Automated Inverse Design Framework for Traveling-Wave Electrode Electro-Optic Modulators with Discrete Fabrication Constraints

Abstract

The utilization of electro-optic modulators in engineering is progressively expanding. In this paper, an automated inverse design framework is proposed for traveling-wave electrode electro-optic modulators (EOM). It addresses key challenges in modulator design, such as multi-parameter coupling and discrete fabrication constraints. Applied to a segmented electrode lithium niobate modulator, the framework achieves a 100 GHz electro-optic (EO) bandwidth and a half-wave voltage-length product of $V\pi ·L=1 \mathrm{V}·\mathrm{c}\mathrm{m}$ at 5 mm length, with all 20 independent runs converging successfully under the tested conditions. The framework is further validated on four modulator structures and six engineering conditions, consistently yielding EO bandwidth > 40 GHz and $V\pi ·L<5 \mathrm{V}·\mathrm{c}\mathrm{m}$. This work offers a practical and adaptable solution for the automated design of high-performance electro-optic modulators under realistic fabrication constraints.

1. Introduction

In recent years, optical computing has attracted widespread attention due to its low power consumption and high parallelism [1,2,3,4,5]. As a fundamental component in this field [6,7,8,9,10,11,12,13,14], the electro-optic modulator industry is evolving rapidly. Advances in EOM architecture—such as T-shaped electrodes [15,16,17], substrate etching [17], and hybrid material stacking [12,18]—confirm that structural modifications are key to enhancing modulation efficiency and bandwidth. However, these performance gains come with increased design complexity due to intricate parameter coupling. To address this, automation tools are essential to minimize manual effort and maximize design performance [19].

To address the limitations of traditional design approaches, researchers have explored various inverse designs and automated optimization methods [20,21,22,23,24,25,26,27,28,29]. The current mainstream methods include the adjoint/topology method [20,23,24,25], the machine learning-assisted agent model [26,27,28] and Bayesian optimization [30,31,32]. The adjoint method leverages Lorentz reciprocity to overcome the limitation tying gradient computation to parameter dimensionality, requiring only two electromagnetic simulations to obtain high-dimensional parameter gradients [22]. Limitations include susceptibility to local optima when initial designs are far from the global optima, and significantly increased complexity in deriving adjoint equations for nonlinear devices with phase-change or Kerr effect materials. To overcome these limitations, surrogate model-based alternatives have emerged. Machine learning-assisted surrogate approaches construct neural networks, random forests, or other surrogate models to replace computationally expensive simulations, enabling efficient prediction and optimization [22]. However, the performance is highly dependent on the quality and coverage of training data [27]. Bayesian optimization represents another important class of methods. It constructs Gaussian process surrogates to approximate objective functions. Recent advances have extended its applicability. Research has shown that physics-informed Bayesian optimization can improve optimization efficiency [30]. Nevertheless, Bayesian optimization does not scale well to applications with more than approximately 10 design parameters.

However, the optimization design of traveling-wave electrode electro-optic modulators faces multiple challenges stemming from intense multi-physics coupling, highly correlated parameters, and strict fabrication constraints [21,33]. These challenges limit the application of conventional optimization methods. While the adjoint method offers potential for efficient gradient computation, its optimization results are highly dependent on the initial value, leading to poor stability. And the free-form geometries often fail to meet design rules and fabrication constraints for tape-out. Surrogate-based optimization methods can reduce computational costs to some extent, but their performance relies heavily on sampling density. Constructing a high-fidelity surrogate model still requires significant computational expense. And coupled with systematic bias in prediction accuracy, it becomes difficult to fully meet the requirements of engineering applications. To the best of our knowledge, there have been no reports on the application of inverse design to the optimization of traveling-wave electrode electro-optic modulators. It provides a clear entry point and research opportunity for the work presented in this paper.

This paper proposes an automated inverse design framework for traveling-wave electrode electro-optic modulators. In this framework, global high-potential region identification is achieved through Latin hypercube sampling (LHS) and k-nearest neighbors (KNN) interpolation. It helps guide the search process away from local optima. Meanwhile, a penalty function method is introduced to handle the discrete manufacturing constraints and multi-objective optimization limitations, significantly improving convergence efficiency. Compared to adjoint-based methods, this framework is far less sensitive to the initial design point and more readily satisfies the design rules and manufacturing constraints. Compared to surrogate modeling, it keeps computational costs in check and directly leverages real simulation data, avoiding errors from model approximation. With impedance matching, velocity matching, low microwave loss, and low half-wave voltage as optimization objectives, it achieves the mapping from desired performance to specific geometric parameters under the constraints of geometric feasibility and discrete manufacturing requirements. The framework is applied to various structures and matching conditions, demonstrating its robustness. The framework significantly improves engineering applicability and design efficiency.

2. Materials and Methods

The overall framework of the proposed method is shown in Figure 1. The method includes three stages.

2.1. Design Space Preprocessing

We use a Python-based framework that has data interfaces with HFSS (version 19.5) and COMSOL (version 6.2) to drive an efficient simulation-driven design process. The established interfaces automatically transfer design parameters and return key simulation results: HFSS returns high-frequency transmission characteristics, while COMSOL provides optoelectronic interaction metrics. After implementing the data communication, we apply Spearman correlation analysis. This method quantifies the impact of each input parameter on the output performance. It helps us select the key design variables, as shown in Figure 2a. The Spearman correlation analysis method is shown in Equation (1).

$$\rho =1-{\displaystyle \frac{6\sum {d_i}^2}{n\left(n^2-1\right)}}$$

(1)

where $d_i$ denotes the level difference between two variables and $n$ denotes the number of observed samples. The Spearman correlation analysis significantly improves the computational efficiency of the subsequent optimization process.

2.2. High-Potential Region Identification

In the reduced-dimensional design space constructed after preprocessing, we employ the Latin hypercube sampling (LHS) method. It generates an initial set of sample points with good spatial coverage. The method of LHS is shown in Equation (2).

$$x_{ij}={\displaystyle \frac{{\pi }_j\left(i\right)-u_{ij}}{N}}$$

(2)

where $u_{ij}$ are random numbers sampled from a uniform distribution U(0,1).

By calculating the corresponding physical field performance metrics for each sample point, we used KNN interpolation to build a model for the global scan. In this work, KNN interpolation is implemented as a distance-weighted averaging scheme. For the query point $x_q$, its interpolated value $\widehat{y}\left(x_q\right)$ is computed as:

$$\widehat{y}\left(x_q\right) ={\displaystyle \frac{{\displaystyle {\sum }_{i=1}^K}{w}_{i}y\left(x_i\right)}{{\displaystyle {\sum }_{i=1}^K}{w}_i}}$$

(3)

where $x_i$ are the K nearest neighbors in the sampled set, $y\left(x_i\right)$ are their corresponding objective values, and $w_i=1/\parallel x_q-x_i\parallel$ is the inverse distance weight. This formulation provides a smooth and differentiable interpolant suitable for gradient estimation.

The number of nearest neighbors K is set to 5 as a balance between local accuracy and robustness to noise. To validate this choice, a sensitivity analysis was conducted on a representative 5-dimensional case. Among the tested values, K = 5 yielded the best overall performance, with Root Mean Square Error (RMSE) values of 0.019 dB for microwave loss, 1.67 Ω for characteristic impedance, and 0.0448 for the microwave refractive index. Furthermore, K = 5 demonstrated robust performance across design spaces ranging from 4 to 7 dimensions, making it a suitable default choice for the varied dimensionality encountered in this work. The sample size for the Latin hypercube sampling is chosen within the range of 200 to 500, depending on the dimensionality of the reduced design space, aiming to balance sampling coverage and computational cost.

It is worth noting that this part does not employ a KNN surrogate model in the conventional sense but merely uses KNN interpolation to generate a dense grid. KNN interpolation is chosen because it directly performs interpolation using existing data, eliminating the need for complex global model training. Additionally, its small number of hyperparameters offers better applicability and lower tuning costs. Meanwhile, since this part is only concerned with identifying high-potential regions and computing gradients, constructing a complete KNN surrogate model is unnecessary. It has been found in our work that employing KNN interpolation alone is sufficient to effectively accomplish the aforementioned objectives.

Furthermore, we interpolate this set of sample points to provide gradients for subsequent optimization, as shown in Figure 2b. The gradient information is derived from the interpolated grid data via numerical differentiation.

2.3. Fine-Grained Optimization

We concentrate the optimization on the high-potential region, using a combined approach of the external penalty function method and the Adagrad adaptive gradient algorithm to identify the optimal solution, as shown in Figure 2c. The external penalty function method can transform a constrained optimization problem into an unconstrained optimization problem and is widely used in various engineering optimization designs [34,35,36,37,38,39]. Meanwhile, the AdaGrad adaptive gradient algorithm has become a commonly used optimizer choice in practical applications due to its faster convergence speed and stronger adaptability to prior knowledge and has achieved good results in many aspects such as image processing, position control and fault diagnosis [40,41,42,43].

Firstly, the constraints $g_i\left(X\right)\left(i=\mathrm{1,2}\right)$ are set (Equation (4)), and the constrained optimization is transformed into an unconstrained optimization $\mathit{min}\Phi \left(X,{\sigma }_k\right)$ by using the external penalty function method, as shown in Equation (5).

$$\left\{\begin{array}{l}{g}_1\left(X\right)=\left|Zo\left(X\right)-R\right|-∆Z\le 0\\ g_2\left(X\right)=\left|n_e\left(X\right)-n_o\left(X\right)\right|-∆n\le 0\end{array} \right.$$

(4)

$$\mathit{min}\Phi \left(X,{\sigma }_k\right)=f\left(X\right)+{\sigma }_k{\displaystyle \sum _{i=1}^2}{\left[\mathit{max}\left(0,g_i\left(X\right)\right)\right]}^2$$

(5)

$$f\left(X\right)={\displaystyle \sum _{i=1}^4}{k}_i{f}_i\left(X\right)$$

(6)

$$\left\{\begin{array}{l}{f}_1\left(X\right)=\left|n_e\left(X\right)-n_o\left(X\right)\right|\\ f_2\left(X\right)=\left|Zo\left(X\right)-R\right|\\ f_3\left(X\right)=\left|Loss\left(X\right)\right|\\ f_4\left(X\right)=\left|V\pi \left(X\right)\right| \end{array}\right.$$

(7)

where $Zo\left(x\right)$ is the characteristic impedance, $R$ is the target characteristic impedance value, $∆Z$ is the characteristic impedance mismatch tolerance, $n_e\left(X\right)$ is the microwave refractive index, $n_o\left(X\right)$ is the group refractive index, $∆n$ is the velocity mismatch tolerance, $V\pi \left(X\right)$ is the half-wave voltage, and $Loss\left(X\right)$ is the microwave loss.

The $\mathit{min}\Phi \left(X,{\sigma }_k\right)$ is optimized iteratively using the Adagrad method with iterations as shown in Equations (8) and (9).

$$X_{t+1}=X_t-{\displaystyle \frac{\eta }{\sqrt{G_t+ϵ}}}·h_t$$

(8)

$$h_t=\nabla \Phi \left(X_t,{\mu }_k\right)$$

(9)

where $\eta$ is the basic learning rate, $h_t$ is the gradient, and $G_t$ is a diagonal matrix.

The optimal solution for the variable X obtained by the kth outer penalty iteration is Equation (10). The penalty factor is continuously updated during the iteration process as shown in Equations (11) and (12). The penalty factor is updated according to Equation (12) when Equation (11) is not satisfied.

$$X=X\left({\sigma }_k\right)$$

(10)

$${\sigma }_k{\displaystyle \sum _{i=1}^4}{\left[\mathit{max}\left(0,g_i\left(X\right)\right)\right]}^2<\epsilon$$

(11)

$${\sigma }_{k+1}=\alpha {\sigma }_k$$

(12)

The Adagrad algorithm projects the parameters to the nearest discrete point after each gradient update to satisfy the discrete nature of the engineering optimization problem.

Finally, the optimization results should be tested for feasibility, and the design parameters $X$ are used as the design parameters of the model. The main parameter settings of the optimization algorithm are shown in

Table A2. Optimization model hyperparameters.

Property	Value
Learning rate	0.02
Convergence threshold	30
Smooth item	1 × 10−8
Amplification factor	1.3
Maximum iterations	300

.

In this work, an iteration is defined as one update of the design variables $X$ using the AdaGrad update rule (Equation (8)). Each iteration corresponds to a single gradient descent step, regardless of whether the penalty factor ${\sigma }_k$ is updated. The objective function value $f\left(X\right)$ (Equation (6)) refers to the weighted sum of the four performance metrics, evaluated using the original high-fidelity simulators (HFSS and COMSOL). The convergence criterion is based on the value of $f\left(X\right)$; optimization terminates when $f\left(X\right)$ falls below a predefined threshold.

2.4. Computational Details

This work relies on numerical simulation and modeling as its important methodology. This subsection details the computational methods used and places them in the context of current numerical techniques for optical and photonic structures.

The numerical simulations presented in this manuscript were performed using two commercial software packages based on the finite element method (FEM), each tailored to the specific physics involved in the electro-optic device analysis. The FEM discretizes complex continua into small elements to numerically solve partial differential equations for approximate solutions, with a wide range of applications. For the analysis of the quasi-static electric field distribution and optical field distribution, we utilized COMSOL Multiphysics (version 6.2). The subsequent computation of the half-wave voltage used an electro-optic overlap calculation. The calculation method for half-wave voltage is consistent with [44]. The high-frequency electromagnetic parameters were calculated using ANSYS HFSS (version 19.5). The terminals were excited using wave ports, whose dimensions were set in accordance with the official standard. The simulation data has been normalized to the low-frequency (100 MHz) reference point.

While the FEM is the primary numerical technique used in this work for both quasi-static and full-wave analysis, it is important to acknowledge the broader landscape of computational methods widely adopted for modeling optical and photonic structures. A prominent alternative is the finite-difference time-domain (FDTD) method, which solves Maxwell’s equations directly in the time domain. The FDTD method is highly intuitive and powerful for analyzing broadband responses, light propagation, and scattering in complex photonic structures, though it can be computationally intensive for structures with high quality factors or fine geometrical details [45,46]. Another fundamental technique is the transfer matrix method (TMM), which is extensively used for analyzing layered media, such as Bragg mirrors, multi-layer waveguides, and thin-film filters. The TMM provides a simple and efficient way to compute reflection and transmission spectra through stratified structures. For periodic structures like photonic crystals, the scattering matrix method (SMM) is often employed to relate incoming and outgoing waves, providing a rigorous framework for understanding resonance and bandgap properties. The choice of the appropriate numerical method depends on the specific physical phenomena under investigation and the structural characteristics of the device. Numerical analysis methods, such as the spectral domain method (SDA) [47] and Dirichlet-to-Neumann (DtN) [48], are also commonly used.

3. Results

3.1. Optimization of LiNbO₃ Segmented Electrodes Modulators

We developed a model of a LiNbO₃ electro-optic modulator with segmented electrodes, on which the optimization was subsequently performed. The lithium niobate film has a thickness of 600 nm and an etching depth of 300 nm. The device structure is illustrated in Figure 3, with the corresponding parameters summarized in

Table 1. Optimization parameters of the LiNbO₃ electro-optic modulator with segmented electrodes.

Optimization Parameter	Minimum Parameter Value	Maximum Parameter Value	Parameter Meaning	Typical Value [16]
ws	20 μm	100 μm	Signal electrode width	50 μm
wg	50 μm	300 μm	Ground electrode width	/
gapin	3 μm	5 μm	Inter-electrode spacing	3 μm
gapout	7 μm	20 μm	Outer electrode spacing	15 μm
hsio2	100 nm	800 nm	SiO2 thickness	100 nm
height	1 μm	5 μm	Electrode height	1.4 μm
T	30 μm	80 μm	T-shaped electrode period	50 μm
q	0.7	0.95	Duty cycle	0.9

.

Stage 1: We established an interface between HFSS and COMSOL using Python (version 3.13), which provides a quantitative foundation for device design optimization and performance prediction. The parameter settings used in the model are listed in

Table A1. HFSS and Comsol model settings.

Property	Value
Relative Permittivity of Si	11.9
Electrical Conductivity of Si	0.01 S/m
Relative Permittivity of SiO2	4
Relative Permittivity of LiNbO3	ε11 = 44.2 ε33 = 28.5
Refractive Index of LiNbO3 (1550 nm)	$n_o$ = 2.21 $n_e$ = 2.14
Refractive Index of SiO2	1.44
Pockels Index of LiNbO3	r13 = 8.6 pm/V r33 = 30.8 pm/V

. This interface enables visual analysis of complex electromagnetic field distributions, including optical and microwave mode fields, as shown in Figure 4. To identify the key design parameters, we performed a statistical analysis using Spearman’s correlation coefficient. Figure 5a,b present the Spearman correlation coefficients corresponding to the different parameters, where the gray bars indicate no significant correlation. Based on the analysis in Figure 5a,b, we identified seven parameters as key design variables affecting modulator performance.

Stage 2: The key design variables were normalized within their feasible ranges (mapped to the interval [0, 1]), and multiple sample points were generated using Latin hypercube sampling (LHS). LHS significantly reduces the number of required samples while ensuring a uniform coverage of the sample space. Additionally, KNN interpolation is used to build a model for the global scan. Figure 5c illustrates the distribution of feasible solutions within the design space, providing an intuitive visualization of high-potential design subspaces, thereby identifying target regions for subsequent refined optimization. The plot presents a projection of hypercube sampling points across multiple dimensions, wherein red dots denote high-potential designs and gray dots represent other designs. Additionally, the diagonal elements of the figure show the density distribution of feasible solutions over each key parameter.

Stage 3: Based on the high-potential design region, we searched for the optimal solution using a hybrid optimization strategy that combines the external penalty function method and the Adagrad adaptive gradient algorithm. This approach achieves efficient gradient-based optimization of design parameters by leveraging the constraint-handling capability of the external penalty function method and the adaptive learning rate property of the Adagrad algorithm.

Figure 6 presents the optimized design obtained using the proposed method. It features a compact 5 mm modulator length. This configuration ensures minimal mismatch in both the microwave refractive index and the characteristic impedance. Consequently, it enables a simulated EO bandwidth of 100 GHz and $V\pi ·L=1 \mathrm{V}·\mathrm{c}\mathrm{m}$. These simulated outcomes underscore the method’s capability to effectively optimize multiple physical constraints for superior performance.

We set up comparison experiments: the gradient descent algorithm offers better convergence, the quasi-Newton algorithm provides a faster convergence rate, and the hierarchical optimization method serves as a commonly used benchmark. These methods were applied to iteratively solve the optimization problem within the identified high-potential design region. To ensure a fair comparison, all algorithms were configured under identical conditions. They shared the same initialization points, stopping criteria, and feasible regions. Additional hyperparameters for the comparative models are listed in

Table A3. Hyperparameters of Gradient Descent.

Property	Value
Learning rate	0.02
Gradient norm threshold	1 × 10−5
Maximum iterations	300

and

Table A4. Hyperparameters of Quasi-Newton.

Property	Value
Gradient norm threshold	1 × 10−6
Line search parameters	1 × 10−4, 0.9
Maximum iterations	300

. The results, obtained from 20 independent runs of each type, are summarized in

Table 2. A comparison of several optimization methods.

Optimization Method	Average Number of Iterations	Minimum Iteration	Maximum Iteration	Average Number of Final f(X)	Converged Runs
Proposed Method	96.7	8	238	28.5	20
Gradient Descent Algorithm	181.5	24	371	30.8	16
Quasi-Newton Algorithm	43	4	109	40.7	10
Sequential Optimization *	2096	/	/	27	/

* sequential optimization: benchmark strategy.

and Figure 7a,b.

Table 2 compares the optimization performance across different methods. Here, iteration denotes the total number of variable updates performed by the optimizer, while the average number of the final $f\left(X\right)$ reports the final converged value of the objective function defined in Equation (6). The experimental results show that the proposed method demonstrates competitive performance in both convergence efficiency and computational cost. Among the 20 independent runs, all converged successfully under the tested conditions. This outperforms the gradient descent algorithm (80%) and the quasi-Newton algorithm (50%). Although the quasi-Newton method requires the fewest average iterations (43), it exhibits a lower convergence rate and a higher average number of $f\left(X\right)$ (40.7). This indicates potential inefficiency in handling complex or noisy landscapes. In contrast, the proposed method maintains a better balance. It averages 96.7 iterations and requires the lowest average number of $f\left(X\right)$ (28.5). This suggests higher computational efficiency. These results further highlight the robustness and practicality of our approach to engineering applications. The excellent results are partly due to the effectively constrained search space (Figure 7c). This constraint is enforced through a progressively increasing penalty factor. It leads to an improved convergence profile, as shown in Figure 7d.

3.2. Effectiveness Analysis in Engineering

To evaluate the effectiveness of the proposed framework, we designed and conducted a series of systematic experiments. These experiments focused on the following two aspects:

(1)

Different Modulator Structures: The proposed framework is applied to the design of modulators with different model structures, verifying its generalization capability. Variations in etching depth and film thickness also represent different modulator configurations.

(2)

Different Engineering Conditions: We varied the matching impedance to assess the method’s adaptability under parametric perturbations.

Based on the systematic optimization conducted across four modulator structures (Figure 8), we further evaluate the performance of the proposed method on different modulators. The conventional structure in Figure 8a,c,d is consistent with the standard traveling-wave modulator layouts reported in the literature [11,16,49]. Figure 8b represents a dielectric-over-cladded electrode configuration considered as an additional validation case. This configuration is included as an additional validation case to test the adaptability of the optimization framework, rather than to claim a newly proposed fabrication process. Similar over-cladded configurations have been reported in the literature [44], where the SiO2 layer is used to tune the microwave refractive index and improve structural protection. In practical implementation, RF access remains feasible because the contact pads can remain uncovered [44] or limit the SiO₂ coverage to the modulation region only [49].

In the validation phase, we selected the LiNbO₃ material system. The design of modulators possesses well-recognized representativeness and is supported by the abundant literature data. This provides a solid benchmark for evaluating the performance of our inverse design framework. It should be noted that this choice implies our analytical conclusions are primarily applicable to the LiNbO₃ platform in quantitative terms.

The thickness of LiNbO₃ is 800 nm, and the etching depth is 400 nm. The results are summarized in

Table 3. The different modulator structures optimized with the proposed method.

Model	Optimized Results		Optimized Indicators (40 GHz)		Iteration Count
Figure 8a	ws	21 μm	Zo	50 Ω	97
	gap	10 μm	ne	2.25
	hsio2	0.1 μm	Loss	3.3 dB/cm
	height	1 μm	$V\pi ·L$	4 V·cm
Figure 8b	ws	11.5 μm	Zo	48 Ω	129
	gap	5 μm	ne	2.25
	hsio2	1.4 μm	Loss	4.6 dB/cm
	height	1.4 μm	$V\pi ·L$	2.2 V·cm
Figure 8c	ws	80 μm	Zo	55 Ω	302
	gapin	3 μm	ne	2.24
	gapout	14 μm	Loss	4.7 dB/cm
	hsio2	0.1 μm	$V\pi ·L$	1.4 V·cm
	height	1.2 μm
	T	30 μm
	q	0.9
Figure 8d	ws	16 μm	Zo	50 Ω	107
	gapITO	3 μm	ne	2.25
	gapAu	7 μm	Loss	4.5 dB/cm
	hsio2	0.1 μm	$V\pi ·L$	1.5 V·cm
	height	2.8 μm

. It lists the optimized parameters, key performance indicators, and the number of iterations required for the convergence of each structure.

All four modulator types were successfully optimized to meet target performance metrics. The conventional structures in Figure 8a,b achieved performance goals with high computational efficiency. They converged within 97 and 129 iterations, respectively. Their low microwave loss values, around 4 dB/cm, further confirm the approach’s robustness for standard designs. The segmented electrode modulator, shown in Figure 8c, demanded the highest computational cost. It required 302 iterations to navigate its more complex parameter space. However, this yielded a significant performance payoff. It achieved a notably lower $V\pi ·L$ of 1.4 V·cm among all designs. This highlights the method’s capability to handle complex geometries for superior performance. The ITO-based modulator in Figure 8d also achieved the target impedance and index. It did so with good efficiency, converging in 107 iterations. This demonstrates the method’s applicability to advanced material systems.

Based on a typical Mach–Zehnder electro-optic modulator as the benchmark model (structure shown in Figure 8b), we systematically evaluated the robustness of the proposed method under different engineering conditions. To emulate different application scenarios in engineering, two key conditions were intentionally varied:

Matching Impedance: Values were perturbed within the range of [35 Ω, 50 Ω, 75 Ω] to simulate different driving circuit conditions.

Etching Depth: Values were altered between [300 nm, 400 nm] to represent variations arising from the nanoscale fabrication processes.

The objective was to verify whether the proposed framework could converge rapidly and stably identify design parameters that satisfy predefined performance targets across these diverse external conditions. As summarized in

Table 4. The different engineering conditions optimized with the proposed method.

Match Parameters		Optimized Results				Optimize Indicators				Other Indicators
Matching impedance	Etching Depth	ws (μm)	gap (μm)	hsio2 (μm)	height (μm)	|∆Z| (Ω)	|∆n|	Loss (dB/cm)	$V\pi ·L$ (V·cm)	Iteration count
35 Ω	300 nm	23	4	1.3	2	1	0.02	4.5	1.7	43
50 Ω	300 nm	8	5	0.1	1.5	2	0.02	4.8	2.6	101
75 Ω	300 nm	6	10	1.1	2	1	0.02	3.7	4.3	139
35 Ω	400 nm	22	4	1.3	1.8	0	0.01	4.1	1.7	77
50 Ω	400 nm	11.5	5	1.4	1.4	2	0	4.6	2.2	129
75 Ω	400 nm	7	12	0.9	1.3	1	0	3.6	4.9	121

, all test cases successfully achieve convergence. This demonstrates the method’s robustness under varying external constraints. The data indicate that optimized designs under all tested conditions successfully meet the predefined performance targets. They achieve $V\pi ·L<5 \mathrm{V}·\mathrm{c}\mathrm{m}$ and EO bandwidth exceeding 40 GHz. Furthermore, the consistent convergence within a practical number of iterations (ranging from 43 to 139) confirms the computational efficiency and stability of the proposed approach, making it suitable for engineering applications characterized by parameter uncertainties.

4. Discussion

The consistent performance achieved across the diverse modulator structures and under varying engineering conditions verifies the effectiveness of the proposed framework. This successful application in complex scenarios affirms its practical value and strong adaptability to a wide range of application needs. This work not only provides an efficient automated design solution for electro-optic modulators, but also its core framework provides a useful reference for the inverse design of other complex photonic devices.

4.1. Analysis of Microwave Loss Mechanisms

In traveling-wave electro-optic modulators, microwave loss is a critical factor limiting high-frequency performance. In our optimization framework, microwave loss is directly incorporated as an objective function $f_3\left(X\right)=\left|Loss\left(X\right)\right|$.

The loss mechanisms can be broadly categorized into three types: conductor loss, dielectric loss, and radiation loss. Conductor loss arises from the finite conductivity of the electrode materials. At high frequencies, the skin effect confines the current flow to a thin surface layer. This increases the effective resistance and thus the loss. This effect is exacerbated in segmented or T-shaped electrodes. Dielectric loss originates from the complex permittivity of the materials surrounding the electrodes, such as SiO₂ and LiNbO₃. This loss is frequency-dependent. It can be modeled using the imaginary part of the permittivity or an effective loss tangent. Radiation loss becomes significant when the electrode structure supports leaky modes. Although typically smaller than conductor and dielectric losses in well-designed modulators, radiation loss can dominate in certain cases. This occurs at frequencies approaching the terahertz range or in structures with poor mode confinement.

Future work could benefit from a more explicit decomposition of loss components. This would help guide geometry selection.

4.2. Experimental Validation and Fabrication Feasibility

All results in this manuscript are numerical. We recognize the importance of addressing the feasibility and experimental viability of the proposed structure. Experimental validation is absent here because our focus was on establishing a robust optimization framework prior to costly fabrication cycles. Nevertheless, to ensure the optimized design is practically realizable, we discuss the potential fabrication methods and measurement configurations below.

The proposed modulator structure is compatible with standard microfabrication processes. Fabrication can proceed on a lithium-niobate-on-insulator wafer. Electron beam lithography or deep ultraviolet (DUV) lithography defines the waveguide and electrode patterns. The rib waveguide with 300 nm etching depth can be realized through inductively coupled plasma (ICP) etching. This offers precise control over sidewall angle and etch depth uniformity—critical factors affecting optical propagation loss. For the electrodes, a lift-off process or electroplating of gold or copper can be employed. The electrode thickness is optimized to minimize microwave loss while maintaining velocity matching with the optical wave.

For experimental characterization, the measurement configuration would involve optical testing and electrical testing. Optically, a tunable laser couples into the waveguide via lensed fibers to measure insertion loss and optical mode profiles. Electrically, a vector network analyzer (VNA) connects to ground-signal-ground (GSG) probes to extract the microwave S-parameters of the electrodes. From these, characteristic impedance, microwave index, and attenuation constant can be derived. Half-wave voltage is measured using a sawtooth modulation signal applied to the electrodes. The modulated optical output is monitored on a photodetector connected to an oscilloscope.

Furthermore, discrepancies may exist between simulated predictions and experimentally realized devices. These differences can impact device performance. Firstly, numerical simulations rely on idealized material properties and boundary conditions. These assumptions may not fully capture real-world complexities. For instance, refractive index and microwave dielectric constants are often assumed to be uniform and wavelength-independent. In reality, materials exhibit variations due to crystal orientation, fabrication-induced defects, and dispersion. Secondly, manufacturing tolerances play a critical role in determining the final device performance. Key geometric parameters—such as etching depth, waveguide width, sidewall angle, and electrode dimensions—are subject to microfabrication imperfections.

4.3. Stability Challenges

Although the current design framework has achieved satisfactory results, future research can still build on it to further deepen its foundation. While the current optimization focuses on electromagnetic performance, more complex physical fields can be incorporated into the joint optimization model to more accurately optimize device performance in the future.

Beyond electromagnetic performance, stability is critical for practical modulator deployment. Long-term thermal stability, in particular, poses significant challenges. Temperature fluctuations induce refractive index changes in lithium niobate through the thermo-optic effect. It collectively causes drift in the half-wave voltage, degradation of the extinction ratio, and shifts in the operating point, ultimately compromising system reliability. Similar thermal stability concerns are also critical in other optoelectronic devices such as organic light-emitting diodes and perovskite solar cells [50,51].

Furthermore, the multi-physics optimization model can be expanded to simultaneously account for electromagnetic, thermal, and mechanical phenomena. This holistic approach would enable the identification of design trade-offs and synergies between competing performance metrics. Such integration paves the way for next-generation electro-optic modulators that are not only high-performing but also stable and reliable under diverse operating conditions.

5. Conclusions

In this paper, an automated inverse design framework for traveling-wave electrode electro-optic modulator is proposed, effectively solving the optimization problems caused by multi-parameter coupling, nonlinear constraints and discrete processes in traditional design. The proposed framework employs four key objective functions: characteristic impedance, refractive index matching, microwave loss, and half-wave voltage. Under explicit geometric and process constraints—accounting for both the geometric feasibility and discrete manufacturing requirements—the framework automatically searches for the optimal combination of geometric parameters. It has been proven to exhibit stable convergence and good adaptability under various modulator structures and different engineering conditions, highlighting its engineering value in practical applications.