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Optical tweezers use the radiation pressure to trap and manipulate the microscopic particles. Using various algorithms multiple traps are being formed which can trap a number of particles simultaneously. In contrast to multiple traps, many particles can be trapped at a single trap position. It is known that when two or more particles are trapped in a single trap they align themselves in axial direction and it appears as if only one particle is trapped. We present a study of the dependence of the optical trapping force on the number of particles in a single trap using equipartition method; the study was carried out for particles of different sizes. The trapping force was first found to increase then decrease with number of particles in trap for all particle sizes. We feel that our studies will be useful in applications of optical tweezers involving trapping of multiple particles in a single trap.

Since their invention in 1980s [

Apart from single beam optical trap, scientists are using multiple optical traps for a diversity of applications [

In contrast to it, multiple particles in a trap were not so much of a popular topic to study. In fact it was thought more of a nuisance. However some studies have been done on the multiple particles in a single trap [10,11]. It has been found that when two particles are trapped in an optical trap they do not move independently but there exists a binding between them [

Our experimental setup has been shown in

The beam expanding assembly was used so that the laser beam could fill the back aperture of the objective lens completely. The refractive index of the immersion oil is 1.515. The laser power before the dichroic mirror is

around 10.7 mW. The trap formed was elliptical in shape and trap size (1/e type) along its two axes is 0.8 µm (y-direction) and 0.62 µm (x-direction) respectively. The particles used in this experiment are silica latex beads (from Bang’s Laboratory). These particles are spherical in shape and they were suspended in double distilled water. We have used particles of different sizes (0.33 µm, 0.6 µm, 0.97 µm, 1.86 µm and 2.47 µm) in this experiment. The specimen sample was illuminated by a halogen lamp assembly. The light from illuminated particles was sent to the cooled color CCD camera (Evolution VF, Q-Imaging) connected to the computer through the objective lens, dichroic mirror and a beam steering prism. The same objective lens is used for the viewing and trapping of the particles. A color glass filter is used before the CCD camera to block the laser to avoid saturation of the camera. The stage movement with a knob was used to bring the trap at the different parts of the sample.

Upto three particles could be trapped stably in the trap. But stable trapping for four particles was not possible for all particle sizes. Though for particle size of 0.97 µm even four particles could be trapped, for other particle sizes especially 0.33 µm and 2.47 µm trapping of four particles in a single trap was not stable. The graphs for multiple particle trapping have been plotted for five different particle sizes viz. 0.33 µm, 0.6 µm, 0.97 µm, 1.86 µm, 2.47 µm (Figures 2 to 6). The plots show the variation of the trapping force with the number of particles in the trap as well as the variation of the trapping force with the size of particles. The method used for calculating the strength of the force is Brownian motion method (equipartition method). It is clear that the trapping force first

increases with the number of particles in the trap, reaches a maximum and then decreases. The increase in the trap strength with the increase in the number of particles can be understood by looking into fundamental mechanism

of optical trapping which tells that trapping requires refraction of light, more the refraction of light from the particle the more will be trapping strength [

and so does the trapping power [

From Ashkin [

The total force on the sphere is the sum of contributions due to the reflected ray of power PR and infinite number of emergent refracted rays of successively decreasing power The quantities R and T are the Fresnel reflection and transmission coefficients of the surface at θ_{i}. The net force acting through the origin can be broken into F_{Z} and F_{Y} such that

and

where θ_{r} is the angle of refraction.

If we consider the case of two particles trapped together and for simplicity consider that both the particles are of same size and are in contact and also trapping of 2^{nd} particle is only due rays refracted through first particle.

Let any ray of power P falls on the 1^{st} particle at an angle of incidence θ_{i}, (^{st} particle.

The 2^{nd} outgoing ray which is the first outgoing refracted ray will make an angle from the direction of the incident ray. And from the

Similarly n^{th} ray coming out 1^{st} particle will make an angle from the direction of incident ray (

Not all rays scattered from first sphere will hit 2nd sphere. If we consider the lower half of the second sphere (^{nd} sphere. This condition puts a constraint on the values of α for which the outgoing rays from first sphere will hit the second sphere.

From trigonometry it can be shown that

Rest of the calculations remain the same, we only have to replace θ_{r} with −θ_{r}.

If we put value of β equal to 90 degrees in above two equations, we will get two critical values for α. The outgoing ray from first sphere will hit second sphere only if for that ray the value of α is between the two critical values. We have prepared a program for calculating the value of force for second and third sphere. The program first calculates the values of α for successive reflections. If the value of α is between two critical values it calculates the angle of incidence on 2^{nd} sphere i.e. β. Then using Equations (1) and (2) it calculates the force on 2^{nd} sphere due to that ray. It does this for all those rays for which the ratio of power to the incoming ray is greater than a specified value. Finally it adds the contribution of all such outgoing rays which hit the 2^{nd} sphere. It should be noted that in our calculations we have not considered the effect of the rays which are reflected back from second particle and enter the first particle and also rays which are reflected back from third particle and enter the second particle. Therefore presence of second particle will not have any effect on trapping of first particle and trapping of third particle will have no effect on trapping of second particle which actually will not be the case.

If we study the trapping force on a single particle from Ashkin [

Using our program we have calculated the values of trapping force for one, two and three particles in a trap. Total trapping force (resultant of gradient and scattering force) for first particle with S has been plotted in

taneously is slightly unstable but possible.

We have experimentally and theoretically studied trapping of 1 to 3 particles in a single trap. Theoretical results for trapping of one particle are qualitatively same as obtained by Ashkin [

greater than that on the second particle, on the third particle it was unidirectional hence destabilizing.

It should be noted that in our calculations we have not considered the effect of the rays which are reflected back from second particle and enter the first particle and also rays which are reflected back from third particle and enter the second particle. Therefore by our calculations presence of second particle will not have any effect on trapping of first particle and trapping of third particle will have no effect on trapping of second particle which actually will not be the case. Considering the effect of all the rays will provide the clearer picture.