on the two edges of the beam inside the slab; while its horizontal components are aligned (and, therefore, do not give rise to a torque), its vertical components F edge)= F ledge)sine B produce a torque. The product of the area of each edge of the beam and the distance between the edges is(d/cos0'B(A/cos0B)=Ad(n+ 1 )cosey/cos0'B, and the e-field inside the medium is E =En, so the torque produced by F edge) is given by V4EEo Ad (n-n)coseB This is exactly equal and opposite to the torque exerted on the induced surface charges by the vertical component of the E-field Note: In a 1912 paper, G. Barlow(stimulated by J H. Poynting) reports on an experiment similar to that described in this section, although incidence is not at Brewster's angle [16] Barlow claims that his results confirm Poynting's previous theory, which apparently supports Minkowski's form of momentum inside the plate. His contention, however, cannot be correct as the torque is entirely determined by the momentum of the light outside the plate. Any momentum p assigned to the light inside the glass medium results in the same overall torque 8. Force experienced by an anti-reflection coating layer To our knowledge, the discussion in this section has not appeared in the open literature. Aside from the practical significance of a force that can tear off a coating layer from its substrate. the results obtained below will be useful in balancing the electromagnetic and mechanical momenta of a beam of light that enters a dielectric slab from the free-space without any reflection losses at the interface Ho=EnZO X EnZo E2 d E2/Zo E1=iE。N nE。/Z oated by a quarter-wave layer of index Vn, a perfect substrate. TH coating layer produces an upward force equal to the time rate of change of the field's momentum in the free-space minus that in the substrate #5025-S1500US Received 10 August 2004; revised 13 October 2004; accepted 20 October 2004 (C)2004OSA November 2004/Vol 12. No 22/OPTICS EXPRESS 5389on the two edges of the beam inside the slab; while its horizontal components are aligned (and, therefore, do not give rise to a torque), its vertical components Fz (edge) = F (edge) sinθ′B produce a torque. The product of the area of each edge of the beam and the distance between the edges is (d/cosθ′B)(A/cosθB) = Ad(n2 + 1)cosθB/cosθ′B, and the E-field inside the medium is Et = Eo/n, so the torque produced by Fz (edge) is given by T = ¼εoEo 2 Ad (n − n–3)cosθB. (20) This is exactly equal and opposite to the torque exerted on the induced surface charges by the vertical component of the E-field. Note: In a 1912 paper, G. Barlow (stimulated by J. H. Poynting) reports on an experiment similar to that described in this section, although incidence is not at Brewster’s angle [16]. Barlow claims that his results confirm Poynting's previous theory, which apparently supports Minkowski’s form of momentum inside the plate. His contention, however, cannot be correct as the torque is entirely determined by the momentum of the light outside the plate. Any momentum p assigned to the light inside the glass medium results in the same overall torque. 8. Force experienced by an anti-reflection coating layer To our knowledge, the discussion in this section has not appeared in the open literature. Aside from the practical significance of a force that can tear off a coating layer from its substrate, the results obtained below will be useful in balancing the electromagnetic and mechanical momenta of a beam of light that enters a dielectric slab from the free-space without any reflection losses at the interface. Fig. 7. Semi-infinite medium of index n, coated by a quarter-wave layer of index √n, a perfect anti-reflection layer. A normally incident plane wave is fully transmitted to the semi-infinite substrate. The standing wave within the coating layer produces an upward force equal to the time rate of change of the field’s momentum in the free-space minus that in the substrate. Ho=Eo/Zo Eo Et = iEo /√n Ht = i√n Eo/Zo X Z -√n E2/Zo √n E E2 1/Zo E1 d √n n (C) 2004 OSA 1 November 2004 / Vol. 12, No. 22 / OPTICS EXPRESS 5389 #5025- $15.00 US Received 10 August 2004; revised 13 October 2004; accepted 20 October 2004