can be related to the electric field energy density by n-E We may also write the momentum flux vector T=gy (19) and the momentum in terms of the momentum density (17)with a concrete coefficient 3. Slowly Varying Refractive Index A spatially inhomogeneous medium can be thought of as a sequence of spatially homogeneous media of vanishingly small width. Then the Fresnel relations can be employed in a WKB treat- ment of electromagnetic momentum in an inhomogeneous medium for which the refractive index varies sufficiently slowly that the reflection is negligible. Expanding the Fresnel relation (2)in a power series, the refracted field can be written in terms of the incident field as neT=NiE for the case in which An=n2-nI is sufficiently small that reflection can be neglecte (21)represents the continuity of the flux T=avnE (22) at the interface between the two materials. Starting from the vacuum and repeatedly applying the boundary condition Eq (21), we obtain the WKB results (23) g(x)=T(x)/(x)=n32(x)E() Integrating the momentum density ge: over the volume, we obtain the conserved quantity /国体 as the momentum of the field in an inhomogeneous linear medium in the slowly varying index Conservation of momentum requires spatial invariance and we should not necessarily expect momentum formula to apply in all cases. The significance of the variant momentum(25)is that it provides a clear demonstration that momentum conservation depends on the inhomo- geneity of the medium and that momentum conservation laws need to be tested for media with different types of inhomogeneity #77863·S1500USD Received 18 December 2006, accepted 7 January 2007 (C)2007OSA 22 January 2007/Vol 15, No. 2/OPTICS EXPRESS 718can be related to the electric field energy density by ue = 1 8π n 2 |E| 2 = |g| 2 2ρ0 . (18) We may also write the momentum flux vector T = |g|v = r ρ0 4π n|E| c n ez (19) and the momentum G = Z V gdv = r ρ0 4π Z V n|E|dvez (20) in terms of the momentum density (17) with a concrete coefficient. 3. Slowly Varying Refractive Index A spatially inhomogeneous medium can be thought of as a sequence of spatially homogeneous media of vanishingly small width. Then the Fresnel relations can be employed in a WKB treat￾ment of electromagnetic momentum in an inhomogeneous medium for which the refractive index varies sufficiently slowly that the reflection is negligible. Expanding the Fresnel relation (2) in a power series, the refracted field can be written in terms of the incident field as √ n2Et = √ n1Ei (21) for the case in which ∆n = n2−n1 is sufficiently small that reflection can be neglected. Equation (21) represents the continuity of the flux T = α √ n|E| (22) at the interface between the two materials. Starting from the vacuum and repeatedly applying the boundary condition Eq. (21), we obtain the WKB results T(z) = α p n(z)|E(z)|ez (23) g(z) = T(z)/v(z) = α c n 3/2 (z)|E(z)|. (24) Integrating the momentum density gez over the volume, we obtain the conserved quantity G = Z V gdvez = r ρ0 4π Z V n 3/2 |E|dvez (25) as the momentum of the field in an inhomogeneous linear medium in the slowly varying index limit. Conservation of momentum requires spatial invariance and we should not necessarily expect a momentum formula to apply in all cases. The significance of the variant momentum (25) is that it provides a clear demonstration that momentum conservation depends on the inhomo￾geneity of the medium and that momentum conservation laws need to be tested for media with different types of inhomogeneity. #77863 - $15.00 USD Received 18 December 2006; accepted 7 January 2007 (C) 2007 OSA 22 January 2007 / Vol. 15, No. 2 / OPTICS EXPRESS 718