Comment on “Trapping force, force constant, and potential depths for dielectric spheres in the presence of spherical aberrations”

Patrick C. Chaumet

I point out a confusion that is rather common in optical forces, i.e., that the time average of the Lorentz force on a dipolefor a harmonic time-varying ﬁeldis sometimes assumed to be a gradient force that is due to omission of the radiative reaction term in the polarizability of the dipole.© 2004 Optical Society of America OCIS codes:290.0290, 180.0180, 260.0260.

1. Introduction made by those authors was to interpretmr,ttas the difference between the time-averaged modulus of 1 In a recent paper Rohrbach and Stelzerderived the the Poynting vector scattered by the particle and the optical force acting on a lossless dielectric particle incident Poynting vector; they wrote illuminated by a harmonic time-varying ﬁeld.The authors started by calculating the electromagnetic density forcei.e., the force per unit volumefr,tfrfr,t acting on a small polarizable particle with dipole mo- mr,taftermr,tbefore mentpr,tdensity force was written as. Thefor Ir, (3) 2ct more details of their method, see Ref. 2 where the subscripts “before” and “after” refer to pr,t fr,tpr,tEr,tBr,t. (1)times before and after scattering, respectively.Note t that the ﬁrst term in Eq.3is the gradient force; after some tedious calculations Rohrbach and Stelzer This is the Lorentz force.Using the relationspr,t showed that the second term is the scattering force. Er,tandEr,t Br,tt, they obtained They supported their reasoning by citing the work of 3 Gordon. However,Gordon used this reasoning for fr,tEr,tEr,t2 Er,t a pulse, and he emphasized that, for a standing wave, t mr,tis a well-known fact thatvanishes. This Br,t, (2)has been pointed out elsewheresee, for example, Ref. 4when the intensity of the ﬁeld is modulated. Only which is the total force experienced by the particle. at a low frequency can one hope to measure the op-Notice thatis the polarizability of the particle and4 – 6 tical force produced by this term.Unlike for a thatmr,t Er,t Br,tis proportional to the pulse, for which a harmonic ﬁeld is assumed, there is Poynting vector.At optical frequencies one needs to no after or before.In that case the scattering force work with the time average of Eq.2ﬁrst error. The may be derived from an erroneous computation.In fact, their assumption is that the difference between the momentum carried by the scattered light and The authorpatrick.chaumet@fresnel.fris with the Institut that by the incident light is directly related to the Fresnel, Unite´ Mixte de Recherche 6133, Faculte´ des Sciences et second law of Newton and hence gives the scattering TechniquesdeStJe´rˆome,AvenueEscadrille,Normandie-Niemen, force. Infact, for a standing wave the total time-F-13397 Marseille Cedex 20, France. averaged force is given by the time average of the ﬁrst Received 7 April 2003; revised manuscript received 9 June 2003; term of Eq.2as the second term vanishes: accepted 10 December 2003. 0003-693504091825-02$15.000 © 2004 Optical Society of Americafr12Er,tEr,t. (4)

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This expression seems far from that of a total force acting on a small polarizable particle, as the scatter ing force does not appear explicitly.The second con fusion of the authors of Ref. 1 is rather common, as they wrote that the time average of Eq.4is the gradient force, i.e.,Iris no longer the. This case. Themistake lies in assuming thatpr,t Er,t, whereis related to the Clausius– 0 0 Mossotti relation, which for a lossless material yields 1,2 real polarizability.In fact, one must not forget that the totalﬁeld at the position of a polarizable particle is the sum of incidentﬁeldEr,tand the ﬁeld that is due to the particle at its own location, 7 Er,t i.e., the radiative reaction term. Forsmall s polarizable particle, this radiationreactionﬁeld can 7,8 be written as 3 Esr,ti23kpr,t, (5) wherekis the modulus of the wave vector.There fore the correct dipole moment for a small polarizable 9 particle is given by pr,tEr,t0Er,tEsr,t, (6) which gives the following wellknown form for the 9 polarizability : 3 0123ik0. (7) It is important to make the correction to the Clausius–Mossotti relation to satisfy the optical the orem and derive the correct expression of the optical 9 force. Thenet force, from Eq.4, is then given 10,11 by j fir12ReEjriEr*, (8) which contains the gradient and the scattering force. For example, if we compute the net force on a minute sphere, using Eq.8, when the incident wave is a plane waveEEexpikz, weﬁnd thatf x0z 4 22 kE 3, which is the scattering force for a small 0 0 sphere. In conclusion, the expression used by Rohrbach and Stelzer to compute the optical force on their ob ject, although it led to the correct resultthe gradient force plus the scattering force, because of two mis

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takes that compensate for each other is based on ﬂFirst, there is no branching of theawed reasoning. interaction for harmonicﬁelds and hence there can be no splitting of the interaction into before and after events; hence the time average of the Poynting vector vanishes. Second,it is essential to include radiation reaction to satisfy the law of energy conservation. Equation4, which is the total force, gives only the gradient force:Theﬁrst error, which gives the scat tering force, compensates for the omission of the scat tering force in Eq.4.

References and Notes 1. A.Rohrbach and E. H. K. Stelzer,“Trapping force, force con stant, and potential depths for dielectric spheres in the pres ence of spherical aberrations,”Appl. Opt.41,2494–2507 2002. 2. A.Rohrbach and E. H. K. Stelzer,“Optical trapping of dielec tric particles in arbitraryﬁelds,”J. Opt. Soc. Am. A18,839– 8532001. 3. J.P. Gordon,“Radiation forces and momenta in dielectric me dia,”Phys. Rev. A8,14–211973. 4. I.Brevik,“Experiments in phenomenological electrodynamics and the electromagnetic energymomentum tensor,”Phys. Rep.52,133–2011979. 5. S.Antoci and L. Mihich,“Detecting Abraham’s force of light by the Fresnel–Fizeau effect,”Eur. Phys. J. D3,205–2101998. 6. S. Antoci and L. Mihich,“Does light exert Abraham’s force in a transparent medium?”arXiv.org ePrint archive,ﬁle 9808002, http://arxive.org/abs/physics/9808002. 7. J.D. Jackson,Classical Electrodynamics, 2nd ed.Wiley, New York, 1980. 8. Onecanﬁnd Eq.5by taking the transverse imaginary part of T3 the freespace Green function, ImGr,r 23k, as de scribed in S. M. Barnett, B. Huttner, R. Loudon, and R. Mat loob,“Decay of excited atoms in absorbing dielectrics,”J. Phys. B29,3763–37811996. 9. B. T. Draine,“The discrete dipole approximation and its ap plication to interstellar graphite grains,”Astrophys. J.333, 848–8721988. 10. P.C. Chaumet and M. NietoVesperinas,“Timeaveraged total force on a dipolar sphere in an electromagneticﬁeld,”Opt. Lett.25,1065–10672000. 11. P. C. Chaumet and M. NietoVesperinas,“Coupled dipole method determination of the electromagnetic force on a parti cle over aﬂat dielectric substrate,”Phys. Rev. B61,14,119– 14,1272000.