Direct measurement of axial optical forces

Gregor Thalhammer; Lisa Obmascher; Monika Ritsch-Marte

doi:10.1364/OE.23.006112

1. Introduction

Optical tweezers are a not only a powerful tool for contact-less manipulation of micron-sized particles such as cells or micro-organisms, their ability to measure tiny exerted forces also provides quantitative information about mechanical properties on the micro-scale. This gives deeper insights into processes taking place on the molecular or cellular level. Accurate methods for measuring the exerted optical force play a crucial role for such applications.

1.1. Direct force measurement

In this work we will study an approach for measuring optical forces acting on small particles that is based on conservation of momentum: The exerted optical force is equal to the change in the momentum flux of the trapping light. Analyzing the momentum of the scattered light directly yields the optical force. This idea has already been used in the seminal paper of Ashkin [1] to explain the optical forces on a dielectric sphere in a ray-optics picture. In the far-field the momentum density of the light scattered by the particle points radially outward, and its magnitude is directly related to the intensity I(ϑ, φ) scattered into direction ϑ, φ (in spherical coordinates), see Fig. 1. Consequently, the cartesian components of the momentum flux density ṗ of the outgoing light is given by

\dot{p} = \frac{1}{c} I (ϑ, φ) (\begin{matrix} sin ϑ cos φ \\ sin ϑ sin φ \\ cos ϑ \end{matrix})

The total momentum flux is obtained by integrating over the full sphere. The optical force exerted on the particle is thus given as the difference between the total momentum flux of ingoing and outgoing light, where the ingoing momentum flux density is also described by Eq. (1), but with opposite sign. If an external force acts on an optically trapped particle, it quickly moves to a new equilibrium position, where the optical trapping force and the external force cancel each other. Therefore measuring the optical force in equilibrium also discerns the strength of an external force. As a very appealing property, this method for a direct determination of optical forces is independent of the shape, size and refractive index of the particle. Also, this method is insensitive to changes to the trap shape, e.g., due to aberrations induced by refraction at interfaces.

Fig. 1 Principle of the direct force measurement method. The optical force exerted by transfer of momentum from the light field to the particle can be directly deduced from the angular or the far-field intensity distribution of the in- and outgoing trapping light. Here ϑ and φ denote the usual polar and azimuthal angles of spherical coordinates for a ray, which is mapped to a position x, y (in cartesian coordinates) in the back focal plane of the condenser.

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Widely applied methods for indirect force measurements rely on observing the displacement of a particle from its centre position in an optical trap, e.g., via imaging or back focal plane interferometry [2, and references therein]. To relate the particle displacement to the optical force, a calibration of the trap stiffness is needed. Due to the fact that the trap stiffness depends on a multitude of parameters, such as particle size and shape, refractive index of particle and medium, and properties of the optical trap (power, beam shape), a frequent recalibration of the trap stiffness is required. Typically this is done by applying a known force to the particle, such as a drag force in a fluid or gas, by displacing the trap or the probe chamber (active calibration). Another possibility is to observe the effects of the random forces exerted by thermal fluctuations (passive calibration). Performing an accurate calibration is a non-trivial task, as testified by the large amount of literature published on this topic [3, 4]. Often prior knowledge such as particle shape and size, viscosity of the surrounding medium, distance to a interface boundary, or temperature is required. Alternatively, information about some key parameters needs to be obtained, e.g., by a combination of active and passive calibration [4].

Commonly these indirect methods rely on the assumption of a linear relationship between particle displacement and force. This is prone to fail for the large particle displacements needed to achieve maximum forces, which is often necessary due to the weak nature of optical forces. In particular, passive force calibration methods based on thermal fluctuations explore only a limited region around the equilibrium position and are therefore not able to deliver information about the entire force profile.

A crucial point for the experimental implementation of direct force measurements is the requirement to detect all of the scattered light. In a strict sense this is experimentally not feasible, therefore some error will be introduced in detecting the optical force. However, for typical experimental situations, i.e., particles with quite similar refractive index like water like polystyrene beads or cells in water, most of the trapping light is only slightly deflected, easing the collection of the scattered light. This important finding will be presented in more detail later. Accordingly, a first experimental implementation [5,6] of the direct force measurement method was mentioned for a dual-beam setup, with a trapping laser beam underfilling the aperture of the focusing objective lenses such that nearly all of the deflected light is collected. For the more common single beam optical tweezers the direct force measurement method has been largely disregarded with the argument that it is “impractical to collect the entirety of the scattered light” [2, 7].

Only much later has the applicability of the direct force measurement method for single beam tweezers been revisited [8, 9]. The authors showed that for typical experimental settings (polystyrene beads in water) indeed most of the light (> 95 %) is scattered in the forward direction, where it can be efficiently collected by using a high-NA (NA> 1.33) objective lens or an oil condenser. Independent of bead size and refractive index, the direct force measurements yield good agreement with measurements based on drag forces. Recently, the direct force measurement method has been successfully applied to in vivo measurements of forces due to active transport processes of lipid droplets in cells [10] with favorable results compared to conventional force calibration methods. However, these publications [6, 8–10] presented detailed investigations only for direct force measurements in the radial direction. The apparent neglect of axial force measurements is possibly explained by the fact that the light scattered into the backward direction, which goes undetected, has a larger impact on the observed axial force than on the radial force, leading to larger experimental errors in the axial direction. Another reason may be that in these previous investigations a single position sensitive detector (PSD) was used as a sensor, which is well suited for radial, but not for axial force measurements, as explained later in Sec. 2.3.

In this work we validate the accuracy of the direct force measurement method for axial forces in a single beam tweezers setup. By demonstrating that in situations where the true force is known, the direct force measurement method indeed provides accurate results, independent of the size and refractive index of the trapped particle, we aim to establish it as a reliable method that is applicable also in challenging situations, such as for particles of unknown shape embedded in a medium of unknown properties.

Our investigations are based on simulations of the measurement procedure and experimental tests of the achievable accuracy. As a key result of our numerical calculations we find that the error, which is introduced by only analyzing the angular distribution of the light scattered in the forward direction, can be largely compensated for by also taking into account the total amount of light scattered in the backward direction. Experimentally this is easy to implement by looking at the image plane of the objective lens. There the total amount of light, which is scattered by the particle in the backward direction, is concentrated in a small spot, which is easy to distinguish from the, in total, much stronger reflections of the trapping light from interfaces and lenses. We find that for typical experimental settings an accurate measurement of the axial force is possible (with less than 1 % error), independent of the size and refractive index of the trapped particle, see Sec. 3.2– 3.5.

In this paper, with its intent of validating the approach, we restrict ourselves to spherical particles, mainly for two practical reasons: scattering by homogeneous spherical particles can be simulated precisely and efficiently with generalized Lorenz-Mie theory; for an experimental comparison the drag force acting on spherical particles is well known (Stokes drag). Nevertheless, our results are not limited to spherical particles. To demonstrate the inherent shape independence of the direct force measurement method, we show experimentally that for two simultaneously trapped microspheres the direct force measurement method yields the total force acting on both particles together, see Sec. 3.6.

2. Experimental implementation of the direct force measurement method

In this section we describe the implementation of the direct force measurement method with a single beam optical tweezers setup. This defines the base for our further investigation of the achievable accuracy in the subsequent sections.

2.1. Description of the experimental setup

The setup we use to implement the direct force measurement method for a single beam, holo-graphic optical tweezers setup is shown in Fig. 2. It comprises a modified inverted microscope (Zeiss Axioscope 135) with a high-NA aplanatic oil immersion condenser lens (Zeiss, NA 1.4, focal length approx. 7 mm), which collects the light scattered in the forward direction. The back focal plane of the condenser is imaged on a digital camera (Teledyne DALSA Genie HM1024) with a standard camera lens (f = 25 mm, Fujinon HF25HA-1B). For focusing the trapping light we use a water immersion objective lens (Olympus UPlanSApo 60x, NA 1.2). To control the spot position we holographically shape the trapping laser beam with a liquid crystal based spatial light modulator (Boulder Nonlinear Systems, 512 × 512 pixel). A fiber laser (IPG Laser PYL-10-1064-LP, max. 10 W) provides the trapping light with a wavelength of 1064 nm. The optics of the microscope is also used for illuminating and imaging the trapped particles, the optical paths for the trapping and illumination light are combined by non-polarizing beamsplitter cubes, in particular one with about 50 % reflectivity (Thorlabs BS013) on the condenser side, and another one with 90 % reflectivity for near infrared light (Thorlabs BS029) before the objective lens. Another digital camera (Matrix-Vision mvBlueFOX 120aG) takes images of the trapped particles and is also used to record the total amount of back-scattered trapping light. As a light source for imaging of the particles we use a blue LED. To enable determination of particle sizes via inline holography [11], the sample is alternatively illuminated with collimated coherent light from a fiber coupled laser diode at 640 nm.

Fig. 2 Experimental setup.

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2.2. Calculation of the optical force from the intensity distribution in the back focal plane

To determine the optical force we rely on Eq. (1). For this we need to deduce the angular light distribution from the image of the intensity distribution in the back focal plane of the condenser, see Fig. 1. The aplanatic condenser lens obeys the Abbe sine condition, therefore light propagating at an angle ϑ relative to the optical axis is imaged at a radial distance r ∝ sinϑ from the optical axis. Since refraction at a plane interface leaves nsinϑ invariant, where n is the refractive index, light scattered at an angle ϑ (measured in the medium surrounding the particle) intersects the back focal plane of the condenser at a distance r = Rsinϑ. Imaging the back focal plane onto the camera leads to a further scaling of the light pattern. As a result, we obtain the optical force F_f from the observed intensity distribution I(x, y) of the light collected in the forward direction ( $ϑ < \frac{π}{2}$ ) by

F_{f} = (\begin{matrix} F_{f, x} \\ F_{f, y} \\ F_{f, z} \end{matrix}) = \frac{1}{c_{m}} \int I (x, y) (\begin{matrix} x / R \\ y / R \\ \sqrt{1 - (x^{2} + y^{2}) / R^{2}} \end{matrix}) d x d y - F_{0} .

Here, c_m = c/n_m is the speed of light in the medium surrounding the particle. Furthermore, F₀ describes the total momentum flux of the incoming light. During an experimental run this is typically kept constant, and F₀ can be conveniently obtained by evaluating the integral in Eq. (2) from a separate measurement of the outgoing light for an empty trap. In this case the in-and outgoing light has the same angular distribution. Alternatively, one can turn to a situation of a trapped particle (in equilibrium) without external force to determine F₀, since in this case the (average) optical force is zero. The important parameter R, which designates the maximum radius of the detected light pattern, corresponding to light scattered to the side, depends on the focal length of the condenser lens and the magnification of the relay optics used to image the back focal plane. We deduce it directly from the observed intensity distribution of a scattering particle. [To conveniently determine R and the center of the observed light pattern during alignment of the setup, we position a fixed bead slightly before the focus of the trapping light, which enhances scattering at large angles and also increases the amount of back scattered light, shown in Fig. 3(b).]

Fig. 3 Calculated axial force profile for a 3 μm diameter polystyrene bead. (a) True axial force Q_z (solid blue line) and estimate Q_z,f from forward scattered light only (red dashed line). Also shown (black dotted line) is the axial BFPI signal (for a condenser with NA = 0.8), scaled and shifted such that value and slope coincides with the true force at the zero crossing. (b) Difference ΔQ of the estimated and the true force (red dashed line), and normalized total power P_r of the light that is scattered in the backward direction and that is collected by the objective lens with NA = 1.2.

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In the experimental implementation we obtain all the components of the optical force (radial and axial) from a single image by a discretized version of Eq. (2) as weighted sums of all the pixel values. However, we do not precisely know the conversion factor that relates the intensity to the camera signal. Instead of determining the individual contributions such as camera sensitivity and transmission losses of the optical path, we once determine a calibration factor by applying a well defined drag force, see Sec. 3.4. We also compensate for inhomogeneous transmission losses of the condenser lens, which we have characterized by directing a narrow collimated laser beam at the back focal plane and recording the power after reflection by a silver mirror in front of the condenser. We find that in the direction normal to the polarization of the trapping light the transmission losses are well described by a parabolic shape (expressed in terms of the radial distance in the back focal plane) with a maximum transmission loss of 21 % at the border (for single pass transmission), whereas in the other direction the losses appear to be independent of the radial position. All the calculations are performed in real time, employing a consumer graphics card (AMD HD6850). The maximum frame rate of our camera (approx. 400 frames per second for an image size of 400 × 400 pixels) limits the maximum acquisition rate.

2.3. Relationship to back focal plane interferometry

There exists a close relationship between the direct force measurement method and back focal plane interferometry (BFPI). In both cases the scattered light is collected by a condenser lens, and the light intensity distribution in the back focal plane, i.e., the far-field pattern, is imaged on a detector. For BFPI commonly a quadrant photodiode (QPD) is used, whereas for the direct force measurement method mostly a position sensitive detector (PSD) is employed, which measures the center of mass of the light distribution more accurately than a QPD. As another difference, for BFPI typically a standard condenser with medium NA (0.5–0.8) is used, whereas for the direct force measurement method a high-NA condenser is required.

BFPI is generally regarded as a method to measure the position of a trapped particle (relative to the trapping beam), but, as pointed out by [8, 12] for measurements in the radial direction, the detector signal actually more closely reflects the optical force than the particle position, especially when using a high-NA condenser and a PSD. The signal interpreted as a force is more robust against variations, e.g., of particle size and refractive index. Furthermore, as noticed experimentally [13], the radial BFPI signal agrees with the optical force well beyond the range with a linear relationship between force and position.

For measuring the position (or force) in the axial direction with BFPI, the total collected laser power is detected, which is easily available as the sum signal output of the detector. However, if a high-NA condenser is used, which is beneficial for the robustness of radial force measurements, there is only little variation in the total power, and hence a poor signal is obtained, since most of the light is scattered in the forward direction (see also Sec. 3.1). Therefore, to obtain a strong axial signal with BFPI, a NA of about 0.5–0.8 has been recommended [14, 15]. Alternatively, for setups with a high-NA condenser, the use of a modified QPD with an additional radial segmentation has been proposed [16], which provides separate information about light collected at small and large angles. These detection schemes can be seen as an approximation to Eq. (2), where the axial weight $\sqrt{1 - r^{2} / R^{2}}$ with a spherical shape is replaced by a box-like shape. Not too surprisingly, this gives only a modestly accurate result for the force (for a quantitative comparison, see Sec. 3.1 and 3.3), and a calibration is required.

2.4. Sensor temporal bandwidth considerations

For conventional force measurement methods, which are based on position measurements and hence require a calibration, a high temporal bandwidth in the order of at least 10 kHz is desireable to enable passive calibration by analyzing the power spectrum of the Brownian motion. However, with the direct force measurement method such a high bandwidth is actually not necessary, in particular, when the trapped particle acts as a force transducer to measure an external force. The particle needs a timescale $τ = \frac{γ}{k}$ to adapt to changes of an external force, where k is the trap stiffness, and γ the drag coefficient. For a spherical particle of radius r in a medium with viscosity η the drag coefficient is given by Stokes law γ = 6πηr, leading to a timescale τ of about 2 ms for typical settings (particle radius r = 1 μm, water with viscosity η = 10⁻³ Pa s, trap stiffness k = 10pN/μm). With digital cameras such a temporal resolution is accessible, we for instance use a camera at a framerate of 400 Hz. Force measurements at a shorter timescale do not reveal a better time resolution for changes of external forces. In contrast, since the random Brownian motion of a trapped particle introduces noise, for a precise measurement of external forces some averaging is recommended. A measurement of the intensity distribution by a camera with an extended exposure time directly yields the average force during exposure.

3. Achievable accuracy of direct force measurements in axial direction

To quantify the achievable accuracy we rely on numerical simulations and experimental comparison with known drag forces. We will show that it is possible to attain an accuracy for axial measurements of a few percent of the maximum force. For many applications this is sufficient; due to the presence of noise (Brownian motion) or large individual variations in biological applications a higher accuracy is often not meaningful.

3.1. Numerical simulations

Performing numerical simulations for the measurement procedure has the advantage that the true optical force is accessible for comparison. For that reason we first present a characterization of the achievable accuracy for axial force measurements based on simulations. To summarize briefly, we simulate the intensity distribution in the far-field due to scattering of the trapping light by a spherical particle, and from this we calculate the true optical force and the estimated force, deduced from the light collected in forward direction by Eq. (2). Details on the procedure are given in Appendix A.

To start with, we assume somewhat idealized conditions, e.g., neglecting losses when collecting the transmitted light even up to angles of ϑ = π/2 (in water). This yields limits for the achievable accuracy under ideal experimental conditions.

Figure 3 shows the calculated axial force profile for a 3 μm diameter polystyrene bead in water. For the simulations the results are given as the scaled force $Q = \frac{F}{P_{0} / c_{m}}$ , where P₀ denotes the total light power. The observable axial force Q_z,f resembles the true force Q_z quite well at all positions, also outside the region of stable trapping. The largest error between the estimated and true force of about 13 % occurs at the maximum positive (pushing) force. In general the difference is always positive, as can be expected from the fact that the condenser collects only light in the forward direction, thus missing some light that pushes the particle in the forward direction.

For comparison, also the simulated axial BFPI signal is shown, i.e., the total light power collected by a condenser with a NA of 0.8, scaled and shifted such that it matches the true force near the equilibrium position. We checked that this choice for the value of NA is close to the optimum value that minimizes the (average) error for the range of particle sizes we studied. Only in a small region around equlibrium the BFPI signal comes close to the true force. Similarly, a linear approximation of the force profile around the equilibrium position gives only a rather poor approximation of the force profile.

3.2. Improving the accuracy with the aid of the total amount of back scattered light

As shown above, collecting only the light in the forward direction leads to an evident error in measuring the force. In principle one could further improve the accuracy by also analyzing the angular distribution of the backward scattered light. But experimentally this is difficult, since the amount of light of a few percent due to back scattering of the particle is overwhelmed by additional reflections, especially from the many lens surfaces within the objective lens. We expect them to be in the order of several percent per surface because the anti-reflective coating designed for visible light is not effective at 1064 nm anymore.

However, we noticed that it is sufficient to determine the total power of the backward scattered light, instead of measuring the full angular distribution in the backward direction, to obtain a striking improvement of the accuracy. Experimentally this is much easier to accomplish: on the camera, which we use to image the particles, all the light that is back-scattered and collected by the objective lens is detected as a localized spot at the particle image position.

The simulations shown in Fig. 3(b) reveal that there is a striking similarity between the error of the normalized force −ΔQ_z = Q_z − Q_z,f and the total normalized power P_r/P₀ of the collected back-scattered light. Accordingly, the amount of backward light P_r can be used to obtain an improved estimate

{\hat{F}}_{z} = F_{z, f} + k_{P} P_{r} / c_{m}

for the axial optical force. One simply has to add the scaled backward power P_r to the estimate F_f,z obtained from the angular distribution of the forward scattered light. The value of the scaling factor k_P in Eq. (3) that gives the best correction depends on the NA and the collection efficiency of the objective lens. Specifically, for the data presented in Fig. 3 we find an optimum value of k_P = 0.98, which reduces the maximum error to only 0.3 % of the maximum force after applying the correction.

In principle the amount of back-scattered light can also be deduced from the amount of missing light detected in the forward direction, i.e., the difference P_f − P₀ of the total transmitted light with and without particle. An advantage of this is that it relies only on data acquired in the forward direction, but on the other hand it requires knowledge of the incoming laser power. But the amount of missing light also includes contributions from light scattered at about ϑ = π/2, which according to Eq. (1) does only slightly contribute to F_z because of the weighting factor cosϑ. Therefore we observe that, in general, a correction based on P_r clearly gives better results.

3.3. Independence of particle size

A fundamental strength of the direct force measurement method is the independence of the measured force from the particle size. In this section we will demonstrate, now based on simulations and experiments, that this is indeed the case for axial force measurements, provided the simple correction procedure as outlined in the previous section is applied.

We start with numerical simulations, similar to Fig. 3, but now for polystyrene beads with various sizes in the range from 0.5 μm – 4 μm diameter. We present the results in Fig. 4(a). Here we have chosen a different type of presentation, showing the difference between the true force and the estimated force from the direct force measurement method versus the true force, both with and without the correction based on the back scattered light, see Eq. (3). Additionally these simulations are more realistic and also include an uncompensated transmission loss due to (Fresnel) reflections at the water glass interface at large incidence angles. For these settings we use a correction factor of k_P = 0.87, which minimizes the average error after correction.

Fig. 4 Calculation of the accuracy for polystyrene beads from 0.5 μm to 4 μm diameter. (a) Measurement error ΔQ_z = Q_z,f − Q_z for the direct force measurement method versus true force, without correction (blue lines) and with additional correction (green lines) based on the total amount of backward scatterd light. The results for some specific bead sizes are highlighted. (b) Error for force measurements based on BFPI (for a condenser with NA = 0.8), assuming a linear relationship between force and axial BFPI signal. For each bead size the slope and value at equilibrium was matched to the true force, mimicking a force calibration.

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Without correction there is a significant measurement error, which depends quite sensitively on the bead size. For some sizes Mie-resonances lead to a stronger scattering, in particular in the backward direction. The induced increased errors are more pronounced around the equilibrium position. Besides this, the error increases for large positive forces, especially for larger beads. We observe that in general small beads, alongside their reduced maximum force, show a reduced error. At any size the error before correction is limited to about 15 % of the maximum force. In many cases a substantial part of the error can be removed by adding an offset such that at equilibrium the estimated axial force is set to zero.

The results for the direct force measurement without additional correction are already much better than the accuracy that can be achieved with the axial BFPI signal based on the assumption of a linear relationship between BFPI signal and force (i.e., matching value and slope of the signal with the true force, but without a calibration based on a full force profile). In BFPI, outside the equilibrium position errors quickly amount to 50 %. Only for some selected sizes (e.g., for a precisely 1 μm large bead) good accuracy can be achieved.

With the correction method applied, the errors are strongly reduced. No systematic offset is visible. Independent of the bead size, the maximum error is always less than 1.2 % of the maximum force, and in most cases less than 0.5 %. For comparison, if the correction is based only on the missing light 1 − P_f, the error is reduced to 1 %–3 %, depending on the particle size, except for the smallest beads of about 500 nm diameter, where the error reaches about 5.5 %.

3.4. Experimental validation of direct force measurement method with drag forces

In this section we present experimental results, comparing the observed axial force obtained with the direct force measurement method with an applied drag force for several bead sizes. We observe that the measured force (after applying the correction procedure) and the applied force agree nicely, independent of the particle size.

To apply a well defined force on the trapped particle, we move the trapped particle by repeatedly translating the objective lens with a piezo focus stage (PI P-721.SL2 with driver E-709), which we control with a waveform generator (Agilent AWG-33220A). By this we impose a drag force F_d = 6πηrv proportional to the translation velocity v. Our goal is to explore most of the force range where stable trapping is possible. Since the force profile in the axial direction has an asymmetric shape (see Fig. 3), where the maximum pushing force is about 2.5–3 times larger than the maximum pulling force, we have chosen an asymmetric movement pattern that is composed of sinusoidal shapes (half a period) with different durations for moving in the forward or backward direction. To adjust the maximum drag force to about 80 % of the maximum optical force, we change the total period of the movement, typically in a range of 1–3 s. The maximum translation distance is set to 60 μm, centered in a 125 μm thick probe chamber. We have experimentally checked that modifications of the Stokes law due to hydrodynamic proximity effects near a surface (Faxén’s law) are negligible small for this configuration. For these measurements the use of a water-immersion microscope is crucial, since it allows to focus deep into the water filled probe chamber without introducing aberrations due to refractive index mismatch. We simultaneously record the observed optical force, the total reflected power, and the position of the objective lens for about 10 s. Since the piezo stage shows some lag, we calculate the velocity from the measured position data by numerical differentiation, using a Savitzky-Golay filter for noise reduction. We also correct for small discontinuities of the digital oscilloscope (Tektronix MSO3014) that records the analog position sensor signal of the piezo stage, which otherwise appear as artefacts (spikes) in the velocity data.

As test particles we use polystyrene beads with sizes of 1 μm, 1.85 μm, and 3 μm, the size of which we checked with inline holography [11]. Some results for the remaining difference between the observed optical force and the applied drag force are shown in Fig. 5. Indeed we observe that the direct force measurement method with a correction based on the total backscattered light gives good agreement with the applied drag force, independent of the particle size. Without correction noticeable deviations occur. The observed behaviour of the force and the amount of back scattered light is well represented by the corresponding simulations. For a 1 μm diameter bead (data not shown) only minor deviations of the observed force from the drag force are recognizable, even without correction (except for an offset), as expected from the simulations (see Fig. 4).

Fig. 5 Experimental comparison of the direct force measurement method with drag forces for 1.8 μm and 3 μm diameter polystyrene beads (left column), and corresponding simulations (right column).

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In Fig. 5 we also present data (from experiment and simulation) for 1 − P_f/P₀, the amount of light missing in the forward direction due to scattering. As we described in Sec. 3.2, using this for correction would lead to results that are not quite as good. Although 1 − P_f/P₀ shows a similar behaviour as the back scattered light P_r/P₀, significant differences can be observed.

As already explained in Sec. 2.2, we adjusted the scaling factor, which relates the camera signal to the intensity in the back focal plane, and the correction factor k_P (Sec. 3.2) such that the difference between observed and true force is minimized. To reduce the noise present in the data due to thermal fluctuations, we applied a smoothing to the data with a Savitzky-Golay filter, effectively averaging over 31 data points, corresponding to a time interval of 75 ms. When axially translating the microscope objective lens, we observe (even without trapped particle) a small modulation (typically 1-2 %) of the total transmitted light intensity, which is proportional to the lens displacement. Therefore for calculating the optical force by Eq. (2) we take F₀ from a measurement at the same position without the particle. Due to intensity drifts of our laser, the results, in particular the force and the amount 1 − P_f/P₀ of light missing in forward direction, often show some small offsets after subtracting the reference measurement without particle. In practice, for the force such an offset can be easily removed by adjusting the zero point at equilibrium position for a free particle. However, for the data presented here we did not perform such an offset compensation.

3.5. Influence of the refractive index of the trapped particle on the measurement error

The optical force depends, apart from particle size, on the refractive index of the particle. In this section we study how the achievable accuracy is influenced by the refractive index.

In general, a particle with larger contrast of the refractive index to the surrounding buffer shows a stronger scattering of light. Consequently, optical forces typically increase, which is often desired. But for the direct force measurement we expect larger errors due to non-detected scattered light. In fact, our simulations shown in Fig. 6 support this belief. Here we calculate for several values of the refractive index of the particle in the range between 1.4 – 1.6 the maximum error relative to the maximum force for particles sizes of 0.5 μm– 4 μm. Polystyrene beads with n = 1.57 lie in the upper range, while PMMA beads (n = 1.48) or silica beads (n = 1.44) have a smaller refractive index. Cells or micro-organisms typically have a rather low refractive index of about n = 1.4. But for biological samples, sub-structures of the cell also contribute to scattering. Without correction, see Fig. 6(a), the maximum error increases with larger refractive index, but for low refractive index n < 1.45 it is low enough (maximum error < 5%) for many applications. With a simple offset compensation at equilibrium in many cases a significant reduction of the error is achieved, see Fig. 6(b). With correction based on the total reflected power, see Fig. 6(c), we observe a strong improvement of the maximum error to less than 1 %, independent of the particle size and refractive index. For particles in the range between n = 1.55 and n = 1.60 in most cases the error is even lower. This behaviour is not too surprising, since the correction factor k_P = 0.87 of Eq. (3), used for all data of Fig. 4 and Fig. 6, has been optimized for n = 1.57.

Fig. 6 Simulation of the accuracy of the direct force measurement method depending on the particles refractive index. Every single point gives the maximum error relative to the maximum force for a certain particle size in the range from 0.5 μm to 4 μm. (a) Error without correction, (b) with an offset compensation at the equlibrium position, (c) with correction based on the total power of the back scattered light (note the different scale).

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Experimental results for polystyrene beads with n = 1.57 have already been shown in the previous section and in Fig. 5. For this material a correction based on the amount of back scattered light is necessary to obtain an accurate direct measurement for the optical force. In contrast, as demonstrated in Fig. 7, silica beads (diameter 2.1 μm) with low refractive index (n = 1.44) exhibit much less (back-)scattering and we get accurate results even without correction. On the other hand, silica beads exhibit about half the maximum force compared to polystyrene beads.

Fig. 7 Measurement error for a 2.1 μm diameter silica bead. Due to the low refractive index (n = 1.44) scattering of the trapping light is reduced, and the direct force measurement method yields accurate results even without correction.

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3.6. Measuring the total force acting on several trapped particles

In this section we present an experimental demonstration of the fact that the direct force measurement method provides the total optical force, independent of the shape of the particle. However, for particles of arbitrary shape we cannot follow the same approach for validation as above, that is using the drag force as a reference, because for non-spherical particles calculating the drag force is not straightforward and would require a detailed knowledge of the shape of the particle. Instead we simultaneously trap two spherical particles, for which the individual drag forces can be calculated easily, and check that the total optical force obtained with the direct force measurement method equals the sum of the drag forces acting on the single particles. Indeed, the experimental results for the direct measurement of the total force (see Fig. 8) show a good agreement with the total drag force.

Fig. 8 Direct measurement of the total optical force acting on two simultaneously trapped beads (polystyrene, 3 μm diameter).

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To perform this experiment, we create two optical traps with our holographic optical tweezers setup, employing the SLM as a diffractive beam splitter. We simultaneously trap two 3 μm diameter polystyrene beads at a distance of about 12 μm. Similar to the measurements for a single particle, we translate the traps by moving the microscope objective lens, thereby exerting well defined drag forces. The observed far-field intensity distribution of the transmitted trapping light is not the mere sum of the intensity distribution due to scattering by the single particles, but also shows a fringe pattern due to interference of the individual traps. But, when calculating the optical force by Eq. (2), the narrow spaced fringes are averaged out, and the measurement yields the total optical force, which equals the sum of the drag forces acting on the particles.

4. Discussion

The results presented in the previous section show that omitting the light scattered in the backward direction affects the achievable accuracy. The induced error depends in particular on the refractive index difference of the particle and the surrounding medium. However, for biological samples with small difference, such as cells, micro-organisms or their constituents, the maximum error is less than 5 % of the maximum force, which for many applications is insignificant. This makes the direct force measurement method a prime choice for biological samples with large individual variations in size and shape, and for in vivo experiments of sub-cellular structures, such as probing the forces arising due to active transport processes, where conventional force calibration methods are not feasible. Our results suggest that accurate measurements are possible under such circumstances.

Particles with larger refractive index such as polystyrene beads, which are widely used for optical trapping, since they offer large optical forces, show stronger scattering in the backward direction and hence are affected by a larger measurement error. As a key result we find that in this case the error can be strongly reduced, to typically less than 1 %, by also considering the total amount of back scattered light. Experimentally this is very easy to implement, since the standard imaging path present in every single beam optical tweezers setup can be used. For applications, where large optical forces are needed, the direct force measurement method has the advantage that it provides accurate force measurements across the full range up to maximum force.

We observe that the measurement of the axial forces is rather robust against experimental imperfections, such as (axial) misalignment of the condenser, loss of light due to a limited transmission of the condenser, or reflections at the water glass interface. This prevailingly affects the collection of light scattered at a large angle. Due to the cosϑ weighting in calculating the axial force, the result is only altered slightly if the light scattered near $ϑ \approx \frac{π}{2}$ goes undetected. In this respect measuring the axial force is experimentally less demanding than direct force measurements in the radial direction.

The trapping properties in the axial direction often pose critical limitations for the manipulation of particles, especially regarding the maximum trapping or escape force. They are more easily affected by spherical aberrations, which typically constitute the largest contribution of aberrations present in a setup, in particular when using a high-NA oil immersion microscope objective for focusing into a water filled probe chamber, or due to poorer correction of the objective lens at near infrared wavelengths. Direct force measurements in the axial direction provide a simple means to experimentally characterize the full force profile and in turn optimize the trapping properties. This is also applicable to characterization of forces in Bessel or Airy beams that do not sustain stable axial trapping, but have useful properties for transport or sorting of particles.

For future applications we expect benefits from a combination of the direct force measurement method with advanced particle tracking methods, e.g. with stereoscopic video microscopy or inline holography, which provide precise 3D position data.

The inherent property of the direct force measurement to reveal the total optical force, which makes it independent of the particle or beam shape, is a drawback if one is interested in the individual forces acting on several simultaneously trapped particles, especially when using holo-graphic optical trapping. Several approaches are possible to overcome this limitation, e.g., time multiplexing for having only one trap at a time, the use of two orthogonally polarized trapping beams [17], or placing a pinhole in an image plane to spatially select the light scattered by an individual particle [18]. We envisage that a holographic measurement of amplitude and phase in the back focal plane of the condensor, which allows for a complete reconstruction of the forward scattered field, will give detailed information not only about individual forces, but also about optical torque, which is not accessible when only looking at the intensity distribution. This is of interest when working with non-spherical particles, which in general experience torques in an optical trap. Preliminary results have shown that this is feasible.

5. Conclusions

To summarize, in this work we analyze in theory and experiment the achievable accuracy of direct force measurements in the axial direction for single beam optical tweezers. As a key result we observe that when in addition to the far-field light distribution in the forward direction also the total amount of backward scattered light is analyzed, the measurement error can be reduced across the full trapping range to less than 1 % of the maximum force, independent of particle size and refractive index. For biological samples with typically a small difference of the refractive index to the surrounding, a good accuracy of a few percent can be obtained even without a correction based on the backward scattered light. Our results, together with the previous studies on direct force measurements in the radial direction [6, 8, 9], confirm the good reliability and accuracy of this method for 3D calibration free force measurements. This makes the direct force measurement method a prime choice, e.g., for in vivo measurements in biological samples, or for photonic force microscopy with nano-fabricated tools.

A. Appendix: Numerical simulations of far-field intensity distribution for light scattering by spherical particles

To calculate the light forces and the angular distribution of the transmitted light, we numerically simulate the scattering of a focused beam by a homogeneous spherical particle, employing generalized Lorenz-Mie theory. A comprehensive presentation is contained e.g. in [19–21].

Starting point is a description of the trapping beam at the back focal plane of the objective lens with focal length f. There we introduce normalized cartesian coordinates x₁, y₁ and spherical coordinates ϑ, φ, which are related by $x_{1} = \frac{x}{f} = sin ϑ cos φ$ and $y_{1} = \frac{y}{f} = sin ϑ sin φ$ . Here we assume that the focusing lens (and also the condensor lens) obeys the Abbe sine condition. For our setup the intensity distribution I is described by a Gaussian with width σ = 0.87 (normalized by the focal length f),

I (x_{1}, y_{1}) = I_{0} exp (\frac{x_{1}^{2} + y_{1}^{2}}{σ^{2}}),

which is further clipped by the aperture stop with (normalized) radius NA/n, with NA = 1.2 the numerical aperture of our water immersion objective lens and n = 1.33 the refractive index of water. The field in the back focal plane is then described by

E_{x} = N \sqrt{I} exp (i ψ)

with an additional phase modulation ψ, which we apply with an SLM. We choose the normalization factor N such that the total incoming power equals 1. For simplicity we omit extra conversion factors when relating intensity and field amplitude, i.e., we set I = |E|². For linearly polarized light, as we use, the other field component is E_y = 0, whereas E_y = ±iE_x for circularly polarized light. To displace the focus to a position (x₀, y₀, z₀) we apply a phase modulation

\begin{array}{l} ψ (x, y) & = k (x_{0} x_{1} + y_{0} y_{1} + z_{0} \sqrt{1 - x_{1}^{2} - y_{1}^{2}}) \\ = k (x_{0} sin ϑ cos φ + y_{0} sin ϑ sin φ + z_{0} cos ϑ) . \end{array}

We model the focusing by the objective lens as a transformation of the nearly plane wave in the back focal plane into a spherical wave, where the tangential components E_ϑ, E_φ are obtained by

(\begin{matrix} E_{ϑ} \\ E_{φ} \end{matrix}) = \sqrt{cos ϑ} (\begin{matrix} cos φ & sin φ \\ - sin φ & cos φ \end{matrix}) (\begin{matrix} E_{x} \\ E_{y} \end{matrix}) .

The additional apodization factor

\sqrt{cos ϑ}

is needed to ensure energy conservation for a lens obeying the Abbe sine condition.

The next step is to express the far-field as a multipole expansion of an ingoing electromagnetic wave

E = \sum_{l = 1}^{l_{max}} \sum_{m = - l}^{l} s_{lm} S_{lm} (ϑ, φ) + t_{lm} T_{lm} (ϑ, φ)

with the vector spherical harmonics S_lm = r∇Y_lm and T_lm = ∇Y_lm × r derived from the standard spherical harmonics

Y_{lm} = \sqrt{\frac{2 l + 1}{4 π}} \sqrt{\frac{(l - m)!}{(l + m)!}} P_{l}^{m} (cos ϑ) exp (i m φ),

where the associated Legendre functions are defined by

P_{l}^{m} (x) = {(- 1)}^{m} {(1 - x^{2})}^{m / 2} \frac{d^{m}}{d x^{m}} P_{l} (x) .

Since the vector spherical harmonics are orthogonal, the expansion coefficients s_lm and t_lm can be found by projection, integrating over the sphere

\begin{array}{l} s_{lm} = \frac{1}{l (l + 1)} \int S_{lm}^{*} \cdot E d Ω \\ t_{lm} = \frac{1}{l (l + 1)} \int T_{lm}^{*} \cdot E d Ω \end{array}

Here we used the orthogonality of the vector spherical harmonics S_lm,

\int S_{lm}^{*} \cdot S_{l^{'} m^{'}} d Ω = l (l + 1) δ_{l l^{'}} δ_{m m^{'}},

and same for T_lm.

To numerically perform the discretized integrals of Eq. (11), we rely on the library SHTns [22], which provides an highly optimized implementation. For performing the integration along the φ coordinate, it employs a fast Fourier transform, and a Gauss-Legendre quadrature along ϑ. This library only supports transforms for real valued fields, for which the symmetry relation $s_{l - m} = s_{lm}^{*}$ is exploited to store only coefficients for m ≥ 0. Therefore expansion coefficients s_lm,R and s_lm,I for the real and imaginary parts of the field are calculated separately, and then combined via

s_{lm} = {\begin{matrix} s_{lm, R} + i s_{lm, I} & if m \geq 0 \\ s_{l | m |, R}^{*} + i s_{l | m |, I}^{*} & if m < 0 \end{matrix}

with the inverse relation for m ≠ 0

\begin{matrix} ℜ s_{lm, R} = & \frac{1}{2} (ℜ s_{lm} + ℜ s_{l - m}), \\ ℑ s_{lm, R} = & \frac{1}{2} (ℑ s_{lm} - ℑ s_{l - m}), \\ ℜ s_{lm, I} = & \frac{1}{2} (ℑ s_{lm} + ℑ s_{l - m}), \\ ℑ s_{lm, I} = & - \frac{1}{2} (ℜ s_{lm} - ℜ s_{l - m}) . \end{matrix}

With the incoming light field described by the multipole expansion in spherical waves, calculating the outgoing light scattered by a spherical particle centered at the origin is very simple. To obtain the expansion coefficients for the outgoing field

\begin{array}{l} s_{lm}^{out} = {(- 1)}^{l} (1 - 2 α_{l}) s_{lm}, \\ t_{lm}^{out} = {(- 1)}^{l + 1} (1 - 2 β_{l}) t_{lm}, \end{array}

one only needs to multiply the expansion coefficients of the ingoing field s_lm, t_lm with a factor which contains the Lorenz-Mie scattering coefficients

\begin{array}{l} α_{l} = \frac{n_{r} ψ_{l} (n_{r} ρ) ψ_{l}^{'} (ρ) - ψ_{l} (ρ) ψ_{l}^{'} (n_{r} ρ)}{n_{r} ψ_{l} (n_{r} ρ) ξ_{l}^{'} (ρ) - ξ_{l} (ρ) ψ_{l}^{'} (n_{r} ρ)}, \\ β_{l} = \frac{ψ_{l} (n_{r} ρ) ψ_{l}^{'} (ρ) - n_{r} ψ_{l} (ρ) ψ_{l}^{'} (n_{r} ρ)}{ψ_{l} (n_{r} ρ) ξ_{l}^{'} (ρ) - n_{r} ξ_{l} (ρ) ψ_{l}^{'} (n_{r} ρ)}, \end{array}

where ρ = kr is the size parameter of the particle, n_r the relative refractive index of particle and surrounding medium, and ψ_l and ξ_l denote the Riccati-Bessel functions.

Again, we use the SHTns library to finally calculate the far-field E^out by Eq. (8). From the outgoing field we are now able to find all the results needed to simulate the experimental realization of the direct force measurement method. Specifically, the (normalized) axial force follows from Eq. (1) by integrating over the full sphere

Q_{z} = \int ({| E^{out} |}^{2} - {| E^{in} |}^{2}) cos ϑ d Ω,

whereas the axial force as obtained in an experiment from the light collected only in forward direction can be calculated by restricting the integration to the half sphere. Additionally, for a more realistic modelling, we include Fresnel transmission factors for the water/glass interface.

To simulate force profiles, we repeat the calculations, modifying the focus position with help of Eq. (6). We observed that this approach is in our implementation actually faster and more robust than other algorithms that achieve translation of the field by a linear transformation of the multipole expansion coefficients. For an improved speed for the calculation of axial force profiles we also exploit the fact that for a rotationally symmetric beam profile only coefficients with m = ±1 are non-zero.

Acknowledgments

The present work was supported by the ERC Advanced Grant catchIT (No. 247024).

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Direct measurement of axial optical forces

Abstract

1. Introduction

1.1. Direct force measurement

2. Experimental implementation of the direct force measurement method

2.1. Description of the experimental setup

2.2. Calculation of the optical force from the intensity distribution in the back focal plane

2.3. Relationship to back focal plane interferometry

2.4. Sensor temporal bandwidth considerations

3. Achievable accuracy of direct force measurements in axial direction

3.1. Numerical simulations

3.2. Improving the accuracy with the aid of the total amount of back scattered light

3.3. Independence of particle size

3.4. Experimental validation of direct force measurement method with drag forces

3.5. Influence of the refractive index of the trapped particle on the measurement error

3.6. Measuring the total force acting on several trapped particles

4. Discussion

5. Conclusions

A. Appendix: Numerical simulations of far-field intensity distribution for light scattering by spherical particles

Acknowledgments

References and links

Cited By

Figures (8)

Equations (17)

Optics Express