I the normal components of the E-field on the liquid and solid sides of the interface are used to compute the normal component of the average E-field as well as the induced charge density at the interface. In other words, the charges on the solid and liquid sides of the interface are lumped together, then subjected to the forces of the tangential E-field, which is continuous at the interface, and the perpendicular E-field, which is discontinuous at the interface(and therefore, in need of averaging) Method 1, which corresponds to Eq (27)in Ref [2], is also the method used in the case of a cylindrical rod immersed in water discussed earlier in conjunction We use a Gaussian beam(2o=0.65um)having a full-width of 7 8um at the 1/e point of the field amplitude, incident on the prism at Brewster's angle 0B =arctan(n/ninc); see Fig. 2. In the case of incidence from the free-space, n inc 1.0, the prism has refractive index n= 1.5, while for nine=1.33(water)the prism index is n=1995. The incident field amplitude's peak value is Eo=10/ninc V/m. Table 2 compares the total radiation force(consisting of the surface force density integrated over both surfaces, and the volume force density integrated over the prism volume)exerted on the prism, obtained with the exact method and with the FDTD simulations As shown in Fig. 2, the total force has a component Fy directed along the bisector of the prisms apex, and a second component F: perpendicular to this bisector in the plane of incidence. (The exact value of Fi is zero in all cases. For a prism immersed in water(ninc=1.33)the results obtained with the aforementioned methods 1, 11, and Ill of force computation are tabulated The numerical results are generally in good agreement with the exact solutions, the largest disagreements occur in the case of Method I used in conjunction with the immersed prism, where(due to the specific set of parameters chosen for the simulation) the exact value of F happens to be so small (less than 10-pN/ m) that it falls below the numerical accuracy of our simulations. In the computations with method lll the separation of the charges on the liquid and solid sides of the interface was simulated using an actual air-gap(width=2A). The numerical and exact solutions in general are in good agreement with less than 1% error in the net force magnitude when△≤20nm 3. Single-beam optical trap in the air We present computed results of the electromagnetic force exerted on a dielectric micro-sphere at and near the focus of a laser beam in the free space(n inc= 1.0). The incident beam, obtained by focusing a 20=532nm(or 1064nm) plane-wave through a 0.9NA, 5mm focal-length objective lens, propagates in the negative =-direction, as shown in Fig. 3. The plane-wave entering the abjective's pupil is linearly polarized along either the x-axis or the y-axis. The total power of the incident beam is P=S_dxdy= 1.oW. Figure 4 shows the Poynting vector distribution (S-field) in the XZ-plane for a micro-sphere of refractive index n=1.5 and diameter d 460nm, offset by(x,y, =)-offset=(250, 0, 50)nm from the focal point of the y-polarized incident beam. Positive(negative)values of the z-offset represent the particle being displaced into the converging(diverging) half-space of the focused beam. Upon scattering from the micro-sphere the large positive momentum acquired by the incident light along the x-axis, seen in Fig. 4, results in a net Fr force directed towards the beam axis. Figure 5 shows the computed(Fr, F) force components experienced by the micro-sphere as funct ere cente from the focal point of the lens ( -offset=0). The top( bottom) row shows the case of an x- polarized (y-polarized)incident beam. Due to symmetry, y-offset is set to zero, and offsets along the positive half of the x-axis are the only ones considered. The y component of the force was found to be zero(within the numerical accuracy of the simulations) for these linearly-polarized beams. Figure 5 indicates that the particle is trapped laterally, but not vertically along the z axis, since F- <0 for the entire range of offsets shown. (Note: A 460nm-diameter glass bead weighs approximately 1.0fN. If the focused laser beam shines on the bead from below, and #67575-$15.00USD Received 15 February 2006, revised 7 April 2006, accepted 10 April (C)2006OSA 17 April 2006/Vol 14, No 8/OPTICS EXPRESSI the normal components of the E-field on the liquid and solid sides of the interface are used to compute the normal component of the average E-field as well as the induced charge density at the interface. In other words, the charges on the solid and liquid sides of the interface are lumped together, then subjected to the forces of the tangential E-field, which is continuous at the interface, and the perpendicular E-field, which is discontinuous at the interface (and, therefore, in need of averaging). Method I, which corresponds to Eq.(27) in Ref. [2], is also the method used in the case of a cylindrical rod immersed in water discussed earlier in conjunction with Fig. 1. We use a Gaussian beam (λ0 = 0.65μm) having a full-width of 7.8μm at the 1/e point of the field amplitude, incident on the prism at Brewster’s angle θ B = arctan(n/ninc); see Fig. 2. In the case of incidence from the free-space, ninc = 1.0, the prism has refractive index n = 1.5, while for ninc = 1.33 (water) the prism index is n = 1.995. The incident field amplitude’s peak value is E0 = 103/ninc V/m. Table 2 compares the total radiation force (consisting of the surface force density integrated over both surfaces, and the volume force density integrated over the prism volume) exerted on the prism, obtained with the exact method and with the FDTD simulations. As shown in Fig. 2, the total force has a component Fy directed along the bisector of the prism’s apex, and a second component Fz perpendicular to this bisector in the plane of incidence. (The exact value of Fz is zero in all cases.) For a prism immersed in water (ninc = 1.33) the results obtained with the aforementioned methods I, II, and III of force computation are tabulated. The numerical results are generally in good agreement with the exact solutions; the largest disagreements occur in the case of Method I used in conjunction with the immersed prism, where (due to the specific set of parameters chosen for the simulation) the exact value of Fy happens to be so small (less than 10−2 pN/m) that it falls below the numerical accuracy of our simulations. In the computations with method III the separation of the charges on the liquid and solid sides of the interface was simulated using an actual air-gap (width = 2Δ). The numerical and exact solutions in general are in good agreement with less than 1% error in the net force magnitude when Δ ≤ 20nm. 3. Single-beam optical trap in the air We present computed results of the electromagnetic force exerted on a dielectric micro-sphere at and near the focus of a laser beam in the free space (ninc = 1.0). The incident beam, obtained by focusing a λ0 = 532nm (or 1064nm) plane-wave through a 0.9NA, 5mm focal-length objective lens, propagates in the negative z-direction, as shown in Fig. 3. The plane-wave entering the objective’s pupil is linearly polarized along either the x-axis or the y-axis. The total power of the incident beam is P = Szdxdy = 1.0W. Figure 4 shows the Poynting vector distribution (S-field) in the XZ-plane for a micro-sphere of refractive index n = 1.5 and diameter d = 460nm, offset by (x,y,z)-offset=(250,0,50)nm from the focal point of the y-polarized incident beam. Positive (negative) values of the z-offset represent the particle being displaced into the converging (diverging) half-space of the focused beam. Upon scattering from the micro-sphere, the large positive momentum acquired by the incident light along the x-axis, seen in Fig. 4, results in a net Fx force directed towards the beam axis. Figure 5 shows the computed (Fx,Fz) force components experienced by the micro-sphere as functions of the sphere center’s offset from the focal point of the lens (y-offset = 0). The top (bottom) row shows the case of an x￾polarized (y-polarized) incident beam. Due to symmetry, y-offset is set to zero, and offsets along the positive half of the x-axis are the only ones considered. The y component of the force was found to be zero (within the numerical accuracy of the simulations) for these linearly-polarized beams. Figure 5 indicates that the particle is trapped laterally, but not vertically along the z￾axis, since Fz < 0 for the entire range of offsets shown. (Note: A 460nm-diameter glass bead weighs approximately 1.0 f N. If the focused laser beam shines on the bead from below, and #67575 - $15.00 USD Received 15 February 2006; revised 7 April 2006; accepted 10 April 2006 (C) 2006 OSA 17 April 2006 / Vol. 14, No. 8 / OPTICS EXPRESS 3664