Because the change of the momentum of the material is due to reflection, radiation pressure is deemed to be a surface force acting over the illuminated area of the material and we show that the lorentz volume force is ni 2. Piecewise homogeneous media A piecewise homogeneous linear medium represents the special case of a material that, neglect- g absorption, can be represented as a composite of finite translationally invariant spacetimes In a previous short communication [17], we derived a conserved electromagnetic momentum for a piecewise homogeneous medium from the macroscopic effective Hamiltonian and used the result to derive the fresnel relations. However. the derivation of an effective longitudinal momentum by averaging the dynamics of the harmonic oscillators that comprise the model of the electromagnetic field might be viewed as disconcerting. Here, we take the opposite ap- proach and show that the Fresnel relations imply the continuity of two quantities: the elec tromagnetic energy flux and the flux of something else. We show that the unknown conserved electromagnetic quantity has the characteristics of a linear momentum. Because the electromag- netic momentum is only determined to within a constant of proportionality, this derivation is not as complete as the Hamiltonian-based theory. However, the Fresnel-based derivation com- plements the prior work[17], in which it is implicit, by providing a simple direct derivation of the electromagnetic momentum in terms of familiar continuum electrodynamic concepts that can serve as a platform for extensions of the theory We consider the boundary conditions at the interface of two homogeneous linear media. A plane electromagnetic wave is normally incident from a medium VI with refractive index nI into a medium V2 with index n >nI, where ni and n2 are real. The fields are assumed to be monochromatic plane waves polarized in the x-direction and we write Ei=erie Er=e ere-i(ot+kri Er e tei(or-ka as the respective amplitudes of the incident, reflected, and refracted waves. If we assume the Fresnel relations E.-n2-nI +n2 then equivalent Fresnel equations n e=mE+ne Ei=et+er can be derived algebraically. The Fresnel equations (3)and (4)are recognized as continuity equations in which the rate at which some electromagnetic quantity arrives at the boundary is equal to the rate at which that quantity leaves the boundary. Equation(3)expresses continuity of a flux S=ynE with an undetermined constant y. The Fresnel continuity equation(4) represents continuity of #77863·S1500USD Received 18 December 2006, accepted 7 January 2007 (C)2007OSA 22 January 2007/ Vol 15, No. 2/OPTICS EXPRESS 716Because the change of the momentum of the material is due to reflection, radiation pressure is deemed to be a surface force acting over the illuminated area of the material and we show that the Lorentz volume force is nil. 2. Piecewise Homogeneous Media A piecewise homogeneous linear medium represents the special case of a material that, neglect￾ing absorption, can be represented as a composite of finite translationally invariant spacetimes. In a previous short communication [17], we derived a conserved electromagnetic momentum for a piecewise homogeneous medium from the macroscopic effective Hamiltonian and used the result to derive the Fresnel relations. However, the derivation of an effective longitudinal momentum by averaging the dynamics of the harmonic oscillators that comprise the model of the electromagnetic field might be viewed as disconcerting. Here, we take the opposite ap￾proach and show that the Fresnel relations imply the continuity of two quantities: the elec￾tromagnetic energy flux and the flux of something else. We show that the unknown conserved electromagnetic quantity has the characteristics of a linear momentum. Because the electromag￾netic momentum is only determined to within a constant of proportionality, this derivation is not as complete as the Hamiltonian-based theory. However, the Fresnel-based derivation com￾plements the prior work [17], in which it is implicit, by providing a simple direct derivation of the electromagnetic momentum in terms of familiar continuum electrodynamic concepts that can serve as a platform for extensions of the theory. We consider the boundary conditions at the interface of two homogeneous linear media. A plane electromagnetic wave is normally incident from a medium V1 with refractive index n1 into a medium V2 with index n2 > n1, where n1 and n2 are real. The fields are assumed to be monochromatic plane waves polarized in the x-direction and we write Ei = exEie −i(ωt−kiz) Er = exEre −i(ωt+krz) Et = exEte −i(ωt−ktz) as the respective amplitudes of the incident, reflected, and refracted waves. If we assume the Fresnel relations Er = n2 −n1 n1 +n2 Ei (1) Et = 2n1 n1 +n2 Ei (2) then equivalent Fresnel equations n1E 2 i = n2E 2 t +n1E 2 r (3) Ei = Et +Er (4) can be derived algebraically. The Fresnel equations (3) and (4) are recognized as continuity equations in which the rate at which some electromagnetic quantity arrives at the boundary is equal to the rate at which that quantity leaves the boundary. Equation (3) expresses continuity of a flux S = γn|E| 2 (5) with an undetermined constant γ. The Fresnel continuity equation (4) represents continuity of a flux T = α|E| (6) #77863 - $15.00 USD Received 18 December 2006; accepted 7 January 2007 (C) 2007 OSA 22 January 2007 / Vol. 15, No. 2 / OPTICS EXPRESS 716