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

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 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 interpretmr,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,taftermr,tbefore mentpr,tdensity force was written as. Thefor Ir, (3) 2ct more details of their method, see Ref. 2where the subscripts “before” and “after” refer to pr,tfr,tpr,tEr,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,tshowed 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,tEr,tEr,t2 Er,ta pulse, and he emphasized that, for a standing wave, t mr,tis 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 Americafr12Er,tEr,t. (4)
20 March 2004Vol. 43, No. 9APPLIED OPTICS1825
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 Clausius0 0 Mossotti relation, which for a lossless material yields 1,2 real polarizability.In fact, one must not forget that the totaleld at the position of a polarizable particle is the sum of incidenteldEr,tand the eld that is due to the particle at its own location, 7 Er,t i.e., the radiative reaction term. Forsmall s polarizable particle, this radiationreactioneld 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 wellknown form for the 9 polarizability : 3 0123ik0. (7) It is important to make the correction to the ClausiusMossotti 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 fir12ReEjriEr*, (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, wend thatfx0z 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
1826 APPLIEDOPTICSVol. 43, No. 920 March 2004
takes that compensate for each other is based on First, there is no branching of theawed reasoning. interaction for harmonicelds 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:Therst 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.41,24942507 2002. 2. A.Rohrbach and E. H. K. Stelzer,Optical trapping of dielec tric particles in arbitraryelds,J. Opt. Soc. Am. A18,8398532001. 3. J.P. Gordon,Radiation forces and momenta in dielectric me dia,Phys. Rev. A8,14211973. 4. I.Brevik,Experiments in phenomenological electrodynamics and the electromagnetic energymomentum tensor,Phys. Rep.52,1332011979. 5. S.Antoci and L. Mihich,Detecting Abrahams force of light by the FresnelFizeau effect,Eur. Phys. J. D3,2052101998. 6. S. Antoci and L. Mihich,Does light exert Abrahams force in a transparent medium?arXiv.org ePrint archive,le 9808002, http://arxive.org/abs/physics/9808002. 7. J.D. Jackson,Classical Electrodynamics, 2nd ed.Wiley, New York, 1980. 8. Onecannd Eq.5by taking the transverse imaginary part of T3 the freespace 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. B29,376337811996. 9. B. T. Draine,The discrete dipole approximation and its ap plication to interstellar graphite grains,Astrophys. J.333, 8488721988. 10. P.C. Chaumet and M. NietoVesperinas,Timeaveraged total force on a dipolar sphere in an electromagneticeld,Opt. Lett.25,106510672000. 11. P. C. Chaumet and M. NietoVesperinas,Coupled dipole method determination of the electromagnetic force on a parti cle over aat dielectric substrate,Phys. Rev. B61,14,11914,1272000.