Cite this article as:

Tsoy V. I. Electromagnetic Momenta at Wave Refraction into Dispersive Medium. Izvestiya of Saratov University. New series. Series Physics, 2018, vol. 18, iss. 1, pp. 23-31. DOI:


Electromagnetic Momenta at Wave Refraction into Dispersive Medium


Background and Objectives: There are two types of a field momentum in the classical electrodynamics, namely, the Abraham momentum, and the Minkowski momentum. The question arises how these momenta are conserved in the refraction on the boundary surface of a medium.

Methods: The electromagnetic stress tensor and momentum density in a dispersive medium, including the case of negative permittivity and negative permeability, are derived and used in this paper.

Results and Conclusion: It is shown that the momentum of an incidence wave is conserved as the sum of the Abraham momentum, the Abraham force momentum, and the momentum of the electromagnetic pressure on the boundary surface of the medium. In the same time, the tangential component of the incidence momentum is conserved as the Minkowski momentum. The reason is that there is an invariance of the "vacuum–plane–matter" system in the transfer along the boundary surface of the medium. Consequently, a quasi-momentum should be conserved in this direction. On the other hand, there is no such symmetry in the direction perpendicular to the boundary surface of the medium. Consequently, any quasi-momentum in this direction does not exist. This shows that the Abraham momentum and the Minkowski momentum can work together, although these quantities corresponds to the different expansions of the total momentum into field and media parts.


1. Griffits D. J. Resource Letter EM-1: Electromagnetic Momentum. Amer. J. Phys., 2012, vol. 80, pp. 7–18. DOI:

2. Landau L. D., Lifshitz E. M., Pitaevskii L. P. Electrodynamics of Continuous Media. 2nd ed. ButterworthHeinemann, 1984. Vol. 8. 460 p. (Russ. ed.: Landau L. D., Lifshits E. M. Elektrodinamika sploshnykh sred [Electrodynamics of Continuous Media]. Moscow, Nauka Publ., 1982. 620 p.). 

3. Walker G. B., Lahoz D. G. Experimental observation of Abraham force in a dielectric. Nature, 1975, vol. 253, pp. 339–340. 

4. Jones R. V., Leslie B. The measurement of optical radiation pressure in dispersive media. Proc. R. Soc. Lond. A, 1978, vol. 360, pp. 347–363. DOI:

5. Polevoi V. G., Rytov S. M. The four-dimensional group velocity. Sov. Phys. Usp., 1978, vol. 21, pp. 630–638. DOI:

6. Makarov V. P., Rukhadze A. A. Negative group velocity electromagnetic waves and the energy-momentum tensor. Phys. Usp., 2011, vol. 54, pp. 1285–1296. DOI:

7. Toptygin I. N., Levina K. Energy-momentum tensor of the electromagnetic field in dispersive media. Phys. Usp., 2016, vol. 59, pp. 141–152. DOI:

8. Veselago V. G. The electrodynamics of substances with simultaneously negative values of ε and μ . Sov. Phys.Usp., 1968, vol. 10, pp. 509–514. DOI:

9. Mansuripur M. Resolution of the Abraham-Minkowski Controversy. Opt. Commun., 2010, vol. 283, pp. 1997–2006. 

10. Barnett S. M. Resolution of the Abraham–Minkowski Dilemma. Phys. Rev. Lett., 2010, vol. 104, pp. 070401. DOI:

11. Peierls R. Impul’s i kvaziimpuls sveta i zvuka [Momentum and Pseudomomentum of Light and Sound]. UFN, 1991, vol. 161, pp. 161–176. DOI: (in Russian). 

12. Landau L. D., Lifshitz E. M. The classical theory of fields. Oxford, Pergamon Press, 1971. 374 p. (in Russ. ed.: Landau L. D., Lifshits E. M. Teoriya polya [The classical theory of fields]. Moscow: Nauka Publ., 1988. 509 p.). 

13. Mandelshtam L. I. Lektsii po teorii otnositel’nosti i kvantovoy mekhanike [Lectures on the relativity and quantum mechanics]. Moscow, Nauka Publ., 1972. 440 p. (in Russian). 

14. Pafomov V. E. Transition Radiation and Cerenkov Radiation. Soviet Physics JETF, 1959, vol. 36, pp. 1321–1324. 

15. Veselago V. G. Electrodynamics of materials with negative index of refraction, Phys.Usp., 2003, vol. 46, pp. 764–768. DOI:

16. Veselago V. G. Energy, linear momentum and mass transfer by an electromagnetic wave in a negative refraction medium. Phys. Usp., 2009, vol. 52, pp. 649–654. DOI:

17. Rautian S. G. Reflection and refraction at the boundary of a medium with negative group velocity. Phys. Usp., 2008, vol. 51, pp. 981–988. DOI:

18. Ginzburg V. L. The laws of conservation of energy and momentum in emission on electromagnetic waves (photons) in a medium and the energy-momentum tensor in macroscopic electrodynamics Sov. Phys. Usp., 1973, vol. 16, pp. 434–439. DOI:

19. Shеng Xi, Hongsheng Chen, Tao Jian, Lixin Ran, Jiangtao Huangfu, Bac-Ian Wu, Jin Au Kong, Min Chen. Experimental Verification of Reversed Cherenkov Radiation in Left-Handed Metamaterial. Phys. Rev. Lett., 2009, vol. 103, pp. 194801. DOI:

20. Campbell G. K., Leanhardt A. E., Mun J., Boyd M., Streed W., Ketterle W., Pritchhard D. E. Photon Recoil Momentum in Dispersive Media. Phys. Rev. Lett., 2005, vol. 94, pp. 170403. DOI:

21. Frank I. M. Photon momentum in a medium with negative group velocity. JETP Lett., 1978, vol. 28, pp. 446–448. 

22. Davidovich M. V. On energy and momentum conservation laws for an electromagnetic field in a medium or at diffraction on a conducting plate. Phys. Usp., 2010, vol. 180, pp. 623–638. DOI:

23. Chu C. Upravleniye neyitralnymi chastitsami [The Manipulation of Neutral Particles]. UFN, 1999, vol. 169, pp. 274–291. DOI: (in Russian).

Short text (in English): 
Full text (in Russian):