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PHYSICAL REVIEW LETTERS week ending PRL97,133902(2006) 29 SEPTEMBER 2006 The connection of (1) and(2)to momentum conserva- and free currents is found by integrating a surface in(5) tion can be shown by considering a normally incident that just encloses the entire particle so that T is evaluated at electromagnetic wave with complex wave number k r= a as shown in Fig 1(a), and the tensor in(6) reduces ker+ ik, transmitted into a medium occupying the half- to the free-space Maxwell stress tensor [19]. The force on pace>0. Substitution of the transmitted field E free currents Fc is found by integrating the stress tensor(6) sEe kul-ekzrz into(1)and(2) yields along the interior of the particle boundary at r= aas f6=-ifk,[ER-Eo)IE12+(uR-Fo Fb=F-F f=器kR[E|E2+1|HP (3b) The choice of integration paths for F, F, and Fc allow for the description of electromagnetic forces in media consis- where H is the magnetic field determined from Faradays tent with the direct application of the Lorentz force in(1) law.The negative sign leading the left-hand side of (a) and(2) the incident wave propagation direction when the medium The two methods are applied to model known experi- is optically dense. The force density on free currents can be ments by considering the general solution of a TE electro- written as liquely incident on the surface of infinite nonmagnetic medium (u= uo) occupying the 7.-110[|BP+AB]-=2Re{",s region z >0. The radiation pressure on bound currents is found by integrating the Maxwell stress tensor in(6)along (4) a path that Just encloses the boundary as in Ref [10] and for a weakly absorbing dielectric, is where n= ckeR/a is the index of refraction, c is the speed of light in vacuum, and 5=EXH is the complex Fb=2E? E0(+1R12)cos0: -ER ITIcos2e,(7) Poynting vector resulting from the application Poynting's theorem to the second equality. The result of where r is the reflection coefficient. t is the transmission (4)is interpreted as two means of momentum transfer. First coefficient, and 0, and 0, are the incident and transmitted the transfer of momentum at the boundary is due entirely to angles, respectively. Thus, the total force on bound currents tion or transmission (i.e, Rev. S=0).Second, the is normal to the surface and directed toward the ine wave for ER Eo, while the tangential component of wave transfer of momentum to free currents due to the attenu- momentum is conserved across the boundary due to phase ation of the wave in the medium is given by the divergence matching. To formally prove this assertion, it is necessary of the momentum p=nS/c in (4). Recent experiments to consider the tangential component of the Lorentz force onfirm this result by showing that the observed transfer of density momentum to an atom in a dilute directly propor tional to the macroscopic refractive index [16]. This de R·fb=是Re{-ieR-∈0)kEP2T2e-2k3,(8) pendence on n has also been observed in the photon drag which is due to the transmitted field E neasurements [17 and was recentl Refs.[15, 18]. We conclude that the direct dependence of yEiTe kile'ir-ekg. The tangential absorbed momentum on the refractive index n holds for bound currents in (8)is zero when k x ielding a both dielectric and magnetic media normal pressure on a half-space void of free currents. The The connection of momentum transfer to bound and free total radiation pressure on an absorbing half-space is found currents is formalized by applying the momentum conser- from (5)with the tensor integration extended to z=+oo vation theorem via the maxwell stress tensor The time- average force on currents and charges enclosed by a sur face of area a with unit normal n is F=-Re∮dAti where the complex stress tensor is given by [ll] T=IDE+B. HY-DE*-BH. (6) In(6), DE and B"A are dyadic products and I is the(3X FIG. 1. Integration path for(5)applied to a lossy particle with 3)identity matrix. The tensor(6) can be applied to distin- radius a and (6, u) in a background of (Eo, uo).(a)An guish between Fb and Fc by noting the relationship in(4). integration path that completely encloses the particle gives the As an example, we consider the force calculation on a total Lorentz force F.(b)The integration path just inside the particle of radius a. The total Lorentz force F on bound boundary gives the force on the free carriers Fc 133902-2The connection of (1) and (2) to momentum conservation can be shown by considering a normally incident electromagnetic wave with complex wave number kz kzR ikzI transmitted into a medium occupying the halfspace z > 0. Substitution of the transmitted field E yE^ 0ekzIzeikzRz into (1) and (2) yields f b z^ 1 2kzI R 0jEj 2 R 0jHj 2 ; (3a) f c z^ 1 2kzR IjEj 2 IjHj 2 ; (3b) where H is the magnetic field determined from Faraday’s law. The negative sign leading the left-hand side of (3a) indicates that the force on the bound currents is opposite to the incident wave propagation direction when the medium is optically dense. The force density on free currents can be written as f c z^ 1 2 n! c IjEj 2 IjHj 2 z^ 1 2 Re n c r S ; (4) where n ckzR=! is the index of refraction, c is the speed of light in vacuum, and S E H is the complex Poynting vector resulting from the application of Poynting’s theorem to the second equality. The result of (4) is interpreted as two means of momentum transfer. First the transfer of momentum at the boundary is due entirely to F b since electromagnetic power is conserved in the reflection or transmission (i.e., Refr Sg 0). Second, the transfer of momentum to free currents due to the attenuation of the wave in the medium is given by the divergence of the momentum p nS=c in (4). Recent experiments confirm this result by showing that the observed transfer of momentum to an atom in a dilute gas is directly proportional to the macroscopic refractive index [16]. This dependence on n has also been observed in the photon drag measurements [17] and was recently analyzed in Refs. [15,18]. We conclude that the direct dependence of absorbed momentum on the refractive index n holds for both dielectric and magnetic media. The connection of momentum transfer to bound and free currents is formalized by applying the momentum conservation theorem via the Maxwell stress tensor. The timeaverage force on currents and charges enclosed by a surface of area A with unit normal n^ is F 1 2 ReI A dA n^ T r ; (5) where the complex stress tensor is given by [11] T 1 2D E B H I DE BH: (6) In (6), DE and BH are dyadic products and I is the (3 3) identity matrix. The tensor (6) can be applied to distinguish between Fb and Fc by noting the relationship in (4). As an example, we consider the force calculation on a particle of radius a. The total Lorentz force F on bound and free currents is found by integrating a surface in (5) that just encloses the entire particle so that T is evaluated at r a as shown in Fig. 1(a), and the tensor in (6) reduces to the free-space Maxwell stress tensor [19]. The force on free currents Fc is found by integrating the stress tensor (6) along the interior of the particle boundary at r a as shown in Fig. 1(b) such that all free currents are enclosed. The force on bound currents and charges is F b F Fc. The choice of integration paths for F, F b, and F c allow for the description of electromagnetic forces in media consistent with the direct application of the Lorentz force in (1) and (2). The two methods are applied to model known experiments by considering the general solution of a TE electromagnetic wave obliquely incident on the surface of an infinite nonmagnetic medium ( 0) occupying the region z > 0. The radiation pressure on bound currents is found by integrating the Maxwell stress tensor in (6) along a path that just encloses the boundary as in Ref. [10] and, for a weakly absorbing dielectric, is F b zE^ 2 i 0 2 1 jRj 2cos2i R 2 jTj 2cos2t ; (7) where R is the reflection coefficient, T is the transmission coefficient, and i and t are the incident and transmitted angles, respectively. Thus, the total force on bound currents is normal to the surface and directed toward the incoming wave for R > 0, while the tangential component of wave momentum is conserved across the boundary due to phase matching. To formally prove this assertion, it is necessary to consider the tangential component of the Lorentz force density x^ f b 1 2 RefiR 0k xjEij 2jTj 2e2kzIzg; (8) which is due to the transmitted field E yE^ iTekzIzeikzRzeikxx. The tangential force density on bound currents in (8) is zero when kx is real, yielding a normal pressure on a half-space void of free currents. The total radiation pressure on an absorbing half-space is found from (5) with the tensor integration extended to z ! 1 FIG. 1. Integration path for (5) applied to a lossy particle with radius a and (, ) in a background of (0, 0). (a) An integration path that completely encloses the particle gives the total Lorentz force F. (b) The integration path just inside the boundary gives the force on the free carriers Fc. PRL 97, 133902 (2006) PHYSICAL REVIEW LETTERS week ending 29 SEPTEMBER 2006 133902-2