Abstract

I compute from first principles the local heating rate q (the amount of electromagnetic energy converted to heat per unit time per unit volume) for electromagnetic waves propagating in magnetically and electrically polarizable media. I find that, in magnetic media, this rate has two separate contributions, q(V) and q(S), the first coming from the volume of the medium and the second from its surface. I argue that the second law of thermodynamics requires that the volume contribution be positive and that this requirement, in turn, prohibits negative refraction. This result holds for active or passive media and in the presence of anisotropy and spatial dispersion.

© 2008 Optical Society of America

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References

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  1. V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of ? and ?." Physics-Uspekhi 10, 509-514 (1968).
    [CrossRef]
  2. J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
    [CrossRef] [PubMed]
  3. A. A. Ramakrishna, "Physics of negative refractive index materials," Rep. Prog. Phys. 68, 449-521 (2005).
    [CrossRef]
  4. C. M. Soukoulis, S. Linden, and M. Wegener, "Negative refraction index at optical wavelengths," Science 315, 47-49 (2007).
    [CrossRef] [PubMed]
  5. V. M. Shalaev, "Optical negative-index metamaterials," Nature Photonics 1, 41-48 (2007).
    [CrossRef]
  6. M. I. Stockman, "Criterion for negative refraction with low optical losses from a fundamental principle of causality," Phys. Rev. Lett. 98, 177404 (2007).
    [CrossRef]
  7. W. Gough, "Poynting in the wrong direction?" Eur. J. Phys. 3, 83-87 (1982).
    [CrossRef]
  8. R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics, vol. 2 (Addison-Wesley, 1964).
  9. D. F. Nelson, "Generalizing the Poynting vector," Phys. Rev. Lett. 76, 4713-4716 (1996).
    [CrossRef] [PubMed]
  10. I. Campos and J. L. Jimenez, "About Poynting’s theorem," Eur. J. Phys. 13, 117-121 (1992).
    [CrossRef]
  11. F. Richter, M. Florian, and K. Henneberger, "Poynting’s theorem and energy conservation in the propagation of light in bounded media," Europhys. Lett. 81, 67005 (2008).
    [CrossRef]
  12. R. A. Depine and A. Lakhtakia, "Comment I on ’Resonant and antiresonant frequency dependence of the effective parameters of metamaterial’," Phys. Rev. E 70, 048601 (2004).
    [CrossRef]
  13. A. L. Efros, "Comment II on ’Resonant and antiresonant frequency dependence of the effective parameters of metamaterial’," Phys. Rev. E 70, 048602 (2004).
    [CrossRef]
  14. J. Schwinger, L. L. DeRaad, K. A. Milton, and W. Tsai, Classical Electrodynamics (Perseus Books, Reading, MA, 1998).
  15. L. D. Landau and L. P. Lifshitz, Electrodynamics of Continuous Media (Pergamon Press, Oxford, 1984).
  16. In the zero-frequency limit, we must replace E? by 2E, where E is the real-valued electric field, and similarly for the magnetic field.
  17. J. B. Pendry, A. J. Holden, D. J. Robbins, andW. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory and Techniques 47, 2075-2084 (1999).
    [CrossRef]
  18. V. A. Podolskiy, A. K. Sarychev, and V. M. Shalaev, "Plasmon modes and negative refraction in metal nanowire composites," Opt. Express 11, 735 (2003).
    [CrossRef] [PubMed]
  19. A. K. T. Assis, W. A. Rodrigues, and A. J. Mania, "The electric field outside a stationary resistive wire carrying a constant current," Foundations of Physics 29, 729-753 (1999).
    [CrossRef]
  20. P. S. Pershan, "Nonlinear optical properties of solids: energy considerations," Phys. Rev. 130, 919-929 (1963).
    [CrossRef]
  21. R. Marques, F. Martin, and M. Sorolla, Metamaterials with Negative Parameters (Wiley, 2008).
  22. I. Tsukerman, "Negative refraction and the minimum lattice cell size," J. Opt. Soc. Am. B 25, 927-936 (2008).
    [CrossRef]
  23. V. M. Agranovich, Y. N. Gartstein, and A. A. Zakhidov, "Negative refraction in gyrotropic media," Phys. Rev. B 73, 045114 (2006).
    [CrossRef]
  24. A. V. Kildishev, V. P. Drachev, U. K. Chettiar, V. M. Shalaev, D. H. Werner, and D. H. Kwon, "Comment on ’Negative refractive index in artificial metamaterials’," Opt. Lett. 32, 1510-1511 (2007).
    [CrossRef] [PubMed]
  25. A. N. Grigorenko, "Reply to comment on ’Negative refractive index in artificial metamaterials’," Opt. Lett. 32, 1512-1514 (2007).
    [CrossRef] [PubMed]
  26. C. R. Simovski and S. A. Tretyakov, "Local constitutive parameters of metamaterials from an effective-medium perspective," Phys. Rev. B 75, 195111 (2007).
    [CrossRef]
  27. R. J. Deissler, "Dipole in a magnetic field, work, and quantum spin," Phys. Rev. E 77, 036609 (2008).
    [CrossRef]

2008 (3)

F. Richter, M. Florian, and K. Henneberger, "Poynting’s theorem and energy conservation in the propagation of light in bounded media," Europhys. Lett. 81, 67005 (2008).
[CrossRef]

R. J. Deissler, "Dipole in a magnetic field, work, and quantum spin," Phys. Rev. E 77, 036609 (2008).
[CrossRef]

I. Tsukerman, "Negative refraction and the minimum lattice cell size," J. Opt. Soc. Am. B 25, 927-936 (2008).
[CrossRef]

2007 (6)

C. R. Simovski and S. A. Tretyakov, "Local constitutive parameters of metamaterials from an effective-medium perspective," Phys. Rev. B 75, 195111 (2007).
[CrossRef]

A. V. Kildishev, V. P. Drachev, U. K. Chettiar, V. M. Shalaev, D. H. Werner, and D. H. Kwon, "Comment on ’Negative refractive index in artificial metamaterials’," Opt. Lett. 32, 1510-1511 (2007).
[CrossRef] [PubMed]

A. N. Grigorenko, "Reply to comment on ’Negative refractive index in artificial metamaterials’," Opt. Lett. 32, 1512-1514 (2007).
[CrossRef] [PubMed]

C. M. Soukoulis, S. Linden, and M. Wegener, "Negative refraction index at optical wavelengths," Science 315, 47-49 (2007).
[CrossRef] [PubMed]

V. M. Shalaev, "Optical negative-index metamaterials," Nature Photonics 1, 41-48 (2007).
[CrossRef]

M. I. Stockman, "Criterion for negative refraction with low optical losses from a fundamental principle of causality," Phys. Rev. Lett. 98, 177404 (2007).
[CrossRef]

2006 (1)

V. M. Agranovich, Y. N. Gartstein, and A. A. Zakhidov, "Negative refraction in gyrotropic media," Phys. Rev. B 73, 045114 (2006).
[CrossRef]

2005 (1)

A. A. Ramakrishna, "Physics of negative refractive index materials," Rep. Prog. Phys. 68, 449-521 (2005).
[CrossRef]

2004 (2)

R. A. Depine and A. Lakhtakia, "Comment I on ’Resonant and antiresonant frequency dependence of the effective parameters of metamaterial’," Phys. Rev. E 70, 048601 (2004).
[CrossRef]

A. L. Efros, "Comment II on ’Resonant and antiresonant frequency dependence of the effective parameters of metamaterial’," Phys. Rev. E 70, 048602 (2004).
[CrossRef]

2003 (1)

2000 (1)

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

1999 (2)

J. B. Pendry, A. J. Holden, D. J. Robbins, andW. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory and Techniques 47, 2075-2084 (1999).
[CrossRef]

A. K. T. Assis, W. A. Rodrigues, and A. J. Mania, "The electric field outside a stationary resistive wire carrying a constant current," Foundations of Physics 29, 729-753 (1999).
[CrossRef]

1996 (1)

D. F. Nelson, "Generalizing the Poynting vector," Phys. Rev. Lett. 76, 4713-4716 (1996).
[CrossRef] [PubMed]

1992 (1)

I. Campos and J. L. Jimenez, "About Poynting’s theorem," Eur. J. Phys. 13, 117-121 (1992).
[CrossRef]

1982 (1)

W. Gough, "Poynting in the wrong direction?" Eur. J. Phys. 3, 83-87 (1982).
[CrossRef]

1968 (1)

V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of ? and ?." Physics-Uspekhi 10, 509-514 (1968).
[CrossRef]

1963 (1)

P. S. Pershan, "Nonlinear optical properties of solids: energy considerations," Phys. Rev. 130, 919-929 (1963).
[CrossRef]

Agranovich, V. M.

V. M. Agranovich, Y. N. Gartstein, and A. A. Zakhidov, "Negative refraction in gyrotropic media," Phys. Rev. B 73, 045114 (2006).
[CrossRef]

Assis, A. K. T.

A. K. T. Assis, W. A. Rodrigues, and A. J. Mania, "The electric field outside a stationary resistive wire carrying a constant current," Foundations of Physics 29, 729-753 (1999).
[CrossRef]

Campos, I.

I. Campos and J. L. Jimenez, "About Poynting’s theorem," Eur. J. Phys. 13, 117-121 (1992).
[CrossRef]

Chettiar, U. K.

Deissler, R. J.

R. J. Deissler, "Dipole in a magnetic field, work, and quantum spin," Phys. Rev. E 77, 036609 (2008).
[CrossRef]

Depine, R. A.

R. A. Depine and A. Lakhtakia, "Comment I on ’Resonant and antiresonant frequency dependence of the effective parameters of metamaterial’," Phys. Rev. E 70, 048601 (2004).
[CrossRef]

Drachev, V. P.

Efros, A. L.

A. L. Efros, "Comment II on ’Resonant and antiresonant frequency dependence of the effective parameters of metamaterial’," Phys. Rev. E 70, 048602 (2004).
[CrossRef]

Florian, M.

F. Richter, M. Florian, and K. Henneberger, "Poynting’s theorem and energy conservation in the propagation of light in bounded media," Europhys. Lett. 81, 67005 (2008).
[CrossRef]

Gartstein, Y. N.

V. M. Agranovich, Y. N. Gartstein, and A. A. Zakhidov, "Negative refraction in gyrotropic media," Phys. Rev. B 73, 045114 (2006).
[CrossRef]

Gough, W.

W. Gough, "Poynting in the wrong direction?" Eur. J. Phys. 3, 83-87 (1982).
[CrossRef]

Grigorenko, A. N.

Henneberger, K.

F. Richter, M. Florian, and K. Henneberger, "Poynting’s theorem and energy conservation in the propagation of light in bounded media," Europhys. Lett. 81, 67005 (2008).
[CrossRef]

Holden, A. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, andW. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory and Techniques 47, 2075-2084 (1999).
[CrossRef]

Jimenez, J. L.

I. Campos and J. L. Jimenez, "About Poynting’s theorem," Eur. J. Phys. 13, 117-121 (1992).
[CrossRef]

Kildishev, A. V.

Kwon, D. H.

Lakhtakia, A.

R. A. Depine and A. Lakhtakia, "Comment I on ’Resonant and antiresonant frequency dependence of the effective parameters of metamaterial’," Phys. Rev. E 70, 048601 (2004).
[CrossRef]

Linden, S.

C. M. Soukoulis, S. Linden, and M. Wegener, "Negative refraction index at optical wavelengths," Science 315, 47-49 (2007).
[CrossRef] [PubMed]

Mania, A. J.

A. K. T. Assis, W. A. Rodrigues, and A. J. Mania, "The electric field outside a stationary resistive wire carrying a constant current," Foundations of Physics 29, 729-753 (1999).
[CrossRef]

Nelson, D. F.

D. F. Nelson, "Generalizing the Poynting vector," Phys. Rev. Lett. 76, 4713-4716 (1996).
[CrossRef] [PubMed]

Pendry, J. B.

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

J. B. Pendry, A. J. Holden, D. J. Robbins, andW. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory and Techniques 47, 2075-2084 (1999).
[CrossRef]

Pershan, P. S.

P. S. Pershan, "Nonlinear optical properties of solids: energy considerations," Phys. Rev. 130, 919-929 (1963).
[CrossRef]

Podolskiy, V. A.

Ramakrishna, A. A.

A. A. Ramakrishna, "Physics of negative refractive index materials," Rep. Prog. Phys. 68, 449-521 (2005).
[CrossRef]

Richter, F.

F. Richter, M. Florian, and K. Henneberger, "Poynting’s theorem and energy conservation in the propagation of light in bounded media," Europhys. Lett. 81, 67005 (2008).
[CrossRef]

Robbins, D. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, andW. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory and Techniques 47, 2075-2084 (1999).
[CrossRef]

Rodrigues, W. A.

A. K. T. Assis, W. A. Rodrigues, and A. J. Mania, "The electric field outside a stationary resistive wire carrying a constant current," Foundations of Physics 29, 729-753 (1999).
[CrossRef]

Sarychev, A. K.

Shalaev, V. M.

Simovski, C. R.

C. R. Simovski and S. A. Tretyakov, "Local constitutive parameters of metamaterials from an effective-medium perspective," Phys. Rev. B 75, 195111 (2007).
[CrossRef]

Soukoulis, C. M.

C. M. Soukoulis, S. Linden, and M. Wegener, "Negative refraction index at optical wavelengths," Science 315, 47-49 (2007).
[CrossRef] [PubMed]

Stockman, M. I.

M. I. Stockman, "Criterion for negative refraction with low optical losses from a fundamental principle of causality," Phys. Rev. Lett. 98, 177404 (2007).
[CrossRef]

Tretyakov, S. A.

C. R. Simovski and S. A. Tretyakov, "Local constitutive parameters of metamaterials from an effective-medium perspective," Phys. Rev. B 75, 195111 (2007).
[CrossRef]

Tsukerman, I.

Veselago, V. G.

V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of ? and ?." Physics-Uspekhi 10, 509-514 (1968).
[CrossRef]

Wegener, M.

C. M. Soukoulis, S. Linden, and M. Wegener, "Negative refraction index at optical wavelengths," Science 315, 47-49 (2007).
[CrossRef] [PubMed]

Werner, D. H.

Zakhidov, A. A.

V. M. Agranovich, Y. N. Gartstein, and A. A. Zakhidov, "Negative refraction in gyrotropic media," Phys. Rev. B 73, 045114 (2006).
[CrossRef]

Eur. J. Phys. (2)

W. Gough, "Poynting in the wrong direction?" Eur. J. Phys. 3, 83-87 (1982).
[CrossRef]

I. Campos and J. L. Jimenez, "About Poynting’s theorem," Eur. J. Phys. 13, 117-121 (1992).
[CrossRef]

Europhys. Lett. (1)

F. Richter, M. Florian, and K. Henneberger, "Poynting’s theorem and energy conservation in the propagation of light in bounded media," Europhys. Lett. 81, 67005 (2008).
[CrossRef]

Foundations of Physics (1)

A. K. T. Assis, W. A. Rodrigues, and A. J. Mania, "The electric field outside a stationary resistive wire carrying a constant current," Foundations of Physics 29, 729-753 (1999).
[CrossRef]

IEEE Trans. Microwave Theory and Techniques (1)

J. B. Pendry, A. J. Holden, D. J. Robbins, andW. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory and Techniques 47, 2075-2084 (1999).
[CrossRef]

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

Nature Photonics (1)

V. M. Shalaev, "Optical negative-index metamaterials," Nature Photonics 1, 41-48 (2007).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. (1)

P. S. Pershan, "Nonlinear optical properties of solids: energy considerations," Phys. Rev. 130, 919-929 (1963).
[CrossRef]

Phys. Rev. B (2)

V. M. Agranovich, Y. N. Gartstein, and A. A. Zakhidov, "Negative refraction in gyrotropic media," Phys. Rev. B 73, 045114 (2006).
[CrossRef]

C. R. Simovski and S. A. Tretyakov, "Local constitutive parameters of metamaterials from an effective-medium perspective," Phys. Rev. B 75, 195111 (2007).
[CrossRef]

Phys. Rev. E (3)

R. J. Deissler, "Dipole in a magnetic field, work, and quantum spin," Phys. Rev. E 77, 036609 (2008).
[CrossRef]

R. A. Depine and A. Lakhtakia, "Comment I on ’Resonant and antiresonant frequency dependence of the effective parameters of metamaterial’," Phys. Rev. E 70, 048601 (2004).
[CrossRef]

A. L. Efros, "Comment II on ’Resonant and antiresonant frequency dependence of the effective parameters of metamaterial’," Phys. Rev. E 70, 048602 (2004).
[CrossRef]

Phys. Rev. Lett. (3)

M. I. Stockman, "Criterion for negative refraction with low optical losses from a fundamental principle of causality," Phys. Rev. Lett. 98, 177404 (2007).
[CrossRef]

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

D. F. Nelson, "Generalizing the Poynting vector," Phys. Rev. Lett. 76, 4713-4716 (1996).
[CrossRef] [PubMed]

Physics-Uspekhi (1)

V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of ? and ?." Physics-Uspekhi 10, 509-514 (1968).
[CrossRef]

Rep. Prog. Phys. (1)

A. A. Ramakrishna, "Physics of negative refractive index materials," Rep. Prog. Phys. 68, 449-521 (2005).
[CrossRef]

Science (1)

C. M. Soukoulis, S. Linden, and M. Wegener, "Negative refraction index at optical wavelengths," Science 315, 47-49 (2007).
[CrossRef] [PubMed]

Other (5)

R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics, vol. 2 (Addison-Wesley, 1964).

J. Schwinger, L. L. DeRaad, K. A. Milton, and W. Tsai, Classical Electrodynamics (Perseus Books, Reading, MA, 1998).

L. D. Landau and L. P. Lifshitz, Electrodynamics of Continuous Media (Pergamon Press, Oxford, 1984).

In the zero-frequency limit, we must replace E? by 2E, where E is the real-valued electric field, and similarly for the magnetic field.

R. Marques, F. Martin, and M. Sorolla, Metamaterials with Negative Parameters (Wiley, 2008).

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Figures (1)

Fig. 1.
Fig. 1.

A cyclic process involving a negative refractive index material that violates the Carnot theorem. The black oval represents a negative refraction medium and the white oval is an ideal Carnot engine.

Equations (59)

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Im ( ε μ ) < 0
J = P t + c × M
S = c 4 π E × B
S = c 4 π E × H .
Q = V q ( V ) ( r ) d 3 r + S q ( S ) ( r ) d 2 r ,
q ( conv ) = 1 4 π E · D t + H · B t ,
q ( conv ) = ω 4 π [ ε ' ' ( ω ) E 2 + μ ' ' ( ω ) H 2 ] .
E = Re [ E 0 e i ( k · r ω t ) ] , H = Re [ H 0 e i ( k · r ω t ) ] ,
q ( conv ) = ω 8 π [ ε ' ' ( ω ) E 0 2 + μ ' ' ( ω ) H 0 2 ] e 2 k ' ' · r .
q ( conv ) = ω 8 π [ ε ' ' ( ω ) + ε ( ω ) μ ( ω ) μ ' ' ( ω ) ] E 0 2 e 2 k ' ' · r .
q = J · E ,
J = Re [ J 0 e i ( k · r ω t ) ] .
q ( V ) = 1 2 Re ( J 0 · E 0 * ) e 2 k ' ' · r .
c × E = B t , c × B = E t + 4 π J ,
4 π J t = c 2 × × E + 2 E t 2 .
J 0 = ( c 2 4 π i ω ) [ k × k × E 0 + ( ω c ) 2 E 0 ] .
J 0 = ω 4 π i [ μ ( ω ) ε ( ω ) 1 ] E 0 .
q ( V ) = ω E 0 2 8 π Im [ μ ( ω ) ε ( ω ) ] e 2 k · r .
ε > 0 , μ > 0 ,
μ ε + ε μ > 0 .
ε μ + μ ε > 0 .
q = c 4 π H · ( × E ) E · ( × H ) .
c 4 π · ( E × H ) + 1 4 π ( E · D t + H · B t ) = 0
s = c 4 π e × h .
S s ¯ = c 4 π [ E × B + δ e ¯ × B + E × δ h ¯ + δ e × δ h ¯ ] .
E = Re [ E ω ( r ) e i ω t ] , D = Re [ D ω ( r ) e i ω t ] , H = Re [ H ω ( r ) e i ω t ] , B = Re [ B ω ( r ) e i ω t ] ,
S = c 8 π E ω * × B ω
q ( V ) = · S = c 8 π Re [ · ( E ω * × B ω ) ] = c 8 π Re [ E ω * · ( × B ω ) B ω · ( × E ω * ) ]
= c 8 π Re [ E ω * · ( × μ ( ω ) H ω ) + i ω c B ω · B ω * ] = c 8 π Re [ i ω c μ ( ω ) E ω * · D ω ]
= ω E ω 2 8 π Im [ μ ( ω ) ε ( ω ) ] .
q ( V ) = ω E ω 2 8 π Im [ μ ( ω ) ε ( ω ) ] .
J = Re [ J ω ( r ) e i ω t ] ,
J ω = 1 4 π ( c × B ω + i ω E ω ) .
q ( V ) = J · E = 1 2 Re ( J ω · E ω * ) = 1 8 π Re [ i ω E ω · E ω * + c ( × B ω ) · E ω * ]
= c 8 π Re [ μ ( ω ) ( × H ω ) · E ω * ] = c 8 π Re [ i ω c μ ( ω ) ε ( ω ) E ω · E ω * ]
= ω E ω 2 8 π Im [ μ ( ω ) ε ( ω ) ] .
J · E + · S = 0 .
× B ω = × μ ( r ) H ω = μ ( r ) × H ω + [ μ ( r ) ] × H ω ,
J ω ( S ) = c n ̂ × M ω ,
q ( s ) = 1 2 Re ( J ω ( S ) · E ω * ) .
q ( S ) = c 8 π Re [ ( 1 μ ) ( n ̂ × H ω ) · E ω * ] .
q ( S ) = c 8 π Re [ ( 1 μ ) ( H ω × E ω * ) · n ̂ ] .
Q ( V ) L = π a 2 σ μ E 2
Q ( S ) L = π a 2 σ ( 1 μ ) E 2 .
Q L = Q ( V ) L + Q ( S ) L = π a 2 σ E 2 ,
0 2 π q ( S ) ( φ ) d φ = 0 ,
Q ( S ) S q ( S ) ( r ) d 2 r = Q + ( S ) Q ( S ) .
Q ( V ) V q ( V ) ( r ) d 3 r
Q = Q ( V ) + Q + ( S ) Q ( S )
Q + ( S ) = Q + Q ( V ) + Q ( S ) > Q .
η C = 1 T L T H
η = A θ = η C θ + ( S ) θ + ( S ) θ ( V ) = η C 1 1 θ ( V ) θ + ( S ) > η C
det ε ̂ 1 k × μ ̂ 1 k × + ( ω c ) 2 = 0
det k 2 T ̂ + ( ω c ) 2 = 0 ,
E = Re [ E 0 e i ( k · r ω t ) ] , B = Re [ B 0 e i ( k · r ω t ) ] , J = Re [ J 0 e i ( k · r ω t ) ] ,
4 π J 0 = i ω E 0 + i c k × B 0 .
q ( V ) = c e 2 k · r 8 π Im [ ( k × B 0 ) · E 0 * ] .
q ( V ) = ω e 2 k . r 8 π ( ω c ) 2 Im [ E 0 2 ( k · k ) ( k · E 0 ) ( k · E 0 * ) ] .
q ( V ) = ω E 0 2 e 2 k · r 8 π sin 2 θ Im ( k 2 ) ( ω c ) 2 .

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