The force in Eq (8)is orthogonal to the beams propagation direction, sine'x+ cos8z Moreover, the magnitude of AF is proportional to sine, the cross-sectional area of a segment from the edge of the beam just below the interface; see Fig 4. The force per unit area of the beam's edge is thus F(edge)=VE(E-1)E, F( For example, if the optical power density inside a glass medium having n= 1.5 happens to be 1.0 W/mm, the lateral pressure on each edge of the beam will be 1.39 nN/mm" )This force, which the light exerts on the dielectric at both edges of the beam, is orthogonal to the edge and compressive in this case of s-polarized light (We will see in the next section that the force at the edges of a p-polarized beam has exactly the same magnitude but opposite direction, tending to expand the dielectric medium. )Once the component of the force acting on the beam's edge has been subtracted from Eq (7), the remaining force turns out to be in the propagation direction and in full agreement with the esults of the preceding section 6. Oblique incidence with p-polarized light The case of p-polarized light depicted in Fig. 5 differs somewhat from the case of s-polarized light in that, in addition to the magnetic Lorentz force on its bulk, the medium also experiences an electric Lorentz force on the(bound) charges induced at its interface with the free space Inside the medium the field distributions are H,(x, =)=(I-Ip)Ho exp[i2r(x sine +ive-sin'e nl E(x, 3)=(VE-Sin-01eZ Hu, (r, 3), E1:(x,-)=-(sin/)Z。Hny(x,2) Here /o=(vE-sinF0-Ecos0)/(vE-sin 0+Ecos)is the Fresnel reflection coefficient at the interface for p-light. Since there are no net bound charges inside the medium, the only relevant force in the bulk is the Lorentz force of the H-field on the dipolar current density J(x, =)=-ioE(E-1)Ex, =) Following the same procedure as before, we find the force components exerted on the bulk along the x- and z-axes as follows Frbulk)=v sine Real(VE-sin'0)ler2ll-IpPHo (10a) F:(uk)=vlo( 0+ le-sineD)lerl1-IpPHo2 Next we consider the density of bound charges at the interface with the free space. In the region just above the interface the =-component of E is E(x, ==0)=(1-Ip)sine Eo exp(itX sine/o). Immediately below the interface, the continuity of D, requires that the above E: be divided by e. The surface charge density a, being equal to the discontinuity in E,Er, is thus given by o=E(1-1E )(1-rpsineEoexp(itx sine/o (11) The force components felt by these charges are Fx= Real [o(x)Er(x, ==0)1, where Ex is continuous across the interface, and F:=vReal lo(xe: (,=0)l, where E, is the average Er across the boundary, namely, E: =V(1+llExI-rpsin Eo exp(i2Tr sine/no).Thus Fx surface)= vll osine Real [(e*-1NVE-Sin 01e(1-r PHo e(1-|E)|1-rpPH2 The total force, the sum of the bulk and surface forces of Eqs.(10)and(12), is given by #5025-S1500US Received 10 August 2004; revised 13 October 2004; accepted 20 October 2004 (C)2004OSA November 2004/Vol 12. No 22/OPTICS EXPRESS 5385The force in Eq. (8) is orthogonal to the beam’s propagation direction, sinθ′x + cosθ′z . Moreover, the magnitude of ∆F is proportional to sinθ′, the cross-sectional area of a segment from the edge of the beam just below the interface; see Fig. 4. The force per unit area of the beam’s edge is thus F (edge) = ¼εo(ε − 1)|Et |2 . (For example, if the optical power density inside a glass medium having n = 1.5 happens to be 1.0 W/mm2 , the lateral pressure on each edge of the beam will be 1.39 nN/mm2 .) This force, which the light exerts on the dielectric at both edges of the beam, is orthogonal to the edge and compressive in this case of s-polarized light. (We will see in the next section that the force at the edges of a p-polarized beam has exactly the same magnitude but opposite direction, tending to expand the dielectric medium.) Once the component of the force acting on the beam’s edge has been subtracted from Eq. (7), the remaining force turns out to be in the propagation direction and in full agreement with the results of the preceding section. 6. Oblique incidence with p-polarized light The case of p-polarized light depicted in Fig. 5 differs somewhat from the case of s-polarized light in that, in addition to the magnetic Lorentz force on its bulk, the medium also experiences an electric Lorentz force on the (bound) charges induced at its interface with the free space. Inside the medium the field distributions are Ht y (x, z) = (1 − rp) Ho exp[i2π(x sinθ + z√ε − sin2 θ )/λo], (9a) Et x (x, z) = (√ε − sin2 θ /ε)Zo Ht y (x, z), (9b) Et z (x, z) = − (sinθ /ε) Zo Ht y (x, z). (9c) Here rp = (√ε – sin2 θ − ε cosθ) / (√ε – sin2 θ + ε cosθ) is the Fresnel reflection coefficient at the interface for p-light. Since there are no net bound charges inside the medium, the only relevant force in the bulk is the Lorentz force of the H-field on the dipolar current density J(x, z) = −iω εo(ε − 1)Et(x, z). Following the same procedure as before, we find the force components exerted on the bulk along the x- and z-axes as follows: Fx (bulk) = ½µ osinθ Real(√ε – sin2 θ ) |ε |−2 |1 − rp | 2 Ho 2 , (10a) Fz (bulk) = ¼µ o(|ε | 2 − sin2 θ + |ε – sin2 θ|)|ε | −2 |1 − rp | 2 Ho 2 . (10b) Next we consider the density of bound charges at the interface with the free space. In the region just above the interface the z-component of E is Ez (x, z= 0) = (1 − rp) sinθ Eo exp(i2πx sinθ/λo). Immediately below the interface, the continuity of D⊥ requires that the above Ez be divided by ε. The surface charge density σ, being equal to the discontinuity in εoEz, is thus given by σ = εo(1 − 1/ε )(1 − rp)sinθEoexp(i2πx sinθ/λo). (11) The force components felt by these charges are Fx = ½Real [σ (x)Ex*(x, z =0)], where Ex is continuous across the interface, and Fz = ½Real [σ (x)Ez*(x, z = 0)], where Ez is the average E⊥ across the boundary, namely, Ez = ½(1 + 1/ε )(1 − rp)sinθ Eo exp(i2πx sinθ/λo). Thus Fx (surface) = ½µ osinθ Real [(ε* − 1)√ε – sin2 θ ] |ε |−2 |1 − rp | 2 Ho 2 (12a) Fz (surface) = −¼µ osin2 θ (1 − |ε | −2 ) |1 − rp | 2 Ho 2 (12b) The total force, the sum of the bulk and surface forces of Eqs. (10) and (12), is given by (C) 2004 OSA 1 November 2004 / Vol. 12, No. 22 / OPTICS EXPRESS 5385 #5025- $15.00 US Received 10 August 2004; revised 13 October 2004; accepted 20 October 2004