## Radiation torque on nonspherical particles in the transition matrix formalism

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

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]

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

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]

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

## 2. Radiation torque

*n*. Then, the conservation of angular momentum for the combined system of field and particle decrees that the particle experiences a torque given by

**n̂**is the unit outward normal to an arbitrary surface

*S*including the particle and

**E**and

**B**are the superposition of the incident and of the scattered fields.

*S*to be a sphere of arbitrary radius with its center within the particle so that Eq. (1) becomes

**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

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]

**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).

**u**û

_{η}, such that

**û**

_{1}×

**û**

_{2}=

**k̂**

_{I},

**k̂**

_{I}is the direction of the propagation of the incident plane wave. Note that the vectors

**u**û

_{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.

*k*=

*nk*

_{v}, with

*k*

_{v}=

*ω*/

*c*, is the propagation constant of the incident plane wave. We also define the multipole fields

**X**

_{lm}(

**r̂**) denotes vector spherical harmonics [1]: The multipole fields

**r**,

*k*) are identical to the

**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

*πi*

^{p+l-1}

**u**û

_{η}·

**k̂**

_{I}),

**k̂**)=

**X**

_{lm}(

**k̂**),

**k̂**)=

**X**

_{lm}(

**k̂**)×

**k̂**

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

*𝓢*

^{(pp′)}

_{lml′m′}=

*δ*

_{pp′}

*δ*

_{ll′}

*δ*

_{mm′}.

*b*

_{l},

*a*

_{l},

*a*

_{l}and

*b*

_{l}being the well known Mie coefficients [9].

**H**-multipole fields we get [16]

**J**-multipole fields we get [16]

**H**- and

**J**-fields enter the integrand in (3) through the dot products

*I*

_{Iη̄η}=

*E**

_{0η}̄

*E*

_{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

*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

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

*α*,

*ᾱ*, and

*p̄*or

*p̄*

^{′}. The same is true for

*η*and

*η̄*.

*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.

*α*=

*ᾱ*=2 vanishes both for

*p̄*=1 and for

*p̄*′=2.

*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.

*̄*are identical to the

*̄*above, except for the substitution of

*c*

_{Γ}=

*n*

^{2}/8

*πk*

^{3}and

*s*

_{µ;lm}

**k̂**

_{I}but it is immediately obtained in the case

**k̂**

_{I}=

**ê**

_{z}. As a consequence,

**k̂**

_{I}is parallel to its axis. In Eqs. (15) and (16) we define the quantities

*I*

_{I}=Σ

_{η}|

*E*

_{0η}|

^{2}. Moreover, the extinction and the scattering cross sections, in case of radiation polarized along

**u**û

_{η}, turn out to be

_{Tη}=Re(

_{Tηη}), σS

_{η}=

_{ηη}.

**k̂**

_{I}is parallel to its axis the contribution for α=2,

*p̄*=2 and that for α=1,

*p̄*=1 are equal; the same is also true for α=2,

*p̄*′=1 and for α=1,

*p̄*

^{′}=2.

_{T1}=σ

_{T2}, σ

_{S1}=σ

_{S2}, and

_{Tηη}̄=

_{Sηη}̄=0 for

*η*≠

*η̄*, the result is

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]

*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

*axial average*of the radiation torque.

*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

^{(1)}, i.e., as a first rank tensor, we get

_{µ;ηη̄}=

^{(ext)}

_{µ;η̄η}-

^{(sca)}

_{µ;η̄η},

*c*

_{Γ}=

*n*

^{2}/8

*πk*

^{3}and

*s*

_{µ;lm}are given by (14).

## 4. Radiation torque on aggregated spheres

*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]

_{Radη}is the radiation torque calculated for incident light with polarization

*η*, and

*n*=1.

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

*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.

**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

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

*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.

_{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*

_{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

*µ*m and circular polarization with

*η*=1 is assumed. Note that the axes of

**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.

*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.

*µ*m and circular polarization with

*η*=1, the contour plot of

_{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

*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

_{I}=0° and ϑ

_{I}=180° are reported in Table 2 for the four configurations considered.

_{1}=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

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]

## References and links

1. | J. D. Jackson, |

2. | E. M. Purcell, “Suprathermal rotation of interstellar grains,” Astrophys. J. |

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 |

4. | B. T. Draine and J. C. Weingartner, “Radiative torques on interstellar grains. I. Superthermal spin-up,” Astrophys. J. |

5. | B. T. Draine and J. C. Weingartner, “Radiative torques on interstellar grains. II. Grain alignment,” Astrophys. J. |

6. | E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. |

7. | B. T. Draine and P. J. Flatau, “Discrete dipole approximation for scattering calculations,” J. Opt. Soc. Am. A |

8. | P. C. Waterman, “Symmetry, unitarity and geometry in electromagnetic scattering,” Phys. Rev. D |

9. | F. Borghese, P. Denti, and R. Saija, |

10. | P. L. Marston and J. H. Crichton, “Radiation torque on a sphere caused by a circularly-polarized electromagnetic wave,” Phys. Rev. A |

11. | F. J. García de, “Momentum transfer to small particles by passing electron beams,” Phys. Rev. B |

12. | F. J. García de Abajo, “Electromagnetic forces and torques in nanoparticles irradiated by plane waves,” J. Quant. Spectrosc. Radiat. Transfer |

13. | M. I. Mishchenko, “Radiation force caused by scattering, absorption and emission of light by nonspherical particles,” J. Quant. Spectrosc. Radiat. Transfer |

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 |

15. | E. M. Rose, |

16. | M. Abramowitz and I. Stegun, |

17. | H. Goldstein, C. Poole, and J. Safko, |

18. | B. T. Draine and H. M. Lee, “Optical properties of interstellar graphite and silicate grains,” Astrophys. J. |

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 |

20. | P. Galajda and P. Ormos, “Rotation of microscopic propellers in laser tweezers,” J. Opt. B: Quantum Semiclass. Opt. |

**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

- J. D. Jackson, Classical electrodynamics, 2d edition (Wiley, New York, 1975)
- E. M. Purcell, "Suprathermal rotation of interstellar grains," Astrophys. J. 231, 404-416 (1979) [CrossRef]
- 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]
- B. T. Draine and J. C. Weingartner, "Radiative torques on interstellar grains. I. Superthermal spin-up," Astrophys. J. 470, 551-565 (1996) [CrossRef]
- B. T. Draine and J. C. Weingartner, "Radiative torques on interstellar grains. II. Grain alignment," Astrophys. J. 480, 633-646 (1997) [CrossRef]
- E. M. Purcell and C. R. Pennypacker, "Scattering and absorption of light by nonspherical dielectric grains," Astrophys. J. 186, 705-714 (1973) [CrossRef]
- B. T. Draine and P. J. Flatau, "Discrete dipole approximation for scattering calculations," J. Opt. Soc. Am. A 11, 1491-1499 (1994) [CrossRef]
- P. C. Waterman, "Symmetry, unitarity and geometry in electromagnetic scattering," Phys. Rev. D 4, 825-839 (1971) [CrossRef]
- F. Borghese, P. Denti and R. Saija, Scattering from model nonspherical particles (Springer, Heidelberg, 2002)
- 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]
- F. J. García de Abajo, "Momentum transfer to small particles by passing electron beams," Phys. Rev. B 70, 115422 (2004) [CrossRef]
- 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]
- 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]
- 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]
- E. M. Rose, Elementary theory of angular momentum, (Wiley, New York, 1956)
- M. Abramowitz and I. Stegun, Handbook of mathematical functions (Dover, New York, 1970)
- H. Goldstein, C. Poole and J. Safko, Classical Mechanics 3d edition (Addison-Wesley, Reading, Mass., 2002)
- B. T. Draine and H. M. Lee, "Optical properties of interstellar graphite and silicate grains," Astrophys. J. 285, 89-108 (1984) [CrossRef]
- 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]
- 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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