In averaging F(n)of Eq (9)over the time interval T, we neg but the first term. This is easily justified for the third, fourth, and fifth terms, which exhibit rapid oscillations. However, the second term is harder to neglect, especially when n=0(i.e for a nearly dispersionless medium), because the cosine term under these circumstances is not rapidly oscillating. However, the magnitude of the second term is proportional to AnAf, which reduces the term's significance when the cosine is weakly oscillating. All in all, the time average of the second term in Eq (9)turns out to be negligible even when the medium is dispersionless(or nearly so) Adding <F> of Eq (11)to the rate of flow of the electromagnetic (i.e, Abraham) momentum inside the dielectric given by Eq (7) yields PV+<F>=2[(m2+1)(n+1)]eE This is identical to the total momentum per unit area per unit time imparted to the dielectric as given by Eq (4b). We have thus confirmed the conservation of momentum in the system of Fig. I by deriving the expression for the radiation pressure, Eq (11), and by requiring propagation at the group velocity Vg for the electromagnetic momentum inside the medium; see Eq (7) If mechanical momentum is assumed to travel through the dielectric with the group velocity Vs a mechanical momentum density P(ech)=<F:>/Vs can be defined which with the electromagnetic momentum density p total optical momentum residing in the medium. In the limit when n0, for each frequency component of the beam (i.e, fi,f2), p mech -V(E-1)<S >/c2, in agreement with the result obtained in [1] for dispersionless media 4. The photon drag effect An intriguing experimental observation in certain (weakly absorbing) semiconductors notably Si and Ge, is the photon drag effect [7]. When a photon of energy hf from a monochromatic beam(vacuum wavelength Mo=c/f) is absorbed within a semiconductor of refractive index n(i. e, the real part of the complex refractive index n+ ik, whose imaginary part is the absorption coefficient K), the excited charge carrier acquires a momentum equal to thf/c, the so-called Minkowski momentum of the photon. In contrast, the photons Abraham momentum-seen from the preceding sections discussions to be hf/[(n+nf)cl-is clearly different from the Minkowski momentum. Combining the photons electromagnetic and mechanical momenta does not resolve the discrepancy either, as the total photon momentum v[n+(I/n)] hf/c, differs from the Minkowski momentum as well Loudon et al [ 8] have given a comprehensive theory of the photon drag effect, arguing that the"transparent part"of the semiconductor(associated with the real part of the complex refractive index)takes up the difference between the photon's momentum and the Minkowski momentum, when the latter is transferred to the "absorbing part"of the material (i.e, the part associated with the imaginary component of the complex refractive index). We present a similar(though by no means identical)explanation of the photon drag effect by showing that the momentum picked up by a thin absorbing layer embedded in a transparent dielectric is equal to the minkowski momentum of the incident photon With reference to Fig. 2. the reflection and transmission coefficients of g lay of thickness d and complex index n+ix, in the limit of d<<no(where A is the vacuum wavelength of the incident beam), can be shown to be p=-[1+(k/2n)](2xxdD) (13a) 1-(2兀kd)+i[2n-(x2n)] (13b) 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 2249In averaging Fz(t) of Eq.(9) over the time interval T, we neglected the contributions of all but the first term. This is easily justified for the third, fourth, and fifth terms, which exhibit rapid oscillations. However, the second term is harder to neglect, especially when n′ ≈ 0 (i.e., for a nearly dispersionless medium), because the cosine term under these circumstances is not rapidly oscillating. However, the magnitude of the second term is proportional to ∆n∆f, which reduces the term’s significance when the cosine is weakly oscillating. All in all, the time average of the second term in Eq.(9) turns out to be negligible even when the medium is dispersionless (or nearly so). Adding < Fz > of Eq.(11) to the rate of flow of the electromagnetic (i.e., Abraham) momentum inside the dielectric given by Eq.(7) yields: pzVg + <Fz > = 2[(n2 + 1)/(n + 1)2 ]εoEo 2 . (12) This is identical to the total momentum per unit area per unit time imparted to the dielectric as given by Eq.(4b). We have thus confirmed the conservation of momentum in the system of Fig. 1 by deriving the expression for the radiation pressure, Eq.(11), and by requiring propagation at the group velocity Vg for the electromagnetic momentum inside the medium; see Eq.(7). If mechanical momentum is assumed to travel through the dielectric with the group velocity Vg, a mechanical momentum density pz (mech) = <Fz >/Vg can be defined which, combined with the electromagnetic momentum density pz (Abraham) = < Sz >/c 2 , accounts for the total optical momentum residing in the medium. In the limit when n′ →0, for each frequency component of the beam (i.e., f1, f2), pz (mech) →½(ε – 1) < Sz >/c 2 , in agreement with the result obtained in [1] for dispersionless media. 4. The photon drag effect An intriguing experimental observation in certain (weakly absorbing) semiconductors, notably Si and Ge, is the photon drag effect [7]. When a photon of energy hf from a monochromatic beam (vacuum wavelength λo = c/f ) is absorbed within a semiconductor of refractive index n (i.e., the real part of the complex refractive index n + iκ, whose imaginary part is the absorption coefficient κ ), the excited charge carrier acquires a momentum equal to nhf /c, the so-called Minkowski momentum of the photon. In contrast, the photon’s Abraham momentum – seen from the preceding section’s discussions to be hf/[(n+ n′ f ) c] – is clearly different from the Minkowski momentum. Combining the photon’s electromagnetic and mechanical momenta does not resolve the discrepancy either, as the total photon momentum, ½[n + (1/n)] hf /c, differs from the Minkowski momentum as well. Loudon et al [8] have given a comprehensive theory of the photon drag effect, arguing that the “transparent part” of the semiconductor (associated with the real part of the complex refractive index) takes up the difference between the photon’s momentum and the Minkowski momentum, when the latter is transferred to the “absorbing part” of the material (i.e., the part associated with the imaginary component of the complex refractive index). We present a similar (though by no means identical) explanation of the photon drag effect by showing that the momentum picked up by a thin absorbing layer embedded in a transparent dielectric is equal to the Minkowski momentum of the incident photon. With reference to Fig. 2, the reflection and transmission coefficients of an absorbing layer of thickness d and complex index n + iκ, in the limit of d << λo (where λo is the vacuum wavelength of the incident beam), can be shown to be ρ = − [1 + i(κ /2n)](2πκ d/λo), (13a) τ = 1 − (2πκ d/λo) + i[2n – (κ2 /n)] πd/λo. (13b) (C) 2005 OSA 21 March 2005 / Vol. 13, No. 6 / OPTICS EXPRESS 2249 #6629 - $15.00 US Received 18 February 2005; revised 14 March 2005; accepted 15 March 2005