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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 20 — Sep. 26, 2011
  • pp: 19613–19626
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Static and dynamic behavior of two optically bound microparticles in a standing wave

O. Brzobohatý, V. Karásek, M. Šiler, J. Trojek, and P. Zemánek  »View Author Affiliations


Optics Express, Vol. 19, Issue 20, pp. 19613-19626 (2011)
http://dx.doi.org/10.1364/OE.19.019613


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Abstract

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 OSA

1. Introduction

In this paper we focus on a longitudinal optical binding occurring along the propagation of two counter-propagating interfering zero-order Bessel beams [9

9. T. Čižmár, M. Šiler, and P. Zemánek, “An optical nanotrap array movable over a milimetre range,” Appl. Phys. B 84, 197–203 (2006). [CrossRef]

, 11

11. T. Čižmár, O. Brzobohatý, K. Dholakia, and P. Zemánek, “The holographic optical micro-manipulation system based on counter-propagating beams,” Laser Phys. Lett. 8, 50–56 (2011). [CrossRef]

, 30

30. O. Brzobohatý, T. Čižmár, and P. Zemánek, “High quality quasi-Bessel beam generated by round-tip axicon,” Opt. Express 16, 12688–12700 (2008). [PubMed]

, 31

31. D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46, 15–28 (2005). [CrossRef]

]. 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

16. V. Karásek, T. Čižmár, O. Brzobohatý, P. Zemánek, V. Garcés-Chávez, and K. Dholakia, “Long-range one-dimensional longitudinal optical binding,” Phys. Rev. Lett. 101, 143601 (2008). [CrossRef] [PubMed]

, 32

32. V. Karásek and P. Zemánek, “Analytical description of longitudinal optical binding of two spherical nanoparticles,” J. Opt. A: Pure Appl. Opt. 9, S215–S220 (2007). [CrossRef]

, 33

33. V. Karásek, O. Brzobohatý, and P. Zemánek, “Longitudinal optical binding of several spherical particles studied by the coupled dipole method,” J. Opt. A: Pure Appl. Opt. 11, 034009 (2009). [CrossRef]

] 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

9. T. Čižmár, M. Šiler, and P. Zemánek, “An optical nanotrap array movable over a milimetre range,” Appl. Phys. B 84, 197–203 (2006). [CrossRef]

, 34

34. P. Zemánek, A. Jonáš, L. Šrámek, and M. Liška, “Optical trapping of Rayleigh particles using a Gaussian standing wave,” Opt. Commun. 151, 273–285 (1998). [CrossRef]

41

41. M. Šiler and P. Zemánek, “Particle jumps between optical traps in a one-dimensional optical lattice,” New. J. Phys. 12, 083001 (2010). [CrossRef]

].

It was shown, that a single particle of certain size [34

34. P. Zemánek, A. Jonáš, L. Šrámek, and M. Liška, “Optical trapping of Rayleigh particles using a Gaussian standing wave,” Opt. Commun. 151, 273–285 (1998). [CrossRef]

, 35

35. P. Zemánek, A. Jonáš, and M. Liška, “Simplified description of optical forces acting on a nanoparticle in the Gaussian standing wave,” J. Opt. Soc. Am. A 19, 1025–1034 (2002). [CrossRef]

] can be strongly trapped axially while particles of other sizes can move freely along the standing wave [37

37. T. Čižmár, V. Garcés-Chávez, K. Dholakia, and P. Zemánek, “Optical conveyor belt for delivery of submicron objects,” Appl. Phys. Lett. 86, 174101 (2005). [CrossRef]

]. 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 [42

42. T. M. Grzegorczyk, B. A. Kemp, and J. A. Kong, “Trapping and binding of an arbitrary number of cylindrical particles in an in-plane electromagnetic field,” J. Opt. Soc. Am. A 23, 2324–2330 (2006). [CrossRef]

] 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 U(z)=z0zF(z)dz, where z 0 is any starting axial position.

Fig. 1 Single particle optically trapped in two counter-propagating interfering Bessel beams. (a) Calculated force F(z) acting on the particle and optical potential U(z) for the symmetrical case (both laser beams have identical optical intensities 1.82 mW/μm2 on the beam axis). (b) Calculated force F(z) acting on the particle and optical potential U(z) for the asymmetrical case. In this case the optical intensity of the beam propagating along positive direction of z is higher by 5% relatively to the counter-propagating beam. The following parameters were used in the calculations: the Bessel beam core radii were equal to 2.15 μm, laser wavelength in vacuum 1064 nm, refractive index of the polystyrene particle 1.59 and water 1.32.

2.1. Single particle in the standing wave

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

40. M. Šiler, T. Čižmár, A. Jonáš, and P. Zemánek, “Surface delivery of a single nanoparticle under moving evanescent standing-wave illumination,” New J. Phys. 10, 113010 (2008). [CrossRef]

, 45

45. J. E. de Oliveira Rodrigues and R. Dickman, “Asymmetric exclusion process in a system of interacting Brownian particles,” Phys. Rev. E 81, 061108 (2010). [CrossRef]

].

2.2. Two particles arranged axially in the standing wave

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
U(zL,z)=0zFbind(zL,z)dz+U0.
(1)
The binding force between two particles keeps the convention that negative (positive) binding force pushes the particles closer to (further from) each other. We choose the potential offset U 0 in such a way that the global potential minimum has a zero value. Consequently, two studied particles are stably bound for a separation where U(z L, z′) takes its minimum, which is close to z′ = 10μm in our case.

Fig. 2 Two particles optically trapped in two counter-propagating interfering/non-interfering Bessel beams, respectively. (a) Behavior of two particles trapped in non-interfering beams depending on theirs separation. In this case the particles are stably optically bound at the mutual distance close to 10 μm. (b) Particles behavior in interfering beams. All dependencies are calculated for three slightly different positions zL (−121 nm (blue), 0 nm (green), 121 nm (red)) of the left-hand particle with respect to an intensity maximum of the standing wave. In the second row the calculated force acting on each particle (F L(z L, z′) and F R(z L, z′)) is shown. The third and fourth rows show the binding force (F bind(z L, z′) = F R(z L, z′) – F L(z L, z′)) and the optical potential U(z L, z′) (see Eq. (1)) related to the force F bind.The input parameters for the calculation were the same as in Fig. 1.

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 [11

11. T. Čižmár, O. Brzobohatý, K. Dholakia, and P. Zemánek, “The holographic optical micro-manipulation system based on counter-propagating beams,” Laser Phys. Lett. 8, 50–56 (2011). [CrossRef]

]. 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 [46

46. T. Čižmár, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photonics 4, 388–394 (2010). [CrossRef]

] 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.

Fig. 3 A collimated Gaussian beam coming from the fiber laser IPG ILM-10-1070-LP (wavelength 1064 nm) is expanded on the telescope made of lenses L1 (f 1 = 150 mm) and L2 (f 2 = 300 mm) and projected on the SLM (Hamamatsu LCOS X10468-07). Encoded phase at the SLM produces two BBs in the focal plane of the lens L3 (f 3 = 400 mm) above the undiffracted zero-order beam. Unwanted higher diffraction orders and the zero-order are blocked by a dual aperture spatial filter placed into a focal plane of L3. The passed beams are separated by prisms P1 and P2, sent in opposite directions and collimated by lenses L4 and L7 (both with f 4 = 200 mm), respectively. Each of the lenses L4 and L7 forms a telescope with L3 projecting the SLM plane on mirrors M2 and M4, respectively. The SLM plane is imaged onto the back focal plane of aspherical lenses AS1 (AS2) (both f = 8 mm) by a telescope consisting of lenses L5 (L8) (f 5 = 100 mm) and L6 (L9) (f 6 = 150 mm). AS1 and AS2 focuses both beams into a capillary (Vitrocells 8510) containing the sample (SC). A half-wave plate is 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 (not shown). To reach sub-micron alignment precision and stability of the system we omitted translational stages and properly placed the mirrors M2 and M4 to ensure lateral positioning of the focal points along the sample plane and the mirrors M3 and M5 to center the beams at the back aperture of the aspherical lenses. The right-hand side inset shows the spatial BB optical intensity between AS1 and AS2.

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 [47

47. M. K. Cheezum, W. F. Walker, and W. H. Guilford, “Quantitative comparison of algorithms for tracking single fluorescent particles,” Biophys. J. 81, 2378–2388 (2001). [CrossRef] [PubMed]

] and inter-particle distances were analyzed.

Fig. 4 (a) Time record of the formation of an optically bound structure in two counter-propagating interfering BBs. A single isolated polystyrene particle of diameter 1070 nm is stably trapped in the interfering structure and afterwards (30 sec) the second polystyrene particle of the same size is trapped at the distance of about 25 μm apart from the first particle. Both particles start to move over the interfering structure to their new equilibrium positions and after about 50 s they are stably bound. (b) Experimental investigation of the dynamics of two optically bound particles forming the stable optically bound structure in two counter-propagating interfering BBs. Three different observations of two particles approaching their stationary inter-particle separation in the optically bound structure are presented. These experimental data were fitted using an exponential decay function (see Eq. (2)) where τ is the time constant of the structure formation.

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
f(t)=C1exp(t/τ)+C2,
(2)
where τ is the time constant of the structure formation. We have found that τ was the same (within the error) for attracting particles. Such an attraction of particles was about two times faster comparing to formation of the stable optical bound structure from repelling particles.

4. Discussion

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

16. V. Karásek, T. Čižmár, O. Brzobohatý, P. Zemánek, V. Garcés-Chávez, and K. Dholakia, “Long-range one-dimensional longitudinal optical binding,” Phys. Rev. Lett. 101, 143601 (2008). [CrossRef] [PubMed]

,33

33. V. Karásek, O. Brzobohatý, and P. Zemánek, “Longitudinal optical binding of several spherical particles studied by the coupled dipole method,” J. Opt. A: Pure Appl. Opt. 11, 034009 (2009). [CrossRef]

,48

48. V. Karásek, K. Dholakia, and P. Zemánek, “Analysis of optical binding in one dimension,” Appl. Phys. B 84, 149–156 (2006). [CrossRef]

].

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.

Fig. 5 Comparison of the optical potential (separation work) and theoretical probability density of two identical polystyrene particles placed on the optical axis at different inter-particle distances if particles of different diameters are considered. The red (resp. green) curves correspond to the left particle placed at the intensity maximum (resp. minimum). The blue curves show the results for non-interfering beams and lay in between the cases mentioned above. The radius of the BB used in the simulations was ρ 0 = 2.15nm. The motion of 1150 nm sized particles is minimally influenced by the standing wave and therefore its behavior is nearly identical to the case of two non-interfering beams.

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 [9

9. T. Čižmár, M. Šiler, and P. Zemánek, “An optical nanotrap array movable over a milimetre range,” Appl. Phys. B 84, 197–203 (2006). [CrossRef]

]. 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

Fig. 6 A particle moving in a tilted periodic potential having several local energy minima - optical traps (black line). The particle located at the point z can reach the left trap edge (a) after the average time T while it is reflected back into the trap on its right edge (b). Similarly, T + is the average time needed to leave the trap over the right edge (b) while the particle is reflected back on the left edge (a). The average time needed to leave the well over any of its edges is denoted as T ±. Note, that the trap edges may be moved into any other points (e.g. a′ or b′).

So called Mean First Passage Time (MFPT) [43

43. C. W. Gardiner, Handbook of Stochastic Methods (Springer-Verlag, 2004).

] 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 azb. Such particle leaves this regino after the average time T given by the MFPT. We may distinguish three different ways of the escape:
  1. The particle can leave only through the boundary a and it is reflected back when it reaches the boundary b (the blue arrow in Fig. 6). In this case the MFPT is [43

    43. C. W. Gardiner, Handbook of Stochastic Methods (Springer-Verlag, 2004).

    ]
    T(z)=γkBTazdxψ(x)xbdxψ(x),whereψ(z)=exp[U(z)kBT]
    (3)
    and z is the initial position of particle at time t = 0 and γ is the Stokes drag coefficient.
  2. The particle can leave only through the boundary b (the green arrow in Fig. 6) and it is reflected back when it reaches the boundary a. The MFPT is done as
    T+(z)=γkBTzbdxψ(x)axdxψ(x).
    (4)
  3. The particle can leave either through a or b (the red arrows in Fig. 6). In this case the MFPT is
    T±(z)=γkBTabdyψ(y)[(azdyψ(y))zbdxψ(x)axdxψ(x)(zbdyψ(y))azdxψ(x)axdxψ(x)].
    (5)

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.

Fig. 7 (a) The average time that the right particle spends in a certain local potential energy minimum before it jumps to the neighboring potential minimum. We assume that the left particle is fixed in the intensity maximum of the standing wave and the dependence on the inter-particle separation is shown. We consider, that the particle starts its motion in one of the minimum of the potential energy profile shown by the green curve on the third line of Fig. 5. The blue, green, and red curves consider particles jumps over the local potential maximum located to the left, right, and both boundaries, respectively. (b) The average time required by the particle to reach the global minimum of the potential energy profile for various temperatures of water.

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 [49

49. Y. Seol, A. E. Carpenter, and T. T. Perkins, “Gold nanoparticles: enhanced optical trapping and sensitivity coupled with significant heating,” Opt. Lett. 31, 2429–2431 (2006). [CrossRef] [PubMed]

]. 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 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).

Fig. 8 (a) Probability density of the pair of polystyrene particles of diameter 1070 nm being separated by a given distance at different times. (b)The time evolution of the mean inter-particle separation.

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.

Fig. 9 The influence of the laser beam optical intensity on the characteristic time τ needed to settle the particles into the stable optically bound structure.

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.

Fig. 10 The comparison of experimentally obtained distributions of particles separations with results of the Monte-Carlo simulations. The radius of BB used in the simulations was ρ 0 = 2.150nm.

5. Conclusion

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.

Acknowledgments

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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A. E. Chiou, W. Wang, G. J. Sonek, J. Hong, and M. W. Berns, “Interferometric optical tweezers,” Opt. Commun. 133, 7–10 (1997). [CrossRef]

6.

A. Casaburi, G. Pesce, P. Zemánek, and A. Sasso, “Two-and three-beam interferometric optical tweezers,” Opt. Commun. 251, 393–404 (2005). [CrossRef]

7.

E. Schonbrun, R. Pistun, P. Jordan, J. Cooper, K. D. Wulff, J. Courtial, and M. Padgett, “3D interferometric optical tweezers using a single spatial light modulator,” Opt. Express 13, 3777–3786 (2005). [CrossRef] [PubMed]

8.

P. Zemánek, A. Jonáš, L. Šrámek, and M. Liška, “Optical trapping of nanoparticles and microparticles using Gaussian standing wave.” Opt. Lett. 24, 1448–1450 (1999). [CrossRef]

9.

T. Čižmár, M. Šiler, and P. Zemánek, “An optical nanotrap array movable over a milimetre range,” Appl. Phys. B 84, 197–203 (2006). [CrossRef]

10.

M. Šiler, T. Čižmár, M. Šerý, and P. Zemánek, “Optical forces generated by evanescent standing waves and their usage for sub-micron particle delivery,” Appl. Phys. B 84, 157–165 (2006). [CrossRef]

11.

T. Čižmár, O. Brzobohatý, K. Dholakia, and P. Zemánek, “The holographic optical micro-manipulation system based on counter-propagating beams,” Laser Phys. Lett. 8, 50–56 (2011). [CrossRef]

12.

M. M. Burns, J.-M. Fournier, and J. A. Golovchenko, “Optical binding,” Phys. Rev. Lett. 63, 1233–1236 (1989). [CrossRef] [PubMed]

13.

M. M. Burns, J.-M. Fournier, and J. A. Golovchenko, “Optical matter: crystallization and binding in intense optical fields,” Science 249, 749–754 (1990). [CrossRef] [PubMed]

14.

S. A. Tatarkova, A. E. Carruthers, and K. Dholakia, “One-dimensional optically bound arrays of microscopic particles,” Phys. Rev. Lett. 89, 283901 (2002). [CrossRef]

15.

W. Singer, M. Frick, S. Bernet, and M. Ritsch-Marte, “Self-organized array of regularly spaced microbeads in a fiber-optical trap,” J. Opt. Soc. Am. B 20, 1568–1574 (2003). [CrossRef]

16.

V. Karásek, T. Čižmár, O. Brzobohatý, P. Zemánek, V. Garcés-Chávez, and K. Dholakia, “Long-range one-dimensional longitudinal optical binding,” Phys. Rev. Lett. 101, 143601 (2008). [CrossRef] [PubMed]

17.

Z. H. Hang, J. Ng, and C. T. Chan, “Stability of extended structures stabilized by light as governed by the competition of two length scales,” Phys. Rev. A 77, 063838 (2008). [CrossRef]

18.

R. Gómez-Medina and J. J. Sáenz, “Usually strong optical interaction between particles in quasi-one-dimensional geometries,” Phys. Rev. Lett. 93, 243602 (2004). [CrossRef]

19.

J. Ng and C. T. Chan, “Localized vibrational modes in optically bound structures,” Opt. Lett. 31, 2583–2585 (2006). [CrossRef] [PubMed]

20.

F. J. G. de Abajo, “Collective oscillations in optical matter,” Opt. Express 15, 11082–11094 (2007). [CrossRef]

21.

N. K. Metzger, R. F. Marchington, M. Mazilu, R. L. Smith, K. Dholakia, and E. M. Wright, “Measurement of the restoring forces acting on two optically bound particles from normal mode correlations,” Phys. Rev. Lett. 98, 068102 (2007). [CrossRef] [PubMed]

22.

C. D. Mellor, T. A. Fennerty, and C. D. Bain, “Polarization effects in optically bound particle arrays,” Opt. Express 14, 10079–10088 (2006). [CrossRef] [PubMed]

23.

P. J. Reece, E. M. Wright, and K. Dholakia, “Experimental observation of modulation instability and optical spatial soliton arrays in soft condensed matter,” Phys. Rev. Lett. 98, 203902 (2007). [CrossRef] [PubMed]

24.

J.-M. Fournier, J. Rohner, P. Jacquot, R. Johann, S. Mieas, and R.-P. Salathé, “Assembling mesoscopic partices by various optical schemes,” Proc. SPIE 14, 59300Y (2005). [CrossRef]

25.

O. Brzobohatý, T. Čižmár, V. Karásek, M. Šiler, K. Dholakia, and P. Zemánek, “Experimental and theoretical determination of optical binding forces,” Opt. Express 18, 25389–25402 (2010). [CrossRef] [PubMed]

26.

S. K. Mohanty, K. S. Mohanty, and M. W. Berns, “Organization of microscale objects using a microfabricated optical fiber,” Opt. Lett. 33, 2155–2157 (2008). [CrossRef] [PubMed]

27.

Y. Liu and M. Yu, “Optical manipulation and binding of microrods with multiple traps enabled in an inclined dual-fiber system,” Biomicrofluidics 4, 043010 (2010). [CrossRef]

28.

K. Dholakia and P. Zemánek, “Gripped by light: optical binding,” Rev. Mod. Phys. 82, 1767–1791 (2010). [CrossRef]

29.

T. Čižmár, L. C. D. Romero, K. Dholakia, and D. L. Andrews, “Multiple optical trapping and binding: new routes to self-assembly,” J. Phys. B 43, 102001 (2010). [CrossRef]

30.

O. Brzobohatý, T. Čižmár, and P. Zemánek, “High quality quasi-Bessel beam generated by round-tip axicon,” Opt. Express 16, 12688–12700 (2008). [PubMed]

31.

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46, 15–28 (2005). [CrossRef]

32.

V. Karásek and P. Zemánek, “Analytical description of longitudinal optical binding of two spherical nanoparticles,” J. Opt. A: Pure Appl. Opt. 9, S215–S220 (2007). [CrossRef]

33.

V. Karásek, O. Brzobohatý, and P. Zemánek, “Longitudinal optical binding of several spherical particles studied by the coupled dipole method,” J. Opt. A: Pure Appl. Opt. 11, 034009 (2009). [CrossRef]

34.

P. Zemánek, A. Jonáš, L. Šrámek, and M. Liška, “Optical trapping of Rayleigh particles using a Gaussian standing wave,” Opt. Commun. 151, 273–285 (1998). [CrossRef]

35.

P. Zemánek, A. Jonáš, and M. Liška, “Simplified description of optical forces acting on a nanoparticle in the Gaussian standing wave,” J. Opt. Soc. Am. A 19, 1025–1034 (2002). [CrossRef]

36.

P. Zemánek, A. Jonáš, P. Jákl, M. Šerý, J. Ježek, and M. Liška, “Theoretical comparison of optical traps created by standing wave and single beam,” Opt. Commun. 220, 401–412 (2003). [CrossRef]

37.

T. Čižmár, V. Garcés-Chávez, K. Dholakia, and P. Zemánek, “Optical conveyor belt for delivery of submicron objects,” Appl. Phys. Lett. 86, 174101 (2005). [CrossRef]

38.

X. Yu, T. Torisawa, and N. Umeda, “Manipulation of particles with counter-propagating evanescent waves,” Chin. Phys. Lett 24, 2833–2835 (2007). [CrossRef]

39.

J. M. Taylor, L. Y. Wong, C. D. Bain, and G. D. Love, “Emergent properties in optically bound matter,” Opt. Express 16, 6921–6928 (2008). [CrossRef] [PubMed]

40.

M. Šiler, T. Čižmár, A. Jonáš, and P. Zemánek, “Surface delivery of a single nanoparticle under moving evanescent standing-wave illumination,” New J. Phys. 10, 113010 (2008). [CrossRef]

41.

M. Šiler and P. Zemánek, “Particle jumps between optical traps in a one-dimensional optical lattice,” New. J. Phys. 12, 083001 (2010). [CrossRef]

42.

T. M. Grzegorczyk, B. A. Kemp, and J. A. Kong, “Trapping and binding of an arbitrary number of cylindrical particles in an in-plane electromagnetic field,” J. Opt. Soc. Am. A 23, 2324–2330 (2006). [CrossRef]

43.

C. W. Gardiner, Handbook of Stochastic Methods (Springer-Verlag, 2004).

44.

H. A. Kramers, “Brownian motion in the field of force and the diffusion model of chemical reactions,” Physica 7, 284–304 (1940). [CrossRef]

45.

J. E. de Oliveira Rodrigues and R. Dickman, “Asymmetric exclusion process in a system of interacting Brownian particles,” Phys. Rev. E 81, 061108 (2010). [CrossRef]

46.

T. Čižmár, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photonics 4, 388–394 (2010). [CrossRef]

47.

M. K. Cheezum, W. F. Walker, and W. H. Guilford, “Quantitative comparison of algorithms for tracking single fluorescent particles,” Biophys. J. 81, 2378–2388 (2001). [CrossRef] [PubMed]

48.

V. Karásek, K. Dholakia, and P. Zemánek, “Analysis of optical binding in one dimension,” Appl. Phys. B 84, 149–156 (2006). [CrossRef]

49.

Y. Seol, A. E. Carpenter, and T. T. Perkins, “Gold nanoparticles: enhanced optical trapping and sensitivity coupled with significant heating,” Opt. Lett. 31, 2429–2431 (2006). [CrossRef] [PubMed]

OCIS Codes
(070.6110) Fourier optics and signal processing : Spatial filtering
(140.3300) Lasers and laser optics : Laser beam shaping
(170.4520) Medical optics and biotechnology : Optical confinement and manipulation
(350.4855) Other areas of optics : Optical tweezers or optical manipulation

ToC Category:
Optical Trapping and Manipulation

History
Original Manuscript: August 3, 2011
Manuscript Accepted: August 28, 2011
Published: September 22, 2011

Virtual Issues
Vol. 6, Iss. 10 Virtual Journal for Biomedical Optics

Citation
O. Brzobohatý, V. Karásek, M. Šiler, J. Trojek, and P. Zemánek, "Static and dynamic behavior of two optically bound microparticles in a standing wave," Opt. Express 19, 19613-19626 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-20-19613


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References

  1. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett.11, 288–290 (1986). [CrossRef] [PubMed]
  2. K. Sasaki, M. Koshioka, H. Misawa, N. Kitamura, and H. Masuhara, “Pattern formation and flow control of fine particles by laser-scanning micromanipulation,” Opt. Lett.16, 1463–1465 (1991). [CrossRef] [PubMed]
  3. T. Čižmár, D. I. C. Dalgarno, P. C. Ashok, F. J. Gunn-Moore, and K. Dholakia, “Interference-free superposition of nonzero order light modes: Functionalized optical landscapes,” Appl. Phys. Lett.98, 081114 (2011). [CrossRef]
  4. G. Spalding, J. Courtial, and R. D. Leonardo, “Holographic optical trapping,” in Structured Light and Its Applications: An Introduction to Phase-Structured Beams and Nanoscale Optical Forces (Elsevier, Academic Press, 2008). [PubMed]
  5. A. E. Chiou, W. Wang, G. J. Sonek, J. Hong, and M. W. Berns, “Interferometric optical tweezers,” Opt. Commun.133, 7–10 (1997). [CrossRef]
  6. A. Casaburi, G. Pesce, P. Zemánek, and A. Sasso, “Two-and three-beam interferometric optical tweezers,” Opt. Commun.251, 393–404 (2005). [CrossRef]
  7. E. Schonbrun, R. Pistun, P. Jordan, J. Cooper, K. D. Wulff, J. Courtial, and M. Padgett, “3D interferometric optical tweezers using a single spatial light modulator,” Opt. Express13, 3777–3786 (2005). [CrossRef] [PubMed]
  8. P. Zemánek, A. Jonáš, L. Šrámek, and M. Liška, “Optical trapping of nanoparticles and microparticles using Gaussian standing wave.” Opt. Lett.24, 1448–1450 (1999). [CrossRef]
  9. T. Čižmár, M. Šiler, and P. Zemánek, “An optical nanotrap array movable over a milimetre range,” Appl. Phys. B84, 197–203 (2006). [CrossRef]
  10. M. Šiler, T. Čižmár, M. Šerý, and P. Zemánek, “Optical forces generated by evanescent standing waves and their usage for sub-micron particle delivery,” Appl. Phys. B84, 157–165 (2006). [CrossRef]
  11. T. Čižmár, O. Brzobohatý, K. Dholakia, and P. Zemánek, “The holographic optical micro-manipulation system based on counter-propagating beams,” Laser Phys. Lett.8, 50–56 (2011). [CrossRef]
  12. M. M. Burns, J.-M. Fournier, and J. A. Golovchenko, “Optical binding,” Phys. Rev. Lett.63, 1233–1236 (1989). [CrossRef] [PubMed]
  13. M. M. Burns, J.-M. Fournier, and J. A. Golovchenko, “Optical matter: crystallization and binding in intense optical fields,” Science249, 749–754 (1990). [CrossRef] [PubMed]
  14. S. A. Tatarkova, A. E. Carruthers, and K. Dholakia, “One-dimensional optically bound arrays of microscopic particles,” Phys. Rev. Lett.89, 283901 (2002). [CrossRef]
  15. W. Singer, M. Frick, S. Bernet, and M. Ritsch-Marte, “Self-organized array of regularly spaced microbeads in a fiber-optical trap,” J. Opt. Soc. Am. B20, 1568–1574 (2003). [CrossRef]
  16. V. Karásek, T. Čižmár, O. Brzobohatý, P. Zemánek, V. Garcés-Chávez, and K. Dholakia, “Long-range one-dimensional longitudinal optical binding,” Phys. Rev. Lett.101, 143601 (2008). [CrossRef] [PubMed]
  17. Z. H. Hang, J. Ng, and C. T. Chan, “Stability of extended structures stabilized by light as governed by the competition of two length scales,” Phys. Rev. A77, 063838 (2008). [CrossRef]
  18. R. Gómez-Medina and J. J. Sáenz, “Usually strong optical interaction between particles in quasi-one-dimensional geometries,” Phys. Rev. Lett.93, 243602 (2004). [CrossRef]
  19. J. Ng and C. T. Chan, “Localized vibrational modes in optically bound structures,” Opt. Lett.31, 2583–2585 (2006). [CrossRef] [PubMed]
  20. F. J. G. de Abajo, “Collective oscillations in optical matter,” Opt. Express15, 11082–11094 (2007). [CrossRef]
  21. N. K. Metzger, R. F. Marchington, M. Mazilu, R. L. Smith, K. Dholakia, and E. M. Wright, “Measurement of the restoring forces acting on two optically bound particles from normal mode correlations,” Phys. Rev. Lett.98, 068102 (2007). [CrossRef] [PubMed]
  22. C. D. Mellor, T. A. Fennerty, and C. D. Bain, “Polarization effects in optically bound particle arrays,” Opt. Express14, 10079–10088 (2006). [CrossRef] [PubMed]
  23. P. J. Reece, E. M. Wright, and K. Dholakia, “Experimental observation of modulation instability and optical spatial soliton arrays in soft condensed matter,” Phys. Rev. Lett.98, 203902 (2007). [CrossRef] [PubMed]
  24. J.-M. Fournier, J. Rohner, P. Jacquot, R. Johann, S. Mieas, and R.-P. Salathé, “Assembling mesoscopic partices by various optical schemes,” Proc. SPIE14, 59300Y (2005). [CrossRef]
  25. O. Brzobohatý, T. Čižmár, V. Karásek, M. Šiler, K. Dholakia, and P. Zemánek, “Experimental and theoretical determination of optical binding forces,” Opt. Express18, 25389–25402 (2010). [CrossRef] [PubMed]
  26. S. K. Mohanty, K. S. Mohanty, and M. W. Berns, “Organization of microscale objects using a microfabricated optical fiber,” Opt. Lett.33, 2155–2157 (2008). [CrossRef] [PubMed]
  27. Y. Liu and M. Yu, “Optical manipulation and binding of microrods with multiple traps enabled in an inclined dual-fiber system,” Biomicrofluidics4, 043010 (2010). [CrossRef]
  28. K. Dholakia and P. Zemánek, “Gripped by light: optical binding,” Rev. Mod. Phys.82, 1767–1791 (2010). [CrossRef]
  29. T. Čižmár, L. C. D. Romero, K. Dholakia, and D. L. Andrews, “Multiple optical trapping and binding: new routes to self-assembly,” J. Phys. B43, 102001 (2010). [CrossRef]
  30. O. Brzobohatý, T. Čižmár, and P. Zemánek, “High quality quasi-Bessel beam generated by round-tip axicon,” Opt. Express16, 12688–12700 (2008). [PubMed]
  31. D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys.46, 15–28 (2005). [CrossRef]
  32. V. Karásek and P. Zemánek, “Analytical description of longitudinal optical binding of two spherical nanoparticles,” J. Opt. A: Pure Appl. Opt.9, S215–S220 (2007). [CrossRef]
  33. V. Karásek, O. Brzobohatý, and P. Zemánek, “Longitudinal optical binding of several spherical particles studied by the coupled dipole method,” J. Opt. A: Pure Appl. Opt.11, 034009 (2009). [CrossRef]
  34. P. Zemánek, A. Jonáš, L. Šrámek, and M. Liška, “Optical trapping of Rayleigh particles using a Gaussian standing wave,” Opt. Commun.151, 273–285 (1998). [CrossRef]
  35. P. Zemánek, A. Jonáš, and M. Liška, “Simplified description of optical forces acting on a nanoparticle in the Gaussian standing wave,” J. Opt. Soc. Am. A19, 1025–1034 (2002). [CrossRef]
  36. P. Zemánek, A. Jonáš, P. Jákl, M. Šerý, J. Ježek, and M. Liška, “Theoretical comparison of optical traps created by standing wave and single beam,” Opt. Commun.220, 401–412 (2003). [CrossRef]
  37. T. Čižmár, V. Garcés-Chávez, K. Dholakia, and P. Zemánek, “Optical conveyor belt for delivery of submicron objects,” Appl. Phys. Lett.86, 174101 (2005). [CrossRef]
  38. X. Yu, T. Torisawa, and N. Umeda, “Manipulation of particles with counter-propagating evanescent waves,” Chin. Phys. Lett24, 2833–2835 (2007). [CrossRef]
  39. J. M. Taylor, L. Y. Wong, C. D. Bain, and G. D. Love, “Emergent properties in optically bound matter,” Opt. Express16, 6921–6928 (2008). [CrossRef] [PubMed]
  40. M. Šiler, T. Čižmár, A. Jonáš, and P. Zemánek, “Surface delivery of a single nanoparticle under moving evanescent standing-wave illumination,” New J. Phys.10, 113010 (2008). [CrossRef]
  41. M. Šiler and P. Zemánek, “Particle jumps between optical traps in a one-dimensional optical lattice,” New. J. Phys.12, 083001 (2010). [CrossRef]
  42. T. M. Grzegorczyk, B. A. Kemp, and J. A. Kong, “Trapping and binding of an arbitrary number of cylindrical particles in an in-plane electromagnetic field,” J. Opt. Soc. Am. A23, 2324–2330 (2006). [CrossRef]
  43. C. W. Gardiner, Handbook of Stochastic Methods (Springer-Verlag, 2004).
  44. H. A. Kramers, “Brownian motion in the field of force and the diffusion model of chemical reactions,” Physica7, 284–304 (1940). [CrossRef]
  45. J. E. de Oliveira Rodrigues and R. Dickman, “Asymmetric exclusion process in a system of interacting Brownian particles,” Phys. Rev. E81, 061108 (2010). [CrossRef]
  46. T. Čižmár, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photonics4, 388–394 (2010). [CrossRef]
  47. M. K. Cheezum, W. F. Walker, and W. H. Guilford, “Quantitative comparison of algorithms for tracking single fluorescent particles,” Biophys. J.81, 2378–2388 (2001). [CrossRef] [PubMed]
  48. V. Karásek, K. Dholakia, and P. Zemánek, “Analysis of optical binding in one dimension,” Appl. Phys. B84, 149–156 (2006). [CrossRef]
  49. Y. Seol, A. E. Carpenter, and T. T. Perkins, “Gold nanoparticles: enhanced optical trapping and sensitivity coupled with significant heating,” Opt. Lett.31, 2429–2431 (2006). [CrossRef] [PubMed]

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