not purely real, but has a small imaginary part. The complex refractive index of the material is and the reflection coefficient is r=(1-VE V(+ve) Inside the dielectric, the E-field is E(=)=Er exp(inve:o), where E,=(1+r)Eo,the field is H(E)=vE(E, /Z exp(inve:/o), the D-field is D(E)=E()+P()=EEE(),and the dipolar current density is J)=-ioP()=-ioE(E-1)E(), where @=2If= 2nc/o is the optical frequency. The force per unit volume is thus given by F:=%2 Real(xB")=y(2/o)Real [-ive"(E-1)) EolE, Pexp(-4x-/o %(2)(n2+x2+1) EoE, Fexp(-4rx2) The total force per unit surface area is obtained by integrating the above F: from==0 to The multiplicative coefficient x disappears after integration, and the force per unit area becomes F:=4(+K+1)EE, I. Upon substitution for E, and r, this expression for F: turns out to be identical to that obtained earlier based on momentum considerations We now let x-0 and write the radiation force per unit surface area of the dielectric as F:=v4(n+ 1)EE,F(A similar trick has been used by R. Loudon in his calculation of the photon momentum inside dielectrics [7].)Considering that H,=nE,/Zo, one may also write F=V4 EoE, F+4 HolH, P. This must be equal to the rate of the momentum entering the medium at ==0. Since the speed of light in the medium is c/n, the momentum density(per unit volume)within the dielectric may be expressed as follor P:=7(n+ I)nEeR /c= v4(E+ l)EoJE, BrI Equation(4), the fundamental expression for the momentum density of plane waves in dielectrics, may also be written as p= 4(DXB)+4(Ex Hye. Historically, there has been a dispute as to whether the proper form for the momentum density of light in dielectrics is Minkowski's DxB or Abraham's Ex H/c2 [5]. The above discussion leads to the conclusion that neither form is appropriate; rather, it is the average of the two that yields most plausible expression for P In the limit when 2-1, the two terms in the expression for p become identical, and the familiar form for the free-space, P=S/c2, emerges Replacing D with EE+ Pand B with uH, we obtain P=y4(Px B)+ExH)c,which shows the separate contributions to a plane-wave's momentum density by the medium and by the radiation field. The mechanical momentum of the medium. 4P xB arises from the interaction between the induced polarization density P and the light's B-field. The contribution of the radiation field. ExH/C2 has the same form. S/c. as the momentum density of electromagnetic radiation in free space. Since P=E(E-D)E, the mechanical momentum density may be written as 4PxB=/dE-1)S/c. For a dilute medium having refractive index n= l, the coefficient of S/c in the above formula reduces to vE-1=n-I which leads to the expression(n-1)S/c derived in [5] for the mechanical momentum of dilute gases. The physical basis for the separation of the momentum density into electromagnetic and mechanical contributions will be further elaborated in Section 12 Note: In a recent paper [151, Obukhov and Hehl argue, as we do here, that the correct interpretation of the electromagnetic momentum in dielectric media must be based on the standard form of the Lorentz force, taking into account both free and bound charges and currents. In their discussion of the case of normal incidence from vacuum onto a semi-infinite dielectric, however, they neglect to account for the mechanical momentum imparted to the dielectric medium. As a result, they find only the electromagnetic part of the momentum density;their Eq( 27)is in fact identical to ExH/c, where E and H are evaluated inside the dielectric. In contrast, our approach in the present section, which involves the introduction of a small(but non-zero)K, followed by an integration of the feeble magnetic Lorentz force over the infinite thickness of the dielectric ensures that the mechanical momentum of the #5025-S1500US Received 10 August 2004; revised 13 October 2004; accepted 20 October 2004 (C)2004OSA November 2004/Vol 12. No 22/OPTICS EXPRESS 5382not purely real, but has a small imaginary part. The complex refractive index of the material is n + iκ = √ε , and the reflection coefficient is r = (1 − √ε )/(1 + √ε ). Inside the dielectric, the E-field is E(z) = Et exp(i2π√ε z/λo), where Et = (1 + r )Eo, the H￾field is H(z) = √ε (Et /Zo)exp(i2π√ε z/λ0), the D-field is D(z) = εoE(z) + P(z) = εoεE(z), and the dipolar current density is J(z) = −iω P(z) = −iω εo(ε − 1)E(z), where ω = 2πf = 2πc/λo is the optical frequency. The force per unit volume is thus given by Fz = ½ Real (J × B*) = ½(2π/λo) Real [−i√ε*( ε − 1)] εo|Et |2 exp(−4πκ z/λo) = ½(2π/λo) (n2 + κ2 + 1)κ εo|Et |2 exp(−4πκ z/λo). (3) The total force per unit surface area is obtained by integrating the above Fz from z = 0 to ∞. The multiplicative coefficient κ disappears after integration, and the force per unit area becomes Fz = ¼ (n2 + κ2 + 1) εo|Et |2 . Upon substitution for Et and r, this expression for Fz turns out to be identical to that obtained earlier based on momentum considerations. We now let κ → 0 and write the radiation force per unit surface area of the dielectric as Fz = ¼ (n2 + 1)εo|Et |2 . (A similar trick has been used by R. Loudon in his calculation of the photon momentum inside dielectrics [7].) Considering that Ht = nEt /Zo, one may also write Fz = ¼ εo|Et |2 + ¼ µo|Ht |2 . This must be equal to the rate of the momentum entering the medium at z = 0. Since the speed of light in the medium is c/n, the momentum density (per unit volume) within the dielectric may be expressed as follows: pz = ¼ (n2 + 1) nεo|Et |2 /c = ¼(ε + 1)εo|Et Bt |. (4) Equation (4), the fundamental expression for the momentum density of plane waves in dielectrics, may also be written as p = ¼ (D × B ) + ¼ (E × H )/c2 . Historically, there has been a dispute as to whether the proper form for the momentum density of light in dielectrics is Minkowski’s ½ D × B or Abraham’s ½ E × H /c2 [5]. The above discussion leads to the conclusion that neither form is appropriate; rather, it is the average of the two that yields the most plausible expression for p. In the limit when ε → 1, the two terms in the expression for p become identical, and the familiar form for the free-space, p = S/c2 , emerges. Replacing D with εoE + P and B with µ oH, we obtain p = ¼(P × B) + ½(E × H )/c2 , which shows the separate contributions to a plane-wave’s momentum density by the medium and by the radiation field. The mechanical momentum of the medium, ¼P × B, arises from the interaction between the induced polarization density P and the light’s B-field. The contribution of the radiation field, ½ E × H /c 2 , has the same form, S/c 2 , as the momentum density of electromagnetic radiation in free space. Since P = εo(ε − 1)E, the mechanical momentum density may be written as ¼P × B = ½(ε − 1)S/c2 . For a dilute medium having refractive index n ≈ 1, the coefficient of S/c2 in the above formula reduces to ½(ε − 1) ≈ n – 1, which leads to the expression (n − 1)S/c2 derived in [5] for the mechanical momentum of dilute gases. The physical basis for the separation of the momentum density into electromagnetic and mechanical contributions will be further elaborated in Section 12. Note: In a recent paper [15], Obukhov and Hehl argue, as we do here, that the correct interpretation of the electromagnetic momentum in dielectric media must be based on the standard form of the Lorentz force, taking into account both free and bound charges and currents. In their discussion of the case of normal incidence from vacuum onto a semi-infinite dielectric, however, they neglect to account for the mechanical momentum imparted to the dielectric medium. As a result, they find only the electromagnetic part of the momentum density; their Eq. (27) is in fact identical to ½ E × H /c 2 , where E and H are evaluated inside the dielectric. In contrast, our approach in the present section, which involves the introduction of a small (but non-zero) κ, followed by an integration of the feeble magnetic Lorentz force over the infinite thickness of the dielectric, ensures that the mechanical momentum of the (C) 2004 OSA 1 November 2004 / Vol. 12, No. 22 / OPTICS EXPRESS 5382 #5025- $15.00 US Received 10 August 2004; revised 13 October 2004; accepted 20 October 2004