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

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 refractive index in the Wentzel–Kramers–Brillouin (WKB) limit. Both generalized 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 material 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 tensor,” 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, condmat/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 relativistic 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 element 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 inconsistency 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 momentum 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, neglecting 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 approach and show that the Fresnel relations imply the continuity of two quantities: the electromagnetic 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 electromagnetic 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 complements 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 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 properties 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 momentum 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. However, 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