1. Introduction

Optical tweezers were conceived as tools capable of harvesting radiation pressure to hold and manipulate tiny objects [1,2]. Throughout the past three decades, optical trapping has encountered a vast number of applications in different research fields [3]. They might, for instance, be used in biology to measure physical properties of cell membranes [4,5] or manipulate living microorganisms [6,7]; in chemistry, to build and trap single molecules [8,9], and in physics, as precise force sensors [10,11] or to achieve quantum ground state cooling [12,13]. To perform all of these tasks, it is necessary to quantitatively know the force exerted by the light beam on the trapped particle.

While the behaviour of the force as a function of particle displacement can be predicted for different trapping beams [14–16], its numerical value is sensitive to a number of experimental variables, such as the medium’s viscosity [17] and spherical aberrations [18,19]. This often makes it hard to exactly calculate the relevant forces from first principles [20,21]. Instead, these are experimentally measured during the tweezer calibration. For a Gaussian beam the trapping potential is harmonic and one of the goals of calibration is to find the trap’s spring constant.

Due to randomness introduced by the medium in which the trapped particle finds itself, the spring constant is usually obtained by measuring statistical quantities associated to the particle’s position. Among these quantities there are the autocorrelation function [22] and the power spectral density [23] of the particle’s position. Whichever quantity is chosen, one needs to be able to measure the particle’s displacement as a function of time. This can be done in a number of ways.

For example, video imaging [24] is useful when dealing with multiple trapped particles in holographic tweezers [25]. The detection of the forward and back-scattered light on the other hand enables measurements of the position at higher sampling frequencies [26,27]. In this paper, we will focus on the use of forward-scattering detection to measure the power spectral density of a trapped particle’s position. In this type of setup the standard experimental procedure consists in collecting the light scattered by the bead using an objective lens and directing it onto a position sensitive detector - denoted throughout this work by the acronym PS. Position sensitive detectors allow monitoring of the position of the light beam with respect to a calibrated center similar to what can be done using a quadrant photo-detector.

The PS detector outputs analog signals as a function of time denoted $X, Y, S$. The signals $X(t)$ and $Y(t)$ vary according to changes in the beam position with respect to the detector’s center, which in turn correspond to motion of the particle in the $x$ and $y$ directions, respectively. Since it is expected that $X(t), Y(t)$ also suffer variations due to fluctuating laser power, it is required to normalize them, sample-by-sample, by the signal $S(t)$, the sum of total intensity at the detector. All signals are provided by the detector in units of mV. The signals can be collected by an oscilloscope or a data acquisition board for post-processing, i.e. dividing $X(t), Y(t)$ by $S(t)$ for each sample point and numerically calculating the resulting auto-correlation function and power spectral density (PSD). These can be used to find the desired spring constants after fitting the data to a Lorentzian function and using the proportionality relation between the trap stiffness and the Lorentzian’s corner frequency [28].

In this work we propose simplifications to this procedure which reduce the standard hardware and computational requirements. First, we will show that under reasonable approximations, the radial particle position can be directly obtained from the signals $X(t)$ and $Y(t)$, making the step of dividing signals at each sampling unnecessary. Next we show that excessive electronic noise in the detection channels can be accounted for by adapting the function used to fit the measured power spectral density. This generalizes a method proposed in the context of video imaging [29], and allows for the use of inexpensive data acquisition devices. Finally, still motivated by the interest in reducing the cost and complexity of an optical tweezer setup [30,31], we propose a knife-edge method that aims to substitute a position sensitive detector by regular power-sensing silicon detectors. These detectors also present the advantage of having an increased bandwidth which can be exploited in statistical mechanics experiments involving trapped particles [32].

In the next sections, we present a brief overview of the analysis of a trapped particle’s motion, followed by the theory behind the proposed calibration procedure and the above-mentioned knife-edge detection system. We then describe our experimental apparatus and use it to verify both the calibration as well as the knife-edge method. We do that by measuring the power spectral density of the particle’s motion and comparing it to the standard calibration of an optical tweezer. We close with a brief discussion and the conclusions of this work.

2. Motion of a trapped particle

In this section we briefly review the theory behind the motion of a trapped dielectric particle immersed in a fluid. With this we establish the relation between corner frequency and the measured parameters in the remaining of this work.

When the particle, considered here as a sphere of radius $r$, is trapped in the focus of an optical tweezer, its motion in a direction $x$ orthogonal to the $z$-axis along which the beam propagates is described by the Langevin equation [23],

For a spherical particle of radius $r$ we have $\gamma _0(T)=6\pi \nu (T) r$, where $\nu (T)$ is the fluid’s viscosity at a temperature $T$. Because the mass of the particle increases with the volume ($m \propto r^3$) and the drag coefficient increases with radius ($\gamma _0(T)\propto r$), the inertial term in 1 is often neglected for small particles [23]. Under this approximation, the PSD of the particle’s position - evaluated over a time interval $\Delta t$ - is given by a Lorentzian function,

In the scenario treated in this work, a quantity proportional to $x_p(t)$ is measured at a sampling frequency $f_s$ during the interval $\Delta t$, yielding a time series $\{\alpha x(t_j)\}$ where $\alpha$ is the conversion factor between units of measurement and the actual displacements of the particle, and $t_j=j/f_s$, with $j \in \mathbb {Z}$ and $0\leq j\leq \Delta t f_s$. This time series is then used to calculate the position PSD of the particle, resulting in an aliased Lorentzian (AL),

In the case of interest $\alpha$ and $f_c$ are unknown quantities which can be found by fitting the experimentally determined PSD to Eq. (3). The coefficient $\gamma _0(T)$ can be obtained given the temperature of the sample and the sphere’s radius. Moreover, to take into account the presence of a cover-slip at a distance $h$ from the sphere’s center, we include corrections to the drag coefficient given by [33],

3. Theory: data analysis and knife-edge detector

We now describe the approximations proposed as simplifications to the optical tweezer setup and data analysis as well as the knife-edge detection method.

In what follows, $\langle A(t)\rangle$ is the time average of $A(t)$ and $\delta _A(t)$ is the instantaneous deviation of $A(t)$ with respect to $\langle A(t)\rangle$. We start by taking a closer look at the division of $X(t)$ by $S(t)$, which yields the particle position in the $x$ direction - i.e. $X(t)/S(t)=\alpha x_p(t)$ - when using standard forward scattering detection [28],

Note the importance of good centralization between the beam and the PS detector when applying this approximation: a non-zero mean value of $X(t)$ would create a non-negligible term $\langle X(t)\rangle \delta _S(t)/\langle S(t)\rangle ^2$. This would cause the PSD in the radial direction to be distorted by frequency components of the PSD in the axial direction, preventing proper calibration.

A direct consequence of Eq. (6) is that the power spectra obtained from the division of $X(t)$ by the instantaneous values of $S(t)$ are approximately the same as obtained by the division of $X(t)$ by the mean value of $S(t)$. We refer to the former PSD as $\rm {PSD_{inst}}$ and to the latter as $\rm {PSD_{mean}}$. From Eq. (3) it can be seen that dividing the measured position by a constant value alters the conversion parameter $\alpha$ while leaving the corner frequency $f_c$ unchanged. Since $f_c$ is proportional to the spring constant, one need only measure $X(t)$ and $Y(t)$ in order to obtain the radial trap stiffness. This is useful when dealing with Data Acquisition boards (DAq) that have sampling frequency or data transfer rate limited by the number of active channels in the equipment. Also, it greatly reduces the computational cost of data analysis.

Consider the following explicit models for $X(t)$ and $S(t)$,

Since the noise in the $X$ channel and the particle position are uncorrelated, the $\rm {PSD}$ can be separated in two parts. The first is an Aliased Lorentzian (AL), characteristic of discrete sampling of the particle position [23]. The second is a term proportional to the noise PSD in the $X$ channel. Moreover, this last terms is inversely proportional to the squared value of the trapping power, since $\alpha _S \langle P(t)\rangle$ is proportional to the latter. This provides a method for dealing with noisy detection systems: instead of fitting the $\rm {PSD}$ to a pure AL, one can measure the power spectrum of the noise in the $X$ channel when the trapping laser is off, which we call $\rm {PSD_{dark}}$, and fit the measured PSD to an AL added to a term proportional to $\rm {PSD_{dark}}$. This can also be easily extended to the PSD of motion in the axial direction, with the noise in the $X$ channel replaced by the noise in the $S$ channel. As a final remark, note that this is different from the whitening procedure used by [35] to deal with detectors that have a frequency dependent response.

One interesting alternative to a PS sensor is what we will call a knife-edge detector. In such a setup, the collected scattered light is divided into two beams. One of the beams is focused on a regular photo-detector yielding a signal $S(t)$, proportional to the collected power $P(t)$. The other beam is partially blocked by a knife in such a way that when the particle is at the center of the trap, half of the light passes by the knife and is focused on a second photo-detector.

The movement of the particle in the $xy$-plane causes the beam to be deviated by an angle proportional to the particle’s position. The second detector thus gives a signal $X_k(t)$ containing two terms: one proportional to $P(t)$ relative to the detected power when the particle is not radially displaced; and another proportional to both $P(t)$ and $x_p(t)$. In practice, due to electronic noise, the ratio between $X_k(t)$ and $S(t)$ reads,

Since $x_P(t)$, $\eta _X(t)$ and $\eta _S(t)$ are all independent of each other, the PSD of $X_k(t)/S(t)$ - which we’ll refer to as $\rm {PSD_{knife}}$ - is the sum of the power spectra of each of these signals weighted by different constants, with the constant term $\alpha _0/\alpha _S$ in Eq. (9) making no contributions to the spectrum. This knife-edge setup allows one to measure the position of a trapped particle - and beam deviations in general - using detectors sensitive only to the overall light power. This is useful since not only PS detectors are often not readily available in a laboratory, but they also have limited bandwidth [36]; with the knife-edge detector regular silicon detectors can be used in combination with low noise electronics when high frequencies must be accessed. This is an extension of the use of knife-edge detection, proposed by [37] in the field of high-speed atomic force microscopy, for situations in which the detected power varies with time.

As a final remark consider the expansion of $X_k(t)$:

The approximations used in the derivation of Eq. (6), Eq. (8) and Eq. (9) were not arrived at on theoretical grounds, but based on empirical evidence. Next we describe an experimental setup which implements the data analysis and knife-edge detection discussed in this section and justifies these approximations by comparing it to the standard calibration procedure.

4. Experiment

To test the validity of the above considerations for trap stiffness measurements, an optical tweezer with multiple data acquisition channels was assembled, as in Fig. 1. A 55 mW laser beam at 780 nm (Toptica DL-pro) is focused by a high numerical aperture objective (Olympus UPlanFLN 100x, NA = 1.3) creating an optical trap for a 1.15 µm silica bead (microParticles GmbH) in a water immersion between two cover-slips. The scattered light from the particle is collected by a second objective lens (Olympus PlanN 10x, NA = 0.25) and divided into two beams by a beam splitter. The reflected light is collected by a PS detector (New Focus 2931) generating the signals $X(t)$ and $S(t)$, which are simultaneously monitored by an oscilloscope (sampling frequency $f_s=10$kHz) and a simple data acquisition board (DataQ 1100, sampling frequency $f_s=20$kHz). At the same time the transmitted beam is partially blocked by a knife and collected by a silicon photo-detector (Thorlabs PDA100A2), creating the signal $X_k(t)$, also acquired by the oscilloscope.

 

Fig. 1. Experimental setup used to test the derived approximations: a position detector (PS) and a knife-edge detector - assembled using a silicon detector (Si) - are used to measure the beam deviation. The data is recorded by an oscilloscope (Scope) and a data acquisition board (DAq) and used to calculate the particle’s position and its PSD in four different ways. For auxiliary purposes, the image of the sample is formed by focusing the light from a light emitting diode (LED) onto a camera (CAM). As usual, BS and PBS stand for beam splitter and polarizing beam splitter, respectively. The wave plates used to control the polarization of the beam have been omitted from the figure for simplicity.

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The data simultaneously recorded by the DAq and the oscilloscope was used to obtain the radial displacement of the trapped particle in four different ways, as summarized in the table from Fig. 1. Due to the frequency independent electronic noise of the oscilloscope, an AL added to a constant value was used to fit $\rm {PSD_{inst}}$, resulting in Fig. 2(a). Since this fitting procedure is described in the literature by [29] in the context of position detection by video imaging, the corner frequency $f_c=742.8\pm 6.7 \; \textrm{Hz}$ of the fitted AL was taken to be the comparison standard for the other three methods. We do not attempt to translate the corner frequency into a spring constant value yet, since this would introduce additional errors relative to the medium’s viscosity and temperature and to the particle’s radius that would obscure the results of the intended comparison. For completeness, the values of the spring constants are presented later on.

 

Fig. 2. Measured PSD and the resultant fit for (a) $\rm {PSD_{inst}}$ (reduced $\chi ^2=1.02$) and (b) $\rm {PSD_{mean}}$ (reduced $\chi ^2=1.02$). Each point/error bar is the mean value/standard deviation obtained from 4 PSD’s. Each of these PSD’s is obtained by averaging 50 spectra calculated using data sets of 0.25 seconds (2500 points). Typical examples of these time series are shown in (c) for the case of $X(t)$, and in (d) for the case of S(t).

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To test Eq. (6), an AL added to a constant value was also fitted to the data from $\rm {PSD_{mean}}$, resulting in Fig. 2(b). In this case a corner frequency of $f_c = 740.4\pm 6.3 \; \textrm{Hz}$ was obtained, in agreement to the standard value. This is consistent with the expectation that $\rm {PSD_{inst}}$ and $\rm {PSD_{mean}}$ are almost identical and that the radial trap stiffness can be measured using either of them. Moreover, since dividing by a constant value does not change the corner frequency, one can obtain it directly from the spectral density of the signal $X(t)$ alone, reducing the amount of data processing required to measure trap stiffness in the radial direction. This enables real time measurements of the spring constant; one no longer needs to do the post-processing step of dividing $X(t)$ by $S(t)$.

The fulfilment of the conditions $\langle X(t)\rangle \approx 0$ and $\delta _S(t)/\langle S(t)\rangle \ll 1$ used in Eq. (6) is indicated by Fig. 2(c) and Fig. 2(d), which show time series of $X(t)$ and $S(t)$ for a duration of 0.25 seconds. A more detailed discussion regarding these conditions and the remaining ones employed in the derivation of Eq. (8) and Eq. (9) can be found in the Appendix.

The spectrum $\rm {PSD_{noise}}$ was used to test Eq. (8). The power spectrum $\rm {PSD_{dark}}$ of the electronic noise in the DAq’s $X$ channel, measured when the laser was off, is displayed in Fig. 3(a). As it can be seen, it is not frequency independent, making an AL added to a constant value insufficient as the fitting function. Instead, as prescribed by Eq. (8), we used

 

Fig. 3. (a) PSD estimate for the electronic noise in the $X$ and $S$ channels. (b) Measured PSD and resultant fit for $\rm {PSD_{noise}}$ (reduced $\chi ^2=1.51$). Each point/error bar is the mean value/standard deviation obtained from 5 PSD’s. Each of these PSD’s was obtained by blocking a spectrum calculated with data sets of approximately 25 seconds ($5\times 10^5$ points) using blocks of 250 points. (c) AL and noise components of the resultant fit.

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Using an AL added to a constant value instead of an AL added to $\rm {PSD_{dark}}$ to fit the data resulted in a 8 times larger $\chi ^2$ and a corner frequency of $694.3\pm 4.7$Hz, significantly worst when compared to our standard value. This corroborates with the result in Eq. (8), that can be seen as a generalization of the prescription presented in [29], which deals with the particular case of white noise in the detection channel. With this result, one can use “simple” acquisition boards with complex electronic noise structure and yet perform quality measurements of the spring constant of an optical tweezer.

Finally, the knife-edge method was tested using $\rm {PSD_{knife}}$. Since the noise in each of the channels of the PS detector combined with the oscilloscope is taken to be an independent white noise, the third and fourth terms in Eq. (9) results in a constant term in the PSD of the displacement in the $x$ direction. Therefore, $\rm {PSD_{knife}}$ was fitted to an AL added to a constant term, resulting in Fig. 4, for which $f_c=749.3\pm 6.4$Hz. To avoid the low frequency noise introduced by the mechanical vibrations of the knife, the data below 30Hz was not included in the fit. This demonstrates that a regular silicon detector combined with a knife can indeed be used to measure a trapped particle’s displacement in the radial direction. Moreover, axial displacements can be measured using the total value of the forwardly scattered power, while displacements in a radial direction orthogonal to the first one can be measured by dividing the beam once again, before it hits the total power detector, and partially blocking it with a knife positioned in a direction orthogonal to that of the first knife. This allows for full trap stiffness calibration using three regular silicon detectors.

 

Fig. 4. Knife-edge detection: (a) Measured $\rm {PSD_{knife}}$ and resultant fit (reduced $\chi ^2=1.03$); (b) PSD of $X_k(t)$ and $S(t)$ displayed separately. Each point/error bar is the mean value/standard deviation obtained from 4 PSD’s. Each of these PSD’s is obtained by averaging 50 spectra calculated using data sets of 0.25 seconds (2500 points).

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The necessity of dividing $X_k(t)$ by $S(t)$ when using the knife-edge method becomes clear when comparing Fig. 4(a) and Fig. 4(b), in which the PSD of $X_k(t)$ and $S(t)$ are displayed separately. The PSD of $X_k(t)$ clearly doesn’t resemble $\rm {PSD_{knife}}$, being dominated by the PSD of $S(t)$, as suggested by Eq. (10) for the case in which the ratio $\alpha _X/\alpha _0$ is small. Note that this is analogous to what would happen, due to the term $\langle X(t)\rangle \delta _S(t)/\langle S(t) \rangle ^2$ in Eq. (6), if the beam is not properly centralized with respect to the sensor when using a PS detector.

Since the approximations made in the derivations of Eq. (6), Eq. (8) and Eq. (9) depend on the mean value of $S(t)$ being much larger than the variations in $S(t)$, it is expected some of them to be invalid as the trapping power gets small and $\eta _S(t)$ stops being negligible. To show the wide range of validity of the derived expressions, the four methods of corner frequency measurement were applied for trapping powers of 15mW, 25mW, 35mW and 45mW other than 55mW. The results were used to calculate the spring constants $\kappa$ for each trapping power using the definition of the drag coefficient and its correction due to the presence of the coverslip introduced in Section 2. The results for each method are displayed in Fig. 5. Since it is expected a linear dependence of $\kappa$ on the trapping power - i.e $\kappa (P) = aP$ - a straight line was used to fit the spring constants [28]. The linear coefficients of each straight line, which are displayed in Fig. 5, shows that the expressions derived in this work are valid throughout the range of tested trapping powers.

 

Fig. 5. Spring constant as a function of trapping power extracted from (a) $\rm {PSD_{inst}}$; (b) $\rm {PSD_{mean}}$; (c) $\rm {PSD_{noise}}$ and (d) $\rm {PSD_{knife}}$. Reduced $\chi ^2$ values for the linear fits are 0.51, 0.35, 0.60 and 0.35, respectively, and the linear coefficients are given in fN/nmW. The calculation of the spring constant used Eq. (4) and Eq. (5) considering the sphere’s diameter to be $1.15\pm 0.04\mu$m (microParticles GmbH), the medium’s temperature to be $294\pm 1$K and the distance from the cover slip to the bead to be $10\pm 2 \ \mu$m, with the uncertainty coming from the micrometric translation stage used to control the position of the sample.

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5. Conclusions

Using suitable approximations, we have derived mathematical expressions for the position of a trapped particle measured by forward scattering detection. The first of these approximations imply that the PSD calculated from the division of $X(t)$ by $S(t)$ at each sample is almost identical to that obtained directly from $X(t)$. This is useful when dealing with acquisition devices that have a sampling frequency and data transfer rate limited by the number of active channels. Also, it reduces the need of post-processing, allowing for real time PSD visualization.

The second derived expression resulted in a generalization of the procedure used in [29], in which a constant value is added to the Lorentzian function used to fit the experimental PSD. We concluded that for acquisition systems having frequency dependent electronic noise, the constant value can be substituted by the power spectral estimate of the noise in the $X$ channel, in the case of radial calibration, or in the $S$ channel, in the case of axial calibration. This, together with the whitening methods used in [35], allow for the use of devices having complex response and noise spectral structure.

Finally, the third expression suggested a method for using knife-edge detection in optical tweezers. Dividing the power detected after the knife by the total detected power, one can extend the use of this kind of detection to situations in which the beam power varies in time, in contrast to what was proposed in [37]. Since knife-edge detector can be implemented using detectors sensitive only to light power, such as regular silicon detectors, this can be useful when high frequencies are of interest. Also, we believe this kind of detector can be useful to the general optics community in situations in which position sensitive detectors are unavailable or when reducing costs and complexity is necessary [30,31].

All of the above expressions were experimentally tested and compared using a custom optical tweezer setup, equipped both with a position sensitive detector and the proposed knife-edge detector connected to two different acquisition devices. Good agreement between the three methods and the standard method was found when calculating the corner frequency of the PSD in the radial direction for different trapping powers.

Appendix: Error estimate

The validity of the approximations proposed in this paper was verified by comparing the corner frequencies obtained from $\rm {PSD_{mean}}$, $\rm {PSD_{noise}}$ and $\rm {PSD_{knife}}$ and the one obtained from $\rm {PSD_{inst}}$. We now use the experimental data to better access the error introduced by the approximations in the derived expressions. In what follows, the values of $X(t)$, $S(t)$ and $X_k(t)$ taken from 2500 points data sets (0.25 seconds) acquired with the oscilloscope are used in the analysis of Eqs. (6) and 9, while 5000 points data sets (0.25 seconds) acquired with the DAq are used in the case of Eq. (8). In both cases, the trapping power is 55mW. Finally, values for $\eta _X(t)$ and $\eta _S(t)$ are taken from data sets recorded by the oscilloscope and by the DAq when the trapping laser was off.

In Eq. (6) we introduce the Taylor expansion of $1/(1+\delta _S(t)/\langle S(t)\rangle )$ to first order in $\delta _S(t)/\langle S(t)\rangle$. This - and other assumptions and typical values used in the estimations - is motivated by the values presented in Table 1. The Taylor approximation is equivalent to dropping the term

(12)$$\epsilon_A(t)=\frac{X(t)}{\langle S(t)\rangle}\sum_{n=2}^\infty\left(-\frac{\delta_S(t)}{\langle S(t)\rangle}\right)^n.$$

Using the typical values from Table 1, this introduces a relative error of

(13)$$\frac{\epsilon_A(t)}{X(t)/S(t)}\approx\sum_{n=2}^\infty\left(-\frac{\delta_S(t)}{\langle S(t)\rangle}\right)^n\approx \frac{0.103^2}{4.27^2}\approx0.1\%.$$

Next we drop the term

(14)$$\epsilon_B(t)={-}\frac{\langle X(t)\rangle \delta_S(t)}{\langle S(t)\rangle^2}-\frac{ \delta_X(t)\delta_S(t)}{\langle S(t)\rangle^2}$$

based on the fact that $\langle X(t)\rangle \approx 0$ and $\delta _S(t)\delta _X(t)/\langle S(t)\rangle ^2\ll \delta _X(t)/\langle S(t)\rangle$, since $\delta _S(t)/\langle S(t)\rangle ~\ll ~1$. The former is true when the beam is well centralized with respect to the detector. The fulfilment of this requirement by our setup can be verified by noting that $\vert \langle X(t)\rangle \vert \ll 5$V, which is the maximum attainable voltage for $X(t)$.

Table 1. Typical values used to estimate the error in the approximations used in this work. For channels $X$ and $S$, they were taken from time series (0.25 seconds, $f_s$=10kHz for the scope and $f_s$=20kHz for the DAq) recorded when the trapping power is 55mW. For the electronic noises, they were taken from time series (same specifications) recorded when the trapping laser was off.

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The resultant relative error is

(15)$$\begin{aligned} \frac{\epsilon_B(t)}{X(t)/S(t)}&\approx\left(\frac{7.45\times10^{{-}4}\times0.103}{4.27^2}+\frac{7.99\times10^{{-}3}\times0.103}{4.27^2}\right)\\ &\times\frac{4.27+0.103}{(0.745+7.99)\times10^{{-}3}} \approx2\% \end{aligned}$$

Therefore the expected accumulated error in Eq. (6) is about 2%, having a minor contribution from the Taylor expansion to first order. Note that the error in $\rm {PSD_{inst}}$ and $\rm {PSD_{mean}}$ is actually smaller, due to the need of averaging over several spectra in order to get the PSD [23]. This can be seen in Fig. 6, which shows the ratio between $X(t)/S(t)$ and $X(t)/\langle S(t)\rangle$ for the time series used here to estimate the errors and the ratio between $\rm {PSD_{inst}}$ and $\rm {PSD_{mean}}$.

 

Fig. 6. Ratios between (a) the particle position calculated as $X(t)/S(t)$ and as $X(t)/\langle S(t)\rangle$ over a time period of 0.25 seconds (2500 points); (b) $\rm {PSD_{mean}}$ and $\rm {PSD_{inst}}$. The error in $X(t)/\langle S(t)\rangle$ is clearly larger than the error in $\rm {PSD_{mean}}$.

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The derivation of Eq. (8) starts with the approximation derived in Eq. (6), which introduces an error of 2%. Then, the mean value $\langle S(t) \rangle$ is approximated to $\alpha _S\langle P(t)\rangle$. This introduces the error

(16)$$\frac{1/\alpha_S\langle P(t)\rangle -1/\langle \alpha_SP(t)+\eta_S(t)\rangle}{1/\langle \alpha_SP(t)+\eta_S(t)\rangle}=\frac{\langle \eta_S(t)\rangle}{\langle \alpha_SP(t)\rangle}\approx \frac{\langle \eta_S(t)\rangle}{\langle S(t)\rangle}\approx 0.2\%$$

where we have used the values recorded by the DAq shown in Table 1. Then, we drop the second order term $\alpha _X\delta _P(t)\delta _{x_p}(t)/\alpha _S\langle P(t)\rangle$, while maintaining $\alpha _X\langle P(t)\rangle \delta _{x_p}(t)/\alpha _S\langle P(t)\rangle$. From the values of $\sqrt {\langle \delta _S(t)^2\rangle }$ and $\langle S(t)\rangle$ - which are approximately proportional to $\langle \delta _P(t)\rangle$ and $\langle P(t)\rangle$, the relative error caused by this approximation can estimated to be $0.116/4.29\approx 2.7\%$. However, note that, because we also maintain the term $\eta _X(t)/\alpha _S(t)\langle P(t)\rangle$, the relative error of this last approximation is expected to be smaller than $2.7\%$.

Overall, the accumulated error in Eq. (8) is estimated to be 5%. This is the error in treating $X(t)/S(t)$ as the sum of a term proportional to the particle’s position and another one proportional to the noise in the $X$ channel at each instant $t$.

As for Eq. (9), we start by expanding $1/S(t)$ for small $\eta _S(t)/\alpha _SP(t)$ and retain only the terms up to first order. Like in Eq. (12), this is similar to dropping the term

(17)$$\epsilon_C(t)=\frac{X_k(t)}{\alpha_SP(t)}\sum_{n=2}^\infty\left(-\frac{\eta_S(t)}{\alpha_SP(t)}\right)^n.$$

which introduces the error

(18)$$\frac{\epsilon_C(t)}{X_k(t)/S(t)}=\frac{S(t)}{\alpha_SP(t)}\sum_{n=2}^\infty\left(-\frac{\eta_S(t)}{\alpha_SP(t)}\right)^n\approx \frac{(2.02\times10^{{-}2})^2}{4.27^2}\approx0.002\%.$$

Then we neglect the term

(19)$$\epsilon_D(t)=\frac{\eta_X(t)\eta_S(t)}{\alpha_S^2P(t)^2}+\frac{\alpha_Xx_p(t)P(t)\eta_S(t)}{\alpha_S^2P(t)^2}$$

which yields a relative error of

(20)$$\begin{aligned} \frac{\epsilon_D(t)}{X_k(t)/S(t)}&\approx \frac{\eta_X(t)\eta_S(t)}{\alpha_SP(t)X_k(t)}+\frac{\delta_{X_k}(t)\eta_S(t)}{\alpha_SP(t)X_k(t)}\\ &\approx \frac{(9.36\times10^{{-}3})\cdot(2.02\times10^{{-}2})}{4.27\times3.51}+\frac{(8.36\times10^{{-}2})\cdot(2.02\times10^{{-}2})}{4.27\times3.51}\\ &\approx 0.01\%, \end{aligned}$$

using that $\alpha _X x_p(t)P(t)\approx \alpha _X x_p(t)\langle P(t)\rangle \approx \delta _{X_k}(t)$.

Finally, the terms $\eta _X(t)/\alpha _S P(t)-\alpha _0\eta _S(t)/\alpha _S^2P(t)$ are approximated to $\eta _X(t)/\alpha _S\langle P(t)\rangle -\alpha _0\eta _S(t)/\alpha _S^2\langle P(t)\rangle$, leading to a relative error given by

(21)$$\begin{aligned} &\, \frac{(\eta_X(t)/\alpha_S P(t)-\alpha_0\eta_S(t)/\alpha_S^2P(t))-(\eta_X(t)/\alpha_S\langle P(t)\rangle-\alpha_0\eta_S(t)/\alpha_S^2\langle P(t)\rangle)}{X_k(t)/S(t)}\\ &\approx \frac{-\eta_X(t)\delta_P(t)/\langle P(t)\rangle+\alpha_0\eta_S(t)\delta_P(t)/\alpha_S\langle P(t)\rangle}{X_k(t)}\\ &\approx \frac{-\eta_X(t)\delta_S(t)/\langle S(t)\rangle+\langle X_k(t)\rangle\eta_S(t)\delta_S(t)/\langle S(t)\rangle^2}{X_k(t)}\approx0.02\% \end{aligned}$$

where we have used $S(t)\approx \alpha _S P(t)\approx \langle S(t)\rangle \approx \langle \alpha _S P(t)\rangle$, $\alpha _S \delta _P(t)\approx \delta _S(t)$ and $\alpha _0P(t)\approx \alpha _0\langle P(t)\rangle \approx \langle X_k(t)\rangle$.

The accumulated error we get from treating $X_k(t)/S(t)$ as the sum of a constant term, a term proportional to the particle position and two terms proportional to the noise in the detection channels is estimated to be 0.03%. This small error is due to the fact that $X_k(t)/S(t)$ is dominated by the constant term. Therefore, the error in eliminating oscillating terms is expected to be small. To better quantify the effect of the approximations, it is useful to compare the neglected oscillating terms with the oscillating terms in $X_k(t)/S(t)$. We can do that by replacing $X_k(t)/S(t)$ with $\delta _{X_k}(t)/S(t)$ in the calculation of the relatives errors. This gives the value $(X_k(t)/\delta _{X_k}(t))\times 0.03\%\approx (3.51/0.0836)\times 0.03\%\approx 1\%$.

The error in Eq. (10) can be easily estimated by noting that we neglect the term $\alpha _Sx_p(t)\delta _P(t)$ while maintaining $\alpha _Sx_p(t)\langle P(t)\rangle$. This introduces a relative error of about $\delta _P(t)/\langle P(t)\rangle \approx \delta _S(t)/\langle S(t)\rangle \approx 0.103/4.27\approx 2\%$. Since we also keep other oscillating terms, the error is expected to be smaller than $2\%$.

In summary, we estimated that, in the time domain:

• (i) the error obtained from calculating the particle position as $X(t)/\langle S(t)\rangle$ instead of $X(t)/S(t)$ is 2%;
• (ii) the error in treating $X(t)/S(t)$ as the sum of terms proportional to the particle position and the noise in the $X$ channel is 5%;
• (iii) the error in treating the oscillating terms in $X_k(t)/S(t)$ as the sum of terms proportional to the particle position and the noise in the $X$ and $S$ channels is 1%;
• (iv) the error in treating the oscillating terms of $X_k(t)$ as the sum of terms proportional to the particle’s position in the radial and axial directions and a term equal to the noise in the $X$ channel is $2\%$.

Funding

Instituto Serrapilheira (Serra-1709-21072); Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (001); Conselho Nacional de Desenvolvimento Científico e Tecnológico.

Acknowledgments

The authors would like to thank Paulo Américo, Natan Viana and Luis Pires from UFRJ for useful discussions and advice during the development of this work.

Disclosures

The authors declare no conflicts of interest.

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