complex amplitudes such as E are expressed as elexp(idE). Time-averaged products of two fields, say, Real (A)=| I cos(@t-oA) and Real(B)=B I cos(@t-oB), given by v2) AB I cos(OA-oB), may also be written /Real(AB") To compute the force exerted by the electromagnetic field on a given medium, we use Maxwell's equations to determine the distributions of the E-and H-fields both inside and outside the medium. We then apply the lorentz law F=q(E+vx B), which gives the electromagnetic force on a particle of charge q and velocity V. The magnetic induction B is assumed to be related to the H-field via B=loH, where lo=4I x 10 henrys/meter is the permeability of free space, in other words, any magnetic moments that might exist in the medium and their interactions with the radiation field at optical frequencies are ignored Typically, there are no free charges in the system, so V D=0, where D=EE+ P is the electric displacement vector, E =8.82 x 10 farads/meter is the permittivity of free space, and P is the local polarization density within the medium. In linear media, D=Eoe E, where E is the mediums relative permittivity; hence, P=E(E-lE. We ignore the frequency- dependence of e throughout the paper and treat the media as non-dispersive When V D=0, the density of bound charges P,=-V P may be expressed as Pb=EV- E Inside a homogeneous and isotropic medium, E being proportional to D and V.D=0 imply that PB=0; no bound charges, therefore, can exist inside such media. However, at the interface between two different media, the component of d perpendicular to the interface, Di, must be continuous. The implication is that El is discontinuous and therefore, bound charges can exist at such interfaces; the interfacial bound charges will thus have an areal density o=E(E21- Eln. Under the influence of the local E-field, these charges give rise to an electric Lorentz force F=VReal(oE), where F is the force per unit area of there is no ambiguity as to which field must be used in conjunction with the lorentz law s the interface. Since the tangential E-field, El, is generally continuous across the interi for the perpendicular component, the average E across the boundary, 2(E11+ E21), must be used in calculating the interfacial force. (The use of the av erage El in this context is not a matter of choice; it is the only way to get the calculated force at the boundary to agree with the time rate of change of the momentum that passes through the interface. From a physical andpoint, the interfacial charges produce a local Ei that has the same magnitude but opposite directions on the two sides of the interface. It is this locally-generated Ei that is responsible for the E-fields discontinuity. Averaging El across the interface eliminates the local field, as it should, since the charge cannot exert a force on itself. Since V. B=0 and B=uoh, the perpendicular H-field, Hi, at the interface between adjacent media must remain continuous. The tangential H-field at such interfaces, however, may be discontinuous. This, in accordance with Maxwells equation VxH=J+ aD/a, gives rise to an interfacial current density J, =H21-HIl Such currents can exist on the surfaces of ood conductors, where Eyl is negligible, yet the high conductance of the medium permits the flow of the surface current. Elsewhere, the only source of electrical currents are bound charges, with the bound current density being Jp=ap/dt=E(e-1)dE/ar. Assuming time harmonic fields with the time-dependence factor exp(ian), we can write J=-ioE(E-lE The H-field of the electromagnetic wave then exerts a force on the bound current according to the Lorentz law, namely, F=Real ( bx B), where F is force per unit volume Note: For time-harmonic fields, the contribution of conduction electrons to current density may be combined with that of bound electrons. Since J=o E, where O is the conductivity of the medium, the net current density Jc Jb may be attributed to an effective dielectric constant E+i(oc/E@). In general, since E is complex-valued, there is no need to distinguish conduction electrons from bound electrons, and e may be treated as an effective dielectric constant that contains both contributions. An exception will be made in Section 3 in the case #5025-S1500US Received 10 August 2004; revised 13 October 2004; accepted 20 October 2004 (C)2004OSA November 2004/Vol 12. No 22/OPTICS EXPRESS 5378complex amplitudes such as E are expressed as | E | exp(iφE). Time-averaged products of two fields, say, Real (A ) = | A | cos(ω t – φA) and Real (B ) = |B | cos(ω t – φB), given by ½| AB | cos(φA – φB), may also be written ½Real (AB*). To compute the force exerted by the electromagnetic field on a given medium, we use Maxwell’s equations to determine the distributions of the E- and H-fields both inside and outside the medium. We then apply the Lorentz law F = q (E + V × B), which gives the electromagnetic force on a particle of charge q and velocity V. The magnetic induction B is assumed to be related to the H-field via B = µ oH, where µ o = 4π × 10−7 henrys/meter is the permeability of free space; in other words, any magnetic moments that might exist in the medium and their interactions with the radiation field at optical frequencies are ignored. Typically, there are no free charges in the system, so ∇ · D = 0, where D = εoE + P is the electric displacement vector, εo = 8.82 × 10−12 farads/meter is the permittivity of free space, and P is the local polarization density within the medium. In linear media, D =εoε E, where ε is the medium’s relative permittivity; hence, P =εo(ε – 1)E. We ignore the frequency￾dependence of ε throughout the paper and treat the media as non-dispersive. When ∇ · D = 0, the density of bound charges ρb = −∇ · P may be expressed as ρb =εo∇ · E. Inside a homogeneous and isotropic medium, E being proportional to D and ∇ · D = 0 imply that ρb = 0; no bound charges, therefore, can exist inside such media. However, at the interface between two different media, the component of D perpendicular to the interface, D⊥, must be continuous. The implication is that E⊥ is discontinuous and, therefore, bound charges can exist at such interfaces; the interfacial bound charges will thus have an areal density σ =εo(E2⊥ − E1 ⊥). Under the influence of the local E-field, these charges give rise to an electric Lorentz force F = ½Real(σ E*), where F is the force per unit area of the interface. Since the tangential E-field, E| | , is generally continuous across the interface, there is no ambiguity as to which field must be used in conjunction with the Lorentz law. As for the perpendicular component, the average E across the boundary, ½(E1 ⊥ + E2 ⊥), must be used in calculating the interfacial force. (The use of the average E⊥ in this context is not a matter of choice; it is the only way to get the calculated force at the boundary to agree with the time rate of change of the momentum that passes through the interface. From a physical standpoint, the interfacial charges produce a local E⊥ that has the same magnitude but opposite directions on the two sides of the interface. It is this locally-generated E⊥ that is responsible for the E-field’s discontinuity. Averaging E⊥ across the interface eliminates the local field, as it should, since the charge cannot exert a force on itself.) Since ∇ · B = 0 and B = µ oH, the perpendicular H-field, H⊥ , at the interface between adjacent media must remain continuous. The tangential H-field at such interfaces, however, may be discontinuous. This, in accordance with Maxwell’s equation ∇ × H = J + ∂D/∂t, gives rise to an interfacial current density Js = H2 | | − H1 | | . Such currents can exist on the surfaces of good conductors, where E| | is negligible, yet the high conductance of the medium permits the flow of the surface current. Elsewhere, the only source of electrical currents are bound charges, with the bound current density being Jb = ∂P/∂t = εo(ε – 1)∂E/∂t. Assuming time￾harmonic fields with the time-dependence factor exp(−iω t), we can write Jb = −iω εo(ε – 1)E. The H-field of the electromagnetic wave then exerts a force on the bound current according to the Lorentz law, namely, F = ½Real (Jb × B*), where F is force per unit volume. Note: For time-harmonic fields, the contribution of conduction electrons to current density may be combined with that of bound electrons. Since Jc = σcE, where σc is the conductivity of the medium, the net current density Jc + Jb may be attributed to an effective dielectric constant ε + i(σc /εoω ). In general, since ε is complex-valued, there is no need to distinguish conduction electrons from bound electrons, and ε may be treated as an effective dielectric constant that contains both contributions. An exception will be made in Section 3 in the case (C) 2004 OSA 1 November 2004 / Vol. 12, No. 22 / OPTICS EXPRESS 5378 #5025- $15.00 US Received 10 August 2004; revised 13 October 2004; accepted 20 October 2004