law takes the form E1-+(2 (28) +n Conservation of momentum is guaranteed by matching the boundary condition on the momen- 5. Energy Conservation It is straightforward to numerically integrate the field at any chosen time in the propagation and show that the electromagnetic energy U (nE2+B2) and the gordon total momentum [22] G nE×Hd are conserved. A more general proof of the conservation of Gr can be provided by writing a continuity law. Denoting the magnitude of the electromagnetic momentum density nEXT (31) as gx allows one to write the continuity law (7)as The velocity of the field in the linear medium is c/n in the direction of E X H. Substituting the electromagnetic momentum density (31)into Eq (32)results in a momentum conservation law dg that is redundant with Poynting's theorem, where S=(c/4E x H is the Poynting vector and u=cgx is the energy density. Therefore, Gx is conserved, but is redundant with the electromag etic energy from which it was derived [22. As a matter of linear algebra, the electromagnetic momenta that are quadratic in the field are either redundant or inconsistent with the electro magnetic energy. 6. Radiation Pressure In a pedagogical example, Stone [23] describes the difference between momentum and pseu domomentum in terms of transverse waves on a string momentum is conserved whenever the string, together with any disturbance on it, is translated. When the string is left fixed, but the disturbance is translated, the conserved quantity is pseudomomentum. By analogy, pseudoen- ergy and pseudomomentum travel with the electromagnetic field as the excitation of spacetime degrees of freedom. Then, because spacetime is not in motion, the electromagnetic momen tum and energy that appear in this article in their conventional usage should be reinterpreted as pseudomomentum and pseudoenergy of the electromagnetic field. Pseudoenergy and pseudo- momentum can sometimes be converted to real energy and real momentum [23]. In particular, charges, associated with matter, can couple into the internal degrees of freedom and evince eal-momentum effects such as radiation pressure #77863·S1500USD Received 18 December 2006, accepted 7 January 2007 (C)2007OSA 22 January 2007/Vol 15, No. 2/OPTICS EXPRESS 723law takes the form Eiw = 2n1n2 n2 +n1 Ei w n2 + (2−1) n2 −n1 n2 +n1 Eiw. (28) Conservation of momentum is guaranteed by matching the boundary condition on the momen￾tum flux. 5. Energy Conservation It is straightforward to numerically integrate the field at any chosen time in the propagation and show that the electromagnetic energy U = 1 8π Z V (n 2E 2 +B 2 )dv (29) and the Gordon total momentum [22] Gx = 1 4πc Z V nE×H dv, (30) are conserved. A more general proof of the conservation of Gx can be provided by writing a continuity law. Denoting the magnitude of the electromagnetic momentum density gx = 1 4πc nE×H (31) as gx allows one to write the continuity law (7) as ∇· gxv = − ∂gx ∂t . (32) The velocity of the field in the linear medium is c/n in the direction of E×H. Substituting the electromagnetic momentum density (31) into Eq. (32) results in a momentum conservation law ∇· c 4π E×H = −c ∂gx ∂t (33) that is redundant with Poynting’s theorem, where S = (c/4π)E×H is the Poynting vector and u = cgx is the energy density. Therefore, Gx is conserved, but is redundant with the electromag￾netic energy from which it was derived [22]. As a matter of linear algebra, the electromagnetic momenta that are quadratic in the field are either redundant or inconsistent with the electro￾magnetic energy. 6. Radiation Pressure In a pedagogical example, Stone [23] describes the difference between momentum and pseu￾domomentum in terms of transverse waves on a string. Momentum is conserved whenever the string, together with any disturbance on it, is translated. When the string is left fixed, but the disturbance is translated, the conserved quantity is pseudomomentum. By analogy, pseudoen￾ergy and pseudomomentum travel with the electromagnetic field as the excitation of spacetime degrees of freedom. Then, because spacetime is not in motion, the electromagnetic momen￾tum and energy that appear in this article in their conventional usage should be reinterpreted as pseudomomentum and pseudoenergy of the electromagnetic field. Pseudoenergy and pseudo￾momentum can sometimes be converted to real energy and real momentum [23]. In particular, charges, associated with matter, can couple into the internal degrees of freedom and evince real-momentum effects such as radiation pressure. #77863 - $15.00 USD Received 18 December 2006; accepted 7 January 2007 (C) 2007 OSA 22 January 2007 / Vol. 15, No. 2 / OPTICS EXPRESS 723