## Total longitudinal momentum in a dispersive optical waveguide |

Optics Express, Vol. 19, Issue 25, pp. 25263-25278 (2011)

http://dx.doi.org/10.1364/OE.19.025263

Acrobat PDF (2362 KB)

### Abstract

Using the Lorentz force law, we derived simpler expressions for the total longitudinal (conserved) momentum and the mechanical momentums associated with an optical pulse propagating along a dispersive optical waveguide. These expressions can be applied to an arbitrary non-absorptive optical waveguide having continuous translational symmetry. Our simulation using finite difference time domain (FDTD) method verified that the total momentum formula is valid in a two-dimensional infinite waveguide. We studied the conservation of the total momentum and the transfer of the momentum to the waveguide for the case when an optical pulse travels from a finite waveguide to vacuum. We found that neither the Abraham nor the Minkowski momentum expression for an electromagnetic wave in a waveguide represents the complete total (conserved) momentum. Only the total momentum as we derived for a mode propagating in a dispersive optical waveguides is the ‘true’ conserved momentum. This total momentum can be expressed as **P ^{Tot}** = –U

^{Die}/v

_{g}+ n

_{eff}U/c. It has three contributions: (1) the Abraham momentum; (2) the momentum from the Abraham force, which equals to the difference between the Abraham momentum and the Minkowski momentum; and (3) the momentum from the dipole force which can be expressed as –U

^{Die}/v

_{g}. The last two contributions constitute the mechanical momentum. Compared with FDTD-Lorentz-force method, the presently derived total momentum formula provides a better method in terms of analyzing the permanent transfer of optical momentum to a waveguide.

© 2011 OSA

## 1. Introduction

1. R. N. C. Pfeifer, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Colloquium: momentum of an electromagnetic wave in dielectric media,” Rev. Mod. Phys. **79**(4), 1197–1216 (2007). [CrossRef]

2. I. Brevik, “Experiment in phenomenological electrodynamics and the electromagnetic energy-momentum tensor,” Phys. Rep. **52**(3), 133–201 (1979). [CrossRef]

**p**=

_{M}**D**×

**B**[3,4], and expressed an optical pulse’s total momentum as P

_{M}= n

_{p}U/c [5

5. P. W. Milonni and R. W. Boyd, “Momentum of light in a dielectric medium,” Adv. Opt. Photonics **2**(4), 519–553 (2010). [CrossRef]

6. C. Baxter and R. Loudon, “Radiation pressure and the photon momentum in dielectrics,” J. Mod. Opt. **57**(10), 830–842 (2010). [CrossRef]

**p**=

_{A}**E**×

**H**/c

^{2}[2

2. I. Brevik, “Experiment in phenomenological electrodynamics and the electromagnetic energy-momentum tensor,” Phys. Rep. **52**(3), 133–201 (1979). [CrossRef]

6. C. Baxter and R. Loudon, “Radiation pressure and the photon momentum in dielectrics,” J. Mod. Opt. **57**(10), 830–842 (2010). [CrossRef]

7. V. M. Abraham, “Zur Elektrodynamik bewegter Körper,” Rend. Circ. Matem. Palermo **28**(1), 1–28 (1909). [CrossRef]

_{A}= U/(n

_{g}c) [5

5. P. W. Milonni and R. W. Boyd, “Momentum of light in a dielectric medium,” Adv. Opt. Photonics **2**(4), 519–553 (2010). [CrossRef]

6. C. Baxter and R. Loudon, “Radiation pressure and the photon momentum in dielectrics,” J. Mod. Opt. **57**(10), 830–842 (2010). [CrossRef]

_{p}and n

_{g}are respectively the phase and group refractive index of the dispersive bulk medium. Noether theorem [8

8. D. F. Nelson, “Momentum, pseudomomentum, and wave momentum: Toward resolving the Minkowski-Abraham controversy,” Phys. Rev. A **44**(6), 3985–3996 (1991). [CrossRef] [PubMed]

9. A. Feigel, “Quantum vacuum contribution to the momentum of dielectric media,” Phys. Rev. Lett. **92**(2), 020404 (2004). [CrossRef] [PubMed]

11. R. Loudon, “Radiation pressure and momentum in dielectrics,” Fortschr. Phys. **52**(11-12), 1134–1140 (2004). [CrossRef]

12. R. Loudon, S. M. Barnett, and C. Baxter, “Radation pressure and momentum transfer in dielectrics: The photon drag effect,” Phys. Rev. A **71**(6), 063802 (2005). [CrossRef]

13. M. Mansuripur, “Radiation pressure and the linear momentum of the electromagnetic field,” Opt. Express **12**(22), 5375–5401 (2004). [CrossRef] [PubMed]

14. S. M. Barnett and R. Loudon, “On the electromagnetic force on a dielectric medium,” J. Phys. At. Mol. Opt. Phys. **39**(15), S671–S684 (2006). [CrossRef]

15. B. A. Kemp, T. Grzegorczyk, and J. A. Kong, “Ab initio study of the radiation pressure on dielectric and magnetic media,” Opt. Express **13**(23), 9280–9291 (2005). [CrossRef] [PubMed]

16. 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**(2), 026608 (2009). [CrossRef] [PubMed]

17. E. A. Hinds and S. M. Barnett, “Momentum exchange between light and a single atom: Abraham or Minkowski?” Phys. Rev. Lett. **102**(5), 050403 (2009). [CrossRef] [PubMed]

18. S. M. Barnett, “Resolution of the abraham-minkowski dilemma,” Phys. Rev. Lett. **104**(7), 070401 (2010). [CrossRef] [PubMed]

19. P. L. Saldanha, “Division of the momentum of electromagnetic waves in linear media into electromagnetic and material parts,” Opt. Express **18**(3), 2258–2268 (2010). [CrossRef] [PubMed]

20. A. R. Zakharian, M. Mansuripur, and J. V. Moloney, “Radiation pressure and the distribution of electromagnetic force in dielectric media,” Opt. Express **13**(7), 2321–2336 (2005). [CrossRef] [PubMed]

1. R. N. C. Pfeifer, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Colloquium: momentum of an electromagnetic wave in dielectric media,” Rev. Mod. Phys. **79**(4), 1197–1216 (2007). [CrossRef]

2. I. Brevik, “Experiment in phenomenological electrodynamics and the electromagnetic energy-momentum tensor,” Phys. Rep. **52**(3), 133–201 (1979). [CrossRef]

1. R. N. C. Pfeifer, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Colloquium: momentum of an electromagnetic wave in dielectric media,” Rev. Mod. Phys. **79**(4), 1197–1216 (2007). [CrossRef]

*et al.*[21

21. W. She, J. Yu, and R. Feng, “Observation of a push force on the end face of a nanometer silica filament exerted by outgoing light,” Phys. Rev. Lett. **101**(24), 243601 (2008). [CrossRef] [PubMed]

22. M. Mansuripur and A. R. Zakharian, “Theoretical analysis of the force on the end face of a nanofilament exerted by an outing light pulse,” Phys. Rev. A **80**(2), 023823 (2009). [CrossRef]

*et al.*[23

23. H. Yu, W. Fang, F. Gu, M. Qiu, Z. Yang, and L. Tong, “Longitudinal Lorentz force on a subwavelength-diameter optical fiber,” Phys. Rev. A **83**(5), 053830 (2011). [CrossRef]

24. I. Brevik and S. A. Ellingsen, “Transverse radiation force in a tailored optical fiber,” Phys. Rev. A **81**(1), 011806 (2010). [CrossRef]

*et al.*'s conclusion is questioned [25

25. I. Brevik, “Comment on “Observation of a push force on the end face of a nanometer silica filament exerted by outgoing light”,” Phys. Rev. Lett. **103**(21), 219301, author reply 219302 (2009). [CrossRef] [PubMed]

27. M. Mansuripur, “Comment on “Observation of a push force on the end face of a nanometer silica filament exerted by outgoing light”,” Phys. Rev. Lett. **103**(1), 019301 (2009). [CrossRef] [PubMed]

5. P. W. Milonni and R. W. Boyd, “Momentum of light in a dielectric medium,” Adv. Opt. Photonics **2**(4), 519–553 (2010). [CrossRef]

**57**(10), 830–842 (2010). [CrossRef]

28. J. Yu, R. Feng, and W. She, “Low-power all-optical switch based on the bend effect of a nm fiber taper driven by outgoing light,” Opt. Express **17**(6), 4640–4645 (2009). [CrossRef] [PubMed]

## 2. Abraham and Minkowski momentums in a dispersive waveguide

29. L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature **426**(6968), 816–819 (2003). [CrossRef] [PubMed]

30. L. Tong, J. Lou, and E. Mazur, “Single-mode guiding properties of subwavelength-diameter silica and silicon wire waveguides,” Opt. Express **12**(6), 1025–1035 (2004). [CrossRef] [PubMed]

31. P. Russell, “Photonic crystal fibers,” Science **299**(5605), 358–362 (2003). [CrossRef] [PubMed]

32. B. J. Eggleton, C. Kerbage, P. S. Westbrook, R. S. Windeler, and A. Hale, “Microstructured optical fiber devices,” Opt. Express **9**(13), 698–713 (2001). [CrossRef] [PubMed]

33. V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. **29**(11), 1209–1211 (2004). [CrossRef] [PubMed]

34. Q. Xu, V. R. Almeida, R. R. Panepucci, and M. Lipson, “Experimental demonstration of guiding and confining light in nanometer-size low-refractive-index material,” Opt. Lett. **29**(14), 1626–1628 (2004). [CrossRef] [PubMed]

_{g}c), and the Minkowski momentum is n

_{eff}U/c, where U denotes the total energy of the optial pulse, n

_{eff}and n

_{g}are the effective and group refractive index respectively, and c is the speed of light in vacuum. In the following, nonlinear effect of the waveguide medium is ignored, and Maxwell equations are therefore assumed linear.

**p**=

_{A}**E**×

**H**/c

^{2}[1

**79**(4), 1197–1216 (2007). [CrossRef]

**52**(3), 133–201 (1979). [CrossRef]

**79**(4), 1197–1216 (2007). [CrossRef]

**52**(3), 133–201 (1979). [CrossRef]

_{eff}(ω) with respect to ω, the total Minkowski momentum becomes

_{eff}dispersion, iswhere

_{eff}dispersion term

_{eff}is neglected, the Minkowski momentum of a single photon equals to Ñ

*β*. Here Ñ

*β*is a canonical momentum of a single photon which roots in the continuous translational symmetry of the waveguide and the wave property of photons since

*β*is corresponding to the eigenvalue of translational operator of the waveguide supported mode (see Appendix A). This result shows that the Minkowski momentum is not identical to canonical momentum except that the contribution

_{eff}dispersion in Eq. (11) can be neglected.

## 3. Mechanical and total momentum in a dispersive waveguide

13. M. Mansuripur, “Radiation pressure and the linear momentum of the electromagnetic field,” Opt. Express **12**(22), 5375–5401 (2004). [CrossRef] [PubMed]

17. E. A. Hinds and S. M. Barnett, “Momentum exchange between light and a single atom: Abraham or Minkowski?” Phys. Rev. Lett. **102**(5), 050403 (2009). [CrossRef] [PubMed]

18. S. M. Barnett, “Resolution of the abraham-minkowski dilemma,” Phys. Rev. Lett. **104**(7), 070401 (2010). [CrossRef] [PubMed]

17. E. A. Hinds and S. M. Barnett, “Momentum exchange between light and a single atom: Abraham or Minkowski?” Phys. Rev. Lett. **102**(5), 050403 (2009). [CrossRef] [PubMed]

**52**(3), 133–201 (1979). [CrossRef]

**2**(4), 519–553 (2010). [CrossRef]

_{eff}in Eq. (28) is replaced with the phase index n

_{p}of a bulk medium. However, it should be noted that the dispersion relationship of a waveguide depends not only on the properties of the waveguide material, but also on the transverse geometrical structure of the waveguide [31

31. P. Russell, “Photonic crystal fibers,” Science **299**(5605), 358–362 (2003). [CrossRef] [PubMed]

34. Q. Xu, V. R. Almeida, R. R. Panepucci, and M. Lipson, “Experimental demonstration of guiding and confining light in nanometer-size low-refractive-index material,” Opt. Lett. **29**(14), 1626–1628 (2004). [CrossRef] [PubMed]

13. M. Mansuripur, “Radiation pressure and the linear momentum of the electromagnetic field,” Opt. Express **12**(22), 5375–5401 (2004). [CrossRef] [PubMed]

## 4. Numerical simulations

_{y}(x,z,t) at 93.5fs is shown in Fig. 1 . The material dispersion is assumed to be zero and the waveguide dispersion is automatically accounted for in the FDTD simulation. Here, the grid size Δx = Δz = 0.02μm and the simulated region size is 40μm × 13μm with a perfect match layer of 1μm thickness.

### 4.1 Verification of the momentum formulas in an infinite waveguide

_{g}of 1.3920 and a group velocity v

_{g}of 215.37μm/ps. The corresponding effective refractive index n

_{eff}for TE fundamental mode at the central wavelength of 800nm is found to be 1.1938, which agreed well with a theoretical computation result of 1.1939 reported in Ref. [37]. Due to the waveguide dispersion, the pulse width increases as the pulse propagates along the waveguide, and the FWHM are 4.38fs, 4.80fs, 5.20fs respectively at the three positions [see Fig. 2]. A strong group-velocity dispersion of ~656.44fs

^{2}/mm is obtained according to the formula in Ref. [37]. Since the medium is assumed lossless, the energy of the pulse, obtained by integrating the instant power over time, is found to be U = 187.27fN⋅um/m and is unchanged along the waveguide.

### 4.2 Conservation and transfer of optical momentum in a finite waveguide

**P**

^{FDTD}_{i}with a value of 0.5712fN⋅ps/m as computed by Eq. (29) using FDTD, is conserved. After leaving the waveguide at ~130fs, the pulse spreads out, with only a little amount of the pulse emerging out at the top and bottom boundaries of the black integral region at 145fs. After 145fs, the total momentum declines slowly as shown by the black solid line in Fig. 5(c). The decline becomes quicker after 180fs due to fact that the pulse is now leaving the right boundary of the simulated region. After the pulse completely leaves the integral regions (z∈ [15, 38]μm) at ~188fs, the total momentum does not vanish as there is a small amount of momentum

**P**

^{FDTD}_{w}with a value of 0.0497 fN⋅ps/m, that is transferred to the waveguide. This can be seen by the overlap of the red dotted line with the black solid line after 188fs. The reflected pulse reaches the region (z∈ [12, 17]μm) at ~145fs and leaves it at ~175fs. Accompanied with the pulse is a reflected total momentum

**P**

^{FDTD}_{r}of −0.0151fN⋅ps/m, as shown by the cyan dotted line in Fig. 5(c). The transmitted total momentum

**P**

^{FDTD}_{t}can be found to be 0.5366fN⋅ps/m from the blue dash-dot line in Fig. 5(c). Thus, the identity,

**P**

^{FDTD}_{i}=

**P**

^{FDTD}_{r}+

**P**

^{FDTD}_{t}+

**P**

^{FDTD}_{w}, is confirmed. This shows that the total momentum is the true conserved momentum when considering an optical pulse propagating along a waveguide. Using Eq. (28), we can conveniently determine that the permanent transfer of momentum to the waveguide can be given bywhere ρ = 19.22% is the ratio of the dielectric energy over the total energy of the pulse, and R and T are the reflectance and transmittance of the energy of the pulse at the waveguide end face, respectively. In our simulation, R = 6.87% and T = 91.00% and they are computed based on the instant optical powers obtained at z = 20μm and z = 22.02μm in Fig. 7 . Note that 2.13% of incident energy is scattered into other directions. By inserting these quantities into Eq. (30), the permanently transferred momentum is found to be

**P**

^{Eq}_{w}= 0.0490fN⋅ps/m. Compared with FDTD-computed transfer of the momentum

**P**

^{FDTD}_{w}, Eq. (30) has a small relative error of 1.4%. We therefore believe that Eq. (30) provide a relatively precise expression for the momentum transferred to a finite waveguide.

*et al.*carried out [21

21. W. She, J. Yu, and R. Feng, “Observation of a push force on the end face of a nanometer silica filament exerted by outgoing light,” Phys. Rev. Lett. **101**(24), 243601 (2008). [CrossRef] [PubMed]

## 5. Conclusions

_{eff}times the momentum in vacuum. However, neither Abraham nor Minkowski momentum represents the correct total (conserved) momentum in the waveguide. Only the total momentum is ‘truely’ conserved along a waveguide. The total momentum is expressed as

**P**= -U

^{Tot}^{Die}/v

_{g}+ n

_{eff}U/c, which has three contributions: (1) the Abraham momentum in waveguide,

**P**= (1/n

^{A}_{g})U/c; (2) the momentum from the Abraham force, i.e.

**P**= (n

_{2}^{AF}_{eff}−1/n

_{g})U/c, which is the difference of the Minkowski and Abraham momentum; and (3) the momentum due to the dipole force,

**P**= -U

^{can}_{mech}^{Die}/v

_{g}, which equals to the negative dielectric energy divided by the group velocity. In Abraham (kinetic) terms, the mechanical momentum carried by the dielectric medium of a waveguide equals to the sum of the last two contributions, i.e.

**P**= -U

^{kin}_{mech}^{Die}/v

_{g}+ (n

_{eff}−1/n

_{g})U/c. By using FDTD, we verified that the momentum formulas without dispersive term can exactly give the total (conserved) momentum in a waveguide. A simpler formula for the force exerted by an outgoing optical pulse from a waveguide is also derived. This shows the convenience of total momentum formula for the analysis of optical forces and the transfer of optical momentum in a finite waveguide.

## Appendix A

^{T}are one of the possible solutions. Translational operator

^{T}are the solutions of Eq. (A1) with the same eigenvalue as the mode fields [

^{T}. According to Maxwell equations, it is necessary to ensure that the electric and magnetic fields can be excited by each other mutually and thus the electromagnetic wave can propagate along the waveguide steadily, the field mode must be a periodic function along the waveguide. Assuming that the spatial period of field mode along the waveguide is λ

_{eff}for a given optical frequency ω, we have

*l*=±1, ±2, ±3…, ±n

_{max}).

*β*corresponds to the propagation constant for

_{l}*l*th mode at a given optical frequency ω, where the sign of

*l*indicates the direction of light wave propagation, and n

_{max}is determined by eigenequations of

## Acknowledgments

## References and links

1. | R. N. C. Pfeifer, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Colloquium: momentum of an electromagnetic wave in dielectric media,” Rev. Mod. Phys. |

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

3. | H. Minkowski, ““Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern,” Nachr. Ges. Wiss. Gottn Math.-Phys. Kl. |

4. | J. A. Kong, |

5. | P. W. Milonni and R. W. Boyd, “Momentum of light in a dielectric medium,” Adv. Opt. Photonics |

6. | C. Baxter and R. Loudon, “Radiation pressure and the photon momentum in dielectrics,” J. Mod. Opt. |

7. | V. M. Abraham, “Zur Elektrodynamik bewegter Körper,” Rend. Circ. Matem. Palermo |

8. | D. F. Nelson, “Momentum, pseudomomentum, and wave momentum: Toward resolving the Minkowski-Abraham controversy,” Phys. Rev. A |

9. | A. Feigel, “Quantum vacuum contribution to the momentum of dielectric media,” Phys. Rev. Lett. |

10. | C. Wang, “ Wave four-vector in a moving medium and the Lorentz covariance of Minkowski's photon and electromagnetic momentums,” arXiv.org, arXiv:1106.1163v11 (2011). |

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

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

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

14. | S. M. Barnett and R. Loudon, “On the electromagnetic force on a dielectric medium,” J. Phys. At. Mol. Opt. Phys. |

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

16. | 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. |

17. | E. A. Hinds and S. M. Barnett, “Momentum exchange between light and a single atom: Abraham or Minkowski?” Phys. Rev. Lett. |

18. | S. M. Barnett, “Resolution of the abraham-minkowski dilemma,” Phys. Rev. Lett. |

19. | P. L. Saldanha, “Division of the momentum of electromagnetic waves in linear media into electromagnetic and material parts,” Opt. Express |

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

21. | W. She, J. Yu, and R. Feng, “Observation of a push force on the end face of a nanometer silica filament exerted by outgoing light,” Phys. Rev. Lett. |

22. | M. Mansuripur and A. R. Zakharian, “Theoretical analysis of the force on the end face of a nanofilament exerted by an outing light pulse,” Phys. Rev. A |

23. | H. Yu, W. Fang, F. Gu, M. Qiu, Z. Yang, and L. Tong, “Longitudinal Lorentz force on a subwavelength-diameter optical fiber,” Phys. Rev. A |

24. | I. Brevik and S. A. Ellingsen, “Transverse radiation force in a tailored optical fiber,” Phys. Rev. A |

25. | I. Brevik, “Comment on “Observation of a push force on the end face of a nanometer silica filament exerted by outgoing light”,” Phys. Rev. Lett. |

26. | W. She, J. Yu, and R. Feng, “She et al. Reply,” Phys. Rev. Lett. |

27. | M. Mansuripur, “Comment on “Observation of a push force on the end face of a nanometer silica filament exerted by outgoing light”,” Phys. Rev. Lett. |

28. | J. Yu, R. Feng, and W. She, “Low-power all-optical switch based on the bend effect of a nm fiber taper driven by outgoing light,” Opt. Express |

29. | L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature |

30. | L. Tong, J. Lou, and E. Mazur, “Single-mode guiding properties of subwavelength-diameter silica and silicon wire waveguides,” Opt. Express |

31. | P. Russell, “Photonic crystal fibers,” Science |

32. | B. J. Eggleton, C. Kerbage, P. S. Westbrook, R. S. Windeler, and A. Hale, “Microstructured optical fiber devices,” Opt. Express |

33. | V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. |

34. | Q. Xu, V. R. Almeida, R. R. Panepucci, and M. Lipson, “Experimental demonstration of guiding and confining light in nanometer-size low-refractive-index material,” Opt. Lett. |

35. | A. W. Snyder and J. D. Love, |

36. | A. Talflove and S. C. Hagness, |

37. | A. Yariv and P. Yeh, |

**OCIS Codes**

(000.2690) General : General physics

(230.7370) Optical devices : Waveguides

(260.2110) Physical optics : Electromagnetic optics

**ToC Category:**

Physical Optics

**History**

Original Manuscript: September 2, 2011

Revised Manuscript: October 3, 2011

Manuscript Accepted: November 13, 2011

Published: November 23, 2011

**Citation**

Jianhui Yu, Chunyan Chen, Yanfang Zhai, Zhe Chen, Jun Zhang, Lijun Wu, Furong Huang, and Yi Xiao, "Total longitudinal momentum in a dispersive optical waveguide," Opt. Express **19**, 25263-25278 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-25-25263

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### References

- R. N. C. Pfeifer, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Colloquium: momentum of an electromagnetic wave in dielectric media,” Rev. Mod. Phys.79(4), 1197–1216 (2007). [CrossRef]
- I. Brevik, “Experiment in phenomenological electrodynamics and the electromagnetic energy-momentum tensor,” Phys. Rep.52(3), 133–201 (1979). [CrossRef]
- H. Minkowski, ““Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern,” Nachr. Ges. Wiss. Gottn Math.-Phys. Kl.1908, 53–111 (1908); reprinted in Math Ann 68, 472–525 (1910).
- J. A. Kong, Electromagnetic Wave Theory (Wiley, 1986), Chap. VII.
- P. W. Milonni and R. W. Boyd, “Momentum of light in a dielectric medium,” Adv. Opt. Photonics2(4), 519–553 (2010). [CrossRef]
- C. Baxter and R. Loudon, “Radiation pressure and the photon momentum in dielectrics,” J. Mod. Opt.57(10), 830–842 (2010). [CrossRef]
- V. M. Abraham, “Zur Elektrodynamik bewegter Körper,” Rend. Circ. Matem. Palermo28(1), 1–28 (1909). [CrossRef]
- D. F. Nelson, “Momentum, pseudomomentum, and wave momentum: Toward resolving the Minkowski-Abraham controversy,” Phys. Rev. A44(6), 3985–3996 (1991). [CrossRef] [PubMed]
- A. Feigel, “Quantum vacuum contribution to the momentum of dielectric media,” Phys. Rev. Lett.92(2), 020404 (2004). [CrossRef] [PubMed]
- C. Wang, “ Wave four-vector in a moving medium and the Lorentz covariance of Minkowski's photon and electromagnetic momentums,” arXiv.org, arXiv:1106.1163v11 (2011).
- R. Loudon, “Radiation pressure and momentum in dielectrics,” Fortschr. Phys.52(11-12), 1134–1140 (2004). [CrossRef]
- R. Loudon, S. M. Barnett, and C. Baxter, “Radation pressure and momentum transfer in dielectrics: The photon drag effect,” Phys. Rev. A71(6), 063802 (2005). [CrossRef]
- M. Mansuripur, “Radiation pressure and the linear momentum of the electromagnetic field,” Opt. Express12(22), 5375–5401 (2004). [CrossRef] [PubMed]
- S. M. Barnett and R. Loudon, “On the electromagnetic force on a dielectric medium,” J. Phys. At. Mol. Opt. Phys.39(15), S671–S684 (2006). [CrossRef]
- B. A. Kemp, T. Grzegorczyk, and J. A. Kong, “Ab initio study of the radiation pressure on dielectric and magnetic media,” Opt. Express13(23), 9280–9291 (2005). [CrossRef] [PubMed]
- 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(2), 026608 (2009). [CrossRef] [PubMed]
- E. A. Hinds and S. M. Barnett, “Momentum exchange between light and a single atom: Abraham or Minkowski?” Phys. Rev. Lett.102(5), 050403 (2009). [CrossRef] [PubMed]
- S. M. Barnett, “Resolution of the abraham-minkowski dilemma,” Phys. Rev. Lett.104(7), 070401 (2010). [CrossRef] [PubMed]
- P. L. Saldanha, “Division of the momentum of electromagnetic waves in linear media into electromagnetic and material parts,” Opt. Express18(3), 2258–2268 (2010). [CrossRef] [PubMed]
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