Abstract

We propose a set of postulates to describe the mechanical interaction between a plane-wave electromagnetic pulse and a dispersive, dissipative slab having a refractive index of arbitrary sign. The postulates include the Abraham electromagnetic momentum density, a generalized Lorentz force law, and a model for absorption-driven mass transfer from the pulse to the medium. These opto-mechanical mechanisms are incorporated into a one-dimensional finite-difference time-domain algorithm that solves Maxwell’s equations and calculates the instantaneous force densities exerted by the pulse onto the slab, the momentum-per-unit-area of the pulse and slab, and the trajectories of the slab and system center-of-mass. We show that the postulates are consistent with conservation of global energy, momentum, and center-of-mass velocity at all times, even for cases in which the refractive index of the slab is negative or zero. Consistency between the set of postulates and well-established conservation laws reinforces the Abraham momentum density as the one true electromagnetic momentum density and enables, for the first time, identification of the correct form of the electromagnetic mass density distribution and development of an explicit model for mass transfer due to absorption, for the most general case of a ponderable medium that is both dispersive and dissipative.

© 2012 OSA

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  1. J. C. Maxwell, A Treatise on Electricity and Magnetism (Dover, 1891), Vol. 2.
  2. P. N. Lebedev, “Untersuchungen über die druckkräfte des lichtes,” Ann. Phys. 6, 433–458 (1901).
    [CrossRef]
  3. E. F. Nichols and G. F. Hull, “A preliminary communication on the pressure of heat and light radiation,” Phys. Rev. 13, 307–320 (1901).
  4. H. Minkowski, “Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern,” Nachr. Ges. Wiss. Goettingen, Math.-Phys. Kl.53–111 (1908).
  5. K. J. Chau and H. J. Lezec, “Re-visiting the Balazs thought experiment in the presence of loss: electromagnetic-pulse-induced displacement of a positive-index slab having arbitrary complex permittivity and permeability,” Appl. Phys. A 105, 267–281 (2011).
    [CrossRef]
  6. M. Abraham, “Zur Elektrodynamik bewegter Körper,” R. C. Circ. Mat. Palermo 28, 1–28 (1909).
    [CrossRef]
  7. R. N. C. Pfeifer, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Momentum of an electromagentic wave in dielectric media,” Rev. Mod. Phys. 79, 1197–1216 (2007).
    [CrossRef]
  8. P. Penfield and H. A. Haus, Electrodynamics of Moving Media (M.I.T., 1960).
  9. S. R. dr Groot and L. G. Suttorp, Foundations of Electrodynamics (North Holland, 1972).
  10. R. Peierls, “The momentum of light in a refracting medium,” Proc. R. Soc. Lond. A 347, 475–491 (1976).
    [CrossRef]
  11. P. W. Milonni and W. Boyd, “Recoil and photon momentum in a dielectric,” Laser Phys. 15, 1–7 (2005).
  12. S. Stallinga, “Energy and momentum of light in dielectric media,” Phys. Rev. E 73, 026606 (2006).
    [CrossRef]
  13. B. A. Kemp, J. A. Kong, and T. M. Grzegorczyk, “Reversal of wave momentum in isotropic left-handed media,” Phys. Rev. A 75, 053810 (2007).
    [CrossRef]
  14. P. L. Saldanha, “Division of the momentum of electromagnetic waves in linear media into electromagnetic and material parts,” Opt. Express 18, 2258–2268 (2010).
    [CrossRef] [PubMed]
  15. J. P. Gordon, “Radiation forces and momenta in dielectric media,” Phys. Rev. A 8, 14–21 (1973).
    [CrossRef]
  16. N. L. Balazs, “The energy-momentum tensor of the electromagnetic field inside matter,” Phys. Rev. 91, 408–411 (1953).
    [CrossRef]
  17. A. Einstein, “The principle of conversation of motion of the center of gravity and the inertia of energy,” Ann. Phys. 20, 627–633 (1906).
    [CrossRef]
  18. V. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
    [CrossRef]
  19. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (1999).
    [CrossRef]
  20. G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780 nm wavelength,” Opt. Lett. 32, 53–55 (2007).
    [CrossRef]
  21. V. M. Shalaev, W. Cai, U. K. Chettiar, H.-K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. 30, 3356–3358 (2005).
    [CrossRef]
  22. H. J. Lezec, J. A. Dionne, and H. A. Atwater, “Negative refraction at visible frequencies,” Science 316, 430–432 (2007).
    [CrossRef] [PubMed]
  23. M. Mansuripur and A. R. Zakharian, “Energy, momentum, and force in classical electrodynamics: application to negative-index media,” Proc. SPIE OP101, 73920Q-1 (2009).
  24. M. Mansuripur and A. R. Zakharian, “Energy, momentum, and force in classical electrodynamics: application to negative-index media,” Opt. Commun. 283, 4594–4600 (2010).
    [CrossRef]
  25. R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64, 056625 (2001).
    [CrossRef]
  26. R. W. Ziolkowski, “Superluminal transmission of information through an electromagnetic metamaterial,” Phys. Rev. E 63, 046604 (2001).
    [CrossRef]
  27. R. A. Depine and A. Lakhtakia, “A new condition to identify isotropic dielectric-magnetic materials displaying negative phase velocity,” Microw. Opt. Technol. Lett. 41, 315–316 (2004).
    [CrossRef]
  28. A. Einstein and J. Laub, “On the ponderomotive forces exerted on bodies at rest in the electromagnetic field,” Ann. Phys. 26, 541–550 (1908).
    [CrossRef]
  29. M. Mansuripur, “Radiation pressure and the linear momentum of the electromagnetic field in magnetic media,” Opt. Express 15, 13502–13518 (2007).
    [CrossRef] [PubMed]
  30. M. Mansuripur and A. R. Zakharian, “Maxwell’s macroscopic equations, the energy-momentum postulates, and the Lorentz law of force,” Phys. Rev. E 79, 026608 (2009).
    [CrossRef]
  31. M. Mansuripur, “Resolution of the Abraham-Minkowski controversy,” Opt. Commun. 283, 1997–2005 (2010).
    [CrossRef]
  32. W. Shockley and R. P. James, ““Try simplest cases” discovery of “hidden momentum” forces on “magnetic currents”,” Phys. Rev. Lett. 18, 876–879 (1967).
    [CrossRef]
  33. I. Brevik, “Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor,” Phys. Rep. 52, 133–201 (1979).
    [CrossRef]
  34. B. A. Kemp, “Resolution of the Abraham-Minkowski debate: implications for the electromagnetic wave theory of light in matter,” J. Appl. Phys. 109, 111101 (2011).
    [CrossRef]
  35. R. Ruppin, “Electromagnetic energy density in a dispersive and absorptive material,” Phys. Lett. A 299, 309–312 (2002).
    [CrossRef]
  36. R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A 3, 233–245 (1970).
    [CrossRef]

2011

K. J. Chau and H. J. Lezec, “Re-visiting the Balazs thought experiment in the presence of loss: electromagnetic-pulse-induced displacement of a positive-index slab having arbitrary complex permittivity and permeability,” Appl. Phys. A 105, 267–281 (2011).
[CrossRef]

B. A. Kemp, “Resolution of the Abraham-Minkowski debate: implications for the electromagnetic wave theory of light in matter,” J. Appl. Phys. 109, 111101 (2011).
[CrossRef]

2010

M. Mansuripur, “Resolution of the Abraham-Minkowski controversy,” Opt. Commun. 283, 1997–2005 (2010).
[CrossRef]

M. Mansuripur and A. R. Zakharian, “Energy, momentum, and force in classical electrodynamics: application to negative-index media,” Opt. Commun. 283, 4594–4600 (2010).
[CrossRef]

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

2009

M. Mansuripur and A. R. Zakharian, “Energy, momentum, and force in classical electrodynamics: application to negative-index media,” Proc. SPIE OP101, 73920Q-1 (2009).

M. Mansuripur and A. R. Zakharian, “Maxwell’s macroscopic equations, the energy-momentum postulates, and the Lorentz law of force,” Phys. Rev. E 79, 026608 (2009).
[CrossRef]

2007

M. Mansuripur, “Radiation pressure and the linear momentum of the electromagnetic field in magnetic media,” Opt. Express 15, 13502–13518 (2007).
[CrossRef] [PubMed]

H. J. Lezec, J. A. Dionne, and H. A. Atwater, “Negative refraction at visible frequencies,” Science 316, 430–432 (2007).
[CrossRef] [PubMed]

G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780 nm wavelength,” Opt. Lett. 32, 53–55 (2007).
[CrossRef]

B. A. Kemp, J. A. Kong, and T. M. Grzegorczyk, “Reversal of wave momentum in isotropic left-handed media,” Phys. Rev. A 75, 053810 (2007).
[CrossRef]

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

2006

S. Stallinga, “Energy and momentum of light in dielectric media,” Phys. Rev. E 73, 026606 (2006).
[CrossRef]

2005

2004

R. A. Depine and A. Lakhtakia, “A new condition to identify isotropic dielectric-magnetic materials displaying negative phase velocity,” Microw. Opt. Technol. Lett. 41, 315–316 (2004).
[CrossRef]

2002

R. Ruppin, “Electromagnetic energy density in a dispersive and absorptive material,” Phys. Lett. A 299, 309–312 (2002).
[CrossRef]

2001

R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64, 056625 (2001).
[CrossRef]

R. W. Ziolkowski, “Superluminal transmission of information through an electromagnetic metamaterial,” Phys. Rev. E 63, 046604 (2001).
[CrossRef]

1999

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (1999).
[CrossRef]

1979

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

1976

R. Peierls, “The momentum of light in a refracting medium,” Proc. R. Soc. Lond. A 347, 475–491 (1976).
[CrossRef]

1973

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

1970

R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A 3, 233–245 (1970).
[CrossRef]

1968

V. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

1967

W. Shockley and R. P. James, ““Try simplest cases” discovery of “hidden momentum” forces on “magnetic currents”,” Phys. Rev. Lett. 18, 876–879 (1967).
[CrossRef]

1953

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

1909

M. Abraham, “Zur Elektrodynamik bewegter Körper,” R. C. Circ. Mat. Palermo 28, 1–28 (1909).
[CrossRef]

1908

H. Minkowski, “Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern,” Nachr. Ges. Wiss. Goettingen, Math.-Phys. Kl.53–111 (1908).

A. Einstein and J. Laub, “On the ponderomotive forces exerted on bodies at rest in the electromagnetic field,” Ann. Phys. 26, 541–550 (1908).
[CrossRef]

1906

A. Einstein, “The principle of conversation of motion of the center of gravity and the inertia of energy,” Ann. Phys. 20, 627–633 (1906).
[CrossRef]

1901

P. N. Lebedev, “Untersuchungen über die druckkräfte des lichtes,” Ann. Phys. 6, 433–458 (1901).
[CrossRef]

E. F. Nichols and G. F. Hull, “A preliminary communication on the pressure of heat and light radiation,” Phys. Rev. 13, 307–320 (1901).

Abraham, M.

M. Abraham, “Zur Elektrodynamik bewegter Körper,” R. C. Circ. Mat. Palermo 28, 1–28 (1909).
[CrossRef]

Atwater, H. A.

H. J. Lezec, J. A. Dionne, and H. A. Atwater, “Negative refraction at visible frequencies,” Science 316, 430–432 (2007).
[CrossRef] [PubMed]

Balazs, N. L.

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

Boyd, W.

P. W. Milonni and W. Boyd, “Recoil and photon momentum in a dielectric,” Laser Phys. 15, 1–7 (2005).

Brevik, I.

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

Cai, W.

Chau, K. J.

K. J. Chau and H. J. Lezec, “Re-visiting the Balazs thought experiment in the presence of loss: electromagnetic-pulse-induced displacement of a positive-index slab having arbitrary complex permittivity and permeability,” Appl. Phys. A 105, 267–281 (2011).
[CrossRef]

Chettiar, U. K.

Depine, R. A.

R. A. Depine and A. Lakhtakia, “A new condition to identify isotropic dielectric-magnetic materials displaying negative phase velocity,” Microw. Opt. Technol. Lett. 41, 315–316 (2004).
[CrossRef]

Dionne, J. A.

H. J. Lezec, J. A. Dionne, and H. A. Atwater, “Negative refraction at visible frequencies,” Science 316, 430–432 (2007).
[CrossRef] [PubMed]

Dolling, G.

dr Groot, S. R.

S. R. dr Groot and L. G. Suttorp, Foundations of Electrodynamics (North Holland, 1972).

Drachev, V. P.

Einstein, A.

A. Einstein and J. Laub, “On the ponderomotive forces exerted on bodies at rest in the electromagnetic field,” Ann. Phys. 26, 541–550 (1908).
[CrossRef]

A. Einstein, “The principle of conversation of motion of the center of gravity and the inertia of energy,” Ann. Phys. 20, 627–633 (1906).
[CrossRef]

Gordon, J. P.

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

Grzegorczyk, T. M.

B. A. Kemp, J. A. Kong, and T. M. Grzegorczyk, “Reversal of wave momentum in isotropic left-handed media,” Phys. Rev. A 75, 053810 (2007).
[CrossRef]

Haus, H. A.

P. Penfield and H. A. Haus, Electrodynamics of Moving Media (M.I.T., 1960).

Heckenberg, N. R.

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

Heyman, E.

R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64, 056625 (2001).
[CrossRef]

Hull, G. F.

E. F. Nichols and G. F. Hull, “A preliminary communication on the pressure of heat and light radiation,” Phys. Rev. 13, 307–320 (1901).

James, R. P.

W. Shockley and R. P. James, ““Try simplest cases” discovery of “hidden momentum” forces on “magnetic currents”,” Phys. Rev. Lett. 18, 876–879 (1967).
[CrossRef]

Kemp, B. A.

B. A. Kemp, “Resolution of the Abraham-Minkowski debate: implications for the electromagnetic wave theory of light in matter,” J. Appl. Phys. 109, 111101 (2011).
[CrossRef]

B. A. Kemp, J. A. Kong, and T. M. Grzegorczyk, “Reversal of wave momentum in isotropic left-handed media,” Phys. Rev. A 75, 053810 (2007).
[CrossRef]

Kildishev, A. V.

Kong, J. A.

B. A. Kemp, J. A. Kong, and T. M. Grzegorczyk, “Reversal of wave momentum in isotropic left-handed media,” Phys. Rev. A 75, 053810 (2007).
[CrossRef]

Lakhtakia, A.

R. A. Depine and A. Lakhtakia, “A new condition to identify isotropic dielectric-magnetic materials displaying negative phase velocity,” Microw. Opt. Technol. Lett. 41, 315–316 (2004).
[CrossRef]

Laub, J.

A. Einstein and J. Laub, “On the ponderomotive forces exerted on bodies at rest in the electromagnetic field,” Ann. Phys. 26, 541–550 (1908).
[CrossRef]

Lebedev, P. N.

P. N. Lebedev, “Untersuchungen über die druckkräfte des lichtes,” Ann. Phys. 6, 433–458 (1901).
[CrossRef]

Lezec, H. J.

K. J. Chau and H. J. Lezec, “Re-visiting the Balazs thought experiment in the presence of loss: electromagnetic-pulse-induced displacement of a positive-index slab having arbitrary complex permittivity and permeability,” Appl. Phys. A 105, 267–281 (2011).
[CrossRef]

H. J. Lezec, J. A. Dionne, and H. A. Atwater, “Negative refraction at visible frequencies,” Science 316, 430–432 (2007).
[CrossRef] [PubMed]

Linden, S.

Loudon, R.

R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A 3, 233–245 (1970).
[CrossRef]

Mansuripur, M.

M. Mansuripur, “Resolution of the Abraham-Minkowski controversy,” Opt. Commun. 283, 1997–2005 (2010).
[CrossRef]

M. Mansuripur and A. R. Zakharian, “Energy, momentum, and force in classical electrodynamics: application to negative-index media,” Opt. Commun. 283, 4594–4600 (2010).
[CrossRef]

M. Mansuripur and A. R. Zakharian, “Maxwell’s macroscopic equations, the energy-momentum postulates, and the Lorentz law of force,” Phys. Rev. E 79, 026608 (2009).
[CrossRef]

M. Mansuripur and A. R. Zakharian, “Energy, momentum, and force in classical electrodynamics: application to negative-index media,” Proc. SPIE OP101, 73920Q-1 (2009).

M. Mansuripur, “Radiation pressure and the linear momentum of the electromagnetic field in magnetic media,” Opt. Express 15, 13502–13518 (2007).
[CrossRef] [PubMed]

Maxwell, J. C.

J. C. Maxwell, A Treatise on Electricity and Magnetism (Dover, 1891), Vol. 2.

Milonni, P. W.

P. W. Milonni and W. Boyd, “Recoil and photon momentum in a dielectric,” Laser Phys. 15, 1–7 (2005).

Minkowski, H.

H. Minkowski, “Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern,” Nachr. Ges. Wiss. Goettingen, Math.-Phys. Kl.53–111 (1908).

Nichols, E. F.

E. F. Nichols and G. F. Hull, “A preliminary communication on the pressure of heat and light radiation,” Phys. Rev. 13, 307–320 (1901).

Nieminen, T. A.

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

Peierls, R.

R. Peierls, “The momentum of light in a refracting medium,” Proc. R. Soc. Lond. A 347, 475–491 (1976).
[CrossRef]

Penfield, P.

P. Penfield and H. A. Haus, Electrodynamics of Moving Media (M.I.T., 1960).

Pfeifer, R. N. C.

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

Rubinsztein-Dunlop, H.

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

Ruppin, R.

R. Ruppin, “Electromagnetic energy density in a dispersive and absorptive material,” Phys. Lett. A 299, 309–312 (2002).
[CrossRef]

Saldanha, P. L.

Sarychev, A. K.

Schultz, S.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (1999).
[CrossRef]

Shalaev, V. M.

Shelby, R. A.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (1999).
[CrossRef]

Shockley, W.

W. Shockley and R. P. James, ““Try simplest cases” discovery of “hidden momentum” forces on “magnetic currents”,” Phys. Rev. Lett. 18, 876–879 (1967).
[CrossRef]

Smith, D. R.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (1999).
[CrossRef]

Soukoulis, C. M.

Stallinga, S.

S. Stallinga, “Energy and momentum of light in dielectric media,” Phys. Rev. E 73, 026606 (2006).
[CrossRef]

Suttorp, L. G.

S. R. dr Groot and L. G. Suttorp, Foundations of Electrodynamics (North Holland, 1972).

Veselago, V.

V. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

Wegener, M.

Yuan, H.-K.

Zakharian, A. R.

M. Mansuripur and A. R. Zakharian, “Energy, momentum, and force in classical electrodynamics: application to negative-index media,” Opt. Commun. 283, 4594–4600 (2010).
[CrossRef]

M. Mansuripur and A. R. Zakharian, “Maxwell’s macroscopic equations, the energy-momentum postulates, and the Lorentz law of force,” Phys. Rev. E 79, 026608 (2009).
[CrossRef]

M. Mansuripur and A. R. Zakharian, “Energy, momentum, and force in classical electrodynamics: application to negative-index media,” Proc. SPIE OP101, 73920Q-1 (2009).

Ziolkowski, R. W.

R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64, 056625 (2001).
[CrossRef]

R. W. Ziolkowski, “Superluminal transmission of information through an electromagnetic metamaterial,” Phys. Rev. E 63, 046604 (2001).
[CrossRef]

Ann. Phys.

P. N. Lebedev, “Untersuchungen über die druckkräfte des lichtes,” Ann. Phys. 6, 433–458 (1901).
[CrossRef]

A. Einstein and J. Laub, “On the ponderomotive forces exerted on bodies at rest in the electromagnetic field,” Ann. Phys. 26, 541–550 (1908).
[CrossRef]

A. Einstein, “The principle of conversation of motion of the center of gravity and the inertia of energy,” Ann. Phys. 20, 627–633 (1906).
[CrossRef]

Appl. Phys. A

K. J. Chau and H. J. Lezec, “Re-visiting the Balazs thought experiment in the presence of loss: electromagnetic-pulse-induced displacement of a positive-index slab having arbitrary complex permittivity and permeability,” Appl. Phys. A 105, 267–281 (2011).
[CrossRef]

J. Appl. Phys.

B. A. Kemp, “Resolution of the Abraham-Minkowski debate: implications for the electromagnetic wave theory of light in matter,” J. Appl. Phys. 109, 111101 (2011).
[CrossRef]

J. Phys. A

R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A 3, 233–245 (1970).
[CrossRef]

Laser Phys.

P. W. Milonni and W. Boyd, “Recoil and photon momentum in a dielectric,” Laser Phys. 15, 1–7 (2005).

Microw. Opt. Technol. Lett.

R. A. Depine and A. Lakhtakia, “A new condition to identify isotropic dielectric-magnetic materials displaying negative phase velocity,” Microw. Opt. Technol. Lett. 41, 315–316 (2004).
[CrossRef]

Nachr. Ges. Wiss. Goettingen, Math.-Phys. Kl.

H. Minkowski, “Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern,” Nachr. Ges. Wiss. Goettingen, Math.-Phys. Kl.53–111 (1908).

Opt. Commun.

M. Mansuripur and A. R. Zakharian, “Energy, momentum, and force in classical electrodynamics: application to negative-index media,” Opt. Commun. 283, 4594–4600 (2010).
[CrossRef]

M. Mansuripur, “Resolution of the Abraham-Minkowski controversy,” Opt. Commun. 283, 1997–2005 (2010).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Lett. A

R. Ruppin, “Electromagnetic energy density in a dispersive and absorptive material,” Phys. Lett. A 299, 309–312 (2002).
[CrossRef]

Phys. Rep.

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

Phys. Rev.

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

E. F. Nichols and G. F. Hull, “A preliminary communication on the pressure of heat and light radiation,” Phys. Rev. 13, 307–320 (1901).

Phys. Rev. A

B. A. Kemp, J. A. Kong, and T. M. Grzegorczyk, “Reversal of wave momentum in isotropic left-handed media,” Phys. Rev. A 75, 053810 (2007).
[CrossRef]

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

Phys. Rev. E

M. Mansuripur and A. R. Zakharian, “Maxwell’s macroscopic equations, the energy-momentum postulates, and the Lorentz law of force,” Phys. Rev. E 79, 026608 (2009).
[CrossRef]

S. Stallinga, “Energy and momentum of light in dielectric media,” Phys. Rev. E 73, 026606 (2006).
[CrossRef]

R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64, 056625 (2001).
[CrossRef]

R. W. Ziolkowski, “Superluminal transmission of information through an electromagnetic metamaterial,” Phys. Rev. E 63, 046604 (2001).
[CrossRef]

Phys. Rev. Lett.

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Figures (15)

Fig. 1.
Fig. 1.

Formulation of the Balazs thought experiment. Two identical enclosures each contain a photon of mass m and a non-dispersive, lossless slab of mass M and length L. (a) In enclosure 1, the photon propagates in a straight line above the slab through only vacuum. (b) In enclosure 2, the photon propagates in a straight line through both vacuum and the slab.

Fig. 2.
Fig. 2.

(a) Power spectrum of the incident electromagnetic pulse. The complex refractive index of the slab is set by adjusting ε, μ, ωpe, ωpm, Γe, and Γm to vary the values of ε̲r and μ̲r over the bandwidth of the incident pulse. Under the assumption of impedance matching, ε̲r = μ̲r = . A positive-index is realized by setting Re[ε̲r] > 0 and Re[μ̲r] > 0 (right-handed material), a negative-index by setting Re[ε̲r] < 0 and Re[μ̲r] < 0 (left-handed material) and a zero-index by setting Re[ε̲r] ≃ 0 and Re[μ̲r] ≃0. The real (solid) and imaginary (dash) parts of ε̲r, μ̲r, and for (b) test case 1 (blue) and test case 2 (red), (c) test case 3 and test case 4, and (d) test case 5. The material parameters corresponding to each of the test cases are summarized in Table 2.

Fig. 3.
Fig. 3.

Spatio-temporal grid used for the finite-difference time-domain calculations highlighting the discretization of the electric field, magnetic field, electric current density, and magnetic current density.

Fig. 4.
Fig. 4.

Time sequence of the FDTD-calculated (a) electric field and (b) force density for a pulse incident onto a positive-index slab without loss (test case 1). FDTD-calculated (c) electric field and (d) force density for a pulse incident onto a positive-index slab with loss (test case 2). For clarity, the curves have been offset such that the horizontal asymptotic value of each curve corresponds to zero values. The pulse amplitude is normalized such that the total pulse power is 1W. The slab has a length L = 750nm and a mass M = 1kg. The dashed lines indicate the edges of the slab.

Fig. 5.
Fig. 5.

Time sequence of the FDTD-calculated (a) electric field and (b) force density for a pulse incident onto a negative-index slab without loss (test case 1). FDTD-calculated (c) electric field and (d) force density for a pulse incident onto a negative-index slab with loss (test case 2). For clarity, the curves have been offset such that the horizontal asymptotic value of each curve corresponds to zero values. The pulse amplitude is normalized such that the total pulse power is 1W. The dashed lines indicate the edges of the slab.

Fig. 6.
Fig. 6.

Time sequence of the FDTD-calculated (a) electric field and (b) force density for a pulse incident onto a zero-index slab without loss (test case 5). For clarity, the curves have been offset such that the horizontal asymptotic value of each curve corresponds to zero electric field. The pulse amplitude is normalized such that the total pulse power is 1W. The dotted lines indicate the edges of the slab.

Fig. 7.
Fig. 7.

(a) Instantaneous force-per-unit-area and (b) momentum-per-unit-area of the slab (red), pulse (blue), and system (black) for the case of a positive-index, lossless slab (test case 1). (c) Instantaneous force-per-unit-area and (d) momentum-per-unit-area of the slab (red), pulse (blue), and system (black) for the case of a positive-index, lossy slab (test case 2).

Fig. 8.
Fig. 8.

(a) Instantaneous force-per-unit-area and (b) momentum-per-unit-area of the slab (red), pulse (blue), and system (black) for the case of a negative-index, lossless slab (test case 3). (c) Instantaneous force-per-unit-area and (d) momentum-per-unit-area of the slab (red), pulse (blue), and system (black) for the case of a negative-index, lossy slab (test case 4).

Fig. 9.
Fig. 9.

(a) Instantaneous force-per-unit-area and (b) momentum-per-unit-area of the slab (red), pulse (blue), and system (black) for the case of a zero-index, lossless slab (test case 5).

Fig. 10.
Fig. 10.

Slab center-of-mass displacement for a pulse incident onto a slab for all five test cases studied.

Fig. 11.
Fig. 11.

System center-of-mass displacement for a pulse incident onto a slab for all five test cases studied.

Fig. 12.
Fig. 12.

Time sequence of the FDTD-calculated (a) electric field and (b) force density for a pulse incident onto an impedance-matched, non-dispersive, positive-index slab without loss. The force density is calculated using the form of the force density given by Eq. (54), which corresponds to the Minkowski momentum density. For clarity, the curves have been offset such that the horizontal asymptotic value of each curve corresponds to a zero quantity. The pulse amplitude is normalized such that the total pulse power is 1W. The slab has a length L = 750nm and a mass M = 1kg. The dotted lines indicate the edges of the slab.

Fig. 13.
Fig. 13.

(a) Instantaneous force density, (b) momentum-per-unit-area, (c) slab center-of-mass displacement, and (d) system center-of-mass displacement calculated for a pulse incident onto a lossless, non-dispersive, non-impedance-matched slab having a positive refractive index, using the Minkowski momentum density given by Eq. (50) and the force density given by Eq. (54).

Fig. 14.
Fig. 14.

Time sequence of the FDTD-calculated (a) electric field and (b) force density for a pulse incident onto a non-impedance-matched, negative-index slab with loss. For clarity, the curves have been offset such that the horizontal asymptotic value of each curve corresponds to zero values. The pulse amplitude is normalized such that the total pulse power is 1W. The slab has a length L = 750nm and a mass M = 1kg. The dotted lines indicate the edges of the slab.

Fig. 15.
Fig. 15.

(a) Instantaneous force density, (b) momentum-per-unit-area, (c) slab center-of-mass displacement, and (d) system center-of-mass displacement calculated for a pulse incident onto a dissipative, dispersive, non-impedance-matched slab having a negative real part of its refractive index.

Tables (2)

Tables Icon

Table 1. Assumptions and postulates used in our analysis of electromagnetic pulse interaction with a slab. The quantities in the equations are defined in the text.

Tables Icon

Table 2. Parameters used in the FDTD simulations for the five test cases.

Equations (54)

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× H = D t
× E = B t .
D = ε 0 E + P = ε 0 E + P s + P d = ε 0 E + ( ε 1 ) ε 0 E + P d ,
B = μ 0 H + M = μ 0 H + M s + M d = μ 0 H + ( μ 1 ) μ 0 H + M d ,
J e = P t = D t ε 0 E t
J m = M t = B t μ 0 H t .
J e = J es + J ed = P s t + P d t = ε 0 ( ε 1 ) E t + P d t
J m = J ms + J md = M s t + M d t = μ 0 ( μ 1 ) H t + M d t ,
J ed t + Γ e J ed = ε 0 ω pe 2 E
J md t + Γ m J md = μ 0 ω pm 2 H ,
J _ ed = ε 0 ω pe 2 Γ e i ω E _
J _ md = μ 0 ω pm 2 Γ m i ω H _ ,
× H _ = i ω D _ .
J _ e = i ω D _ + i ω ε 0 E _
J _ e = i ω ε 0 ( ε 1 ) E _ + J _ ed ,
× H _ = i ω ε ε 0 E _ + ε 0 ω pe 2 Γ e i ω E _ = i ω ε 0 ( ε ω pe 2 ω ( ω + i Γ e ) ) E _ = i ω ε 0 ε _ r ( ω ) E _ ,
ε _ r ( ω ) = ε ω pe 2 ω ( ω + i Γ e ) .
× E _ = i ω B _ .
J _ m = i ω B _ + i ω μ 0 H _
J _ m = i ω μ 0 ( μ 1 ) H _ + J _ md ,
× E _ = i ω μ μ 0 H _ μ 0 ω pm 2 Γ m i ω H _ = i ω μ 0 ( μ ω pm 2 ω ( ω + i Γ m ) ) H _ = i ω μ 0 μ _ r ( ω ) H _ ,
μ _ r ( ω ) = μ ω pm 2 ω ( ω + i Γ m ) .
n _ ( ω ) = Sign [ Re [ ε _ r ( ω ) ] | μ _ r ( ω ) | + Re [ μ _ r ( ω ) ] | ε _ r ( ω ) | ] ε _ r ( ω ) μ _ r ( ω ) ,
G = E × H c 2 .
f = ( P ) E + ( M ) H + J e + μ 0 H J m × ε 0 E ,
f = J e × μ 0 H J m × ε 0 E .
T ¯ ¯ + G t = f
T ¯ ¯ = ( P ) E ( M ) H J e × μ 0 H + J m × ε 0 E t ( E × H c 2 ) .
T ¯ ¯ = ( P E + M H ) P t × μ 0 H + M t × ε 0 E ε 0 E t × μ 0 H + μ 0 H t × ε 0 E .
T ¯ ¯ = ( D E ε 0 E E + B H μ 0 H H ) D t × μ 0 H + B t × ε 0 E .
T ¯ ¯ = ( D E B H ) + 1 2 ( μ 0 H H ) + 1 2 ( ε 0 E E ) .
T ¯ ¯ = D E B H + 1 2 ( μ 0 H 2 + ε 0 E 2 ) I ¯ ¯ ,
S = E × H .
S ( E × H ) d S = V ( E × H ) dV = V [ H ( × E ) E ( × H ) ] dV = V ( μ μ 0 H H t + H J md + ε ε 0 E E t + E J ed ) dV .
S ( E × H ) d S = V [ μ μ 0 H H t + 1 μ 0 ω p m 2 ( J m d t + Γ m J md ) J md + ε ε 0 E E t + 1 ε 0 ω pe 2 ( J ed t + Γ e J ed ) J ed ] dV .
S ( E × H ) d S + V ( Γ e J ed 2 ε 0 ω pe 2 + Γ m J md 2 μ 0 ω pm 2 ) dV = V t ( μ μ 0 H 2 2 + ε ε 0 E 2 2 + J ed 2 2 ε 0 ω pe 2 + J md 2 2 μ 0 ω pm 2 ) dV = V W t dV ,
W = μ μ 0 H 2 2 + ε ε 0 E 2 2 + J ed 2 2 ε 0 ω pe 3 + J md 2 2 μ 0 ω pm 2 ,
ρ = W c 2 .
H y n + 1 / 2 ( i + 1 / 2 ) = H y n 1 / 2 ( i + 1 / 2 ) Δ t μ 0 Δ z [ E x n ( i + 1 ) E x n ( i ) + J m , y n ( i + 1 / 2 ) Δ z ] J m , y n + 1 ( i + 1 / 2 ) = 1 0.5 Γ m Δ t 1 + 0.5 Γ m Δ t J m , y n ( i + 1 / 2 ) + μ 0 ω pm 2 Δ t 1 + 0.5 Γ m Δ t H y n + 1 / 2 ( i 1 / 2 ) E x n + 1 ( i ) = E x n ( i ) Δ t ε 0 Δ z [ H y n + 1 / 2 ( i + 1 / 2 ) H y n + 1 / 2 ( i 1 / 2 ) ] 1 2 Δ t ε 0 [ J e , x n + 1 / 2 ( i + 1 / 2 ) + J e , x n + 1 / 2 ( i 1 / 2 ) ] J e , x n + 3 / 2 ( i + 1 / 2 ) = 1 0.5 Γ e Δ t 1 + 0.5 Γ e Δ t J e , x n + 1 / 2 ( i + 1 / 2 ) + 1 2 ε 0 ω pe 2 Δ t 1 + 0.5 Γ e Δ t [ E x n + 1 ( i ) + E x n + 1 ( i + 1 ) ]
F ( t ) = 0 L f ( z , t ) dz .
p s ( t ) = t F ( τ ) d τ .
p p ( t ) = G ( z , t ) dz .
m ( t ) = A ρ ( z , t ) dz .
z p ( t ) = A ρ ( z , t ) zdz m ( t ) .
m 0 + M 0 = m ( t ) + M ( t )
ρ a ( z , t + Δ t ) = Δ m ( t + Δ t ) ρ ( z , t + Δ t ) m ( t + Δ t ) .
ρ s ( z , t + Δ t ) = ρ s ( z , t ) + ρ a ( z , t + Δ t )
z s ( t ) = A t p s ( τ ) d τ + A ρ s ( z , t ) ( z L / 2 ) dz M ( t ) ,
z sys ( t ) = m ( t ) z p ( t ) + M ( t ) z s ( t ) M ( t ) + m ( t ) .
G = D × B .
T ¯ ¯ = D E B H + 1 2 ( E D + H B ) I ¯ ¯ .
f = t ( D × B ) + ( D E + B H ) 1 2 ( E D + H B ) .
f = 1 2 [ E × ( × D ) + D × ( × E ) ( E ) D + ( D ) E H × ( × B ) + B × ( × H ) ( H ) B + ( B ) H ] .
f = 1 2 [ E × ( × D ) + D × ( × E ) H × ( × B ) + B × ( × H ) ] .

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