,1={2mE[Ln1+1)}sin{2rf(n/c)-l]} -{2nE[Zm2+1)]}sin{2xf2[(n2z/c)-l} As before, the Poynting vector may be calculated and time-averaged over one beat period T Once again, ignoring the terms in A and higher order(or assuming= NAf so that these terms would automatically vanish), we find the average rate of flow of energy(per unit area per unit time)within the dielectric medium to be independent of z and given by <S(=,1)>=%{4n(m1+1)+[4m(m2+1)}E21Z乙 The electromagnetic(Abraham) momentum inside the dielectric whose volumetric density P=<S >/c must propagate with the group velocity V,=c(2-fiV(2f2-nifi)=c/(n+n) Here n=7(m+n2),f=2(1+ 2), and n=(n2-n)/(fi-fi is the derivative of n with respect to frequency. In the limit i-2, the refractive indices n, and n2 appearing in Eq(6) are nearly identical, and the time rate of flow of the electromagnetic mor medium is given by dq /dt=p k=4nE Eo/[(n+nf)(n+1)] (7) Clearly, this is not equal to the momentum flux across the vacuum-dielectric interface given by Eq (4b). The reason for the discrepancy is that some of the incoming momentum has been converted into mechanical force exerted on the dielectric medium. To compute this force w use the Lorentz law, taking into account the fact that the polarization density within the medium is P(=,t)=E(E-lE(, n), where E=n is a function of the frequency f The bound current density is thus given by JME, t=aP(a, t/ar=-4Ifi(n-1)Eo Eo cos 2I fi [(n=/c)-1 +4If2(n -1) Eo cos(2f(n =/c)-1) The Lorentz force density F=, t=JxXHo Hy is obtained by multiplying Eqs. 8)and(5b) At time t the leading edge of the beat"waveform"(see Fig. 1) has penetrated a distance =o=Vgt into the dielectric. The integrated force density from ==0 to =o thus yields the force per unit cross-sectional area, F:(0, exerted at time t on the medium. We find F(1)(26E2)={[n2-(mh-m2f)(m22-n/[(m+1)n2+1)}{1-cos2m(f-fn]} -{(m2-m)f2-f)/[(n1+1)n2+1)mnf+n2f2)}cos[4an2-n)t/m2-n1f +{[lnn2-(mf2+n2fn1f+n2/2)/[(n1+1)n2+1)}cos{2m(f1+f) [(n1-1)(n1+1)]cos(4xft)-%(n2-1)(n2+1)cos(4f1 Averaging F:(n) over the duration of a single beat waveform, T=1/(f2-fi), we find that second, third, fourth, and fifth terms in the above expression contribute very little to average force, and that the only significant contribution arises from the first term, namely, <F2>=(1/T)/F0dr=2eE2[mn2-(n-n2f(n2压-m)/[(m+1)m2+1)](10) In the limit f2-fi the above expression simplifies to yield the net(average) force per unit cross-sectional area exerted on the dielectric medium as follow {n2-[(n-nn+nf)]}E。n+1) 6629-$1500US Received 18 February 2005; revised 14 March 2005; accepted 15 March 2005 (C)2005OSA 21 March 2005/VoL 13. No 6/ OPTICS EXPRESS 2248Hy(z, t) = {2n1Eo/[Zo(n1 + 1)]} sin{2πf1 [(n1 z/c) – t ]} – {2n2Eo/[Zo(n2 + 1)]}sin{2π f2 [(n2 z/c) – t ]} (5b) As before, the Poynting vector may be calculated and time-averaged over one beat period T. Once again, ignoring the terms in ∆f and higher order (or assuming f = N∆f so that these terms would automatically vanish), we find the average rate of flow of energy (per unit area per unit time) within the dielectric medium to be independent of z and given by < Sz (z, t) > = ½{[4n1/(n1 + 1)2 ]+[4n2 /(n2 + 1)2 ]}Eo 2 /Zo. (6) The electromagnetic (Abraham) momentum inside the dielectric whose volumetric density pz = < Sz >/c 2 must propagate with the group velocity Vg = c( f2 – f1)/(n2 f2 – n1 f1) = c/(n + n′ f ). Here n = ½(n1 + n2), f = ½( f1 + f2), and n′ = (n2 – n1)/( f2 – f1) is the derivative of n with respect to frequency. In the limit f1 → f2, the refractive indices n1 and n2 appearing in Eq. (6) are nearly identical, and the time rate of flow of the electromagnetic momentum through the medium is given by d qz /dt = pzVg = 4nεoEo 2 /[(n + n′ f )(n + 1)2 ]. (7) Clearly, this is not equal to the momentum flux across the vacuum-dielectric interface given by Eq.(4b). The reason for the discrepancy is that some of the incoming momentum has been converted into mechanical force exerted on the dielectric medium. To compute this force we use the Lorentz law, taking into account the fact that the polarization density within the medium is P(z, t) = εo(ε – 1)E(z, t), where ε = n2 is a function of the frequency f. The bound current density is thus given by Jx(z, t) = ∂Px(z, t)/∂t = – 4πf1 (n1 – 1)εoEo cos{2π f1 [(n1z /c) – t]} + 4π f2 (n2 – 1)εoEo cos{2π f2 [(n2 z /c) – t]}. (8) The Lorentz force density Fz(z, t)= Jx ×µoHy is obtained by multiplying Eqs.(8) and (5b). At time t the leading edge of the beat “waveform” (see Fig. 1) has penetrated a distance zo = Vg t into the dielectric. The integrated force density from z = 0 to zo thus yields the force per unit cross-sectional area, Fz(t), exerted at time t on the medium. We find Fz(t)/(2εoEo 2 ) = {[n1n2 – (n1 f2 – n2 f1)/(n2 f2 – n1 f1)]/[(n1+ 1)(n2 + 1)]}{1 – cos[2π( f2 – f1)t]} – {(n2 – n1)( f2 – f1)/[(n1+ 1)(n2 + 1)(n1 f1+ n2 f2)]}cos[4π(n2 – n1) f1 f2 t /(n2 f2 – n1 f1)] + {[n1n2 – (n1 f2+ n2 f1)/(n1 f1+ n2 f2)]/[(n1 + 1)(n2+ 1)]}cos[2π( f1 +f2)t] –½[(n1 – 1)/(n1 + 1)] cos(4π f1 t) –½[(n2 – 1)/(n2 + 1)] cos(4π f2 t). (9) Averaging Fz(t) over the duration of a single beat waveform, T= 1/( f2 – f1), we find that the second, third, fourth, and fifth terms in the above expression contribute very little to the average force, and that the only significant contribution arises from the first term, namely, <Fz > = (1/T )∫ Fz(t) dt = 2εoEo 2 [n1 n2 – (n1 f2 – n2 f1)/(n2 f2 – n1 f1)]/[(n1 + 1)(n2 + 1)]. (10) In the limit f2 → f1 the above expression simplifies to yield the net (average) force per unit cross-sectional area exerted on the dielectric medium as follows: <Fz > = 2{n2 – [(n – n′ f )/(n + n′ f )]}εoEo 2 /(n+1)2 . (11) 0 T (C) 2005 OSA 21 March 2005 / Vol. 13, No. 6 / OPTICS EXPRESS 2248 #6629 - $15.00 US Received 18 February 2005; revised 14 March 2005; accepted 15 March 2005