vall osine Real(eve-sin0)lEr|1 - pho F: total)= yuo(le 0+|- el )le |1-rpPHo (13b) These results are valid for all values of e, real or complex. Considering that the incident beams cross-sectional area must be A= cose to yield a unit-area footprint at the interface, the time rate of change of the incident beams momentum upon reflection gives rise to Fr=v2 sine cose(1-PpluoHo and F:= Vcos"0(1+lpluoHo These can easily be shown to agree with Eqs. (13), obtained by a direct calculation of the force from the Lorentz lav E H Eo ++ +++--+++ X E H,=(I-Ip)Ho n+ik= va A p-polarized plane wave is obliquely incident at the surface of a semi-infinite medium of(complex) ctric constant E The Fresnel reflection coefficient is denoted by re- It is of some interest to compare the forces of s-and p-beams on the bulk of the medium given by Eqs. (6)and (10). In Eq (6) the E-field inside the medium is Et=(1+rs Eo, whereas in Eq (10)the H-field in the medium is H,=(1-rp)Ho. From Eqs. (5)and(9)we find E/Z=VHE+H2 Thus, even when E, I is the same in both cases, the force components turn out to be different This is caused by the direction of the force on the beams edges being different in the two cases, as will become clear below when we analyze the case of media with real-valued a We now allow e to be real, and set the refractive index n=vE. Since H, P=|l-rpPHo2 H,=nE, /Zo, sine =n sine, and ve-sine=n cose, Eq ( 10)can be written Fx E,R F: buk)=VEE-sine+cose)E,P Note that the cross-sectional area A of the transmitted beam is not unity, but cose. From the know that, inside the dielectric, the force per unit cross-sectional area #5025-S1500US Received 10 August 2004; revised 13 October 2004; accepted 20 October 2004 (C)2004OSA November 2004/Vol 12. No 22/OPTICS EXPRESS 5386Fx (total) = ½µ osinθ Real (ε*√ε – sin2 θ ) |ε |−2 |1 − rp | 2 Ho 2 (13a) Fz (total) = ¼µ o(|ε | 2 cos2 θ + |ε – sin2 θ|)|ε | −2 |1 − rp | 2 Ho 2 (13b) These results are valid for all values of ε, real or complex. Considering that the incident beam’s cross-sectional area must be A = cosθ to yield a unit-area footprint at the interface, the time rate of change of the incident beam’s momentum upon reflection gives rise to Fx = ½ sinθ cosθ (1 − |rp | 2 )µ oHo 2 and Fz = ½cos2 θ (1 + |rp| 2 )µ oHo 2 . These can easily be shown to agree with Eqs. (13), obtained by a direct calculation of the force from the Lorentz law. Fig. 5. A p-polarized plane wave is obliquely incident at the surface of a semi-infinite medium of (complex) dielectric constant ε. The Fresnel reflection coefficient is denoted by rp. It is of some interest to compare the forces of s- and p-beams on the bulk of the medium given by Eqs. (6) and (10). In Eq. (6) the E-field inside the medium is Et = (1 + rs )Eo, whereas in Eq. (10) the H-field in the medium is Ht = (1 − rp )Ho. From Eqs. (5) and (9) we find √ε Et y /Zo = √Htx 2 + Htz 2 (14a) Ht y = √εEtx 2 + εEtz 2 /Zo . (14b) Thus, even when |Et | is the same in both cases, the force components turn out to be different. This is caused by the direction of the force on the beam’s edges being different in the two cases, as will become clear below when we analyze the case of media with real-valued ε. We now allow ε to be real, and set the refractive index n = √ε. Since |Ht |2 = |1 − rp | 2 Ho 2 , Ht = nEt /Zo, sinθ = n sinθ′, and √ε – sin2 θ = n cosθ′, Eq. (10) can be written Fx (bulk) = ½εo sinθ′ cosθ′ |Et |2 , (15a) Fz (bulk) = ¼εo(ε − sin2 θ′ + cos2 θ′) |Et |2 . (15b) Note that the cross-sectional area A of the transmitted beam is not unity, but cosθ′. From the earlier discussions we know that, inside the dielectric, the force per unit cross-sectional area Ho Eo θ θ′ Et X Z rp Eo -rpHo Ht = (1 − rp)Ho − − +++ − − +++ − − +++ − − +++ σ n + iκ = √ε (C) 2004 OSA 1 November 2004 / Vol. 12, No. 22 / OPTICS EXPRESS 5386 #5025- $15.00 US Received 10 August 2004; revised 13 October 2004; accepted 20 October 2004