диафрагмированные волноводные фильтры / 2fb6bb1f-b73f-464b-bb4a-f78efd67023d

considered frequencies, the loaded quality factor declines significantly for the lower values of the graphene chemical potential. The loaded quality factor is calculated as QL = frez /BW3 dB, with BW3 dB denoting the 3 dB bandwidth of the S21 parameter. The graphene-on-quartz resonators are, in any case, characterized by the notably lower loaded quality factors than their pure metallic counterparts, and this is particularly pronounced for the lower chemical potential and larger impedances. Increase in the graphene surface impedance produces effects similar to the effects of varying the effective resonator length by changing the distance, lrez, between the E-plane inserts. The same holds for the total length of the E-plane insert which appears effectively shorter and leads to the decline in the resonator quality factor. We address this issue by introducing the graphene–metal combined waveguide resonators. The graphene stripes are now located along the inner edges of the E-plane inserts, at the lrez side, whereas the remaining parts of the inserts are covered by the Cu or Au thin film. Such a configuration is expected to have less influence on the resonator quality factor; however, the decrease in tunability is also expected and has to be investigated.

The concept of the graphene–metal combined waveguide resonators is presented in figure 6, using the WR-2.2 waveguide section with the resonator length lrez  =  210 µm. The ratio lT/ a is varied as denoted in the figures. The S21-parameter curves for the purely metallic resonators of the same sizes are denoted using the dash–dot–dot lines, with the resonant frequencies at about 480 GHz. In figure 6(a) we investigate the

effect on the resonator parameters of the width of the graphene stripe, expressed as the percentage graphene with respect to the E-plane insert length. To keep the graph less complicated, for each of the five compared graphene stripe widths, we only show the S21-parameter for µc = 0.4 eV and µc = 1.0 eV. Each pair of lines corresponding to the same percent of graphene, pg, is presented in distinctive color (shade of gray) in different line style. For better legibility, the data shown in figure 6(a) is also summarized in table 3.

As can be concluded from figure 6(a) as well as table 3, a graphene percentage larger than 50% does not further improve the tunability; however, the quality factor decreases and the loss increases. Therefore, less than half of an insert could be covered by graphene. Please note, that the tunability range presented in table 3 differs from the one used throughout the paper due to the use of the (0.4–1.0) eV range for the chemical potential. There already is a significant quality factor deterioration for µc = 0.4 eV, in comparison with the one for µc = 1.0 eV, with the increase in the graphene stripe width. This is even more pronounced for µc (0.2–1.0) eV, whereas the tunability is approximately doubled.

As can be inferred from the above observations, the gra- phene–metal combined waveguide resonators can be carefully designed to meet the specific design requirements. Adjustment of the design parameters starts with the choice of the graphene stripe width, lG, in accordance with the desired tunability range. This is illustrated by choosing lG = 80 µm for the figure 6(b) and lG = 50 µm for the figure 6(c). After

that, the tunability remains almost the same for different total insert lengths, lT. Optimizing a trade-off between the quality factor and the loss is the next step, in which the total E-plane insert length, lT, is defined. With the increase of lT, both the quality factor and the insertion loss increase. Relevant data for these two cases is summarized in table 4. We can conclude from table 4 that the obtained tunability percentages Tp (cal-

culated very strictly, i.e. with respect to f Mrez, which is always well above the device tunable bandwidth), in the considered examples range from around 5% to higher than 8%. Noting that a wide tunability range is hard to obtain in submillimeter applications, and that even better tunability can be achieved using stripe widths wider than 80 µm, the reported tunability percentages are on the same order of magnitude as those in the state-of-the-art receivers [45], where several receivers are employed to cover a wider submillimeter bandwidth.

Reasons for the higher insertion loss of the graphene–metal combined waveguide resonators, in comparison with the initially analyzed graphene-on-quartz resonators, stem from the different boundary conditions at the E-plane inserts. Different boundary conditions mandate different EM field distribution in the vicinity of the inserts, as well as in the graphene stripe.

Along the part of an E-plane insert covered by graphene, EM field complies with equations (16a) and (16b), whereas in the metallic part of an insert the field vanishes. For a narrow graphene stripe and a longer metallic part of an insert, a stronger field is generated. Simulation results for different graphene percentages, pg, are presented in figure 7.

Dependence of the tunability, quality factor, and insertion loss on the graphene percentage, illustrated in figure 6(a), has been studied in detail taking the values from zero to pg  =  100%, with the step of 2.5%, as an input parameter. The results are presented in figure 8. For the relatively narrow graphene stripes, with pg 20%, the loaded quality factor remains close to the one corresponding to purely metallic inserts, QML = 42.44. At the same time, insertion loss is low and varies only slightly for the considered graphene chemical potential interval from 0.2 to 1.0 eV. Tunability of up to 5% can be achieved. These results agree with the third row describing the lG = 50 µm case in table 4, where pg = 21.3%. At about 50% graphene coverage of an insert, 10% tunability can be expected with the still very good quality factor of about 70% of QML . In this case loss is the highest. According to figure 8,

the optimal width of the graphene stripe is from about 15% to approximately 40%, where the quality factor is high and consistent throughout the achieved tunability range.

Frequency dependence of the key representative para­ meters of the graphene–metal combined waveguide resonators is given in figure 9. We again analyze the graphene-based waveguide resonators corresponding to the WR-6, WR-4, WR-3, WR-2.8, WR-2.2, WR-1.5, WR-1.2, and WR-1.0 waveguides. Resonator length has been kept the same as in the previous analyses; it is listed in table 5, along with

the chosen insert length, lT = 0.42a. Quartz thickness is d = a/16, whereas the graphene stripe width is kept at 25% of an E-plane insert length in all the analyses. Resonant frequencies are listed in table 5, whereas the other results are shown in figure 9. There is a significant decrease of the insertion loss with frequency in the investigated frequency range. Quality factors at higher frequencies are excellent, represented in comparison with the QML. Tunability range in between 5% and 6% is obtained with the graphene–metal combined waveguide resonators, supporting the proposed concept as one of

the possible solutions for the millimeter and submillimeter wave applications.

4.  Conclusion

A novel concept of tunable waveguide resonators for submillimeter wave applications has been studied in detail, following our promising preliminary investigation of WR-3 waveguide. Theoretical analysis has been presented, which could be used to develop customized software tools for the design of this type of waveguide resonator. Thorough full-wave numerical simulations covering several waveguide sections and frequencies ranging from 100 to 1100 GHz confirmed the possibility of obtaining 5% tunability with excellently preserved resonator loaded quality factors, as well as larger tunability ranges, where the trade-off with quality factor and insertion loss is carried out. Tunable waveguide resonators are important basic building blocks of tunable filter devices, and the proposed concept is of great significance for the development of the compact and flexible components in the submillimeter wave spectral region.

Acknowledgments

This work was supported by the Serbian Ministry of Education, Science, and Technological Development under grant III-45003 and in part by the EU—Erasmus Mundus Action 2 project EUROWEB.

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