From Photon Momentum Transfer to Accelerometer Based on Optical Levitated Microsphere at Dynamic Input from 0.1 μg to 1 g

Jan. 03 , 2025photonics1

Abstract

As a typical application of photon momentum transfer, optically levitated systems are known for their ideal isolation from mechanical dissipation and thermal noise. These characteristics offer extraordinary potential for acceleration precision sensing and have attracted extensive attention in both fundamental and applied physics. Although considerable improvements of optically levitated accelerometers have been reported, the dynamic testing of the sensing performance remains a crucial challenge before utilization in practical application scenarios. In this work, we present a dual-beam optically levitated accelerometer and demonstrate a test with dynamic inputs for the first time. An acceleration sensing sensitivity of 0.1 μg (g = 9.8 m/s²) and a measurement range of 1 g are achieved. These advancements solidify the potential of optically levitated accelerometers for deployment in practical domains, including navigation, intelligent driving, and industrial automation, building a bridge between laboratory systems and real-world applications.

Introduction

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Acceleration, a crucial physical quantity, has been frequently measured in the advancement of science and technology. As the properties of conventional quartz accelerometers are now limited by the material and the manufacturing processes, (1) there is a rapidly increasing demand of new-type accelerometers across a spectrum of applications. Accelerometers based on MEMS, (2,3) electrostatic levitation, (4,5) and cold atom interferometry (6) have been developed to fit different demanding situations. However, before any new-type accelerometer can be put into real-world applications, it is essential to test the accelerometer’s performance with dynamic tests. (7,8) Different from static tests, the dynamic test refers to the application of inputs with varying frequencies or amplitudes. Dynamic testing is crucial for ensuring the reliability of the device, offering key insights into how it performs under real-world conditions. It uncovers potential weaknesses, driving the development of robust systems that can withstand demanding environments and ultimately ensuring safety and performance in applications.

Photon momentum transfer leads to optical forces, a concept exploited by Ashkin, who pioneered the development of optically levitated systems in 1976. (9) These systems have made great progress and are now widely used in precision sensing applications. (10−14) Thanks to their exceptional isolation from mechanical and thermal noise, the vacuum optically levitated systems are particularly suitable for acceleration sensing, (15−18) offering theoretical sensitivities lower than 1 ng. (19) At present, however, optically levitated accelerometers are still in the experimental stage inside laboratories, and the journey toward their real-world application is still in its infancy. (20) The main challenge is that most laboratory systems are now designed for static testing environments and cannot maintain their precision under dynamic inputs, or even failing to function. Besides, many optically levitated accelerometers face structural limitations that prevent them from achieving a broad attitude tolerance, which refers to the ability of a device to maintain proper functionality despite changes in orientation or position. For example, in vertical optical traps, the need for the scattering force to precisely balance gravity restricts their ability to function across a range of orientations. (15,16) Achieving wide attitude tolerance is essential for devices to adapt to varying environmental conditions, especially in dynamic fields like aerospace. Furthermore, some accelerometers have a limited measurement range, which also constrains their use. Together, these factors restrict the applicability of such devices in many scenarios. Therefore, to bridge the gap between laboratory systems and real-world applications, it is essential to design an optically levitated accelerometer with high sensitivity, a large measurement range, and a large attitude tolerance, whose properties can be verified under dynamic inputs.

In this paper, we present a dual-beam optically levitated system that levitates a 25 μm diameter silica microsphere, thereby enabling high-sensitivity acceleration sensing. This system is designed to maintain stability across a large range of attitude tolerance of ±90° through a close-loop method. We have conducted a thorough evaluation of the system’s sensing properties using dynamic input methods for the first time in the optically levitated accelerometer field, which has substantiated the system’s capability of achieving a sensing sensitivity of 0.1 μg (@19 Hz) with a measurement range that extends beyond ±1 g. The outstanding sensing performance of our system builds a robust bridge between the laboratory systems and a large spectrum of real-world applications, particularly in aerospace, (21) navigation systems, (22) intelligent driving, (23) industrial automation, (24) and the proliferation of smart devices. (25)

Experimental System

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The sensing sensitivity of the optically levitated accelerometer is influenced by factors such as 1/f noise, ground vibrations, airflow fluctuations, laser pointing instability, etc. If this noise is effectively minimized, the ultimate limit on sensitivity is set by the acceleration noise due to Brownian motion acting on the levitated mass, which corresponds to the theoretical sensing sensitivity. The root-mean-square magnitude of the noise floor can be described by the following equation: (26)

S_{a} = \sqrt{\frac{4 k_{B} T Γ b}{m}}

$S_{a} = \sqrt{\frac{4 k_{B} T Γ b}{m}}$

(1)

where k_B is the Boltzmann constant, T is the bath temperature, Γ is the damping rate, b is the effective bandwidth of the measurement, and m is the mass of the microsphere. It should be noted that because we need to conduct tests in the flat region of the response function, our system performs measurements below the resonant frequency of the levitated particle, rather than using it as a resonant force sensor. The equation suggests that an enhancement in the system’s sensitivity can be achieved by increasing the mass of the levitated microsphere or decreasing the damping rate Γ by working at low pressure.

As depicted in Figure 1, the experimental setup levitates a silica microsphere with a diameter of 25 μm, serving as the sensing oscillator. The microsphere reaches a mass of 18 ng, enabling it to achieve ultrahigh acceleration sensing. The mass of the levitated microsphere is primarily limited by the microsphere loading process. The system utilizes two 1064 nm laser beams, focused by two aspheric lenses with an 18.4 mm focal length within a vacuum chamber, to form a dual-beam optical trap with a numerical aperture (NA) of 0.10. We use the axial direction of the trapping light (z-axis) as the sensitive axis for acceleration sensing. The two beams have a combined power of 400 mW, with a power discrepancy of not exceeding 5% between them. The corresponding eigenfrequency of the optically levitated resonator is 95 Hz. Compared with the vertical optical trap, this setup is designed to achieve a broader range of attitude tolerance as the counter-propagating beam trap is a real three-dimensional trap that can maintain stability even when the system’s orientation changes.

Figure 1. Simplified schematic of the experimental setup. A microsphere with a diameter of 25 μm is levitated within a vacuum chamber by a counter-propagating optical trap. Two 532 nm lasers irradiate the levitated microsphere in the x and y directions, functioning as the feedback cooling beams (y-axis beam has been omitted for a more clear structure). Additionally, two high-power 532 nm laser beams aligned with the trapping beam serve as the acceleration modulation beams for measurement range testing.

The 1064 nm laser beam, after passing through a pointing stabilization system, is then split by a λ/2 plate and polarizing beam splitter (PBS) into two trapping beams with orthogonal polarization directions. One of the two beams, modified by an AOM (Acousto Optic Modulator), forms the optical trap by the +1st diffracted order. The AOM fulfills two functions: it adjusts the beam’s power for axial (z-direction) feedback cooling and alters the frequency of the beam to prevent interference. (27) The s-polarized beam, after passing through the microsphere, is partially reflected by a beam splitter toward a quadrant photodetector (QPD), which detects the microsphere’s motion. Two 532 nm laser beams, each with 80 mW power, are modulated with power by AOMs. These beams irradiate the levitated microsphere along the x and y axes, providing feedback cooling on each axis. The feedback cooling was kept on during the test and cooled the center of mass (COM) temperature of the microsphere down to below 150 μK. Therefore, feedback cooling suppresses Brownian motion, the primary source of system noise, thus significantly improving acceleration sensing sensitivity. Moreover, two high-power 532 nm laser beams with a maximum power of 500 mW for each are directed toward the levitated microsphere along both axial directions, acting as simulation inputs. The power of these beams is modulated by two electronically controlled λ/2 plates and PBSs. During the experiment, a high-vacuum environment of a pressure of 1 × 10^–7 mBar was maintained in the vacuum chamber. According to eq 1, the optimal acceleration sensing noise floor achievable by the system under ideal conditions is 5.89 ng/

\sqrt{Hz}

$\sqrt{Hz}$ .

Measurement Range

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Closed-Loop Control

Upon input of external acceleration to the system, the resulting displacement of levitated microsphere in the absence of feedback control is governed by the following equation: (26)

Δ x = \frac{4 π^{2} a_{in}}{Ω^{2}}

$Δ x = \frac{4 π^{2} a_{in}}{Ω^{2}}$

(2)

where a_in is the input acceleration and Ω is the natural resonant frequency of the levitated microsphere in the optical trap. According to eq 2, a system acceleration input of 1 g causes the microsphere displacement to exceed 1000 μm, significantly surpassing the effective capture range of the optical trap and leading to microsphere escape. Furthermore, significant deviation of the microsphere from its equilibrium position can lead to obvious nonlinear effects within the detection system. This not only influences the system’s sensing performance but also decreases its sensitivity.

To address this, we implemented a closed-loop control scheme. The closed-loop control has been widely applied in many conventional MEMS or capacitive accelerometers. (28−30) The direct aim of this control scheme is to maintain the trapped microsphere at the original equilibrium position, that is, to prevent large displacements of the microsphere by modulating one of the trapping beam powers via an AOM. While the microsphere shifts toward the +z (−z) axis, the AOM correspondingly reduces (increases) the trapping beam’s power to recenter the microsphere. Activation of the closed-loop control depends on the microsphere’s displacement exceeding 10 times the standard deviation of Brownian motion, thus preventing unnecessary interference with the feedback cooling process. For a detailed description of the inner workings of the closed-loop control, refer to Supporting Information Figure S1. With the closed-loop control engaged, the system is capable of maintaining the microsphere’s positional fluctuation within 10 times the standard deviation of its equilibrium state, even under ±1 g external acceleration inputs. The readout acceleration of the system can then be calculated as

a_{output} = β_{vol} (β_{pow} Δ P + V_{detect})

$a_{output} = β_{vol} (β_{pow} Δ P + V_{detect})$

(3)

where ΔP is the change of the trapping power modulated by closed-loop control. β_pow = 127.8 V/W is the calibration factor that relates the QPD detected voltage to the variation in the closed-loop control power. β_vol = 226.2 mg/V is the calibrated detection factor of the QPD obtained by calibration at 10 mbar according to the thermal equilibrium. (31) And V_detect is the signal of the QPD.

Thus, there are two different control schemes in our system: feedback cooling control and closed-loop control. The feedback cooling, as the name implies, reduces the center-of-mass (COM) temperature of the trapped particle, suppressing Brownian motion, which is a primary source of system noise. Therefore, feedback cooling is essential for achieving high-precision acceleration sensing. In addition, according to the simulation results, feedback cooling increases the system’s 3 dB bandwidth from 55 to 139 Hz by suppressing the microsphere’s response near the resonance peak. Band-pass filters were applied to the feedback cooling signal around the resonant frequency to optimize the cooling effect. However, the band-pass filters also prevent the feedback cooling from effectively responding to the microsphere’s slow drifts or quasi-static displacements. On the other hand, the closed-loop control has different functions as

1.	Keep the trapped particle stable. As calculated, the displacement of the trapped particle will significantly surpass the effective capture range of the optical trap under large input without closed-loop control. We apply the closed-loop control to maintain the particle around the origin of the optical trap, preventing the particle from being lost under large acceleration.
2.	Optimize linearity. This is also why many conventional accelerometers employ closed-loop control. The nonlinearity of the detection system and the nonlinearity of the gradient force become significant under large acceleration. (32) On the other hand, the radiation pressure force applied through the closed-loop control is equal to the external input and proportional to the laser power, which has excellent linearity. Therefore, the closed-loop control can optimize the system’s linearity under large acceleration.
3.	Optimize sensitivity. The sensing sensitivity of optically levitated accelerometers directly depends on the feedback cooling performance. In order to achieve the best performance, the optimal parameters need to be determined through numerous experiments. However, under large displacements of the trapped particle, the original optimal parameters may fail, and the feedback cooling may lose its effect. We apply closed-loop control to keep the particle around the origin, which maintains the best performance of the feedback cooling. Therefore, we can optimize the sensing sensitivity under large acceleration.

In summary, we achieve the best sensing performance of our system by combining feedback cooling control and closed-loop control. In fact, these two control schemes (feedback cooling along the z-axis) depend on the same AOM to realize modulation of the particle motion. Therefore, they actually belong to the same generalized feedback control scheme.

Dynamic Test

The introduction of dynamic acceleration input into the system is realized by mounting it on a precision tilt table, as illustrated in Figure 2. This setup aligns the tilt table’s rotational axis with the system’s x-axis. The alignment is precisely achieved by orienting two apertures, centered on the rotation axis, to coincide with the 532 nm feedback cooling beam along the x-axis. This alignment, assured by the precision of machining, maintains an error below 0.001°. The tilt table’s rotation, ranging from −90 to +90°, introduces the gravitational acceleration component as the dynamic acceleration input along the axial (z-axis) direction. Figure 3 (red dots) shows the system output variation as the tilt table rotated from the horizontal position (0 g) to +90° (+1 g), then to −90° (−1 g), and back to 0°. This procedure, performed under a chamber pressure of 1 × 10^–7 mBar and with rotation parameter set to 0.3°/s speed and 0.05°/s² acceleration, ensures the system remains in a quasi-static state to avoid the effects of centripetal acceleration. The system’s output, presenting a sinusoidal response pattern during rotation, accurately reflects the expected characteristics of the input acceleration (Figure 3, top right), demonstrating the reliability of the closed-loop control mechanism. Additionally, it has also been demonstrated that the system can maintain stability under a wide range of attitude tolerance up to ±90°.

Figure 2. Entire optically levitated accelerometer system is mounted on a tilt table with its rotation axis aligned with the x-axis of the optical system. For reference, an accelerometer is also affixed to the tilt table’s edge.

Figure 3. Measurement range. The red scatter plot represents the system’s input–output curve when the tilt table is rotated to input ±1 g, while the blue scatter plot depicts the input–output relationship curve when a simulated acceleration is applied to the microsphere using high-power 532 nm laser beams. As the tilt table’s rotational input approaches ±1 g, the output error increases due to the rise in optical path pointing noise. The increasing of output error can also be observed in the upper left inset, which shows the system’s corresponding signal output curve during the tilt table’s rotation. Additionally, the initial state of the high-power 532 nm laser cannot achieve a 100% extinction ratio, resulting in a deviation of the simulated input curve near zero.

To further ascertain the system’s measurement range, two high-power 532 nm laser beams, coaxially aligned with the trapping beam, are introduced along the optical axis to simulate acceleration inputs through optical force. Incrementally increasing the power of one of the green beams until achieving a system output equivalent to ±1 g facilitated the calibration of the relationship between the 532 nm laser power and input acceleration. Further increasing the green laser power until the microsphere approaches its escape threshold, after which power reduction and repetition of the procedure on the opposite side afford evaluation on both sides. The dynamic range of the system, as demonstrated in Figure 3 (blue dots), spans from −3.2 to +3.5 g, with a nonlinearity of 3%. The system’s measurement range is primarily limited by changes in the trapping force caused by variations in the trapping beam power, which are induced by the closed-loop control method. Because when the trapping force becomes too strong or too weak, it affects system stability and can lead to noise or escape of the levitated microsphere. The deviation near the zero point is because the high-power 532 nm laser beam cannot achieve a 100% extinction through λ/2 plates and PBSs. The asymmetric dynamic range results from a slight imbalance in the 1064 nm trapping beams. Although the measurement range of the system could only be tested under quasi-static conditions due to the modulation frequency limitations of our equipment, the closed-loop control system can achieve a bandwidth of up to 60 Hz. Therefore, we can infer that the measurement range results can be extended to 60 Hz.

Sensitivity

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The sensing sensitivity we evaluated here can also be called resolution, which is the minimum input that the system can sense. The definition of this property in IEEE standard for inertial sensor terminology is the largest value of the minimum change in input, for inputs greater than the noise level, that produces a change in output equal to some specified percentage (at least 50%) of the change in output expected using the nominal scale factor. (33) In real-world applications, this sensitivity is a crucial criterion to evaluate the precision of accelerometers. And only through dynamic input can we evaluate it accurately and reliably.

To evaluate the sensing sensitivity of the system, we input a weak acceleration signal into the system and test its response. We employ the method of rotating the tilt table to input acceleration signals, too. However, constraints imposed by the limited control precision of the tilt table necessitated a different approach. A rigid base was affixed directly beneath the edge of the tilt table, separated by a piezoelectric transducer (PZT), which, through its deformation, induces a subtle rotational adjustment of the tilt table. By modulating the PZT driving voltage, we can achieve a slight modulation of the tilt table’s angle, thereby inputting a weak acceleration into the system. The inputs through the PZT are calibrated using a accelerograph (Namometrics, TitanSMA) installed on the tilt table. (34) According to the calibration, the relationship between the PZT driving voltage V and the input acceleration a_in in the system is a_in = ηV – 0.17 m/s², η = 0.20 m/V·s².

To mitigate the influence of 1/f noise and ground vibration noise, the PZT was sinusoidally modulated at 19 Hz, inputting a continuous sine signal into the system. The alignment of the tilt table axis of rotation with the system x-axis ensures that the microsphere is positioned on the tilt table’s axis, avoiding vertical displacement of the microsphere. Thereby potential interference is obviated from extraneous acceleration inputs and guarantees the exclusive input of the gravitational acceleration component. The system’s output was inferred from the peak on the motion PSD spectrum of the microsphere, as delineated in eq 4:

a_{output} = a_{peak} \sqrt{\frac{2 f_{sample}}{N_{FFT}}}

$a_{output} = a_{peak} \sqrt{\frac{2 f_{sample}}{N_{FFT}}}$

(4)

where a_peak is the modulated peak intensity of the power spectrum, f_sample is the sampling rate, and N_FFT is the number of the FFT points. As depicted in Figure 4b, over a continuous testing period of 1370 s with an input signal of 0.1 μg, identifying a motion peak at 19 Hz. The integration time is chosen to balance the spectral resolution and the Allan deviation (Figure 4a). According to eq 4, the output acceleration value can be derived as 0.116 ± 0.014(syst.) ± 0.003(stat.) μg. As a conclusion, the sensitivity of 0.1 μg is successfully achieved as the result is within the tolerance given by the IEEE standard. (33) The noise floor corresponding to the optimal sensitivity is 3.7 μg/

\sqrt{Hz}

$\sqrt{Hz}$ , which is very close to the actual noise floor of 0.74 μg/

\sqrt{Hz}

$\sqrt{Hz}$ shown in Figure 4b. Furthermore, as illustrated in Figure 5, the relationship observed between diminishing input signals and the consequent reduction in system output signals within the 5 μg range, with a nonlinearity of 17%, (33) underscores the system sensing reliability.

Figure 4. (a) Allan deviation. The bias stability at zero input is 13.2 μg at 334 s. (b) Power spectral density of acceleration modulation. By sinusoidal modulation of the tilt table, a weak acceleration of 0.1 μg with 19 Hz in frequency is input into the system, resulting in a distinct motion peak at the modulation frequency.

Figure 5. System’s corresponding output values when inputting different accelerations. The relationship within the 5 μg range has a nonlinearity of 17%. The primary source of error in the output is misalignment between the axis of rotation and the x-axis of the optical trap system.

Based on the signal-to-noise ratio, the system could theoretically achieve a higher sensing sensitivity. However, under the current testing methods and environment, the minimum acceleration input for the system is 0.1 μg. When the input acceleration drops below 0.1 μg, the nonlinearity in the relationship between the input acceleration and the PZT driving voltage becomes significant. (34) Moreover, the nominal maximum sensitivity of the accelerograph used for calibration is 0.03 μg. Therefore, we conclude that for inputs below 0.1 μg, the input accuracy is insufficient. As a result, the system’s maximum sensitivity test is limited to 0.1 μg.

Discussion and Conclusions

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To actualize the application of optically levitated accelerometers in applications such as navigation, intelligent driving, and industrial automation, it is essential that these systems not only exhibit high sensing sensitivity but also encompass a suitable measurement range and a broad spectrum of attitude tolerance. Figure 6 delineates a comparative analysis between various systems utilized for acceleration sensing. (3,8,15,16,19,35−38) Those encased in dashed outlines signify the methods that have a narrow attitude tolerance range. The asterisk (*) in Figure 6 denotes results obtained from simulated calculations. In our system, the sensitivity simulation assumes that all other noise is effectively minimized, and the noise floor is ultimately limited by the Brownian noise. Then, the corresponding optimal sensitivity was obtained by incorporating the optimal noise floor and the same integration time (1370 s) as in our dynamic tests into eq 1. The optimal measurement range was determined based on the input acceleration corresponding to the point where the trapping light power on one side reaches zero during closed-loop control. The sensitivity of other optically levitated systems shown in Figure 6 was calculated based on their noise floor as well. The shaded areas in different colors represent the performance ranges of accelerometers based on different sensing technologies. This clearly demonstrates the trade-off between sensitivity and measurement range.

Figure 6. Comparison of acceleration measurement ranges and sensitivity among different systems. The dashed outlines signify the methods that have a narrow attitude tolerance range. And the asterisk (*) in the figure denotes results obtained from simulated calculations. The sensitivity of the optical trap methods was assessed using the same method as presented in this paper. Note that for better visualization, the y-axis in the figure has been inverted.

Our system stands out by offering both high sensitivity and a wide dynamic range, along with excellent attitude tolerance, overcoming the typical limitations of narrow measurement range and limited adjustability found in high-sensitivity optical traps. Additionally, the measurement range has been expanded by 3 orders of magnitude compared to previous works, highlighting the system’s significant improvement. This combination of high sensitivity and a broad measurement range also aligns with the critical needs and future development directions in fields such as aerospace, navigation, and intelligent driving, where advanced accelerometers are required to meet demanding performance criteria in terms of both sensitivity and range. A comparative assessment against the well-established MEMS technology reveals that our system not only matches the measurement range but also surpasses the majority of MEMS systems in terms of sensing sensitivity. Moreover, by optimizing the system, the acceleration sensing sensitivity can reach 159 pg, significantly surpassing current standards. This positions our work as a pivotal advancement for propelling the application of optically levitated accelerometer into a broader spectrum of real-world applications across the aforementioned fields and helps to fulfill the escalating demand for heightened sensitivity in those acceleration sensing applications.

The sensing properties of our system are currently influenced by two factors. The first pertains to the challenge of maintaining the microsphere’s stability during extensive attitude modifications. This challenge arises from the system’s current design, which predominantly employs a spatial optical path configuration with all optical components vertically anchored to an optical plate. This mounting strategy, although standard, leads to an increased vibration amplitude of the components when the system’s attitude is significantly altered. This, in turn, amplifies the laser pointing noise, adversely impacting the system stability. A promising solution to this challenge lies in the transition to a fiber-optic system, which not only promises to mitigate these stability issues but also offers the advantages of system miniaturization and enhanced robustness.

The second challenge involves reducing the noise floor, especially at low frequencies. The current system’s acceleration sensing sensitivity, although significantly improved, still falls short of its theoretical potential, as delineated in eq 1. This discrepancy can largely be attributed to the high noise floor at low frequencies. The main sources of low-frequency noise are 1/f noise, ground vibrations, air turbulence, and fluctuations in temperature and humidity. Based on experience in conventional accelerometer design, these noise sources are expected to be mitigated or even eliminated through packaging, integration design, or by replacing the free-space optical path with optical fibers. (36,39,40) Therefore, these solutions are the key directions for our future work. After adopting these mature industrial designs, the system’s sensitivity and Allan deviation are both expected to improve significantly, with a lower and flatter noise floor, bringing it much closer to practical applications.

This paper presents a novel optically levitated accelerometer utilizing a dual-beam trap with a 25 μm diameter microsphere that is stabilized during substantial acceleration inputs through the implementation of a closed-loop control strategy. Utilizing dynamic inputs, we have convincingly confirmed the system’s outstanding sensing performance, with an acceleration sensing sensitivity of 0.1 μg (@19 Hz), and a measurement range that surpasses ±1 g. Furthermore, the system has demonstrated remarkable stability across a wide spectrum of attitude tolerances, up to ±90°, which is a critical requirement for various technological applications. The contributions of this work are significant, as they establish a robust foundation for the practical applications of optically levitated systems with ultrahigh acceleration sensing capabilities into a myriad of fields. These include but are not limited to aerospace, navigation, intelligent driving, industrial automation, and smart devices. The system’s exceptional sensing performance, combined with its stability and broad applicability, positions it as a promising tool for advancing the state-of-the-art in precision sensing technology.