Despite the tremendous success of force-measuring optical traps in recent years, the calibration methods most commonly used in the field have been plagued with difficulties and limitations. Force sensing based on direct measurement of light momentum changes stands out among these as an exception. Especially significant is this method’s potential for working within living cells, with non-spherical particles or with non-Gaussian beams. However, so far, the technique has only been implemented in counter-propagating dual-beam traps, which are difficult to align and integrate with other microscopy techniques. Here, we show the feasibility of a single-beam gradient-trap system working with a force detection technique based on this same principle.
© 2010 OSA
Since its discovery in the early 1970s , the trapping of micron-sized particles by radiation pressure has attracted the interest of a growing number of research laboratories, especially those involved in molecular and cellular biology. The use of the technique has now become routine .
As first noticed by Ashkin , the existence of momentum transfer between photons and material particles provides light beams with the ability to exert contactless forces in the piconewton scale under certain conditions. Using this principle, optical traps operate by means of a tightly focused laser beam, which is capable of stably trapping microscopic structures near its focal region due to the balance of forces in the trap . Furthermore, modern photonics technology enables dynamic and precise 3D positioning of the trapped particle, as well as remarkable flexibility in generating structured traps  or complex light patterns [6,7], for a wide variety of manipulation experiments.
The technique finds its main application, however, in the measurement of the piconewton forces that dominate many processes at the microscopic scale: that is, as a quantitative tool. The confirmation of kinesin stepping , analysis of dsDNA elasticity , or the study of the DNA packaging in bacteriophage ϕ29  are some dramatic examples where the optical tweezers’ capability of measuring forces was of primary importance.
However, the way these forces are usually detected has not changed considerably since the method was proposed independently by Svoboda et al.  and Ghislain et al.  in 1993. The technique, initially motivated by the high spatial and temporal resolution achieved by laser differential interferometry , uses the position of the sample in the trap, x, to indirectly measure the optical force, F. When the trapped particle remains in the vicinity of the equilibrium position, the restoring force that maintains the object in a stable location exhibits a linear response to displacements in x, so that F can be readily obtained from an optical equivalent to Hooke’s law: F = -kx. The performance of the system is reflected in the trap stiffness k, by means of a complex dependence on several experimental parameters, such as particle size, the objective numerical aperture, or the laser power [13,14].
Throughout the 1990s and in to the 21st century, improvement of the technique has focused on looking for alternatives means of accurate position detection, since this lies at the heart of precise force measurements. Differential interference contrast [8,12], CCD imaging  and back-focal-plane (BFP) interferometry [7,16,17] are some of the methods that have been satisfactorily used thus far. In particular, the last of these has achieved wide acceptance due to its simple implementation, and both its large temporal bandwidth (kHz) and high spatial resolution (< 1 nm). Similarly, a precise determination of stiffness, k, has played a central role in the development of the technique. Different external forces, especially those of thermal or hydrodynamic origin, have been used to calibrate optical traps [18,19]. As an appreciable discrepancy between different methods has been repeatedly reported [20,21], the power spectrum analysis of the Brownian motion of a trapped bead has often been considered the reference calibration procedure [22,23].
Making use of the harmonic approximation, high-precision force measurements have been routinely carried out. Despite its simplicity, this technique provides a powerful method of obtaining accurate information, since the instrument is fine-tuned for a certain experiment, through the specific calibration of both the trap stiffness and the position sensor sensitivity. In this manner, the measurement of minute forces in the femtonewton range has been reported .
Ironically, this is, in turn, one of the major drawbacks of the system; the value of k is only valid for a given configuration and it demands recalibration when any of the parameters changes. This makes the detection of forces impossible when properties such as the temperature or the medium viscosity fluctuate in time and/or space. The limitation arises because the harmonic approximation does not provide a direct measurement of the force. Instead it is estimated according to a delicate relation with position, which is extremely sensitive to changes in the conditions.
Furthermore, the mere existence of a stiffness constant which characterizes the trap is intimately tied to meeting some pre-requisites. In particular, it requires the use of non-aberrated Gaussian beams, spherical particles, and optically- and mechanically-homogeneous viscous media. Thus, use of a stiffness constant is inevitably restricted to certain rules, and, hence, to certain experimental conditions. In parallel, a similar situation exists for position detection through BFP interferometry, where a meaningful conversion between the electric signal, S, from the detector and the displacement of the sample, x, exhibits identical restrictions.
Experiments are designed, when possible, so that these requirements are met. In general, polystyrene or silica microbeads are used as ‘handles’ to study the force or the response generated by the sample in a controlled buffer. Nonetheless, a broad spectrum of experiments, ranging from the manipulation of irregular samples to the intracellular trapping of vesicles or organelles, does not meet such requirements. In these cases, more complicated or even unpredictable time-dependent relations between F and x, and between S and x, can appear.
Some calibration methods have addressed these questions and enabled the use of non-harmonic potentials  and buffers with arbitrary viscosities . Similarly, some progress has been made to allow BFP interferometry to work with non-spherical particles or in the presence of additional structures that interfere with the beam [27,28]. However, the problem has not been completely solved and optical tweezers still encounter many difficulties when working under certain conditions. Particularly critical are cellular experiments, which have mainly been restricted to in vitro conditions, or to prior calibration in a viscous buffer followed by a later correction of k during the experiment [29,30], which might introduce a large error. Recent work has shed light into this problem by enabling the calibration of traps in viscoelastic media [31,32], although the model must still be proved accurate inside cells .
The main problem with solutions that try to circumvent the drawbacks of the harmonic approximation is that, as a rule, they are clever but limited modifications of existing calibration methods, which largely rely on the same or similar assumptions. Consequently, they are doomed, to a greater or lesser extent, to suffer the same deficiencies.
In response to this situation, Smith et al.  developed a new technique following a completely different approach. The method was based on the detection of the momentum change in the trapping beam, which is directly related to the optical force created by the trap. This is a powerful way of obtaining force information, since it is based on first principles (Newton’s second and third laws) and thus the measurement is not dependent on any experimental condition. It may allow the use of both non-spherical particles and trapping beams with arbitrary intensity profiles, the measurement of forces in homogeneous buffers with unknown viscosity and/or refractive index, and it does not require continued recalibration . Experiments in living cells may also benefit from the features of this technique since no in situ calibration is needed. However, the accuracy of the force detection will still be subject to the non-homogeneities of the surrounding medium since these might change the angular distribution or intensity of the scattered light, and therefore the momentum of the beam. This is something that will need to be addressed in the future if the method is to be used in such more complex environments.
Unfortunately, this promising alternative has not yet been used with single-beam optical traps since the technique is sensitive to losses of light in the detection process. All the light scattered at the sample must be collected in order to derive the value of the force, which, as has been pointed out in several occasions [18,34,35], is very difficult or impossible with this configuration. This led to the design of an experimental setup where two counter-propagating laser beams were used to trap the sample by the effect of the scattering force. The system cannot be easily implemented in a microscope, as it requires duplicated optical trains, so both the optical alignment and the use of more advanced imaging techniques, such as fluorescence or TIRF, can be considerably cumbersome. As a result, it has not been as widely accepted as other methods.
Here, we show the conditions necessary for using such a force detection technique based on the measurement of the light momentum change in a single-beam configuration, bringing together the advantages of the two techniques .
2. Experimental setup
We use a modified inverted microscope as the basis for our setup. The detailed layout is depicted in Fig. 1 . The force sensor apparatus required herein is akin to that used in BFP interferometry , albeit not identical because of some specific features of our method. Despite the apparent theoretical differences between the two approaches, this shared setup suggests an intimate connection between them, which we discuss below.
In BFP interferometry, the calibration of the system provides, when possible, a conversion of the diode electric signal into sample displacements. The relation between the magnitudes then enables the determination of the position of particles with high accuracy. However, the detection process is more closely related to the measurement of forces than to that of positions. Indeed, the momentum structure of the trapping beam and hence the light momentum change responsible for the optical force in the trap becomes visible at the BFP of the condenser. Thus, under the right conditions the position sensing detector (PSD) signal gives a reading of the applied force.
This same idea was used by Smith et al.  to measure forces through the beam momentum change in their counter-propagating dual-beam apparatus. Unfortunately, the constraints on the collection of light reported by those authors and others, prevented the development of the single-beam version of this versatile force detection method, leaving the setup used in BFP interferometry to the detection of sample displacements, x. Some years later, Gittes et al.  provided the first-order interference theory for the pattern formation at the BFP of the condenser for position detection. They pointed out that their model for small particles was also able to describe the lateral trapping force in this regime. In a similar vein, based on experimental data, Barlett et al.  suggested that BFP interferometry may be better understood as a force-sensing rather than as a position–sensing technique.
We will show that only some additional requirements in the setup are necessary to allow direct measurement of forces with the instrument used ordinarily. The determination of position, which is actually feasible because the information at the BFP provides a measurement of the force, which in turn is linear with x, is less strict because the linear conversion from volts to nanometers is still valid despite the loss of light. In this case, the conditions of the setup are strongly relaxed, and it can easily be implemented with low-NA lenses . By contrast, the apparatus that we show here necessarily includes a high-NA condenser lens (Nikon, oil immersion, NA = 1.4). In particular, the numerical aperture must be chosen higher than the refractive index of the medium used to suspend the particle, nm< NA, and the microchamber containing the sample should not be thick, ~100 µm in our case. Also, a PSD (Pacific Silicon Sensor, DL100-7PCBA3), instead of a quadrant photodiode, is necessary for correctly reading the force, while the relay lens that projects the light pattern from the BFP of the condenser onto that sensor, needs to be carefully selected to avoid vignetting the beam. The use of a PSD provides, on the other hand, the advantage of little parasitic filtering [18,39], which might be useful when a high temporal bandwidth is required. The experimental value for the rolloff frequency that has been previously reported for our PSD  is similar to the nominal value provided by the manufacturer (f3dB = 257 kHz), and larger than the typical frequency limit used in the calibration through analysis of the power spectrum of the Brownian motion of a trapped bead. In addition, PSDs seem also slightly less noisy than QPDs as recently pointed out in .
With these constraints, the detector can provide a signal proportional to the force exerted on the sample, and the constant relating the two becomes independent of the experimental conditions.
The force in an optical trap arises from a rather complex interaction between the laser beam and the sample. Diffraction, internal reflections, scattering and absorption are some of the optical phenomena taking place in this process. As an example, movie 1 shows an FDTD simulation of a 1.3-NA Gaussian beam interacting with a 1-µm polystyrene microsphere. A frame of this movie together with the numerical results of the light intensity scattered by the bead in the same conditions appear in Fig. 2 .
The detection of this force relies on analysis of the changes in the angular distribution of the light. Any net variation in the propagation direction of the photons in the beam gives rise to a force, F, on the particle. Thus, correct measurement of F demands, in principle, the capture of all light interacting with the sample, in order to collect all the information about the changes in angular distribution.
However, detecting the light over the whole solid angle generally proves to be unfeasible. Fortunately, as can be observed in the example, the light travelling in the backward direction usually corresponds to a small fraction of all the emitted light. As a result, the information about the momentum change is mainly concentrated on the forward-scattered light, and, therefore, only one detection lens is needed to collect the light and retrieve the force with reasonable accuracy.
The forward-scattered electric field, u, at the sample plane can be decomposed into a set of plane waves weighted with the corresponding 2D-Fourier transform of u :
This is relevant because the photons in a plane wave all have the same elementary momentum, p = ħk, so Eq. (1) can also be regarded as the momentum spectrum of the field u. That is, weights U(kx, ky) are related to the number of photons in u having transverse momentum components (ħkx, ħky).
After being collected by the condenser lens, the electric field at the BFP is projected onto the PSD by means of a relay lens (Fig. 1). In the paraxial approximation, disregarding some unimportant phase terms, this light corresponds to the Fourier transform of the field at the sample. That is, to factors, U, of the plane wave decomposition , so that at the PSD the momentum structure of the beam becomes visible, as mentioned. Any force exerted will cause a change in this structure that may thus be easily detected.
Some care must be taken if, by contrast, a high-NA lens is used to collect the light. The same result can be extended to large angles only if the lens fulfills the Abbe sine condition , otherwise, the pattern at the BFP will be warped, more difficult to interpret as a momentum decomposition of the beam, and will cause the PSD to give a wrong result.
If the optical system is also well corrected , according to the sine condition the plane wave with momentum pr = p0n sinθ before the lens focuses at a position:
The intensity pattern I(x,y), given by U, is then projected onto the PSD which, according to , produces an electric signal given by:
Since I(x,y)dxdy is the radiant power at point (x,y) and thus proportional to the number of photons per unit time having transverse momentum (px, py), the integral in Eq. (3) represents the orderly addition of the x-component of all the momenta (similarly for y). Change in signals Sx and Sy before and after the light passes through the sample are thus proportional to the light force. Signals without a trapped sample, Sx empty, are usually zero since the trapping laser profile is often center-symmetric, which further simplifies the measurement.
Furthermore, the conversion factor, α, between V and pN does not depend on the experimental parameters:
In this equation, E and px are the energy and the x-component of the momentum per photon, and c is the speed of light in vacuum.
Although the theory is straightforward, the experimental implementation is difficult, since many problems may arise with the collection of the light. We believe that a previous quantitative analysis of the light patterns formed at the BFP of the condenser is enlightening and provides insights into the main issues concerning light capture and utilization. The reason for this is that they provide an explicit and intuitive picture of the interaction between the trap and sample. All the information about the light momentum change is comprehended in a single clear image, where it is possible to visualize the formation of the measurement. In particular, the loss of light can be readily observed and quantified.
Our first step is to calibrate the BFP, that is, to measure the relation between positions on that plane and the momentum of the photons at the sample. This is important to better understand the light patterns but also to check the correct operation of the technique: a linear map between positions and momenta establishes a globally valid p-coordinate system on the BFP. This is necessary if the momentum structure, that is, the Fourier transform, of the beam must appear here undistorted. Otherwise, the PSD will not provide a correct measurement of the force, according to Eq. (4). This is equivalent to showing that the lens fulfills the Abbe sine condition [Eq. (2)].
The placement of a Ronchi ruling at the sample plane provides a particularly simple and intuitive way to determine this relation. In this experiment, the objective is removed, so the source of light interacting with the grating is a collimated beam [Fig. 3(a) ]. After diffraction by the ruling, the electric field, u, given by Eq. (1) turns into a discrete sum of plane waves (the Fraunhofer diffraction orders). In particular, only those waves fulfilling the relation:
Once collected by the lens, the waves are focused at different locations on the BFP. In general, these positions, rm, will be related to the angle of propagation before the condenser, θm, according to a certain expression r = f’G(θ), where f’ is the focal length. Different shapes of the condenser principal surface, S, lead to different expressions and hence, to different focusing positions  [Fig. 3(b)]. Specifically, when the lens fulfills the Abbe sine condition, its principal surface is spherical, and every point on the plane is assigned to a single transverse component of the momentum through a linear relation. In this case, the light pattern provided by the diffraction grating would correspond to an array of equi-distributed spots at positions:
The analysis of the CCD images showed that a similar distribution of light is obtained with our setup [Fig. 3(c)]. The positions of the peaks were measured and plotted against G(θ), where the expressions for the conditions were extracted from , and the angles were computed according to Eq. (5). In this plot, the real G(θ) that best fits the condenser behavior should appear as a straight line with slope f’. The data obtained with the Abbe sine condition provided the best results, whereas other options, such as Herschel or Lagrange, clearly deviated from linear behavior at large angles [Fig. 3(d)]. The fitting was performed at low values of θ, where all G(θ) collapse to the Lagrange condition, and the free parameter (the focal length f’ of the system formed by the condenser plus the relay lens) gave us the calibration of the plane. We can also check that the maximum angle accepted by the lens did indeed correspond to the NA of the condenser, 1.4, so that there is no vignetting elsewhere in the system.
From the results, we can safely assume that the condenser provides the Fourier transform of the field over its whole BFP and, in doing so, discloses the beam momenta.
Also, the known relation between the transverse components of the momentum (and thus angles of propagation prior to focusing) and the positions in the plane allows us to easily estimate the amount of light that the condenser lens is collecting, which is the most critical part of the technique.
If no beads are trapped and the NA of the objective is smaller than that of the condenser the light cone coming out from the former is completely captured by the latter and, therefore, all the light from the trap reaches the detector. A circular pattern appears with a diameter determined by the angle of propagation of its marginal ray, as shown in Figs. 4 and 5(a) .
Also, in a different interpretation, that circle corresponds to the image of the entrance pupil of the objective. In general, when the system is well adjusted throughout its whole length, a set of planes is conjugate to that of the PSD (or the CCD), as illustrated in Fig. 4. The image shows the simultaneous conjugation of the spatial light modulator (our system is holographic), the circular entrance pupil of the microscope objective, its phase plate (it is a phase contrast lens), the annulus of the (phase-contrast) condenser, and, finally, the condenser aperture stop. This property is useful for setting and tuning the relay lens.
We studied the set of patterns created with different numerical aperture objectives, focusing the trap inside a sample chamber filled with water buffer. Using the calibration of the plane, the effective NA associated with the radius of these circular patterns was computed, and it matched the nominal values given by the manufacturer in all cases. Figure 5(a) shows the example of a 1.2 water immersion objective.
The main difficulty regarding the collection of light arises when the trap is drawn near to a sample, because the particle deflects the beam. In principle, the scattered photons are distributed over a 4π surface, and they may not all enter the condenser. However, the lens is capable of collecting nearly all the forward-scattered light when certain precautions are taken. Specifically, the sample must be close to the upper cover-slip (exit surface) of the suspension chamber, for the light to quickly reach the buffer-glass interface, and the buffer must have a refractive index lower than the numerical aperture of the condenser. Under these constraints, shortly after being scattered by the sample, the beam refracts at the interface between the sample medium and the coverslip, so the rays propagating almost parallel to this surface (α~90°) are collected by the condenser [Fig. 5(b)] . Mathematically, the following condition must be met:
It is worth pointing out that the change of propagation direction at the interface does not affect the lateral momentum of the beam, since the transverse components of the wave vector are preserved by Snell’s law (kr’ = kr). So the detector still provides a correct measurement.
The images of the light patterns confirm that we are capturing all the light when the trapped particle is in contact with the interface [Fig. 5(c)]. The calibration of the plane gave us an effective numerical aperture of 1.32 for the light scattered at largest angles, which is equal to the refractive index of water in the infrared. This fulfills Eq. (7) for our condenser of NA = 1.4, and proves that the light was travelling at right angles to the surface normal inside the chamber.
The amount of light collected depended weakly on the size and optical properties of the sample but it always remained a high percentage: between 95% and 99%. On the contrary, this result relied heavily upon the axial positions of both the sample in the flow chamber and the condenser. We needed to finely adjust these two distances in order to recover all the light. However, this was facilitated by the calibration of the BFP.
Experiments were performed in a range of 10-30 µm from the upper surface of the microchamber. The results indicate that the method provides a robust measurement of the force when the sample is maintained in a region of some tens of micrometers below the slide, even if the axial position is not kept constant. This small separation of the trap from the coverglass is required to collect a large percentage of the light scattered by the sample, as discussed above, and, to a lesser degree, to avoid the effects of the aberrations of the condenser lens. In particular, spherical aberrations induce a distortion of the light pattern at the BFP of the condenser when the sample is moved down into the chamber, with detrimental effects on the measurement of the force.
As a result of these restrictions in the particle axial position, it becomes necessary the use of aberration-free objectives with long working distances (water-immersion typically providing better performance than oil-immersion lenses). Alternatively, if the thickness of the construction is not a concern, one can directly build a thin flow chamber (~50 µm) so any accessible location in the sample fulfills the requirement.
Finally, we calibrated the sensor, that is, we determined the constant, α, which relates the PSD signal in V and the applied force in pN. To this effect, we used the hydrodynamic forces created by the surrounding fluid on a trapped bead. A piezoelectric stage (Piezosystem Jena, TRITOR 102 SG) was moved sinusoidally and the PSD response was plotted against the force applied to the sample. We repeated the experiment with different values of the laser power, the refractive index and the radius of the particle (Fig. 6 shows a typical run). All the data fitted a single straight line with slope 5.18 mV/pN almost perfectly. The value of the calibration factor was then αx = (0.193 ± 0.007) pN/mV (two different constants along the perpendicular axes, x and y, of the detector were found), where the error is the standard deviation of α. Importantly, the method does not measure the force in the axial direction, but it may be modified to do so by incorporating a second detector, following .
The important result of this experiment is that α can then be used regardless of the experimental conditions, since the detection of the force does not depend on them: it is an absolute measurement. As we mentioned in the Introduction, this has enormous advantages with respect to other calibration methods for single-beam traps.
5. Discussion and conclusions
The results shown here indicate that our method for measuring forces through the beam momentum change in a single-beam trap provides a robust measurement of the force despite the changes in the experimental conditions. The error in the calibration factor α, coming from the standard deviation is around 4%, which, despite limited experimental data, is similar to that reported in the counter-propagating trap configuration and in the conventional calibration techniques. There still exist, however, different questions that will need to be addressed in order to improve the accuracy.
Some preliminary experiments indicate the reliability of the calibration factor, α, under both the loss of light in the measurement process and the effect of the aberrations of the condenser lens. However, a more systematic analysis of these effects would allow us to quantify the actual robustness of the method. In particular, we want to explore the variability of α with the use of different numerical apertures of the condenser and different axial positions of the sample, and evaluate their relation with the accuracy in the measurement of α. Likewise, it is necessary to check the contribution of the backscattered light to the precision of the method, although we envision a small effect due to the low percentage of light reflected back at the sample, as shown in the simulations.
The calibration of the sensor should also be extended to greater forces. The maximum values used in each experiment, so far, do not correspond to the escape force, Fmax, of the trap . They only represent a fraction of it, around 0.6 Fmax. The maximum force of the trap is achieved when moving the bead along a certain curve in the x-z plane. The calibration experiments were performed by moving the sample chamber only in one dimension, so the particle was pulled away from the trap before reaching the maximum force value (escaping in the axial direction). An experiment that compensates for this effect would allow us to explore the linearity of the conversion between V and pN at larger forces.
The study of the system at large forces will also include a deeper analysis of the bending of the calibration curves (Fig. 6). A small curvature effect at the ends of the straight lines has been systematically observed, leading to a non-linear relation between the PSD signal and the force. We believe that the effect is related to the calibration method itself (of hydrodynamic origin, for example) rather than to light loss in the detection step, because the bending in this case is in the opposite direction to the one previously reported . Also, the images of the light pattern at the BFP of the condenser are strong evidence in favor of the collection of most, if not all, forward-scattered light.
Finally, the angle dependency of the Fresnel coefficients at the water-glass interface of the microchamber should also be taken into account when studying the precision of this method. The correction of this effect should be small, but it could be compensated for by adding a filter at the BFP with an appropriate radial transmission profile.
We would like to thank J. Mas and A. Carnicer for their help with the programming of the software used to acquire the PSD data. This work has been funded by the Spanish Ministry of Education and Science, under grant FIS2007-65880, as well as by the Agency for Assessing and Marketing Research Results (AVCRI) of the University of Barcelona. A. Farré is sponsored by a doctoral research fellowship from the Generalitat de Catalunya (grant FI).
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