Simulation in this section is based on the powerful mathematical simulation platform- Maple. Maple is commonly used mathematical engineering software, powerful both in numerical simulation as well as the derivation of analytical equations, which allows us easily to complete tasks including simplification, iteration, differentiation, integration and so on. The complexity of derivation and numerical solution is no longer a hindrance to a simulation.

Previous section detailed the origin and physical meaning of required formulae, which is divided into three parts: Mie particles in the optical trap in the force equation, the beam intensity distribution of equation. In addition, more equations are needed to link the relations between two parts [14]:

where (3.27), (3.28), (3.31) and (3.32) are geometric constraints; (3.29) is the relationship between light intensity and power, and (3.30) is abide by refraction law on interface. So far, we have all the formulae needed, the final force expressions can be derived after inputting these formulas into software Maple.

Being focused by the fiber axicon, a four-spot LP_{2}i mode beam with a diameter of about 1 ^m can be obtained. Four spots are no longer round spots any more, but the relative symmetric position of four spots remains unchanged. We use a plane 90 ^m away from focal plane for manipulation and rotation of cells, in a way to allow beam spot size to fit cell diameter. Microscopic diagram of beam intensity at the two planes are shown in Fig. 3.15. This position is not uniquely determined; we can adjust the distance between the focal plane of the particles and to adjust the relative size and position between the particles and four spots, which helps to improve the optical trapping power in manipulation and rotation. Relative position between particle and four-spot beam strongly affects optical trapping force, especially in the rotation effect (Fig. 3.21).

Assume that each of the four spots in LP_{2}i can be approximated by a single Gaussian beam. For cylindrical symmetry of a Gaussian beam, no angular momentum can be generated for the rotation of the particles. In order to manipulate particles to rotate a center of rotation is required and a suitable layout for cells and beam spots needs to be planned. A model is constructed as shown in Fig. 3.22. In an XYZ coordinate system, Y-axis is the beam propagation direction, XZ plane is the plane where translation and rotation of particles occur. Through focusing of a fiber axicon, four spots of LP_{2}i mode are turned into four elliptical spots, as shown in Fig. 3.15b. According to measurement result, either single or two cells (dimer) can be conveniently confined in the four-lobed light trap. Oftentimes one cell is observed

Fig. 3.21 Parameters to define a Gaussian beam

to locate in the center of LP_{21} speckle gram, while the other cell stays next to the one at center side by side, in the middle of two laser spots. This configuration agrees with low energy distribution given the potential wells. In Fig.3.22, a large dashed circle indicates boundary area for effective cell rotation, while two small dashed circles stands for two extreme positions where slippery of rotation will take place if angular acceleration is too high for cells to follow the laser speckle gram. Torque due to Stokes dragging force hinders the rotation. In this model the rotating cell(s) are sandwiched by two laser spots, while the other two spots are far from the cell, their force are almost negligible. To find rotation torque, we therefore only need to super impose the electromagnetic fields of the two spots.

It is important to note that, since the operating plane for cell manipulation can be selected by moving the tapered fiber, particles, relative positioning and size of the spot can be adjusted freely, Fig. 3.22 displays only one of the allocations.

To model the “chuc” capture force, the intensity distribution of the four lobes of LP21 mode were simplified as a superposition of four Gaussian beams, and the optical force acting on the biological particle can be approximated with a set of parameters: with P = 10mW, z_{0}= 15 ^m, a beam waist w_{0} = 0.5 ^m, and radius of particle a = 5 ц-m, a mathematical symbolic derivation was conducted using MATLAB and MAPLE. The modeled optical capture force as a function of lateral deviation from symmetric axis d is shown in Fig. 3.22, where the dashed and dot- dashed lines are the force generated by two of four beams in LP21 mode, with two zero-force points positioned at 0 and 7 ц-m, respectively. The net force acting on a particle captured inside the LP21 beam spot is the sum of two forces, modeled as the superposition of two curves, yielding a constraining repelling force: when the bioparticle deviates from the zero-force equilibrium point (d = 3.5 ц-m) toward either

Fig. 3.22 Schematic of a dimer of yeast cells rotated by a LP21 beam

d < 3.5 ^m or d > 3.5 ц-m, the target particle is experiences a restoring force back to the equilibrium point d = 3.5 ц-m. From the force distribution it can be seen that attraction force maximizes at d = -2.0 ^m and decreases to zero at d = 3.5 ц-m, whereas the repulsive force gradually increases to reach a maximum at d = 9 ц-m. The peak attractive/repulsive force due to one LP_{2}i mode lobe was calculated to be 0.9pN with the above parameters; the net combined force from two 90° lobes is calculated to be 1.2pN.

The twisting and bending properties of LP_{21} propagation in optical fiber were used to model the rotational forces of the system. Previous literature documents that fiber twisting causes the LP_{21} mode distribution to rotate around the central fiber axis with a scale factor of 0.9112 due to geometric and opto-elastic effects, while fiber bending causes no deformation or rotational changes to the output LP_{21 }distribution [22]. Therefore, once a cell or a cell cluster with a suitable shape (e.g. a dimer) is captured by a LP_{21} beam the cell/cluster orientation can be adjusted in space by twisting a fiber segment to rotate the LP_{21} beam.

As shown in Fig. 3.23, for a particle,there are three equilibrium positions in an individual beam, namely zero-force positions. But only at the optical axis it is a stable equilibrium, small perturbations from equilibrium position allow the particle restore to its original position. The other two equilibrium positions are unstable, even small perturbations can cause its deviation from its equilibrium position. As we know, the stable equilibrium is at the center of an equilibrium system, unstable equilibrium is on the boundary of an equilibrium system.

Fig. 3.23 Modeled forces induced by a focused LP21 mode beam acting on a dielectric particle. Dotted lines A and B: forces acting on a dielectric particle by individual lobes of LP21 mode centered at 0 and 7 respectively. Solid line total force acting on the dielectric particle as a function of deviation from equilibrium point at 3.5

By applying model of a single Gaussian beam spot to four Gaussian beam spots, forces are superimposed to obtain resultant trapping force as shown in solid black line in Fig. 3.23 [20]. In the model, when the particle deviates from the optical axis for greater than 12 ^m, it can be inferred that the particle is no longer affected by the optical trap. Thus, in the fourth-spot model, when an individual particle is away from the two beams for more than 12 ^m, the particle will be no longer under the influence of the trapping force. Suppose center of two beam spots are located at 7 ^m and 0 ^m, respectively, then in the middle of the two beams, i.e. r_{0} = 3.5 ^m a particle will be at a zero-force point; the particle moving to either sides will experience a restoring force, bringing the particle back to r_{0} = 3.5 ^m, reaching a stable equilibrium. It is found that resultant force field of two beam spots has a similar graph of the sine function as an individual Gaussian beam, greater than the gradient slope of a single beam force, which also shows the use of LP21 mode to manipulate in rotation, the combined four-spot beam can produce greater effect than that of a single Gaussian beam.