induced surface charges, computed in FDTD with Ay= A =5.0 nm, is(Fy, F-)=(1.696, 2. 468)N/m*, versus the exact value of(1.699, 2.467)pN/m. For n=2.0, FDTD simulations yield(Fr F)=(1.5903, 1.6475)pN/m2 versus the exact value of (1.5911, 1.6493)pN/m2 Figure 2(b)corresponds to a semi-infinite dielectric with a rapid (linear) transition of the refractive index from no= 1.0 to n=3.4 over a 40 nm-thick region. In this case, the force per unit sur face area computed in FDTD is(Fys F: )=(1.744, 2.638)pN/m We mention in passing that the case of a finite-diameter beam at Brewster's incidence on a dielectric wedge is also amenable to analytical as well as numerical solution, and that the numerically computed forces are in excellent agreement with the theoretical values [8] s(a) .s(b) 2^ ^^ n=3.4 n=1.0 3.4 n=10 Fig. 2. Time-snapshots of the E, component of a p-polarized plane-wave, Ao=0.65 um -<0. (a) Abrupt transition of the refractive index at ==0. (b)Linear transition of the refractive index from n,=1.0 to n,=3.4 over a 40 nm-thick region. The above results establish the accuracy of our numerical calculations, especially when transitions at the boundaries between regions of differing refractive indices, as was done in the case of Fig. 2(b), is a tool that is available in numerical simulations. Such smooth transitions at the boundaries may represent physical reality, or they may be used as an artificial tool to eliminate sharp discontinuities and singularities of the equations. To the extent that such smoothing operations do not modify the actual physics of the problem under consideration, they may be used with varying degrees of effectiveness 4. Force exerted by beams edge on the host medium We have shown in [1] that, among other things, the magnetic Lorentz force is responsible for a lateral pressure exerted on the host medium at the edges of a finite-diameter beam; the force per unit area at each edge (i.e, side-wall) of the beam is given by Here lEol is the magnitude of the E-field of a(finite-diameter) plane-wave in a medium of dielectric constant E If the E-field is parallel(perpendicular)to the beams edge, the force is ompressive(expansive); in other words, the opposite side-walls of the beam tend to push the medium toward(away from) the beam center. The"edge force" does not appear to be sensitive to the detailed structure of the beams edge, in particular, a one-dimensional Gaussian beam exhibits the edge force described by Eq(9)when its(magnetic)Lorentz force on the host medium is integrated laterally on either side of the beams center[1] Consider a one-dimensional Gaussian beam (uniform along x, Gaussian along y, and propagating in the negative z-direction) in a homogeneous host medium of refractive index n=2.0 the free-space wavelength of the cw beam is no=0.65um. Figure 3 shows time snapshots of the field profile(first row) and time-averaged force density distributions( second #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 2325induced surface charges, computed in FDTD with ∆y = ∆z = 5.0 nm, is (Fy, Fz) = (1.696, 2.468) pN/m2 , versus the exact value of (1.699, 2.467) pN/m2 . For n = 2.0, FDTD simulations yield (Fy, Fz) = (1.5903, 1.6475) pN/m2 versus the exact value of (1.5911, 1.6493) pN/m2 . Figure 2(b) corresponds to a semi-infinite dielectric with a rapid (linear) transition of the refractive index from no = 1.0 to n1 = 3.4 over a 40 nm-thick region. In this case, the force per unit surface area computed in FDTD is (Fy, Fz) = (1.744, 2.638) pN/m2 . We mention in passing that the case of a finite-diameter beam at Brewster’s incidence on a dielectric wedge is also amenable to analytical as well as numerical solution, and that the numerically computed forces are in excellent agreement with the theoretical values [8]. Fig. 2. Time-snapshots of the Ez component of a p-polarized plane-wave, λo = 0.65 µm, incident at θ = 50° on a semi-infinite dielectric of refractive index n = 3.4, located in the region z < 0. (a) Abrupt transition of the refractive index at z = 0. (b) Linear transition of the refractive index from no = 1.0 to n1 = 3.4 over a 40 nm-thick region. The above results establish the accuracy of our numerical calculations, especially when the surface charge is limited to a single pixel, as was the case in Fig. 2(a). Making smooth transitions at the boundaries between regions of differing refractive indices, as was done in the case of Fig. 2(b), is a tool that is available in numerical simulations. Such smooth transitions at the boundaries may represent physical reality, or they may be used as an artificial tool to eliminate sharp discontinuities and singularities of the equations. To the extent that such smoothing operations do not modify the actual physics of the problem under consideration, they may be used with varying degrees of effectiveness. 4. Force exerted by beam’s edge on the host medium We have shown in [1] that, among other things, the magnetic Lorentz force is responsible for a lateral pressure exerted on the host medium at the edges of a finite-diameter beam; the force per unit area at each edge (i.e., side-wall) of the beam is given by F (edge) = ¼εo(ε − 1)|Eo| 2 . (9) Here |Eo| is the magnitude of the E-field of a (finite-diameter) plane-wave in a medium of dielectric constant ε. If the E-field is parallel (perpendicular) to the beam’s edge, the force is compressive (expansive); in other words, the opposite side-walls of the beam tend to push the medium toward (away from) the beam center. The “edge force” does not appear to be sensitive to the detailed structure of the beam’s edge; in particular, a one-dimensional Gaussian beam exhibits the edge force described by Eq. (9) when its (magnetic) Lorentz force on the host medium is integrated laterally on either side of the beam’s center [1]. Consider a one-dimensional Gaussian beam (uniform along x, Gaussian along y, and propagating in the negative z-direction) in a homogeneous host medium of refractive index n = 2.0; the free-space wavelength of the cw beam is λo = 0.65µm. Figure 3 shows time snapshots of the field profile (first row) and time-averaged force density distributions (second (a) n = 3.4 n = 1.0 n = 3.4 n = 1.0 (b) (C) 2005 OSA 4 April 2005 / Vol. 13, No. 7 / OPTICS EXPRESS 2325 #6863 - $15.00 US Received 14 January 2005; revised 15 March 2005; accepted 15 March 2005