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   . However, a TE20-like mode, which is propagated in the 2nd band, is an odd mode, and generation systems for odd modes have seldom been reported.
In this investigation, we fabricate a mode converter, which passes through the TE10 mode for the low frequency range and efficiently converts TE10 to TE20 mode (or TE20 to TE10 mode) for the high frequency range. Typical metallic waveguide mode converters convert electromagnetic waves from one mode to another mode for some frequency range. However, most electromagnetic waves outside of the specified frequency range are reflected   .
In a rectangular metallic waveguide, only the TE10 mode propagates for 0.5 < ωw/2πc < 1 (the 1st band) for a given frequency ω, which is normalized using the width of the waveguide w. On the other hand, both the TE10 and TE20 modes can propagate for 1 < ωw/2πc < 1.5 (the 2nd band), where c is the velocity of light in a vacuum. The TE10 and TE20 modes have the same frequency ω, but different wave vectors k and group velocity vg = (dk/dω)−1. When a mode in the 2nd band is converted from TE10 to TE20, the group velocity of the electromagnetic wave changes. In the 1st band, it cannot be changed.
We have previously reported a waveguide which restricts certain modes  and a prototype mode converter  . These devices consist of periodic dielectric rod arrays in a waveguide similar to that shown in Figure 1. However, it is not easy to determine the group velocity in the waveguide. The propagation modes in a waveguide having in-line dielectric rods with period a can be calculated using a supercell approach  by applying appropriate periodic Bloch conditions at the boundary of the unit cell  . When the location of the dielectric rods is fixed at a distance d from the sidewall, the group velocity vg for both the 1st and 2nd bands can be changed by varying the radius r. However, the group velocities for both bands are changed at the same time and cannot be changed individually. For d and r fixed to certain values, vg/c can be calculated for the periodic structure of the dielectric rods at a specific frequency  .
After calculating vg with pairs of d and r, each pair of d and r is combined.
2. Initial Design
We consider a metallic waveguide of size 17.4 × 8 mm2 (cutoff frequency fc ≈ 8.62 GHz). The period a is fixed at 7.2 mm for target frequencies of 10 GHz (1st band) and 20 GHz (2nd band). Figure 2 shows a sample of the calculated normalized velocities along the axis of the waveguide at 10 GHz and 20 GHz for dielectric rods (LaAlO3: εr = 24, radius r (mm)) aligned at a distance from the sidewall d (mm). It is desirable that the normalized velocity vg/c = 0.902 (TE10: point A) monotonically decrease to vg/c = 0.507 (TE20: point B) at 20 GHz, and that the normalized velocity vg/c = 0.507 (TE10: point C) be unchanged at 10 GHz. However, at 20 GHz, such a condition is not found around d = 6 - 7 mm, because the placement of the dielectric rods at the center of the waveguide minimizes the electric field. On the other hand, placing the dielectric rods near the sidewall of the waveguide maximizes the electric field. At the transition region, around d = 6 - 7 mm, the characteristics are complex. A priority of the design is that the
Figure 1. Metallic waveguide with in-line dielectric rods.
Figure 2. Group velocity in a metallic waveguide with a periodic array of dielectric rods for various distances from the sidewall d and various radii of the rods r, at 20 and 10 GHz.
group velocity at 10 GHz not change. Since the group velocity decreases for dielectric material at 10 GHz and becomes slowest at d = 3 mm, the design employs the 20 GHz frequency for mode conversion, because both the 20 GHz and 10 GHz conditions cannot be satisfied at the same time.
Figure 3 shows the layout of the rods (though note that rods 0 and 10 are not included in the prototype, but are additions, as detailed below). Since the mode profile gradually shifts from TE10 to TE20, the dielectric rods are placed progressively away from near the sidewall to the center of the waveguide, as illustrated in the figure. The nine dielectric rods (i = 1 to 9) have decreasing radii ri and are placed constant distance apart a = 7.2 mm. Three dielectric rods (i = 7, 8, and 9) are located at the center of the waveguide. Table 1 shows distances di and radii ri of the rods.
The S parameters between the input port (Port 1) and output port (Port 2) are calculated using the HFSS software by Ansys  , and the results are shown as dotted lines in Figure 4. The electromagnetic waves enter as the TE10 mode for 9.1 - 14.4 GHz and are converted to the TE20 mode with 18.3 - 21.0 GHz with over 95% efficiency. However, reflection of the TE10 mode at low frequencies and the TE20 mode at high frequencies range are significant under −15 dB, and optimization of the design is necessary to reduce reflection.
3. Decreasing the Reflection 
As shown in Figure 4, the reflection of the TE10 mode is significant around 10 to 14 GHz. This is because the rods located at the center of the waveguide are truncated at Port 2. Another dielectric rod located at the center of the waveguide (rod 10) is needed to decrease electromagnetic reflection. The rod has a smaller cross-section radius of r = 0.29 mm compared with rods 8 and 9. The reflection of the TE20 mode is also significant around 18 to 20 GHz. This is because the rods are located asymmetrically at Port 1. To decrease reflection of the TE20 mode, another dielectric rod (rod 0) is placed on the opposite side of rod 1. The structure of the final design is shown in Figure 3. Rods 0 and 10, whose dimensions and locations are shown in Table 1, have been added to the structure.
Figure 3. Proposed structure of the TE10 to TE20 mode converter (the prototype structure does not have rod 0 (i = 0) and rod 10 (i = 10)).
Figure 4. S parameters for the mode converter, |S21| and |S11| (the dotted lines show the case without rods 0 and 10. The solid lines are for the case with rods 0 and 10.).
Table 1. Location and radii of the dielectric rods.
The S parameters between the input port (Port 1) and output port (Port 2) calculated using the HFSS are shown as solid lines in Figure 4.
4. Fabrication and Experimental Results
In fabricating a mode converter, such as the Type A illustrated in Figure 5(a), it is necessary to install the dielectric rods in the waveguide without a gap at the top or bottom. Such a structure can be difficult to fabricate. In our approach, holes are machined at the top of the waveguide and the dielectric rods are inserted. For low-cost fabrication, a rectangular dielectric rod quarried from a slab is more suitable than a column (Type B; Figure 5(b)). The rectangular dielectric rods are fabricated from slabs of LaAlO3 with thicknesses of 0.5, 1.0, and 1.5 mm, and are cut to give cross sections the same as would be the case for columns. The dimensions of the dielectric rectangular rods are shown in Table 2.
A problem arises in that a typical coaxial-waveguide converter cannot detect the TE20 mode since an electrode is placed at the center of the waveguide. Our procedure to detect the TE20 mode is to first confirm that the TE10 mode does not propagate to the
Figure 5. (a) Dielectric rod located in a waveguide without gaps at top and bottom; (b) Dielectric rectangular rod inserted in a hole made at the top of the waveguide with a space of 0.02 mm between the rods and the top wall of the waveguide. (a) Type A; (b) Type B.
Table 2. Locations and sizes of the dielectric rods arranged in Type B.
end of the TE10 to TE20 mode converter at mode conversion frequency and then to connect another mode converter of the same type to the original converter in the opposite direction and measure the reconverted TE10 from TE20 mode.
Fabricated mode converters have a full length of 130 mm. The outside appearance is shown in Figure 6 and the experimental setups are shown in Figures 7(a)-(c). Figure 7(a) is the setup for confirming that the TE10 mode has passed through the mode converter at the low frequency range. The mode converter was connected to a HP-8510B
Figure 6. Frequency range dependent mode converter.
Figure 7. Experimental setup for measurement of S parameters. (a) |S21| and |S11| at low frequencies; (b) |S21| (TE10 - TE10), |S11| (TE10 - TE20 - TE10) and |S22| (TE10 - TE10) at high frequencies. (c) |S21| (TE10 - TE20 - TE10) and |S11| (TE10 - TE10, TE10 - TE20 - TE10 and TE10 - TE20 - TE10 - TE20 - TE10).
network analyzer by a coaxial-waveguide converter (Maury Microwave X200A2: 8.2 - 12.4 GHz, VSWR max = 1.05) through a tapered waveguide of 170 mm length.
The experimental results are shown in Figure 8(a) as solid lines. The dotted lines indicate the calculated results and are sufficiently consistent with the experimental re-
Figure 8. Experiment results of S parameters: (a) |S21| and |S11| in Figure 7(a); (b) |S21|, |S11| and |S22| in Figure 7(b); (c) |S21| and |S11| in Figure 7(c).
sults. Figure 7(b) is the setup for confirming that the TE10 mode is converted to the TE20 and that the TE10 is not propagated (the coaxial-waveguide converter is a Fairview Microwave Inc. 62AC206: 12.4 - 18.0 GHz, VSWR max = 1.25). The experimental results are shown in Figure 8(b) as solid lines. The figures show that S21 for the TE10 mode propagation is very small.
To measure the conversion efficiency from the TE10 to TE20 mode, another of the same type of mode converter is connected in the opposite direction to the previous mode converter. The values of the S parameter S21 is the conversion efficiency after converting from TE10 to TE20 and reconverting from TE20 to TE10. Figure 8(c) shows the experimental results for the setup shown in Figure 7(c). The solid lines are experimental results and are almost the same as the calculation results shown as dotted lines.
We have proposed a frequency range dependent TE10 to TE20 mode converter that efficiently converts the low frequency TE10 mode to the high frequency TE20 mode by small variations of the group velocity of the TE10 mode. The mode converter allows low frequency electromagnetic waves to pass through but converts higher frequency electromagnetic waves from the TE10 mode to TE20 mode without reflection. We demonstrate that electromagnetic waves pass through in the TE10 mode at 9.7 - 14.6 GHz and are converted to the TE20 mode at 19.1 - 20.3 GHz at over 95% efficiency. Such frequency range dependent mode converters are useful for applications that require single-mode propagation at two frequencies.
The authors thank Mr. H. Miyazaki for help with the experiment set-up.
 Zhang, Q., Yuan, C.W. and Liu, L. (2012) Theoretical Design and Analysis for TE20-TE10 Rectangular Waveguide Mode Converters. IEEE Transactions on Microwave Theory and Techniques, 60, 1018-1026.
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