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# Gradient and Scattering Force of Optical Tweezers

In geometrical optics model, we normally use a transparent dielectric bead for analysis in the interaction of light with a bioparticle. In Fig. 3.1, assuming that refractive indices of the surrounding medium and the bead are n and n2 (n2 > ni), respectively, we use two light rays, ray a and b, to analyze optical force acting on the bead. A series of refraction (solid line) and reflection (dotted line) of light rays occurs on bead surface. By momentum conservation, Fa and Fb are generated by the two light rays passing through the bead,as shown in Fig.3.1a, b. Figure3.1c displays how a refracted beam has a momentum change Ap via a ray deflection. Due to Newton’s third law, there should be an equal and opposite momentum change onto the bead (particle).

Suppose the bead is located in a uniform light field (Fig. 3.1a), then light ray a is the same as ray b, therefore forces Fa and Fb in the transverse plane are equal; However, when the bead is in a non-uniform light field (Fig.3.1b), assuming light intensity on the right side is stronger than that on the left, namely Fb > Fa, for a light momentum is proportional to light intensity. Therefore in this case two forces in the lateral direction are no longer equal, pulling the bead toward right side where intensity is higher. This type of force, originated from light field gradient, is called the gradient force. Fig. 3.1 Scattering and gradient force in the geometrical optics model analysis

On the surface of particles, in addition to refraction of light, there are still forces due to reflection, scattering and absorption and other effects. Generally these effects tend to push the particles along the optical axis direction, referred to scattering force. Thus, the gradient force makes the particles tend to be at maximum light intensity gradient, but the scattering force acting upon particles along the beam propagation direction, when the particle is in the vicinity of where the resultant force is zero, then theparticle is said to be captured. In the case of Fig.3.1a, theparticle is in equilibrium in lateral direction, remaining in the symmetric axis of the beam; however, in the direction of beam propagation the scattering force cannot be balanced with small axial component of gradient force, causing the particle to move along the direction of beam propagation.

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