Abstract

An exact theory describing the electromagnetic plane-wave force density on a scattering slab having various constitutive parameters and embedded in a background material with complex impedance is presented. It is shown that the constitutive parameters of the background medium contribute to the force density only through the impedance and not the refractive index. Asymptotic expressions show that the total force per unit area for sufficiently thick slabs having overall loss or gain remains positive, irrespective of the refractive index sign in the slab. However, example material responses indicate that for thin slabs the total force per unit area can be negative for both positive and negative refractive index slabs with gain.

© 2012 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. A. Lorentz, The Theory of Electrons, 2nd ed. (Dover, 1952). These are notes from lectures given at Columbia University in the spring of 1906, as collected by H. A.Lorentz in 1909 and then in revised form in 1915.
  2. A. Einstein and J. Laub, “Über die im elektromagnetischen Felde auf ruhende Körper ausgübten ponderomotorischen Kräfte,” Ann. Phys. 331, 541–550 (1908). English commentary on this paper and a reprint of the original paper appears in The Collected Papers of Albert Einstein (Princeton University, 1989), Vol. 2.
    [CrossRef]
  3. P. Penfield and H. A. Haus, Electrodynamics of Moving Media (MIT, 1967).
  4. J. P. Gordon, “Radiation forces and momenta in dielectric media,” Phys. Rev. A 8, 14–21 (1973).
    [CrossRef]
  5. I. Brevik, “Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor,” Phys. Rep. 52, 133–201 (1979).
    [CrossRef]
  6. M. Mansuripur, “Radiation pressure and the linear momentum of the electromagnetic field,” Opt. Express 12, 5375–5401 (2004).
    [CrossRef]
  7. 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]
  8. 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]
  9. 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]
  10. M. Mansuripur, “Resolution of the Abraham-Minkowski controversy,” Opt. Commun. 283, 1997–2005 (2010).
    [CrossRef]
  11. C. Baxter and R. Loudon, “Radiation pressure and photon momentum in dielectrics,” J. Mod. Opt. 57, 830–842 (2010).
    [CrossRef]
  12. S. M. Barnett, “Resolution of the Abraham-Minkowski dilemma,” Phys. Rev. Lett. 104, 070401 (2010).
    [CrossRef]
  13. K. J. Webb and Shivanand, “Negative electromagnetic plane-wave force in gain media,” Phys. Rev. E 84, 057602 (2011).
    [CrossRef]
  14. K. J. Webb and Shivanand, “Electromagnetic plane-wave forces on homogeneous material,” J. Opt. Soc. Am. B 29, 1904–1910 (2012).
  15. K. J. Chau and H. J. Lezec, “Revisiting the Balazs thought experiment in the case of a left-handed material: electromagnetic-pulse-induced displacement of a dispersive, dissipative negative-index slab,” Opt. Express 20, 10138–10162 (2012).
    [CrossRef]
  16. M. Mansuripur, “Trouble with the Lorentz law of force: incompatibility with special relativity and momentum conservation,” Phys. Rev. Lett. 108, 193901 (2012).
    [CrossRef]
  17. E. F. Nichols and G. F. Hull, “The pressure due to radiation,” Phys. Rev. 17, 26–50 (1903).
    [CrossRef]
  18. R. V. Jones and B. Leslie, “The measurement of optical radiation pressure in dispersive media,” Proc. Roy. Soc. A 360, 347–363 (1978).
    [CrossRef]
  19. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11, 288–290 (1986).
    [CrossRef]
  20. D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
    [CrossRef]
  21. G. K. Campbell, A. E. Leanhardt, J. Mun, M. Boyd, E. W. Streed, W. Ketterle, and D. E. Pritchard, “Photon recoil momentum in dispersive media,” Phys. Rev. Lett. 94, 170403 (2005).
    [CrossRef]
  22. V. G. Veselagao, “Energy, linear momentum, and mass transfer by an electromagnetic wave in a negative-refraction medium,” Phys. Usp. 52, 649–654 (2009).
    [CrossRef]
  23. V. Yannopapas and P. G. Galiatsatos, “Electromagnetic forces in negative-refractive-index metamaterials: a first-principles study,” Phys. Rev. A 77, 043819 (2008).
    [CrossRef]
  24. R. W. Ziolkowski, “Superluminal transmission of information through an electromagnetic metamaterial,” Phys. Rev. E 63, 046604 (2001).
    [CrossRef]
  25. R. Loudon and S. M. Barnett, “Theory of the radiation pressure on dielectric slabs, prisms and single surfaces,” Opt. Express 14, 11855–11869 (2006).
    [CrossRef]
  26. 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]

2012 (3)

2011 (1)

K. J. Webb and Shivanand, “Negative electromagnetic plane-wave force in gain media,” Phys. Rev. E 84, 057602 (2011).
[CrossRef]

2010 (4)

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

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

S. M. Barnett, “Resolution of the Abraham-Minkowski dilemma,” Phys. Rev. Lett. 104, 070401 (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]

2009 (1)

V. G. Veselagao, “Energy, linear momentum, and mass transfer by an electromagnetic wave in a negative-refraction medium,” Phys. Usp. 52, 649–654 (2009).
[CrossRef]

2008 (1)

V. Yannopapas and P. G. Galiatsatos, “Electromagnetic forces in negative-refractive-index metamaterials: a first-principles study,” Phys. Rev. A 77, 043819 (2008).
[CrossRef]

2007 (2)

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]

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]

2006 (1)

2005 (2)

G. K. Campbell, A. E. Leanhardt, J. Mun, M. Boyd, E. W. Streed, W. Ketterle, and D. E. Pritchard, “Photon recoil momentum in dispersive media,” Phys. Rev. Lett. 94, 170403 (2005).
[CrossRef]

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]

2004 (1)

2003 (1)

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[CrossRef]

2001 (1)

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

1986 (1)

1979 (1)

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

1978 (1)

R. V. Jones and B. Leslie, “The measurement of optical radiation pressure in dispersive media,” Proc. Roy. Soc. A 360, 347–363 (1978).
[CrossRef]

1973 (1)

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

1908 (1)

A. Einstein and J. Laub, “Über die im elektromagnetischen Felde auf ruhende Körper ausgübten ponderomotorischen Kräfte,” Ann. Phys. 331, 541–550 (1908). English commentary on this paper and a reprint of the original paper appears in The Collected Papers of Albert Einstein (Princeton University, 1989), Vol. 2.
[CrossRef]

1903 (1)

E. F. Nichols and G. F. Hull, “The pressure due to radiation,” Phys. Rev. 17, 26–50 (1903).
[CrossRef]

Ashkin, A.

Barnett, S. M.

S. M. Barnett, “Resolution of the Abraham-Minkowski dilemma,” Phys. Rev. Lett. 104, 070401 (2010).
[CrossRef]

R. Loudon and S. M. Barnett, “Theory of the radiation pressure on dielectric slabs, prisms and single surfaces,” Opt. Express 14, 11855–11869 (2006).
[CrossRef]

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]

Baxter, C.

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

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]

Bjorkholm, J. E.

Boyd, M.

G. K. Campbell, A. E. Leanhardt, J. Mun, M. Boyd, E. W. Streed, W. Ketterle, and D. E. Pritchard, “Photon recoil momentum in dispersive media,” Phys. Rev. Lett. 94, 170403 (2005).
[CrossRef]

Brevik, I.

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

Campbell, G. K.

G. K. Campbell, A. E. Leanhardt, J. Mun, M. Boyd, E. W. Streed, W. Ketterle, and D. E. Pritchard, “Photon recoil momentum in dispersive media,” Phys. Rev. Lett. 94, 170403 (2005).
[CrossRef]

Chau, K. J.

Chu, S.

Dziedzic, J. M.

Einstein, A.

A. Einstein and J. Laub, “Über die im elektromagnetischen Felde auf ruhende Körper ausgübten ponderomotorischen Kräfte,” Ann. Phys. 331, 541–550 (1908). English commentary on this paper and a reprint of the original paper appears in The Collected Papers of Albert Einstein (Princeton University, 1989), Vol. 2.
[CrossRef]

Galiatsatos, P. G.

V. Yannopapas and P. G. Galiatsatos, “Electromagnetic forces in negative-refractive-index metamaterials: a first-principles study,” Phys. Rev. A 77, 043819 (2008).
[CrossRef]

Gordon, J. P.

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

Grier, D. G.

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[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 (MIT, 1967).

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]

Hull, G. F.

E. F. Nichols and G. F. Hull, “The pressure due to radiation,” Phys. Rev. 17, 26–50 (1903).
[CrossRef]

Jones, R. V.

R. V. Jones and B. Leslie, “The measurement of optical radiation pressure in dispersive media,” Proc. Roy. Soc. A 360, 347–363 (1978).
[CrossRef]

Kemp, B. 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]

Ketterle, W.

G. K. Campbell, A. E. Leanhardt, J. Mun, M. Boyd, E. W. Streed, W. Ketterle, and D. E. Pritchard, “Photon recoil momentum in dispersive media,” Phys. Rev. Lett. 94, 170403 (2005).
[CrossRef]

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]

Laub, J.

A. Einstein and J. Laub, “Über die im elektromagnetischen Felde auf ruhende Körper ausgübten ponderomotorischen Kräfte,” Ann. Phys. 331, 541–550 (1908). English commentary on this paper and a reprint of the original paper appears in The Collected Papers of Albert Einstein (Princeton University, 1989), Vol. 2.
[CrossRef]

Leanhardt, A. E.

G. K. Campbell, A. E. Leanhardt, J. Mun, M. Boyd, E. W. Streed, W. Ketterle, and D. E. Pritchard, “Photon recoil momentum in dispersive media,” Phys. Rev. Lett. 94, 170403 (2005).
[CrossRef]

Leslie, B.

R. V. Jones and B. Leslie, “The measurement of optical radiation pressure in dispersive media,” Proc. Roy. Soc. A 360, 347–363 (1978).
[CrossRef]

Lezec, H. J.

Lorentz, H. A.

H. A. Lorentz, The Theory of Electrons, 2nd ed. (Dover, 1952). These are notes from lectures given at Columbia University in the spring of 1906, as collected by H. A.Lorentz in 1909 and then in revised form in 1915.

Loudon, R.

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

R. Loudon and S. M. Barnett, “Theory of the radiation pressure on dielectric slabs, prisms and single surfaces,” Opt. Express 14, 11855–11869 (2006).
[CrossRef]

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]

Mansuripur, M.

M. Mansuripur, “Trouble with the Lorentz law of force: incompatibility with special relativity and momentum conservation,” Phys. Rev. Lett. 108, 193901 (2012).
[CrossRef]

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

Mun, J.

G. K. Campbell, A. E. Leanhardt, J. Mun, M. Boyd, E. W. Streed, W. Ketterle, and D. E. Pritchard, “Photon recoil momentum in dispersive media,” Phys. Rev. Lett. 94, 170403 (2005).
[CrossRef]

Nichols, E. F.

E. F. Nichols and G. F. Hull, “The pressure due to radiation,” Phys. Rev. 17, 26–50 (1903).
[CrossRef]

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]

Penfield, P.

P. Penfield and H. A. Haus, Electrodynamics of Moving Media (MIT, 1967).

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]

Pritchard, D. E.

G. K. Campbell, A. E. Leanhardt, J. Mun, M. Boyd, E. W. Streed, W. Ketterle, and D. E. Pritchard, “Photon recoil momentum in dispersive media,” Phys. Rev. Lett. 94, 170403 (2005).
[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]

Shivanand,

K. J. Webb and Shivanand, “Electromagnetic plane-wave forces on homogeneous material,” J. Opt. Soc. Am. B 29, 1904–1910 (2012).

K. J. Webb and Shivanand, “Negative electromagnetic plane-wave force in gain media,” Phys. Rev. E 84, 057602 (2011).
[CrossRef]

Streed, E. W.

G. K. Campbell, A. E. Leanhardt, J. Mun, M. Boyd, E. W. Streed, W. Ketterle, and D. E. Pritchard, “Photon recoil momentum in dispersive media,” Phys. Rev. Lett. 94, 170403 (2005).
[CrossRef]

Veselagao, V. G.

V. G. Veselagao, “Energy, linear momentum, and mass transfer by an electromagnetic wave in a negative-refraction medium,” Phys. Usp. 52, 649–654 (2009).
[CrossRef]

Webb, K. J.

K. J. Webb and Shivanand, “Electromagnetic plane-wave forces on homogeneous material,” J. Opt. Soc. Am. B 29, 1904–1910 (2012).

K. J. Webb and Shivanand, “Negative electromagnetic plane-wave force in gain media,” Phys. Rev. E 84, 057602 (2011).
[CrossRef]

Yannopapas, V.

V. Yannopapas and P. G. Galiatsatos, “Electromagnetic forces in negative-refractive-index metamaterials: a first-principles study,” Phys. Rev. A 77, 043819 (2008).
[CrossRef]

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]

Ziolkowski, R. W.

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

Ann. Phys. (1)

A. Einstein and J. Laub, “Über die im elektromagnetischen Felde auf ruhende Körper ausgübten ponderomotorischen Kräfte,” Ann. Phys. 331, 541–550 (1908). English commentary on this paper and a reprint of the original paper appears in The Collected Papers of Albert Einstein (Princeton University, 1989), Vol. 2.
[CrossRef]

J. Mod. Opt. (1)

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

J. Opt. Soc. Am. B (1)

Nature (1)

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[CrossRef]

Opt. Commun. (2)

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 (3)

Opt. Lett. (1)

Phys. Rep. (1)

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

Phys. Rev. (1)

E. F. Nichols and G. F. Hull, “The pressure due to radiation,” Phys. Rev. 17, 26–50 (1903).
[CrossRef]

Phys. Rev. A (4)

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. 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]

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

V. Yannopapas and P. G. Galiatsatos, “Electromagnetic forces in negative-refractive-index metamaterials: a first-principles study,” Phys. Rev. A 77, 043819 (2008).
[CrossRef]

Phys. Rev. E (2)

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

K. J. Webb and Shivanand, “Negative electromagnetic plane-wave force in gain media,” Phys. Rev. E 84, 057602 (2011).
[CrossRef]

Phys. Rev. Lett. (3)

S. M. Barnett, “Resolution of the Abraham-Minkowski dilemma,” Phys. Rev. Lett. 104, 070401 (2010).
[CrossRef]

M. Mansuripur, “Trouble with the Lorentz law of force: incompatibility with special relativity and momentum conservation,” Phys. Rev. Lett. 108, 193901 (2012).
[CrossRef]

G. K. Campbell, A. E. Leanhardt, J. Mun, M. Boyd, E. W. Streed, W. Ketterle, and D. E. Pritchard, “Photon recoil momentum in dispersive media,” Phys. Rev. Lett. 94, 170403 (2005).
[CrossRef]

Phys. Usp. (1)

V. G. Veselagao, “Energy, linear momentum, and mass transfer by an electromagnetic wave in a negative-refraction medium,” Phys. Usp. 52, 649–654 (2009).
[CrossRef]

Proc. Roy. Soc. A (1)

R. V. Jones and B. Leslie, “The measurement of optical radiation pressure in dispersive media,” Proc. Roy. Soc. A 360, 347–363 (1978).
[CrossRef]

Rev. Mod. Phys. (1)

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]

Other (2)

H. A. Lorentz, The Theory of Electrons, 2nd ed. (Dover, 1952). These are notes from lectures given at Columbia University in the spring of 1906, as collected by H. A.Lorentz in 1909 and then in revised form in 1915.

P. Penfield and H. A. Haus, Electrodynamics of Moving Media (MIT, 1967).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1.
Fig. 1.

Infinite slab geometry used in the simulations. Region 2 is the slab having impedance η, and regions 1 and 3 are the background having impedance ηb.

Fig. 2.
Fig. 2.

Time-averaged force density in a slab in vacuum and composed of material having (a) μ=1, ϵ=4+i2, (b) μ=1, ϵ=1+i, (c) μ=1, ϵ=4i2, (d) μ=1, ϵ=1i, (e) μ=1, ϵ=4, and (f) μ=ϵ=1. A plane wave with ω0=2π×108 and E0=1V/m is normally incident on the slab (d=0.75λ, with λ the wavelength in the slab) from the left in Fig. 1. The exact numerical results (solid lines) are in excellent agreement with the analytical results (circles) obtained from Eq. (9).

Fig. 3.
Fig. 3.

Time-averaged force density in a dielectric slab (ϵ=4, μ=1) in (a) a lossless negative index (μ=2, ϵ=2) and (b) a lossy positive index (μ=1, ϵ=1+i0.2) background. A plane wave with ω0=2π×108 and E0=1V/m is normally incident on the slab from the left in Fig. 1, and d=0.75λ. Both the exact numerical results (solid lines) and the analytical results (circles) obtained from Eq. (9) are shown.

Fig. 4.
Fig. 4.

Total time-averaged force per unit area Fz for the slab placed in vacuum and composed of material having (a) μ=1, ϵ=4+i2, (b) μ=1, ϵ=1+i, (c) μ=1, ϵ=4i2, (d) μ=1, ϵ=1i, (e) μ=1, ϵ=4, and (f) μ=ϵ=1. The plane wave normally incident from the left in Fig. 1 has ω0=2π×108 and E0=1V/m. The exact numerical results (solid lines) obtained from a series of FEM simulations are in excellent agreement with the analytical results (circles) obtained from Eq. (11). Note the asymptotic behavior of the total force for loss and gain materials.

Equations (17)

Equations on this page are rendered with MathJax. Learn more.

f=Pt×μ0Hμ0Mt×ϵ0E,
E=y^12π[E(z,ω0)eiω0t+c.c.],
H=x^12π[H(z,ω0)eiω0t+c.c.],
P=y^ϵ02π[χE(ω0)E(z,ω0)eiω0t+c.c.],
M=x^12π[χH(ω0)H(z,ω0)eiω0t+c.c.].
f=z^μ0ϵ0ω02π2I[(χEχH*)EH*],
E=2ηE0[(ηb+η)eik(zd)+(ηbη)eik(zd)(ηb+η)2eikd(ηbη)2eikd],
H=2E0[(ηb+η)eik(zd)(ηbη)eik(zd)(ηb+η)2eikd(ηbη)2eikd],
f=z^2μ0ϵ0ω0E02π2|(ηb+η)2eikd(ηbη)2eikd|2{2[(χEχH)η(χE+χH)η][2(ηbηηbη)cos[2k(zd)]+(|η|2|ηb|2)sin[2k(zd)]+[(χEχH)η+(χE+χH)η][|ηb+η|2e2k(zd)|ηbη|2e2k(zd)]}.
f=z^μ0ϵ0ω0E022π2|η|2[(χEχH)η+(χE+χH)η]e2kz,
F=0dfdz=z^2μ0ϵ0ω0E02π2|(ηb+η)2eikd(ηbη)2eikd|2{2[(χEχH)η(χE+χH)η][2(ηbηηbη)×sin(2kd)2k+(|η|2|ηb|2)cos(2kd)12k]+[(χEχH)η+(χE+χH)η][|ηb+η|2e2kd12k|ηbη|21e2kd2k]}.
f=z^ω0μ0ϵ0E02(χEχH)η(|η|2|ηb|2)sin[2k(zd)]π2[4η2ηb2cos2(kd)+(ηb2+η2)2sin2(kd)],
F=z^ω0μ0ϵ0E02(χEχH)η(|η|2|ηb|2)[cos(2kd)1]π2[4η2ηb2cos2(kd)+(ηb2+η2)2sin2(kd)]2k.
|(ηb+η)2eikd(ηbη)2eikd|2{|ηb+η|4e2|k|dfor loss|ηbη|4e2|k|dfor gain.
|ηb+η|2e2kd12k|ηbη|21e2kd2k{|ηb+η|2e2|k|d2|k|for loss|ηbη|2e2|k|d2|k|for gain.
[(χEχH)η+(χE+χH)η]={|(χEχH)η+(χE+χH)η|for loss|(χEχH)η+(χE+χH)η|for gain.
F{z^ω0μ0ϵ0E02π2|k||ηb+η|2|(χEχH)η+(χE+χH)η|for lossz^ω0μ0ϵ0E02π2|k||ηbη|2|(χEχH)η+(χE+χH)η|for gain.

Metrics