# Modeling of Optical Trapping Force

This section analyzes force acting on Mie particles using the geometrical optics model. Light rays are analyzed using MAPLE to find optical force in differential form of geometrical optics model theory. This section also builds model for equilibrium position of the particles in a four-lobed laser beam, obtaining binding properties of Gaussian beam through numerical simulation in terms of force distribution. Yeast cells with a diameter of about 3-6 ^m are used in test experiments for convenience. At a wavelength around 650 nm, the particle size is much greater than the wavelength of light, making the yeast cell a typical Mie’s particle. In the following sections light trapping force will be calculated and analyzed using geometrical optics model.

## Force Analysis of Mie Particles in Optical Trap

Since most biological cells and cell walls have a good light transmission and their refractive indices are slightly greater than the surrounding solution, so it is usually regarded cells as a uniform transparent media spheres when we analyze forces acting on the cells in the optical trap. Because the size of cell is much greater than the wavelength of light, as light propagates through a cell, a series of reflection, refraction and diffraction occurred can be approximated as independent processes in calculation, among them the diffraction belongs to the second-order effects in the model, and can be ignored for its little contribution to the forces. Therefore, we only need to super impose reflection on refraction forces to obtain total force acting upon particles. In electromagnetic wave theory, the radiation pressure P can be expressed as:

where ^_{0} is the permeability of vacuum, *c* is the speed of light, and *E* is the electric field strength. For a specific set of a light irradiation onto a surface element *dA* on the surface of a particle, with *i* being the angle between incident light and surface normal vector, the radiation force is:

The incident light experiences three processes on the surface of the medium sphere: absorption, reflection and refraction. Resultant force per unit area can be expressed as:

where *F*_{a}, *F*_{r}, *F _{t}* represent the forces of light absorption, reflection and refraction, respectively.

A transparent medium sphere (radius a, refractive index *n _{1})* is placed into isotropic refractive index medium n

_{2}(n

_{1}> n

_{2}), as shown in Fig. 3.19. When a light ray incidents on the sphere surface with an angle of

*a*between light ray and the optical axis Z, We calculate lateral component, for example, of each of the incident, reflected and refracted light in change of momentum per unit time, thus to obtain axial force acting on the sphere. Each component of the lateral force on a unit surface element

*dA*of

*f*components shown in Fig. 3.1 can be expressed in the following form:

_{y}

where *f*_{yai}, *f*_{yn}, *f _{yti}* respectively represents the y-direction component force caused by the

*i*th (i = 0, 1,

*2...)*absorption, reflection, refraction (subscript letter

*t*refers to

**Fig. 3.19 ****Light force analysis of a Light ray undergoing one refraction and two reflections on surface of a medium sphere**

transmission) process of the incident light in on sphere surface. For a non-absorbing medium, according to the conservation of energy and momentum conservation, we can get:

By substituting formula (3.11) and (3.12) into the formula (3.10) we can obtain: Three terms can be calculated respectively [14]:

Similarly, we can obtain the radiation force in the axial direction (z-direction):

Finally we can get force acting on the particle on *dA* in the transverse *y _{1}* and longitudinal

*z*directions:

_{1}

where *n _{2}* is the refractive index of the surrounding medium,

*R*is the reflection coefficient,

*T*is the transmission coefficient.

Using (3.20) and (3.21), lateral force *F _{y}* and the vertical force

*F*can be obtained by surface integration:

_{z}

where *6 _{m}* is the angle when the light ray is tangential to the particle surface, used as the upper limit for integration.

*6*value depends on the relative position of the light source and radius of the particle.

_{m}To study the optical trapping/dragging force with which LP21 mode beam can capture bioparticles in either translation or rotation, we combine Mie theory (for particles with scale much larger than the light wavelength) with geometrical optics (RO model for optical field force on particle) for the analysis of manipulation physics.

As shown in Fig.3.20, when a spherical target particle of radius a is located at off- axis coordinate *P =* (0, *d _{0},* zo), and the optical axis is along the z-axis, the optical force acting on the unit area

*dA*of the target surface can be modeled in terms of

**Fig. 3.20 ***(Left)* A ray of light originating from C has refracts at point M on the water/subject interface; *(Right)* The principle plane containing CP, and a ray trace in the *Z**1* — *P* — *У**1* coordinate system, where M is a point on the girdle of the subject reachable by the ray

lateral force *F _{yi}* and the longitudinal Fz1generated by the refraction of each light beam on the water/target boundary. Thus, the total lateral optical force

*F*on the particle can be calculated by integrating

_{y}*dF*and

_{yi}*dF*over the area S (shown in Fig.3.20) that illuminated by rays originating from point C, and

_{Z1}*dF*is the resultant force of dF

_{y}_{y1}and

*dF*projecting onto the y-axis.

_{Z1}Where *n _{2}* is the index of refraction of the medium enclosed in the sphere,

*a*the angle of the incident ray,

_{i}*a*the angle of refraction of the ray,

_{r}*в*the angle between the C

_{Z1}direction and the incident ray originating from position

*C*, в

_{0}the angle between the CP and CM direction,

*в*the limit angle of в

_{т}_{0}when CM is tangent to the sphere at

*P*.

*R*and

*T*denote reflection and transmission coefficients of light intensity respectively, and

*c*is the speed of light in vacuum. The polarization effects of

*R*and

*T*are considered negligible over the two polarization states.