Design of High-Precision Terahertz Filter Based on Directional Optimization Correction Method

Abstract

The directional optimization correction (DOC) method is proposed to reduce the performance error between desired and fabricated terahertz (THz) devices. Three 340 GHz terahertz filters with a bandwidth of 20 GHz are designed and fabricated. The traditional global optimization correction (GOC) method and the proposed DOC method are used to optimize and reduce the performance error, respectively. It is garnered that the center frequency error and bandwidth error of the fabricated terahertz filter optimized by the GOC method are reduced to 3.5 GHz (~1.03%) and 2.2 GHz (~11%), respectively. Meanwhile, the center frequency error and bandwidth error of the fabricated terahertz filter optimized by the DOC method are reduced to 0.2 GHz (~0.06%) and 0.4 GHz (~2.0%), respectively, which has fewer optimization parameters and higher accuracy than the GOC method. Furthermore, the in-band return loss (RL) of two optimized terahertz filters based on the DOC and GOC methods is less than 15 dB, and the in-band insertion loss (IL) is less than 2.3 dB.

1. Introduction

Currently, with the rapid development of a new generation of information technologies, such as the air–space–ground integrated information communication network, the intelligent connection of all things, integrated sensing and communication, 5 G/6 G technology, etc., this radio frequency (RF) front-end system is becoming increasingly complex and the performance requirements for RF devices are becoming more stringent. In order to achieve rapid design and fabrication of RF devices, such as filters, antennas, power dividers, couplers, etc., researchers and engineers usually use a combination of theoretical calculations and EDA software to achieve rapid design and performance optimization of RF devices. However, due to factors such as the simulation accuracy of EDA software, the accuracy of fabrication technology, materials, and complex structures of RF devices, there is always a certain performance error between the designed and fabricated RF devices. As the operating frequency of RF devices increases and as the RF device size becomes smaller and smaller, the performance error will become increasingly significant without significant improvement in the fabrication accuracy. As an important device of the terahertz (0.1 THz~10 THz) RF front-end system, the terahertz filter directly affects the performance of the terahertz transceiver and communication system [1]. Meanwhile, because the wavelength of the terahertz wave is at the 0.03 mm~3 mm level, their physical size is smaller, and they are sensitive to the device structure, it is extremely difficult to obtain high-performance and high-precision terahertz filters.

At present, in order to reduce performance errors between fabricated and desired RF devices, the usual solution is to manually introduce design offsets based on the measured and simulation results of the first fabricated RF device and optimize new devices using EDA software. This method of using EDA software to scan all device structure parameters to obtain the best performance can be called the global optimization correction (GOC) method, which is very helpful to eliminate the linear error generated in the fabrication process of RF devices. This GOC method is commonly used for the design and optimization of terahertz filters based on low-temperature co-fired ceramics (LTCC) [2,3] and CNC [4,5,6,7,8] technologies, as these fabrication techniques have low costs and can achieve multiple device design and fabrication iterations. However, the GOC method not only requires a long optimization time but also struggles to eliminate nonlinear errors caused by the steepness and roughness of the device structure, making it impossible to reduce device performance errors to a relatively low level.

On the other hand, there is also much literature based on advanced high-precision semiconductor technologies, such as deep reactive ion etching (DRIE) [9,10], silicon via (TSV) [11,12], and microelectromechanical systems (MEMS) [13,14,15,16,17], to achieve the manufacturing of terahertz filters. Ideally, with the improvement of fabrication accuracy, performance errors can be reduced to a lower level, but due to the maturity of advanced semiconductor technology and material limitations, the current performance of terahertz devices is not excellent. The expensive fabrication technology has also led to a significant increase in device fabrication costs.

In addition, some researchers have recently used Surface Plasma Polaritons [18], Frequency Selective Surface [19], and Planar Goubau Line [20] to design planar terahertz filters, which can effectively avoid errors caused by manufacturing complex filter structures. However, due to the high transmission line loss and low Q value of planar terahertz filters, this method is still far from practical application. Therefore, it is very valuable to study a low-cost and high-performance THz front-end RF device fabrication technology or rapid optimization method for THz communication systems, imaging systems, radar systems, etc.

In this paper, the DOC method is proposed by analyzing the coupling window structure, resonator structure, and asymmetry parameters of the THz bandpass filter. The center frequency and bandwidth errors of the terahertz filter are quickly corrected by adjusting the coupling window length and resonator width based on the DOC method. Compared with the traditional GOC method, the DOC method does not need to optimize all parameters of the filter. It can optimize the single performance of the terahertz filter by changing a single parameter, which greatly reduces the design and optimization time of the terahertz device. This DOC method is also of great significance for the rapid design of high-precision terahertz front-end devices and for improving the design efficiency of terahertz error-sensitive devices.

2. Design and Fabrication of 340 GHz Terahertz Filter

Terahertz filters play a role in filtering out harmonics and noise signals in terahertz transceiver front-end systems. Therefore, it is particularly important to design and fabrication high-performance filters with the required center frequency, bandwidth, and out-of-band rejection. As one of the core devices of a 340 GHz high-speed communication system foundation program, a high-precision 330–350 GHz filter with rejection greater than 40 dB at 320 GHz and 360 GHz is necessary for the THz front-end system of the program. As shown in Figure 1, in order to obtain the corresponding design indicators, a 7-order Chebyshev response low-pass filter with an in-band ripple of 0.5 dB was selected as the design prototype.

As a basic knowledge, it is easy to find that the normalization coefficients g_i (i = 1, 2,… 9) in Figure 1b are g1 = g9 = 1, g2 = g8 = 1.7372, g3 = g7 = 1.2583, g4 = g6 = 2.6381, and g5 = 1.3444, respectively.

How to convert the prototype circuit of the 7-order Chebyshev response low-pass filter in Figure 1b into a 7-order induction diaphragm waveguide filter is shown in Figure 1d.

Firstly, use the K-invert converters to convert the normalization coefficient g_i into a K-coefficient K_i in Figure 1c, as follows:

$$K_{\mathrm{i}}=\sqrt{\frac{1}{g_{\mathrm{i}}\times g_{\mathrm{i}+1}}}\times \frac{\pi \times BW}{2\times f}\times {\left(\frac{{\lambda }_{g0}}{{\lambda }_0}\right)}^2, \mathrm{i} = 2,3,4\dots 7$$

(1)

$$K_{\mathrm{i}}=\sqrt{\frac{1}{g_{\mathrm{i}}\times g_{\mathrm{i}+1}}}\times \sqrt{\frac{\pi \times BW}{2\times f}}\times \left(\frac{{\lambda }_{g0}}{{\lambda }_0}\right), \mathrm{i} = 1,8$$

(2)

$${\lambda }_{g0}=\frac{{\lambda }_0}{\sqrt{1-(\frac{{\lambda }_0}{2a})2}}$$

(3)

Here, BW = 20 GHz is the operating bandwidth of the THz filter, f = 340 GHz is the center frequency of the THz filter, λ₀ = 0.8824 mm is the operating wavelength of the THz filter, and a is the width of rectangular waveguide. The characteristic wavelength of the rectangular waveguide can be calculated as λ_g0 = 1.149 mm according to Formula (3). If the WR-2.2 (a = 0.56 mm; b = 0.28 mm) standard waveguide structure is used to design the 340 GHz 7-order inductive diaphragm waveguide filter in Figure 1d, the K coefficients K_i (i = 1, 2,… 8) in Figure 1c can be calculated as follows: K1 = K8 = 0.449, K2 = K7 = 0.237, K3 = K6 = 0.192, and K4 = K5 = 0.186 in Figure 1c.

Secondly, ideally, the length of seven resonators in the 7-order inductive diaphragm waveguide filter is half wavelength λ₀/2; however, due to the negative length on both sides of eight symmetric inductor diaphragms, it should be included in the electrical length of the adjacent waveguide resonator. Therefore, it can be calculated that the physical length Li and K-coefficient K_i of seven resonators satisfy the following formula:

$${\theta }_{\mathrm{i}}=\pi -\arctan K_{\mathrm{i}}-\arctan K_{\mathrm{i}+1}, \mathrm{i} = 1,3,4\dots 7$$

(4)

$$l_{\mathrm{i}}=\frac{{\lambda }_{g0}}{2\pi }{\theta }_{\mathrm{i}}, \mathrm{i} = 1,7$$

(5)

Similarly, based on the equivalent admittance of eight symmetric inductive diaphragms in rectangular waveguide, the coupling window width w_i and K-coefficients K_i can be calculated to satisfy the following formula:

$$w_{\mathrm{i}}=\frac{2\mathrm{a}}{\pi }\arctan \sqrt{\frac{K_{\mathrm{i}}}{1-{(K_{\mathrm{i}})}^2}\frac{{\lambda }_{g0}}{\mathrm{a}}}, \mathrm{i} = 1,3,4\dots 8$$

(6)

Finally, substituting the K_i coefficients into the above Formulas (4)–(6), the resonant cavity length Li and coupling window width w_i of the 340 GHz 7-order inductive diaphragm waveguide filter can be calculated, as shown in

Table 1. Theoretical values of structural parameters of 340 GHz filter.

Li	L1	L2	L3	L4	L5	L6	L7
Theory/mm	0.567	0.620	0.631	0.6323	0.631	0.620	0.567
wi	w1	w2	w3	w4	w5	w6	w7	w8
Theory/mm	0.319	0.2494	0.229	0.2263	0.2263	0.229	0.2494	0.319

. It is worth noting that due to the symmetry of g_i, the K_i, Li, and w_i parameters all have symmetry, so in subsequent design, only the values of w1–w4 and L1–L4 need to be considered.

According to the above theoretical analysis, the influence of the coupling window length t in Figure 1d is ignored when solving the resonant cavity length Li and coupling window width w_i of the 340 GHz 7-order inductance diaphragm waveguide filter. Therefore, only when t < < w_i can the above theoretical calculation results be consistent, so t = 0.02 mm is set.

As shown in Figure 2 and Figure 3, the 340 GHz 7-order inductive diaphragm waveguide filter in Figure 1d was modeled and simulated in HFSS software. The actual physical structure is shown in Figure 2a, and the HFSS simulation results of a 340 GHz terahertz filter based on t = 0.02 mm and the theoretical parameters in Table 1 are shown in Figure 3a, which can obtain a center frequency of 343.1 GHz and a bandwidth of 20.6 GHz. The out-of-band rejection is 84.3 dB and 35.9 dB at 320 GHz and 360 GHz, respectively. The in-band return loss is close to 10 dB, and the in-band insertion loss is close to 1 dB.

There is still a significant performance error between Figure 3a and the expected technical indicators in Figure 1a. To improve the simulation performance, the optimization ability of HFSS is utilized to scan and simulate all parameters of the filter to obtain the expected technical indicators. Through extensive simulation optimization, the simulation results shown in Figure 2b were finally obtained. The optimized filter parameters for HFSS are a = 0.56, b = 0.28, t = 0.18, d = 0.56, L1 = 0.4363, L2 = 0.5277, L3 = 0.548, L4 = 0.551, w1 = 0.4232, w2 = 0.3498, w3 = 0.3288, and w4 = 0.325. The coupling window length t of the 340 GHz THz filter significantly increased to make terahertz filters easier to manufacture.

As shown in Figure 3b, the optimized HFSS filter has a bandwidth of 330~350 GHz, an in-band insertion loss (IL) of less than 1 dB, and an in-band return loss (RL) of less than 20 dB. The out-of-band rejection is 73.4 dB and 40.7 dB at 320 GHz and 360 GHz, respectively, meeting practical application technical indicators in Figure 1a. Therefore, the most commonly used CNC technology (manufacturing accuracy = 5 µm) is used to manufacture the 340 GHz THz filter.

As shown in Figure 4a, its overall structure is red copper with a conductivity of 3.5 × 10⁷ S/m, copper surface electroplated with 2 µm gold. The test environment is shown in Figure 4b, and the simulation and measurement results of the terahertz filter are shown in Figure 4c. The measurement results indicate that the center frequency (CF) is 330 GHz, the bandwidth (BW) is 12 GHz, the RL is less than 15 dB, and the in-band IL is less than 1.8 dB. Compared with the simulation results or the needed technical indicators in Figure 1a, the CF and BW errors of the terahertz filter are 10 GHz (~3.03%) and 8 GHz (~40%), respectively, and there is a very large performance error.

Due to the manufacturing technology itself having a precision of 5 µm, HFSS tolerance analysis shows that a structural error of 5 µm is not enough to cause such a large error. Therefore, considering that this error is mainly affected by factors such as the accuracy of HFSS simulation software, structural steepness, surface roughness, etc., these types of errors only have a small degree of randomness. Therefore, optimization and performance correction can be carried out by introducing a design offset.

3. Optimized 340 GHz Terahertz Filter by GOC Method

As mentioned above, in order to obtain the required 20 GHz BW and 340 GHz center frequency terahertz filters in Figure 1a, a center frequency offset of 10 GHz and a bandwidth offset of 8 GHz should be introduced based on the errors in the simulation and measurement results in Figure 4c. Based on the above design method, the design and optimization of 350 GHz center frequency and 28 GHz bandwidth terahertz filters should be carried out. By introducing f = 350 GHz and BW = 28 GHz into Formulas (1)–(6), the resonant cavity length and coupling window width of the new 336–364 GHz Thz filter can be calculated as shown in

Table 2. Theoretical values of structural parameters of 336–364 GHz filter.

Li	L1	L2	L3	L4	L5	L6	L7
Theory/mm	0.5064	0.555	0.568	0.569	0.568	0.555	0.5064
wi	w1	w2	w3	w4	w5	w6	w7	w8
Theory/mm	0.321	0.257	0.236	0.232	0.232	0.236	0.257	0.321

.

Similarly, based on the parameters in Table 2, performance optimization is achieved by scanning all parameters of the filter using HFSS software. The optimized filter parameters were (in mm) t = 0.22, L1 = 0.2667, L2 = 0.3562, L3 = 0.3803, L4 = 3846, w01 = 0.47, w12 = 0.384, w23 = 0.358, and w34 = 0.3514. The process of introducing a design offset to redesign and optimize the filter mentioned above can be referred to as the GOC method, as shown in Figure 3a. The new 350 GHz terahertz filter was fabricated and measured using the same CNC technology and testing environment as Figure 4.

As shown in Figure 5b, the measured results indicate that the center frequency of the second fabricated terahertz filter optimized by the GOC method is 343.5 GHz, with a bandwidth of 22.2 GHz, an in-band return loss of less than 15 dB, and an in-band insertion loss of less than 1.9 dB. Compared with the simulation results, the center frequency error is 6.5 GHz, and the bandwidth error is 5.8 GHz. However, compared to the expected results in Figure 1a, the center frequency error and bandwidth error are 3.5 GHz (~1.03%) and 2.2 GHz (~11%), respectively.

It can be seen that based on the GOC method, the performance error between the measurement results and the expected results is significantly reduced. However, due to the need to optimize all filter parameters and the presence of certain randomness, it is difficult to further reduce the error of terahertz filters. The GOC method is equivalent to redesigning the RF device, which not only has a long simulation time but also makes it difficult to maintain linear correction of the center frequency and bandwidth due to significant changes in the structural parameters.

4. Optimized Terahertz Filter by DOC Method

In order to further reduce the performance error between the designed and manufactured filters, the best path is to find the independent influence parameters of the center frequency and bandwidth to achieve precise directional correction of the filter. Meanwhile, according to Formulas (3) and (4) in the filter design process in Section 2, it can be found that it is difficult to directly derive the formula between the center frequency or bandwidth and the structural parameters of the filter. Therefore, the following uses simulation experiments to analyze the coupling window structure, resonator structure, and asymmetry of the 340 GHz terahertz filters on their performance in detail, and the DOC method are proposed on this basis.

4.1. Structural Parameters Analysis of Terahertz Filter

The terahertz filter shown in Figure 2a mainly includes seven resonators and eight coupling windows.

As shown, S₂₁ changes with the length t of eight coupling windows in Figure 6a; if the t changes between 0.1 and 0.3 mm in Figure 2a, the bandwidth will change between 33.4 and 8.8 GHz; and the center frequency will change from 340.5 to 340.9 GHz. As shown in Figure 6b, S₂₁ changes with the width of the eight coupling windows w1~w4. If w1~w4 increases by −0.03~0.03 mm synchronously in Figure 2a, the bandwidth will change between 13.5 and 25.6 GHz, and the center frequency will change between 347.3 and 333.9 GHz.

Conspicuously, it can be concluded that the bandwidth of the filter is mainly affected by the coupling window. Here, t will change the bandwidth of the filter while changes to the center frequency are small; w1~w4 will change the bandwidth of the filter, but the center frequency will be significantly affected.

As shown, S₂₁ changes with the length L1~L4 of the seven resonators in Figure 7a; if L1~L4 increase synchronously by −0.05~0.05 mm in Figure 2a, the bandwidth of the filter will change from 24 GHz to 16.2 GHz, and the center frequency will change between 350.4 and 332.55 GHz. As shown, S₂₁ changes with the width d of the seven resonators in Figure 7b; if d is changed between 0.5 and 0.6 mm in Figure 2a, the center frequency of the filter will change between 367.65 and 329.85 GHz, and the bandwidth will change between 20.5 and 19.7 GHz.

It can be concluded that the center frequency of the filter is mainly affected by the resonators. If the resonators’ width d is changed, the center frequency of the filter can be adjusted, while the bandwidth remains relatively constant. If the lengths L1~L4 are changed, the center frequency can still be adjusted, but the bandwidth will be significantly affected.

As shown, the filter structures and S₂₁ change with m2 and m4 in Figure 8. If the THz filter has a horizontal shift of m2 = 0~0.2 mm, the bandwidth will change between 20 and 26 GHz, and the center frequency will change between 340.6 and 349 GHz. If the filter has a vertical shift of m4 = 0~0.05 mm, the bandwidth of the filter will change between 20 and 24.5 GHz, and the center frequency will change between 340.6 and 327.35 GHz.

It can be concluded that the asymmetry in the horizontal direction of the filter will increase both bandwidth and center frequency. The asymmetry in the vertical direction of the filter, on the other hand, will increase bandwidth but reduce center frequency. It is noteworthy that the filter performance error can be slightly optimized by changing the asymmetry during filter assembly.

Based on the above structural parameters analysis of the terahertz filter, it can be concluded that the DOC method enables independent adjustment of the bandwidth of the filter by synchronously and equivalently changing the length t of all coupling windows and the center frequency of the filter can be independently adjusted by synchronously and equivalently changing the width d of all resonators.

4.2. Third Fabricated Terahertz Filter

As shown in Figure 1c, the terahertz filter is optimized by the DOC method in Figure 2a, and the optimized filter parameters are t = 0.13 mm, d = 0.531 mm. As shown in Figure 9a, the new 350 GHz terahertz filter is fabricated and measured by the same CNC technology as Figure 4a.

The simulation and measured results of the third fabricated terahertz filter optimized by the GOC method are shown in Figure 9b,c. The measured center frequency of the filter is about 339.8 GHz, and the measured bandwidth is 19.6 GHz. Compared with the simulation results in Figure 9b, the center frequency and bandwidth of the filter have large errors. However, compared with the desired results in Figure 9c, the center frequency error and bandwidth error are 0.2 GHz (~0.06%) and 0.4 GHz (~2%), respectively. The performance errors are greatly reduced based on the DOC method. In addition, the return loss of the filter is better than 17 dB, and the insertion loss is about 2.3 dB.

Finally, compared with the related works in

Table 3. Comparisons with Other Previous Works.

Ref.	Fabrication Technology	Desired CF (GHz)	Desired BW (GHz)	CF Error *	BW Error *	IL (dB)	RL (dB)
[4]	CNC	220	30	−0.91%	+3.4%	0.6	—
[5]	CNC	260	22.1	−0.88%	+2.26%	0.7	15
[6]	CNC	340	18	−0.29%	−11.1%	0.6	20
[8]	CNC	255	28	−1.17%	3.57%	3.9	15
[10]	DRIE	400	32.2	−1.23%	7.76%	2.84	16
[11]	TSV	337	46	−1.78%	+10.87%	1.5	15
[13]	MEMS	350	45.5	−0.86%	2.2%	2.5	—
[15]	MEMS	185	18	−0.27%	5.56%	1.55	15
[16]	MEMS	450	4.5	−0.67%	+11.2%	2.6	16
GOC method	CNC	350 (simulation)	28 (simulation)	−1.86%	−20.7%	1.9	15
		340	20	+1.03%	+11%
DOC method	CNC	350 (simulation)	28 (simulation)	−2.9%	−30%	2.3	17
		340	20	−0.06%	−2.0%

* CF error = 100%∗(measured CF − desired CF)/desired CF. * BW error = 100%∗(measured BW − desired BW)/desired BW.

, our work shows that the terahertz filter based on the DOC method has a lower center frequency (CF) error and bandwidth (BW) error.

5. Conclusions

In this paper, the DOC method is proposed to reduce the performance errors between desired and fabricated THz filters. According to the analysis of the filter structure parameters, we obtained the directional optimization parameters of the filter center frequency and bandwidth, which are the width of the resonators and the length of the coupling windows, respectively. On this basis, a 340 GHz filter with excellent performance and small performance errors is realized. Compared with the normal GOC method, the proposed DOC method can quickly and greatly reduce performance errors between desired and fabricated THz filters by adjusting fewer parameters. Furthermore, it is also applicable to the faster design and optimization of other terahertz front-end devices.