Introduction
As its ability to confine the electromagnetic wave over sub-wavelength distances, the surface plasmon polaritons (SPPs), which offer localized plasmons at metal/dielectric interfaces, are widely used to design various optical devices. However, the operational frequency of the nature SPP, which is restricted by the intrinsic plasma frequency of the metal are almost at the optical and tera-hertz regime [Reference Maier, Andrews, Martín-Moreno and García-Vidal1]. In order to imitate the SPP feature at the radio frequency or microwave regime, the spoof surface plasmon polariton (SSPP) is invented by setting particular periodic stubs over the substrate [Reference Pendry, Martín-Moreno and Garcia-Vidal2–Reference Shen and Cui5]. Various circuits are designed based on the SSPP structure to employ the SSPPs slow wave and low-pass feature [Reference Gao, Zhou, Yu, Cao, Li, Ma and Cui6–Reference Zhao, Zhang, Zhao and Tang9].
The multi-band rejection filter (MBRF), which is an indispensable device in multi-band/multi-mode communication systems can reject several unwanted spectrums simultaneously. The popular design methodology of the MBRFs is to put different resonators into a low-pass filter. The SSPP just performs a low-pass feature so it is suitable for designing MBRFs [Reference Pan, Liao, Zhao and Cui10–Reference Tian, Xu, Xu, Zhang and Shi14]. Ref. [Reference Pan, Liao, Zhao and Cui10,Reference Xiao, Kong and Xiao11] achieve the MBRFs by placing the rejection resonators next to the SSPP units with low-pass frequency characteristics, which offer narrow rejection bands and bulky sizes. The low-pass SSPP units are also etched into rejection ones, hence the filter size is reduced [Reference Qian, Hao, Jia, Bai and Cui12,Reference Xu, Li, Liu, Xu, Chen, Ning, Chen and Gu13]. But the rejection bandwidth is still narrow. High-order dispersion modes of the SSPP unit are analyzed in Ref. [Reference Tian, Xu, Xu, Zhang and Shi14] and the rejection bands among the dispersion modes are applied to realize the MBRF, where the number of rejection bands is increased and the rejection bandwidths are also broadened, but the rejection bands cannot be controlled independently. Furthermore, none of these MBRFs discuss how to adjust the rejection bandwidth.
A novel dualband rejection filter (DBRF) and a triband rejection filter (TBRF) are proposed in this letter, both of which are implemented by cascading SSPP units with diverse rejection bands. Frequency characteristics of the SSPP units are mutually independent, thus the different rejection bands of one filter can be independently controlled. Both proposed filters provide more compact structures, wide rejection bands and favorable independent adjustability. The method of changing the rejection bandwidths is also introduced. The comparison between the DBRF and the TBRF demonstrates the effect of the rejection bands' number on the filters' performance. Measurement S 21 also agrees well with the dispersion of SSPP units.
Dispersion analysis
The SSPP unit is the core of the MBRFs. Its schematic and structural parameters are displayed in Fig. 1(a) and Table 1, respectively. The dispersion of the SSPP unit is simulated in CST Microwave Studio and the corresponding results are presented in Fig. 1(b). The SSPP unit is designed on RT/duroid 5880, whose permittivity and thickness are 2.2 and 0.508 mm, respectively.
Schematic and dispersion simulation result of the proposed SSPP unit. (a) Schematic, (b) Dispersion.
Parameters of the SSPP unit (unit:mm)
In Fig. 1(b), Mode 0 is the fundamental mode and Mode 1 is a high-order mode. The yellow and blue regions are passbands of the modes where SSPP wave can be transmitted. Conversely, the gray region is a rejection band where SSPP wave cannot be transferred. Overall, the dispersion of the unit has a pass-reject-pass characteristic in the frequency domain.
In order to study how L s controls the rejection band, dispersions of the SSPP units with different L s are also simulated, where L s is set to 0.94, 0.99, 1.24, and 1.54 mm, respectively. Simulation results are displayed in Fig. 2 and Table 2, respectively, where f 0 is the center frequency of the rejection band, RBW is the rejection bandwidth and f c1 is the cutoff frequency of mode 1, respectively. As shown in Table 2, as L s increases, f 0 is largely reduced, meanwhile f c1 and RBW are slightly decreased. In summary, adjusting L s can effectively change the center frequency of the rejection band meanwhile keep the rejection bandwidth and the cutoff frequency of mode 1 almost unchanged.
Simulated dispersion of the proposed SSPP unit with diverse L s.
Dispersion frequency with diverse L s
Filters design
Due to the favorable frequency adjustability in the rejection band, the proposed SSPP unit is applied to achieve the DBRF and the TBRF. Schematics of the filters are illustrated in Fig. 3. Both filters are composed of three parts: Part I, Part II, and Part III. Part I is the SSPP part, which transmits TM mode wave and is crucial to the filter's final performance. The identical SSPP units are periodically arrayed into blocks and the blocks with diverse rejection bands are cascaded together. Between diverse blocks, only L s is different and other structural parameters are identical. In the DBRF, L s is 0.99 and 1.24 mm and each rejection band corresponds to 6 SSPP units. In the TBRF, L s is 0.94, 1.24, and 1.54 mm and each rejection band corresponds to 4 SSPP units. Part III is the micro-strip connecting with other microwave devices, which transmits quasi-TEM wave. The characteristic impedance of the micro-strip is 50 Ω and its width is 1.54 mm. Part II is the mode conversion part transforming quasi-TEM wave and TM wave bidirectionally, which effectively improves the transmission efficiency of the filters.
Schematics of the proposed dualband rejection filter (DBRF) and trilband rejection filter (TBRF).
The filters are also simulated in CST Microwave Studio. Simulated S parameters are illustrated in Fig. 4 and Table 3, respectively, where f 0 is the center frequency of the rejection band, BW −3 and BW −15 are −3 and −15 dB rejection bandwidth, respectively. And Depth is the rejection depth.
Simulation results of the proposed filters: (a) the DBRF, (b) the TBRF.
Simulated rejection performance of the proposed filters
As Fig. 4 shows, the DBRF provides two rejection bands and the TBRF offers three. Average BW −3 and BW −15 of all the rejeciton bands are 1.43 and 0.98 GHz, respectively. Average rejection depths of the DBRF and the TBRF are 61.5 and 43.6 dB, respectively. Consequently, the rejection bands are wide, steep, and deep. Cutoff frequencies of the DBRF and the TBRF are 27.67 and 27.66 GHz, respectively. Average insertion losses of the passbands are 0.9, 1.7, and 2.3 dB, respectively, in the DBRF and 0.8, 1.2, 1.9, and 2.7 dB, respectively, in the TBRF. Out-of-band rejections of both filters are <− 80 dB.
In Fig. 4, the gray bands are rejection bands of dispersion and the vertical lines are cutoff frequencies of mode 1 (f c1). A comparison between cutoff frequencies of the dispersion and the filters are listed in Table 4, where f dis is the cutoff frequency of the dispersion, f −3dB is the − 3 dB cutoff frequency of the filter. 
$\vert\triangle f\vert= \vert f_{-3dB} - f_{dis} \vert$, whose values are pretty small at the lower side-bands and the cutoff bands, so the cutoff effect of dispersion agrees with the filters in these bands. However, values of 
$\vert\triangle f\vert$ are larger at the upper side-bands, which may be caused by the transition effect from rejection to pass meanwhile they keep in reasonable ranges. In conclusion, the frequency characteristics of the dispersion agree with those of the filters.
Comparison of cutoff frequencies of dispersion and Filters
As the dispersion of the SSPP unit indicates, rejection bands of different SSPP units are mutually independent. Since frequency characteristics of the dispersion agree with those of the filters, center frequencies of the rejection bands in one filter are also independent of each other and the different rejection bands in each filter can be independently controlled.
Both the DBRF and the TBRF are composed of 12 SSPP units. But the number of SSPP units in the TBRF is 2 less than that in the DBRF, which is the reason why the average rejection depth of the TBRF is shallower than that of the DBRF. In each filter, the insertion loss and the passband fluctuation increase with increasing frequency and these phenomena are more pronounced in the TBRF, which demonstrates that an increase in the number of rejection bands results in an increase in the insertion loss and the fluctuation of the passbands.
Electromagnetic (EM) field simulation results of the TBRF are illustrated in Fig. 5. As shown in Figs 5(a) and 5(b), the EM energy transmission is blocked by the SSPP part at the rejection band. In contrast, as Figs 5(c) and 5(d) display, the EM energy of the passband can be transmitted in the SSPP part. The EM filed of the SSPP part is also stronger than that of the micro-strip, thus the SSPP has better constraints to the EM field than the micro-strip. In the SSPP part of Fig. 5(d), the EM field is tightly bound at the interface between the metal and the substrate. Meanwhile, peaks of the field also locate on the metal strips in the SSPP. It can be seen that the SSPP can achieve the control and constraint on the EM field.
EM field simulation results of the TBRF: (a)/(b) Top/side view of the rejection band (f = 15.22 GHz); (c)/(d) Top/side view of the passband (f = 21 GHz).
Discussion
The method of changing the center frequency of the rejection band has been illustrated in the previous section. Approaches of controlling the rejection bandwidth, the rejection depth and the filter's cutoff frequency are continued in this section. In addition, there is only one sort of SSPP units in the SSPP parts of traditional SSPP rejection filters, thus the entire SSPP part is periodical (Fig. 7). But the proposed filters contain more than one sort of SSPP units, hence the SSPP part becomes a non-periodical one (Fig. 3). This section also compares periodical and non-periodical SSPP rejection filters.
Rejection bandwidth tunability
Perhaps due to the limitations of the filter structure, the methodology of controlling the rejection bandwidth by the structural parameters is rarely mentioned in traditional SSPP rejection filters, which is achieved in the proposed filters. Based on the DBRF, L s and s of the SSPP units are increased by 0.2 mm simultaneously and a new DBRF is produced. Simulated S 21 curves of the DBRFs are displayed in Fig. 6(a) and Table 5, respectively, where 
$ \triangle $BW is the difference between the rejection bandwidths of the two DBRFs. When L s and s in the SSPP unit increase simultaneously, the shape of the FSRR becomes higher and thinner meanwhile the rejection bandwidth is also greatly reduced. Consequently, higher and thinner FSRR will result in a narrower rejection band.
Simulated S 21. (a) S 21 of the DBRF with different rejection bandwidths. (b) S 21 of the DBRF with different rejection depths. (c) S 21 of the DBRF with different cutoff frequencies. (d) S 21 of the TBRF and the periodical SBRFs.
Schematics of the periodical single band rejection filters.
Simulated rejection bandwidth of the DBRF
In the two DBRFs of Fig. 6(a), although shapes of the FSRRs are different, their total lengths are the same. However, in addition to the bandwidth change in the above variation, the center frequency of the rejection band is also increased by 1.3 GHz, which may be caused by the shape variation of the FSRR, but this effect can be compensated by adjusting the values of L s or g. Besides, the rejection depth and the filter's cutoff frequency hardly change during this process. Accordingly, the adjustment of the rejection bandwidth does not affect performances of the rejection depth and the filter's cutoff frequency.
Rejection depth tunability
In the proposed MBRFs, the depth of the rejection band depends on the number of SSPP units. In order to evaluate the effect of the SSPP units' number on the rejection depth, this section simulates multiple DBRFs whose SSPP units' numbers of each rejection band are different. The simulation results are illustrated in Fig. 6(b).
In Fig. 6(b), the filter with more SSPP units produces deeper rejection bands. The blue curve (2 units) provides narrower rejection bandwidth than others, while the red and black curves (6 and 10 units) have approximate rejection bandwidths. It can be seen that sufficient SSPP unit is indispensable for ensuring the rejection width. Besides, filters with more SSPP units provide better roll-off factors, but also have larger insertion losses and areas. Therefore, the trade-off between the number of SSPP units and the rejection bandwidth is worth considering.
Filter's cutoff frequency tunability
The simulation results where diverse slot depth H g controls the filter's cutoff frequency f c are plotted in Fig. 6(c). When H g increases by 0.2 mm, f c decreases by 2.5 GHz and S 21 curves of the rejection bands are almost unchanged. As a result, the filter's cutoff frequency can be controlled by H g and the control process is independent of the rejection bands.
Depending on the above results, the parameter adjustment methods can be concluded. Firstly, approaches of changing the rejection bandwidth, the rejection depth and the filter's cutoff frequency are independent of each other. Secondly, the different rejection bands of one filter are independently adjustable. Finally, although the center frequency and the bandwidth adjustment approaches of the rejection bands are not completely independent, the resulting changes can be compensated by fine-tuning other parameters. In summary, the proposed filters have a favorable independent tunability.
Comparison with periodical ones
In order to study the performance variation when the periodic SSPP units are cascaded into non-periodic ones, the SSPP part of the TBRF is divided into three different single band rejection filters (SBRFs) according to the different rejection bands, where only identical SSPP units with the same rejection band are placed in the same filter (Fig. 7). Compared with the TBRF, the SSPP parts of the SBRFs are periodical and total numbers of the SSPP units in the SBRFs are also reduced. However, numbers of the SSPP units are the same for the rejection bands of the same SSPP units but of different filters. Mode conversion parts and micro-strips of the SBRFs are also the same as the TBRF.
Simulation results of the SBRFs and the TBRF are compared in Fig. 6(d). S 21 curves of the SBRFs and the TBRF roughly coincide, especially in the rejection bands. But the insertion loss and the fluctuation of the TBRF are greater than the SBRFs in the passbands. The insertion loss of the TBRF is also greater than the SBRF at the lower side-band of the third rejection band, which may be caused by the further distance between the SSPP units. The different cutoff frequencies of the SBRFs agree with f c1 respectively, and the cutoff frequency of the TBRF is amongst f c1. Accordingly, the cutoff frequency of the TBRF depends on f c1 of the diverse SSPP units.
Based on the comparison result between the SBRFs and the TBRF, performance differences between the cascaded non-periodical and the periodical SSPP parts can be summarized. Firstly, rejection performances of two types of filters are approximate and dispersion analysis of SSPP units is equally applicable for cascaded ones. Secondly, roll-off factors of the cascaded ones are better due to their more SSPP units, but the insertion loss and the fluctuation in passbands are greater. Finally, the further distance between SSPP units increases their insertion loss.
Experimental results
The fabricated filters are displayed in Fig. 8(a), which are put in the aluminum box to be measured by the Ceyear AV3672C vector network analyzer. The height of the box is 3.0 mm and sizes of the filters are 40.0 mm×3.0 mm, which is 3.57λg × 0.29λg for the DBRF and 3.45λg × 0.28λg for the TBRF (λg is the guided wavelength of the average center frequency of the respective rejection bands). Besides, a micro-strip with the same substrate is also fabricated for the sake of determining the insertion loss from the welding, connectors, and the substrate. The characteristic impedance of the micro-strip is 50 Ω, accordingly its width is 1.54 mm. The length of the micro-strip is the same as the filters, which is 40.0 mm, too.
Photographs and measurement S parameters of the fabricated micro-strip and MBRFs. (a) The fabricated micro-strip and filters. (b) Measured micro-strip. (c) Measured DBRF. (d) Measured TBRF.
In the measurement, after an open-short-load calibration on the vector network analyzer, the micro-strip is placed in the aluminum box to be measured. Next, measured S 21 is normalized into 0 dB and the filters are also put in the same box to be measured by the normalized vector network analyzer. Measurement results of the filters are illustrated in Figs 8(c), 8(d) and Table 6, respectively.
Measured rejection performance of the proposed filters
Measurement and simulation S parameters of the micro-strip are plotted in Fig. 8(b). Both return loss and insertion loss in the measurement are greater than the simulation. And the insertion loss also increases as the frequency grows, which is more obvious in the measurement. Within the frequency range of 10—30 GHz, average insertion losses of simulation and measurement are 0.33 and 1.78 dB respectively, thus the loss of the welding, connectors and the substrate is about 1.45 dB.
As Figs 8(c), 8(d) and Table 6 demonstrate, both measured S 11 and S 21 agree well with the simulated ones. Measured rejection bands have an average bandwidth of 1.49 GHz and an average depth of 42.1 dB. Measured f c of the DBRF and the TBRF both are 27.66 GHz. Average insertion losses of passbands are 0.5, 1.1, and 2.6 dB, respectively, in the DBRF and 0.7, 1.6, 1.6, and 2.8 dB, respectively, in the TBRF. Therefore, measured f c and insertion loss also approximate to the simulation. Overall, the fabricated filters agree well with the simulation and the proposed filters are available for microwave engineering.
A comparison between the proposed filters and traditional SSPP MBRFs are illustrated in Table 7. Compared with traditional ones, the proposed filters not only provide wide rejection bands, they also offer more compact sizes and better tunability. In this work, the rejection bandwidth can be controlled by the filters' structural parameters. And the rejection bandwidth, the rejection depth, and the filter's cutoff frequency can be independently controlled. Moreover, the insertion loss and the rejection depth are also promoted. And the correspondence between the dispersion and the filter is also presented.
Performances comparison with traditional SSPP MBRFs
aNumber is the rejection bands' number.
bFBW is the fractional rejection bandwidth.
cIL is the insertion loss.
dIPA is the independent control between the rejection bands in the same filter.
eRBWT is the rejection bandwidth tunability.
The comparison between non-SSPP MBRFs and our SSPP MBRFs are listed in Table 8, where FBW −3 and FBW −15 are −3 and −15 dB fractional bandwidths, respectively. According to the table, the SSPP ones have longer sizes and narrower fractional bandwidths than non-SSPP ones. However, in a whole view, the differences between FBW −3 and FBW −15 of non-SSPP ones are larger than those of SSPP ones, which indicates that the SSPP ones have steeper rejections. Besides, the rejection depths of the non- and SSPP ones are approximate in the TBRFs, but which of the SSPP one is better than non-SSPP ones in the DBRFs, which may be caused by the more SSPP units in the SSPP DBRF. Hence the rejection level of the SSPP ones can behave better than non-SSPP ones.
Performances comparison with non-SSPP MBRFs
Conclusion
By cascading SSPP units of different rejection bands, a dual-band rejection filter and a tri-band rejection filter are achieved in this letter. The filters offer simple structures, compact areas, controllable rejection bandwidths, good independent adjustability, and wide rejection bands at the same time, which broaden the range of applications for multi-band SSPP rejection filters. Measurement results also support the design approach.
Author ORCIDs
Luping Li, 0000-0001-9294-6730; Lijuan Dong, 0000-0002-1773-0777; Peng Chen, 0000-0003-1444-5097.
Acknowledgments
This work was supported by the National Natural Science Foundation of China 61601088 and 61571093.
Luping Li received his Bachelor degree in electronic information engineering from Chengdu University of Information Technology, Chengdu, China in 2016. Now he is studying his Master degree in University of Electronic Science and Technology of China, Chengdu, China since September 2016. His research interests are RF/microwave filters.
Lijuan Dong received her Bachelor degree in biomedical engineering from Chengdu University of Information Technology, Chengdu, China in 2017. She is studying her master degree in University of Electronic Science and Technology of China, Chengdu, China since September 2017. Her research interests are RF/microwave filters and electroencephalogram signal processing.
Peng Chen received Bachelor in electronic science and technology and Ph.D. degree in circuits and systems from University of Electronic Science and Technology of China, Chengdu, China in 2009 and 2015, respectively, where he became a lecture in 2015. He was also a visiting scholar in Tohoku University, Sendai, Japan from November 2017 to November 2018. His main research interests are design and optimization RF and microwave devices and systems.
Kai Yang became an associate professor and a professor of University of Electronic Science and Technology of China, Chengdu, China in 2001 and 2007, respectively. He became a senior member of Chinese Institution of Electronics in 2006. His research interests are RF/microwave circuits and systems, superconducting microwave components and circuit theory.
