p=<S(2=%Re(2E1×2H)2=%n2EHc2 This is the Minkowski momentum of the plane-wave in the dielectric. A similar analysis for the case depicted in Fig 3(b), where the plane of the gap is parallel to xz, yield k/k=± E= 72EO 12)E。Z=%2(y Here the m um density in the gap is clearly that of Abraham, as it is given by >/c-=VRe(2Ex X 2H, *)c-=vEoHo (10) a) (b) n=n n- no k =noko k:=n0k。 贴(E四P in(b). The electromagnetic field inside the gap is the superposition of two evanesce waves whose combined fields satisfy the boundary conditions on both walls of the gap In the above example, the results depended on the choice of the gap orientation, as the symmetry of space was broken by the linear polarization of the beam. The field momentum, therefore, must be averaged over all possible gap orientations, leading to the mean value of the Abraham and Minkowski momenta. If, however, the beam happens to be circularly polarized, the results become independent of the gap orientation and yield the same mean value for the momentum density in all cases. For states of polarization other than circular, perhaps a better choice of the gap would be one in the form of a thin cylindrical shell of arbitrary radius, thickness 8<< /o, and cylinder axis aligned with the :-axis (i. e, the direction of propagation of the beam). All possible momentum densities will then occur at different locations around the circumference of this cylindrical shell, and the overall momentum density in the shell will coincide with the aforementioned average of the Minkowski and Abraham momenta Acknowledgements The author is grateful to Ewan Wright and Pavel Polynkin for helpful discussions. This work has been supported by the Air Force Office of Scientific Research(AFOSR) under contract number fa9550-04-1-0213 #79094$1500USD Received 17 January 2007; accepted 22 February 2007 (C)2007OSA 5 March 2007/ VoL 15. No 5/ OPTICS EXPRESS 2682p = < Sz >/c 2 = ½ Re(2Ex× 2Hy*)/c2 = ½ no 2EoHo/c2 . (8) This is the Minkowski momentum of the plane-wave in the dielectric. A similar analysis for the case depicted in Fig. 3(b), where the plane of the gap is parallel to xz, yields k± /ko = ± i√ no 2 – 1 y + no z (9a) E± = ½Eo x (9b) H± = ½(no y + i√ no 2 – 1 z)Eo/Zo = ½( y + i√ 1 – no –2 z ) Ho. (9c) Here the momentum density in the gap is clearly that of Abraham, as it is given by p = < Sz >/c 2 = ½Re (2Ex × 2Hy*)/c2 = ½EoHo/c 2 . (10) Fig. 3. A linearly polarized plane-wave propagates along the z-axis in a dielectric host of refractive index no; inside the medium, the field amplitudes are (Ex, Hy) = (Eo, Ho). A narrow gap of width δ << λo is assumed to exist in this medium; the plane of the gap is yz in (a) and xz in (b). The electromagnetic field inside the gap is the superposition of two evanescent plane￾waves whose combined fields satisfy the boundary conditions on both walls of the gap. In the above example, the results depended on the choice of the gap orientation, as the symmetry of space was broken by the linear polarization of the beam. The field momentum, therefore, must be averaged over all possible gap orientations, leading to the mean value of the Abraham and Minkowski momenta. If, however, the beam happens to be circularly polarized, the results become independent of the gap orientation and yield the same mean value for the momentum density in all cases. For states of polarization other than circular, perhaps a better choice of the gap would be one in the form of a thin cylindrical shell of arbitrary radius, thickness δ << λo, and cylinder axis aligned with the z-axis (i.e., the direction of propagation of the beam). All possible momentum densities will then occur at different locations around the circumference of this cylindrical shell, and the overall momentum density in the shell will coincide with the aforementioned average of the Minkowski and Abraham momenta. Acknowledgements The author is grateful to Ewan Wright and Pavel Polynkin for helpful discussions. This work has been supported by the Air Force Office of Scientific Research (AFOSR) under contract number FA9550-04-1-0213. δ x z H Eo o kz = noko n = no z x δ H Eo o H Eo o kz = noko n = no (a) (b) ^ ^ ^ ^ ^ ^ ^ #79094 - $15.00 USD Received 17 January 2007; accepted 22 February 2007 (C) 2007 OSA 5 March 2007 / Vol. 15, No. 5 / OPTICS EXPRESS 2682