ENERGY AND MOMENTUM OF LIGHT IN DIELECTRIC PHYSICAL REVIEW E 73. 026606(2006) Helmholtz force expression when the latter is restricted to An important result of this paper is that the ratio of the static incompressible media [27, 28] total energy and total momentum is given by the phase ve- Of particular importance is the case of an interface be- locity. This is consistent with the assignment of an energy tween two otherwise homogeneous media. According to E=ho and momentum p=hk to a single photon(k is the (116)the stress tensor must be discontinuous across the in- wave vector in the medium), which gives E/p=olk=c/n terface. This discontinuity is restricted to the flux across the This suggests that in the quantized version of the auxiliary interface of the momentum component normal to the inter- eld model the total momentum proposed here should corre- face. The fiux across the interface of momentum components spond to a momentum hk per quantum. Garrison and Chiao parallel to the interface is continuous due to the continuity requirements of the different fields(parallel components ofe argue that this is the case based on the requirement that the and h nuous, normal component of D and B continu- total electromagnetic momentum operator should be the gen- ous). The discontinous flux of normal momentum must be erator of translations [4]. The ratio between energy and mo- balanced by a mechanical flux of normal momentum, such as mentum being the phase velocity agrees with the phase a pressure difference between the two media. The total force matching condition in spontaneous down conversion [3, 4] exerted by the first medium on the adjacent second medium ith experiments on the photon drag effect in semiconduc tors(see discussion in [6]), and recently with experiments on first medium. A similar view is found in Landau and Lifshitz the recoil of an atom in a Bose-Einstein condensate when it [291. where it is shown that in the electrostatic limit the absorbs a photon [31]. This perspective on photon momen Minkowski stress tensor must be used to calculate the ther- tum may be tested theoretically by using the well-developed modynamic equilibrium forces on a dielectric body. The quantum theory of light in dielectric media [10, 14-17, 32,331 same conclusion is obtained by Gordon for optical frequen- to find out if indeed the proposed total momentum corre cies and negligible dispersion [2]. Both results are general- sponds to hk per photon. It is mentioned that this also offers ized to arbitrary dispersive and absorbing media if the flux a look on the Casimir effect in general linear dielectrics. In density of the total momentum(50) is used to calculate the contrast to the use of the Minkowski or Maxwell stress ten- force on a dielectric bod sor [34], it may well be that the Alux density of the total According to a different point of view, the lorentz forc ce momentum is the relevant quantity for calculating the Ca- is the basic quantity, and the force on a dielectric body is simir force found by integrating this force over the volume of the body Finally, it is mentioned that the auxiliary field model can [5-8. This approach is equivalent to using the Abraham mo- be extended in a number of ways. a generalization to de- mentum and the Maxwell stress tensor. A variation of this formable media has been mentioned already a derivation of approach is due to Mansuripur [8] who argues that a me. the model from microscopical principles, including statistical mechanical principles to incorporate dissipation, would jus small dispersion and dissipation, accompanies a pulse of tify the use of the auxiliary field model and elucidate the light in a dielectric, and that this contribution should count as ircumstances under which the model can be applied to de- electromagnetic momentum as well. It follows that the total scribe experimental results. Other applications of the auxil- momentum is then the average of the Abraham and Minkowski forms. The evaluation of radiation forces fron ary field model are in the description of other types of me- the lorentz force does not depend on this interpretation of spatially dispersive media, and nonlinear media. on how radiation forces should be calculated are incompat ible in some cases, notably the case of a dielectric slab im- ACKNOWLEDGMENTS mersed in a different dielectric. and can thus be tested ex- perimentally. This variation on the Jones-Leslie experiment I thank Mischa Megens, Martin van der Mark, and 30] will be discussed in a separate paper. Gert 't Hooft for useful discussions and suggestions [1]I. Brevik, Phys. Rep. 52, 133(1979). [11 A. Figotin and J. E. Schenker, e-print physics/0410127 [2JJ. P. Gordon, Phys. Rev. A8, 14(1973) [12]J. D. Jackson, Classical Electrodynamics ( John Wiley Sons B3]D. F. Nelson, Phys. Rev. A 44, 3985(1991) New York, 1975) 14]J. C. Garrison and R. Y. Chiao, Phys. Rev. A 70, 053826 [13] H. Goldstein, Classical Mechanics(Addison-Wesley, Reading, (2004) [5]R. Loudon, J Mod. Opt. 49, 821(2002) [14] T. Gruner and D - G. Welsch, Phys. Rev. A 51, 3246(1995) 6]R Loudon, S M. Barnet C. Baxter, Phys. Rev. A 71, [15]T Gruner and D. G Welsch, Phys. Rev. A 53, 1818(1996) 63802(2005) [16]H. T. Dung, L. Knoll, and D.-G. Welsch, Phys. Rev. A 57, [ 7Y. N. Obukhov and F. W. Hehl, Phys. Lett. A 311, 277(2003) 3931(1998) [8]M. Mansuripur, Opt. Express 12, 5375(2004) [17]S. Scheel, L. Knoll, and D -G. Welsch, Phys. Rev. A 58, 700 J A Tip, Phys. Rev. A 56, 5022(1997) [0]A.Tip.Phys.Rev.A57,4818(1998) [18]S. Glasgow, M. Ware, and J. Peatross, Phys. Rev. E 64Helmholtz force expression when the latter is restricted to static incompressible media 27,28. Of particular importance is the case of an interface be￾tween two otherwise homogeneous media. According to
116 the stress tensor must be discontinuous across the in￾terface. This discontinuity is restricted to the flux across the interface of the momentum component normal to the inter￾face. The flux across the interface of momentum components parallel to the interface is continuous due to the continuity requirements of the different fields
parallel components of E and H continuous, normal component of D and B continu￾ous. The discontinous flux of normal momentum must be balanced by a mechanical flux of normal momentum, such as a pressure difference between the two media. The total force exerted by the first medium on the adjacent second medium is found by evaluating the stress tensor at the interface in the first medium. A similar view is found in Landau and Lifshitz 29, where it is shown that in the electrostatic limit the Minkowski stress tensor must be used to calculate the ther￾modynamic equilibrium forces on a dielectric body. The same conclusion is obtained by Gordon for optical frequen￾cies and negligible dispersion 2. Both results are general￾ized to arbitrary dispersive and absorbing media if the flux density of the total momentum
50 is used to calculate the force on a dielectric body. According to a different point of view, the Lorentz force is the basic quantity, and the force on a dielectric body is found by integrating this force over the volume of the body 5–8. This approach is equivalent to using the Abraham mo￾mentum and the Maxwell stress tensor. A variation of this approach is due to Mansuripur 8 who argues that a me￾chanical momentum density, equal to PB/ 2 in media with small dispersion and dissipation, accompanies a pulse of light in a dielectric, and that this contribution should count as electromagnetic momentum as well. It follows that the total momentum is then the average of the Abraham and Minkowski forms. The evaluation of radiation forces from the Lorentz force does not depend on this interpretation of what the total electromagnetic momentum is. The two views on how radiation forces should be calculated are incompat￾ible in some cases, notably the case of a dielectric slab im￾mersed in a different dielectric, and can thus be tested ex￾perimentally. This variation on the Jones-Leslie experiment 30 will be discussed in a separate paper. An important result of this paper is that the ratio of the total energy and total momentum is given by the phase ve￾locity. This is consistent with the assignment of an energy E=
and momentum p=k to a single photon
k is the wave vector in the medium, which gives E/ p=
/k=c/n. This suggests that in the quantized version of the auxiliary field model the total momentum proposed here should corre￾spond to a momentum k per quantum. Garrison and Chiao argue that this is the case based on the requirement that the total electromagnetic momentum operator should be the gen￾erator of translations 4. The ratio between energy and mo￾mentum being the phase velocity agrees with the phase￾matching condition in spontaneous down conversion 3,4, with experiments on the photon drag effect in semiconduc￾tors
see discussion in 6, and recently with experiments on the recoil of an atom in a Bose-Einstein condensate when it absorbs a photon 31. This perspective on photon momen￾tum may be tested theoretically by using the well-developed quantum theory of light in dielectric media 10,14–17,32,33 to find out if indeed the proposed total momentum corre￾sponds to k per photon. It is mentioned that this also offers a look on the Casimir effect in general linear dielectrics. In contrast to the use of the Minkowski or Maxwell stress ten￾sor 34, it may well be that the flux density of the total momentum is the relevant quantity for calculating the Ca￾simir force. Finally, it is mentioned that the auxiliary field model can be extended in a number of ways. A generalization to de￾formable media has been mentioned already. A derivation of the model from microscopical principles, including statistical mechanical principles to incorporate dissipation, would jus￾tify the use of the auxiliary field model and elucidate the circumstances under which the model can be applied to de￾scribe experimental results. Other applications of the auxil￾iary field model are in the description of other types of me￾dia, in particular anisotropic and bianisotropic media, spatially dispersive media, and nonlinear media. ACKNOWLEDGMENTS I thank Mischa Megens, Martin van der Mark, and Gert ’t Hooft for useful discussions and suggestions. 1 I. Brevik, Phys. Rep. 52, 133
1979. 2 J. P. Gordon, Phys. Rev. A 8, 14
1973. 3 D. F. Nelson, Phys. Rev. A 44, 3985
1991. 4 J. C. Garrison and R. Y. Chiao, Phys. Rev. A 70, 053826
2004. 5 R. Loudon, J. Mod. Opt. 49, 821
2002. 6 R. Loudon, S. M. Barnett, and C. Baxter, Phys. Rev. A 71, 063802
2005. 7 Y. N. Obukhov and F. W. Hehl, Phys. Lett. A 311, 277
2003. 8 M. Mansuripur, Opt. Express 12, 5375
2004. 9 A. Tip, Phys. Rev. A 56, 5022
1997. 10 A. Tip, Phys. Rev. A 57, 4818
1998. 11 A. Figotin and J. E. Schenker, e-print physics/0410127. 12 J. D. Jackson, Classical Electrodynamics
John Wiley & Sons, New York, 1975. 13 H. Goldstein, Classical Mechanics
Addison-Wesley, Reading, 1950. 14 T. Gruner and D.-G. Welsch, Phys. Rev. A 51, 3246
1995. 15 T. Gruner and D.-G. Welsch, Phys. Rev. A 53, 1818
1996. 16 H. T. Dung, L. Knoll, and D.-G. Welsch, Phys. Rev. A 57, 3931
1998. 17 S. Scheel, L. Knoll, and D.-G. Welsch, Phys. Rev. A 58, 700
1998. 18 S. Glasgow, M. Ware, and J. Peatross, Phys. Rev. E 64, ENERGY AND MOMENTUM OF LIGHT IN DIELECTRIC... PHYSICAL REVIEW E 73, 026606
2006 026606-11