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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 5381downward 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
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