1. Introduction

It will be soon four decades since the discovery of optical trapping of micron- sized particles by means of laser light by Arthur Ashkin [1]. Nowadays, optical traps are used in many different scientific areas due to their great versatility [2]. They can be applied to study a wide variety of physical and biological systems, such as molecular motors and cells [3, 4], soft condensed matter, and also to study the properties of light itself [5–7]. Optical traps can be used as well for technological purposes, like the development of micromachines [8,9] and microfluidic devices [10]. Whereas the single beam optical trap or optical tweezers is the suitable choice for individual particle manipulation in 3D [11], there are many different possibilities when the simultaneous manipulation of many particles is the aim [12]. The most common alternatives are traps based on interference patterns [13–15], dynamical digital holography [16, 17], phase contrast techniques [18] and scanning beam traps [19]. These have been successfully applied, for instance, in particle organization in periodic patterns [13, 14], in the development of optical sorting methods [10, 20–23], and in the study of brownian particle dynamics [24, 25].

Although the basic physics underlying the optical trapping phenomenon is well understood, many different theoretical models of the light-particle interaction have been developed. The main reason for the lack of consensus about a unique general description is that most of the models developed so far have limited applicability, either because of restrictions in the size or geometry of the trapped objects or because the description of the light distribution is idealized, or both. In this context, direct measurements of the optical forces represent a central issue, not only for testing theoretical models but also for practical applications of optical micromanipulation systems as force transducers. A well established experimental calibration method in the case of spherical particles in individual traps consists in quantifying the displacement of the particle from its equilibrium position due to the viscous drag while it is in relative motion with respect to the host medium at a known constant velocity [26–28].

For extended light patterns the paraxial beam description is appropriated; models using either the generalized Lorenz-Mie theory [22, 29] or a simpler ray tracing approach [21, 29, 30], in the proper limit, have given results that compare well with experiments. In this case, interesting phenomena arise due to a strong dependence of the effective optical potential on the size and/or shape of the particle. As this is the basis of several optical sorting devices [10, 20–22], a characterization of the optical forces and/or potentials associated to extended light patterns is particularly important in all the different particle size regimes. In the stochastic regime, the optical potential associated with a periodic pattern of light has been characterized by studying the thermally activated escape from the potential wells. McCann and coworkers obtained in this way a full three dimensional mapping of a double-well optical potential confining submicron particles using Kramer’s theory [31]. Similarly, Blickle et al. studied the motion of brownian particles in a tilted periodic potential, which led to an approximate map of the optical potential by measuring the stationary probability distribution and the current of particles in the system [32]. These tilted periodic potentials are usually known as washboard potentials [33] and, in optics, these have also been created with alternative light distributions, for instance, with a tilted Bessel beam, and used as well to study the stochastic dynamics of brownian particles [34]. In other studies including larger particles (diameters of up to 5 microns), the existence of different size-dependent types of behavior has been drawn from the particle dynamics and compared with theory for a Bessel beam profile [29, 30], and also for other extended periodic potentials [21, 22]. However, in the deterministic regime, for which the thermal noise can be practically neglected (particle diameter > 5 microns), we are not aware of any report on a detailed quantitative characterization of the optical forces and/or potentials in this regime.

We present here a full quantitative mapping of the trapping forces acting on Mie particles (8, 10 and 14.5 microns diameters) as a function of the position for an optical potential of interference fringes that extends over distances of up to 400 microns. In the experiment, we study the dynamics of a particle in a washboard potential. We physically tilt the sample cell by a controlled small angle and let the bead run through the light pattern of fringes by effect of gravity. From the experimental data, we are able to determine the particle’s velocity as a function of position, which is directly proportional to the force acting on the particle for an overdamped regime, as it is our case. The period of the fringes is varied in order to investigate the dependence of the maximum force on this parameter. We compare our experimental results with theoretical curves obtained with a ray optics model of the optical forces, appropriated in this case.

2. Particle dynamics in a tilted extended optical potential of interference fringes

The simplest physical model describing our experiments corresponds to a deterministic spherical particle moving across a washboard potential, as schematized in Fig. 1. The radius, density and refractive index of the dielectric sphere are denoted, respectively, by R ₀, ρ_p and n_p. The host medium has density ρ_m and refractive index n_m. The particle moves along the x axis due to its effective weight component along this direction F_g = W sin ψ, where W = (ρ_p - ρ_m)(4π R ₀ ³/3)g and g is the acceleration of gravity. There is a dissipative force, opposite to the direction of motion, owed to viscous drag and surface interaction effects, which will be simply considered as proportional to particle’s velocity. We have experimentally verified that the system is over-damped, and thus the inertial term is not considered in the equation of motion for the particle. Thus the dynamics of the system is described by

where V(x) represents an extended optical potential and the prime denotes the derivative with respect to x. Notice that γ is different from the conventional Stokes coefficient for a sphere, but the Faxen’s correction should be applied, due to the close proximity of the particle to the bottom of the glass cell [35]. Furthermore, there might be additional effects of interaction between the sphere and the surface on which it moves down, that can also contribute to the dissipation or effective friction coefficient γ.

Our aim is to experimentally determine the optical force F(x) = -V′(x) by analyzing the dynamics of the particle and compare it with a theoretical model. Although the light pattern and thus the optical potential extend over the transverse plane in two dimensions, we are only interested in the motion of the particle along the direction of the spatial periodicity x. In the experiments, the sphere naturally moves across the pattern along the x axis due to the inclination of the sample cell, but also because the light pattern is narrower in the y direction.

 

Fig. 1. Schematic of the physical system. A spherical particle is moving down a inclined plane in a washboard potential in the overdamped regime. The y axis is normal to the plane of the figure according to a right-hand system. W is the effective weight of the particle, N is the normal force, γẋ is the damping force, V(x) is an optical potential and ψ is the inclination angle.

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The parameter β = 2πn_mR ₀/λ, typically used for characterizing the size of a particle with respect to the wavelength ranges in our case from 63 (for R ₀ = 4.0μm,) to 118 (for R0 = 7.25μm,), hence β ≫ 1. In this regime, geometrical optics provides a good alternative for calculating the optical force [36]. In this scheme, the x component of the optical force exerted by a collimated light field propagating vertically upwards on a spherical particle located at the position (x ₀,y ₀) in a tranverse plane (z = constant) can be written as [21],

I(x,y) denotes the transverse intensity distribution of the light field, c is the speed of light in vacuum, and the integration is performed over the illuminated hemisphere of the particle, where φ is the azimuthal angle and θ is complementary to the polar angle. The incidence angle coincides with θ at each point on the sphere’s surface, θ_t is the transmitted angle and R and T are, respectively, the average of the reflectance and transmittance over the two transverse polarizations. The coordinates of each point at the particle’s surface, (R ₀, φ, θ), and the position of the center of the particle with respect to the beam axis, (x ₀ y ₀), are related by means of x = x ₀+R ₀ cos φ sin θ and y = y ₀+R ₀ sin φ sin θ (we set y ₀ = 0). The assumption that the x component of the optical force, described by Eq. (2), is conservative is based on the fact that there is no propagation of the beam along this direction [36].

The intensity distribution is, according to our experimental conditions,

which corresponds to the interference of two elliptical gaussian beams, whose major and minor semiaxes at the plane of interest are w_x and w_y, respectively. P is the power of the laser at the same plane and L is the spatial periodicity of the interference fringes. The coordinates of the center of the gaussian envelope of the intensity distribution in the laboratory reference frame are denoted by (ξ, η); we set η =0.

Models for the trapping forces merging ray optics with a paraxial gaussian beam description, as in Eq. (2), have been considered unsuitable for the case of tightly focused beams [2]. However, we are dealing with extended light patterns, for which the parameter s_i = λ/2πw_i (i = x,y), used by Davis in an expansion to account for corrections to the paraxial theory [37], gives s_x < 0.0004 and s_y < 0.0009.

In order to model the dynamics of the particle, we are assuming that the optical force given by Eq. (2) is the same force given as the derivative of the potential in Eq. (1). The effective friction coefficient γ can be experimentally determined in an independent way by letting the particle move down the inclined plane in absence of the light field. In this case, we verified that the system is overdamped and so the particle moves with a constant velocity, denoted by ẋ ₀, that is directly proportional to the force F_g; therefore γ= F_g/ẋ ₀.

3. Experiment

The experimental setup is illustrated in Fig. 2. We used a doubled Nd:YAG laser with a wavelength of 532 nm (Coherent, Verdi V5). A polarizing beam splitter (PBS) diverted part of the beam into a standard optical tweezers, while the other part was introduced into a Mach-Zehnder interferometer. The output interference pattern was reduced with a telescope (lenses L3 and L4) and directed to the sample cell from below. An additional cylindrical lens (CL) was used to concentrate the interfering beams in the y direction, giving rise to dimensions of w_x = 232±2μm and w_y = 95±2μm at the sample plane, measured with the moving knife edge method. The spatial period of the interference fringes and the position of the pattern in the sample plane were controlled by tilting and displacing the mirrors M2 and M3 along the directions indicated in Fig. 2. We explored eleven values for the period in the range from L = 9μm to L = 40μm. On the other hand, for the optical tweezers we used a low NA microscope objective (×20, NA = 0.4), which allowed us to place the particle of interest at the same initial position each time while keeping a wide field of view. The tweezers beam was blocked during the travel of the particle across the interference pattern. Two CCD cameras were used in the experiment; in one of them the laser beam was blocked to record the path of the particle (CCD1), whereas the other one was used to monitor the interference pattern (CCD2).

 

Fig. 2. Experimental setup. Mirrors and lenses are represented by M and L, respectively; CL denotes a cylindrical lens and DM are dichroic mirrors. PBS are polarizing beam splitter cubes and NPBS is a non-polarizing beam splitter. Half-wave plates (λ/2) are used to control the laser intensity. A microscope objective (×20) is used to generate an optical tweezers and to observe the sample. The tweezers beam is blocked while the particle travels down the interference pattern with a shutter modulus (SM). CCD1 records the path of the particle, while CCD2 monitors the interference fringes.

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Our samples consisted of monodisperse borosilicate glass microspheres with diameters of 8±1μm, 10±1μm and 14.5±1μm, density of ρ_p = 2.5g/cm ³ and refractive index n = 1.56 (Duke Scientific). The particles were dispersed in deionized water and put into hermetically sealed cells (~1.2cm ² area, ~100μm depth) made with two cover slips, which were previously cleaned with a free rinsing surfactant (LiquiNox) and then with acetone. The values for the laser power at sample plane were 183, 281 and 521 mW for the samples of 8, 10 and 14.5 μm beads, respectively. The sample cell was placed in an XYZ translation stage, and it was inclined with a micro motorized stage. The inclination angle was fixed at ψ = 5.6° for spheres of 14.5 μm, and φ =6.7° for spheres of 8 and 10 μm. The particles are sufficiently large and dense to respond to the gravity force in the way described in Section 2. The experimental conditions naturally leave the particle of interest alone in the monitored region. The position of the sphere as a function of time was measured using standard digital video microscopy, using the method of centroid determination [38]. The imaging system yields a magnification of 0.86 μm/ pixel and a field of view of 401 × 534 μm ².

4. Results and Discussion

Figure 3(a) shows the motion tracking (position as a function of time) for spheres of 8μm, 10μm and 14.5μm of diameter, in all the cases the spatial period of the fringes was L = 20.2μm. For each particle size and each spatial period we did at least three repetitions of the experiment under the same conditions, but the results presented in Fig. 3(a) were obtained with eleven repetitions. It can be seen from the figure that, in the case of the largest sphere, there arise a delay of some of the trajectories with respect to others for t > 40s, which means that the particle remained slightly different times in each potential well for the different repetitions of the experiment. We believe this may be due to the interaction of the sphere with the bottom surface of the sample cell, whose effect is more noticeable for heavier and larger particles. In other words, the interaction between the particle and the surface is stronger for larger spheres. This might be a glass-glass type interaction, but may also involve other effects such as surface roughness at a microscopic scale. In addition, a change in the slope of the curves when t ~ 40s, which is also more evident for the largest sphere, is due to the gaussian envelope modulation of the pattern of fringes. The particle speeds up when going towards the region of maximum intensity and gets slower when moving outwards from that region. In Fig. 3(b) we show a close up of three paths (in color) for the 14.5μm diameter particle illustrating the noise in the data associated with the spatial resolution of the imaging system and tracking process. This noise arises because digitized images usually suffer from imperfections such as nonuniform contrast, geometric distorsion and the electronic noise inherent to the detection system (dark and leakage currents in the CCD) [38], giving rise to an uncertainty in the determination of the position of the particle.

As we have mentioned, our goal is to experimentally determine the optical force from the dynamics of the particle moving across the optical pattern. According to Eq. (1), this can be accomplished by obtaining the effective friction coefficient and the velocity as a function of position. Firstly, the effective friction coefficient was independently determined, as described in Section 2, from the relation γ = F_g/ẋ ₀. Here ẋ ₀ is the constant velocity of the particle in the absence of the light field and F_g can be directly calculated using the characteristics of the particle specified by the manufacturer (radios and density). Since we verified that the position of the particle in this case varies linearly with time, we obtained the constant velocity ẋ ₀ simply as the slope of the corresponding line. The values for the friction coefficients found in this way were γ = 0.146 pN · s/μm for the particle of 8/μm, γ = 0.175pN · s/μm for the particle of 10μm, and γ = 0359 pN · s/μm for the particle of 14.5μm.

On the other hand, the velocity of the particle as a function of position in the presence of the light field was determined using the same probe particles. The original data of position as a function of time (Fig. 3(a)) were smoothed using a Savitzky-Golay filter [39], which consists in performing a local polynomial regression; we used cubic polynomials. An example of the smoothed data is illustrated with the black curves in Fig. 3(b). The velocity is obtained as the derivative of these polynomials. Figs. 4(a) and 4(b) show the curves of the velocity as a function of position for particles of 10μm and 14.5μm diameters, respectively, corresponding to the data illustrated in Fig. 3; the curve in black is the average. For calculating the average, we joined the discrete points of each set of data for the independent experimental realizations by means of Piecewise Cubic Hermite interpolation, and then we averaged the resulting curves. Since the velocity as a function of position is a local property, the time delay in Fig. 3 does not affect the correlation among the velocity curves. The gaussian modulation of the interference pattern becomes evident from Figs. 4(a) and 4(b).

 

Fig. 3. (a) Experimental data: Curves of the position as a function of time for eleven repetitions of the experiment for each particle size: 8μm, 10μm and 14.5μm of diameter. In all cases, the spatial period of the interference pattern of fringes was L = 20.2μm and the recording rate was 30 frames/s. (b) Close up of three paths for the 14.5μm diameter particle illustrating the original experimental data (color curves) in comparison with the curves obtained after performing a smoothing process (in black).

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The results for the averaged velocity (left hand side scale) and the corresponding optical force (right hand side scale) are presented in Fig. 5 (red markers), for the same cases of Fig. 4. The blue curves represent the optical force calculated with the theoretical model described in Section 2. Figure 6 shows the comparison between experimental results (red markers) and the theoretical model (blue curves) for other spatial periods of the interference fringes and for the 10μm and 14.5μm particles (left and right columns, respectively). It is worth to point out that our experimental data comprises between two and three thousand experimental points over the whole range of the extended pattern we analyzed.

 

Fig. 4. Curves of velocity as a function of position for the same data of Fig. 3: (a) 2R ₀ = 10μm and (b) 2R ₀ = 14.5μm. Plot in black is the resulting average.

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Fig. 5. Experimental force profile (red circles) of a particle ((a) 2R ₀ = 10μm and (b) 2R ₀ = 14.5μm) running in a fringes pattern with spatial periodicity L = 20.2μm. Solid blue lines are the corresponding theoretical predictions according to Eqs. (2) and (3). Experimental results correspond to the average in black in Fig. 4.

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Each of the plots presented in Figs. 5 and 6 correspond to a mapping of the optical force over the extended light pattern, from which the respective optical potential can be obtained. In general, we found very good quantitative agreement between theory and experiment. It is worth to stress that we are not adjusting any parameter; we are simply using the quantities assumed to be known (particle radius and relative refractive index) and the experimentally determined values (beam spot size, spatial period of the fringes and effective friction coefficient) in our theoretical model. The best agreement was found, as expected, for the largest particle (14.5μm), in accord with the ray optics theoretical description. Differences of up to 10% can be accounted for from the uncertainty in the diameter of the particles provided by the manufacturer, which is also larger for the smaller particles. On the other hand, the comparison of the theoretical and experimental results from Fig. 6 is better for the larger periods, Figs. 6(e) and 6(f). Nevertheless, we believe that the main source of error in the case of small periods is rather associated with the spatial and time resolution of our image and tracking systems. Namely, the curves of position as a function of time have a stair-like structure associated with the periodic pattern of fringes (see Fig. 3(a)). The steps in the stair are better defined for the case of larger periods for a given particle size. The uncertainty in the position of the particle associated with the spatial resolution of the imaging system is considerably smaller than the scale of these steps for the case of large periods, but the difference between both magnitudes reduces in the case of smaller periods. In other words, the noise in the tracking and imaging system becomes more relevant for smaller periods. A higher frame rate and/or an improved spatial resolution would reduce the noise and make the tracking process more accurate, improving the results in consequence. This was also an issue in the case of the 8μm diameter sphere, which was more difficult to track, giving rise to a difference between the theoretical and experimental results (not shown) that exceeded 20%. Although in the case of smaller particles, of course, the geometrical optics approximation is also expected to deteriorate.

 

Fig. 6. Comparison of experimental force with the theoretical model for spheres of 10μm of diameter (left column) and 14.5μm of diameter (right column), for spatial periods of: (a) and (b) L = 13.25μm; (c) and (d) L = 17.16μm; (e) and (f) L = 25.21μm

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To finish this section, in Fig. 7 we show the theoretical predictions (lines) and experimental results (markers) for the dependence of the maximum optical force on the period of the fringes for the microspheres of 10μm (red) and 14.5μm (blue). The larger uncertainties for the smaller periods of the fringes are associated with the resolution of the tracking system, as mentioned before. As the power used in the experiment for the two cases was different, we rescaled the plots for comparative purposes, considering the same power in both cases (P = 508mW). It can be seen from the figure that, in the region explored in our experiments, the curves have a maximum. In practice, the maxima correspond to a range of periods rather than a single well defined value. This dependence of the force on the period has been previously observed, and it is the basis for some optical sorting devices [21]. For example, in the range 8.5μm < L < 11μm, the force is stronger for the 10μm-diameter particle, whereas for L > 11μm it occurs the opposite. As illustrated from Fig. 7, the method we propose here can be used for calibration of optical sorting devices.

 

Fig. 7. Comparison between the theoretical predictions (lines) and experimental results (markers) for the maximum optical force as a function of the period of the interference fringes for the particles of 10μm diameter (red) and 14.5μm diameter (blue).

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5. Conclusions

We have performed an experiment to fully map the optical trapping force as a function of the position exerted by an extended light pattern of interference fringes on glass microspheres. In contrast with other studies, in our experiment the sample cell was literally tilted by a small angle to generate a washboard potential that combines the constant force of gravity and the optical force of the interference fringes. We put a glass microsphere on the washboard potential and directly measured its position as a function of time as it moved down the inclined plane. With this information, we determined the velocity as a function of the position. Since we are in a deterministic and overdamped regime, this spatially dependent velocity is proportional to the optical force. Therefore, we were able to determine the optical force as a function of the position of the microsphere on the extended interference landscape. On the other hand, we calculated the optical force using a ray-optics model and found very good quantitative agreement between theory and experiment.

The basic assumptions are that the ray-optics model is valid and that the dynamics occurs in a deterministic and overdamped regime. These assumptions are justified, since we are using microspheres of diameters on the order of tens of microns, which are much larger than the wavelength of the laser light in the host medium. Due to this size, thermal fluctuations are very small, and we experimentally verified that the inertial forces are negligible in comparison with dissipative forces.

Specifically, we used three sizes of microspheres: 8, 10 and 14.5 μm of diameter. All the quantities involved in our model, such as the effective friction coefficient and the gravitational force along the direction of motion, were directly determined from the experiment. We repeated the experiment for more than ten different values of the spatial period of the interference fringes. We obtained curves for the optical force as a function of position covering distances of up to 400μm and comprising between two and three thousand points for each of the cases we analyzed. This allowed us to establish a neat comparison between experiment and theory, in which there are no free parameters.

The dependence of the maximum force on the period of the fringes was also investigated. It was verified that there is an optimum force for a given range of the period of the fringes. In this sense, the method presented here can be used for practical calibration of optical sorting devices in the case of particles in the Mie regime. This is important because much of the optically manipulated biological material, like some cells, lie in the size regime of applicability of the technique proposed here. In addition, our method can be used for mapping arbitrary one-dimensional optical potentials.

To our knowledge, this is the first instance in which a non-linear optical force is fully mapped as a function of the position over extendend distances using very simple deterministic dynamics.