In section 2 we use the example of an idealized mirror, one with a negative value of the dielectric constant E, to show that, when no* l, the exactly calculated radiation pressure will o. Within the ideal flat with 100% reflectance), sets up a perfect standing wave between the incident and reflected plane-waves. When o=1800, the mirror's surface will be at the null point of the standing E- field; this is essentially the situation when conventional mirrors are used in the experiment, and the results of our calculations for the case of o=180 confirm the well-documented experimental findings [1, 2]. However, when o deviates from 1800, the calculated radiation pressure drops; all else being the same, the ratio of the pressures on two mirrors, one with o= 180 and the other with o=00, is found to be equal to no In section 3 we introduce a novel physical argument to demonstrate that the results of section 2 are not limited to certain(idealized) types of mirror, but are a general property of nding waves in a dielectric host. It will be shown that the radiation pressure, if observed locally within a standing wave, would be a function of location within the interference fringe A submerged pressure sensor would thus detect either the Abraham or the Minkowski momentum depending on whether the sensor is located at the peak or the valley of interference fringe. Since all locations within a fringe are equally accessible, the average photon momentum associated with a standing wave in a dielectric will thus have the arithmetic mean value of the abraham and Minkowski momenta q The argument of section 3 is employed in a different guise in section 4 in conjunction with a single plane-wave traveling in a dielectric host. The inescapable conclusion, once again, is that the plane-wave's momentum density is halfway between the Abraham and Minkowski values 2. Radiation pressure on an ideal submerged mirror The diagram in Fig. I depicts the interaction of light with a perfectly reflecting mirror whose index of refraction is the purely imaginary number in, (i.e, the mirror's dielectric constant. E=-nl, is a negative real number). The incidence medium is a transparent dielectric of refractive index no. The normally incident plane-wave has frequency fo, free-space wavelength Mo=cfo, wave-number k:=noko= 2/no, and electromagnetic field amplitudes (Ex, H )=(Eo, noEJZ ) where Zo=(Eo)is the impedance of the free space [5, 6] The Fresnel reflection coefficient p=(no-in)/(n, in1) of the submerged mirror has unit magnitude for all values of n but its phase angle, o=-2 arctan(n/no), can be anywhere in the range from 0 to 180 depending on the value of n Beneath the surface of the mirror the transmitted beam is an inhomogeneous plane-wave with an imaginary propagation vector k=i(2Tn/ao)z, which causes the beam amplitude to drop exponentially along the z-axis[6] The transmitted E- and H-fields have a relative phase of 90o, yielding a time-averaged Poynting vector <S>=72 Re(e x h)=0 everywhere inside the mirror; this, of course, is consistent with the mirror surface's 100% reflectance (i.e, PF=1) The radiation force per unit area of the mirror surface may be computed using the Lorentz force density F=psE+Jb x B exerted on the bound charge density Pb=-V P= -(E 1)V E=0, and also on the bound current density Jb=aP/at=-io E(E-D)E[7]. The force density F, when integrated over the penetration depth of the light beam, yields <F2>=/%Re-i(E-1)E2()H*()d v2 Re[ia E (ni+1(1+p(1-p*)n.] /exp(-2nko =)d= no2(1+n2)(n2+n12lE Received 17 January 2007, accepted 22 February 2007 (C)2007OSA 5 March 2007/ VoL 15. No 5/ OPTICS EXPRESS 2678In section 2 we use the example of an idealized mirror, one with a negative value of the dielectric constant ε, to show that, when no ≠ 1, the exactly calculated radiation pressure will have a strong dependence on φ. Within the submerging liquid, an ideal flat mirror (i.e., one with 100% reflectance), sets up a perfect standing wave between the incident and reflected plane-waves. When φ = 180°, the mirror’s surface will be at the null point of the standing E￾field; this is essentially the situation when conventional mirrors are used in the experiment, and the results of our calculations for the case of φ = 180° confirm the well-documented experimental findings [1, 2]. However, when φ deviates from 180°, the calculated radiation pressure drops; all else being the same, the ratio of the pressures on two mirrors, one with φ = 180° and the other with φ = 0°, is found to be equal to no 2 . In section 3 we introduce a novel physical argument to demonstrate that the results of section 2 are not limited to certain (idealized) types of mirror, but are a general property of standing waves in a dielectric host. It will be shown that the radiation pressure, if observed locally within a standing wave, would be a function of location within the interference fringe. A submerged pressure sensor would thus detect either the Abraham or the Minkowski momentum depending on whether the sensor is located at the peak or the valley of an interference fringe. Since all locations within a fringe are equally accessible, the average photon momentum associated with a standing wave in a dielectric will thus have the arithmetic mean value of the Abraham and Minkowski momenta. The argument of section 3 is employed in a different guise in section 4 in conjunction with a single plane-wave traveling in a dielectric host. The inescapable conclusion, once again, is that the plane-wave’s momentum density is halfway between the Abraham and Minkowski values. 2. Radiation pressure on an ideal submerged mirror The diagram in Fig. 1 depicts the interaction of light with a perfectly reflecting mirror whose index of refraction is the purely imaginary number in1 (i.e., the mirror’s dielectric constant, ε = −n1 2 , is a negative real number). The incidence medium is a transparent dielectric of refractive index no. The normally incident plane-wave has frequency fo, free-space wavelength λo = c/fo, wave-number kz = noko = 2π no/λo, and electromagnetic field amplitudes (Ex, Hy) = (Eo, noEo/Zo), where Zo = (μo/εo) ½ is the impedance of the free space [5, 6]. The Fresnel reflection coefficient ρ = (no – in1)/(no + in1) of the submerged mirror has unit magnitude for all values of n1, but its phase angle, φ = −2 arctan (n1/no), can be anywhere in the range from 0° to 180° depending on the value of n1. Beneath the surface of the mirror, the transmitted beam is an inhomogeneous plane-wave with an imaginary propagation vector k = i(2π n1/λo) z, which causes the beam amplitude to drop exponentially along the z-axis [6]. The transmitted E- and H-fields have a relative phase of 90°, yielding a time-averaged Poynting vector < S > = ½ Re (E × H*) = 0 everywhere inside the mirror; this, of course, is consistent with the mirror surface’s 100% reflectance (i.e., |ρ |2 = 1). The radiation force per unit area of the mirror surface may be computed using the Lorentz force density F = ρ bE + Jb × B exerted on the bound charge density ρ b = −∇ · P = −εo(ε – 1)∇ · E = 0, and also on the bound current density Jb = ∂P/∂t = −iω εo(ε – 1)E [7]. The force density F, when integrated over the penetration depth of the light beam, yields < Fz > = ∫ ½ Re[−iω εo μo (ε – 1) Ex (z)Hy*(z)] dz = ½ Re[iω εo μo (n1 2 + 1)(1 + ρ)(1 – ρ *) noEo 2 /Zo] ∫ exp(−2n1ko z) dz = [no 2 (1 + n1 2 )/(no 2 + n1 2 )]εoEo 2 . (1) ^ ∞ 0 0 ∞ #79094 - $15.00 USD Received 17 January 2007; accepted 22 February 2007 (C) 2007 OSA 5 March 2007 / Vol. 15, No. 5 / OPTICS EXPRESS 2678