《电动力学》课程参考文献：Electromagnetic Momentum and Radiation Pressure derived from the Fresnel Relations

Electromagnetic Momentum and Radiation pressure derived from the Fresnel relations Michael e. crenshaw AMSRD-AMR-WS-ST. USA RDECOM. Redstone ArsenaL. Alabama 35898 Abstract: Using the Fresnel relations as axioms, we derive a generalized electromagnetic momentum for a piecewise homogeneous medium and a different generalized momentum for a medium with a spatially varying re- fractive index in the Wentzel-Kramers-Brillouin(WKB)limit. Both gener- alized momenta depend linearly on the field, but the refractive index appears to different powers due to the difference in translational symmetry. For the case of the slowly varying index, it is demonstrated that there is negligible transfer of momentum from the electromagnetic field to the material Such a transfer occurs at the interface between the vacuum and a homogeneous ma- terial allowing us to derive the radiation pressure from the Fresnel reflection formula. The Lorentz volume force is shown to be nil OCIS codes: (260.2110) Electromagnetic theory; (260.2160) Energy transfer References and links I. H. Minkowski, Natches. Ges. Wiss. Gottingen 53(1908): Math. Ann. 68, 472(1910) 2. M. Abraham, Rend Circ. Mat. Palermo 28, 1(1909); 30, 33 (1910) 3. A Einstein and J. Laub, Ann. Phys. (Leipzig)26, 541(1908) 4. R Peierls, "The momentum of light in a refracting medium, "Proc R Soc. Lond. A 347, 475-491(1976). 5. M. Kranys, "The Minkowski and Abraham Tensors, and the non-uniqueness of non-closed systems. In Engng.sci20,1193-1213(1982 6. G. H. Livens, The Theory of Electricity.( Cambridge University Press, Cambridge, 1908) 7. Y. N. Obukhov and F. w. Hehl, ""Electromagnetic energy-momentum and forces in matter, "Phys. Lett. A 311 277-284(2003) 8. J C. Garrison and R. Y Chiao. "Canonical and kinetic forms of the electromagnetic momentum in quantization scheme for a dispersive dielectric, "Phys. Rev. A 70, 053826-1-8 9. S. Antoci and L. Mihich, "A forgotten argument by gordon uniquely selects momentum tensor of the electromagnetic field in homogeneous, isotropic matte Cim.B112,991-1001 10. R. V. Jones and J. C. S. Richards, "The pressure of radiation in a refracting medium. "Proc. R. Soc. London A 221,480(1954) 11. A. Ashkin and (1973) 12. A. F. Gibson, M. E. Kimmitt, A O. Koohian, D. E. Evans, and G. F D. Levy, "A Study of Radiation Pressure in a Refractive Medium by the Photon Drag Effect, "Proc. R Soc. London A 370, 303-318(1980). 13. D. G. Lahoz and G. M. Graham, ""Experimental decision on the electromagnetic momentum. "J. Phys. A 15, 303-318(1982 14. I Brevik, "Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum ten- sor”"Phys.Rep.52,133-201(1979) 15. I Brevik, "Photon-drag experiment and the electromagnetic momentum in matter, "Phys. Rev. B 33, 1058-106 (1986). H. Goldstein. Classical 17. M. E Crenshaw, "Generalized electromagnetic momentum and the Fresnel relations "Phys. Lett. A 346, 249- #77863·S1500USD Received 18 December 2006, accepted 7 January 2007 (C)2007OSA January 2007/Vol 15, No. 2/OPTICS EXPRESS 714

Electromagnetic Momentum and Radiation Pressure derived from the Fresnel Relations Michael E. Crenshaw AMSRD-AMR-WS-ST, USA RDECOM, Redstone Arsenal, Alabama 35898 michael.crenshaw@us.army.mil Abstract: Using the Fresnel relations as axioms, we derive a generalized electromagnetic momentum for a piecewise homogeneous medium and a different generalized momentum for a medium with a spatially varying re￾fractive index in the Wentzel–Kramers–Brillouin (WKB) limit. Both gener￾alized momenta depend linearly on the field, but the refractive index appears to different powers due to the difference in translational symmetry. For the case of the slowly varying index, it is demonstrated that there is negligible transfer of momentum from the electromagnetic field to the material. Such a transfer occurs at the interface between the vacuum and a homogeneous ma￾terial allowing us to derive the radiation pressure from the Fresnel reflection formula. The Lorentz volume force is shown to be nil. OCIS codes: (260.2110) Electromagnetic theory; (260.2160) Energy transfer References and links 1. H. Minkowski, Natches. Ges. Wiss. Gottingen 53 (1908); Math. Ann. ¨ 68, 472 (1910). 2. M. Abraham, Rend. Circ. Mat. Palermo 28, 1 (1909); 30, 33 (1910). 3. A. Einstein and J. Laub, Ann. Phys. (Leipzig) 26, 541 (1908). 4. R. Peierls, “The momentum of light in a refracting medium,” Proc. R. Soc. Lond. A 347, 475–491 (1976). 5. M. Kranys, “The Minkowski and Abraham Tensors, and the non-uniqueness of non-closed systems,” Int. J. Engng. Sci. 20, 1193–1213 (1982). 6. G. H. Livens, The Theory of Electricity, (Cambridge University Press, Cambridge, 1908). 7. Y. N. Obukhov and F. W. Hehl, “Electromagnetic energy–momentum and forces in matter,” Phys. Lett. A 311, 277-284 (2003). 8. J. C. Garrison and R. Y. Chiao, “Canonical and kinetic forms of the electromagnetic momentum in an ad hoc quantization scheme for a dispersive dielectric,” Phys. Rev. A 70, 053826-1–8 (2004). 9. S. Antoci and L. Mihich, “A forgotten argument by Gordon uniquely selects Abraham’s tensor as the energy– momentum tensor of the electromagnetic field in homogeneous, isotropic matter,” Nuovo Cim. B112, 991–1001 (1997). 10. R. V. Jones and J. C. S. Richards, “The pressure of radiation in a refracting medium,” Proc. R. Soc. London A 221, 480 (1954). 11. A. Ashkin and J. M. Dziedzic, “Radiation Pressure on a Free Liquid Surface,” Phys. Rev. Lett. 30, 139–142 (1973). 12. A. F. Gibson, M. F. Kimmitt, A. O. Koohian, D. E. Evans, and G. F. D. Levy, “A Study of Radiation Pressure in a Refractive Medium by the Photon Drag Effect,” Proc. R. Soc. London A 370, 303–318 (1980). 13. D. G. Lahoz and G. M. Graham, “Experimental decision on the electromagnetic momentum,” J. Phys. A 15, 303–318 (1982). 14. I. Brevik, “Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum ten￾sor,” Phys. Rep. 52, 133–201 (1979). 15. I. Brevik, “Photon-drag experiment and the electromagnetic momentum in matter,” Phys. Rev. B 33, 1058–1062 (1986). 16. H. Goldstein, Classical Mechanics, 2nd Ed., (Addison-Wesley, Reading, MA, 1980). 17. M. E. Crenshaw, “Generalized electromagnetic momentum and the Fresnel relations,” Phys. Lett. A 346, 249– 254, (2005). #77863 - $15.00 USD Received 18 December 2006; accepted 7 January 2007 (C) 2007 OSA 22 January 2007 / Vol. 15, No. 2 / OPTICS EXPRESS 714

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 715

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 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

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 716

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 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

that agai ins an unknown constant. For a generic property, the continuity law has the form of dp where p is the property density and pv is the property flux vector. It is then a simple matter to derive the conservation laws that correspond to the continuous fluxes. We define flux vectors S=mEle T=aee such that S=S andT=T. Denoting the respective property densities as u and g, we have S/V=-ymEL (11) Integrating the property densities over the appropriate volume, we obtain the conservation laws 1听E如= nE dv+/n2 Eidv n Erd+/neRdy We identify Eq(12)as the conservation law for electromagnetic energy for a monochromatic lane wave. Equation(13)is the conservation law for the property n (14) We only need to show that the conserved quantity G, taken as a vector G= Gez, has prop- erties of linear momentum. The second Fresnel continuity equation, Eq (4), is algebraically equivalent t Likewise, the conservation law (13)can be written as ne二几ne一n小e+2nE Then the conserved quantity has the characteristics of linear momentum in which the momen tum of the reflection is in the negative direction and twice the momentum of the reflection is imputed to the material in the forward direction The constants of proportionality for the conserved quantities cannot be determined by the current procedure due to the nature of the Fresnel relations as linear boundary conditions. How ever,we can identify y=c/(4) based on the known form for the electromagnetic energy for a monochromatic plane wave. By comparison with the prior work [17], the value of a is given in terms of a unit mass density po as a=vc2po/(47). Then the momentum density #77863·S1500USD Received 18 December 2006, accepted 7 January 2007 (C)2007OSA 22 January 2007/ Vol 15, No. 2/OPTICS EXPRESS 717

that again contains an unknown constant. For a generic property, the continuity law has the form of ∇· ρv = − ∂ ρ ∂t (7) where ρ is the property density and ρv is the property flux vector. It is then a simple matter to derive the conservation laws that correspond to the continuous fluxes. We define flux vectors S = γn|E| 2 ez (8) T = α|E|ez (9) such that S = |S| and T = |T|. Denoting the respective property densities as u and g, we have u = S/v = n c γn|E| 2 (10) g = T/v = n c α|E|. (11) Integrating the property densities over the appropriate volume, we obtain the conservation laws Z V1 n 2 1E 2 i dv = Z V1 n 2 1E 2 r dv+ Z V2 n 2 2E 2 t dv (12) Z V1 n1Eidv = Z V1 n1Erdv+ Z V2 n2Etdv. (13) We identify Eq. (12) as the conservation law for electromagnetic energy for a monochromatic plane wave. Equation (13) is the conservation law for the property G = α c Z V n|E|. (14) We only need to show that the conserved quantity G, taken as a vector G = Gez , has prop￾erties of linear momentum. The second Fresnel continuity equation, Eq. (4), is algebraically equivalent to Ei = Et −Er +2Er . (15) Likewise, the conservation law (13) can be written as Z V1 n1Eidvez = Z V2 n2Etdvez − Z V1 n1Erdvez +2 Z V1 n1Erdvez . (16) Then the conserved quantity has the characteristics of linear momentum in which the momen￾tum of the reflection is in the negative direction and twice the momentum of the reflection is imputed to the material in the forward direction. The constants of proportionality for the conserved quantities cannot be determined by the current procedure due to the nature of the Fresnel relations as linear boundary conditions. How￾ever, we can identify γ = c/(4π) based on the known form for the electromagnetic energy for a monochromatic plane wave. By comparison with the prior work [17], the value of α is given in terms of a unit mass density ρ0 as α = p c 2ρ0/(4π). Then the momentum density g = r ρ0 4π n|E|ez (17) #77863 - $15.00 USD Received 18 December 2006; accepted 7 January 2007 (C) 2007 OSA 22 January 2007 / Vol. 15, No. 2 / OPTICS EXPRESS 717

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 718

can 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

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 719

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￾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

5 0.6 00t Propagation of the vector potential from vacuum into a linear medium with a ent-index antireflection layer on thethe entry and exit faces. The shaded region is file of the index of refraction The field travels to the right and the horizontal axis is scaled to the wavelengt 70-50-30·101030507 Fig. 2. Same as Fig. I after the field has propagated out of the medium through the anti- flection #77863·S1500USD Received 18 December 2006, accepted 7 January 2007 (C)2007OSA 22 January 2007/ Vol 15, No. 2/OPTICS EXPRESS 720

-70 -50 -30 -10 10 30 50 70 DISTANCE (wavelengths) 0.0 0.2 0.4 0.6 0.8 1.0 AMPLITUDE (arb. units) 1.0 1.1 1.2 1.3 1.4 1.5 1.6 REFRACTIVE INDEX (no units) Fig. 1. Propagation of the vector potential from vacuum into a linear medium with a gradient-index antireflection layer on the the entry and exit faces. The shaded region is the profile of the index of refraction. The field travels to the right and the horizontal axis is scaled to the wavelength. -70 -50 -30 -10 10 30 50 70 DISTANCE (wavelengths) 0.0 0.2 0.4 0.6 0.8 1.0 AMPLITUDE (arb. units) 1.0 1.1 1.2 1.3 1.4 1.5 1.6 REFRACTIVE INDEX (no units) Fig. 2. Same as Fig. 1 after the field has propagated out of the medium through the anti￾reflection layer. #77863 - $15.00 USD Received 18 December 2006; accepted 7 January 2007 (C) 2007 OSA 22 January 2007 / Vol. 15, No. 2 / OPTICS EXPRESS 720

We now extend this result to the more usual or realistic case of a step change in the refractive index for a field entering a homogeneous linear medium from the vacuum. The main differ ence is that a backward traveling wave is generated at the vacuum/dielectric interface and the resulting change in momentum must be incorporated into the conservation law. Using the same incident field as in Fig. 1, Fig 3 shows the field entering the linear medium. The leading part of the field has entered the medium and a portion of that field has been reflected. The oscillations that are seen in the figure represent the interference of the reflection with the trailing part of the incident field. In Fig. 4, the forward propagating refracted field is entirely within the medium and the reflection has separated from the interface. Figure 5 shows the fields after the refracted field has exited the medium through a gradient-index layer that suppresses the secondary reflec- tion. at the end of the calculation the momentum of the transmitted field is found to be smaller than that of the incident field by the momentum of the reflection. In order to satisfy conservation of momentum, a permanent forward momentum of twice the momentum of the refected field must by imputed to the material. In this regard, conservation of electromagnetic momentum is analogous to momentum conservation in the elastic collision of a small object with a wall DISTANCE (wavelengths) Fig 3. Propagation of the vector potential from vacuum into a linear medium through a step increase in the refractive index. The shaded region is the profile of the index of refraction The field travels to the right and the horizontal axis is scaled to the wavelength. At any point in the calculation, conservation of the momentum(20)can be demonstrated if a forward momentum of twice the momentum of the reflected field is contributed by the material As soon as the field starts to enter the medium the momentum of the forward traveling field begins to decrease and the momentum of the backward traveling wave, initially zero, begins to grow as does the momentum of the material. Once the refracted field is entirely within the medium. the momentum of the refracted field remains the same until the field exits the medium hrough the antireflective layer, and thereafter. Meanwhile, the process of reflection is complete and the reflected field travels through the vacuum and is unchanged. We can conclude that there is no additional transfer of momentum to the material once the field is no longer incident on its surface. Consequently, the permanent transfer of momentum from the field to the material occurs at the point of reflection, the surface of the medium, and only while the field is present at the boundary and undergoing Fresnel reflection. There is no temporary material momentum because the momentum is fully accounted for at any time, particularly as the field exits the medium. For the approximate square pulse of vacuum width w, the momentum conservation #77863·S1500USD Received 18 December 2006, accepted 7 January 2007 (C)2007OSA 22 January 2007/ Vol 15, No. 2/OPTICS EXPRESS 721

We now extend this result to the more usual or realistic case of a step change in the refractive index for a field entering a homogeneous linear medium from the vacuum. The main differ￾ence is that a backward traveling wave is generated at the vacuum/dielectric interface and the resulting change in momentum must be incorporated into the conservation law. Using the same incident field as in Fig. 1, Fig. 3 shows the field entering the linear medium. The leading part of the field has entered the medium and a portion of that field has been reflected. The oscillations that are seen in the figure represent the interference of the reflection with the trailing part of the incident field. In Fig. 4, the forward propagating refracted field is entirely within the medium and the reflection has separated from the interface. Figure 5 shows the fields after the refracted field has exited the medium through a gradient-index layer that suppresses the secondary reflec￾tion. At the end of the calculation, the momentum of the transmitted field is found to be smaller than that of the incident field by the momentum of the reflection. In order to satisfy conservation of momentum, a permanent forward momentum of twice the momentum of the reflected field must by imputed to the material. In this regard, conservation of electromagnetic momentum is analogous to momentum conservation in the elastic collision of a small object with a wall. -30 -20 -10 0 10 20 30 DISTANCE (wavelengths) 0.0 0.3 0.6 0.9 1.2 AMPLITUDE (arb. units) 1.0 1.1 1.2 1.3 1.4 1.5 1.6 REFRACTIVE INDEX (no units) Fig. 3. Propagation of the vector potential from vacuum into a linear medium through a step increase in the refractive index. The shaded region is the profile of the index of refraction. The field travels to the right and the horizontal axis is scaled to the wavelength. At any point in the calculation, conservation of the momentum (20) can be demonstrated if a forward momentum of twice the momentum of the reflected field is contributed by the material. As soon as the field starts to enter the medium, the momentum of the forward traveling field begins to decrease and the momentum of the backward traveling wave, initially zero, begins to grow as does the momentum of the material. Once the refracted field is entirely within the medium, the momentum of the refracted field remains the same until the field exits the medium through the antireflective layer, and thereafter. Meanwhile, the process of reflection is complete and the reflected field travels through the vacuum and is unchanged. We can conclude that there is no additional transfer of momentum to the material once the field is no longer incident on its surface. Consequently, the permanent transfer of momentum from the field to the material occurs at the point of reflection, the surface of the medium, and only while the field is present at the boundary and undergoing Fresnel reflection. There is no temporary material momentum because the momentum is fully accounted for at any time, particularly as the field exits the medium. For the approximate square pulse of vacuum width w, the momentum conservation #77863 - $15.00 USD Received 18 December 2006; accepted 7 January 2007 (C) 2007 OSA 22 January 2007 / Vol. 15, No. 2 / OPTICS EXPRESS 721

DISTANCE (wavelengths) Fig. 4. Propagation of the vector potential from vacuum into a linear medium with a step- yer on th region is the profile of the index of refraction. The field travels to the right and the horizontal tis is scaled to the wavelength. DISTANCE (wavelengths) Fig. 5. Same as Fig 4 after the field has propagated out of the medium through the anti- reflection layer. #77863·S1500USD Received 18 December 2006, accepted 7 January 2007 (C)2007OSA 22 January 2007/ Vol 15, No. 2/OPTICS EXPRESS 722

-70 -50 -30 -10 10 30 50 70 DISTANCE (wavelengths) 0.0 0.2 0.4 0.6 0.8 1.0 AMPLITUDE (arb. units) 1.0 1.1 1.2 1.3 1.4 1.5 1.6 REFRACTIVE INDEX (no units) Fig. 4. Propagation of the vector potential from vacuum into a linear medium with a step￾index for the entry and a gradient-index antireflection layer on the exit face. The shaded region is the profile of the index of refraction. The field travels to the right and the horizontal axis is scaled to the wavelength. -70 -50 -30 -10 10 30 50 70 DISTANCE (wavelengths) 0.0 0.2 0.4 0.6 0.8 1.0 AMPLITUDE (arb. units) 1.0 1.1 1.2 1.3 1.4 1.5 1.6 REFRACTIVE INDEX (no units) Fig. 5. Same as Fig. 4 after the field has propagated out of the medium through the anti￾reflection layer. #77863 - $15.00 USD Received 18 December 2006; accepted 7 January 2007 (C) 2007 OSA 22 January 2007 / Vol. 15, No. 2 / OPTICS EXPRESS 722

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 723

law 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