFR is composed of top plasmonic electrodes on BCZT ferroelectric substrate, as illustrated in Fig. 1. The plasmonic electrode structure is designed as a ring shape composed of two Archimedean spiral not intersecting with each other on the CPFR surface. The two Archimedean spiral are mirror symmetry connected with the two semicircular electrode and they always maintain equal distance during spreading. The cylindrical substrate is a voltage controllable material with a radius of 5.075 mm, a thickness of M2 = 1 mm and a variable permittivity, which can be regulated in the range of 3 to 4 by changing the voltage applied to the Archimedean spiral electrode. The two semicircular electrodes, labeled as S and D with a gap of 0.1 mm between them, are used to apply a DC bias voltage, which is denoted as Vdc, between the electrodes (V + and V−). Of particular relevance to this work is the plasmonic electrode pattern, in which a localized surface plasmonic resonator can be created with a dispersion behavior. The dispersion behavior of the CPFR can be considered as the structural deformation from that of the periodically textured SLSPs cylinder particle proposed by Anders Pors [23], where the straight spoke groove is deformed into the Archimedean spiral curved groove in the present design, as depicted in Fig. 1(b). The depth of the straight spoke groove equals the length of the Archimedean spiral. It can be seen that the slot depth of the present resonator with Archimedean spiral is equivalently increased without increasing the volume of the resonator, which is beneficial for the miniaturization of the resonator. Also, the mode and frequency of the CPFR resonator can be regulated by the length of the Archimedean spiral. All the structural dimensions in the article are listed in Table 1.
Fig. 1. (a) Schematic diagram of the resonator. (b) the diagrammatic sketch of the equivalent LSP with straight spoke groove.
Table 1. The denotation and physical dimensions of each part of the designe CPFR loaded SSPPs waveguide
Table 2. Comparison with different works of active controllable notch band SSPPs filters
3. Results and discussion
After setting the Archimedean spiral electrode, the effect of bias voltage on cylindrical voltage controllable CPFR can be investigated. The ECS of the CPFR resonator was simulated using commercial software CST microwave studio. The horizontal plane waves were used to excite from the side of the resonator for the calculation of ECS spectra, ensuring that the direction of the electric field is perpendicular to the rectangular groove in the middle of the resonator. The boundary condition was set as an open boundary, The boundary condition was set as an open boundary, select time domain solver with a grid resolution of 1000 nm in every direction of the mesh cell. A plane electromagnetic wave in the frequency range of 0 to 7 GHz, propagating in the x direction and with an electric field amplitude of 1 V/m, was employed as an incident source in the ECS investigation. A far-field/RCS monitor was selected to record the field information with the sampling of 700 frequency points. The simulated electric field distribution of the first four resonant peaks under different spiral slot depths of the resonator is investigated.
3.1 Resonant ordering mechanism of FDR
Since two long and continuous Archimedean spirals are adopted in the present resonator model, the first magnetic resonant modes have been pushed to lower frequencies than that of the first electric one (electric dipole mode). However, in terms of the previously discussed behavior for corrugated PEC SLSPs with straight spoke groove, the electric plasmonic mode commonly resonate at lower frequencies than that of the magnetic dipole mode [48]. In order to gain a deeper insight into this contrary resonant phenomenon of the present SLSPs, we employ the following electrodynamics function to elucidate:
Fig. 2. (a) the ECS spectrum of the CPFR when the depth of the spiral groove h at 28.5 mm. (b) the internal electric field distribution of the CPFR when the depth of the spiral groove h at 28.5 mm. (c) the internal magnetic field distribution of the CPFR when the depth of the spiral groove h at 28.5 mm. (d) the ECS spectrum of the CPFR when the depth of the spiral groove h at 37.9 mm. (e) the internal electric field distribution of the CPFR when the depth of the spiral groove h at 37.9 mm. (f) the internal magnetic field distribution of the CPFR when the depth of the spiral groove h at 37.9 mm. The colour scales are saturated in each case to make the field profiles more apparent.
3.2 Resonant engineering of CPFR using spiral groove depth h and refraction index
In terms of the modal expansion equation for the resonance condition of SLSPs in the straight spoke groove CPFR, as illustrated in Fig. 1(b), the dispersion equation can be described as:
Firstly, we will investigate the effect of Archimedean spiral groove depth h on the resonant frequency of the SLSPs partical and under the circumstances, the permittivity of the resonator was set to be a fixed value of 4. Using the center of the resonator as the pole, the Archimedean spiral can be depicted by Eq. (3), where ρ represents the polar diameter of the spiral, θ is the pole angle of the spiral, u is the spacing of the two spirals, r is the inner radius of the spiral and r = 1.2 mm. As the pole angle θ increases from 3.5 π to 4.5 π, u decreases from 2.4 mm to 1.8 mm, and correspondingly, the depth h of the spiral groove can be calculated as 28.5 mm and 37.9 mm respectively from Eq. (4). As the depth h increases from 28.5 mm to 37.9 mm, the entire ECS spectrum of the resonator shifts towards lower frequency, while the electric field distribution diagram show that the resonance mode ordering remains unchanged, as illustrated in Fig. 2.
As discussed above, the resonance frequency is dictated by the length of the grooves. Since the grooves in CPFR structure are Archimedes curved, which equals the h is elongated. According to Eq. (2), since the geometry parameter a/d is almost invariable under the circumstances, the wave vector kg should be reduced. Then, in terms of Eq. (5), as the spiral length h increases, the elongated grooves impose a decrease value in resonant frequency in the CPFR subwavelength structure.
To facilitate the design of subsequent CPFR structure, we set the spiral groove depth h to 37.9 mm for the refraction index ng, or the permittivity investigation. As shown in Fig. 3, as the permittivity of the resonator being adjusted from 4 to 3 by the biased voltage in real-time, the resonance peaks undergo an overall blue shift. Among them, the resonance peak of the first magnetic dipole mode shifts from 2.07 GHz to 2.27 GHz, with a blue shift of 200 MHz, or a controllable range of 9.7%. Meanwhile, the resonant peak of the first electrical dipole mode shifts from 2.32 GHz to 2.58 GHz, with a blue shift of 260 MHz or a regulation range of 11%.
Fig. 3. (a) The ECS spectra of the resonator regulated by the variation of the dielectric permittivity of the BCZT ferroelectric material with the dielectric constant of 4, the frequency of the first four resonance peaks is 2.11 GHz, 2.33 GHz, 4.62 GHz and 4.87 GHz. (b) the internal electric field distribution of the CPFR resonator with the permittivity of the BCZT ferroelectric being 3. (c) the internal magnetic field distribution of the CPFR with the permittivity of the BCZT being 3. The colour scales are saturated in each case to make the field profiles more apparent.
As mentioned above, the dielectric permittivity of the CPFR resonator is also an important parameter for the entire ECS spectra tunability. From the point view of effective medium theory, we can depict the region with SLSPs grooves by an anisotropic and inhomogeneous layer in CPFR metamaterial, where positive and negative values of the permittivity and permeability tensors with off-diagonal elements are exhibited. When the CPFR metamaterial is irradiated under TM incidence, the radial grooves will force electric field as Eρ = Ez = 0 and magnetic field with Hφ = 0 in terms of the PEC boundary condition. Then, the effective permittivity parameters can be deduced as ɛρ = ɛz = ∞ and ${\varepsilon _\varphi } = {\varepsilon _r}\textrm{d/a}$ by averaging ɛ in the azimuthal direction φ in the subwavelength structure. As EM propagates in the grooves in the ρ or z directions with the velocity c/ng, the effective material must satisfy the equations: $\sqrt {{\varepsilon _\phi }{\mu _z}} = \sqrt {{\varepsilon _\phi }{\mu _\rho }} = {n_g}$, and thus, µρ = µz = a/d. Since the effective medium in CPFR can be viewed as a cylindrical resonator, thus, as the permittivity ɛr within the grooves increases, the effective permittivity ɛφ in CPFR resonator will also increase while the effective permeability µρ = µz = a/d and µφ = ∞ remain invariable. If we model the CPFR resonator as an equivalent LC circuit with resonance frequency ω, ω2 = 1/LC, it will be reduced due to the increase of the capacitance C when the effective permittivity ɛφ increases, as illustrated in Fig. 3.
However, it is worthy to mention that for the above two crucial factors, the former is static, while the latter is more flexible since it can control the CPFR in a real time manner.
Finally, it should be point out that due to the influence of the rectangular groove introduced in the middle of the Archimedean spiral, the resonator is a polarization sensitive structure. When the depth of the spiral groove h = 37.9 mm remains unchanged and a plane wave is irradiated from the side, the variation of the polarized electric field direction, will result in different ECS spectra, as shown in Fig. 4. When the plane wave irradiates from different direction, the position of the resonant peaks remains unchanged but the intensity of the electric modes varies significantly. Meanwhile, the magnetic modes appear more stable than the electric modes with different irradiation polarization, which is more beneficial for working in polarization insensitive environments.
Fig. 4. ECS spectra of ferroelectric composite resonators under different irradiation direction of plane wave.
3.3 Transmission characteristics of SSPPs notch filter based on CPFR loading
Based on the controllable resonant properties of the CPFR, a center frequency adjustable SSPPs notch filter is designed. The filter structure consists of a coplanar waveguide, an SSPPs transmission line with a pair of resonators mirror symmetrically loaded at the center. Annealed copper with a thickness of M1 = 0.018 mm is selected as the metal material, which is deposited on Rogers RT5880 substrate, with the thickness of S = 0.5 mm, the permittivity of 2.2 and the loss tangent of 0.0009, as shown in Fig. 5. In the SSPPs transmission line, the composite “H” type periodic structure reported in our previous work is employed [49], in which, the cutoff frequency of SSPPs transmission lines can be significantly reduced without changing the slot depth, beneficial for constructing miniaturization. By properly designing the geometric parameters of the waveguide, the dispersion characteristics of SSPPs unit structure, as depicted in Fig. 5(b) and the transmission properties of the waveguide, the S parameters, can be well adjusted.
Fig. 5. (a) The overall diagram of the proposed filter structure. (b) Local magnification plot of the SSPPs transmission line.
After loading CPFR, we conducted numerical simulation on the S parameter of the filter, and the results are shown in Fig. 6. The waveguide exhibits favorable transmission characteristics, as evidenced by the observed data. When the dielectric permittivity of the resonator is varied from 3 to 4, the transmission properties and electric field distribution of the CPFR at the resonant frequencies are shown in Fig. 6(a) and (b) respectively. When the dielectric permittivity of the resonator changes from 4 to 3, the entire resonance peak position undergoes a blue shift, where the resonant frequency of the magnetic dipole mode shifts from 2.06 GHz to 2.24 GHz, with a blue shift of 180 MHz or an 8.7% controllable range. Meanwhile, the resonant frequency of the electric dipole mode shifts from 2.28 GHz to 2.53 GHz, with a blue shift of 250 MHz or a regulation range of 11%, which is well consistent with the variation pattern of the ECS spectra of the resonator illustrated in Fig. 2.
Fig. 6. (a) the S21 parameters of the SSPPs waveguide loaded with ferroelectric composite resonators with their dielectric permittivity varied from 4 to 3 under bias voltage. (b) When the dielectric constant is 4, the internal electric field distribution of the resonator at the resonant frequencies of 2.06 GHz, 2.28 GHz, 4.53 GHz and 4.76 GHz.
It should be noted that in the SSPPs transmission line, since the electric field both have components on the perpendicular and the parallel direction to the groove, regardless of the angle between the CPFR and the SSPPs transmission line, thus, the electric mode marked as 2 and 3 can be excited in the resonator, as shown in Fig. 6. . We compare our work with previous work, including passband, number of transmission zeros, minimum insertion loss, etc., as shown in Table 2.
Finally, after applying voltage to the top two electrodes of the CPFR, it is worth noting that the potential vector penetrates into the interior of the resonator, however, its distribution is uneven as shown in Fig. 7(a). The simulation results depict that the area covered by the electrode exhibits a distribution with dense field in the above area, sparse field in the below, dense field in the middle area and sparse field on both sides. Therefore, the value of the dielectric permittivity of the whole resonator employed in the simulation is actually an equivalent average value. The controllable resonator can be achieved using functional materials such as nematic liquid crystals, or using composite ferroelectric resonators, whose equivalent dielectric permittivity can be adjusted by its composite structure. For example, for composite dielectric resonators with stacked structures, the final permittivity can be designed by controlling the relative thickness of each layer, as depicted in Eq. (6)–(7):
where, ɛi, hi, Pe,i (i = 1, 2, 3) refer to the permittivity, thickness, and filling coefficient of the ith layer in the CPFR respectively, Pe,i value locates between 0 and 1 and it can be described as:Fig. 7. (a) The distribution of internal potential vector of the resonator under the biased voltage. (b) the comparison of ECS spectra of the ferroelectric composite dielectric resonator vs the effective proposed CPFR resonator. The left inset is the electric field distribution of each resonant point of the resonator, the right inset is the schematic diagram of the composite resonator. The composite resonator consists of three layers, with the upper and lower layers being dielectrics exhibiting a permittivity of 2.2, and the middle layer being the ferroelectric with spiral electrodes on its surface, and the permittivity of the ferroelectric film is 190.
From the above equation, the above equation reveals that precise adjustment of the equivalent dielectric permittivity of the resonator can be achieved by judiciously designing the thickness and filling coefficient of each layer in the composite CPFR. For example, the composite resonator is composed of three layers with the permittivity of upper and lower dielectric layers being 2.2 and thickness of h1 = h3 = 0.5 mm. The middle layer is the BCZT ferroelectric layer with spiral electrodes on its surface. The ferroelectric layer is 0.003 mm thick with the permittivity being 190 in the working frequency range. Then, the permittivity of the composite resonator can be calculated from Eq. (6)–(7), and the result exhibits an equivalent value of 4. From Fig. 7(b), The observation reveals that the composite CPFR possesses equivalent ECS properties as a single layer resonator with its permittivity also being 4. Therefore, the permittivity of the resonator employed in the above simulation can be realized by manipulating this laminated composite structure shown in Fig. 7(b). Moreover, different from nematic liquid crystal materials, the composite ferroelectric resonators proposed in this paper have a larger and more flexible regulation range than that of the nematic liquid crystals whose permittivity can only be regulated in the range of 2.5 to 3.5.
4. Conclusion
In this study, we have conceived and achieved a real-time adjustable notch filter. The notching functionality is achieved by incorporating a localized spoof plasmonic voltage-controllable resonator, which is meticulously constructed using a ferroelectric substrate in conjunction with a plasmonic structure. This flexibility proves beneficial for the development of ultra-wideband microwave devices. When a pair of mirror-symmetric CPFR resonators is integrated onto the SSPPs transmission line, the central frequency of the notch can be dynamically controlled in real-time by varying the applied voltage on the resonator. Capitalizing on its notable advantages, including low insertion loss, swift response kinetics, and an expansive adjustable range, this SSPPs notch filter presents a novel avenue for the design of real-time adjustable devices and holds promising prospects for microwave sensing and energy harvesting applications in the future.
Funding
Guizhou Provincial Department of Science and Technology (GCC2023086); Natural Science Foundation of Guizhou Province (ZK2021YB301); The innovation talent project of Guizhou Provincial Science and Technology Department (GCC2023086); Key Project of the Education Department of Guizhou Province (KY2021045); National Natural Science Foundation of China (62261008).
Acknowledgements
This work was supported in part by the National Science Foundation of China under Grant (62261008), in part by the Key Project of the Education Department of Guizhou Province (KY2021045), the innovation talent project of Sci&Tech Department of Guizhou Province (GCC2023086), the National Science Foundation of Guizhou Province (ZK2021YB301).
Disclosures
The authors declare no conflict of interest.
Data availability
Data available on request from the authors.
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