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88    3 Optical Tweezers
                                      (a)      Laser        (b)      Laser


                                         b              a      b        F    a
                                                 F at                    at
                                                        Objective
                                                f                     f
                                                                         F a
                                             F a  F  F b  Microsphere  F  O
                                                O                       F b
                                        F ao                    F ao


                            Fig. 3.8. Qualitative description of optical trap. The microsphere is transparent
                            and its refractive index is slightly higher than that of the surrounding medium [3.4]


                            and F b is shown pointingin the direction of the laser focus f from the center
                            of the sphere. As a result, the sphere is trapped stably at the point where the
                            optical pressure force F balances the difference between the gravity force and
                            the buoyant force.
                               We can see, for the arbitrary displacement of the sphere center O from the
                            focus f, that the vector sum of F a and F b gives a net optical pressure force
                            F directs to the focus as shown in Fig. 3.8b. Since the optical pressure F is
                            expressed as (3.4), the axial and transverse trappingforce analysis results the
                            simulation of the nondimensional Q of the object, beam convergence angle,
                            sphere size, shape and relative refractive index with respect to the surrounding
                            medium, polarization state, and beam profile as parameters.
                               First, consider a ray of the power P incident to a microsphere at the
                            angle θ 1 . Figure 3.9 shows the geometry for this model and trapping force
                            is calculated accordingto Ashkin [3.4]. The expressions of the net optical
                            pressure by the emerging rays in the direction parallel (F s : scatteringforce)
                            and perpendicular (F g : gradient force) to the incident ray can be expressed
                            (Example 3.3) as
                                                  T {cos 2(θ 1 − θ 2 )+ R cos 2θ 1 } n 1 P  n 1 P
                                                    2
                               F s = 1+ R cos 2θ 1 −        2                      = Q s   ,
                                                       1+ R +2R cos 2θ 2        c        c
                                                                                           (3.5)
                                                2
                                               T {sin 2(θ 1 − θ 2 )+ R sin 2θ 1 } n 1 P  n 1 P
                               F g = R sin 2θ 1 −                              = Q g   ,   (3.6)
                                                        2
                                                   1+ R +2R cos 2θ 2        c        c
                            where θ 1 is the incident angle, θ 2 is the refracted angle, T and R are the Fresnel
                            transmission and reflection coefficients, and Q s and Q g are the scatteringand
                            gradient trapping efficiency, respectively. As a result, total Q and total F are

                                                          2
                                                               2
                                                 Q t =  Q + Q ,                            (3.7)
                                                          s    g
                                                                     n 1 P

                                                          2
                                                               2
                                                  F t =  F + F = Q t     .                 (3.8)
                                                          s
                                                               g
                                                                      c
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