The ability to characterize static and time-dependent electric fields
The levitated nanomechanical resonator exhibits unique advantages in the development of precise force [20–22] and acceleration sensors [23,24] at the micro- and nanoscale, attributed to its high-sensitivity and potential for miniaturization . The nanomechanical system optically levitates the charged dielectric nanoparticle in high vacuum, thus making it a harmonic oscillator sensitive to the surrounding electric field. In case of a weak electric field, the harmonically driven response of the oscillator’s displacement is directly proportional to the electric intensity at its location and the net charge it carries. Therefore, on the premise of knowing the net charge, ultra-high force detection sensitivity means ultra-high electric field detection sensitivity.
In the present study, we extend previous works on highly sensitive force detection using an optically levitated nano-resonator  to a novel, three-dimensional, high-sensitivity electric field measurement technology. Using the parallel plate electrodes as the source of the electric field with known frequency, motion signals of the nanoparticle in the three orthogonal directions are used to measure the electric field vectors of the corresponding axis. By changing the relative positions of the nanoparticle and the electrodes, the electric field of the electrodes is scanned point-by-point, and the three-dimensional electric field mapping ability of the scheme is demonstrated. By applying parametric feedback at (), the force and electric intensity detection sensitivity equivalent from the measured displacement spectral density reach the orders of and , respectively. In addition, we demonstrate the measurement of a near-resonance frequency electric signal with a linear range of more than 91 dB. This work may provide an avenue for developing optically levitated nano-resonators into high-precision, continuous broadband electric field sensors.
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2. EXPERIMENTAL SETUP
As shown in Fig. 1, the predetermined electric field is generated by applying sinusoidal voltage onto the simplest parallel plate electrodes, and the optically levitated nanoparticle placed within the electric field produces a displacement response to the field. Though this experimental apparatus of the present study is similar to that in Refs. [26,27], it differs in that its electrodes are composed of two horizontal steel (40CrMoV5) needles that are 1 mm in diameter and placed 2.52 mm apart. This allows for producing more distinguishing changes in electric field distribution around the light field. Similar to most previously published studies, the electrodes in Refs. [28,29] are used to calibrate the nanomechanical parameters such as particle mass and the conversion factor from detection voltage to displacement, where the FDTD numerically simulated value of the electric intensity is employed as a known constant. In this study, however, electric intensity generated by the electrodes is no longer a presumed parameter, but a parameter to be detected. To obtain triaxial electric intensity components at each point, an independent triaxial position detection scheme is built to obtain the motion signal along each axis. The electric driving signal is then loaded onto the electrodes, while being synchronously input into the phase locked loops (PLLs) as a reference signal. The PLL extracts the signal components with the same frequency from the input motion signals of three axes. For stable levitation and suppression of frequency fluctuation in high vacuum, a triaxial parametric feedback scheme sums up all the feedback signals and drives a single acousto-optic modulator to cool the center of mass motion of nanoparticles.
Figure 1.Schematic of the experimental setup. The setup consisted of a single-beam optical trap, triaxial position detection and parametric feedback scheme, electric driving, and field measurement circuit. OBJ, NA microscope objective; SNE, horizontally placed steel needles; AOM, acousto-optic modulator; AL, aspheric lens; QPD, self-developed quadrant photodetector. Here shows the top view of the setup, and the
A. Three-Dimensional Electric Field Vector Detection
The electric intensity can be deduced from the driven displacement response and the parameter of the nano-resonator. The relationship between electric field component and displacement response of the corresponding axis (taking the axis for example) is as follows:
We first moved the electrodes with a nano-positioning stage to place the particle at the symmetrical midpoint of the electrodes and measured the electric intensity at that point. As shown in Fig. 2, the normalized measured value of three orthogonal components are , , and , respectively, corresponding to the case where the voltage amplitude applied at the electrodes is 1 V. The electrodes are basically capacitive, and the measured equivalent impedance is about 3 pF, which means that the response electric field within 1 MHz is almost frequency independent. According to the simulation result of COMSOL, three electric intensity components at this point are , , and , respectively, which deviate slightly from the measured values. The discrepancy between the measured and theoretical values may be a result of manufacturing error and alignment error of two steel needles, as well as the relative position error between the symmetrical midpoint and the nanoparticle.
Figure 2.Measured electric intensity at the symmetrical midpoint of the electrodes. A nanoparticle with a diameter of 142.8(33) nm and charge of
B. Three-Dimensional Electric Field Mapping
Taking the above position as the center point, we moved the relative positions of the nanoparticle and the electrodes along three orthogonal axes and obtained the component of electric intensity at each point, as shown in Fig. 3(a). The variation trend of along each axis is consistent with the theoretical simulation results. In addition, the other two components can also be measured by using the motion signals of the other two axes in the same way and comparing them with the simulation results.
There may be two causes for the deviations of the theory to the measurements near the edges of the vertical and horizonal positions. First, there is a deviation between the simulation and the actual situation. The simple simulation is based on the case that the relative position of the electric field and the center of the optical trap is fixed. However, the relative position of the moving electrodes with the objective and lens changes in experiment, and the measured electric field may be affected slightly by the zero potential of these metal devices.
Second, the initial position error of the nanoparticle in the electric field may be another cause. The nanoparticle should be placed initially in the symmetric center of the electrodes. In the plane, we adjusted the position of the particle as close to the symmetric center of the electrodes as possible by the CCD imaging above the chamber. However, in the plane, the imaging method could not work for alignment, so we measured at different positions by changing the electrode position along the axis. Based on the trend of simulated in the axis as shown in Fig. 3(a), the position with the largest measured was regarded as the middle position in the axis. However, when there was alignment error (either translation error or rotation error) between electrodes, the method would cause certain error to the initial position.
Three-dimensional electric field mapping was realized by obtaining the resultant vectors of each component at different array points. Taking the case of the plane (section ), the resultant vectors of and components in this plane were measured, as shown in Fig. 3(b), together with the results of COMSOL simulation. The parameters of the simulation model and the simulation process are detailed in Appendix B.
C. Noise Equivalent Electric Intensity
The electric intensity detection sensitivity of the nano-resonator depends on its force detection sensitivity, which can be improved by restraining thermal noise in high vacuum. But the accompanying frequency fluctuation in high vacuum would increase the complexity of model fits from the thermal noise response and the electric driven response near the resonance frequency . Both issues can result in significant inaccuracies in the conversion from displacement to electric intensity with a calibrated transfer function. By applying feedback cooling, the nano-resonator can be levitated stably in high vacuum, and the frequency drift effect caused by the non-linearity of the optical trap can be suppressed, which makes the nano-resonator more stable in response to near-resonance driving forces. Therefore, feedback cooling is indispensable for realizing ultrasensitive electric field detection, although theoretically it does not improve the detection sensitivity at certain pressure conditions (see Appendix C). The displacement noise floors of nano-resonators were measured at different pressures, where the electrodes and other metal structures in the chamber were grounded to isolate the residual electric field. The resulting displacement spectral densities in high vacuum for two nano-resonators with parametric feedback cooling are shown in Fig. 4(a). The fits of the displacement spectral density to the expected thermomechanical noise response superimposed on the detection noise for the nano-resonator with different feedback damping show close agreement. Beside the axis resonant frequency, the additional modes originating from cross talk of other axes are generally visible in the thermomechanical noise response.
Figure 4.(a) Displacement spectral densities for nano-resonators in high vacuum. Gray dashed line, detection noise; light dashed line, fit to the thermomechanical noise model; dark solid line, superposition of thermomechanical noise and detection noise, that is, theoretical transfer function of nano-resonator. At
The thermal noise dominates over frequency range near resonance while the noise floor closely approaches the optical shot noise limit over that far from resonance. Comparing the displacement spectral density at and , a reduction in gas damping, due to the balance between the thermomechanical noise and shot noise, the frequency range over which the spectral density is thermal noise limited, is clearly narrowed. The theoretical force transfer function of the nano-resonator can be calculated based on the resonant frequency and damping as follows:
The displacement spectral densities in Fig. 4(a) are converted to a noise equivalent force (NEF) by dividing the displacement spectral densities by the amplitude of theoretical transfer function above, as shown in the right side of Fig. 4(b). Further, the noise equivalent electric intensity (NEEF) can be obtained by dividing the NEF by the charges as shown in the left side.
As expected, the NEF and NEEF both reach the thermal noise limit near resonance frequency. When the damping is lower in higher vacuum, a lower thermodynamic limit could be provided, meanwhile, which is more difficult to reach since the thermomechanical noise must be above the shot noise. The minimum NEEF reaches the order of at , corresponding to the case of , which is lower than that at by more than 1 order of magnitude, and can be further reduced by increasing the net charge or the vacuum level. The bandwidth over which the NEEF is within 3 dB above the thermodynamic limit is 48.6 kHz and 1.1 kHz for two pressures, respectively. This could be further broadened by 1 order of magnitude by adopting a heterodyne detection scheme and optimizing the detection noise to approach the standard quantum limit .
As a comparison, the achieved minimum detectable field is superior to the reported performance using Rb Rydberg atoms by 1 order of magnitude , approaching the equivalent performance for an antenna dipole electronic sensor with length of 1 cm . One benefit of optically levitated nano-resonators is that the bandwidth of interest within which the thermal noise is above or equal to the shot noise is tunable, and the tunable bandwidth can reach the order of tens of kHz or even hundreds of kHz by tuning the power of trapping beam. For example, realistically one could trap with as little as 50 mW and up to 1 W of laser power, and the corresponding resonant frequency could change from about 100 kHz to about 400 kHz. In contrast, traditional passive dipole electronics usually need to change the structure size to achieve similar effects.
D. Linearity and Linear Range Analysis
As a test of sensing performance for linearity and linear range of electric field sensing in axis, the nano-resonator was moved back to the center point and charged with a high value of (see Appendix D). The measurement was performed at and by applying a sinusoidal electric field with frequency of 140 kHz, which is 10 kHz offset from the resonance to reduce the effect of frequency instability on the measurement. We set a specific driving voltage as a reference first (e.g., 5 V) and recorded the corresponding response of the nanoparticle, as a fiducial value for judging whether the charges would change during the experiment. After applying different driving voltages each time, we changed the driving voltage back to the reference voltage and observed whether the response is different from the fiducial value. When changing the pressure, we also applied a similar method that used a monitoring response value to make sure the charges remained constant. At , we applied driving voltages ranging from 1 V to 160 V and for higher electric field sensitivity at , the same operation was conducted except that the range of driving voltages ranged from 5 mV to 500 mV. Finally, the measurement results at different driving voltages were converted to the measured electric intensities by using Eq. (1) and measurement time of 1 s, as shown in Fig. 5(a).
Figure 5.(a) Linearity and linear range measurement of the nano-resonator. The
Within the corresponding measured electric intensities ranging from 1.03 V/m to 36.2 kV/m, which span over 4 orders of magnitude (91 dB), the linearity of electric field sensing was within 10%. The minimum detectable electric field is mainly limited by the detection sensitivity of the nano-resonator at . While further reduction in pressure can lead to higher detection sensitivity, the accompanying instability and drift of resonance frequency become more pronounced, resulting in increased uncertainty and deviation of the measured value. The maximum detectable electric field in this measurement was merely limited by the maximum output of the high-voltage amplifier (Aigtek ATA-2031), and its theoretical limit is related to the linear range of optical force and the capture region of trap  and is ultimately limited by the response range of the detection scheme to nanoparticle displacement. Figure 5(b) shows the simulated detector signal in terms of balanced power with particle displacement. For the maximum electric intensity measured in experiment, the amplitude of the electric field force acting on the nanoparticle is about 0.57 pN. Combined with the stiffness of the trap, the corresponding maximum amplitude of particle displacement reached about 0.21 μm, which is still within the linear region of the detector with nonlinearity of . The linear range of the detector with 10% nonlinearity would be about 0.38 μm, corresponding to a larger detectable electric field of value 65.5 kV/m. Theoretically, we can further improve the upper range of the detectable electric field by reducing the charges of the particle. For example, if the charge is reduced to , the corresponding maximum electric field may reach 6.5 MV/m. This is an extreme situation where the escape of nanoparticle, air breakdown, and other issues need to be considered.
We have demonstrated a high-sensitivity electric field measurement technology using optically levitated nano-resonators. By scanning the electric field distribution between parallel electrodes, the three-dimensional electric field mapping capability of the system was demonstrated. Its measuring spatial resolution depends on the motion amplitude of the nanoparticle in the equilibrium position and the manipulation accuracy of the equilibrium position, which can reach the order of nanometers. Broadband measurement at the thermodynamic limit yields a noise equivalent detection resolution of the order of in high vacuum, which is competitive to that of previously reported electric field detection schemes. Linearity analysis near resonance shows a linear range of more than 4 orders of magnitude.
Having higher net charges is the key to further improve the detection resolution of nano-resonators. Although this can be achieved simply by using larger particles, for example, the net charge of the micron-sized particle can reach the order of , and the resulting force detection sensitivity is worse due to larger mass. Therefore, the size of the particle needs to be optimized according to these two factors to obtain the optimal electric field detection sensitivity. Although this work is based on optical levitation systems, charged particles in other levitation systems are eligible to be developed into highly sensitive electric field sensors. The advantage of the levitated resonator is that its resonant frequency can be adjusted from Hz to MHz according to size of particle and stiffness of the potential well to meet the application requirements of different frequency bands, especially low-frequency submarine communication.
Our method is most similar to electric field sensing with trapped ions that use mechanical oscillators as exquisite quantum tools to measure small displacements due to weak forces and electric fields, albeit with a very different charge to mass ratio. As schemes with optically levitated nanoparticles do not need an additional DC or AC electric field for a stable trap that is generally utilized in ions schemes, it can eliminate the influence of the existing electric field in the device on the electric field measurement as much as possible. Furthermore, the ion trap scheme is generally sensitive to the electric field in a certain direction, such as the direction of the magnetic field used to generate the cyclotron motion, while a single nanoparticle can be used to measure the three-dimensional vector electric field at the same time, which would be an obvious advantage of this method.
APPENDIX A: ELECTRIC FIELD SENSING WITH A HARMONIC OSCILLATOR
A major benefit of the electric field sensor described in this work is that its dynamic response closely follows that of a one-dimensional viscously damped harmonic oscillator, making it possible to convert from measured nano-resonator displacement to an equivalent electric intensity using a low-order model. In this section, we describe the harmonic oscillator model and the conversion between displacement and electric intensity. Much of the analysis in this section follows directly from the work of Ricci
The simplified diagram of electric field sensing with a optically levitated nano-resonator is described in Fig.
Figure 6.Using optically levitated nanoparticle as nano-resonator to measure the electric field intensity near beam focus.
Figure 7.PSD of motion signal along the
APPENDIX B: ERROR ESTIMATION AND COMSOL SIMULATION OF ELECTRIC INTENSITY
Obviously, reducing the uncertainty of the nanoparticle mass can improve the accuracy of the electric field measurement. It was observed that both the density and radius of nanoparticles may vary when pressure changing (especially near 1 mbar), which was mainly caused by the separation of surface functional groups [
Figure 8.Partial display of TEM result of particles from Nanocym. The measured diameter of each particle is indicated in the figure, and the mean value and deviation is 142.8(33) nm.
Figure 9.Pseudo-color maps of the potential applied to the electrodes and the intensity of the generated electric field. (a), (b) and (c), (d) correspond to the simulation results in the horizontal section (
Figure 10.Noise equivalent displacement and electric intensity for varying optical shot noise level. (a) Noise equivalent displacement combining thermomechanical noise and optical shot noise at three different shot noise levels.
In experiment, we used an open-loop electromotive positioning stage (Newport AG-LS25), whose travel range is 12 mm with an absolute positioning accuracy of 100 μm. The step displacement is 50 nm, and we moved 1200 steps and 2000 steps, respectively, to obtain 60 μm and 100 μm sampling interval in two directions. Therefore, the cumulative errors for the actual position of nanoparticles were 0.5 μm and 0.83 μm, respectively.
APPENDIX C: NOISE EQUIVALENT DISPLACEMENT AND ELECTRIC INTENSITY
By extending the drive frequency to the broadband, the relationship between the harmonically driven displacement, , and driving electric intensity, , as a function of frequency, , can be determined from Eq. (
Here, represents the transfer function between the displacement of the nano-resonator and the driving electric field, which is related to the general force transfer function of the nano-resonator via its net charge:
Further, similar to the force detection sensitivity of the nano-resonator, the equivalent electric intensity due to thermal noise is then
Obviously, is only a function of the resonator parameters (, , , and ) and not a function of frequency, meaning that the thermomechanical noise floor in terms of electric intensity is flat.
When parametric feedback cooling is applied to the nano-resonator, the feedback cooling term is added to Eq. (
The equivalent electric intensity due to thermal noise with feedback cooling is rewritten as
Here, represents a lower equivalent temperature noted as
Interestingly, can be deduced from Eqs. (
Here, and are the transimpedance gain and responsivity of the photodetector, while is the calibration factor. The shot noise in terms of electric intensity is
Unlike the equivalent electric intensity in terms of thermal noise, is a function of frequency and gets the minimum value at the eigenfrequency of the resonator. Since the thermomechanical noise and shot noise are uncorrelated, they can be summed to get the total noise equivalent displacement or electric intensity . Although the optical shot noise does not represent real motion of the nano-resonator, it is detection noise that analytically refers to either displacement or electric force. The best-case scenario for a nano-resonator with fixed parameters is for the thermomechanical noise to be higher than the optical shot noise, which can be done by tuning the power of trapping beam. Within the bandwidth of interest, the optical readout will measure the motion of the resonator with minimal contribution from shot noise. This is shown in Fig.
APPENDIX D: CONTROLLING THE NET CHARGE ON THE NANOPARTICLE
The net charge on the nanoparticle was controlled by corona discharge generated by high-voltage () electrodes placed about 5 cm away from the nanoparticle in the chamber. A plasma of both positive and negative ions was created and adsorbed on the nanoparticle to change the charge [
Figure 11.Controlling the net charge on nanoparticle. Top, to obtain the voltage step of a single charge by multiple short discharge processes. At 1 mbar, a drive signal with an amplitude of
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