exerts a force on the induced current density. The net force can then be obtained by integrating these forces over the entire volume of the dielectric The above method, although involving no approximations, differs from the so-called rigorous"methods commonly used in computing the force of radiation [2-4]. The primary difference is that we sidestep the use of Maxwells stress tensor by reaching directly to bound electrons and their associated currents, treating them as the localized sources of electro nagnetic force under the influence of the lights E-and B-fields. We also avoid the use of the two-component approach, where scattering and gradient forces are treated separately, either in the geometric-optical regime of ray-tracing [5], or in the electromagnetic regime where the light's momentum and intensity gradient are tracked separately [6]. Unlike some of the other approaches, our method computes the force density distribution and not just the total force In this paper we use Finite-Difference-Time-Domain(FDTD) computer simulations to obtain the electromagnetic field distribution in and around several dielectric media. The theoretical underpinnings of our computational method are described in Section 2. Section 3 is devoted to comparisons with exact solutions in the case of a dielectric slab illuminated at normal incidence, and also in the case of a semi-infinite medium under a p-polarized plane- wave at oblique incidence. In Section 4 we show that a one-dimensional gaussian beam propagating in an isotropic, homogeneous medium, exerts either an expansive or a compressive lateral force on its host medium, depending on the beams polarization state. The case of a top-hat-shaped beam entering a semi-infinite dielectric medium at oblique incidence is also covered in Section 4. In Section 5 we study the behavior of a cylindrical glass ro illuminated by a(one-dimensional) Gaussian beam, paying particular attention to the effects of polarization on the force-density distribution. In Section 6 we analyze the single-beam trapping of a small spherical bead, immersed in a liquid, by a sharply focused laser beam both linear and circular polarization states of the beam are shown to result in strong trapping forces. a dielectric half-slab under a one-dimensional gaussian beam centered on one edge of the slab is studied in Section 7. General remarks and conclusions are the subject of Section 8 2. Theoretical considerations To compute the force of the electromagnetic radiation on a given medium, we solve Maxwells equations numerically to determine the distributions of the E-and H-fields(both inside and outside the medium). We then apply the Lorentz law F=pbE+Jb×B, where F is the force density, and Pb and Jb are the bound charge and current densities, respectively [7]. The magnetic induction B is related to the H-field via H. where uo=4I x 10 henrys/meter is the permeability of free space. In the absence of free charges V.D=0, where D=EE+P is the displacement vector, E =88542 x 10 farads/meter is the free-space permittivity, and P is the local polarization density. In linear media, D=EE, where a is the mediums relative permittivity; hence, P=E(E-DE When V D=0, the bound-charge density P,=-V P may be expressed as P,=EVE Inside a homogeneous, isotropic medium, E being proportional to D and V d=0 imply that P=0: no bound charges, therefore, exist inside such media. However, at the interface between two adjacent media, the component of d perpendicular to the interface, Di, must be continuous. The implication is that El is discontinuous and, therefore, bound charges exist at such interfaces; the interfacial bound charges will thus have areal density ob=E(E21-E11 Under the influence of the local E-field, these charges give rise to a Lorentz force densi F=2 Real(o E), where F is the force per unit area of the interface. Since the tangential E- field, Ell, is continuous across the interface, there is no ambiguity as to the value of eythat should be used in computing the force. As for the perpendicular component, the average El across the boundary, 2(E11+ E21), must be used in calculating the interfacial force [1, 8] #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 2322exerts a force on the induced current density. The net force can then be obtained by integrating these forces over the entire volume of the dielectric. The above method, although involving no approximations, differs from the so-called “rigorous” methods commonly used in computing the force of radiation [2-4]. The primary difference is that we sidestep the use of Maxwell’s stress tensor by reaching directly to bound electrons and their associated currents, treating them as the localized sources of electro￾magnetic force under the influence of the light’s E- and B-fields. We also avoid the use of the two-component approach, where scattering and gradient forces are treated separately, either in the geometric-optical regime of ray-tracing [5], or in the electromagnetic regime where the light’s momentum and intensity gradient are tracked separately [6]. Unlike some of the other approaches, our method computes the force density distribution and not just the total force. In this paper we use Finite-Difference-Time-Domain (FDTD) computer simulations to obtain the electromagnetic field distribution in and around several dielectric media. The theoretical underpinnings of our computational method are described in Section 2. Section 3 is devoted to comparisons with exact solutions in the case of a dielectric slab illuminated at normal incidence, and also in the case of a semi-infinite medium under a p-polarized plane￾wave at oblique incidence. In Section 4 we show that a one-dimensional Gaussian beam propagating in an isotropic, homogeneous medium, exerts either an expansive or a compressive lateral force on its host medium, depending on the beam’s polarization state. The case of a top-hat-shaped beam entering a semi-infinite dielectric medium at oblique incidence is also covered in Section 4. In Section 5 we study the behavior of a cylindrical glass rod illuminated by a (one-dimensional) Gaussian beam, paying particular attention to the effects of polarization on the force-density distribution. In Section 6 we analyze the single-beam trapping of a small spherical bead, immersed in a liquid, by a sharply focused laser beam; both linear and circular polarization states of the beam are shown to result in strong trapping forces. A dielectric half-slab under a one-dimensional Gaussian beam centered on one edge of the slab is studied in Section 7. General remarks and conclusions are the subject of Section 8. 2. Theoretical considerations To compute the force of the electromagnetic radiation on a given medium, we solve Maxwell’s equations numerically to determine the distributions of the E- and H-fields (both inside and outside the medium). We then apply the Lorentz law F = ρb E + Jb × B, (1) where F is the force density, and ρb and Jb are the bound charge and current densities, respectively [7]. The magnetic induction B is related to the H-field via B = µ oH, where µ o = 4π × 10−7 henrys/meter is the permeability of free space. In the absence of free charges ∇ · D = 0, where D = εoE + P is the displacement vector, εo = 8.8542 × 10−12 farads/meter is the free-space permittivity, and P is the local polarization density. In linear media, D =εoε E, where ε is the medium’s relative permittivity; hence, P =εo(ε – 1)E. When ∇ · D = 0, the bound-charge density ρb = −∇ · P may be expressed as ρb = εo∇ · E. Inside a homogeneous, isotropic medium, E being proportional to D and ∇ · D = 0 imply that ρb = 0; no bound charges, therefore, exist inside such media. However, at the interface between two adjacent media, the component of D perpendicular to the interface, D⊥, must be continuous. The implication is that E⊥ is discontinuous and, therefore, bound charges exist at such interfaces; the interfacial bound charges will thus have areal density σ b =εo(E2⊥ − E1 ⊥). Under the influence of the local E-field, these charges give rise to a Lorentz force density F = ½ Real(σb E*), where F is the force per unit area of the interface. Since the tangential E￾field, E| |, is continuous across the interface, there is no ambiguity as to the value of E| | that should be used in computing the force. As for the perpendicular component, the average E⊥ across the boundary, ½(E1 ⊥ + E2 ⊥), must be used in calculating the interfacial force [1,8]. (C) 2005 OSA 4 April 2005 / Vol. 13, No. 7 / OPTICS EXPRESS 2322 #6863 - $15.00 US Received 14 January 2005; revised 15 March 2005; accepted 15 March 2005