Three-dimensional (3D) particle tracking of microscopic samples is an invaluable technique for understanding various mechanisms in biophysics and cell biology [1]. The technique typically utilizes optical tweezers to actively trap and manipulate the locations of trapped particles [2].

To measure the precise positions of microscopic particles in three dimensions, various techniques have been developed including quadrant photodiode detection [3,4], stereomicroscopy [5 –7], and V-shaped micro-mirrors [8,9]. Among the various methods, digital holographic microscopy has been widely implemented for tracking the 3D positions of microscopic particles because a single hologram contains information about the axial position of an object [10]. Using the measured holograms, the 3D positions of samples are traced by volumetric reconstructions using Rayleigh–Sommerfeld backpropagation [11 –13]. Alternatively, Fresnel particle tracing [14] and Lorentz–Mie fitting [12,15,16] can achieve high-resolution measurement of 3D particle tracking by direct fitting of the measured holograms with calculated theoretical holograms.

However, these existing methods for tracking particles are limited to applications involving samples with a simple geometry, such as single-layered microspheres, and usually exhibit poor axial reconstructions that cannot be applied to complex systems, including axially stacked particles or particles interacting with biological cells and tissues [12]. Furthermore, most existing visualization methods used with the optical tweezers cannot be applied to samples that have complex shapes, such as biological cells, because they require a priori structural information about the objects for tracking, or are incapable of considering light diffraction. However, a direct visualization of complex objects should be highly desirable in many studies including interactions of microspheres [17] and biological samples [18]. Recently, several volumetric methods that are compatible with optical trapping have been developed [19 –22]. However, they require translational motions of an objective lens [19], fluorescence labeling [20], or the use of optical trapping for tomographic measurements [21].

Optical diffraction tomography (ODT) techniques have been introduced to measure the 3D refractive index (RI) distributions of biological [23 –29] and colloidal samples [30]. Multiple 2D complex optical fields diffracted by a sample are acquired via interferometry for various illumination angles, from which the 3D RI distribution of the sample is numerically reconstructed. Because ODT considers light diffraction inside samples for tomogram reconstruction, the reconstructed 3D RI distribution reveals the correct shapes and 3D positions of complex samples at a defocused plane as well as the focused plane. Furthermore, ODT also provides unique advantages including the quantitative RI measurements of samples, noninvasiveness, and the ability for samples to remain stationary for tomographic measurements.

In this Letter, we propose and experimentally demonstrate a novel technique capable of simultaneous 3D visualization and tracking of optically trapped particles with complex geometry. Spherical particles are trapped by the holographic optical tweezers (HOT) system, and ODT reconstructs the 3D RI distribution of the samples with high lateral and axial resolution. We validate the capability of the proposed method for 3D tracking of micrometer-sized particles from reconstructed tomograms, and compare the results obtained with axial reconstructions using the conventional angular spectrum method (ASM). In addition, 3D tracking and visualization of a colloidal particle in the vicinity of biological cells is also demonstrated.

The experimental system consists of a HOT system (the red boxes in Fig. 1) and an ODT system (the green boxes in Fig. 1). The positions of microscopic particles are manipulated using the HOT system. A collimated beam from a high-power diode-pumped solid-state (DPSS) laser (${\lambda }_T=1064\mathrm{nm}$, 10 W, MATRIX 1064-10-CW, Coherent, Inc.) is expanded to overfill the active area of a spatial light modulator (SLM, X10468-07, Hamamatsu Photonics K.K.). The SLM generates phase-only holograms to modulate the wavefront of a laser beam incident to the samples to be trapped. By adding a phase grating pattern on the SLM, the unmodulated beam (zeroth-order diffraction) is spatially separated from the modulated beam (first-order diffraction) in the Fourier plane of the SLM. Then the unmodulated beam is blocked by a spatial filter (SF in Fig. 1). The modulated patterns on the SLM are demagnified by an additional $4-f$ telescopic system in order to overfill the back aperture of a high numerical aperture (NA) objective lens (OL in Fig. 1, UPLSAPO $100\times$, oil immersion, $\mathrm{NA}=1.4$, Olympus, Inc.) equipped in an inverted microscope (IX71, Olympus Inc.)

 

Fig. 1. Experimental setup. SLM, spatial light modulator; SF, spatial filter; TL, tube lens; OL, objective lens; CL, condenser lens; GM, galvanomirror; BS, beam splitter; DM, dichroic mirror. Holographic optical tweezers trap spherical particles and manipulate their 3D positions, while optical diffraction tomography measures 3D RI tomograms of samples with high speed.

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The 3D positions of the optical traps are manipulated using phase-only holograms on the SLM, which are generated using the superposition of lenses and gratings (S algorithm) [31,32]. Using ODT, the 3D positions and RI distributions of trapped particles are simultaneously measured (the green boxes in Fig. 1). A sample is illuminated with plane waves from a monochromatic laser (${\lambda }_D=532\mathrm{nm}$, 100 mW, SDL-532-100T, Shanghai Dream Laser Co.) with various incident angles by tilting a 2D galvanomirror (GVS012, Thorlabs Inc.). Then, multiple full-field holograms of the sample obtained with various illumination angles are recorded via Mach–Zehnder interferometery.

In ODT, from the measured 2D holograms, complex optical fields of the sample are retrieved using the field retrieval algorithm [33,34], from which the 3D RI distribution of the samples is reconstructed via the Fourier diffraction theorem [24,35]. To solve the inversion problem of light diffraction, the first Rytov approximation was implemented, which is valid when the spatial distribution of RI values does not vary significantly, and this condition is suited for most biological and colloidal samples [24,36]. Due to the sparse illumination angles and the finite NA of the objective lens, there exists missing information in the 3D Fourier space, which is filled by the iterative nonnegativity constraint algorithm [30,37]. The detailed optical setup and the MATLAB code used can be found elsewhere [30].

To demonstrate the 3D particle tracking capability of ODT, we performed simultaneous optical trapping and tomographic measurements of colloidal microspheres (Fig. 2). Silica beads with a diameter of 2 μm (81108, Sigma-Aldrich Co.) in a 40% (w/w) sucrose solution were used. As shown in Figs. 2(a)–2(c), using the HOT system, eight beads were first aligned in the focal plane as a $2\times 4$ formation $(t=0\mathrm{s})$, and arranged to the vertices of a 6-μm-side cube $(t=0.5\mathrm{s})$. The cube was then rigidly rotated with respect to the $y$ axis $(t=1.5\mathrm{s})$. The galvanomirror circularly scanned 10 illumination beams with various azimuthal angles, followed by one beam at the normal angle with a scanning rate of 16.7 ms/cycle. The acquisition of 11 complex optical fields from various illumination angles is enough for a robust tomogram reconstruction; the tomograms reconstructed from 11 viewing angles and 500 full viewing angles have a correlation coefficient of 0.95 [30].

 

Fig. 2. 3D tracking of eight optically trapped silica beads forming a rotating cube. (a) Bright-field images of silica beads. (b) Cross-sectional slices of fields in the $x–z$ plane reconstructed using the ASM. (c) Cross-sectional slices of reconstructed RI tomograms in the $x–z$ plane. Each scale bar indicates 10 μm. (d) Axial profiles of the reconstructed volumes of individual silica beads at $t=0$ sec as reconstructed by either conventional ASM (red lines) or ODT (blue lines). For comparison purposes, the axial profile of an ideal bead is shown as a dotted gray line. (e) 3D rendered isosurfaces of the RI maps of the beads after the release of the trapping beam. (f) Time-lapse trajectories of the beads for 1.7 s, measured at the frame rate of 60 Hz. Rendered RI isosurfaces of the trapped beads at three consecutive time frames (top). 3D trajectories of the beads under Brownian motion (bottom). Time-lapse isosurfaces in (c) and (e) are presented in Media 1 and Media 2, respectively.

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For systematic comparison with previous methods, the reconstructed tomograms using ODT were compared with volumetric fields. In order to obtain a volumetric field using the ASM based on Rayleigh–Sommerfeld backpropagation [38], the measured optical field with a normal illumination was numerically propagated along the axial direction.

The 3D volumetric reconstructions of the trapped silica beads obtained by ASM and ODT are shown in Fig. 2. Because silica is optically transparent in the visible range, it is difficult to identify each bead in intensity images obtained with bright-field microscopy [Fig. 2(a)]. In addition, cross-sectional slice images of volumetric reconstructions using the ASM exhibit poor axial reconstruction in the $x–z$ plane [Fig. 2(b)] In particular, when two beads are aligned along the optical axis as shown in the center panel in Fig. 2(b), the ASM fails to determine the axial position of the beads because the ASM assumes that the surrounding medium is optically homogeneous [39]. In contrast, the cross-sectional slices of reconstructed RI tomograms [the $x–z$ plane in Fig. 2(c)] clearly demonstrate that ODT can precisely determine the 3D positions of the trapped beads without image elongation or deterioration along the axial direction.

The difference in volumetric reconstruction between the ASM and ODT can also be seen clearly in the axial profiles of trapped beads at the focus plane [Fig. 2(d)]. The profiles of reconstructed tomograms using ODT exhibit a trapezoidal rather than a rectangular shape due to finite lateral and axial resolution. By analyzing the edge of the profiles, the lateral and axial resolution were calculated to be 373 nm and 496 nm, respectively, which is well consistent with the theoretical expectation.

To demonstrate the applicability of the present method, 2-μm-diameter silica beads were optically trapped in a cube formation ($t=0\mathrm{ms}$), and then consecutive 3D RI tomograms of the beads were measured, after releasing the optical trap, with the frame rate of 60 Hz for 1.7 s, as shown in Fig. 2(e). 3D positions of individual beads were determined as the local maxima of the imaginary parts of reconstructed fields [12].

In order to further demonstrate the applicability of the present technique in the field of biophysics and cell biology, we performed the 3D position tracking of an optically trapped silica bead approaching a macrophage (Fig. 3). The macrophages were collected from mice peritoneal cavity. Ice-cold phosphate buffered saline (PBS) with 3% fetal bovine serum (FBS) was injected into the peritoneal cavity of euthanized mice. Then, a fluid was collected from the same peritoneal cavity position and centrifuged at 1500 RPM for 8 min. Isolated macrophages were maintained in Dulbecco’s modified Eagles’ medium (DMEM, Gibco) supplemented with 10% heat-inactivated FBS for two days before experiments. The sample preparation procedures and the methods were approved by the KAIST Institutional Review Board (IRB project No. KA2014-01). Macrophages were sandwiched between two cover slips, and imaged at room temperature. For the bead trapping experiments, silica beads with a diameter of 2 μm were added to a medium of cultured macrophages 1 h before measurements.

 

Fig. 3. 3D tracking of a macrophage and an optically trapped silica bead. (a) Bright-field images. (b) Cross-sectional slices of field in the $x–z$ plane reconstructed using the ASM. (c) Cross-sectional slices of 3D RI tomograms in the $x–z$ plane. The locations of the bead are indicated with black arrows. Time-lapse slice images are also presented in Media 3. (d) Consecutive time-lapse cross-sectional slices in (c).

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A silica bead in the vicinity of a macrophage was trapped by the HOT system, and the trapped bead was set to approach the position of the macrophage at the speed of 25 μm/s. The axial position of the bead was adjusted to be between the bottom cover glass and the highest position of the cell. The approaching bead first touched the cell membrane of the macrophage, and climbed the cell in response to the optical trap position.

Figures 3(b) and 3(c) show the 3D volumetric reconstruction of the silica bead approaching the macrophage, as obtained by the ASM and ODT, respectively. Because most biological cells are transparent in the visible range, the intensity images of the macrophage in Fig. 3(a) do not have enough imaging contrast, and that results in the poor volumetric reconstructions of the macrophage using the ASM in Fig. 3(b). When the bead touched the cellular membrane and climbed the cell [the right panel of Fig. 3(b)], the cross-sectional slice of field in the $x–z$ plane of the bead was further degraded. This is because the overlapping complex optical fields of the bead and the macrophage produce an effect similar to those of the stacked beads in a cube.

In contrast, Fig. 3(c) and Media 3 clearly demonstrate that ODT can precisely determine the 3D positions of microscopic particles in the presence of complex biological cells. Importantly, the reconstructed tomograms obtained with ODT clearly demonstrate the shape change of the macrophage, indicated as the blue arrows in Fig. 3(d), in response to the collision of the silica bead [the dotted lines in Fig. 3(d) and Media 3]. The 3D imaging capability of ODT also enables visualization of subcellular organelles including vesicles inside the macrophages.

In conclusion, we present a novel technique for the 3D visualization and position tracking of optically trapped particles. Employing the combined system, 3D RI tomograms of optically trapped particles were simultaneously and precisely measured with high lateral and axial resolution. In comparison to the conventional ASM, the present approach provides the 3D visualization and position tracking of complex samples. Specifically, the present approach can localize the 3D positions of multiply stacked particles without shape distortion, whereas conventional ASM fails to determine the localization of the particles. For demonstration, a cubic formation of optically trapped silica beads was clearly visualized using the proposed 3D RI tomographic measurements. The precise 3D localization of an optically trapped microsphere in the vicinity of a macrophage was also demonstrated.

Because ODT does not require a priori structural information of samples for tomogram reconstruction, the present approach can trace the 3D positions of samples that have complex geometry, including biological cells. Moreover, ODT provides additional quantitative phase information about the samples [40,41], which allows the assessment of other important properties including dry mass [42], protein concentration [43,44], and visualization of subcellular organelles [45].

The present method, however, has technical limitations in acquisition speed and spatial resolution. Reconstructed 3D RI tomograms have diffraction-limited spatial resolution, which is determined by the NAs of the used condenser and objective lens. In order to reconstruct a 3D RI tomogram, ODT needs to measure multiple 2D optical fields with various illumination angles, which can limit the acquisition speed [39]. This work employed a sparse scanning scheme in order to reduce the number of illumination angles [30], and the acquisition rate was 60 tomograms/s. Because the repetition rate of commercial SLMs is a few tens of hertz, the present acquisition rate is sufficient for measuring particles trapped using SLM. In the future, rapid wavefront shaping devices, such as a dynamic mirror device, could be used to investigate complex and rapid dynamic systems.

In addition, the complicated optical setup used in this work can be simplified using a recently developed quantitative phase imaging unit [26,46]. We envision that the proposed method can pave the way for real-time optical simultaneous manipulation and visualization of diverse systems including complex colloids and biological cells and tissue.