Abstract
This paper presents a novel procedure to design a compact and miniaturized unit cell in the substrate integrated plasmonic waveguide (SIPW) structures. The slot of the unit cell is automatically obtained in this method without needing any prior knowledge of its shape. The method is based on topology optimization through pixelization of the slot surface in the unit cell and uses a combination of binary particle swarm optimization (BPSO) and commercial electromagnetic (EM) software. With this method, first, the shapes of the two unit cells are engineered to arbitrarily reduce the high cut-off frequency. Then, two low-pass filters consisting of the proposed unit cells are designed and simulated. To further verify the proposed procedure, one of the filters is fabricated, whose measurement results are in good agreement with the simulation results. This filter provides a pass band of 5.6 to 6.3 GHz and has dimensions of . The proposed method has the potential to minimize microwave and terahertz devices.
1. Introduction
Substrate integrated plasmonic waveguides (SIPWs) which have attracted attention in recent years are formed by etching slots on metal layers of the substrate integrated waveguide (SIW) to support the spoof surface plasmon polaritons (SSPP) mode at microwave and terahertz (THz) frequencies [1–19]. The SIPW inherits the characteristics of both SIW and SSPP and exhibits bandpass behavior; its low cut-off frequency depends on the width of the SIW, and its high cut-off frequency can be individually adjusted with the shape and the dimension of the slots. Simple rectangular-shaped and comb-shaped slots have first been proposed in [1–11]. However, to reduce the high cut-off frequency and design more compact microwave and THz devices, the length of slots has to be longer. It is possible to extend the slots transversely up to a certain limit due to the finite width of the SIW. In addition, the longer the slots, the more difficult the impedance matching. Therefore, different slot shapes have been proposed whose asymptotic frequency is lower than that of the traditional rectangle slot. Dumbbell-shaped and T-shaped slots have been presented in [12–15], respectively, and have better field confinement than the traditional rectangle-shaped slot. Furthermore, an on-chip terahertz SIW filter with the Yagi-Uda antenna-like slots has been proposed in [16] which effectively reduces asymptotic frequency compared to previous structures. Asymptotic frequency reduction using meander slot on SIW is also shown in [17]. In [18], a compact bandpass filter with a fishbone-shaped and hourglass-shaped slot has been designed with better electromagnetic (EM) field confinement than that of the comb-shaped slot.
In this paper, we introduce a new procedure to design the slot shape of a SIPW unit cell automatically. Unlike previous traditional methods in which the predetermined shape of the unit cell is applied and its dispersion curve is controlled by varying geometrical parameters, this novel method is based on slot shape extraction with a connection between the binary particle swarm optimization (BPSO) algorithm and a full-wave EM simulator. This procedure has the advantage of designing the SIPW unit cell with the desired reduction at asymptotic frequency to achieve strong field confinement. First, the design procedure is discussed, and then, two SIPW unit cells with different values of asymptotic frequencies are achieved. Finally, to verify the proposed method, a SIPW waveguide prototype of the proposed unit cell is fabricated and measured.
2. Unit Cell Design
The substrate is RO4003 with a thickness of 0.8 mm and a relative permittivity of 3.55. Figure 1 depicts a SIPW unit cell in which the location of the slot is known, but its shape is unknown. To determine the slot shape, the location of the slot is pixelated as illustrated in Figure 1. Each pixel can be either “1” or “0” which corresponds to the absence or presence of the metal, respectively. In fact, each combination of binary values of pixels forms a binary matrix that is equivalent to a certain shape of the slot. Since the “0” or “1” status of each pixel affects the dispersion curve, by optimizing their different compositions, the desired dispersion curve can be achieved. In this paper, the BPSO algorithm is applied to minimize the asymptotic frequency to a desired value. Among the various optimization methods with binary solution space, it has been shown that the BPSO method is much more efficient and has fewer parameters [20, 21]. It is also found that the BPSO algorithm converges faster than other algorithms because it is more targeted in each searching direction and avoids getting stuck in local optima [20, 22]. The BPSO algorithm has also been successfully used in pixelated structures [23–27]. The BPSO algorithm has a population (called a swarm) of candidate solutions (called particles) which move in the search space of the problem. The movement of particles in each iteration is based on the own best-known position and the swarm’s best-known position. In each iteration of the optimization, the cost function of each particle is calculated using its dispersion curve obtained from the eigenmode solver of CST Microwave Studio (CST MWS). Therefore, it is necessary to appropriately link MATLAB (in which the optimization algorithm is written) and CST MWS. In addition, the initial solution can be selected from a unit cell that has already been proposed. The fitness function in the th iteration of the optimization is defined as follows: where is the maximum frequency of the dispersion curve in the th iteration and is the desired maximum frequency under which we want the dispersion curve to be located. To increase the convergence rate, the number of pixels to be optimized can be reduced by considering two-fold symmetry with respect to the -axis. It is clear that the low cut-off frequency depends only on the dimension of the SIW (especially the parameter of ) and cannot be changed with slot shape optimization. To clarify the optimization process, the flowchart of the program execution steps is shown in Figure 2. The main optimization program is implemented in MATLAB (green boxes), which is connected with CST to calculate the dispersion curves (blue box).
The design procedure has been examined with two examples. In the first design, we have considered the T-shaped slot as the initial solution. As shown in Figure 3, the asymptotic frequency of the T-shaped slot is about 8.8 GHz. The desired maximum frequency is also selected as 7 GHz. Concerning the symmetry of the slot shape, the dimensions of the half-shape binary matrix are . In the BPSO algorithm, the population size of particles is 20, and every particle includes 75 pixels to assign a pattern to the slot shape. After 23 iterations, the optimization process reaches the termination criteria, and the dispersion curve is placed under 7 GHz, as shown in Figure 3(c). The optimized shape of the slot is illustrated in Figure 3(b).
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A Yagi-Uda antenna-like slot, shown in Figure 4(a), has been assumed as the initial guess in the second design. According to our knowledge, this slot provides the lowest asymptotic frequency compared to the previous works. Its asymptotic frequency is obtained as 7.56 GHz with the dimensions mentioned in the caption of Figure 4. Note that the unit cell dimensions and pixelated region dimensions have changed with respect to the previous example. Considering bilateral symmetry, the number of pixels to be optimized in this example is 70, and the optimization parameters are the same as in the previous case. The desired maximum frequency is also supposed to be 6.2 GHz. The optimization is stopped after 50 iterations, and the asymptotic frequency reaches 6.3 GHz. The optimized shape of the unit cell and its dispersion curve are shown in Figure 4. It is clear that the optimized unit cell is more compact than the initial one. To visualize the compactness of the presented unit cell compared to the previous structures, Figure 5 shows the dimension of the slots in different unit cells which provide the same high cut-off frequency of 6.2 GHz. The length of the proposed slot reduces up to 42% compared with the length of the conventional rectangular-shaped slot. Based on the comparison between the unit cells of Figure 5, it is clear that to reduce the transverse frequency and concentrate more waves in the conventional unit cells, their dimensions should be larger. This causes the slots to be closer to the vias. Therefore, one should not expect suitable behavior from such a structure. In fact, instead of increasing the dimensions of the slots, we engineer them to reduce the transverse frequency.
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To demonstrate the field confinement of the proposed unit cell, the electric field distributions of the unit cells with the conventional slot and the optimized slot having the same length are compared in Figure 6. It can be observed that the proposed structure demonstrates higher field concentration than that of the traditional unit cell.
Figure 7 displays the convergence curves of the BPSO algorithm in these two examples. The convergence rate of the second example is slower than that of the first example due to the compactness of its initial solution.
3. Simulation and Measurement Results
To verify the transmission characteristics of the proposed unit cells, in this section, two SIPW filters consisting of the proposed unit cells are designed. The configuration of the first one is depicted in Figure 8(a). The gradual matching technique of the transverse EM field to the transverse magnetic field is applied to the input and output of the structure. The simulated S-parameters are also shown in Figure 8(b). As predicted by the dispersion curve, this filter provides transmission between the frequencies of 5.6 GHz and 7.3 GHz. The configuration of the second one and its fabricated prototype are illustrated in Figures 9(a) and 9(b), respectively. As shown in Figure 9(c), the simulated and measured results, which are in good agreement, indicate transmission between 5.6 GHz and 6.3 GHz, which is consistent with the dispersion curve.
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To get insight into propagation characteristics, the simulated electric field distributions of the second proposed structure at 4, 6, and 8 GHz are shown in Figure 10. At low and high stop bands, wave propagation is prohibited by SIW structure and slots, respectively. However, at the passband, the wave can effectively propagate through the structure and is highly localized.
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In Table 1, the second designed filter is compared with similar structures in other references. All structures in this table are based on the combination of SIW and SSPP structures, whose low cut-off frequency is controlled by the width of the SIW waveguide and the high cut-off frequency is controlled by the SSPP slots. Since the bandwidth obtained in the proposed structure is less than that of other structures, the reduction of the high cut-off frequency by the slots is also greater in this structure. Also, the dimensions of this structure are smaller than others.
4. Conclusion
A novel and different procedure for the design of SIPW unit cells dependent on the dispersion curve optimization was proposed. Compared with the conventional design procedure, the SIPW unit cells designed with this procedure had high field confinement. To verify the proposed design, two unit cells with a miniaturized shape slot were designed which provided lower high cut-off frequencies than those of traditional unit cells. A 5.6-6.3 GHz bandpass filter with the proposed unit cell was fabricated. The measured results were in good agreement with the simulated ones. The method can be applied to the design of compact microwave and terahertz devices with adjustable bandwidths.
Data Availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Conflicts of Interest
We have no conflicts of interest to disclose.