High-speed video stereo-microscopy relies on illumination from two distinct angles to create two views of a sample from different directions. The 3D trajectory of a microscopic object can then be reconstructed using parallax to combine 2D measurements of its position in each image. In this work, we evaluate the accuracy of 3D particle tracking using this technique, by extending the number of views from two to four directions. This allows us to record two independent sets of measurements of the 3D coordinates of tracked objects, and comparison of these enables measurement and minimisation of the tracking error in all dimensions. We demonstrate the method by tracking the motion of an optically trapped microsphere of 5 μm in diameter, and find an accuracy of 2–5 nm laterally, and 5–10 nm axially, representing a relative error of less than 2.5% of its range of motion in each dimension.
© 2014 Optical Society of America
High-resolution measurement of the three-dimensional trajectories of microscopic objects is important in a wide range of fields from colloid science to biophysics [1, 2]. It is often used in conjunction with optical tweezers, where the motion of optically trapped objects can be used to recover a wealth of detailed information about their surroundings . A variety of techniques have been developed to achieve three dimensional tracking, such as using a quadrant photodiode to observe the interference pattern generated in a laser beam as it is focussed through a sample , digital holographic microscopy , image correlation  and stereo-microscopy . Each of these is suited to particular applications, but in all cases it is important to quantify the quality of the tracking in terms of its accuracy.
A stereo-microscope enables three dimensional visualisation of a sample by providing two views of it from different directions. An object that moves axially in the sample is observed to move laterally in each image by an amount and direction that is dependent upon the viewing angle. By illuminating from two distinct directions simultaneously, two views of the sample from different angles can be imaged using a single objective lens . The superimposed images can be separated by channel colour , or by redirecting the light in the Fourier plane of the sample with a bi-prism . By performing two dimensional tracking of the position of a particle in each image, its three dimensional coordinates can be reconstructed using parallax. High-speed video stereo-microscopy has been used in a variety of situations, including the axial position clamping of optically trapped objects [10, 11], the observation of optically stacked particles , tracking of live algae and artificial non-spherical structures [13, 14], and in an optically controlled scanning probe microscope .
The functionality of stereo-microscopy has been further expanded by replacing the bi-prism with a spatial light modulator (SLM) [16, 17]. The SLM separates the images by diffracting the light from each illumination direction to a different position on the camera. The flexibility of an SLM enables the incorporation of stereoscopic imaging with additional modalities such as dark-field imaging  and dynamic refocussing to track particles over an increased depth of field . A stereoscopic version of digital holography has also recently been demonstrated, improving the axial resolution of holographic images .
In this work we demonstrate a stereo-microscope with four viewing directions, therefore providing two independent measurements of the 3D position of an optically trapped microsphere. Comparison of these measurements yields an estimate of the accuracy of the tracking in each dimension, in real-time. Previous work has provided an estimate of the accuracy in a single lateral dimension  or the noise on a static object (i.e. the precission or repeatability of the technique) , but our goal here is to measure the accuracy of axial tracking of an optically trapped object while it is undergoing Brownian motion, as this is representative of the situation in which the technique is most often used. We investigate how tracking errors depend on the degree of spatial filtering and exposure time of each image, and optimise the system.
2.1. Experimental set-up
Figure 1 shows a schematic of the experimental set-up. We use a compact design for stability, based on the microscope described by Lee et al. , with a modified illuminator which holds four optical fibre light guides each illuminating the sample from a different direction. We use an SLM to diffract each of the four views of the sample to a different location on the camera. The optical layout is shown in Fig. 1(a), and the four illumination paths through the system are shown in Fig. 1(b). Each illumination direction corresponds to a unique position in the Fourier plane of the sample where the SLM is positioned. The pattern displayed on the SLM is shown in Fig. 1(c), it consists of four apertures, each centred on a position corresponding to one of the illumination directions. Within each aperture, a phase grating diffracts the light to a separate region on the camera chip, resulting in an image such as that shown in Fig. 1(d).
2.2. 3D coordinate calculation using parallax
The position of a particle within each image is tracked in two dimensions by finding its centre of symmetry . The particle’s apparent position is a projection of its 3D position onto the focal plane from a direction defined by the angle of illumination. Figure 2(a) shows how the projected 2D image of a microsphere is related to its 3D position. The Cartesian lab. frame in the schematic is defined with the x–y plane parallel to the focal plane. Each illumination direction is defined by azimuthal and polar angles, θ and ϕ respectively. A microsphere at position a results in a projected image at location ai in the focal plane. If the microsphere moves vertically up to position b, the corresponding projected image moves to position bi. If the microsphere moves parallel to y (e.g. from position b to c), the corresponding image moves the same distance along the y axis (bi to ci), and likewise for motion parallel to the x axis. Therefore it is straight forward to show that for an arbitrary 3D translation of the microsphere described by vector v = [δx, δy, δz], its measured motion in the focal plane is described by a 2D vector vm = [xm, ym], where the x-component xm = δx + δztanϕ cosθ, and the y-component ym = δy + δztanϕ sinθ. The 3 components of v cannot be uniquely identified from a single illumination channel, but can be recovered by combining tracking measurements of the particle from any two arbitrary channels:Eqs. (3) and (4). These are chosen or weighted based on their signal to noise ratio, which depends upon the illumination directions relative to the x and y axis. When the azimuthal angles are separated by 180° and aligned with the lab. x-axis (i.e. θi = 0°, θj = 180°), and the polar angles of the two illuminators are equal, then Eqs. (1)–(3) give those described in .
For accurate tracking using Eqs. (1)–(4), the angles of the illumination of each channel must be known. This can be achieved by aligning each channel through an aperture at a preset position in the focal plane. The light is centred on the aperture when the image of a microsphere appears most symmetrical. When using an SLM to separate the images, the alignment can also be fine tuned by adjusting the position of the aperture on the hologram. Once the system is aligned in this way, the azimuthal and polar angles of each illumination channel can be calculated from the location of the centre of each aperture on the SLM, without the need for calibration. If necessary, the illumination angles can also be directly measured by monitoring the translation of a microsphere fixed to the substrate as it is stepped axially by a known distance.
Figures 3(a)–3(c) show a comparison of the two independent measurements of the trajectory of an optically trapped microsphere recorded using our four-directional stereo-microscope. The difference between each measurement is shown by the red trace. In this case the azimuthal illumination angles were separated by ∼ 90°, and the polar angles were all ∼ 25°. We found no strong dependence of the tracking error with variation in polar illumination angles between 20 – 30°. The RMS of the differences between each measurement in this case are 2.6 nm in x, 4.5 nm in y and 5.1 nm in z, using data recorded over a 30 s period with an exposure time of 1877 μs (corresponding to a frame rate of 532 Hz if not limited by other factors). Figure 3(d) shows a zoom in on part of the axial position trace to more clearly reveal the similarity of the measurements. As shown in Fig. 2, off-axis illumination introduces distortions in the images of a microsphere as the horizontal focal plane images at an angle to the illuminating light. These aberrations can be reduced by low-pass spatially filtering the images (the degree of spatial filtering is dictated by the aperture size on the SLM), which removes some of the high spatial frequency distortions. Figure 3(e) shows how the tracking errors depend upon the degree of spatial filtering. If the spatial filter frequency is set too low, the amount of light reaching the camera is significantly reduced, causing the signal to noise ratio to fall, increasing the tracking error. Conversely, if the spatial filter frequency is too high, the distortions in the image of the microsphere caused by off-axis illumination change asymmetrically as it moves axially, as shown in Fig. 2(c), which is problematic for the centre of symmetry tracking algorithm, also increasing the tracking error. A balance between these effects results in an optimum tracking error (see Figs. 2(b) and 2(d)). The reduction in the signal to noise ratio due to low light levels is also evident as the light reaching the camera is reduced by shortening the exposure time, as shown in Fig. 3(f). In our set-up the light level was limited by the diffraction efficiency of the SLM, and the 10 nm bandpass filter used to reduce dispersion of light diffracted from the SLM. Despite this, we still find an error of less than 10 nm in each dimension down to an exposure time of 663 μs, corresponding to a frame rate of 1508 Hz. This can be further improved by increasing the illumination intensity . Figures 3(e) and 3(f) show a small difference in the minimum error for each dimension. This is representative of the variation in measurements taken with our system, where the measured accuracy was dependent upon system alignment, and the state of the sample (for example, how perfectly spherical and optically clear the microsphere of interest was). Throughout our measurements we obtained accuracies of 2–5 nm laterally, and 5–10 nm axially on a 5 μm diameter optically trapped microsphere.
Submerged microscopic particles undergo stochastic thermal motion, resulting in a tracking signal that is inherently difficult to separate from random noise. In this work, we have shown how an SLM microscope can be used to provide a real-time evaluation of the 3D particle tracking accuracy in high-speed video stereo-microscopy. We have demonstrated an accuracy of less than 2.5% of the range of motion of a 5 μm diameter optically trapped microsphere, and shown how the error depends upon the degree of spatial filtering, and camera exposure time, allowing us to optimise the system. Knowledge of this error can also be factored into the stiffness calibration of optical tweezers for accurate force measurements, and used to more accurately recover microrheological properties of a sample. As the error signal is generated in real-time, it can also be used to alert an operator during an experiment if an unwanted piece of detritus is attracted into an optical trap.
This work was supported by the EPSRC (grant number EP/I007822/1). MJP acknowledges support from the Royal Society. We would like to thank Richard Bowman for useful discussions.
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