High-performance electro-optic switches with compact multimode interference and Bezier S-bend waveguides

High-speed and energy-efficient electro-optic (EO) switches are crucial for next-generation data communication systems. This paper presents a novel EO switch utilizing a multimode interference (MMI) coupler-based approach on a lithium-niobate (LiNbO3) platform. The switch is designed with Bezier S-bend waveguides, leading to a significant reduction in bending loss to 0.18 dB. This Bezier-bend EO switch shows an excess loss of 0.33 dB and crosstalk of −20.44 dB for the bar port switching, while the cross port switching exhibits an excess loss of 0.64 dB and crosstalk of −13.66 dB. Applying a 3.27 V voltage achieves a balanced splitting ratio of 29:29:29 between the three outputs. The length of this EO switch is 4.09 mm, which requires 4.1 V to create a phase shift, showing a voltage efficiency of 1.68 V.cm. This paper presents a promising pathway for a novel EO switch design and introduces improved signal management for next-generation computing systems.


Introduction
Next-generation computing systems demand high-speed data transmission, reduced signal degradation, and optimal energy utilization.This technological evolution highlights the need for efficient switches, which are crucial components for data communication networks, quantum computing, and sensing applications [1,2].Optical switches have emerged as essential solutions to meet these escalating demands and are typically categorized into two primary types based on their operational principles: electro-optic (EO) and thermo-optic (TO) switches [1][2][3][4].EO switches generally outperform TO switches in terms of lower power consumption, higher modulation efficiency, faster switching speeds, and reduced insertion losses (ILs) [5,6].As a result, EO switches realized with various material platforms have garnered significant attention.
Conventional silicon-based EO switches suffer from high power consumption and large physical dimensions due to refractive index modifications caused by optical Kerr nonlinearity resulting from two-photon absorption [6].Their substantial physical size and high-energy signal modulation present challenges for integration within densely packed photonic circuits, limiting scalability and hindering the development of efficient and high-performance systems [7].To address these limitations, lithium niobate on insulators (LNOI) emerged as a promising photonic platform, offering reduced optical loss and linear EO effects [7][8][9].Leveraging established lithium niobate (LiNbO 3 ) nanofabrication techniques, optical modes, and electric fields can be confined to the sub-micrometer scale, resulting in lower modulation voltages compared to traditional material waveguide structures [10,11].The seamless integration of LNOI with current fabrication processes enhances scalability and user-friendliness, enabling the effective deployment of high-speed LNOI photonic devices paired with CMOS drivers on a microchip [12].The distinctive attributes of LiNbO 3 have significantly improved EO switching performance, efficiency, and versatility compared to silicon-based alternatives.
Selecting an appropriate switch architecture is essential for designing advanced EO devices.Utilizing a LiNbO 3 -based EO switch with either a directional coupler (DC) or a Mach-Zehnder interferometer (MZI) has become increasingly popular due to their distinct characteristics [13,14].Although the MZI is commonly employed to construct high-performance optical switches with wide optical bandwidth and low driving voltage, random phase errors arising from fabrication complexity and inherent polarization sensitivity make the DC a more appealing choice for modern optical switches [5,7].Employing strategic manipulation in the design process is crucial to position the DC as a viable alternative to the MZI for versatile switching functionality (SF).When modifying the DC configuration for switching applications, the issue of loss holds significant importance, as it directly affects performance [15].Mao et al suggest integrating DC arms with adjustable widths using adiabatic tapering [16].Adiabatic tapering in DC arms demonstrates a uniform SF and IL between 0.3 and 1 dB.Additionally, integrating sub-wavelength gratings (SWGs) into the DC component has enabled significant broadband SF [17] with low IL and acceptable spectrum stability, maintaining deviations from the intended splitting ratio of less than 5% [18].While this strategy may help lower overall loss, it fails to effectively address bending loss, which occurs when light escapes due to abrupt changes in the waveguide shape.In the context of DC-based optical switches, strategically integrating Bezier bends can be a more effective method for mitigating bending losses [19].These bend structures feature continuously changing curvature, allowing for smooth mode transition throughout the waveguide bends.Bogaerts and Selvaraja introduce a single-mode waveguide comprised of a Bezier (adiabatic) bend, demonstrating its efficiency in mitigating bending loss [20].Adiabatic manipulation ensures the preservation of the light's mode profile, thereby reducing the mode mismatch and bending losses.This strategic integration makes the DC a more favorable choice for modern EO switches compared to alternatives such as the MZI.In telecommunications and optical sensing, such low-loss switches play a pivotal role in signal routing and switching, enabling simultaneous monitoring of multiple optical signals and facilitating efficient data transmission [21].However, achieving versatile SF with the Bezier bend-enhanced DC necessitates incorporating numerous outputs in the switch configuration's bar or cross port.Therefore, further research is necessary to develop compact and efficient EO switches with multiple outputs that can route optical signals with minimal loss and distortion, thereby advancing signal processing technologies.
This paper demonstrates a Bezier-bend EO switch, that exhibits a versatile SF, offering a compact and efficient solution for routing optical signals in photonic integrated circuits.We introduce an EO switch based on multimode interference (MMI) couplers on the LNOI platform with silicon dioxide (SiO 2 ) as the substrate and LiNbO 3 as the core waveguide material.We replace the traditional S-bend structure with a Bezier bend to minimize the bending losses.This device demonstrates switching properties between the bar and cross ports or equal splitting, depending on the applied voltage.Integrating these switches into photonic integrated circuits facilitates the development of advanced optical computing and signal processing systems.

Theory and simulation setup
The rapid evolution of integrated optics has driven the development of innovative components that facilitate efficient light manipulation within the photonic circuits.This theory elucidates the creation of a dynamic DC by synergistically integrating Bezier bend waveguides and MMI structures.Incorporating MMI with DC improves the switching performances of the DCs.These DCs inherently function as wavelength-dependent power-splitting devices.Through precise selection of the two DC lengths and time delay, we can adjust the device's free spectral range (FSR) and center wavelength to achieve a wavelength-independent response, exemplified by considering the input E in in equation ( 1) and the corresponding transfer function [22] as follows: [ here κ represents the coupling coefficient, τ denotes the through-coupling coefficient, while ϕ L and ϕ R correspond to the induced phase shifts of individual pulses propagating through the left and right arms, respectively.Given the assumption of minimal DC loss and |E| 2 input = 1, the field coupling coefficient κ can be equated to the electric field of the DC cross port [23], as expressed in equation (2): where, as L is the length of the straight waveguide, the resulting induced phase is ϕ = βL, where β = 2π n eff λ signifies the propagation constant, n eff represents the effective index of the guided mode, and λ corresponds to the wavelength.Consequently, when considering two identical cross-sections of waveguides, the disparity in phase between the two arms can be depicted using equation (3): where ∆L signifies the discrepancy in path length.The suggested couplers deviate from conventional straight waveguides, resulting in distinct effective indices for each propagating mode.The standard S-bend arrangement encompasses circular arcs with uniform radii throughout the core.The equation of traditional S-bend [24] is represented by equation (4): , and y (t) = y 0 + b here x 0 and y 0 represents the starting point of the S-bend, a and b determine the horizontal and vertical displacement of the bend section.At the same time, t ranges from 0 to 1, representing the parametric value along the curve.We know that for the Euler curve, the curvature K displays a linear increment in correlation with the propagation length x, following this pattern in equation ( 5): here, k denotes the scaling factor, which is typically assumed to be 1 in most Euler curves [23,24].Now, in the case of the Bezier S-bend, when we introduce an adjustable parameter n, the relationship becomes as follows in equation ( 6): The bend parameter, n, significantly influences the geometry of a flexible curve.Adjusting the value of n yields an array of curves with diverse geometries.Considering the curvature definition, the associated differential expression for the curvature at a point on the curve is outlined in equation (7): In this equation, θ (x, n) signifies the angle of the tangent direction at a bending point, and x represents the path length along the bend.For a given curve point P, where the path length from the initial point O to point P is denoted as x p .The tangent direction angle at point P can be determined through integration.When k equals 1, and K (x) = x n , integrating within the interval 0 to x p results in the following expression [25] in equation (8): Subsequently, the coordinates of the waveguide can be represented as integrals involving the cos and sin part of the tangent angle, following equation (9): ) dx, and y here, A stands for constant dictating the curve's size.The integrals themselves cannot be analytically resolved; hence, a series expansion technique can be employed to derive a solution with sufficient precision [24].A cubic Bezier curve [26] is generally described by equation ( 10) as: where (x 0 , y 0 ) and (x 3 , y 3 ) represent the initial and final coordinates of the bend, and (x 1 , y 1 ) and (x 2 , y 2 ) are the control points influencing the curvature.The curvature radius can be calculated using these equations, aiding in designing smooth Bezier bends that minimize radiation losses.For the Bezier bend arm, the transfer matrix [27] T B can be represented by equation ( 11) as: here ϕ B represents the phase shift introduced by the Bezier bend arm due to its curvature and design parameters.The MMI couplers showcase the characteristic integral correlation among the propagation constants of various modes.This relationship results in the self-imaging characteristics of the input signal as it traverses through the length of multimode waveguides [28].The transfer matrix for the MMI device, frequently employed for power splitting and combining within integrated photonics, is depicted by equation ( 12) as follows: here t MMI and r MMI are transmission and reflection coefficients of the MMI structure, respectively.The overall transfer matrix T for the entire DC, considering the cascading of the arms, is given by equation ( 13): The matrix encapsulates how the phase shifts and power distributions from the individual arms interact to govern the overall behavior of the coupler.The suggested coupler operates with two mutually perpendicular modes, and the unitary matrix operates on these optical modes.During the coupling region, the input light divides, and the output power collected by the through and cross output [29] can be expressed as per equation ( 14): ) , In this equation, P in represents the input power, L g stands for the separation gap and L c signifies the coupling length required for total power transfer.The value of the coupling length is determined by equation ( 15): here n even and n odd denote the effective indices of the even and odd symmetric modes [15].The RSoft CAD BPM software is used for device simulation, leveraging the beam propagation method (BPM) to design and simulate integrated switching systems.We develop the DC waveguide structure within the CAD environment using built-in components.Subsequently, the waveguide's refractive index profile and EO coefficient are defined, followed by the establishment of electrode configuration and voltage via the Electrode Editor.
Transfer matrices are pivotal in simulating a DC-based EO switch using the RSoft CAD BPM solver.A comprehensive model is created by defining these matrices to encapsulate DC, waveguide, and EO behaviors.These matrices accurately depict light propagation, power division, and phase modulation.The transfer matrix elements are integrated into the layout, replicating real-world configurations and enabling dynamic simulations of the switch behavior and performance analysis across the input and operational conditions.This approach empowers precise parameter refinement, optimizing switch functionality and forming efficient photonic devices.

Waveguide width optimization
We investigate the core width variation from 3 µm to 9 µm as a function of the output intensity at 1.55 µm, as shown in figure 1.We found that the waveguides demonstrate maximum output intensity within the core width range of 5.5 µm to 6.5 µm.The normalized output intensity consistently exceeded to 0.88 arbitrary units (a.u.).We obtain the maximum output intensity with a value of 0.91 ± 0.01 a.u. at the core width of 6 µm, and we confirm that the device ensures optimal performance at the core width of 6 µm.To achieve design simplicity and coherence, we decided to maintain uniformity in both the width and height of the component.Hence, we set the core width to 6 µm, resulting a symmetrical square cross-sectional profile of 6 × 6 µm 2 for this waveguide.

Bezier S-bend
This section introduces and compares Bezier S-bend waveguides with traditional S-bend waveguides.The traditional S-bend waveguide maintains a constant core width along its propagation path, and the curvature radius of the S-bend waveguide induces bending loss and mode mismatch.Conversely, the Bezier S-bend waveguide gradually varies its width, reducing modal mismatch and bending loss.The superior performance of the Bezier S-bend over the conventional design is attributed to reduced mode mismatch and bending loss,  facilitated by a smoother mode transition enabled by width multiplication, leading to lower overall loss and higher intensity within the waveguide.We systematically varied the scaling factor from 1 to 1.8, achieving maximum output at a scaling factor of 1.3.The proposed Bezier S-bend waveguide includes width variations as follows: W (6 µm) in the starting section, W × 1.3 (scaling factor) in the middle section, and W (6 µm) in the final output section.This optimization technique ensures efficient light confinement at the Bezier bending section, thereby enhancing the overall transmission efficiency of the waveguide.The loss comparison of the Bezier S-bend and traditional S-bend is shown in figure 2, and the output intensity of the Bezier S-bend (0.90 ± 0.06 a.u.) is better than that of the conventional S-bend (0.87 ± 0.07 a.u.) in a wavelength range of 1.30 µm to 1.70 µm.Notably, the traditional S-bend yields an output normalized intensity of 0.89 a.u., whereas the Bezier S-bend excels with a value of 0.93 a.u. at 1.55 µm.The bending losses of the traditional S-bend demonstrate a value of 0.62 ± 0.34 dB, while the Bezier S-bend exhibits a lower value of 0.46 ± 0.30 dB.Specifically, the conventional S-bend experiences a bending loss of 0.48 dB, whereas the Bezier-S bend demonstrates a notably reduced loss of 0.30 dB.This detailed analysis underscores the distinct advantages of the Bezier S-bend configuration in terms of enhanced output intensity and minimized bending losses.

MMI splitter
We present a schematic of 1 × 2 MMI waveguide splitters, as depicted in figure 3(a).The characteristic of this MMI splitter is based on the self-imaging principle [30].The width (W mmi ) and length (L mmi ) of the multimode waveguide are the main parameters to optimize this device with minimum loss.Another crucial factor is the core spacing in the MMI waveguide, which refers to the gap between the outputs on either side of the MMI central position.Customizing this core spacing is essential for MMI splitter optimization.Maintaining a uniform intensity distribution within the MMI splitters depends on meticulous control of core spacing, as any alteration affects the splitting ratio.Notably, the core spacing remains consistent across the MMI splitters.In configuring the output cores, we align them with the multimode waveguide's width while maintaining uniform spacing between the outputs, denoted as -W mmi /3.2 and W mmi /3.2, as visually represented in figure 3(a).This strategy dictates a core spacing of 6.9 µm between the two outputs for the splitter.We conducted several numerical simulations to determine the optimal dimensions of this multimode waveguide.During these trials, we systematically altered both parameters while monitoring the corresponding output intensity.Subsequently, we graphically represented the output intensity as a function of W mmi and L mmi simultaneously, as depicted in figure 3(b).This figure displays a range of color combinations corresponding to varying intensity levels, with the intensity scale indicated along the vertical axis.Red denotes the maximum intensity, whereas purple signifies the minimum intensity.The span of W mmi is from 10 µm to 30 µm, while the L mmi ranges from 100 µm to 600 µm.From the color bar, it is evident that the red color region corresponds to W mmi in the range of 15 µm to 25 µm and L mmi in the range of 250 µm to 500 µm, exhibits the highest intensity.There are two notable areas of heightened intensity; one corresponds to L mmi in the range of 250 µm-300 µm and W mmi in the range of 17 µm-18 µm, while the other spans L mmi in the range of 410 µm-425 µm and W mmi in the range of 20 µm-24 µm.However, it is essential to note that the first region poses challenges in achieving equal splitting ratios, and the width limitations hinder accommodating two output cores effectively.Following comprehensive preliminary inquiries, an optimized configuration for the MMI splitters is determined, resulting in the core spacing of 6.9 µm, W mmi of 22 µm, and L mmi of 419 µm.Upon determining the parameter configuration, we simulated the rib MMI structure, where the core, cladding, and cover materials are LiNbO 3 , SiO 2 , and air.The outcomes of these simulations are illustrated in figure 4, offering comprehensive insights into the MMI splitter's performance.In figure 4(a), the field intensity distribution between the two outputs showcases an approximately equal intensity.Figure 4(b) presents a modal analysis of the rib waveguide, shedding light on the modal properties of the MMI splitter.We explored the full width at half maximum (FWHM) of the MMI waveguide by measuring the wavelength range where the mode amplitude reaches half of its peak value.We evaluated the FWHM across the various phase angles (φ) to analyze the response of the MMI, depicted in figure 4(c).At a wavelength of 1.55 µm, our proposed splitter demonstrates FWHM values of 8.70 and 21.81 for φ = 0 and φ = 90•, respectively.
Next, we investigate the output intensity as a function of the wavelength of this splitter, with the normalized output intensities are shown in figure 5(a).The MMI splitter demonstrates a splitting ratio between 37:37 and 42:42 at a wavelength range of 1.30 µm-1.70 µm, reaching a peak of nearly 44:44 at 1.55 µm.We explore light propagation through the MMI-based waveguide splitters, particularly examining the multimode waveguide's length and its self-imaging properties at 1.55 µm, illustrated in figure 5(b).This depiction introduces the concept of light propagation guided by the self-imaging length [31], represented with two distinct colors to differentiate the output fields.The overlapping of these fields signifies that both outputs possess an equal field intensity splitting ratio.Additionally, figure 5(b) highlights that within the MMI splitter, the uniform output splitting ratio aligns with the completion of the periodic field interference at the end of the self-imaging length, an anticipated outcome of the MMI configuration.The self-imaging length for the MMI splitter is 619 µm at 1.55 µm.The sharp intensity fluctuations in the MMI splitter are due to mode coupling-induced interference effects and modal dispersion resulting from different propagation speeds of guided modes.Figure 5(c) provides insights into the splitter's output efficiency across  the 1.30 µm-1.70 µm wavelength spectrum.The results reveal a fluctuation in efficiency, ranging from 73% to 83%, with a peak efficiency of 88% observed at 1.55 µm.This comprehensive analysis underscores the remarkable capabilities of the MMI splitter, demonstrating consistent and wavelength-independent splitting ratios, emphasizing its efficiency across varying wavelengths.

EO switch with Bezier bend
The EO switch design comprises a sequence of integral components working in tandem to enable efficient signal control.In particular, this configuration includes two essential Bezier S-bends, a 1 × 2 MMI splitter, and a strategically positioned electrode that induces a phase change in the waveguide arm.The schematic of the EO switch-based coupler is shown in figure 6.The working mechanism of this Bezier bend switch is as follows: an incident light enters through the input port and propagates along the input waveguide.Then, the input light propagates into the distinctive Bezier-S bend, and its trajectory is redirected.The coupling region, indicated by a distinctive black dotted box, plays a vital role in coupling the light effectively.We use an electrode/heater element at a suitable position to influence the phase of the propagating light.The voltage variation applied to the electrode induces phase modulation, enabling the light to follow distinct paths through either the bar or cross port of the switch.This configuration leverages phase manipulation to achieve desired signal routing outcomes with efficient EO switching.
The separation gap at the coupling region emerges as a critical parameter influencing the efficacy of the EO switch.To comprehensively analyze its impact, we systematically varied the gap within the coupling region, ranging from 6 µm to 9 µm.The results of these investigations are depicted in figure 7, where the output intensity is plotted against the separation gap. Figure 7 reveals essential insights into the relationship between the gap and output performance.Notably, a separation gap of approximately 7 µm, with a precision of ±0.4 µm, yields a normalized output intensity exceeding 0.80 a.u.Moreover, a noteworthy observation is made at the 7 µm separation gap, where the normalized output intensity significantly increases to nearly 0.92 a.u.These findings demonstrate the importance of separation gap in determining the switch's performance, providing crucial guidance for optimizing its operational parameters.
The fabrication procedure for the proposed EO switch design involves several crucial steps.Initially, preparation of a SiO 2 wafer substrate is required.A layer of LiNbO 3 will then be deposited onto the substrate using techniques such as sputtering or chemical vapor deposition [32].Subsequently, waveguide patterns are defined through photolithography, followed by etching of the LiNbO 3 layer to form waveguide structures and the Bezier S-bend using reactive ion etching [33].Integration of the coupler region can be achieved by depositing additional LiNbO 3 or through etching.Metal electrodes are then deposited onto the LiNbO 3 layer for voltage application, potentially employing methods electron beam evaporation or sputtering [34].The introduction of air as the cover material is accomplished by creating an air gap above the waveguides by controlled deposition and removal of sacrificial material.Individual switches on the chip must be isolated to prevent interference, and the devices must be tested and characterized to assess their performance [35].Finally, the devices are packaged for integration into larger photonic circuits or systems, ensuring proper alignment and protection.

Results and discussion
After determining the optimum configuration of the EO switch-based compact splitter, we investigate its optical characteristics.Figure 8(a) provides a multi-dimensional perspective of the intensity peak distribution of the EO switch performances.This visual representation illustrates the intricate journey of light, commencing from the origin (0,0) coordinate.It traverses through the input waveguide, navigating the Bezier S-bend waveguide, effectively coupling in the coupling region, and eventually returning to the bar port into the second Bezier S-bend waveguide.Notably, the dark blue ripples surrounding the waveguide indicate losses in the electromagnetic field.Figure 8(b) demonstrates the rib waveguide's mode profile, revealing the spatial distribution of the guiding mode.Figure 8(c) provides the intensity distribution through various ports, including the bar port, cross port 1, and cross port 2. This analysis reveals that light enters with a normalized intensity of 1 a.u., traversing the coupling region and returning to the bar port with an intensity of 0.92 a.u.The cross ports, on the other hand, exhibit significantly lower normalized intensities (both are lower than 0.001 a.u.), underscoring the effective control achieved through this design.Figure 8(d) introduces the intensity variation at the bar port, showcasing a range of 0.76 ± 0.18 a.u., with an intensity of 0.92 a.u. at 1.55 µm.Similarly, for cross port 1, an intensity range of 0.07 ± 0.02 a.u. is observed, while cross port 2 demonstrates an intensity range of 0.06 ± 0.05 a.u.These results underscore the switch's consistent and reliable performance across various wavelengths, confirming its potential for effective EO signal manipulation.
We employ a heater to modulate the refractive index of the waveguide material, leveraging the EO effect.In our simulation process, we define electrodes and specify materials with EO parameters: r 13 = 9.6 pm V −1 , r 22 = 6.8 pm V −1 , and r 33 = 30.9pm V −1 , d 31 = −4.8pm V −1 , d 33 = −25 pm V −1 [7].We meticulously detail electrodes in the simulation setup for accuracy, as illustrated in figure 6.Additionally, we configure the index profile type to remain inactive, ensuring its sole contribution to computing the DC electric field without directly altering the index profile of the structure.Furthermore, we precisely set the control parameter to the applied voltage (V) to finely control the switch behavior.We then calculate the half-wave voltage (V π ), a vital parameter for investigating the efficiency of the EO switch [36].V π represents the required voltage to change the phase by π radians.Its value is determined by a combination of factors, notably influenced by the design of optical components and electrodes [7,[33][34][35][36].In the absence of applied voltage, the light remains confined within the bar port of the switch.However, upon applying an external voltage, the light's path diverges based on the voltage magnitude, with the input light transitioning through either the bar or cross port or a combination of both.To illustrate this interplay, figure 9 is introduced, which sheds light on the voltage-dependent behavior.In figure 9(a), it is apparent that a phase shift in the light requires a voltage of approximately 4.1 ± 0.02 V.The subfigure in figure 9(a) provides an insightful demonstration of how the index profile and electric potential alter along the y-plane due to the applied voltage by the heater.This figure further outlines three distinct scenarios determined by the applied voltage.Firstly, the light may remain solely within the bar port.Alternatively, it could split equally across all three output ports or exclusively traverse the two cross ports.Figure 8 corresponds to the bar port switching, while figures 9(b) and (c) depict equal switching and cross-port switching, respectively.All three ports obtain nearly identical splitting ratios for an applied voltage of 3.27 ± 0.02 V (figure 9(b)).In this case, all cross ports exhibit an output intensity of approximately 0.29 a.u.On the other hand, when the applied voltage reaches 4.1 ± 0.02 V (figure 9(c)), the light selectively traverses into the two cross ports, bypassing the bar port.This results in an output intensity of about 0.40 ± 0.02 a.u.
The length of this switch is precisely 4.09 mm, a compact dimension that aligns with the demand for miniaturized photonic components.To induce the crucial first-order phase shift necessary for efficient switching, a voltage of 4.1 V is applied.This voltage translates to an impressive voltage efficiency of 1.68 V.cm.These findings collectively underscore the switch's inherent switching property, facilitated by the innovative Bezier S-bend induced EO design.We develop the three distinct voltage settings to provide a comprehensive view of the field profile and output intensity across various scenarios: bar port propagation, cross port 1 propagation, and cross port 2 propagation.This meticulous analysis introduces data tables, which illustrate the diverse combinations of field profiles for different wavelengths and voltages.By scrutinizing these data, we gain valuable insights into the intricate interplay between voltage, wavelength, and the resulting optical behaviors within each propagation scenario, as shown in table 1.
We investigate the performance of the EO switch by calculating the excess loss [37], which significantly impacts the overall effectiveness of this switch.Another crucial aspect is the occurrence of crosstalk, a phenomenon in which the input signal intrudes into the output signal path, resulting in signal degradation, interference, and distortion [38,39].To achieve high-quality output signals, reducing crosstalk becomes crucial, which involves minimizing the presence of unwanted signals at the output of the switch, thereby ensuring a desired and untainted transmission.Moreover, another crucial metric is output efficiency, which encapsulates the ability of the switch to effectively deliver the desired output signal, gauging its overall  efficiency in channeling light to the designated output ports.By judiciously considering and optimizing these variables, the EO switch can achieve its desired performance characteristics and seamlessly function in intricate optical networks.To ensure a comprehensive performance analysis, we calculated the loss function and efficiency by treating two cross ports (cross port 1 and cross port 2) as a single entity.We investigate the excess loss of this EO switch varying the wavelength, and the loss results are presented in figure 10(a), where the excess loss showcases a noteworthy variation, ranging from 1.13 dB to 0.21 dB across the spectrum of 1.30 µm-1.70 µm wavelength for the bar port.Notably, the excess loss stands at 0.33 dB at 1.55 µm, indicating an optimal operational point.Similarly, this analysis extended to the cross port, where excess loss demonstrates a varying trend, ranging from 1.73 dB to 0.73 dB over the same wavelength range.At the crucial wavelength of 1.55 µm, the excess loss for the cross port is 0.64 dB.We evaluate the crosstalk of this EO switch, as shown in figure 10(b).Here, the variation of crosstalk is revealed, with values ranging from −7.11 dB to −12 dB across the wavelength spectrum of 1.30 µm-1.70 µm for the bar port.At 1.55 µm, the crosstalk of the bar port is −20.44 dB, underlining the switch's ability to maintain signal isolation.The cross port also exhibits a similar pattern, with crosstalk varying from −12.66 dB to −9.07 dB for the same wavelength range, and at 1.55 µm, it shows −13.66 dB of crosstalk.
For the efficiency analysis, figure 10(c) encompasses a range of scenarios showcasing output efficiency trends.When the light solely traverses through the bar port, the efficiency remains consistent at 86 ± 9% over the wavelength range of 1.30 µm-1.70 µm, culminating an efficiency peak of 93% 1.55 µm.Furthermore, when the light undergoes equal splitting between the outputs, the efficiency showcases variation, with values ranging from 79 ± 9% for the wavelength range of 1.30 µm-1.70 µm and a peak efficiency of 88% at 1.55 µm.Lastly, the cross-port efficiency is 77 ± 10% across the 1.30 µm to 1.70 µm spectrum, with a peak efficiency of 86% at 1.55 µm.This rigorous evaluation unveils the EO switch's diverse performance aspects, corroborating its proficiency in multifaceted operational scenarios.We conducted a comparative analysis of the performance of various EO switches, comparing them with our proposed design and presented the findings in table 2.

Conclusion
In summary, we have presented an MMI coupler-based EO switch realized with a TFLN platform and investigated the efficiency and functionality.We developed this EO switch using unique Bezier S-bend waveguides with minimal bending loss, specifically 0.18 dB.The EO switch's performance reveals an excess loss of 0.33 dB and crosstalk of −20.44 dB for bar port switching, while cross-port switching showcases an excess loss of 0.64 dB and crosstalk of −13.66 dB.When the applied voltage is 3.27 V, this EO switch demonstrates an equal splitting ratio of 29:29:29 among the output ports, exemplifying its adept signal distribution management.Notably, our developed compact EO switch has a length of 4.09 mm and requires 4.1 V for a first-order phase shift, achieving a commendable voltage efficiency of 1.68 V.cm.Therefore, this EO switch underscores a promising pathway for pioneering EO switch design to enhance signal manipulation in next-generation computing systems.

Figure 2 .
Figure 2. (a) Schematic comparison of traditional and Bezier S-Bend.(b) Wavelength-dependent variation in output intensity for both S-Bends.(c) Wavelength-dependent bending loss analysis.

Figure 3 .
Figure 3. (a) Schematic of an MMI splitter.(b) Optimization of W mmi as a function of L mmi .

Figure 4 .
Figure 4. (a) 3D intensity pattern of the MMI splitter.(b) Modal analysis of the MMI cross-section.(c) FWHM analysis from the polar projection of far-field intensity.

Figure 5 .
Figure 5. (a) Variation in output intensity with wavelength for both outputs of the MMI.(b) Normalized intensity along the propagation direction of the MMI.(c) Efficiency variation of the MMI output with wavelength.

Figure 6 .
Figure 6.Schematic of an EO switch.(a) 3D view of the switch.(b) Top view with the parameters indication.(c) Cross-section of the switch.

Figure 7 .
Figure 7.The Separation gap optimization as a function of the total output intensity.

Figure 8 .
Figure 8. Multifaceted analysis of switch performance and characteristics.(a) Simulation result: optical field distribution of the switch.(b) Mode distribution (x-y plane).(c) Light propagation: outputs along the propagation direction.(d) Intensity variation: function of wavelength.

Figure 9 .
Figure 9. Voltage-dependent characteristics of the switch.(a) Response change of the EO switch with applied voltage, including variations in the index profile and electric potential due to applied voltage.(b) Light propagation for equal switching under identical splitting ratios.(c) Light propagation for cross-port switching subset, illustrating 2D light propagation along the xz direction.

Figure 10 .
Figure 10.Performance metrics variation with wavelength (a) variation of excess loss with wavelength.(b) Crosstalk variation with wavelength.(c) Output efficiency across different wavelengths.

Table 1 .
The interplay between voltage, wavelength, and the optical behaviors of this device.

Table 2 .
Comparative analysis of EO switch performance a. CSb: conventional S bend, ASb: asymmetric S bend, BSb: Bezier S bend, SOI: silicon on insulator. a