Comment on fiTrapping force, force constant, and potential depths for dielectric spheres in the presence

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Comment on “Trapping force, force constant, andpotential depths for dielectric spheres in thepresence of spherical aberrations”Patrick C. ChaumetI point out a confusion that is rather common in optical forces, i.e., that the time average of the Lorentzforce on a dipole for a harmonic time-varying field is sometimes assumed to be a gradient force that isdue to omission of the radiative reaction term in the polarizability of the dipole. © 2004 Optical Societyof AmericaOCIS codes: 290.0290, 180.0180, 260.0260.1. Introduction made by those authors was to interpret mr, tt as1 the difference between the time-averaged modulus ofIn a recent paper Rohrbach and Stelzer derived thethe Poynting vector scattered by the particle and theoptical force acting on a lossless dielectric particleincident Poynting vector; they wroteilluminated by a harmonic time-varying field. Theauthors started by calculating the electromagneticfr fr, tdensity force i.e., the force per unit volume fr, tacting on a small polarizable particle with dipole mo- mr, t mr, t after beforement pr, t. The density force was written as for Ir , (3)2c tmore details of their method, see Ref. 2where the subscripts “before” and “after” refer topr, tfr, t pr, t Er, t Br, t. (1) times before and after scattering, respectively. Notetthat the first term in Eq. 3 is the gradient force;after some tedious calculations Rohrbach and StelzerThis is the ...

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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 dipolefor a harmonic time-varying fieldis 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 interpretmr,ttas 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 forcei.e., the force per unit volumefr,t frfr,t acting on a small polarizable particle with dipole mo-  mr,t after mr,t before  mentpr,tdensity force was written as. Thefor Ir, (3) 2ct more details of their method, see Ref. 2 where the subscripts “before” and “after” refer to pr,t fr,tpr,tEr,tBr,t. (1) times before and after scattering, respectively.Note t that the first term in Eq.3is the gradient force; after some tedious calculations Rohrbach and Stelzer This is the Lorentz force.Using the relationspr,t showed that the second term is the scattering force.  Er,tandEr,t  Br,tt, they obtained They supported their reasoning by citing the work of 3 Gordon. However,Gordon used this reasoning for  fr,tEr,tEr,t2 Er,t a pulse, and he emphasized that, for a standing wave, t mr,tis a well-known fact thatvanishes. This Br,t, (2) has been pointed out elsewheresee, for example, Ref. 4when 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 thatis the polarizability of the particle and 4 – 6 tical force produced by this term.Unlike for a thatmr,t Er,t Br,tis 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.2first 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 author[e-mail]is 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.2as the second term vanishes: accepted 10 December 2003. 0003-693504091825-02$15.000 © 2004 Optical Society of America fr12Er,tEr,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.4is the gradient force, i.e.,Iris no longer the. This case. Themistake lies in assuming thatpr,t  Er,t, whereis 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 incidentfieldEr,tand the field that is due to the particle at its own location, 7 Er,t i.e., the radiative reaction term. Forsmall s polarizable particle, this radiationreactionfield can 7,8 be written as 3 E s r,ti23kpr,t, (5) wherekis the modulus of the wave vector.There fore the correct dipole moment for a small polarizable 9 particle is given by pr,tEr,t 0 Er,tE s r,t, (6) which gives the following wellknown form for the 9 polarizability : 3  0 123ik 0 . (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 f i r12ReE j r i Er*, (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 waveEEexpikz, wefind thatf x0z 4 22 kE 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 resultthe 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. Equation4, 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– 8532001. 3. J.P. Gordon,“Radiation forces and momenta in dielectric me dia,”Phys. Rev. A8,14–211973. 4. I.Brevik,“Experiments in phenomenological electrodynamics and the electromagnetic energymomentum tensor,”Phys. Rep.52,133–2011979. 5. S.Antoci and L. Mihich,“Detecting Abraham’s force of light by the Fresnel–Fizeau effect,”Eur. Phys. J. D3,205–2101998. 6. S. Antoci and L. Mihich,“Does light exert Abraham’s force in a transparent medium?”arXiv.org ePrint archive,file 9808002, http://arxive.org/abs/physics/9808002. 7. J.D. Jackson,Classical Electrodynamics, 2nd ed.Wiley, New York, 1980. 8. Onecanfind Eq.5by taking the transverse imaginary part of T3 the freespace Green function, ImGr,r  23k, as de scribed in S. M. Barnett, B. Huttner, R. Loudon, and R. Mat loob,“Decay of excited atoms in absorbing dielectrics,”J. Phys. B29,3763–37811996. 9. B. T. Draine,“The discrete dipole approximation and its ap plication to interstellar graphite grains,”Astrophys. J.333, 848–8721988. 10. P.C. Chaumet and M. NietoVesperinas,“Timeaveraged total force on a dipolar sphere in an electromagneticfield,”Opt. Lett.25,1065–10672000. 11. P. C. Chaumet and M. NietoVesperinas,“Coupled dipole method determination of the electromagnetic force on a parti cle over aflat dielectric substrate,”Phys. Rev. B61,14,119– 14,1272000.