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Abstract
Recently, ultrathin localized spoof surface plasmon (LSSP) resonators are found to have intrinsic defects of relatively low quality factors (Q-factors) because of unavoidable material and radiation losses. In this paper, multilayer structures of planar-circular-grating resonators and their magnetic-coupling schemes are proposed to achieve effective excitation of high-Q LSSPs modes. By adopting the multilayer structures with air between the layers, the power dissipation effected by both material and radiation losses is significantly suppressed. Experimental results show that the Q-factors could reach more than 200 and the excitation efficiencies could reach more than 90%. Numerical simulations show the distribution of the electromagnetic field and illustrate the principle of magnetic coupling. Besides, the Q-factors of resonators with different structural parameters were measured and analyzed. This study aims to provide some inspirations on planar gyro-devices and to improve the performance of existing applications, such as sensors and filters.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Localized surface plasmons (LSPs), which exist in structures with closed surfaces in the optical band, have been proven to have outstanding subwavelength confinement and near-field enhancement characteristics [1,2], making them great candidates for near-field optics [3], surface-enhanced spectroscopy [4–6], plasmonic antennas [7,8], biological sensors [6,9], and hyper lenses [10,11]. To mimic the behaviors and inherit the characteristics of natural LSPs, localized spoof surface plasmons (LSSPs) have been proposed, which reduce the effective plasma frequency of metal to millimeter-wave and terahertz-wave bands by using textured closed surfaces [12]. Later, an ultrathin structure for LSSPs was proposed and experimentally demonstrated [13], which has many promising applications in sensors [14], bandpass filters [15,16], and topological photonics [17].
However, the quality factors (Q-factors) of LSSP resonators are relatively low compared with those of large-electrical-size resonators such as traditional metallic cavities [18]. Therefore, different excitation schemes have been proposed to improve the Q-factors. In early research, probe-excitation schemes have been widely used to excite LSSPs, which verified the existence of LSSPs but presented low excitation efficiencies and low Q-factors [13,19–21]. Later, excitation from free space [22] and excitation via spoof surface plasmons (SSPs) [23] were reported, but these showed only minor improvements. Microstrip line excitation is a state-of-the-art method with high efficiency, high reliability, and compatibility with printed circuit boards (PCBs) [16,24,25]. Excitation via a microstrip line was first proposed in [24] and verified the existence of high-order LSSPs modes. In a given corrugated ring structure, the Q-factor was increased from 11.4 for probe excitations to 69.6. Later, a similar excitation scheme via a microstrip line was designed, and the Q-factors of the TE1,1, TE2,1, and TE3,1 LSSPs modes were measured as 148, 64.5, and 78.5, respectively [25]. Recently, derivations based on microstrip line excitation have been proposed to increase the Q-factors. A slit perturbation was induced to the top spoke to break the in-plane symmetry, and a trapped mode was excited. Owing to strong field confinement, the radiation loss was suppressed to 0.04% and the Q-factor was increased to 104.8 [26]. A hybrid plasmonic resonator composed of a fan-shaped perturbing resonator and an LSSP resonator overlapping each other was proposed. A hybrid mode formed by modal interference was excited, whose Q-factor was enhanced 3.4-fold from 44.8 to 153 [27].
Planar gyro-devices are new conceptions of vacuum electron devices which follow the principle of electron cyclotron maser (ECM) and are expected to be made on chips [28]. The development of them has two requirements for LSSP resonators [29]. First, the Q-factor should be as high as possible, to reduce the starting current, suppress the ohmic loss, and improve the efficiency. Second, the electric field should be parallel to the grating (TE mode) and should be exposed outside the dielectric. Although the above microstrip line coupling seems to be the perfect solution in the field of microwave components, it is not suitable for vacuum electronic devices. Because the electric field is vertical to the grating owing to the incorporation of the ground plane and is mainly distributed in the dielectric substrate owing to its large permittivity, the cyclotron electrons cannot interact with the LSSPs. Therefore, there is a strong need to develop a novel coupling method with high-Q resonators.
In this paper, an efficient magnetic-coupling excitation scheme of LSSPs on high-Q planar-circular-grating resonators is proposed, which not only satisfies all the abovementioned requirements for gyro-devices, but also shows better intrinsic performance as a microwave component. Multilayer structures of resonators including mono, double, and quadra layers are designed step by step to reduce loss and increase the Q-factor. Owing to the absence of the ground plane, the magnetic-coupling method was employed, which showed better efficiency than the counterparts coupled by microstrip lines. A prototype of the system was manufactured, and an experiment was conducted to measure the S parameters in the microwave band. The experimental results verify the superiority of the proposed method, such that the Q-factors can reach more than 200. By changing different types of multilayer structures and different parameters such as grating structure, grating spacing, and coupler structure, various Q-factors were measured and analyzed comprehensively. Numerical simulations based on the finite element method (FEM) were also performed to obtain the distribution of the electromagnetic field. The electric field is demonstrated to be parallel to the planar circular grating, which provides the possibility of electron-wave interaction. The distribution of the magnetic field vividly illustrates the principle of magnetic coupling. We anticipate that our work will be a starting point for developing planar gyro-devices and to be an improvement for existing applications, such as sensors and filters.
2. Results and discussions
There are three potential parts of an LSSP resonator and its coupler: planar circular grating, reflecting disk, and magnetic coupler [Fig. 1(a)–(c)]. They are all ultrathin structures with a thickness of 0.035 mm (1 oz) and were printed on a dielectric substrate (Rogers RO4003C) with a thickness of 0.203 mm, relative permittivity of 3.38, and loss tangent of 0.0027. To study the performance change under different parameters, a series of planar circular gratings and magnetic couplers was designed and manufactured. For planar circular gratings, R1, R2, and R3 are constants, whereas N and a/d are variables. R1 was fixed at 5 mm, R2 was fixed at 15 mm, and R3 was fixed at 16 mm. h and b, two driven dimensions, were fixed at 10 and 1 mm, respectively. N has two values: 60 and 36, a/d has two values: 0.4 and 0.2. Four types of planar circular gratings with all possible combinations of N and a/d were produced. For magnetic couplers, L and w are constants, and D is a variable. L was fixed at 20 mm, and w was fixed at 2.54 mm ( = 100 mil) to be compatible with the width of the pins on the subminiature version A (SMA) connector. Four types of magnetic couplers were produced with different D values of 5, 6, 7, and 8 mm. In addition, organic solderability preservatives (OSPs) were applied to protect copper from oxidation or corrosion.
Fig. 1. (a) Schematic of the planar circular grating with an inset showing the unit cell of the grating. R1 is the inner radius of the grating, R2 is the outer radius of the grating, and R3 is the radius of the edge. h = R2 − R1 is the depth of the grating, b = R3 – R2 is the width of the edge. a/d is the duty cycle of the grating. N in the number of the gratings. The yellow areas represent copper and the blue areas represent dielectric substrate. (b) Schematic of the reflecting disk. R3 is the radius of the disk and is unified with (a). (c) Schematic of the magnetic coupler. L is the length of the two-conductor line and w is the width of the two-conductor line. D is the diameter of the magnetic probe. The end of the two-conductor line is connected to a SMA connector. (d) Photograph of the experiment. The coaxial line (black part) is connected to a VNA. (e) A side view of the experiment. (f) A close-up shot illustrating how to keep the fixed spacing between the different layers. (g) A side view of the wooden base. By turning the handle, the lead screw pushes the wood bricks forward to tighten the LSSP resonator.
Three types of multilayer structures of resonators, mono layer, double layer, and quadra layer, are proposed step by step. The latter two types involve the guarantee of the spacing between layers, and thus standard wood dielectric strips with precise thickness (under the verification of a Vernier caliper) were placed between different layers at the top and bottom of the resonator [Fig. 1(f)]. Its advantages are low relative permittivity and low loss tangent, which have little influence on electromagnetic waves. A wooden base was employed to fix the bottom of the resonator, and a rubber band was used to fix the top of the resonator. In the relaxed state, the rubber band has a width of 3 mm, thickness of 1.4 mm, and diameter of 19 mm. The S parameters were tested using a vector network analyzer (VNA, model: Agilent E5071C). The magnetic couplers were connected to the VNA via SMA connectors, which converted the two-conductor line mode to the coaxial line mode.
To quantify the performance of the LSSP resonator, the Q-factor and figure of merit (FoM) are induced [30]. The Q-factor is an important parameter that indicates the energy loss in the resonator. It is generally affected by radiation loss related to the electrical size and material loss, including metal loss and dielectric loss [31]. Here, the Q-factor is defined as
where f0 is the resonant frequency and Δf3dB is the 3-dB bandwidth, which is equivalent to the full-width at half-maximum of power on a linear scale. FoM is another important parameter that synthetically indicates the total effect of interaction with the environment [32]. It is defined as where δI is the excitation efficiency, i.e., the dip of S11 on a linear scale.In this section, the mono-layer structure of the resonator is first proposed as a preliminary design of an LSSP resonator, which verifies the suitability of the magnetic-coupling method. The measured S11 spectra show that the maximum Q-factor appearing in the TE7,1 mode is 198.0, owing to the suppression of material loss. The electromagnetic field is simulated, the electric field is demonstrated to be parallel to the planar circular grating, and the principle of magnetic coupling is illustrated through the magnetic distribution. Later, the double-layer structure of the resonator is proposed as an advanced design, in which two layers of planar circular gratings are employed. The maximum Q-factor, which also appears in the TE7,1 mode, is enhanced to 294.4, resulting from the suppression of radiation loss. Finally, the quadra-layer structure of the resonator is proposed as an additional design, in which two layers of reflecting disks are added. The radiation loss is further reduced. The measured S11 spectra illustrate a compensation effect of Q-factors of lower-order modes and show that the maximum Q-factor appearing in TE3,1 mode is 250.0.
2.1 Preliminary design of LSSP resonator
The preliminary design of the LSSP resonator is a mono-layer structure, i.e. a planar circular grating layer. In addition, a magnetic coupler is employed to excite LSSPs modes, as shown in Fig. 2(a) and (b). The planar circular grating is parallel to the magnetic coupler with a spacing of d1. The projection of the magnetic coupling ring (at the front of the magnetic coupler) on the grating is tangential to the edge of the grating.
Fig. 2. Schematic of the mono-layer structure of LSSP resonator from (a) side view and (b) perspective view. (c) Measured S11 spectrum under parameters N = 60 and a/d = 0.4. The modes and corresponding calculated Q-factors are labeled next to the dips. (d) Measured S11 spectra under different parameter settings. Simulated (e) electric field distribution and (f) magnetic field distribution of TE5,1 in the cross-section of magnetic coupler (on logarithmic scale). (g) Simulated normal electric field distribution of TE5,1 in the cross-section of x = 0 (on linear scale).
The measured S11 spectrum under N = 60 and a/d = 0.4 is shown in Fig. 2(c). Multiple modes from TE1,1 to TE8,1 were successfully excited. We can also perceive directly from the S11 spectrum that the higher-order modes dip deeper, such that they have a higher excitation efficiency. The dips of higher-order modes are narrow, which correspond to narrower 3-dB bandwidths. Not surprisingly, the calculated Q-factors of higher-order modes are higher than those of lower-order modes. The maximum Q-factor, which appears in TE7,1 mode, reaches 198.0, and exceeds those of most of the previous studies. However, the Q-factors of lower-order modes are worse than those of previous works based on microstrip lines. This phenomenon satisfies the need of gyro-devices for high-order modes with high Q-factors. Owing to the high-harmonic [28,33,34] and large-orbit [35] technologies, high-order modes are generally adopted at present. In the proposed resonators, high-order modes have relatively high Q-factors and can improve the efficiencies of gyro-devices and enhance the sensitivity of sensors. Figure 2(d) shows the measured S11 spectra under different grating parameters N and a/d, and shows that the resonant frequency of the corresponding mode varies slightly under different values of N and a/d. This may appear to contradict [12], where the eigenfrequencies vary with grating parameters according to an eigenequation, but it actually does not. We adopted ultrathin structures, where the electromagnetic field extends into the air on both sides of the grating and has little contact with the grating, whereas [12] analyzed a two-dimensional structure, where the electromagnetic field is distributed in the grooves and has great contact with the grating, such that the resonant frequency of the proposed structure is less sensitive to the changes of the grating parameters compared with the two-dimensional structure in [12].
Numerical simulations based on FEM were carried out using CST Microwave Studio. A tetrahedral mesh was adopted, and an open boundary was set. A perfectly matched layer (PML) [36] was chosen to realize the open boundary with low artificial reflection, because the free space standard impedance boundary condition (Free Space SIBC) did not work well. The losses of the copper and the dielectric substrate were considered. An electric field monitor and a magnetic field monitor were placed, and the results of the TE5,1 mode under N = 60 and a/d = 0.4 are shown in Fig. 2(e)–(g) as examples to analyze the field distributions. The electric field presents the pattern of the TE5,1 mode and is mostly distributed on the inner side of the grating, as shown in Fig. 2(e). It is parallel to the grating; in other words, it is the TE mode, because it is mainly distributed in the normal direction in Fig. 2(g). The magnetic field is mostly distributed on the outer side of the grating, which is why the coupling ring is placed at the edge of the grating, and is vertical to the grating as well as the magnetic coupler [Fig. 2(f)]. Thus, the ring at the front of the magnetic coupler could capture the magnetic field passing through the ring and, as a result, magnetic coupling is realized. Simulated S11 spectra were also obtained, as shown in Fig. S1 in Supplement 1.
The reason for the high Q-factors is the reduction of the material loss, and more precisely, the dielectric loss. The metal loss of copper is inevitable, but the dielectric loss of the substrate is reduced. Because the electric field is mainly exposed in the air, which is intrinsically lossless, rather than confined in the dielectric substrate such as in microstrip line coupling, the energy absorbed by the dielectric is reduced. To reduce the radiation loss and further increase the Q-factor, a double-layer structure is proposed.
2.2 Advanced design of LSSP resonator
The advanced design of the LSSP resonator is a double-layer structure consisting of two layers of planar circular gratings. In addition, a magnetic coupler is employed to excite LSSPs modes, as shown in Figs. 3(a) and (b). The planar circular gratings and magnetic coupler are parallel to each other. The spacing between the gratings is d1, and because of the symmetry of the structure, the spacing between the magnetic coupler and either of the gratings is d1/2. The projection of the ring at the front of the magnetic coupler on the grating is tangential to the edge of the grating.
Fig. 3. Schematic of the double-layer structure of LSSP resonator from (a) side view and (b) perspective view. (c) Measured S11 spectra, (d) Calculated Q-factors, and (e) Calculated FoMs with different grating parameters.
To verify the effect of the double-layer structure, the S11 spectra of the mono-and double-layer structures with the same grating parameters N = 60 and a/d = 0.4 are analyzed. The Q-factors of all the corresponding modes were improved by adding an extra grating. The maximum Q-factor appearing in the TE7,1 mode reaches 203.7, indicating that the double-layer scheme is effective. This agrees with the results of the mono-layer structure, in which higher-order modes have higher Q-factors.
The performance of the resonators under different parameters was studied. When d1 = 2 mm is unchanged, the measured S11 spectra with different grating parameters are shown in Fig. 3(c) and illustrate some changes on resonant frequency with different values of N and a/d, because the electromagnetic field has more contact with the grating in the case of the double layer. From Fig. 3(d), we found that the resonators with a/d = 0.4 performed better in Q-factors than those with a/d = 0.2, but found little effect of changing N on performance. The calculated FoMs (= Q × δI) seem to have the same distributions as the Q-factors, owing to the high excitation efficiency (δI), which is generally greater than 0.9. For the same reason, the maximum FoM reaches 206.6, which exceeds those of almost all previous studies.
With the grating parameters N = 60 and a/d = 0.4 unchanged, the measured S11 spectra with different grating spacings d1 are shown in Fig. 4(a). With an increase in the grating spacing, the resonant frequency of the corresponding mode increases, resulting in a blue shift, and the maximum excitation efficiency shifts to the higher-order mode. The calculated Q-factors and FoMs are shown in Fig. 4(b) and (c), which are distributed in order with respect to the change in d1. The resonator with closer spacing has a higher Q-factor and a higher FoM, owing to a lower radiation loss from the side. In the case of d1 = 1.5 mm, the maximum Q-factor and FoM, which appear in the TE7,1 mode, reach 278.4 and 206.8, respectively.
Fig. 4. (a) Measured S11 spectra, (b) Calculated Q-factors, and (c) Calculated FoMs with different grating spacings.
With N = 60, a/d = 0.4, and d1 = 2 mm unchanged, the measured S11 spectra with different coupler parameters D are shown in Fig. 5(a). With an increase in the diameter of the magnetic coupler ring, the maximum excitation efficiency shifts to lower-order mode and the higher-order mode vanishes; for example, the TE8,1 mode cannot be excited under D = 8 mm. To explain this phenomenon, we introduce the equivalent wavelength ${\lambda _e}$:
where m is the azimuthal mode index (e.g., m = 4 represents the TE4,1 mode) and substitute (R2 − D/2) with R, where the magnetic coupler ring is placed. When the equivalent wavelength of a mode is approximately equal to or less than the diameter of the magnetic coupler ring, it is difficult to excite the mode. Because all phases of one wavelength from 0 to $2\pi $ enter the magnetic coupler, the magnetic flux cancels each other out and the magnetic coupling cannot be realized. For example, in the case of D = 8 mm, the equivalent wavelength of TE8,1 is 8.63 mm, which is approximately equal to D; therefore, the TE8,1 mode vanished.Fig. 5. (a) Measured S11 spectra, (b) Calculated Q-factors, and (c) Calculated FoMs with different coupler parameters.
The calculated Q-factors and FoMs are shown in Fig. 5(b) and (c), respectively. It seems that the resonator with a smaller coupler has a higher Q-factor and a higher FoM, but it is unusual that the Q-factor and FoM of TE7,1 are extremely high in the case of D = 8 mm, reaching 294.4 and 211.3 respectively. The reason for this is the absence of TE8,1. Without the interference of TE8,1, the excitation of TE7,1 is intensified.
To study the field distributions of the double-layer structure, we take the TE4,1 mode under N = 60, a/d = 0.4, and d1 = 2 mm as an example (Fig. 6). The distributions of the electric and magnetic fields are similar to those of the mono-layer structure; however, because of the symmetry of the structure, the electric field in the cross-section of the magnetic coupler is strictly in the transverse direction without a normal component. Both the electric and magnetic fields between the gratings are enhanced, and the intensity of the magnetic coupling is improved; thus, the excitation is more efficient. The Q-factors are improved because each grating radiates energy only from the outer side, the energy radiated from the inner side of one grating is received by the other grating, and the radiation loss is greatly reduced. To further reduce the radiation loss, a quadra-layer structure is proposed.
Fig. 6. Simulated electric field distribution of TE4,1 in the cross-section of (a) magnetic coupler and (b) planar circular grating (on logarithmic scale). Simulated magnetic field distribution of TE4,1 in the cross-section of (c) magnetic coupler and (d) planar circular grating (on logarithmic scale). Simulated (e) normal electric field distribution and (f) magnetic field distribution of TE4,1 in the cross-section of x = 0 (on linear scale).
2.3 Additional design of LSSP resonator
The additional design of the LSSP resonator is a quadra-layer structure consisting of two layers of planar circular gratings and two layers of reflecting disks. In addition, a magnetic coupler is employed to excite LSSPs modes, as shown in Fig. 7(a) and (b). The planar circular gratings, reflecting disks, and magnetic coupler are parallel to each other. The spacing between the gratings is d1, and because of the symmetry of the structure, the spacing between the magnetic coupler and either of the gratings is d1/2. The spacing between the planar circular grating and magnetic coupler is d2. The projection of the ring at the front of the magnetic coupler on the grating is tangential to the edge of the grating.
Fig. 7. Schematic of quadra-layer structure of LSSP resonator from (a) side view and (b) perspective view. (The broken part is to show the structure of the resonator, which is actually complete.) (c) Measured S11 spectra with different d2. The modes and corresponding calculated Q-factors are labeled near the dips.
With N = 60, a/d = 0.4, d1 = 2 mm, and D = 5 mm unchanged, the measured S11 spectra with different spacing d2 are shown in Fig. 7(c). Surprisingly, we found that the Q-factors of the lower-order modes were compensated by the two extra reflecting disks. The maximum Q-factor appearing in the TE3,1 mode reaches 250.0, which is 11.5 times greater than that of the double-layer structure. However, the Q-factors of the higher-order modes are not higher than those of the double layer. In addition, fewer modes were excited, too high- and low-order modes vanished.
To study the field distributions of the quadra-layer structure, we take the TE5,1 mode under N = 60, a/d = 0.4, and d1 = d2 = 2 mm as an example (Fig. 8). The distributions of the electric and magnetic fields are similar to those of the quadra-layer structure, including the transverse electric characteristic. However, owing to the reflection of the copper reflective disks, the field was confined between the reflecting disks. Strictly speaking, no electromagnetic field passes through the reflectors, and the radiation loss is greatly reduced.
Fig. 8. Simulated electric field distribution of TE5,1 in the cross-section of (a) magnetic coupler and (b) planar circular grating (on logarithmic scale). Simulated magnetic field distribution of TE5,1 in the cross-section of (c) magnetic coupler and (d) planar circular grating (on logarithmic scale). Simulated (e) normal electric field distribution and (f) magnetic field distribution of TE5,1 in the cross-section of x = 0 (on linear scale).
Thus far, material loss and radiation loss have been suppressed step by step. By employing different types of multilayer structures, the high Q-factors of all modes can be achieved. The proposed multilayer resonators are compared with the LSSP resonators in previous studies in Table 1. The maximum Q-factor of our work, which is 294.4 and appears in the TE7,1 mode in the case of the double layer, is approximately twice as much as the maximum Q-factor of the counterparts coupled by microstrip lines, which is 153.0 in [27]. This work shows good performance in Q-factors, besides the efficient and simple magnetic-coupling excitation method.
Table 1. Comparison with Previous LSSP Resonators
3. Summary
In summary, a series of high-Q multilayer planar-circular-grating resonators was designed, and an efficient magnetic-coupling method to excite LSSPs was proposed for the first time. Experiments showed that Q-factors were doubled to more than 200 by suppressing material loss as well as radiation loss, exceeding almost all previous studies. The Q-factors of resonators with different structural parameters were measured and calculated, and the advantages of different types of multilayer structures were analyzed. Through numerical simulations, the principle of magnetic coupling was explained, the electric field was demonstrated to be transverse, and the reasons for the high Q-factor were revealed. This work is intended to be a pioneering study on the development of planar gyro-devices and the reform of existing applications such as sensors and filters.
Funding
National Natural Science Foundation of China (NSAF-U1830201, 61861130367, 61531002, 61971013); Royal Society Newton Advanced Fellowship (NAF/R1/180121).
Acknowledgments
The authors thank Dr. Bao-Liang Hao, Beijing Vacuum Electronics Research Institute, China, for his support with the CST software simulation. The authors are grateful to Ph.D. student Han-Ting Zhao, Department of Electronics, Peking University, China, for his help on the operation of the VNA. The authors are grateful to master student Yi-Dong Wang, Department of Electronics, Peking University, China, for his help on the design of the PCB.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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