row) in the y= cross-sectional plane. The beams full-width at half-maximum-amplitude measured at the waist (located at ==+0.3 um)is FWHM= 1.5 um (d) 0.1 2 y uum y lum] The beam waist is at ==0.3um, and the beams propagation direction is along the negative xis.(a) Time snapshot of H, distribution for p-light, with superposed arrows depicting the Er) vector field. (b)Time snapshot of Er distribution for s-light, with the(H, Hs)vector eld superposed. (c)Force density distribution of the Fy component in the case of p larization, with the (Fn F:)vector field superposed. (d)Distribution of the force density e (Fn Fs)vector field superpose In Fig 3 the field is p-polarized on the left-and s-polarized on the right-hand-side. The upper-left frame shows a color-coded plot of the H distribution, on which the E-field vector (Ey, Es)is superposed. The frame below it shows the force density plot of Fy, together with the (F Fs) vector field in the case of p-polarization. The upper-right frame is a time snapshot of the Er distribution, with the H-field vector(Hy, H)superposed. The frame below it shows Fy for the s-polarized beam, with the(Fy F:) vector field superposed. These force-density fields are computed using time-averages(over one period of the optical wave, T=nc)of the time and space-dependent force density. Note that Fy is expansive for p-light and compressive for s-light. The integrated lateral force per unit area on each half of the beam, J F(x,y, =)dy, is 1.6577 pN/m* for p-light and 1.6581 pN/m- for s-light. The theoretical value of the force per unit area on each edge of the beam given by Eq (9)with E=0.5V/m is Fledge)=+1.66 pN/m2 Another example of the same phenomenon(lateral force at the beams edge)is exhibited by a finite-diameter beam that enters a semi-infinite dielectric at oblique incidence. Figure 4 #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 2326
row) in the y z cross-sectional plane. The beam’s full-width at half-maximum-amplitude measured at the waist (located at z = +0.3 µm) is FWHM = 1.5 µm. Fig. 3. One-dimensional Gaussian beam in a homogeneous medium of refractive index n = 2.0. The beam waist is at z = 0.3µm, and the beam’s propagation direction is along the negative zaxis. (a) Time snapshot of Hx distribution for p-light, with superposed arrows depicting the (Ey, Ez) vector field. (b) Time snapshot of Ex distribution for s-light, with the (Hy, Hz) vector field superposed. (c) Force density distribution of the Fy component in the case of ppolarization, with the (Fy, Fz) vector field superposed. (d) Distribution of the force density component Fy for the s-polarized beam, with the (Fy, Fz) vector field superposed. In Fig. 3 the field is p-polarized on the left- and s-polarized on the right-hand-side. The upper-left frame shows a color-coded plot of the Hx distribution, on which the E-field vector (Ey, Ez) is superposed. The frame below it shows the force density plot of Fy, together with the (Fy, Fz) vector field in the case of p-polarization. The upper-right frame is a time snapshot of the Ex distribution, with the H-field vector (Hy, Hz) superposed. The frame below it shows Fy for the s-polarized beam, with the (Fy, Fz) vector field superposed. These force-density fields are computed using time-averages (over one period of the optical wave, T = λo/c) of the timeand space-dependent force density. Note that Fy is expansive for p-light and compressive for s-light. The integrated lateral force per unit area on each half of the beam, ∫ F(x, y, z) dy, is 1.6577 pN/m2 for p-light and 1.6581 pN/m2 for s-light. The theoretical value of the force per unit area on each edge of the beam given by Eq.(9) with Eo= 0.5V/m is F (edge) = ±1.66 pN/m2 . Another example of the same phenomenon (lateral force at the beam’s edge) is exhibited by a finite-diameter beam that enters a semi-infinite dielectric at oblique incidence. Figure 4 (a) (b) (c) (d) (C) 2005 OSA 4 April 2005 / Vol. 13, No. 7 / OPTICS EXPRESS 2326 #6863 - $15.00 US Received 14 January 2005; revised 15 March 2005; accepted 15 March 2005
shows time-snapshots of the field distributions for a linearly-polarized wave(o=0.65um) having a top-hat cross-sectional profile with smooth edges. The beam is incident from the free-space at inc=50 onto a semi-infinite dielectric of index ny=2.0; the dielectric fills the half-space :<0. The case of p-polarization is shown on the left-, that of s-polarization on the ght- hand side For the p-polarized beam, its angle of incidence being close to Brewster's angle BB=63.43, there is little reflectivity at the interface and most of the light is transmitted through to the dielectric medium, whereas for the s-polarized beam a goo fraction of the incident light is being reflected Fig. 4. Time snapshots of the field components for a linearly-polarized wave (o=0.65um) having a top-hat cross-sectional profile with smooth edges, incident at Binc=50% from free- dielectric of refractive index n,=2.0. located in the half-s field H, in the case of p-polarization, the superposed arrows represent the electric-field(En Er).(Right) Electric field Ex in the case of s-polarization, the superposed Computed force densities(Fy F: )inside the dielectric medium are displayed in Fig. 5; the interface region has been excluded to avoid, in the case of p-light, the high force region of the induced surface charges(For s-light the E-field is continuous at the boundary and, therefore no surface charges are induced. In both cases the force fields near the leading edge of the beam show oscillatory behavior, whereas the trailing edge is fairly smooth. Despite oscillations near the edge, the force is seen to be generally expansive for p-light and compressive for s-light, that is, the sign of the integrated lateral force on each side of the center is consistent with the theoretical arguments presented in [1]. In units of pN/m, the integrated force(JFy dy, JF: dy) for p-light is(-2.64, -1. 13)for the left-edge and(2.59, 1.096) for the right-edge. These edge forces are nearly identical in strength(to better than +1.5%) are orthogonal to the propagation direction within the dielectric, and are in fair agreement with the theoretical value of +(2.51, 1.04)PN/m2 obtained from Eq (9)with E=0.64 V/m The corresponding edge forces for the s-light depicted in Fig. 5, right-hand column,are (1.957, 0.821) for the left edge and(1.939,-0815)for the right edge. Again, this compressive force is orthogonal to the propagation direction, and is in reasonable agreement with the theoretical value of+(1.92, 0.8)pN/m2 obtained from Eq (9)with E.=0.56 V/m 5. Cylindrical rod illuminated by Gaussian beam Figure 6 shows time-snapshot plots of Hx, Ey, E: components of a p-polarized, one- dimensional Gaussian beam(no=0.65 um, amplitude FWHM=0.5 um), propagating in free- space along the negative :-direction the beams waist is at ==0.5 um. Given the amplitude profile as H( z=0.5um)=Ho exp[-wvlyo)"], the value of yo at the waist is 0.3um, which corresponds to a divergence half-angle 8=arctan(dIy)=35%. The small diameter of the beam at the waist thus results in its rapid divergence along the propagation direction. The cone angle of the beam, although large enough to exhibit lateral trapping of small dielectric objects, is not sufficient to produce vertical trapping [5], as will be seen below #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 2327
shows time-snapshots of the field distributions for a linearly-polarized wave (λo = 0.65µm) having a top-hat cross-sectional profile with smooth edges. The beam is incident from the free-space at θ inc = 50° onto a semi-infinite dielectric of index ns = 2.0; the dielectric fills the half-space z < 0. The case of p-polarization is shown on the left-, that of s-polarization on the right-hand side. For the p-polarized beam, its angle of incidence being close to Brewster’s angle θ B = 63.43°, there is little reflectivity at the interface and most of the light is transmitted through to the dielectric medium, whereas for the s-polarized beam a good fraction of the incident light is being reflected. Fig. 4. Time snapshots of the field components for a linearly-polarized wave (λo = 0.65µm) having a top-hat cross-sectional profile with smooth edges, incident at θ inc = 50° from freespace onto a semi-infinite dielectric of refractive index ns = 2.0, located in the half-space z < 0. (Left) Magnetic field Hx in the case of p-polarization; the superposed arrows represent the electric-field (Ey, Ez). (Right) Electric field Ex in the case of s-polarization; the superposed arrows represent the magnetic-field (Hy, Hz). Computed force densities (Fy, Fz) inside the dielectric medium are displayed in Fig. 5; the interface region has been excluded to avoid, in the case of p-light, the high force region of the induced surface charges. (For s-light the E-field is continuous at the boundary and, therefore, no surface charges are induced.) In both cases the force fields near the leading edge of the beam show oscillatory behavior, whereas the trailing edge is fairly smooth. Despite oscillations near the edge, the force is seen to be generally expansive for p-light and compressive for s-light, that is, the sign of the integrated lateral force on each side of the center is consistent with the theoretical arguments presented in [1]. In units of pN/m2 , the integrated force ( ∫Fy dy, ∫Fz dy) for p-light is (−2.64, −1.13) for the left-edge and (2.59, 1.096) for the right-edge. These edge forces are nearly identical in strength (to better than ±1.5%), are orthogonal to the propagation direction within the dielectric, and are in fair agreement with the theoretical value of ±(2.51, 1.04) pN/m2 obtained from Eq. (9) with Eo = 0.64 V/m. The corresponding edge forces for the s-light depicted in Fig. 5, right-hand column, are (1.957, 0.821) for the left edge and (−1.939, −0.815) for the right edge. Again, this compressive force is orthogonal to the propagation direction, and is in reasonable agreement with the theoretical value of ±(1.92, 0.8) pN/m2 obtained from Eq. (9) with Eo = 0.56 V/m. 5. Cylindrical rod illuminated by Gaussian beam Figure 6 shows time-snapshot plots of Hx, Ey, Ez components of a p-polarized, onedimensional Gaussian beam (λo = 0.65 µm, amplitude FWHM = 0.5 µm), propagating in freespace along the negative z-direction; the beam’s waist is at z = 0.5 µm. Given the amplitude profile as Hx( y,z = 0.5µm) = Ho exp[−(y/yo) 2 ], the value of yo at the waist is 0.3µm, which corresponds to a divergence half-angle θ = arctan (λo/πyo) ≈ 35°. The small diameter of the beam at the waist thus results in its rapid divergence along the propagation direction. The cone angle of the beam, although large enough to exhibit lateral trapping of small dielectric objects, is not sufficient to produce vertical trapping [5], as will be seen below. (C) 2005 OSA 4 April 2005 / Vol. 13, No. 7 / OPTICS EXPRESS 2327 #6863 - $15.00 US Received 14 January 2005; revised 15 March 2005; accepted 15 March 2005