Simulation-Based Approach to the Matching of a Dielectric-Filled Circular Waveguide Aperture

The circular waveguide aperture or open-end radiator, one of the canonical antenna elements, can be filled with a dielectric material for miniaturization. With dielectric filling, the aperture reflection increases and impedance matching is necessary. This paper presents a simple but innovative simulation-based approach to the aperture matching of a dielectric-filled circular waveguide aperture. By properly loading the aperture with two- or three-section dielectric rings, the impedance matching is possible over a wide frequency range starting slightly above the TE11-mode cutoff and continuing upward. The material for the aperture matching is the same as that filling the waveguide. The proposed matching structure is analyzed and optimized using a simulation tool for the dielectric constant εr of the filling material ranging from 1.8 to 10. For εr ≥ 5, the unmatched reflection coefficient ranges from −6.0 dB to −0.9 dB while the matched reflection coefficient is from −20.4 dB to −10.0 dB. The impedance matching has been achieved over more than an octave bandwidth.


Introduction
The circular waveguide aperture or open-end radiator is useful as an element in phased arrays [1,2] and as a reflector antenna feed [3,4].Waveguides can be filled with a low-loss dielectric material for miniaturization [5,6], for protection from the environment [7,8], for feeding a dielectric rod antenna [9] and for some other purposes.Dielectric-filled waveguides can be used in the realization of antennas for radar sensors.For example, in [8], Xu and co-workers presented a dielectric-filled waveguide antenna element for a 3D imaging radar operating in high-temperature and excessive dust conditions.
In filling a waveguide with a dielectric material, it is often advantageous to use materials of different dielectric constants (ε r ) for different applications.The operating frequency can be lowered for a given waveguide size, or the waveguide size can be reduced for a given frequency by a factor of 1/ √ ε r .Traditionally, the Rexolite 1422 TM , a cross-linked polystyrene plastic (with dielectric constant ε r = 2.54 and loss tangent tanδ = 0.00066 at 10 GHz [10]), and the Teflon TM (ε r = 2.02 and tanδ = 0.0006 [11]) have been employed in dielectric-filled waveguides and in dielectric rod antennas.For ε r < 2, Eccostock LoK TM , a low-loss and low-weight thermosetting plastic by Laird Technologies (ε r = 1.7 and tanδ = 0.004) can be employed [12].For ε r from 3 up to 30, Eccostock ® HIK500F, a series of low-loss, high-temperature, adjusted-dielectric-constant materials, can be used [13].Table 1 shows the physical properties of the Eccostock ® LoK and Eccostock ® HIK500F.In the Eccostock ® HIK500F series, dielectric rods with ε r of 3, 4, Avient [20] EDGETEK ® 7500 Series 1 3.0, 3.3, 3.4, 3.6, 3.8, 4.4, 4.8, 5.3, 5.9, 6.0, 7.0, 7.3, 9.0 0.0002-0.0012 The aperture reflection coefficient of a dielectric-filled waveguide aperture increases rapidly with increasing dielectric constant of the filling material, making the aperture matching necessary.A simple approach to the aperture matching of a dielectric-filled waveguide aperture is to extend the dielectric material a little distance beyond the aperture [4,6,21].The shaping of the extended dielectric material reduces the reflection to some degree [6,8].Other existing methods include a groove in the dielectric [21], an inductive iris [22], an airgap or low-k insert [22] and high-k low-k insert [23,24].In previous works, the design goal has been the aperture matching for a specific frequency range and thus those works do not show broadband performance.
For some applications, it is necessary to make a dielectric-filled circular waveguide aperture work over a broad frequency range.Examples include broadband phased arrays and wideband reflector antennas.For bandwidth extension, it is to necessary to make a circular waveguide aperture operate from slightly above the first or fundamental TE 11mode cutoff (f cTE11 ) to slightly below the third TM 11 -mode cutoff (f cTM11 ).The second TM 01 mode with cutoff at 1.66f cTE11 can be suppressed by using symmetrical structures in the circular waveguide.Since f cTM11 /f cTE11 is 2.08, a bandwidth close to 2:1 is possible with a circular waveguide aperture.
In this paper, we present a simple but innovative technique for the broadband matching of a dielectric-filled circular waveguide aperture.A special emphasis is placed on making the lower operating frequency limit or the start frequency of the aperture as close to the TE 11 -mode cutoff as possible.The smaller the start frequency, the larger the size reduction ratio and the wider the bandwidth.
The proposed method is validated with simulation by CST Studio Suite TM V2023, a widely used simulation tool in the antenna and RF community.We believe that the accuracy of this simulation tool is good enough to prove the proposed technique.All the dimensions of the proposed designs are given, and anyone can verify the results presented in this paper.To present experimental data, we need to design a high-performance coaxial-to-circular waveguide transition with a ratio bandwidth in excess of 2:1.The transition should work in a dielectric-filled circular waveguide.This is clearly another topic for research.
We acknowledge that there are certain design problems where the simulation-based design alone is not enough and an experimental verification is necessary.Examples may include structures that require new and critical fabrication methods and devices for which the dimensional accuracy is critical to the performance.We can find many papers where the design method is validated with simulation alone [24][25][26][27][28][29][30][31][32][33][34].In our opinion, the aperture matching structure proposed in this paper is not about such a problem.In the next section, we will show the proposed technique with simulation-based design examples.

New Innovative Aperture-Matching Method
Figure 1 shows an unmatched circular waveguide aperture.The waveguide inner wall diameter is 2a and the wall thickness is t.The waveguide is filled with a low-loss material with dielectric constant ε r and loss tangent tanδ.The dielectric filling lowers the operating frequency or reduces the waveguide diameter for a given operating frequency.For high size-reduction ratios, the dielectric constant of the filling material needs to be high, and this will greatly increase the reflection at the aperture.
Sensors 2024, 24, x FOR PEER REVIEW 3 In this paper, we present a simple but innovative technique for the broadband ma ing of a dielectric-filled circular waveguide aperture.A special emphasis is placed on m ing the lower operating frequency limit or the start frequency of the aperture as clos the TE11-mode cutoff as possible.The smaller the start frequency, the larger the size re tion ratio and the wider the bandwidth.
The proposed method is validated with simulation by CST Studio Suite TM V202 widely used simulation tool in the antenna and RF community.We believe that the a racy of this simulation tool is good enough to prove the proposed technique.All th mensions of the proposed designs are given, and anyone can verify the results prese in this paper.To present experimental data, we need to design a high-performance c ial-to-circular waveguide transition with a ratio bandwidth in excess of 2:1.The trans should work in a dielectric-filled circular waveguide.This is clearly another topic fo search.
We acknowledge that there are certain design problems where the simulation-b design alone is not enough and an experimental verification is necessary.Examples include structures that require new and critical fabrication methods and devices for w the dimensional accuracy is critical to the performance.We can find many papers w the design method is validated with simulation alone [24][25][26][27][28][29][30][31][32][33][34].In our opinion, the aper matching structure proposed in this paper is not about such a problem.In the next sec we will show the proposed technique with simulation-based design examples.

New Innovative Aperture-Matching Method
Figure 1 shows an unmatched circular waveguide aperture.The waveguide i wall diameter is 2a and the wall thickness is t.The waveguide is filled with a low material with dielectric constant εr and loss tangent tanδ.The dielectric filling lowers operating frequency or reduces the waveguide diameter for a given operating freque For high size-reduction ratios, the dielectric constant of the filling material needs t high, and this will greatly increase the reflection at the aperture.Figure 2 shows the reflection coefficient of the apertures of an unfilled circular w guide and of a circular waveguide filled with a material of εr = 5.0.The waveguide d eter 2a is 9.20 mm and the wall thickness t is 0.80 mm.In Figure 2  Figure 2 shows the reflection coefficient of the apertures of an unfilled circular waveguide and of a circular waveguide filled with a material of ε r = 5.0.The waveguide diameter 2a is 9.20 mm and the wall thickness t is 0.80 mm.In Figure 2, one can observe a dramatic increase in the reflection coefficient when the waveguide is filled with a material of ε r = 5.0.A simple and innovative method of the aperture matching proposed in this paper is depicted in Figure 3.A circular waveguide is filled with a dielectric material of dielectric constant εr and loss tangent tanδ in the same way as in Figure 1.The same dielectric material as the filling material is placed in the waveguide open end for aperture matching.Broadband impedance matching is obtained by forming the matching material in a shape of two or three stacked rings which can be created by carving out concentric cylindrical volumes in the center and filling the carved-out space with air.Carving out the edge portion of a dielectric cylinder, i.e., solid coaxial dielectric cylinders of varying diameter, does not give a broadband aperture matching.More than three stacked rings can yield a wider bandwidth, but two or three rings give a bandwidth large enough for practical applications.
The matching structure can be thought of as a three-section (M0, M1 and M2) impedance transformer that converts the circular waveguide TE11-mode wave impedance into the free-space planewave impedance.The function of the matching rings is to gradually transform the effective dielectric constant of the medium from εr of the dielectric-filled waveguide to 1 of the free space.The amount of the removed portion in the dielectric ring is increased as the wave propagates from the waveguide to the air, transforming the effective dielectric constant from εr to 1.In this way, the guided TE11-mode wave in the circular waveguide is smoothly transformed to the TEM-mode planewave in the air.A broadband impedance matching is possible by proper dimensioning of the two or three concentric rings.
(a) (b) Figure 4 shows a simplified equivalent circuit model of the proposed aperture matching structure.The parameter ZL is the characteristic impedance of the free space in front of the first matching element M0.The parameters Z0, Z1, Z2 and β0, β1, β2 and L0, L1, L2 are A simple and innovative method of the aperture matching proposed in this paper is depicted in Figure 3.A circular waveguide is filled with a dielectric material of dielectric constant ε r and loss tangent tanδ in the same way as in Figure 1.The same dielectric material as the filling material is placed in the waveguide open end for aperture matching.Broadband impedance matching is obtained by forming the matching material in a shape of two or three stacked rings which can be created by carving out concentric cylindrical volumes in the center and filling the carved-out space with air.Carving out the edge portion of a dielectric cylinder, i.e., solid coaxial dielectric cylinders of varying diameter, does not give a broadband aperture matching.More than three stacked rings can yield a wider bandwidth, but two or three rings give a bandwidth large enough for practical applications.A simple and innovative method of the aperture matching proposed in this paper is depicted in Figure 3.A circular waveguide is filled with a dielectric material of dielectric constant εr and loss tangent tanδ in the same way as in Figure 1.The same dielectric material as the filling material is placed in the waveguide open end for aperture matching.Broadband impedance matching is obtained by forming the matching material in a shape of two or three stacked rings which can be created by carving out concentric cylindrical volumes in the center and filling the carved-out space with air.Carving out the edge portion of a dielectric cylinder, i.e., solid coaxial dielectric cylinders of varying diameter, does not give a broadband aperture matching.More than three stacked rings can yield a wider bandwidth, but two or three rings give a bandwidth large enough for practical applications.
The matching structure can be thought of as a three-section (M0, M1 and M2) impedance transformer that converts the circular waveguide TE11-mode wave impedance into the free-space planewave impedance.The function of the matching rings is to gradually transform the effective dielectric constant of the medium from εr of the dielectric-filled waveguide to 1 of the free space.The amount of the removed portion in the dielectric ring is increased as the wave propagates from the waveguide to the air, transforming the effective dielectric constant from εr to 1.In this way, the guided TE11-mode wave in the circular waveguide is smoothly transformed to the TEM-mode planewave in the air.A broadband impedance matching is possible by proper dimensioning of the two or three concentric rings.Figure 4 shows a simplified equivalent circuit model of the proposed aperture matching structure.The parameter ZL is the characteristic impedance of the free space in front of the first matching element M0.The parameters Z0, Z1, Z2 and β0, β1, β2 and L0, L1, L2 are The matching structure can be thought of as a three-section (M 0 , M 1 and M 2 ) impedance transformer that converts the circular waveguide TE 11 -mode wave impedance into the freespace planewave impedance.The function of the matching rings is to gradually transform the effective dielectric constant of the medium from ε r of the dielectric-filled waveguide to 1 of the free space.The amount of the removed portion in the dielectric ring is increased as the wave propagates from the waveguide to the air, transforming the effective dielectric constant from ε r to 1.In this way, the guided TE 11 -mode wave in the circular waveguide is smoothly transformed to the TEM-mode planewave in the air.A broadband impedance matching is possible by proper dimensioning of the two or three concentric rings.
Figure 4 shows a simplified equivalent circuit model of the proposed aperture matching structure.The parameter Z L is the characteristic impedance of the free space in front of the first matching element M 0 .The parameters Z 0 , Z 1 , Z 2 and β 0 , β 1 , β 2 and L 0 , L 1 , L 2 are the characteristic impedance, propagation constant and length in the equivalent transmission line representation of the matching sections M 0 , M 1 and M 2 , respectively.The parameters Z 3 , β 3 and L 3 are the characteristic impedance, propagation constant and length of the dielectric-filled circular waveguide, respectively.The parameter Z i3 is the input impedance at the circular waveguide port.From the equivalent circuit model, the input reflection coefficient Γ at the circular waveguide port is obtained using The reflection coefficient Γ0 after the matching section M0 is given by where ΓL is the reflection coefficient at the input of M0 given by The impedance Zi0 at the input of the second matching element M1 is obtained from Γ0 using By repeatedly applying Equations ( 2)-( 4), we calculate Zi1, Zi2, and Zi3, from which we obtain the input reflection coefficient Γ given by (1).
The equivalent circuit model of Figure 4 will give accurate results if true values of the model parameters are used and the dielectric junction effect is included.A rigorous evaluation of the model parameters and the junction effect is very complicated if not impossible.Firstly, for a non-TEM transmission line, the characteristic impedance can be defined using the VI-definition, or the PV-definition or the PI-definition [35].One has to determine on a theoretical basis which definition to use.Secondly, a simple transmission line model is not good enough for the proposed matching structure due to the fringing field in the aperture.The field near the aperture does not conform to that of a uniform transmission line where the transverse distribution of the electric and magnetic fields is invariant along the direction of wave propagation.Thirdly, the discontinuity effect of the dielectric step junction is difficult to model due to the complicated field structure of the aperture.Therefore, it is better to use an equivalent circuit model for the elucidation of the operating principles and to employ the numerical simulation and optimization for the actual design of the proposed matching structure.
Figure 5 shows the dimensional parameters of the proposed matching structure.Fixed values of 9.20, 20.00 and 0.90 mm have been used for the inner wall diameter 2a, length L3 and wall thickness t of the circular waveguide.The parameter S is the position From the equivalent circuit model, the input reflection coefficient Γ at the circular waveguide port is obtained using The reflection coefficient Γ 0 after the matching section M 0 is given by where Γ L is the reflection coefficient at the input of M 0 given by The impedance Z i0 at the input of the second matching element M 1 is obtained from By repeatedly applying Equations ( 2)-( 4), we calculate Z i1 , Z i2 , and Z i3 , from which we obtain the input reflection coefficient Γ given by (1).
The equivalent circuit model of Figure 4 will give accurate results if true values of the model parameters are used and the dielectric junction effect is included.A rigorous evaluation of the model parameters and the junction effect is very complicated if not impossible.Firstly, for a non-TEM transmission line, the characteristic impedance can be defined using the VI-definition, or the PV-definition or the PI-definition [35].One has to determine on a theoretical basis which definition to use.Secondly, a simple transmission line model is not good enough for the proposed matching structure due to the fringing field in the aperture.The field near the aperture does not conform to that of a uniform transmission line where the transverse distribution of the electric and magnetic fields is invariant along the direction of wave propagation.Thirdly, the discontinuity effect of the dielectric step junction is difficult to model due to the complicated field structure of the aperture.Therefore, it is better to use an equivalent circuit model for the elucidation of the operating principles and to employ the numerical simulation and optimization for the actual design of the proposed matching structure.
Figure 5 shows the dimensional parameters of the proposed matching structure.Fixed values of 9.20, 20.00 and 0.90 mm have been used for the inner wall diameter 2a, length L 3 and wall thickness t of the circular waveguide.The parameter S is the position of the interface between the second (M 1 ) and third (M 2 ) matching sections relative to the end of the waveguide wall.It is positive if the M 1 -M 2 interface is into the circular waveguide and negative otherwise.D 0 is the diameter of the matching dielectric's portion that is outside the waveguide.D 0 is fixed at 2a + 2t.The remaining parameters are the hole diameters D 1 , D 2 , D 3 and the lengths L 0 , L 1 and L 2 of the matching sections M 0 , M 1 and M 2 , respectively.The loss tangent (tanδ) of the filling and matching dielectric materials is fixed at 0.0008.A higher value of the loss tangent will not affect the aperture matching as far as it is small enough, for example, less than 0.008.It will, however, increase the attenuation of the waveguide.Power loss in a uniform transmission line is given by where z is the propagation distance and β is the propagation constant of the waveguide TE11 mode.The power loss factor P/P0 of a one-wavelength long waveguide is 0.975 and 0.997 for tanδ of 0.008 and 0.0008, respectively.The impedance matching is controlled by the effective dielectric constant of the matching rings, which is in turn determined by the carved-out volume.Since the length of the impedance matching structure is not largeless than 1.35 times the waveguide diameter, as shown below in the design examplesthe impedance matching performance is virtually the same for a loss tangent of 0.008 and 0.0008.This can easily be checked by simulation.
In designing an aperture-matched circular waveguide radiator, it is necessary to analyze the higher-order modes generated along with the fundamental TE11 mode.In a circular waveguide, the first 15 higher-order modes in the order of increasing cutoff frequencies are TE11, TM01, TE21, TE01/TM11, TE31, TM21, TE41, TE12, TM02, TM31, TE51, TE22 and TE02/TM12.The modes with the electric field being symmetric in the E and H planes, nonzero at the waveguide center and irrotational in the waveguide transverse plane, are easily excited when the fundamental TE11 mode is launched by a probe, a slot or other methods.They include TM11, TE12 and TM12 modes whose electric fields at 73 GHz have been obtained using CST Studio Suite TM V2023 and are shown in Figure 6.The cutoff frequency of these modes in a 9.20-mm diameter waveguide is given in Table 3, including the εr =1 case.The loss tangent (tanδ) of the filling and matching dielectric materials is fixed at 0.0008.A higher value of the loss tangent will not affect the aperture matching as far as it is small enough, for example, less than 0.008.It will, however, increase the attenuation of the waveguide.Power loss in a uniform transmission line is given by where z is the propagation distance and β is the propagation constant of the waveguide TE 11 mode.The power loss factor P/P 0 of a one-wavelength long waveguide is 0.975 and 0.997 for tanδ of 0.008 and 0.0008, respectively.The impedance matching is controlled by the effective dielectric constant of the matching rings, which is in turn determined by the carved-out volume.Since the length of the impedance matching structure is not large-less than 1.35 times the waveguide diameter, as shown below in the design examples-the impedance matching performance is virtually the same for a loss tangent of 0.008 and 0.0008.This can easily be checked by simulation.
Starting from initial values D 1 = 0.75(2a), D 2 = 0.5(2a), D 3 = 0.25(2a), L 0 = L 1 = L 2 = 0.5(2a), S = 0, these parameters are optimized for ε r = 1.8, 2.5, 5.0, 7.5 and 10.0 using CST Studio Suite TM V2023 for low reflection over as broad a frequency range as possible with the start frequency (reflection < −10 dB) as close as possible to the TE 11 -mode cutoff frequency of the dielectric-filled circular waveguide.For the two-section matching structure, the first matching section length L 0 is set to zero.
In designing an aperture-matched circular waveguide radiator, it is necessary to analyze the higher-order modes generated along with the fundamental TE 11 mode.In a circular waveguide, the first 15 higher-order modes in the order of increasing cutoff frequencies are TE  By employing symmetric structures, it is possible to operate a circular-wavegu based device from the TE11-mode cutoff to the TM11-mode cutoff.The cutoff freque fc,TM11 of the TM11 mode is 2.08 times the fundamental TE11-mode cutoff frequency fc Allowing for a guard band of 5% at fc,TE11 and fc,TM11, the frequency range will be from fc,TE11 to 0.95 fc,TM11 (=1.98 fc,TE11) or a ratio bandwidth of 1.98/1.05= 1.89.Since the leve the TM11-mode generation is dependent on a specific geometry of structures to be use the waveguide, it is possible to operate a circular waveguide device above the TM11 m cutoff.For example, Bang and Ahn have proposed a coaxial-to-circular waveguide tra tion operating with a ratio bandwidth of 2.18 [36].
The modes with the electric field being antisymmetric in the E and H planes, or at the waveguide center, or rotational in the waveguide transverse plane, are not ea excited when the fundamental TE11 mode is launched, the first three of which are T TE21 and TE01, whose cutoff frequencies are 1.31, 1.66 and 2.08 times the fundamental T mode cutoff frequency, respectively.With εr = 1 and 2a = 9.20 mm, they are 24.94, 3 and 39.74 GHz. Figure 7 shows the electric field of these three modes along with the mode at 44 GHz.By employing symmetric structures, it is possible to operate a circular-waveguidebased device from the TE 11 -mode cutoff to the TM 11 -mode cutoff.The cutoff frequency f c,TM11 of the TM 11 mode is 2.08 times the fundamental TE 11 -mode cutoff frequency f c,TE11 .Allowing for a guard band of 5% at f c,TE11 and f c,TM11 , the frequency range will be from 1.05 f c,TE11 to 0.95 f c,TM11 (=1.98 f c,TE11 ) or a ratio bandwidth of 1.98/1.05= 1.89.Since the level of the TM 11 -mode generation is dependent on a specific geometry of structures to be used in the waveguide, it is possible to operate a circular waveguide device above the TM 11 mode cutoff.For example, Bang and Ahn have proposed a coaxial-to-circular waveguide transition operating with a ratio bandwidth of 2.18 [36].
The modes with the electric field being antisymmetric in the E and H planes, or zero at the waveguide center, or rotational in the waveguide transverse plane, are not easily excited when the fundamental TE 11 mode is launched, the first three of which are TM 01 , TE 21 and TE 01 , whose cutoff frequencies are 1.31, 1.66 and 2.08 times the fundamental TE 11 -mode cutoff frequency, respectively.With ε r = 1 and 2a = 9.20 mm, they are 24.94,31.68 and 39.74 GHz. Figure 7 shows the electric field of these three modes along with the TE 11 mode at 44 GHz.In simulating a circular waveguide over a broad frequency range, the inclusion o higher-order modes whose cutoff is in the frequency range of analysis has an effect on reflection coefficient of the fundamental TE11 mode. Figure 8 shows the effect on th flection coefficient of the higher-order modes in the frequency-domain analysis of a section matching structure with a filling material of εr = 5.0.When only the TE11 fu mental mode is terminated properly (i.e., no reflection) but all higher-order modes are included (i.e., total reflection), many resonance spikes appear due to multiple reflect of the higher-order modes between the aperture and the port plane.If we include modes (TE11 mode + the first four higher-order modes TM01, TE21 and TE01/TM11), the sp in the reflection coefficient will disappear.In the actual implementation of the prop aperture matching structure, the termination condition of the higher-order modes pends on a device connected to the circular waveguide port.
In Figure 8, the resonance spikes occur at four frequencies; 18.27, 18.77, 19.48 20.36 GHz.The cutoff frequencies of the TM01, TE21, TE01/TM11, TE31 and TM21 mod this waveguide are 11.16, 14.17, 17.77/17.77,19.49 and 23.82 GHz, respectively.It is portant to note that the resonance spikes occur at frequencies larger than the TM11 cu In the design of the proposed matching structure, the frequency upper limit is set to onset of the first resonance spike.The proposed matching structure can be used bey the first resonance spike frequency if the generation of the TM11 mode is properly pressed.In simulating a circular waveguide over a broad frequency range, the inclusion of the higher-order modes whose cutoff is in the frequency range of analysis has an effect on the reflection coefficient of the fundamental TE 11 mode.Figure 8 shows the effect on the reflection coefficient of the higher-order modes in the frequency-domain analysis of a two-section matching structure with a filling material of ε r = 5.0.When only the TE 11 fundamental mode is terminated properly (i.e., no reflection) but all higher-order modes are not included (i.e., total reflection), many resonance spikes appear due to multiple reflections of the higher-order modes between the aperture and the port plane.If we include five modes (TE 11 mode + the first four higher-order modes TM 01 , TE 21 and TE 01 /TM 11 ), the spikes in the reflection coefficient will disappear.In the actual implementation of the proposed aperture matching structure, the termination condition of the higher-order modes depends on a device connected to the circular waveguide port.The optimization of the matching structure dimensions is performed in the following steps.The target frequency range for impedance matching optimization is set from the start frequency f1 = 1.01fc,TE11 to the end frequency f2 = 2fc,TE11, where fc,TE11 is the dominant TE11-mode cutoff frequency in the dielectric-filled circular waveguide given by fc,TE11 = 299.792/(1.70629(2a)√εr)(6) In the first step of the design, a parametric analysis is carried out for the reflection coefficient versus dimensional parameters to find out the range of parameter values to be set in the optimization.The dimensions of the designed matching structure have been In Figure 8, the resonance spikes occur at four frequencies; 18.In the design of the proposed matching structure, the frequency upper limit is set to the onset of the first resonance spike.The proposed matching structure can be used beyond the first resonance spike frequency if the generation of the TM 11 mode is properly suppressed.
The optimization of the matching structure dimensions is performed in the following steps.The target frequency range for impedance matching optimization is set from the start frequency f 1 = 1.01f c,TE11 to the end frequency f 2 = 2f c,TE11 , where f c,TE11 is the dominant TE 11 -mode cutoff frequency in the dielectric-filled circular waveguide given by f c,TE11 = 299.792/(1.70629(2a) In the first step of the design, a parametric analysis is carried out for the reflection coefficient versus dimensional parameters to find out the range of parameter values to be set in the optimization.The dimensions of the designed matching structure have been obtained using the 'Trust Region Framework' optimization algorithm provided by CST Studio Suite TM V2023.The optimized design is typically obtained after 200 to 300 hundred iterations, which takes several hours on a desk-top computer.
Next, a first round of optimization is carried out for a target reflection coefficient of −10 dB, and optimum dimensions are obtained.With the dimensions obtained in the first round of optimization and the target reflection coefficient set at −15 dB, a second-round optimization is completed.Using the dimensions obtained in the second optimization round and with the reflection coefficient set at −20 dB, a third round of optimization is carried out.After the third round of optimization, good results are usually obtained.
Following the aforementioned procedures, we were successful in obtaining good aperture matching for ε r from 1.8 to 10. Figure 9 shows the reflection coefficient versus dimensional parameters of a two-section aperture matching design for ε r = 5.0.The curve in blue is for the reflection coefficient after optimization.In Figure 9, the spikes in the reflection coefficient at f > 18 GHz are caused by the exclusion of the higher-order modes in the waveguide port.As stated in the above, the resonance spikes are not generated if higher-order modes are included in the simulation.One design may require the lowest operating frequency with reflection < −10 dB and another design the lowest reflection at a specified frequency range.In Figure 9, one can see that this design flexibility is possible with a proper choice of dimensional parameters.
Figure 10 shows the change in the reflection coefficient during an optimization by CST Studio Suite TM V2023.Starting from a high reflection coefficient, an optimization process finds the dimensions of the matching structure for a target reflection coefficient over a specified frequency range.
We have carried out aperture matching designs for ε r = 1.8, 2.5, 5.0, 7.5 and 10.0.We have made the lower limit of the operating frequency as close to the TE 11 -mode cutoff frequency as possible while maintaining a low reflection coefficient over a broad frequency range.Tables 4 and 5 show the dimensions of the designed two-and three-section matching structures, respectively.The cross-section of the respective matching structure is also shown in Figures 11-15.
Table 4. Dimensions of the two-section aperture matching structures (mm).We have carried out aperture matching designs for εr = 1.8, 2.5, 5.0, 7.5 and 10.0.We have made the lower limit of the operating frequency as close to the TE11-mode cutoff frequency as possible while maintaining a low reflection coefficient over a broad frequency range.Tables 4 and 5 show the dimensions of the designed two-and three-section  ranges from −6.0 to −0.9 dB, while with the proposed aperture matching the reflection coefficient is reduced to the −20 to −10 dB level.

Matching
(   Figures 11-15 show the cross-section and the reflection coefficient of the aperturematching structure with ε r = 1.8, 2.5, 5.0, 7.5 and 10.0, respectively.The frequency range in the figures is from a little below the TE 11 -mode cutoff to the onset of the spikes in the reflection coefficient.In all the cases, the reflection coefficient is dramatically reduced with the three-section (N = 3) matching as well as with the two-section (N = 2) matching.The three-section matching makes a starting frequency closer to the TE 11 -mode cutoff than the two-section matching and thus offers a wider bandwidth.The aperture matching can also be designed for a smaller reflection coefficient over a narrower frequency range.
Tables 6 and 7 summarize the performance of the proposed aperture matching design with two-and three-section structures for ε r = 1.8, 2.5, 5.0, 7.5 and 10.0.The unmatched circular waveguide aperture has a reflection coefficient of less than −10 dB for ε r = 1.8 and 2.5.With dielectric constants of 5.0, 7.5 and 10.0, the unmatched reflection coefficient ranges from −6.0 to −0.9 dB, while with the proposed aperture matching the reflection coefficient is reduced to the −20 to −10 dB level.
In order to ascertain the simulation accuracy, we compared, in Figure 16, the reflection coefficients of the three-section matching structure computed using the frequency-and timedomain solvers for ε r from 1.8 to 10.0.The agreement between two simulation methods is excellent.
In Figures 11-15, one can observe that the length of the matching structure is not large compared with the waveguide diameter.Tables 8 and 9 summarize the matching structure lengths outside and inside the waveguide for the two-and three-section cases, respectively.The length of the outside portion ranges from 0.20 to 0.37 times the waveguide diameter (2a) in the two-section matching (N = 2) and from 0.30 to 0.87 times 2a in the three-section matching (N = 3), while the length of the inside portion ranges from 0.28 to 0.54 times 2a for N = 2 and from 0.19 to 0.48 times 2a for N = 3.The total length of the matching structure ranges from 0.48 to 0.91 times 2a for N = 2 and from 0.49 to 1.35 times 2a for N = 3.The compactness is one of the merits of the proposed matching structure.Table 10 summarizes the impedance matching performance of the proposed threesection matching structures.As the dielectric constant ε r varies from 1.8 to 10.0, the reflection coefficient (|S 11 |) plateau increases from −31.7 dB to −10.0 dB.The ratio bandwidth for the reflection coefficient of less than the plateau reflection coefficient ranges from 1.68 to 2.01.A special merit of the proposed matching technique is that the start frequency (f S ) for |S 11 | < −10 dB is very close to the TE 11 -mode cutoff, ranging from 1.015-1.051.In Figures 11-15, one can observe that the length of the matching structure is not large compared with the waveguide diameter.Tables 8 and 9 summarize the matching structure lengths outside and inside the waveguide for the two-and three-section cases, respectively.The length of the outside portion ranges from 0.20 to 0.37 times the waveguide diameter (2a) in the two-section matching (N = 2) and from 0.30 to 0.87 times 2a in the three-section matching (N = 3), while the length of the inside portion ranges from 0.28 to 0.54 times 2a for N = 2 and from 0.19 to 0.48 times 2a for N = 3.The total length of the matching structure ranges from 0.48 to 0.91 times 2a for N = 2 and from 0.49 to 1.35 times 2a for N = 3.The compactness is one of the merits of the proposed matching structure.The reflection coefficient of the unmatched circular waveguide aperture at f S ranges from −10.7 dB to −0.93 dB for ε r from 1.8 to 10.If the aperture matching is carried out inside the waveguide to match an aperture with a reflection coefficient in the order of −3 dB to −1 dB, it will inevitably result in a very narrow bandwidth.In the proposed technique, however, the aperture matching starts outside the waveguide and ends inside the waveguide.The outside portion significantly reduces the aperture reflection, and a further reduction in reflection is carried out by the inside portion.
Figures 17-21 show the Eand H-plane gain patterns of the designed matching structures with three-section matching, where f a is the start frequency for the reflection coefficient plateau and f m and f b are the middle and end frequencies of the plateau.For comparison, we present Figure 22    The aperture-matched dielectric-filled waveguide radiator shows gain patterns similar to those of an unfilled case.The maximum gain decreases as the dielectric constant increases, since the aperture dimension decreases compared to the wavelength in the air.For a given dielectric constant, the aperture's gain increases as the frequency increases, except in the cases with ε r = 5.0, where the gain at f a is slightly larger than the gain at f m .At the start frequency, the gain in the backward direction (θ = 180 • ) is relatively large; −3.3 dBi with ε r = 1.0, and −5.0 dBi to 1.7 dBi with ε r from 1.8 to 10.0.Table 11 summarizes the waveguide diameter 2a normalized by the waveleng vacuum at the start frequency fa for the plateau reflection coefficient.Table 11 also sh the gain of the waveguide aperture at the start (fa), middle (fm) and end (fb) frequencie Table 11 summarizes the waveguide diameter 2a normalized by the wavelength in vacuum at the start frequency f a for the plateau reflection coefficient.Table 11 also shows the gain of the waveguide aperture at the start (f a ), middle (f m ) and end (f b ) frequencies for the plateau reflection coefficient.The frequencies f a and f b are listed in Table 10.The aperture's gain ranges from 6.7 dBi to 1.0 dBi at f a , from 10.1 dBi to 4.0 dBi at f m and from 12.1 dBi to 6.5 dBi at f b for ε r from 1.8 to 10. Table 11 also lists the gain of an air-filled waveguide aperture of the same diameter for comparison.The gain in this case is greater than that of the dielectric-filled aperture, since the operating frequency is higher.In Table 11, we note that with ε r = 10, the waveguide diameter is only 0.19 times the wavelength at f a , where the matched reflection coefficient is −10 dB, while the unmatched reflection coefficient is only −0.19 dB.We think that this is a remarkable result, which has not been achieved until now.Table 12 compares the bandwidth performance of the proposed matching structures with previous works.First, we should mention that we could not compare the bandwidth under an equal condition since sufficient data have not been provided in previous works.In the proposed matching structures, the plateau reflection coefficient ranges from −31.7 dB to −13.1 dB for ε r from 1.8 to 7.5.If we use the condition |S 11 | < −10 dB in these cases, the bandwidth will be increased a little further.In [4], the frequency range for |S 11 | < −10 dB is not fully drawn.Therefore, bandwidth for |S 11 | < −20 dB is used in Table 12.
Figure 2 shows the reflection coefficient of the apertures of an unfilled circular w guide and of a circular waveguide filled with a material of εr = 5.0.The waveguide d eter 2a is 9.20 mm and the wall thickness t is 0.80 mm.In Figure 2, one can obser dramatic increase in the reflection coefficient when the waveguide is filled with a mat of εr = 5.0.In an unfilled circular waveguide, the dominant TE11-mode cutoff freque (fcTE11) is 19.10 GHz, and the reflection coefficient reaches −10 dB at 20.43 GHz (1.070f and a minimum value of −38.4 dB at 37.36 GHz.With εr = 5.0, the dominant TE11-m cutoff frequency (fcTE11) is 8.54 GHz and the reflection coefficient reaches −2.11 dB at GHz (1.015fcTE11) and a plateau value of −6.9 dB at 22.48 GHz.

Figure 3 .
Figure 3. Aperture matching structure proposed in this paper: (a) transparent view and (b) cutaway view.

Figure 2 .
Figure 2. Reflection coefficient increase in the open end of a circular waveguide due to dielectric filling (ε r = 1 and 5.0, 2a = 9.20 mm, t = 0.80 mm).

Figure 3 .
Figure 3. Aperture matching structure proposed in this paper: (a) transparent view and (b) cutaway view.

Figure 3 .
Figure 3. Aperture matching structure proposed in this paper: (a) transparent view and (b) cutaway view.

Sensors 2024 ,
24,  x FOR PEER REVIEW 5 of 20 the characteristic impedance, propagation constant and length in the equivalent transmission line representation of the matching sections M0, M1 and M2, respectively.The parameters Z3, β3 and L3 are the characteristic impedance, propagation constant and length of the dielectric-filled circular waveguide, respectively.The parameter Zi3 is the input impedance at the circular waveguide port.

Figure 4 .
Figure 4. Equivalent circuit model of the proposed aperture matching structure.

Figure 4 .
Figure 4. Equivalent circuit model of the proposed aperture matching structure.

Sensors 2024 ,
24,  x FOR PEER REVIEW 6 of 20 of the interface between the second (M1) and third (M2) matching sections relative to the end of the waveguide wall.It is positive if the M1-M2 interface is into the circular waveguide and negative otherwise.D0 is the diameter of the matching dielectric's portion that is outside the waveguide.D0 is fixed at 2a + 2t.The remaining parameters are the hole diameters D1, D2, D3 and the lengths L0, L1 and L2 of the matching sections M0, M1 and M2, respectively.

Figure 5 .
Figure 5. Dimensional parameters of the proposed aperture matching structure.

Figure 5 .
Figure 5. Dimensional parameters of the proposed aperture matching structure.

Sensors 2024 ,Figure 8 .
Figure 8.Effect of the higher-order mode inclusion on the reflection coefficient of a dielectric-filled circular waveguide aperture with εr = 5.0.

Figure 8 .
Figure 8.Effect of the higher-order mode inclusion on the reflection coefficient of a dielectric-filled circular waveguide aperture with ε r = 5.0.

Figure 10
Figure 10 shows the change in the reflection coefficient during an optimization by CST Studio Suite TM V2023.Starting from a high reflection coefficient, an optimization process finds the dimensions of the matching structure for a target reflection coefficient over a specified frequency range.

Figure 9 .Figure 10 .
Figure 9. Reflection coefficient versus dimensional parameters at the aperture of a circular waveguide filled with material of ε r = 5.0 with (a) D 0 , (b) D 1 , (c) D 2 , (d) L 0 and (e) L 2 in mm.Sensors 2024, 24, x FOR PEER REVIEW 11 of 20

Figure 10 .
Figure 10.Changes in the reflection coefficient during an optimization of an aperture-matched dielectric-filled circular waveguide aperture with ε r = 5.0.The intermediate reflection coefficient curves are drawn in gray and the final one is in blue.

Table 3 .
Cutoff frequencies of the modes excitable along with the TE11 mode in a dielectric-fi circular waveguide.

Table 3 .
Cutoff frequencies of the modes excitable along with the TE 11 mode in a dielectric-filled circular waveguide.
27, 18.77, 19.48 and 20.36 GHz.The cutoff frequencies of the TM 01 , TE 21 , TE 01 /TM 11 , TE 31 and TM 21 modes in this waveguide are 11.16, 14.17, 17.77/17.77,19.49 and 23.82 GHz, respectively.It is important to note that the resonance spikes occur at frequencies larger than the TM 11 cutoff.

Table 5 .
Dimensions of the three-section aperture matching structures (mm).

Table 6 .
Reflection coefficient of the two-section matching structure (N = 2).

Table 7 .
Length of the three-section matching structure (N = 3).

Table 8 .
Length of the two-section matching structure (N = 2) normalized by the waveguide diameter.

Table 9 .
Length of the three-section matching structure (N = 3) normalized by the waveguide diameter.

Table 8 .
Length of the two-section matching structure (N = 2) normalized by the waveguide diameter.

Table 10 .
Impedance matching performance of the proposed three-step aperture-matching structure.

Table 11 .
Gain of the aperture-matched dielectric-filled waveguide open end.