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Abstract
How far a particle moves along the optical axis in a holographic optical trap is not simply dictated by the programmed motion of the trap, but rather depends on an interplay of the trap’s changing shape and the particle’s material properties. For the particular case of colloidal spheres in optical tweezers, holographic video microscopy reveals that trapped particles tend to move farther along the axial direction than the traps that are moving them and that different kinds of particles move by different amounts. These surprising and sizeable variations in axial placement can be explained by a dipole-order theory for optical forces. Their discovery highlights the need for real-time feedback to achieve precise control of colloidal assemblies in three dimensions and demonstrates that holographic microscopy can meet that need.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Holographic optical traps use the forces and torques exerted by computer-generated holograms to localize and manipulate micrometer-scale objects [1,2]. In principle, holographic traps can move colloidal particles along arbitrary paths in three dimensions and can arrange multiple particles into precisely specified three-dimensional configurations [3–5]. In practice, however, where a trap places a particle depends on details of the particle’s interaction with the light field. Here, we use Lorenz-Mie microscopy to measure colloidal spheres’ trajectories in holographic traps and thereby to demonstrate that particles with different sizes and compositions not only reside at different axial positions within coplanar traps, but indeed travel substantially different distances when the traps are displaced along the optical axis. This surprising observation can be explained by considering how an optical trap’s structure depends on its axial position. Even so, axial displacements pose a practical challenge because variations from particle to particle can be large and are difficult to predict quantitatively. We demonstrate that Lorenz-Mie microscopy can provide the real-time feedback needed to achieve precise three-dimensional control over colloidal assemblies with holographic optical traps.
2. Holographic optical trapping
The holographic trapping technique, depicted schematically in Fig. 1(a), uses computer-generated holograms to structure the wavefronts of a laser beam so that the modified beam forms the desired configuration of optical traps when brought to a focus by a strongly converging objective lens [1,2]. The implementation used for this study is driven by a 10 W fiber laser (IPG Photonics, YLR-10-LP) operating at a vacuum wavelength of $\lambda _0 = 1064\;\textrm{nm}$. Holograms are imprinted on this laser’s wavefronts using a liquid-crystal spatial light modulator (Holoeye Pluto). The modified beam is then projected into the sample by a $100\times$ oil-immersion objective lens with a focal length of $f = 200\;$µm and a numerical aperture of $\mathrm {NA} = 1.4$. (Nikon S-Plan Apo).
Fig. 1. (a) Schematic representation of the combined instrument for holographic optical trapping and in-line holographic video microscopy. Holographic traps are projected into the sample cell by imprinting a phase hologram onto the wavefronts of an infrared laser beam and relaying the hologram to the input pupil of an objective lens with a dichroic beamsplitter. In-line holograms of trapped particles are recorded with a blue laser beam that passes through the dichroic to a video camera. (b) Typical holograms of a polystyrene sphere (PS) and a silica sphere (SiO$_2$) displaced to specified axial positions, $z_j$, by adjusting the phase hologram. (c) Measured axial positions, $z_p(z_j)$, of a polystyrene sphere (PS) and a silica sphere (SiO$_2$) as a function of specified trap position, $z_j$. Large circles depict the holographically measured radii, $a_p$, of the two spheres and are positioned at the measured plateau heights of their trajectories. The two particles thus agree on the height, $z_{\textrm{wall}}$, of the upper glass wall of their sample cell. Shading between the traces emphasizes the spheres’ increasing axial separation.
The ideal scalar hologram encoding $N$ point-like optical tweezers [6,7],
Equation (1) can be generalized to project line traps [8,9], ring traps [10,11], knotted traps [11,12] and tractor beams [13,14], among many other modalities. Hardware-accelerated algorithms can perform the necessary field calculations in real time, permitting dynamic interaction with trapped materials [15].
Although the focal point of a trap may be located at $\boldsymbol {r}_j$, the point of mechanical equilibrium for a particle localized in the trap typically is displaced by radiation pressure [16] as well as by external forces such as gravity. Measuring this effect with Lorenz-Mie microscopy [17] not only reveals material-dependent displacements, but also shows that the displacement depends strongly on the trap’s axial position, which means that the particle systematically moves either more or less than the trap that is moving it.
3. Lorenz-Mie microscopy
The instrument depicted in Fig. 1(a) creates in-line holograms of optically trapped particles [17–19] by illuminating the sample with a collimated laser beam at vacuum wavelength $\lambda _1 = 447\;\textrm{nm}$ (Coherent Cube) aligned with the optical axis of the objective lens. Light scattered by a particle interferes with the rest of the beam in the focal plane of the objective lens and is relayed by a tube lens to a monochrome video camera (Flir Flea3) that records the intensity of the magnified interference pattern. The imaging subsystem is separated from the trapping subsystem by a dichroic beamsplitter (Semrock). The images in Fig. 1(b) show holograms of a polystyrene sphere and a silica sphere recorded at three different axial displacements, $z_j$.
Lorenz-Mie microscopy is distinguished from other holographic microscopy techniques by the approach used to extract information from recorded holograms. The incident field is modeled as a monochromatic plane wave linearly polarized along $\hat {x}$ and propagating along $-\hat {z}$:
where $q = 2 \pi n_m/\lambda _1$ is the wavenumber of the imaging illumination and $\omega = 2\pi c/\lambda _1$ is its frequency. A particle located at position $\boldsymbol {r}_p$ within this beam creates a scattered field,The only instrumental calibration constants for Lorenz-Mie microscopy are the vacuum wavelength of the laser, the magnification of the microscope and the refractive index of the medium. Hardware-accelerated computation of the Lorenz-Mie scattering function enables complete fits to be performed in tens of milliseconds [24–26], which is fast enough for interactive operation and instrumental feedback.
4. Above and beyond: axial displacements
The representative data plotted in Fig. 1(c) illustrate the challenge posed by axial displacements in holographic optical traps. The two traces show the axial positions of a polystyrene sphere (PS, ThermoFisher Scientific, catalog number 4202A) and a silica sphere (SiO$_2$, Bangs Laboratories, catalog number SS05N) localized in water by an optical tweezer and translated straight upward along the optical axis in a sample cell composed of two parallel glass surfaces separated by $15\;$µm. The two spheres were selected from two populations co-dispersed in the same sample at a concentration of 105 particles/mL. The polystyrene sphere has a measured radius of $a_p = (0.989 \pm 0.001)\;$µm and refractive index $n_p = 1.612 \pm 0.001$. The silica sphere has a radius of $a_p = (1.184 \pm 0.031)\;$µm and refractive index $n_p = 1.396 \pm 0.003$. The two particles are trapped and translated one at a time by the same optical tweezer located at a fixed in-plane position within the sample cell. The trapped sphere is raised in discrete steps of 750 nm. Its characteristics and mean position at each step are obtained by analyzing 50 holograms recorded at 24 frames/s to average over thermal fluctuations. The holograms in Fig. 1(b) show individual snapshots from each of three stages in this process.
The trapped sphere eventually collides with the upper glass wall of the sample cell and so stops rising, even as the trap moves further upward. The height of the plateau augmented by the holographically measured radius of the sphere provides an estimate for the axial position of the wall, $z_{\textrm{wall}} = \max (z_p(z_j)) + a_p$, as indicated in Fig. 1(c). Although the two spheres’ plateau heights differ by $(240 \pm 40)\;\textrm{nm}$, the associated estimates for $z_{\textrm{wall}}$ differ by just 40 nm, which is comparable to the uncertainty in the radius of the silica particle. Comparably good agreement is obtained consistently with different pairs of particles and usefully validates the precision and accuracy of Lorenz-Mie microscopy for measuring particles’ axial positions and radii.
Using $z_{\textrm{wall}}$ as a fiducial point, the substantial difference of $1.2\;$µm in the spheres’ axial positions just before they reach the wall at $z_j = 14.5\;$µm cannot be ascribed to measurement error. Instead, this discrepancy shows that the two spheres rise through the cell with significantly different values of the axial scale factor,
The polystyrene sphere rises faster than the trap that is translating it, with $m_p = (1.068 \pm 0.007)\;$µm µm−1. The silica sphere sits consistently lower in the trap and moves upward in significantly smaller steps, with $m_p = (0.967 \pm 0.007)\;$µm µm−1.Comparable discrepancies in $m_p$ are observed reproducibly, not just from particle to particle at the same position in the same sample cell, but also from position to position and even in different sample cells. Figure 2(a) presents axial translation data from three types of colloidal spheres measured in multiple sample cells. These samples consist of $1.0\;$µm-radius polystyrene (PS) spheres, $0.8\;$µm-radius 3-(trimethoxysilyl)propyl methacrylate (TPM) spheres [26] and $0.8\;$µm-radius silica (SiO$_2$) spheres. Each type of sphere translates with a distinctive axial scale factor. The solid lines in Fig. 2(a) represent linear least squares fits whose slopes yield $m_p$. Their residuals, plotted in Fig. 2(b), display no significant trends.
Fig. 2. (a) Axial displacements of colloidal spheres in holographic optical traps, including polystyrene (PS), 3-(trimethoxysilyl)propyl methacrylate (TPM) and silica (SiO$_2$). Data for polystyrene and TPM are displaced upward for clarity. Each composition is represented by data from multiple spheres (11 for PS, 10 for SiO$_2$ and 4 for TPM) of nominally identical radii measured in different sample cells. (b) Residuals of displacements from linear fits for each of the three classes of spheres. (c) Experimental measurements of particles’ axial scale factors, $m_p(a_p)$, as a function of radius, $a_p$, for spheres made of polystyrene, silica and TPM. Values from (a) are plotted as crosses.
The axial scale factors from Fig. 2(a) are plotted as crosses in Fig. 2(c), where they are compared with corresponding results from other populations of spheres made of the same three materials. Micrometer-scale spheres made of polystyrene and TPM both rise faster than their traps ($m_p\,>\,1$), with axial scale factors that depend only weakly on size. Silica spheres have axial scale factors that decrease significantly with increasing particle radius, smaller spheres rising faster than their traps and larger spheres rising more slowly.
5. Modeling axial displacements
The observed deviations of $m_p$ from unity cannot arise from a simple scaling error because different types of spheres consistently deviate by different characteristic amounts. Nor can these deviations be attributed to aberrations in the trapping beam because both measurements reported in Fig. 1(c) were performed at the same position in the same sample cell, without any intervening mechanical adjustments that could have changed the shape of the trap.
To explain these observations, we model the intensity profile of an optical tweezer as a Gaussian beam brought to a focus at $\boldsymbol {r}_j$ with intensity profile, $\left \vert {\boldsymbol {E}(\boldsymbol {r} - \boldsymbol {r}_j)} \right \vert ^{2}$, given by
Moving the trap along the axis by $z_j$ changes $z_0$ by changing the shape of the beam’s focus. Specifically, the Rayleigh range depends on $z_j$ as [28]
where $z_R(0) \approx 8 / (k {\textrm{NA}}^{2})$ sets the scale for a Gaussian beam focused by a lens of numerical aperture NA. The derivation of Eq. (11) requires the Rayleigh range of the beams diffracted by the SLM to be much larger than the focal length of the objective lens, a condition that generally is met in holographic trapping systems.In the absence of other forces, a trapped particle comes to mechanical equilibrium at
Particles therefore tend to be displaced along the optical axis by more than the displacement of the optical trap, with the extra displacement depending on the particle’s size and refractive index. The upper trace in Fig. 3 shows $z_0(z_j)$ for a 120 nm-diameter polystyrene sphere in an optical trap modeled by Eq. (8) with parameters for our instrument. Contours of the intensity distribution show that the focal spot expands along the axial direction as $z_j$ increases, thereby increasing the particle’s displacement from the focal point.Fig. 3. Particle position, $z_0$, within an optical trap as a function of the trap’s axial displacement, $z_j$ for 120 nm-diameter polystyrene (PS, magenta, upper) and silica (SiO$_2$, yellow, lower) spheres dispersed in water and trapped in a diffration-limited optical tweezer at $\lambda _0 = 1064\;\textrm{nm}$. Colored contours reveal how the trap’s intensity profile, $\left \vert {\boldsymbol {E}(\boldsymbol {r})} \right \vert ^{2}$, broadens and elongates with increasing $z_j$. The polystyrene sphere rises in the trap, while the silica sphere sinks.
Displacing a trap along the axial direction decreases its stiffness through the dependence of $\boldsymbol {F}_e(\boldsymbol {r})$ on $z_R$, as shown approximately in Eq. (9). The scale of a trapped particle’s thermal fluctuations about $z_p(z_j)$ therefore should increase as $z_j$ increases. This trend is reflected in the increasing standard deviation of $\Delta z_p(z_j)$ plotted in Fig. 2(b).
For displacements smaller than the focal length of the objective lens, $z_j\,<\,f$, the dependence of $z_p$ on $z_j$ is roughly linear and is characterized by the axial scale factor
For our instrument, $m_p \approx 1 + 0.037 \frac {\alpha _e''}{\alpha _e'}$. Observing the nonlinear scaling predicted by Eqs. (10) through (12) is not feasible because particles escape their traps when $z_0(z_j)$ becomes large. Indeed, material-dependent displacements set an upper limit on the axial range through which colloidal particles can be translated with holographic optical tweezers.For the particular case of a small dielectric sphere of radius $a_p$ and refractive index $n_p$ immersed in a medium of refractive index $n_m$, the electric dipole polarizability is [29]
where the Clausius-Mossotti polarizability isEquation (16) is valid for particles that are substantially smaller than the wavelength of light $k a_p\,<\,1$. Holographic tracking and characterization, however, requires particles that are larger than the wavelength of light. The data for polystyrene and TPM in Fig. 2(c) therefore all have $k a_p\,>\,1$ and show a comparatively weak dependence on particle size, presumably because the particles are larger than the radius of the beam waist, $w_0 \approx 550\;\textrm{nm}$.
The data for silica spheres in Fig. 2(c) show a downward trend in $m_p(a_p)$ that runs counter to the prediction of Eq. (13) and can be explained by the influence of gravity. The weight of a sphere of density $\rho _p$ dispersed in a fluid of density $\rho _m$ shifts the point of mechanical equilibrium by a distance
Equation (17) is valid for particles whose density mismatch is small enough that $z_0 + \Delta z_0\,<\,z_R$. Within this range, Eqs. (14), (15) and (17) suggest that $\Delta z_0$ depends only weakly on $a_p$. In fact, the data for silica spheres in Fig. 2(c) shows that gravity causes larger spheres to sit lower in their traps and to rise more slowly. This is consistent with the observation of weaker-than-predicted size scaling for comparably sized polystyrene and TPM spheres. Although the dipole-order theory accounts for qualitative features of the observed displacements, quantitative predictions for larger spheres presumably require a higher-order treatment. Limitations of the analytically tractable dipole-order theory highlight the value of holographic microscopy for providing in situ experimental feedback, particularly for particles that are larger than the wavelength of light.
6. Discussion
We have used Lorenz-Mie microscopy to demonstrate that holographically trapped colloidal spheres are displaced within their traps by amounts that depend substantially on the traps’ axial positions. These displacements, often amounting to more than the wavelength of light, are explained by changes in the traps’ Rayleigh ranges as they move along the optical axis. How this affects the position of a particle within a trap depends on the particle’s size and refractive index, and also can be influenced by external forces such as gravity. Whereas in-plane displacements can be measured and calibrated with conventional microscopy and standard techniques of image analysis [30], axial displacements have been much more challenging to measure and so have been largely overlooked. The present study demonstrates that ignoring material-dependent axial displacements can lead to large errors in particle placement, not only for heterogeneous assemblies, but also for nominally identical spheres arranged in three-dimensional patterns. The comparative difficulty of predicting and correcting these offsets a priori creates a need for the real-time feedback that can be provided by quantitative holographic video microscopy.
Funding
National Science Foundation (DMR-0922680, DMR-1420073).
Acknowledgments
This work was supported by the MRSEC program of the National Science Foundation through Award Number DMR-1420073. The instrument used for this work was constructed with support of the MRI program of the NSF under Award Number DMR-0922680. We gratefully acknowledge the support of the nVidia Corporation through the donation of the Titan Xp GPU used for this research.
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