Inverse‐Design Methodology for Generating Twisted Ultrasonic Motor in Free Space

A new level of freedom is made possible by orbital angular momentum (OAM), which can be used to create twisted ultrasonic motors for particle manipulation and control. For steady and free‐flowing particle management, it is essential to produce acoustic OAM beams of high order and high quality. However, this task is difficult due to the phased array's fundamental restriction of insufficient resolution or aperture. Herein, a generic approach is presented to inverse design that may be used to improve the OAM beam quality and produce twisted ultrasonic motors in free space. In comparison to the previous direct‐design method, up to a 14% improvement in amplitude uniformity and a 12 dB improvement in signal‐to‐noise ratio (SNR) of OAM modes are achieved. These notable benefits enable the ultrasonic motor to operate with a straightforward and compact system while enabling steady particle rotation in free space with an undersampled phased array, which is not possible with the conventional technique. This study paves the way for effective particle control, high‐fidelity communication, and other OAM‐based technologies by providing an inverse‐design framework for optimizing OAM.


Introduction
In recent years, there has been a growing interest in the study of orbital angular momentum (OAM) beams.[3][4][5][6] Acoustic OAM beams exhibit a central dark region and helical wavefronts, resulting in the creation of an acoustic potential well and the transfer of angular momentum to matter.[9] The radiation force and torque associated with the transfer of linear and angular momentum from wave to matter are particularly significant in maneuvering and controlling objects in diverse fields. [10]The latest experiments involving acoustic OAM beams have showcased their ability to transfer momentum to matter.[13] By introducing the acoustic virtual vortex, stable trapping, and spin control have been achieved for Mie particles using a ring-shaped array. [14]The manipulation of rotational movement in enclosed cavities has been successfully showcased through the development of meta-engine blocks. [15]However, the stable rotation control of particles in unbounded free space is challenging, due to the stringent requirements of the OAM qualities in the aspects of OAM purity, shape, and symmetry of the amplitude profile.The generation of high-order and high-quality OAM beams has been a longstanding objective in the field of acoustics.While both active and passive arrays have been proposed for this purpose, [16] challenges persist in achieving this goal, particularly in scenarios where there are constraints on control channels and spatial resolution.[19] For generating OAM in free space and at ultrasonic frequency, the spiral structures [20][21][22] and uniform circular array [23] have been developed, which however have no or limited dynamic control ability.In addition, the classical phased array has been widely used in various application scenarios [24] for dynamic control with the compatibility to integrate with other advanced technologies. [25]However, when the number of control channels and spatial resolution deteriorates, the generation surfers from the inevitable low OAM purity and mode crosstalk responsible for the instability and high bit-error-rate (BER) in practical applications. [26]In light of the increasing demand for OAM-based technologies, it is imperative to improve the quality of OAM beams using a straightforward and compact facility.This is essential for the successful implementation of OAM-based control systems and the enhancement of various applications, such as freewheeling particle control and high-capacity wireless communication.
The current method for generating OAM relies on classical formulas to directly calculate the phase of the helical emission.However, this approach is not capable of optimizing the phase distribution specific for the undersampled or sparse emitting surface.This limitation results in suboptimal performance and restricts the full utilization of OAM beam in fabricating the precise and efficient ultrasonic motor.To overcome these limitations, the inverse-design method has been introduced as a promising approach for achieving OAM beam optimization.The inverse-design methodology is highly suitable for addressing complex problems that lack well-defined relationships between the arguments and desired objectives.[29] Optimizing OAM beams involves the inverse design of phase distribution across an emission array.However, tackling this issue is challenging due to the highly nonlocal and nonlinear nature of the field distribution.It is crucial to consider the complex coupling between different control channels.Another arduous task is the large parameter space for optimization, making it difficult to go through the entire phase space effectively, especially considering the issue of phase wrapping.In addition, it is essential to have a clearly defined evaluation method and an appropriate objective function.
The selection of various protocols can significantly impact the overall quality of the constructed field. [30]Therefore, both an efficient algorithm and a well-defined evaluation method are highly required to optimize the phase distribution on the emitting surface, ultimately maximizing the quality of OAM beams, which thereby leads to the enhanced performance in constructing the twisted ultrasonic motor and opens up new avenues for technological advancements in OAM-based applications.
This study introduces a comprehensive inverse-design methodology for enhancing the quality of OAM beams.Additionally, we showcase the successful achievement of stable particle rotation in free space through the construction of the twisted ultrasonic motor.The methodology is specialized to perform a global optimization for the nonlocal phase distribution on the emitting surface.The optimization process has been specifically designed to handle OAM and effectively handle phase wrapping.This enables the implementation of personalized search strategies, updated rules, and evaluation criteria.We use the coefficient of variation (COV) that is employed to evaluate the uniformity of acoustic pressure amplitude in OAM beams throughout the optimization process.This assessment is crucial in determining the stability of particle control using the twisted ultrasonic motor.In addition, the signal-to-noise ratio (SNR) is employed as a comprehensive indicator of the OAM beam quality, which is a basic index and affects the accuracy in the OAM-based communication.These performance metrics play as critical evaluation indicators in the OAM optimization.Acoustic OAM beams with topological charges up to 8 are generated experimentally using an undersampled 16 Â 16 array.However, the direct-design method faces challenges and may even be inaccessible in this scenario.In addition, compared to the previous direct-design method, a 14% improvement in amplitude uniformity and a 12 dB improvement in SNR of OAM modes are achieved.These improvements on the OAM quality further enable the efficient particle rotation by constructing the optimized twisted ultrasonic motor, where the particle motion is found to be more stable and consistent with the desired trajectory, as shown in Figure 1a.Moreover, the improved SNR serves to emphasize the effectiveness of our methodology in enhancing the operational capabilities of acoustic communication systems based on OAM.The utilization of inverse-design methodology to optimize OAM beams indicates that our approach holds promise for wider applications in the optimization of various complex systems across diverse fields.

Theory
Figure 1b depicts the schematic diagram of the direct-designed OAM, while Figure 1c showcases the inverse-designed OAM based on global optimization.OAM beams have the helical phase profile associated with azimuth angle.The generation of acoustic vortex beams with varying topological charge m at the distance F requires the determination of the theoretical phase distribution on the emitting plane using the following formula where k represents the acoustic wavenumber in free space, φ = Arg(x þ iy) is the azimuthal angle.Figure 1b illustrates the two primary terms of the theoretical phase distribution: the focus phase and the vortex phase. [31]Equation ( 1) is suitable for generating OAM beams of different topological charge m using both active phased array or passive metasurface approaches.[34][35] Passive metasurface systems involve the design of phase distribution by incorporating resonant structures on the surface.These structures are patterned in a way that imparts the desired phase shift to the incident wave. [36,37]By controlling the phase of the waves, these arrays can manipulate the wavefront to generate OAM beams of specific topological charge.The qualities of directdesigned OAM beams are significantly impacted in practice due to limited control channels and a noncircular arrangement.This leads to mode crosstalk and a decrease.The OAM qualities further deteriorate when using the undersampled or sparse array (active and passive) which are frequently encountered in realworld scenarios.To tackle these challenges and enhance performance, a proposed approach involves the utilization of an inverse-design methodology that relies on global optimization.The initial phased profile on the array is theoretically calculated based on the desired topological charge and array arrangement, as shown in Figure 1c.Following the process of forwardpropagation, the quantitative assessment of the OAM beams' quality is carried out using two indicators, namely, the COV and SNR.The performance of the phase profile is improved and the limitations of direct-design methods are overcome through the iterative updating process, which involves the application of customized search strategies, update rules, and evaluation criteria.Additional information can be found in the Supporting Information.

Models and Experiments
The inverse-design methodology offers several advantages over traditional direct approaches.It enables the construction of a twisted ultrasonic motor in free space, providing superior performance in terms of improved amplitude uniformity and enhanced SNR.Additionally, this methodology facilitates stable particle manipulation, further highlighting its benefits.As a typical example, we focus on generating the high-quality OAM beams and constructing the twisted ultrasonic motor with an undersampled active phased array.A 16 Â 16 array of transducers with a diameter of 10 mm is created to produce OAM beams with topological charges ranging from 1 to 8. The array operates at a frequency of 40 kHz for airborne sound.).The acoustic pressure is normalized with its respective maximum value.Additionally, the measured phase distributions can be found in Figure S1, Supporting Information.The optimized phase profiles on the phased array are available in the Supporting Information.The OAM beams designed with both the direct and inverse methods exhibit the expected helical wavefronts and central dark regions.The amplitude profiles of the desired doughnut-shaped beams in direct-designed OAM beams exhibit noticeable deformations, resulting in distorted central singularities.This distortion has a significant impact on the manipulation of particles with varying acoustic impedance, whether they are soft or hard.In contrast, the amplitude profiles of the inverse-designed OAM beams exhibit a consistent doughnut shape, while the central dark region appears uniformly dark.This indicates a higher level of purity and improved performance when compared to the direct-designed beams.The contrast between the direct and inverse design methods is more obvious in generating the high-order OAM beams.The exceptional performance highlights the potential of the inverse-design methodology in surpassing the limitations of the direct-design method for generating OAM beams with high purity and high order.This is especially beneficial when faced with constraints such as limited control channels and spatial resolution.

Evaluating the OAM Quality
The enhancements obtained through the inverse-design method are measured by assessing the qualities of the OAM beams using both simulation (Figure 3a-d) and experimental analysis (Figure 3e-h).The evaluation process involves selecting the analysis of the OAM beam with m = 4 as a representative example for visualization purposes.Acoustic pressure amplitude profiles along the peak amplitude ring are depicted in Figure 3a,e, where the proposed inverse-design methodology yields a remarkably uniform distribution, while the direct-designed beam exhibits a pronounced fluctuation.The COV is introduced to provide a more precise quantification of the improvement.It is determined by dividing the standard deviation (σ) to the mean value (μ) of acoustic pressure where r max denotes the radius of the acoustic pressure amplitude peak.The COV value measures the relative variability of the amplitude profile, with lower value indicating a more uniform distribution and, consequently, a higher quality OAM beam.
The COV values for OAM beams with various topological charges ranging from 1 to 8 are calculated, as shown in Figure 3b,f, where a significant enhancement in uniformity up to 14% is experimentally observed for the considered topological charges.
The OAM mode purity and mode crosstalk are quantified by calculating the angular spectrum of the inverse-designed OAM beams.This calculation takes into account the orthogonality of OAM modes 1 2π The angular spectrum is calculated by the Fourier transform of the pressure field p(r, φ)  This transformation provides the composition of various OAM components within the beams that have been designed.The direct-designed OAM beams exhibit nonnegligible spectrum leakage to other modes, as shown in Figure 3c,g.Conversely, the inverse-designed OAM beams demonstrate localized angular spectrum specifically in the desired m = 4 mode.This spectrum leakage also indicates crosstalk among different OAM modes.To quantify the crosstalk, the SNR is calculated as the ratio between the powers of the desired mode m and the powers of the other modes.This ratio is expressed as The SNR is a metric used to evaluate the purity of OAM and the level of crosstalk in an OAM-based communication system.It is also directly related to the transmission efficiency of the system.The SNR of OAM beams is computed for topological charges (m) ranging from 1 to 8.This analysis is illustrated in Figure 3d,h, for both the direct and inverse-designed beams.An improvement up to 12 dB is achieved in the inverse optimized results, indicating a significant reduction in mode crosstalk and the potential advantage in the particle control and OAM communication.The decrease in COV and SNR values with topological charge is attributed to the limited spatial resolution on the array.The inverse-designed OAM beams consistently demonstrate superior performance in terms of both COV and SNR.This emphasizes the effectiveness of the inverse-design methodology in generating high-purity OAM beams, particularly when utilizing the undersampled phased array.The disparity between the experimental and simulated values can be ascribed to the heterogeneity of the elements present on the phased array.The evaluation methods utilized in this study, such as the analysis of COV and SNR, enable a quantitative assessment of the quality of the generated OAM beams.This assessment is essential for the effective implementation of the inverse-design methodology for OAM generation.It is noteworthy that these two indicators, COV and SNR, are cooperative during the inversedesign process.In addition to quantifying beam quality, they directly influence the optimization trajectory and thereby significantly impact the final solution.The collaborative utilization of these factors contributes to avoid the local minima and nonconvergence problems, which enable the development of a comprehensive and precise inverse-design methodology.Additional information can be found in the Supporting Information.

Twisted Ultrasonic Motor
The stable control of particles in free space has long been a challenge.However, we have successfully achieved this by utilizing the significant advancements in the quality of OAM through inverse-designed beams.Our approach involves the construction of a twisted ultrasonic motor.Figure 4a displays a photograph of the experimental setup of the twisted ultrasonic motor, where the OAM beam is generated to induce rotation of the trapped particle.The phased array is responsible for generating an OAM beam with a uniform doughnut-shaped amplitude profile and a helical wavefront on the desired focal plane at z = 200 mm.Consequently, the interaction between waves and matter leads to the formation of a ring-shaped acoustic potential well in close proximity to the boundary.This well serves as a confinement mechanism, effectively trapping the particle within it.Additionally, the transfer of angular momentum from the wave to the particle induces rotational motion.The expanded polypropylene (EPP) particle of diameter 2.2 mm is used in our experiment, and a 6 Vpp voltage is applied to the array.
The swinging radius is determined by the topological charge, and the stability of particle rotation is closely linked to the uniformity of the potential well.This information has recently been uncovered.The motion trajectory is captured by recording the particle rotation at a rate of 60 frames per second.The video files are provided in the Supporting Information.The perspective correction technique and multiple frame synthesis are introduced to extract the trajectory from the video.Figure 4b,c displays the particle trajectories controlled by the direct-designed OAM beams with m being 1 and 2, overlaid with the semitransparent amplitude fields for better visualization.The phenomenon of particle rotation being impeded and eventually ceasing is attributed to the instability of the potential well produced by the conventional direct-design approach.The trajectories exhibit irregular and inconsistent patterns in each cycle.On the contrary, the particle controlled by the inverse-designed OAM beams rotates quietly and the trajectories in different rotation cycles are nearly overlapped, as shown in Figure 4d,e, proving the improved stability and precision in the generated twisted ultrasonic motor.Additionally, it aids in decreasing the likelihood of particles escaping from the intended potential well or encountering unforeseen disruptions, thereby enhancing the dependability of the control system.The enhanced performance can be attributed to the optimization process employed in the inverse-design methodology.This process improves the purity of OAM beams, resulting in a more efficient transfer of angular momentum from the wave to the object.The enhanced stability is particularly important in free space control since there are no external boundaries to constrain the particle, relieving the limitation of the traditional methods and providing the potential of freewheeling object control in various practical applications such as precise drug delivery, radiation force measurement, and microassembly processes, where the ability to maneuver particles with high precision in free space could offer unprecedented advantages and breakthroughs.

Discussion
The research conducted has yielded a new and innovative inverse-design methodology for producing high-quality OAM beams.These beams are crucial in the development of twisted ultrasonic motors that operate in open space.The presented methodology successfully addresses the limitations faced by conventional direct-design methods and can be easily applied to passive metasurfaces.This opens up possibilities for enhanced performance in various applications that rely on OAM, including particle control and wireless communication.
Significant enhancements in amplitude uniformity and SNR have been experimentally observed during the generation of high-order (m = 8) OAM beams using an undersampled active phased array.The utilization of these advancements has facilitated the construction of a twisted ultrasonic motor, enabling the attainment of stable particle control in free space.The inverse-design methodology we employ enhances particle control, resulting in improved stability and precision.This advancement has the potential to greatly impact the field of OAM-based applications.
It is promising to further advance the inverse optimization method by integrating the deep learning techniques.There are challenges that need to be addressed before fully utilizing those techniques, include constructing a high-quality training dataset and acquiring massive amounts of computing resources.With these questions solved, deep neural networks [38,39] could help to understand the intricate connections between the design parameter and the resultant complex-valued 2D or 3D-acoustic fields, which would enable the development of an enhanced and streamlined approach for the inverse design of OAM beam and other freewheeling wave manipulation.

Figure 1 .
Figure1.Generating high-quality OAM for twisted ultrasonic motor.a) Stable particle control in free space with the optimized twisted ultrasonic motor.b) Schematic illustration of the direct-design method, which is the linear superposition of the focus phase and the vortex phase.c) Schematic diagram of the inverse-design methodology, which overcomes the limitations in the direct-design method by utilizing global optimization and iterative updates of the phase profile to enhance OAM beam quality.

Figure 2
displays the simulated and experimentally measured OAM beams at the plane z = 200 mm for different values of m (2, 4, 6, and 8

Figure 3 .
Figure 3. Quantitative evaluation of OAM beam quality.The simulated and experimental results are plotted in a-d) and e-h), respectively.a,e) Acoustic pressure amplitude profiles along the peak amplitude ring for m = 4.The corresponding angular spectra are plotted in (c,g).b,f ) COV values calculated from the amplitude profiles (Equation (4)) for OAM beams of topological charge m ranging from 1 to 8, with the corresponding SNR values shown in (d,h), respectively.

Figure 2 .
Figure 2. Comparison of the OAM beams generated with the direct and inverse methods, in both the simulation and experiment.a-d) OAM beams with the topological charge m being 2, 4, 6, and 8, respectively.The phase distributions of OAM beams are plotted in Figure S1, Supporting Information.

Figure 4 .
Figure 4. Experimental construction of twisted ultrasonic motor for stable particle control in free space.a) Photograph of the experimental setup for the twisted ultrasonic motor.b-e) Particle trajectories controlled by OAM beams, generated by the direct and inverse-design methods.The trajectories are derived from the recorded videos of particle rotation over 4 s (240 frames).The measured acoustic fields (semitransparent) are overlaid to clearly visualize the rotation radius.Video files of the particle motion are available in the Supporting Information.