week ending PRL97,133902(20 PHYSICAL REVIEW LETTERS 29 SEPTEMBER 2006 Optical Momentum Transfer to absorbing Mie Particles Brandon A. Kemp, Tomasz M. Grzegorczyk, and Jin Au Kon Research laboratory of electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA (Received 12 April 2006: revised manuscript received 26 June 2006; published 26 September 2006) The momentum transfer to absorbing particles is derived from the Lorentz force density without prior assumption of the momentum of light in media. We develop a view of momentum conservation rooted in the stress tensor formalism that is based on the separation of momentum contributions to bound and free currents and charges consistent with the Lorentz force density. This is in contrast with the usual separation of material and field contributions. The theory is applied to predict a decrease in optical momentum transfer to Mie particles due to absorption, which contrasts the common intuition based on the scattering and absorption by rayleigh particles. PACS numbers: 41.20Jb The momentum of light in optically dense media has lent interpretation in terms of momentum conservation that been the center of a debate in physics for nearly a century distinguishes two processes of momentum transfer result- [1, 2]. Although the So-called Abraham- Minkowski contro- ing from the wave refiection or transmission at the bound versy originated out of relativistic formulations, the pri- ary and the attenuation in the medium. Our proposed view mary issue of the radiation pressure exerted on the renders a more direct description of experiments than the interface of a dielectric boundary can be studied indepen- usual separation of wave momentum into electromagnetic dently of material motion [3]. The momentum density and material contributions [3, 5, 12-14. The Lorentz force vector derived from the macroscopic electromagnetic density and momentum conservation are equivalently ap- ave theory [4] for a nonmagnetic medium is G=Dx plied to explain relevant experimental observations and to B= EOuOE X H+ PX uoH, where the wave momentum calculate the radiation pressure on absorbing Mie particles density is expressed as the sum of the electromagnetic In contrast to the scattering plus absorption forces derived momentum density EouoE X H and a mechanical momen- for small particles, we predict that absorption can reduce tum density resulting from the dielectric polarization P= the total optical momentum transfer to certain particles due D-EoE in the presence of a field [5]. The debate of the to the balance between the force on free currents and the of normally incident light from free force on bound currents and charg space onto a dielectric interface can be demonstrated by The Lorentz force is applied directly to bound and free momentum conservation at the interface. The difference in currents and charges, which are used to model lossy media the radiation pressure resulting from either D X B or with complex permittivity E= ER iEr and permeability ∈00E× H transmitted into the dielectric is significant;μ=R+ iui in a background of(∈0,μo). The time. an outward force results from the former, while an inward average Lorentz force density on bound currents and force results from the latter [6] charges due to harmonic excitation with e -ior dependence Recently, the pressure of light on lossless media as is [10 directly to bound currents and charges [7-9) and the b =RelEo(V B)E+ Ko(V H)h. momentum conservation theorem [4] were shown to be io(ER-E0)EXB+io(pR-uo)H×D”}.(1) in agreement [10]. Application of the Lorentz force di- rectly may be regarded as more fundamental, but it here Ref represents the real part of a complex quantity computationally expensive compared to momentum con- and denotes the complex conjugate. The leading two servation [11]. However, there still exist questions as terms in (D)contribute via a surface force density on bound to the application of momentum conservation to the radia- electric and magnetic charges, while the final two terms tion pressure in lossy materials represent the volume force density on bound electric and In this Letter, we rigorously treat the optical momentum magnetic currents [10, 11. In addition to(1), the force transfer to lossy media in the framework of the macro- ensity on free currents scopic electromagnetic theory. We apply the Lorentz force directly to bound and free currents and charges, thus fc=Re{ WEE X B”-o1H×D” avoiding a priori assumptions regarding the form of extends the recent analysis of the photon drag effect [15]to wave momentum. The separation of the total Lorentz force include magnetization, oblique incidence, and arbitrary in terms of forces on bound currents and charges(Fb) and polarization. The total time-average force on the material on free currents(Fc) provides insight into the mechanisms F= F Fb results from integration of the time-average of momentum transfer in lossy media. We show an equiva- force densities over the entire medium. 0031-9007/06/97(13)/133902(4) 133902-1 o 2006 The American Physical Society
Optical Momentum Transfer to Absorbing Mie Particles Brandon A. Kemp,* Tomasz M. Grzegorczyk, and Jin Au Kong Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA (Received 12 April 2006; revised manuscript received 26 June 2006; published 26 September 2006) The momentum transfer to absorbing particles is derived from the Lorentz force density without prior assumption of the momentum of light in media. We develop a view of momentum conservation rooted in the stress tensor formalism that is based on the separation of momentum contributions to bound and free currents and charges consistent with the Lorentz force density. This is in contrast with the usual separation of material and field contributions. The theory is applied to predict a decrease in optical momentum transfer to Mie particles due to absorption, which contrasts the common intuition based on the scattering and absorption by Rayleigh particles. DOI: 10.1103/PhysRevLett.97.133902 PACS numbers: 41.20.Jb, 42.25.Fx The momentum of light in optically dense media has been the center of a debate in physics for nearly a century [1,2]. Although the so-called Abraham-Minkowski controversy originated out of relativistic formulations, the primary issue of the radiation pressure exerted on the interface of a dielectric boundary can be studied independently of material motion [3]. The momentum density vector derived from the macroscopic electromagnetic wave theory [4] for a nonmagnetic medium is G D B 00E H P 0H, where the wave momentum density is expressed as the sum of the electromagnetic momentum density 00E H and a mechanical momentum density resulting from the dielectric polarization P D 0E in the presence of a field [5]. The debate of the radiation pressure of normally incident light from free space onto a dielectric interface can be demonstrated by momentum conservation at the interface. The difference in the radiation pressure resulting from either D B or 00E H transmitted into the dielectric is significant; an outward force results from the former, while an inward force results from the latter [6]. Recently, the pressure of light on lossless media as calculated by both the application of the Lorentz force directly to bound currents and charges [7–9] and the momentum conservation theorem [4] were shown to be in agreement [10]. Application of the Lorentz force directly may be regarded as more fundamental, but it is computationally expensive compared to momentum conservation [11]. However, there still exist some questions as to the application of momentum conservation to the radiation pressure in lossy materials. In this Letter, we rigorously treat the optical momentum transfer to lossy media in the framework of the macroscopic electromagnetic theory. We apply the Lorentz force directly to bound and free currents and charges, thus avoiding a priori assumptions regarding the form of wave momentum. The separation of the total Lorentz force in terms of forces on bound currents and charges (Fb) and on free currents (Fc) provides insight into the mechanisms of momentum transfer in lossy media. We show an equivalent interpretation in terms of momentum conservation that distinguishes two processes of momentum transfer resulting from the wave reflection or transmission at the boundary and the attenuation in the medium. Our proposed view renders a more direct description of experiments than the usual separation of wave momentum into electromagnetic and material contributions [3,5,12–14]. The Lorentz force density and momentum conservation are equivalently applied to explain relevant experimental observations and to calculate the radiation pressure on absorbing Mie particles. In contrast to the scattering plus absorption forces derived for small particles, we predict that absorption can reduce the total optical momentum transfer to certain particles due to the balance between the force on free currents and the force on bound currents and charges. The Lorentz force is applied directly to bound and free currents and charges, which are used to model lossy media with complex permittivity R iI and permeability R iI in a background of (0, 0). The timeaverage Lorentz force density on bound currents and charges due to harmonic excitation with ei!t dependence is [10] f b 1 2Ref0r EE 0r HH i!R 0E B i!R 0H D g; (1) where Refg represents the real part of a complex quantity and denotes the complex conjugate. The leading two terms in (1) contribute via a surface force density on bound electric and magnetic charges, while the final two terms represent the volume force density on bound electric and magnetic currents [10,11]. In addition to (1), the force density on free currents, f c 1 2Ref!IE B !IH D g; (2) extends the recent analysis of the photon drag effect [15] to include magnetization, oblique incidence, and arbitrary polarization. The total time-average force on the material F F c Fb results from integration of the time-average force densities over the entire medium. PRL 97, 133902 (2006) PHYSICAL REVIEW LETTERS week ending 29 SEPTEMBER 2006 0031-9007=06=97(13)=133902(4) 133902-1 © 2006 The American Physical Society
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 133902-2
The 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
PRL97,133902(2006) PHYSICAL REVIEW LETTERS week ending 29 SEPTEMBER 2006 300 diameter 0.5 um. The choice of an infinite dielectric cyl- inder incident by a TE wave allows for a complete descrip- E tion of the Lorentz force by the distribution inside the particle since the force on bound charges at the boundary is zero. The total electric field intensity is shown in Fig. 2(a) for the lossless dielectric particle. The total force s the force on bound currents F= Fb=84.02X 108N/m, which is computed by integrating the force density (1)over the area of the particle or, equivalently, by FIG. 2 Lorentz force density on bound currents (arrows)over- integrating the stress tensor of(6)along the circular path lain on electric field intensity E2I[(V/m)]resulting from ai shown in Fig. 1(a). The integration is performed by simple polarized plane wave of unit amplitude incident from free space numerical integration as in Ref [11]. Although equivalent with wavelength Ao =1064 nm onto a dielectric cylinder 1.25x 10-6[N/m )(b)The lossy cylinder, described by e= the particle is pulled toward the resulting high intensity ents.(max(f6=3.00×10-9N/m3) being pushed by the transfer of wave momentum. The total electric field intensity and force density on bound currents for a lossy particle is shown in Fig. 2(b). The resulting 2F=E2(1+|R2cos2 (9a) force density Fb=-x205 X 10-8N/m indicates that CEC-IRI)cose, sin;.(9b) which is offset by a positive momentum transfer to free currents represented by F=A5.80 X 108N/m, found which is in agreement with the Lorentz force density equivalently by applying an integration path to the stress integrated over the region z E [O, oo)as in Refs. [7,9]. tensor shown in Fig. 1(b). The total pressure on the particle The result(9b) was originally demonstrated in 1905 by is F=x3.75 X 10-18 N/m, which is less than the total Poynting [20], who observed a tangential force given by force on the transparent particle E sing, cose, for a nearly perfect absorbing medium a physically realistic situation of an electromagnetic (R=0 and Er + O)and zero tangential force for the wave impinging on a spherical particle is studied using reflection from a mirror [RI= 1. Thus, the radiation pres- Mie theory. For ER/Eo=2, Fig 3(a)shows that a maxi- sure is normal to the surface of a perfect reflector and mum optical momentum transfer occurs for a value of e given by the force on free currents at the surface F=F= near maximum absorption(i.e the penetration depth is on ZEolE 2cos20 If the dielectric constant of the background the order of the particle diameter). In contrast, a particle medium is increased by Vn, then the force on the reflector with large value for Er can exhibit reduced momentum increases EolEFcos0;=4;, which has been transfer due to significant wave attenuation in the sphere as observed for mirrors submerged in dielectric liquids shown in Fig 3(b). A further decrease in F for the high [21,22 contrast sphere is observed as e, approaches the limit of a The total fields due to plane-wave incidence on a 2D perfect reflector. The later point is made by comparing the particle are found from Mie theory applied to an absorbing adiabatic momentum transfers to the transparent dielectric infinite cylinder as in Refs. [23, 24]. We consider two sphere and to the reflecting sphere of equal size. However, separate problems of Einc melkor incident from free space the combined effect of Fb and Fc is required to explain the onto a lossless cylinder and onto a lossy cylinder, each of radiation pressure increase of Fig. 3(a) or decrease of f(b log1o(EI/Eo) FIG 3. Forces on a 2 um diameter sphere due to a plane wave of unit amplitude. The wave is incident from free space with 1064 nto a nonmagnetic sphere with(a)∈=2∈o+ iEr and(b)∈=16∈0+i∈p 133902-3
z^ F 0 2 E2 i 1 jRj 2cos2i; (9a) x^ F 0 2 E2 i 1 jRj 2 cosi sini; (9b) which is in agreement with the Lorentz force density integrated over the region z 2 0; 1 as in Refs. [7,9]. The result (9b) was originally demonstrated in 1905 by Poynting [20], who observed a tangential force given by 0 2 E2 i sini cosi for a nearly perfect absorbing medium (R 0 and I 0) and zero tangential force for the reflection from a mirror jRj 1. Thus, the radiation pressure is normal to the surface of a perfect reflector and is given by the force on free currents at the surface F Fc z^ 0jEij 2cos2i. If the dielectric constant of the background medium is increased by n p , then the force on the reflector increases to n p 0jEij 2cos2i n c Scos2i, which has been observed for mirrors submerged in dielectric liquids [21,22]. The total fields due to plane-wave incidence on a 2D particle are found from Mie theory applied to an absorbing infinite cylinder as in Refs. [23,24]. We consider two separate problems of E inc ze^ ik0x incident from free space onto a lossless cylinder and onto a lossy cylinder, each of diameter 0:5 m. The choice of an infinite dielectric cylinder incident by a TE wave allows for a complete description of the Lorentz force by the distribution inside the particle since the force on bound charges at the boundary is zero. The total electric field intensity is shown in Fig. 2(a) for the lossless dielectric particle. The total force is the force on bound currents F F b x^4:02 1018 N=m, which is computed by integrating the force density (1) over the area of the particle or, equivalently, by integrating the stress tensor of (6) along the circular path shown in Fig. 1(a). The integration is performed by simple numerical integration as in Ref. [11]. Although equivalent in results, the former approach provides the viewpoint that the particle is pulled toward the resulting high intensity focus, while the latter gives the usual intuition of a particle being pushed by the transfer of wave momentum. The total electric field intensity and force density on bound currents for a lossy particle is shown in Fig. 2(b). The resulting force density Fb x^2:05 1018 N=m indicates that the bound currents are pulled toward the incident wave, which is offset by a positive momentum transfer to free currents represented by Fc x^5:80 1018 N=m, found equivalently by applying an integration path to the stress tensor shown in Fig. 1(b). The total pressure on the particle is F x^3:75 1018 N=m, which is less than the total force on the transparent particle. A physically realistic situation of an electromagnetic wave impinging on a spherical particle is studied using Mie theory. For R=0 2, Fig. 3(a) shows that a maximum optical momentum transfer occurs for a value of I near maximum absorption (i.e. the penetration depth is on the order of the particle diameter). In contrast, a particle with large value for R can exhibit reduced momentum transfer due to significant wave attenuation in the sphere as shown in Fig. 3(b). A further decrease in F for the high contrast sphere is observed as I approaches the limit of a perfect reflector. The later point is made by comparing the adiabatic momentum transfers to the transparent dielectric sphere and to the reflecting sphere of equal size. However, the combined effect of Fb and Fc is required to explain the radiation pressure increase of Fig. 3(a) or decrease of FIG. 2. Lorentz force density on bound currents (arrows) overlain on electric field intensity jEzj 2 V=m 2 resulting from a z^ polarized plane wave of unit amplitude incident from free space with wavelength 0 1064 nm onto a dielectric cylinder. (a) The lossless cylinder is defined by 160. ( maxjfbj 1:25 106 N=m3 ) (b) The lossy cylinder, described by 16 10i0, contains an additional force density on free currents. maxjfbj 3:00 109 N=m3). −3 −2 −1 0 1 2 3 −1 0 1 2 3 x 10−23 log10( I 0) F [N] F Fc Fb (a) −3 −2 −1 0 1 2 3 −1 0 1 2 3 x 10−23 log10( I 0) F [N] F Fc Fb (b) FIG. 3. Forces on a 2 m diameter sphere due to a plane wave of unit amplitude. The wave is incident from free space with wavelength 0 1064 nm onto a nonmagnetic sphere with (a) 20 iI and (b) 160 iI. PRL 97, 133902 (2006) PHYSICAL REVIEW LETTERS week ending 29 SEPTEMBER 2006 133902-3
week ending PRL97,133902(20 PHYSICAL REVIEW LETTERS 29 SEPTEMBER 2006 No. FA8721-05-C-0002, and by the Chinese National Foundation under Contract Nos. 60371010 and No. 60531020. Opinions, interpretations, conclusions, and recommendations are those of the author and are not necessarily endorsed by the United States Government [1 H. Minkowski, Nachr. Akad. Wiss. Gottingen 1, 53 2000 (1908). diameter [nm [2 M. Abraham, Rend. Circ. Mat. Palermo 28, 1(1909) [3] D F. Nelson, Phys. Rev. A 44, 3985(1991) FIG4. Force versus diameter for a dielectric sphere(E/Eo= [4] J.A. Kong, Electromagnetic Wave Theory (EMW 16+ i)incident by a unit amplitude plane wave. The free-space Publishing. Cambridge. MA. 2005 wavelength of the incident wave is o =1064 nm. 5]J.P. Gordon, Phys. Rev. A 8, 14(1973) [6 R. Loudon, Fortschr. Phys. 52, 1134(2004) [7]R Loudon, J Mod. Opt. 49, 821(2002) Fig 3(b)due to absorption. This separation of F into Fb [81 Y.N. Obukhov and F.w. Hehl, Phys. Lett. A 311, 277 and Fc is further investigated by plotting the force versus (2003) sphere diameter for constant material parameters in Fig. 4. [9] M. Mansuripur, Opt. Express 12, 5375(2004) For small spheres, the power absorption is small since the [10] B.A. Kemp, T M. Grzegorczyk, and J.A.Kong, Opt diameter is much less than the penetration depth. When the Express13.9280(2005) diameter is of the order of the penetration depth, the force [I] B.A. Kemp, T.M. Grzegorczyk, and J.A. Kong, on free currents becomes significant due to the direct J. Electromagn. Waves. Appl. 20, 827(2006 dependence upon n given by (4) [12] P. Penfield and H. A. Haus, Electrodynamics of Moving In this Letter, we have provided a perspective of mo- Media (m.I.t. Press, Cambridge, MA, 1967) mentum transfer in lossy media in agreement with the [13]R. Loudon, L. Allen, and D F. Nelson, Phys. Rev. E55, distribution of Lorentz force and relevant experiments. In 071(1997). the case of an absorbing Mie particle, the contributions [14]S. Stallinga, Phys. Rev. E 73. 026606(2006 from Fh and Fc sum to give the total force on the particle. [15] R Loudon, S M. Barnett, and C. Baxter, Phys. Rev.a71 063802(2005) The particles we consider consist of ER-16Eo, a value [16] G K. Campbell, A.E. Leanhardt, J. Mun, M. Boyd, EW typical for semiconductors, and ER= 2, which is repre treed, w. Ketterle, and D. E. Pritchard, Phys. Rev. Lett. sentative of many insulators. A novelty of our results is the 94,170403(2005) reduction of optical momentum transfer to particles due to [17] A.E. Gibson, M.E. Kimmitt, and A C. Walker, Appl bsorption, which requires high dielectric contrast with the Phys.Let.17,75(1970). background medium and an attenuation length on the order [18] M. Mansuripur, Opt. Express 13, 2245(2005) of particle diameter. These results differ from the expected [19] J.A. Stratton, Electromagnetic Theory(McGraw-Hill, esult of scattering plus absorption forces resulting from New York, 1941) Rayleigh particles [25]. Because a detailed understanding [21] R V Jones and J.C. S Richards, ProcRSoc. A221,480 of both Fb and Fc are required to describe the physics involved, the theory presented here is fundamental to the (221 R.V. Jones and B. Leslie, Proc R Soc. A 360, 347(1978) understanding of optical momentum transfer to absorbing [23] T M. Grzegorczyk,BA. Kemp, and J.A.Kong,J.Opt particles Soc.Am.A23,2324(2006) This work is sponsored by NASA-USRA under [24] T.M. Grzegorczyk, B A. Kemp, and J.A. Kong, Phys Contracts No. NAS5-03110 and No. 07605-003-055, by Rev.Let.96,113903(2006) the Department of the Air Force under Contract [25]K Svoboda and S.M. Block, Opt Lett. 19, 930(1994) 133902-4
Fig. 3(b) due to absorption. This separation of F into Fb and Fc is further investigated by plotting the force versus sphere diameter for constant material parameters in Fig. 4. For small spheres, the power absorption is small since the diameter is much less than the penetration depth. When the diameter is of the order of the penetration depth, the force on free currents becomes significant due to the direct dependence upon n given by (4). In this Letter, we have provided a perspective of momentum transfer in lossy media in agreement with the distribution of Lorentz force and relevant experiments. In the case of an absorbing Mie particle, the contributions from F b and F c sum to give the total force on the particle. The particles we consider consist of R 160, a value typical for semiconductors, and R 2, which is representative of many insulators. A novelty of our results is the reduction of optical momentum transfer to particles due to absorption, which requires high dielectric contrast with the background medium and an attenuation length on the order of particle diameter. These results differ from the expected result of scattering plus absorption forces resulting from Rayleigh particles [25]. Because a detailed understanding of both F b and Fc are required to describe the physics involved, the theory presented here is fundamental to the understanding of optical momentum transfer to absorbing particles. This work is sponsored by NASA-USRA under Contracts No. NAS5-03110 and No. 07605-003-055, by the Department of the Air Force under Contract No. FA8721-05-C-0002, and by the Chinese National Foundation under Contract Nos. 60371010 and No. 60531020. Opinions, interpretations, conclusions, and recommendations are those of the author and are not necessarily endorsed by the United States Government. *Electronic address: bkemp@mit.edu [1] H. Minkowski, Nachr. Akad. Wiss. Go¨ttingen 1, 53 (1908). [2] M. Abraham, Rend. Circ. Mat. Palermo 28, 1 (1909). [3] D. F. Nelson, Phys. Rev. A 44, 3985 (1991). [4] J. A. Kong, Electromagnetic Wave Theory (EMW Publishing, Cambridge, MA, 2005). [5] J. P. Gordon, Phys. Rev. A 8, 14 (1973). [6] R. Loudon, Fortschr. Phys. 52, 1134 (2004). [7] R. Loudon, J. Mod. Opt. 49, 821 (2002). [8] Y. N. Obukhov and F. W. Hehl, Phys. Lett. A 311, 277 (2003). [9] M. Mansuripur, Opt. Express 12, 5375 (2004). [10] B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, Opt. Express 13, 9280 (2005). [11] B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, J. Electromagn. Waves. Appl. 20, 827 (2006). [12] P. Penfield and H. A. Haus, Electrodynamics of Moving Media (M.I.T. Press, Cambridge, MA, 1967). [13] R. Loudon, L. Allen, and D. F. Nelson, Phys. Rev. E 55, 1071 (1997). [14] S. Stallinga, Phys. Rev. E 73, 026606 (2006). [15] R. Loudon, S. M. Barnett, and C. Baxter, Phys. Rev. A 71, 063802 (2005). [16] G. K. Campbell, A. E. Leanhardt, J. Mun, M. Boyd, E. W. Streed, W. Ketterle, and D. E. Pritchard, Phys. Rev. Lett. 94, 170403 (2005). [17] A. F. Gibson, M. F. Kimmitt, and A. C. Walker, Appl. Phys. Lett. 17, 75 (1970). [18] M. Mansuripur, Opt. Express 13, 2245 (2005). [19] J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941). [20] J. H. Poynting, Philos. Mag. 9, 169 (1905). [21] R. V. Jones and J. C. S. Richards, Proc. R. Soc. A 221, 480 (1954). [22] R. V. Jones and B. Leslie, Proc. R. Soc. A 360, 347 (1978). [23] T. M. Grzegorczyk, B. A. Kemp, and J. A. Kong, J. Opt. Soc. Am. A 23, 2324 (2006). [24] T. M. Grzegorczyk, B. A. Kemp, and J. A. Kong, Phys. Rev. Lett. 96, 113903 (2006). [25] K. Svoboda and S. M. Block, Opt. Lett. 19, 930 (1994). 0 500 1000 1500 2000 −1 0 1 2 3x 10−23 diameter [nm] F [N] F Fc Fb FIG. 4. Force versus diameter for a dielectric sphere (=0 16 i) incident by a unit amplitude plane wave. The free-space wavelength of the incident wave is 0 1064 nm. PRL 97, 133902 (2006) PHYSICAL REVIEW LETTERS week ending 29 SEPTEMBER 2006 133902-4