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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 3 — Jan. 31, 2011
  • pp: 2466–2475
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Discrete dipole approximation for time-domain computation of optical forces on magnetodielectric scatterers

Patrick C. Chaumet, Kamal Belkebir, and Adel Rahmani  »View Author Affiliations


Optics Express, Vol. 19, Issue 3, pp. 2466-2475 (2011)
http://dx.doi.org/10.1364/OE.19.002466


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Abstract

We present a general approach, based on the discrete dipole approximation (DDA), for the computation of the exchange of momentum between light and a magnetodielectric, three-dimensional object with arbitrary geometry and linear permittivity and permeability tensors in time domain. The method can handle objects with an arbitrary shape, including objects with dispersive dielectric and/or magnetic material responses.

© 2011 Optical Society of America

1. Introduction

The discrete dipole approximation (DDA, also called coupled dipole method) is a general computational method that is widely used to study scattering problems in electrodynamics and photonics. The method was introduced by Purcell and Pennypacker [1

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

, 2

2. B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988). [CrossRef]

] to address the scattering of light by interstellar grains with arbitrary shapes. Since its initial formulation, the DDA been generalized to study the scattering of light by periodic structures [3

3. P. C. Chaumet, A. Rahmani, and G. W. Bryant, “Generalization of the coupled dipole method to periodic structure,” Phys. Rev. B 67, 165,404–5 (2003). [CrossRef]

, 4

4. P. C. Chaumet and A. Sentenac, “Numerical simulations of the electromagnetic field scattered by defects in a double-periodic structure,” Phys. Rev. B 72, 205,437–8 (2005). [CrossRef]

], spontaneous emission [5

5. A. Rahmani, P. C. Chaumet, and F. de Fornel, “Enrironment-induced modification of spontaneous emission: Single-molecule near-field probe,” Phys. Rev. A 63, 023,819–11 (2001). [CrossRef]

, 6

6. A. Rahmani and G. W. Bryant, “Spontaneous emission in microcavity electrodynamics,” Phys. Rev. A 65, 033,817–12 (2002). [CrossRef]

, 7

7. F. Bordas, N. Louvion, S. Callard, P. C. Chaumet, and A. Rahmani, “Coupled dipole method for radiation dynamics in finite photonic crystal structures,” Phys. Rev. E 73, 056,601 (2006). [CrossRef]

, 8

8. A. Rahmani, P. C. Chaumet, and G. W. Bryant, “Discrete dipole approximation for the study of radiation dynamics in a magnetodielectric environment,” Opt. Express 18, 8499–8504 (2010). [CrossRef] [PubMed]

], optical forces [9

9. B. T. Draine and J. C. Weingartner, “Radiative Torques on Interstellar Grains: I. Superthermal Spinup,” Astrophys. J. 470, 551–565 (1996). [CrossRef]

, 10

10. A. G. Hoekstra, M. Frijlink, L. B. F. M. Waters, and P. M. A. Sloot, “Radiation forces in the discrete-dipole approximation,” J. Opt. Soc. Am. A 18, 1944–1953 (2001). [CrossRef]

, 11

11. P. C. Chaumet, A. Rahmani, and M. Nieto-Vesperinas, “Photonic force spectroscopy on metallic and absorbing nanoparticles,” Phys. Rev. B 71, 045,425 (2005). [CrossRef]

, 12

12. P. C. Chaumet, A. Rahmani, A. Sentenac, and G. W. Bryant, “Efficient computation of optical forces with the coupled dipole method,” Phys. Rev. E 72, 046,708–6 (2005). [CrossRef]

, 13

13. A. Rahmani and P. C. Chaumet, “Optical Trapping near a Photonic Crystal,” Opt. Express 14, 6353–6358 (2006). [CrossRef] [PubMed]

] and optical torques [14

14. P. C. Chaumet and C. Billaudeau, “Coupled dipole method to compute optical torque: Application to a micro-propeller,” J. Appl. Phys. 101, 023,106–6 (2007). [CrossRef]

]. A recent overview of the DDA can be found in Ref. [15

15. M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: An overview and recent developments,” J. Quant. Spect. Rad. Transf. 106, 558–589 (2007). [CrossRef]

].

In the standard formulation of the DDA the electromagnetic fields are computed in the frequency domain. While this is most appropriate for finding the steady state solution of an electromagnetic problem, this restriction means that one cannot exploit the advantages of the DDA (ability to consider arbitrary scatterers, spatial discretization limited to the scatterer and its immediate neighborhood) to explore transient phenomena. This is a particularly limiting factor for optomechanical studies where one might be interested in modelling the optical force experienced by a scatterer illuminated by a time-varying electromagnetic (EM) field.

Previously we developed a formulation of the DDA to study light scattering, in time domain, by a linear, dispersive, lossy objects [16

16. P. C. Chaumet, K. Belkebir, and A. Rahmani, “Coupled-dipole method in time domain,” Opt. Express 16, 20,157–20,165 (2008). [CrossRef]

]. More recently, we generalized the time-domain DDA to address optical forces in non-magnetic systems [17

17. P. C. Chaumet, K. Belkebir, and A. Rahmani, “Optical forces in time domain on arbitrary objects,” Phys. Rev. A (2010). [CrossRef]

]. In this article we describe how these two aspects (time varying fields and optical forces on magnetodielectric scatterers) can be combined in a more general formulation of the DDA, that can be used to study optical forces on arbitrary three-dimensional magnetodielectric object with material dispersion and/or losses, in time domain.

2. DDA for optical force in time domain

3. Advantages of the DDA for the computation of optical forces in time domain

Traditionally, the finite difference time domain (FDTD) method has been used to study electromagnetic scattering problems in time domain [28

28. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propagat. 14, 302–307 (1969).

, 29

29. A. Taflove and M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell’s equations,” IEEE Transactions on Microwave Theory and Techniques 23, 623–630 (1975). [CrossRef]

, 30

30. A. Taflove, “Application of the finite-difference time-domain method to sinusoidal steady state electromagnetic penetration problems,” IEEE Trans. Antennas Propagat. 22, 191–202 (1975).

]. In the FDTD, the differential forms of the time-dependent Maxwell equations are discretized in both space and time. As a result, the entire computational window must be discretized and not just the scatterer. Moreover, in the FDTD, boundary conditions at the edges of the computational window must be handled carefully (usually using perfectly matched layer techniques). Furthermore, numerical dispersion and/or instabilities may occur when the fields are propagated over large distances (large object compared to the wavelength).

By contrast, the DDA circumvent many of these issues. First, there is no computational window per se since only the scatterer is discretized. Once the fields inside the scatterer are computed, the fields anywhere outside can be computed in a straightforward way using a free-space susceptibility tensor that is known analytically. In the FDTD, the fields are computed either within the computational window or in the far-field limit using near-to-far-field transformations. Furthermore, since unlike the FDTD, the DDA is based on the integral form of Maxwell’s equations and, therefore, is “built” around the concept of field susceptibility tensor, certain type of geometries such as a semi-infinite substrate, or a multilayer stack can be handled analytically which is not possible with the FDTD.

Finally, let us mention that in the DDA one needs to solve linear systems of size 6N × 6N where N is the number of discretization subunits for the scatterers under study. Except for the smallest values of N these systems cannot be solved by direct matrix inversion, rather, one should use an efficient iterative scheme [26

26. P. C. Chaumet and A. Rahmani, “Efficient iterative solution of the discrete dipole approximation for mag neto-dielectric scatterers,” Opt. Lett. 34, 917–919 (2009). [CrossRef] [PubMed]

] with an appropriate initial estimate of the solution [16

16. P. C. Chaumet, K. Belkebir, and A. Rahmani, “Coupled-dipole method in time domain,” Opt. Express 16, 20,157–20,165 (2008). [CrossRef]

].

4. How many contributions to the time-dependant optical force?

5. Time-dependant optical force on a discretized object

5.1. Contributions to the force: single dipole versus discretized scatterer.

As we discussed, the foundation of the DDA is the description of the various scattering processes at the level of a single dipole (subunit). However, most of the time the underlying structure of the scatterer as a collection of dipoles must be viewed as a convenient numerical description as opposed to a true representation of the internal geometry of the scatterer. Because of this, the self-term in Eq. (3) will lose its significance when a finite object is discretized into a large collection of dipoles. To illustrate this point, consider a spherical scatterer made of a double-negative medium for which the material responses are described by lossy Drude models [33

33. R. W. Ziolkowski, “Pulsed and CW Gaussian beam interactions with double negative metamaterial slabs,” Opt. Express 11, 662–681 (2003). [CrossRef] [PubMed]

]. In the frequency domain, the permittivity and permeability are given by:
ε(ω)=1(ωpe)2ω(ω+iΓe),μ(ω)=1(ωpm)2ω(ω+iΓm)
(17)
where ωpe, ωpm, Γe and Γm denote the corresponding plasma and damping frequencies respectively. Let us start by assuming that the incident field is a plane wave with a Gaussian time envelop of the form:
(t)=exp[16(tττ)2]sin(2πf0t).
(18)
where f0 = ω0/2π = c/λ0 is the central frequency of the pulse τ = 8/f0 is the duration of the pulse. The spectral and time profiles of the incident pulse are plotted in Figs. 1(a) and 1(b). Let us first consider a sphere with radius a = λ0 discretized into N = 33552 subunits. The parameters for this computation are f0 chosen such that Re[ε(ω0)]=Re[μ(ω0)]= −1, i.e. ωpe=ωpm=w02, and damping terms Γe = Γm = ωpe/10. This corresponds to a sphere made of a lossy, negative-index material and exhibiting a resonance at frequency ω0, i.e. in the time-harmonic case the time average of the optical force would be maximum for ω = ω0. Notice that due to the large size of the sphere the resonance does not occur for Re[ε(ω0)]=Re[μ(ω0)]=−2 (plasmon resonance of a sphere much smaller than the wavelength) but is shifted toward Re[ε(ω0)]=Re[μ(ω0)]−1 (surface plasmon resonance). We plot in Fig. 2(a) the total optical force and its different contribution, as a function of time. Because ε(ω) = μ(ω), and the equivalence between the plasma frequencies and damping terms for the electric and magnetic parts of the material response, we have ℱpm(t) = ℱpe(t) and ℱhm(t) = ℱhe(t). Note that, in the present configuration, the oscillations of the total force are mainly due to the term associated with the Poynting vector. In Fig. 2(b) we can see that the contribution of this term to the transfer of momentum from the EM field to the object vanishes (as expected) at the end of the pulse and only the harmonic contributions to the force remain. The convergence of the method is illustrated in Fig. 2(c) which shows the momentum imparted by the EM wave to the object, versus the number of subunits. The computed value of the momentum only changes by 1% when the number of subunits is increased from N = 4224 to N = 113104.

Fig. 1 Spectral (a) and time (b) profiles of the incident field.
Fig. 2 (a) Total force (solid line) versus time and its different contribution, i.e.pm = ℱpe (dashed line), ℱhm = ℱhe (dot-dashed line). (b) Total momentum imparted to the object and its contribution associated. (c) Total momentum imparted to the object by the pulse versus the numbers of subunits to represent the sphere. (d) In solid line (dashed line) ℱrecoil with N = 33552 (N = 4224) subunits to represent the object. (e) Momentum imparted to the object due to ℱrecoil versus the time. (f) Momentum imparted to the object due to ℱrecoil versus the numbers of subunits to represent the sphere.

The main difference between the optical force on a single dipole and on one subunit of a discretized object comes from the recoil (self-interaction) of Eq. (3). The recoil (radiation reaction) force showed Fig. 2(d) is very weak compare to the other terms. In fact the magnitude of this term decreases as the number of subunit N increases as illustrated by the two curves plotted for different values of N, Figs. 2(d) and 2(e) for the force and the momentum respectively. This is due to the fact that the radiation reaction force for each subunit scales as the volume of the subunit squared (cross product of the electric and magnetic dipole moments), hence when N, the number of subunits, increases the contribution to the total force on the object of this recoil term (summed over all the subunits) decreases like 1/N as illustrated in Fig. 2(f). Note that this term would vanish in the limit where N tends to infinity, however, its contribution to the force would then be taken into account through the other contributions of the force and the multiple scattering between the subunits. In other words, the separation of the total optical force into 5 terms, while helpful in understanding the origin of the force experienced by a small particle, is somewhat artificial in the case of a discretized object. As a result, as the number of discretized subunits tend to infinity, the various contributions of the total forces can be grouped into four terms instead of the five terms, which is consistent with the expression for the generalized Lorentz force derived by Mansirupur [22

22. M. Mansuripur and A. R. Zakharian, “Maxwell’s macroscopic equations, the energy-momentum postulates, and the Lorentz law of force,” Phys. Rev. E 79, 026,608–10 (2009). [CrossRef]

].

However, we emphasize again that if one is interested in calculating the optical forces on a single, or a collection of particles treated in the dipole approximation it is essential to take into account the recoil term explicitly. This was illustrated in Sect. 4, where in Eq. (16) the recoil term gives the term involving the product of the electric and magnetic polarizabilities which is of the same magnitude as the other two terms in the square brackets.

5.2. Influence of losses and plasmon resonances

If a small sphere is illuminated with a plane wave at frequency f0, the spectrum of the optical force would exhibit peaks at two frequencies: the zero frequency and 2f0. This can be seen in Eq. (16) where there is a term cos2(kzωt) in the expression of the total force. Accordingly, for an homogeneous sphere with no dispersion and an illumination given by Eq. (18) we observe two peaks: one at zero frequency and the second at 2f0. As showed in Fig. 3(b), an increase in absorption (damping term) produces a slight redshift of the maximum around 2f0, and a decrease of the magnitude of the two peaks, confirmed by Fig. 3(a) where the total momentum imparted to the object is weaker for higher absorption. This decrease of the momentum transfer with material losses is due to the fact that an increase of the damping term weakens the resonance.

Fig. 3 (a) Total momentum imparted to the object and (b) spectrum of the force, for different values of the damping term Γ.

5.3. Influence of the value of the plasma frequency

As the resonance of the sphere is around ε = μ = −1 when ωp = ωpe = ωpm go far 2ω0 the total momentum imparted to the sphere decrease particularly when ωp is shifted toward the low frequency, Fig. 4(a), as for the low frequencies (ε,μ) are close to one. If we compare the ration between the maximum at the frequency 2 f0 and the maximum at the null frequency is higher when the central frequency of the pulse correspond to the resonance of the sphere showing that the observation of oscillation of the force with the frequency of the pulse should be done at a resonance.

Fig. 4 (a) Total momentum imparted to the object and (b) spectrum of the force, for different of ωp = ωpe = ωpm.

6. Conclusion

In conclusion we have presented a general framework based on the discrete dipole approximation (DDA) for the computation of optical force on arbitrary magnetodielectric, three-dimensional objects in time domain. The principal advantage of the method is that only the scatterer and its immediate neighborhood need to be discretized, allowing analytic expressions of the incident fields to be used. From its DDA foundation our approach inherits the ability to handle any material with a linear response, including dispersive, anisotropic and/or lossy magnetodielectric scatterers.

References and links

1.

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

2.

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988). [CrossRef]

3.

P. C. Chaumet, A. Rahmani, and G. W. Bryant, “Generalization of the coupled dipole method to periodic structure,” Phys. Rev. B 67, 165,404–5 (2003). [CrossRef]

4.

P. C. Chaumet and A. Sentenac, “Numerical simulations of the electromagnetic field scattered by defects in a double-periodic structure,” Phys. Rev. B 72, 205,437–8 (2005). [CrossRef]

5.

A. Rahmani, P. C. Chaumet, and F. de Fornel, “Enrironment-induced modification of spontaneous emission: Single-molecule near-field probe,” Phys. Rev. A 63, 023,819–11 (2001). [CrossRef]

6.

A. Rahmani and G. W. Bryant, “Spontaneous emission in microcavity electrodynamics,” Phys. Rev. A 65, 033,817–12 (2002). [CrossRef]

7.

F. Bordas, N. Louvion, S. Callard, P. C. Chaumet, and A. Rahmani, “Coupled dipole method for radiation dynamics in finite photonic crystal structures,” Phys. Rev. E 73, 056,601 (2006). [CrossRef]

8.

A. Rahmani, P. C. Chaumet, and G. W. Bryant, “Discrete dipole approximation for the study of radiation dynamics in a magnetodielectric environment,” Opt. Express 18, 8499–8504 (2010). [CrossRef] [PubMed]

9.

B. T. Draine and J. C. Weingartner, “Radiative Torques on Interstellar Grains: I. Superthermal Spinup,” Astrophys. J. 470, 551–565 (1996). [CrossRef]

10.

A. G. Hoekstra, M. Frijlink, L. B. F. M. Waters, and P. M. A. Sloot, “Radiation forces in the discrete-dipole approximation,” J. Opt. Soc. Am. A 18, 1944–1953 (2001). [CrossRef]

11.

P. C. Chaumet, A. Rahmani, and M. Nieto-Vesperinas, “Photonic force spectroscopy on metallic and absorbing nanoparticles,” Phys. Rev. B 71, 045,425 (2005). [CrossRef]

12.

P. C. Chaumet, A. Rahmani, A. Sentenac, and G. W. Bryant, “Efficient computation of optical forces with the coupled dipole method,” Phys. Rev. E 72, 046,708–6 (2005). [CrossRef]

13.

A. Rahmani and P. C. Chaumet, “Optical Trapping near a Photonic Crystal,” Opt. Express 14, 6353–6358 (2006). [CrossRef] [PubMed]

14.

P. C. Chaumet and C. Billaudeau, “Coupled dipole method to compute optical torque: Application to a micro-propeller,” J. Appl. Phys. 101, 023,106–6 (2007). [CrossRef]

15.

M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: An overview and recent developments,” J. Quant. Spect. Rad. Transf. 106, 558–589 (2007). [CrossRef]

16.

P. C. Chaumet, K. Belkebir, and A. Rahmani, “Coupled-dipole method in time domain,” Opt. Express 16, 20,157–20,165 (2008). [CrossRef]

17.

P. C. Chaumet, K. Belkebir, and A. Rahmani, “Optical forces in time domain on arbitrary objects,” Phys. Rev. A (2010). [CrossRef]

18.

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

19.

B. D. H. Tellegen, “Magnetic-Dipole models,” Am. J. Phys. 30, 650–652 (1962). [CrossRef]

20.

L. Vaidman, “Torque and force on a magnetic dipole,” Am. J. Phys. 58, 978–983 (1990). [CrossRef]

21.

M. Mansuripur, “Radiation pressure and the linear momentum of the electromagnetic field in magnetic media,” Opt. Express 15, 13,502–13,518 (2007). [CrossRef]

22.

M. Mansuripur and A. R. Zakharian, “Maxwell’s macroscopic equations, the energy-momentum postulates, and the Lorentz law of force,” Phys. Rev. E 79, 026,608–10 (2009). [CrossRef]

23.

E. E. Radescu and G. Vaman, “Exact calculation of the angular momentum loss, recoil force, and radiation intensity for an arbitrary source in terms of electric, magnetic, and toroid multipoles,” Phys. Rev. A 65, 046,609–47 (2002).

24.

E. E. Radescu and G. Vaman, “Toroid moments in the momentum and angular momentum loss by a radiating arbitrary source,” Phys. Rev. A 65, 035,601–3 (2002).

25.

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

26.

P. C. Chaumet and A. Rahmani, “Efficient iterative solution of the discrete dipole approximation for mag neto-dielectric scatterers,” Opt. Lett. 34, 917–919 (2009). [CrossRef] [PubMed]

27.

P. C. Chaumet and A. Rahmani, “Coupled-dipole method for magnetic and negative refraction materials,” J. Quant. Spect. Rad. Transf. 110, 22–29 (2009). [CrossRef]

28.

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propagat. 14, 302–307 (1969).

29.

A. Taflove and M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell’s equations,” IEEE Transactions on Microwave Theory and Techniques 23, 623–630 (1975). [CrossRef]

30.

A. Taflove, “Application of the finite-difference time-domain method to sinusoidal steady state electromagnetic penetration problems,” IEEE Trans. Antennas Propagat. 22, 191–202 (1975).

31.

P. C. Chaumet, “Comment on “Trapping force, force constant, and potential depths for dielectric spheres in the presence of spherical aberrations”,” Appl. Opt. 43, 1825–1826 (2004). [CrossRef] [PubMed]

32.

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

33.

R. W. Ziolkowski, “Pulsed and CW Gaussian beam interactions with double negative metamaterial slabs,” Opt. Express 11, 662–681 (2003). [CrossRef] [PubMed]

OCIS Codes
(290.5850) Scattering : Scattering, particles
(350.4855) Other areas of optics : Optical tweezers or optical manipulation

ToC Category:
Optical Trapping and Manipulation

History
Original Manuscript: June 3, 2010
Revised Manuscript: September 13, 2010
Manuscript Accepted: October 4, 2010
Published: January 25, 2011

Virtual Issues
Vol. 6, Iss. 2 Virtual Journal for Biomedical Optics

Citation
Patrick C. Chaumet, Kamal Belkebir, and Adel Rahmani, "Discrete dipole approximation for time-domain computation of optical forces on magnetodielectric scatterers," Opt. Express 19, 2466-2475 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-3-2466


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References

  1. E. M. Purcell and C. R. Pennypacker, "Scattering and absorption of light by nonspherical dielectric grains," Astrophys. J. 186, 705-714 (1973). [CrossRef]
  2. B. T. Draine, "The discrete-dipole approximation and its application to interstellar graphite grains," Astrophys. J. 333, 848-872 (1988). [CrossRef]
  3. P. C. Chaumet, A. Rahmani, and G. W. Bryant, "Generalization of the coupled dipole method to periodic structure," Phys. Rev. B 67(165), 404-405 (2003). [CrossRef]
  4. P. C. Chaumet and A. Sentenac, "Numerical simulations of the electromagnetic field scattered by defects in a double-periodic structure," Phys. Rev. B 72(205), 437-438 (2005). [CrossRef]
  5. A. Rahmani, P. C. Chaumet, and F. de Fornel, "Environment-induced modification of spontaneous emission: Single-molecule near-field probe," Phys. Rev. A 63(023), 819-11 (2001). [CrossRef]
  6. A. Rahmani and G. W. Bryant, "Spontaneous emission in microcavity electrodynamics," Phys. Rev. A 65(033), 817-12 (2002). [CrossRef]
  7. F. Bordas, N. Louvion, S. Callard, P. C. Chaumet, and A. Rahmani, "Coupled dipole method for radiation dynamics in finite photonic crystal structures," Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(056), 601 (2006). [CrossRef]
  8. A. Rahmani, P. C. Chaumet, and G. W. Bryant, "Discrete dipole approximation for the study of radiation dynamics in a magneto dielectric environment," Opt. Express 18, 8499-8504 (2010). [CrossRef] [PubMed]
  9. B. T. Draine and J. C. Weingartner, "Radiative Torques on Interstellar Grains: I. Superthermal Spinup," Astrophys. J. 470, 551-565 (1996). [CrossRef]
  10. A. G. Hoekstra, M. Frijlink, L. B. F. M. Waters, and P. M. A. Sloot, "Radiation forces in the discrete-dipole approximation," J. Opt. Soc. Am. A 18, 1944-1953 (2001). [CrossRef]
  11. P. C. Chaumet, A. Rahmani, and M. Nieto-Vesperinas, "Photonic force spectroscopy on metallic and absorbing nanoparticles," Phys. Rev. B 71(045), 425 (2005). [CrossRef]
  12. P. C. Chaumet, A. Rahmani, A. Sentenac, and G. W. Bryant, "Efficient computation of optical forces with the coupled dipole method," Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(046), 708-6 (2005). [CrossRef]
  13. A. Rahmani and P. C. Chaumet, "Optical Trapping near a Photonic Crystal," Opt. Express 14, 6353-6358 (2006). [CrossRef] [PubMed]
  14. P. C. Chaumet and C. Billaudeau, "Coupled dipole method to compute optical torque: Application to a micropropeller," J. Appl. Phys. 101(023), 106-6 (2007). [CrossRef]
  15. M. A. Yurkin and A. G. Hoekstra, "The discrete dipole approximation: An overview and recent developments," J. Quant. Spect. Rad. Transf. 106, 558-589 (2007). [CrossRef]
  16. P. C. Chaumet, K. Belkebir, and A. Rahmani, "Coupled-dipole method in time domain," Opt. Express 16, 20,157-20,165 (2008). [CrossRef]
  17. P. C. Chaumet, K. Belkebir, and A. Rahmani, "Optical forces in time domain on arbitrary objects," Phys. Rev. A (2010). [CrossRef]
  18. P. C. Chaumet, and M. Nieto-Vesperinas, "Time-averaged total force on a dipolar sphere in an electromagnetic field," Opt. Lett. 25, 1065-1067 (2000). [CrossRef]
  19. B. D. H. Tellegen, "Magnetic-Dipole models," Am. J. Phys. 30, 650-652 (1962). [CrossRef]
  20. L. Vaidman, "Torque and force on a magnetic dipole," Am. J. Phys. 58, 978-983 (1990). [CrossRef]
  21. M. Mansuripur, "Radiation pressure and the linear momentum of the electromagnetic field in magnetic media," Opt. Express 15, 13,502-13,518 (2007). [CrossRef]
  22. M. Mansuripur and A. R. Zakharian, "Maxwell’s macroscopic equations, the energy-momentum postulates, and the Lorentz law of force," Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(026), 608-610 (2009). [CrossRef]
  23. E. E. Radescu and G. Vaman, "Exact calculation of the angular momentum loss, recoil force, and radiation intensity for an arbitrary source in terms of electric, magnetic, and toroid multipoles," Phys. Rev. A 65(046), 609-647 (2002).
  24. E. E. Radescu and G. Vaman, "Toroid moments in the momentum and angular momentum loss by a radiating arbitrary source," Phys. Rev. A 65(035), 601-603 (2002).
  25. P. C. Chaumet and A. Rahmani, "Electromagnetic force and torque on magnetic and negative-index scatterers," Opt. Express 17, 2224-2234 (2009). [CrossRef] [PubMed]
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