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Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 20 — Oct. 2, 2006
  • pp: 9508–9521
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Radiation torque on nonspherical particles in the transition matrix formalism

Ferdinando Borghese, Paolo Denti, Rosalba Saija, and Maria Antonia Iatì  »View Author Affiliations


Optics Express, Vol. 14, Issue 20, pp. 9508-9521 (2006)
http://dx.doi.org/10.1364/OE.14.009508


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Abstract

The torque exerted by radiation on small particles is recognized to have a considerable relevance, e.g., on the dynamics of cosmic dust grains and for the manipulation of micro and nanoparticles under controlled conditions. In the present paper we derive, in the transition matrix formalism, the radiation torque applied by a plane polarized wave on nonspherical particles. In case of circularly polarized waves impinging on spherical particles our equations reproduce the findings of Marston and Crichton [Phys. Rev. A30, 2508–2516 (1984)]. Our equations were applied to calculate the torque on a few model particles shaped as aggregates of identical spheres, both axially symmetric and lacking any symmetry, and the conditions for the stability of the induced rotational motion are discussed.

© 2006 Optical Society of America

1. Introduction

The interaction of the radiation field with particles, besides the conservation of energy and linear momentum, also implies the conservation of angular momentum [1

1. J. D. Jackson, Classical electrodynamics, 2d edition (Wiley, New York, 1975)

]. As a result, particles illuminated by a radiation field may experience a torque which contributes in determining their dynamical behavior. As an example, the radiation torque is considered as a relevant agent of the partial alignment of the interstellar dust grains [2

2. E. M. Purcell, “Suprathermal rotation of interstellar grains,” Astrophys. J. 231, 404–416 (1979) [CrossRef]

]. The radiation torque has also useful application to the manipulation of micro and nanoparticles under controlled laboratory conditions [3

3. M. E. J. Friese, T.A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394, 348–349 (1998); “Erratum”, ibid. 395, 621 (1998) [CrossRef]

].

The actual calculation of the radiation torque requires that the electromagnetic field in the presence of the particles be known. Draine and Weingartner [4

4. B. T. Draine and J. C. Weingartner, “Radiative torques on interstellar grains. I. Superthermal spin-up,” Astrophys. J. 470, 551–565 (1996) [CrossRef]

, 5

5. B. T. Draine and J. C. Weingartner, “Radiative torques on interstellar grains. II. Grain alignment,” Astrophys. J. 480, 633–646 (1997) [CrossRef]

] calculated through the discrete dipole approximation (DDA) [6

6. E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973) [CrossRef]

, 7

7. B. T. Draine and P. J. Flatau, “Discrete dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (1994) [CrossRef]

] the field scattered by model cosmic grains and the torque experienced by the latter. Unfortunately, DDA required to repeat calculations ex novo for each choice of the orientation of the particles and for each choice of the polarization of the incident field. Therefore, it is not surprising that the above authors dealt with a few simple model particles only.

The computational situation is more favorable when the multipole expansion of the field in the framework of the transition matrix approach is used [8

8. P. C. Waterman, “Symmetry, unitarity and geometry in electromagnetic scattering,” Phys. Rev. D 4, 825–839 (1971) [CrossRef]

]. In fact, the transition matrix of any particle does not depend on the polarization of the incident field and is endowed of well defined transformation properties under rotation [9

9. F. Borghese, P. Denti, and R. Saija, Scattering from model nonspherical particles (Springer, Heidelberg, 2002)

]. As a result, the computational demand becomes more sustainable even for particles with complex morphology, as the transition matrix need to be calculated only for a single, arbitrarily chosen orientation of the particle concerned.

As far as we know, the first complete solution to the the problem of radiation torque in terms of multipole fields is due to Marston and Crichton [10

10. P. L. Marston and J. H. Crichton, “Radiation torque on a sphere caused by a circularly-polarized electromagnetic wave,” Phys. Rev. A 30, 2508–2516 (1984) [CrossRef]

] who dealt with the torque exerted by an elliptically polarized plane wave on a homogeneous sphere. More recently, de Abajo [11

11. F. J. García de, “Momentum transfer to small particles by passing electron beams,” Phys. Rev. B 70, 115422 (2004) [CrossRef]

, 12

12. F. J. García de Abajo, “Electromagnetic forces and torques in nanoparticles irradiated by plane waves,” J. Quant. Spectrosc. Radiat. Transfer 89, 3–9 (2004) [CrossRef]

] exploited the multipole field approach to calculate the torque exerted by radiation on model nonspherical particles but he does not report the derivation of the formulas he uses. Since only these three papers are reported in the literature, one cannot say that the subject is a well-known one. Therefore, in the present paper we deal with the actual derivation of the equations for the radiation torque produced by a plane wave using the expansion of the electromagnetic field in terms of vector multipole fields in the framework of the transition matrix approach. Assuming the incident field to be a plane wave is in no way restrictive. In fact, when the incident field can be described as a superposition of plane waves, the extension of the formalism is trivial; otherwise, it is only necessary to appropriately define the multipole amplitudes of the incident field [11

11. F. J. García de, “Momentum transfer to small particles by passing electron beams,” Phys. Rev. B 70, 115422 (2004) [CrossRef]

]. Our formulas will be applied to model particles shaped as aggregates of spheres. Due to the length and complexity of the derivations we report the main steps only, but in such a way that the interested reader can easily perform the missing steps.

2. Radiation torque

Let us consider an electromagnetic plane wave impinging on a particle embedded into a homogeneous, isotropic, nonmagnetic medium with a real refractive index n. Then, the conservation of angular momentum for the combined system of field and particle decrees that the particle experiences a torque given by

ΓRed=n̂·TM×rdS,
(1)

where is the unit outward normal to an arbitrary surface S including the particle and

TM=18πRe[n2EE*+BB*12(n2E·E*+B·B*)I]
(2)

is the time averaged Maxwell stress tensor [1

1. J. D. Jackson, Classical electrodynamics, 2d edition (Wiley, New York, 1975)

], I being the unit dyadic. Of course, in Eq. (2)

E=EI+ES,B=BI+BS,

i.e., E and B are the superposition of the incident and of the scattered fields.

We found convenient to choose the integration surface S to be a sphere of arbitrary radius with its center within the particle so that Eq. (1) becomes

ΓRed=r3Ωr̂·TM×r̂dΩ.
(3)

Since ·I×=0, the last two terms in Eq. (2) give no contribution to the integral, and by choosing the radius of the integration sphere to be large, possibly infinite, we can resort to the asymptotic expression of the fields. Nevertheless, the reader is warned that, as regards the scattered field, the customary far zone expression in terms of the scattering amplitude

ES=exp(ikr)rf(k̂S,k̂I)

cannot be used lest to get a vanishing result [13

13. M. I. Mishchenko, “Radiation force caused by scattering, absorption and emission of light by nonspherical particles,” J. Quant. Spectrosc. Radiat. Transfer 70, 811–816 (2001) [CrossRef]

] because ·f=0. The correct result is obtained by solving the problem of scattering, then by expanding the fields for large r and retaining all terms that give contributions of order 1/r 3 to the integrand in Eq. (3).

Since the polarization of the field is relevant, we introduce a pair of mutually orthogonal unit vectors, say uûη, such that

û 1×û 2=I,

where I is the direction of the propagation of the incident plane wave. Note that the vectors uûη may be either real (linear polarization basis) or complex (circular polarization basis). In case the linear polarization basis is used the subscripts η=1, 2 denote polarization parallel and perpendicular to a fixed plane of reference through I, respectively, whereas, when the circular basis is used, η=1, 2 denote left and right polarization, respectively.

We expand both the incident and the scattered field in a series of vector multipole fields [9

9. F. Borghese, P. Denti, and R. Saija, Scattering from model nonspherical particles (Springer, Heidelberg, 2002)

]. Accordingly,

EI=ηE0ηûηexp(ikIr)=ηE0ηplmJlm(p)(r,k)WIηlm(p),

where k=nk v, with k v=ω/c, is the propagation constant of the incident plane wave. We also define the multipole fields

ES=ηE0ηplmHlm(p)(r,k)Aηlm(p),

where X lm () denotes vector spherical harmonics [1

1. J. D. Jackson, Classical electrodynamics, 2d edition (Wiley, New York, 1975)

]: The multipole fields Hlm(p) (r,k) are identical to the Jlm(p) (r,k) fields except for the substitution of the spherical Hankel functions of the first kind hl (kr) in place of the spherical Bessel functions j l(kr). The multipole amplitudes of the incident field are defined as

WIηlm(p)=4πi p+l-1 uûη·Zlm(p) (I),

where

Zlm(1) ()=X lm (), Zlm(2) ()=X lm (

Jlm(1)(r,k)=jl(kr)Xlm(r̂),Jlm(2)(r,k)=1kJlm(1)(r,k),

that defines the elements of the transition matrix of the particle. It may be useful to note that for axially symmetric particles the transition matrix has the property

𝓢lmlm(pp)=𝓢ll;m(pp)δmm
(4)

provided the z axis is along the cylindrical axis. In particular, for a sphere

𝓢 (pp) lmlm=S1(p) δ pp δ ll δ mm.

We notice that Sl(1) =b l , Sl(2) =a l , a l and b l being the well known Mie coefficients [9

9. F. Borghese, P. Denti, and R. Saija, Scattering from model nonspherical particles (Springer, Heidelberg, 2002)

].

Hlm(1)(i)l+1eikrkrZlm(1),
Hlm(2)(i)lk2r2l(l+1)eikrYlmr̂(i)l+1k2r2eikrZlm(2)(i)lkreikrZlm(2),
(5)

whereas, for the J-multipole fields we get [16

16. M. Abramowitz and I. Stegun, Handbook of mathematical functions (Dover, New York, 1970)

]

Jlm(1)1krsin(krlπ2)Zlm(1),
Jlm(2)ik2r2l(l+1)sin(krlπ2)Ylmr̂1k2r2sin(krlπ2)Zlm(2)
1krsin[kr(l1)π2]Zlm(2).
(6)

Now, the H- and J-fields enter the integrand in (3) through the dot products

r̂·Hlm(2)=l(l+1)(i)leikrk2r2Ylm=cr2lYlm,

r̂·Jlm(2)=il(l+1)sin(krlπ2)k2r2Ylm=cr1lYlm,

and through the cross products

Hlm(p)*r̂=il+peikrkrZlm(p)*r̂=ct2l(p)Zlm(p)*r̂,

Jlm(p)*r̂=sin[kr(l+p1)π2]krZlm(p)*r̂=ct1l(p)Zlm(p)*r̂,

where we neglected the further terms that would come from (5) and (6) because they vanish at infinity to an order higher than the order we have to retain.

ΓRadx=Re[12(Γ1Γ1)],ΓRady=Re[i2(Γ1+Γ1)],ΓRadz=Re(Γ0),
(7)

where

Γμ=ηηIIηηΓμ;ηη
(8)

with I Iη̄η=E*0η̄E 0η. In the following, on account that all the fields are given in terms of vector spherical harmonics, we found convenient to focus just on the spherical components Γ µ , i.e., actually, on Γµ;η̄η. Accordingly, we note that the integrand in (3) gets contributions that can be written as

Kμ;ααlmlm(pp)=δp2dαlmvμ;αlm(p),
(9)

where

d1lm=r̂·Jlm(2),d2lm=r̂·Hlm(2),

The subscripts α and in (9) denote terms coming either from the incident field (α, =1) or from the scattered field (α, =2). We thus get

vμ;1lm(p)=(Jlm(p)*r̂)μ,vμ;2lm(p)=(Hlm(p)*r̂)μ.

K1;ααlmlm(21)=icrαlctαl(1)Ylm4π3[Y10C(1,l,l;1,m1)Yl,m1*Y11C(1,l,l;0,m)Ylm*],

K0;ααlmlm(21)=icrαlctαl(1)Ylm4π3[Y1,1C(1,l,l;1,m1)Yl,m1*Y11C(1,l,l;1,m+1)Ylm+1*],

K1;ααlmlm(21)=icrαlctαl(1)Ylm4π3[Y1,1C(1,l,l;0,m)Yl,m*Y10C(1,l,l;1,m+1)Yl,m+1*],

where the C’s denote Clebsch-Gordan coefficients [15

15. E. M. Rose, Elementary theory of angular momentum, (Wiley, New York, 1956)

]. When these contributions, according to

Eq. (3), are integrated we get

Kμ;ααlmlm(22)=crαlctαl(2)YlmC(1,l,l;μ,mμ)Yl,mμ*,

I1;ααlmlm(21)=icrαlCtαl(1)2l+12l+1C(1,l,l;0,0)δm,m1

[C(1,l,l;1,m1)C(1,l,l;0,m,m1)C(1,l,l;0,m)C(1,l,l;1,m,m)],

I0;ααlmlm(21)=icrαlCtαl(1)2l+12l+1C(1,l,l;0,0)δm,m

[C(1,l,l;1,m1)C(1,l,l;1,m,m1)

C(1,l,l;1,m+1)C(1,l,l;1,m,m+1)],

and

Γμ;ηη=n2r38παα[ppplmlmIμααlmlm(2p)(aαηlm(2)aαηlm(p)*+aαηlm(1)aαηlm(p)*)],
(10)

Finally, collecting all the terms one obtains

Γμ;ηη=n2r38παα[pp'plmlmIμααlmlm(2p)(aαηlm(2)aαηlm(p)*+aαηlm(1)aαηlm(p')*)],
(11)

where

a1ηlm(p)=WIηlm(p), and a2ηlm(p)=Aηlm(p).

At this stage a number of remarks are in order. Equation (11) could be used for a brute force calculation of the spherical components of the radiation torque, but we can obtain a more efficient formula through a detailed inspection of the terms present. In fact, Eq. (11) is built as a sum of contributions with different α, , and or . The same is true for ΓRad on account of (7) and (8). A careful analysis and rather long manipulations then lead to the following conclusions that refer just to the contributions to ΓRad for each η and η̄.

• The contributions with α==1 are just vanishing both for =2 or =1 and for =1 or =2. This result, that is identical to the one of Marston and Crichton, comes from the fact that these terms describe the flux of the momentum of the Maxwell stress tensor through a closed surface in the absence of any scatterer.

• The contribution with α==2 vanishes both for =1 and for ′=2.

• The contributions for α depend on r, i.e. on the radius of the spherical surface of integration. Nevertheless, those with =1 are identical but of opposite sign and cancel each other; on the other hand, the sum of those with =2 turns out to be independent of r. Analogous considerations hold true for the contributions with =2 and =1, respectively. This result is a consequence of the fact that the torque cannot depend on the choice of the surface of integration.

Now, separating in Eq. (11) the sum over α from the one over α=, we can write

Γμ;ηη=Γμ;ηη(ext)Γμ;ηη(sca).
(12)

In (12)

Γμ;ηη(ext)=cΓplmsμ;lmWIηl,mμ(p)Aηlm(p)*,
(13)

whereas the quantities Γµ;ηη(sca) ̄ are identical to the Γµ;ηη(ext) ̄ above, except for the substitution of Aηlm(p) in place of WIηlm(p) and for the change of the sign. In Eq. (13) we define c Γ=n 2/8πk 3 and

s1;lm=(lm)(l+1+m)2,s0;lm=m,s1;lm=(l+m)(l+1m)2.
(14)

The formulas above, when applied to a sphere, yield just the results of Marston and Crichton. In fact, let us write (13) in a compact form as

Γμ;ηη(ext)=cΓplmVμ;ηlm(p)Aηlm(p)*,

where Vµ;ηlm(p)=s µ;lm WIηl,mµ(p). Considering the vector Vηlm(p) with spherical components Vµ;ηlm(p), we find for a circular polarization basis

Vηlm(p)k̂I=μ()μVμ;ηlm(p)k̂Iμ=()ηWIηlm(p).

Note that this equation does not hold true in a linear polarization basis. In practice, obtaining this result requires long manipulations for general direction of I but it is immediately obtained in the case I=êz. As a consequence,

Γηη(ext)k̂I=n2()n18πkσˇTηη*.
(15)

On the contrary,

Γηη(sca)k̂In2()η18πkσˇη*
(16)

unless the scatterer is axially symmetric and I is parallel to its axis. In Eqs. (15) and (16) we define the quantities

σˇTηη=1k2plmAηlm(p)WIηlm(p)*,σˇSηη=1k2plmAηlm(p)Aηlm(p)*, σT=1IIηη'Re(IIηη'σˇTηη'),σS=1IIηη'Re(IIηη'σˇSηη'),

σTη=Re(σˇTηη),σSη=σˇSηη.

where I Iη |E |2. Moreover, the extinction and the scattering cross sections, in case of radiation polarized along uûη, turn out to be

σ=Re(σˇ Tηη), σSη=σˇ Sηη.

• When the scatterer is axially symmetric and I is parallel to its axis the contribution for α=2, α¯ =1 with =2 and that for α=1, α¯ =2 with =1 are equal; the same is also true for α=2, α¯ =1 with ′=1 and for α=1, α¯ =2 with =2.

On account that in the case above, using a circular polarization basis, one finds σT1T2, σS1S2, and σˇ Tηη̄=σˇ Sηη̄=0 for ηη̄, the result is

ΓRadk̂I=n28πk(II11II22)σA,
(17)

whereas there is no component of the torque perpendicular to the axis, for obvious symmetry reasons. Thus, a plane wave propagating along the axis can apply a torque to the axially symmetric particle only when the latter is absorbing and the wave is elliptically polarized. This is just the result that Marston and Crichton obtained for a spherical scatterer.

3. Axial average of radiation torque

The radiation torque depends on the orientation of the particle with respect to the incident field, i.e. with respect to the direction of incidence and to the polarization. Nevertheless, it is not to be expected that a dispersion of particles shows a collective response to the torque exerted by the incident field: from a dynamical point of view each particle has its own response. However, since the free rotation of a particle around a principal axis of inertia through its center of mass is a stable one [17

17. H. Goldstein, C. Poole, and J. Safko, Classical Mechanics3d edition (Addison-Wesley, Reading, Mass., 2002)

], it may be interesting to consider the average of the radiation torque applied to a particle in a complete rotation. In other words it is meaningful calculating the axial average of the radiation torque.

To this end let us fix the particle in a local frame of reference Σ¯ whose orientation with respect to a frame Σ fixed in the laboratory is given by the appropriate Eulerian angles α, β, γ that we denote collectively as Θ for short. Then, let us assume that the particle rotates around the local z axis which remain always parallel to the z axis of the laboratory frame. Let us also assume that the calculation of the radiation torque has been performed in the local frame. Then, denoting with an overbar the quantities calculated in Σ¯ , the same quantities can be transformed into the laboratory frame Σ by exploiting the transformation rules that are reported in Ref. [9

9. F. Borghese, P. Denti, and R. Saija, Scattering from model nonspherical particles (Springer, Heidelberg, 2002)

]. In particular, since the spherical components of the radiation torque transform according to D(1), i.e., as a first rank tensor, we get

Γμ';ηη=μΓμ;ηημμ'(1),
(18)

whereΓ¯ µ;ηη̄=Γ¯ (ext) µ;η̄η-Γ¯ (sca) µ;η̄η,

and, according to Eq. (13),

Γμ;ηη(ext)=cΓplmsμ;lmWIηl,mμ(p)Aηlm(p)*,Γμ;ηη(sca)=cΓplmsμ;lmAηl,mμ(p)Aηlm(p)*.
(19)

Moreover,

WIηlμ(p)=m𝒟μm(l)(Θ)WIηlm(p),Aηlm(p)=p'l'm'm'𝒮lml'm'(pp')𝒟m'm'(pp')WIηl'm'(p').

Substitution into Eq. (19) leads us to the conclusion that the axial average of Eq. (18) requires calculating the integrals

𝒟μμ'(1)𝒟mμ,m(l)𝒟m'm'(l')*P(Θ)dΘ=δμμ'δmμ,mδm'm'δmm'δmm'

and

𝒟μμ'(1)𝒟m'm'(l')𝒟m"m"(l")*P(Θ)dΘ=δμμ'δm'm'δm"m"δμ+m',m"',

where

P(Θ)=12πδ(β)sinβδ(γ),

as we are dealing with Case 4 in Table 4.1 in Ref. [9

9. F. Borghese, P. Denti, and R. Saija, Scattering from model nonspherical particles (Springer, Heidelberg, 2002)

]. Ultimately,

Γμ;ηη(ext)=cΓplmsμ;lmplWIηl,mμ(p)lmlm(pp)*WIηlm(pp)*WIηlm(p)*,

Γμ;ηη(sca)=cΓplmsμ;lmpp'll'm'𝒮lml,m'μ(pp)WIηl,m'μ(p)𝒮lml'm'(pp')*WIηlm(p')*,

where again c Γ=n 2/8πk 3 and s µ;lm are given by (14).

4. Radiation torque on aggregated spheres

The medium we referred to in Sects. 2 and 3 is typically the vacuum or a homogeneous fluid. Since in this section we apply the theory of the preceding sections to model cosmic dust grains, the refractive index is assumed to be n=1. Actually, the torque experienced by particles as a result of the interaction with electromagnetic radiation is believed to be the main agent of the (at least partial) alignment of the grains. In principle, the alignment should also occur for the particles that compose the atmospheric aerosols. Nevertheless, the frictional forces that oppose the alignment are comparatively stronger in the atmosphere than they are in the interstellar medium, where they are due to collision of the particles with the atoms and molecules of the interstellar gas [4

4. B. T. Draine and J. C. Weingartner, “Radiative torques on interstellar grains. I. Superthermal spin-up,” Astrophys. J. 470, 551–565 (1996) [CrossRef]

]. In any case, since we are here interested in the electromagnetic aspects of the radiation torque, we upright neglect the frictional forces. We stress, however, that the mechanical effects of the embedding medium should be taken into account when a realistic description of the dynamics of the particles is required.

The quantity whose components we actually report in the following figures is the adimensional vector

Tη=8πkn2IIσTηΓRadη,

where Γ⃗Radη is the radiation torque calculated for incident light with polarization η, and n=1.

First, we consider a binary aggregate of identical spheres with radius 50 nm, composed of astronomical silicates [18

18. B. T. Draine and H. M. Lee, “Optical properties of interstellar graphite and silicate grains,” Astrophys. J. 285, 89–108 (1984) [CrossRef]

]. The origin of the frame of reference lies at the point of contact of the spheres. Since, according to Sect. 2, the torque depends on the absorptivity of the particles, we report in Fig. 1 the quantity T ηk =T η·I as a function of the wavelength for the cluster considered with its axis perpendicular to k I itself, both in the case in which the imaginary part of the dielectric function is set to zero and in the case it assumes its actual value. We assume the incident field to be circularly polarized and show the results for η=1 only, because, on account of the symmetry of the scatterer, the results for η=2 are identical, except for the sign. As expected, the transverse components of Γ⃗Radη, i.e., those in the plane orthogonal to k I, were found to be zero. We do not report the results for the case in which the axis of the aggregate is parallel to k I, because they are a direct consequence of Eq. (17), i.e., Γ⃗Radη·k I is nonvanishing only for complex refractive index, and changes its sign with the change of polarization. Even in this case the transverse components are rigorously zero for symmetry reasons. Anyway, the results in Fig. 1 show the great importance of the absorptivity of the particles on the value of the torque, and in our opinion do not deserve further comments.

In the second example, we consider a more complex cluster composed of five spheres identical to each other and to the spheres we used above to compose the binary cluster. The geometry is chosen so that no symmetry is present, and the coordinates of the centers of the spheres are listed in Table 1; a sketch of the geometry is reported in Fig. 4 (a). The incident wavevector k I is parallel to the z axis (Iêz) and circular polarization is assumed. In Fig. 2 we report the cartesian components of T η both for η=1 and for η=2. Since the role of the absorptivity has been already clarified in discussing the binary cluster, we assume here that the refractive index is just the one of astronomical silicates [18

18. B. T. Draine and H. M. Lee, “Optical properties of interstellar graphite and silicate grains,” Astrophys. J. 285, 89–108 (1984) [CrossRef]

]. We first notice that the x and y components of Γ⃗Rad do not change their sign with changing polarization, unlike the z component which does change sign. Of course, the lack of symmetry prevents the curves of each component to coincide with changing polarization. Nevertheless, the most striking result is the coincidence of the axial average around the z axis, 〈Γ⃗Radη·I〉, with Γ⃗Radη·I. This result would be quite evident for the axial average of the binary cluster with its axis along k I. In the present case, because of the lack of symmetry, the occurrence of this coincidence deserves a few additional comments. It is not easy to extract any conclusion from the results on the basis of the formulas of Sect. 2. Nevertheless, a few heuristic remarks can be drawn with the help of a complete set of calculations, in the sense that they were also performed using linearly polarized incident radiation. Both when we assume circular and when we assume linear polarization, the components of Γ⃗Radη in a plane orthogonal to k I are found, in general, nonzero and different from each other, whereas their axial averages around k I do vanish. Actually, averaging around k I makes the average particle akin to an axially symmetric particle, for which the mentioned result is to be expected. Furthermore, assuming circular polarization, we found 〈Γ⃗Radη·I〉=Γ⃗Radη·I that follows at once, if one considers the behavior of the field of a circularly polarized wave. On the other hand, in case linear polarization is assumed, the axial average of Γ⃗Radη around k I turns out to be independent of η. In fact, as regards the axial averaging, the different polarization is seen only as a different orientation of the particle around . Of course, in general, the axial average 〈Γ⃗Radη·I〉≠Γ⃗Radη·I for linear polarization.

Table 1. Coordinates of the centers of the spheres in nm (see Fig. 4 (a))

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Fig. 1. Component of T 1 along k I for a binary cluster composed of astronomical silicates both when the imaginary part of their dielectric function has its actual value [18] and when it is arbitrarily set to zero. The axis of the cluster is orthogonal to k I

At this stage, since Γ⃗Rad depends on the orientation of the particle, the problem arises of the stability of the rotational motion driven by the electromagnetic torque. In order to asses this point we report in Fig. 3 the quantities T2=Tx2 +Ty2 and T z as a function of the polar angles ϑI and φ I of the wavevector kI incident on the binary cluster with its axis along the y axis of Σ¯ (see Sect 3). The wavelength is λ=0.3µm and circular polarization with η=1 is assumed. Note that the axes of Σ¯ are principal axes of inertia for the aggregate. The results reported in Fig. 3 show that when the incidence is along the axis of highest moment of inertia, the transverse components of the torque become zero, whereas the driving torque is appreciably nonzero. Obviously, similar results hold for k I parallel to the axis of the aggregate. In conclusion, a radiation field impinging on an aggregate along any of the principal axes of inertia drives it to rotate with stability as long as no transverse torque is exerted to deviate the axis of rotation.

The results discussed above are a direct consequence of the symmetry of the binary aggregate. Therefore, it may be illuminating to discuss the analogous results for the 5-spheres aggregate the coordinates of whose centers are given in Table 1, as it has been built so as to have no symmetry. A sketch of the geometry is drawn in Fig. 4 (a), whereas in (b), (c) and (d) we sketch the geometry of the same aggregate in a frame of reference with origin at the center of mass and the coordinate axes along the principal axes of inertia. Configurations (b), (c) and (d) differ for the choice of the z axis. In fact, the aggregate is so oriented that in (b) the principal moments of inertia, in units of the maximum moment, are I x=0.777, I y =1.000, I z =0.326, in (c) they are I x =1.000, I y =0.777, I z =0.326 and in (d) I x =0.326, I y =0.777, I z =1.000.

Fig. 2. Cartesian components of T 1 and T 2 for the five-spheres cluster of Fig. 4 (a). The direction of the incident wavevector is along the z axis
Fig. 3. Contour plot of T2=Tx2 +Ty2 (left panel) and of T z (right panel) as a function of the polar angles of the wavevector k I incident on the binary cluster with its axis along the y axis of Σ¯ (see Sect. 3). The incident wave has λ=0.3µm and circular polarization with η=1 is assumed. At ϑ I=0° we find T2=0 for any φ I, and T z=0.2196, and at ϑ I=180°, T2=0 and T z=-0.2196, as expected
Fig. 4. Sketch of the geometry of the 5-spheres aggregate. In (a) is reported the geometry of the aggregate the coordinates of whose centers are given in Table 1. In (b), (c) and (d) is reported the sketch of the same aggregate referred to a coordinate frame with origin in the center of mass and the z axis along one of the principal axes of inertia. The yellow sphere in (b), (c), and (d) is the same that in (a) has its center at the origin of the coordinates

Table 2. Values of T2 for the configurations sketched in Fig. 4

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In Fig. 5 we report, for λ=0.3µm and circular polarization with η=1, the contour plot of T2 as a function of the polar angles ϑI and φI of the wavevector incident on the 5-spheres aggregates in the configurations sketched in Fig. 4. When ϑI=0° and ϑI=180°, the quantity T2 gives information on the torques that tend to deviate the aggregate from the rotation around the z axis. Note that for configurations (b), (c), and (d) the z axis is a principal axis of inertia through the center of mass of the aggregate. The calculated values of T2 at ϑI=0° and ϑI=180° are reported in Table 2 for the four configurations considered.

Fig. 5. Contour plot of T2 as a function of the polar angles of the wavevector k I incident on the 5-sphere aggregate in the configurations shown in Fig. 4. λ=0.3µm and circular polarization with η=1 is assumed.

The only case in which there is an appreciable deviating torque is the one for ϑ1=180° in configuration (a). This is not surprising because the z axis is not a principal axis of inertia for the 5-spheres aggregate in configuration (a). On the contrary no appreciable deviating torque exists for configurations (b), (c) and (d). Note that all the values reported in Table 2 are independent of φ I as a result of the circular polarization of the incident wave. Quite analogous results are obtained by assuming circular polarization with η=2.

5. Conclusive remarks

The results shown in Sect. 4 deserve some specific remarks. When a circularly polarized plane wave impinges on a cluster with a general direction of incidence, there are significant components of the applied torque that tend to deviate the axis of rotation. On the contrary, the rotation driven by a plane wave incident along one of the principal axes of inertia may be nearly stable, even when the aggregate lacks any symmetry.

There are a few other points, however, that it may be useful to stress.

We mentioned in Sect. 1 that the theory of Sect. 2 applies also to any superposition of plane waves constituting the field that impinges onto the particles, as it happens, e.g., in the focal region of a lens [19

19. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. Roy. Soc. London A 253, 358–379 (1959) [CrossRef]

]. Thus, the theory applies to the study of the torques exerted on particles trapped within the focal region.

References and links

1.

J. D. Jackson, Classical electrodynamics, 2d edition (Wiley, New York, 1975)

2.

E. M. Purcell, “Suprathermal rotation of interstellar grains,” Astrophys. J. 231, 404–416 (1979) [CrossRef]

3.

M. E. J. Friese, T.A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394, 348–349 (1998); “Erratum”, ibid. 395, 621 (1998) [CrossRef]

4.

B. T. Draine and J. C. Weingartner, “Radiative torques on interstellar grains. I. Superthermal spin-up,” Astrophys. J. 470, 551–565 (1996) [CrossRef]

5.

B. T. Draine and J. C. Weingartner, “Radiative torques on interstellar grains. II. Grain alignment,” Astrophys. J. 480, 633–646 (1997) [CrossRef]

6.

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973) [CrossRef]

7.

B. T. Draine and P. J. Flatau, “Discrete dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (1994) [CrossRef]

8.

P. C. Waterman, “Symmetry, unitarity and geometry in electromagnetic scattering,” Phys. Rev. D 4, 825–839 (1971) [CrossRef]

9.

F. Borghese, P. Denti, and R. Saija, Scattering from model nonspherical particles (Springer, Heidelberg, 2002)

10.

P. L. Marston and J. H. Crichton, “Radiation torque on a sphere caused by a circularly-polarized electromagnetic wave,” Phys. Rev. A 30, 2508–2516 (1984) [CrossRef]

11.

F. J. García de, “Momentum transfer to small particles by passing electron beams,” Phys. Rev. B 70, 115422 (2004) [CrossRef]

12.

F. J. García de Abajo, “Electromagnetic forces and torques in nanoparticles irradiated by plane waves,” J. Quant. Spectrosc. Radiat. Transfer 89, 3–9 (2004) [CrossRef]

13.

M. I. Mishchenko, “Radiation force caused by scattering, absorption and emission of light by nonspherical particles,” J. Quant. Spectrosc. Radiat. Transfer 70, 811–816 (2001) [CrossRef]

14.

E. Fucile, F. Borghese, P. Denti, R. Saija, and O. I. Sindoni, “General reflection rule for electromagnetic multipole fields on a plane interface,” IEEE Trans. Antennas Propag. AP 45, 868–875 (1997) [CrossRef]

15.

E. M. Rose, Elementary theory of angular momentum, (Wiley, New York, 1956)

16.

M. Abramowitz and I. Stegun, Handbook of mathematical functions (Dover, New York, 1970)

17.

H. Goldstein, C. Poole, and J. Safko, Classical Mechanics3d edition (Addison-Wesley, Reading, Mass., 2002)

18.

B. T. Draine and H. M. Lee, “Optical properties of interstellar graphite and silicate grains,” Astrophys. J. 285, 89–108 (1984) [CrossRef]

19.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. Roy. Soc. London A 253, 358–379 (1959) [CrossRef]

20.

P. Galajda and P. Ormos, “Rotation of microscopic propellers in laser tweezers,” J. Opt. B: Quantum Semiclass. Opt. 4, S78–S81 (2002) [CrossRef]

OCIS Codes
(260.2110) Physical optics : Electromagnetic optics
(260.2160) Physical optics : Energy transfer
(290.0290) Scattering : Scattering

ToC Category:
Physical Optics

History
Original Manuscript: June 30, 2006
Revised Manuscript: September 11, 2006
Manuscript Accepted: September 20, 2006
Published: October 2, 2006

Citation
Ferdinando Borghese, Paolo Denti, Rosalba Saija, and Maria A. Iatì, "Radiation torque on nonspherical particles in the transition matrix formalism," Opt. Express 14, 9508-9521 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-20-9508


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References

  1. J. D. Jackson, Classical electrodynamics, 2d edition (Wiley, New York, 1975)
  2. E. M. Purcell, "Suprathermal rotation of interstellar grains," Astrophys. J. 231, 404-416 (1979) [CrossRef]
  3. M. E. J. Friese, T.A. Nieminen, N. R. Heckenberg and H. Rubinsztein-Dunlop, "Optical alignment and spinning of laser-trapped microscopic particles," Nature 394, 348-349 (1998); "Erratum", ibid. 395, 621 (1998) [CrossRef]
  4. B. T. Draine and J. C. Weingartner, "Radiative torques on interstellar grains. I. Superthermal spin-up," Astrophys. J. 470, 551-565 (1996) [CrossRef]
  5. B. T. Draine and J. C. Weingartner, "Radiative torques on interstellar grains. II. Grain alignment," Astrophys. J. 480, 633-646 (1997) [CrossRef]
  6. E. M. Purcell and C. R. Pennypacker, "Scattering and absorption of light by nonspherical dielectric grains," Astrophys. J. 186, 705-714 (1973) [CrossRef]
  7. B. T. Draine and P. J. Flatau, "Discrete dipole approximation for scattering calculations," J. Opt. Soc. Am. A 11, 1491-1499 (1994) [CrossRef]
  8. P. C. Waterman, "Symmetry, unitarity and geometry in electromagnetic scattering," Phys. Rev. D 4, 825-839 (1971) [CrossRef]
  9. F. Borghese, P. Denti and R. Saija, Scattering from model nonspherical particles (Springer, Heidelberg, 2002)
  10. P. L. Marston and J. H. Crichton, "Radiation torque on a sphere caused by a circularly-polarized electromagnetic wave," Phys. Rev. A 30, 2508-2516 (1984) [CrossRef]
  11. F. J. García de Abajo, "Momentum transfer to small particles by passing electron beams," Phys. Rev. B 70, 115422 (2004) [CrossRef]
  12. F. J. García de Abajo, "Electromagnetic forces and torques in nanoparticles irradiated by plane waves," J. Quantum Spectrosc. Radiat. Transfer 89, 3-9 (2004) [CrossRef]
  13. M. I. Mishchenko, "Radiation force caused by scattering, absorption and emission of light by nonspherical particles," J. Quantum Spectrosc. Radiat. Transfer 70, 811-816 (2001) [CrossRef]
  14. E. Fucile, F. Borghese, P. Denti, R. Saija and O. I. Sindoni, "General reflection rule for electromagnetic multipole fields on a plane interface," IEEE Trans. Antennas Propag. AP 45, 868-875 (1997) [CrossRef]
  15. E. M. Rose, Elementary theory of angular momentum, (Wiley, New York, 1956)
  16. M. Abramowitz and I. Stegun, Handbook of mathematical functions (Dover, New York, 1970)
  17. H. Goldstein, C. Poole and J. Safko, Classical Mechanics 3d edition (Addison-Wesley, Reading, Mass., 2002)
  18. B. T. Draine and H. M. Lee, "Optical properties of interstellar graphite and silicate grains," Astrophys. J. 285, 89-108 (1984) [CrossRef]
  19. B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system," Proc. Roy. Soc. London A 253, 358-379 (1959) [CrossRef]
  20. P. Galajda and P. Ormos, "Rotation of microscopic propellers in laser tweezers," J. Opt. B: Quantum Semiclass. Opt. 4, S78-S81 (2002) [CrossRef]

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