It is generally accepted that the interaction between particles mediated by the scattered light (called optical binding) is very weak. Therefore, the optical binding is usually neglected in a multi-particle trapping in distinct optical traps. Here we show that even the presence of only two dielectric particles confined in the standing wave leads to their significantly different behavior comparing to the case of a single trapped particle. We obtained persuading coincidence between our experimental records and the results of the deterministic and stochastic theoretical simulations based on the coupled dipole method.
©2011 Optical Society of America
A single optical trap usually exists in the region of high optical intensity surrounded by strong intensity gradients. Optical tweezers represent such well-known example . Optical trapping of more particles can be reached either by the scanning of one or more beams over the desired positions of several optical traps (so called time-sharing trap) [2, 3] or by using many optical traps existing in parallel and usually represented by holographic optical tweezers , interferometric tweeezers [5–7], or a standing wave trapping [8–11]. Obviously the phenomenon of the inter-particle interaction mediated by light exists only in the case of the multiple parallel trapping. Surprisingly, this type of the inter-particle interaction has been observed shortly after the optical tweezers announcement  because it led to the self-arrangement of microparticles along the interference fringe. This was probably the reason why this interaction is now frequently called the optical binding and the particles self-arranged this way as the optically bound matter . This intriguing self-arrangement is based upon delicate equilibrium between the optical forces resulting both from the incident beam and from the light re-scattered by the objects which then results in well defined stable equilibrium positions for the particles concerned. Such a self-arrangement is not initiated by predefined optical traps existing before the objects are placed to the field. The mere presence of the particles in the field scatters the light and thus modifies the field spatial distribution. Under certain circumstances it leads to creation of stable configurations of interacting particles even in the beams where no optical traps exist in advance. For about a decade this phenomenon has not been studied deeply but recently several interesting observations attracted new attention to this topic. One of them was a self-arrangement of particles along the propagation axes of two incoherent counter-propagating beams (also called longitudinal optical binding) [14, 15]. Laterally the objects are confined by the mechanism of optical trapping in the high intensity core of the beam. The longitudinal binding forms optically bound matter in three dimensions far from any surface and has the potential for long-range particle arrangement [16,17] with an interesting dynamic behavior [18–21]. The second observation concerned the formation of self-arranged two dimensional structures near the surface illuminated by two counter-propagating evanescent waves [22, 23], wide Gaussian beams [24, 25], or near the optical fibre tips [26, 27]. The overview of related activities is summarized in two recent review papers [28, 29].
In this paper we focus on a longitudinal optical binding occurring along the propagation of two counter-propagating interfering zero-order Bessel beams [9, 11, 30, 31]. We choose Bessel beams because their lateral intensity profile does not change along the beam propagation and, therefore, the axial position of the particle does not modify the axial optical force acting upon the particle. However, in contrast to the previously considered configurations of longitudinal optical binding [16, 32, 33] we use coherent laser beams and thus a standing wave exists in the case of parallel beam polarization. Behavior of a single particle illuminated by Gaussian, Bessel, or evanescent standing wave is well understood and described in literature [9, 34–41].
It was shown, that a single particle of certain size [34, 35] can be strongly trapped axially while particles of other sizes can move freely along the standing wave . Based on these results one would expect that the addition of another particle to the standing wave should not significantly modify the trapping conditions of individual particles. However, theoretical simulations of T. Grzegorczyk  demonstrated that the placement of 20 cylindrical particles into three-plane-wave interference pattern leads to rearrangement of the particles to new equilibrium positions that differed from the original positions of the standing wave intensity maximum without particles. Up to our best knowledge no quantitative experimental and theoretical comparison of optical binding in interfering waves has been presented yet.
2. Basics of particles behavior in the standing wave
We will focus on the behavior of two polystyrene particles (1070 nm in diameter) optically trapped in two counter-propagating linearly polarized Bessel beams (BBs) with parallel or perpendicular polarization. The BB core radii of both beams were set to 2.15 μm. Let us start with the simplest case of a single particle trapped in the standing wave (parallel polarization), see Fig. 1. If the particle is of higher refractive index than the liquid, the particle is trapped on the optical axis. Since in the rest of the paper we will focus only on the axial (one-dimensional) behavior of the particle, we will consider only the axial forces transporting the particles along or against z axis. In such one-dimensional case this force F(z) is conservative and allows us to (use the term optical potential , where z 0 is any starting axial position.
2.1. Single particle in the standing wave
If both identical counter-propagating beams are of the same optical intensity (see Fig. 1a), the axial radiation pressures acting by both beams on the single particle compensate each other. Therefore, the final optical (gradient) force comes from the periodic (sinusoidal) interference fringes of the standing wave and the related optical potential U is sinusoidal, too. Consequently, the particle is trapped axially due to the gradient force in the standing wave node or antinode . Since we consider optical trapping in liquid, the particle is influenced by the thermal movement of the surrounding molecules (Brownian motion). It leads to particle jumps between the neighboring stable positions (corresponding to optical traps or minimum of the U) [41, 43, 44]. In the considered case the optical trap is axially symmetric and the particle jumps with the same probability in both axial directions.
However, if the intensities of both counter-propagating beams are unbalanced, e.g. the intensity of the beam propagating along the positive direction of z axis is higher, there exists an extra axial force pushing the particle in the positive direction of z axis. This force tilts the original sinusoidal optical potential U and, consequently, the work needed to push the particle to the neighboring optical trap along the positive direction of z axis is lower comparing to the particle transport in opposite direction (see Fig. 1b). Consequently, the probability of the particle to jump along the positive direction of z axis is higher comparing to the opposite direction and we may observe macroscopic particle current in one direction [40, 45].
2.2. Two particles arranged axially in the standing wave
Let us consider now more complex case of two polystyrene particles optically trapped in counter-propagating beams where both beams are of equal optical intensity. Comparing it to the previous case of the single optically trapped particle, the system of two particles is disturbed by the inter-particle interactions (longitudinal optical binding) which is caused by the field re-scattered by the particles. Up to now the longitudinal optical binding was studied solely in the non-interfering counter-propagating beams because it was not expected that such subtle inter-particle interaction can significantly modify trapping conditions of particles in the standing wave.
Previous studies [9, 36] focused on the behavior of a single particle trapped in the standing wave have shown that the particle size determines the extremal axial force acting on the particle placed in the standing wave, axial trap stiffness, and minimal trapping power. These quantities differ significantly for optical trapping in a standing wave and a single beam.
Proper combination of relevant parameters such as particle size, beam intensity, and temperature of the surrounding liquid medium can even lead to an observation of the optical binding of particles in the standing wave, where particles have to overcome the barriers between neighboring optical traps (i.e. interfering fringes) to form the optically bound structure. In the case of non-interfering beams the binding force and potential depend only on the inter-particle separation z′ (see Fig. 2a). On the other hand in the case of particles trapped in the standing wave we have to consider the position of the left particle z L as well. For better insight into the problem of two particles we assume general forces F L(z L, z′) and F R(z L, z′) acting on the left and the right particle, respectively, depending on the left particle position z L and the inter-particle separation z′. Further we define the binding force as F bind(z L, z′) = F R(z L, z′) – F L(z L, z′) and the optical potential
In Fig. 2b we present binding force and potential for three slightly different positions z L within one chosen interference fringe. The forces F L(z L, z′) and F R(z L, z′) acting on the particles are stronger about an order of magnitude due to the presence of the standing wave. The standing wave strongly localizes the particles in the individual optical traps (interference fringes). However, the presence of the second particle changes the proportion between the optical powers incident on the particle from the left and the right laser beams. This situation is similar to the case of a single particle placed in two counter-propagating beams of different optical intensities (described in Fig. 1b) where the particle moves in the periodical but slightly tilted optical potential. Due to the thermal activation the particle can easily jump over the lower potential barrier between neighboring optical traps until it reaches the axial region where this tilt disappears. This corresponds to the global minimum of the optical potential which is placed almost at the same axial region as in the case of non-interfering counter-propagating beams. The corresponding inter-particle distance will be constant in time at the scale of several laser wavelengths but, of course, the stochastic jumps will occur to the closes axial standing wave traps in both axial direction. This stationary inter-particle distance will be the subject of the studies in the following sections from the theoretical and experimental point of view. We will also address the transition of the particles to this state.
3. Experimental setup and observations
Two counter-propagating BBs were generated using a dual-beam configuration that allowed the dynamic modification of the beams parameters . This system employs a diffractive optical element that is imprinted on a single spatial light modulator (SLM) and dynamically addressed from a computer interface (see Fig. 3). The system is based on the standard Fourier holographic encoding and spatial filtering of the beams in both counter-propagating arms and enhanced by a recently developed in situ wave-front optimization method  eliminating aberrations introduced in the optical pathways. A half-wave plate was inserted into one of the arms to control the polarization of the beam and, thereby, to switch between the cases of interfering or non-interfering counter-propagating beams.
The counter-propagating BBs were focused into a square capillary with the 100 μm inner cross-section filled with a water suspension of polystyrene spheres of diameter equal to 1070 nm. We used calibrated CCD camera and measured the lateral intensity profile of both BBs in the air. The radii of both BBs cores were ρ 0 = (2.1 ± 0.1)μm and the intensity in the BB core was equal to (0.44± 0.05) mW/μm2.
Our experimental activities were focused on the investigation of the behavior of an optically bound structure of two particles in the cases of interfering and non-interfering counter-propagating beams, respectively. Particle positions were determined from bright field images recorded by fast CCD camera (IDT XS3). The individual particles positions were obtained from each frame by a correlation algorithm  and inter-particle distances were analyzed.
Figure 4a presents formation of an optically bound structure composed from two polystyrene particles trapped in the standing wave. The experiment started with the trapping of a single particle in the standing wave. During several seconds this particle jumps repeatedly between neighboring interference fringes in both axial directions due to the symmetry of the optical potential (see the red record in Fig. 4a). The second particle entered the beams at a distance 26 μm apart from to the first particle. Consequently, the optical potential was modified according to Fig. 2. Both particles started to jump in this optical potential overcoming the interference fringes of the standing wave and after about 50 s they reached the stationary optically bound structure. The inter-particle separation in this configuration was about 10 μm. This behavior and inter-particle separation fit well with the description and preliminary results presented in the previous section.
Figure 4b presents three different observations of two particles approaching their stationary inter-particle separation in the optically bound structure. The red and black dependences demonstrate how the particles approach each other in series of jumps across individual standing wave traps, the blue dependence illustrates the opposite process where the particles are repelled from each other. In all cases the stationary inter-particle distances are close to each other. These processes were fitted by the exponential decay functions
4.1. Computer modeling of inter-particle distances in optically bound structures
We started our computer modeling by calculations of the binding forces F bind between two polystyrene particles placed on the axis of the interfering and non-interfering counter-propagating BBs, respectively. The axial optical intensity and the BB core radius correspond to the experimental values, i.e., I = 0.44 mW/μm2 and ρ 0 = 2.15μm. The binding forces were calculated employing our numerical model based upon a coupled dipole method CDM [16,33,48].
As we showed in the Section 2 the optical potential can reveal directly the possible stable configuration of the optically bound structure, therefore, we present here only the optical potential U instead of the binding force. Figure 5 summarizes the calculated results for various particle sizes and interfering/non-interfering beams. We present here the optical potentials together with probability densities of distribution of the inter-particle separation which gives us a simple visualization of particles occurrence near the stable configuration.
In the calculations we fixed the position of one particle (the left one) while the position of the other determined the inter-particle separation. The dashed blue curves in Fig. 5 denote the calculated optical potentials and probability densities in the case of non-interfering counter-propagating beams corresponding to the previous studies . The red (resp. green) curves correspond to the left particle placed at the intensity maximum (resp. minimum) of the interference fringe of the standing wave and they illustrate that the position of this fixed particle significantly affects the profile of the optical potential and consequently particles behavior. It is seen, especially from the probability density results, that the fixed position of the second particle with respect to the intensity maximum significantly influences the final inter-particle separation in the optically bound structure. Moreover, the larger particles are localized closer to each other in the stable optically bound structure.
The fifth row in Fig. 5 demonstrates the situation when the diameter of both particles is properly selected (1150 nm in our case) so that the motion of such particle is not influences by the standing wave and the particles behave like trapped in the non-interfering laser beams . Indeed all the curves are very close to each other here.
We conclude that the coupled-dipole method provides results that are comparable to the experimentally observed particles behavior in the deterministic regime where the Brownian motion is ignored.
4.2. Particles dynamics towards the stable optically bound structure: I
The method presented in the previous section ignored the stochastic nature of the process when the particles jump over the potential barrier between neighboring optical traps. Therefore, it can not be used to study the dynamics of the optically bound structure formation. In this section we will consider that one particle is fixed and the other moves in the tilted periodic optical potential U(z L, z′) that consists of several neighboring local minimum - axial optical traps (see Fig. 6). Since the movement of this Brownian particle is random by its nature, the particle may jump between neighboring traps. The time the particle spends in each trap is random quantity as well, however, its average value can be quantified. Similar problem of particle escape over the potential barrier was studied in 1940 by H.A. Kramers . However, the Kramers’ approach represents the approximation for the potential barrier higher than kBT, where kB is the Boltzmann constant and T the thermodynamic temperature. Since we consider potential minimum close to each other that may be shallow as well, more precise theoretical must be used in our case [41, 43].
So called Mean First Passage Time (MFPT)  provides an exact value of the average time that a particle spends in a certain part of space. Let us limit the particle motion to a single optical trap in the region a ≤ z ≤ b. Such particle leaves this regino after the average time T given by the MFPT. We may distinguish three different ways of the escape:
Further, let us consider the situation that the left particle is fixed at the center of bright interference fringe and the right hand side particle moves freely in a potential energy profile that is created by the presence of the standing wave which is, moreover, modified by the mutual optical binding interaction. The potential energy profile of such interaction corresponds to the red curve on the third row of Fig. 5, i.e. the optical potential calculated using our CDM model for polystyrene patricles of diameter 1070 nm while the left particle is located at the intensity maximum.
Consider first that this particle jumps from the interference fringe only to the neighboring trap on the left. The MFPT is then given by Eq. (3) and its quantification for inter-particle separations between 5 to 15 μm is presented by the blue curve in Fig. 7a. In analogous way we quantify the MFPT for the case where the particle jumps only to the neighboring optical trap on the right. The MFPT is given by Eq. (4) and the results is denoted by the green curve in the same figure. From this simple model of the particle behavior we conclude that a free particle which moves in the standing wave modified by the presence of another particle jumps over the array of interference fringes to the global minimum z min of the optical potential. In another words, the particles form an optically bound structure despite the fact that they are trapped for a while in the standing wave.
The MFPT, which particle needs to move from various separations to this state, can be calculated employing combination of both Eqs. (3) and (4) if the particle position meets the condition z > z min and z < z min, respectively. The result of this simulation is presented in Fig. 7b by the blue curve. These results reveal that the average MFPT is much longer than the values we observed experimentally in Fig. 4b.
However, our approach above considered room temperature of the surrounding liquid and did not take into account the water heating due to the absorption of laser beam power. The imaginary part of the water refractive index is relatively low, in the order of 10−6, however the energy absorbed in water for the typical laser powers used in optical trapping can locally increase the water temperature on the order of tens of Kelvins . Such heating would increase the probability that the particle overcomes the potential barrier and at the same time it decreases the viscosity of water.
We have calculated the average time needed for the particle to reach the global potential minimum also for higher temperatures of water reaching 30, 40 and 50°C. The results are shown by green, red and cyan colors in Fig. 7b, while the room temperature result is blue. One can clearly see the strong decrease of the average time to reach the global potential minimum. However, this time is still longer than the experimentally observed values and, therefore, we will apply a computer model employing Monte Carlo method to describe the behavior of both stochastically moving particles.
4.3. Particles dynamics towards the stable optically bound structure: II
The principle difference between the model presented in the previous section and the experiment is that both particles moves stochastically. Therefore, the potential profile changes rapidly in time depending not only on the inter-particle distance but also on the position of the left particle in the standing wave formed by the incident beams. Therefore, one cannot use the probability density function based on the Boltzmann distribution for the comparison between the calculated potential energy (force) profiles and the experimental data. Therefore, here we consider a pair of particles (coordinates z L and z R) moving along the BB’s core and the stochasticity of the their motion is modeled using the Monte Carlo Simulation (MCS). We assume that such particles moves in the 1D force field F that comprehends again both the presence of the standing wave as well as the mutual optical binding force between the particles, calculated by the coupled dipole method. We used the over-damped Langevin equation to study the particles dynamics:43] that describes the time evolution of the probability density function.
The MCS evolved in times steps of 50 μs and we considered the motion of 104 particles at once. The particles initial positions were distributed randomly using the uniform distribution function in the interval 19 μm ≤ z(0) ≤ 19.5 μm. The left particle was always kept confined within the extent of the single standing wave fringe (optical trap) of the length L. If such particle was about to leave the considered volume, it was translated to the other edge of the trap (because of the trap symmetry) and the right particle was moved in the appropriate direction over distance L in order to keep the particle separation constant. This transformation does not influence the particle dynamics because we assume particles motion in BBs that have non-varying axial properties. During the simulation the histogram of the particle separations was assembled until the steady state was reached.
Figure 8a shows the simulated probability density at different times. We can see that the “equilibrium” state obtained at t = 480 s is rather similar to the red curve in the right column of Fig. 5 for corresponding particle diameter (1070 nm). However, the probability density shown in Fig. 8a consists of slightly wider peaks due to the fact that both particles are in motion. We have also calculated the average particle separations at each time step. These are shown in Fig. 8b. We parametrize these curves by Eq. (2) and we obtained the characteristic decay time τ = 114 s. Obviously, the motion of both particles speeds the formation of the optically bound structure at least 4.8 times comparing to the theoretical description based on the MFPT (decay time to reach the equilibrium was τ = 550 s).
In order to study the dependence of particles behavior on the total laser power in both BBs we performed a new set of MCS. We constructed the time evolutions of the mean particle distances similar to the one shown in Fig. 8. We parametrized these curves again by Eq. (2) and we obtained the characteristic decay times τ, plotted in Fig. 9. It is clearly seen that such time increases exponentially as the laser optical intensity increases. Furthermore, if the temperature of the surrounding water increases from initial 293 K upto 313 K, it causes significant decrease of the characteristic time τ. This trend is partly given by the decrease of the water viscosity decrease and partly by the stronger thermal activation coming from the Brownian motion.
The experimentally expected value of the optical intensity corresponds to (0.44 ± 0.05) mW/μm2 and the average time τ obtained from Fig. 4 is 24.5 s. The results presented in Fig. 9 by the red curve are still about three times longer. However, considering the complexity of this process we can conclude that the observed dynamics of the optically bound structure formation can be adequately modeled by a combination of the coupled dipoles model with the Monte Carlo simulation of the particles stochastic motion.
Finally, we present the comparison of experimental and theoretical probability densities of inter-particle separation distributions in Fig. 10. During the experiment a pair of particles was observed and its mutual separation was evaluated. Once the equilibrium distance was approximately reached, the histogram of particles separation was assembled from about 105 data points. Such measurements were performed both in the case of non-interfering BBs (Fig. 10a) and interfering BBs (Fig. 10b). The results of MCS are shown by the blue region and they coincide very well with the experimental data.
We dealt with an optically induced longitudinal self-arrangement of two polystyrene particles placed into two counter-propagating linearly polarized and interfering BBs forming an axial standing wave. The BBs ensured negligible variations of the axial intensity profile of both incident beams and, therefore, only the axial placement of a pair of particles with respect to the formed standing wave must be considered. We observed experimentally that the particles formed a stable optically bound structure in axial direction with inter-particle distances similar to the case of two counter-propagating non-interfering beams. We offered simplified physical picture of the process and compared the observed inter-particle distances in such a structure with the theoretical model based on the coupled-dipoles with very persuading coincidence. We also focused on the dynamics of the process and we studdied the average time needed to form such an optically bound structure if the particles are placed further from each other. We included the particles jumps over the potential barries between individual optical traps formed by the standing wave and also the stochastic motion of both particles. Taking the optical forces from the coupled dipole model and inluding the Stochastic motion by the Langevin equations and Monte Carlo simulation we studied the influence of the particles sizes, temperature of the surrounding water, and the optical intensity of incident beams on the average time to settle the stable optically bound structure. The found results encourrage to conclude that the developed model can also describe the dynamics of the formation of such structures and estimate the order of the average time needed to form the stable optically bound structure.
The authors acknowledge the support from Czech Science Foundation ( 202/09/0348; P205/11/P294), Institutional Research Plan of the Institute of Scientific Instruments of the ASCR, v.v.i. ( AV0Z20650511), Ministry of Education, Youth and Sports of the Czech Republic ( LC06007) together with the European Commission (ALISI No. CZ.1.05/2.1.00/01.0017).
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