Abstract

Optical tweezers are a powerful tool to hold and manipulate particles on the microscale. The ability to measure tiny forces enables detailed investigations, e.g., of the mechanical properties of biological systems. Here we present a generally applicable method to simultaneously measure all components of the force applied to a specific particle in a trapped ensemble, or to a specific site of an extended object. This holographic force measurement relies on a detailed analysis of a single interference pattern formed in the far field to recover amplitude and phase of the field. It requires no information about size, shape, or optical properties of the particles and can be scaled to many traps—we show individual force measurements for up to 10 particles. In addition, we demonstrate force measurements when stretching a red blood cell, held directly by four traps. This method opens up a wealth of new opportunities made possible by localized quantitative force measurements in complex biological settings.

Published by The Optical Society under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

1. INTRODUCTION

Light carries energy and momentum, and thus exerts force and torque when interacting with matter. Although optical forces are typically too weak to be significant for macroscopic objects, at the microscopic scale, they can have dominant impact.

Optical forces are employed in a wide variety of research fields, e.g.,  in biology and medicine to hold cells or microorganisms in a noncontact manner, or in physics to handle ultracold atoms, molecules, or quantum optomechanical devices in vacuum (see [1] for an overview).

The simultaneous manipulation of several particles as, e.g.,  enabled by using holographic optical trapping [2,3], provides extended possibilities to probe particle interactions, or to squeeze or stretch microorganisms or biomolecules such as DNA with the help of optically trapped particles that act as handles [4,5].

It is of fundamental interest to quantitatively determine the exerted optical forces. A commonly used method to measure forces [6] relies on the back-action of the trapped particle on the trapping light. The displacement of a particle from equilibrium due to an external force induces a deflection of the trapping light, which can be detected by measuring the (average) beam direction or, equivalently, the beam position in the far field. This can be performed with high sensitivity by a quadrant photodiode or a position-sensitive detector placed at the back focal plane (BFP) of a lens, which collects the transmitted trapping light. This allows one to detect weak forces (sub-piconewton) or displacements in the nanometer range, which are difficult to detect by direct observation.

This conventional force detection method requires a calibration to relate the detector signal (proportional to beam position in the BFP) to the optical force. However, one is able to directly deduce the total exerted optical force by a measurement of the change in momentum flux of the trapping light [4,79]. Experimentally, this is given by the center-of-mass position of the far-field distribution of all of the deflected trapping light. This requires collection of the outgoing trapping light with, e.g., a high-NA condenser lens. This direct force measurement method has the outstanding property that it is calibration free. No information about the particle shape and size, trap shape, or surrounding medium is required. Especially for biological specimens, such data are usually not available with sufficient accuracy due to large individual variations.

But these force measurement methods are not immediately applicable to the simultaneous trapping of several particles, because in the far field, the trapping light overlaps. Thus, one needs a method to disentangle the individual scattered fields. Some approaches to split up the individual contributions rely on using two orthogonal polarization states and also two or more lasers with different wavelengths [5]. However, when scaling up to simultaneous measurements on many particles, these methods impose a rather large experimental burden. Time multiplexing, i.e., rapidly switching between several trapping positions with synchronized force detection, is experimentally easier to scale to many particles. However, as the beam scanning is commonly achieved by using acousto-optic deflectors (AODs), particle positions are controlled only within a single plane, which significantly reduces the flexibility compared to holographic optical tweezers (HOTs). Furthermore, time multiplexing inherently does not allow for truly simultaneous measurements.

Our approach to disentangle the individual contributions to the force is to recover the complex field, i.e., amplitude and phase, in the far field, as sketched in Fig. 1 and described in detail in Section 2. This provides us with full information about the light field, allowing us to computationally propagate the retrieved field into the focal region. There the particles and the optical traps are usually spatially well separated, and the total field can be easily split up into the individual contributions to determine the individual forces. This is related to the idea in [10] to select a single contribution from a conjugate (image) plane of the object plane with a pinhole.

 

Fig. 1. Idea of individual direct force measurements for optically trapped particles: the superposition of all the single scattered light fields is collected by a high-NA condenser lens. From a single interference pattern occurring in the back focal plane, we numerically reconstruct the full complex light field in the focal plane, where the individual light fields are well separated, allowing us to determine the forces individually.

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Our computational approach provides a simple setup with high flexibility, not constraining the possibilities of HOTs. The method is able to follow dynamically changing trap positions without difficulties, even for axial displacements. Exactly simultaneous direct force measurements of all components (radial, axial) for all particles are enabled.

In this work, we present an experimental implementation of this idea that is suited for practical applications in terms of simplicity and robustness. The main innovations are the following:

  • • To measure the complex field, we rely on common-path interferometry. The reference beams pass nearby the trapped particles and closely follow the same optical path as the trapping beams. By this we get rid of the high susceptibility of interferometric measurements to aberrations or fluctuations if both arms of the interferometer are separated.
  • • We deduce the complex field from a single image of the far-field intensity distribution with the help of an iterative field-retrieval algorithm, making use of the known positions of the trapping spots. This makes this method suitable for dynamic settings.
  • • To obtain accurate results, we rely on a detailed model that predicts phase and amplitude of the reference beams from the known phase pattern on the accurately calibrated spatial light modulator (SLM), including a model for the SLM response including pixel crosstalk and information about obstructions (apertures) in the optical path.

Our method conserves the benefits of direct force measurements, i.e., it is calibration free and provides correct measurements for particles of arbitrary shape, as has been demonstrated earlier [8,9,11]. It adds little additional experimental effort to a holographic optical trapping setup with conventional force detection. Nevertheless, it requires a careful characterization of the optical train (e.g.,  condenser distortion, aperture size, and position), for which we have developed automated procedures. As a drawback, the iterative field retrieval requires high computational power. This is mitigated by using an optimized implementation that makes use of GPU acceleration. The source code of our software implementation for holographic force measurements is publicly available [12].

We demonstrate simultaneous force measurements with up to 10 particles, and confirm that the disentangled individual forces well match control measurements obtained with a single trap. Simultaneous measurements of thermally induced forces (Brownian motion), in our case for six microspheres, show that this method is able to detect small forces, way below the average thermal noise floor. To establish that this method is well suited for applications, we measure the forces exerted on red blood cells (RBCs), which are optically trapped with four spots without using additional handles, such as microspheres attached to the cells.

2. PRINCIPLE OF INDIVIDUAL DIRECT FORCE MEASUREMENTS

As a main contribution, we present here the basic principles and describe our algorithm to retrieve individual optical forces from a single image of the far-field intensity distribution.

A. Basic Optical Setup

To explain the basic principle of our method, we first present a simplified scheme of the optical setup (Fig. 2). A more detailed description of the actual setup is given in Section 3. Beam shaping takes place with the help of a SLM. With two lenses in a $4f$-configuration, the SLM is imaged to the back focal or pupil plane of the objective lens. In the intermediate focal plane between the two lenses, some unwanted diffraction orders are blocked, i.e., undiffracted zero-order light and higher orders are removed. A central optical element for the force measurements is the high-NA condenser lens, which collects all forward scattered light. The intensity distribution in its BFP, which is equivalent to the far field, is detected with a digital camera. In this configuration, the light fields in consecutive planes (Fig. 2) are related by a Fourier transform.

 

Fig. 2. Simplified optical setup for optical trapping and force detection.

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B. Direct Force Measurements

The idea of direct force measurements relies on the basic principle of conservation of momentum. When a particle interacts with a laser beam, optical forces occur due to transfer of momentum from light to particle. The exerted optical force is given by the difference between total ingoing and outgoing momentum flux. The momentum flux, invariant when light propagates freely, can be easily determined from the far-field intensity distribution. On the far-field sphere, light stemming from a small scatterer travels radially. The local momentum flux is proportional to the intensity and points in the radial direction. Hence, the components of the total momentum flux can be obtained by summing all contributions over the far-field sphere. In the BFP of the condenser, the far-field intensity distribution is mapped to a circular distribution $I(x,y)$. Assuming that the condenser (with focal length ${f_{\text{C}}}$) obeys the Abbe–Sine condition, i.e., a central ray at angle $\vartheta$ to the optical axis is imaged at a distance $\rho = {f_{\text{C}}}\sin \vartheta$, all components of the force can be calculated from the BFP by

$${\bf F} = \frac{1}{c}\int I(x,y){\left({x,y,\sqrt {1 - ({x^2} + {y^2})}} \right)^T}\text{d}A - {{\bf F}_0} ,$$
where ${{\bf F}_0}$ denotes the ingoing momentum flux. For simplicity, here we use dimensionless coordinates $x$ and $y$, which are normalized by ${f_{\text{C}}}$.

In practice, one has to once determine a single calibration factor to relate the detector signal to forces [8,9,13]. For this, we apply a known drag force to a trapped particle, similar to the procedure described in Section 4.B.1.

C. Reconstruction of Individual Forces from a Single Interferometric Measurement

A key step for the measurement of individual forces from the far-field intensity pattern is the reconstruction of the full complex field, i.e., amplitude and phase of the light. To reveal the hidden phase information from the far-field intensity measurements, we rely on interference with a reference light field with known phase and amplitude (see Section 2.C.1). Once the complex field is known in one of the planes, it can be numerically propagated to other planes.

Here, and also in several other respects, we count on the fact that in optical trapping, the light configuration typically is sparse: a few particles are held by separated, tightly focused beams of known locations, immersed in an optically clear environment (buffer solution). In the far field, all scattered fields overlap and contribute to the intensity distribution. Due to the typically small size of a scatterer and the focused optical trap, the individual far field of a single scatterer shows only smooth variations. Interference with the reference field and other scatterers at distances larger than the spot size leads to fringes in the BFP, with each contribution possessing a distinct spatial frequency. This enables the disentanglement of the individual contributions, as described in more detail in the following.

Our approach to recover the individual complex light fields of the scatterers, i.e., phase and amplitude of the individual contributions, is to seek a solution to the following inverse problem: determine the fields in small patches around the traps such that the calculated total far-field intensity pattern agrees with the measured one.

The patches are chosen such that local redistribution of light due to the scattering by the object is well contained within. Typically, we use patch diameters of 4–5 µm. For close distances down to 3 µm, we crop the patches to make them disjoint. For even smaller distances, the fields start to overlap in the object plane, and it becomes difficult to split up the light.

1. Reference Field for Interferometric Reconstruction of Phase and Amplitude

As a reference for the interferometric measurement of phase and amplitude in the far field, we use additional trapping beams that are placed such that they pass unobstructed through the sample chamber. Except for a short distance around the focal plane, the reference beams follow mostly the same optical path as the trapping beams. Aberrations present in the optical train, in particular stemming from the simple optical design of the condenser lens, are cancelled to a large extent, rendering the method robust against changes in alignment of the optics.

As a slight complication, one has to take into account that splitting up the trapping light into several beams with a phase-only SLM in a holographic optical trapping setup also generates additional, weaker spots (“ghost traps”), which typically contain about 10% to 20% of the total power. In the far field, they all overlap and contribute to the reference field, creating an intricate pattern.

To obtain full information about the reference field, we rely on calculations. We start from the known phase pattern of the SLM and numerically propagate the light field (see Section 2.C.3) to the intermediate focal plane, where we take into account clipping and beam obstruction by the aperture placed there. Next, in the object plane, we set the field in small areas around the occupied trapping sites, where the field is initially unknown, to zero, keeping only the known reference beams. Then we calculate the far field of the modified object plane field, to be used as the reference field for interferometry in the BFP.

Since wrong assumptions about amplitude and phase of the reference field translate into errors in the results of the unknown scattered field, we have taken care to obtain accurate results about the reference field. We use a carefully calibrated SLM, for which we characterized the phase response versus control voltage for every pixel across the SLM surface. Additionally, we use our refined model for the pixel crosstalk to accurately predict the actually realized phase pattern [14]. Especially for patterns with small periods, this leads to significant improvements compared to simple models that neglect crosstalk, assuming uniform response across a single pixel with sudden transitions.

Furthermore, we observe the actual position of the obstructions in the intermediate focal plane (see Section 3.B.2 for details of the measurement procedure) and use this information in the numerical light propagation.

2. Field Reconstruction from Single Far-Field Intensity Data

In general, in interferometry, a single intensity measurement of the interference pattern is not enough to uniquely determine the unknown source fields. A source distribution that is mirrored about the origin would yield the same far-field intensity pattern. However, in our setting the mirror solution would not coincide with the known location of the scatterers and reference, unless for a highly symmetric configuration, and can thus be ruled out. It is known [15] that sparse spatial support of the unknown complex field in the focal plane without symmetries contributes to the existence of a unique solution. For this reason, it is possible to retrieve phase and amplitude from a single intensity measurement. This is highly favorable for dynamically changing settings, as mostly present in optical trapping, when thermal forces acting on optically trapped particles lead to fluctuating conditions.

3. Numerical Model for Light Field Propagation

Numerical propagation of the light field between the different planes, as depicted in Fig. 2, plays a central part in our method. Essentially, the light fields in consecutive planes are related by a Fourier transform. We rely on a slightly more general algorithm based on the Collins integral [16]; details of the propagator are given in Supplement 1.

A convenient feature of the algorithm is that propagation between planes of any axial position is possible. We use an implementation using fast Fourier transforms (FFTs) in a way that the discretization grid spacing and size can be chosen at will, i.e., it has a “zoom-in” capability that allows one to calculate the target field in a selected sub-region only, similar to that shown in [17].

We do not account for the vectorial nature of light; our model based on the paraxial approximation turns out to be sufficient. We checked that with our samples, polarization effects are negligibly small; therefore, considering only a single polarization direction is enough.

Correction of condenser distortion. As explained above, the field retrieval problem requires to relate the measured intensity in the BFP with the simulated one. For this one needs to properly align the data. Additionally, we observe that the condenser lens induces a significant barrel distortion, which also needs to be corrected. As explained in Supplement 1 in more detail, we characterize the distortions of the condenser and calculate undistorted BFP images for further processing.

Empirical model for detector. We observe that the measured intensity distribution in the BFP shows a reduced contrast compared to what we expect from the numerical simulations. This affects the field retrieval and might lead to underestimation of the amplitude of the recovered field. While we were not able to locate the origins of this issue in all detail, we could identify some major contributions, which can be divided into two groups: effects that lead to smearing of the image, which reduces contrast especially for patterns with small details, such as the finite pixel size of the camera and the interpolation used to correct for the condenser distortion, and effects that lead to a rather uniform background signal, such as stray light. To include this contrast reduction in our model for calculating the intensity in the BFP, we use an empirical model that convolves the calculated intensity pattern with a kernel that consists of the sum of two Gaussians, one with a narrow peak (width ${\sigma _1} = 0.9$ pixels), which contains most (88%) of the energy, and a much broader (${\sigma _2} = 40$ pixels) pedestal, which models the stray light.

Angle-dependent transmission losses in detection path. Light scattered at large angles experiences enhanced losses in the detection path. These transmission losses need to be corrected before calculating forces according to Eq. (1). We found that the main contributions stem from reflections at the boundary from water to the mounting glass and within the condenser, especially at the back surface of the strongly curved front element. To model the losses, we combine the Fresnel coefficients for the water–glass interface, which exhibit a rather steep decrease at the border, with an empirical model for the condenser transmission. Based on measurements, we found that the transmission through the condenser for the radial polarization component is nearly constant, whereas the tangential component shows a more pronounced, approximately quadratic fall-off of about 20% at the border.

4. Field Retrieval by Iterative, Gradient-Based Optimization

In the presence of noise and experimental imperfections, the abovementioned task for the field retrieval from a single far-field intensity image needs to be restated as an optimization problem: find values for the field in patches in the object plane close to the scatterers such that the mean squared difference $R = \sum {(I - {I_0})^2}$ between the calculated intensity $I$ and the observed one ${I_0}$ in the BFP is minimized.

To find such an optimum, we rely on iterative, gradient-based optimization, in particular on a variation of steepest descent. The forward model consists of numerically propagating the combined individual patches ${E_{\text{obj},k}}({{\bf r}_{\text{obj}}}) = {m_k}({{\bf r}_{\text{obj}}}){E_{\text{obj}}}({{\bf r}_{\text{obj}}})$ in the object plane to the BFP, where ${m_k}$ denotes the mask function for patch $k$, and adding them with the reference field to get the far field:

$${E_{\text{bfp}}} = {E_{\text{bfp},\text{ref}}} + {\cal P}\left(\sum\limits_k {E_{\text{obj},k}}\right),$$
where ${\cal P}$ denotes the forward propagation. The simulated intensity at the camera $I = (|{E_{\text{bfp}}}{|^2}T) * {K_{\text{cam}}}$ is given as the far-field intensity $|{E_{\text{bfp}}}{|^2}$, multiplied with the transmission function $T$ of the detection optics, and convolved with a kernel ${K_{\text{cam}}}$ describing the (incoherent) point spread function of the detector (see Section 2.C.3).

For gradient-based optimization, we need to calculate the gradient ${\bar E_{\text{obj},k}} = \frac{{\partial\! R}}{{\partial {E_{\text{obj},k}}(i,j)}}$ of the cost function $R$ with respect to each value of the light field close to the scatterers in the object plane ${E_{\text{obj},k}}(i,j)$ at grid point $(i,j)$. With help of the rules for automatic differentiation [18], this is straightforward to accomplish: first we calculate the gradient for the electric field in the BFP: ${\bar E _{\text{bfp}}} = \frac{{\partial\! R}}{{\partial {E_{\text{bfp}}}(i,j)}} = 2{E_{\text{bfp}}}({T(I - {I_0}) * {K_{\text{cam}}}})$. We then propagate the field gradient ${\bar E _{\text{bfp}}}$ in the BFP back to the object plane via inverse propagation: ${\bar E _{\text{obj}}} = {{\cal P}^{- 1}}({\bar E _{\text{bfp}}})$. The gradient for an individual patch ${\bar E _{\text{obj},k}}$ is then obtained by keeping only the values within the individual patch region by multiplying with the mask ${m_k}$, i.e., ${\bar E _{\text{obj},k}} = {\bar E _{\text{obj}}}{m_k}$.

To find optimal values for the patch fields from the gradients ${\bar E _{\text{obj},k}}$, we use a simplified version of Nesterov’s accelerated gradient method [19]. This method is similar to gradient descent, but keeps a memory of previous steps. At each iteration, we first calculate the forward model and then the back propagated gradient ${\bar E _{\text{obj},k}}$ starting from the modified patch fields ${E_{\text{obj},k}} +\mu v_k^{(n)}$. This we use to apply an update to the step direction ${v_k}$ and the patch fields by

$$v_k^{(n + 1)} =\mu v_k^{(n)} - \alpha {\bar E _{\text{obj},k}}\quad \text{and}\quad E_{\text{obj},k}^{(n + 1)} = E_{\text{obj},k}^{(n)} + v_k^{(n + 1)}.$$
Here $\alpha$ denotes the step size, which we choose manually for optimal convergence speed, and $0 \le \mu \le 1$ controls the rate at which the actual step ${v_k}$ is modified by the gradient. We typically set $\mu= 0.8$ for all iterations. By this, the step gains some momentum in the direction of previous steps, significantly enhancing the rate of convergence.

As a starting point for the iteration, we use the incident field in the object plane close to the scatterer, which we obtain from calculating the reference field (see Section 2.C.1). We stop the iteration after a predetermined number of iterations. With an optimized step size, 20 iterations are typically sufficient.

Data processing speed. The iterative field retrieval from a single BFP intensity image is computationally intensive. Our implementation utilizes GPU acceleration for the core operations, i.e., forward and backward gradient propagation. With our current implementation, we achieve a processing speed of about 50 ms for the field retrieval, valid for 20 iterations and a BFP image size of $512 \times 512$, using a computer equipped with a quad-core processor (Intel Xeon E5-1607v3, 3.1 GHz) and an AMD Radeon VII graphics card.

3. EXPERIMENTAL IMPLEMENTATION

In this section, we give a more in-depth description of the experimental setup and procedures we use to perform the measurement of individual forces.

A. Experimental Setup

Our experimental setup resembles a standard HOTs setup with conventional force detection. It basically consists of a laser, a SLM for beam shaping, some relay optics, and a high-NA objective lens to focus the trapping beams to narrow spots on the sample. The transmitted trapping light is recollected with a high-NA condenser lens, and its far-field intensity distribution is detected with a digital camera. We introduced a few addition optical elements for better control and monitoring of the trapping beams. A complete description of the elements used in the setup as well as a schematic drawing is given in Supplement 1.

We use a 10 W fiber laser at 1064 nm. To clean up the polarization state, we pass the light through a polarizer. Since our laser shows optimal noise conditions when running at elevated power levels (${\gt}{1}\;\text{W}$), which is much more than typically needed for trapping, we use only about 10% by reflection off a glass wedge.

To create a beam that uniformly illuminates only the active area of the SLM, we first use a $2 \times$ beam expander and additionally a telescope lens assembly with $2 \times$ magnification. The latter sharply images an adjustable aperture onto the SLM surface and thereby creates a well-defined illumination of the active area only. This reduces stray light and suppresses unwanted side effects on the electrical response of the SLM when illuminating the whole surface at high power levels.

The SLM introduces a modulation of the phase with a pattern controlled by a computer. We use mostly superpositions of blazed gratings to split up the beam to create several optical traps. Furthermore, adding a spherical lens pattern allows us to axially displace the focus positions. For our irregular trap positions (see Section 2.C.2), we obtain with this approach comparable trapping performance as with more sophisticated iterative algorithms [20]. Aberrations present in the laser beam or introduced by the curved SLM surface are also compensated for by appropriate patterns.

A relay optical system consisting of two achromatic doublets in a $4f$ configuration images the SLM onto the entrance pupil of the microscope objective lens. Close to the intermediate focal plane of the relay optics, we place an adjustable aperture and a removable metal stripe. By this, we are able to block the strong zero diffraction order and some of the unwanted higher diffraction orders. To monitor the ingoing trapping light, we use a beam sampler and an auxiliary digital camera with a lens, which images the intermediate focal plane. This enables us to conveniently check the placement of the beam stops.

With the help of a beam splitter cube, we combine the trapping light with the optical path of an inverted microscope. The beam splitter reflects most of the near-infrared trapping light, while it allows the light (green LED) used for illumination to enter the observation path.

For focusing the trapping light and for observation of the sample, we use a high-NA water immersion objective lens (Olympus UPLSAPO60XW, $60 \times$, NA 1.2). The trapped particles, mostly silica beads of various diameters ranging from 1 µm to 10 µm, are suspended in water. A small probe chamber is formed between a mounting glass slide (1 mm thick) and a glass cover slip (170 µm thick), with spacers made of double-sided adhesive tape (${\sim}90$ µm thick).

To collect most of the light scattered in the forward direction, we use a high-NA oil condenser (NA 1.35). Its rather long focal length of 8 mm reduces the effort needed for alignment, but implies a rather large exit beam size (diameter 14 mm). We directly add a lens at the back side of the condenser, which makes the beam converge and avoids clipping at the rim of the following optical elements, namely, a beam splitter to separate illumination and trapping light, some attenuators, and a camera with a lens to image the BFP of the condenser.

B. Calibration and Measurement Procedures

The method relies on accurate knowledge of amplitude and phase of the reference field at the detector plane. Furthermore, it is crucial to experimentally determine which parts of the diffraction pattern of the SLM are blocked by the beam stop in the intermediate focal plane.

1. Experimental Characterization of Ingoing Intensity Distribution

The intensity of the trapping laser at the SLM surface can in principle be easily measured by recording its image with the BFP camera for a flat SLM pattern. However, in this situation, we observe additional contributions, e.g., from light reflected at the SLM’s front surface, and light scattered by dust particles. Instead, we use an improved procedure, where we average several BFP images taken with blazed gratings SLM patterns. For details, refer to Supplement 1.

2. Measurement of Aperture Placement in Intermediate Focal Plane

The beam stop and the aperture in the intermediate focal plane (Fourier plane of the SLM) block some parts of the diffraction pattern of the SLM. To precisely determine the region where light is allowed to pass, we use the auxiliary camera that images the intermediate focal plane. For this, we display a phase mask with a randomly chosen phase for each pixel, which distributes the light roughly uniformly across the focal plane. To attenuate the contrast of the light speckles due to the coherent illumination, we average over several random patterns. This experimentally measured aperture location and shape are then included in our retrieval algorithm as a mask for the intermediate focal plane.

4. RESULTS AND DISCUSSION

It has been demonstrated previously that direct force measurements give correct results for a variety of situations, e.g.,  for optically trapped microspheres [7,8], including measurements of the axial force [9], or various particles of non-spherical shape such as rods [11]. Our method inherits all these beneficial properties, in particular to provide calibration-free force measurements for particles of arbitrary shape. Here we concentrate on a demonstration of the ability of our method to disentangle individual forces. For the measurements, the traps are placed within an area of ${80}\;{\unicode{x00B5}\text{m}} \times {80}\;{\unicode{x00B5}\text{m}}$, which corresponds to a minimum grating period of four pixels on the SLM. For larger distances of the traps from the optical axis, we observe that the results of the retrieval are less reliable due to a decrease in overall image quality for minute details.

 

Fig. 3. (a) Comparison of individual force profiles obtained from simultaneous measurements with 10 traps (solid lines), with data from single-trap measurements (dashed lines). Shown is the force component directed along the trajectory of each trap. (b) All components of the force profiles for one of the microspheres. (c) Image of the silica microspheres, which are firmly attached to a glass surface. The colored arrows depict the trajectories of the individual optical traps.

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A. Simultaneous Force Measurements for 10 Microspheres

We demonstrate disentangling of individual forces for 10 microspheres (silica, about 3 µm diameter), which are attached to the probe chamber glass surface. This allows us to compare the results for the individual forces, when we simultaneously scan 10 optical trapping spots in various directions across the ensemble, with repeated direct force measurements, where a single trap is used.

Comparison of force profiles. A comparison of the results is shown in Fig. 3(a) for the force component along the scan direction. We observe a very good agreement of the force profiles between multi-trap and single-trap measurements. In Fig. 3(b), we additionally show all components of the retrieved optical force for one of the individual microspheres, which also agree nicely with the single-trap measurement. This shows that we are indeed able to extend the direct force measurement method from a single particle to many more traps. It also demonstrates that the method can be easily used for dynamically changing spot configurations. One trap [marked by an asterisk in Fig. 3(a)] was unintentionally defocused by 5 µm, leading to weak forces. To get in this case correct results in the field retrieval, we had to increase the patch size in the object plane to 10 µm for this trap to encompass nearly all of the light.

For both configurations we determine the ingoing momentum flux ${{\bf F}_0}$ of Eq. (1) by repeating the measurements without particles. Furthermore, for the single-spot measurements, the laser power is concentrated in a single trap, thus leading to higher forces. To enable a comparison with the multi-spot data, we rescaled the forces according to the calculated ratio of spot intensities.

The axial force, which is given by the difference of the in- and outgoing axial momentum flux with similar magnitude, is more susceptible to errors introduced, e.g., by laser fluctuations or limitations of our numerical model, as we discuss in more detail in Supplement 1. We found that we get an improved agreement between multi- and single-trap axial force profiles if we rescale the total power of each individual scattered field such that it matches the respective value of the measurement without particles. A simpler but slightly less accurate approach, which does not require an additional measurement with empty traps, is to use the calculated ingoing power in each trap. Here we rely on the assumption that (nearly) all of the ingoing light is scattered in the forward direction and thus the total detected light resembles the ingoing power, which is well fulfilled for silica beads with relatively low refractive index [9]. However, for strong scatterers, this approach introduces a bias if a significant amount of light is lost due to scattering in the backward direction.

Supplement 1 includes additional results on simultaneous measurements with four traps. We leave two of the traps empty, allowing us to assess more accurately how well our method is able to independently measure the individual forces. We find that in this situation, our method provides reliable results within 3% of the maximum exerted force.

Comparison of far-field patterns. To provide deeper insight into the reconstruction of individual forces, we show in Fig. 4 an image of the far-field intensity pattern. From this and the known pattern on the SLM, we calculate the individual scattered light fields, and compare it with the far-field patterns for single-trap measurements. For a comparison of BFP patterns on equal footing, we insert the retrieved patches in the object plane from the multi-trap measurement into the calculated reference field for the single-trap configuration and propagate it to the far field. By this, we also incorporate features stemming from higher diffraction orders that are present in the single-trap measurements. We observe a good agreement of the retrieved and measured individual far-field intensity patterns.

 

Fig. 4. (a) Observed far-field intensity pattern for simultaneous measurement with 10 microspheres. (b) Single-trap far-field intensity pattern for two of the 10 traps. The comparison between measured (top) and reconstructed (bottom) intensity patterns is shown.

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Notably, due to the limited spatial support in the object plane, the retrieved data show only smooth features. Narrow features present in the measured patterns, e.g., due to localized reflections (“ghosts”) by the optics or dust are filtered out.

B. Measurements with Optically Trapped Particle Ensembles

In this section, we present simultaneous force measurements for ensembles of optically trapped microspheres in water: First, we recover the individual drag forces on four simultaneously trapped microspheres with various diameters when exposing them to an oscillating flow of the surrounding liquid. In this test setting, we find that the forces scale with the bead diameters as expected, which confirms that our method provides correct results. With this kind of measurement, we have also determined the single calibration factor that relates the optical force to the detector signal for all presented measurements. Second, we analyze the thermal forces acting on six simultaneously trapped microspheres. The signal clearly stands out from the noise floor. This demonstrates that we are able to detect tiny forces with our method.

1. Drag Forces

Here we simultaneously apply drag forces to four differently sized polystyrene microspheres by moving the probe with the help of a piezo stage. Since the travel range is limited, we move the stage back and forth in a sinusoidal oscillation. According to Stokes’ law, the drag force is given by ${{\bf F}_{\text{drag}}} = - \gamma {\bf v}$, where ${\bf v}$ is the velocity relative to the fluid, and $\gamma = 6\pi \eta r/(1 - \frac{9}{{16}} \frac{r}{d})$ is the friction coefficient for a sphere of radius $r$ in a fluid of viscosity $\eta \sim 1 \,\text{mPa s}$, including the first-order Faxén correction [1] for hydrodynamic interaction with the cover slip at a distance $d$. For our setting of $d = 30 \,{\unicode{x00B5}\text{m}}$, the correction is only 3% for the largest microsphere with a diameter of 3 µm. The retrieved individual forces show the expected linear scaling with the microsphere diameter (Fig. 5).

 

Fig. 5. Retrieved individual forces (dots) as well as the expected drag force (solid line) based on the sinusoidal movement of the stage and scaling of the drag force with particle size. For each trace, the diameter of the bead is annotated, and pictures of the beads are shown on the right.

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2. Characterization of Thermal Forces

To demonstrate that our method is able to measure small forces with a good signal-to-noise ratio, we studied the thermal forces acting on trapped particles.

We measured time sequences of the individual forces for six simultaneously trapped microspheres with various diameters from 2 µm to 3 µm. Typically, we record data for 10 s at the maximum acquisition rate of ${f_{\text{acq}}} = 400\,\text{Hz}$ of our camera for an image size of $512 \times 512$. In Fig. 6, we show the power spectral density (PSD) of the radial and axial individual forces for the largest and smallest particles (green and blue markers, respectively).

 

Fig. 6. Power spectral density of the individual force time sequences for two of the six simultaneously trapped microspheres (solid dots), for (a) radial and (b) axial directions. Also shown is the expected behavior $P_F^{\text{aliased}}(f)$ from Eq. (4) with aliasing included (solid lines). The noise floor of the measurement, as obtained by a measurement without trapped particles, is given by open dots. (c) Image of the six trapped silica microspheres with overlays denoting their diameters.

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For analysis of the results and a comparison with theoretical expectations, we assume that the optical force points along the displacement direction and the force strength $F = - \kappa (x - {x_0})$ depends linearly on the displacement, where $\kappa$ denotes the trap stiffness. This is well justified since the rather weak thermal forces lead only to small displacements $x - {x_0}$ from the equilibrium position ${x_0}$. Drag forces lead to a damping of the motion with drag coefficient $\gamma$ as described above.

Due to the equipartition theorem [1], the trap stiffness $\kappa = \frac{{{k_{\text{B}}}T}}{{\sigma _F^2}}$ can be obtained from the standard deviation ${\sigma _F}$ of the force data, with $T \sim 300 \,\text{K}$ the temperature of the fluid and ${k_{\text{B}}}$ the Boltzmann constant. Together with the radii of the particles, which we determined from particle images [Fig. 6(c)], this allows us to compare the data with the theoretical expression for the PSD ${P_F}(f)$ for the optical forces acting on a trapped particle [1], given by

$$\begin{array}{*{20}{l}}{{P_F}(f) = {\kappa ^2} {P_r}(f) = {\kappa ^2} \frac{{D/(2{\pi ^2})}}{{f_{\text{c}}^2 + {f^2}}},}\end{array}$$
where $D = \frac{{{k_{\text{B}}}T}}{\gamma}$ is the bulk diffusion coefficient, and ${f_{\text{c}}} = \frac{\kappa}{{2\pi \gamma}}$ denotes the corner frequency. In Fig. 6, the measured PSD as well as the theoretical curve for the $x$ and $z$ forces are shown. We observe good agreement between experiment and modeled behavior $P_F^{\text{aliased}}(f) = \sum\nolimits_n {P_F}(f + n{f_{\text{acq}}}) + {P_0}$, where we also include the effect of aliasing [21] and the noise floor. Here ${P_0}$ describes the approximately frequency-independent noise floor of our setup, which we characterized by repeating the measurement with empty traps. It lies well below the signal of thermally induced forces, showing that even very small forces can be detected with this measurement method. In our setup, the noise floor is dominated by intensity fluctuations within the beam profile of our laser. Noise due to fluctuations of the total power, which are also present in our setup and which affect in particular the axial force component of the ingoing momentum flux significantly, has been suppressed by normalizing the force data by the total power in the individual retrieved far fields. A detailed discussion of this is given in Supplement 1.

C. Force Measurements on Optically Stretched Red Blood Cells

Optical tweezers are widely used for probing cell mechanics, in particular of RBCs; see the recent review in [22] for details. Here we demonstrate that our method enables individual measurements of forces exerted on a RBC, which is held directly and stretched by four optical traps in a novel configuration without attachment of microspheres [Fig. 7(a)]. Imaging the cells with simultaneous force measurements allows us to gather information about cell deformation and mechanical stress with unprecedented detail.

 

Fig. 7. (a) Schematic figure of a red blood cell held by four optical traps. (b), (c) Image of the cell for the initial and maximally stretched states. The colored arrows indicate the individual applied forces and the effective stretching force ${S_\parallel}$. (d) Applied optical forces $|{S_\parallel}|$ and $|{S_ \bot}|$ as defined by equations Eq. (5), and cell extension in stretching direction and perpendicular to it.

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The blood cells were freshly collected from a healthy adult female and diluted in phosphate-buffered saline (PBS) with 0.5% bovine serum albumin (BSA). To avoid the cells adhering to the cover slip, the glass surface was rendered hydrophobic by short exposure to the vapor of dimethyldichlorosilane.

The four traps were arranged in a rectangular pattern. Traps at opposing sides were moved along the $x$ direction between three discrete positions to stretch the cell. Figures 7(b) and 7(c) show images of the RBC for initial and maximally stretched positions, where we displaced the traps by 0.6 µm. We subsume the measured individual forces into two quantities:

$$\begin{split}{{{\bf S}_\parallel}}&{= {\textstyle{1 \over 2}}(({{\bf F}_1} + {{\bf F}_2}) - ({{\bf F}_3} + {{\bf F}_4}))\quad \text{and}}\\{{{\bf S}_ \bot}}&{= {\textstyle{1 \over 2}}(({{\bf F}_1} + {{\bf F}_3}) - ({{\bf F}_2} + {{\bf F}_4}))},\end{split}$$
which represent the stretching force parallel and perpendicular to the stretching direction, respectively. Additionally, we analyze the cell shape, determining the size $L$ along central lines in $x$ and $y$ directions, and from it the extension $\Delta L = L - {L_0}$ from the initial size ${L_0}$ of 8 µm in both directions. The measurement results are presented in Fig. 7(d). Visualization 1 shows all the data, including the cell image, far-field trapping light distribution, and retrieved forces. We see that the extent of the cell in the stretching direction increases, while in the perpendicular direction, it decreases. At the end, the cell returns to its initial shape. The reaction of the cell to sudden displacements of the traps takes place at two different timescales: after a quick response within ${\lt}50$ ms, we also observe a slower adaptation of the shape and forces on the timescale of seconds, which is generally attributed to active processes in the cell.

5. CONCLUSION

Here we have presented a novel measurement approach that allows us to deduce individual forces for several optical traps in a truly simultaneous way. It is scalable to many traps, as demonstrated by the measurement on 10 microspheres. Our results confirm that we are able to retain the advantages and high sensitivity of direct force measurements, e.g., we are able to detect tiny forces, well below the thermal fluctuations. A key feature of the measurement method is that it is calibration free and gives correct results for arbitrarily shaped particles. This makes the method a prime choice for studying biological samples with unknown or strongly varying shapes and properties with more sophisticated optical trapping landscapes.

In this work, we considered configurations where all particles are trapped within a single plane. Nevertheless, our method can be extended to recover forces for particles placed at different axial positions. We expect this to be highly beneficial, as it allows one to address even more complex situations where the manipulation of a sample as well as the forces are truly three dimensional. This will also enable us to cope with localized obstructions in out-of-focus planes, which are difficult to steer clear of, enhancing the accuracy and robustness of force measurements in crowded particle configurations. However, in such settings, the data analysis requires a significantly larger computational effort, as we need to numerically propagate the light field to each plane where a particle is located. We foresee that further optimization of our software implementation will improve the processing speed and allow us to tackle more demanding configurations.

In the future, we plan to use the detailed information in the scattered light field, which we gain by phase retrieval, to uncover not only the individual forces, but also the optical torques exerted on trapped particles. This will allow us to gain an even more complete picture of the mechanical properties of trapped particles. Polarization-sensitive detection will allow us to measure optical torque due to transfer of both spin and orbital angular momenta.

Funding

Austrian Science Fund (P29936-N36).

Disclosures

The authors declare no conflicts of interest.

Supplemental Documents

See Supplement 1 for supporting content.

REFERENCES

1. P. H. Jones, O. M. Maragò, and G. Volpe, Optical Tweezers: Principles and Applications (Cambridge University, 2015).

2. J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207, 169–175 (2002). [CrossRef]  

3. M. Padgett and R. Di Leonardo, “Holographic optical tweezers and their relevance to lab on chip devices,” Lab Chip 11, 1196–1205 (2011). [CrossRef]  

4. S. B. Smith, Y. Cui, and C. Bustamante, “Overstretching b-DNA: the elastic response of individual double-stranded and single-stranded DNA molecules,” Science 271, 795–799 (1996). [CrossRef]  

5. R. T. Dame, M. C. Noom, and G. J. L. Wuite, “Bacterial chromatin organization by H-NS protein unravelled using dual DNA manipulation,” Nature 444, 387–390 (2006). [CrossRef]  

6. F. Gittes and C. F. Schmidt, “Interference model for back-focal-plane displacement detection in optical tweezers,” Opt. Lett. 23, 7–9 (1998). [CrossRef]  

7. A. Farré and M. Montes-Usategui, “A force detection technique for single-beam optical traps based on direct measurement of light momentum changes,” Opt. Express 18, 11955 (2010). [CrossRef]  

8. A. Farré, F. Marsà, and M. Montes-Usategui, “Optimized back-focal-plane interferometry directly measures forces of optically trapped particles,” Opt. Express 20, 12270–12291 (2012). [CrossRef]  

9. G. Thalhammer, L. Obmascher, and M. Ritsch-Marte, “Direct measurement of axial optical forces,” Opt. Express 23, 6112–6129 (2015). [CrossRef]  

10. D. Ott, S. Nader, S. Reihani, and L. B. Oddershede, “Simultaneous three-dimensional tracking of individual signals from multi-trap optical tweezers using fast and accurate photodiode detection,” Opt. Express 22, 23661–23672 (2014). [CrossRef]  

11. F. Català, F. Marsà, M. Montes-Usategui, A. Farré, and E. Martín-Badosa, “Extending calibration-free force measurements to optically-trapped rod-shaped samples,” Sci. Rep. 7, 42960 (2017). [CrossRef]  

12. G. Thalhammer and F. Strasser, “Direct measurement of individual optical forces: data analysis software,” 2020, https://github.com/geggo/holographic-force-measurements.

13. A. A. M. Bui, A. V. Kashchuk, M. A. Balanant, T. A. Nieminen, H. Rubinsztein-Dunlop, and A. B. Stilgoe, “Calibration of force detection for arbitrarily shaped particles in optical tweezers,” Sci. Rep. 8, 10798 (2018). [CrossRef]  

14. S. Moser, M. Ritsch-Marte, and G. Thalhammer, “Model-based compensation of pixel crosstalk in liquid crystal spatial light modulators,” Opt. Express 27, 25046–25063 (2019). [CrossRef]  

15. J. R. Fienup, “Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint,” J. Opt. Soc. Am. A 4, 118–123 (1987). [CrossRef]  

16. S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics*,” J. Opt. Soc. Am. A 60, 1168–1177 (1970). [CrossRef]  

17. A. S. Jurling and J. R. Fienup, “Phase retrieval with unknown sampling factors via the two-dimensional chirp z-transform,” J. Opt. Soc. Am. A 31, 1904–1911 (2014). [CrossRef]  

18. A. S. Jurling and J. R. Fienup, “Applications of algorithmic differentiation to phase retrieval algorithms,” J. Opt. Soc. Am. A 31, 1348–1359 (2014). [CrossRef]  

19. Y. Nesterov, Introductory Lectures on Convex Optimization, Vol. 87, of Applied Optimization (Springer, 2004).

20. J. E. Curtis, C. H. Schmitz, and J. P. Spatz, “Symmetry dependence of holograms for optical trapping,” Opt. Lett. 30, 2086–2088 (2005). [CrossRef]  

21. K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75, 594–612 (2004). [CrossRef]  

22. T. Avsievich, R. Zhu, A. Popov, A. Bykov, and I. Meglinski, “The advancement of blood cell research by optical tweezers,” Rev. Phys. 5, 100043 (2020). [CrossRef]  

References

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  1. P. H. Jones, O. M. Maragò, and G. Volpe, Optical Tweezers: Principles and Applications (Cambridge University, 2015).
  2. J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207, 169–175 (2002).
    [Crossref]
  3. M. Padgett and R. Di Leonardo, “Holographic optical tweezers and their relevance to lab on chip devices,” Lab Chip 11, 1196–1205 (2011).
    [Crossref]
  4. S. B. Smith, Y. Cui, and C. Bustamante, “Overstretching b-DNA: the elastic response of individual double-stranded and single-stranded DNA molecules,” Science 271, 795–799 (1996).
    [Crossref]
  5. R. T. Dame, M. C. Noom, and G. J. L. Wuite, “Bacterial chromatin organization by H-NS protein unravelled using dual DNA manipulation,” Nature 444, 387–390 (2006).
    [Crossref]
  6. F. Gittes and C. F. Schmidt, “Interference model for back-focal-plane displacement detection in optical tweezers,” Opt. Lett. 23, 7–9 (1998).
    [Crossref]
  7. A. Farré and M. Montes-Usategui, “A force detection technique for single-beam optical traps based on direct measurement of light momentum changes,” Opt. Express 18, 11955 (2010).
    [Crossref]
  8. A. Farré, F. Marsà, and M. Montes-Usategui, “Optimized back-focal-plane interferometry directly measures forces of optically trapped particles,” Opt. Express 20, 12270–12291 (2012).
    [Crossref]
  9. G. Thalhammer, L. Obmascher, and M. Ritsch-Marte, “Direct measurement of axial optical forces,” Opt. Express 23, 6112–6129 (2015).
    [Crossref]
  10. D. Ott, S. Nader, S. Reihani, and L. B. Oddershede, “Simultaneous three-dimensional tracking of individual signals from multi-trap optical tweezers using fast and accurate photodiode detection,” Opt. Express 22, 23661–23672 (2014).
    [Crossref]
  11. F. Català, F. Marsà, M. Montes-Usategui, A. Farré, and E. Martín-Badosa, “Extending calibration-free force measurements to optically-trapped rod-shaped samples,” Sci. Rep. 7, 42960 (2017).
    [Crossref]
  12. G. Thalhammer and F. Strasser, “Direct measurement of individual optical forces: data analysis software,” 2020, https://github.com/geggo/holographic-force-measurements .
  13. A. A. M. Bui, A. V. Kashchuk, M. A. Balanant, T. A. Nieminen, H. Rubinsztein-Dunlop, and A. B. Stilgoe, “Calibration of force detection for arbitrarily shaped particles in optical tweezers,” Sci. Rep. 8, 10798 (2018).
    [Crossref]
  14. S. Moser, M. Ritsch-Marte, and G. Thalhammer, “Model-based compensation of pixel crosstalk in liquid crystal spatial light modulators,” Opt. Express 27, 25046–25063 (2019).
    [Crossref]
  15. J. R. Fienup, “Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint,” J. Opt. Soc. Am. A 4, 118–123 (1987).
    [Crossref]
  16. S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics*,” J. Opt. Soc. Am. A 60, 1168–1177 (1970).
    [Crossref]
  17. A. S. Jurling and J. R. Fienup, “Phase retrieval with unknown sampling factors via the two-dimensional chirp z-transform,” J. Opt. Soc. Am. A 31, 1904–1911 (2014).
    [Crossref]
  18. A. S. Jurling and J. R. Fienup, “Applications of algorithmic differentiation to phase retrieval algorithms,” J. Opt. Soc. Am. A 31, 1348–1359 (2014).
    [Crossref]
  19. Y. Nesterov, Introductory Lectures on Convex Optimization, Vol. 87, of Applied Optimization (Springer, 2004).
  20. J. E. Curtis, C. H. Schmitz, and J. P. Spatz, “Symmetry dependence of holograms for optical trapping,” Opt. Lett. 30, 2086–2088 (2005).
    [Crossref]
  21. K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75, 594–612 (2004).
    [Crossref]
  22. T. Avsievich, R. Zhu, A. Popov, A. Bykov, and I. Meglinski, “The advancement of blood cell research by optical tweezers,” Rev. Phys. 5, 100043 (2020).
    [Crossref]

2020 (1)

T. Avsievich, R. Zhu, A. Popov, A. Bykov, and I. Meglinski, “The advancement of blood cell research by optical tweezers,” Rev. Phys. 5, 100043 (2020).
[Crossref]

2019 (1)

2018 (1)

A. A. M. Bui, A. V. Kashchuk, M. A. Balanant, T. A. Nieminen, H. Rubinsztein-Dunlop, and A. B. Stilgoe, “Calibration of force detection for arbitrarily shaped particles in optical tweezers,” Sci. Rep. 8, 10798 (2018).
[Crossref]

2017 (1)

F. Català, F. Marsà, M. Montes-Usategui, A. Farré, and E. Martín-Badosa, “Extending calibration-free force measurements to optically-trapped rod-shaped samples,” Sci. Rep. 7, 42960 (2017).
[Crossref]

2015 (1)

2014 (3)

2012 (1)

2011 (1)

M. Padgett and R. Di Leonardo, “Holographic optical tweezers and their relevance to lab on chip devices,” Lab Chip 11, 1196–1205 (2011).
[Crossref]

2010 (1)

2006 (1)

R. T. Dame, M. C. Noom, and G. J. L. Wuite, “Bacterial chromatin organization by H-NS protein unravelled using dual DNA manipulation,” Nature 444, 387–390 (2006).
[Crossref]

2005 (1)

2004 (1)

K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75, 594–612 (2004).
[Crossref]

2002 (1)

J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207, 169–175 (2002).
[Crossref]

1998 (1)

1996 (1)

S. B. Smith, Y. Cui, and C. Bustamante, “Overstretching b-DNA: the elastic response of individual double-stranded and single-stranded DNA molecules,” Science 271, 795–799 (1996).
[Crossref]

1987 (1)

1970 (1)

S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics*,” J. Opt. Soc. Am. A 60, 1168–1177 (1970).
[Crossref]

Avsievich, T.

T. Avsievich, R. Zhu, A. Popov, A. Bykov, and I. Meglinski, “The advancement of blood cell research by optical tweezers,” Rev. Phys. 5, 100043 (2020).
[Crossref]

Balanant, M. A.

A. A. M. Bui, A. V. Kashchuk, M. A. Balanant, T. A. Nieminen, H. Rubinsztein-Dunlop, and A. B. Stilgoe, “Calibration of force detection for arbitrarily shaped particles in optical tweezers,” Sci. Rep. 8, 10798 (2018).
[Crossref]

Berg-Sørensen, K.

K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75, 594–612 (2004).
[Crossref]

Bui, A. A. M.

A. A. M. Bui, A. V. Kashchuk, M. A. Balanant, T. A. Nieminen, H. Rubinsztein-Dunlop, and A. B. Stilgoe, “Calibration of force detection for arbitrarily shaped particles in optical tweezers,” Sci. Rep. 8, 10798 (2018).
[Crossref]

Bustamante, C.

S. B. Smith, Y. Cui, and C. Bustamante, “Overstretching b-DNA: the elastic response of individual double-stranded and single-stranded DNA molecules,” Science 271, 795–799 (1996).
[Crossref]

Bykov, A.

T. Avsievich, R. Zhu, A. Popov, A. Bykov, and I. Meglinski, “The advancement of blood cell research by optical tweezers,” Rev. Phys. 5, 100043 (2020).
[Crossref]

Català, F.

F. Català, F. Marsà, M. Montes-Usategui, A. Farré, and E. Martín-Badosa, “Extending calibration-free force measurements to optically-trapped rod-shaped samples,” Sci. Rep. 7, 42960 (2017).
[Crossref]

Collins, S. A.

S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics*,” J. Opt. Soc. Am. A 60, 1168–1177 (1970).
[Crossref]

Cui, Y.

S. B. Smith, Y. Cui, and C. Bustamante, “Overstretching b-DNA: the elastic response of individual double-stranded and single-stranded DNA molecules,” Science 271, 795–799 (1996).
[Crossref]

Curtis, J. E.

J. E. Curtis, C. H. Schmitz, and J. P. Spatz, “Symmetry dependence of holograms for optical trapping,” Opt. Lett. 30, 2086–2088 (2005).
[Crossref]

J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207, 169–175 (2002).
[Crossref]

Dame, R. T.

R. T. Dame, M. C. Noom, and G. J. L. Wuite, “Bacterial chromatin organization by H-NS protein unravelled using dual DNA manipulation,” Nature 444, 387–390 (2006).
[Crossref]

Di Leonardo, R.

M. Padgett and R. Di Leonardo, “Holographic optical tweezers and their relevance to lab on chip devices,” Lab Chip 11, 1196–1205 (2011).
[Crossref]

Farré, A.

Fienup, J. R.

Flyvbjerg, H.

K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75, 594–612 (2004).
[Crossref]

Gittes, F.

Grier, D. G.

J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207, 169–175 (2002).
[Crossref]

Jones, P. H.

P. H. Jones, O. M. Maragò, and G. Volpe, Optical Tweezers: Principles and Applications (Cambridge University, 2015).

Jurling, A. S.

Kashchuk, A. V.

A. A. M. Bui, A. V. Kashchuk, M. A. Balanant, T. A. Nieminen, H. Rubinsztein-Dunlop, and A. B. Stilgoe, “Calibration of force detection for arbitrarily shaped particles in optical tweezers,” Sci. Rep. 8, 10798 (2018).
[Crossref]

Koss, B. A.

J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207, 169–175 (2002).
[Crossref]

Maragò, O. M.

P. H. Jones, O. M. Maragò, and G. Volpe, Optical Tweezers: Principles and Applications (Cambridge University, 2015).

Marsà, F.

F. Català, F. Marsà, M. Montes-Usategui, A. Farré, and E. Martín-Badosa, “Extending calibration-free force measurements to optically-trapped rod-shaped samples,” Sci. Rep. 7, 42960 (2017).
[Crossref]

A. Farré, F. Marsà, and M. Montes-Usategui, “Optimized back-focal-plane interferometry directly measures forces of optically trapped particles,” Opt. Express 20, 12270–12291 (2012).
[Crossref]

Martín-Badosa, E.

F. Català, F. Marsà, M. Montes-Usategui, A. Farré, and E. Martín-Badosa, “Extending calibration-free force measurements to optically-trapped rod-shaped samples,” Sci. Rep. 7, 42960 (2017).
[Crossref]

Meglinski, I.

T. Avsievich, R. Zhu, A. Popov, A. Bykov, and I. Meglinski, “The advancement of blood cell research by optical tweezers,” Rev. Phys. 5, 100043 (2020).
[Crossref]

Montes-Usategui, M.

Moser, S.

Nader, S.

Nesterov, Y.

Y. Nesterov, Introductory Lectures on Convex Optimization, Vol. 87, of Applied Optimization (Springer, 2004).

Nieminen, T. A.

A. A. M. Bui, A. V. Kashchuk, M. A. Balanant, T. A. Nieminen, H. Rubinsztein-Dunlop, and A. B. Stilgoe, “Calibration of force detection for arbitrarily shaped particles in optical tweezers,” Sci. Rep. 8, 10798 (2018).
[Crossref]

Noom, M. C.

R. T. Dame, M. C. Noom, and G. J. L. Wuite, “Bacterial chromatin organization by H-NS protein unravelled using dual DNA manipulation,” Nature 444, 387–390 (2006).
[Crossref]

Obmascher, L.

Oddershede, L. B.

Ott, D.

Padgett, M.

M. Padgett and R. Di Leonardo, “Holographic optical tweezers and their relevance to lab on chip devices,” Lab Chip 11, 1196–1205 (2011).
[Crossref]

Popov, A.

T. Avsievich, R. Zhu, A. Popov, A. Bykov, and I. Meglinski, “The advancement of blood cell research by optical tweezers,” Rev. Phys. 5, 100043 (2020).
[Crossref]

Reihani, S.

Ritsch-Marte, M.

Rubinsztein-Dunlop, H.

A. A. M. Bui, A. V. Kashchuk, M. A. Balanant, T. A. Nieminen, H. Rubinsztein-Dunlop, and A. B. Stilgoe, “Calibration of force detection for arbitrarily shaped particles in optical tweezers,” Sci. Rep. 8, 10798 (2018).
[Crossref]

Schmidt, C. F.

Schmitz, C. H.

Smith, S. B.

S. B. Smith, Y. Cui, and C. Bustamante, “Overstretching b-DNA: the elastic response of individual double-stranded and single-stranded DNA molecules,” Science 271, 795–799 (1996).
[Crossref]

Spatz, J. P.

Stilgoe, A. B.

A. A. M. Bui, A. V. Kashchuk, M. A. Balanant, T. A. Nieminen, H. Rubinsztein-Dunlop, and A. B. Stilgoe, “Calibration of force detection for arbitrarily shaped particles in optical tweezers,” Sci. Rep. 8, 10798 (2018).
[Crossref]

Thalhammer, G.

Volpe, G.

P. H. Jones, O. M. Maragò, and G. Volpe, Optical Tweezers: Principles and Applications (Cambridge University, 2015).

Wuite, G. J. L.

R. T. Dame, M. C. Noom, and G. J. L. Wuite, “Bacterial chromatin organization by H-NS protein unravelled using dual DNA manipulation,” Nature 444, 387–390 (2006).
[Crossref]

Zhu, R.

T. Avsievich, R. Zhu, A. Popov, A. Bykov, and I. Meglinski, “The advancement of blood cell research by optical tweezers,” Rev. Phys. 5, 100043 (2020).
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Supplementary Material (2)

NameDescription
» Supplement 1       Supplement 1
» Visualization 1       (a) Far-field intensity distribution of the trapping light (b) Red blood cell, stretched by four optical traps, the arrows indicate the optical forces. Video shows real-time data, acquired at 50 Hz.

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Figures (7)

Fig. 1.
Fig. 1. Idea of individual direct force measurements for optically trapped particles: the superposition of all the single scattered light fields is collected by a high-NA condenser lens. From a single interference pattern occurring in the back focal plane, we numerically reconstruct the full complex light field in the focal plane, where the individual light fields are well separated, allowing us to determine the forces individually.
Fig. 2.
Fig. 2. Simplified optical setup for optical trapping and force detection.
Fig. 3.
Fig. 3. (a) Comparison of individual force profiles obtained from simultaneous measurements with 10 traps (solid lines), with data from single-trap measurements (dashed lines). Shown is the force component directed along the trajectory of each trap. (b) All components of the force profiles for one of the microspheres. (c) Image of the silica microspheres, which are firmly attached to a glass surface. The colored arrows depict the trajectories of the individual optical traps.
Fig. 4.
Fig. 4. (a) Observed far-field intensity pattern for simultaneous measurement with 10 microspheres. (b) Single-trap far-field intensity pattern for two of the 10 traps. The comparison between measured (top) and reconstructed (bottom) intensity patterns is shown.
Fig. 5.
Fig. 5. Retrieved individual forces (dots) as well as the expected drag force (solid line) based on the sinusoidal movement of the stage and scaling of the drag force with particle size. For each trace, the diameter of the bead is annotated, and pictures of the beads are shown on the right.
Fig. 6.
Fig. 6. Power spectral density of the individual force time sequences for two of the six simultaneously trapped microspheres (solid dots), for (a) radial and (b) axial directions. Also shown is the expected behavior $P_F^{\text{aliased}}(f)$ from Eq. (4) with aliasing included (solid lines). The noise floor of the measurement, as obtained by a measurement without trapped particles, is given by open dots. (c) Image of the six trapped silica microspheres with overlays denoting their diameters.
Fig. 7.
Fig. 7. (a) Schematic figure of a red blood cell held by four optical traps. (b), (c) Image of the cell for the initial and maximally stretched states. The colored arrows indicate the individual applied forces and the effective stretching force ${S_\parallel}$. (d) Applied optical forces $|{S_\parallel}|$ and $|{S_ \bot}|$ as defined by equations Eq. (5), and cell extension in stretching direction and perpendicular to it.

Equations (5)

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F = 1 c I ( x , y ) ( x , y , 1 ( x 2 + y 2 ) ) T d A F 0 ,
E bfp = E bfp , ref + P ( k E obj , k ) ,
v k ( n + 1 ) = μ v k ( n ) α E ¯ obj , k and E obj , k ( n + 1 ) = E obj , k ( n ) + v k ( n + 1 ) .
P F ( f ) = κ 2 P r ( f ) = κ 2 D / ( 2 π 2 ) f c 2 + f 2 ,
S = 1 2 ( ( F 1 + F 2 ) ( F 3 + F 4 ) ) and S = 1 2 ( ( F 1 + F 3 ) ( F 2 + F 4 ) ) ,