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Abstract
This paper presents a method to directly calibrate the position of a trapped micro-sphere in optical tweezers utilizing its interference pattern formed at the back focal plane (BFP). Through finite difference time domain (FDTD) and scalar diffraction theorem, the scattering field complex amplitude of the near and far fields can be simulated after interference between the trapped sphere and focus Gaussian beam. The position of the trapped sphere can be recovered and calibrated based on a back focal plane interferometry (BFPI) algorithm. Theoretical results demonstrate that optical tweezers with a larger numerical aperture (NA) Gaussian beam will yield a better detection sensitivity but with a smaller linear range. These results were experimentally validated by trapping a microsphere in a single beam optical tweezer. We used an extra focused laser to manipulate the trapped sphere and then compared its position in the images and that obtained using the BFP method. The interference pattern from simulation and experiments showed good agreement, implying that the calibration factor can be deduced from simulation and requires no intermediate calculation process. These results provide a pathway to obtain the calibration factor, enable a faster and direct measurement of the sphere position, and show possibilities for adjusting the crosstalk and nonlinearity inside an optical trap.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Since Ashkin demonstrated that micrometer-sized dielectric particles can be accelerated and trapped by the gradient force field originated from radiation pressure [1,2], optical trapping technology and its applications have been developed extensively. Optical tweezers are versatile to trap and manipulate a wide range of objects ranging from atoms, nano- or micro-sized particles to living organism owing to a high level of interaction between light and matter [3–9]. Naturally particles levitated by gradient radiation pressure in air or vacuum are physically isolated from most sources of environmental disturbance, such as mechanical, thermal, and electrical noise from residual gas molecules [10]. This allows for a variety of applications including ultra-sensitive measurements [11–13], biological studies [14–16], and fundamental physics [17–19].
The afore-mentioned applications of optical tweezers typically require a precise position detection of trapped particles. For instance, the measurement of the power spectrum of the microsphere movement could be utilized to detect ultra-sensitive force and acceleration [20,21]. Some basic physical studies such as the Brownian motion measurement or macroscopic quantum realization require precise measurement of the sphere motion [22,23]. The conventional position measurement method deduces the position of sphere according to the beam profile detected at the back-focal-plane [24–26]. It can achieve a position detection sensitivity of the order of nanometers or even lower [27]. Previous studies utilized Brownian motion or exerted an extra drag force to calibrate the position of the sphere, which depends on the surrounding condition and needs to be recalibrated when the temperature or the surrounding condition changes [28]. As an indirect measurement method, it assumes that the detection signal is proportional to the movement of the sphere and ignores the co-axis crosstalk in optical tweezers, which is usually not the case [29–31].
Hence, it is desirable to calibrate the position of the microsphere in an optical tweezer in a more accurate and reliable way. One natural approach is to simulate the scattering beam-field complex at the back focal plane (BFP) and establish the relationship between the intensity distribution at the BFP and the position of the sphere. In the scattering field analysis, the Rayleigh scattering model is only suitable for the analysis of the field intensity distribution for nanosized particles [32]. The generalized Lorenz–Mie theory (GLMT) method can extend this range to micron-sized particles; however, this method requires a complex numerical calculation process and significant computer resources [33–36].
In this study, we simulated the beam profiles both in the near and far fields after interference between the trapped microsphere and Gaussian beam. Therefore, the relationship between the position of the sphere and the corresponding beam profile can be established and hence, the 3D position of the sphere can be directly deduced from the detected beam intensity distribution. The method proposed in this study does not require an indirect process from thermal motion or the use of drag forces [24], which offers a direct and convenient method to calibrate the position of the sphere. Moreover, our method can be applied to calibrate arbitrary shaped particles and anisotropic particles.
By using the FDTD method, the beam intensity distribution of the forward-scattering field in the near field can be simulated. The beam profile in any plane can be determined through the Fresnel diffraction method, which can extend the simulation planes to the BFP detection plane. There are two parameters that are of utmost concern in optical tweezer calibration methods: the linear range and detection sensitivity. Our simulation results indicate that a Gaussian beam with a larger numerical aperture (NA) would have a larger detection sensitivity but its linearity will be degraded. Additionally, the cross talk of various axes was analyzed to correct for actual detection. Simulation results show that a Gaussian beam with a lager NA would have at most 20% crosstalk on the x axis, and an NA of 0.05 would only cause a crosstalk of less than 10% uncertainty. To validate the simulation results, we built a single beam optical tweezer setup and manipulated the position of the sphere with an extra focused laser beam. The beam profile at the back focal plane was detected and compared with the position acquired in image. Finally, the simulation and experimental results at various sphere positions both on the horizontal and vertical positions were compared and we found that the results were consistent.
2. Theoretical analysis
A key step to calibrate the position of the trapped sphere is to reconstruct its scattering full complex field at the detection plane after interference between the trapping laser beam and the sphere. For a dielectric particle such as that typically trapped in optical tweezers, its interaction with focused light beam can be analyzed using the FDTD method [37]. This technique solves Maxwell’s equations in the time domain by separating the computation domain into a discrete finite fine mesh, which was first introduced by Lee in 1966 [38]. Here, the computation process is briefly introduced. The incident wave interacts with the sphere and generates an electromagnetic field inside and around the sphere, as shown in Fig. 1. In FDTD, these fields are calculated with separated finite fine meshes. Each mesh represents one electromagnetic parameter, which can be acquired by applying the finite difference method to solve the Maxwell equation. The space arrangement of a discrete mesh containing electric and magnetic fields is called a Yee cell, as shown in Fig. 1. In each Yee cell, one magnetic component is surrounded by four electric components. Similarly, each electric component is also surrounded by four magnetic components. For a given mesh size and original boundary condition, the subsequent electromagnetic field can be derived step by step.
Fig. 1. Schematic diagram of the FDTD computation process showing a single sphere trapped in the source plane.
In the FDTD method, the Maxwell curl equation was transformed into a discrete differential equation. In the Cartesian coordinate system, the electric or magnetic component in the Maxwell curl equation can be rewritten as a discrete expression:
Here, f represents the electric field or magnetic field component in time and space domains, and i, j, and k represent the number of cells in the x, y, and z directions, respectively.Once the beam electromagnetic field distribution in the near field is obtained, the diffractive field in the far field can be deduced using the scalar diffraction method. Assuming that U0(x0,y0) is the scattering wave complex amplitude in the near field, according to Fresnel diffraction integral, the light wave complex amplitude U(x,y) arriving at the observation plane with a diffractive distance d can be expressed as [39,40]:
To obtain full information about the position of the sphere, we used on the BFPI position detection method owing to its high time and spatial resolution [26]. The schematic diagram of the principles of BFP is shown in Fig. 2. One focusing lens is used to create an optical trap, and a sphere is trapped near the focus point. Another condenser is placed behind the sphere to collect the light deflected by trapped particles and the beam intensity distribution it projected onto the detection plane. A quadrant photodetector (QPD) or a D-shaped mirror combined with a balance photodetector is placed at the BFP to detect the field intensity distribution on the BFP and measure the spheres’ positional signal. The positional change in the sphere deflects the beam profile on the BFP, resulting in signal changes on the QPD detector. The complex amplitude in the near and far fields can be calculated using the above method, and the positional signal on the BFP can be simulated. On the other hand, the detection sensitivity and nonlinearity can be simulated and experimentally verified.
Fig. 2. Simplified schematic diagram of the position measurement principles in optical tweezers. One microsphere is trapped at the focus point, and one condenser collects the scattering beam and projects the beam intensity distribution onto the BFP, while a quadrant photodetector is used to detect the intensity distribution at the BFP.
To describe the detection sensitivity obtained in the real BFPI method, we simulated the actual detection process and disintegration of the beam into four parts. The beam propagation direction is defined as z-axis, and the horizontal direction of the beam is defined as x-axis. The total optical power collected by an arbitrary QPD part can be calculated using the simulated beam profile in the QPD. Once a sphere moves in the optical trap, the optical beam is deflected and the beam intensity distribution changes in the QPD. As a result, the power distribution of the four parts in the QPD halves also changes, which can be deduced to trace the position of the sphere in the optical trap in nanometer scale and time resolution of up to several MHz. Usually, the position of the sphere can be expressed as the following equations:
3. Simulation results
The scattering field in the near and far fields can be derived using the FDTD and scalar diffraction theorem based on the above formulas, and the simulated intensity distribution in the far field can be utilized to deduce the position of the trapped particle. Figure 3 illustrates the typical field distributions of the near and far fields scattered by a 10 μm sphere in a Gaussian beam with 5 μm waist. In Fig. 3, the x or y axis represents the lateral direction as shown in Fig. 2. Figure 3(a) shows a beam profile 10 μm behind the sphere when the microsphere is located at the center of the axis. According to the near field electric and magnetic field complex distributions, the field distribution in the far field can be simulated using the Fresnel diffraction with Eq. (2). Here, we selected the diffractive distance as 50 mm, the same as the focus length of the condenser we used in the experimental setup. The interference beam pattern in the far field is shown in Fig. 3(b). The longitudinal movement of the sphere is also of interest. As shown in Figs. 3(c) and (d), the beam intensity distribution shifts when the sphere moves by 1 μm on the x-axis. The sphere deflects the beam momentum, and the beam intensity profile is no long axially symmetrical, which can be used to calibrate the position of the sphere.
Fig. 3. Typical scattering field intensity distribution of the near and far fields when the sphere is on z axis and has a misalignment of 1 µm on lateral direction. (a), (b) The near and far field intensity distributions after interference between the sphere and Gaussian beam when the sphere is on z axis. (c), (d) The near and far field intensity distributions when the sphere has a 1 μm lateral movement.
Relating the sphere’s motion with the subsequent signal detected by the QPD is an essential step in calibrating the position of the sphere. Using the simulated laser beam intensity distribution in the far field and substituting it in Eq. (3), the lateral positional signal can be calculated as a function of the sphere’s lateral movement. Figure 4 shows a typical signal-to-position curve for a 10 μm microsphere trapped in a Gaussian beam with various NAs and axial positions. Figures 4(a) and (b) show the results of the signal-to-μm curve when the sphere is located at the focus (z = 0 µm) in a Gaussian with NA = 0.3. We further simulated the detection signal when the sphere is located 10 μm off the focus because in most cases, the trapped microsphere cannot be trapped precisely at the focus. In addition, the signal-to-μm as a function of the position of the sphere was also simulated with various Gaussian NAs. Figures 4(c) and 4(d) depict the simulated detection results when the NA decreased to 0.1. Under such condition, the positional signal is less sensitive to the lateral movement of the trapped microsphere than the results shown in Figs. 4(a) and 4(b). However, it will have a larger linear range, implying that a smaller Gaussian waist size will have a better detection sensitivity but will degrade the linearity. This trend is further confirmed when the Gaussian beam is decreased to 0.05 NA. The axial positions of the sphere are set as 0 µm and 50 µm, respectively, as shown in Figs. 4(e) and 4(f).
Fig. 4. Simulation of the lateral signal-to-position curve in the BFPI model. The Gaussian beam NA and the axial position of the sphere are set as (a) NA = 0.3, z = 0 μm, (b) NA = 0.3, z = 10 μm, (c) NA = 0.1, z = 0 μm, (d) NA = 0.1, z = 50 μm, (d) NA = 0.05, z = 0 μm, (f) NA = 0.05, z = 50 μm. Several typical intensity distributions are also shown as inset figures.
The simulation results of three different NAs of Gaussian beams are shown in Fig. 4. A typical far field intensity distribution beam profile is also shown as inset figure. It was observed that the particle lateral motion generates a sine wave signal response, and the position of the sphere can be calibrated using the positional signal. For a highly focused Gaussian beam, the beam momentum would be more deflected by the sphere’s lateral motion and it has a larger detection sensitivity. In the experimental validation, two characteristics of BFPI were investigated: linear range and sensitivity in the linear range. These are defined as follows:
Linear range: the restricted area where the displacement of the sphere produces linear and direct output values of the positional signal.
Sensitivity: ability of the detection system to react to nanometer-scale displacement, which is denoted by the ratio of the detection signal to the sphere’s lateral motion $\beta \textrm{ = }\frac{{{S_x}}}{{{x_0}}}$.
For clarity, a linearity factor is used to evaluate the linearity of the detection model, which is defined as:
Here, $\beta _{vx}^{}(x)$ is the detection sensitivity when the sphere has a lateral movement of x, and $\beta _{vx}^{}(0)$ is the detection sensitivity near the axial center x = 0. With this equation, we can define the linear range as the area where $f(x)$<10%. The linear range of the sphere is shown in Fig. 4 as viewed in the optical tweezer.The region in which the linearity is valid depends on the NA of the Gaussian as well as the axial position of the trapped sphere inside the Gaussian beam. We define the linear range over the lateral direction when the linearity factor is less than 10% using the definition in Eq. (4). Detection sensitivity is the ratio of the detected signal over the x movement in the linear range. As shown in Fig. 5, the detection sensitivity will decrease when the sphere moves away from the focus. On the other hand, the linear range will increase when the sphere moves away from the focus.
Fig. 5. Simulation of the linear range (LR) and detection sensitivity (DS) as a function of the axial position of the sphere with Gaussian numerical apertures of (a) 0.3, (b) 0.1, and (c) 0.05.
In the experiments, the axial or y lateral direction displacement affects the x position detection output as a type of crosstalk that inherently exists. Here, we analyzed the crosstalk while tracking the x direction displacements with the y direction displacement. Figure 6 shows the beam profile in the far field beam profile with and without lateral displacement. Here, the sphere was set to move 500 nm on the x-axis. Figure 6(a) shows the lateral beam intensity distribution on the x-axis without misalignment. Figures 6(b)–(d) show the horizontal intensity distribution with y direction misalignments of 0.4 μm, 0.8 μm, and 1.2 μm.
Fig. 6. Horizontal beam intensity distribution on x-axis with y=0, y = 0.4, y=0.8, y=1.2 μm misalignment.
We can analyze the influence of lateral crosstalk on the detection sensitivity using the detection model given in Eq. (3). Here, we define crosstalk as the detection error of the sphere due to misalignment from other axes, which can be expressed as:
Here, ${S_y}(x)$ is the detection sensitivity when the sphere has a misalignment on the y-axis, and ${S_0}(x)$ is the detection sensitivity when there is no misalignment on the y-axis.Figure 7 shows the crosstalk while tracking x detected signals with y direction displacement. As shown in Fig. 7(a), when the bead’s displacement in the y direction is 400 nm, the crosstalk of the x direction output at x = 0 is less than 1%, which can be ignored. However, when the sphere has a displacement of 1200 nm in the y direction, the crosstalk can be as large as 12%, which is intolerable. Figure 7(b) shows the crosstalk simulation result with various Gaussian NAs as a function of y misalignments. It can be observed that the crosswalk increases with an increase in NA and misalignments in the y-axis.
4. Experimental results
4.1 Experimental setup
The experimental setup consists of three modules, as shown in Fig. 8. The levitation module serves to create a stable optical trap in the vacuum chamber. The trapping beam is generated by a continuous wave trapping laser with a wavelength of 1064 nm. The output beam is split by a 1:99 beam splitter (BS) and a power meter is placed at Port 1 to monitor the laser power. The transmitted beam passes through two lenses to be collimated. Two high reflection mirrors are utilized to adjust the beam direction and angle. The laser beam power is stabilized using an acoustic optical module (AOM). A dichroic mirror and an aspherical lens serve to combine and focus the trapping laser beam and controlling laser beam into the vacuum chamber. The size of the propagation beam and the focus length of the lens are carefully selected to ensure that the focused Gaussian beam has an NA of 0.05. The controlling module is used to control the position of the trapped sphere. Similarly, two lenses are set to collimate the 532 nm laser beam. Two half waveplates (HW) and polarization beam splitters (PBS) are used to adjust the laser power exerted on the sphere and, thus, control the force acting on the sphere. In this way, the position of the sphere can be manipulated, which can be measured using two horizontal and vertical CCDs. Finally, the detection module is used to detect and analyze the scatter beam profile using a beam profile analyzer. A BFP detection system is built with a D-shaped mirror and a balance photodetector to compare with the conventional measurement method.
The main feature of our optical tweezers (OT) compared with the conventional OT is that we used an extra laser beam to exert force to the sphere. This has the advantage of controlling the position of the sphere without changing the trapping laser beam. Thus, the scattering beam profile is merely affected by the position of the sphere, which can be used to analyze the sphere’s position. Hence, the scattering beam profile properties can be characterized using the known position in two CCDs.
4.2 Calibration of the trapped particle position
In this section, we compared the simultaneous position calibration of trapped particles from CCD images and the scattering beam profile at the BFP. First, the position of the sphere from a custom image system was controlled when it is exposed it to an extra focused laser beam. Second, the movement of the trapped particle from the image and beam profile was analyzed. The movement of the sphere can be determined from an image with a known image magnification factor and pixel changes in the CCD. The scattering beam profile was also monitored in a beam analyzer and the position of the sphere was obtained using applying Eq.(3). It was found that these two methods provide consistent results, which confirms the accuracy of the proposed method.
Figure 9 shows the top image of the trapped 10μm sphere in the horizontal plane in CCD1. The x and z-axis positions of the sphere are controlled by applying extra radiation force to the sphere using a 532 nm focused laser beam. Figure 9(a) shows its position with no controlling laser focus on it. Under this condition, the sphere is merely trapped by a vertical trapping beam and this position is defined as the original sphere position. Figure 9(b) depicts the movement of the sphere when it is hit by an extra 532 nm laser beam. We can compare its position with the original position and characterized its movement using an image system magnification factor.
Fig. 9. Top view of the trapped microsphere in an optical microscope image in CCD1. (a) Original position of the sphere when it is trapped merely using the trapping beam. (b) Movement of the sphere when it is hit by an extra 532 nm focus laser beam in the horizontal direction.
Figure 10 shows the beam profile collected at the BFP when the particle is trapped at various horizontal positions in the experimental validation. Figures 10(a)–(c) show typical intensity distributions of the beam profile when the sphere performs a horizontal movement from 0 to 2 µm. As the trapped sphere moves on the x-axis, the beam profile collected at the beam analyzer also has a momentum shift in the x direction. The experimental results were then compared with the simulated beam profile distribution. The horizontal laser power collected on the x-axis was calculated and compared with the simulation result, as shown in Figs. 10(d)–(f). The blue solid line represents the summed laser power on the x-axis from experimental validation and the red dashed lines are the respective simulation results. There is reasonable agreement between the simulation prediction and the measured beam profile.
Fig. 10. Beam profile collected at the back focal plane and its intensity distribution on the x-axis. The intensity distribution is shown with (a) 0 µm, (b) 1 µm, and (c) 2 µm x displacements and the total intensity distribution of the scattering.
Similarly, changes in the beam intensity distribution were further analyzed when the trapped sphere moved in the axial direction. Figure 11 shows the experimental results of the beam profiles as the sphere moves on the z-axis. Figures 11(a)–(c) show the beam profile when the sphere performs an axial movement from 50 µm to 150 µm from the focus. To compare the results with our simulation results, we compared the total laser power collected on the x-axis. In Figs. 10(d)–(f), the blue solid line is the summed laser power on the x-axis and the red dashed lines are the respective simulation results. Note that the total power that passes through the sphere changes with the position of the sphere, which can then be utilized to track the vertical position of the sphere.
Fig. 11. Beam profile at the back focal plane and its intensity distribution on the x-axis. The x–y plane beam intensity distributions are shown with (a) 50 µm, (b) 100 µum, (c) 150 µum z displacements. The total intensity distribution of the scattering beam intensity is shown in (d)–(f).
Once the beam profile at the BFP has been obtained, the positional signal ratio can be determined using Eq. (3). Figure 12 shows a representative signal–position curve as the sphere moves on the x or z-axis. The blue lines represent the simulated signals as a function of the sphere’s position, and the red dots are the experimental results. In the experimental validation, uncertainty arises from the imperfections of the image system, such as the performance of the condense lens and the limiting pixel size of the CCD. The positional signals measured in the experiments are consistent with the simulated results. This implies that the position of the sphere can be calibrated from the simulated signal curve results with a known sphere size and beam characteristics.
Fig. 12. Comparison of the 3D position calibration results obtained from experiments (blue line) and simulations (red dot). (a) Lateral signal curve as a function of the sphere movement on the x-axis. (b) Axial signal curve as a function of the sphere movement on the z-axis.
5. Conclusion
In this study, we have proposed a method to directly calibrate the position of the sphere in an optical trap and experimentally validated this method based on BFPI method. We extended the theory proposed by Yee [38] to acquire the complex field distribution both in the near and far fields after interference between the sphere and trapping beam. The signal-to-position can be calculated based on the algorithm used in the BFP measurement, providing a faster method to calibrate the position of the sphere in optical tweezers. This method does not require calibration of the position of the sphere using thermal theorem or exerting a drag force to the sphere.
To validate this direct calibration method, we constructed an experimental setup that can be used to efficiently trap, manipulate and measure a micron-sized particle. An extra focused laser beam was used to exert a force on the sphere and control its position inside the optical tweezers. Through this approach, the position of the sphere can be manipulated to obtain a direct voltage–position transfer ratio. The results are consistent with theoretical predictions, validating the proposed approach. Based on this model, the position of the sphere could be calibrated and measured in a direct and faster way in optical tweezers. In addition, our findings indicate the potential applications of position calibration in optical tweezers using construction laser beam types such as those used in Ref. [41,42].
Funding
National Natural Science Foundation of China (11304282, 61601405); Major Scientific Project of Zhejiang Laboratory (2019MB0AD01); Fundamental Research Funds for the Central Universities (2016XZZX004-01, 2018XZZX001-08); Joint Fund of Ministry of Education (6141A02011604).
Acknowledgment
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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