The main contribution of this work is not the design of the filter unit so that we just discuss the brief working mechanism. The slot coupling between the top and bottom lines excites weak odd and even modes with opposite propagating directions and similar magnitude. Two slots with π/4 phase delay form a directional coupler, where the odd modes are out-of-phase and canceled, and the even modes are in phase and added, as shown in the inset of Figure 1A. The DF unit is composed of two directional couplers and one loop resonator. By distributing L ₁₁, L ₂₁, L ₂₂, and L ₂₃ as π/4, π/4, π/2, and π, respectively, the desired spectra are directionally filtered into port 3, as illustrated in Figure 1C. The design and optimization details can be found in our previous work (Zhang et al., 2017).

Here, the filter units at 92, 95, and 97 GHz are designed for the cascaded DFs in the following sections, the dimensions of which are shown in Table 1 and the simulated passbands of which are shown in Figure 2. It can be seen that the bandwidth of a filter unit ranges from 92.6 GHz to 97.6 GHz, i.e., a fractional bandwidth of only 5.5%, the insertion loss (|S ₃₁|) is around 4.6 dB in the passband, and the reflection attenuation (|S ₁₁|) is larger than 10 dB at the W-band. As can be seen in Figure 1C, the through loss (|S ₂₁|) is less than 10 dB in one filter unit so that cascading filter units can enlarge the through loss and reduce the insertion loss.

As discussed above, one DF unit cannot sufficiently filter the desired signals, having large insertion loss (|S ₃₁|) and small through loss (|S ₂₁|) at the resonant frequency. In this section, we will discuss how to suppress the insertion loss and broaden the bandwidth by cascading filter units.

First, a two-stage DF with the identical units is investigated as the simplest example, as shown in Figure 3A. Port 1 and port 3 of the second unit connect to port 2 and port 4 of the first unit, respectively. Due to the symmetry of the device, the S-parameters of the first unit satisfy the following relations (Pozar, 2012):   S 21 1 = S 12 1 = S 34 1 = S 43   1   , (2.1a) S 41 1 = S 14 1 = S 23 1 = S 32 1 . (2.1b)

S ₂₁ and S ₄₁ of the first unit feed the second unit as the input. The transmission can be described by the following formula (Zhang et al., 2017): S i n ′ ≈ S 21 1 e − j θ , (2.2a) S i n ″ ≈ S 41 1 e − j θ , (2.2b) where S 21 1 and S 41 1 are the through and isolated signals of the first DF unit, respectively, θ is the phase delay between filter units, and S i n ′ and S i n ″ are the input signals to port 1 and port 3 of the second unit, respectively. Thus, the S-parameters of the two-stage DF can be obtained by cascading the microwave networks. As S 11 1 , S 21 1 , and S 41 1 are much smaller than S 31 1 at the resonant frequency, it is fair to neglect their high-order terms for simplification. The simplified formula can be expressed as follows:   S 11 T ≈ S 11 1 + S 11 2 e − j θ S 12 1 + S 31 2 e − j θ S 14 1 , = S 11 1 + S 21 1 S 21 1 S 11 1 + S 41 1 S 31 1 e − j 2 θ + S 41 1 S 21 1 S 31 1 + S 41 1 S 11 1 e − j 2 θ , ≈ S 11 1 + 2 S 21 1 S 41 1 S 31 1 e − j 2 θ , (2.3a) S 21 T ≈ S 21 2 = S 21 1 S 21 1 + S 41 1 S 41 1 e − j θ , (2.3b) S 31 T ≈ S 31 1 + S 31 2 e − j θ S 34 1 + S 11 2 e − j θ S 32 1 , = S 31 1 + S 21 1 S 21 1 S 31 1 + S 41 1 S 11 1 e − j 2 θ + S 41 1 S 21 1 S 11 1 + S 41 1 S 31 1 e − j 2 θ , ≈ S 31 1 + S 21 1 S 21 1 S 31 1 e − j 2 θ + S 41 1 S 41 1 S 31 1 e − j 2 θ , (2.3c) S 41 2 = 2 S 21 1 S 41 1 e − j θ . (2.3d)

S 31 T is maximum at θ = nπ (n is an integer) and is minimum at θ = (2n + 1) π/2. Limited by the dimensions of the quarter-wavelength directional couplers, θ = 2π and P _r = 1.9 mm is chosen to reduce the insertion loss. The simulated S-parameters of the two-stage DF with the identical units are illustrated in Figure 3B, where the insertion loss |S ₃₁| is suppressed to 2.8 dB at 95 GHz and the through loss |S ₂₁| increases to 17 dB. The 3-dB bandwidth is improved to 8% due to the in-phase coupling between two units. As depicted in Eqs 2.3a, 2.3b, 2.3c, ans 2.3d, S 21 1 is the main contribution factor to broaden the bandwidth of S 31 T .

Next, a three-stage DF with identical units is investigated based on the above analysis of the two-stage DF, as illustrated in Figure 4A. In addition, the high-order terms of S 11 T , S 21 T , and S 41 T are neglected in the equation derivation. According to the Eqs 2.3a, 2.3b, 2.3c, and 2.3d, the simplified passband response of the three-stage DF is as follows: S 31 T ′ ≈ S 31 T + S 31 3 e − j θ ′ S 34 T + S 11 3 e − j θ ′ S 32 T , = S 31 1 + S 21 T S 21 T S 31 1 + S 41 T S 11 1 e − j 2 θ ′ + S 41 T S 21 T S 11 1 + S 41 T S 31 1 e − j 2 θ ′ , ≈ S 31 T + S 21 T S 21 T S 31 1 e − j 2 θ ′ + S 41 T S 41 T S 31 1 e − j 2 θ ′ , ≈ S 31 1 + S 21 1 S 21 1 S 31 1 e − j 2 θ + S 21 1 S 21 1 S 21 1 S 21 1 S 31 1 e − j 2 θ + θ ′ , (2.4) where θ′ is the phase delay between the second and third units. ST’ 31 is maximum at θ = nπ and θ' = mπ (m and n are integers), and the corresponding P _r and Pʹ _r are chosen as 1.9 mm. Figure 4B illustrates the simulated S-parameters of the three-stage DF. It can be seen that the insertion loss |S ₃₁| is reduced to 2.6 dB, and the through loss |S ₂₁| is increased to 26 dB at 95 GHz. The 3-dB bandwidth is broadened up to 10.2%. The main contribution factors are S1 21, θ, and θ'.

Adding more filter units can further increase the bandwidth slightly. However, due to the increasing through loss |S ₂₁|, this approach is limited. Figure 5A illustrates the simulated S-parameters of a four-stage DF with the identical units at 95 GHz, whose bandwidth is slightly increased to 11.3%. The bandwidth and insertion loss performances of the cascaded DFs with respect to the filter unit numbers are shown in Figure 5B and Table 2. The DFs can achieve lower insertion loss and wider bandwidth at a cost of unit numbers and device profile. However, the improved efficiency reduces significantly as the unit number increases.

In the reported DF designs, all resonant units are designed at the same frequency to get consistent phase configuration (Cameron and Yu, 2011; Lobato-Morales et al., 2011; Lobato-Morales et al., 2013; Kim, 2011; Zinka et al., 2003; Cohn and Coale, 1956; Coale, 1956; Uvasl, 1997; Cheng et al., 2007; Walker, 1978; Uvsal, 2003; Sarkar et al., 2007; Tanaka et al., 1988; Zhang et al., 2017). In contrast, classic Chebyshev-type filters may use resonant units designed at adjacent frequencies to broaden the filter bandwidth (Pozar, 2012). In this section, this approach will be investigated in the cascaded DFs to broaden the passband for the first time.

First, a two-stage DF with different units is investigated, whose configuration is illustrated in Figure 3A. Empirically, the insertion loss of filters and propagation loss of microstrip lines increase as the frequency increases. To compensate the dispersive loss and achieve broadband, the filter units are cascaded in an order from high to low frequency. The second filter unit is designed at f ₂ = 95 GHz, and the frequency of the first unit f ₁ sweeps from 99 to 95 GHz. P _r remains 1.9 mm initially, as in the last section. Figure 6A illustrates the simulated S-parameters with various f ₁, where the bandwidth enlarges as f ₁ increases. However, a transmission dip occurs between f ₁ and f ₂ as these two resonant frequencies diverge, which may damage the bandwidth. Therefore, f ₁ is chosen as 97 GHz to suppress the dip. Next, the phase delay θ and pitch P _r are swept for optimization according to Equations 2.3a, 2.3b, 2.3c, and 2.3d, as shown in Figure 6B. As P _r gets smaller, the bandwidth gets larger, and the transmission dip gets stronger. In this case, P _r is chosen as 1.85 mm. The simulated S-parameters of the optimized two-stage DF with different units are illustrated in Figure 6C. The 3-dB bandwidth of the passband is increased to 10%, which is similar to the three-stage DF with the identical units. The insertion loss (|S ₃₁|) is 2.6 dB at 96 GHz, and the corresponding through loss (|S ₂₁|) is around 17 dB.

Then, we investigate a three-stage DF with different units based on the above two-stage DF with 97-GHz and 95-GHz units. To broaden the bandwidth and suppress the transmission dip, the third filter unit is designed at f ₃ = 92 GHz, whose dimensions can be found in Table 1. Initially, P _r remains 1.85 mm, and Pʹ _r is set as 1.9 mm. According to Eq. 2.4, both θ and θ′ affect the passband, so we need to optimize both P _r and Pʹ _r. Figure 7A illustrates the simulated S-parameters with various Pʹ _r. As Pʹ _r increases, the passband red shifts and enlarges slightly. However, a transmission dip will be induced at large Pʹ _r, which may damage the bandwidth. In this case, Pʹ _r is chosen as 1.7 mm to compensate the bandwidth and transmission dip. Next, parameter variation of P _r is studied again, which is shown in Figure 7B. Smaller P _r leads to the blue shift of the passband and a stronger transmission dip between f ₂ and f ₃. To avoid strong transmission dips, both P _r and Pʹ _r are optimized as 1.7 mm. Finally, the simulated S-parameters of the optimized three-stage DF are illustrated in Figure 7C. The insertion loss (|S ₃₁|) is 2.6 dB at 95 GHz, and the out-of-band attenuation is −20 dB at 85 and −13 dB at 110 GHz, respectively. The 3-dB bandwidth of S ₃₁ is 14.26 GHz, and the corresponding fractional bandwidth is 15% centered at 95 GHz, which is significantly improved using different filter units.

As comparison, a three-stage DF with 95-, 97-, and 92-GHz units cascaded in series is investigated, whose S-parameters are illustrated in Figure 8. The unit pitch P _r and Pʹ _r are kept at 1.7 mm. With respect to Figure 7C, the high-frequency performance of S ₃₁ is sacrificed significantly. The bandwidth of the passband is significantly reduced to 11%, which is similar to the three-stage DF with the identical units, and the insertion loss is slightly reduced to 2.4 dB. Therefore, it should be concluded that the first unit is more dominant for the cascaded DF design and is more important for high frequencies. The second filter unit has an input signal around 3-dB smaller than the first unit; see S ₂₁ at 97 GHz in Figure 1C. The bandwidth and insertion loss parameters of the proposed DFs with various units are listed in Table 2. On using different filter units, the bandwidth of the three-stage DF is much larger than that of the four-stage DF with the identical units. In contrast, the different filter units show little loss discrepancy to the identical filter units.

As can be seen in Table 2, the insertion loss can be minimized by cascading filter units, which shows decreasing efficiency at large unit numbers. The propagation loss of microstrip lines on a 100-μm LCP substrate is 0.156 dB/mm, and the averaged radiation loss is approximately 0.065 dB/mm in the loop resonator (Zhang et al., 2017). In this case, the absolute insertion loss induced by each directional coupler is just around 0.8 dB. The main contributing factor for the insertion loss of the cascaded DF is the wave propagation loss on the microstrip lines, which is induced by the loss tangent of the LCP substrate and leaky radiation.