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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 537612. 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
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