1.

Introduction

Active and tunable silicon photonic devices have received tremendous attention over the past few decades. Photonic devices could combine into microelectromechanical systems (MEMS) to make reconfigurable devices and address several functions simultaneously for different applications such as optical network components and sensors.^1–5 Mechanical tuning of photonic devices is a favorable method for controlling photonic circuits due to its compatibility with fabrication.⁶

In the last few decades, MEMS have represented a prosperous technology for miniaturizing devices. Also, their well-developed fabrication technology could be integrated with other existing platforms such as silicon photonics. With MEMS technology, reconfigurable photonic-integrated circuits (PICs) allow for robust and low-power platforms for tunable photonic devices such as phase shifters, couplers, switches, or resonators.⁷ The main MEMS actuation methods are electrostatic, electromagnetic, piezoelectric, and electrothermal.⁸ Choosing the best way of actuation depends directly on the PIC design and its requirements. Three primary criteria for actuation method selection are (a) required range of traveling, (ii) demanded force magnitude, and (iii) power consumption limitations.^8,9 Thus, according to the required force and displacement, a range of suitable actuators can be selected, and the final option would be determined based on other criteria such as power consumption and footprint.

Electrostatic actuation is the most commonly used actuation method for PICs because of its advantages, such as power consumption, speed, and ease of control. More importantly, the simplicity of combining fabrication methods makes electrostatic actuators more compatible with silicon photonics devices.¹⁰ Electrostatic actuators usually utilize two charged electrodes, with a dielectric gap in between, that electrostatically attract each other to make displacement. The created force directly relates to the dielectric constant of the gap and electric field between electrodes. This method’s advantages include excellent scaling, low power operation, PIC compatibility, wavelength insensitivity, low crosstalk, fast switching, and simple fabrication.^11,12 Also, PICs are mainly fabricated with semiconductors such as silicon that can be either a conductor electrode or a reliable mechanical layer as a suspended structure.

In this study, we focus on the comparative analysis of electrostatic actuators within the framework of thicker silicon platforms, which are chosen for their superior mechanical robustness. For thinner silicon layers, more specific considerations would be required. Thinner electrode layers, such as those using the silicon waveguide layer for MEMS actuators, introduce new constraints and effects. For example, devices using the same layer as the waveguide core for the electrodes, such as silicon-on-insulator (SOI) silicon photonics with thin silicon layers, may face specific challenges related to stiffness and fringing fields. These issues necessitate careful design considerations to maintain the mechanical integrity and functionality of the devices.

In our study, we focused on comparing three primary types of electrostatic actuators: parallel plate (PP), rectangular (REC), and triangular (TRI) designs. These designs represent the key categories of electrostatic actuation:

This comparative analysis aimed to highlight the pros and cons of each design in terms of force density, defined as the amount of force per unit area of the actuator’s footprint, and force magnitude, which refers to the total force generated by the actuator. By investigating these aspects alongside the traveling range, we provide a comprehensive understanding of the suitability of different actuator designs for photonic applications.

There are studies that compare specific designs within the area overlap category, particularly focusing on combinations of gap-closing and area-overlap mechanisms. A notable example is the study by Phuc Hong Pham et al.,¹³ which presents a trapezoidal-shaped electrostatic comb-drive actuator (TECA) that outperforms the conventional rectangular-shaped actuator (RECA) in both force magnitude and displacement. TECA achieved up to 2.2 times greater displacement than RECA, thanks to optimized comb finger geometry. Incorporating these findings into our paper broadens the perspective on alternative comb finger shapes, enhancing the discussion on electrostatic actuator performance in silicon photonics.

Although this study focuses on comparing three primary electrostatic actuator designs of PP, REC, and TRI, it is important to acknowledge that other finger designs fall within the third category (area overlap and gap closing). Notable examples include the step finger structure, as explored by Ref. 14, curved finger geometries, as well as the triangular step geometry discussed by Refs. 1516.–17 and the double-tilted finger shape.¹⁸ These designs can influence force magnitude and traveling range.

However, in this work, we selected the TRI design as a representative of the area overlap and gap-closing category to provide a baseline comparison among the main types of electrostatic actuators. It is also worth noting that other studies have not considered the footprint parameter to be a figure of merit. Future research could explore these alternative designs to further enhance our understanding of their performance characteristics and optimization potential in terms of force density and traveling range.

The use of electrostatic actuators also has drawbacks. Parallel plate (PP) actuators suffer from a nonlinear force-displacement relationship leading to pull-in instability, a limited traveling range, a higher risk of mechanical failure due to stiction, and higher voltage requirements for significant displacement.^12,19 Rectangular (REC) actuators are limited in force density due to a constant electrode gap and have lower efficiency in space utilization when high force (on the order of a few mN) is required for some photonics device actuation.²⁰ Also, it faces potential performance reductions at higher traveling ranges due to side pull-in issues and requires a complex design for making very large actuators to generate a few mN force to ensure alignment and gap consistency. Triangular (TRI) actuators, while offering a hybrid approach, present increased fabrication complexity due to the need for precise triangular profiles, a larger footprint required for high force designs with smaller angles, and nonlinear behavior and necessitate optimization to balance force density and traveling range efficiently.^18,21–23 By addressing these key drawbacks, our comparative analysis aims to highlight the pros and cons of each design in terms of force density and traveling range, providing a comprehensive understanding of photonic applications.

2.

Mechanical Requirements of Actuators for Photonic Applications

In the ever-evolving landscape of PICs, some applications necessitate actuators calibrated for low traveling ranges and variable force magnitudes. These requirements cater to the precise tuning and stabilization of photonic elements, where excessive movement is not desirable, and the actuation must be executed with controlled and moderate force.

Tunable photonic devices such as couples, switches, and resonators, as detailed in Refs. 1 and 24, incorporate MEMS/NEMS actuators to enable the mechanical tuning of photonic devices. The emphasis here is on precision over a minimal range, ensuring that devices such as tunable lasers and filters can swiftly and accurately adapt to the required optical characteristics.

Silicon photonic-integrated devices and circuits, highlighted in Refs. 7 and 25, use MEMS technology to bring about reconfigurability in optical networks. Actuators in this space provide moderate force to facilitate the tuning of resonators and couplers over short ranges, which is essential for dynamic network management and signal processing.

In the area of optical MEMS for telecommunication, as per Refs. 10, 26, and 27, the importance of moderate force magnitude is underscored in applications such as optical switches and variable optical attenuators. These devices rely on actuators capable of precise micro-movements to modulate the light path or intensity, fundamental for maintaining signal integrity in optical communication systems.

Furthermore, tunable coupling regimes of silicon microdisk or ring resonators, as explored in Ref. 27, necessitate actuators that can finely adjust the coupling strength between resonant structures without requiring extensive traveling range. This allows for fine-tuning the photonic devices to desired resonance conditions, which is crucial for filtering and wavelength multiplexing applications.

Finally, programmable photonic circuits, as envisioned in Refs. 28 and 29, are an emerging area where the integration of MEMS with photonic circuits allows for the agile reconfiguration of optical pathways. The moderate force magnitude and low traveling range are critical here, as the actuators must execute minute adjustments to waveguides and resonators, enabling a new degree of flexibility in photonic circuit functionality.

These applications, from reconfigurable optical networks to dynamic photonic circuits, underline the tailored requirement for actuators that combine various force magnitudes with low traveling ranges. This combination is pivotal in the precise and delicate control of photonic components, ensuring high performance and reliability in silicon photonics. Specific silicon photonics applications require high-force actuators over minimal traveling ranges. These applications, often characterized by the need for fine, sub-micrometer adjustments, demand actuators that can provide precise control without the necessity for large-scale movements. In a study detailed in Ref. 30, a novel approach utilizing nanoelectromechanical systems (NEMS) in silicon waveguides demonstrates the effectiveness of electrostatic actuation in creating tunable photonic devices. This method leverages the concept that applying a voltage across a freestanding slot waveguide alters the slot width, inducing a change in the effective refractive index and consequently modulating the phase of light. The electrostatic force generated between the beams of the waveguide adjusts the slot width in response to the applied voltage, providing a mechanism for dynamic, precise control over the optical properties of the waveguide. This innovative technique highlights the significant impact of mechanical alterations, even on a minuscule scale, on the optical characteristics of silicon waveguides, showcasing the potential for highly responsive and compact photonic devices. As discussed in Refs. 25 and 31, MEMS gratings are another application in which high force magnitude is required to fine-tune the grating positions, affecting the diffraction and propagation of light through the photonic circuit with minimal physical displacement.

Nano-opto-electro-mechanical systems (NOEMS), highlighted in Ref. 32, are at the forefront of integrating mechanical actuation with optical and electrical components at the nanoscale. Here, the force magnitude must be considerable to influence the mechanical properties of nanoscale devices without necessitating significant travel ranges. Finally, the strain engineering of germanium nanobeams by electrostatic actuation, detailed in Ref. 33, requires actuators that can precisely apply force to modulate the mechanical strain, directly translating to changes in optical properties. The actuation must be highly controlled and localized, emphasizing a high force magnitude over a short range.

These applications, ranging from modulators to MEMS gratings and NOEMS, underscore the critical requirement for actuators capable of high force at low traveling ranges, enabling precise manipulation of photonic components within compact spaces of integrated photonics systems.

In addition, specific silicon photonic applications stand out for their stringent demands on actuator performance, specifically requiring both high force magnitude and a significant traveling range. These applications, documented in a range of studies, span from tunable laser sources and optical MEMS spectrometers to advanced optical switching systems and adaptive optics applications, each with unique operational landscapes that push the limits of MEMS technology. For instance, developing tunable laser sources, as documented in Ref. 34, requires actuators capable of an extensive tuning range and resisting vibration while providing fine control over the laser tuning. Similarly, the wide-band silicon photonic micro-opto-electro mechanical system (MOEMS) spectrometers described in Ref. 35 depend on actuators that must operate across a broad bandwidth with a single photodetector, necessitating a high force magnitude to maintain precision across this range. Grating couplers and resonators, pivotal in determining coupling efficiency and resonant frequencies, also require actuators with substantial force magnitude to achieve the desired mechanical tuning.³¹ Optical coherence tomography (OCT), a critical tool in biomedical imaging, necessitates actuators that can achieve fine-tuning, challenging the actuators to maintain stability and precision over microscales.³⁶

Furthermore, applications that integrate rotational MEMS mirrors for beam steering, as outlined in Ref. 37, require actuators to have a high force magnitude and traveling range to rotate the mirrors with the precision needed for advanced optical communications systems. Compact MEMS platforms for planar optical switching, such as those in Ref. 5, exemplify the necessity for both a small footprint and precise displacement capabilities, stressing the need for actuators that offer high force magnitude for efficient optical switching. Silicon photonic switches, cornerstones of modern optical networks, need actuators capable of rapid and precise switching operations. This is evident in works such as Ref. 38, which explores silicon nanowire waveguide couplers, and Ref. 39, which details low-power optical beam steering technologies. Finally, the mechanical tuning of multi-channel optical components, especially those that are wavelength-tunable, relies on actuators with high force magnitude to adjust the components accurately, as seen in Refs. 20, 40, and 41. Each of these applications exemplifies the critical need for actuators that provide the requisite force magnitude for precise control and maintain their performance over the demanding traveling ranges required by the complex mechanisms of silicon photonics.

Figure 1 is a strategic guide to MEMS tuning concepts for PICs, providing a clear framework for understanding the mechanical modulation techniques applicable in optical microsystems. It systematically outlines the tuning methods, categorizes them based on the interaction that they employ, and aligns them with specific applications within photonic systems. The figure further details the mechanical requirements in terms of force magnitude ($F$) and traveling range ($\delta$), which are crucial for the proper functioning of each application.

Fig. 1

MEMS tuning concepts for PICs.

The tuning methods are grouped into three distinct categories. Evanescent field tuning focuses on finely adjusting component spacing to manipulate the evanescent field for phase control and light coupling. Strain tuning leverages physical deformation to alter the optical properties, such as refractive index and absorption coefficients, which are essential for laser and filter operation. Waveguide steering switching encompasses methods that directly change the light path within the circuit, which is pivotal for switching and modulating photonic signals.

Each category is assessed for its force magnitude and traveling range needs, indicating the relative importance of these parameters for the tuning method to be effective. The figure also provides a visual schematic for each method, enhancing comprehension of the mechanical action involved and referencing key works for further exploration. This structured arrangement thus offers a concise reference for selecting and designing MEMS actuators tailored to the specific requirements of various photonic components.

3.

Methodology

3.1.

General Comb Geometry and Unit Footprint Definition

Figure 2 illustrates the comb drive structure designed to actuate photonic components. The photonic component is depicted within a red box and is anchored using stators, the gray fixed beams on either side. Integral to this structure is the electrostatic actuator cell, highlighted in the green box, which is the repetitive unit within the comb drive. This cell comprises arms connected to electrodes, configurable into various finger shapes, thus simplifying the design to a modular cell concept.

Fig. 2

Schematic and integration examples of MEMS actuators with photonic components. (a) Details of the mechanical platform for the comb drive actuator for photonic components. The red box illustrates the position of a free-standing photonic component that will be actuated by the electrostatic comb drive via a beam transferring mechanical force. The green box shows the position of a unit electrostatic actuator cell, including two pairs of electrodes with an arbitrary shape. (b)–(d) Three main categories of the comb derive unit cell are demonstrated. Typical forms of gap closing or parallel plate (PP) (b), area overlap or rectangular (REC) (c), and a combination of gap closing and area overlap or triangular (TRI) (d) designs. The green dotted box in each design shows a unit cell consisting of a pair of arms with electrodes, a gap between the pair of arms and the next cell, and half of the stator and moving rail. (e), (f) Cross-sectional view of the MEMS layer with integrated photonic components, highlighting the layered structure including the substrate, sacrificial layer, MEMS layer, and photonic core. (e) Example of an integrated silicon nitride waveguide on an SOI layer used as a MEMS platform. (f) An example of a silicon waveguide (SOI) in which the silicon layer functions as both the waveguide core layer and the electrode layer.

A unit footprint with a consistent spring structure and a specified restoring force is defined to facilitate an equitable comparison of the three electrostatic actuator finger types of parallel plate (PP), rectangular (REC), and triangular (TRI) electrostatic actuators. This setup fairly compares each design’s spatial efficiency and force generation. Specifically, the evaluation of force per traveling range hinges on the precise measurement of the restoring force as it directly influences the calculation of the actuator’s traveling range, a crucial parameter for real-world applications.

The proposed mechanical platform and unit footprint, detailed in Fig. 2, include a movable beam, the actuator’s central element, designed for precise directional movement. This beam is supported by single-beam springs and provides stability and electrical isolation while minimizing lateral stiffness to enhance actuation accuracy. These springs also address the mechanical requirement of having minimum stiffness in the moving direction and maximum stiffness laterally, which is important, especially in the design with triangular (TRI) fingers. To meet specific design requirements for the TRI finger design, we selected a single beam spring with a clamped-clamped configuration due to its high lateral stiffness. The TRI finger design necessitates a small initial electrode gap ($\sim 3\mu \mathrm{m}$), which unlike the REC finger design reduces further near the pull-in point, making lateral stability critical. Slight lateral deviations can cause side pull-in. In addition, one of the main focuses of our study is force density, defined as the force per footprint occupied by the actuator for a certain traveling range. To ensure a fair comparison of the traveling range and footprint among the three designs, we used the same frame (stators, moving beam, single beam spring, and serpentine springs) for all three designs. This approach ensures consistent restoring force and footprint across all designs. Furthermore, a constant area for all compositions is maintained to ensure design comparability.

For electrical connectivity, serpentine springs are employed to connect the moving electrodes. These springs are wide enough to meet fabrication limitations and provide a conductor coating, thus fulfilling key electrical considerations. They ensure that their stiffness does not impede the actuator’s movement and provide proper electrical insulation to prevent current leakage between electrodes.

3.3.

Simulation of Comb Drive Mechanical Frame Stiffness

In this section, we employ a simplified version of the general mechanical platform for the comb drive, as delineated in Fig. 2, to facilitate a focused comparison of the three distinct finger shapes (PP, REC, and TRI) under identical simulation conditions (Fig. 3). The following describes the simulation detail to determine the stiffness profile of the comb drive’s mechanical frame. The design parameters, listed in Table 1, were selected based on the following criteria.

Fig. 3

Three-dimensional view of the comb drive support structure and associated design parameters, detailing key components such as the fixed beams, mobile rail, single beam, and serpentine springs and the dimensions that govern their design.

Table 1

Mechanical frame design parameters for electrode shape comparison.

Parameter	Description	Value
$W_{fb}$	Fixed beam width (stator)	$200\mu \mathrm{m}$
$W_{mb}$	Mobile beam width (moving beam)	$40\mu \mathrm{m}$
$H$	Element thickness	$10\mu \mathrm{m}$
$L_{\text{arm}}$	Arm length	$\sim 100\mu \mathrm{m}$
$W_{\text{area}}$	The width of the area available for electrostatic actuator cells	$1063\mu \mathrm{m}$
$W_{s1}$	Width of single beam springs	$2\mu \mathrm{m}$
$W_{s2}$	Width of serpentine springs	$9\mu \mathrm{m}$
$l_{s2}$	Length of serpentine springs	$293\mu \mathrm{m}$
$l_{s1}$	Length of single-beam springs	$160\mu \mathrm{m}$
$P_{s2}$	The pitch of the serpentine springs	$19\mu \mathrm{m}$

3.3.1.

Fabrication constraints

In this study, our electrostatic comb actuator designs and analysis were based on the MEMSCAP PiezoMUMPS process design kit (PDK).⁴⁸ Utilizing this PDK ensured that our designs were both realistic and fabricable within the constraints and standards provided by the MEMSCAP PiezoMUMPS process. This process includes a standard SOI process, allowing us to skip the piezo layer.

Fabrication constraint considerations include the minimum widths mandated by the foundry used for MEMS device fabrication. For instance, the minimum stator width for free-standing structures is $200\mu \mathrm{m}$, considering the need for deep trenching and undercuts. The minimum silicon layer width is set at $2\mu \mathrm{m}$, and the minimum width for free-standing structures with a conductor layer on top is $\sim 9\mu \mathrm{m}$, factoring in the conductor layer width and the necessary separation from the silicon pattern.

3.3.2.

Design parameters

The middle moving beam, holding the free-standing comb arms, must be sufficiently wide to prevent lateral bending during actuation. In addition, the design of the serpentine springs should introduce minimal stiffness along the moving direction. Table 1 summarizes the design parameters for the mechanical frame supporting the comb drive.

The comb drive support frame is simulated using the finite element method with COMSOL Multiphysics using the parameters presented in Table 1. A force was applied to the end of the moving beam in the actuator’s direction of movement to calculate the overall frame stiffness, as depicted in Fig. 4. This stiffness profile is crucial for understanding the actuator’s force output and traveling range. The restoring force generated by the structure is assumed to be proportional to the displacement at the tip of the moving beam, exhibiting a nonlinear stiffness as the force escalates. The nonlinear stiffness observed in our comb drive’s mechanical frame is primarily due to the stress stiffening effect in the clamped-clamped single-beam spring configuration. As the beam undergoes large deformations that are comparable to its width (i.e., not much smaller than the beam’s width), tensile forces develop along the length of the beam. These forces lead to a progressive increase in the overall stiffness of the structure as displacement increases. This stress stiffening phenomenon is common in MEMS structures with clamped boundaries, where the large deformation significantly influences the mechanical response, resulting in the nonlinear stiffness profile depicted in Fig. 4.

Fig. 4

Finite element analysis of comb drive frame stiffness and actuation behavior. Left: The stiffness profile of the comb drive’s mechanical frame as a function of displacement, highlighting a nonlinear increase in stiffness with increased displacement. Right: A simulated visualization of the mechanical frame’s deformation under actuation, with the color scale representing the degree of displacement in $\mu \mathrm{m}$.

Accurately grasping this stiffness profile is essential for predicting the actuator’s efficiency under various load conditions and across its entire motion range. The mechanical restoring force is defined as follows:

Eq. (1)

$$F_m=K_m(\delta ).\delta ,$$

where $F_m$ is the mechanical restoring force, $K_m$ is the lumped stiffness of the comb support frame, and $\delta$ is the displacement.

4.

Comparative Analysis of Electrode Designs

This section advances the comparative study of force densities among three distinct electrostatic actuator configurations of PP, REC, and TRI finger designs. The foundation of this analysis is an optimization framework aimed at maximizing force magnitude against the actuator footprint (i.e., maximizing force density) while maintaining a given traveling range. Throughout this process, we applied consistent design criteria to each actuator type to ensure comparability and to adhere to practical fabrication limits as follows:

The implementation strategy for the PP and REC finger designs was straightforward, keeping in line with these constraints and efficiently utilizing the frame’s footprint. However, optimizing the triangular-shaped finger design was more intricate due to its geometric variables and is part of ongoing research. Nevertheless, the preliminary outcomes from this research have been integrated into the comparative analysis presented here, offering a preview of the potential advantages of each design within the ambit of force density optimization.

The comparative analysis presented in this paper centers on the relative performance of each design, offering insights into their efficiencies and potential applications in electrostatic actuator technology. This approach facilitates drawing meaningful conclusions about the advantages and disadvantages of each finger type and guides future design choices in this field.

4.1.

Geometry and Design Criteria Overview

A critical evaluation of three electrostatic actuator designs (PP, REC, and TRI) is conducted with a focus on optimizing force density. The design parameters were systematically chosen to fulfill the mechanical stability requirements and fabrication capabilities, ensuring a uniform $8\text{-}\mu \mathrm{m}$ travel range and 500-nm traveling range across all actuator types.

Figure 5 shows the geometries of actuator finger types. This figure visually contrasts the geometrical configurations of the PP, REC, and TRI actuator fingers. It highlights each design’s unique structural features and dimensions, providing a clear visual reference for the earlier discussed design considerations and their impact on actuator performance.

Fig. 5

Geometries of actuator finger types. The geometrical distinctions of the (a) parallel plate, (b) rectangular, and (c) triangular actuator fingers. Each configuration specifies essential design parameters influencing actuator efficacy, showcasing the spatial arrangement and physical dimensions critical to functionality. The red dashed outlines highlight the individual repeating unit cell, representing the basic building block for the actuator arm and finger assemblies in each respective design.

Table 2 compiles the optimum design parameters for the PP, REC, and TRI electrostatic actuators, reflecting the earlier defined constraints such as the arm width, electrostatic cell separation, and MEMSCAP PDK⁴⁸ fabrication standards. This table emphasizes the dimensional specifications methodically selected to achieve the desired mechanical performance, such as the arm lengths, beam widths, and electrode gaps, ensuring that each design meets the uniform $8\text{-}\mu \mathrm{m}$ or 500-nm travel range criterion. These parameters form the foundation of each design’s functionality and are integral to the actuator’s mechanical integrity and operational capabilities within the optical microsystem domain.

Table 2

Parameters and comparative metrics for actuator designs.

Parameter	Description	Value
	Parallel plate
$d$	Plate gap for $8\mu \mathrm{m}/500$ TRa	$15\mu \mathrm{m}/1.5\mu \mathrm{m}$
$d_0$	Plate width	$15\mu \mathrm{m}$
$d_1$	Spacing between cells	$18\mu \mathrm{m}$
$L_{\text{arm}}$	Plate length	$100\mu \mathrm{m}$
$L_p$	Overlap length of electrode	$90\mu \mathrm{m}$
$L_c$	Separation of arm tip to the stator or moving beam	$5\mu \mathrm{m}$
$h$	Plate height (fabrication restricted)	$10\mu \mathrm{m}$
	Rectangular finger
$b_0$	Fingertip and the opposite arm spacing	$9\mu \mathrm{m}$
$c$	Finger overlap (fabrication restricted)	$2\mu \mathrm{m}$
$a$	Finger length	$11\mu \mathrm{m}$
$d_0$	Arm width	$15\mu \mathrm{m}$
$d_1$	Spacing between cells	$18\mu \mathrm{m}$
$m$	finger width (fabrication restricted)	$2\mu \mathrm{m}$
$g$	Gap between fingers (fabrication restricted)	$2\mu \mathrm{m}$
$p$	Finger pitch	$4\mu \mathrm{m}$
$N_{\mathrm{rec}}$	Number of fingers per arm	12
$l_{\mathrm{arm}}$	Arm length	$100\mu \mathrm{m}$
$L_c$	Separation of arm tip to the stator or moving beam	$5\mu \mathrm{m}$
$h$	Beam thickness (fabrication restricted)	$10\mu \mathrm{m}$
	Triangular finger
$\alpha$	Triangular finger angle (fabrication restricted)	10.16 deg
$d$	Finger separation along $y$ for $8\mu \mathrm{m}/500\mathrm{nm}$ TR	$15\mu \mathrm{m}/1.5\mu \mathrm{m}$
$d_0$	Fingertip height	$15\mu \mathrm{m}$
$d_1$	The separation between two cell	$18\mu \mathrm{m}$
$L$	Length of one finger	$15.38\mu \mathrm{m}$
$N_{\mathrm{tri}}$	Number of fingers per arm	6
$l_{\text{arm}}$	Arm length	$\sim 100\mu \mathrm{m}$
$L_c$	Separation of arm tip to the stator or moving beam	$5\mu \mathrm{m}$
$h$	Beam thickness (fabrication restricted)	$10\mu \mathrm{m}$

^aTR: traveling range

4.2.

Electrostatic Actuation Principle

In the literature, electrostatic actuator electrode shapes are described as (i) gap closing, (ii) area overlap, and (iii) area overlap and gap closing.²³ The PP electrostatic actuator and the conventional comb drive actuator (with REC comb fingers) are the primary approaches in designing actuators for active photonics. The significant difference between these two designs is the actuation force dependency on the electrode gap.

4.2.1.

Parallel plate electrostatic electrodes (gap-closing)

In the gap-closing configuration, the space between electrodes changes equally with the motion generated by the actuator, such as PP combs [Fig. 5(a)]. Changing the electrode gap causes the actuator to have nonlinear behavior. The PP actuator design features two opposing flat surfaces that generate an electrostatic force when a voltage is applied. This design is characterized by its direct force application and simplicity.

Force magnitude

In PP actuators, the force magnitude can be relatively high due to the direct attraction between the large surface areas of the plates. Nevertheless, the separation between the plates substantially affects the force magnitude and, as a result, the achievable travel distance. This dependency may restrict the usable force or travel distance due to the pull-in effec:⁴⁹

Eq. (2)

$$F_{pp}=\frac{{\epsilon }_0{\epsilon }_{r}A{V}^2}{2d^2};A=h{L}_p,$$

where ${ϵ}_0$ is the vacuum permittivity, ${ϵ}_r$ is the relative permittivity of the dielectric medium, $h$ is the thickness of the electrode, $V$ is the voltage between the fixed and moving electrodes, $d$ is the gap between electrodes that is equal to the traveling range, $L_p$ is the overlap length of the electrode, and $A$ is the active area for electrode force.

Traveling range

The traveling range for PP actuators is typically less than that of other designs as the force decreases rapidly with increased distance, and stability issues arise due to the pull-in effect.

Footprint

PP designs often have a smaller footprint due to their simple geometry, which can be advantageous in densely packed silicon photonics for which space is at a premium.

Considering the force relation for gap closing configuration, as the electrostatic force in the parallel plates varies inversely with the square of the electrode distance, reducing electrode separation in the PP design would significantly boost electrostatic force. However, the electrode separation reduction could not be used as a parameter to increase the force because the d is limited by the traveling range, which is the design parameter.

4.2.2.

Rectangular finger electrodes (area overlap)

Unlike gap-closing, the area overlap configuration, such as REC finger comb drives, maintains electrode distances during actuation regardless of the actuator motion, and the force inversely varies with the gap [Fig. 5(b)]. Due to this, the comb drives follow a linear behavior, but their force remains constant during actuation. REC electrode designs, commonly found in comb drives, consist of interdigitated fingers that move laterally. The electrostatic force is generated across the overlapping area of the fingers.⁴⁹

Force magnitude

The force magnitude for REC designs for a given traveling range tends to be higher than that of parallel plates as the gap between electrodes does not need to be proportional to the traveling range.⁴⁹

Eq. (3)

$$F_{\mathrm{rec}}=\frac{{\epsilon }_0{\epsilon }_{r}h{V}^2}{g},$$

where the $g$ is the gap between fingers. The $F_{\mathrm{rec}}$ is inversely proportional to the gap between fingers, and the gap between fingers is limited by the fabrication limitations. So, $g$ is also a fabrication-limited parameter to increase the force.

Traveling range

These actuators can offer a more extensive traveling range due to the fixed electrode separation over the entire traveling range.

Footprint

The footprint for REC electrode designs is more extensive than the parallel plate, given the extended structure of the comb fingers. This may require more space in the photonic circuit layout. However, it would also depend on the traveling range.

Considering these two actuators, a PP comb drive will produce a large force with a small motion range, and a conventional comb drive will generate a constant electrostatic force with a more extensive range of motion. However, there is a need for a large force with a broader range of motion to extend the tunability range for some photonic devices. In this case, a TRI finger comb design could take advantage of the abovementioned designs.

In the gap-closing and area overlap configuration, the gap between electrodes will change with a specific motion generated by the actuator, and design parameters should be selected in a way to achieve the desired performance [Fig. 5(c)].

4.2.3.

Triangular finger (gap-closing and area-overlap)

TRI electrodes are a novel approach that combines the benefits of PP and REC designs. The TRI shape allows for a varying gap and overlapping area, which can optimize force and displacement.

Force magnitude

TRI designs aim to maximize force magnitude by increasing the area of attraction as the electrodes move closer together, leveraging both gap-closing and area overlap benefits. To determine the force produced by a TRI finger, imagine it as a segment of a PP with a central angle. Consequently, the force between two TRI electrodes can be represented by the component of the force along the $y$-axis from each segment of the parallel plates, illustrated in Fig. 6. The force component $F_y$ is described as

Eq. (4)

$$F_y=2F\sin \alpha ,$$

where $F$ is the electrostatic force acting between two parallel electrodes on one flank of the TRI electrode and $\alpha$ is the vertex half-angle of the TRI finger.

Fig. 6

TRI electrode of a comb drive actuator, depicting the geometric parameters and force components.

To calculate the electrostatic force, the electrode overlap $b$, as indicated in Fig. 6, is considered equivalent to the effective length of the electrode overlap

Eq. (5)

$$b=\frac{L}{2\sin \alpha }-d\cos \alpha ,$$

where $b$ represents the overlapped segment of the electrode on one side of the TRI finger, $L$ denotes the projected length of the electrode along the $x$-axis, and $d$ is the tip-to-tip separation of the TRI fingers. Subsequently, the electrostatic force for each side of the TRI electrode is expressed as

Eq. (6)

$$F=\frac{{\epsilon }_0{\epsilon }_{r}hb{V}^2}{2c^2}.$$

The separation between the electrodes, $c$, is quantified by

Eq. (7)

$$c=d\sin \alpha .$$

Incorporating the values of $b$ and $c$ from Eqs. (5) and (7) into Eq. (6) yields the electrostatic force for each side of the TRI electrodes and

Eq. (8)

$$F=\frac{{\epsilon }_0{\epsilon }_{r}hL{V}^2}{2d^2}(\frac{1}{2{\sin }^3\alpha }-\frac{d\cos \alpha }{L{\sin }^2\alpha }).$$

Using Eq. (4), the total force in the $y$-direction that drives the movement in the comb drive is expressed as

Eq. (9)

$$F_y=\frac{{\epsilon }_0{\epsilon }_{r}hL{V}^2}{2d^2}(\frac{1}{{\sin }^2\alpha }-\frac{2d\cot \alpha }{L}).$$

Traveling range

The traveling range can be optimized in TRI designs by adjusting the angle of the fingers, potentially offering a middle ground between PP and REC designs. At an angle of $\alpha =90\mathrm{deg}$, the force from the TRI electrodes ($F_{\mathrm{tri}}$) matches that of a standard PP actuator ($F_{\mathrm{pp}}$). By adjusting $\alpha$, the electrode gap can be fine-tuned to achieve the desired motion range while the gap kept minimum (depend on the limits of fabrication processes) to maximize the force. The flexibility of this design demonstrates the practicality of TRI fingers for precise actuation in MEMS.

Footprint considerations

Although potentially offering an improved performance, the TRI design can be more complex and may result in a larger footprint due to the geometric intricacies of the fingers.

Based on Eq. (9), we introduce a gain factor $(\gamma )$ to quantify the increase in force for the TRI design compared with the PP design:

Eq. (10)

$$F_{\mathrm{tri}}=\gamma F_{pp},$$

Eq. (11)

$$\gamma =\left[\right(\frac{1}{{\sin }^2\alpha })-(\frac{2d}{L}\left)\cot \alpha \right],$$

where $\gamma$ is the gain factor. $F_{\mathrm{tri}}$, similar to $F_{pp}$, is inversely proportional with the $d$ (here equal to the traveling range), but the coefficient is a key parameter that could increase $F_{\mathrm{tri}}$ for a given traveling range.

Typically, the coefficient of $\cot \alpha$ in Eq. (11), which is ($\frac{2d}{L}$), is less than one as the gap between electrodes is smaller than the length of the finger. Consequently, the force of the TRI design would always be more than the PP force with the same electrode length in smaller finger angles. Figure 7 shows the gain factor to compare the TRI finger and PP designs of the same finger length ($L$). The gain factor is demonstrated for a range of electrodes when the ratio of the traveling range ($d$) to the finger length ($L$) is 1% to 50% which is the typical range for TRI finger electrostatic actuators. The graph demonstrates that the gain factor increases significantly at smaller angles.

Fig. 7

Transition from parallel plate to triangular finger design and gain factor analysis. (a)–(d) The transformation from a parallel plate (PP) design to a triangular (TRI) finger design by folding the electrode from the middle while maintaining the initial gap d at the tip of the TRI finger. As the angle $\alpha$ increases, the gap between the TRI electrodes decreases. (e) Gain factor ($\gamma$) versus angle ($\alpha$) for triangular (TRI) electrostatic actuators compared with parallel plate (PP) actuators. The plot demonstrates the gain factor for different ratios of traveling range $d$ to finger length $L$. The divergence at small $\alpha$ is due to the decreasing gap, which theoretically approaches zero, increasing the force to infinity. The shaded region indicates the non-functional area where pull-in instability occurs, maintaining a minimum electrode gap to avoid divergence. The minimum gap could be different based on the design and the restoring force of the system.

As we have shown and is corroborated by the literature,^23,50 the TRI finger design is known for its ability to produce the highest electrostatic force; however, its overall effectiveness in terms of force magnitude and the efficient use of the traveling range and actuator footprint are not yet fully understood.

This design merges gap-closing and area-overlap features, which theoretically suggests benefits such as a wide traveling range and a smaller electrode gap, potentially leading to stronger forces. However, whether it represents the optimal choice in terms of balancing force magnitude and effective utilization of the traveling range is a question that remains open for investigation. The remaining sections of this paper aim to explore and provide clarity on these aspects, determining if the TRI finger design can indeed claim superiority in these critical performance metrics.

4.3.

Comparative Analysis of Actuator Designs for Performance Efficiency

Building on the conditions outlined in Sec. 4, we now present a detailed comparative analysis of the force magnitude and force density for PP, REC, and TRI electrostatic actuators. For an in-depth understanding of the comparison conditions and the uniformity in design parameters such as arm width, electrostatic cell separation, fabrication constraints, and travel range, refer to the beginning of Sec. 4.

4.3.1.

Force magnitude and traveling range analysis

This section presents the analytical outcomes of the mechanical and electrostatic force calculations for all three finger shapes within the defined unit footprint of the comb structure. The mechanical force calculations are derived from the numerical computation of frame stiffness detailed in the previous section, which was simulated in COMSOL Multiphysics. In addition, our analytical approach incorporates the influence of adjacent cells to ensure a comprehensive understanding of the forces at play. This integration of numerical simulations with analytical calculations allows for a robust evaluation of different comb geometries’ performance within the comb drive actuator.

To provide a thorough analysis of the actuator performance, we considered two different traveling ranges: a small traveling range of 500 nm and a large traveling range of $8\mu \mathrm{m}$. This dual-range analysis is essential due to the wide dynamic range of applications for these actuator types, for which both small and large displacements are required depending on the specific photonic application. Figures 8Fig. 9Fig. 10–11 illustrate the force-displacement characteristics and performance metrics for the PP, REC, and TRI designs across these two traveling ranges. By examining both extremes and the range in between, we ensure a comprehensive evaluation of each design’s capabilities and limitations, providing a clearer understanding of their suitability for various operational scenarios.

Fig. 8

Electrostatic forces and mechanical restoring forces for REC (a), PP (b), and TRI (c) designs for a large traveling range of $8\mu \mathrm{m}$. The mechanical restoring force along with the electrostatic force generated by each design at different voltages. The pull-in state occurs when the electrostatic force surpasses the mechanical restoring force. The TRI design achieves $8\mu \mathrm{m}$ of displacement at a lower voltage compared with the PP and REC designs.

Fig. 9

Electrostatic forces and mechanical restoring forces for REC (a), PP (b), and TRI (c) designs for a small traveling range of 500 nm. The mechanical restoring force along with the electrostatic force generated by each design at different voltages. The pull-in state occurs in TRI and PP designs when the electrostatic force exceeds the mechanical restoring force. The TRI design can generate the required force to achieve displacements of 500 nm with the minimum voltage.

Fig. 10

Direct comparison of PP, REC, and TRI electrostatic force magnitude in large and small traveling ranges at constant voltages. (a) At a constant voltage of 150 V, the TRI design consistently produces more force and larger displacement than the PP and REC designs in large traveling ranges. (b) At a constant voltage of 1.7 V, the TRI design also outperforms the other two designs in terms of force and displacement in small traveling ranges. However, unlike in large traveling ranges, the PP design shows superior performance over the REC design in small traveling ranges due to the smaller gap in the PP design, which increases the force more effectively than in the REC design, giving it a constant minimum fabricable gap of $2\mu \mathrm{m}$.

Fig. 11

Force density and traveling range for PP, REC, and TRI designs across a range of voltages for both large traveling ranges (TR) (a), (b) and small traveling ranges (c), (d). (a) Force density for PP, REC, and TRI designs at large traveling ranges (up to $8\mu \mathrm{m}$). In medium voltages, the REC design demonstrates superior force density compared with the other two designs. However, at higher voltages, the TRI design surpasses both REC and PP in force density. The PP design performs poorly at high traveling ranges. (b) Displacement for PP, REC, and TRI designs at large traveling ranges (up to $8\mu \mathrm{m}$). In medium voltages, the REC design achieves greater displacement than the other designs. At higher voltages, the TRI design outperforms both REC and PP in terms of displacement. (c) Force density for PP, REC, and TRI designs at small traveling ranges (up to 500 nm). The TRI design is superior in force density at small traveling ranges, followed by the PP design. The REC design performs poorly for small traveling ranges. (d) Displacement for PP, REC, and TRI designs at small traveling ranges (up to 500 nm). The TRI design consistently achieves greater displacement across all voltages, with the PP design performing better than the REC design at small traveling ranges.

Figures 8 and 9 show that the intersection of the electrostatic force with the restoring mechanical force determines the actuator deflection at a certain voltage. The PP and TRI finger designs exhibit exponential growth in electrostatic force with electrode displacement.

Figure 8 shows the results for the large traveling range ($8\mu \mathrm{m}$); the TRI design achieves this range at 160 V, the PP design at 650 V, and the REC design at 260 V. The TRI finger design generates a higher electrostatic force than the PP design, reaching its electrostatic force equilibrium with the mechanical force at larger displacements when the voltage constant is kept constant.

Figure 9 illustrates the force-displacement curves for the three designs at a low traveling range (500 nm). The small traveling range study reveals quasi-linear behavior for the spring, with the PP actuator pull-in occurring after 500 nm displacement at 7.5 V, the TRI design reaching pull-in just above 500 nm at 1.7 V, and the REC design requiring 23 V for 500 nm displacement. These findings highlight the dynamic performance and suitability of each design under different operational conditions, reinforcing the TRI design’s advantage in both force magnitude and traveling range.

In a further analysis, Fig. 10 presents a comparison of the electrostatic force magnitude and the traveling range generated by the PP, REC, and TRI actuators at a uniform voltage of 150 V for the large traveling range and at 1.7 V for the small traveling range to ensure comparability. This comparison elucidates the relative efficiencies of each actuator design in a common operational scenario, allowing for a clear differentiation between the force capabilities of each configuration when exposed to identical electrical conditions. It becomes evident from this comparative perspective that the TRI design exhibits a superior force characteristic and consequently larger displacement in large and small traveling ranges (Fig. 10), aligning with our previous findings on its heightened force magnitude potential. In small traveling ranges, the PP design would be the second-best option after the TRI design.

However, in large traveling ranges, the REC design would be the second-best option after the TRI design. The reason is that the REC design, being an area overlap system, maintains a constant minimum fabrication gap of $2\mu \mathrm{m}$, allowing it to perform better than the PP design in large traveling ranges. Conversely, in small ranges, the PP design is superior due to its higher force generation at smaller gaps.

4.3.2.

Force density and traveling range analysis

To further understand the relationship between force density (force per unit footprint of the actuator) and traveling range across different voltages, we performed an in-depth analysis of PP, REC, and TRI designs in both small and large traveling ranges. This analysis incorporates the actuator’s footprint to provide a more comprehensive view of performance metrics, particularly in scenarios demanding efficient spatial utilization.

Based on the defined unit cell for each of the three actuator types shown in Fig. 5 and the parameters listed in Tables 1 and 2, the area of each unit cell is calculated. The force density, $\sigma$, for each design is then derived as

Eq. (12)

$$\sigma =\frac{F}{A},$$

where $F$ represents the force magnitude, as determined by Eqs. (2), (3), and (9), and $A$ is the respective area of the actuator. These force density formulas are represented as ${\sigma }_{pp}$, ${\sigma }_{\mathrm{rec}}$, and ${\sigma }_{\mathrm{tri}}$, respectively, as follows:

Eq. (13)

$${\sigma }_{pp}=\frac{{\epsilon }_0{\epsilon }_{r}h{L}_p{V}^2}{2d^2(\frac{W_{fb}}{2}+\frac{W_{mb}}{2}+2L_c+L_p)(2d_0+d+d_1)},$$

Eq. (14)

$${\sigma }_{rec}=\frac{{\epsilon }_0{\epsilon }_{r}h{V}^2}{g(\frac{W_{fb}}{2}+\frac{W_{mb}}{2}+2L_c+N_{\mathrm{rec}}p)(d_1+2d_0+b_0+a)},$$

Eq. (15)

$${\sigma }_{tri}=\frac{{\epsilon }_0{\epsilon }_{r}hL{V}^2(\frac{1}{{\sin }^2\alpha }-\frac{2d\cot \alpha }{L})}{2d^2(\frac{W_{fb}}{2}+\frac{W_{mb}}{2}+2L_c+N_{\mathrm{tri}}L)(2d_0+d+d_1+\frac{L}{2\tan \alpha })}.$$

The comprehensive analysis illustrated in Fig. 11 reveals key insights into the performance of PP, REC, and TRI electrostatic actuator designs across varying voltages and traveling ranges. For large traveling ranges (up to $8\mu \mathrm{m}$), the TRI design consistently demonstrates superior force density and displacement capabilities, especially at higher voltages [Figs. 11(a) and 11(b)].

The REC design shows competitive performance at medium voltages but is outperformed by the TRI design as voltage increases. The PP design, although effective at producing force in smaller ranges, shows limited performance in high-traveling ranges. In small traveling ranges (up to 500 nm), the TRI design again excels, providing the highest force density and displacement across all voltages [Figs. 11(c) and 11(d)]. The PP design outperforms the REC design in small traveling ranges due to its higher force generation, attributable to the smaller gap between electrodes compared with the REC design’s fabrication constraints. These findings underscore the versatility and superiority of the TRI design, which achieves high force density and displacement across both large and small traveling ranges, making it the most suitable choice for applications requiring robust and efficient actuation.

4.3.3.

Numerical simulation of rectangular and triangular electrostatic actuator designs

To further enrich our comparative study of electrostatic actuator configurations, we conducted detailed numerical simulations using COMSOL multiphysics. These simulations aimed to validate the earlier analytical calculations, focusing on the REC and TRI designs. Using identical mechanical structures and restoring forces for each design, we aimed to closely analyze the patterns of force density and compare them with our theoretical model. The following subsections describe the approach used in these simulations, present the data obtained, and examine its importance in the wider scope of electrostatic actuator design.

Figure 12 captures select snapshots from a series of numerical simulations of REC and TRI electrostatic actuator designs, conducted over a bias voltage range from 0 to 150 V using COMSOL multiphysics. This series illustrates the performance characteristics of REC and TRI actuators at two distinct voltage levels, 100 and 140 V, to exemplify the simulation results within the broader voltage spectrum.

Fig. 12

Numerical simulations of REC and TRI electrostatic actuator designs at different voltages. Subfigure (a) illustrates the mesh quality of REC design at 100 V, and subfigure (d) shows the same for TRI design, reflecting the precision of the finite element analysis. Subfigures (b) and (e) represent the electric field norm across the actuators for REC and TRI designs at 100 and 140 V, respectively, providing a comparative view of the electric field intensity and distribution. Subfigures (c) and (f) display the resulting displacement from the applied voltage for REC and TRI designs, respectively. The displacement is visualized at 100 volts in subfigure (c) for REC and at 140 V in subfigure (f) for TRI, highlighting the mechanical response of each design under electrostatic actuation.

4.3.4.

Integration of analytical and numerical simulation insights

Analytical and numerical simulations show consistent force density trends for both TRI and REC electrostatic actuators, with a notable emphasis on the superior performance of the TRI design under high-voltage scenarios. Figure 13 presents a comparative analysis of the force density for TRI versus REC electrostatic actuator designs. Subfigure (a) shows the results from analytical calculations, and subfigure (b) displays the outcomes of the numerical simulations.

Fig. 13

Comparative force density of TRI versus REC electrostatic actuator designs. This graph depicts the force density as a function of applied voltage, illustrating the performance of TRI and REC actuator designs from analytical calculation (a) and numerical simulation (b). In both subfigures, the TRI design demonstrates a steeper increase in force density with voltage, surpassing the REC design beyond a threshold voltage. The side-by-side comparison reinforces the validity of the results, showing that the numerical simulations agree with the analytical calculations.

The analytical findings for the PP actuator show a reduced force density. This trend was not examined numerically, given the PP’s simpler geometry and well-known behavior that typically allows for predictability without in-depth numerical confirmation. This decreased force density is consistent with the inherent features of the PP design, especially the pull-in limitation, which limits its effective force output when the electrode separation is larger. By contrast, the REC design displays competitive force density at lower voltages, but this advantage wanes at higher voltages where the TRI design becomes superior. This change is consistently observed in both sets of simulations in Fig. 13, clearly identifying the voltage level above which the TRI design proves to be more beneficial.

The agreement between analytical and numerical data underscores the reliability of these findings, providing a solid basis for selecting actuator designs. The TRI design emerges as the preferable choice for systems that require robust actuation capabilities across a spectrum of voltages.

5.

Dynamic Force Density: Impact of Traveling Range and Voltage

Up to this point in our analysis, the focus has been on comparing the force densities of the PP, REC, and TRI electrostatic actuators within two fixed traveling ranges of $8\mu \mathrm{m}$ and 500 nm. This constraint has allowed us to evaluate the performance of each actuator type under uniform conditions. However, in practical applications, the traveling range is a variable factor significantly impacting actuator performance. Recognizing this, we extend our analysis to explore how varying traveling ranges influence the force densities of these actuators. This broader perspective is crucial for understanding each design’s capabilities and limitations within diverse operational contexts.

Figure 14 offers a compelling visualization of the force density landscapes for PP, REC, and TRI electrostatic actuators, considering the variations in traveling range and applied voltage. This analysis is pivotal in highlighting how the performance of each actuator type adapts to different operational ranges, providing a comprehensive view that is essential for informed actuator selection in optical microsystems. The figure is segmented into parts (a) to (c), displaying the distinct force density profiles for PP, REC, and TRI electrostatic actuators with respect to travel range, with each actuator type assessed on its own merits. This distinction clarifies their unique operational traits. Subsection (d) merges these individual profiles into a unified graph, providing a side-by-side comparison that clearly delineates the operational domains and relative advantages of each actuator type when viewed together.

Fig. 14

Operational characterization of electrostatic actuators that quantifies the correlation between force density and travel range for PP, REC, and TRI actuators. Sections (a) through (c) provide isolated assessments of each actuator variant, mapping out the force capabilities and movement extents within their operational confines, which are highlighted in yellow. Section (d) integrates these profiles into a comparative framework, depicting the optimal performance zones for each actuator type and identifying the voltage thresholds that mark the limit of effective actuation.

The PP design is characterized by high force density at low traveling ranges, as indicated by the vertical rise in the leftmost region of the graph. However, its practical application is limited to a narrow traveling range, beyond which the force density rapidly diminishes, making it less suitable for applications demanding high force and extended displacement.

The TRI actuator emerges as the most versatile, with a broad operational window regarding both force density and traveling range. The transitional area “TRI/REC” illustrates the voltage-dependent crossover point where the TRI design outperforms the REC in force density. This region is particularly critical for applications that require a balance between force density and traveling range at moderate voltage levels.

Conversely, the REC design maintains a consistent force density over a wide traveling range, as denoted by the extensive green area. Its performance remains relatively stable across varying voltages, making it an ideal candidate for applications in which consistency and reliability of actuation are paramount.

The black zone represents the parameter space where no actuators can function effectively, serving as a boundary for design considerations. Within this region, the required force density or the traveling range is too great for the actuators to operate within their mechanical and electrical constraints. It should be noted that the graph in Fig. 14 assumes a maximum actuation voltage of 150 V. Elevating this voltage threshold could potentially extend the operational areas into the black region, depending on the desired increase in voltage and the actuator’s capacity to handle higher electrical stresses.

Therefore, in summary, selecting an actuator must be a deliberate decision that aligns with the photonic application’s specific force and displacement requirements. The PP design is optimal for applications with minimal travel requirements and a premium on force density. The TRI actuator offers a robust solution across a spectrum of voltage inputs, providing high force density and displacement. Finally, the REC design is the most appropriate for applications in which uniform force over an extended range is needed without the constraints of higher voltage operation. Figure 14 is a comparative tool and a decision-making aid that captures the nuances of electrostatic actuator performance in an intuitive visual format. It allows designers to extrapolate the potential behavior of actuators under different operational scenarios, ensuring that the most effective and efficient actuator is selected to meet the exacting demands of advanced photonic systems.

5.1.

Incorporating Mechanical Actuation into Tunable Photonic Device Design

The preceding discussion outlined the significance of force density and traveling range in the context of electrostatic actuators. To refine our understanding of actuator performance further, Fig. 14 integrates mechanical actuation methods into the visualization, depicting regions where specific actuators are most advantageous for tunable photonic devices.

Figure 15 builds upon the foundational analysis provided in Fig. 14 by superimposing the mechanical actuation methods relevant to the design of tunable photonic components. This overlay directly correlates the physical actuation mechanisms and the specific photonic applications that they enable, as categorized in Fig. 1. Adding these actuation methods clarifies the practical implementation of electrostatic actuators in silicon photonic systems and bridges the gap between theoretical analysis and applied technology. The zones labeled A to D in Fig. 15 correspond to distinct actuation techniques as follows:

• A. Evanescent field tuning: This region highlights where actuators excel in applications requiring delicate adjustments to the evanescent field. These actuators operate with low to moderate force density but limited traveling range and are suitable for phase shifters, couplers, switches, and resonators. The associated references detail the nuanced applications of these tunable components within PICs.

• B. Strain tuning: The area designated for strain tuning underscores the method’s low to high force density and small traveling range capabilities. It is particularly relevant for applications such as external cavity diode lasers (ECDLs), resonators, and filters. This method leverages the material strain to alter optical properties, enabling fine-tuning of the photonic device’s performance.

• C. Waveguide steering switching: This sector represents the application of displacement or deformation techniques and the introduction of absorbers/reflectors or phase shifters. The highlighted applications include switches, filters, ECDLs, and phase shifters, requiring substantial force density and small to large traveling ranges. These actuators are pivotal in systems in which reconfiguration and adaptability are critical.

• D. Waveguide steering switching (extended traveling range): Expanding on the previous category, this domain shows actuators suitable for steering waveguide switches over even more extensive traveling ranges. This highlights the peak of actuator performance, suitable for demanding applications that require a combination of moderate force density and extensive displacement.

Fig. 15

Operational domains for PP, REC, and TRI electrostatic actuators in relation to mechanical actuation methods for tunable photonic devices. The force density and traveling range suitable for different actuators and their corresponding photonic tuning methods are highlighted. The operational regions are color-coded, illustrating where each actuator design excels and how it aligns with specific photonic applications such as evanescent field tuning, strain tuning, and waveguide steering.

Each designated region within Fig. 15 offers a snapshot of the potential applications and their respective operational requirements, serving as a comprehensive map for designers to navigate the complex landscape of photonic actuator selection. This detailed approach ensures that the chosen actuator is theoretically capable and practically aligned with the intended application’s force and range requirements.

Ultimately, Fig. 15 builds on the previous analysis; it serves as a useful resource that bridges mechanical actuation methods with practical applications in photonics. Although this graphical representation is useful for designers and engineers in the field of photonics, helping them to choose suitable actuation methods, it does have limitations, such as its reliance on specific configurations discussed earlier in the paper. The figure offers guidance but should be considered alongside other design considerations to ensure optimal performance, reliability, and functionality of photonic devices within their specific operational environments.

6.

Conclusion

This study conducted an exploration of electrostatic actuator designs, primarily focusing on their applicability in the evolving field of optical microsystems. By analyzing parallel plate, rectangular, and triangular electrostatic actuator configurations, we have uncovered critical insights into the interplay between force density, traveling range, and footprint—pivotal in PICs.

Our findings reveal the key differences between these actuator designs, emphasizing the importance of selecting the appropriate configuration based on specific application needs. The parallel plate design, characterized by its simplicity and high force density at low traveling ranges, proves advantageous in applications with minimal travel requirements. Conversely, the rectangular design is a reliable choice for applications demanding consistent force over an extended range, particularly at lower voltage levels. The triangular finger design provides enhanced force density across a wide voltage range, making it particularly effective in achieving both high force output and extended displacement. This design’s ability to optimize force production while maintaining a compact footprint positions it as a strong candidate for applications requiring robust performance in varying operational conditions.

Our study also highlights the dynamic nature of force density in relation to traveling range and voltage. By presenting a visual representation of these relationships, we have provided a valuable tool for designers and engineers in the field of optical microsystems. This tool aids in the informed selection of actuators, ensuring that the chosen design aligns with the operational demands of specific photonic applications.

This work’s analysis is based on silicon platforms featuring a dedicated $10\mu \mathrm{m}$ MEMS layer, where optical waveguides are fabricated on top, ensuring enhanced mechanical robustness. Although our analytical models offer valuable insights for these platforms, they may require adjustments for photonic MEMS platforms utilizing a thinner silicon layer, where the waveguide layer also serves as the MEMS actuator, presenting different mechanical challenges and constraints.

In conclusion, the advancement of optical microsystems greatly benefits from the careful consideration and improvement of MEMS actuators. Our research provides guidance in this regard, highlighting the capabilities and limitations of different electrostatic actuator designs. By understanding these details, designers can make informed decisions, contributing to the development of more efficient and compact PICs. This study not only contributes to the field of optical microsystems but also supports future innovations in MEMS actuator technology.