Given the electromagnetic fields E and H as functions of spatial coordinates and time, we compute the force density distribution by time-averaging the Lorentz equation, namel <F>=(/TV(EV EE+JxuH)dr density P=VEE), and by the magnetic-field, IxB, can be readily identified in Ea, (op a% The time integral in the above equation is taken over one period of the(time-harmonic)li field. Contributions to the Lorentz force by the electric-field, pE (where the total charge 3. Comparison to exact solutions The case of a dielectric slab of index n and thickness d illuminated by a plane-wave at normal incidence was analyzed in our previous paper [1]. To fix the frame of reference, we assume that the slab is parallel to the xy-plane and the beam propagates in the -z direction. At normal incidence, there are no induced charges, and the electromagnetic pressure in its entirety may be attributed to the magnetic component of the Lorentz force. Inside the slab, a pair of counter-propagating plane-waves(along +=)interfere with each other and set up a system of fringes. The force is distributed within the fringe pattern, being positive (ie, in the +2 direction) over one-half of each fringe and negative over the other half. The net force is obtained by integrating the local force density through the thickness of the slab Figure 1 shows the computed force density F: inside a slab illuminated at normal incidence by a plane-wave of wavelength no=640nm and E-field amplitude Eo=1.0 V/m (Computed values of F and Fy were zero, as expected. The slab is suspended in free-space, and has refractive index n=2.0 and thickness d= 110nm. The total force along the =-axis(per unit cross-sectional area)is found to be F()d==-2481 pN/m2(4-=5.0 nm in our FDTD simulations)versus the exact value of.479 pN/m, obtained from theoretical considerations [1]. For a quarter-wave-thick slab, d= 80nm, the simulation yields j F()d==-3192 pN where the exact solution is. 188 pN/m2 1.0 n=2.0 n=1.0 Incident beam z lum] Fig. 1. Computed force density F:(per unit cross-sectional area) versus inside a dielectric slab illuminated with a normally incident plane-wave(o=0.64 um). The slab, suspended free-space, has n=2.0, d=110nm. The incident beam propagates along the negative =-axis As another example, consider a p-polarized plane-wave(1=650 nm)incident at 0=500 on a semi-infinite dielectric of refractive index n=3. 4. located in the region =<0. Figure 2 hows computed time-snapshots of the E: component of the field In Fig. 2(a)the transition of the refractive index from no=1.0 to n, =3. 4 is abrupt, and the force per unit area due to the #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 2324Given the electromagnetic fields E and H as functions of spatial coordinates and time, we compute the force density distribution by time-averaging the Lorentz equation, namely, <F> = (1/T )∫ (E ∇ ·εoE + Ĵ ×µoH ) dt. (8) The time integral in the above equation is taken over one period of the (time-harmonic) light field. Contributions to the Lorentz force by the electric-field, ρE (where the total charge density ρ = ∇ ·εoE ), and by the magnetic-field, J ×B, can be readily identified in Eq.(8). 3. Comparison to exact solutions The case of a dielectric slab of index n and thickness d illuminated by a plane-wave at normal incidence was analyzed in our previous paper [1]. To fix the frame of reference, we assume that the slab is parallel to the xy-plane and the beam propagates in the –z direction. At normal incidence, there are no induced charges, and the electromagnetic pressure in its entirety may be attributed to the magnetic component of the Lorentz force. Inside the slab, a pair of counter-propagating plane-waves (along ± z) interfere with each other and set up a system of fringes. The force is distributed within the fringe pattern, being positive (i.e., in the +z direction) over one-half of each fringe and negative over the other half. The net force is obtained by integrating the local force density through the thickness of the slab. Figure 1 shows the computed force density Fz inside a slab illuminated at normal incidence by a plane-wave of wavelength λo = 640nm and E-field amplitude Eo = 1.0 V/m. (Computed values of Fx and Fy were zero, as expected.) The slab is suspended in free-space, and has refractive index n = 2.0 and thickness d = 110nm. The total force along the z-axis (per unit cross-sectional area) is found to be ∫ Fz(z) dz = −2.481 pN/m2 (∆z = 5.0 nm in our FDTD simulations) versus the exact value of −2.479 pN/m2 , obtained from theoretical considerations [1]. For a quarter-wave-thick slab, d = 80nm, the simulation yields ∫ Fz(z) dz = −3.192 pN/m2 , where the exact solution is −3.188 pN/m2 . Fig. 1. Computed force density Fz (per unit cross-sectional area) versus z inside a dielectric slab illuminated with a normally incident plane-wave (λo = 0.64 µm). The slab, suspended in free-space, has n = 2.0, d = 110nm. The incident beam propagates along the negative z-axis. As another example, consider a p-polarized plane-wave (λo = 650 nm) incident at θ = 50° on a semi-infinite dielectric of refractive index n = 3.4, located in the region z < 0. Figure 2 shows computed time-snapshots of the Ez component of the field. In Fig. 2(a) the transition of the refractive index from no = 1.0 to n1 = 3.4 is abrupt, and the force per unit area due to the 0 T n = 1.0 n = 2.0 n = 1.0 Incident beam (C) 2005 OSA 4 April 2005 / Vol. 13, No. 7 / OPTICS EXPRESS 2324 #6863 - $15.00 US Received 14 January 2005; revised 15 March 2005; accepted 15 March 2005