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

The actual calculation of the radiation torque requires that the electromagnetic field in the presence of the particles be known. Draine and Weingartner [4, 5] calculated through the discrete dipole approximation (DDA) [6, 7] 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]. 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]. 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] who dealt with the torque exerted by an elliptically polarized plane wave on a homogeneous sphere. More recently, de Abajo [11, 12] 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]. 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

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

is the time averaged Maxwell stress tensor [1], I being the unit dyadic. Of course, in Eq. (2)

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

Since r̂·I×r̂=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

${\mathbf{E}}_S=\frac{\exp \left(\mathit{ikr}\right)}{r}\mathbf{f}({\hat{\mathbf{k}}}_S,{\hat{\mathbf{k}}}_I)$

cannot be used lest to get a vanishing result [13] because r̂·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 uû_η, such that

û ₁×û ₂=k̂_I,

where k̂_I is the direction of the propagation of the incident plane wave. Note that the vectors uû_η 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 k̂_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]. Accordingly,

${\mathbf{E}}_I=\sum _{\eta }{E}_{0\eta }{\hat{\mathbf{u}}}_{\eta }\exp \left(i{\mathbf{k}}_I\mathbf{r}\right)=\sum _{\eta }{E}_{0\eta }\sum _{\mathit{plm}}{\mathbf{J}}_{\mathit{lm}}^{\left(p\right)}(\mathbf{r},k)W_{I\eta \mathit{lm}}^{\left(p\right)},$

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

${\mathbf{E}}_S=\sum _{\eta }{E}_{0\eta }\sum _{\mathit{plm}}{\mathbf{H}}_{\mathit{lm}}^{\left(p\right)}(\mathbf{r},k)A_{\eta \mathit{lm}}^{\left(p\right)},$

where X _lm (r̂) denotes vector spherical harmonics [1]: The multipole fields ${\mathbf{H}}_{\mathit{\text{lm}}}^{\left(p\right)}$ (r,k) are identical to the ${\mathbf{J}}_{\mathit{\text{lm}}}^{\left(p\right)}$ (r,k) fields except for the substitution of the spherical Hankel functions of the first kind h_l (kr) in place of the spherical Bessel functions j _l(kr). The multipole amplitudes of the incident field are defined as

$W_{\mathrm{I}\eta \mathit{\text{lm}}}^{\left(p\right)}$=4πi ^p+l-1 uû_η·${\mathbf{Z}}_{\mathit{\text{lm}}}^{\left(p\right)}$ (k̂_I),

where

${\mathbf{Z}}_{\mathit{\text{lm}}}^{\left(1\right)}$ (k̂)=X _lm (k̂), ${\mathbf{Z}}_{\mathit{\text{lm}}}^{\left(2\right)}$ (k̂)=X _lm (k̂)×k̂

are transverse vector harmonics [14], and the amplitudes of the scattered field $A_{\eta \mathit{\text{lm}}}^{\left(p\right)}$ are calculated by imposing the customary boundary conditions across the surface of the particle. The amplitudes $A_{\eta \mathit{\text{lm}}}^{\left(p\right)}$ bear the index η to recall the polarization of the incident field, and are related to the $W_{\mathrm{I}\eta \mathit{\text{lm}}}^{\left(p\right)}$ by the equation [9]

${\mathbf{J}}_{\mathit{lm}}^{\left(1\right)}(\mathbf{r},k)=j_l\left(\mathit{kr}\right){\mathbf{X}}_{\mathit{lm}}\left(\hat{\mathbf{r}}\right),{\mathbf{J}}_{\mathit{lm}}^{\left(2\right)}(\mathbf{r},k)=\frac{1}{k}\nabla {\mathbf{J}}_{\mathit{lm}}^{\left(1\right)}(\mathbf{r},k),$

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

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

𝓢 ^(pp′) _lml^′m^′=${\mathcal{S}}_1^{\left(p\right)}$ δ _pp^′ δ _ll^′ δ _mm^′.

We notice that ${\mathcal{S}}_l^{\left(1\right)}$ =b _l , ${\mathcal{S}}_l^{\left(2\right)}$ =a _l , a _l and b _l being the well known Mie coefficients [9].

Assuming the scattered field has been calculated, let us search for its expression in the far zone. For the H-multipole fields we get [16]

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

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

$\hat{\mathbf{r}}·{\mathbf{H}}_{\mathit{lm}}^{\left(2\right)}=\sqrt{l\left(l+1\right)}{(-i)}^l\frac{e^{\mathit{ikr}}}{k^2{r}^2}{Y}_{\mathit{lm}}=c_{r2l}{Y}_{\mathit{lm}},$

$\hat{\mathbf{r}}·{\mathbf{J}}_{\mathit{lm}}^{\left(2\right)}=i\sqrt{l\left(l+1\right)}\frac{\sin \left(\mathit{kr}-l\pi ⁄2\right)}{k^2{r}^2}{Y}_{\mathit{lm}}=c_{r1l}{Y}_{\mathit{lm}},$

and through the cross products

${\mathbf{H}}_{\stackrel{}{l}\stackrel{}{m}}^{\left(\stackrel{}{p}\right)*}\hat{\mathbf{r}}=i^{\stackrel{}{l}+\stackrel{}{p}}\frac{e^{-\mathit{ikr}}}{\mathit{kr}}{\mathbf{Z}}_{\stackrel{}{l}\stackrel{}{m}}^{\left(\stackrel{}{p}\right)*}\hat{\mathbf{r}}=c_{t2\stackrel{}{l}}^{\left(\stackrel{}{p}\right)}{\mathbf{Z}}_{\stackrel{}{l}\stackrel{}{m}}^{\left(\stackrel{}{p}\right)*}\hat{\mathbf{r}},$

${\mathbf{J}}_{\stackrel{}{l}\stackrel{}{m}}^{\left(\stackrel{}{p}\right)*}\hat{\mathbf{r}}=\frac{\sin [\mathit{kr}-\left(\stackrel{}{l}+\stackrel{}{p}-1\right)\pi ⁄2]}{\mathit{kr}}{\mathbf{Z}}_{\stackrel{}{l}\stackrel{}{m}}^{\left(\stackrel{}{p}\right)*}\hat{\mathbf{r}}=c_{t1l}^{\left(\stackrel{}{p}\right)}{\mathbf{Z}}_{\stackrel{}{l}\stackrel{}{m}}^{\left(\stackrel{}{p}\right)*}\hat{\mathbf{r}},$

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.

Now, we define the vector Γ⃗ such that its spherical components [15] are related to the cartesian components of Γ⃗_Rad according to

where

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 Γ_µ;η̄η. Accordingly, we note that the integrand in (3) gets contributions that can be written as

where

$d_{1lm}=\hat{\mathbf{r}}·{\mathbf{J}}_{lm}^{\left(2\right)},d_{2lm}=\hat{\mathbf{r}}·{\mathbf{H}}_{lm}^{\left(2\right)},$

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

$v_{\mu ;1\stackrel{}{l}\stackrel{}{m}}^{\left(\stackrel{}{p}\right)}=\left({\mathbf{J}}_{\stackrel{}{l}\stackrel{}{m}}^{\left(\stackrel{}{p}\right)*}\hat{\mathbf{r}}\right)\mu ,v_{\mu ;2lm}^{\left(\stackrel{}{p}\right)}=\left({\mathbf{H}}_{\stackrel{}{l}\stackrel{}{m}}^{\left(\stackrel{}{p}\right)*}\hat{\mathbf{r}}\right)\mu .$

$K_{1;\alpha \stackrel{}{\alpha }lm\stackrel{}{l}\stackrel{}{m}}^{\left(21\right)}={\mathit{ic}}_{r\alpha l}{c}_{t\stackrel{}{\alpha }\stackrel{}{l}}^{\left(1\right)}{Y}_{\mathit{lm}}\sqrt{\frac{4\pi }{3}}\left[Y_{10}C(1,\stackrel{}{l},\stackrel{}{l;}1,\stackrel{}{m}-1)Y_{\stackrel{}{l},\stackrel{}{m}-1}^{*}-Y_{11}C(1,\stackrel{}{l},\stackrel{}{l;}0,\stackrel{}{m})Y_{\stackrel{}{l}\stackrel{}{m}}^{*}\right],$

$K_{0;\alpha \stackrel{}{\alpha }lm\stackrel{}{l}\stackrel{}{m}}^{\left(21\right)}={\mathit{ic}}_{r\alpha l}{c}_{t\stackrel{}{\alpha }\stackrel{}{l}}^{\left(1\right)}{Y}_{\mathit{lm}}\sqrt{\frac{4\pi }{3}}\left[Y_{1,-1}C(1,\stackrel{}{l},\stackrel{}{l;}1,\stackrel{}{m}-1)Y_{\stackrel{}{l},\stackrel{}{m}-1}^{*}-Y_{11}C(1,\stackrel{}{l},\stackrel{}{l;}-1,\stackrel{}{m}+1)Y_{\stackrel{}{l}\stackrel{}{m}+1}^{*}\right],$

$K_{-1;\alpha \stackrel{}{\alpha }lm\stackrel{}{l}\stackrel{}{m}}^{\left(21\right)}=i{c}_{r\alpha l}{c}_{t\stackrel{}{\alpha }\stackrel{}{l}}^{\left(1\right)}{Y}_{lm}\sqrt{\frac{4\pi }{3}}\left[Y_{1,-1}C(1,\stackrel{}{l},\stackrel{}{l};0,\stackrel{}{m})Y_{\stackrel{}{l},\stackrel{}{m}}^{*}-Y_{10}C(1,\stackrel{}{l},\stackrel{}{l};-1,\stackrel{}{m}+1)Y_{\stackrel{}{l},\stackrel{}{m}+1}^{*}\right],$

where the C’s denote Clebsch-Gordan coefficients [15]. When these contributions, according to

Eq. (3), are integrated we get

$K_{\mu ;\alpha \stackrel{}{\alpha }lm\stackrel{}{l}\stackrel{}{m}}^{\left(22\right)}=c_{r\alpha l}{c}_{t\stackrel{}{\alpha }\stackrel{}{l}}^{\left(2\right)}{Y}_{lm}C(1,\stackrel{}{l},\stackrel{}{l};\stackrel{}{\mu },\stackrel{}{m}-\mu )Y_{\stackrel{}{l},\stackrel{}{m}-\mu }^{*},$

$I_{1;\alpha \stackrel{}{\alpha }lm\stackrel{}{l}\stackrel{}{m}}^{\left(21\right)}=i{c}_{r\alpha l}{C}_{t\stackrel{}{\alpha }\stackrel{}{l}}^{\left(1\right)}\sqrt{\frac{2l+1}{2l+1}}C(1,l,\stackrel{}{l};0,0){\delta }_{m,\stackrel{}{m}-1}$

$\left[C(1,\stackrel{}{l},\stackrel{}{l};1,\stackrel{}{m}-1)C(1,l,\stackrel{}{l};0,m,\stackrel{}{m}-1)-C(1,\stackrel{}{l},\stackrel{}{l};0,\stackrel{}{m})C(1,l,\stackrel{}{l};1,m,\stackrel{}{m})\right],$

$I_{0;\alpha \stackrel{}{\alpha }\mathit{l}\mathit{m}\stackrel{}{l}\stackrel{}{\mathit{m}}}^{\left(21\right)}=i{c}_{r\alpha l}{C}_{t\stackrel{}{\alpha }\stackrel{}{l}}^{\left(1\right)}\sqrt{\frac{2l+1}{2l+1}}C(1,l,\stackrel{}{l};0,0){\delta }_{m,\stackrel{}{m}}$

$\left[C(1,\stackrel{}{l},\stackrel{}{l};1,\stackrel{}{m}-1)C\right(1,l,\stackrel{}{l};-1,m,\stackrel{}{m}-1)$

$-C(1,\stackrel{}{l},\stackrel{}{l};-1,\stackrel{}{m}+1)C(1,l,\stackrel{}{l};1,m,\stackrel{}{m}+1)],$

and

Finally, collecting all the terms one obtains

where

$a_{1\eta \mathit{\text{lm}}}^{\left(p\right)}$=$W_{\mathrm{I}\eta \mathit{\text{lm}}}^{\left(p\right)}$, and $a_{2\eta \mathit{\text{lm}}}^{\left(p\right)}$=$A_{\eta \mathit{\text{lm}}}^{\left(p\right)}$.

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, ᾱ, and p̄ or p̄^′. The same is true for ${\mathrm{\Gamma }}_{\text{Rad}}^{⃗}$ 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 ${\mathrm{\Gamma }}_{\text{Rad}}^{⃗}$ for each η and η̄.

• The contributions with α=ᾱ=1 are just vanishing both for p̄=2 or p̄=1 and for p̄^′=1 or p̄^′=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=ᾱ=2 vanishes both for p̄=1 and for p̄′=2.

• The contributions for α≠ᾱ depend on r, i.e. on the radius of the spherical surface of integration. Nevertheless, those with p̄=1 are identical but of opposite sign and cancel each other; on the other hand, the sum of those with p̄=2 turns out to be independent of r. Analogous considerations hold true for the contributions with p̄^′=2 and p̄^′=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≠ᾱ from the one over α=ᾱ, we can write

In (12)

whereas the quantities ${\mathrm{\Gamma }}_{\mathit{\mu }\mathit{;}\eta \eta }^{\left(\text{sca}\right)}$ ̄ are identical to the ${\mathrm{\Gamma }}_{\mathit{\mu }\mathit{;}\eta \eta }^{\left(\text{ext}\right)}$ ̄ above, except for the substitution of $A_{\eta \mathit{\text{lm}}}^{\left(p\right)}$ in place of $W_{\mathrm{I}\eta \mathit{\text{lm}}}^{\left(p\right)}$ and for the change of the sign. In Eq. (13) we define c _Γ=n ²/8πk ³ and

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

${\Gamma }_{\mu ;\stackrel{}{\eta }\eta }^{\left(\mathrm{ext}\right)}=c_{\Gamma }\sum _{\mathit{plm}}{V}_{\mu ;\eta lm}^{\left(p\right)}{A}_{\eta lm}^{\left(p\right)*},$

where $V_{\mathit{\mu }\mathit{;}\eta \mathit{\text{lm}}}^{\left(p\right)}$=s _µ;lm $W_{\mathrm{I}\eta l,m-\mathit{\mu }}^{\left(p\right)}$. Considering the vector ${\mathbf{V}}_{\eta \mathit{\text{lm}}}^{\left(p\right)}$ with spherical components $V_{\mathit{\mu }\mathit{;}\eta \mathit{\text{lm}}}^{\left(p\right)}$, we find for a circular polarization basis

${\mathbf{V}}_{\eta lm}^{\left(p\right)}{\hat{\mathbf{k}}}_I=\sum _{\mu }{(-)}^{\mu }{V}_{-\mu ;\eta lm}^{\left(p\right)}{\hat{k}}_{I\mu }={(-)}^{\eta }{W}_{I\eta lm}^{\left(p\right)}.$

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 k̂_I but it is immediately obtained in the case k̂_I=ê_z. As a consequence,

On the contrary,

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

${\check{\sigma }}_{T\eta \stackrel{}{\eta }}=\frac{-1}{k^2}\sum _{\mathit{plm}}{A}_{\stackrel{}{\eta }lm}^{\left(p\right)}{W}_{I\eta lm}^{\left(p\right)*},{\check{\sigma }}_{S\eta \stackrel{}{\eta }}=\frac{1}{k^2}\sum _{\mathit{plm}}{A}_{\stackrel{}{\eta }lm}^{\left(p\right)}{A}_{\eta lm}^{\left(p\right)*}$, ${\sigma }_T=\frac{1}{I_I}\sum _{\eta \eta \text{'}}\mathrm{Re}\left(I_{I\eta \eta \text{'}}{\check{\sigma }}_{T\eta \eta \text{'}}\right),{\sigma }_S=\frac{1}{I_I}\sum _{\eta \eta \text{'}}\mathrm{Re}\left(I_{I\eta \eta \text{'}}{\check{\sigma }}_{S\eta \eta \text{'}}\right),$

which are related to the extinction and scattering cross sections of the particle through the relations

${\sigma }_{T\eta }=\mathrm{Re}\left({\check{\sigma }}_{T\eta \eta }\right),{\sigma }_{S\eta }={\check{\sigma }}_{S\eta \eta }.$

where I _I=Σ_η |E _0η|². Moreover, the extinction and the scattering cross sections, in case of radiation polarized along uû_η, turn out to be

σ_Tη=Re($\check{\sigma }$ _Tηη), σS_η=$\check{\sigma }$ S_ηη.

We stress that (16) becomes a true equality for whatever direction of incidence when the scatterer is a sphere. This result is related to the property (4) of the transition matrix and to the remark that

• When the scatterer is axially symmetric and k̂_I is parallel to its axis the contribution for α=2, $\overline{\alpha }$ =1 with p̄=2 and that for α=1, $\overline{\alpha }$ =2 with p̄=1 are equal; the same is also true for α=2, $\overline{\alpha }$ =1 with p̄′=1 and for α=1, $\overline{\alpha }$ =2 with p̄^′=2.

On account that in the case above, using a circular polarization basis, one finds σ_T1=σ_T2, σ_S1=σ_S2, and $\check{\sigma }$ _Tηη̄=$\check{\sigma }$ _Sηη̄=0 for η≠η̄, the result is

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.

We remark that our Eq. (13) differs from the equations of de Abajo (equation (11) of Ref. [11] and equation (4) of Ref. [12]), because in the latter the amplitudes $A_{\stackrel{}{\eta }lm}^{\left(p\right)*}$ and $W_{\mathrm{I}\eta l\prime ,m-\mathit{\mu }}^{\left(p\right)}$, in our notation, bear the different indexes l and l ^′, and one has to sum both over l and l ^′. According to Eq. (10) and to our previous analysis of the contributions to Γ⃗_Rad, such a double sum should be reduced to a single sum. In fact, we were kindly informed that this discrepancy is a mere accident, since it was due to a misprint, i.e., to a missing factor δ_ll^′ in de Abajo’s equations.

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], 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 $\overline{\Sigma }$ 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 $\overline{\Sigma }$ , the same quantities can be transformed into the laboratory frame Σ by exploiting the transformation rules that are reported in Ref. [9]. In particular, since the spherical components of the radiation torque transform according to D⁽¹⁾, i.e., as a first rank tensor, we get

where$\overline{\Gamma }$ _µ;ηη̄=$\overline{\Gamma }$ ^(ext) _µ;η̄η-$\overline{\Gamma }$ ^(sca) _µ;η̄η,

and, according to Eq. (13),

Moreover,

${\stackrel{}{W}}_{I\eta l\mu }^{\left(p\right)}=\sum _m{𝒟}_{\mu m}^{\left(l\right)}\left(\Theta \right)W_{I\eta lm}^{\left(p\right)},{\stackrel{}{A}}_{\eta lm}^{\left(p\right)}=\sum _{p\text{'}l\text{'}m\text{'}}\sum _{\stackrel{}{m}\text{'}}{\stackrel{}{𝒮}}_{lml\text{'}m\text{'}}^{\left(pp\text{'}\right)}{𝒟}_{m\text{'}\stackrel{}{m}\text{'}}^{\left(pp\text{'}\right)}{W}_{I\eta l\text{'}\stackrel{}{m}\text{'}}^{\left(p\text{'}\right)}.$

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

$\int {𝒟}_{\mu \mu \text{'}}^{\left(1\right)}{𝒟}_{m-\mu ,\stackrel{}{m}}^{\left(l\right)}{𝒟}_{m\text{'}\stackrel{}{m\text{'}}}^{\left(l\text{'}\right)*}P\left(\Theta \right)d\Theta ={\delta }_{\mu \mu \text{'}}{\delta }_{m-\mu ,\stackrel{}{m}}{\delta }_{m\text{'}\stackrel{}{m}\text{'}}{\delta }_{m\stackrel{}{m}\text{'}}{\delta }_{mm\text{'}}$

and

$\int {𝒟}_{\mu \mu \text{'}}^{\left(1\right)}{𝒟}_{m\text{'}\stackrel{}{m}\text{'}}^{\left(l\text{'}\right)}{𝒟}_{m"\stackrel{}{m}"}^{\left(l"\right)*}P\left(\Theta \right)d\Theta ={\delta }_{\mu \mu \text{'}}{\delta }_{m\text{'}\stackrel{}{m}\text{'}}{\delta }_{m"\stackrel{}{m}"}{\delta }_{\mu +m\text{'},m"\text{'}},$

where

$P\left(\Theta \right)=\frac{1}{2\pi }\frac{\delta \left(\beta \right)}{\sin \beta }\delta \left(\gamma \right),$

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

$〈{\Gamma }_{\mu ;\stackrel{}{\eta }\eta }^{\left(\mathrm{ext}\right)}〉=c_{\Gamma }\sum _{\mathrm{plm}}{s}_{\mu ;lm}\sum _{\stackrel{}{p}l}{W}_{I\eta l,m-\mu }^{\left(p\right)}{\stackrel{}{풟}}_{l\stackrel{}{m}l\stackrel{}{m}}^{\left(p\stackrel{}{p}\right)*}{W}_{I\stackrel{}{\eta }l\stackrel{}{m}}^{\left(p\stackrel{}{p}\right)*}{W}_{I\stackrel{}{\eta }l\stackrel{}{m}}^{\left(\stackrel{}{p}\right)*},$

$〈{\Gamma }_{\mu ;\stackrel{}{\eta }\eta }^{\left(\mathrm{sca}\right)}〉=c_{\Gamma }\sum _{\mathit{plm}}{s}_{\mu ;lm}\sum _{\stackrel{}{p}\stackrel{}{p}\text{'}}\sum _{\stackrel{}{l}\stackrel{}{l}\text{'}}\sum _{m\text{'}}{𝒮}_{lml,m\text{'}-\mu }^{\left(p\stackrel{}{p}\right)}{W}_{I\eta \stackrel{}{l},m\text{'}-\mu }^{\left(\stackrel{}{p}\right)}{𝒮}_{lml\text{'}m\text{'}}^{(p\stackrel{}{p}\text{'})*}{W}_{I\stackrel{}{\eta }l\stackrel{}{m}}^{(\stackrel{}{p}\text{'})*},$

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]. 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

$T_{\eta }=\frac{8\pi k}{n^2{I}_I{\sigma }_{T\eta }}{\overrightarrow{\Gamma }}_{\mathrm{Rad}\eta },$

where Γ⃗_Radη 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]. 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·k̂_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 Γ⃗_Radη, 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., Γ⃗_Radη·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 (k̂_I≡ê_z) 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]. We first notice that the x and y components of Γ⃗_Rad 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, 〈Γ⃗_Radη·k̂_I〉, with Γ⃗_Radη·k̂_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 Γ⃗_Radη 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 〈Γ⃗_Radη·k̂_I〉=Γ⃗_Radη·k̂_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 Γ⃗_Radη 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 z̄. Of course, in general, the axial average 〈Γ⃗_Radη·k̂_I〉≠Γ⃗_Radη·k̂_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

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At this stage, since Γ⃗_Rad 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 $T_{\perp }^2$=$T_x^2$ +$T_y^2$ and T _z as a function of the polar angles ϑ_I and φ _I of the wavevector kI incident on the binary cluster with its axis along the y axis of $\overline{\Sigma }$ (see Sect 3). The wavelength is λ=0.3µm and circular polarization with η=1 is assumed. Note that the axes of $\overline{\Sigma }$ 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

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Fig. 3. Contour plot of $T_{\perp }^2$=$T_x^2$ +$T_y^2$ (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 $\overline{\Sigma }$ (see Sect. 3). The incident wave has λ=0.3µm and circular polarization with η=1 is assumed. At ϑ _I=0° we find $T_{\perp }^2$=0 for any φ _I, and T _z=0.2196, and at ϑ _I=180°, $T_{\perp }^2$=0 and T _z=-0.2196, as expected

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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

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Table 2. Values of $T_{\perp }^2$ 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 $T_{\perp }^2$ as a function of the polar angles ϑ_I and φ_I of the wavevector incident on the 5-spheres aggregates in the configurations sketched in Fig. 4. When ϑ_I=0° and ϑ_I=180°, the quantity $T_{\perp }^2$ 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 $T_{\perp }^2$ at ϑ_I=0° and ϑ_I=180° are reported in Table 2 for the four configurations considered.

 

Fig. 5. Contour plot of $T_{\perp }^2$ 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.

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

The application of the theory to a randomly generated structure had the declared purpose of avoiding unwanted effects due to some symmetry of the 5-sphere cluster of Sect. 4. Obviously, we could assume any other structure convenient for pursuing specific effects. For instance, one could devise clusters reproducing the overall shape of a propeller, a shape of interest, on account of the induced rotational motion, in some problems of microrheology [20]. In fact, according to our previous experience [9], the usefulness of such a flexible model for small particles as the cluster of spheres model stems from the remark that their scattering properties, in many instances, depend more on the quantity of refractive material and on the overall shape than on the details of the structure.