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Abstract
Displacement measurement using a D-shaped mirror is a key technology in optical tweezers, which have emerged as an important tool for precision measurement. In this paper, we first study the influences of installation errors for the D-shaped mirror on the displacement measurement. The calibration factor and sensitivity of the different installation parameters are quantified. The results show that the variation of the calibration factor obeys the cosine curve with the angle error, and the sensitivity increases exponentially with the translation error. Besides, we find that the translation error will also lead to crosstalk between transverse and axial displacement. Our work will contribute to improving the performance of optical tweezers for the application in precision measurement and basic physics.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Following the pioneering work of Ashkin [1–3], optical tweezers (OTs) have been made great progress in precision measurement research [4–7]. Owing to the decoupling from the environment noise such as vibrations and collisions with residual gas molecules, the levitated particles in the high vacuum have been used to achieve zeptonewton force detection [8], or nano-g acceleration sensing [9]. In addition, femtogram mass measurement [10], millicharged particle detection [11], and photon recoil measurement [12] have also been accomplished. These advances rely on a high-performance displacement measurement technique for the levitated particles.
One particularly common method to detect particle displacement is to send the scattered beam onto a four-quadrant detector. The method has made the sensing accuracy reach sub-piconewton and sub-nanometer regimes [13–15]. Another method is using a spatial homodyne detection that splits the scattered beam into halves. Then they are focused onto both ends of the balanced detector. This method has demonstrated that the displacement sensitivity was improved over one order of magnitude than that of the four-quadrant detector [16]. Meanwhile, the sub-nanometer spatial resolution has been achieved for Brownian motion of particles [17]. Tongcang Li et.al. first have measured the instantaneous velocity of a Brownian particle using this method based on a sharp edge mirror (hereinafter referred to as a D-shaped mirror) [18]. That has enabled the displacement sensitivity to fm·Hz-1/2 magnitude [19] and promoted the rapid development of OTs in vacuum [20]. However, the installation errors of the D-shaped mirror may amplify the displacement measurement noise. And these errors will be detrimental to feedback cooling and precision measurement of the levitated particle.
In this paper, we first study the influences of the D-shaped mirror on the displacement measurement for the spatial homodyne detection. We further quantify the variations of the calibration factor and sensitivity caused by the installation errors of the D-shaped mirror based on T-matrix method. The simulation results are proved through trapping and rotating the particle in the OTs. Furthermore, we find that the installation errors of the D-shaped mirror will also introduce crosstalk between the degrees of motion.
2. Principle
2.1 Measurement setup
Figure 1(a) is a schematic of the displacement measurement setup. A mesoscopic particle is trapped in a counter-propagating dual-beam traps composed of the focused S- and P- polarized beams. A polarizing beam splitter (PBS) reflects the forward scattered beam from the trapped particle, and then the beam is divided into three probing beams by two beam splitters (BS1, BS2). These probing beams provide information about the particle’s 3D displacements. Two D-shaped mirrors (DX and DY) equally split the probing beams into halves, which are focused onto both ends of the balanced detectors for measuring transverse displacements. In addition, two ends of PDBz are located on the focal plane of lens L6 and outside that of L5 to measure the axial displacement, respectively.
Fig. 1. (a) The schematic of displacement measurement. (b) The diagram of the D-mirror splitting the probing beam. The black thick and dotted lines denote the actual and ideal installation positions for the D-shaped mirrors, respectively.
Angle and translation errors are inevitable when we install the D-shaped mirrors to split the probing beam, as shown in Fig. 1(b). Here, the rotation and translation of the D-shaped mirrors are only considered in the same plane. The Gaussian-like profiles represent the cross-sections of the probing beams, and the black solid and dashed lines are the actual and ideal installation positions, respectively. α and d denote the angle error and translation error.
2.2 Principle of the displacement measurement
The incident optical field is decomposed into the expansion of the vector spherical wave functions given by a series of coefficients [21,22]. We apply the T-matrix method to calculate the expansion coefficients and transformation matrices [23]. Then the probing optical field is determined by these coefficients and matrices, and as [24]
Measurement sensitivity of the detector is usually limited by the electronic and shot noises. Some methods can reduce the electronic noise, such as the differential measurement means [17,26], and the optical lock-in particle tracking scheme [25]. In the shot-noise limit, a gain Gx is given by the coefficients of an N-th order polynomial, which fits the responded photocurrent ix in a least-squares method for small displacements along the x-direction [27]. Combining the signal-to-noise ratio, the minimum resolvable displacement in units of m·Hz−1/2 is
where $\langle {{i_T}} \rangle$ is the mean total photocurrent generated by the beam hitting the detector.3. Simulation and experiment
3.1 Simulation results
A numerical model is applied for the laser’s diameter of ω = 5 mm, the wavelength of λ = 1064 nm, the laser power of P = 100 mW, the condenser and trapping lens of numerical aperture NA = 0.34, the particle radius of r = 2 µm and the particle refractive index of n = 1.46. For the particle’s linear displacement, calibration factor C = |I / x | [30] is introduced to quantify the D-shaped mirror. Where I and x are the photocurrent and the particle’s displacement.
Figure 2(a) shows the photocurrent as a function of the particle’s displacement along the x-direction for the different angle errors. The calibration factor gradually declines with α, as indicated in the green circle of Fig. 2(b). The angle error can cause the photocurrent to be coupled onto the y-direction, as exhibited in Fig. 1 (b), resulting in the decrease of the calibration factor in the x-direction. The x- and y-directions are orthogonal when we use the D-shaped mirror to split the probing beam. Therefore, we fit the calibration factor using a cosine formula $C = {C_0}cos(\alpha )$, where C0 is the calibration factor when α = 0. The simulation result is consistent with the cosine function curve.
Fig. 2. Influences of the angle error α on the displacement measurement. (a) The photocurrent as a function of the particle’s displacement along the x-direction. (b) The calibration factor under the different angle errors.
The variation of photocurrent concerning the translation error d is indicated in Fig. 3(a). Here, d is normalized by the probing beam size, with a dimensionless unit. To quantify the relationship between C and d, the relative deviation of the calibration factor is introduced and written as
where C0 and C1 are the calibration factors when d = 0 (a.u.) and d ≠ 0 (a.u.), respectively. And the relative deviation is shown in the illustration of Fig. 3(a). The relative deviation is increased and is a similar quadratic curve with the absolute d. The translation error will enhance the laser common-mode noise of the photocurrent, leading to a rise in the relative deviation.Fig. 3. Influences of the translation error d on the displacement measurement. (a) Photocurrent as a function of the particle’s displacement along the x-direction. (b) Crosstalk between the transverse and axial displacement when the particle moves along the z-direction.
Furthermore, we simulate the photocurrent versus the particle displacement along the z-direction when x = 0, as exhibited in Fig. 3(b). The results show that the translation error can cause crosstalk between the axial and transverse photocurrents, and the crosstalk is raised due to the increased d. The translation error prevents the probing beam from being evenly split by D-shaped mirror, resulting in the slope of photocurrent at x = 0 and the crosstalk between the degrees of motion. However, the D-shaped mirror with the angle error will split the probing beam evenly and therefore introduce no crosstalk.
3.2 Experiment results
In our experiment, a silica particle (r = 2 µm) is stably trapped in the dual-beam trap. The displacement of the Brownian particle is measured by a balanced detector, whose output voltage is recorded at a sampling rate of 2 MHz. We adjust the installation distance of the D-shaped mirror in the x-direction.
Figure 4(a) shows the power spectral density (PSD) of the output voltage when d = 0 (a.u.) and 0.4 (a.u.). The illustration of Fig. 4(a) is a magnification of sensitivity in high frequency. The calibration factor between voltage and displacement is acquired by fitting the PSD with the Lorentzian function [28]. Then the sensitivity of the displacement measurement with the unit nm·Hz-1/2 can be obtained. We utilize Eq. (3) to simulate the theoretical sensitivity. The experimental and simulation results are exhibited in Fig. 4(b). The results show that the sensitivities decline when the absolute d increases. That is attributed to increasing common-mode noise of the output voltage. But there are errors between the experiment and simulation due to the experimental electronic noises.
Fig. 4. The Experimental results of the restricted Brownian particle. (a) PSD of the output voltage when d = 0 (a.u.) and 0.4 (a.u.). (b) The sensitivity of displacement measurement in the simulation and experiment.
The influences of the translation error are demonstrated by using the orbiting particle, which is trapped in the dual-beam trap with a transverse offset [29]. The output voltage is recorded at a sampling rate of 500 kHz, as shown in Fig. 5(a). It reveals the rotating period in the time domain when d = 0 (a.u.). The noise of the voltage is enhanced as d increases to 0.4 (a.u.). Figure 5(b) indicates the power spectral density of the voltage. The result finds that the rotating peaks above the double frequency (2f0 = 9600 Hz) are covered by the noise when d increases. Where, f0 is the fundamental frequency of the orbital motion for the particle. Besides, the amplitude of the fundamental frequency for d = 0 (a.u.) is higher than that for d = 0.4 (a.u.) and vice versa in the amplitude of the double frequency. These results arise from the laser common-mode noise enhanced by the translation error.
Fig. 5. The Experimental results of the orbital rotating particle. (a) Two typical output voltages in the time domain. (b) PSD of four typical output voltages. (c) The filtered displacement. (d) The relative deviation of the calibration factor in the experiment and simulation.
For small displacement of the trapped particle, the output voltage U is proportional to the particle displacement (x), i.e. U = ζx, where ζ is the calibration factor between the voltage and displacement, with a unit of V/ m. ζ is obtained directly by the energy equipartition theorem [30]
where M and v are the mass and velocity of the particle, kB is the Boltzmann constant, and T0 is the environment temperature. Due to the surface morphology characteristics of the particle such as rough, irregular, the orbital rotating motion appears the phenomenon of multiple frequencies, which are superimposed on the fundamental frequency motion. The calibration factor is calculated by the one-dimensional function between the voltage and displacement. Hence the displacement signal of Fig. 6(a) retains the rotation feature with fundamental frequency (f0 ∼ 4800 Hz). Combining Fig. 5(a) with Eq. (5), we get filtered displacement with the fundamental frequency, as demonstrated in Fig. 5(c). The relative deviation of the calibration factor is denoted in Fig. 5(d). These results prove that the relative deviation of the experiment agrees well with that of the simulation in Fig. 3(a). The errors between simulation and experiment results are mainly caused by the optical components such as lenses, mirrors, polarizing beam-splitter prisms and detectors etc, which will introduce low-frequency noise and electronic noise, etc. in the actual experiments. In the above discussions, we quantify the influence of the displacement detection caused by the D-shaped mirror. That can guide our experiment to debug the displacement detection system. We can confirm the minimum sensitivity or the relative deviation R by adjusting the angle and distance of the D-shaped mirrors, leading to the best installation position of the D-mirror in the experiment. As a result, it will improve the accuracy of the trapped particle’s displacement measurement to the greatest extent.4. Conclusion
In this paper, the installation errors of D-shaped mirrors are quantified in our simulation and experiment for the first time. We fully discuss the influences of the angle and translation errors on the displacement measurement by using the T-matrix method.
The angle error can flatten the photocurrent curve of the displacement measurement, resulting in the reduction of the calibration factor. The variation of the calibration factor obeys the cosine curve. Moreover, the translation error will introduce a crosstalk between the axial and transverse displacements. This crosstalk is enhanced with the translation error. Finally, we establish a displacement system for demonstrating the simulation results of the translation error. The experimental results indicate that the sensitivity increases exponentially with the translation error. Besides, the relative deviation of the experimental calibration factor agrees well with that of theoretical simulation.
Our work has strong practical value for improving the sensitivity and reducing the error of the displacement measurement. It will benefit the further applications of OTs in precision measurement and basic physics.
Funding
National Natural Science Foundation of China (11904405, 61975237); Scientific Research Project of National University of Defense Technology (ZK20-14); Independent Scientific Research Project of National University of Defense Technology (ZZKY-YX-07-02).
Acknowledgments
The authors wish to thank the anonymous reviewers for their valuable suggestions.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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