of the beam is F=vE(E+ 1E, F(sine'x cose'z), aligned with the propagation direction Multiplying this force by the beams cross-sectional area A= cose, then subtracting it from the previously calculated force in Eq (15), yields the following residual force AF=-4E(E-1)sine(cos0'x-sinAz)Er The force in Eq (16)is orthogonal to the beam,s propagation direction, sinex+ cosea Moreover, its magnitude 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 beams edge is thus F(edge)=Eo(2-1)E, L. This force, exerted on the dielectric at both edges of the beam, is orthogonal to the edge and expansive in this case of p-polarized light. Once the force component acting on the beams edge has been subtracted from Eq (15), the remaining force turns out to be in the propagation direction and in full agreement with the results of Section 4 7. Dielectric slab illuminated at Brewsters angle This problem has been studied by Barlow [16], although not at Brewsters angle. The simplification afforded by our choice of the incidence angle is the elimination of multiple reflections within the slab. Unlike Barlow, we focus our attention on the radiation pressure at the edges of the beam inside the dielectric. and obtain the same result as in Section 6 from a completely new and unexpected perspective a p-polarized plane wave is incident on a dielectric slab of thickness d and refractive index n at the Brewsters angle AB(tan0B=n); the refracted angle inside the slab is given by tanAB=1/n. Since the reflectivity at Brewsters angle is zero, the only beams in this system are the incident beam, the refracted beam inside the slab, and the transmitted beam; Fig 6(a). Inside the slab, the H-field is the same as that outside, as required by the contin of Hy at the interfaces. Similarly, the continuity of En and Di at the interfaces require inside the slab and just beneath the surface, Ex EocosBB, and E:=(Eon)sineB, which means that the magnitude of E inside the slab is Eon 6+6′=90° A:E/n d H En ++++-++ +++-- E。 Brewster's angle BB. The H-field's magnitude inside the slab is the same as that outside but he E-field inside is reduced by a factor of n compared to the outside field. The bound charges n the upper and lower surfaces feel the force of the E-field. The transmitted beam is displaced by d/n horizontally and by d(1-1/r) vertically. The force Frx+F: z on the cancelled out by the force on the lower surface, but the slab experiences a net torque from Fr The torque of F: is cancelled out by the forces exerted at the beams edges within the dielectric.(b) Tilted cylinder of base area a and length d/sine, aligned with internal E-field. #5025-S1500US Received 10 August 2004; revised 13 October 2004; accepted 20 October 2004 (C)2004OSA November 2004/Vol 12. No 22/OPTICS EXPRESS 5387of the beam is F = ¼εo(ε + 1)|Et |2 (sinθ′x + cosθ′z), aligned with the propagation direction. Multiplying this force by the beam’s cross-sectional area A = cosθ′, then subtracting it from the previously calculated force in Eq. (15), yields the following residual force: ∆F = −¼εo(ε − 1) sinθ′(cosθ′x − sinθ′z) |Et |2 . (16) The force in Eq. (16) is orthogonal to the beam’s propagation direction, sinθ′x + cosθ′z . Moreover, its magnitude 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 . This force, exerted on the dielectric at both edges of the beam, is orthogonal to the edge and expansive in this case of p-polarized light. Once the force component acting on the beam’s edge has been subtracted from Eq. (15), the remaining force turns out to be in the propagation direction and in full agreement with the results of Section 4. 7. Dielectric slab illuminated at Brewster’s angle This problem has been studied by Barlow [16], although not at Brewster’s angle. The simplification afforded by our choice of the incidence angle is the elimination of multiple reflections within the slab. Unlike Barlow, we focus our attention on the radiation pressure at the edges of the beam inside the dielectric, and obtain the same result as in Section 6 from a completely new and unexpected perspective. A p-polarized plane wave is incident on a dielectric slab of thickness d and refractive index n at the Brewster’s angle θB (tanθB = n); the refracted angle inside the slab is given by tanθ′B = 1/n. Since the reflectivity at Brewster’s angle is zero, the only beams in this system are the incident beam, the refracted beam inside the slab, and the transmitted beam; see Fig. 6(a). Inside the slab, the H-field is the same as that outside, as required by the continuity of H| | at the interfaces. Similarly, the continuity of E|| and D⊥ at the interfaces require that, inside the slab and just beneath the surface, Ex = EocosθB, and Ez = (Eo/n2 )sinθB, which means that the magnitude of E inside the slab is Eo/n. Fig. 6. (a) Dielectric slab of thickness d and index n, illuminated with a p-polarized plane wave at Brewster’s angle θB. The H-field’s magnitude inside the slab is the same as that outside, but the E-field intside is reduced by a factor of n compared to the outside field. The bound charges on the upper and lower surfaces feel the force of the E-field. The transmitted beam is displaced by d/n horizontally and by d(1 – 1/n2 ) vertically. The force Fx x +Fz z on the upper surface is cancelled out by the force on the lower surface, but the slab experiences a net torque from Fx. The torque of Fz is cancelled out by the forces exerted at the beam’s edges within the dielectric. (b) Tilted cylinder of base area a and length d/sinθ′B , aligned with internal E-field. Ho Eo θ θ −−+++− −+++− −+++−− + d θ′ E /n θ +θ′ = 90° −−+++− −+++− −+++−− + X Z d/n 2 Eo Ho Ho d/n (a) θ′ +++ a E /n θ′ −−− (b) (C) 2004 OSA 1 November 2004 / Vol. 12, No. 22 / OPTICS EXPRESS 5387 #5025- $15.00 US Received 10 August 2004; revised 13 October 2004; accepted 20 October 2004