Introduction

Band-pass filters (BPFs) are the essential front-end component for RF/microwave systems. Recent advancements in modern wireless and radar communication applications demand high-performance and dynamically controlled RF subsystems. Hence, electronically tunable microwave filters are gaining more attention for research and development. Conventional microwave filters generally use traditional microstrip technology which suffers from radiation losses, crosstalk, and mutual coupling problems. Crosstalk involves in-between transmission line results in signal integrity issues and thus limits the performance of RF systems [1]. The solution to overcome this problem lies in using spoof surface plasmon polariton (spoof SPP)-based RF systems [2, 3].

Natural SPP are unique surface waves that are highly localized at the interface of metal-dielectric and found at optical and near-infrared frequencies. It decays exponentially in the vertical direction of the interface hence exhibit subwavelength field confinement and enhancement [4, 5]. Since metal behaves like plasma with negative permittivity at optical frequencies, hence SPP wave is supported by the metal-dielectric interface. However, metals show characteristics like a perfect conductor (PEC) at microwave and THz frequencies. Hence, subwavelength features are usually not found at these frequency regimes. Recently, it is shown that semimetal like graphene and semiconductors also support SPPs at THz frequency naturally which enables its applications in optoelectronic devices like solar cell, graphene as sensors, and other plasmonic and nano-photonic devices [6,7,8,9,10,11,12,13,14,15,16,17,18]. However, to realize the SPP-like characteristics in the vicinity of the metal-dielectric interface at THz and microwave frequencies, plasmonic metamaterial [19] that are the periodically corrugated metallic surfaces with 2D hole or 1D groove has been proposed, also called as spoof or designer SPP. The dispersion curve and cutoff frequency for such a structure can be altered through its physical parameter. Thus, these structures have capability of guiding and manipulating the EM waves at subwavelength scale, which can be applied to design transmission lines [20,21,22], and components like filters [23,24,25], for the excitation of antenna [26], amplifier [27], switches [28], and tunable filters [29,29,30,31,33]. In [29, 30], band-stop tunable behavior has been shown by changing the physical dimension. In [32], capacitors with different values have been introduced in between the two-unit cell through which dispersion and cutoff frequency of the low-pass filter can be tuned. However, structures demonstrated in [29,29,30,32] required manual fabrication each time with different parameter values to demonstrate tunable feature. In [33], researchers have demonstrated passband tunability of spoof surface plasmon polaritons. They have used 24 varactor diodes and 25 lumped inductors in different positions of the spoof SPP-based transmission line to control the higher and lower cutoff frequency of the filter which makes this arrangement very bulky and very complex for a practical prototype.

In this report, an electronically reconfigurable center frequency as well a bandwidth BPF is proposed using T-shape spoof SPP-based resonator through varactor diodes. Three varactors are introduced at different positions of the T-shape resonator which can be controlled individually by DC bias voltage applied across them. Transition from QTEM mode of microstrip to spoof SPP mode has been used to excite the SSPP-based T-shape resonator with coupling gap g. The prototype has been fabricated and measured; results have been presented for the validation of the design concept.

The organization of the paper is as follows: the theory and design principle of the reconfigurable center frequency and bandwidth band-pass filter based on the spoof SPP compact T-shaped resonator has been discussed in “Theory and Design Principle.” The fabricated prototype and characterization of the designed filters has been provided in “Results and Discussion.” Finally, the conclusion has been drawn in “Conclusions.”

Theory and Design Principle

Design of T-Shaped Spoof SPP Resonator

The proposed T-shape spoof SPP resonator along with its even- and odd-mode equivalent circuits is shown in Fig. 1. This T-shape resonator can be interpreted as two λg/4 resonators coupled to each other via K-invertor implemented by stubbed SSPP transmission line section. Electrical length (in degree) and admittance (1/Ω) of two of line sections are (ϕ1/2, Y1) while other one has (ϕk, Y1). Each transmission line is corrugated with metal strip on it with electrical length and admittance of △ϕ′ and △Y′, respectively. By adjusting the length of the tapped open stub, which can be longer or shorter than λg/4, a transmission zero can be created at a frequency lower or higher than the desired passband. Since, the length of the open stub is less than λg/4, hence it creates a transmission zero at upper stopband as shown in Fig. 3. As the structure of the resonator is symmetric;,it will excite multimode frequency and its operating mechanism can be explained in terms of its even-odd mode analysis.

Fig. 1
figure 1

Schematic of a proposed T-shape resonator and its even- and odd-mode equivalent circuits and b spoof SPP-based band-pass filter

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Resonance condition for the designed resonator can be derived as:

$$ Img\left[{Y}_{ine}\right]=0 $$
(1)

where Yine is the even-mode admittance and expressed as:

$$ {Y}_{ine}={Y}_1\left[\frac{Y_M+j{Y}_1\tan \left({\varPhi}_1/2\right)}{Y_1+j{Y}_M\tan \left({\varPhi}_1/2\right)}\right]+{n}^{\ast}\left({jY}^{\hbox{'}}\tan {\varDelta \varPhi}^{\hbox{'}}\right) $$

with \( {Y}_M=j\frac{Y_1}{2}\tan {\varPhi}_k+{m}^{\ast}\left({j\varDelta Y}^{\hbox{'}}\tan {\varDelta \varPhi}^{\hbox{'}}\right) \), n = 4, m = 2

and

$$ Img\left[{Y}_{ino}\right]=0 $$
(2)

where Yino is the odd-mode admittance and it is defined as:

$$ {Y}_{ino}=\Big(-j{Y}_1\cot \left(\frac{\varPhi_1}{2}\right)+{n}^{\ast}\left({\varDelta Y}^{\hbox{'}}\tan {\varDelta \varPhi}^{\hbox{'}}\right) $$

where Yine, Yino are the even- and odd-mode admittances.

From Eqs. (1) and (2), it is clear that Φ1 affects both the even and odd modes while Φk influences even mode only. Hence, by tailoring the physical length the equivalent electrical path for different even and odd modal frequencies can be changed.

Design of Feeding Network

The proposed T-shape SSPP resonator is directly coupled with spoof SPP transmission feed line at input and output as shown in Fig. 1b. This transmission line consists of periodical array of unit cell as shown in the inset of Fig. 2a. The behavior of the spoof SPP has been given by its dispersion diagram as shown in Fig. 2a in which the blue-colored line is the propagation wave vector (k0) for the freely propagation wave and the green-colored curve is the propagation wave vector (ky) for the spoof SPP unit cell. The propagation characteristics of the guided modes of the SSPP can be controlled by its structural parameters, which are the separation between two grooves d, groove height h, thickness of metal t, and lattice constant p. The dispersion characteristics of the spoof SPP unit cell have been analyzed through numerical simulation using CST MWS. Since ky deviates from k0, hence there is a mismatch of momentum and polarization between them. This mismatch leads to bad efficiency hence a conversion region needs to be designed which is fulfilled by the gradual conversion as shown in the inset of Fig. 2b, in which the height of the groove has been varied from h1–h8 (0.25 to 2 mm) with equal step of h0 (0.25 mm). The optimized structural parameters of the proposed unit cell are as follows: w = 2 mm, h = 4 mm, d = 4 mm, a = 1 mm, and p = 5 mm. Dielectric substrate having dielectric constant of 3.2, loss tangent of 0.0024, and a height of 1.52 mm with a conducting ground plane has been used for analysis and implementation. CST microwave studio has been used to obtain the numerical simulations.

Fig. 2
figure 2

Dispersion diagram a for double-sided grooved spoof SPP unit cell (inset: schematic of unit cell) and b relating the gradual conversion

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Analysis of T-Shape Resonator with Different Horizontal and Vertical Electrical Length

The designed filter has a dual-mode frequency response for which parametric analysis with respect to l1 and lk has been done in Fig. 3a, b, respectively, to verify Eqs. (1) and (2) in EM simulation. In Fig. 3a, as we change the l1 with lk and all other physical parameter fixed, both the even- and odd-mode frequencies are varying hence overall center frequency of BPF is moving towards the lower side. Similarly, in Fig. 3(b), by varying the lk with l1 and all other physical parameters fixed, it has been observed that the even mode is shifting towards the lower frequency side whereas the odd mode is fixed at a particular frequency. Figure 4 shows the reflection and transmission coefficient for the designed band-pass filter with different coupling gap g. It can be observed that as coupling gap increases, the input-output coupling decreases due to small value of gap capacitance and consequently a poor matching is obtained.

Fig. 3
figure 3

Simulated S-parameter of the T-shape SSPP resonator–based band-pass filter with respect to change in al1 and blk

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Fig. 4
figure 4

Simulated S-parameter characteristics for the designed SSPP-based T-shape band-pass filter with different coupling gap g

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Design of Reconfigurable T-Shape SSPP Resonator with Even-Odd Mode Analysis

Figure 5 depicts the schematic of the proposed reconfigurable T-shape resonator along with its even- and odd-mode equivalent circuit configurations. Two varactor diodes (C) and one varactor diode (C1) are symmetrically placed in the X and Y direction of XY plane in the resonator, respectively.

Fig. 5
figure 5

Schematic of the proposed T-shape resonator with varactor diodes and its equivalent even- and odd-mode circuits

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Resonance schemes for the designed resonator can be derived as:

$$ Img\left[{Y}_{ine}\right]=0 $$
(3)

and

$$ Img\left[{Y}_{ino}\right]=0 $$
(4)

where Yine and Yino are the even- and odd- mode admittance and expressed as

$$ {Y}_{in}={Y}_1\left[\frac{Y{\hbox{'}}_M+j{Y}_1\tan \left({\varPhi}_A\right)}{Y_1+ jY{\hbox{'}}_M\tan \left({\varPhi}_A\right)}\right]+{n}^{\ast}\left({j\varDelta Y}^{\hbox{'}}\tan {\varDelta \varPhi}^{\hbox{'}}\right) $$
$$ Yine= Yin $$

when

$$ Y{\prime}_M=\frac{Y_M\ast j\omega C}{Y_M+ j\omega C} $$

with

$$ {Y}_M=\frac{Y_1}{2}\left[\frac{j\omega {C}_1/2+j\frac{Y_1}{2}\mathit{\tan}\left({\varPhi}_k\right)}{\frac{Y_1}{2}+j\Big( j\omega {C}_1/2\mathit{\tan}\left({\varPhi}_k\right)}\right]+\left({j\varDelta Y}^{\prime }{tan\varDelta \varPhi}^{\prime}\right) $$
$$ {Y}_{in o}={Y}_{in} $$

when

$$ Y{\prime}_M= j\omega C $$

For the given even- and odd-mode admittances, admittance matrix can be expressed as:

$$ Y=\left[\begin{array}{cc}{Y}_{11}& {Y}_{12}\\ {}{Y}_{21}& {Y}_{22}\end{array}\right]=\left[\begin{array}{cc}\left({Y}_{ine}-{Y}_{ino}\right)/2& \left({Y}_{ine}+{Y}_{ino}\right)/2\\ {}\left({Y}_{ine}+{Y}_{ino}\right)/2& \left({Y}_{ine}-{Y}_{ino}\right)/2\end{array}\right] $$

Further the transmission zeros in terms of admittance parameter can be expressed as:

$$ {Y}_{12}={Y}_{21}=\left({Y}_{ine}+{Y}_{ino}\right)/2=0 $$

According to Eqs. (3) and (4), it can be observed that varactor (C) affects both the even and odd modes whereas varactors (C1) affect the even mode only.

Figure 6 depicts the simulated E- and H-field distribution for the designed T-shaped spoof SPP resonator–based BPF at 3.23 GHz of frequency. In the passband of the designed filter, electric field is highly confined near the resonator and allows the EM field to propagate through it.

Fig. 6
figure 6

(i–ii) E-field and (iii–iv) H-field distribution for designed band-pass filter at 3.23 GHz frequency

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Results and Discussions

Figure 7a, b show the layout and fabricated prototype of the proposed spoof SPP-based varactor loaded tunable filter which simultaneously shows the tuning of the center frequency as well as the bandwidth. A coupling gap g = 0.12 mm has been chosen for fabricated prototype due to the fabrication tolerance.

Fig. 7
figure 7

a Schematic, b fabricated prototype of the proposed reconfigurable BPF

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We have chosen skyworks SMV1232-079LF as a varactor diode for our design and implementation. The simple model of a packaged varactor diode is shown in Fig. 8a. This varactor diode has junction capacitance CJ0 = 3.43 pF, parasitic inductance LS = 0.7 nH, parasitic capacitance CP = 0.68 pF, and series resistance RS = 1.5 Ω. Capacitance vs reverse bias voltage graph has been shown in Fig. 8b for SMV1232. It can be observed from Fig. 8b that as bias voltage increases, the junction capacitance decreases. In hardware prototype, the SMV1232-079LF package is used as a variable capacitor device (C, C1) whereas Coilcraft 33 nH inductors and ATC 100 pF capacitors are used as RF choke and DC block, respectively. Corresponding to the proposed configuration, measured results have been illustrated in Figs. 9 and 10.

Fig. 8
figure 8

a Model for varactor diode with Cj0 = 3.43 pF, Ls = 0.7 nH, CP = 0.68 pF, Rs = 1.5 Ω and b capacitance variation with frequency for Varactor SMV 1232 [35]

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Fig. 9
figure 9

Measured S-parameters of the proposed center frequency tunable BPF obtained with fixed VC1 = 0 V and varying VC

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Fig. 10
figure 10

Measured S-parameters of proposed bandwidth reconfigurable BPF obtained with VC = 5.19 V and varying VC1

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Since varactor C is affecting the even and odd mode simultaneously, hence by changing the bias voltage across the varactor C, the overall center frequency of degenerative mode can be varied. It is observed from Fig. 9 that keeping the VC1 fixed at 0 V, when we change the bias (VC) across the varactor diodes (C) from 4.59 to 7.09 V, the overall center frequency of the band shifts from 4.2 to 4.4 GHz (200 MHz). Insertion loss of 4.2 dB is observed during the tuning range. Figure 10 depicts the tuning of bandwidth due to the change in the bias VC1 across the varactor diodes (C1) for the fixed VC = 5.19 V. As C1 affects only the even-mode wave, hence degenerative mode splitting took place here and resulted in bandwidth tuning. In this case, odd-mode frequency is fixed to 4.12 GHz while even-mode frequency varies from 4.12 to 4.52 GHz. Correspondingly, − 3 dB impedance bandwidth for the proposed configuration is varying from 500 to 630 MHz (130 MHz). Insertion loss around 3.9 dB is observed due to the little loose input-output coupling because of fabrication tolerance and SMA connector losses, also, due to the finite Q value of varactors which have inherent resistance that causes losses.

Also, intermodulation analysis has been performed for the designed band-pass filter. For that, signal distortion has been investigated using a two-tone method [35]. Simulated intermodulation product has been obtained using Harmonic Balance simulation ADS around the lowest frequency f0 of center frequency tuning range. Two tones are separated by ∆f and located at frequencies, f1 = f0−∆f/2 and f2 = f0 + ∆f/2. The intermodulation product at different input power s of 0, 10 and 15 dBm are calculated and the corresponding IIP3 for ∆f = 100 KHz is obtained 27.7, 28, and 28.5 dBm, respectively, for designed reconfigurable filter. A state-of-art comparison of the proposed work is given in Table 1.

Table 1 Comparison of state-of-the-art
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Conclusions

In this paper, simultaneous center frequency and bandwidth reconfigurable T-shape spoof SPP BPF have been designed and analyzed by incorporating varactor diodes. This configuration is further verified by measured results of a fabricated prototype. A slightly higher insertion loss is observed which may be minimized by maintaining external Q value with capacitance change, using high-Q varactors, low-loss dielectric material, and CPW transmission line. The developed T-shape spoof SPP resonator–based reconfigurable band-pass filter has potential application in integrated and flexible plasmonic circuits in microwave frequency regimes.