Here F: is the time-averaged force per unit area of the mirror surface. For typical metallic mirrors,n >>no and the above formula reduces to <F:>=E/lo Eo. This is the expected result of assigning a Minkowski momentum density, Px=72 Red x B l, to the incident and reflected beams[7-10]. However, in the limit of small nI, Eq. (1)yields <F:>=E0E0, which is consistent with the presence of the Abraham momentum density, PA=v Re[*yc,in the dielectric medium. Although ordinary metals at visible wavelengths have a large value of nl,at higher frequencies (i.e, just below the plasma resonance frequency @p [ 5])the metals dielectric constant E assumes small negative values, leading to small(imaginary) values for the refractive index Hy PEx k2=nok。 (+pEx (1-p)H k ini ko Fig. 1. Reflection of light from a mirror having a purely imaginary refractive index of i n The cidence medium is a transparent dielectric of refractive index no. The normally incident plar wave has frequency fo free-space wavelength d=clfo, wa er k=nk=2rndao, an field amplitudes(En H,)=(Eo, noEdzo). The Fresnel reflection coefticient of the submerged (ig)=exp(2i arctan(n/no)). Beneath the mirror's surface, the transmitted inhomogeneous plane-wave whose imaginary propagation vector k=i(2n/2)2 eam ampl itude to drop exponentially along the =-axis We believe the above result is independent of the nature of the mirror, depending solely on the phase angle o of the Fresnel coefficient. In fact, Eq (1) may be written exclusively in terms of o( with no explicit reference to the mirror's refractive index in)as follow <F2>=[1+(n2-1)sin?(2)lE2 Since multilayer dielectric mirrors having large reflectance and arbitrary values of o can be readily designed, we expect the entire range of radiation pressures between EEo and Eno Eo predicted by Eq (2)to be amenable to experimental verification 3. Electromagnetic momentum in standing waves We present a general argument concerning the nature of the electromagnetic momentum in standing waves formed within transparent dielectric media. With reference to Fig. 2, two identical plane-waves propagating in opposite directions(along =) create a standing-wave inside a dielectric medium of refractive index n. For each wave the e- and H-field amplitudes are(Ex, Hy)=(Eo, noEZo The standing waves may thus be expressed as follows E, n=2E sin(n, k=)sin(an), H3(=,D)=2(nE。)cos(nok。=)co(OD #79094$1500USD Received 17 January 2007, accepted 22 February 2007 (C)2007OSA 5 March 2007/ VoL 15. No 5/ OPTICS EXPRESS 2679Here < Fz > is the time-averaged force per unit area of the mirror surface. For typical metallic mirrors, n1 >> no and the above formula reduces to < Fz > ≈ εono 2 Eo 2 . This is the expected result of assigning a Minkowski momentum density, pM = ½ Re[D ×B*], to the incident and reflected beams [7-10]. However, in the limit of small n1, Eq. (1) yields < Fz > ≈ εoEo 2 , which is consistent with the presence of the Abraham momentum density, pA = ½ Re[E × H*]/c2 , in the dielectric medium. Although ordinary metals at visible wavelengths have a large value of n1, at higher frequencies (i.e., just below the plasma resonance frequency ωp [5]) the metal’s dielectric constant ε assumes small negative values, leading to small (imaginary) values for the refractive index. Fig. 1. Reflection of light from a mirror having a purely imaginary refractive index of i n1. 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 field amplitudes (Ex, Hy) = (Eo, noEo/Zo). The Fresnel reflection coefficient of the submerged mirror is ρ = exp (iφ ) = exp [−2i arctan (n1/no)]. Beneath the mirror’s surface, the transmitted beam is an inhomogeneous plane-wave whose imaginary propagation vector k = i (2π n1/λo) z causes the beam amplitude to drop exponentially along the z-axis. We believe the above result is independent of the nature of the mirror, depending solely on the phase angle φ of the Fresnel coefficient. In fact, Eq. (1) may be written exclusively in terms of φ (with no explicit reference to the mirror’s refractive index in1) as follows: < Fz > = [1 + (no 2 – 1) sin2 (φ /2) ]εoEo 2 . (2) Since multilayer dielectric mirrors having large reflectance and arbitrary values of φ can be readily designed, we expect the entire range of radiation pressures between εoEo 2 and εono 2 Eo 2 predicted by Eq. (2) to be amenable to experimental verification. 3. Electromagnetic momentum in standing waves We present a general argument concerning the nature of the electromagnetic momentum in standing waves formed within transparent dielectric media. With reference to Fig. 2, two identical plane-waves propagating in opposite directions (along ± z) create a standing-wave inside a dielectric medium of refractive index no. For each wave the E- and H-field amplitudes are (Ex, Hy) = (Eo, noEo/Zo). The standing waves may thus be expressed as follows: Ex(z, t) = 2Eo sin(no ko z) sin(ω t), (3a) Hy(z, t) = 2(noEo/Zo) cos(no ko z) cos(ω t). (3b) n = no n = i n1 z x Ex Hy kz = no ko ρEx −ρ Hy × (1 + ρ)Ex kz = i n1 ko (1 − ρ) Hy ^ #79094 - $15.00 USD Received 17 January 2007; accepted 22 February 2007 (C) 2007 OSA 5 March 2007 / Vol. 15, No. 5 / OPTICS EXPRESS 2679