## Theory of the radiation pressure on dielectric slabs, prisms and single surfaces

Optics Express, Vol. 14, Issue 24, pp. 11855-11869 (2006)

http://dx.doi.org/10.1364/OE.14.011855

Acrobat PDF (391 KB)

### Abstract

Two alternative formulations of the Lorentz force theory of radiation pressure on macroscopic bodies are reviewed. The theories treat the medium respectively as formed from individual dipoles and from individual charges. The former theory is applied to the systems of dielectric slab and dielectric prism, where it is shown that the total torque and force respectively agree with the results of the latter theory. The longitudinal shift of the slab caused by the passage of a single-photon pulse is calculated by Einstein box and Lorentz force theories, with identical results. The Lorentz forces on a single dielectric surface are shown to differ in the two theories and the basic reasons for the discrepancy are discussed. Both top-hat and Gaussian transverse beam profiles are considered.

© 2006 Optical Society of America

## Introduction

1. I. Brevik, “Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor,” Phys. Reps. **52**, 133–201 (1979). [CrossRef]

*ħω*/

*cη*and the Minkowski expression

*ħωη*/

*c*, where

*η*is the refractive index of the medium. The few available experiments seem to favor the latter, while many theories favor the former.

2. J. P. Gordon, “Radiation forces and momenta in dielectric media,” Phys. Rev. A **8**, 14–21 (1973). [CrossRef]

**E**and

**B**are respectively the electric field strength and the magnetic induction associated with the radiation, while

**P**is the electric polarization in the medium. Thus, the linear forces Ref. [3

3. R. Loudon, “Theory of the radiation pressure on dielectric surfaces,” J. Mod. Opt. **49**, 812–836 (2002). [CrossRef]

4. R. Loudon, “Theory of the forces exerted by Laguerre-Gaussian light beams on dielectrics,” Phys. Rev. A **68**, 013806 (2003). [CrossRef]

6. 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]

7. R. Loudon, “Radiation pressure and momentum in dielectrics,” Fortschr. Phys. **52**, 1132–1140 (2004). [CrossRef]

*ħω*/

*cη*to the total momentum in the bulk dielectric but the Minkowski value of

*ħωη*/

*c*to the momentum transferred to objects immersed in the dielectric, for example mirrors or charge carriers. The (negative) difference between these two momenta is taken up by the host dielectric.

8. M. Mansuripur, “Radiation pressure and the linear momentum of the electromagnetic field,” Opt. Express **13**, 5375–5401 (2004). [CrossRef]

8. M. Mansuripur, “Radiation pressure and the linear momentum of the electromagnetic field,” Opt. Express **13**, 5375–5401 (2004). [CrossRef]

9. M. Mansuripur, A. R. Zakharian, and J. V. Moloney, “Radiation pressure on a dielectric wedge,” Opt. Express **13**, 2064–2074 (2005). [CrossRef] [PubMed]

10. M. Mansuripur, “Radiation pressure and the linear momentum of light in dispersive dielectric media,” Opt. Express **13**, 2245–2250 (2005). [CrossRef] [PubMed]

11. M. Mansuripur, “Angular momentum of circularly polarized light in dielectric media,” Opt. Express **13**, 5315–5324 (2005). [CrossRef] [PubMed]

13. S. M. Barnett and R. Loudon, “On the electromagnetic force on a dielectric medium,” J. Phys. B **39**, S671–S684 (2006). [CrossRef]

*total*forces and torques on dielectric samples, and this conclusion has been confirmed Ref. [14

14. A. R. Zakharian, P. Polynkin, M. Mansuripur, and J. V. Moloney, “Single-beam trapping of micro-beads in polarized light: numerical simulations,” Opt. Express **14**, 3660–3676 (2006). [CrossRef] [PubMed]

8. M. Mansuripur, “Radiation pressure and the linear momentum of the electromagnetic field,” Opt. Express **13**, 5375–5401 (2004). [CrossRef]

9. M. Mansuripur, A. R. Zakharian, and J. V. Moloney, “Radiation pressure on a dielectric wedge,” Opt. Express **13**, 2064–2074 (2005). [CrossRef] [PubMed]

## 1. Radiation torque and force on transparent dielectric slab

*i*on the surface of a dielectric slab of thickness

*D*and refractive index

*η*. The Brewster incidence ensures that no reflections occur at either the entrance or exit surfaces of the slab. The lateral shift of the beam on its passage through the slab results in a torque that can be calculated in two ways, (i) by evaluation of the effective torque on the beam and (ii) by evaluation of the Lorentz forces applied to the slab by the light beam. The two quantities should be equal in magnitude but opposite in sign.

### 2.1 Effective torque on the light beam

*i*and the angle of refraction

*r*by

*i*+

*r*=

*π*2 and

*f*photons of energy

*ħω*per second, the effective torque on the light is anticlockwise for the arrangement shown in Fig. 1, with value

**13**, 5375–5401 (2004). [CrossRef]

### 1.2 Torque on the slab

*T*

_{L}given by Eq. (6). There are three Lorentz force contributions at the entrance surface of the slab:

- The magnetic term of the Lorentz force Eq. (1) in the bulk dielectric, which contributes a force parallel to the refracted beam as the optical pulse passes through the surface Ref. [3]. This is the only force contribution in conditions of normal incidence.
3. R. Loudon, “Theory of the radiation pressure on dielectric surfaces,” J. Mod. Opt.

**49**, 812–836 (2002). [CrossRef] - A surface Lorentz force from the electric term in Eq. (1) directed perpendicular to the slab into the free space above it.
- A further electric force directed normal to the refracted beam edge within the dielectric. The three contributions, illustrated in Fig. 1, all occur with the same magnitudes but opposite signs at the exit surface of the slab.

*A*of the incident beam becomes

*Aη*in the dielectric, so that the Poynting vector is conserved. The illuminated surface area is

*F*

_{1}for contribution 1 is evaluated in subsection 2.3 as

*z*=0 with the

*z*axis pointing away from the slab. The tangential components of the

*E*field are continuous across the surface but the normal field changes discontinuously from

*E*sin

*i*at

*z*>0 to

*E*sin

*r*/

*η*at

*z*<0. It follows with the use of Eq. (3) that

*D*tan

*r*=

*D*/

*η*, and the clockwise torque on the slab is therefore

4. R. Loudon, “Theory of the forces exerted by Laguerre-Gaussian light beams on dielectrics,” Phys. Rev. A **68**, 013806 (2003). [CrossRef]

7. R. Loudon, “Radiation pressure and momentum in dielectrics,” Fortschr. Phys. **52**, 1132–1140 (2004). [CrossRef]

**13**, 5375–5401 (2004). [CrossRef]

*ζ*is an inwards-pointing coordinate normal to the beam propagation direction, the analogues of Eq. (10) and Eq. (11) are

*A*, and the resulting edge force is

*D*/cos

*r*. Their contribution to the torque is again clockwise, with value

*T*

_{L}given by Eq. (6), as expected. Alternatively, the forces can first be added, with components parallel and perpendicular to the incident beam direction given by

**13**, 5375–5401 (2004). [CrossRef]

**13**, 5375–5401 (2004). [CrossRef]

15. G. Barlow, “On the torque produced by a beam of light in oblique refraction through a glass plate,” Proc. Roy. Soc. Lond. A **87**, 1–16 (1912). [CrossRef]

### 1.3 Slab displacement

*ħω*/

*c*parallel to the incident light. This applies to the three stages of incident pulse in free space, refracted pulse within the dielectric, and transmitted pulse in free space. The conservation is demonstrated below, with particular emphasis on the second of the three stages.

*τ*given by

*Z*of the slab is now easily determined by the principle of the Einstein box Refs. [5,16

16. A. Einstein, “Das Prinzip von der Erhaltung der Schwerpunktsbewegung und die Trägheit der Energie,” Ann. Phys. (Leipzig) **20**, 627–633 (1906). [CrossRef]

*M*is the mass of the slab. Note that the displacement always occurs in the propagation direction of the incident and transmitted light. The assumption of Brewster-angle incidence in an Einstein-box theory of slab displacement was previously made by Balazs Ref. [17

17. N. L. Balazs, “The energy-momentum tensor of the electromagnetic field inside matter,” Phys. Rev. **91**, 408–411 (1953). [CrossRef]

*F*

_{2}and

*F*

_{3}exerted by a photon flux are converted to the momentum transfers from a single-photon pulse by the simple removal of

*f*. The sum force given by Eq. (18) and Eq. (19) thus corresponds to a slab momentum transfer

*p*

_{1}parallel to the refracted beam. It results from the magnetic term in the Lorentz force Eq. (1) within the dielectric and its magnitude is

**49**, 812–836 (2002). [CrossRef]

*f*, it provides the force component

*F*

_{1}quoted in Eq. (9). The bulk contribution has the Abraham form of photon momentum in the dielectric. The total momentum has components

*F*

_{2}and

*F*

_{3}given in Eq. (24) and the parallel component ensures momentum conservation with the incident light. The surface part of Eq. (25) represents a momentum transfer to the slab, which is cancelled after a time

*τ*as the pulse passes through the exit surface to leave the slab at rest again, with a displacement

**F**

_{2}+

**F**

_{3}, given by Eq. (18) and Eq. (19), acts perpendicular to the incident light beam and it determines the torque on the slab given by Eq. (17). For a single-photon incident pulse, it provides the momentum transfer

*p*

_{2}+

*p*

_{3}to the slab given by Eq. (24). The force

**F**

_{1}, with magnitude given by Eq. (9), acts parallel to the refracted light beam and it makes no contribution to the torque on the slab. For a single-photon incident pulse, it provides the slab displacement Eq. (27) in agreement with Einstein box theory. In combination with the photon momentum

*ħω*/

*cη*, also in the direction of the refracted beam, it cancels the momentum

*p*

_{2}+

*p*

_{3}perpendicular to the incident beam. In accordance with Eq. (26), it ensures that the linear momentum

*ħω*/

*c*is conserved throughout the transmission process.

## 2. Radiation force on transparent dielectric prism

*i*on the surface of a dielectric prism or wedge with semi-angle

*r*such that

*i*+

*r*=

*π*/2. The refracted beam in these conditions is directed perpendicular to the prism axis and the light also exits the prism in Brewster configuration, with no reflected components. The forces

**F**

_{2}and

**F**

_{3}at the entrance and exit surfaces considered individually are identical to those for the slab shown in Fig. 1. However, the reversal of the exit angle of the transmitted light from the prism results in quite different effects from those in the slab. Thus there is clearly no torque on the prism but there is now a net horizontal force.

*f*, the effective horizontal force on the light is immediately obtained by reference to Fig. 2 as

9. M. Mansuripur, A. R. Zakharian, and J. V. Moloney, “Radiation pressure on a dielectric wedge,” Opt. Express **13**, 2064–2074 (2005). [CrossRef] [PubMed]

*A*, with zero intensity outside this area. We now generalize the prism calculations to a more physical light beam of Gaussian intensity profile and show how the above results are substantiated. The position dependence of the incident field is now given by

*w*is the beam waist,

*p*is the

*x*coordinate of the intersection of the incident beam axis with the prism axis, and

**x̂**′ is a unit polarization vector. The coordinates

*z*′ and

*x*′ point parallel and perpendicular to the incident beam propagation in the plane of Fig. 2, with

*y*perpendicular to the plane. For the coordinate system defined in the figure,

*c*/

*ω*≪

*w*. It is also assumed that essentially all of the light passes through the prism, expressed in the condition

*x*coordinate of the prism apex.

*T*, with a cycle-averaged momentum flux density given by Ref. [18

_{ij}18. S. M. Barnett, “Optical angular-momentum flux,” J. Opt. B Quantum Semiclassical Opt. **4**, S7–S16 (2002). [CrossRef]

*i*per unit area in direction

*j*. The rate of change of the momentum of the incident light is obtained by integration over a plane perpendicular to its propagation direction, with use of the electric field in Eq. (30) and the corresponding magnetic field. This quantity is equivalent to an effective force exerted on the incident beam at the entrance surface of the prism, and the result is

**F**

_{out}associated with the transmitted beam leaving the exit surface of the prism is given by the same expression but with the opposite sign of

*z*component. The total effective force on the beam thus agrees with Eq. (28) when expressed in terms of the photon flux by means of Eq. (34).

13. S. M. Barnett and R. Loudon, “On the electromagnetic force on a dielectric medium,” J. Phys. B **39**, S671–S684 (2006). [CrossRef]

**F**

_{2}but the second form is adequate for calculation of the edge force

**F**

_{3}. The force

**F**

_{2}at the entrance surface is calculated by the same method as used for the slab surface in Eq. (10) to Eq. (11). A coordinate

*n*normal to the surface with its origin at the point where the axis of the incident beam intersects the entrance surface is defined by

*z*integration is performed first with use of the delta function, the

*y*integration is straightforward, and the range for the remaining

*x*integral is

*z*component in the unit vector

*n̂*in Eq. (39) is reversed. The total surface force on the prism is thus parallel to the

*x*axis and its magnitude agrees with the

*F*

_{2}contribution in Eq. (29).

*x*axis and it is given by the second form of force density in Eq. (37) as

*z*>0 and the field expression is taken from Eq. (32). The

*x*integration, with limits

*y*integration is again straightforward, with the result

*F*

_{3}contribution in Eq. (29).

*ħω*/

*c*of an incident photon pulse is conserved throughout the transmission process into and through the prism, when account is taken of a contribution analogous to that shown in Eq. (25) for the slab system, but we do not consider this aspect here.

## 3. Radiation pressure on a free liquid surface

19. A. Ashkin and J. M. Dziedzic, “Radiation pressure on a free liquid surface,” Phys. Rev. Lett. **30**, 139–142 (1973). [CrossRef]

*µ*N. The effect was interpreted as caused by a longitudinal force associated with the change of photon momentum as the light passes from air into the liquid dielectric.

2. J. P. Gordon, “Radiation forces and momenta in dielectric media,” Phys. Rev. A **8**, 14–21 (1973). [CrossRef]

*F*

_{3}calculated in Eq. (15) for a top-hat profile and in Eq. (46) for a Gaussian profile. In conditions of normal incidence, the force is readily calculated for a Gaussian beam profile. The effective outward force on the liquid surface is found to have a magnitude Refs. [2

2. J. P. Gordon, “Radiation forces and momenta in dielectric media,” Phys. Rev. A **8**, 14–21 (1973). [CrossRef]

7. R. Loudon, “Radiation pressure and momentum in dielectrics,” Fortschr. Phys. **52**, 1132–1140 (2004). [CrossRef]

*Q*is the beam power. In contrast to the Brewster-angle incidence assumed in sections 2 and 3, there is now significant surface reflection, whose effects are included. The calculated value Eq. (47) of the force thus agrees with the experimental estimate quoted above. The radial force has significant contributions from both the electric and magnetic components of the Lorentz force given by Eq. (1). Although the physical origin of the radial force is quite distinct, its qualitative effect on the liquid surface is the same as that of a hypothetical longitudinal force.

*F*

_{1}given by Eq. (9), and its associated momentum

*p*

_{1}given by Eq. (25) is important in determining the displacement of a dielectric slab. However, it has been shown Ref. [7

**52**, 1132–1140 (2004). [CrossRef]

19. A. Ashkin and J. M. Dziedzic, “Radiation pressure on a free liquid surface,” Phys. Rev. Lett. **30**, 139–142 (1973). [CrossRef]

*F*

_{2}given by Eq. (12) and Eq. (43), arises from the electric field term in Eq. (1) as a result of the discontinuities in

*P*and

_{z}*E*at

_{z}*z*=0. We calculate its contribution for a light beam incident from free space at

*z*>0 with a field profile described by the normalized mode function

**E**mainly parallel to

*x*, but there is also a small

*z*component associated with the finite cross section of the beam. The analogous small component parallel to

**ẑ**′ is ignored in Eq. (30). The

*E*fields can be calculated from the classical versions of Eq. (6.4) and Eq. (6.7) of Ref. [4

4. R. Loudon, “Theory of the forces exerted by Laguerre-Gaussian light beams on dielectrics,” Phys. Rev. A **68**, 013806 (2003). [CrossRef]

*E*component of the incident light and a surface charge density is calculated in §3 of Ref. [12]. The calculation is based on the charge form of Lorentz force given in Eq. (2). A focused Gaussian laser beam of elliptical intensity profile is assumed and the magnitude of the force depends on the optical polarization state. With conversion to the notation used here and simplification for an incident beam of circular intensity profile and linear polarization, the force is

_{z}19. A. Ashkin and J. M. Dziedzic, “Radiation pressure on a free liquid surface,” Phys. Rev. Lett. **30**, 139–142 (1973). [CrossRef]

**.P**. The magnetic component also vanishes for a current P

^{Ý}that is

*π*/2 out of phase with

*B*, leading to a zero cycle average of their product for the monochromatic light assumed in the derivation. The nonzero magnetic force is, however, verified Ref. [8

**13**, 5375–5401 (2004). [CrossRef]

**8**, 14–21 (1973). [CrossRef]

**52**, 1132–1140 (2004). [CrossRef]

**30**, 139–142 (1973). [CrossRef]

*P*and

_{z}*E*are the fields in the liquid adjacent to its surface. This relation is similar to Eq. (15) of Ref. [13

_{z}13. S. M. Barnett and R. Loudon, “On the electromagnetic force on a dielectric medium,” J. Phys. B **39**, S671–S684 (2006). [CrossRef]

## 4. Discussion

**49**, 812–836 (2002). [CrossRef]

**13**, 5375–5401 (2004). [CrossRef]

*ħω*and much of the discussion of radiation pressure phenomena is couched in terms of these momenta. However, with the relations given in Eq. (7) and Eq. (34), a classical

*E*field is readily converted to a photon flux

*f*in expressions for the radiation pressure forces, and removal of

*f*from these same expressions provides the corresponding single-photon momentum transfers to a dielectric sample.

**8**, 14–21 (1973). [CrossRef]

**13**, 5375–5401 (2004). [CrossRef]

**39**, S671–S684 (2006). [CrossRef]

14. A. R. Zakharian, P. Polynkin, M. Mansuripur, and J. V. Moloney, “Single-beam trapping of micro-beads in polarized light: numerical simulations,” Opt. Express **14**, 3660–3676 (2006). [CrossRef] [PubMed]

*F*

_{3}point into the beam for the theory based on Eq. (1) but out of the beam for the theory based on Eq. (2). The final example treated here, of light incident normally on a liquid surface, treats only part of the total system and finds the difference between the results of the two theories shown in Eq. (54). This discrepancy between the predictions of the theories can be resolved only by a decision as to the appropriate choice of dipolar or charge-based model for the dielectric material. In the context of the experiment reported in Ref. [19

**30**, 139–142 (1973). [CrossRef]

**8**, 14–21 (1973). [CrossRef]

**13**, 5375–5401 (2004). [CrossRef]

**13**, 5375–5401 (2004). [CrossRef]

*f*of such wave packets. In contrast to the zero cycle average of the magnetic term in Eq. (1) or Eq. (2) for a monochromatic wave, this does not vanish when the fields have a distribution of frequencies

*ω*, as in the Gaussian distribution defined in Eq. (33) of Ref. [3

**49**, 812–836 (2002). [CrossRef]

*l*/

*c*is much longer than the optical period at the central frequency. The spatial and temporal integrals of the force in a bulk dielectric both vanish Refs. [2

**8**, 14–21 (1973). [CrossRef]

**49**, 812–836 (2002). [CrossRef]

**49**, 812–836 (2002). [CrossRef]

*l*→∞, while the Poynting vector in Eq. (35) of Ref. [3

**49**, 812–836 (2002). [CrossRef]

**30**, 139–142 (1973). [CrossRef]

*F*

_{1}produced by the magnetic term in Eq. (1). Further, the dominant effective force Eq. (47) on a free liquid surface receives significant contributions from both the electric and magnetic terms of the Lorentz force Eq. (1). These contributions are absent from the theory for a monochromatic wave based on the form Eq. (2) of Lorentz force.

## Acknowledgments

## References and links

1. | I. Brevik, “Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor,” Phys. Reps. |

2. | J. P. Gordon, “Radiation forces and momenta in dielectric media,” Phys. Rev. A |

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

4. | R. Loudon, “Theory of the forces exerted by Laguerre-Gaussian light beams on dielectrics,” Phys. Rev. A |

5. | M. Padgett, S. M. Barnett, and R. Loudon, “The angular momentum of light inside a dielectric,” J. Mod. Opt. |

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

7. | R. Loudon, “Radiation pressure and momentum in dielectrics,” Fortschr. Phys. |

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

9. | M. Mansuripur, A. R. Zakharian, and J. V. Moloney, “Radiation pressure on a dielectric wedge,” Opt. Express |

10. | M. Mansuripur, “Radiation pressure and the linear momentum of light in dispersive dielectric media,” Opt. Express |

11. | M. Mansuripur, “Angular momentum of circularly polarized light in dielectric media,” Opt. Express |

12. | M. Mansuripur, “Radiation pressure and the distribution of electromagnetic force in dielectric media,” in |

13. | S. M. Barnett and R. Loudon, “On the electromagnetic force on a dielectric medium,” J. Phys. B |

14. | A. R. Zakharian, P. Polynkin, M. Mansuripur, and J. V. Moloney, “Single-beam trapping of micro-beads in polarized light: numerical simulations,” Opt. Express |

15. | G. Barlow, “On the torque produced by a beam of light in oblique refraction through a glass plate,” Proc. Roy. Soc. Lond. A |

16. | A. Einstein, “Das Prinzip von der Erhaltung der Schwerpunktsbewegung und die Trägheit der Energie,” Ann. Phys. (Leipzig) |

17. | N. L. Balazs, “The energy-momentum tensor of the electromagnetic field inside matter,” Phys. Rev. |

18. | S. M. Barnett, “Optical angular-momentum flux,” J. Opt. B Quantum Semiclassical Opt. |

19. | A. Ashkin and J. M. Dziedzic, “Radiation pressure on a free liquid surface,” Phys. Rev. Lett. |

**OCIS Codes**

(260.2110) Physical optics : Electromagnetic optics

**ToC Category:**

Physical Optics

**History**

Original Manuscript: September 26, 2006

Revised Manuscript: November 3, 2006

Manuscript Accepted: November 6, 2006

Published: November 27, 2006

**Citation**

Rodney Loudon and Stephen M. Barnett, "Theory of the radiation pressure on dielectric slabs, prisms and single surfaces," Opt. Express **14**, 11855-11869 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-24-11855

Sort: Year | Journal | Reset

### References

- I. Brevik, "Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor," Phys. Rep. 52, 133-201 (1979). [CrossRef]
- J. P. Gordon, "Radiation forces and momenta in dielectric media," Phys. Rev. A 8, 14-21 (1973). [CrossRef]
- R. Loudon, "Theory of the radiation pressure on dielectric surfaces," J. Mod. Opt. 49, 812-836 (2002). [CrossRef]
- R. Loudon, "Theory of the forces exerted by Laguerre-Gaussian light beams on dielectrics," Phys. Rev. A 68, 013806 (2003). [CrossRef]
- M. Padgett, S. M. Barnett and R. Loudon, "The angular momentum of light inside a dielectric," J. Mod. Opt. 50, 1555-1562 (2003).
- 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]
- R. Loudon, "Radiation pressure and momentum in dielectrics," Fortschr. Phys. 52, 1132-1140 (2004). [CrossRef]
- M. Mansuripur, "Radiation pressure and the linear momentum of the electromagnetic field," Opt. Express 13, 5375-5401 (2004). [CrossRef]
- M. Mansuripur, A. R. Zakharian and J. V. Moloney, "Radiation pressure on a dielectric wedge," Opt. Express 13, 2064-2074 (2005). [CrossRef] [PubMed]
- M. Mansuripur, "Radiation pressure and the linear momentum of light in dispersive dielectric media," Opt. Express 13, 2245-2250 (2005). [CrossRef] [PubMed]
- M. Mansuripur, "Angular momentum of circularly polarized light in dielectric media," Opt. Express 13, 5315-5324 (2005). [CrossRef] [PubMed]
- M. Mansuripur, "Radiation pressure and the distribution of electromagnetic force in dielectric media," in Optical Trapping and Optical Micromanipulation II, K. Dholakia, and G. C. Spalding, eds., Proc. SPIE 5930, 154-160 (2005).
- S. M. Barnett and R. Loudon, "On the electromagnetic force on a dielectric medium," J. Phys. B 39, S671-S684 (2006). [CrossRef]
- A. R. Zakharian, P. Polynkin, M. Mansuripur and J. V. Moloney, "Single-beam trapping of micro-beads in polarized light: numerical simulations," Opt. Express 14, 3660-3676 (2006). [CrossRef] [PubMed]
- G. Barlow, "On the torque produced by a beam of light in oblique refraction through a glass plate," Proc. Roy. Soc. Lond. A 87, 1-16 (1912). [CrossRef]
- A. Einstein, "Das Prinzip von der Erhaltung der Schwerpunktsbewegung und die Trägheit der Energie," Ann. Phys. (Leipzig) 20, 627-633 (1906). [CrossRef]
- N. L. Balazs, "The energy-momentum tensor of the electromagnetic field inside matter," Phys. Rev. 91, 408-411 (1953). [CrossRef]
- S. M. Barnett, "Optical angular-momentum flux," J. Opt.B: Quantum Semiclassical Opt. 4, S7-S16 (2002). [CrossRef]
- A. Ashkin and J. M. Dziedzic, "Radiation pressure on a free liquid surface," Phys. Rev. Lett. 30, 139-142 (1973). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.