no Eo cos(no ko =o)=Eg cos(ko =o+ y) These equations, when solved for the phase angle yand the amplitude ratio Ey Eo, yield an(k。0+y)=(l/m)tan(nk。 E2/E2=1+(m2-1)cos(nk=0) Thus, according to Eq (6b), if =o is varied from 0 to 1/4no, the value of Eg would range from noEo to Eo. The gap field, of course, is the superposition of two identical counter-propagating beams in free space, each with its own(well-defined) momentum density +/c)z.t the null and peak positions of the E-field intensity within the dielectric, the plane-waves of the gap carry, respectively, the Minkowski and Abraham momenta of the dielectric medium When the Eg of Eq (6b)is averaged over all values of =o, each of the gap's plane-waves is seen to have an average momentum density equal to the arithmetic mean of the Abraham and Minkowski values associated with each plane-wave of the dielectric host. This is essentially the same conclusion as reached in our earlier papers [7, 11] via a different line of argument Returning now to the submerged mirror discussed in section 2, it must be clear that phase o of the reflection coefficient p plays a role similar to that of =o in Eq(6b). Thus when o varies from 180 to 0, it is as though the position of the mirror within the standing wave produced by interference between incident and reflected beams) has shifted from =0=0 toward =o=1/4no. This is tantamount to substituting sin(o/2)for cos(noko=o)in Eq (6b) which would then reproduce the result in Eq(2). The radiation pressure on the submerged mirror, derived in section 2 by a direct application of the lorentz law of force, is now seen to be identical with the pressure exerted by the counter-propagating plane-waves within the (fictitious)gap. Needless to say, the gap must be at the same location relative to the standing wave of Fig. 2 as is the mirror surface with respect to the standing wave created by the uperposition of the incident and reflected beams. 4. Momentum of a plane-wave The introduction of a narrow gap inside a dielectric medium is a useful theoretical device that may be employed in an alternative way to provide further insight. Figure 3 shows a linearly polarized plane-wave propagating along the --axis in a dielectric host of refractive index no the field amplitudes inside the medium are (Er, Hy)=(Eo, Ho)=(Eo, n, EZo). Let us now magine a narrow gap of width 8<< no in the host medium, and examine the nature of the fields inside this gap. The plane of the gap is y= in(a)and xz in(b). The electromagnetic field inside the gap is the superposition of two evanescent plane-waves, each of which must satisfy the constraints kk=ko, k. E=0, and (k/ko)xe= ZH imposed by Maxwells equations The combined E- and H-fields of these evanescent waves must also satisfy the boundary conditions on both walls of the gap In the case depicted in Fig. 3(a), continuity is required of the perpendicular D-field, D=Enlo Eo, and the tangential H-field, Hy=Ho, whereas in the case of Fig 3(b)it is Ex and B, =uoHy that must be continuous. In Fig 3(a) the two(co- propagating)gap fields have the following k-vector and field amplitudes Vn 2=1x+n2 E4=%(n02x z)E。 H=%(E。Z)y=%H。 It is readily verified that, in the limit when 8-0, superposition of the above fields satisfies the continuity requirements at both boundaries. The momentum density p in the gap is derived from the Poynting vector component S; along the propagation direction, namely #79094s1500USD Received 17 January 2007; accepted 22 February 2007 (C)2007OSA 5 March 2007/ VoL 15. No 5/ OPTICS EXPRESS 2681no Eo cos(no ko zo) = Eg cos(ko zo + ψ ). (5b) These equations, when solved for the phase angle ψ and the amplitude ratio Eg/Eo, yield tan (ko zo + ψ ) = (1/no) tan (no ko zo), (6a) Eg 2 /Eo 2 = 1 + (no 2 – 1) cos2 (no ko zo) . (6b) Thus, according to Eq. (6b), if zo is varied from 0 to λo/4no, the value of Eg would range from noEo to Eo. The gap field, of course, is the superposition of two identical counter-propagating beams in free space, each with its own (well-defined) momentum density ±½(εoEg 2 /c) z. At the null and peak positions of the E-field intensity within the dielectric, the plane-waves of the gap carry, respectively, the Minkowski and Abraham momenta of the dielectric medium. When the Eg 2 of Eq. (6b) is averaged over all values of zo, each of the gap’s plane-waves is seen to have an average momentum density equal to the arithmetic mean of the Abraham and Minkowski values associated with each plane-wave of the dielectric host. This is essentially the same conclusion as reached in our earlier papers [7, 11] via a different line of argument. Returning now to the submerged mirror discussed in section 2, it must be clear that the phase φ of the reflection coefficient ρ plays a role similar to that of zo in Eq. (6b). Thus when φ varies from 180° to 0°, it is as though the position of the mirror within the standing wave (produced by interference between incident and reflected beams) has shifted from zo = 0 toward zo = λo/4no. This is tantamount to substituting sin(φ /2) for cos(no ko zo) in Eq. (6b), which would then reproduce the result in Eq. (2). The radiation pressure on the submerged mirror, derived in section 2 by a direct application of the Lorentz law of force, is now seen to be identical with the pressure exerted by the counter-propagating plane-waves within the (fictitious) gap. Needless to say, the gap must be at the same location relative to the standing wave of Fig. 2 as is the mirror surface with respect to the standing wave created by the superposition of the incident and reflected beams. 4. Momentum of a plane-wave The introduction of a narrow gap inside a dielectric medium is a useful theoretical device that may be employed in an alternative way to provide further insight. Figure 3 shows a linearly polarized plane-wave propagating along the z-axis in a dielectric host of refractive index no; the field amplitudes inside the medium are (Ex, Hy) = (Eo, Ho) = (Eo, noEo/Zo). Let us now imagine a narrow gap of width δ << λo in the host medium, and examine the nature of the fields inside this gap. 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, each of which must satisfy the constraints k · k = ko 2 , k · E = 0, and (k/ko) × E = ZoH imposed by Maxwell’s equations. The combined E- and H-fields of these evanescent waves must also satisfy the boundary conditions on both walls of the gap. In the case depicted in Fig. 3(a), continuity is required of the perpendicular D-field, Dx = εono 2 Eo, and the tangential H-field, Hy = Ho, whereas in the case of Fig. 3(b) it is Ex and By = μoHy that must be continuous. In Fig. 3(a) the two (copropagating) gap fields have the following k-vector and field amplitudes: k± /ko = ±i√ no 2 – 1 x + no z (7a) E± = ½(no 2 x + i no√ no 2 – 1 z)Eo (7b) H± = ½(noEo/Zo) y = ½Ho y. (7c) It is readily verified that, in the limit when δ → 0, superposition of the above fields satisfies the continuity requirements at both boundaries. The momentum density p in the gap is derived from the Poynting vector component Sz along the propagation direction, namely, ^ ^ ^ ^ ^ ^ ^ #79094 - $15.00 USD Received 17 January 2007; accepted 22 February 2007 (C) 2007 OSA 5 March 2007 / Vol. 15, No. 5 / OPTICS EXPRESS 2681