## Are optical forces derived from a scalar potential?

Optics Express, Vol. 15, Issue 15, pp. 9817-9830 (2007)

http://dx.doi.org/10.1364/OE.15.009817

Acrobat PDF (268 KB)

### Abstract

The expression of optical forces provoked by an incident light illuminating particles can be deduced from the Lorentz law. It is shown that these forces derive from a scalar potential in the 2D problem and s-polarization, with light propagating in the cross-section plane of the particles, a fact which shows that the separation between gradient and scattering forces could be questioned. This property does not extend to the p-polarization and 3D problem. In the general case, it is shown that one of the components of the optical force is intimately linked with the reactive energy inside the particle. A possible application is given.

© 2007 Optical Society of America

## 1. Introduction

## 2-Expression of the optical force.

*iωt*). The 3D non-magnetic particles of index

*ν*are placed in vacuum. These particles are illuminated by an incident light of wavelength

_{r}*λ*. We call

_{0}=2π/k_{0}*ε*the permittivity of the particles,

_{r}=ε_{0}ν^{2}_{r}*µ*being the permeability of both media.

_{0}*∂*Ω, with a normal vector

**n**oriented towards the exterior is the consequence of the existence of a polarization vector

**P**inside the dielectric, where no free current or free charge can exist. However, it is equivalent to represent the effects of the polarization vector in terms of bound currents and charges [23

23. M. Mansuripur, “Radiation pressure and the linear momentum of the electromagnetic field,” Opt. Express **12**, 5375–5401 (2004). [CrossRef] [PubMed]

26. B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, “Optical momentum transfer to absorbing Mie particles,” Phys. Rev. Lett. **97**, 133902 (2006). [CrossRef] [PubMed]

*ε*of space coordinates(

*x,y,z*), with

*ε(x,y,z)*=

*ε*outside Ω. One can imagine for example that Ω is divided in two regions: an homogeneous internal region where the permittivity is equal to

_{0}*ε*and a transition region of width

_{r}*η*, close to the boundary ∂Ω, where the permittivity goes continuously from

*ε*to

_{r}*ε*. The case of homogeneous particles in vacuum is obtained by imposing

_{0}*η*to tend to zero. The optical force can be deduced from Maxwell stress tensor [28–30]. Here, we achieve a direct and rigorous derivation from the Lorentz law.

### 2.1 The optical force in a continuous medium.

**j**placed in vacuum satisfies the equation:

*ε*

**E**)=∇.

**D**=0 it comes out that the bound volume charge density is given by:

**f**

_{E}, viz. the force generated by the electric field on the bound charges, from the magnetic forces

**f**

*, viz. those generated by the magnetic field on the bound currents.*

_{M}### 2.2 The optical force on homogeneous particles in vacuum.

**E**=0. As a consequence, the electric force vanishes outside the transition region and tends to a distribution when

*η*tends to zero [23

23. M. Mansuripur, “Radiation pressure and the linear momentum of the electromagnetic field,” Opt. Express **12**, 5375–5401 (2004). [CrossRef] [PubMed]

26. B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, “Optical momentum transfer to absorbing Mie particles,” Phys. Rev. Lett. **97**, 133902 (2006). [CrossRef] [PubMed]

**E**whose support lies on the boundary of the particle by the vector

**E*** which has a normal component which is discontinuous on the same boundary. In order to remove this ambiguity, we separate the tangential and normal fields in the transition region:

*η*tends to zero. Inserting the right-hand member of Eq. (11) in Eq. (9) yields:

*η*tends to zero. Using the theory of distributions, we can write that at the limit:

*uδ*defined (incorrectly from a mathematical point of view but conveniently for the physicist) by:

_{∂Ω}*dV*a volume element of space,

*dS*a surface element around M, and [

**E**] the jump of the electric field on ∂Ω in the direction of the normal. From Eqs. (13) and (14), we deduce that the total contribution of the electric tangential forces on ∂Ω is given by:

*η*tends to zero, ∇.(

**E**

*) remains a function and since this function is summed on a volume which tends to zero, its contribution to the force tends to zero as well, which entails:*

_{t}*η*tends to zero, let us re-write this equation in the form:

*a*∇.V+∇

*a*.V and bearing in mind that

*εE*remains continuous as

_{n}*η*tends to zero, thus that its divergence tends to a function, we can write:

*η*tends to zero,

*εE*and its conjugate remain continuous and the gradient tends to a distribution, in such a way that the right hand member of Eq. (21) is no more ambiguous, and since at the limit:

_{n}*E*〉 stands for the mean value of the normals of the electric fields on both sides of the surface. Remarking that

_{n}*ε*(

_{0}*E*) is the surface charge density, it comes out that the force is given by:

^{out}_{n}-E^{in}_{n}23. M. Mansuripur, “Radiation pressure and the linear momentum of the electromagnetic field,” Opt. Express **12**, 5375–5401 (2004). [CrossRef] [PubMed]

26. B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, “Optical momentum transfer to absorbing Mie particles,” Phys. Rev. Lett. **97**, 133902 (2006). [CrossRef] [PubMed]

## 3. Expression of the optical force in the 2D problem

### 3.1 Calculation of the optical force in the 2D problem.

*z*axis of a cartesian coordinates system

*xyz*. In order to keep the same notations as in the last section, Ω and ∂Ω will denote the cross section of a cylinder and its boundary respectively,

*dV*and

*dS*becoming a small surface element in Ω and a small curvilinear element on ∂Ω respectively. In these conditions, Eq. (28) holds for the force exerted by the incident field on a unit length of the cylinder. It must be noticed that the force exerted along the

*z*axis is contained in the first and second terms of the right-hand member of Eq. (28).

*z*components of the electric and magnetic fields. Thus we separate the

*xy*(transverse) components

**E**

*xy*and

**H**

*xy*from the

*z*(longitudinal) components

*E*

_{z}ẑ and

*H*ẑ of the fields. It can be shown easily from Maxwell equations and vector differential relationships that inside Ω :

_{z}**f**

*can also be written in the form:*

_{xy}**f**

*=-*

_{xy}*∇U*d

*V*applied inside Ω.

*H*or

_{z}*E*vanishes, thus the first term in the right-hand member of Eq. (34) vanishes. This conclusion holds for the second term since in

_{z}*E*and

^{in}_{n}*E**are equal to zero for s and p-polarization respectively, thus

_{z}*f*=0. In these two fundamental cases of polarization, the total optical force reduces to

_{z}**f**

*. Moreover, for s-polarization, the normal component of the electric field*

_{xy}*E*=0 and thus the total optical force derives from the potential

^{in}_{n}*U*is continuous across ∂Ω (

*E*and its normal derivative are continuous), one can state that a particle moves towards the spots of electric power density outside Ω. For p-polarization, the magnetic component of the force (first term in the right-hand member of Eq. (32)) derives from the potential

_{z}*U*is not continuous across Ω since the normal derivative of

*H*is not continuous but the normal derivatives inside and outside the particle are proportional and thus one can state again that this component of the force attracts a particle towards regions of small magnetic power density. Due to the existence of two additive surface integrals (electric force) in Eq. (32), it is not possible to state a simple rule for the motion of particles for that polarization.

_{z}*z*component of the force is the sum of two terms, a volume term of magnetic origin and a surface term of electric origin. Remembering the symmetry with respect to the

*xy*plane of the structure, this result seems surprising. In fact, it should be noticed that the incident field is not symmetrical with respect to the

*xy*plane. A similar transverse effect can be found in [31

31. Ch. Imbert, “Calculation and Experimental Proof of the Transverse Shift Induced by Total Internal Reflection of a Circularly Polarized Light Beam,” Phys. Rev. D **5**, 787–796 (1972). [CrossRef]

*z*component of the force does not vanish in our problem, but shows that this result is not contradictory with the laws of Physics. We have not been able to show theoretically that the two terms representing this force cancel each other, nor the opposite. Moreover, we have not at hand numerical tools for computing both terms. Thus, we have conjectured

*a priori*that

*f*≠0.

_{z}### 3.2 About gradient and scattering optical forces.

**97**, 133902 (2006). [CrossRef] [PubMed]

*xz*axis and illuminated in normal incidence, the direction of incidence being thus the y axis (Fig. 2). Due to the transfer of momentum the vertical optical force on the slab must be oriented in the same direction as the incident wavevector, i.e. towards

*y*=-∞, except in the particular case where all the light is transmitted by the slab. Thus from Eq. (35), the value of the potential at the bottom of the slab (shadow side of the slab) must be smaller than that on the front of the slab (illuminated side). Since the problem is invariant by rotation of polarization, the value of

**E.E*** should be greater at the bottom of the slab than at the front. This elementary property can be verified by using the analytic formulae giving the amplitudes of the plane waves in the three different regions of space [33

33. D. Maystre, “Getting effective permittivity and permeability equal to -1 in 1D dielectric photonic crystals,” J. Mod. Opt. **53**, 1901–1917 (2006). [CrossRef]

*w*and index

*ν*placed in vacuum and illuminated in normal incidence by a plane wave of unit amplitude, with the electric field parallel to the

_{r}*z*axis. The system of cartesian coordinates is chosen in such a way that the

*xz*plane coincides with the bottom of the slab.

*a*and thus the amplitude

*b*of the reflected plane waves is given by

*ρ*being the reflection coefficient given by Fresnel formula:

*z*axis,

**E**=

*E*and the value of

_{z}ẑ*E*inside the slab is given by:

_{z}*ν*>1, the coefficient

_{r}**E**|

^{2}is reached when exp(+2

*ik*)=1, or equivalently:

_{0}ν_{r}y*p*=0,1,2,…. In that case, the square modulus at the top and at the bottom of the slab are equal, and it can be shown that the reflected field vanishes, which explains why the optical forces generated by scattering vanish too.

*ν*=3 and radius

_{r}*R*=0.3

*µm*illuminated in the external medium of index

*ν*=1.3 by a s (resp. p) polarized plane wave of unit amplitude and wavelength

*λ*=0.546

_{0}*µm*in vacuum. In that case, the index of the external medium is not equal to unity but the conclusions of the present study remain identical.

*y*=-∞. Thus, from Eqs. (32) and (33), the integral

**J**given by:

*y*=-∞ for s-polarization. Taking into account the symmetry of the problem,

**J**is parallel to the y axis, thus, after projecting

**J**on the y axis and simplification, it emerges that the projection

*J*given by:

*J*=3.9 10

^{-19}in the International System of Units (SI). This property is explained by the fact that the largest values of the electric field are obtained in the shadow side (

*θ*close to 0) while those of the magnetic field are reached on the illuminated side (

*θ*close to 180°). For large values of the radius, this result seems to be linked to that obtained for the parallel slab, but it is worth to notice that it holds in the resonance domain. Even though the optical force does not derive from a potential for p-polarization, we have found

*J*=-4.3 10

^{-19}(SI) in that case by replacing

*E*by

_{z}*H*. However, this result does not allow one to believe that the potential term is the predominant one in the expression of the force. It would be interesting to evaluate numerically the importance of the gradient and surface terms in that case.

_{z}### 3.3 About the optical force in the 3D problem.

**P**is linked to the time average of the difference between the total magnetic and electric powers inside Ω, generally denoted as reactive energy:

### 3.4 A possible application of optical forces on 2D particles.

*E*of the optical force, which vanishes for s and p-polarizations. Let us suppose that a particle obtained by truncating a circular cylinder is allowed to move along the axis of the cylinder only and that this particle is illuminated by an unpolarized incident light propagating in the cross-section of the cylinder and converging on the cylinder. We can imagine for example that the particle is maintained on the same axis by a mechanical means (insertion in a transparent micropipe). If a polarizer is placed on the incident beam, the longitudinal optical force will vanish for s or p polarized light. Now, rotating the polarizer from this equilibrium position will create a longitudinal optical force which can move the particle in the longitudinal direction on both sides, according to the sense of rotation. Bearing in mind that the direction of polarization of some kinds of polarizers can be changed very rapidly using for example quartz oscillations in MOEMS, such a device could be used in order to generate rapid liquid flows in microfluidics or rapid arbitrary translations of cylinders in micromechanics. It is worth noting that when the particle is placed in a background medium, a mechanical contribution to the force should be added [34

_{z}34. J. P. Gordon, “Radiation Forces and Momenta in Dielectric Media,” Phys. Rev. A **8**, 14–21 (1973). [CrossRef]

## 4. Conclusion

## Acknowledgements

## References and links

1. | A. Ashkin, “Acceleration and Trapping of Particles by Radiation Pressure,” Phys. Rev. Lett. |

2. | A. Ashkin, “Optical trapping and manipulation of neutral particles using lasers,” Proc. Natl. Acad. Sci. USA |

3. | M. Burns, J-M. Fournier, and J. Golovshenko, “Optical Binding,” Phys. Rev. Lett. |

4. | M. Burns, J-M. Fournier, and J. Golovshenko, “Lateral binding effect, due to particle’s optical interaction,” Science |

5. | J-M. Fournier, G. Boer, G. Delacrétaz, P. Jacquot, J. Rohner, and R. Salathé, “Building Optical Matter with Binding and Trapping Forces,” Proc. SPIE |

6. | W. Singer, M. Frick M, S. Bernet, and M. Ritsch-Marte, “Self-organized array of regularly spaced microbeads in a fiber-optical trap,” J. Opt. Soc. Am. B |

7. | S. Tatarkova, A. Carruthers, and K. Dholakia, “One-Dimensional Optically Bound Arrays of Microscopic Particles,” Phys. Rev. Lett. |

8. | N. Metzger, K. Dholakia, and E. Wright, “Observation of Bistability and Hysteresis in Optical Binding of Two Dielectric Spheres,” Phys. Rev. Lett. |

9. | C. Mellor and C. Bain Chem., “Array Formation in Evanescent Waves,” Phys. Chem. |

10. | N. Metzger, E. Wright, W. Sibbett, and K. Dholakia, “Visualization of optical binding of microparticles using a femtosecond fiber optical trap,” Opt. Express |

11. | P. Chaumet and M. Nieto-Vesperinas, “Time averaged total force on a dipolar sphere in an electromagnetic field,” Opt. Lett. |

12. | T. Grzegorczyk, B. Kemp, and J. Kong, “Trapping and binding of an arbitrary number of cylindrical particles in an in-plane electromagnetic field,” J. Opt. Soc. Am. A |

13. | T. Grzegorczyk, B. Kemp, and J. Kong, “Stable optical trapping based on optical binding forces, “Phys. Rev. Lett. |

14. | M. Povinelli, S. Johnson, M. Lonèar, M. Ibanescu, E. Smythe, F. Capasso, and J. Joannopoulos, “High-Q enhancement of attractive and repulsive optical forces between coupled whispering-gallery- mode resonators,” Opt. Express |

15. | D. McGloin, A. Carruthers, K. Dholakia, and E. Wright, “Optically bound microscopic particles in one. dimension,” Phys. Rev. E |

16. | A. Rohrbach and E. Stelzer, “Trapping forces and potentials of dielectric spheres in the presence of spherical aberrations,” J. Opt. Soc. Am. A |

17. | E. Lidorikis, Q. Li, and C. Soukoulis, “Optical Bistability in Colloidal Crystals,” Phys. Rev. E |

18. | M. Antonoyiannakis and J. Pendry, “Electromagnetic forces in photonic crystals,” Phys. Rev. B |

19. | J. Ng, C. Chan, Z. Sheng, and Lin, “Strong optical force induced by morphology-dependent resonances,” Opt. Lett. |

20. | D. Maystre and P. Vincent, “Making photonic crystals using trapping and binding optical forces on particles,” J. Opt. A: Pure Appl. Opt. |

21. | D. Maystre and P. Vincent, “Phenomenological study of binding in optically trapped photonic crystals,” submitted to the J. Opt. Soc. Am. A. |

22. | M. Tomasz, Jin Grzegorczyk, and Au Kong, “Analytical expression of the force due to multiple TM plane wave incidences on an infinite lossless dielectric circular cylinder of arbitrary size,” to be published in the J. Opt. Soc. Am. B. |

23. | M. Mansuripur, “Radiation pressure and the linear momentum of the electromagnetic field,” Opt. Express |

24. | A. R. Zakharian, M. Mansuripur, and J. V. Moloney, “Radiation pressure and the distribution of electromagnetic force in a dielectric media,” Opt. Express |

25. | B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, “Ab initio study of the radiation pressure on dielectric and magnetic media,” Opt. Express |

26. | B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, “Optical momentum transfer to absorbing Mie particles,” Phys. Rev. Lett. |

27. | L. Novotny and B. Hecht, “Principles of Nano-Optics,” (Cambridge University Press, Cambridge) (2006). |

28. | J D Jackson Classical Electrodynamics, 2nd edition (New-York-Wiley) (1975). |

29. | J.A. Kong, Maxwell Equations (EMW Publishing: Cambridge, MA) (2002). |

30. | J. Van Bladel Electromagnetic Fields (Mc Graw-Hill: New York) (1964). |

31. | Ch. Imbert, “Calculation and Experimental Proof of the Transverse Shift Induced by Total Internal Reflection of a Circularly Polarized Light Beam,” Phys. Rev. D |

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

33. | D. Maystre, “Getting effective permittivity and permeability equal to -1 in 1D dielectric photonic crystals,” J. Mod. Opt. |

34. | J. P. Gordon, “Radiation Forces and Momenta in Dielectric Media,” Phys. Rev. A |

35. | Y. N. Obukhov and F. W. Hehl, “Electromagnetic energy-momentum and forces in matter,” Phys. Lett. A |

36. | R. Loudon, “Theory of the radiation pressure on dielectric surfaces,” J. Mod. Opt. |

37. | R. Loudon, S. M. Barnett, and C. Baxter, “Radiation pressure and momentum transfer in dielectrics: the photon drag effect,” Phys. Rev. A |

38. | C. Raabe and D. G. Welsch, “Casimir force acting on magnetodielectric bodies embedded in media,” Phys. Rev. A |

39. | L. P. Pitaevskii, “Why and when the Minkowski’s stress tensor can be used in the problem of Casimir force acting on bodies embedded in media,” Cond-mat, 0505754 (2005). |

**OCIS Codes**

(030.1670) Coherence and statistical optics : Coherent optical effects

(140.7010) Lasers and laser optics : Laser trapping

(260.2110) Physical optics : Electromagnetic optics

(290.5850) Scattering : Scattering, particles

**ToC Category:**

Physical Optics

**History**

Original Manuscript: December 19, 2006

Revised Manuscript: February 22, 2007

Manuscript Accepted: April 2, 2007

Published: July 20, 2007

**Virtual Issues**

Vol. 2, Iss. 8 *Virtual Journal for Biomedical Optics*

**Citation**

Daniel Maystre and Patrick Vincent, "Are optical forces derived from a scalar potential?," Opt. Express **15**, 9817-9830 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-15-9817

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### References

- A. Ashkin,"Acceleration and Trapping of Particles by Radiation Pressure," Phys. Rev. Lett. 24, 156-159 (1970). [CrossRef]
- A. Ashkin,"Optical trapping and manipulation of neutral particles using lasers,"Proc. Natl. Acad. Sci. USA 94, 4853-4860 (1997). [CrossRef] [PubMed]
- M. Burns, J-M. Fournier and J. Golovshenko,"Optical Binding,"Phys. Rev. Lett. 63, 1233-1236 (1989). [CrossRef] [PubMed]
- M. Burns, J-M. Fournier and J. Golovshenko, "Lateral binding effect, due to particle's optical interaction,"Science 289, 749-754 (1990). [CrossRef]
- J-M. Fournier, G. Boer, G. Delacrétaz, P. Jacquot, J. Rohner and R. Salathé,"Building Optical Matter with Binding and Trapping Forces," Proc. SPIE 5514, 309-317 (2004). [CrossRef]
- W. Singer, M. Frick, S. Bernet and M. Ritsch-Marte,"Self-organized array of regularly spaced microbeads in a fiber-optical trap," J. Opt. Soc. Am. B 20, 1568-1574 (2003). [CrossRef]
- S. Tatarkova, A. Carruthers and K. Dholakia,"One-Dimensional Optically Bound Arrays of Microscopic Particles," Phys. Rev. Lett. 89, 283901 (2002). [CrossRef]
- N. Metzger, K. Dholakia and E. Wright,"Observation of Bistability and Hysteresis in Optical Binding of Two Dielectric Spheres," Phys. Rev. Lett. 96, 068102 (2006). [CrossRef] [PubMed]
- C. Mellor, C. Bain Chem., "Array Formation in Evanescent Waves," Phys. Chem. 7, 329-332 (2006). [CrossRef]
- N. Metzger, E. Wright, W. Sibbett and K. Dholakia," Visualization of optical binding of microparticles using a femtosecond fiber optical trap," Opt. Express 14, 3677-3687 (2006). [CrossRef] [PubMed]
- P. Chaumet and M. Nieto-Vesperinas,"Time averaged total force on a dipolar sphere in an electromagnetic field," Opt. Lett. 25, 1065-1067 (2000). [CrossRef]
- T. Grzegorczyk, B. Kemp, and J. Kong, "Trapping and binding of an arbitrary number of cylindrical particles in an in-plane electromagnetic field," J. Opt. Soc. Am. A 23, 2324-2330 (2006). [CrossRef]
- T. Grzegorczyk, B. Kemp, and J. Kong, "Stable optical trapping based on optical binding forces, "Phys. Rev. Lett. 96, 113903 (2006). [CrossRef] [PubMed]
- M. Povinelli, S. Johnson, M. Lonèar, M. Ibanescu, E. Smythe, F. Capasso and J. Joannopoulos,"High-Q enhancement of attractive and repulsive optical forces between coupled whispering-gallery- mode resonators,"Opt. Express 13, 8286-8295 (2005). [CrossRef] [PubMed]
- D. McGloin, A. Carruthers, K. Dholakia and E. Wright,"Optically bound microscopic particles in one. dimension," Phys. Rev. E 69,021403 (2004). [CrossRef]
- A. Rohrbach and E. Stelzer, "Trapping forces and potentials of dielectric spheres in the presence of spherical aberrations,"J. Opt. Soc. Am. A 18, 839-853 (2001). [CrossRef]
- E. Lidorikis, Q. Li and C. Soukoulis, "Optical Bistability in Colloidal Crystals," Phys. Rev. E 55, 3613-3618 (1997). [CrossRef]
- M. Antonoyiannakis and J. Pendry, "Electromagnetic forces in photonic crystals, "Phys. Rev. B 60, 2363-2374 (1999). [CrossRef]
- J. Ng, Chan, C. Sheng and Z. Lin, "Strong optical force induced by morphology-dependent resonances," Opt. Lett. 30, 1956-1958 (2005). [CrossRef] [PubMed]
- D. Maystre and P. Vincent, "Making photonic crystals using trapping and binding optical forces on particles," J. Opt. A: Pure Appl. Opt. 8, 1059-1066 (2006). [CrossRef]
- D. Maystre and P. Vincent, "Phenomenological study of binding in optically trapped photonic crystals," submitted to the J. Opt. Soc. Am. A.
- T. M. Grzegorczyk and Jin Au Kong, "Analytical expression of the force due to multiple TM plane wave incidences on an infinite lossless dielectric circular cylinder of arbitrary size," to be published in the J. Opt. Soc. Am. B.
- M. Mansuripur, "Radiation pressure and the linear momentum of the electromagnetic field," Opt. Express 12, 5375-5401 (2004). [CrossRef] [PubMed]
- A. R. Zakharian, M. Mansuripur, and J. V. Moloney, "Radiation pressure and the distribution of electromagnetic force in a dielectric media," Opt. Express 13, 2321-2336 (2005). [CrossRef] [PubMed]
- B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, "Ab initio study of the radiation pressure on dielectric and magnetic media," Opt. Express 13, 9280-9291 (2005). [CrossRef] [PubMed]
- B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, "Optical momentum transfer to absorbing Mie particles," Phys. Rev. Lett. 97, 133902 (2006). [CrossRef] [PubMed]
- L. Novotny and B. Hecht, "Principles of Nano-Optics," (Cambridge University Press, Cambridge) (2006).
- JacksonJ D Classical Electrodynamics, 2nd edition (New-York-Wiley) (1975).
- J.A. Kong, Maxwell Equations (EMW Publishing: Cambridge, MA) (2002).
- J. Van Bladel Electromagnetic Fields (Mc Graw-Hill: New York) (1964).
- Ch. Imbert, "Calculation and Experimental Proof of the Transverse Shift Induced by Total Internal Reflection of a Circularly Polarized Light Beam," Phys. Rev. D 5, 787 - 796 (1972). [CrossRef]
- B.T. Draine, "The discrete-dipole approximation and its application to interstellar graphite grains,"Astrophys. J. 333, 848-872 (1988). [CrossRef]
- D. Maystre, "Getting effective permittivity and permeability equal to −1 in 1D dielectric photonic crystals," J. Mod. Opt. 53, 1901-1917 (2006). [CrossRef]
- J. P. Gordon, "Radiation Forces and Momenta in Dielectric Media," Phys. Rev. A 8,14-21 (1973). [CrossRef]
- Y. N. Obukhov and F. W. Hehl, "Electromagnetic energy-momentum and forces in matter," Phys. Lett. A 311, 277-284 (2003). [CrossRef]
- R. Loudon, "Theory of the radiation pressure on dielectric surfaces," J. Mod. Opt. 49, 812-836 (2002). [CrossRef]
- R. Loudon, S. M. Barnett and C. Baxter, "Radiation pressure and momentum transfer in dielectrics: the photon drag effect," Phys. Rev. A 71, 063802 (2005). [CrossRef]
- C. Raabe and D. G. Welsch, "Casimir force acting on magnetodielectric bodies embedded in media," Phys. Rev. A 71, 013814 (2005). [CrossRef]
- L. P. Pitaevskii, "Why and when the Minkowski’s stress tensor can be used in the problem of Casimir force acting on bodies embedded in media," Cond-mat, 0505754 (2005).

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