Abstract

Maxwell’s macroscopic equations combined with a generalized form of the Lorentz law of force are a complete and consistent set of equations. Not only are these five equations fully compatible with special relativity, they also conform with conservation laws of energy, momentum, and angular momentum. We demonstrate consistency with the conservation laws by showing that, when a beam of light enters a magnetic dielectric, a fraction of the incident linear (or angular) momentum pours into the medium at a rate determined by the Abraham momentum density, E × H/c 2, and the group velocity V g of the electromagnetic field. The balance of the incident, reflected, and transmitted momenta is subsequently transferred to the medium as force (or torque) at the leading edge of the beam, which propagates through the medium with velocity V g. Our analysis does not require “hidden” momenta to comply with the conservation laws, nor does it dissolve into ambiguities with regard to the nature of electromagnetic momentum in ponderable media. The linear and angular momenta of the electromagnetic field are clearly associated with the Abraham momentum, and the phase and group refractive indices (n p and n g) play distinct yet definitive roles in the expressions of force, torque, and momentum densities.

© 2008 Optical Society of America

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References

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  1. J. D. Jackson, Classical Electrodynamics, 2nd edition, Wiley, New York, 1975.
  2. R.P.  Feynman, R.B.  Leighton, M. Sands, The Feynman Lectures on Physics, Addison-Wesley, Reading (1964).
  3. B. D. H. Tellegen, "Magnetic-Dipole Models," Am. J. Phys. 30, 650 (1962).
    [CrossRef]
  4. 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]
  5. W. Shockley, "Hidden linear momentum related to the ?·E term for a Dirac-electron wave packet in an electric field," Phys. Rev. Lett. 20, 343-346 (1968).
    [CrossRef]
  6. P. Penfield and H. A. Haus, Electrodynamics of Moving Media, MIT Press, Cambridge (1967).
  7. L. Vaidman, "Torque and force on a magnetic dipole," Am. J. Phys. 58, 978-983 (1990).
    [CrossRef]
  8. T. B.  Hansen and A. D.  Yaghjian, Plane-Wave Theory of Time-Domain Fields, IEEE Press, New York (1999).
    [CrossRef]
  9. A. D. Yaghjian, "Electromagnetic forces on point dipoles," IEEE Anten. Prop. Soc. Symp. 4, 2868-2871 (1999).
  10. M. Mansuripur, "Radiation pressure and the linear momentum of the electromagnetic field in magnetic media," Opt. Express 15, 13502-13518 (2007).
    [CrossRef] [PubMed]
  11. R. Loudon, "Theory of the forces exerted by Laguerre-Gaussian light beams on dielectrics," Phys. Rev. A 68, 013806 (2003).
    [CrossRef]
  12. M. Padgett, S. Barnett, and R. Loudon, "The angular momentum of light inside a dielectric," J. Mod. Opt. 50, 1555-1562 (2003).
  13. S. M. Barnett and R. Loudon, "On the electromagnetic force on a dielectric medium," J. Phys. B: At. Mol. Opt. Phys. 39, S671-S684 (2006).
    [CrossRef]
  14. 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] [PubMed]
  15. A. R. Zakharian, P. Polynkin, M. Mansuripur, and J. V. Moloney, "Single-beam trapping of micro-beads in polarized light: Numerical simulations," Opt. Express 14, 3660-3676 (2006).
    [CrossRef] [PubMed]
  16. R. N. C. Pfeifer, T. A. Nieminen, N. R Heckenberg, and H. Rubinsztein-Dunlop, "Momentum of an electro-magnetic wave in dielectric media," Rev. Mod. Phys. 79, 1197-1216 (2007).
    [CrossRef]
  17. M. Mansuripur, "Angular momentum of circularly polarized light in dielectric media," Opt. Express 13, 5315-5324 (2005).
    [CrossRef] [PubMed]
  18. M. Kristensen and J. P. Woerdman, "Is photon angular momentum conserved in a dielectric medium?" Phys. Rev. Lett. 72, 2171-2174 (1994).
    [CrossRef] [PubMed]

2007

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

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

2006

2005

2003

R. Loudon, "Theory of the forces exerted by Laguerre-Gaussian light beams on dielectrics," Phys. Rev. A 68, 013806 (2003).
[CrossRef]

M. Padgett, S. Barnett, and R. Loudon, "The angular momentum of light inside a dielectric," J. Mod. Opt. 50, 1555-1562 (2003).

1999

A. D. Yaghjian, "Electromagnetic forces on point dipoles," IEEE Anten. Prop. Soc. Symp. 4, 2868-2871 (1999).

1994

M. Kristensen and J. P. Woerdman, "Is photon angular momentum conserved in a dielectric medium?" Phys. Rev. Lett. 72, 2171-2174 (1994).
[CrossRef] [PubMed]

1990

L. Vaidman, "Torque and force on a magnetic dipole," Am. J. Phys. 58, 978-983 (1990).
[CrossRef]

1968

W. Shockley, "Hidden linear momentum related to the ?·E term for a Dirac-electron wave packet in an electric field," Phys. Rev. Lett. 20, 343-346 (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]

1962

B. D. H. Tellegen, "Magnetic-Dipole Models," Am. J. Phys. 30, 650 (1962).
[CrossRef]

Barnett, S.

M. Padgett, S. Barnett, and R. Loudon, "The angular momentum of light inside a dielectric," J. Mod. Opt. 50, 1555-1562 (2003).

Barnett, S. M.

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] [PubMed]

S. M. Barnett and R. Loudon, "On the electromagnetic force on a dielectric medium," J. Phys. B: At. Mol. Opt. Phys. 39, S671-S684 (2006).
[CrossRef]

Heckenberg, N. R

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

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]

Kristensen, M.

M. Kristensen and J. P. Woerdman, "Is photon angular momentum conserved in a dielectric medium?" Phys. Rev. Lett. 72, 2171-2174 (1994).
[CrossRef] [PubMed]

Loudon, R.

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] [PubMed]

S. M. Barnett and R. Loudon, "On the electromagnetic force on a dielectric medium," J. Phys. B: At. Mol. Opt. Phys. 39, S671-S684 (2006).
[CrossRef]

R. Loudon, "Theory of the forces exerted by Laguerre-Gaussian light beams on dielectrics," Phys. Rev. A 68, 013806 (2003).
[CrossRef]

M. Padgett, S. Barnett, and R. Loudon, "The angular momentum of light inside a dielectric," J. Mod. Opt. 50, 1555-1562 (2003).

Mansuripur, M.

Moloney, J. V.

Nieminen, T. A.

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

Padgett, M.

M. Padgett, S. Barnett, and R. Loudon, "The angular momentum of light inside a dielectric," J. Mod. Opt. 50, 1555-1562 (2003).

Pfeifer, R. N. C.

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

Polynkin, P.

Rubinsztein-Dunlop, H.

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

Shockley, W.

W. Shockley, "Hidden linear momentum related to the ?·E term for a Dirac-electron wave packet in an electric field," Phys. Rev. Lett. 20, 343-346 (1968).
[CrossRef]

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]

Tellegen, B. D. H.

B. D. H. Tellegen, "Magnetic-Dipole Models," Am. J. Phys. 30, 650 (1962).
[CrossRef]

Vaidman, L.

L. Vaidman, "Torque and force on a magnetic dipole," Am. J. Phys. 58, 978-983 (1990).
[CrossRef]

Woerdman, J. P.

M. Kristensen and J. P. Woerdman, "Is photon angular momentum conserved in a dielectric medium?" Phys. Rev. Lett. 72, 2171-2174 (1994).
[CrossRef] [PubMed]

Yaghjian, A. D.

A. D. Yaghjian, "Electromagnetic forces on point dipoles," IEEE Anten. Prop. Soc. Symp. 4, 2868-2871 (1999).

Zakharian, A. R.

Am. J. Phys.

B. D. H. Tellegen, "Magnetic-Dipole Models," Am. J. Phys. 30, 650 (1962).
[CrossRef]

L. Vaidman, "Torque and force on a magnetic dipole," Am. J. Phys. 58, 978-983 (1990).
[CrossRef]

IEEE Anten. Prop. Soc. Symp.

A. D. Yaghjian, "Electromagnetic forces on point dipoles," IEEE Anten. Prop. Soc. Symp. 4, 2868-2871 (1999).

J. Mod. Opt.

M. Padgett, S. Barnett, and R. Loudon, "The angular momentum of light inside a dielectric," J. Mod. Opt. 50, 1555-1562 (2003).

J. Phys. B: At. Mol. Opt. Phys.

S. M. Barnett and R. Loudon, "On the electromagnetic force on a dielectric medium," J. Phys. B: At. Mol. Opt. Phys. 39, S671-S684 (2006).
[CrossRef]

Opt. Express

Phys. Rev. A

R. Loudon, "Theory of the forces exerted by Laguerre-Gaussian light beams on dielectrics," Phys. Rev. A 68, 013806 (2003).
[CrossRef]

Phys. Rev. Lett.

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]

W. Shockley, "Hidden linear momentum related to the ?·E term for a Dirac-electron wave packet in an electric field," Phys. Rev. Lett. 20, 343-346 (1968).
[CrossRef]

M. Kristensen and J. P. Woerdman, "Is photon angular momentum conserved in a dielectric medium?" Phys. Rev. Lett. 72, 2171-2174 (1994).
[CrossRef] [PubMed]

Rev. Mod. Phys.

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

Other

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

T. B.  Hansen and A. D.  Yaghjian, Plane-Wave Theory of Time-Domain Fields, IEEE Press, New York (1999).
[CrossRef]

J. D. Jackson, Classical Electrodynamics, 2nd edition, Wiley, New York, 1975.

R.P.  Feynman, R.B.  Leighton, M. Sands, The Feynman Lectures on Physics, Addison-Wesley, Reading (1964).

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Figures (1)

Fig. 1.
Fig. 1.

A semi-infinite slab having material parameters (ε, µ) is illuminated at normal incidence with an elliptically-polarized, finite diameter light pulse. The incident E and H amplitudes are (E x o, E y o, E z o) and (H x o, H y o, H z o), respectively. The isotropic material is transparent and free from dispersion, i.e., (ε, µ) are real-valued and frequency-independent. The refractive index n = μ ε thus determines the speed of the leading edge of the pulse within the material as c/n.

Equations (114)

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ε t = E · J free + E · D t + H · B t
F 1 ( r , t ) = ( P · ) E + ( M · ) H + ( P t ) × μ 0 H ( M t ) × ε 0 E .
· D = ρ free , × H = J free + D t , × E = B t , · B = 0 .
D = ε 0 E + P = ε 0 ( 1 + χ e ) E = ε 0 ε E , B = μ 0 H + M = μ 0 ( 1 + χ m ) H = μ 0 μ H .
F 2 ( r , t ) = ( · P ) E ( · M ) H + ( P t ) × μ 0 H ( M t ) × ε 0 E .
T 1 ( r , t ) = r × F 1 ( r , t ) + P ( r , t ) × E ( r , t ) + M ( r , t ) × H ( r , t ) .
T 2 ( r , t ) = r × F 2 ( r , t ) .
f p = ( p · E ) = ( p · ) E + p × ( × E ) = ( p · ) E p × B t ,
f m = ( m · H ) = ( m · ) H + m × ( × H ) = ( m · ) H + m × D t .
F 1 ( r , t ) = ( P · E + M · H ) + ( D × B E × H c 2 ) t .
F 1 ( r , t ) = 1 2 ε 0 ( ε 1 ) ( E x 2 + E y 2 + E z 2 ) + 1 2 μ 0 ( μ 1 ) ( H x 2 + H y 2 + H z 2 ) + ( μ ε 1 ) t ( E × H c 2 ) .
F 1 ( t ) = 1 2 [ ε 0 ( ε 1 ) ( E x 2 + E y 2 + E z 2 ) z = 0 + dxdy + μ 0 ( μ 1 ) ( H x 2 + H y 2 + H z 2 ) z = 0 + dxdy ] z ̂
+ ( c n ) ( μ ε 1 ) ( E × H c 2 ) z = 0 + dxdy .
r = ( 1 ε μ ) ( 1 + ε μ ) ,
< F z > = 1 4 [ ε 0 ( ε 1 ) ( E x 2 + E y 2 ) + μ 0 ( μ 1 ) ( H x 2 + H y 2 ) 2 ( c n ) 1 ( μ ε 1 ) ( E x H y E y H x ) ] ( x = 0 , y = 0 , z = 0 + )
= 1 4 ε o [ ( ε 1 ) ( 1 + r ) 2 + ( μ 1 ) ( 1 r ) 2 2 n 1 ( μ ε 1 ) ( 1 r ) 2 ] ( E x o 2 + E y o 2 )
= 1 4 o n 1 ( μ + ε 2 ) ( 1 r 2 ) ( E x o 2 + E y o 2 ) .
T ( t ) = 1 2 ε o ( ε 1 ) r × ( E x 2 + E y 2 + E z 2 ) d x d y d z + 1 2 μ o ( μ 1 ) r × ( H x 2 + H y 2 + H z 2 ) d x d y d z
( c n ) ( μ ε 1 ) r × z ( E × H c 2 ) d x d y d z
= 1 2 ε o ( ε 1 ) [ ( y z z y ) x ̂ + ( z x x z ) y ̂ + ( x y y x ) z ̂ ] ( E x 2 + E y 2 + E z 2 ) d x d y d z
+ 1 2 μ o ( μ 1 ) [ ( y z z y ) x ̂ + ( z x x z ) y ̂ + ( x y y x z ̂ ) ] ( H x 2 + H y 2 + H z 2 ) d x d y
( cn ) 1 ( μ ε 1 ) { [ y z ( E x H y E y H x ) z z ( E z H x E x H z ) ] x ̂
+ [ z z ( E y H z E z H y ) x z ( E x H y E y H x ) ] y ̂
+ [ x z ( E z H x E x H z ) y z ( E y H z E z H y ) ] z ̂ } d x d y d z .
< T z > = 1 2 ( c n ) 1 ( μ ε 1 ) [ x ( E z H x E x H z ) z = 0 + y ( E y H z E z H y ) z = 0 + ] d x d y
= 1 2 ( c n ) 1 ( μ ε 1 ) { x [ ε 1 ( 1 r ) 2 E z o H x o μ 1 ( 1 + r ) 2 E x o H z o ]
y [ μ 1 ( 1 + r ) 2 E y o H z o ε 1 ( 1 r ) 2 E z o H y o ] } d x d y
= 1 2 c 1 [ ( μ ε 1 ) μ ε ] ( 1 r 2 ) [ x ( E z o H x o E x o H z o ) y ( E y o H z o E z o H y o ) ] d x d y
= ( 1 n 2 ) ( 1 r 2 ) ( < x S y o y S x o > c ) d x d y .
E x o x ̂ + E y o y ̂ = 1 2 [ ( E x o i E y o ) x ̂ + i ( E x o i E y o ) y ̂ ] + 1 2 [ ( E x o + i E y o ) x ̂ i ( E x o + i E y o ) y ̂ ] .
r 1 = μ y ε x μ y + ε x , r 2 = μ x ε y μ x + ε y .
L z = ( ε o k o ) Im [ ( 1 r 1 * r 2 ) E x o * E y o ] = 2 ( ε o k o ) Im { [ μ x ε x * + μ y * ε y ] E x o * E y o ( μ x + ε y ) ( μ y * + ε x * ) } .
E x ( z , t ) = ( 1 + r 1 ) E x o exp ( i k o μ y ε x z i ω t ) ,
H y ( z , t ) = ( 1 r 1 ) Z o 1 E x o exp ( i k o μ y ε x z i ω t ) ,
E y ( z , t ) = ( 1 + r 2 ) E y o exp ( i k o μ x ε y z i ω t ) ,
H x ( z , t ) = ( 1 r 2 ) Z o 1 E y o exp ( i k o μ x ε y z i ω t ) .
< T z > = 1 2 0 Re [ ( P x × E y * + P y × E x * + M x × H y * + M y × H y * ) ] dz
= 1 2 0 Re { ε o [ ( ε x * 1 ) ( ε y 1 ) ] E x * E y + μ o [ ( μ x 1 ) ( μ y * 1 ) ] H x H y * } dz
= 1 2 ε o Re { [ ( ε x * ε y ) ( 1 + r 1 * ) ( 1 + r 2 ) ( μ x μ y * ) ( 1 r 1 * ) ( 1 r 2 ) ]
× E x o * E y o 0 exp [ i k o ( μ x ε y μ y * ε x * ) z ] dz }
= 2 ( ε o k o ) Re { i [ ( ε x * ε y ) μ x μ y * ( μ x μ y * ) ε x * ε y ] ( μ x ε y μ y * ε x * ) ( μ x + ε y ) ( μ y * + ε x * ) E x o * E y o }
= 2 ( ε o k o ) Im { μ x ε x * + μ y * ε y ( μ x + ε y ) ( μ y * + ε x * ) E x o * E y o } .
E ( r , t ) = 1 2 ± ω 1 , 2 ε ( k x , k y , ω ) exp [ i ( k x x + k y y + k z z ω t ) ] d k x d k y ,
H ( r , t ) = 1 2 ± ω 1 , 2 H ( k x , k y , ω ) exp [ i ( k x x + k y y + k z z ω t ) ] d k x d k y .
ε ( k x , k y , ω ) = ε * ( k x , k x , ω ) , K ( k x , k y , ω ) = H * ( k x , k y , ω ) .
ε ( k x , k y , ω ) = ε ( k x , k y , ω ) , K ( k x , k y , ω ) = H ( k x , k y , ω ) .
ε ( k x , k y ± ω 1 ) = ε ( k x , k y , ± ω 2 ) , K ( k x , k y , ± ω 1 ) = H ( k x , k y , ± ω 2 ) .
V g = c ( ω 2 ω 1 ) ω 2 μ 2 ε 2 ω 1 μ 1 ε 1 .
S ( r , t ) = E ( r , t ) × H ( r , t ) = 1 4 Σ ε ( k x , k y , ω ) × H ( k x , k y , ω ) exp [ i ( k x + k x ) x ]
× exp [ i ( k y + k y ) y ] exp [ i ( k z + k z ) z ] exp [ i ( ω + ω ) t ] d k x d k y d k x d k y .
exp [ i ( k + k ) ζ ] d ζ = δ ( k + k ) ,
p EM ( z = 0 , t ) = ( 1 A c 2 ) S ( x , y , z = 0 , t ) d x d y
= ( 4 A c 2 ) 1 Σ ε ( k x , k y , ω ) × H ( k x , k y , ω ) exp [ i ( ω + ω ) t ] d k x d k y .
< p EM ( z = 0 , t ) > = ( 2 A c 2 ) 1 ω 1 , 2 Real { ε ( k x , k y , ω ) × H * ( k x , k y , ω ) d k x d k y } .
L z ( z = 0 , t ) = ( A c 2 ) 1 [ x S y ( x , y , z = 0 , t ) y S x ( x , y , z = 0 , t ) ] d x d y .
ζ exp [ i ( k + k ) ζ ] d ζ = i δ ( k + k ) ,
< L z ( z = 0 , t ) > = ( 4 i A c 2 ) 1 ± ω 1 , 2 [ H x ( k x , k y , ω ) ( k x ) ε z * ( k x , k y , ω ) H z ( k x k y , ω ) ( k x ) ε x * ( k x , k y , ω )
H z ( k x , k y , ω ) ( k y ) ε y * ( k x , k y , ω ) + H y ( k x , k y , ω ) ( k y ) ε z * ( k x , k y , ω ) ] d k x d k y .
k y ε z k z ε y = ω μ o μ H x , k z ε x k x ε z = ω μ 0 μ H y , k x ε y k x ε x = ω μ o μ H z .
< L z ( z = 0 , t ) > = ( ε O 2 A ) ω 1 , 2 ( ω μ k z 2 ) 1 Imag { 2 ( ω c ) 2 μ ε ε x * ε y + k y [ ( ω c ) 2 μ ε k y 2 ] ε x ε x * k x
+ k y [ ( ω c ) 2 μ ε k x 2 ] ε y ε y * k x + k x k y 2 ε x ε y * k x + k x k y 2 ε y ε x * k x k x [ ( ω c ) 2 μ ε k x 2 ] ε y ε y * k y
k x [ ( ω c ) 2 μ ε k y 2 ] ε x ε x * k y k x 2 k y ε y ε x * k y k x 2 k y ε x ε y * k y } d k x d k y .
< L z ( z = 0 , t ) > ( ε ο A ) ω 1 , 2 ( ω μ ) 1 ε x ( k x , k y , ω ) 2 d k x d k y .
F z ( x , y , z , t ) = P x ( E z x ) + P y ( E z y ) + P z ( E z z ) + ( P x t ) μ ο H y ( P y t ) μ ο H x
+ M x ( H z x ) + M y ( H z y ) + M z ( H z z ) ( M x t ) ε ο E y + ( M y t ) ε ο E x .
F z ( x , y , z , t ) d x d y = 1 2 I mag ω 1 , 2 { ε ο ( ε 1 ) ( k x ε x + k y ε y + k z ε z ) ε z * + μ ο ( μ 1 ) ( k x x + k y y + k z z ) z *
+ [ k z ε ο ( ε 1 ) μ ] ( ε x 2 + ε y 2 + ε z 2 ) + [ k z μ ο ( μ 1 ) ε ] ( x 2 + y 2 + z 2 ) } d k x d k y
+ 1 4 i ω ω { ε ο ( ε 1 ) [ k x ε x ( k x , k y , ω ) k y ε y ( k x , k y , ω ) + k z ε z ( k x , k y , ω ) ] ε z ( k x , k y , ω )
+ μ ο ( μ 1 ) [ k x x ( k x , k y , ω ) k y y ( k x , k y , ω ) + k z z ( k x , k y , ω ) ] z ( k x , k y , ω )
ω μ ο ε ο ( ε 1 ) [ ε x ( k x , k y , ω ) y ( k x , k y , ω ) ε y ( k x , k y , ω ) x ( k x , k y , ω ) ]
ω μ ο ε ο ( μ 1 ) [ ε x ( k x , k y , ω ) y ( k x , k y , ω ) ε y ( k x , k y , ω ) x ( k x , k y , ω ) ] }
× exp [ i ( k z + k z ) z ] exp [ i ( ω + ω ) t ] d k x d k y .
( 1 T ) 0 T exp { i [ ( k z + k z ) V g ( ω + ω ) ] t } d t = 1 .
< F z > = ( 1 AT ) 0 T 0 V g t F z ( x , y , z , t ) d x d y d z d t
= ( 1 4 A ) ± ω 1 , 2 { ε o ( ε 1 ) ε x ( k x , k y , ω ) ε z * ( k x , k y , ω ) + μ o ( μ 1 ) H z ( k x , k y , ω ) H z * ( k x , k y , ω )
ω μ o ε 0 ( ε 1 ) ( k z + k z ) 1 [ ε x ( k x , k y , ω ) H y * ( k x , k y , ω ) ε y ( k x , k y , ω ) H x * ( k x , k y , ω ) ]
ω μ o ε 0 ( μ 1 ) ( k z + k z ) 1 [ ε x * ( k x , k y , ω ) H y ( k x , k y , ω ) ε y * ( k x , k y , ω ) H x ( k x , k y , ω ) } d k x d k y .
< F z > = ( 2 A ) 1 Real ω 1 , 2 [ ε 0 ( ε 1 ) ε z 2 + μ 0 ( μ 1 ) H z 2 ω μ 0 ε 0 ( ε + μ 2 ) k z + k z ( ε x y * ε y x * ) ] d k x d k y .
< F z > = ( 1 + r 2 ) Real { ( Ac ) 1 ( ε x o H y o * ε y o H x o * ) d k x d k y }
V g Real { ( A c 2 ) 1 ( ε x H y * ε y H x * ) d k x d k y } .
T z ( x , y , z , t ) = x [ P x ( E y x ) + P y ( E y y ) + P z ( E y z ) + ( P z t ) μ o H x ( P x t ) μ o H z
+ M x ( H y x ) + M y ( H y y ) + M z ( H y z ) ( M 2 t ) ε o E x + ( M x t ) ε o E z ]
y [ P x ( E x x ) + P y ( E x y ) + P z ( E x z ) + ( P y t ) μ o H z ( P z t ) μ o H y + M x ( H x x )
+ M y ( H x y ) + M z ( H x z ) ( M y t ) ε o E z + ( M z t ) ε o E y ]
+ ( P x E y P y E x ) + ( M x H y M y H x ) .
T z ( x , y , z , t ) dxdy = 1 4 ω 1 , 2 { ( k y k x k x k y ) [ ε o ( ε 1 ) ( ε x 2 + ε y 2 + ε z 2 )
+ μ o ( μ 1 ) ( x 2 + y 2 + z 2 ) ] } d k x d k y
+ 1 4 ω ω { ε o ( ε 1 ) [ k y ε x ε x * k x k x ε x ε x * k x + k y ε y ε y * k y k x ε y ε y * k y ( k z k z ) 1
× ( k x ε x + k y ε y ) ( k y ε x * k x ε y * + k x k y ε x * k x + k y 2 ε y * k x k x 2 ε x * k y k x k y ε y * k y ) ]
+ μ o ( μ 1 ) [ k y x x * k x k x x x * k y + k y y y * k x k x y y * k y ( k z k z ) 1
× ( k x x + k y y ) ( k y x * k x y * + k x k y x * k x + k y 2 y * k x k x 2 x * k y k x k y y * k y ) ]
+ ε o ( ε 1 ) ( ε x ε y * ε y ε x * ) ε o ( μ 1 ) ( μ o ε o μ μ ω ω k z k z ) 1 [ k x k y ( k z 2 k z 2 ) ( ε x ε x * ε y ε y * )
( k z 2 k z 2 + k x 2 k z 2 + k y 2 k z 2 ) ε x ε y * + ( k z 2 k z 2 + k x 2 k z 2 + k y 2 k z 2 ) ε y ε x * ]
+ ε o ( μ ε + 1 ) ( ω + ω ) ( μ ω ) 1 { ( k x ε y k y ε x ) ( ε x * k x + ε y * k y ) + ( k z k z 3 ) 1
× [ k x k y ( k z 2 k z 2 ) ( ε x ε x * ε y ε y * ) ( k z 2 k z 2 + k x 2 k z 2 + k y 2 k z 2 ) ε x ε y *
+ ( k z 2 k z 2 + k x 2 k z 2 + k y 2 k z 2 ) ε y ε x * ] + ( k z k z ) 1 [ k x k y ε x + ( k y 2 + k z 2 ) ε y ] ( k x ε x * k x + k y ε y * k x )
( k z k z ) 1 [ ( k x 2 + k z 2 ) ε x + k x k y ε y ] ( k x ε x * k y + k y ε y * k y ) } }
× exp [ i ( k z + k z ) z ] exp [ i ( ω + ω ) t ] d k x d k y .
< T z > = ( 1 A T ) 0 T 0 V g t T z ( x , y , z , t ) d x d y d z d t = ( 2 A ) 1 Imag ω 1 , 2 ( k z + k z ) 1 { ε 0 ( ε 1 )
× [ k y ε x ε x * k x k x ε x ε x * k y + k y ε y ε y * k x k x ε y ε y * k y ( k z k z ) 1 ( k x ε x + k y ε y )
× ( k y ε x * k x ε y * + k x k y ε x * k x + k y 2 ε y * k x k x 2 ε x * k y k x k y ε y * k y ) ]
+ μ 0 ( μ 1 ) [ k y x x * k x k x x x * k y + k y y y * k x k x y y * k y ( k z k z ) 1
× ( k x x + k y y ) ( k y x * k x y * + k x k y x * k x + k y 2 y * k x k x 2 x * k y k x k y y * k y ) ]
+ ε 0 ( ε 1 ) ( ε x ε y * ε y ε x * ) ε 0 ( μ 1 ) ( μ 0 ε 0 μ μ ω ω k z k z ) 1 [ k x k y ( k z 2 k z 2 ) ( ε x ε x * ε y ε y * )
( k z 2 k z 2 + k x 2 k z 2 + k y 2 k z 2 ) ε x ε y * + ( k z 2 k z 2 + k x 2 k z 2 + k y 2 k z 2 ) ε y ε x * ]
+ ε 0 ( μ ε 1 ) ( ω + ω ) ( μ ω ) 1 { ( k x ε y k y ε x ) ( ε x * k x ) + ( ε y * k y ) + ( k z k z 3 ) 1 [ k x k y ( k z 2 k z 2 )
× ( ε x ε x * ε y ε y * ) ( k x 2 k z 2 + k x 2 k z 2 + k y 2 k z 2 ) ε x ε y * + ( k z 2 k z 2 + k x 2 k z 2 + k y 2 k z 2 ) ε y ε x * ]
+ ( k z k z ) 1 [ k x k y ε x + ( k y 2 + k z 2 ) ε y ] ( k x ε x * k x + k y ε y * k x )
( k z k z ) 1 [ ( k x 2 + k z 2 ) ε x + k x k y ε y ] ( k x ε x * k y + k y ε y * k y ) } } d k x d k y .
< T z > ( ε 0 2 A ) Imag ω 1 , 2 ( k z + k z ) 1 { ( ε 1 ) ( ε x ε y * ε y ε x * )
+ ( μ ω k z k z 3 ) 1 [ ( μ ε 1 ) ( ω + ω ) k z 2 ( μ 1 ) ( μ 0 ε 0 μ ω ) 1 ] [ k x k y ( k z 2 k z 2 ) ( ε x ε x * ε y ε y * )
( k z 2 k z 2 + k x 2 k z 2 + k y 2 k z 2 ) ε x ε y * + ( k z 2 k z 2 + k x 2 k z 2 + k y 2 k z 2 ) ε y ε y * ] } d k x d k y .
< T z > 2 c ( ε 0 A ) { μ ε + 1 2 ω [ ε ( d μ d ω ) + μ ( d ε d ω ) ] } 1 ω μ [ d ( ω μ ε ) d ω ] ε x ( k x , k y , ω ) 2 d k x d k y .
< T z > 2 ( ε 0 A ω ) ( c ε μ μ 1 V g ) ε x ( k x , k y , ω ) 2 d k x d k y .

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