Abstract

A widely held viewpoint in optics, namely, that dynamic magnetic effects are extremely weak at optical frequencies, is re-examined. Nonlinear charge motion induced by the optical magnetic field in dielectric systems is analyzed, is predicted to be resonantly enhanced, and is observed experimentally in CCl4, C6H6, and H2O at the fundamental input frequency. Excellent agreement is obtained with a classical magnetic harmonic oscillator model, which shows that the maximum dynamic magnetic dipole (MD) moment at optical frequencies is one half the electric dipole (ED) moment. As a consequence, magnetic dipole radiation generated by the optical magnetic field with an intensity one fourth that of ED radiation, as well as unanticipated nonlinear optical effects such as magnetic white-light generation, can arise in homogeneous transparent dielectrics. The mechanism of MD formation is confirmed experimentally to be second order in the input field, and the strength of the radiation is accounted for as a first-order contribution to the vector potential. Predictions are made of optical magnetic resonance, negative permeability, self-induced magnetic birefringence, and optically induced Faraday rotation.

© 2008 Optical Society of America

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References

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  1. L. P. Pitaevskii, “Electric forces in a transparent dispersive medium,” Sov. Phys. JETP 12, 1008-1013 (1961).
  2. Y. R. Shen and N. Bloembergen, “Interaction between light waves and spin waves,” Phys. Rev. 143, 372-384 (1966).
    [Crossref]
  3. P. S. Pershan, J. P. Van der Ziel, and L. D. Malmstrom, “Theoretical discussion of the inverse Faraday effect, Raman scattering, and related phenomena,” Phys. Rev. 143, 574-583 (1966).
    [Crossref]
  4. A. V. Kimel, A. Kirilyuk, F. Hansteen, R. V. Pisarev, and T. Rasing, “Non-thermal optical control of magnetism and ultrafast laser-induced spin dynamics in solids,” J. Phys.: Condens. Matter 19, 043201 (2007).
    [Crossref]
  5. G. A. Mourou, C. P. J. Barty, and M. D. Perry, “Ultrahigh-intensity lasers: Physics of the extreme on a tabletop,” Phys. Today 51, 22-28 (1998).
    [Crossref]
  6. W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism, 2nd ed. (Addison-Wesley, 1962), p. 132.
  7. L. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, 1984), pp. 268-269.
  8. See, for example, R. W. Boyd, Nonlinear Optics (Academic, 1992).
  9. W. M. Fisher and S. C. Rand, “Dependence of optical magnetic response on molecular electronic structure,” J. Lumin (to be published).
  10. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley & Sons, 1975).
  11. W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism, 2nd ed. (Addison-Wesley, 1962), pp. 245-248.
  12. See, for example, A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations (Wiley, 1979).
  13. S. L. Oliveira and S. C. Rand, “Intense nonlinear magnetic dipole radiation at optical frequencies: Molecular scattering in a dielectric liquid,” Phys. Rev. Lett. 98, 093901 (2007).
    [Crossref] [PubMed]
  14. F. Courvoisier, V. Boutou, C. Favre, S. C. Hill, and J.-P. Wolf, “Plasma formation dynamics within a water microdroplet on femtosecond time scales,” Opt. Lett. 28, 206-208 (2003).
    [Crossref] [PubMed]
  15. A. Brodeur and S. L. Chin, “Ultrafast white-light continuum generation and self-focusing in transparent condensed media,” J. Opt. Soc. Am. B 16, 637-650 (1999).
    [Crossref]
  16. N. L. Sharma, “Nondipole scattering from liquids and nanoparticles,” Phys. Rev. Lett. 98, 217402 (2007).
    [Crossref] [PubMed]
  17. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966-3969 (2000).
    [Crossref] [PubMed]
  18. D. Sievenpiper, L. Zhang, R. F. J. Broas, N G. Alexopolous, and E. Yablonovitch, “High-impedance, electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microwave Theory Tech. 47, 2059-2074 (1999).
    [Crossref]
  19. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780-1782 (2006).
    [Crossref] [PubMed]
  20. See, for example, G. Arfken, Mathematical Methods for Physicists, 2nd ed. (Academic, 1970), p. 136.

2007 (3)

A. V. Kimel, A. Kirilyuk, F. Hansteen, R. V. Pisarev, and T. Rasing, “Non-thermal optical control of magnetism and ultrafast laser-induced spin dynamics in solids,” J. Phys.: Condens. Matter 19, 043201 (2007).
[Crossref]

S. L. Oliveira and S. C. Rand, “Intense nonlinear magnetic dipole radiation at optical frequencies: Molecular scattering in a dielectric liquid,” Phys. Rev. Lett. 98, 093901 (2007).
[Crossref] [PubMed]

N. L. Sharma, “Nondipole scattering from liquids and nanoparticles,” Phys. Rev. Lett. 98, 217402 (2007).
[Crossref] [PubMed]

2006 (1)

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780-1782 (2006).
[Crossref] [PubMed]

2003 (1)

2000 (1)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966-3969 (2000).
[Crossref] [PubMed]

1999 (2)

D. Sievenpiper, L. Zhang, R. F. J. Broas, N G. Alexopolous, and E. Yablonovitch, “High-impedance, electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microwave Theory Tech. 47, 2059-2074 (1999).
[Crossref]

A. Brodeur and S. L. Chin, “Ultrafast white-light continuum generation and self-focusing in transparent condensed media,” J. Opt. Soc. Am. B 16, 637-650 (1999).
[Crossref]

1998 (1)

G. A. Mourou, C. P. J. Barty, and M. D. Perry, “Ultrahigh-intensity lasers: Physics of the extreme on a tabletop,” Phys. Today 51, 22-28 (1998).
[Crossref]

1966 (2)

Y. R. Shen and N. Bloembergen, “Interaction between light waves and spin waves,” Phys. Rev. 143, 372-384 (1966).
[Crossref]

P. S. Pershan, J. P. Van der Ziel, and L. D. Malmstrom, “Theoretical discussion of the inverse Faraday effect, Raman scattering, and related phenomena,” Phys. Rev. 143, 574-583 (1966).
[Crossref]

1961 (1)

L. P. Pitaevskii, “Electric forces in a transparent dispersive medium,” Sov. Phys. JETP 12, 1008-1013 (1961).

Alexopolous, N G.

D. Sievenpiper, L. Zhang, R. F. J. Broas, N G. Alexopolous, and E. Yablonovitch, “High-impedance, electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microwave Theory Tech. 47, 2059-2074 (1999).
[Crossref]

Arfken, G.

See, for example, G. Arfken, Mathematical Methods for Physicists, 2nd ed. (Academic, 1970), p. 136.

Barty, C. P. J.

G. A. Mourou, C. P. J. Barty, and M. D. Perry, “Ultrahigh-intensity lasers: Physics of the extreme on a tabletop,” Phys. Today 51, 22-28 (1998).
[Crossref]

Bloembergen, N.

Y. R. Shen and N. Bloembergen, “Interaction between light waves and spin waves,” Phys. Rev. 143, 372-384 (1966).
[Crossref]

Boutou, V.

Boyd, R. W.

See, for example, R. W. Boyd, Nonlinear Optics (Academic, 1992).

Broas, R. F. J.

D. Sievenpiper, L. Zhang, R. F. J. Broas, N G. Alexopolous, and E. Yablonovitch, “High-impedance, electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microwave Theory Tech. 47, 2059-2074 (1999).
[Crossref]

Brodeur, A.

Chin, S. L.

Courvoisier, F.

Favre, C.

Fisher, W. M.

W. M. Fisher and S. C. Rand, “Dependence of optical magnetic response on molecular electronic structure,” J. Lumin (to be published).

Hansteen, F.

A. V. Kimel, A. Kirilyuk, F. Hansteen, R. V. Pisarev, and T. Rasing, “Non-thermal optical control of magnetism and ultrafast laser-induced spin dynamics in solids,” J. Phys.: Condens. Matter 19, 043201 (2007).
[Crossref]

Hill, S. C.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley & Sons, 1975).

Kimel, A. V.

A. V. Kimel, A. Kirilyuk, F. Hansteen, R. V. Pisarev, and T. Rasing, “Non-thermal optical control of magnetism and ultrafast laser-induced spin dynamics in solids,” J. Phys.: Condens. Matter 19, 043201 (2007).
[Crossref]

Kirilyuk, A.

A. V. Kimel, A. Kirilyuk, F. Hansteen, R. V. Pisarev, and T. Rasing, “Non-thermal optical control of magnetism and ultrafast laser-induced spin dynamics in solids,” J. Phys.: Condens. Matter 19, 043201 (2007).
[Crossref]

Landau, L.

L. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, 1984), pp. 268-269.

Lifshitz, E. M.

L. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, 1984), pp. 268-269.

Malmstrom, L. D.

P. S. Pershan, J. P. Van der Ziel, and L. D. Malmstrom, “Theoretical discussion of the inverse Faraday effect, Raman scattering, and related phenomena,” Phys. Rev. 143, 574-583 (1966).
[Crossref]

Mook, D. T.

See, for example, A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations (Wiley, 1979).

Mourou, G. A.

G. A. Mourou, C. P. J. Barty, and M. D. Perry, “Ultrahigh-intensity lasers: Physics of the extreme on a tabletop,” Phys. Today 51, 22-28 (1998).
[Crossref]

Nayfeh, A. H.

See, for example, A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations (Wiley, 1979).

Oliveira, S. L.

S. L. Oliveira and S. C. Rand, “Intense nonlinear magnetic dipole radiation at optical frequencies: Molecular scattering in a dielectric liquid,” Phys. Rev. Lett. 98, 093901 (2007).
[Crossref] [PubMed]

Panofsky, W. K. H.

W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism, 2nd ed. (Addison-Wesley, 1962), pp. 245-248.

W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism, 2nd ed. (Addison-Wesley, 1962), p. 132.

Pendry, J. B.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780-1782 (2006).
[Crossref] [PubMed]

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966-3969 (2000).
[Crossref] [PubMed]

Perry, M. D.

G. A. Mourou, C. P. J. Barty, and M. D. Perry, “Ultrahigh-intensity lasers: Physics of the extreme on a tabletop,” Phys. Today 51, 22-28 (1998).
[Crossref]

Pershan, P. S.

P. S. Pershan, J. P. Van der Ziel, and L. D. Malmstrom, “Theoretical discussion of the inverse Faraday effect, Raman scattering, and related phenomena,” Phys. Rev. 143, 574-583 (1966).
[Crossref]

Phillips, M.

W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism, 2nd ed. (Addison-Wesley, 1962), p. 132.

W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism, 2nd ed. (Addison-Wesley, 1962), pp. 245-248.

Pisarev, R. V.

A. V. Kimel, A. Kirilyuk, F. Hansteen, R. V. Pisarev, and T. Rasing, “Non-thermal optical control of magnetism and ultrafast laser-induced spin dynamics in solids,” J. Phys.: Condens. Matter 19, 043201 (2007).
[Crossref]

Pitaevskii, L. P.

L. P. Pitaevskii, “Electric forces in a transparent dispersive medium,” Sov. Phys. JETP 12, 1008-1013 (1961).

Rand, S. C.

S. L. Oliveira and S. C. Rand, “Intense nonlinear magnetic dipole radiation at optical frequencies: Molecular scattering in a dielectric liquid,” Phys. Rev. Lett. 98, 093901 (2007).
[Crossref] [PubMed]

W. M. Fisher and S. C. Rand, “Dependence of optical magnetic response on molecular electronic structure,” J. Lumin (to be published).

Rasing, T.

A. V. Kimel, A. Kirilyuk, F. Hansteen, R. V. Pisarev, and T. Rasing, “Non-thermal optical control of magnetism and ultrafast laser-induced spin dynamics in solids,” J. Phys.: Condens. Matter 19, 043201 (2007).
[Crossref]

Schurig, D.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780-1782 (2006).
[Crossref] [PubMed]

Sharma, N. L.

N. L. Sharma, “Nondipole scattering from liquids and nanoparticles,” Phys. Rev. Lett. 98, 217402 (2007).
[Crossref] [PubMed]

Shen, Y. R.

Y. R. Shen and N. Bloembergen, “Interaction between light waves and spin waves,” Phys. Rev. 143, 372-384 (1966).
[Crossref]

Sievenpiper, D.

D. Sievenpiper, L. Zhang, R. F. J. Broas, N G. Alexopolous, and E. Yablonovitch, “High-impedance, electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microwave Theory Tech. 47, 2059-2074 (1999).
[Crossref]

Smith, D. R.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780-1782 (2006).
[Crossref] [PubMed]

Van der Ziel, J. P.

P. S. Pershan, J. P. Van der Ziel, and L. D. Malmstrom, “Theoretical discussion of the inverse Faraday effect, Raman scattering, and related phenomena,” Phys. Rev. 143, 574-583 (1966).
[Crossref]

Wolf, J.-P.

Yablonovitch, E.

D. Sievenpiper, L. Zhang, R. F. J. Broas, N G. Alexopolous, and E. Yablonovitch, “High-impedance, electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microwave Theory Tech. 47, 2059-2074 (1999).
[Crossref]

Zhang, L.

D. Sievenpiper, L. Zhang, R. F. J. Broas, N G. Alexopolous, and E. Yablonovitch, “High-impedance, electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microwave Theory Tech. 47, 2059-2074 (1999).
[Crossref]

IEEE Trans. Microwave Theory Tech. (1)

D. Sievenpiper, L. Zhang, R. F. J. Broas, N G. Alexopolous, and E. Yablonovitch, “High-impedance, electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microwave Theory Tech. 47, 2059-2074 (1999).
[Crossref]

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

J. Phys.: Condens. Matter (1)

A. V. Kimel, A. Kirilyuk, F. Hansteen, R. V. Pisarev, and T. Rasing, “Non-thermal optical control of magnetism and ultrafast laser-induced spin dynamics in solids,” J. Phys.: Condens. Matter 19, 043201 (2007).
[Crossref]

Opt. Lett. (1)

Phys. Rev. (2)

Y. R. Shen and N. Bloembergen, “Interaction between light waves and spin waves,” Phys. Rev. 143, 372-384 (1966).
[Crossref]

P. S. Pershan, J. P. Van der Ziel, and L. D. Malmstrom, “Theoretical discussion of the inverse Faraday effect, Raman scattering, and related phenomena,” Phys. Rev. 143, 574-583 (1966).
[Crossref]

Phys. Rev. Lett. (3)

S. L. Oliveira and S. C. Rand, “Intense nonlinear magnetic dipole radiation at optical frequencies: Molecular scattering in a dielectric liquid,” Phys. Rev. Lett. 98, 093901 (2007).
[Crossref] [PubMed]

N. L. Sharma, “Nondipole scattering from liquids and nanoparticles,” Phys. Rev. Lett. 98, 217402 (2007).
[Crossref] [PubMed]

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966-3969 (2000).
[Crossref] [PubMed]

Phys. Today (1)

G. A. Mourou, C. P. J. Barty, and M. D. Perry, “Ultrahigh-intensity lasers: Physics of the extreme on a tabletop,” Phys. Today 51, 22-28 (1998).
[Crossref]

Science (1)

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780-1782 (2006).
[Crossref] [PubMed]

Sov. Phys. JETP (1)

L. P. Pitaevskii, “Electric forces in a transparent dispersive medium,” Sov. Phys. JETP 12, 1008-1013 (1961).

Other (8)

See, for example, G. Arfken, Mathematical Methods for Physicists, 2nd ed. (Academic, 1970), p. 136.

W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism, 2nd ed. (Addison-Wesley, 1962), p. 132.

L. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, 1984), pp. 268-269.

See, for example, R. W. Boyd, Nonlinear Optics (Academic, 1992).

W. M. Fisher and S. C. Rand, “Dependence of optical magnetic response on molecular electronic structure,” J. Lumin (to be published).

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley & Sons, 1975).

W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism, 2nd ed. (Addison-Wesley, 1962), pp. 245-248.

See, for example, A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations (Wiley, 1979).

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Figures (7)

Fig. 1
Fig. 1

Classical charge motion produced by an electromagnetic wave polarized along x and propagating along z. Dashed (solid) vertical arrows schematically indicate motion with (without) the Lorentz force.

Fig. 2
Fig. 2

Illustration of the dependence of restoring forces on direction of classical motion of a charge in a simple molecular potential well V ( r ) . The slopes of the potential (and therefore the restoring forces) are different for motions in the two directions indicated by double-headed arrows.

Fig. 3
Fig. 3

Geometry for integration of Ampère’s law to determine relative magnitudes and phases of electric and magnetic current density.

Fig. 4
Fig. 4

Relative orientations of incident and scattered dipole electromagnetic fields for (a) vertical incident polarization (which yields a maximum intensity for ED radiation detected through a vertical analyzer and an MD null), and (b) horizontal incident polarization (which yields a maximum intensity for MD radiation detected through a horizontal analyzer and an ED null).

Fig. 5
Fig. 5

Polar plots of experimental radiation patterns of magnetic (solid circles) and electric (open circles) scattering intensities obtained using amplified pulses above white-light threshold in (a) CCl 4 and (b) H 2 O .

Fig. 6
Fig. 6

Radiation patterns of MD (solid circles) and ED (open circles) scattering intensities obtained using unamplified pulses below white-light threshold: (a) CCl 4 , (b) magnified view of the magnetic component in CCl 4 , (c) H 2 O , and (d) C 6 H 6 .

Fig. 7
Fig. 7

Experimental intensity of magnetic dipole scattering versus input intensity in CCl 4 . The solid (dashed) curve is a linear (quadratic) regression through the data.

Equations (83)

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4 π μ 0 A ¯ ( r ¯ , t ) = 1 R J ¯ ( r ¯ , t ) d 3 r [ x α ( 1 r ) ] R r α J ¯ ( r ¯ , t ) d 3 r + .
m p = ( m e c a 0 ) 1 ,
A ¯ ω ( r ¯ ) = μ 0 e i k r 4 π r J ¯ E , ω ( r ¯ ) d 3 r + μ 0 e i k r 4 π r J ¯ M , ω ( r ¯ ) d 3 r + .
E ¯ ( z , t ) = 1 2 x ̂ ( E 0 exp [ i ( ω t k z ) ] + c.c. ) ,
B ¯ ( z , t ) = 1 2 y ̂ ( B 0 exp [ i ( ω t k z ) ] + c.c. ) .
2 x ( t ) t 2 + γ 1 x ( t ) t + ω 1 2 x ( t ) = e E ( t ) m e + e B ( t ) m e z ( t ) t ,
2 z ( t ) t 2 + γ 2 z ( t ) t + ω 2 2 z ( t ) = e B ( t ) m e x ( t ) t .
x ( t ) = 1 2 [ x 0 exp ( i ω t ) + x 0 * exp ( i ω t ) ] ,
z ( t ) = 1 2 [ z 0 exp ( i 2 ω t ) + z 0 * exp ( i 2 ω t ) ] .
x 0 = ( e E 0 m e ) ( ω 2 + i ω γ 1 ω 1 2 ) ( 1 + N ( ω , B 0 ) ) ,
z 0 = 1 2 i ω ω c ( e E 0 m e ) ( ω 2 + i ω γ 1 ω 1 2 ) ( 4 ω 2 + 2 i ω γ 2 ω 2 2 ) ( 1 + N ( ω , B 0 ) ) .
N ( ω , B 0 ) = ω 2 ω c 2 2 ( ω 2 + i ω γ 1 ω 1 2 ) ( 4 ω 2 + 2 i ω γ 1 ω 2 2 ) ,
[ J ¯ M ( 2 ω ) ] Cart = J M [ x ̂ cos ω t z ̂ sin ω t ] cos ω t = 1 2 J M [ x ̂ ( 1 + cos 2 ω t ) z ̂ sin 2 ω t ] .
ω = ω 2 2 .
R = J M J E = 2 z ̇ x ̇ = 2 ω ω c [ ( 4 ω 2 ω 2 2 ) 2 + ( 2 ω γ 2 ) 2 ] 1 2 ( 4 ω 2 ω 2 2 ) 2 + 4 ω 2 γ 2 2 .
¯ × H ¯ = ε 0 E ¯ ̇ + J ¯ p + J ¯ M .
J ¯ p = J ¯ p , + J ¯ p , , J ¯ M = J ¯ M , + J ¯ M , ,
S ( ¯ × H ¯ ) d s ¯ = ε 0 S E ¯ ̇ d s ¯ + S J ¯ p , d s ¯ + S J ¯ M , d s ¯ .
0 = S J ¯ p , d s ¯ + S J ¯ M , d s ¯ ,
S J ¯ M , d s ¯ = S J ¯ p , d s ¯ .
S J ¯ M , d s ¯ = S J ¯ p , d s ¯ .
J M = 1 2 [ J p ] tot = 1 2 J E .
E ¯ rad = 1 4 π ε 0 c 2 ( [ J ¯ ̇ ] × r ̂ ) × r ̂ r d V ,
H ¯ rad = 1 4 π c [ J ¯ ̇ ] × r ̂ r d V ,
S M S E = ω ( J ¯ M × r ̂ ) × r ̂ d V r × ω ( J ¯ M × r ̂ ) d V r ω ( J ¯ E × r ̂ ) × r ̂ d V r × ω ( J ¯ E × r ̂ ) d V r = J M 2 J E 2 ( J ̂ M × r ̂ ) × r ̂ d V r × J ̂ M × r ̂ d V r ( J ̂ E × r ̂ ) × r ̂ d V r × J ̂ E × r ̂ d V r = J M 2 J E 2 .
S M S E = R max 2 = 1 4 .
E ¯ = 1 2 [ E 0 x x ̂ + E 0 y y ̂ ] e i ω t + c.c. ,
B ¯ = 1 2 [ B 0 x x ̂ + B 0 y y ̂ ] e i ω t + c.c. ,
r ¯ = [ x 0 ( 0 ) x ̂ + y 0 ( 0 ) y ̂ + z 0 ( 0 ) z ̂ ] + 1 2 [ x 0 ( 1 ) x ̂ + y 0 ( 1 ) y ̂ + z 0 ( 1 ) z ̂ ] e i ω t + c . c . + 1 2 [ x 0 ( 2 ) x ̂ + y 0 ( 2 ) y ̂ + z 0 ( 2 ) z ̂ ] e 2 i ω t + c.c. ,
m e r ¯ ̈ + m e γ r ¯ ̇ + K r ¯ = e E ¯ e r ̇ ¯ × B ¯ ,
m e P ̈ x + m e γ x P ̇ x + K x P x = N e 2 E 0 x e ( P ̇ y B 0 z P ̇ z B 0 y ) ,
m e P ̈ y + m e γ y P ̇ y + K y P y = N e 2 E 0 y e ( P ̇ z B 0 x P ̇ x B 0 z ) ,
m e P ̈ z + m e γ z P ̇ z + K z P z = e ( P ̇ x B 0 y P ̇ y B 0 x ) .
[ χ ( 0 ) ] i j = P i ( 0 ) ε 0 E 0 j ,
[ χ ( ω ) ] i j = P i ( 1 ) ε 0 E 0 j ,
[ χ ( 2 ω ) ] i j = P i ( 2 ) ε 0 E 0 j ,
[ χ ( 0 ; ω , ω ) ] 12 = N e 2 ε 0 [ i e ω B 0 z * 4 K x Δ y F y G y + c.c. ] ,
[ χ ( 0 ; ω , ω ) ] 21 = N e 2 ε 0 [ i e ω B 0 z * 4 K y Δ x F x + c.c. ] ,
[ χ ( 0 ; ω , ω ) ] 13 = N e 2 ε 0 [ i e ω B 0 y * 4 K x Δ z F z G z + c.c. ] ,
[ χ ( 0 ; ω , ω ) ] 31 = N e 2 ε 0 [ i e ω B 0 y * 4 K z Δ x F x + c.c. ] ,
[ χ ( 0 ; ω , ω ) ] 23 = N e 2 ε 0 [ i e ω B 0 x * 4 K y Δ z F z G z + c.c. ] ,
[ χ ( 0 ; ω , ω ) ] 32 = N e 2 ε 0 [ i e ω B 0 x * 4 K z Δ y F y G y + c.c. ] .
[ χ ( ω ; ω ) ] 11 = N e 2 ε 0 [ 1 Δ x F x ] ,
[ χ ( ω ; ω , ω , ω ) ] 12 = N e 2 ε 0 [ e 2 ω 2 B 0 x B 0 y * 2 Δ x Δ y Δ z F x F y G y ] ,
[ χ ( ω ; ω , ω , ω ) ] 21 = N e 2 ε 0 [ e 2 ω 2 B 0 x * B 0 y 2 Δ x Δ y Δ z F x F y G y ] ,
[ χ ( ω ; ω ) ] 22 = N e 2 ε 0 [ 1 Δ y F y G y ] ,
[ χ ( ω ; ω , ω , ω ) ] 23 = N e 2 ε 0 [ e 2 ω 2 B 0 y B 0 z * 2 Δ x Δ y Δ z F y F z G y G z ] ,
[ χ ( ω ; ω , ω , ω ) ] 32 = N e 2 ε 0 [ e 2 ω 2 B 0 y * B 0 z 2 Δ x Δ y Δ z F y F z G y G z ] ,
[ χ ( ω ; ω ) ] 33 = N e 2 ε 0 [ 1 Δ z F z G z ] ,
[ χ ( ω ; ω , ω , ω ) ] 31 = N e 2 ε 0 [ e 2 ω 2 B 0 x * B 0 z 2 Δ x Δ y Δ z F x F z G z ] ,
[ χ ( ω ; ω , ω , ω ) ] 13 = N e 2 ε 0 [ e 2 ω 2 B 0 x B 0 z * 2 Δ x Δ y Δ z F x F z G z ] .
[ χ ( 2 ω ; ω , ω ) ] 12 = N e 2 ε 0 [ i e ω B 0 z 2 Δ x Δ y F y G y ] ,
[ χ ( 2 ω ; ω , ω ) ] 21 = N e 2 ε 0 [ i e ω B 0 z 2 Δ x Δ y F x ] ,
[ χ ( 2 ω ; ω , ω ) ] 13 = N e 2 ε 0 [ i e ω B 0 y 2 Δ x Δ z F z G z ] ,
[ χ ( 2 ω ; ω , ω ) ] 31 = N e 2 ε 0 [ i e ω B 0 y 2 Δ x Δ z F x ] ,
[ χ ( 2 ω ; ω , ω ) ] 32 = N e 2 ε 0 [ i e ω B 0 x 2 Δ y Δ z F y G y ] ,
[ χ ( 2 ω ; ω , ω ) ] 23 = N e 2 ε 0 [ i e ω B 0 x 2 Δ y Δ z F z G z ] .
Δ i m e ω 2 i m e γ i ω + K i ,
Δ i 4 m e ω 2 2 i m e γ i ω + K i .
F i 1 e 2 ω 2 2 Δ i ( B 0 k 2 Δ j + B 0 j 2 Δ k ) ,
G y 1 ( e 2 ω 2 B 0 x B 0 y ) 2 2 Δ x Δ y ( Δ z ) 2 F x F y ,
G z 1 ( e 2 ω 2 B 0 x B 0 z ) 2 4 Δ x ( Δ y ) 2 Δ z F x F z .
χ = [ χ 11 0 i χ 13 0 χ 22 0 i χ 31 0 χ 33 ] .
E z = ( χ 31 1 + χ 11 ) E x .
( J ¯ M ) = R ( J ¯ E ) = R ( P ¯ ) t .
( ¯ × M ¯ ) = ( ¯ × M ¯ ) = R ( P ¯ ) t .
M ¯ ( t ) = c R P ( t ) y ̂ .
¯ S ¯ = E ¯ J ¯ + 1 2 t ( μ 0 H ¯ H ¯ + ε 0 E ¯ E ¯ ) + ( H ¯ t μ 0 M ¯ + E ¯ t P ¯ ) .
4 π R μ 0 A ¯ ω ( r ¯ ) = V J ¯ , ω ( r ¯ ) d V + V J ¯ , ω ( r ¯ ) d V + .
m ¯ = 1 2 V ( r ¯ × J ¯ ) d V .
¯ × M ¯ = ¯ × [ 1 2 ( r ¯ × J ¯ M ) ] .
[ V [ ¯ × M ¯ ] d V ] i = 1 2 V ε i j k x j ε k l m x l ( 2 J M θ ) m d V ,
[ V [ ¯ × M ¯ ] d V ] i = ε i j k ε l m k V x j x l ( J M θ ) m d V ,
[ V [ ¯ × M ¯ ] d V ] i = ( δ i l δ j m δ i m δ j l ) V x j x l ( J M θ ) m d V , = V [ x j x l ( J M θ ) j x j x j ( J M θ ) i ] d V .
[ V [ ¯ × M ¯ ] d V ] i = V [ ( ¯ J ¯ M ) x i + ( J M θ ) i ¯ [ x ¯ ( J M θ ) i ] ] d V .
[ V [ ¯ × M ¯ ] d V ] i = V [ ( J M θ ) i ¯ [ x ¯ ( J M θ ) i ] ] d V .
V [ x ¯ ( J M θ ) i ] d V = ( J M θ ) i S x ¯ d s = 0 ,
[ V [ ¯ × M ¯ ] d V ] i = V ( J M θ ) i d V .
0 = J ¯ sol + J ¯ nonsol + ¯ × M ¯ + P ¯ t ,
J ¯ sol + ¯ × M ¯ = 0 ,
¯ J ¯ sol = 0 ,
J ¯ nonsol + P ¯ t = 0 ,
¯ J ¯ nonsol 0 .

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