Microwave index engineering for slow-wave coplanar waveguides

Microwave index engineering has been investigated in order to properly design slow-wave coplanar waveguides suitable for a wide range of applications in microwave, photonics, plasmonics and metamaterials. The introduction and optimization of novel capacitive and inductive elements is proposed as a design approach to increase the microwave index while keeping the impedance close to 50 Ω to ensure the compatibility with external electronic devices. The contribution of inductive and capacitive elements and their influence on the performance of the slow-wave coplanar waveguide has been systematically analyzed. As a result, a microwave index as high as 11.6 has been experimentally demonstrated in a frequency range up to 40 GHz which is, to the best of our knowledge, the largest microwave index obtained so far in coplanar waveguides.


Microwave theory and proposed design approach
To address the design of slow-wave CPWs, it is necessary to lay down the basis of microwave theory. This includes basically the transmission line theory to understand the transmission behaviour, and the conformal mapping technique to analyze the influence of the different parameters in a CPW. Figure 1(a) shows the equivalent circuit of a CPW while Fig. 1(b) depicts the transversal view of the CPW with the key design parameters.
The CPW has an inductive and capacitive behaviour, as can be seen in Fig. 1(a), so that from the transmission line theory we can obtain the relationship between the microwave index and the impedance with the capacitance, C, and the inductance, L. That relationship is expressed by the following equations 2 : SCIENTIfIC REPORTS | (2018) 8:5672 | DOI: 10.1038/s41598-018-24030-w where Z o is the impedance, N µ is the microwave index, c o is the speed of light in vacuum, L is the inductance and C is the capacitance. On the other hand, the conformal mapping method links these characteristic parameters of the transmission line with the physical parameters of the CPW 28,29 like the central strip width or the gap between signal and ground planes, shown in Fig. 1 where K(k) represents a complete elliptic integral of the first kind 29 , ε r is the relative permittivity of the substrate, ε o is the vacuum permittivity and where W is the gap and S the central strip width of the coplanar waveguide. Therefore, taking into account equations (1)-(5) is possible to obtain the impedance and the microwave index for a regular CPW of given dimensions. Figure 2 shows the influence of W/S on the impedance and microwave index as well as on the inductance and capacitance for a silicon substrate (ε r = 11.9). As can be seen in Fig. 2(a), the impedance increases when W/S increases. However, the microwave index remains constant over the whole W/S range because the reduction of the capacitance is balanced by the larger inductance ( Fig. 2(b)). In previous works 10,11,20,[23][24][25][26] , the introduction of periodic capacitive elements (thin fins) to the CPW has been proposed to increase the capacitance without decreasing the inductance, which results in a microwave index increase and an impedance reduction. This is an effective method but does not take into account the inductance as a parameter that may also be exploited for the design. In order to manipulate the inductance, the CPW strips can be reduced by using periodic thin slots. The proposed slots allow the inductance to be increased while only having a small effect on the capacitance. Therefore, increasing the inductance can be  combined with approaches based on increasing the capacitance. Such a combination will allow us to reaching much higher microwave indices, while keeping the impedance close to 50 Ω.
To design the slow-wave CPW, it is crucial to analyze the thin slots in order to understand their influence on the inductance and the capacitance and therefore on the microwave index. The analysis has been carried out using the electromagnetic simulation software CST microwave studio. CST is a simulation tool that solves Maxwell equations for each point on a 2D or 3D mesh using a finite elements method. The frequency has been fixed at 20 GHz. Figure 3 shows a top view of the regular and slow-wave CPWs with the parameters to be designed. For the regular CPW, S = 15 µm and W = 11 µm so that W/S = 0.73. By properly checking Fig. 2(a), an impedance value close to 50 Ω is achieved. Figure 3(b) depicts the proposed slots to increase the inductance, where w 1 and w 2 are the length and width of the slot. The period of the slots has been previously optimized to 20 µm. The variation of the capacitance and inductance as a function of w 1 is shown in Fig. 4(a) for w 2 = 100 µm. It can be seen that there is an opposite behavior between them. The increase of the inductance is higher than the capacitance reduction, resulting in a microwave index increment ( Fig. 4(b)). However, the increase of the inductance is not constant with w 1 so that the increase of the microwave index is reduced for larger w 1 values. On the other hand, the behavior of the capacitance and the inductance with w 2 is shown in Fig. 4(c), for w 1 = 5 µm. While the inductance increases with w 2 , the capacitance remains constant, which gives rise to higher microwave index with larger w 2 , as it can be seen in Fig. 4(d). Figure 5(a) shows the impedance as a function of w 1 and w 2 . The corresponding microwave index is depicted in Fig. 5(b). It can be seen that larger microwave indices are achieved at the expense of increasing the impedance,  which is in agreement with equation (1) as the inductance increase dominates over the capacitance. Therefore, it is clear that the proposed slots act as enhanced inductive elements in the slow-wave CPW. However, to keep the impedance close to 50 Ω without reducing the microwave index, the introduction of capacitive elements is also required. The capacitance can be increased with the introduction of parallel T-rails, as shown in Fig. 6(a), which are periodically repeated along the propagation direction [21][22][23][24] . A novel approach based on a crossed T-rail, as depicted in Fig. 6(b), is here also proposed to further increase the capacitance. The induced electric field due to the T-rails is represented with blue lines in Fig. 6(a,b).
Parallel T-rails with a length of 18 µm, width of 2 µm and gap of 1 µm have been added on the inductive slow-wave CPW with w 1 = 5 µm and w 2 = 100 µm. The slow-wave CPW is shown in Fig. 6(c) and has been named as SW1. The same inductive slow-wave CPW but with the crossed T-rails has also been considered to evaluate the difference in capacitance and so the influence on the microwave index. In this case, the CPW is depicted in Fig. 6(d) and has been named as SW2. Finally, a slow-wave CPW with a combination of both crossed and parallel T-rails, named as SW3 and shown in Fig. 6(e), has also been designed to further improve the capacitance. The length and width of the T-rails have been optimized to 16 µm and 4 µm, respectively. Furthermore, the inductive performance has been enhanced by changing w 1 to 17 µm and w 2 to 180 µm. As the capacitance of the slow-wave CPW is ruled by the capacitive T-rail elements, the reduction of the capacitance due to the larger w 1 is negligible. Figure 7 shows the simulation results for the different designs of the slow-wave CPWs. SW0 refers to the inductive slow-wave CPW shown in Fig. 3(b) with w 1 = 5 µm and w 2 = 100 µm. The capacitive and inductive behavior of the slow-wave CPWs are depicted in Fig. 7(a). The capacitance is gradually improved up to a factor of four (from 0.2 to 0.8 nF/m) by the introduction of the capacitive T-rail elements in SW1 and their modifications in SW2 and SW3. The inductance is approximately constant for SW0, SW1 and SW2, but it increases from 1.05 to 1.67 µH/m for SW3 due to the changes in the ground slots and the modification of the signal strip. The impedance, Fig. 7(b), and microwave index, Fig. 7(c), will be determined by the capacitive and inductive behavior.  The impedance is decreased to a value close to 50 Ω for the improved designs of the slow-wave CPWs due to the larger capacitance with respect to the original design of SW0. The microwave index is also increased for SW1 and SW2 due to the larger capacitance but the improvement is much larger for SW3 due to the additional increase of  the inductance. Thereby, the microwave index is significantly enhanced from 6.9 in SW0 up to 11 in SW3. The frequency response has also been simulated and is shown in Fig. 7(d). A constant microwave index and therefore wideband operation is achieved due to the low dispersion of the quasi-TEM propagation mode in the slow-wave CPW.

Characterization and Conclusions
In order to demonstrate the simulated performance, some of the previously designed slow-wave CPWs have been fabricated and characterized. The devices were fabricated on a silicon substrate covered with a 300 nm thick SiO 2 layer deposited using a plasma enhanced chemical vapor deposition (PECVD) process. The CPWs where formed with a lift-off process using TI35E photoresist in an image reversal process. The electrodes consists of 40 nm Ti, deposited through sputtering and 1000 nm Au, deposited through thermal evaporation. Figure 8 shows the obtained measurements results and the comparison with simulations. A multiline method has been applied to extract the microwave index from the slow-wave part of the CPW. The method uses a reference CPW (inset of Fig. 8(a)) in addition to the slow-wave CPW. A more detailed description can be found elsewhere 30,31 . Figure 8(a,b) show the variation of the microwave index and impedance in SW0 for values of w 2 varying from 10 µm to 100 µm, and the regular CPW ( Fig. 3(a)), represented at w 2 =0 µm. It can be seen that there is a good agreement with simulations. Figure 8(c) shows the measured microwave index for the regular CPW, SW0, SW2 and SW3. Also these results are in very good agreement with the simulations. The microwave index is improved from 2.36, 3.7 and 6.9 for the regular CPW, SW0 and SW2, respectively, up to 11.6 for SW3. An inset of Fig. 8(c) shows an optical image of the fabricated SW3. Additionally, Fig. 8(d) depicts the frequency response measured for a range up to 40 GHz. Table 1 compares the obtained results in terms of microwave index, propagation losses and impedance with the values reported in the last two decades for slow-wave CPWs. The microwave index is largely increased for SW3 with respect to previous works. Furthermore, though propagation losses increase for SW3 with respect to SW0 and the regular CPW, the value remains comparable and even lower than the ones achieved in CPWs with an impedance of around 50 Ω.
In summary, we have demonstrated a design approach to increase the microwave index in CPWs. The proposed approach is based on the design of periodically distributed inductive and capacitive elements. A microwave index of 4.7 has been achieved by increasing the inductive behavior of the CPW with small slots on the ground and signal planes. In addition, the effect on the impedance has also been considered and it has been shown that it is possible to achieve high microwave indices while keeping the impedance around 50 Ω. Therefore, through the combination of inductive and capacitive elements, a microwave index up to 11.6 has been demonstrated. To the best of our knowledge, such value is the largest to date obtained in planar transmission lines. Data availability. Requests