Plots of the total force components(Fx, F:) versus vertical displacement A- of the focus center from the sphere center are shown in Fig. 10(4>0 when the focused spot is above the sphere center). Different colored curves correspond to different lateral displacements Ax of the particle from the optical axis of the lens(red: Ax=0.25um, green: Ar=0. 5um, blue Ar=0.75um). The top row in Fig. 10 corresponds to a circularly polarized plane-wave entering the pupil of the lens. The middle row corresponds to linear polarization along x, and the bottom row represents the case of linear polarization along the y-axis. Note that for Az>0, F: is positive only when Ar is small; this indicates that the particle is first trapped laterally by the(fairly strong)Fx, then lifted upward until its center nearly coincides with the center of the For linearly polarized light(whether along the direction of the lateral displacement of the bead, x, or perpendicular to that direction, y), Fy was found to be zero, as expected from symmetry considerations. The non-zero values of Fy obtained for circularly polarized light indicate the transfer of a certain amount of angular momentum from the beam to the particle We found Fy to be generally an order of magnitude weaker than Fx, fairly independent of A- and an increasing function of Ax. When the sense of polarization was reversed(say, from right- to left-circular), Fy switched sign, while its magnitude -as well as the signs and magnitudes of Fx and F:- remained unchanged. We thus believe the weak but zero values of Fu predicted to exist under circularly-polarized light, are real and should be subjected to experimental verification. We have also noticed that the inclusion of a thin layer of liquid on the surface of the particle results in somewhat stronger trapping forces. In practice, it is not unreasonable to expect a layer of liquid, with a thickness of several ten nanometers, to stick to the particle's exterior surface. The forces of radiation experienced by this liquid layer may thus have to be included in the overall force calculation in order to obtain more accurate results 7. Dielectric half-slab illuminated near its side-wall The case of a dielectric half-slab illuminated at one edge (i.e, side-wall) by a finite-diameter beam was briefly discussed in our previous paper [1] in conjunction with optical trapping experiments. We argued qualitatively that, even though the edge of a p-polarized beam inside the half-slab tends to exert an expansive force on its host medium(and hence drive it away from the beams center), the net force experienced by the half-slab should be dominated by the surface charges induced on the slab's side-wall, which force tends to draw the slab toward the center of the beam. Attraction of the half-slab to the beam's center is thus expected te occur irrespective of whether the incident beam is p-or s-polarized In the present section we discuss this problem in some detail, using the one-dimensional Gaussian beam of Fig 3(o=0.65um, FWHM=1.5um), normally incident from free-space onto a dielectric half-slab having ns= 2.0 and thickness d= llonm the beam center and the slab-edge are assumed to coincide at y=0. Figure 11 shows the computed field as well as force-density plots when the beam is p-polarized (left column) and s-polarized(right column) In the p-polarized case the color scale has been adjusted to exclude high force-density values at the edges/corners of the half-slab. The horizontal force component Fy(second row) is mostly positive inside the slab but has very strong negative values on the slab's vertical edge where there is a substantial charge accumulation. In the s-polarized case, no surface charges are induced(because the only E-field component, Er is everywhere parallel to the glass surfaces). The only forces in the case of s-polarization, therefore, are due to the bound currents within the medium, acted upon by the H-field of the light beam. Clearly, the force distributions are quite complex, and it is not immediately obvious if the total force in the horizontal direction is pulling the half-slab to the left or pushing it to the right;net (integrated) values of the force are thus needed to settle the question #6863·$1500US Received 14 January 2005; revised 15 March 2005; accepted 15 March 2005 (C)2005OSA 4 April 2005/VoL 13, No. 7/OPTICS EXPRESS 2334Plots of the total force components (Fx, Fz) versus vertical displacement ∆z of the focus center from the sphere center are shown in Fig.10 (∆z > 0 when the focused spot is above the sphere center). Different colored curves correspond to different lateral displacements ∆x of the particle from the optical axis of the lens (red: ∆x = 0.25µm, green: ∆x = 0.5µm, blue: ∆x = 0.75µm). The top row in Fig. 10 corresponds to a circularly polarized plane-wave entering the pupil of the lens. The middle row corresponds to linear polarization along x, and the bottom row represents the case of linear polarization along the y-axis. Note that for ∆z > 0, Fz is positive only when ∆x is small; this indicates that the particle is first trapped laterally by the (fairly strong) Fx, then lifted upward until its center nearly coincides with the center of the focused beam. For linearly polarized light (whether along the direction of the lateral displacement of the bead, x, or perpendicular to that direction, y), Fy was found to be zero, as expected from symmetry considerations. The non-zero values of Fy obtained for circularly polarized light indicate the transfer of a certain amount of angular momentum from the beam to the particle. We found Fy to be generally an order of magnitude weaker than Fx, fairly independent of ∆z, and an increasing function of ∆x. When the sense of polarization was reversed (say, from right- to left-circular), Fy switched sign, while its magnitude – as well as the signs and magnitudes of Fx and Fz – remained unchanged. We thus believe the weak but non-zero values of Fy, predicted to exist under circularly-polarized light, are real and should be subjected to experimental verification. We have also noticed that the inclusion of a thin layer of liquid on the surface of the particle results in somewhat stronger trapping forces. In practice, it is not unreasonable to expect a layer of liquid, with a thickness of several ten nanometers, to stick to the particle’s exterior surface. The forces of radiation experienced by this liquid layer may thus have to be included in the overall force calculation in order to obtain more accurate results. 7. Dielectric half-slab illuminated near its side-wall The case of a dielectric half-slab illuminated at one edge (i.e., side-wall) by a finite-diameter beam was briefly discussed in our previous paper [1] in conjunction with optical trapping experiments. We argued qualitatively that, even though the edge of a p-polarized beam inside the half-slab tends to exert an expansive force on its host medium (and hence drive it away from the beam’s center), the net force experienced by the half-slab should be dominated by the surface charges induced on the slab’s side-wall, which force tends to draw the slab toward the center of the beam. Attraction of the half-slab to the beam’s center is thus expected to occur irrespective of whether the incident beam is p- or s-polarized. In the present section we discuss this problem in some detail, using the one-dimensional Gaussian beam of Fig. 3 (λo = 0.65µm, FWHM = 1.5µm), normally incident from free-space onto a dielectric half-slab having ns = 2.0 and thickness d = 110nm. The beam center and the slab-edge are assumed to coincide at y = 0. Figure 11 shows the computed field as well as force-density plots when the beam is p-polarized (left column) and s-polarized (right column). In the p-polarized case the color scale has been adjusted to exclude high force-density values at the edges/corners of the half-slab. The horizontal force component Fy (second row) is mostly positive inside the slab but has very strong negative values on the slab’s vertical edge, where there is a substantial charge accumulation. In the s-polarized case, no surface charges are induced (because the only E-field component, Ex, is everywhere parallel to the glass surfaces). The only forces in the case of s-polarization, therefore, are due to the bound currents within the medium, acted upon by the H-field of the light beam. Clearly, the force distributions are quite complex, and it is not immediately obvious if the total force in the horizontal direction is pulling the half-slab to the left or pushing it to the right; net (integrated) values of the force are thus needed to settle the question. (C) 2005 OSA 4 April 2005 / Vol. 13, No. 7 / OPTICS EXPRESS 2334 #6863 - $15.00 US Received 14 January 2005; revised 15 March 2005; accepted 15 March 2005