18. J. D. Jackson, Classical Electrodynamics, 2nd Ed, (Wiley, New York, 1975) 19. w.P. Huang, S. T. Chu, A. Goss, and S K. Chaudhuri, A scalar finite-difference time-domain approach uided-wave optics "IEEE Photonics Tech. Lett. 3, 524(1991) 20. A Chubykalo, A. Espinoza, and R. Tzonchev, "Experimental test of the compatibility of the definitions of the romagnetic energy density and the Poynting vector, Eur. Phys. J D 31, 113-120(2004) 21. M. Crenshaw and N. Akozbek, "Electromagnetic energy flux vector for a dispersive linear medium, "Phys. Re E73,056613(206 22. J. P. Gordon, ""Radiation Forces and Momenta in Dielectric Media. "Phys. Rev. A 8, 14-21(1973). 23. M. Stone, Phonons and Forces: Momentum versus Pseudomomentum in Moving Fluids, "ar Xiv. org. cond- mat/0012316(2000),http://arxiv.org/al 24. M. Mansuripur, "Radiation pressure and the linear momentum of the electromagnetic field, Opt. Express 12, 5375-5401(2004,htto//w ct.cfm?URl=oe- 12-22-5375 25. R. Loudon. S. M. Barnett and C. diation pressure and momentum transfer in dielectrics the photon drag effect, "Phys. Rev. A 71, 063802(2005 26. M. Scalora. G. D'Aguanno, N. Mattiucci. M. J. Bloemer, M. Centini, C. Sibilia, and J. W. Haus. "Radi pressure of light pulses and conservation of linear momentum in dispersive media, Phys. Rev. E 73, 05 1. Introduction Since the early years of relativity theory, there has been a controversy regarding the correct rel- ativistic form of the energy-momentum tensor for an electromagnetic field in a linear medium The energy-momentum tensor proposed by Minkowski [1] was faulted for a lack of symmetry leading Abraham[2] to suggest a symmetric form. Einstein and Laub [3], Peierls [4, Kranys [5], Livens [6], and others[7, 8] have proposed variants of the Abraham and Minkowski tensors while still other workers [9] have endorsed one or the other of the principal results. The ele ment of the energy-momentum tensor related to the electromagnetic momentum density is the ain point of contention in the Abraham-Minkowski controversy. While the issue of whether the momentum fux of an electromagnetic field is increased or decreased by the presence of a refractive medium appears to be uncomplicated, experimental measurements[10, 11, 12, 13 have been unable to conclusively identify the electromagnetic momentum density with either the Abraham or the Minkowski formula, or with any of the variant formulas [14, 15]. An in- consistency of this magnitude and persistence in the theoretical and experimental treatment of a simple physical system suggests problems of a fundamental nature Noether's theorem connects conservation laws to symmetries[16]. Conservation of energy is associated with invariance with respect to time translation and conservation of linear momen- tum requires invariance with respect to spatial translation. Because the posited formulas for the momentum density in a dispersionless medium are quadratic in the field, these quantities are as a matter of linear algebra, either inconsistent or redundant with the electromagnetic energy [17]. In particular, the degeneracy of the energy and momentum of the electromagnetic field in the vacuum is implicated as the crux of the Abraham-Minkowski controversy. Since there is no general spatial invariance property for dielectrics, one should ask under what conditions a quantity that behaves like momentum can be derived. We show that, by using the Fresnel boundary conditions to connect spatially invariant regions of linear media, eneralized electromagnetic momentum can be derived in the limiting cases of i)a piecewise homogeneous medium and ii) a medium with a slowly varying refractive index in the WKB limit. Both generalized ta depend linearly on the field but the refractive index appears to different powers due to the difference in the translational symmetry. Momentum conservation is demonstrated numerically and theoretically in both limiting cases. For the case of a material with a slowly varying index, the momentum of the transmitted field is essentially equal to that of the incident field and no momentum is transferred to the material. However, a field entering a homogeneous medium from the vacuum imparts a permanent dynamic momentum to the material that is twice the momentum of the reflected field if momentum is to be conserved #77863·S1500USD Received 18 December 2006, accepted 7 January 2007 (C)2007OSA 22 January 2007/ Vol 15, No. 2/OPTICS EXPRESS 71518. J. D. Jackson, Classical Electrodynamics, 2nd Ed., (Wiley, New York, 1975). 19. W. P. Huang, S. T. Chu, A. Goss, and S. K. Chaudhuri, “A scalar finite-difference time-domain approach to guided-wave optics,” IEEE Photonics Tech. Lett. 3, 524 (1991). 20. A. Chubykalo, A. Espinoza, and R. Tzonchev, “Experimental test of the compatibility of the definitions of the electromagnetic energy density and the Poynting vector,” Eur. Phys. J. D 31, 113–120 (2004). 21. M. Crenshaw and N. Akozbek, “Electromagnetic energy flux vector for a dispersive linear medium,” Phys. Rev. E 73, 056613 (2006). 22. J. P. Gordon, “Radiation Forces and Momenta in Dielectric Media,” Phys. Rev. A 8, 14-21 (1973). 23. M. Stone, “Phonons and Forces: Momentum versus Pseudomomentum in Moving Fluids,” arXiv.org, cond￾mat/0012316 (2000), http://arxiv.org/abs/cond-mat?papernum=0012316. 24. M. Mansuripur, “Radiation pressure and the linear momentum of the electromagnetic field,” Opt. Express 12, 5375–5401 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-22-5375. 25. R. Loudon, S. M. Barnett, and C. Baxter, “Theory of radiation pressure and momentum transfer in dielectrics: the photon drag effect,” Phys. Rev. A 71, 063802 (2005). 26. M. Scalora, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, M. Centini, C. Sibilia, and J. W. Haus, “Radiation pressure of light pulses and conservation of linear momentum in dispersive media,” Phys. Rev. E 73, 056604 (2006). 1. Introduction Since the early years of relativity theory, there has been a controversy regarding the correct rel￾ativistic form of the energy–momentum tensor for an electromagnetic field in a linear medium. The energy–momentum tensor proposed by Minkowski [1] was faulted for a lack of symmetry leading Abraham [2] to suggest a symmetric form. Einstein and Laub [3], Peierls [4], Kranys [5], Livens [6], and others [7, 8] have proposed variants of the Abraham and Minkowski tensors while still other workers [9] have endorsed one or the other of the principal results. The ele￾ment of the energy–momentum tensor related to the electromagnetic momentum density is the main point of contention in the Abraham–Minkowski controversy. While the issue of whether the momentum flux of an electromagnetic field is increased or decreased by the presence of a refractive medium appears to be uncomplicated, experimental measurements [10, 11, 12, 13] have been unable to conclusively identify the electromagnetic momentum density with either the Abraham or the Minkowski formula, or with any of the variant formulas [14, 15]. An in￾consistency of this magnitude and persistence in the theoretical and experimental treatment of a simple physical system suggests problems of a fundamental nature. Noether’s theorem connects conservation laws to symmetries [16]. Conservation of energy is associated with invariance with respect to time translation and conservation of linear momen￾tum requires invariance with respect to spatial translation. Because the posited formulas for the momentum density in a dispersionless medium are quadratic in the field, these quantities are, as a matter of linear algebra, either inconsistent or redundant with the electromagnetic energy [17]. In particular, the degeneracy of the energy and momentum of the electromagnetic field in the vacuum is implicated as the crux of the Abraham–Minkowski controversy. Since there is no general spatial invariance property for dielectrics, one should ask under what conditions a quantity that behaves like momentum can be derived. We show that, by using the Fresnel boundary conditions to connect spatially invariant regions of linear media, a generalized electromagnetic momentum can be derived in the limiting cases of i) a piecewise homogeneous medium and ii) a medium with a slowly varying refractive index in the WKB limit. Both generalized momenta depend linearly on the field but the refractive index appears to different powers due to the difference in the translational symmetry. Momentum conservation is demonstrated numerically and theoretically in both limiting cases. For the case of a material with a slowly varying index, the momentum of the transmitted field is essentially equal to that of the incident field and no momentum is transferred to the material. However, a field entering a homogeneous medium from the vacuum imparts a permanent dynamic momentum to the material that is twice the momentum of the reflected field, if momentum is to be conserved. #77863 - $15.00 USD Received 18 December 2006; accepted 7 January 2007 (C) 2007 OSA 22 January 2007 / Vol. 15, No. 2 / OPTICS EXPRESS 715