## Optical forces on small magnetodielectric particles

Optics Express, Vol. 18, Issue 11, pp. 11428-11443 (2010)

http://dx.doi.org/10.1364/OE.18.011428

Acrobat PDF (1133 KB)

### Abstract

We present a study of the optical force on a small particle with both electric and magnetic response, immersed in an arbitrary non-dissipative medium, due to a generic incident electromagnetic field. This permits us to establish conclusions for any sign of this medium refractive index. Expressions for the gradient force, radiation pressure and curl components are obtained for the force due to both the electric and magnetic dipoles excited in the particle. In particular, for the magnetic force we tentatively introduce the concept of curl of the spin angular momentum density of the magnetic field, also expressed in terms of 3D generalizations of the Stokes parameters. From the formal analogy between the conservation of momentum and the optical theorem, we discuss the origin and significance of the electric-magnetic dipolar interaction force; this is done in connection with that of the angular distribution of scattered light and of the extinction cross section.

© 2010 Optical Society of America

## 1. Introduction

1. A. Ashkin, “Acceleration and Trapping of Particles by Radiation Pressure,” Phys. Rev. Lett. **24**, 156–159 (1970). [CrossRef]

2. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. **11**, 288–290 (1986). [CrossRef] [PubMed]

6. P. C. Chaumet and M. Nieto-Vesperinas, ”Electromagnetic force on a metallic particle in the presence of a dielectric surface,” Phys. Rev.B **62**, 11185–11191 (2000). [CrossRef]

2. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. **11**, 288–290 (1986). [CrossRef] [PubMed]

3. K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. **75**, 2787– 2809(2004). [CrossRef]

9. P. Verkerk, B. Lounis, C. Salomon, and C. Cohen-Tannoudji, “Dynamics and spatial order of cold cesium atoms in a periodic optical potential,” Phys. Rev. Lett. bf **68**, 3861–3864 (1992). [CrossRef]

11. A. Hemmerich and T. W. Hänsch, “Two-dimesional atomic crystal bound by light,” Phys. Rev. Lett. **70**, 410–413 (1993). [CrossRef] [PubMed]

12. M. M. Burns, J. M. Fournier, and J. A. Golovchenko, “Optical Binding,” Phys. Rev. Lett. **63**, 1233–1236 (1989). [CrossRef] [PubMed]

17. M. Mansuripur, “Radiation pressure and the lnear momentum of the electromagnetic field,” Opt. Express **12**, 5375–5401 (2004). [CrossRef] [PubMed]

18. B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, “Ab initio study of the radiation pressure on dielectric and magnetic media,” Opt. Express **13**, 9280–9291 (2005). [CrossRef] [PubMed]

19. M. Mansuripur, “Radiation pressure and the lnear momentum of the electromagnetic field in magnetic media,” Opt. Express **15**, 13502–13517 (2007). [CrossRef] [PubMed]

20. B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, “Optical momentum transfer to absorbing Mie particles,” Phys. Rev. Lett. **97**, 1339021–1339024 (2006). [CrossRef]

21. B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, “Lorentz force on dielectric and magnetic particles,” J. Elec-tromagn. Waves Appl. **20**, 827–839 (2006). [CrossRef]

22. A. Lakhtakia, “Radiation pressure efficiencies of spheres made of isotropic, achiral, passive, homogeneous, negative-phase-velocity materials,” Electromagnetics **28**, 346–353 (2008). [CrossRef]

24. P. C. Chaumet and A. Rahmani, “Electromagnetic force and torque on magnetic and negative-index scatterers,” Opt. Express **17**, 2224–2234 (2009). [CrossRef] [PubMed]

25. S. Albaladejo, M. I. Marques, M. Laroche, and J. J. Saenz, “Scattering Forces from the Curl of the Spin Angular omentum of a Light Field,” Phys. Rev. Lett. **102**, 1136021–1136024 (2009). [CrossRef]

24. P. C. Chaumet and A. Rahmani, “Electromagnetic force and torque on magnetic and negative-index scatterers,” Opt. Express **17**, 2224–2234 (2009). [CrossRef] [PubMed]

24. P. C. Chaumet and A. Rahmani, “Electromagnetic force and torque on magnetic and negative-index scatterers,” Opt. Express **17**, 2224–2234 (2009). [CrossRef] [PubMed]

## 2. Force on a small particle with electric and magnetic response to an electromagnetic wave

*ε*and magnetic permeability

*μ*, subjected to an incident electromagnetic field whose electric and magnetic vectors are

**E**

^{(i)}and

**B**

^{(i)}, respectively. The total time-averaged electromagnetic force acting on the particle is [5

5. P. C. Chaumet and M. Nieto-Vesperinas, “Coupled dipole method determination of the electromagnetic force on a particle over a flat dielectric substrate,” Phys. Rev. B **61**, 14119–14127 (2000). [CrossRef]

*dS*denotes the element of any surface

*S*that encloses the particle. The fields in Eq. (1) are total fields, namely the sum of the incident and scattered (re-radiated) fields:

**E**

^{(i)}+

**E**

^{(r)},

**B**

^{(i)}+

**B**

^{(r)}. s is its local outward unit normal. A time dependence exp(-

*iωt*) is assumed throughout. For a small particle, within the range of validity of the dipolar approximation (see [27

27. J. R. Arias-Gonzalez and M. Nieto-Vesperinas, “Optical forces on small particles. Attractive and repulsive nature and plasmon-resonance conditions,” J. Opt. Soc. Am. A **20**, 1201–1209 (2003). [CrossRef]

**p**and

**m**, respectively. In this case, Eq. (1) leads to the expression

**17**, 2224–2234 (2009). [CrossRef] [PubMed]

*k*=

*nω*/

*c*,

*ω*being the frequency. The symbol ⊗ represents the dyadic product so that the matrix operation:

**W**(

**∇**⊗

**V**) has elements

*W*for

_{j}∂_{i}V_{j}*i*,

*j*= 1,2,3. All variables in Eq. (2) are evaluated at a point

**r**=

**r**

_{0}in the particle. The first term of Eq. (2) is the force <

**F**

_{e}> exerted by the incident field on the induced electric dipole and was obtained in [38

38. P. C. Chaumet and M. Nieto-Vesperinas, “Time-averaged total force on a dipolar sphere in an electromagnetic field,” Opt. Lett. **25**, 1065–1067 (2000). [CrossRef]

**F**

_{m}> and <

**F**

_{e-m}> are the force on the induced magnetic dipole and the force due to the interaction between both dipoles, respectively; they were derived in [24

**17**, 2224–2234 (2009). [CrossRef] [PubMed]

**17**, 2224–2234 (2009). [CrossRef] [PubMed]

*r*≡

*r*

**s**. Choosing

*S*to be a sphere around the dipoles, Eq. (1) can be written as

**S**〉 =

*c*/(8

*π*)ℜ{

**E**×

**H**

^{*}} is the time-averaged Poynting vector, with

**u**is a unit vector characterizing the propagation direction of each plane wave component; the integration is done in the unit sphere

*𝓓*whose solid angle element is

*d*Ω. Equation (9) follows from the

*source*-

*free*condition [29,32

32. G. C. Sherman, “Diffracted wavefields expressible by plane wave expansions containing only homogeneous waves,” Phys. Rev. Lett. **21**, 761–764 (1968). [CrossRef]

*S*in Eq. (4) so large that

*k*∣

*r*-

*r*

_{0}∣ → ∞, introducing Eq. (6) and Eq. (7) into Eq. (4), using Jones’ lemma based on the principle of the stationary phase, (see Appendix XII of [33]), and the source-free condition, Eq. (8)–Eq. (9), the terms that do not become zero after integration in Eq. (4) reduce to

## 3. Optical theorem and forces on a dipolar particle

34. P. Chylek and R. G. Pinnick, “Nonunitarity of the light scattering approximations,” Appl. Opt. **18**, 1123–1124 (1979). [CrossRef] [PubMed]

35. J. A. Lock, J. T. Hodges, and G. Gouesbet, “Failure of the optical theorem for Gaussian-beam scattering by a spherical particle,” J. Opt. Soc. Am. **12**, 2708–2715 (1995). [CrossRef]

36. B. Garcia-Camara, F. Moreno, F. Gonzalez, and J. M. Saiz, “Comment on Experimental Evidence of Zero Forward Scattering by Magnetic Spheres, Phys. Rev. Lett. **98**, 179701–179704 (2007). [CrossRef]

37. A. Alu and N. Engheta, “Cloaking a Sensor,” Pys. Rev. Lett. **102**, 233901–233904 (2009). [CrossRef]

*𝓦*

^{(a)}at which energy is being absorbed by the particle is given by

**s**in the integrand and, as such, using the same calculation procedure leading to Eq. (10), we get the optical theorem for an arbitrary field:

*𝓦*

^{(s)}at which the energy is being scattered, which together with the left hand side of this equation contributes to the rate of energy extinction by the particle

*𝓦*

^{(a)}+

*𝓦*

^{(s)}:

**F**

_{e-m}>. We notice that the power density of the scattered field can be written as the sum of two terms

*S*, that second term does not contribute to the radiated power, while it is the only contribution to the electric-magnetic dipolar interaction term of the force in Eq. (13). Namely, <

**F**

_{e-m}> comes from the interference between the fields radiated by

**p**and

**m**.

### 3.1. Forces on small dielectric magnetic spheres for plane wave incidence.

**E**

^{(i)}=

**e**

^{(i)}

*e*

^{ik·r},

**B**

^{(i)}=

**b**

^{(i)}

*e*

^{ik·r}, with

**e**

^{(i)}= 1/(

*kn*) {

**b**

^{(i)}×

**k**}), on a small dielectric and magnetic spherical particle characterized by its electric and magnetic polarizabilities

*α*and

_{e}*α*. When the induced dipole moments are expressed in terms of the incident field, i.e.

_{m}*c*/

*n*)

*ε*∣

**e**

^{(i)}∣

^{2}/(8

*π*) = (

*c*/

*n*)

*μ*

^{−1}∣

**b**

^{(i)}∣

^{2}/(8

*π*), Eq. (17) becomes after normalization

*σ*

^{(ext)},

*σ*

^{(a)}and

*σ*

^{(s)}being the particle extinction, absorption and scattering cross sections, respectively. Notice that the extinction cross section can be written as the sum of “electric”

*σ*

^{(ext)}

_{e}= 4

*πkℑ*[

*ε*

^{−1}

*α*]) and “magnetic” (

_{e}*σ*

^{(ext)}

_{m}= 4

*πkℑ*[

*μ α*}) contributions,

_{m}*σ*

^{(ext)}=

*σ*

^{(ext)}

_{e}+

*σ*

^{(ext)}

_{m}·

*σ*

^{(s)}, the scattering cross section, is given by

*d*

*σ*^{(S)}/

*d*Ω. is the differential scattering cross section [26,41

41. C. F. Bohren and D. R. Huffman, *Absorption and Scattering of Light by Small Particles*, (John Wiley, New York, 1998). [CrossRef]

*θ*(

**s**·

**k**=

*k*cos

*θ*) and the azimuthal angle

*ϕ*(see Fig. 1):

*σ*

^{(s)}is given by:

**F**

_{e-m}>, is the time-averaged scattered momentum rate, and we shall see below that it also contributes to radiation pressure. Due to the presence of the vector s in the integral, only the interference term, i.e. the last term, of the differential scattering cross section Eq. (22) gives a non-zero contribution. The total force is then given by

### 3.2. Electric-magnetic dipolar interaction contribution of the force

**F**

_{e-m}> of the force has a sign that depends on whether the interference factor ℜ[

*α*

_{e}*α*

_{m}^{*}] from both dipoles is positive or negative, i.e., on whether they are in, or in opposition of, phase. In the first case, <

**F**

_{e-m}> is negative, and thus opposes to the radiation pressure <

**F**

_{e}> + <

**F**

_{m}> of the pure dipoles. However, in the second case, the electric and magnetic polarizabilities have different sign, like for a metallic particle (see below), and <

**F**

_{e-m}> sums to that radiation pressure. It is interesting that this behavior of the force is linked with that of the differential scattering cross section Eq. (22), namely, when

*α*and

_{e}*α*are in phase, the scattering is mainly forward, whereas, there is predominance in the back-scattering region when they are in opposition of phase. For pure dielectric or pure magnetic particles, the force is simply proportional to the extinction cross section as expected from the concept of radiation pressure. However, it should be noticed that the electric-magnetic dipolar interaction, while leading to the term of the force <

_{m}**F**

_{e-m}>, and to the angular distribution of scattered intensity, it does not contribute, as is well known [26], to the extinction or scattering cross-sections. The contribution of these cross terms to the electric-magnetic dipolar interaction force is then connected with the asymmetry that they produce in the angular distribution of scattered power (differential cross section). For instance, for a perfectly conducting sphere, the predominance of the angular distribution of scattered power in the backscattering direction is well known [26] (see Fig. 1 and the discussion in Section 3.4 below).

### 3.3. Forces on Rayleigh particles

*a*, with constants

*ε*and

_{p}*μ*, and in the Rayleigh limit (

_{p}*ka*≪ 1 ,

*n*≪ 1,

_{p}ka39. B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Ap. J. **333**, 848–872 (1988). [CrossRef]

*α*

_{e}^{(0)}being the static polarizability,

*α*

_{e}^{(0)}is a real quantity and Eq. (27) fulfills the lossless version of the optical theorem for dielectric particles,

*α*

_{m}^{(0)}being the static polarizability:

*σ*

^{(ext)}=

*σ*

^{(ext)}

_{e}+

*σ*

^{(ext)}

_{m}=

*σ*

^{(s)}

*a*

^{3}and the total force is mainly given by the first two terms of Eq. (26), (i.e. <

**F**

_{e}> + <

**F**

_{m}> ~

*a*

^{3}). However, in absence of absorption (and assuming

*ε*≠ -2

_{p}*ε*and

*μ*≠ -2

_{p}*μ*),

*ℜα*~

*a*

^{3}while

*ℑα*~

*a*

^{3}(

*ka*)

^{3}≪

*ℜα*, and the three terms of the force scale with ~

*a*

^{6}. For instance, for a non-absorbing dielectric magnetic sphere, using Eq. (27)–Eq. (31) this force becomes:

22. A. Lakhtakia, “Radiation pressure efficiencies of spheres made of isotropic, achiral, passive, homogeneous, negative-phase-velocity materials,” Electromagnetics **28**, 346–353 (2008). [CrossRef]

*ε*~ -2

_{p}*ε*or/and

*μ*~ -2

_{p}*μ*, the radiation pressure can be very high [22

22. A. Lakhtakia, “Radiation pressure efficiencies of spheres made of isotropic, achiral, passive, homogeneous, negative-phase-velocity materials,” Electromagnetics **28**, 346–353 (2008). [CrossRef]

*α*

_{e/m}^{(0)}diverges and the polarizability is then a pure imaginary quantity. The force would then be given by Eq. (26) with

*ℜα*~ 0 and

_{e/m}*ε*

^{−1}

*ℑα*or/and

_{e}*μℑα*replaced by 3/(2

_{m}*k*

^{3}).

### 3.4. Forces beyond the Rayleigh limit

*kan*∣ as long as the particle polarizabilities are given by the first two Mie coefficients

_{p}*a*

_{1}and

*b*

_{1}, (cf. Eq. (4.56) and Eq. (4.57) of [41

41. C. F. Bohren and D. R. Huffman, *Absorption and Scattering of Light by Small Particles*, (John Wiley, New York, 1998). [CrossRef]

*α*

_{e}^{(0)}and

*α*

_{m}^{(0)}now being

*x*=

*ka*and

*j*

_{1}(

*x*),

*y*

_{1}(

*x*) stand for the first order spherical Bessel functions. Using these Mie polarizabilities in Eq. (26), one obtains the first dipolar terms of the Mie expansion of the radiation pressure on a sphere under plane wave illumination [33, 42

42. P. Chýlek, J. T. Kiehl, and K. W. Ko, “Narrow resonance structure in the Mie scattering characteristics,” Appl. Opt. **17**, 3019–3021 (1978). [CrossRef] [PubMed]

*approximations should be made starting from Eq. (34) and Eq. (35)*, or equivalently: Eq. (27) and Eq. (30) with Eq. (36) and Eq. (37); this involves to use the complex spherical Hankel functions

*h*

^{(1)}

_{1}in the denominators of

*a*

_{1}and

*b*

_{1}, (cf. Eq. (4.56) and Eq. (4.57) of [41

41. C. F. Bohren and D. R. Huffman, *Absorption and Scattering of Light by Small Particles*, (John Wiley, New York, 1998). [CrossRef]

27. J. R. Arias-Gonzalez and M. Nieto-Vesperinas, “Optical forces on small particles. Attractive and repulsive nature and plasmon-resonance conditions,” J. Opt. Soc. Am. A **20**, 1201–1209 (2003). [CrossRef]

*Otherwise*, if as commonly made, one directly employes directly Eq. (36) and Eq. (37), which is equivalent to use only the function

*y*

^{(1)}

_{1}in the above mentioned denominators,

*one is led to a result inconsistent with the optical theorem, as pointed out in [42]*.

42. P. Chýlek, J. T. Kiehl, and K. W. Ko, “Narrow resonance structure in the Mie scattering characteristics,” Appl. Opt. **17**, 3019–3021 (1978). [CrossRef] [PubMed]

*μ*

_{p}= 1 and a standard permittivity [33]

*σ*is the conductivity. Figure 2 shows the real and imaginary parts of the static electric and magnetic polarizabilities Eq. (36) and Eq. (37) for a 100 nm radius metallic particle with

*σ*= 5 · 10

^{19}s

^{−1}. The poles of these two coefficients show in Fig. 2 the Mie resonances of these polarizabilities, describing the excitation of the

*localized surface electric and magnetic plasmon*of the particle. On the other hand, Fig. 3 shows the radiation-reaction polarizabilities Eq. (34) and Eq. (35). Notice that a peak in the imaginary part of the polarizabilities corresponds to a maximum in the extinction and, correspondingly, in the radiation pressure terms.

**F**

_{e}> ∣, ∣ <

**F**

_{m}> ∣ and ∣ <

**F**

_{e-m}> ∣, obtained from Eqs. (26) and Eq. (34)–Eq. (38) for the conducting nanoparticle as a function of the wavelength

*λ*. Notice the peaks due to the excitation of both the electric and magnetic localized surface plasmon, described by the

*a*

_{1}and

*b*

_{1}Mie resonances, respectively. The magnetic and the electric-magnetic dipolar interaction force components are smaller, but not at all negligible as compared to the electric one. It is worth noticing that in the region where the real part of both electric and magnetic polarizabilities is positive, the electric-magnetic dipolar interaction term is

*negative*and the scattered light is mainly forward, as indicated by the sign of ℜ[

*α*

_{e}*α*

_{m}^{*}]. At longer wavelengths, however, the real part of the magnetic polarizability is negative and <

**F**

_{e-m}> is positive.

*small perfectly conducting sphere*” limit where

*ka*≪ 1 ≪ ∣

*kan*∣. As discussed in Ref. [43], this limit corresponds to the scattering by a perfectly reflecting dipolar sphere into which neither the electric nor the magnetic field penetrates. In this limit, we can apply Eq. (33) with an effective ∣

_{p}*ε*∣ → ∞ and

_{p}*μ*→ 0 (see Ref. [26,43]). This is of particular interest since then the particle electric and magnetic polarizabilities have the same order of magnitude, differing by a factor -2:

_{p}*α*

_{e}^{(0)}=

*εa*

^{3},

*α*

_{m}^{(0)}= -

*a*

^{3}/(2

*μ*), with

*σ*

^{(ext)}=

*σ*

^{(s)}≈ 10

*πk*

^{4}

*a*

^{6}/3. The differential cross section is then given by [43]

*ϕ*), gives

*θ*=

*π*/2, the scattering is mainly backward as the sign of ℑ[

*α*

_{e}

*α*

_{m}

^{*}] is negative, [cf. Eq. (22)], the ratio of forward and backward scattered intensities being 1/9, (see Fig. 1 and Sections 9.6 and 92 in [26] and [43], respectively). The three terms of the force Eq. (33) then become

### 3.5. Some remarks on radiation pressure forces in negative index media

## 4. Optical forces on dipolar particles in arbitrary fields. Gradient, scattering and curl components

**F**> = <

**F**

_{e}> + <

**F**

_{m}> + <

**F**

_{e-m}> may also be written in terms of the polarizabilities [cf. Eq. (19)] as:

**F**

_{e}> is already well known for an electric-dipole particle [5

5. P. C. Chaumet and M. Nieto-Vesperinas, “Coupled dipole method determination of the electromagnetic force on a particle over a flat dielectric substrate,” Phys. Rev. B **61**, 14119–14127 (2000). [CrossRef]

27. J. R. Arias-Gonzalez and M. Nieto-Vesperinas, “Optical forces on small particles. Attractive and repulsive nature and plasmon-resonance conditions,” J. Opt. Soc. Am. A **20**, 1201–1209 (2003). [CrossRef]

45. V. Wong and M. Ratner, “Gradient and nongradient contributions to plasmon-enhanced optical forces on silver nanoparticles,” Phys. Rev. B **73**, 0754161–0754166 (2006). [CrossRef]

25. S. Albaladejo, M. I. Marques, M. Laroche, and J. J. Saenz, “Scattering Forces from the Curl of the Spin Angular omentum of a Light Field,” Phys. Rev. Lett. **102**, 1136021–1136024 (2009). [CrossRef]

*electric spin density*of the optical field, respectively.

*angular momentum density*of a light field is a well defined quantity [26], its separation in terms of an orbital and a

*spin angular momentum*is not clear except within the paraxial approximation [46

46. S. M. Barnett, “Optical angular-momentum flux,” J. Opt. B: Quantum Semiclass. Opt . **4**, S7–S16 (2002). [CrossRef]

47. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A **45**, 8185–8189 (1992). [CrossRef] [PubMed]

48.
Notice that in ref. [25], the equation giving *L _{Se}* has the opposite sign. However, the actual force (in S.I. units) is exactly the same as the one derived here in Gaussian units.

25. S. Albaladejo, M. I. Marques, M. Laroche, and J. J. Saenz, “Scattering Forces from the Curl of the Spin Angular omentum of a Light Field,” Phys. Rev. Lett. **102**, 1136021–1136024 (2009). [CrossRef]

49. In paraxial beams, the *helicity* is given by the projection of the spin on the propagation direction. If the helicity is not uniform, the curl of the spin would be, in general, different from zero. The spin curl term would then be relevant for optical beams with non-uniform helicity. However, a non-uniform spin density does not imply a non-uniform helicity: we may have non-zero curl force even if the helicity is zero, as it is the case for the standing waves with p-polarized beams discussed in [25]. (Yiqiao Tang, Harvard University, private communication).

**F**

_{m}>, Eq. (43), has an analogous decomposition in terms of a gradient component (first term), a radiation pressure, or scattering, component (second term, which is the only one contributing in the case of a plane wave) and a third term that depends on the polarization and is, analogously to its electric counterpart Eq. (48), linked to the vector

**B**×

**B**

^{*}as a curl:

**B**×

**B**

^{*})=

**B**(∇·

**B**

^{*})-

**B**

^{*}(∇)-·

**B**) + (

**B**

^{*}∇)

**B**-(

**B**·∇)

**B**

^{*}and ∇·

**B**= 0 have been used.

*spin density of angular momentum*of the magnetic field [49

49. In paraxial beams, the *helicity* is given by the projection of the spin on the propagation direction. If the helicity is not uniform, the curl of the spin would be, in general, different from zero. The spin curl term would then be relevant for optical beams with non-uniform helicity. However, a non-uniform spin density does not imply a non-uniform helicity: we may have non-zero curl force even if the helicity is zero, as it is the case for the standing waves with p-polarized beams discussed in [25]. (Yiqiao Tang, Harvard University, private communication).

**E**and

**B**are in the complementary polarization of each other, (say TE or TM), the effect of the gradient and curl components of the electric force <

**F**

_{e}> for one polarization, (say TE), is the same (apart from the polarizability factors) as that of the corresponding quantities of the magnetic force <

**F**

_{m}> for the complementary polarization, (say TM). In particular, results as those obtained in [25

**102**, 1136021–1136024 (2009). [CrossRef]

*the electric-magnetic dipolar interaction force component*<

**F**

_{e-m}>

*also contributes, even though with different weight of the polarizabilities, to the radiation pressure and gradient forces*. In particular, for an incident plane wave, Eq. (44) shows that the whole component <

**F**

_{e-m}> is radiation pressure, and its sign depends on that of ℜ[

*α*

_{e}α_{m}^{*}].

### 4.1. Spin contribution and Stokes parameters

**F**

_{e}> and <

**F**

_{m}> [cf. Eq. (45) and Eq. (50)] as discussed above, suggests the possibility of expressing this term by means of the 3D Stokes parameters. This has an analogy with the generalization to 3D [50

50. T. Setala, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E **66**, 0166151–0166158 (2002). [CrossRef]

50. T. Setala, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E **66**, 0166151–0166158 (2002). [CrossRef]

*=*

^{e}_{ij}*E*

_{i}^{*}

*E*and Φ

_{j}*=*

^{m}_{ij}*B*

_{i}^{*}

*B*,

_{j}*i*,

*j*=

*x*,

*y*,

*z*, which the superscript now used for either the electric or magnetic vector quantities.

## 5. Conclusion

## Acknowledgements

## References and links

1. | A. Ashkin, “Acceleration and Trapping of Particles by Radiation Pressure,” Phys. Rev. Lett. |

2. | A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. |

3. | K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. |

4. | L. Novotny and B. Hecht, |

5. | P. C. Chaumet and M. Nieto-Vesperinas, “Coupled dipole method determination of the electromagnetic force on a particle over a flat dielectric substrate,” Phys. Rev. B |

6. | P. C. Chaumet and M. Nieto-Vesperinas, ”Electromagnetic force on a metallic particle in the presence of a dielectric surface,” Phys. Rev.B |

7. | P. C. Chaumet, A. Rahmani, and M. Nieto-vesperinas, “Optical trapping and manipulation of nano-objects with an apertureless probe,” Phys. Rev. Lett. |

8. | M. Nieto-Vesperinas, P. C. Chaumet, and A. Rahmani, “Near field photonic forces,” Phil. Trans. R. Soc. Lond. A |

9. | P. Verkerk, B. Lounis, C. Salomon, and C. Cohen-Tannoudji, “Dynamics and spatial order of cold cesium atoms in a periodic optical potential,” Phys. Rev. Lett. bf |

10. | P. S. Jessen, C. Gerz, P. D. Lett, W. D. Phillips, S. L. Rolston, R. J. C. Spreeuw, and C. I. Westbrook, “Observation of quantized motion of Rb atoms in an optical field,” Phys. Rev. Lett. |

11. | A. Hemmerich and T. W. Hänsch, “Two-dimesional atomic crystal bound by light,” Phys. Rev. Lett. |

12. | M. M. Burns, J. M. Fournier, and J. A. Golovchenko, “Optical Binding,” Phys. Rev. Lett. |

13. | M. M. Burns, J. M. Fournier, and J. A. Golovchenko, “Optical Matter: Crystallization and Binding in Intense Optical Fields,” Science |

14. | P. C. Chaumet and M. Nieto-Vesperinas, “Optical binding of particles with or without the presence of a flat dielectric surface,” Phys. Rev.B |

15. | S. A. Tatarkova, A. E. Carruthers, and K. Dholakia “One-Dimensional Optically Bound Arrays of Microscopic Particles,” Phys. Rev. Lett. |

16. | R. Gómez-Medina and J. J. Sáenz, “Unusually Strong Optical Interactions between Particles in Quasi-One-Dimensional Geometries,” Phys. Rev. Lett. |

17. | M. Mansuripur, “Radiation pressure and the lnear momentum of the electromagnetic field,” Opt. Express |

18. | B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, “Ab initio study of the radiation pressure on dielectric and magnetic media,” Opt. Express |

19. | M. Mansuripur, “Radiation pressure and the lnear momentum of the electromagnetic field in magnetic media,” Opt. Express |

20. | B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, “Optical momentum transfer to absorbing Mie particles,” Phys. Rev. Lett. |

21. | B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, “Lorentz force on dielectric and magnetic particles,” J. Elec-tromagn. Waves Appl. |

22. | A. Lakhtakia, “Radiation pressure efficiencies of spheres made of isotropic, achiral, passive, homogeneous, negative-phase-velocity materials,” Electromagnetics |

23. | A. Lakhtakia and G. W. Mulholland, “On two numerical techniques for light scattering by dielectric agglomerated structures,” J. Res. Nat. Inst. Stand. Technol. |

24. | P. C. Chaumet and A. Rahmani, “Electromagnetic force and torque on magnetic and negative-index scatterers,” Opt. Express |

25. | S. Albaladejo, M. I. Marques, M. Laroche, and J. J. Saenz, “Scattering Forces from the Curl of the Spin Angular omentum of a Light Field,” Phys. Rev. Lett. |

26. | J. D. Jackson, |

27. | J. R. Arias-Gonzalez and M. Nieto-Vesperinas, “Optical forces on small particles. Attractive and repulsive nature and plasmon-resonance conditions,” J. Opt. Soc. Am. A |

28. | L. Mandel and E. Wolf, |

29. | M. Nieto-Vesperinas, |

30. | N. J. Moore, M. A. Alonso, and C. J. R. Sheppard, “Monochromatic scalar fields with maximum focal irradiance,” J. Opt. Soc. Am. A |

31. | N. J. Moore and M. A. Alonso, “Closed-form bases for the description of monochromatic, strongly focused, electromagnetic fields,” J. Opt. Soc. Am. A |

32. | G. C. Sherman, “Diffracted wavefields expressible by plane wave expansions containing only homogeneous waves,” Phys. Rev. Lett. |

33. | M. Born and E. Wolf, |

34. | P. Chylek and R. G. Pinnick, “Nonunitarity of the light scattering approximations,” Appl. Opt. |

35. | J. A. Lock, J. T. Hodges, and G. Gouesbet, “Failure of the optical theorem for Gaussian-beam scattering by a spherical particle,” J. Opt. Soc. Am. |

36. | B. Garcia-Camara, F. Moreno, F. Gonzalez, and J. M. Saiz, “Comment on Experimental Evidence of Zero Forward Scattering by Magnetic Spheres, Phys. Rev. Lett. |

37. | A. Alu and N. Engheta, “Cloaking a Sensor,” Pys. Rev. Lett. |

38. | P. C. Chaumet and M. Nieto-Vesperinas, “Time-averaged total force on a dipolar sphere in an electromagnetic field,” Opt. Lett. |

39. | B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Ap. J. |

40. | We acknowledge this remark in a private communication from an anonimous reviewer. |

41. | C. F. Bohren and D. R. Huffman, |

42. | P. Chýlek, J. T. Kiehl, and K. W. Ko, “Narrow resonance structure in the Mie scattering characteristics,” Appl. Opt. |

43. | L.D Landau, E.M. Lifshitz, and L.P. Pitaevskii, |

44. | V. G. Veselago, “The electrodynamics of substances with simultanenous negative values of |

45. | V. Wong and M. Ratner, “Gradient and nongradient contributions to plasmon-enhanced optical forces on silver nanoparticles,” Phys. Rev. B |

46. | S. M. Barnett, “Optical angular-momentum flux,” J. Opt. B: Quantum Semiclass. Opt . |

47. | L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A |

48. |
Notice that in ref. [25], the equation giving |

49. | In paraxial beams, the |

50. | T. Setala, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E |

**OCIS Codes**

(170.4520) Medical optics and biotechnology : Optical confinement and manipulation

(260.2110) Physical optics : Electromagnetic optics

(290.4020) Scattering : Mie theory

(350.4855) Other areas of optics : Optical tweezers or optical manipulation

(290.5825) Scattering : Scattering theory

**ToC Category:**

Optical Trapping and Manipulation

**History**

Original Manuscript: March 22, 2010

Revised Manuscript: April 23, 2010

Manuscript Accepted: April 26, 2010

Published: May 14, 2010

**Virtual Issues**

Vol. 5, Iss. 10 *Virtual Journal for Biomedical Optics*

**Citation**

M. Nieto-Vesperinas, J. J. Sáenz, R. Gómez-Medina, and L. Chantada, "Optical forces on small magnetodielectric particles," Opt. Express **18**, 11428-11443 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-11-11428

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### References

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- M. Mansuripur, “Radiation pressure and the linear momentum of the electromagnetic field in magnetic media,” Opt. Express 15, 13502–13517 (2007). [CrossRef] [PubMed]
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- N. J. Moore, and M. A. Alonso, “Closed-form bases for the description of monochromatic, strongly focused, electromagnetic fields,” J. Opt. Soc. Am. A 26, 2211–2218 (2009). [CrossRef]
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- M. Born, and E. Wolf, Principles of Optics, 7 th edition, Cambridge U.P., Cambridge, 1999.
- P. Chylek, and R. G. Pinnick, “Nonunitarity of the light scattering approximations,” Appl. Opt. 18, 1123–1124 (1979). [CrossRef] [PubMed]
- J. A. Lock, J. T. Hodges, and G. Gouesbet, “Failure of the optical theorem for Gaussian-beam scattering by a spherical particle,” J. Opt. Soc. Am. 12, 2708–2715 (1995). [CrossRef]
- B. Garcia-Camara, F. Moreno, F. Gonzalez, and J. M. Saiz, “Comment on Experimental Evidence of Zero Forward Scattering by Magnetic Spheres,” Phys. Rev. Lett. 98, 179701–179704 (2007). [CrossRef]
- A. Alu, and N. Engheta, “Cloaking a Sensor,” Phys. Rev. Lett. 102, 233901–233904 (2009). [CrossRef]
- P. C. Chaumet, and M. Nieto-Vesperinas, “Time-averaged total force on a dipolar sphere in an electromagnetic field,” Opt. Lett. 25, 1065–1067 (2000). [CrossRef]
- B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Ap. J. 333, 848–872 (1988). [CrossRef]
- We acknowledge this remark in a private communication from an anonymous reviewer.
- C. F. Bohren, and D. R. Huffman, Absorption and Scattering of Light by Small Particles, (John Wiley, New York, 1998). [CrossRef]
- P. Chýlek, J. T. Kiehl, and K. W. Ko, “Narrow resonance structure in the Mie scattering characteristics,” Appl. Opt. 17, 3019–3021 (1978). [CrossRef] [PubMed]
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- V. Wong, and M. Ratner, “Gradient and nongradient contributions to plasmon-enhanced optical forces on silver nanoparticles,” Phys. Rev. B 73, 0754161–0754166 (2006). [CrossRef]
- S. M. Barnett, “Optical angular-momentum flux,” J. Opt. B Quantum Semiclassical Opt. 4, S7–S16 (2002). [CrossRef]
- L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992). [CrossRef] [PubMed]
- . Notice that in ref. [25], the equation giving LSe has the opposite sign. However, the actual force (in S.I. units) is exactly the same as the one derived here in Gaussian units.
- In paraxial beams, the helicity is given by the projection of the spin on the propagation direction. If the helicity is not uniform, the curl of the spin would be, in general, different from zero. The spin curl term would then be relevant for optical beams with non-uniform helicity. However, a non-uniform spin density does not imply a non-uniform helicity: we may have non-zero curl force even if the helicity is zero, as it is the case for the standing waves with p-polarized beams discussed in [25]. (Yiqiao Tang, Harvard University, private communication).
- T. Setala, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66, 0166151–0166158 (2002). [CrossRef]

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