Spoof localized surface plasmons (LSPs) in corrugated metal cylinders were proposed by Pendry et al. [1] in 2012, and magnetic spoof LSPs with long curved corrugated grooves were proposed and experimentally demonstrated by Cui et al. [2]. In 2014, Cui et al. realized the design and verification of ultrathin LSPs for the first time [3]. Following that, many different structures of spoof LSPs such as ultrathin corrugated metal–insulator–metal ring resonator [4], compact spoof LSPs [5], spoof LSP hybridization [6], spiral spoof LSPs [7], and meander line structure [8] were investigated and proposed. Spoof LSPs have been proven to be valuable in the design of resonators [9–11], filters [12, 13], sensors [14], biomedical applications [15], etc. Due to the strong confinement of the electromagnetic field [16], the spoof LSP resonance has a high-Q-factor [17] and is sensitive to the surrounding environment. Therefore, spoof LSP structures became popular and widely used in microwave and millimeter wave sensor research.

A metallic ring with corrugated gratings is the elementary structure of spoof LSPs. There have been several reports on spoof LSP sensors, such as quarter-mode spoof LSP microfluidic chemical sensor [18, 19], flexible and printed microwave plasmonic sensor [20], single hybrid plasmonic resonator [21], and effective LSP sensor [22]. Those spoof LSP structures could improve Q-factor and sensitivity significantly, compared to conventional resonator structures [21, 23]. Asymmetric metamaterials [24] or symmetry-broken in toroidal plasmonic resonator [25] can exhibit sharp Fano-resonances or a high-Q trapped mode, which shows excellent performance in resonator and sensor fields. In previous research, hybridization of spoof LSP structures can produce enormous field enhancement and improve the Q-factor [26]. The hybridized spoof LSPs can enhance resonance while preserving the structure area, which is beneficial to on-chip integration design [6].

In addition to simple physical sensors, other dimensions of sensing enhancing methods such as multifactor sensing [27], chemical sensing with the addition of black phosphorus and graphene [28], and index sensors with multimode interference [29] have been carried out. Nevertheless, those methods are limited to the structural level, resulting in high operation difficulty and cost. It would be cheap, convenient, fast, and easy to implement Q-factor and sensing FoM enhancement through back-end data processing.

In this study, we designed a multimode and high-sensitivity sensor in the microwave region, and this sensor is based on hybridized spoof localized surface plasmons and proposes a mixed-resolution algorithm to enhance the Q-factor further. The hybridized spoof LSP sensor is a coupling structure with inner and outer corrugated rings. The sensor exhibits a high-Q-factor and high sensing FoM with samples of different permittivity values on the top of the spoof LSP structure. Measurements and simulations match well, which confirm the theory. It also demonstrated that spoof LSPs could be excited and functional as a microwave sensor with a high-Q-factor and sensing FoM. The proposed mixed-resolution algorithm combines two or more resonant peaks to form one resonant peak with almost doubled Q-factor and sensing FoM than raw data. The mixed-resolution spoof LSP sensor can be extended to the terahertz band and has a wide range of applications in on-chip high-frequency resonators and sensors.

The proposed high-Q concentric corrugated ring-coupled resonator is excited by microstrip lines (MSs), as demonstrated in Figure 1. Both the inner and outer LSP structures contain 24 grooves. The inner radius, the groove height, the groove width, and the strip width of the inner and the outer corrugated rings are r 1 = 9 m m , r 2 = 2 m m , L = 6.3 m m , a 1 = 0.4 m m , a 2 = 0.4 m m , and g = 0.4 m m , respectively. The length and width of the 50Ω microstrip line and the impedance matching strip are L 1 = 7.5 m m , L 2 = 2.5 m m , w 1 = 1.45 m m , and w 2 = 1.15 m m . The angle φ and the width w 3 of the circular arc of the coupling feed structure are 36 ° and 0.3 m m , respectively. The width of the gap ( g a p ) between the coupling feeding structure and the LSPs is 0.2 m m . The LSP resonator and MS coupling structures are etched on a 0.508-mm-thick Rogers RT5880 substrate with 0.018-mm copper. To enhance the resonances of this LSP structure and define the microstrip line’s ground plane, the whole back of the substrate is covered with copper, as shown in Figure 1C. The sample for characterization is placed on the top of this structure, as demonstrated in Figure 1B. Rogers’ plates with very small loss and different permittivities were selected as the samples.

The simulated transmission coefficients of the spoof localized surface plasmon resonator are obtained using the commercial software CST. The experimental results are measured using a vector network analyzer (Agilent N5230C). Both simulated and measured results are plotted in Figure 2A. It shows that the simulated results are in good agreement with the measured results. This structure is designed with 7 resonance peaks, marked as m1 to m7 from the range of 0 to 8 GHz. Those resonance frequencies are located at 2.03, 3.72, 4.29, 4.91, 5.69, 6.46, and 7.82 GHz. In the measurement, the measured frequency responses confirmed those 7 resonance peaks. And those frequency points are located at 2.037, 3.736, 4.27, 4.94, 5.73, 6.43 and 7.77 GHz, respectively, which matches the design. The quality factor of resonator represents the ratio of stored energy to the consumed energy of the resonant circuit, respectively, which can be expressed as follows: Q = f 0 Δ f 3 d B , (1) where f 0 is the resonance frequency and Δ f 3 d B represents the 3dB bandwidth of the resonance peak. In Figure 2A, the Q-factors of simulated modes m 1 to m 7 are 122, 147.6, 98, 186.8, 245, 103, and 122. The Q-factors of measured results are 108, 118, 97.3, 178.4, 160.3, 93.2, and 121.3 at each resonance peak, which is also close to the simulated data. The measured highest Q-factor is 178.4 at 4.94 GHz.

To further study the mechanism of this high-Q-factor resonance of the spoof LSP structure, a 2D near-field electric distribution at the resonance modes m 1 to m 7 on the plane 1 mm above the resonator is illustrated in Figure 2B. The m 1 ,   m 2 ,   m 4 , and m 5 modes in Figure 2B are the odd dipole, quadrupole, hexapole, and octupole, respectively. The inner and outer spoof LSP resonators are resonant in opposite phases in these odd modes. The modes m 3 , m 6 , and m 7 can be seen as resonator’s even modes, with the inner and outer rings resonating in phases. And the Q-factors of odd modes are higher than even modes. And the Q-factors of odd modes are higher than even modes. As can be seen in Figure 2B, the embedded electric field in the odd modes is more complicated than the even modes, so they have different equivalent circuits with fewer losses from the even mode and achieve higher Q-factors.

Previous studies have shown that the Q-factor of outer corrugated spoof LSP resonator is higher than that of inner LSP resonator [5, 30], so only the outer corrugated spoof LSPs are analyzed here. The transmission coefficients of the hybridized spoof LSP resonator and outer corrugated spoof LSP resonators covered with tested samples are compared and analyzed to verify the superiority of the hybridized structure in the sensing test. In this part, only odd modes m 1 , m 2 , and m 5 are analyzed. The curves in Figure 3 show the measured transmission coefficients, which are tested by hybridized spoof LSPs, as shown in Figure 1B, and the outer corrugated spoof LSP structure, as shown in Figure 3. The tested samples are 0.508-mm-thick Rogers RT5880 plates, with a permittivity of 2.2. The black line describes the transmission coefficient of the hybridized spoof LSPs, and the Q-factors of m 1 to m 4 modes are 103.2, 124.6, 71.4, and 141.6. The red dotted line depicts the transmission coefficient of the outer corrugated spoof LSPs, where the Q-factor of the corresponding M 1 and M 2 is 77.2 and 71.2, respectively. The hybridized structure consists of an inner spoof LSPs and an embedded outer corrugated spoof LSPs. By coupling the inner LSPs, outer LSPs, and the metal ground, the electromagnetic field in the hybridized structure is highly concentrated, which reduces the radiation loss and reflection loss, thus improving the resonant Q-factor noticeably. In addition, the complex coupling structure of the hybridized spoof LSPs increases the electromagnetic field transmission path, which reduces the resonant frequencies compared to the outer spoof LSPs. Therefore, the hybridized spoof LSPs show superior advantages over the single outer spoof LSPs in resonance strength and dimension.

The hybridized spoof LSP structure-based resonator with a high-Q-factor is a primary condition for the resonance sensor as the sensitivity requirement. In order to verify the sensing performance, a prototype of this LSP sensor and several samples with different permittivities are fabricated and characterized for demonstration. A spoof LSP sensor and excitation microstrip lines are printed on the substrate of Rogers RT5880, with a relative permittivity of 2.2, as shown in Figure 4. Three Rogers materials, namely, RT5880, RO3003, and RO4003, are applied in the test samples. Those samples are 0.508 mm thickness, and their relative permittivity is 2.2, 3, and 3.55, respectively. The samples are kept in the same size, while their permittivities are different. Those sizes are 20 mm × 25 mm×0.508 mm, and they can be laid on the spoof LSP sensor.

Sensing FoM is an important parameter to characterize the sensor’s sensitivity [31], which synthetically indicates the overall interaction effect with the environment. The sensing FoM is defined as follows [32]: S e n s i n g   F o M = S e n s i t i v i t y Δ f 3 d B , (2) where sensitivity is the frequency offset Δ f when the corresponding refractive index Δ n changes by unit and can be defined as follows [33]: S e n s i t i v i t y = Δ f Δ n , (3) where the refractive index n means the square root of the relative permittivity ε of the tested sample, and it can be calculated by n = ε . From the aforementioned analysis, when the permittivity changes at the same level, the larger the frequency offset and the smaller the 3dB bandwidth, the larger the sensing FoM.

The sample under test is placed directly above the spoof LSP structure in the measurement. The sample and the sensor are fixed together with two identical small iron clips to secure the position and distance, as illustrated in Figure 5A. The spoof LSPs have strong confinement of electromagnetic field, and the field intensity outside of spoof LSPs is weak. Therefore, the small iron clip has little impact on the sensor’s performance, which is also verified in the measurement process. The sensor’s simulated and measured transmission coefficients are plotted in Figure 5A. Figures 5B,C show the detail of the m5’s resonant mode in simulation and in the measurement.

The simulated odd dipole modes locate at 1.6176, 1.6896, and 1.792 GHz with Q-factors of 95.2, 94.5, and 102 when the sample relative permittivity changes from 3.55, 3, to 2.2. The odd quadrupole modes are located at 2.912, 3.0496, and 3.2544 GHz with Q-factors of 112, 116.8, and 124.6 when the sample relative permittivity changes from 3.55, 3, to 2.2. Meanwhile, the odd octupole mode m 5 is located at 4.38, 4.61, and 4.93 GHz with Q-factors of 103, 179.8, and 200. When the permittivity ε changes from 3.55 to 3, the offsets of the dipole mode, quadrupole mode, and octupole mode are 0.072, 0.1344, and 0.2272 GHz, respectively. The corresponding sensitivities are 0.475 GHz ⋅   RIU − 1 , 0.905  GHz ⋅   RIU − 1 , and 1.494 GHz ⋅   RIU − 1 , respectively, while sensing FoMs are 27.05 RIU − 1 , 34.33 RIU − 1 , and 51.43 RIU − 1 , respectively. When the permittivity ε changes from 3 to 2.2, the offsets of dipole mode, quadrupole mode, and octupole mode are 0.1024, 0.2032, and 0.3232 GHz, respectively. The corresponding sensitivities are 0.42  GHz ⋅   RIU − 1 , 0.817  GHz ⋅   RIU − 1 , and 1.299  GHz ⋅   RIU − 1 , respectively, while sensing FoMs are 26.45 RIU − 1 , 31.6 RIU − 1 , and 54.53 RIU − 1 , respectively. Both the Q-factor and sensitivity of this LSP sensor are beyond the state of art in the passive design [21].

The measured results are presented in Figures 5A,C as the dotted line. The measured odd dipole modes locate at 1.6738, 1.724, and 1.835 GHz with Q-factors of 59.8, 76.7, and 81.6 when the sample relative permittivity changes from 3.55 to 3 to 2.2. The odd quadrupole modes locate at 3.015, 3.113, and 3.325 GHz with Q-factors of 70.8, 94.1, and 106.3, while the odd octupole mode locates at 4.546, 4.711, and 5.049 GHz with Q-factors of 44.6, 103, and 111. When the permittivity ε changes from 3.55 to 3, the offset of the dipole mode, quadrupole mode, and octupole mode are 0.05, 0.098, and 0.165 GHz. The corresponding sensitivities are 0.333  GHz ⋅ RIU − 1 , 0.45  GHz ⋅ RIU − 1 , and 1.082  GHz ⋅ RIU − 1 , while sensing FoMs are 15.5 RIU − 1 , 17.16 RIU − 1 , and 14.25 RIU − 1 . When the permittivity ε changes from 3 to 2.2, the offset of dipole mode, quadrupole mode, and octuple mode are 0.111, 0.212, and 0.338 GHz. The corresponding sensitivities are 0.446  GHz ⋅ RIU − 1 , 0.852  GHz ⋅ RIU − 1 , and 1.36  GHz ⋅ RIU − 1 , while sensing FoMs are 19.8 RIU − 1 , 26.46 RIU − 1 , and 30.72 RIU − 1 .

Comparing the simulated results with measured results, it can conclude that the frequencies of the resonant modes are blue-shifted, and both the Q-factor and sensing FoM have been reduced, especially in the odd octuple mode. This shift could be caused by the unavoidable air gap between the samples and the sensor. Also, from the simulation and experiment, it is not difficult to find that the lower the permittivity of the sample, the higher the resonant frequency. The air gap causes the effective permittivity above the sensor to be less than the relative permittivity of the samples, resulting in this blue shift. The odd octuple mode is a resonant mode which is similar to Fano resonance produced by coupling internal and external resonators, requiring higher environment configuration. Manufacturing, material, and measurement errors affect the measured resonant frequency, resonant intensity, and sensitivity.

The proposed hybridized spoof LSP sensor shows excellent performance in simulation and measurement, with Q-factors up to 178.4 and sensing FoM up to 30.72 RIU − 1 . The electrical size of the sensor is very small as the path length of the surface current is compressed, significantly smaller than the size of the spoof LSP sensor. The size of the resonant structure is about one-tenth of the working wavelength. The spoof LSPs of the resonator improve the ability to confine electromagnetic fields and the sensitivity to the surrounding environment. Moreover, the sensor detection can be realized with low cost and simple operation by replacing the sample, which dramatically improves its practical engineering value.

Although the proposed hybridized sensor achieved a very high-Q-factor and sensing FoM, they are expected to be even higher by data processing. Therefore, a mixed-resolution optimization algorithm is proposed to enhance the Q-factor and sensing FoM based on the multi-resonance property. The algorithm is based on the resulting linear data and combines several resonant modes later to form a new resonant mode. The new resonant mode achieves narrower 3dB bandwidth, improving the Q-factor and sensing FoM.

To precisely describe the algorithm for increasing the resonant Q-factor and sensing FoM, a schematic diagram is inserted in Figure 6. Suppose there is a resonant peak, which consists of a curve y 1 with slope 1 and a curve y 1 ' with slope -1. Then, the 3dB bandwidth here is 1, which can be obtained by calculating the bandwidth when the ordinate is half of the peak value, and the corresponding Q-factor is 1. If there are two such resonant peaks, the linear values of the curves are multiplied to the left of the two peaks, which results in a curve of y 2 = x 2 , and the same process is repeated to the right to get the curve of y 2 ' as blue curves shown in the illustration. The 3dB bandwidth and Q-factor here can be calculated as 0.586 and 1.71. Compared with curves y 1 and y 1 ' , the 3dB bandwidth is reduced by 41.4%, and the Q-factor is increased by 71%. Furthermore, the multiplication of two curves composed of y 2 and y 2 ' gives the curves y 3 and y 3 ' illustrated in red lines, whose 3dB bandwidth and Q-factor become 0.32 and 3.14, respectively. The 3dB bandwidth is reduced by 1.83 times, and the Q-factor increases by 1.83 times. It can be seen that when the slopes on both sides of the original resonant peaks are larger, the bandwidth of the composite peak is narrower, and the Q-factor is higher. Also, from Eq. 2, the sensing FoM of the sensor will increase when the 3dB bandwidth is reduced. In the electromagnetic field, the resonant peak cannot reach 1, and there are larger slopes on both sides of the peaks, which further improves the performance of the proposed mix-resolution algorithm.

First, to apply this mixed-resolution algorithm to sensor data processing, 100 linear sampling points at higher frequency and 100 at lower frequency around the dipole and quadrupole resonant frequencies were taken, respectively. The sampling points with different permittivities are shown as the input dotted curves in Figure 6. Then, these three groups of data are processed using the described mixed-resolution algorithm, respectively, and it generated three new groups of resonant peaks, as shown in the solid line in Figure 6. The frequency of the output resonant peak is half the sum of the corresponding frequencies of the dipole and quadrupole. Finally, according to Eqs 1–3, an algorithm for automatically calculating resonant Q-factors and sensing FoM is developed. The calculated input and output Q-factor and sensing FoM are listed in Table 1. As compared in Table 1, the mixed-resolution algorithm increases the Q-factor from 96 by 1.93 times to 185.1, and the sensing FoM increases by 2.17 times, from 23.38 to 50.7. The algorithm can almost double the Q-factor and sensing FoM without changing the sensor structure, which is an important breakthrough in the microwave sensor field [34–37]. With this mixed-resolution algorithm, sensors are expected to achieve multimode enhanced sensing in the future. Also, it can detect more subtle environmental changes to a great extent.

In this work, a hybridized spoof LSP sensor is proposed, implemented, and measured with high environmental sensitivity. A mixed-resolution algorithm was also proposed to further improve the Q-factor and sensing FoM by combining multimode resonant peaks. A prototype was designed, fabricated, and measured for demonstration. High-Q-factor and high sensing FoM are achieved by applying the algorithm. Meanwhile, the mixed-resolution can further double them. The hybridized spoof LSP sensor and the mixed-resolution algorithm have great potential in the field of high-frequency on-chip resonators and sensors.