Location and radii of the dielectric rods
Metallic waveguides have major advantages, such as low propagation loss and high power transmission in the microwave frequency range. However, one disadvantage is that the usable frequency range is restricted to f c< f< 2f c, because the TE20 mode is possible in a frequency region higher than 2f c for rectangular metallic waveguides. A ridge waveguide (Cohn, 1947) (or double-ridge waveguide) has an advantage in that it can spread the propagating frequency range as a result of reduction in the cutoff frequency for the TE10 mode. However, one disadvantage is that the attenuation constant becomes large.
Power sources, such as watt class IMPATT diodes or Gunn diodes, are readily available, and for high frequency use, power sources are sometimes combined, due to their low power rating. However, power combiners consisting of cavity resonators usually have narrow bandwidths (For example, Matsumura et al., 1987). Power dividers and power combiners may be easily setup using mode converters. For example, a TE10–TE30 mode converter easily offers a three-port power divider, and a three-way power combiner can be composed by reversal. A power combiner is useful for application to Gunn diodes in a waveguide array (Bae et al., 2000), because it converts the TE30 mode to the TE10 mode.
2. Design method of the mode converters
We have reported that single-mode propagation is available for a metallic waveguide with dielectric rods arrayed at the center of the waveguide in the frequency under twice the cutoff frequency region using the TE10 mode, and in the frequency over twice the cutoff frequency region using the TE20 mode, because of restrictions of the TE10 mode (Kokubo, 2007; Kokubo & Kawai, 2009 ). However, a TE20-like mode, which is propagated in the second band, is an odd mode, and generation systems for odd modes have seldom been reported. In this investigation, a mode converter is proposed which passes through the TE10 mode for the low frequency range and efficiently converts TE10 to TE20 mode for the high frequency range.
2.1. Design method of the TE10-to-TE20 mode converter
The frequency eigenvalues of a conventional metallic waveguide in a given k wavevector are shown in Fig. 1. In this figure, the wavevector k and frequency ω are normalized using the width of the waveguide w. The electromagnetic wave propagates the TE10 mode only for 0.5<ωw/2πc<1, and can propagate TE10 and TE20 modes for 1<ωw/2πc<1.5. If these two modes are excited by only the TE10 mode, the group velocity of TE10 (A) must be changed to that of TE20 (B) for 1<ωw/2πc<1.5. On the other hand, the group velocity (C) is not changed for 0.5<ωw/2πc<1, because this remains in the TE10 mode. If the distribution of the transverse electromagnetic field is gradually changed from TE10 to TE20, and group velocity (A) is also gradually changed to (B), then the reflection may be reduced for 1<ωw/2πc<1.5. On the other hand, if the group velocity (C) is not significantly changed, the reflection may also be suppressed for 0.5<ωw/2πc<1. Since the mode profile gradually shifts from TE10 to TE20, the dielectric rods are replaced from near the sidewall to the center of the waveguide. In other words, the basic setup is shown in Fig. 2.
The group velocity is given by. However, it is not simple to determine the group velocity in the waveguide shown in Fig. 2. The propagation modes in a waveguide having in-line dielectric rods with period a are calculated using a supercell approach (Benisty, 1996) by application of appropriate periodic Bloch conditions at the boundary of the unit cell (Boroditsky et al.; Kokubo & Kawai, 2008). When the location of the dielectric rods is fixed at a distance d from the sidewall, the group velocity v g at both of the first and the second bands is changed by varying the radius r. However, the group velocities are also changed at the same time and cannot be changed individually.
If the group velocity is normalized using light velocity in a vacuum, v g/c is the same as the gradient of the characteristic curve. Therefore, when d and r are fixed to certain values, v g/c is calculated for the periodic structure of the dielectric rods at a specific frequency. If group velocity (A) is gradually changed to (B) for 1<ωw/2πc<1.5 when d is varied, and group velocity (C) is not changed for 0.5<ωw/2πc<1, then one unit of each pair of d and r connects to its respective pair to form a structure shown in Fig. 2.
The metallic waveguide is assumed to be a WR-90 (22.9×10.2 mm, cutoff frequency f c ≈ 6.55GHz) and period a is fixed at 9.54 mm. Fig. 3 shows a sample of calculated results of normalized velocity along the axis of the waveguide at 15 GHz and 9 GHz for dielectric rods (LaAlO3: εr= 24, radius r [mm]) aligned at a distance from the sidewall d [mm] (Kokubo, 2010). It is desirable that the normalized velocity (A) (TE10: v g/c =0.900) monotonically decreases to (B) (TE20: v g/c =0.487) at 15 GHz and normalized velocity (C) (TE10: v g/c =0.686) is not changed at 9 GHz. However, at 15 GHz, such a condition is not found around d=10 mm, because the placement of the dielectric rod at the center of the waveguide is the same as that where the electric field becomes a minimum. On the other hand, placement of the dielectric rod near the sidewall of the waveguide is the same as that where the electric field becomes a maximum. At the transition region, around d=10 mm, the characteristics are complex. The design takes priority, in order to not vary the group velocity at 9 GHz. Since the group velocity must become slow with dielectric material at 9 GHz and becomes slowest at d=5 mm, the design takes priority at 15 GHz for mode conversion, because both 15 GHz and 9 GHz conditions cannot be satisfied at the same time. The final design of the mode converter is shown in Fig. 2. Three dielectric rods are located at the center of the waveguide. Two of these have r = 0.515 mm and the remainder have a half cross-section radius of 0.36 mm in order to decrease electromagnetic reflection. Nine dielectric rods are placed from the center of the waveguide to near the sidewall with increasing radius of the rods r and with constant a = 9.54 mm. Table 1 shows the relation between the distance d and radius r (Kokubo, 2010). The S parameters between the input port (Port 1) and output. port (Port 2) are calculated using the HFSS software by Ansys Inc. and the results are shown as solid lines in Figs. 4(a) and (b). The electromagnetic waves pass through as the TE10 mode for 7-11.2 GHz and are converted to the TE20 mode for 14.1-16.1 GHz under a condition of over 95% efficiency. However, as the TE20 mode is not sufficiently small under -15 dB, optimization of the design is necessary for reduction of reflections. Though reflection as TE20 is not small at high frequency, if a coaxial-waveguide converter is used for the introduction of electromagnetic waves to a waveguide, odd mode may not have a strong influence on even symmetry structure.
|Rod Number i||Distance di from the Sidewall [mm]||Radius ri of the dielectric rod [mm]|
2.2. Simple fabrication method
For the fabrication of a mode converter, such as the Type A illustrated in Fig. 5(a), it is necessary to locate the dielectric rods in the waveguide without a gap at top and bottom.
Such a structure may be difficult to fabricate. As a solution, holes with diameters 0.2 mm larger than the rods were fabricated at the top of the waveguide and the dielectric rods were inserted (Type B, Fig. 5(b)). The S parameters were calculated using the HFSS software and the results are shown as dotted lines in Figs. 4(a) and (b). The results for these different structural conditions (solid lines and dotted lines) are almost same.
2.3. Reduction of reflection for TE20 mode
As shown in Fig. 4(b), reflection as TE20 mode is not small enough. This reason is dielectric rods are asymmetric arrangement for electromagnetic wave. Fig. 6 shows an improved structure of the TE10 to TE20 mode converter.
|Rod Number i||Distance di from the Sidewall [mm]||Radius ri of the dielectric rod [mm]|
2.4. Design method of the TE10-to-TE40 mode converter
A TE10 to TE40 mode converter can be considered by combination of TE10 to TE20 mode converters. Another structure of the TE10-to-TE20 mode converter is proposed and is shown in Fig. 8. The locations of the dielectric rods are indicated in Table 1. The structure of the proposed TE10-to-TE40 mode converter, which is composed of three TE10-to-TE20 mode converters, is shown in Fig. 9. The S parameters between the input port (port 1) and output port (port 2) calculated using HFSS are shown in Figs. 10(a), (b) and (c).
2.5. Design method of the TE30-to-TE10 mode converter
A structure that contains two arrays of dielectric rods can convert the TE30 mode into the TE10 mode ( Kokubo, 2009 ). The TE30 mode electromagnetic waves in this type of waveguide are converted to the TE10 mode for 7.1–8.9 GHz with over 95% efficiency. However, this structure cannot pass through electromagnetic waves around 2.5–3 GHz without high reflection even if the waveguide is straight. Therefore, a mode converter is proposed that passes the TE10 mode at low frequencies and efficiently converts the TE30 mode into the TE10 mode at high frequencies.
A metallic waveguide that contains two in-line dielectric rods can propagate single modes in two frequency regions (Shibano et al., 2006). The propagation modes in a waveguide with two in-line dielectric rods with period a are calculated using a supercell approach (Benisty, 1996) by applying appropriate periodic Bloch conditions at the boundary of the unit cell (Boroditsky et al., 1998; Kokubo & Kawai, 2008).
The metallic waveguide width is assumed to be w 1 = 21.4 mm, which is 3 times wider than the WR-28 waveguide (7.11× 3.56 mm; f c ≈ 21.1 GHz), and period a is fixed at 5 mm. Fig. 11 shows a sample of the calculated normalized velocity along the waveguide axis at 9 and 27 GHz for dielectric rods (LaAlO3: εr = 24; radius r [mm]) aligned at a distance w 2 [mm] between two arrays. The waveguide must be designed so as not to vary the group velocity (C) (TE10: v g/c = 0.627) at 9 GHz. After calculating v g for pairs of w 2i and r i, if group velocity (A) (TE30: v g/c = 0.627) is gradually changed to (B) (TE10: v g/c = 0.966) at 27GHz, then each pair of w 2i and r i are combined (see Fig. 12).
|Rod Number i||Distance w2i between a pair of the rods [mm]||Radius ri of the dielectric rod [mm]|
The first pair of rods has r 1 = 0.22 mm to reduce electromagnetic reflection at low frequencies. If the first pair of rods is absent, then reflections will be above −10 dB at low frequencies. Twenty dielectric rods are placed from one third of the width of the waveguide to near the sidewall with increasing rod radius r i and constant a (= 5 mm). Table 3 shows the relation between the distance w 2i and radius r i. The S parameters between the input port (port 1) and output port (port 2) are calculated using HFSS software by Ansys Inc., and the results are shown in Figs. 13(a-c). Electromagnetic waves propagate as the TE10 mode for 8.2–14.8 GHz and the TE30 mode is converted into the TE10 mode for 22.2–28.4 GHz at an efficiency of over 95%.
We have previously reported that single-mode propagation is available for a metallic waveguide with dielectric rods arrayed at the center of a waveguide using the TE10 mode, and the TE20 mode. However, a TE20-like mode, which is propagated in the second band, is an odd mode, and generation is not easy. In this investigation, a mode converter is proposed which passes through the TE10 mode for the low frequency range and converts TE10 to the TE20 mode for the high frequency range by small variation of the group velocity. It was shown that the electromagnetic waves pass through as the TE10 mode for 7-11.2 GHz and are converted to the TE20 mode for 14.1-16.1 GHz under a condition of over 95% efficiency.
It was shown that electromagnetic waves propagate as the TE10 mode around 8 GHz and that the TE40 mode is converted into the TE10 mode around 16 GHz.
This work was supported by JSPS KAKENHI 22560336.