Abstract

We review different expressions that have been proposed for the stress tensor and for the linear momentum of light in dielectric media, focusing on the Abraham and Minkowski forms. Analyses of simple models and consideration of available experimental results support the interpretation of the Abraham momentum as the kinetic momentum of the field, while the Minkowski momentum is the recoil momentum of absorbing or emitting guest atoms in a host dielectric. Momentum conservation requires consideration not only of the momentum of the field and of recoiling guest atoms, but also of the momentum the field imparts to the medium. Different model assumptions with respect to electrostriction and the dipole force lead to different expressions for this momentum. We summarize recent work on the definition of the canonical momentum for the field in a dielectric medium.

© 2010 Optical Society of America

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  57. R. V. Jones, “Radiation pressure of light in a dispersive medium,” Proc. R. Soc. London Ser. A 360, 365–371 (1977).
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  58. J. H. Poynting, “Radiation pressure,” Phil. Mag. J. Sci. 9(52), 393–406 (1905).
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  59. M. G. Burt, R. Peierls, “The momentum of a light wave in a refracting medium,” Proc. R. Soc. London Ser. A 333, 149–156 (1973).
    [CrossRef]
  60. A. Ashkin, J. M. Dziedzic, “Radiation pressure on a free liquid surface,” Phys. Rev. Lett. 30, 139–142 (1973).
    [CrossRef]
  61. R. Loudon, “Theory of the radiation pressure on dielectric surfaces,” J. Mod. Opt. 49, 821836 (2002).
    [CrossRef]
  62. G. K. Campbell, A. E. Leanhardt, J. Mun, M. Boyd, E. W. Streed, W. Ketterle, D. E. Pritchard, “Photon recoil momentum in dispersive media,” Phys. Rev. Lett. 94, 170403 (2005).
    [CrossRef] [PubMed]
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  72. D. H. Bradshaw, Z. Shi, R. W. Boyd, P. W. Milonni, “Electromagnetic momenta and forces in dispersive dielectric media,” Opt. Commun. 283, 650–656 (2010).
    [CrossRef]
  73. P. R. Berman, R. W. Boyd, P. W. Milonni, “Polarizability and the optical theorem for a two-level atom with radiative broadening,” Phys. Rev. A 74, 053816 (2006).
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  74. M. Nieto-Vesperinas, J. J. Sáenz, R. Gómez-Medinal, L. Chantada, “Optical forces on small magnetodielectric particles,” Opt. Express 18, 11428–11443 (2010).
    [CrossRef] [PubMed]
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    [CrossRef]
  76. D. F. Nelson, “Momentum, pseudomomentum and wave momentum: toward resolving the Minkowski–Abraham controversy,” Phys. Rev. A 44, 3985–3996 (1991).
    [CrossRef] [PubMed]
  77. H. Washimi, V. I. Karpman, “Ponderomotive force of a high-frequency electromagnetic field in a dispersive medium,” Sov. Phys. JETP 44, 528–534 (1976).
  78. Ref. [32], Eq. (81.13).
  79. R. Loudon, L. Allen, D. F. Nelson, “Propagation of electromagnetic energy and momentum through an absorbing dielectric,” Phys. Rev. E 55, 1071–1085 (1997).
    [CrossRef]
  80. J. P. Gordon, A. Ashkin, “Motion of atoms in a radiation trap,” Phys. Rev. A 21, 1606–1617 (1980).
    [CrossRef]
  81. W. Shockley, “A ‘try simplest cases’ resolution of the Abraham–Minkowski controversy on electromagnetic momentum in matter,” Proc. Natl. Acad. Sci. U.S.A. 60, 807–813 (1968).
    [CrossRef]
  82. H. A. Haus, “Momentum, energy and power densities of TEM wave packet,” Physica (Amsterdam) 43, 77–91 (1969).
    [CrossRef]
  83. M. Mansuripur, “Radiation pressure and the linear momentum of the electromagnetic field,” Opt. Express 12, 5375–5401 (2004).
    [CrossRef] [PubMed]
  84. E. A. Hinds, S. M. Barnett, “Momentum exchange between light and a single atom: Abraham or Minkowski?,” Phys. Rev. Lett. 102, 050403 (2009).
    [CrossRef] [PubMed]
  85. S. E. Harris, “Pondermotive forces with slow light,” Phys. Rev. Lett. 85, 4032–4035 (2000).
    [CrossRef] [PubMed]

2010 (5)

S. M. Barnett, R. Loudon, “The enigma of optical momentum in a medium,” Philos. Trans. R. Soc. London Ser. A 368, 927–939 (2010).
[CrossRef]

C. Baxter, R. Loudon, “Radiation pressure and the 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]

D. H. Bradshaw, Z. Shi, R. W. Boyd, P. W. Milonni, “Electromagnetic momenta and forces in dispersive dielectric media,” Opt. Commun. 283, 650–656 (2010).
[CrossRef]

M. Nieto-Vesperinas, J. J. Sáenz, R. Gómez-Medinal, L. Chantada, “Optical forces on small magnetodielectric particles,” Opt. Express 18, 11428–11443 (2010).
[CrossRef] [PubMed]

2009 (5)

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

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, 219301 (2009).
[CrossRef]

W. She, J. Yu, R. Feng, “Reply to Comment by I. Brevik,” Phys. Rev. Lett. 103, 219302 (2009).
[CrossRef]

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, 019301 (2009).
[CrossRef]

N. Selden, C. Ngalande, N. Gimelshein, A. Ketsdever, “Origins of radiometric forces on a circular vane with a temperature gradient,” J. Fluid Mech. 634, 419–431 (2009).
[CrossRef]

2008 (1)

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

2007 (1)

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

2006 (2)

L. P. Pitaevksii, “Comment on ‘Casimir force acting on magnetodielectric bodies embedded in media’,” Phys. Rev. A 73, 047801 (2006).
[CrossRef]

P. R. Berman, R. W. Boyd, P. W. Milonni, “Polarizability and the optical theorem for a two-level atom with radiative broadening,” Phys. Rev. A 74, 053816 (2006).
[CrossRef]

2005 (2)

P. W. Milonni, R. W. Boyd, “Recoil and photon momentum in a dielectric,” Laser Phys. 15, 1432–1438 (2005).

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

2004 (3)

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

J. C. Garrison, 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 (2004).
[CrossRef]

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

2002 (2)

J. E. Molloy, M. J. Padgett, “Lights, action: optical tweezers,” Contemp. Phys. 43, 241–258 (2002).
[CrossRef]

R. Loudon, “Theory of the radiation pressure on dielectric surfaces,” J. Mod. Opt. 49, 821836 (2002).
[CrossRef]

2000 (2)

P. C. Chaumet, M. Nieto-Vesperinas, “Time-averaged total force on a dipolar sphere in an electromagnetic field,” Opt. Lett. 25, 1065–1067 (2000).
[CrossRef]

S. E. Harris, “Pondermotive forces with slow light,” Phys. Rev. Lett. 85, 4032–4035 (2000).
[CrossRef] [PubMed]

1997 (1)

R. Loudon, L. Allen, D. F. Nelson, “Propagation of electromagnetic energy and momentum through an absorbing dielectric,” Phys. Rev. E 55, 1071–1085 (1997).
[CrossRef]

1992 (1)

B. Huttner, S. M. Barnett, “Quantization of the electromagnetic field in dielectrics,” Phys. Rev. A 46, 4306–4322 (1992).
[CrossRef] [PubMed]

1991 (2)

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

B. Huttner, J. J. Baumberg, S. M. Barnett, “Canonical quantization of light in a linear dielectric,” Europhys. Lett. 16, 177–182 (1991).
[CrossRef]

1982 (1)

M. P. Haugan, F. V. Kowalski, “Spectroscopy of atoms and molecules in gases: corrections to the Doppler-recoil shift,” Phys. Rev. A 25, 2102–2112 (1982).
[CrossRef]

1980 (2)

J. P. Gordon, A. Ashkin, “Motion of atoms in a radiation trap,” Phys. Rev. A 21, 1606–1617 (1980).
[CrossRef]

Y. Gingras, “Mechanical forces acting within non-polar dielectric fluids,” Phys. Lett. 76A, 117–118 (1980).
[CrossRef]

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, B. Leslie, “Measurement of optical radiation pressure in dispersive media,” Proc. R. Soc. London Ser. A 360, 347–363 (1978).
[CrossRef]

1977 (2)

R. V. Jones, “Radiation pressure of light in a dispersive medium,” Proc. R. Soc. London Ser. A 360, 365–371 (1977).
[CrossRef]

R. Peierls, “The momentum of light in a refracting medium. II. Generalization. Application to oblique reflexion,” Proc. R. Soc. London Ser. A 355, 141–151 (1977).
[CrossRef]

1976 (3)

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

C. W. Draper, “The Crookes radiometer revisited. A centennial celebration,” J. Chem. Educ. 53, 356–357 (1976).
[CrossRef]

H. Washimi, V. I. Karpman, “Ponderomotive force of a high-frequency electromagnetic field in a dispersive medium,” Sov. Phys. JETP 44, 528–534 (1976).

1975 (2)

G. B. Walker, D. G. Lahoz, G. Walker, “Measurement of Abraham force in a barium-titanate specimen,” Can J. Phys. 53, 2577–2586 (1975).
[CrossRef]

F. N. H. Robinson, “Electromagnetic stress and momentum in matter,” Phys. Rep. 16, 313–354 (1975).
[CrossRef]

1973 (4)

D. V. Skobel’tsyn, “The momentum-energy tensor of the electromagnetic field,” Sov. Phys. Usp. 16, 381–401 (1973).
[CrossRef]

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

M. G. Burt, R. Peierls, “The momentum of a light wave in a refracting medium,” Proc. R. Soc. London Ser. A 333, 149–156 (1973).
[CrossRef]

A. Ashkin, J. M. Dziedzic, “Radiation pressure on a free liquid surface,” Phys. Rev. Lett. 30, 139–142 (1973).
[CrossRef]

1972 (1)

J.-L. Piqué, J.-L. Vialle, “Atomic-beam deflection and broadening by recoils due to photon absorption or emission,” Opt. Commun. 5, 402–406 (1972).
[CrossRef]

1971 (1)

S. G. Brush, “James Clerk Maxwell and the kinetic theory of gases: a review based on recent historical studies,” Am. J. Phys. 39, 631–640 (1971).
[CrossRef]

1969 (1)

H. A. Haus, “Momentum, energy and power densities of TEM wave packet,” Physica (Amsterdam) 43, 77–91 (1969).
[CrossRef]

1968 (2)

W. Shockley, “A ‘try simplest cases’ resolution of the Abraham–Minkowski controversy on electromagnetic momentum in matter,” Proc. Natl. Acad. Sci. U.S.A. 60, 807–813 (1968).
[CrossRef]

R. P. James, “A ‘simplest case’ experiment resolving the Abraham–Minkowksi controversy on electromagnetic momentum in matter,” Proc. Natl. Acad. Sci. U.S.A. 61, 1149–1150 (1968).

1966 (1)

A. E. Woodruff, “William Crookes and the radiometer,” Isis 57, 188–198 (1966).
[CrossRef]

1954 (1)

R. V. Jones, J. C. S. Richards, “The pressure of radiation in a refracting medium,” Proc. R. Soc. London Ser. A 221, 480–498 (1954).
[CrossRef]

1953 (2)

R. H. Dicke, “The effect of collisions upon the Doppler width of spectral lines,” Phys. Rev. 53, 472–473 (1953).
[CrossRef]

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

1933 (1)

O. Frisch, “Experimental detection of the Einstein recoil radiation,” Z. Phys. 86, 42–48 (1933).
[CrossRef]

1932 (1)

E. Fermi, “Quantum theory of radiation,” Rev. Mod. Phys. 4, 87–132 (1932).
[CrossRef]

1923 (1)

W. Gerlach, A. Golsen, “Investigations with radiometers. II. A new measurement of the radiation pressure,” Z. Phys. 15, 1–7 (1923).
[CrossRef]

1922 (1)

A. Einstein, “The theory of radiometers,” Ann. Phys. 374, 241–254 (1922).
[CrossRef]

1917 (1)

A. Einstein, “On the quantum theory of radiation,” Phys. Z. 18, 121–128 (1917).

1910 (2)

M. Abraham “On Minkowski’s electrodynamics,” Rend. Circ. Mat. Palermo 30, 33–46 (1910).
[CrossRef]

H. Minkowski “Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern,” Math. Ann. 68, 472–525 (1910).
[CrossRef]

1909 (1)

M. Abraham, “On the electrodynamics of moving bodies,” Rend. Circ. Mat. Palermo 28, 1–28 (1909).
[CrossRef]

1908 (2)

H. Minkowski, “The basic equations for electromagnetic processes in moving bodies,” Nachr. Ges. Wiss. Goettingen, Math. Phys. Kl.53–111 (1908).

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

1905 (1)

J. H. Poynting, “Radiation pressure,” Phil. Mag. J. Sci. 9(52), 393–406 (1905).
[CrossRef]

1903 (2)

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

E. F. Nichols, G. F. Hull, “The pressure due to radiation,” Astrophys. J. 57, 315–351 (1903).
[CrossRef]

1901 (1)

P. N. Lebedev, “Investigations on the pressure forces of light,” Ann. Phys. 6, 433–458 (1901).
[CrossRef]

1874 (1)

W. Crookes, “On attraction and repulsion resulting from radiation,” Philos. Trans. R. Soc. London 164, 501–527 (1874).
[CrossRef]

Abraham, M.

M. Abraham “On Minkowski’s electrodynamics,” Rend. Circ. Mat. Palermo 30, 33–46 (1910).
[CrossRef]

M. Abraham, “On the electrodynamics of moving bodies,” Rend. Circ. Mat. Palermo 28, 1–28 (1909).
[CrossRef]

Allen, L.

R. Loudon, L. Allen, D. F. Nelson, “Propagation of electromagnetic energy and momentum through an absorbing dielectric,” Phys. Rev. E 55, 1071–1085 (1997).
[CrossRef]

Ashkin, A.

J. P. Gordon, A. Ashkin, “Motion of atoms in a radiation trap,” Phys. Rev. A 21, 1606–1617 (1980).
[CrossRef]

A. Ashkin, J. M. Dziedzic, “Radiation pressure on a free liquid surface,” Phys. Rev. Lett. 30, 139–142 (1973).
[CrossRef]

Balazs, N. L.

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

Barnett, S. M.

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

S. M. Barnett, R. Loudon, “The enigma of optical momentum in a medium,” Philos. Trans. R. Soc. London Ser. A 368, 927–939 (2010).
[CrossRef]

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

B. Huttner, S. M. Barnett, “Quantization of the electromagnetic field in dielectrics,” Phys. Rev. A 46, 4306–4322 (1992).
[CrossRef] [PubMed]

B. Huttner, J. J. Baumberg, S. M. Barnett, “Canonical quantization of light in a linear dielectric,” Europhys. Lett. 16, 177–182 (1991).
[CrossRef]

Baumberg, J. J.

B. Huttner, J. J. Baumberg, S. M. Barnett, “Canonical quantization of light in a linear dielectric,” Europhys. Lett. 16, 177–182 (1991).
[CrossRef]

Baxter, C.

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

Berman, P. R.

P. R. Berman, R. W. Boyd, P. W. Milonni, “Polarizability and the optical theorem for a two-level atom with radiative broadening,” Phys. Rev. A 74, 053816 (2006).
[CrossRef]

Bowyer, P.

P. Bowyer, “The momentum of light in media: the Abraham–Minkowski controversy,” http://www.peterbowyer.co.uk/purl/abraham-minkowski.

Boyd, M.

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

Boyd, R. W.

D. H. Bradshaw, Z. Shi, R. W. Boyd, P. W. Milonni, “Electromagnetic momenta and forces in dispersive dielectric media,” Opt. Commun. 283, 650–656 (2010).
[CrossRef]

P. R. Berman, R. W. Boyd, P. W. Milonni, “Polarizability and the optical theorem for a two-level atom with radiative broadening,” Phys. Rev. A 74, 053816 (2006).
[CrossRef]

P. W. Milonni, R. W. Boyd, “Recoil and photon momentum in a dielectric,” Laser Phys. 15, 1432–1438 (2005).

Bradshaw, D. H.

D. H. Bradshaw, Z. Shi, R. W. Boyd, P. W. Milonni, “Electromagnetic momenta and forces in dispersive dielectric media,” Opt. Commun. 283, 650–656 (2010).
[CrossRef]

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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, 219301 (2009).
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I. Brevik, “Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor,” Phys. Rep. 52, 133–201 (1979).
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S. G. Brush, “James Clerk Maxwell and the kinetic theory of gases: a review based on recent historical studies,” Am. J. Phys. 39, 631–640 (1971).
[CrossRef]

Burt, M. G.

M. G. Burt, R. Peierls, “The momentum of a light wave in a refracting medium,” Proc. R. Soc. London Ser. A 333, 149–156 (1973).
[CrossRef]

Campbell, G. K.

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

Chantada, L.

Chaumet, P. C.

P. C. Chaumet, M. Nieto-Vesperinas, “Time-averaged total force on a dipolar sphere in an electromagnetic field,” Opt. Lett. 25, 1065–1067 (2000).
[CrossRef]

Chiao, R. Y.

J. C. Garrison, 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 (2004).
[CrossRef]

Chu, S.

J. M. Hensley, A. Wicht, B. C. Young, S. Chu, in Atomic Physics 17, A. Arimondo, P. DeNatale, and M. Inguscio, eds. (American Institute of Physics, 2001).

Crookes, W.

W. Crookes, “On attraction and repulsion resulting from radiation,” Philos. Trans. R. Soc. London 164, 501–527 (1874).
[CrossRef]

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R. H. Dicke, “The effect of collisions upon the Doppler width of spectral lines,” Phys. Rev. 53, 472–473 (1953).
[CrossRef]

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C. W. Draper, “The Crookes radiometer revisited. A centennial celebration,” J. Chem. Educ. 53, 356–357 (1976).
[CrossRef]

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A. Ashkin, J. M. Dziedzic, “Radiation pressure on a free liquid surface,” Phys. Rev. Lett. 30, 139–142 (1973).
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Einstein, A.

A. Einstein, “The theory of radiometers,” Ann. Phys. 374, 241–254 (1922).
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A. Einstein, “On the quantum theory of radiation,” Phys. Z. 18, 121–128 (1917).

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

Feng, R.

W. She, J. Yu, R. Feng, “Reply to Comment by I. Brevik,” Phys. Rev. Lett. 103, 219302 (2009).
[CrossRef]

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

Fermi, E.

E. Fermi, “Quantum theory of radiation,” Rev. Mod. Phys. 4, 87–132 (1932).
[CrossRef]

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O. Frisch, “Experimental detection of the Einstein recoil radiation,” Z. Phys. 86, 42–48 (1933).
[CrossRef]

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J. C. Garrison, 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 (2004).
[CrossRef]

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W. Gerlach, A. Golsen, “Investigations with radiometers. II. A new measurement of the radiation pressure,” Z. Phys. 15, 1–7 (1923).
[CrossRef]

Gimelshein, N.

N. Selden, C. Ngalande, N. Gimelshein, A. Ketsdever, “Origins of radiometric forces on a circular vane with a temperature gradient,” J. Fluid Mech. 634, 419–431 (2009).
[CrossRef]

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Y. Gingras, “Mechanical forces acting within non-polar dielectric fluids,” Phys. Lett. 76A, 117–118 (1980).
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Ginzburg, V. L.

V. L. Ginzburg, Theoretical Physics and Astrophysics (Pergamon, 1960), p. 284.

Golsen, A.

W. Gerlach, A. Golsen, “Investigations with radiometers. II. A new measurement of the radiation pressure,” Z. Phys. 15, 1–7 (1923).
[CrossRef]

Gómez-Medinal, R.

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J. P. Gordon, A. Ashkin, “Motion of atoms in a radiation trap,” Phys. Rev. A 21, 1606–1617 (1980).
[CrossRef]

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

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S. E. Harris, “Pondermotive forces with slow light,” Phys. Rev. Lett. 85, 4032–4035 (2000).
[CrossRef] [PubMed]

Haugan, M. P.

M. P. Haugan, F. V. Kowalski, “Spectroscopy of atoms and molecules in gases: corrections to the Doppler-recoil shift,” Phys. Rev. A 25, 2102–2112 (1982).
[CrossRef]

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H. A. Haus, “Momentum, energy and power densities of TEM wave packet,” Physica (Amsterdam) 43, 77–91 (1969).
[CrossRef]

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

Hensley, J. M.

J. M. Hensley, A. Wicht, B. C. Young, S. Chu, in Atomic Physics 17, A. Arimondo, P. DeNatale, and M. Inguscio, eds. (American Institute of Physics, 2001).

Hinds, E. A.

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

Hull, G. F.

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

E. F. Nichols, G. F. Hull, “The pressure due to radiation,” Astrophys. J. 57, 315–351 (1903).
[CrossRef]

Huttner, B.

B. Huttner, S. M. Barnett, “Quantization of the electromagnetic field in dielectrics,” Phys. Rev. A 46, 4306–4322 (1992).
[CrossRef] [PubMed]

B. Huttner, J. J. Baumberg, S. M. Barnett, “Canonical quantization of light in a linear dielectric,” Europhys. Lett. 16, 177–182 (1991).
[CrossRef]

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J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1975), p. 240.

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R. P. James, “A ‘simplest case’ experiment resolving the Abraham–Minkowksi controversy on electromagnetic momentum in matter,” Proc. Natl. Acad. Sci. U.S.A. 61, 1149–1150 (1968).

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R. V. Jones, B. Leslie, “Measurement of optical radiation pressure in dispersive media,” Proc. R. Soc. London Ser. A 360, 347–363 (1978).
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R. V. Jones, “Radiation pressure of light in a dispersive medium,” Proc. R. Soc. London Ser. A 360, 365–371 (1977).
[CrossRef]

R. V. Jones, J. C. S. Richards, “The pressure of radiation in a refracting medium,” Proc. R. Soc. London Ser. A 221, 480–498 (1954).
[CrossRef]

Karpman, V. I.

H. Washimi, V. I. Karpman, “Ponderomotive force of a high-frequency electromagnetic field in a dispersive medium,” Sov. Phys. JETP 44, 528–534 (1976).

Ketsdever, A.

N. Selden, C. Ngalande, N. Gimelshein, A. Ketsdever, “Origins of radiometric forces on a circular vane with a temperature gradient,” J. Fluid Mech. 634, 419–431 (2009).
[CrossRef]

Ketterle, W.

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

Kowalski, F. V.

M. P. Haugan, F. V. Kowalski, “Spectroscopy of atoms and molecules in gases: corrections to the Doppler-recoil shift,” Phys. Rev. A 25, 2102–2112 (1982).
[CrossRef]

Lahoz, D. G.

G. B. Walker, D. G. Lahoz, G. Walker, “Measurement of Abraham force in a barium-titanate specimen,” Can J. Phys. 53, 2577–2586 (1975).
[CrossRef]

Landau, L. D.

L. D. Landau, E. M. Lifshitz, L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, 1984), Eq. (75.17).

Laub, J.

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

Leanhardt, A. E.

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

Lebedev, P. N.

P. N. Lebedev, “Investigations on the pressure forces of light,” Ann. Phys. 6, 433–458 (1901).
[CrossRef]

Leslie, B.

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

Lifshitz, E. M.

L. D. Landau, E. M. Lifshitz, L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, 1984), Eq. (75.17).

Loudon, R.

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

S. M. Barnett, R. Loudon, “The enigma of optical momentum in a medium,” Philos. Trans. R. Soc. London Ser. A 368, 927–939 (2010).
[CrossRef]

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

R. Loudon, “Theory of the radiation pressure on dielectric surfaces,” J. Mod. Opt. 49, 821836 (2002).
[CrossRef]

R. Loudon, L. Allen, D. F. Nelson, “Propagation of electromagnetic energy and momentum through an absorbing dielectric,” Phys. Rev. E 55, 1071–1085 (1997).
[CrossRef]

Mansuripur, M.

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, 019301 (2009).
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M. Mansuripur, “Radiation pressure and the linear momentum of the electromagnetic field,” Opt. Express 12, 5375–5401 (2004).
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Maxwell, J. C.

J. C. Maxwell, A Treatise on Electricity and Magnetism, 3rd ed. (Clarendon, 1904), Vol. II, p. 441.

Milonni, P. W.

D. H. Bradshaw, Z. Shi, R. W. Boyd, P. W. Milonni, “Electromagnetic momenta and forces in dispersive dielectric media,” Opt. Commun. 283, 650–656 (2010).
[CrossRef]

P. R. Berman, R. W. Boyd, P. W. Milonni, “Polarizability and the optical theorem for a two-level atom with radiative broadening,” Phys. Rev. A 74, 053816 (2006).
[CrossRef]

P. W. Milonni, R. W. Boyd, “Recoil and photon momentum in a dielectric,” Laser Phys. 15, 1432–1438 (2005).

P. W. Milonni, Fast Light, Slow Light, and Left-Handed Light (Institute of Physics, 2005), p. 185.

For a discussion of this aspect of Einstein’s work see, for instance, P. W. Milonni, The Quantum Vacuum. An Introduction to Quantum Electrodynamics (Academic, 1994).

Minkowski, H.

H. Minkowski “Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern,” Math. Ann. 68, 472–525 (1910).
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H. Minkowski, “The basic equations for electromagnetic processes in moving bodies,” Nachr. Ges. Wiss. Goettingen, Math. Phys. Kl.53–111 (1908).

Molloy, J. E.

J. E. Molloy, M. J. Padgett, “Lights, action: optical tweezers,” Contemp. Phys. 43, 241–258 (2002).
[CrossRef]

Mun, J.

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

Nelson, D. F.

R. Loudon, L. Allen, D. F. Nelson, “Propagation of electromagnetic energy and momentum through an absorbing dielectric,” Phys. Rev. E 55, 1071–1085 (1997).
[CrossRef]

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

Newton, R. G.

R. G. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, 1966), Sec. 1.5.

Ngalande, C.

N. Selden, C. Ngalande, N. Gimelshein, A. Ketsdever, “Origins of radiometric forces on a circular vane with a temperature gradient,” J. Fluid Mech. 634, 419–431 (2009).
[CrossRef]

Nichols, E. F.

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

E. F. Nichols, G. F. Hull, “The pressure due to radiation,” Astrophys. J. 57, 315–351 (1903).
[CrossRef]

Nieminen, T. A.

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

Nieto-Vesperinas, M.

M. Nieto-Vesperinas, J. J. Sáenz, R. Gómez-Medinal, L. Chantada, “Optical forces on small magnetodielectric particles,” Opt. Express 18, 11428–11443 (2010).
[CrossRef] [PubMed]

P. C. Chaumet, M. Nieto-Vesperinas, “Time-averaged total force on a dipolar sphere in an electromagnetic field,” Opt. Lett. 25, 1065–1067 (2000).
[CrossRef]

Padgett, M. J.

J. E. Molloy, M. J. Padgett, “Lights, action: optical tweezers,” Contemp. Phys. 43, 241–258 (2002).
[CrossRef]

Pais, A.

A. Pais, Subtle is the Lord. The Science and the Life of Albert Einstein (Oxford Univ. Press, 1982), p. 408.

Panofsky, W. K. H.

W. K. H. Panofsky, M. Phillips, Classical Electricity and Magnetism (Addison-Wesley, 1962), p. 183.

Peierls, R.

R. Peierls, “The momentum of light in a refracting medium. II. Generalization. Application to oblique reflexion,” Proc. R. Soc. London Ser. A 355, 141–151 (1977).
[CrossRef]

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

M. G. Burt, R. Peierls, “The momentum of a light wave in a refracting medium,” Proc. R. Soc. London Ser. A 333, 149–156 (1973).
[CrossRef]

Pfeifer, R. N. C.

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

Phillips, M.

W. K. H. Panofsky, M. Phillips, Classical Electricity and Magnetism (Addison-Wesley, 1962), p. 183.

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J.-L. Piqué, J.-L. Vialle, “Atomic-beam deflection and broadening by recoils due to photon absorption or emission,” Opt. Commun. 5, 402–406 (1972).
[CrossRef]

Pitaevksii, L. P.

L. P. Pitaevksii, “Comment on ‘Casimir force acting on magnetodielectric bodies embedded in media’,” Phys. Rev. A 73, 047801 (2006).
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Pitaevskii, L. P.

L. P. Pitaevskii, “Why and when the Minkowskis stress tensor can be used in the problem of Casimir force acting on bodies embedded in media,” arXiv.org, arXiv:cond-mat/0505754v2 (2005).

L. D. Landau, E. M. Lifshitz, L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, 1984), Eq. (75.17).

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J. H. Poynting, “Radiation pressure,” Phil. Mag. J. Sci. 9(52), 393–406 (1905).
[CrossRef]

Pritchard, D. E.

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

Richards, J. C. S.

R. V. Jones, J. C. S. Richards, “The pressure of radiation in a refracting medium,” Proc. R. Soc. London Ser. A 221, 480–498 (1954).
[CrossRef]

Robinson, F. N. H.

F. N. H. Robinson, “Electromagnetic stress and momentum in matter,” Phys. Rep. 16, 313–354 (1975).
[CrossRef]

Rubinsztein-Dunlop, H.

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

Sáenz, J. J.

Scandurra, M.

M. Scandurra, “Enhanced radiometric forces,” arXiv.org, arXiv:physics/0402011v1 (2004).

Selden, N.

N. Selden, C. Ngalande, N. Gimelshein, A. Ketsdever, “Origins of radiometric forces on a circular vane with a temperature gradient,” J. Fluid Mech. 634, 419–431 (2009).
[CrossRef]

She, W.

W. She, J. Yu, R. Feng, “Reply to Comment by I. Brevik,” Phys. Rev. Lett. 103, 219302 (2009).
[CrossRef]

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

Shi, Z.

D. H. Bradshaw, Z. Shi, R. W. Boyd, P. W. Milonni, “Electromagnetic momenta and forces in dispersive dielectric media,” Opt. Commun. 283, 650–656 (2010).
[CrossRef]

Shockley, W.

W. Shockley, “A ‘try simplest cases’ resolution of the Abraham–Minkowski controversy on electromagnetic momentum in matter,” Proc. Natl. Acad. Sci. U.S.A. 60, 807–813 (1968).
[CrossRef]

Skobel’tsyn, D. V.

D. V. Skobel’tsyn, “The momentum-energy tensor of the electromagnetic field,” Sov. Phys. Usp. 16, 381–401 (1973).
[CrossRef]

Streed, E. W.

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

Vialle, J.-L.

J.-L. Piqué, J.-L. Vialle, “Atomic-beam deflection and broadening by recoils due to photon absorption or emission,” Opt. Commun. 5, 402–406 (1972).
[CrossRef]

Walker, G.

G. B. Walker, D. G. Lahoz, G. Walker, “Measurement of Abraham force in a barium-titanate specimen,” Can J. Phys. 53, 2577–2586 (1975).
[CrossRef]

Walker, G. B.

G. B. Walker, D. G. Lahoz, G. Walker, “Measurement of Abraham force in a barium-titanate specimen,” Can J. Phys. 53, 2577–2586 (1975).
[CrossRef]

Walter, S.

S. Walter, “Minkowski, mathematicians, and the mathematical theory of relativity,” in The Expanding Worlds of General Relativity, H. Goenner, ed. (Birkhäuser, 1999), pp. 45–86.
[CrossRef]

Washimi, H.

H. Washimi, V. I. Karpman, “Ponderomotive force of a high-frequency electromagnetic field in a dispersive medium,” Sov. Phys. JETP 44, 528–534 (1976).

Wicht, A.

J. M. Hensley, A. Wicht, B. C. Young, S. Chu, in Atomic Physics 17, A. Arimondo, P. DeNatale, and M. Inguscio, eds. (American Institute of Physics, 2001).

Woodruff, A. E.

A. E. Woodruff, “William Crookes and the radiometer,” Isis 57, 188–198 (1966).
[CrossRef]

Young, B. C.

J. M. Hensley, A. Wicht, B. C. Young, S. Chu, in Atomic Physics 17, A. Arimondo, P. DeNatale, and M. Inguscio, eds. (American Institute of Physics, 2001).

Yu, J.

W. She, J. Yu, R. Feng, “Reply to Comment by I. Brevik,” Phys. Rev. Lett. 103, 219302 (2009).
[CrossRef]

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

Am. J. Phys. (1)

S. G. Brush, “James Clerk Maxwell and the kinetic theory of gases: a review based on recent historical studies,” Am. J. Phys. 39, 631–640 (1971).
[CrossRef]

Ann. Phys. (1)

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

Ann. Phys. (2)

P. N. Lebedev, “Investigations on the pressure forces of light,” Ann. Phys. 6, 433–458 (1901).
[CrossRef]

A. Einstein, “The theory of radiometers,” Ann. Phys. 374, 241–254 (1922).
[CrossRef]

Astrophys. J. (1)

E. F. Nichols, G. F. Hull, “The pressure due to radiation,” Astrophys. J. 57, 315–351 (1903).
[CrossRef]

Can J. Phys. (1)

G. B. Walker, D. G. Lahoz, G. Walker, “Measurement of Abraham force in a barium-titanate specimen,” Can J. Phys. 53, 2577–2586 (1975).
[CrossRef]

Contemp. Phys. (1)

J. E. Molloy, M. J. Padgett, “Lights, action: optical tweezers,” Contemp. Phys. 43, 241–258 (2002).
[CrossRef]

Europhys. Lett. (1)

B. Huttner, J. J. Baumberg, S. M. Barnett, “Canonical quantization of light in a linear dielectric,” Europhys. Lett. 16, 177–182 (1991).
[CrossRef]

Fortschr. Phys. (1)

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

Isis (1)

A. E. Woodruff, “William Crookes and the radiometer,” Isis 57, 188–198 (1966).
[CrossRef]

J. Chem. Educ. (1)

C. W. Draper, “The Crookes radiometer revisited. A centennial celebration,” J. Chem. Educ. 53, 356–357 (1976).
[CrossRef]

J. Fluid Mech. (1)

N. Selden, C. Ngalande, N. Gimelshein, A. Ketsdever, “Origins of radiometric forces on a circular vane with a temperature gradient,” J. Fluid Mech. 634, 419–431 (2009).
[CrossRef]

J. Mod. Opt. (2)

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Other (18)

For a discussion of this aspect of Einstein’s work see, for instance, P. W. Milonni, The Quantum Vacuum. An Introduction to Quantum Electrodynamics (Academic, 1994).

A. Pais, Subtle is the Lord. The Science and the Life of Albert Einstein (Oxford Univ. Press, 1982), p. 408.

J. C. Maxwell, A Treatise on Electricity and Magnetism, 3rd ed. (Clarendon, 1904), Vol. II, p. 441.

M. Scandurra, “Enhanced radiometric forces,” arXiv.org, arXiv:physics/0402011v1 (2004).

P. Bowyer, “The momentum of light in media: the Abraham–Minkowski controversy,” http://www.peterbowyer.co.uk/purl/abraham-minkowski.

V. L. Ginzburg, Theoretical Physics and Astrophysics (Pergamon, 1960), p. 284.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1975), p. 240.

L. D. Landau, E. M. Lifshitz, L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, 1984), Eq. (75.17).

Ref. [26], p. 285.

Ref. [32], Eq. (75.18).

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W. K. H. Panofsky, M. Phillips, Classical Electricity and Magnetism (Addison-Wesley, 1962), p. 183.

P. W. Milonni, Fast Light, Slow Light, and Left-Handed Light (Institute of Physics, 2005), p. 185.

Ref. [32], Eq. (80.12).

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Ref. [32], Eq. (81.13).

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Figures (3)

Fig. 1
Fig. 1

y = ω ω 0 versus x = k c ω 0 for the two polariton branches given by Eq. (55). In this example ω p ω 0 = 0.1 .

Fig. 2
Fig. 2

A block with refractive index n on a frictionless surface and initially at rest is displaced by Δ x when it transmits a photon.

Fig. 3
Fig. 3

An atom having a transition frequency ω 0 and moving with velocity v away from a source of light of frequency ω. The atom is inside a dielectric medium with refractive index n ( ω ) . Because of the Doppler effect, absorption by the atom requires that ω ω 0 ( 1 + n v c ) .

Equations (114)

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u = 1 2 ( ϵ 0 E 2 + μ 0 H 2 ) ,
g = 1 c 2 E × H = D × B ,
E ( r , t ) = x ̂ E 0 cos ω ( t z c ) , H ( r , t ) = y ̂ ϵ 0 μ 0 E 0 cos ω ( t z c ) ,
u = 1 2 ϵ 0 E 0 2 and g = 1 2 c z ̂ ϵ 0 E 0 2
E ( r , t ) = x ̂ E 0 cos ω ( t n z c ) , H ( r , t ) = y ̂ ϵ μ 0 E 0 cos ω ( t n z c )
u = 1 2 ϵ E 0 2 .
g = 1 2 c 2 z ̂ ϵ μ 0 E 0 2 .
g = q V ω n c z ̂ ,
p A = 1 n ω c .
p M = | k | = n ω c .
D = ρ , B = 0 ,
× E = B t , × H = J + D t ,
d P m d t V d p m d t d V = V [ ρ E + J × B ] d V = V [ ( D ) E + ( × H ) × B D t × B ] d V = V [ ( D ) E + ( × H ) × B + D × B t t ( D × B ) ] d V = V [ ( D ) E + ( H ) B D × ( × E ) B × ( × H ) ] d V d d t V ( D × B ) d V ,
V [ d d t ( p m + D × B ) ] d V = V [ ( D ) E + ( H ) B D × ( × E ) B × ( × H ) ] d V .
d d t ( p m + D × B ) i = j = 1 3 T i j M x j ,
T i j M = E i D j + H i B j 1 2 ( E D + H B ) δ i j ( i , j = 1 , 2 , 3 ) .
g M = D × B ,
( ρ E + J × B + g M t ) i = j = 1 3 T i j M x j .
S = J E E D t H B t .
g A = E × H c 2 .
( ρ E + J × B + f A + g A t ) i = j = 1 3 T i j M x j ,
f A = 1 c 2 ( n 2 1 ) t ( E × H )
( f M ) i = j = 1 3 T i j M x j ( g M t ) i ,
( f A ) i = j = 1 3 T i j M x j ( g A t ) i ,
f A = f M + g M t g A t = f M + f A .
( ρ E + J × B + g M t ) = ( ρ E + J × B + f A + g A t ) .
f M = 1 2 E 2 ϵ .
T i j A = 1 2 ( E i D j + E j D i ) + 1 2 ( H i B j + H j B i ) 1 2 ( E D + H H ) δ i j .
( f A ) i = j = 1 3 x j T i j A ( g A t ) i = ( f M ) i + ( f A ) i ,
f A = 1 2 E 2 ϵ + f A
f = P + [ ρ ϵ ρ 1 2 E 2 ] 1 2 E 2 ϵ + 1 c 2 ( n 2 1 ) t ( E × H ) ,
f = 1 2 E 2 ϵ + 1 c 2 ( n 2 1 ) t ( E × H ) = f A .
F = ( d ) E + d ̇ × B .
F = α [ ( E ) E + E t × B ] = α [ ( 1 2 E 2 ) E × ( × E ) + μ 0 t ( E × H ) μ 0 E × H t ] = α [ ( 1 2 E 2 ) + μ 0 t ( E × H ) ] .
f dipoles = ( ϵ ϵ 0 ) [ ( 1 2 E 2 ) + μ 0 t ( E × H ) ] = ϵ 0 ( n 2 1 ) ( 1 2 E 2 ) + 1 c 2 ( n 2 1 ) t ( E × H ) = ϵ 0 ( n 2 1 ) ( 1 2 E 2 ) + f A ,
f = P + f dipoles + { [ ρ d ϵ d ρ ( ϵ ϵ 0 ) ] 1 2 E 2 } .
ϵ ϵ 0 ϵ + 2 ϵ 0 = A ρ ,
ρ d ϵ d ρ ( ϵ ϵ 0 ) = 1 3 ϵ 0 ( ϵ ϵ 0 ) 2 ,
T i j EL = E i D j + H i B j 1 2 ( ϵ 0 E 2 + μ 0 H 2 ) δ i j
f EL = j = 1 3 T i j EL x j ( g A t ) i ,
f EL = ( P ) P + P ̇ × B = 1 2 ( P E ) 1 2 E 2 ϵ + 1 c 2 ( n 2 1 ) T ( E × H ) = f A + 1 2 ( P E )
g P = [ n 2 + 1 2 σ 2 ( n 2 1 ) 2 ] g A = 1 2 [ 1 n + n σ n ( n 2 1 ) 2 ] g vac ,
T i j P = [ ϵ + σ ϵ 0 ( ϵ ϵ 0 ) 2 ] E i E j + H i B j 1 2 δ i j ( [ ϵ 0 σ ϵ 0 ( ϵ ϵ 0 ) 2 ] E 2 + μ 0 H 2 ) .
p P = 1 2 ( n + 1 n ) ω c = 1 2 ( p M + p A ) .
p P = 1 10 n ( 4 n + 7 n 2 n 4 ) ω c
u = 1 4 [ d d ω ( ϵ ω ) | E ω | 2 + μ 0 | H ω | 2 ] .
u = 1 4 ϵ 0 n n g | E ω | 2 ,
n g = d d ω ( n ω ) = n + ω d n d ω
p A = n c 1 2 ϵ 0 2 ω ϵ 0 n n g V V = 1 n g ω c ,
p M = n 2 n g ω c
n g = 1 2 ϵ ϵ 0 + 1 2 ϵ 0 ϵ d d ω ( ω ϵ ϵ 0 ) .
ϵ R ( ω ) 1 = 2 π 0 ω ϵ I ( ω ) ω 2 ω 2 d ω .
d ϵ R d ω = 4 ω π 0 ω ϵ I ( ω ) ( ω 2 ω 2 ) 2 d ω ,
n 2 ( ω ) = 1 + ω p 2 ω 0 2 ω 2 .
ω 4 ( ω p 2 + ω 0 2 + k 2 c 2 ) ω 2 + k 2 c 2 ω 0 2 = 0 ,
ω ± 2 = 1 2 ( ω 0 2 + ω p 2 + k 2 c 2 ) ± 1 2 ( ω 0 2 + ω p 2 + k 2 c 2 ) 2 4 k 2 c 2 ω 0 2 .
i n ( ω i ) n g ( ω i ) = 1 ,
Δ x = V Δ t = m M ( c v ) a v = ω M c 2 ( n 1 ) a
Δ x = ω M c 2 a n ( 1 n ) .
ω ω 0 ( 1 + n v c ) .
1 2 M ( v 2 v 2 ) M v ( v v ) = M v ( p M ) = ( ω ω 0 ) ,
ω ω 0 + p v M .
p = n ω c = p M .
p med = ( n 1 n ) ω c 1 V
n ω c 1 V p med = 1 n ω c 1 V = p A .
α ( ω ) = μ 2 Δ , Δ = ω 0 ω ,
U ( z ) = Ω 2 Δ sin 2 k z = Ω 2 2 Δ ( 1 cos 2 k z ) ,
e i θ cos ( 2 k z ) = N = ( i ) N J N ( θ ) e 2 i N k z J 0 ( θ ) i J 1 ( θ ) [ e 2 i k z e 2 i k z ]
| ψ ( t p ) = J 0 ( θ ) | 0 i J 1 ( θ ) [ | 2 k | 2 k ] .
| ψ ( t p + τ ) = J 0 ( θ ) | 0 i J 1 ( θ ) [ | 2 k | 2 k ] e 4 i ω r m τ .
| ψ = [ J 0 2 ( θ ) + 2 J 1 2 ( θ ) e 4 i ω r τ ] | 0 + ,
p 0 = J 0 4 ( θ ) + 4 [ J 1 4 ( θ ) + J 0 2 ( θ ) J 1 2 ( θ ) ] cos 2 ( 4 ω r m τ ) .
ω r m = k 2 2 m + Δ E m = n 2 ( ω ) ω 2 2 m c 2 + Δ E m ,
d ( r , t ) = d 0 ( r ) e i ω t
E ( r , t ) = E 0 ( r ) e i ω t , B ( r , t ) = B 0 ( r ) e i ω t .
F z ( r ) = 1 4 α R ( ω ) z | E 0 ( r ) | 2 1 2 α I ( ω ) Im [ E 0 x ( r ) E 0 x * z + E 0 y ( r ) E 0 y * z + E 0 z ( r ) E 0 z * z ] .
F z = 1 2 n ( ω ) ω c α I ( ω ) | E 0 | 2 = n ( ω ) ω c R abs ,
E = E 0 ( r , t ) e i ω t = e i ω t d Δ E ̃ 0 ( r , Δ ) e i Δ t ,
F = ( d ) E + d ̇ × B = ( d ) E + d × ( × E ) + t ( d × B ) F E + F B ,
F E = ( d ) E + d × ( × E ) ,
F B = t ( d × B ) .
d ( r , t ) = d Δ α ( ω + Δ ) E ̃ 0 ( r , Δ ) e i ( ω + Δ ) t d Δ [ α ( ω ) + Δ α ( ω ) ] E ̃ 0 ( r , Δ ) e i ( ω + Δ ) t = [ α ( ω ) E 0 ( r , t ) + i α ( ω ) E 0 t ] e i ω t .
F E = [ 1 4 α ( ω ) | E | 2 ] + 1 4 α ( ω ) k t | E | 2 ,
F E = [ 1 4 α ( ω ) | E | 2 ] + ϵ 0 2 N k n d n d ω t | E | 2 .
W = 0 E d d E = α ( ω ) 0 E E d E = 1 2 α ( ω ) E 2 .
p D = 1 2 ϵ 0 n 2 d n d ω ω c | E | 2 = 1 2 ϵ 0 c n 2 ( n g n ) | E | 2 ,
P A = N d × B .
P A = 1 2 ϵ 0 ( n 2 1 ) k ω | E | 2 , P A = 1 2 ϵ 0 c n ( n 2 1 ) | E | 2 ,
P med = P D + P A = ϵ 0 2 c [ n 2 ( n g n ) + n ( n 2 1 ) ] | E | 2 = ϵ 0 2 c n ( n n g 1 ) | E | 2 .
P A + P D + P A = ϵ 0 2 c [ n + n ( n n g 1 ) ] | E | 2 = ϵ 0 2 c n 2 n g | E | 2
p A + p D + p A = n ω c 1 V ;
p med = p D + p A = ϵ 0 2 c n ( n n g 1 ) 2 ω n n g ϵ 0 V = ( n 1 n g ) ω c 1 V ,
[ ρ ϵ ρ 1 2 E 2 ] = [ ( ϵ ϵ 0 ) 1 2 E 2 ] = [ N α 1 2 E 2 ]
[ N α 1 2 E 2 ] = z [ N α 1 2 E 2 ] z ̂ = n c t [ N α 1 2 E 2 ] z ̂ = n c t 1 2 ( ϵ ϵ 0 ) | E | 2 = ϵ 0 2 c n ( n 2 1 ) t | E | 2 .
P med = ϵ 0 2 c n ( n 2 1 ) | E | 2 ϵ 0 4 c n ( n 2 1 ) | E | 2 = ϵ 0 4 c n ( n 2 1 ) | E | 2 .
P med = 1 2 c 2 ( n 2 1 ) | E × H | ,
p med = ω 2 c ( n 1 n ) ,
p f = | R | 2 ω c + n | T | 2 ω n c + p s ,
p s = 2 ω c n 1 n + 1 ,
f = n | T | 2 = 4 n ( n + 1 ) 2 .
p s f = ω c n 2 1 2 n = ω 2 c ( n 1 n ) .
p = [ 1 2 ( n 1 n ) + 1 n ] ω c = 1 2 ( n + 1 n ) ω c
F = 1 4 α ( ω ) z E 2 + 1 4 α ( ω ) n ( ω ) ω c t E 2 + 1 2 c α ( ω ) n ( ω ) t E 2 ,
E ( z , t ) = E ( t z v g ) cos ( ω t k z ) ,
p = T F d t = 1 4 α T z E 2 ( t z v g ) d t + 1 4 c α n ω T t E 2 ( t z v g ) d t + 1 2 c α n T t E 2 ( t z v g ) d t = 1 4 α 1 v g E 2 + 1 4 c α ω E 2 + 1 2 α n c E 2 = 1 4 c [ ( 2 n n g ) α + n ω α ] E 2 ( T z v g ) .
p = 1 4 c [ α + ω α ] E 2 .
p = 1 2 c [ α + ω α ] ω ϵ 0 V q .
p = 1 2 c [ ϵ 0 ( n 2 1 ) N + 2 ϵ 0 n N ω d n d ω ] ω c q [ n 1 + ω d n d ω ] ω c q K ω c q .
ω c [ 1 K ] ω c 1 1 + K = ω n g c
p = I 2 ϵ 0 c 2 [ ( 2 n g n ) α + ω α ] ,
p = n g 4 c μ 2 E 2 4 ( ω ω 0 ) ,
e i p ̂ a F ( x ̂ ) e i p ̂ a = F ( x ̂ + a )
e i p ̂ Min a A ̂ ( r ) e i p ̂ Min a = A ̂ ( r + a ) ,
[ p ̂ Min j , A ̂ k ( r ) ] = i x j A ̂ k ( r ) ,

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