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 5380area 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