《电动力学》课程参考文献：Radiation pressure and the linear momentum of the electromagnetic field

Radiation pressure and the linear momentum of the electromagnetic field Masud mansuripur Optical Sciences Center. The University of Arona, Tucson, Arisona 85721 Abstract: We derive the force of the electromagnetic radiation on material objects by a direct application of the Lorentz law of classical electro- dynamics. The derivation is straightforward in the case of solid metals and solid dielectrics, where the mass density and the optical constants of the media are assumed to remain unchanged under internal and external pressures, and where material flow and deformation can be ignored. For metallic mirrors, we separate the contribution to the radiation pressure of the electrical charge density from that of the current density of the conduction electrons. In the case of dielectric media. we examine the forces experienced by bound charges and currents, and determine the contribution of each to the radiation pressure. These analyses reveal the existence of a lateral radiation pressure inside the dielectric media, one that is exerted at and around the edges of a finite-diameter light beam. The lateral pressure turns out to be compressive for s-polarized light and expansive for p- polarized light. Along the way, we derive an expression for the momentum density of the light field inside dielectric media, one that has equal contributions from the traditional minkowski and abraham forms this ew expression for the momentum density, which contains both electromagnetic and mechanical terms, is used to explain the behavior of light pulses and individual photons upon entering and exiting a dielectric slab. In all the cases considered, the net forces and torques experienced by material bodies are consistent with the relevant conservation laws. Our method of calculating the radiation pressure can be used in conjunction with numerical simulations to yield the distribution of fields and forces in diverse systems of practical interest C 2004 Optical Society of America OCIS codes:(2602110)Electromagnetic theory; (1407010)Trapping;(0207010) Trapping, (3106860)Thin films, optical properties References 1. H Minkowski, Nachr. Ges. Wiss. Gottingen 53(1908) 2. H Minkowski, Math. Annalon 68, 472(1910) 3. M. Abraham, R C Circ. Mat. Palermo 28, 1(1909) 4. M. Abraham, R C. Circ. Mat. Palermo 30, 33(1910) 5. J. P. Gordon, "Radiation forces and momenta in dielectric media, "Phys. Rev. A8, 14-21(1973) 6. R Loudon. "Radiation Pressure and Momentum in Dielectrics. "De Martini lecture. in Fortschritte der Physik(2004) 7. R. Loudon, "Theory of the radiati re on dielectric surfaces, J Mod. Opt. 49, 821-838(2002 8. L.Landau, E. Lifshitz, Electrodynamics of Continous Media, Pergamon, New York, 1960 9. J.D. Jackson, Classical Electrodynamics, 2 edition, wiley, New York, 1975 10. M. Planck, The Theory of Heat Radiation, translated by M. Masius form the German edition of 1914, Dover Publications, New York(1959) I. R. V Jones and J. C S. Richards, Proc. Roy. Soc. A 221, 480(1954) #5025-S1500US Received 10 August 2004; revised 13 October 2004; accepted 20 October 2004 (C)2004OSA November 2004/Vol 12. No 22/OPTICS EXPRESS 5375

Radiation pressure and the linear momentum of the electromagnetic field Masud Mansuripur Optical Sciences Center, The University of Arizona, Tucson, Arizona 85721 masud@u.arizona.edu Abstract: We derive the force of the electromagnetic radiation on material objects by a direct application of the Lorentz law of classical electro￾dynamics. The derivation is straightforward in the case of solid metals and solid dielectrics, where the mass density and the optical constants of the media are assumed to remain unchanged under internal and external pressures, and where material flow and deformation can be ignored. For metallic mirrors, we separate the contribution to the radiation pressure of the electrical charge density from that of the current density of the conduction electrons. In the case of dielectric media, we examine the forces experienced by bound charges and currents, and determine the contribution of each to the radiation pressure. These analyses reveal the existence of a lateral radiation pressure inside the dielectric media, one that is exerted at and around the edges of a finite-diameter light beam. The lateral pressure turns out to be compressive for s-polarized light and expansive for p￾polarized light. Along the way, we derive an expression for the momentum density of the light field inside dielectric media, one that has equal contributions from the traditional Minkowski and Abraham forms. This new expression for the momentum density, which contains both electromagnetic and mechanical terms, is used to explain the behavior of light pulses and individual photons upon entering and exiting a dielectric slab. In all the cases considered, the net forces and torques experienced by material bodies are consistent with the relevant conservation laws. Our method of calculating the radiation pressure can be used in conjunction with numerical simulations to yield the distribution of fields and forces in diverse systems of practical interest. © 2004 Optical Society of America OCIS codes: (260.2110) Electromagnetic theory; (140.7010) Trapping; (020.7010) Trapping; (310.6860) Thin films, optical properties References 1. H. Minkowski, Nachr. Ges. Wiss. Gottingen 53 (1908). 2. H. Minkowski, Math. Annalon 68, 472 (1910). 3. M. Abraham, R. C. Circ. Mat. Palermo 28, 1 (1909). 4. M. Abraham, R. C. Circ. Mat. Palermo 30, 33 (1910). 5. J. P. Gordon, “Radiation forces and momenta in dielectric media,” Phys. Rev. A 8, 14-21 (1973). 6. R. Loudon, “Radiation Pressure and Momentum in Dielectrics,” De Martini lecture, to appear in Fortschritte der Physik (2004). 7. R. Loudon, “Theory of the radiation pressure on dielectric surfaces,” J. Mod. Opt. 49, 821-838 (2002). 8. L. Landau, E. Lifshitz, Electrodynamics of Continuous Media, Pergamon, New York, 1960. 9. J. D. Jackson, Classical Electrodynamics, 2nd edition, Wiley, New York, 1975. 10. M. Planck, The Theory of Heat Radiation, translated by M. Masius form the German edition of 1914, Dover Publications, New York (1959). 11. R. V. Jones and J. C. S. Richards, Proc. Roy. Soc. A 221, 480 (1954). (C) 2004 OSA 1 November 2004 / Vol. 12, No. 22 / OPTICS EXPRESS 5375 #5025- $15.00 US Received 10 August 2004; revised 13 October 2004; accepted 20 October 2004

12. A. Ashkin, J. M. Dziedzic, J. E Bjorkholm and S. Chu, "Observation of a single-beam gradient force optical trap for dielectric particles, Opt. Lett. 11, 288-290(1986) 13. A. Ashkin and J M. Dziedzic, "Optical trapping and manipulation of viruses and bacteria, "Science 235,1517. 1520(1987) 14. A Rohrbach and E Stelzer, "Trapping force onstants, and potential depths for dielectric sphere presence of spherical aberrations, " AppL. Opt. 41, 2494(2002) 15. Y N Obukhov and F WHehl, "Electromagnetic energy-momentum and forces in matter, "Phys.Lett. A,311 16. G.Barlow, Proc. Roy. Soc. Lond. A 87, 1-16(1912) 17. R. Loudon, "Theory of the forces exerted by Laguerre-Gaussian light beams on dielectrics, "Phys. Rev. A68 013806(2003) 1. Introduction It is well-known that the electromagnetic radiation carries both energy and momentum, that the energy flux is given by the Poynting vector S of the classical electrodynamics, and that, in e-space, the momentum density p(i. e, momentum per unit volume)is given by p=S/c where c is the speed of light in vacuum. What has been a matter of controversy for quite some time now is the proper form for the momentum of the electromagnetic waves in dielectric media. The question is whether the momentum density in a material medium has the form P=DXB, due to Minkowski [1, 2], or p=Ex H/c2, due to Abraham [3, 4). J. P. Gordon [51 attributes the following comment to E. I Blount: "The argument has not, it is true, been carried out at high volume, but the list of disputants is very distinguished. For a historical perspective on the subject and a summary of the relevant experimental results see R. Loudon 6,7] and J. P. Gordon [5 Traditionally, the electromagnetic stress tensor has been used to derive the mechanical force exerted by the radiation field on ponderable media [8, 9]. This approach, while having the advantage of generality, tends to obscure behind complicated mathematics the physical origin of the forces. It is possible, however, to calculate the force of the electromagnetic radiation on various media by direct invocation of the Lorentz law of force. The derivation is especially straightforward in the case of solid metals and solid dielectrics, where the mass tensity and the optical constants of the media may be assumed to remain constant under internal and external pressures, and where material flow and deformation can be ignored Loudon [7] has emphasized "the simplicity and safety of calculations based on the Lorentz force and the dangers of calculations based on derived expressions involving elements of the Maxwell stress tensor, whose contributions may vanish in some situations but not in others In this paper we use the Lorentz law to derive the force of electromagnetic radiation on isotropic solid media in several simple situations. In the case of metallic mirrors, we separate following Planck [101 the contribution to the radiation pressure of the electrical charge tensity from that of the current density(both due to conduction electrons ). In the case of dielectric media, we examine the force experienced by bound charges and currents, an determine the contribution of each to the radiation pressure. Along the way, we derive a new expression for the momentum density of the light field inside dielectric media, one that has equal contributions from the aforementioned Minkowski and Abraham forms. This new expression for the momentum density, which contains both mechanical and electromagnetic terms, is subsequently used to elucidate the behavior of individual photons upon entering and exiting a dielectric slab. With the exception of the semi-quantitative results of Section 13, all the results obtained in this paper are exact, in the sense that no approximations or implifications have been introduced; all derivations are based directly on the Lorentz law of force in conjunction with Maxwells equations, using the standard constitutive relations for homogeneous, isotropic, linear, non-magnetic, and non-dispersive media The organization of the paper is as follows. In Section 2 we describe the notation and define the various parameters used throughout the paper. Section 3 considers the reflection of #5025-S1500US Received 10 August 2004; revised 13 October 2004; accepted 20 October 2004 (C)2004OSA November 2004/Vol 12. No 22/OPTICS EXPRESS 5376

12. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11, 288-290 (1986). 13. A. Ashkin and J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science 235, 1517- 1520 (1987). 14. A. Rohrbach and E. Stelzer, “Trapping forces, force constants, and potential depths for dielectric spheres in the presence of spherical aberrations,” Appl. Opt. 41, 2494 (2002). 15. Y. N. Obukhov and F. W.Hehl, “Electromagnetic energy-momentum and forces in matter,” Phys. Lett. A, 311, 277-284 (2003). 16. G. Barlow, Proc. Roy. Soc. Lond. A 87, 1-16 (1912). 17. R. Loudon, “Theory of the forces exerted by Laguerre-Gaussian light beams on dielectrics,” Phys. Rev. A 68, 013806 (2003). 1. Introduction It is well-known that the electromagnetic radiation carries both energy and momentum, that the energy flux is given by the Poynting vector S of the classical electrodynamics, and that, in free-space, the momentum density p (i.e., momentum per unit volume) is given by p = S/c 2 , where c is the speed of light in vacuum. What has been a matter of controversy for quite some time now is the proper form for the momentum of the electromagnetic waves in dielectric media. The question is whether the momentum density in a material medium has the form p = D × B, due to Minkowski [1,2], or p = E × H/c 2 , due to Abraham [3,4]. J. P. Gordon [5] attributes the following comment to E. I. Blount: “The argument has not, it is true, been carried out at high volume, but the list of disputants is very distinguished.” For a historical perspective on the subject and a summary of the relevant experimental results see R. Loudon [6,7] and J. P. Gordon [5]. Traditionally, the electromagnetic stress tensor has been used to derive the mechanical force exerted by the radiation field on ponderable media [8,9]. This approach, while having the advantage of generality, tends to obscure behind complicated mathematics the physical origin of the forces. It is possible, however, to calculate the force of the electromagnetic radiation on various media by direct invocation of the Lorentz law of force. The derivation is especially straightforward in the case of solid metals and solid dielectrics, where the mass density and the optical constants of the media may be assumed to remain constant under internal and external pressures, and where material flow and deformation can be ignored. Loudon [7] has emphasized “the simplicity and safety of calculations based on the Lorentz force and the dangers of calculations based on derived expressions involving elements of the Maxwell stress tensor, whose contributions may vanish in some situations but not in others.” In this paper we use the Lorentz law to derive the force of electromagnetic radiation on isotropic solid media in several simple situations. In the case of metallic mirrors, we separate, following Planck [10], the contribution to the radiation pressure of the electrical charge density from that of the current density (both due to conduction electrons). In the case of dielectric media, we examine the force experienced by bound charges and currents, and determine the contribution of each to the radiation pressure. Along the way, we derive a new expression for the momentum density of the light field inside dielectric media, one that has equal contributions from the aforementioned Minkowski and Abraham forms. This new expression for the momentum density, which contains both mechanical and electromagnetic terms, is subsequently used to elucidate the behavior of individual photons upon entering and exiting a dielectric slab. With the exception of the semi-quantitative results of Section 13, all the results obtained in this paper are exact, in the sense that no approximations or simplifications have been introduced; all derivations are based directly on the Lorentz law of force in conjunction with Maxwell’s equations, using the standard constitutive relations for homogeneous, isotropic, linear, non-magnetic, and non-dispersive media. The organization of the paper is as follows. In Section 2 we describe the notation and define the various parameters used throughout the paper. Section 3 considers the reflection of (C) 2004 OSA 1 November 2004 / Vol. 12, No. 22 / OPTICS EXPRESS 5376 #5025- $15.00 US Received 10 August 2004; revised 13 October 2004; accepted 20 October 2004

a plane electromagnetic wave from a perfect conductor, and shows that both the Lorentz law of force and the momentum of the electromagnetic field in free-space can consistently account for the radiation pressure exerted on the mirror surface. In Section 4 we use the Lorentz law to determine the radiation pressure on the surface of a semi-infinite dielectric medium at normal incidence. Here we derive an expression for the momentum of the field inside the dielectrics. The results of Section 4 are then extended to cover the case of oblique incidence on a semi-infinite dielectric, first with s-polarized light in Section 5, then with p- polarized light in Section 6. These analyses lead to the discovery of a lateral radiation pressure inside the dielectric medium, exerted at and around the edges of a finite-diameter lane wave. This lateral presond expansive for p-light. To the author's best knowledge, the ressure. while having the same magnitude in both cases turns out to be compressive for s-light expansive lateral force on the dielectric host of p-polarized light has not been discussed in the existing literature, making it a novel prediction that requires experimental verification In Section 7 we examine the torque experienced by a dielectric slab, illuminated at the Brewsters angle by a p-polarized plane wave. The torque is calculated directly from the Lorentz law applied to the induced(bound) charges at the surfaces of the slab, then shown to be consistent with the change in the angular momentum of the incident light. The case of an anti-reflection coated, semi-infinite dielectric medium is taken up in Section 8, where the increase in the momentum of the incident beam upon transmission into the dielectric medium is shown to result in a net force on the anti-reflection coating layer that tends to peel the layer away from its substrate, another new result that requires experimental verification. In Section 9 we analyze the case of a dielectric slab of finite thickness, and show that optical interference within the slab is responsible for the (longitudinal) stress induced by the electromagnetic radiation For a different perspective on the lateral pressure at the edges of a finite-diameter beam a dielectric, Section 10 is devoted to an analysis of the one-dimensional Gaussian beam inside a dielectric medium. Depending on the direction of the E-field, we show that the lateral pressure on the medium can be compressive or expansive, and that the magnitude and direction of this radiation force are in complete accord with the results of Sections 5 and 6 The generality of this lateral pressure(and the dependence of its direction on the state of polarization) are brought to the fore in Section 1l, where the simple fringes produced by the interference between two plane waves are shown to exhibit the same phenomena In Section 12 we extend our method of calculation of force and momentum to(finite- duration) light pulses, where the leading and trailing edges of the pulse are shown to play an important role in exchanging the electromagnetic momentum of the light with the mechanical momentum of the medium of propagation. The physical basis for the designation(in Section 4)of a fraction of the photon's momentum as"mechanical"is clarified in Section 12 The classical experiments pertaining to metallic mirrors immersed in liquid dielectrics [11] are discussed in Section 13, where they are shown to be in complete agreement with our theoretical calculations. It is well known, e.g, from optical tweezers experiments[12-14], that a focused laser beam tends to attract small dielectric beads toward the center of the focused spot. At first glance, this observation might seem at odds with the presence of an expansive lateral pressure inside the dielectric medium of the bead. To resolve this apparent discrepancy, Section 14 offers a semi-quantitative analysis of a simplified model of the optical tweezers experiment. Final remarks and a summary of our important results appear in Section 15 The MKSA system of units is used throughout the paper. Time harmonic fields are written as E(x,,=,0)=E(x,y, =)exp(@r), where @=2f is the angular frequency. For brevity, we omit the explicit dependence of the fields on x, y, = t. To specify their magnitude and phase, #5025-S1500US Received 10 August 2004; revised 13 October 2004; accepted 20 October 2004 (C)2004OSA November 2004/Vol 12. No 22/OPTICS EXPRESS 5377

a plane electromagnetic wave from a perfect conductor, and shows that both the Lorentz law of force and the momentum of the electromagnetic field in free-space can consistently account for the radiation pressure exerted on the mirror surface. In Section 4 we use the Lorentz law to determine the radiation pressure on the surface of a semi-infinite dielectric medium at normal incidence. Here we derive an expression for the momentum of the field inside the dielectrics. The results of Section 4 are then extended to cover the case of oblique incidence on a semi-infinite dielectric, first with s-polarized light in Section 5, then with p￾polarized light in Section 6. These analyses lead to the discovery of a lateral radiation pressure inside the dielectric medium, exerted at and around the edges of a finite-diameter plane wave. This lateral pressure, while having the same magnitude in both cases, turns out to be compressive for s-light and expansive for p-light. To the author’s best knowledge, the expansive lateral force on the dielectric host of p-polarized light has not been discussed in the existing literature, making it a novel prediction that requires experimental verification. In Section 7 we examine the torque experienced by a dielectric slab, illuminated at the Brewster’s angle by a p-polarized plane wave. The torque is calculated directly from the Lorentz law applied to the induced (bound) charges at the surfaces of the slab, then shown to be consistent with the change in the angular momentum of the incident light. The case of an anti-reflection coated, semi-infinite dielectric medium is taken up in Section 8, where the increase in the momentum of the incident beam upon transmission into the dielectric medium is shown to result in a net force on the anti-reflection coating layer that tends to peel the layer away from its substrate; another new result that requires experimental verification. In Section 9 we analyze the case of a dielectric slab of finite thickness, and show that optical interference within the slab is responsible for the (longitudinal) stress induced by the electromagnetic radiation. For a different perspective on the lateral pressure at the edges of a finite-diameter beam in a dielectric, Section 10 is devoted to an analysis of the one-dimensional Gaussian beam inside a dielectric medium. Depending on the direction of the E-field, we show that the lateral pressure on the medium can be compressive or expansive, and that the magnitude and direction of this radiation force are in complete accord with the results of Sections 5 and 6. The generality of this lateral pressure (and the dependence of its direction on the state of polarization) are brought to the fore in Section 11, where the simple fringes produced by the interference between two plane waves are shown to exhibit the same phenomena. In Section 12 we extend our method of calculation of force and momentum to (finite￾duration) light pulses, where the leading and trailing edges of the pulse are shown to play an important role in exchanging the electromagnetic momentum of the light with the mechanical momentum of the medium of propagation. The physical basis for the designation (in Section 4) of a fraction of the photon’s momentum as “mechanical” is clarified in Section 12. The classical experiments pertaining to metallic mirrors immersed in liquid dielectrics [11] are discussed in Section 13, where they are shown to be in complete agreement with our theoretical calculations. It is well known, e.g., from optical tweezers experiments [12-14], that a focused laser beam tends to attract small dielectric beads toward the center of the focused spot. At first glance, this observation might seem at odds with the presence of an expansive lateral pressure inside the dielectric medium of the bead. To resolve this apparent discrepancy, Section 14 offers a semi-quantitative analysis of a simplified model of the optical tweezers experiment. Final remarks and a summary of our important results appear in Section 15. 2. Notation and basic definitions The MKSA system of units is used throughout the paper. Time harmonic fields are written as E (x, y, z, t) = E (x, y, z) exp(−iω t), where ω = 2πf is the angular frequency. For brevity, we omit the explicit dependence of the fields on x, y, z, t. To specify their magnitude and phase, (C) 2004 OSA 1 November 2004 / Vol. 12, No. 22 / OPTICS EXPRESS 5377 #5025- $15.00 US Received 10 August 2004; revised 13 October 2004; accepted 20 October 2004

complex amplitudes such as E are expressed as elexp(idE). Time-averaged products of two fields, say, Real (A)=| I cos(@t-oA) and Real(B)=B I cos(@t-oB), given by v2) AB I cos(OA-oB), may also be written /Real(AB") To compute the force exerted by the electromagnetic field on a given medium, we use Maxwell's equations to determine the distributions of the E-and H-fields both inside and outside the medium. We then apply the lorentz law F=q(E+vx B), which gives the electromagnetic force on a particle of charge q and velocity V. The magnetic induction B is assumed to be related to the H-field via B=loH, where lo=4I x 10 henrys/meter is the permeability of free space, in other words, any magnetic moments that might exist in the medium and their interactions with the radiation field at optical frequencies are ignored Typically, there are no free charges in the system, so V D=0, where D=EE+ P is the electric displacement vector, E =8.82 x 10 farads/meter is the permittivity of free space, and P is the local polarization density within the medium. In linear media, D=Eoe E, where E is the mediums relative permittivity; hence, P=E(E-lE. We ignore the frequency- dependence of e throughout the paper and treat the media as non-dispersive When V D=0, the density of bound charges P,=-V P may be expressed as Pb=EV- E Inside a homogeneous and isotropic medium, E being proportional to D and V.D=0 imply that PB=0; no bound charges, therefore, can exist inside such media. However, at the interface between two different media, the component of d perpendicular to the interface, Di, must be continuous. The implication is that El is discontinuous and therefore, bound charges can exist at such interfaces; the interfacial bound charges will thus have an areal density o=E(E21- Eln. Under the influence of the local E-field, these charges give rise to an electric Lorentz force F=VReal(oE), where F is the force per unit area of there is no ambiguity as to which field must be used in conjunction with the lorentz law s the interface. Since the tangential E-field, El, is generally continuous across the interi for the perpendicular component, the average E across the boundary, 2(E11+ E21), must be used in calculating the interfacial force. (The use of the av erage El in this context is not a matter of choice; it is the only way to get the calculated force at the boundary to agree with the time rate of change of the momentum that passes through the interface. From a physical andpoint, the interfacial charges produce a local Ei that has the same magnitude but opposite directions on the two sides of the interface. It is this locally-generated Ei that is responsible for the E-fields discontinuity. Averaging El across the interface eliminates the local field, as it should, since the charge cannot exert a force on itself. Since V. B=0 and B=uoh, the perpendicular H-field, Hi, at the interface between adjacent media must remain continuous. The tangential H-field at such interfaces, however, may be discontinuous. This, in accordance with Maxwells equation VxH=J+ aD/a, gives rise to an interfacial current density J, =H21-HIl Such currents can exist on the surfaces of ood conductors, where Eyl is negligible, yet the high conductance of the medium permits the flow of the surface current. Elsewhere, the only source of electrical currents are bound charges, with the bound current density being Jp=ap/dt=E(e-1)dE/ar. Assuming time harmonic fields with the time-dependence factor exp(ian), we can write J=-ioE(E-lE The H-field of the electromagnetic wave then exerts a force on the bound current according to the Lorentz law, namely, F=Real ( bx B), where F is force per unit volume Note: For time-harmonic fields, the contribution of conduction electrons to current density may be combined with that of bound electrons. Since J=o E, where O is the conductivity of the medium, the net current density Jc Jb may be attributed to an effective dielectric constant E+i(oc/E@). In general, since E is complex-valued, there is no need to distinguish conduction electrons from bound electrons, and e may be treated as an effective dielectric constant that contains both contributions. An exception will be made in Section 3 in the case #5025-S1500US Received 10 August 2004; revised 13 October 2004; accepted 20 October 2004 (C)2004OSA November 2004/Vol 12. No 22/OPTICS EXPRESS 5378

complex amplitudes such as E are expressed as | E | exp(iφE). Time-averaged products of two fields, say, Real (A ) = | A | cos(ω t – φA) and Real (B ) = |B | cos(ω t – φB), given by ½| AB | cos(φA – φB), may also be written ½Real (AB*). To compute the force exerted by the electromagnetic field on a given medium, we use Maxwell’s equations to determine the distributions of the E- and H-fields both inside and outside the medium. We then apply the Lorentz law F = q (E + V × B), which gives the electromagnetic force on a particle of charge q and velocity V. The magnetic induction B is assumed to be related to the H-field via B = µ oH, where µ o = 4π × 10−7 henrys/meter is the permeability of free space; in other words, any magnetic moments that might exist in the medium and their interactions with the radiation field at optical frequencies are ignored. Typically, there are no free charges in the system, so ∇ · D = 0, where D = εoE + P is the electric displacement vector, εo = 8.82 × 10−12 farads/meter is the permittivity of free space, and P is the local polarization density within the medium. In linear media, D =εoε E, where ε is the medium’s relative permittivity; hence, P =εo(ε – 1)E. We ignore the frequency￾dependence of ε throughout the paper and treat the media as non-dispersive. When ∇ · D = 0, the density of bound charges ρb = −∇ · P may be expressed as ρb =εo∇ · E. Inside a homogeneous and isotropic medium, E being proportional to D and ∇ · D = 0 imply that ρb = 0; no bound charges, therefore, can exist inside such media. However, at the interface between two different media, the component of D perpendicular to the interface, D⊥, must be continuous. The implication is that E⊥ is discontinuous and, therefore, bound charges can exist at such interfaces; the interfacial bound charges will thus have an areal density σ =εo(E2⊥ − E1 ⊥). Under the influence of the local E-field, these charges give rise to an electric Lorentz force F = ½Real(σ E*), where F is the force per unit area of the interface. Since the tangential E-field, E| | , is generally continuous across the interface, there is no ambiguity as to which field must be used in conjunction with the Lorentz law. As for the perpendicular component, the average E across the boundary, ½(E1 ⊥ + E2 ⊥), must be used in calculating the interfacial force. (The use of the average E⊥ in this context is not a matter of choice; it is the only way to get the calculated force at the boundary to agree with the time rate of change of the momentum that passes through the interface. From a physical standpoint, the interfacial charges produce a local E⊥ that has the same magnitude but opposite directions on the two sides of the interface. It is this locally-generated E⊥ that is responsible for the E-field’s discontinuity. Averaging E⊥ across the interface eliminates the local field, as it should, since the charge cannot exert a force on itself.) Since ∇ · B = 0 and B = µ oH, the perpendicular H-field, H⊥ , at the interface between adjacent media must remain continuous. The tangential H-field at such interfaces, however, may be discontinuous. This, in accordance with Maxwell’s equation ∇ × H = J + ∂D/∂t, gives rise to an interfacial current density Js = H2 | | − H1 | | . Such currents can exist on the surfaces of good conductors, where E| | is negligible, yet the high conductance of the medium permits the flow of the surface current. Elsewhere, the only source of electrical currents are bound charges, with the bound current density being Jb = ∂P/∂t = εo(ε – 1)∂E/∂t. Assuming time￾harmonic fields with the time-dependence factor exp(−iω t), we can write Jb = −iω εo(ε – 1)E. The H-field of the electromagnetic wave then exerts a force on the bound current according to the Lorentz law, namely, F = ½Real (Jb × B*), where F is force per unit volume. Note: For time-harmonic fields, the contribution of conduction electrons to current density may be combined with that of bound electrons. Since Jc = σcE, where σc is the conductivity of the medium, the net current density Jc + Jb may be attributed to an effective dielectric constant ε + i(σc /εoω ). In general, since ε is complex-valued, there is no need to distinguish conduction electrons from bound electrons, and ε may be treated as an effective dielectric constant that contains both contributions. An exception will be made in Section 3 in the case (C) 2004 OSA 1 November 2004 / Vol. 12, No. 22 / OPTICS EXPRESS 5378 #5025- $15.00 US Received 10 August 2004; revised 13 October 2004; accepted 20 October 2004

of perfect conductors, where Oc -oo. Here both E-and H-fields inside the medium tend to zero, and the contributions of bound charges/currents become negligible. The effect of the radiation in this case will be the creation of a surface current density Js and a surface charge density or, both of which may be attributed in their entirety to the conduction electrons. The conservation of charge then requires that V J+dg/dt=0 3. Reflection of plane wave from a perfect conductor The material in this section is not new, but the line of reasoning and the methodology will be needed to build the necessary arguments in subsequent sections. The case of normal incidence on perfect conductors is well-known, but the two cases of oblique incidence, originally ublished in [10], have all but vanished from modern textbooks H (a) ++-+++-+++--+++ Js cose J Fig. 1. A linearly-polarized plane wave is reflected from a perfectly conducting mirror Whereas the parallel component of the E-field at the mirror surface is zero, the parallel nt of the H-field is at its maximum. The surface current J, is equal in magnitude cular in direction to the tic field at the surface. (a) Normal incidence que incidence with s-polarization. (c) oblique incidence with p-polarizatie In Fig. 1(a) a plane wave of wavelength no, having E-field amplitude Eo (units=V/m) and H-field amplitude Ho=EZ(units=A/m), where Z=vudEo-37722 is the free-space impedance, is incident on a perfect conductor. The Poynting vector is S=rEal(E xH") and the momentum density (per unit volume)is p=S/c(vacuum speed of light c= I/ueo In unit time, the incoming momentum over a unit area of the reflector is that contained Imn of base A= 1.0 m2 and height c. The same momentum returns to the source after ng reflected from the mirror, so the net rate of change of the field momentum over a unit #5025-S1500US Received 10 August 2004; revised 13 October 2004; accepted 20 October 2004 (C)2004OSA November 2004/Vol 12. No 22/OPTICS EXPRESS 5379

of perfect conductors, where σc→ ∞. Here both E- and H-fields inside the medium tend to zero, and the contributions of bound charges/currents become negligible. The effect of the radiation in this case will be the creation of a surface current density Js and a surface charge density σ, both of which may be attributed in their entirety to the conduction electrons. The conservation of charge then requires that ∇ · Js + ∂σ /∂t = 0. 3. Reflection of plane wave from a perfect conductor The material in this section is not new, but the line of reasoning and the methodology will be needed to build the necessary arguments in subsequent sections. The case of normal incidence on perfect conductors is well-known, but the two cases of oblique incidence, originally published in [10], have all but vanished from modern textbooks. Fig. 1. A linearly-polarized plane wave is reflected from a perfectly conducting mirror. Whereas the parallel component of the E-field at the mirror surface is zero, the parallel component of the H-field is at its maximum. The surface current Js is equal in magnitude and perpendicular in direction to the magnetic field at the surface. (a) Normal incidence. (b) Oblique incidence with s-polarization. (c) Oblique incidence with p-polarization. In Fig. 1(a) a plane wave of wavelength λo, having E-field amplitude Eo (units = V/m) and H-field amplitude Ho = Eo/Zo (units = A/m), where Zo = √µo/εo ~ 377Ω is the free-space impedance, is incident on a perfect conductor. The Poynting vector is S = ½Real (E × H*), and the momentum density (per unit volume) is p = S/c 2 (vacuum speed of light c = 1/√µoεo). In unit time, the incoming momentum over a unit area of the reflector is that contained in a column of base A = 1.0 m2 and height c. The same momentum returns to the source after being reflected from the mirror, so the net rate of change of the field momentum over a unit H E Js H E Js cosθ θ θ (a) (b) H E Js θ θ +++ --- +++ --- +++ --- +++ (c) (C) 2004 OSA 1 November 2004 / Vol. 12, No. 22 / OPTICS EXPRESS 5379 #5025- $15.00 US Received 10 August 2004; revised 13 October 2004; accepted 20 October 2004

area d p/dt= 2S/c, which is equal to the force per unit area, F, exerted on the reflector. The force density of the light on a perfect reflector in free-space is thus given by F=EE The same expression may be derived by considering the Lorentz law F=q(E+vx B) in conjunction with the surface current density Js and the magnetic field H at the surface of the conductor. Here there are neither free nor(unbalanced) bound charges, and the motion of the conduction electrons constitutes a surface current density Js=gl. For time-harmonic fields, the force per unit area may thus be written F=‰2Real(J×B) In the case of a perfect conductor the magnitude of the surface current is equal to the magnetic field at the mirror surface, namely, Js=2H.(because VxH=J+ dp/dt; the factor of 2 arises from the interference between the incident and reflected beams where the two h- fields, being in-phase at the mirror surface, add up. )Since B=uH, one might conclude that F:=2uoHo. The factor of 2 in this formula, however, is incorrect because the magnetic field on the films surface, 2Ho, is assumed to exert a force on the entire s. The problem is that the field is 2Ho at the top of the mirror and zero just under the surface, say, below the skin-depth (Here we are using a limiting argument in which a good conductor, having a finite skin-depth, approaches an ideal conductor in the limit of zero skin-depth. ) Therefore, the average H-field through the"skin-depth"must be used in calculating the force, and this average is Ho not 2Ho The force per unit area thus calculated is F:=uHo =EEo, which is identical to the time rate of change of momentum of the incident beam derived in Eq (1). With 1.0 W/mm- of incident optical power, for example, the radiation pressure on the mirror will be 6.67 nN/mm Next, suppose the beam arrives on the mirror at an oblique angle 0, as in Fig. I(b); here the beam is assumed to be s-polarized. Compared to normal incidence, the component of the magnetic field H on the surface is now multiplied by cose, which requires the surface current density Js to be multiplied by the same factor(remember that s is equal to the magnetic field at the surface). The component of force density along the z-axis, Fs, is thus seen to have been reduced by a factor of cos"0. This result is consistent with the alternative derivation based on the time rate of change of the fields momentum in the z-direction, dp/dt, which is multiplied by cose in the case of oblique incidence. Since the beam has a finite diameter, its foot-print on the mirror is greater than that in the case of normal incidence by 1/cose. Thus the force density Fs, obtained by normalizing d p: /dt by the beams foot-print, is seen once again to be reduced by a factor of cos"0 Figure I(c)shows a p-polarized beam at oblique incidence on a mirror. The magnetic field component at the surface is 2Ho, which means that the surface current Js must also have the same magnitude as in normal incidence. We conclude that the force density on the mirror must be the same as that at normal incidence, namely, F: =EEo. The time rate of change of momentum in the z-direction, however, is similar to that in Fig. I(b), which means that the force density of normal incidence must have been multiplied by cos"0 in the case of oblique incidence. The two methods of calculating F: for p-light thus disagree by a factor of cos"0 The discrepancy is resolved when one realizes that, in addition to the magnetic force, an electric force is acting on the mirror in the opposite direction(-). This additional force pulls on the electric charges induced at the surface by El. Note that E1 2Eosine just above and El=0 just below the surface. The discontinuity in El gives the surface charge density as O= 2EECSine. The perpendicular E-field acting on these charges is the average of the fields just above and just below the surface, namely, E:=Eosine. The electric force density is thus F:=vReal(oE(e*)=EoEo'sine. The upward force on the charges thus reduces the #5025-S1500US Received 10 August 2004; revised 13 October 2004; accepted 20 October 2004 (C)2004OSA November 2004/Vol 12. No 22/OPTICS EXPRESS 5380

area d p/dt = 2S/c, which is equal to the force per unit area, F, exerted on the reflector. The force density of the light on a perfect reflector in free-space is thus given by F = εoEo 2 . (1) The same expression may be derived by considering the Lorentz law F = q (E + V × B) in conjunction with the surface current density Js and the magnetic field H at the surface of the conductor. Here there are neither free nor (unbalanced) bound charges, and the motion of the conduction electrons constitutes a surface current density Js = qV. For time-harmonic fields, the force per unit area may thus be written F = ½Real (Js × B*). (2) In the case of a perfect conductor the magnitude of the surface current is equal to the magnetic field at the mirror surface, namely, Js = 2Ho (because ∇ × H = J + ∂D/∂t; the factor of 2 arises from the interference between the incident and reflected beams where the two H￾fields, being in-phase at the mirror surface, add up.) Since B = µ oH, one might conclude that Fz = 2µ oHo 2 . The factor of 2 in this formula, however, is incorrect because the magnetic field on the film’s surface, 2Ho, is assumed to exert a force on the entire Js. The problem is that the field is 2Ho at the top of the mirror and zero just under the surface, say, below the skin-depth. (Here we are using a limiting argument in which a good conductor, having a finite skin-depth, approaches an ideal conductor in the limit of zero skin-depth.) Therefore, the average H-field through the “skin-depth” must be used in calculating the force, and this average is Ho not 2Ho. The force per unit area thus calculated is Fz = µ oHo 2 = εoEo 2 , which is identical to the time rate of change of momentum of the incident beam derived in Eq. (1). With 1.0 W/mm2 of incident optical power, for example, the radiation pressure on the mirror will be 6.67 nN/mm2 . Next, suppose the beam arrives on the mirror at an oblique angle θ, as in Fig. 1(b); here the beam is assumed to be s-polarized. Compared to normal incidence, the component of the magnetic field H on the surface is now multiplied by cosθ, which requires the surface current density Js to be multiplied by the same factor (remember that Js is equal to the magnetic field at the surface). The component of force density along the z-axis, Fz, is thus seen to have been reduced by a factor of cos2 θ. This result is consistent with the alternative derivation based on the time rate of change of the field’s momentum in the z-direction, dpz/dt, which is multiplied by cosθ in the case of oblique incidence. Since the beam has a finite diameter, its foot-print on the mirror is greater than that in the case of normal incidence by 1/cosθ. Thus the force density Fz, obtained by normalizing d pz/d t by the beam’s foot-print, is seen once again to be reduced by a factor of cos2 θ. Figure 1(c) shows a p-polarized beam at oblique incidence on a mirror. The magnetic field component at the surface is 2Ho, which means that the surface current Js must also have the same magnitude as in normal incidence. We conclude that the force density on the mirror must be the same as that at normal incidence, namely, Fz = εoEo 2 . The time rate of change of momentum in the z-direction, however, is similar to that in Fig. 1(b), which means that the force density of normal incidence must have been multiplied by cos2 θ in the case of oblique incidence. The two methods of calculating Fz for p-light thus disagree by a factor of cos2 θ. The discrepancy is resolved when one realizes that, in addition to the magnetic force, an electric force is acting on the mirror in the opposite direction (−z). This additional force pulls on the electric charges induced at the surface by E⊥. Note that E⊥= 2Eosinθ just above and E⊥ = 0 just below the surface. The discontinuity in E⊥ gives the surface charge density as σ = 2εoEosinθ. The perpendicular E-field acting on these charges is the average of the fields just above and just below the surface, namely, Ez (eff) = Eosinθ. The electric force density is thus Fz = ½Real (σ Ez (eff)*) = εoEo 2 sin2 θ. The upward force on the charges thus reduces the (C) 2004 OSA 1 November 2004 / Vol. 12, No. 22 / OPTICS EXPRESS 5380 #5025- $15.00 US Received 10 August 2004; revised 13 October 2004; accepted 20 October 2004

downward force on the current, leading to a net F: =&Eo(1-sin"0), which is the same as that in the case of normal incidence multiplied by cos"8 The charge density o=2EE sine exp(i2Tx sine/? ple is produced the spatial variations of the current density J,= 2Hoexp(i2Tx SinA/o).Conservation of charge requires V J+ dp/dt=0, which, for time-harmonic fields, reduces to a@o=0 Considering that H.= EdZo and 2nc/Mo, it is readily seen that the above J, and o satisfy the required conservation law Note: The separate contributions of charge and current to the radiation pressure discussed in this section were originally discussed by Max Planck in his 1914 book, The Theory of Heat Radiation [10]. Our brief reconstruction of his arguments here is intended to facilitate the following discussion of electromagnetic force and momentum in dielectric media 4. Semi-infinite dielectric This section presents the core of the argument that leads to a new expression for the momentum of light inside dielectric media. Loudon [6, 7] has presented a similar argument in his quantum mechanical treatment of the problem. Although Loudons final result comes close to ours, there are differences that can be traced to his neglect of the mechanism of photon entry from the free-space into the dielectric medium H E1=(1+p)E H=(1-r)H n+IK Fig. 2. A linearly-polarized plane wave is normally incident on the surface of a sem medium of complex dielectric constant E. The Fresnel reflection coefficient at the surfac Shown are the e- and H-field magnitudes for the incident, reflected and transmitted bean Figure 2 shows a linearly-polarized plane wave at normal incidence on the flat surface of a semi-infinite dielectric. The incident E-and H-fields have magnitudes Eo and Ho= Eo/Zo Assuming a beam cross-sectional area of unity (A=1.0m"), the time rate of flow of momentum onto the surface is 2EE0, of which a fraction rF is reflected back. The net rate of change of linear momentum, which must be equal to the force per unit area exerted on the surface, is thus F:=hE(1+IrP)E. We assume that the mediums dielectric constant g is #5025-S1500US Received 10 August 2004; revised 13 October 2004; accepted 20 October 2004 (C)2004OSA November 2004/Vol 12. No 22/OPTICS EXPRESS 5381

downward force on the current, leading to a net Fz = εoEo 2 (1 − sin2 θ), which is the same as that in the case of normal incidence multiplied by cos2 θ. The charge density σ = 2εoEosinθ exp(i2πx sinθ/λo) in the above example is produced by the spatial variations of the current density Js = 2Hoexp(i2πx sinθ/λo). Conservation of charge requires ∇ · J + ∂ρ/∂t = 0, which, for time-harmonic fields, reduces to ∂Js/∂ x − iω σ = 0. Considering that Ho = Eo/Zo and ω = 2πc/λo, it is readily seen that the above Js and σ satisfy the required conservation law. Note: The separate contributions of charge and current to the radiation pressure discussed in this section were originally discussed by Max Planck in his 1914 book, The Theory of Heat Radiation [10]. Our brief reconstruction of his arguments here is intended to facilitate the following discussion of electromagnetic force and momentum in dielectric media. 4. Semi-infinite dielectric This section presents the core of the argument that leads to a new expression for the momentum of light inside dielectric media. Loudon [6,7] has presented a similar argument in his quantum mechanical treatment of the problem. Although Loudon’s final result comes close to ours, there are differences that can be traced to his neglect of the mechanism of photon entry from the free-space into the dielectric medium. Fig. 2. A linearly-polarized plane wave is normally incident on the surface of a semi-infinite medium of complex dielectric constant ε. The Fresnel reflection coefficient at the surface is r. Shown are the E- and H-field magnitudes for the incident, reflected, and transmitted beams. Figure 2 shows a linearly-polarized plane wave at normal incidence on the flat surface of a semi-infinite dielectric. The incident E- and H-fields have magnitudes Eo and Ho = Eo/Zo. Assuming a beam cross-sectional area of unity (A = 1.0m2 ), the time rate of flow of momentum onto the surface is ½εoEo 2 , of which a fraction | r |2 is reflected back. The net rate of change of linear momentum, which must be equal to the force per unit area exerted on the surface, is thus Fz = ½εo(1 + |r |2 )Eo 2 . We assume that the medium’s dielectric constant ε is Ho Eo Et = (1 + r) Eo Ht = (1 − r) Ho X Z -rHo rEo n + iκ = √ε (C) 2004 OSA 1 November 2004 / Vol. 12, No. 22 / OPTICS EXPRESS 5381 #5025- $15.00 US Received 10 August 2004; revised 13 October 2004; accepted 20 October 2004

not purely real, but has a small imaginary part. The complex refractive index of the material is and the reflection coefficient is r=(1-VE V(+ve) Inside the dielectric, the E-field is E(=)=Er exp(inve:o), where E,=(1+r)Eo,the field is H(E)=vE(E, /Z exp(inve:/o), the D-field is D(E)=E()+P()=EEE(),and the dipolar current density is J)=-ioP()=-ioE(E-1)E(), where @=2If= 2nc/o is the optical frequency. The force per unit volume is thus given by F:=%2 Real(xB")=y(2/o)Real [-ive"(E-1)) EolE, Pexp(-4x-/o %(2)(n2+x2+1) EoE, Fexp(-4rx2) The total force per unit surface area is obtained by integrating the above F: from==0 to The multiplicative coefficient x disappears after integration, and the force per unit area becomes F:=4(+K+1)EE, I. Upon substitution for E, and r, this expression for F: turns out to be identical to that obtained earlier based on momentum considerations We now let x-0 and write the radiation force per unit surface area of the dielectric as F:=v4(n+ 1)EE,F(A similar trick has been used by R. Loudon in his calculation of the photon momentum inside dielectrics [7].)Considering that H,=nE,/Zo, one may also write F=V4 EoE, F+4 HolH, P. This must be equal to the rate of the momentum entering the medium at ==0. Since the speed of light in the medium is c/n, the momentum density(per unit volume)within the dielectric may be expressed as follor P:=7(n+ I)nEeR /c= v4(E+ l)EoJE, BrI Equation(4), the fundamental expression for the momentum density of plane waves in dielectrics, may also be written as p= 4(DXB)+4(Ex Hye. Historically, there has been a dispute as to whether the proper form for the momentum density of light in dielectrics is Minkowski's DxB or Abraham's Ex H/c2 [5]. The above discussion leads to the conclusion that neither form is appropriate; rather, it is the average of the two that yields most plausible expression for P In the limit when 2-1, the two terms in the expression for p become identical, and the familiar form for the free-space, P=S/c2, emerges Replacing D with EE+ Pand B with uH, we obtain P=y4(Px B)+ExH)c,which shows the separate contributions to a plane-wave's momentum density by the medium and by the radiation field. The mechanical momentum of the medium. 4P xB arises from the interaction between the induced polarization density P and the light's B-field. The contribution of the radiation field. ExH/C2 has the same form. S/c. as the momentum density of electromagnetic radiation in free space. Since P=E(E-D)E, the mechanical momentum density may be written as 4PxB=/dE-1)S/c. For a dilute medium having refractive index n= l, the coefficient of S/c in the above formula reduces to vE-1=n-I which leads to the expression(n-1)S/c derived in [5] for the mechanical momentum of dilute gases. The physical basis for the separation of the momentum density into electromagnetic and mechanical contributions will be further elaborated in Section 12 Note: In a recent paper [151, Obukhov and Hehl argue, as we do here, that the correct interpretation of the electromagnetic momentum in dielectric media must be based on the standard form of the Lorentz force, taking into account both free and bound charges and currents. In their discussion of the case of normal incidence from vacuum onto a semi-infinite dielectric, however, they neglect to account for the mechanical momentum imparted to the dielectric medium. As a result, they find only the electromagnetic part of the momentum density;their Eq( 27)is in fact identical to ExH/c, where E and H are evaluated inside the dielectric. In contrast, our approach in the present section, which involves the introduction of a small(but non-zero)K, followed by an integration of the feeble magnetic Lorentz force over the infinite thickness of the dielectric ensures that the mechanical momentum of the #5025-S1500US Received 10 August 2004; revised 13 October 2004; accepted 20 October 2004 (C)2004OSA November 2004/Vol 12. No 22/OPTICS EXPRESS 5382

not purely real, but has a small imaginary part. The complex refractive index of the material is n + iκ = √ε , and the reflection coefficient is r = (1 − √ε )/(1 + √ε ). Inside the dielectric, the E-field is E(z) = Et exp(i2π√ε z/λo), where Et = (1 + r )Eo, the H￾field is H(z) = √ε (Et /Zo)exp(i2π√ε z/λ0), the D-field is D(z) = εoE(z) + P(z) = εoεE(z), and the dipolar current density is J(z) = −iω P(z) = −iω εo(ε − 1)E(z), where ω = 2πf = 2πc/λo is the optical frequency. The force per unit volume is thus given by Fz = ½ Real (J × B*) = ½(2π/λo) Real [−i√ε*( ε − 1)] εo|Et |2 exp(−4πκ z/λo) = ½(2π/λo) (n2 + κ2 + 1)κ εo|Et |2 exp(−4πκ z/λo). (3) The total force per unit surface area is obtained by integrating the above Fz from z = 0 to ∞. The multiplicative coefficient κ disappears after integration, and the force per unit area becomes Fz = ¼ (n2 + κ2 + 1) εo|Et |2 . Upon substitution for Et and r, this expression for Fz turns out to be identical to that obtained earlier based on momentum considerations. We now let κ → 0 and write the radiation force per unit surface area of the dielectric as Fz = ¼ (n2 + 1)εo|Et |2 . (A similar trick has been used by R. Loudon in his calculation of the photon momentum inside dielectrics [7].) Considering that Ht = nEt /Zo, one may also write Fz = ¼ εo|Et |2 + ¼ µo|Ht |2 . This must be equal to the rate of the momentum entering the medium at z = 0. Since the speed of light in the medium is c/n, the momentum density (per unit volume) within the dielectric may be expressed as follows: pz = ¼ (n2 + 1) nεo|Et |2 /c = ¼(ε + 1)εo|Et Bt |. (4) Equation (4), the fundamental expression for the momentum density of plane waves in dielectrics, may also be written as p = ¼ (D × B ) + ¼ (E × H )/c2 . Historically, there has been a dispute as to whether the proper form for the momentum density of light in dielectrics is Minkowski’s ½ D × B or Abraham’s ½ E × H /c2 [5]. The above discussion leads to the conclusion that neither form is appropriate; rather, it is the average of the two that yields the most plausible expression for p. In the limit when ε → 1, the two terms in the expression for p become identical, and the familiar form for the free-space, p = S/c2 , emerges. Replacing D with εoE + P and B with µ oH, we obtain p = ¼(P × B) + ½(E × H )/c2 , which shows the separate contributions to a plane-wave’s momentum density by the medium and by the radiation field. The mechanical momentum of the medium, ¼P × B, arises from the interaction between the induced polarization density P and the light’s B-field. The contribution of the radiation field, ½ E × H /c 2 , has the same form, S/c 2 , as the momentum density of electromagnetic radiation in free space. Since P = εo(ε − 1)E, the mechanical momentum density may be written as ¼P × B = ½(ε − 1)S/c2 . For a dilute medium having refractive index n ≈ 1, the coefficient of S/c2 in the above formula reduces to ½(ε − 1) ≈ n – 1, which leads to the expression (n − 1)S/c2 derived in [5] for the mechanical momentum of dilute gases. The physical basis for the separation of the momentum density into electromagnetic and mechanical contributions will be further elaborated in Section 12. Note: In a recent paper [15], Obukhov and Hehl argue, as we do here, that the correct interpretation of the electromagnetic momentum in dielectric media must be based on the standard form of the Lorentz force, taking into account both free and bound charges and currents. In their discussion of the case of normal incidence from vacuum onto a semi-infinite dielectric, however, they neglect to account for the mechanical momentum imparted to the dielectric medium. As a result, they find only the electromagnetic part of the momentum density; their Eq. (27) is in fact identical to ½ E × H /c 2 , where E and H are evaluated inside the dielectric. In contrast, our approach in the present section, which involves the introduction of a small (but non-zero) κ, followed by an integration of the feeble magnetic Lorentz force over the infinite thickness of the dielectric, ensures that the mechanical momentum of the (C) 2004 OSA 1 November 2004 / Vol. 12, No. 22 / OPTICS EXPRESS 5382 #5025- $15.00 US Received 10 August 2004; revised 13 October 2004; accepted 20 October 2004

expression for the momentum density, which is missing from Obukhov and Hehl's Eq. 74> medium is properly taken into consideration. This yields the term 4 (Px B)in our 5. Oblique incidence with s-polarized light To the best of our knowledge, the momentum of light at oblique incidence has not been discussed previously. This is an extremely important case, since it reveals the existence of a lateral radiation pressure at the edges of the beam within a dielectric medium; consistency with the results of Section 4 simply demands the existence of such lateral pressures. We analyze the case of s-polarization here, leaving a discussion of p-polarized light at oblique incidence for the next section. The two cases turn out to be fundamentally different, although both retain the expression for momentum density derived in the case of normal incidence Figure 3 shows the case of oblique incidence with s-polarized light at the interface between the free-space and a dielectric medium. Again, we assume that the dielectric constant E is complex, allowing it to approach a real number only after calculating the total force by ntegrating through the thickness of the medium. Inside the medium, the E- and H-field distributions are Er (x, =)=(1+r3)E exp[i2T(x sine +NvE-sin e nol (5a) Hx(x,=)=ve-sin0 Err(x, syZ HI: (x, =)=-sine Ety(x,=)Zo Here rs=(cose-Ve-sin 0)/(cos0+VE-sin'e)is the Fresnel reflection coefficient for s- light. Since there are no free charges inside the medium(nor on its surface), the only relevant force here is the magnetic Lorentz force on the dipolar current density Jy(x, = -ioEE-DErv(, =). Following the same procedure as before, we find the net force components along the x-and z-axes to be r X E1=(1+rs)E。 n+ik=ve medium of(complex)dielectric constant E The Fresnel reflection coefficient is denoted by rs #5025-S1500US Received 10 August 2004; revised 13 October 2004; accepted 20 October 2004 (C)2004OSA November 2004/Vol 12. No 22/OPTICS EXPRESS 5383

medium is properly taken into consideration. This yields the term ¼(P × B) in our last expression for the momentum density, which is missing from Obukhov and Hehl’s Eq. (27). 5. Oblique incidence with s-polarized light To the best of our knowledge, the momentum of light at oblique incidence has not been discussed previously. This is an extremely important case, since it reveals the existence of a lateral radiation pressure at the edges of the beam within a dielectric medium; consistency with the results of Section 4 simply demands the existence of such lateral pressures. We analyze the case of s-polarization here, leaving a discussion of p-polarized light at oblique incidence for the next section. The two cases turn out to be fundamentally different, although both retain the expression for momentum density derived in the case of normal incidence. Figure 3 shows the case of oblique incidence with s-polarized light at the interface between the free-space and a dielectric medium. Again, we assume that the dielectric constant ε is complex, allowing it to approach a real number only after calculating the total force by integrating through the thickness of the medium. Inside the medium, the E- and H-field distributions are Et y (x, z) = (1 + rs)Eo exp[i2π(x sinθ + z√ε − sin2 θ )/λo] (5a) Ht x (x, z) = √ε − sin2 θ Et y (x, z)/Zo (5b) Ht z (x, z) = −sinθ Et y (x, z)/Zo (5c) Here rs = (cosθ − √ε – sin2 θ ) / (cosθ + √ε – sin2 θ ) is the Fresnel reflection coefficient for s￾light. Since there are no free charges inside the medium (nor on its surface), the only relevant force here is the magnetic Lorentz force on the dipolar current density Jy(x, z) = −iω εo(ε − 1)Et y (x, z). Following the same procedure as before, we find the net force components along the x- and z-axes to be Fig. 3. Obliquely incident s-polarized plane wave arrives at the surface of a semi-infinite medium of (complex) dielectric constant ε. The Fresnel reflection coefficient is denoted by rs. Ho Eo θ θ′ Ht X Z rs Eo rsHo Et = (1 + rs)Eo n + iκ = √ε (C) 2004 OSA 1 November 2004 / Vol. 12, No. 22 / OPTICS EXPRESS 5383 #5025- $15.00 US Received 10 August 2004; revised 13 October 2004; accepted 20 October 2004

Fr=%Eosine Real(ve-sin0)|+r FEo F:=vE(cos0+lE-sin))|1+rs FEO (6b) The above results are valid for all values of e whether real the incident beams cross-sectional area must be A=cose to produce a unit area footprint at the interface, the time rate of change of the incident beams momentum upon reflection from the surface gives rise to F=hEine cose (1-Ps 2)E. and F:=2Eocos20(1+ks 2E2 These can be readily shown to agree with Eq (6), which has been obtained by direct integration of the force density through the thickness of the medium Next we allow e to be real, and set the refractive index n=VE. Since E1=|I+rs leO sine=nsine, and VE-sin-0=n cose, we may write Eq (6)as follows F=% esine′cose'E1P, (7a) F:=E(l-esin'e+ Ecos)E,R Note that the cross-sectional area A of the transmitted beam is not unity, but cose, which means that the above expressions for Fx and F: must be divided by cose if force per unit area is desired. The direction of the force F=Fxx+ F: z is not the same as the propagation direction (i. e, at angle A to the surface normal); the reason for this will become clear shortly From section 4 we know that. inside the dielectric. the force propagation direction must be F=/4E(E+ D)E, I(sinex cose z). Multiplying this force by the beam's cross-sectional area A=cose, then subtracting it from the previously calculated force in Eq (7)yields the following residual force: AF=E(E-1)sine(cos0x-sinA'z)E,F cOS 6 △F cose 6 Left edge Right edge and cose, respectively. A segment from the beams left edge(area propc a force AF on the dielectric; this force is not compensated by an equal and opposite force the right-hand edge of the beam, as is the case elsewhere at the opposite edges of the beam essive AF shown here corresponds to an s-polarized beam.( For p-light AF retains the same magnitude but reverses direction, so the edge force becomes expa #5025-S1500US Received 10 August 2004; revised 13 October 2004; accepted 20 October 2004 (C)2004OSA November 2004/Vol 12. No 22/OPTICS EXPRESS 5384

Fx = ½εosinθ Real(√ε – sin2 θ ) |1 + rs | 2 Eo 2 (6a) Fz = ¼εo(cos2 θ + |ε – sin2 θ| ) |1 + rs| 2 Eo 2 (6b) The above results are valid for all values of ε, whether real or complex. Considering that the incident beam’s cross-sectional area must be A = cosθ to produce a unit area footprint at the interface, the time rate of change of the incident beam’s momentum upon reflection from the surface gives rise to Fx = ½εosinθ cosθ (1 − |rs | 2 )Eo 2 and Fz = ½εocos2 θ (1 + |rs | 2 )Eo 2 . These can be readily shown to agree with Eq. (6), which has been obtained by direct integration of the force density through the thickness of the medium. Next we allow ε to be real, and set the refractive index n = √ε. Since |Et | = |1 + rs | 2 Eo 2 , sinθ = n sinθ′, and √ε – sin2 θ = n cosθ′, we may write Eq. (6) as follows: Fx = ½εoε sinθ′ cosθ′ |Et |2 , (7a) Fz = ¼εo(1 − ε sin2 θ′ + ε cos2 θ′) |Et |2 . (7b) Note that the cross-sectional area A of the transmitted beam is not unity, but cosθ′, which means that the above expressions for Fx and Fz must be divided by cosθ′ if force per unit area is desired. The direction of the force F = Fx x + Fz z is not the same as the propagation direction (i.e., at angle θ′ to the surface normal); the reason for this will become clear shortly. From Section 4 we know that, inside the dielectric, the force per unit area along the propagation direction must be F = ¼εo(ε + 1)|Et |2 (sinθ′x + cosθ′z). Multiplying this force by the beam’s cross-sectional area A = cosθ′, then subtracting it from the previously calculated force in Eq. (7) yields the following residual force: ∆F = ¼εo(ε − 1)sinθ′(cosθ′x − sinθ′z) |Et | 2 . (8) Fig. 4. Oblique incidence on a semi-infinite dielectric at angle θ. The beam’s footprint at the surface has unit area, while the incident and transmitted beams’ cross-sectional areas are cosθ and cosθ′, respectively. A segment from the beam’s left edge (area proportional to sinθ′) exerts a force ∆F on the dielectric; this force is not compensated by an equal and opposite force on the right-hand edge of the beam, as is the case elsewhere at the opposite edges of the beam. The compressive ∆F shown here corresponds to an s-polarized beam. (For p-light ∆F retains the same magnitude but reverses direction, so the edge force becomes expansive.) θ θ′ cosθ′ sinθ′ cosθ θ′ ∆F n Left edge Right edge (C) 2004 OSA 1 November 2004 / Vol. 12, No. 22 / OPTICS EXPRESS 5384 #5025- $15.00 US Received 10 August 2004; revised 13 October 2004; accepted 20 October 2004