The transmission line equivalent model of hybrid-loaded T type-U type travel-ing-wave electrode structure (TU-TWEs) is shown in Figure 2.

Based on microwave theory, the characteristic impedance and propagation constant formulations derived from the equivalent transmission line model are given by Equations 5, 6 (Fang et al., 2023; Cheng et al., 2024): Z 0 = R e f f + j ω L e f f G e f f + j ω C e f f (5) γ = R e f f + j ω L e f f G e f f + j ω C e f f (6) Where R e f f is the equivalent resistance, G e f f the equivalent conductance, C e f f the equivalent capacitance, and L e f f the equivalent inductance of the transmission line. When analyzing frequency responses beyond 10   G H z , the lossless transmission approximation ( ω C / G ≫ 1 ; ω L / R ≫ 1 ) becomes valid. The expressions then reduce to Equations 7, 8: Z 0 = L e f f C e f f (7) γ = − ω L e f f C e f f (8)

The microwave refractive index is formulated through the propagation constant Equation 9: N p = c * I M γ ω (9)

The equivalent inductance and capacitance extracted from Figure 2 model are expressed as Equations 10, 11: L e f f = L U + L L L U (10) C e f f = C + C T + C U (11)

The equivalent inductance equals the parallel combination of parasitic inductances from main and U-type electrodes, while the equivalent capacitance results from parallel parasitic capacitances of main, T-type, and U-type electrodes.

The U-type structure contributes only parasitic inductance without introducing parasitic capacitance. The velocity synchronization mechanism is achieved through: Compared to conventional CL-TWEs, the integrated U-type structure introduces parallel inductive loading that suppresses equivalent inductance, counterbalancing the microwave refractive index elevation induced by T-type capacitive loading. Although this design increases characteristic impedance as Equation 5, the impedance variation can be mitigated through optimized thickening of the underlying SiO₂ layer (Li et al., 2024).

According to the above analysis, microwave loss is one of the critical factors limiting electro-optic bandwidth. The influence of modulator electrode height on device performance is analyzed via finite element method (FEM) simulations, calculating microwave losses at 75   G H z . Results shown in Figure 3 demonstrate that when electrode height is too small, the skin effect becomes particularly pronounced, confining current propagation predominantly to the conductor’s surface layer. Such concentrated distribution of current will increase the resistance and further significantly increase the loss during the microwave transmission process. When the electrode height is too large, it may have an adverse impact on the distribution of the electric field in the transmission line. Specifically, oversized electrodes may distort normal field distribution patterns, causing partial field energy leakage or radiative emissions. This unintended energy dissipation is termed radiation loss. Minimum microwave loss of 5.6   d B / c m at 75   G H z is achieved with 1   μ m electrode height. Consequently, h 2 = h 3 = 1   μ m is selected as the optimal configuration for microwave loss minimization.

Through systematic analysis of U-type structure’s influence demonstrates that parametric tuning of W r 1 and W u enables optimized inductance-capacitance distribution in the U-type configuration. As shown in Figure 4 simulations at 75   G H z , W u parameter adjustments exhibit limited capacitance variation but substantial inductance modulation. Specifically, increasing W u causes equivalent capacitance to fluctuate slightly around 0.078   f F / m , while equivalent inductance decreases from 0.5   n H / m to 0.4   n H / m ( 20 % reduction). Similarly, increasing W r 1 maintains equivalent capacitance near 0.065   f F / m with minimal variation, while reducing equivalent inductance from 0.375   n H / m to 0.3   n H / m ( 25 % reduction). Extensive simulations confirm the U-type structure’s effective inductance tuning capability with negligible capacitance influence. These findings establish the U-type component as an effective inductive compensation mechanism for microwave-optical velocity matching in modulator electrode design.

We conducted an in-depth analysis of the U-type structure’s effects on microwave refractive indices and radiofrequency attenuation characteristics. Simulation results in Figure 5 show that increasing W u from 4   μ m to 10   μ m reduces RF loss at 75   G H z from 6.5   d B / c m to 3.5   d B / c m , while increasing microwave refractive index from 2.17 to 2.52 . At W u = 4   μ m , optimal refractive index matching with optical group velocity ( 2.25 ) is achieved while maintaining low microwave losses. Consequently, W u = 4   μ m is identified as the design optimum through comprehensive parametric analysis. Similarly, increasing W r 1 from 35   μ m to 45   μ m slightly elevates RF loss from 3.9   d B / c m to 4.5   d B / c m while marginally decreasing microwave refractive index from 2.32 to 2.25 . W r 1 = 45   μ m is chosen to achieve optimal refractive index matching with optical group velocity.

While the T-type structure exhibits negligible impact on equivalent inductance, it substantially modulates equivalent capacitance values (Liu X. C. et al., 2021). The following analysis focuses on the effects of parameters W S , W h , W T , and W R on modulator speed and impedance characteristics.

We conducted systematic parametric analysis of W S and W h dimensions to evaluate their effects on modulator characteristics. Figures 6a,b presents contour mapping results obtained through parametric optimization scanning. Both microwave refractive index and characteristic impedance increase with W S and W h dimensions. The configuration W S = 6   μ m / W h = 2   μ m achieves optimal microwave refractive index ( 2.27 ) and characteristic impedance of 48   Ω . The geometric parameters W S and W h critically alter the parasitic parameters in the transmission-line equivalent model, consequently modifying microwave propagation properties.

Experimental investigations reveal limited parametric influence on overall device characteristics. Figures 6c,d displays contour mapping results obtained through comprehensive parametric optimization. Increasing W r elevates microwave refractive index while reducing characteristic impedance, whereas W t increments decrease both mi-crowave refractive index and impedance. The configuration W r = 40   μ m / W t = 3   μ m is identified as optimal, delivering microwave refractive index 2.24 and characteristic impedance of 48.5   Ω .

Key determinants of electro-optic conversion efficiency include waveguide effective refractive index and electro-optic field overlap factor. Effective refractive index is governed by geometric parameters of the ridge waveguide, specifically ridge width and, etch depth. For defined waveguide architectures and operating wavelengths, the V π L product exhibits inverse proportionality to the overlap factor ( Γ o m ), where enhanced field overlap reduces V π L and improved modulation efficiency. The electro-optic overlap factor is primarily determined by parameters such as electrode spacing and electrode position. Research indicates that enlarging the electrode spacing decreases the half-wave voltage-length product ( V π L ), thereby directly improving electro-optic efficiency. Nevertheless, decreasing the electrode spacing induces a substantial rise in optical absorption loss, requiring careful balancing of modulation efficiency and optical loss in the design process.

Guided by the theoretical framework of the half-wave voltage-length product Equation 5, numerical simulations of the modulator’s optical mode field and electro-static field were conducted using finite element analysis software. The optical mode electric field distribution is depicted in Figure 7a, and the electrostatic field distribution is presented in Figure 7b.

In the T-type structure, G a p 1 exhibits a pronounced impact on modulation efficiency—reducing G a p 1 lowers V π L and enhances modulation efficiency at the expense of elevated optical loss. In contrast, the U-type structure demonstrates limited sensitivity of modulation efficiency to variations in G a p 2 . Simulation results in Figure 8a reveal that increasing G a p 2 from 5   μ m to 8   μ m causes V π L to shift marginally from 1.66   V · c m to 1.72   V · c m , with negligible changes in optical loss. G a p 2 is constrained by the geometric parameters W h and W u , while G a p 1 optimization necessitates a compromise between optical loss and modulation efficiency. As demonstrated in Figure 8b, sweeping G a p 1 from 2.5   μ m to 7   μ m leads to a monotonic increase in the half-wave voltage-length product from 1.3   V · c m to 2.5   V · c m . The optimal G a p 1 value of 3   μ m was chosen, achieving a balanced performance with a half-wave voltage of 1.35   V · c m and an optical loss of 0.08   d B / c m .

According to microwave transmission theory, the optical-electric transmission function can be expressed as Equation 12 (Jing et al., 2020): HF ω = 20 ⁡ log 10 2 Z 0 Z i n Z 0 + Z i n e − α L 2 sin ⁡ h 2 α L 2 + s i n 2 b L 2 α L 2 2 + b L 2 2 (12)

The propagation loss α the deviation between optical and microwave speeds expressed as b , the total length of the modulator L , the input impedance typically set to 50   Ω , the characteristic impedance of the transmission line Z 0 , and the speed matching term b = 2 π f c n m − n g are key parameters in determining the modulator’s performance. Equation 1 provides a unified formulation to evaluate the impact of impedance matching, speed matching, and loss on optical efficiency.

We performed a comprehensive finite-element analysis of the modulator’s performance. We specifically calculated its characteristic impedance, microwave loss, and microwave speed as key parameters. In this study, the modulator length was set to 1   c m , corresponding to a half-wave voltage of 1.35   V . We performed a comprehensive finite-element analysis of the modulator’s performance. The simulation frequency range is from 1 to 110   G H z , and the corresponding microwave loss and microwave refractive index are shown in Figure 9a. Within the entire frequency range, the microwave loss remains below 6.2   d B / c m , while the microwave refractive index gradually approaches 2.25 at higher frequencies. Additionally, we calculated the S parameters to obtain the ABCD matrix and derived the device’s characteristic impedance from this matrix. The results are shown in Figure 9b. As shown in the figure, the device’s characteristic impedance primarily fluctuates around 50   Ω .

Finally, by substituting the calculated frequency-dependent microwave loss, refractive index, and characteristic impedance of the device into Equation 1, the electro-optic response shown in Figure 10 was obtained. Simulation results demonstrate a − 2.8   d B rolloff in the electro-optic response at 110   G H z , with the reflection coefficient maintained below − 10   d B across the operational bandwidth.

Through simulations and analysis, we demonstrated a thin-film lithium niobate electro-optic modulator chip with a half-wave voltage of 1.35   V and an electro-optic response bandwidth exceeding 110   G H z , demonstrating its exceptional electro-optic bandwidth performance. We have summarized the modulator’s performance in Table 1 and conducted comprehensive comparisons with state-of-the-art technologies. This table demonstrates that our work achieves high electro-optic bandwidth on silicon substrate through simulation verification, and successfully realizes a modulator structure design with good matching between optical group velocity and microwave velocity.