Abstract

Recent experiments have found entities in crystals whose behavior is equivalent to magnetic monopoles. In this paper, we explain some optical properties based on the reformulated “Maxwell” equations in material media in which there are equivalent magnetic charges. We calculate the coefficients of reflection and transmission of an electromagnetic wave in a plane interface between the vacuum and a medium with magnetic charges. These results can give a more extended vision of the properties of the materials with magnetic monopoles, since the phase and the amplitudes of the reflected and transmitted waves, differ with and without these magnetic entities.

© 2011 Optical Society of America

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References

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  1. P. Peregrinus, The Letter of Petrus Peregrinus, translated by B. Arnold (McGraw Publishing Company, 1904).
  2. P. A. M. Dirac, "Quantized Singularities in the Electromagnetic Field," Proc. R. Soc. Lond. 133, 60 (1931).
  3. P. A. M. Dirac, "The Theory of Magnetic Poles," Phys. Rev. 74, 817-830 (1948).
    [CrossRef]
  4. G. ’t Hooft, "Magnetic monopoles in unified gauge theories," Nucl. Phys. B 79, 276-284 (1974).
    [CrossRef]
  5. B. Cabrera, "First Results from a Superconductive Detector for Moving Magnetic Monopoles," Phys. Rev. Lett. 48, 1378-1381 (1982).
    [CrossRef]
  6. C. Castelnovo, R. Moessner, and S. L. Sondhi, "Magnetic monopoles in spin ice," Nature 451, 42-45 (2008).
    [CrossRef] [PubMed]
  7. X.-L. Qi, R. Li, J. Zang, and S.-C. Zhang, "Inducing a Magnetic Monopole with Topological Surface States," Science 323, 1184-1187 (2009).
    [CrossRef] [PubMed]
  8. S. T. Bramwell, S. R. Giblin, S. Calder, R. Aldus, D. Prabhakaran, and T. Fennell, "Measurement of the charge and current of magnetic monopoles in spin ice," Nature 461, 956-960 (2009).
    [CrossRef] [PubMed]
  9. D. J. P. Morris, D. A. Tennant, S. A. Grigera, B. Klemke, C. Castelnovo, R. Moessner, C. Czternasty, M. Meissner, K. C. Rule, J.-U. Hoffmann, K. Kiefer, S. Gerischer, D. Slobinsky, and R. S. Perry, "Dirac Strings and Magnetic Monopoles in the Spin Ice Dy2Ti2O7," Science 326, 411-414 (2009).
    [CrossRef] [PubMed]
  10. M. J. Gingrass, "Observing monopoles in a magnetic analog of ice," Science 326, 375-376 (2009).
    [CrossRef]
  11. S. Ladak, D. E. Read, G. K. Perkins, L. F. Cohen, and W. R. Branford, "Direct observation of magnetic monopole defects in an artificial spin-ice system," Nat. Phys. 6, 359-363 (2010).
    [CrossRef]
  12. S. Sondhi, "Wien route to monopoles," Nature 461, 888-889 (2009).
    [CrossRef] [PubMed]
  13. L. J. Onsager, "Deviations from Ohm’s Law in Weak Electrolytes," J. Chem. Phys. 2, 599-615 (1934).
    [CrossRef]
  14. See for example,J. D. Jackson, Classical Electrodynamics, third edition (John Wiley & Sons, Inc., 1999).
  15. K. A. Milton, "Theoretical and experimental status of magnetic monopoles," Rep. Prog. Phys. 69, 1637-1711 (2006).
    [CrossRef]
  16. J. Costa-Quintana, and F. Lopez-Aguilar, "Extended classical electrodynamics with magnetic monopoles," Far East J. Mech. Eng. Phys. 1, 19-56 (2010).
  17. J. Costa-Quintana, and F. Lopez-Aguilar, "Propagation of electromagnetic waves in material media with magnetic monopoles," Prog. Electromagn. Res. 110, 267-295 (2010).
    [CrossRef]
  18. Y. M. Shnir, Magnetic monopoles, (Springer-Verlag, 2005).
  19. G. R. Fowles, Introduction to Modern Optics, (Dover Publications, Inc., 1975).

2010 (3)

S. Ladak, D. E. Read, G. K. Perkins, L. F. Cohen, and W. R. Branford, "Direct observation of magnetic monopole defects in an artificial spin-ice system," Nat. Phys. 6, 359-363 (2010).
[CrossRef]

J. Costa-Quintana, and F. Lopez-Aguilar, "Extended classical electrodynamics with magnetic monopoles," Far East J. Mech. Eng. Phys. 1, 19-56 (2010).

J. Costa-Quintana, and F. Lopez-Aguilar, "Propagation of electromagnetic waves in material media with magnetic monopoles," Prog. Electromagn. Res. 110, 267-295 (2010).
[CrossRef]

2009 (5)

S. Sondhi, "Wien route to monopoles," Nature 461, 888-889 (2009).
[CrossRef] [PubMed]

X.-L. Qi, R. Li, J. Zang, and S.-C. Zhang, "Inducing a Magnetic Monopole with Topological Surface States," Science 323, 1184-1187 (2009).
[CrossRef] [PubMed]

S. T. Bramwell, S. R. Giblin, S. Calder, R. Aldus, D. Prabhakaran, and T. Fennell, "Measurement of the charge and current of magnetic monopoles in spin ice," Nature 461, 956-960 (2009).
[CrossRef] [PubMed]

D. J. P. Morris, D. A. Tennant, S. A. Grigera, B. Klemke, C. Castelnovo, R. Moessner, C. Czternasty, M. Meissner, K. C. Rule, J.-U. Hoffmann, K. Kiefer, S. Gerischer, D. Slobinsky, and R. S. Perry, "Dirac Strings and Magnetic Monopoles in the Spin Ice Dy2Ti2O7," Science 326, 411-414 (2009).
[CrossRef] [PubMed]

M. J. Gingrass, "Observing monopoles in a magnetic analog of ice," Science 326, 375-376 (2009).
[CrossRef]

2008 (1)

C. Castelnovo, R. Moessner, and S. L. Sondhi, "Magnetic monopoles in spin ice," Nature 451, 42-45 (2008).
[CrossRef] [PubMed]

2006 (1)

K. A. Milton, "Theoretical and experimental status of magnetic monopoles," Rep. Prog. Phys. 69, 1637-1711 (2006).
[CrossRef]

1982 (1)

B. Cabrera, "First Results from a Superconductive Detector for Moving Magnetic Monopoles," Phys. Rev. Lett. 48, 1378-1381 (1982).
[CrossRef]

1974 (1)

G. ’t Hooft, "Magnetic monopoles in unified gauge theories," Nucl. Phys. B 79, 276-284 (1974).
[CrossRef]

1948 (1)

P. A. M. Dirac, "The Theory of Magnetic Poles," Phys. Rev. 74, 817-830 (1948).
[CrossRef]

1934 (1)

L. J. Onsager, "Deviations from Ohm’s Law in Weak Electrolytes," J. Chem. Phys. 2, 599-615 (1934).
[CrossRef]

1931 (1)

P. A. M. Dirac, "Quantized Singularities in the Electromagnetic Field," Proc. R. Soc. Lond. 133, 60 (1931).

’t Hooft, G.

G. ’t Hooft, "Magnetic monopoles in unified gauge theories," Nucl. Phys. B 79, 276-284 (1974).
[CrossRef]

Aldus, R.

S. T. Bramwell, S. R. Giblin, S. Calder, R. Aldus, D. Prabhakaran, and T. Fennell, "Measurement of the charge and current of magnetic monopoles in spin ice," Nature 461, 956-960 (2009).
[CrossRef] [PubMed]

Bramwell, S. T.

S. T. Bramwell, S. R. Giblin, S. Calder, R. Aldus, D. Prabhakaran, and T. Fennell, "Measurement of the charge and current of magnetic monopoles in spin ice," Nature 461, 956-960 (2009).
[CrossRef] [PubMed]

Branford, W. R.

S. Ladak, D. E. Read, G. K. Perkins, L. F. Cohen, and W. R. Branford, "Direct observation of magnetic monopole defects in an artificial spin-ice system," Nat. Phys. 6, 359-363 (2010).
[CrossRef]

Cabrera, B.

B. Cabrera, "First Results from a Superconductive Detector for Moving Magnetic Monopoles," Phys. Rev. Lett. 48, 1378-1381 (1982).
[CrossRef]

Calder, S.

S. T. Bramwell, S. R. Giblin, S. Calder, R. Aldus, D. Prabhakaran, and T. Fennell, "Measurement of the charge and current of magnetic monopoles in spin ice," Nature 461, 956-960 (2009).
[CrossRef] [PubMed]

Castelnovo, C.

D. J. P. Morris, D. A. Tennant, S. A. Grigera, B. Klemke, C. Castelnovo, R. Moessner, C. Czternasty, M. Meissner, K. C. Rule, J.-U. Hoffmann, K. Kiefer, S. Gerischer, D. Slobinsky, and R. S. Perry, "Dirac Strings and Magnetic Monopoles in the Spin Ice Dy2Ti2O7," Science 326, 411-414 (2009).
[CrossRef] [PubMed]

C. Castelnovo, R. Moessner, and S. L. Sondhi, "Magnetic monopoles in spin ice," Nature 451, 42-45 (2008).
[CrossRef] [PubMed]

Cohen, L. F.

S. Ladak, D. E. Read, G. K. Perkins, L. F. Cohen, and W. R. Branford, "Direct observation of magnetic monopole defects in an artificial spin-ice system," Nat. Phys. 6, 359-363 (2010).
[CrossRef]

Costa-Quintana, J.

J. Costa-Quintana, and F. Lopez-Aguilar, "Propagation of electromagnetic waves in material media with magnetic monopoles," Prog. Electromagn. Res. 110, 267-295 (2010).
[CrossRef]

J. Costa-Quintana, and F. Lopez-Aguilar, "Extended classical electrodynamics with magnetic monopoles," Far East J. Mech. Eng. Phys. 1, 19-56 (2010).

Czternasty, C.

D. J. P. Morris, D. A. Tennant, S. A. Grigera, B. Klemke, C. Castelnovo, R. Moessner, C. Czternasty, M. Meissner, K. C. Rule, J.-U. Hoffmann, K. Kiefer, S. Gerischer, D. Slobinsky, and R. S. Perry, "Dirac Strings and Magnetic Monopoles in the Spin Ice Dy2Ti2O7," Science 326, 411-414 (2009).
[CrossRef] [PubMed]

Dirac, P. A. M.

P. A. M. Dirac, "The Theory of Magnetic Poles," Phys. Rev. 74, 817-830 (1948).
[CrossRef]

P. A. M. Dirac, "Quantized Singularities in the Electromagnetic Field," Proc. R. Soc. Lond. 133, 60 (1931).

Fennell, T.

S. T. Bramwell, S. R. Giblin, S. Calder, R. Aldus, D. Prabhakaran, and T. Fennell, "Measurement of the charge and current of magnetic monopoles in spin ice," Nature 461, 956-960 (2009).
[CrossRef] [PubMed]

Gerischer, S.

D. J. P. Morris, D. A. Tennant, S. A. Grigera, B. Klemke, C. Castelnovo, R. Moessner, C. Czternasty, M. Meissner, K. C. Rule, J.-U. Hoffmann, K. Kiefer, S. Gerischer, D. Slobinsky, and R. S. Perry, "Dirac Strings and Magnetic Monopoles in the Spin Ice Dy2Ti2O7," Science 326, 411-414 (2009).
[CrossRef] [PubMed]

Giblin, S. R.

S. T. Bramwell, S. R. Giblin, S. Calder, R. Aldus, D. Prabhakaran, and T. Fennell, "Measurement of the charge and current of magnetic monopoles in spin ice," Nature 461, 956-960 (2009).
[CrossRef] [PubMed]

Gingrass, M. J.

M. J. Gingrass, "Observing monopoles in a magnetic analog of ice," Science 326, 375-376 (2009).
[CrossRef]

Grigera, S. A.

D. J. P. Morris, D. A. Tennant, S. A. Grigera, B. Klemke, C. Castelnovo, R. Moessner, C. Czternasty, M. Meissner, K. C. Rule, J.-U. Hoffmann, K. Kiefer, S. Gerischer, D. Slobinsky, and R. S. Perry, "Dirac Strings and Magnetic Monopoles in the Spin Ice Dy2Ti2O7," Science 326, 411-414 (2009).
[CrossRef] [PubMed]

Hoffmann, J.-U.

D. J. P. Morris, D. A. Tennant, S. A. Grigera, B. Klemke, C. Castelnovo, R. Moessner, C. Czternasty, M. Meissner, K. C. Rule, J.-U. Hoffmann, K. Kiefer, S. Gerischer, D. Slobinsky, and R. S. Perry, "Dirac Strings and Magnetic Monopoles in the Spin Ice Dy2Ti2O7," Science 326, 411-414 (2009).
[CrossRef] [PubMed]

Kiefer, K.

D. J. P. Morris, D. A. Tennant, S. A. Grigera, B. Klemke, C. Castelnovo, R. Moessner, C. Czternasty, M. Meissner, K. C. Rule, J.-U. Hoffmann, K. Kiefer, S. Gerischer, D. Slobinsky, and R. S. Perry, "Dirac Strings and Magnetic Monopoles in the Spin Ice Dy2Ti2O7," Science 326, 411-414 (2009).
[CrossRef] [PubMed]

Klemke, B.

D. J. P. Morris, D. A. Tennant, S. A. Grigera, B. Klemke, C. Castelnovo, R. Moessner, C. Czternasty, M. Meissner, K. C. Rule, J.-U. Hoffmann, K. Kiefer, S. Gerischer, D. Slobinsky, and R. S. Perry, "Dirac Strings and Magnetic Monopoles in the Spin Ice Dy2Ti2O7," Science 326, 411-414 (2009).
[CrossRef] [PubMed]

Ladak, S.

S. Ladak, D. E. Read, G. K. Perkins, L. F. Cohen, and W. R. Branford, "Direct observation of magnetic monopole defects in an artificial spin-ice system," Nat. Phys. 6, 359-363 (2010).
[CrossRef]

Li, R.

X.-L. Qi, R. Li, J. Zang, and S.-C. Zhang, "Inducing a Magnetic Monopole with Topological Surface States," Science 323, 1184-1187 (2009).
[CrossRef] [PubMed]

Lopez-Aguilar, F.

J. Costa-Quintana, and F. Lopez-Aguilar, "Extended classical electrodynamics with magnetic monopoles," Far East J. Mech. Eng. Phys. 1, 19-56 (2010).

J. Costa-Quintana, and F. Lopez-Aguilar, "Propagation of electromagnetic waves in material media with magnetic monopoles," Prog. Electromagn. Res. 110, 267-295 (2010).
[CrossRef]

Meissner, M.

D. J. P. Morris, D. A. Tennant, S. A. Grigera, B. Klemke, C. Castelnovo, R. Moessner, C. Czternasty, M. Meissner, K. C. Rule, J.-U. Hoffmann, K. Kiefer, S. Gerischer, D. Slobinsky, and R. S. Perry, "Dirac Strings and Magnetic Monopoles in the Spin Ice Dy2Ti2O7," Science 326, 411-414 (2009).
[CrossRef] [PubMed]

Milton, K. A.

K. A. Milton, "Theoretical and experimental status of magnetic monopoles," Rep. Prog. Phys. 69, 1637-1711 (2006).
[CrossRef]

Moessner, R.

D. J. P. Morris, D. A. Tennant, S. A. Grigera, B. Klemke, C. Castelnovo, R. Moessner, C. Czternasty, M. Meissner, K. C. Rule, J.-U. Hoffmann, K. Kiefer, S. Gerischer, D. Slobinsky, and R. S. Perry, "Dirac Strings and Magnetic Monopoles in the Spin Ice Dy2Ti2O7," Science 326, 411-414 (2009).
[CrossRef] [PubMed]

C. Castelnovo, R. Moessner, and S. L. Sondhi, "Magnetic monopoles in spin ice," Nature 451, 42-45 (2008).
[CrossRef] [PubMed]

Morris, D. J. P.

D. J. P. Morris, D. A. Tennant, S. A. Grigera, B. Klemke, C. Castelnovo, R. Moessner, C. Czternasty, M. Meissner, K. C. Rule, J.-U. Hoffmann, K. Kiefer, S. Gerischer, D. Slobinsky, and R. S. Perry, "Dirac Strings and Magnetic Monopoles in the Spin Ice Dy2Ti2O7," Science 326, 411-414 (2009).
[CrossRef] [PubMed]

Onsager, L. J.

L. J. Onsager, "Deviations from Ohm’s Law in Weak Electrolytes," J. Chem. Phys. 2, 599-615 (1934).
[CrossRef]

Perkins, G. K.

S. Ladak, D. E. Read, G. K. Perkins, L. F. Cohen, and W. R. Branford, "Direct observation of magnetic monopole defects in an artificial spin-ice system," Nat. Phys. 6, 359-363 (2010).
[CrossRef]

Perry, R. S.

D. J. P. Morris, D. A. Tennant, S. A. Grigera, B. Klemke, C. Castelnovo, R. Moessner, C. Czternasty, M. Meissner, K. C. Rule, J.-U. Hoffmann, K. Kiefer, S. Gerischer, D. Slobinsky, and R. S. Perry, "Dirac Strings and Magnetic Monopoles in the Spin Ice Dy2Ti2O7," Science 326, 411-414 (2009).
[CrossRef] [PubMed]

Prabhakaran, D.

S. T. Bramwell, S. R. Giblin, S. Calder, R. Aldus, D. Prabhakaran, and T. Fennell, "Measurement of the charge and current of magnetic monopoles in spin ice," Nature 461, 956-960 (2009).
[CrossRef] [PubMed]

Qi, X.-L.

X.-L. Qi, R. Li, J. Zang, and S.-C. Zhang, "Inducing a Magnetic Monopole with Topological Surface States," Science 323, 1184-1187 (2009).
[CrossRef] [PubMed]

Read, D. E.

S. Ladak, D. E. Read, G. K. Perkins, L. F. Cohen, and W. R. Branford, "Direct observation of magnetic monopole defects in an artificial spin-ice system," Nat. Phys. 6, 359-363 (2010).
[CrossRef]

Rule, K. C.

D. J. P. Morris, D. A. Tennant, S. A. Grigera, B. Klemke, C. Castelnovo, R. Moessner, C. Czternasty, M. Meissner, K. C. Rule, J.-U. Hoffmann, K. Kiefer, S. Gerischer, D. Slobinsky, and R. S. Perry, "Dirac Strings and Magnetic Monopoles in the Spin Ice Dy2Ti2O7," Science 326, 411-414 (2009).
[CrossRef] [PubMed]

Slobinsky, D.

D. J. P. Morris, D. A. Tennant, S. A. Grigera, B. Klemke, C. Castelnovo, R. Moessner, C. Czternasty, M. Meissner, K. C. Rule, J.-U. Hoffmann, K. Kiefer, S. Gerischer, D. Slobinsky, and R. S. Perry, "Dirac Strings and Magnetic Monopoles in the Spin Ice Dy2Ti2O7," Science 326, 411-414 (2009).
[CrossRef] [PubMed]

Sondhi, S.

S. Sondhi, "Wien route to monopoles," Nature 461, 888-889 (2009).
[CrossRef] [PubMed]

Sondhi, S. L.

C. Castelnovo, R. Moessner, and S. L. Sondhi, "Magnetic monopoles in spin ice," Nature 451, 42-45 (2008).
[CrossRef] [PubMed]

Tennant, D. A.

D. J. P. Morris, D. A. Tennant, S. A. Grigera, B. Klemke, C. Castelnovo, R. Moessner, C. Czternasty, M. Meissner, K. C. Rule, J.-U. Hoffmann, K. Kiefer, S. Gerischer, D. Slobinsky, and R. S. Perry, "Dirac Strings and Magnetic Monopoles in the Spin Ice Dy2Ti2O7," Science 326, 411-414 (2009).
[CrossRef] [PubMed]

Zang, J.

X.-L. Qi, R. Li, J. Zang, and S.-C. Zhang, "Inducing a Magnetic Monopole with Topological Surface States," Science 323, 1184-1187 (2009).
[CrossRef] [PubMed]

Zhang, S.-C.

X.-L. Qi, R. Li, J. Zang, and S.-C. Zhang, "Inducing a Magnetic Monopole with Topological Surface States," Science 323, 1184-1187 (2009).
[CrossRef] [PubMed]

Far East J. Mech. Eng. Phys. (1)

J. Costa-Quintana, and F. Lopez-Aguilar, "Extended classical electrodynamics with magnetic monopoles," Far East J. Mech. Eng. Phys. 1, 19-56 (2010).

J. Chem. Phys. (1)

L. J. Onsager, "Deviations from Ohm’s Law in Weak Electrolytes," J. Chem. Phys. 2, 599-615 (1934).
[CrossRef]

Nat. Phys. (1)

S. Ladak, D. E. Read, G. K. Perkins, L. F. Cohen, and W. R. Branford, "Direct observation of magnetic monopole defects in an artificial spin-ice system," Nat. Phys. 6, 359-363 (2010).
[CrossRef]

Nature (3)

S. Sondhi, "Wien route to monopoles," Nature 461, 888-889 (2009).
[CrossRef] [PubMed]

C. Castelnovo, R. Moessner, and S. L. Sondhi, "Magnetic monopoles in spin ice," Nature 451, 42-45 (2008).
[CrossRef] [PubMed]

S. T. Bramwell, S. R. Giblin, S. Calder, R. Aldus, D. Prabhakaran, and T. Fennell, "Measurement of the charge and current of magnetic monopoles in spin ice," Nature 461, 956-960 (2009).
[CrossRef] [PubMed]

Nucl. Phys. B (1)

G. ’t Hooft, "Magnetic monopoles in unified gauge theories," Nucl. Phys. B 79, 276-284 (1974).
[CrossRef]

Phys. Rev. (1)

P. A. M. Dirac, "The Theory of Magnetic Poles," Phys. Rev. 74, 817-830 (1948).
[CrossRef]

Phys. Rev. Lett. (1)

B. Cabrera, "First Results from a Superconductive Detector for Moving Magnetic Monopoles," Phys. Rev. Lett. 48, 1378-1381 (1982).
[CrossRef]

Proc. R. Soc. Lond. (1)

P. A. M. Dirac, "Quantized Singularities in the Electromagnetic Field," Proc. R. Soc. Lond. 133, 60 (1931).

Prog. Electromagn. Res. (1)

J. Costa-Quintana, and F. Lopez-Aguilar, "Propagation of electromagnetic waves in material media with magnetic monopoles," Prog. Electromagn. Res. 110, 267-295 (2010).
[CrossRef]

Rep. Prog. Phys. (1)

K. A. Milton, "Theoretical and experimental status of magnetic monopoles," Rep. Prog. Phys. 69, 1637-1711 (2006).
[CrossRef]

Science (3)

D. J. P. Morris, D. A. Tennant, S. A. Grigera, B. Klemke, C. Castelnovo, R. Moessner, C. Czternasty, M. Meissner, K. C. Rule, J.-U. Hoffmann, K. Kiefer, S. Gerischer, D. Slobinsky, and R. S. Perry, "Dirac Strings and Magnetic Monopoles in the Spin Ice Dy2Ti2O7," Science 326, 411-414 (2009).
[CrossRef] [PubMed]

M. J. Gingrass, "Observing monopoles in a magnetic analog of ice," Science 326, 375-376 (2009).
[CrossRef]

X.-L. Qi, R. Li, J. Zang, and S.-C. Zhang, "Inducing a Magnetic Monopole with Topological Surface States," Science 323, 1184-1187 (2009).
[CrossRef] [PubMed]

Other (4)

P. Peregrinus, The Letter of Petrus Peregrinus, translated by B. Arnold (McGraw Publishing Company, 1904).

Y. M. Shnir, Magnetic monopoles, (Springer-Verlag, 2005).

G. R. Fowles, Introduction to Modern Optics, (Dover Publications, Inc., 1975).

See for example,J. D. Jackson, Classical Electrodynamics, third edition (John Wiley & Sons, Inc., 1999).

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Equations (97)

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

E = ρ e ɛ 0 , × E = K J m B t ,
B = K ρ m , × B = μ 0 J e + μ 0 ɛ 0 E t ,
F = q ( E + v × B ) + K μ 0 g ( B v c 2 × E ) .
κ ɛ 0 c K = K μ 0 c , Ω ( 0 1 1 0 ) ,
Q ( q κ g ) , ρ ( ρ e κ ρ m ) , J ( J e κ J m ) , G ( E c B ) .
G = 1 ɛ 0 ρ , × G = 1 ɛ 0 c Ω J + 1 c t Ω G ,
F = ( q , κ g ) ( 𝟙 v c × Ω ) G .
Q q + i κ g = | Q | e i ζ = | Q | ( cos ζ + i sin ζ ) ρ ρ e + i κ ρ m , J J e + i κ J m , G E + ic B .
G = 1 ɛ 0 ρ , × G = i c ( J ɛ 0 + G t ) ,
F = Re [ Q * ( 1 i v c × ) G ] .
P ( P e κ P m ) 1 Δ 𝒱 Δ 𝒱 r ρ d 3 r , M ( M e κ M m ) 1 2 Δ 𝒱 Δ 𝒱 r × J d 3 r .
d = ρ , d ɛ 0 G + P ,
× h = Ω ( J + d t ) , h 1 μ 0 c G Ω M .
d = ( D ɛ 0 c B ) , h = ( ɛ 0 c E H ) .
J = σ G
σ σ Θ ,
Θ ( cos 2 ζ cos ζ sin ζ cos ζ sin ζ sin 2 ζ ) .
σ = p σ p Θ p ,
σ = ( σ 0 0 σ ¯ ) .
P = ɛ 0 χ Θ G ,
d ɛ 0 G + P = ɛ 0 ( 𝟙 + χ Θ ) G = ɛ G ,
ɛ ɛ 0 ( 𝟙 + χ Θ ) = ɛ 0 ( 1 + χ cos 2 ζ χ cos ζ sin ζ χ cos ζ sin ζ 1 + χ sin 2 ζ ) .
ɛ = ɛ 0 ( 1 + χ e 0 0 1 ) ,
ɛ = ɛ 0 ( 1 0 0 1 + χ ¯ e ) ,
ɛ = ɛ 0 ( 1 + Σ p χ p cos 2 ζ p Σ p χ p cos ζ p sin ζ p Σ p χ p cos ζ p sin ζ p 1 + Σ p χ p sin 2 ζ p ) = ɛ 0 p ( 𝟙 + χ p Θ p ) .
F v = v × ( q B κ c g E ) .
( M e M m ) = n α ( q g ) ( q B κ c g E ) ,
M = ξ μ 0 c ( cos ζ sin ζ cos 2 ζ sin 2 ζ cos ζ sin ζ ) G .
h 1 μ 0 c G Ω M = 1 μ 0 c [ 𝟙 ξ ( sin 2 ζ cos ζ sin ζ cos ζ sin ζ cos 2 ζ ) ] G 1 c μ 1 G .
μ = μ 0 1 ξ ( 𝟙 ξ Θ ) ,
G = c μ h .
μ = μ 0 ( 1 0 0 1 + χ m ) ,
μ = μ 0 ( 1 + χ ¯ m 0 0 1 ) .
μ = μ 0 ( 1 Σ p ξ p sin 2 ζ p Σ p ξ p cos ζ p sin ζ p Σ p ξ p cos ζ p sin ζ p 1 Σ p ξ p cos 2 ζ p ) 1 = μ 0 det | 1 Σ p ξ p Θ p | ( 1 p ξ p Θ p ) .
h = ( f b ) exp [ i ( K r ω t ) ] .
0 = d = c ɛ μ h c ɛ μ ( f b ) exp [ i ( K r ω t ) ] .
ɛ μ ( K f K b ) = 0 ,
K f = K b = 0 .
× h = c Ω ( σ + ɛ t ) μ h ,
K × ( f b ) + ( S R U T ) ( f b ) = 0 ,
( U T S R ) c ( ω ɛ + i σ ) μ .
0 = [ ( S K z K y K z S K x K y K x S ) R ( 1 0 0 0 1 0 0 0 1 ) U ( 1 0 0 0 1 0 0 0 1 ) ( T K z K y K z T K x K y K x T ) ] ( f x f y f z b x b y b z ) .
0 = det | ( S K z K y K z S K x K y K x S ) ( T K z K y K z T K x K y K x T ) + R U ( 1 0 0 0 1 0 0 0 1 ) | .
K = ( k x + i l x ) e x + k y e y = K x e x + k y e y ,
0 = det | ( Δ k y 2 K x k y δ k y K x k y Δ K x 2 δ K x δ k y δ K x Δ K x 2 k y 2 ) | = Δ [ ( Δ K 2 ) 2 + δ 2 K 2 ] ,
K 2 ± i δ K Δ = 0 K = ± [ i δ 2 ± 1 2 ( 4 Δ δ 2 ) 1 / 2 ] ,
K = ± i S T 2 ± 1 2 [ 4 R U ( S + T ) 2 ] 1 / 2 .
K = ± ( R U S 2 ) 1 / 2 .
K 2 = ɛ μ ω 2 [ 1 σ σ ¯ ω 2 ɛ 0 ɛ + i ω ɛ 0 ( σ ɛ r + σ ¯ ɛ ¯ r ) ] ,
K 2 = ɛ μ ω 2 [ 1 + i σ ω ɛ ] ,
K 2 = ɛ ¯ μ ¯ ω 2 [ 1 + i σ ¯ ω ɛ ¯ ] ,
h = h 0 exp [ i ( k 0 r ω t ) ] ,
ω = ω = ω , k y = k y = k y , k z = k z = K z .
K = ( k x + i l x ) e x + k y e y K x e x + k y e y ,
K 2 = K K = K x 2 + k 0 y 2 K x = ( K 2 k 0 2 sin 2 θ ) 1 / 2 ,
K × f = S f + R b b = 1 R ( S + K × ) f
( b x b y b z ) = 1 R ( S 0 k y 0 S K x k y K x S ) ( f x f y f z ) ,
0 = K x f x + k y f y f x = k y K x f y ,
( b y b z ) ( β Ξ Λ β ) ( f y f z ) .
β S R , Ξ K x R , Λ 1 R ( k y 2 K x + K x ) = K 2 R K x ,
( b y b y f z f z b z b z f y f y ) = ( b 0 y f 0 z b 0 z f 0 y )
( Ξ Ξ 0 β 1 1 0 0 0 β Λ Λ 0 0 1 1 ) ( f z f z f y f y ) = ( Ξ 0 f 0 z f 0 z Λ 0 f 0 y f 0 y ) ,
( Ξ Ξ 0 β 1 1 0 0 0 β Λ Λ 0 0 1 1 ) ( f z f z f y f y ) = ( Ξ f z f z 0 0 ) .
f y = f y , f y = β Λ + Λ f z ,
f z = 2 Ξ ( Λ + Λ ) ( Ξ + Ξ ) ( Λ + Λ ) + β 2 f z , f z = ( Ξ Ξ ) ( Λ + Λ ) β 2 ( Ξ + Ξ ) ( Λ + Λ ) + β 2 f z .
f z f z = 2 R k x ( R k 2 / k x + R K 2 / K x ) ( R k x + R K x ) ( R k 2 / k x + R K 2 / K x ) R 2 S 2 , f z f z = ( R k x R K x ) ( R k 2 / k x + R K 2 / K x ) + R 2 S 2 ( R k x + R K x ) ( R k 2 / k x + R K 2 / K x ) R 2 S 2 .
f y f z = β Λ + Λ = R S R k + R K ,
f y f z = f y f z f z f z = 2 β Ξ ( Ξ Ξ ) ( Λ + Λ ) β 2 .
α = tan 1 ( 2 R R S k ( R k R K ) ( R k + R K ) + R 2 S 2 ) .
f z f z = 2 k x / R k x / R + K x / R , f z f z = k x / R K x / R k x / R + K x / R .
K x R = 1 R ( K 2 k 2 sin θ ) 1 / 2 = R * | R | 2 ( U R k 2 sin θ ) 1 / 2 ,
D 2 | R | 2 U R * ( R * ) 2 k 2 sin θ 0 .
( U T S R ) = c μ 0 ( ω ɛ 0 ɛ r + i σ 0 0 ω ɛ 0 ɛ ¯ r + i σ ¯ ) ,
Im D 2 ( μ 0 c ) 2 = | R | 2 ω ɛ 0 ( σ ɛ ¯ r σ ¯ ɛ r ) + 2 ω ɛ 0 ɛ ¯ r σ ¯ k 2 sin θ .
sin θ = ɛ r ɛ ¯ r 2 ( 1 + σ ¯ 2 ω 2 ɛ 0 2 ɛ ¯ r 2 ) ( 1 σ ɛ ¯ r σ ¯ ɛ r ) .
k x R = c k cos θ ω ( 1 + χ m ) = ( 1 + χ e 1 + χ m ) 1 / 2 cos θ = Z 0 Z cos θ ,
f z f z = 2 Z cos θ Z cos θ + Z cos θ , f z f z = Z cos θ Z cos θ Z cos θ + Z cos θ .
K × b = U f + T b f = 1 U ( T K × ) b
( f x f y f z ) = 1 U ( T 0 k y 0 T K x k y K x T ) ( b x b y b z ) .
0 = K x b x + k y b y b x = k y K x b y ,
( f y f z ) = ( β Ξ Λ β ) ( b y b z ) ,
β T U , Ξ K x U , Λ K 2 U K x .
( f y f y b z b z f z f z b y b y ) = ( f 0 y b 0 z f 0 z b 0 y ) ,
b y = b y , b y = β Λ + Λ b z ,
b z = 2 Ξ ( Λ + Λ ) ( Ξ + Ξ ) ( Λ + Λ ) + β 2 b z , b z = ( Ξ Ξ ) ( Λ + Λ ) β 2 ( Ξ + Ξ ) ( Λ + Λ ) + β 2 b z .
b z b z = 2 U k x ( U k 2 / k x + U K 2 / K x ) ( U k x + U K x ) ( U k 2 / k x + U K 2 / K x ) U 2 T 2 , b z b z = ( U k x U K x ) ( U k 2 / k x + U K 2 / K x ) + U 2 T 2 ( U k x + U K x ) ( U k 2 / k x + U K 2 / K x ) U 2 T 2 .
b z = 2 U k x U k x + U K x b z , b z = U k x U K x U k x + U K x b z .
U = μ 0 c ( ω ɛ + i σ )
k x U = k cos θ μ 0 c ω ɛ = ( ɛ r μ r ) 1 / 2 cos θ ɛ r = Z Z 0 cos θ ,
b z b z = 2 Z cos θ Z cos θ + Z cos θ , b z b z = Z cos θ Z cos θ Z cos θ + Z cos θ .
| f | 2 f x 2 + f y 2 + f z 2 = 1 U 2 ( k y 2 + K x 2 ) b z 2 = K 2 U 2 b z 2 ,
| f | | f | = K k U U b z b z = Z Z b z b z .
| f | | f | = 2 Z cos θ Z cos θ + Z cos θ , | f | | f | = Z cos θ Z cos θ Z cos θ + Z cos θ .
μ = μ 0 1 ξ ξ + ξ ξ sin 2 ( ζ ζ ) ( 𝟙 ξ Θ ξ ¯ Θ ¯ ) ,
( U T S R ) = c ( ω ɛ + i σ ) μ = c [ ω ɛ 0 ( 𝟙 + χ Θ + χ ¯ Θ ¯ ) + i σ Θ + i σ ¯ Θ ¯ ] μ 0 1 ξ ξ + ξ ξ sin 2 ( ζ ζ ) ( 𝟙 ξ Θ ξ ¯ Θ ¯ ) .
( U T S R ) = c μ 0 ω ɛ 0 ( 1 ξ ) ( 1 ξ ) [ 𝟙 + ( χ ξ χ ξ ) Θ + ( χ ¯ ξ ¯ χ ¯ ξ ¯ ) Θ ¯ + + i σ ( 1 ξ ) ω ɛ 0 Θ + i σ ¯ ( 1 ξ ¯ ) ω ɛ 0 Θ ¯ ] ,
K 2 = R U T S = ɛ 0 ( 1 + χ ) ( 1 + χ ¯ ) μ 0 ( 1 ξ ) ( 1 ξ ) ω 2 [ 1 + i σ ω ɛ 0 ( 1 + χ ) ] [ 1 + i σ ¯ ω ɛ 0 ( 1 + χ ¯ ) ] = ɛ 0 ɛ r ɛ ¯ r μ 0 μ r μ ¯ r ω 2 [ 1 + i σ ω ɛ 0 ɛ r ] [ 1 + i σ ¯ ω ɛ 0 ɛ ¯ r ] = ɛ μ ω 2 [ 1 σ σ ¯ ω 2 ɛ 0 ɛ + i ω ɛ 0 ( σ ɛ r + σ ¯ ɛ ¯ r ) ] .

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