4. Momentum Conservation Maxwellian continuum electrodynamics describes the interaction of electromagnetic fields and matter in terms of generic material parameters that multiply the fields. As the fields and mat- er are not treated separately, continuum electrodynamics embraces a degree of indeterminacy in the attribution of electromagnetic quantities between field and matter [18]. In this section numerical experiments are used to explore the relationships between electromagnetic quanti- ties in a linear dielectric under controlled conditions. In particular, we numerically demonstrate conservation of the generalized momentum in the two limiting cases e We consider the case of a quasimonochromatic electromagnetic field entering a dielectric edium from the vacuum at normal incidence. It is assumed that dispersion and absorption can reasonably be neglected. The wave equation is solved numerically in one-dimension using a finite-difference time-domain method [19. For the numerical work, the fields are written in terms of an envelope function and a carrier wave. In one dimension, we write the vector poten- tial as A(z, r)=d(z, r)e-i(or-kzer, where d is an envelope function, o is the carrier frequency, and k is the carrier wavenumber. Envelope functions for the electric fieldE=-(1/c)dA/dt. the magnetic induction B=V X A, the displacement field D=n-E, the magnetic field H=B, and other quantities can be defined analogously, as required. The basic phenomenology of a propagating electromagnetic field is demonstrated using the Maxwellian model of a dielectric with a macroscopic refractive index n and numerically solving the wave equation as a2o dd n2 d2d dz cl dr2-2io.d where k=no/c. The approximation of a slowly varying envelope is not made In the first example calculation, antireflective layers are used on the entry and exit faces of the dielectric in order to minimize reflections and thereby simplify the propagation analysis. Figure I shows a typical case in which the electromagnetic field, represented by the envelope of the ector potential Is, starts in vacuum, travels to the right, and enters a linear homogeneous dielectric through a thin gradient-index antireflection layer. The figure shows that the dielectric medium affects the refracted field in two distinct ways. First, the refracted field is reduced width by a factor of the refractive index due to the reduced velocity of the field. Second, the refracted field is reduced in amplitude compared to the incident field due to the creation of the reaction(polarization)field. Both of these effects are reversed upon exiting the medium through a gradient-index antireflection layer, Fig. 2 Momentum is analyzed using the wKB-based formula(25)because the refractive index varies sufficiently slowly that reflections can be neglected. We find that numerical integration of the generalized momentum(25) provides approximate conservation for any chosen time in the propagation. This result is easily confirmed analytically by treating the field as a square pulse, applying the Fresnel boundary condition in the limit of negligible reflection, and scaling the width of the field in the medium. The theoretical conservation law for a square pulse of width w in the vacuum E:w complements the numerical demonstration of momentum conservation in a medium with a slowly varying refractive index. Because the boundary conditions for this exemplar have been devised to minimize reflections, the incident and transmitted fields are essentially identical Then the transmitted field accounts for all the momentum of the incident field and we conclude that no permanent momentum is imparted to a material that does not reflect, or absorb. Further, there is no temporary material momentum because the momentum is fully accounted for at any time that the field is in the medium in whole or in part #77863·S1500USD Received 18 December 2006, accepted 7 January 2007 (C)2007OSA 22 January 2007/Vol 15, No. 2/OPTICS EXPRESS 7194. Momentum Conservation Maxwellian continuum electrodynamics describes the interaction of electromagnetic fields and matter in terms of generic material parameters that multiply the fields. As the fields and mat￾ter are not treated separately, continuum electrodynamics embraces a degree of indeterminacy in the attribution of electromagnetic quantities between field and matter [18]. In this section, numerical experiments are used to explore the relationships between electromagnetic quanti￾ties in a linear dielectric under controlled conditions. In particular, we numerically demonstrate conservation of the generalized momentum in the two limiting cases. We consider the case of a quasimonochromatic electromagnetic field entering a dielectric medium from the vacuum at normal incidence. It is assumed that dispersion and absorption can reasonably be neglected. The wave equation is solved numerically in one-dimension using a finite-difference time-domain method [19]. For the numerical work, the fields are written in terms of an envelope function and a carrier wave. In one dimension, we write the vector poten￾tial as A(z,t) = A (z,t)e −i(ωt−kz) ex, where A is an envelope function, ω is the carrier frequency, and k is the carrier wavenumber. Envelope functions for the electric field E = −(1/c)∂A/∂t, the magnetic induction B = ∇×A, the displacement field D = n 2E, the magnetic field H = B, and other quantities can be defined analogously, as required. The basic phenomenology of a propagating electromagnetic field is demonstrated using the Maxwellian model of a dielectric with a macroscopic refractive index n and numerically solving the wave equation as − ∂ 2A ∂ z 2 −2ik ∂A ∂ z + n 2 c 2 ∂ 2A ∂t 2 −2iω n 2 c 2 ∂A ∂t = 0, (26) where k = nω/c. The approximation of a slowly varying envelope is not made. In the first example calculation, antireflective layers are used on the entry and exit faces of the dielectric in order to minimize reflections and thereby simplify the propagation analysis. Figure 1 shows a typical case in which the electromagnetic field, represented by the envelope of the vector potential |A |, starts in vacuum, travels to the right, and enters a linear homogeneous dielectric through a thin gradient-index antireflection layer. The figure shows that the dielectric medium affects the refracted field in two distinct ways. First, the refracted field is reduced in width by a factor of the refractive index due to the reduced velocity of the field. Second, the refracted field is reduced in amplitude compared to the incident field due to the creation of the reaction (polarization) field. Both of these effects are reversed upon exiting the medium through a gradient-index antireflection layer, Fig. 2. Momentum is analyzed using the WKB-based formula (25) because the refractive index varies sufficiently slowly that reflections can be neglected. We find that numerical integration of the generalized momentum (25) provides approximate conservation for any chosen time in the propagation. This result is easily confirmed analytically by treating the field as a square pulse, applying the Fresnel boundary condition in the limit of negligible reflection, and scaling the width of the field in the medium. The theoretical conservation law for a square pulse of width w in the vacuum Eiw = n 3/2 Ei √ n w n (27) complements the numerical demonstration of momentum conservation in a medium with a slowly varying refractive index. Because the boundary conditions for this exemplar have been devised to minimize reflections, the incident and transmitted fields are essentially identical. Then, the transmitted field accounts for all the momentum of the incident field and we conclude that no permanent momentum is imparted to a material that does not reflect, or absorb. Further, there is no temporary material momentum because the momentum is fully accounted for at any time that the field is in the medium, in whole or in part. #77863 - $15.00 USD Received 18 December 2006; accepted 7 January 2007 (C) 2007 OSA 22 January 2007 / Vol. 15, No. 2 / OPTICS EXPRESS 719