## Discrete dipole approximation for time-domain computation of optical forces on magnetodielectric scatterers |

Optics Express, Vol. 19, Issue 3, pp. 2466-2475 (2011)

http://dx.doi.org/10.1364/OE.19.002466

Acrobat PDF (856 KB)

### 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

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]

*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]

*magnetodielectric*object with material dispersion and/or losses,

*in time domain*.

## 2. DDA for optical force in time domain

*t*) (Fourier transform

*I*(

*ω*) = 𝒢[ℐ(

*t*)]), and a spectrum centered at frequency

*f*

_{0}. The general aspects of the time-domain formulation of the DDA have been detailed in a previous article, therefore only a brief description will be given here [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]

*M*values of the frequency with the incident field

*I*(

*ω*)

**E**(

*ω*), to find the value the electric dipole and magnetic dipole at each subunits location for the time harmonic problems [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]

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]

**r**,

*t*) = 𝒢

^{−1}[

*I*(

*ω*)

**E**(

**r**,

*ω*)] [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]

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

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]

*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.

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]

*N*× 6

*N*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]

**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.*

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

*ω*

^{pe},

*ω*

^{pm}, Γ

^{e}and Γ

^{m}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: where

*f*

_{0}=

*ω*

_{0}/2

*π*=

*c*/

*λ*

_{0}is the central frequency of the pulse

*τ*= 8/

*f*

_{0}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*=

*λ*

_{0}discretized into

*N*= 33552 subunits. The parameters for this computation are

*f*

_{0}chosen such that Re[

*ε*(

*ω*

_{0})]=Re[

*μ*(

*ω*

_{0})]= −1,

*i.e.*

^{e}= Γ

^{m}=

*ω*

^{pe}/10. This corresponds to a sphere made of a lossy, negative-index material and exhibiting a resonance at frequency

*ω*

_{0},

*i.e.*in the time-harmonic case the time average of the optical force would be maximum for

*ω*=

*ω*

_{0}. Notice that due to the large size of the sphere the resonance does not occur for Re[

*ε*(

*ω*

_{0})]=Re[

*μ*(

*ω*

_{0})]=−2 (plasmon resonance of a sphere much smaller than the wavelength) but is shifted toward Re[

*ε*(

*ω*

_{0})]=Re[

*μ*(

*ω*

_{0})]−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 ℱ

^{pm}(

*t*) = ℱ

^{pe}(

*t*) and ℱ

^{hm}(

*t*) = ℱ

^{he}(

*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.

*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]

### 5.2. Influence of losses and plasmon resonances

*f*

_{0}, the spectrum of the optical force would exhibit peaks at two frequencies: the zero frequency and 2

*f*

_{0}. This can be seen in Eq. (16) where there is a term cos

^{2}(

*kz*–

*ω*

*t*) 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 2

*f*

_{0}. As showed in Fig. 3(b), an increase in absorption (damping term) produces a slight redshift of the maximum around 2

*f*

_{0}, 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.

### 5.3. Influence of the value of the plasma frequency

*ε*=

*μ*= −1 when

*ω*=

^{p}*ω*=

^{pe}*ω*go far

^{pm}*ω*is shifted toward the low frequency, Fig. 4(a), as for the low frequencies (

^{p}*ε*,

*μ*) are close to one. If we compare the ration between the maximum at the frequency 2

*f*

_{0}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.

## 6. Conclusion

## References and links

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

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

3. | P. C. Chaumet, A. Rahmani, and G. W. Bryant, “Generalization of the coupled dipole method to periodic structure,” Phys. Rev. B |

4. | P. C. Chaumet and A. Sentenac, “Numerical simulations of the electromagnetic field scattered by defects in a double-periodic structure,” Phys. Rev. B |

5. | A. Rahmani, P. C. Chaumet, and F. de Fornel, “Enrironment-induced modification of spontaneous emission: Single-molecule near-field probe,” Phys. Rev. A |

6. | A. Rahmani and G. W. Bryant, “Spontaneous emission in microcavity electrodynamics,” Phys. Rev. A |

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 |

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 |

9. | B. T. Draine and J. C. Weingartner, “Radiative Torques on Interstellar Grains: I. Superthermal Spinup,” Astrophys. J. |

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 |

11. | P. C. Chaumet, A. Rahmani, and M. Nieto-Vesperinas, “Photonic force spectroscopy on metallic and absorbing nanoparticles,” Phys. Rev. B |

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 |

13. | A. Rahmani and P. C. Chaumet, “Optical Trapping near a Photonic Crystal,” Opt. Express |

14. | P. C. Chaumet and C. Billaudeau, “Coupled dipole method to compute optical torque: Application to a micro-propeller,” J. Appl. Phys. |

15. | M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: An overview and recent developments,” J. Quant. Spect. Rad. Transf. |

16. | P. C. Chaumet, K. Belkebir, and A. Rahmani, “Coupled-dipole method in time domain,” Opt. Express |

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. |

19. | B. D. H. Tellegen, “Magnetic-Dipole models,” Am. J. Phys. |

20. | L. Vaidman, “Torque and force on a magnetic dipole,” Am. J. Phys. |

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

22. | M. Mansuripur and A. R. Zakharian, “Maxwell’s macroscopic equations, the energy-momentum postulates, and the Lorentz law of force,” Phys. Rev. E |

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 |

24. | E. E. Radescu and G. Vaman, “Toroid moments in the momentum and angular momentum loss by a radiating arbitrary source,” Phys. Rev. A |

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

26. | P. C. Chaumet and A. Rahmani, “Efficient iterative solution of the discrete dipole approximation for mag neto-dielectric scatterers,” Opt. Lett. |

27. | P. C. Chaumet and A. Rahmani, “Coupled-dipole method for magnetic and negative refraction materials,” J. Quant. Spect. Rad. Transf. |

28. | K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propagat. |

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 |

30. | A. Taflove, “Application of the finite-difference time-domain method to sinusoidal steady state electromagnetic penetration problems,” IEEE Trans. Antennas Propagat. |

31. | P. C. Chaumet, “Comment on “Trapping force, force constant, and potential depths for dielectric spheres in the presence of spherical aberrations”,” Appl. Opt. |

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 |

33. | R. W. Ziolkowski, “Pulsed and CW Gaussian beam interactions with double negative metamaterial slabs,” Opt. Express |

**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

- 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, "The discrete-dipole approximation and its application to interstellar graphite grains," Astrophys. J. 333, 848-872 (1988). [CrossRef]
- 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]
- 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]
- 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]
- A. Rahmani and G. W. Bryant, "Spontaneous emission in microcavity electrodynamics," Phys. Rev. A 65(033), 817-12 (2002). [CrossRef]
- 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]
- 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]
- B. T. Draine and J. C. Weingartner, "Radiative Torques on Interstellar Grains: I. Superthermal Spinup," Astrophys. J. 470, 551-565 (1996). [CrossRef]
- 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]
- 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]
- 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]
- A. Rahmani and P. C. Chaumet, "Optical Trapping near a Photonic Crystal," Opt. Express 14, 6353-6358 (2006). [CrossRef] [PubMed]
- 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]
- 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]
- P. C. Chaumet, K. Belkebir, and A. Rahmani, "Coupled-dipole method in time domain," Opt. Express 16, 20,157-20,165 (2008). [CrossRef]
- P. C. Chaumet, K. Belkebir, and A. Rahmani, "Optical forces in time domain on arbitrary objects," Phys. Rev. A (2010). [CrossRef]
- P. C. Chaumet, and M. Nieto-Vesperinas, "Time-averaged total force on a dipolar sphere in an electromagnetic field," Opt. Lett. 25, 1065-1067 (2000). [CrossRef]
- B. D. H. Tellegen, "Magnetic-Dipole models," Am. J. Phys. 30, 650-652 (1962). [CrossRef]
- L. Vaidman, "Torque and force on a magnetic dipole," Am. J. Phys. 58, 978-983 (1990). [CrossRef]
- M. Mansuripur, "Radiation pressure and the linear momentum of the electromagnetic field in magnetic media," Opt. Express 15, 13,502-13,518 (2007). [CrossRef]
- 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]
- 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).
- 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).
- P. C. Chaumet and A. Rahmani, "Electromagnetic force and torque on magnetic and negative-index scatterers," Opt. Express 17, 2224-2234 (2009). [CrossRef] [PubMed]
- P. C. Chaumet and A. Rahmani, "Efficient iterative solution of the discrete dipole approximation for magneto dielectric scatterers," Opt. Lett. 34, 917-919 (2009). [CrossRef] [PubMed]
- 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]
- K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media," IEEE Trans. Antenn. Propag. 14, 302-307 (1969).
- A. Taflove and M. E. Brodwin, "Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell’s equations," IEEE Trans. Microw. Theory Tech. 23, 623-630 (1975). [CrossRef]
- A. Taflove, "Application of the finite-difference time-domain method to sinusoidal steady state electromagnetic penetration problems," IEEE Trans. Antenn. Propag. 22, 191-202 (1975).
- 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]
- 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]
- R. W. Ziolkowski, "Pulsed and CW Gaussian beam interactions with double negative metamaterial slabs," Opt. Express 11, 662-681 (2003). [CrossRef] [PubMed]

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