## Electromagnetic Momentum and Radiation Pressure derived from the Fresnel Relations

Optics Express, Vol. 15, Issue 2, pp. 714-725 (2007)

http://dx.doi.org/10.1364/OE.15.000714

Acrobat PDF (191 KB)

### Abstract

Using the Fresnel relations as axioms, we derive a generalized electromagnetic momentum for a piecewise homogeneous medium and a different generalized momentum for a medium with a spatially varying refractive index in the Wentzel–Kramers–Brillouin (WKB) limit. Both generalized momenta depend linearly on the field, but the refractive index appears to different powers due to the difference in translational symmetry. For the case of the slowly varying index, it is demonstrated that there is negligible transfer of momentum from the electromagnetic field to the material. Such a transfer occurs at the interface between the vacuum and a homogeneous material allowing us to derive the radiation pressure from the Fresnel reflection formula. The Lorentz volume force is shown to be nil.

© 2007 Optical Society of America

## 1. Introduction

17. M. E. Crenshaw, “Generalized electromagnetic momentum and the Fresnel relations,” Phys. Lett. A **346**,249–254, (2005). [CrossRef]

*i*) a piecewise homogeneous medium and

*ii*) a medium with a slowly varying refractive index in the WKB limit. Both generalized momenta depend linearly on the field but the refractive index appears to different powers due to the difference in the translational symmetry. Momentum conservation is demonstrated numerically and theoretically in both limiting cases. For the case of a material with a slowly varying index, the momentum of the transmitted field is essentially equal to that of the incident field and no momentum is transferred to the material. However, a field entering a homogeneous medium from the vacuum imparts a permanent dynamic momentum to the material that is twice the momentum of the reflected field, if momentum is to be conserved. Because the change of the momentum of the material is due to reflection, radiation pressure is deemed to be a surface force acting over the illuminated area of the material and we show that the Lorentz volume force is nil.

## 2. Piecewise Homogeneous Media

17. M. E. Crenshaw, “Generalized electromagnetic momentum and the Fresnel relations,” Phys. Lett. A **346**,249–254, (2005). [CrossRef]

17. M. E. Crenshaw, “Generalized electromagnetic momentum and the Fresnel relations,” Phys. Lett. A **346**,249–254, (2005). [CrossRef]

*V*

_{1}with refractive index

*n*

_{1}into a medium

*V*

_{2}with index

*n*

_{2}>

*n*

_{1}, where

*n*

_{1}and

*n*

_{2}are real. The fields are assumed to be monochromatic plane waves polarized in the

*x*-direction and we write

*γ*. The Fresnel continuity equation (4) represents continuity of a flux

*ρ*is the property density and

*ρ*

**v**is the property flux vector. It is then a simple matter to derive the conservation laws that correspond to the continuous fluxes. We define flux vectors

*S*= |

**S**| and

*T*= |

**T**|. Denoting the respective property densities as

*u*and

*g*, we have

*G*, taken as a vector

**G**=

*G*

**e**

_{z}, has properties of linear momentum. The second Fresnel continuity equation, Eq. (4), is algebraically equivalent to

*γ*=

*c*/(4

*π*) based on the known form for the electromagnetic energy for a monochromatic plane wave. By comparison with the prior work [17

**346**,249–254, (2005). [CrossRef]

*α*is given in terms of a unit mass density

*ρ*

_{0}as

## 3. Slowly Varying Refractive Index

*n*=

*n*

_{2}-

*n*

_{1}is sufficiently small that reflection can be neglected. Equation (21) represents the continuity of the flux

*g*

**e**

_{z}over the volume, we obtain the conserved quantity

## 4. Momentum Conservation

19. W. P. Huang, S. T. Chu, A. Goss, and S. K. Chaudhuri, “A scalar finite-difference time-domain approach to guided-wave optics,” IEEE Photonics Tech. Lett. **3**,524 (1991). [CrossRef]

**A**(

*z*,

*t*) =

*A*(

*z*,

*t*)

*e*

^{-i(ωt-kz)}

**e**

_{x}, where

*A*is an envelope function,

*ω*is the carrier frequency, and

*k*is the carrier wavenumber. Envelope functions for the electric field

**E**= -(1/

*c*)

*∂*

**A**/

*∂t*, the magnetic induction

**B**= ∇ ×

**A**, the displacement field

**D**=

*n*

^{2}

**E**, the magnetic field

**H**=

**B**, and other quantities can be defined analogously, as required. The basic phenomenology of a propagating electromagnetic field is demonstrated using the Maxwellian model of a dielectric with a macroscopic refractive index

*n*and numerically solving the wave equation as

*k*=

*nω*/

*c*. The approximation of a slowly varying envelope is not made.

*A*|, starts in vacuum, travels to the right, and enters a linear homogeneous dielectric through a thin gradient-index antireflection layer. The figure shows that the dielectric medium affects the refracted field in two distinct ways. First, the refracted field is reduced in width by a factor of the refractive index due to the reduced velocity of the field. Second, the refracted field is reduced in amplitude compared to the incident field due to the creation of the reaction (polarization) field. Both of these effects are reversed upon exiting the medium through a gradient-index antireflection layer, Fig. 2.

*w*in the vacuum

*w*, the momentum conservation law takes the form

## 5. Energy Conservation

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

**G**

_{x}can be provided by writing a continuity law. Denoting the magnitude of the electromagnetic momentum density

*g*allows one to write the continuity law (7) as

_{x}*c*/

*n*in the direction of

**E**×

**H**. Substituting the electromagnetic momentum density (31) into Eq. (32) results in a momentum conservation law

**S**= (

*c*/4

*π*)

**E**×

**H**is the Poynting vector and

*u*=

*cg*is the energy density. Therefore,

_{x}*G*is conserved, but is redundant with the electromagnetic energy from which it was derived [22

_{x}22. J. P. Gordon, “Radiation Forces and Momenta in Dielectric Media,” Phys. Rev. A **8**,14–21 (1973). [CrossRef]

## 6. Radiation Pressure

23. M. Stone, “Phonons and Forces: Momentum versus Pseudomomentum in Moving Fluids,” arXiv.org, cond-mat/0012316 (2000), http://arxiv.org/abs/cond-mat?papernum=0012316.

23. M. Stone, “Phonons and Forces: Momentum versus Pseudomomentum in Moving Fluids,” arXiv.org, cond-mat/0012316 (2000), http://arxiv.org/abs/cond-mat?papernum=0012316.

*A*of the material. In an increment of time, the change in the pseudomomentum of the reflected field is

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

24. M. Mansuripur, “Radiation pressure and the linear momentum of the electromagnetic field,” Opt. Express **12**,5375–5401 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-22-5375. [CrossRef] [PubMed]

25. R. Loudon, S. M. Barnett, and C. Baxter, “Theory of radiation pressure and momentum transfer in dielectrics: the photon drag effect,” Phys. Rev. A **71**,063802 (2005). [CrossRef]

26. M. Scalora, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, M. Centini, C. Sibilia, and J. W. Haus, “Radiation pressure of light pulses and conservation of linear momentum in dispersive media,” Phys. Rev. E **73**,056604 (2006). [CrossRef]

*ρ*and the charge current

**J**in the derivation and, at the end, takes the limit in which these quantities vanish. Then, using the Maxwell equations to eliminate

*ρ*and

**J**, one finds [18]

**D**=

**E**+ 4

*π*

**P**, we may write

**G**

_{M}as the sum of the Abraham momentum

## 7. Conclusion

10. R. V. Jones and J. C. S. Richards, “The pressure of radiation in a refracting medium,” Proc. R. Soc. London A **221**,480 (1954). [CrossRef]

11. A. Ashkin and J. M. Dziedzic, “Radiation Pressure on a Free Liquid Surface,” Phys. Rev. Lett. **30**,139–142 (1973). [CrossRef]

12. A. F. Gibson, M. F. Kimmitt, A. O. Koohian, D. E. Evans, and G. F. D. Levy, “A Study of Radiation Pressure in a Refractive Medium by the Photon Drag Effect,” Proc. R. Soc. London A **370**,303–318 (1980). [CrossRef]

13. D. G. Lahoz and G. M. Graham, “Experimental decision on the electromagnetic momentum,” J. Phys. A **15**,303–318 (1982). [CrossRef]

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

15. I. Brevik, “Photon-drag experiment and the electromagnetic momentum in matter,” Phys. Rev. B **33**,1058–1062 (1986). [CrossRef]

24. M. Mansuripur, “Radiation pressure and the linear momentum of the electromagnetic field,” Opt. Express **12**,5375–5401 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-22-5375. [CrossRef] [PubMed]

25. R. Loudon, S. M. Barnett, and C. Baxter, “Theory of radiation pressure and momentum transfer in dielectrics: the photon drag effect,” Phys. Rev. A **71**,063802 (2005). [CrossRef]

## References and links

1. | H. Minkowski, Natches. Ges. Wiss. Göttingen53 (1908); Math. Ann.68,472 (1910). |

2. | M. Abraham, Rend. Circ. Mat. Palermo |

3. | A. Einstein and J. Laub, Ann. Phys. (Leipzig)26,541 (1908). |

4. | R. Peierls, “The momentum of light in a refracting medium,” Proc. R. Soc. Lond. A |

5. | M. Kranys, “The Minkowski and Abraham Tensors, and the non-uniqueness of non-closed systems,” Int. J. Engng. Sci. |

6. | G. H. Livens, |

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

8. | J. C. Garrison and R. Y. Chiao, “Canonical and kinetic forms of the electromagnetic momentum in an |

9. | S. Antoci and L. Mihich, “A forgotten argument by Gordon uniquely selects Abraham’s tensor as the energy- momentum tensor of the electromagnetic field in homogeneous, isotropic matter,” Nuovo Cim. |

10. | R. V. Jones and J. C. S. Richards, “The pressure of radiation in a refracting medium,” Proc. R. Soc. London A |

11. | A. Ashkin and J. M. Dziedzic, “Radiation Pressure on a Free Liquid Surface,” Phys. Rev. Lett. |

12. | A. F. Gibson, M. F. Kimmitt, A. O. Koohian, D. E. Evans, and G. F. D. Levy, “A Study of Radiation Pressure in a Refractive Medium by the Photon Drag Effect,” Proc. R. Soc. London A |

13. | D. G. Lahoz and G. M. Graham, “Experimental decision on the electromagnetic momentum,” J. Phys. A |

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

15. | I. Brevik, “Photon-drag experiment and the electromagnetic momentum in matter,” Phys. Rev. B |

16. | H. Goldstein, |

17. | M. E. Crenshaw, “Generalized electromagnetic momentum and the Fresnel relations,” Phys. Lett. A |

18. | J. D. Jackson, |

19. | W. P. Huang, S. T. Chu, A. Goss, and S. K. Chaudhuri, “A scalar finite-difference time-domain approach to guided-wave optics,” IEEE Photonics Tech. Lett. |

20. | A. Chubykalo, A. Espinoza, and R. Tzonchev, “Experimental test of the compatibility of the definitions of the electromagnetic energy density and the Poynting vector,” Eur. Phys. J. D |

21. | M. Crenshaw and N. Akozbek, “Electromagnetic energy flux vector for a dispersive linear medium,” Phys. Rev. E |

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

23. | M. Stone, “Phonons and Forces: Momentum versus Pseudomomentum in Moving Fluids,” arXiv.org, cond-mat/0012316 (2000), http://arxiv.org/abs/cond-mat?papernum=0012316. |

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

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

26. | M. Scalora, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, M. Centini, C. Sibilia, and J. W. Haus, “Radiation pressure of light pulses and conservation of linear momentum in dispersive media,” Phys. Rev. E |

**OCIS Codes**

(260.2110) Physical optics : Electromagnetic optics

(260.2160) Physical optics : Energy transfer

**ToC Category:**

Physical Optics

**History**

Original Manuscript: December 18, 2006

Manuscript Accepted: January 7, 2007

Published: January 22, 2007

**Citation**

Michael E. Crenshaw, "Electromagnetic momentum and radiation pressure derived from the
Fresnel relations," Opt. Express **15**, 714-725 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-2-714

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### References

- 1. H. Minkowski, Natches. Ges. Wiss. Göttingen 53 (1908); Math. Ann. 68, 472 (1910).
- M. Abraham, Rend. Circ. Mat. Palermo 28, 1 (1909); 30, 33 (1910). [CrossRef]
- A. Einstein and J. Laub, Ann. Phys. (Leipzig) 26, 541 (1908).
- R. Peierls, "The momentum of light in a refracting medium," Proc. R. Soc. Lond. A 347, 475-491 (1976). [CrossRef]
- M. Kranys, "The Minkowski and Abraham Tensors, and the non-uniqueness of non-closed systems," Int. J. Engng. Sci. 20, 1193-1213 (1982). [CrossRef]
- G. H. Livens, The Theory of Electricity, (Cambridge University Press, Cambridge, 1908).
- Y. N. Obukhov and F. W. Hehl, "Electromagnetic energy-momentum and forces in matter," Phys. Lett. A 311, 277-284 (2003). [CrossRef]
- J. C. Garrison and R. Y. Chiao, "Canonical and kinetic forms of the electromagnetic momentum in an ad hoc quantization scheme for a dispersive dielectric," Phys. Rev. A 70, 053826-1-8 (2004). [CrossRef]
- S. Antoci and L. Mihich, "A forgotten argument by Gordon uniquely selects Abraham’s tensor as the energy-momentum tensor of the electromagnetic field in homogeneous, isotropic matter," Nuovo Cim. B112, 991-1001 (1997).
- R. V. Jones and J. C. S. Richards, "The pressure of radiation in a refracting medium," Proc. R. Soc. London A 221, 480 (1954). [CrossRef]
- A. Ashkin and J. M. Dziedzic, "Radiation Pressure on a Free Liquid Surface," Phys. Rev. Lett. 30, 139-142 (1973). [CrossRef]
- A. F. Gibson, M. F. Kimmitt, A. O. Koohian, D. E. Evans, and G. F. D. Levy, "A Study of Radiation Pressure in a Refractive Medium by the Photon Drag Effect," Proc. R. Soc. London A 370, 303-318 (1980). [CrossRef]
- D. G. Lahoz and G. M. Graham, "Experimental decision on the electromagnetic momentum," J. Phys. A 15, 303-318 (1982). [CrossRef]
- I. Brevik, "Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor," Phys. Rep. 52, 133-201 (1979). [CrossRef]
- I. Brevik, "Photon-drag experiment and the electromagnetic momentum in matter," Phys. Rev. B 33, 1058-1062 (1986). [CrossRef]
- H. Goldstein, Classical Mechanics, 2nd Ed., (Addison-Wesley, Reading, MA, 1980).
- M. E. Crenshaw, "Generalized electromagnetic momentum and the Fresnel relations," Phys. Lett. A 346, 249-254, (2005). [CrossRef]
- J. D. Jackson, Classical Electrodynamics, 2nd Ed., (Wiley, New York, 1975).
- W. P. Huang, S. T. Chu, A. Goss, and S. K. Chaudhuri, "A scalar finite-difference time-domain approach to guided-wave optics," IEEE Photonics Tech. Lett. 3, 524 (1991). [CrossRef]
- A. Chubykalo, A. Espinoza, and R. Tzonchev, "Experimental test of the compatibility of the definitions of the electromagnetic energy density and the Poynting vector," Eur. Phys. J. D 31, 113-120 (2004). [CrossRef]
- M. Crenshaw and N. Akozbek, "Electromagnetic energy flux vector for a dispersive linear medium," Phys. Rev. E 73, 056613 (2006). [CrossRef]
- J. P. Gordon, "Radiation Forces and Momenta in Dielectric Media," Phys. Rev. A 8, 14-21 (1973). [CrossRef]
- M. Stone, "Phonons and Forces: Momentum versus Pseudomomentum in Moving Fluids," arXiv.org, condmat/0012316 (2000), http://arxiv.org/abs/cond-mat?papernum=0012316.
- M. Mansuripur, "Radiation pressure and the linear momentum of the electromagnetic field," Opt. Express 12, 5375-5401 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-22-5375. [CrossRef] [PubMed]
- R. Loudon, S. M. Barnett, and C. Baxter, "Theory of radiation pressure and momentum transfer in dielectrics: the photon drag effect," Phys. Rev. A 71, 063802 (2005). [CrossRef]
- M. Scalora, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, M. Centini, C. Sibilia, and J. W. Haus, "Radiation pressure of light pulses and conservation of linear momentum in dispersive media," Phys. Rev. E 73, 056604 (2006). [CrossRef]

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