Abstract

In the last ten years, the technology of differential geometry, ubiquitous in gravitational physics, has found its place in the field of optics. It has been successfully used in the design of optical metamaterials through a technique now known as “transformation optics.” This method, however, only applies for the particular class of metamaterials known as impedance matched, that is, materials whose electric permittivity is equal to their magnetic permeability. In that case, the material may be described by a spacetime metric. In the present work we will introduce a generalization of the geometric methods of transformation optics to situations in which the material is not impedance matched. In such situations, the material -or more precisely, its constitutive tensor- will not be described by a metric only. We bring in a second tensor, with the local symmetries of the Weyl tensor, the “W-tensor.” In the geometric optics approximation we show how the properties of the W-tensor are related to the asymmetric transmission of the material. We apply this feature to the design of a particularly interesting set of asymmetric materials. These materials are birefringent when light rays approach the material in a given direction, but behave just like vacuum when the rays have the opposite direction with the appropriate polarization (or, in some cases, independently of the polarization).

© 2017 Optical Society of America

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References

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    [Crossref] [PubMed]
  2. G. Gbur, Chapter 5 - Nonradiating Sources and Other “Invisible” Objects (Elsevier, 2003), pp. 273–315.
  3. A. Greenleaf, M. Lassas, and G. Uhlmann, “Anisotropic conductivities that cannot be detected by eit,” Physiol. Meas. 24, 413 (2003).
    [Crossref] [PubMed]
  4. K. Rosquist, “Letter: A moving medium simulation of schwarzschild black hole optics,” Gen. Rel. Grav. 36, 1977–1982 (2004).
    [Crossref]
  5. A. Alù and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E 72, 016623 (2005).
    [Crossref]
  6. U. Leonhardt, “Notes on conformal invisibility devices,” New J. Phys. 8, 118 (2006).
    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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  19. I. T. Drummond, “Bimetric QED,” Phys. Rev. D95, 025006 (2017).
  20. X. Lu, F.-W. Yang, and Y. Xie, “Strong gravitational field time delay for photons coupled toweyl tensor in a schwarzschild black hole,” Eur. Phys. J. C 357, 76 (2016).
  21. J. Jing, S. Chen, Q. Pan, and J. Wang, “Detect black holes using photons for coupling model of electromagnetic and gravitational fields,” arXiv p. 1704.08794 (2017).
  22. D. Momeni and M. R. Setare, “A note on holographic superconductors with weyl corrections,” Mod. Phys. Lett. A 26, 2889 (2011).
    [Crossref]
  23. C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (W. H. Freeman and Company, 1973).
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    [Crossref]
  25. H. Stephani, Relativity: An Introduction to Special and General Relativity, E-libro (Cambridge University, 2004).
    [Crossref]

2017 (1)

I. T. Drummond, “Bimetric QED,” Phys. Rev. D95, 025006 (2017).

2016 (1)

X. Lu, F.-W. Yang, and Y. Xie, “Strong gravitational field time delay for photons coupled toweyl tensor in a schwarzschild black hole,” Eur. Phys. J. C 357, 76 (2016).

2012 (1)

M. Mutlu, A. E. Akosman, A. E. Serebryannikov, and E. Ozbay, “Diodelike asymmetric transmission of linearly polarized waves using magnetoelectric coupling and electromagnetic wave tunneling,” Phys. Rev. Lett. 108, 213905 (2012).
[Crossref] [PubMed]

2011 (2)

2010 (2)

2007 (1)

2006 (6)

U. Leonhardt, “Notes on conformal invisibility devices,” New J. Phys. 8, 118 (2006).
[Crossref]

D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14, 9794–9804 (2006).
[Crossref] [PubMed]

U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys. 8, 247 (2006).
[Crossref]

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

G. W. Milton and N.-A. P. Nicorovici, “On the cloaking effects associated with anomalous localized resonance,” Proc. Roy. Soc. Lond. A: Math. Phys. Eng. Sci. 462, 3027–3059 (2006).
[Crossref]

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
[Crossref] [PubMed]

2005 (1)

A. Alù and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E 72, 016623 (2005).
[Crossref]

2004 (1)

K. Rosquist, “Letter: A moving medium simulation of schwarzschild black hole optics,” Gen. Rel. Grav. 36, 1977–1982 (2004).
[Crossref]

2003 (1)

A. Greenleaf, M. Lassas, and G. Uhlmann, “Anisotropic conductivities that cannot be detected by eit,” Physiol. Meas. 24, 413 (2003).
[Crossref] [PubMed]

2000 (2)

U. Leonhardt and P. Piwnicki, “Relativistic effects of light in moving media with extremely low group velocity,” Phys. Rev. Lett. 84, 822–825 (2000).
[Crossref] [PubMed]

A. Z. Petrov, “The Classification of spaces defining gravitational fields,” Gen. Rel. Grav. 32, 1661–1663 (2000).
[Crossref]

Akosman, A. E.

M. Mutlu, A. E. Akosman, A. E. Serebryannikov, and E. Ozbay, “Diodelike asymmetric transmission of linearly polarized waves using magnetoelectric coupling and electromagnetic wave tunneling,” Phys. Rev. Lett. 108, 213905 (2012).
[Crossref] [PubMed]

M. Mutlu, A. E. Akosman, A. E. Serebryannikov, and E. Ozbay, “Asymmetric transmission of linearly polarized waves and polarization angle dependent wave rotation using a chiral metamaterial,” Opt. Express 19, 14290–14299 (2011).
[Crossref] [PubMed]

Alù, A.

A. Alù and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E 72, 016623 (2005).
[Crossref]

Chan, C. T.

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nature Materials 9, 387 (2010).
[Crossref] [PubMed]

Chen, H.

Chen, S.

J. Jing, S. Chen, Q. Pan, and J. Wang, “Detect black holes using photons for coupling model of electromagnetic and gravitational fields,” arXiv p. 1704.08794 (2017).

Drummond, I. T.

I. T. Drummond, “Bimetric QED,” Phys. Rev. D95, 025006 (2017).

Engheta, N.

A. Alù and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E 72, 016623 (2005).
[Crossref]

Erices, C.

C. Erices, “Riemannian geometry as a tool for optical metamaterial design,” Master’s thesis, Pontificia Universidad Católica de Chile (2012).

Gbur, G.

G. Gbur, Chapter 5 - Nonradiating Sources and Other “Invisible” Objects (Elsevier, 2003), pp. 273–315.

Greenleaf, A.

A. Greenleaf, M. Lassas, and G. Uhlmann, “Anisotropic conductivities that cannot be detected by eit,” Physiol. Meas. 24, 413 (2003).
[Crossref] [PubMed]

Jing, J.

J. Jing, S. Chen, Q. Pan, and J. Wang, “Detect black holes using photons for coupling model of electromagnetic and gravitational fields,” arXiv p. 1704.08794 (2017).

Lassas, M.

A. Greenleaf, M. Lassas, and G. Uhlmann, “Anisotropic conductivities that cannot be detected by eit,” Physiol. Meas. 24, 413 (2003).
[Crossref] [PubMed]

Leonhardt, U.

U. Leonhardt, “Notes on conformal invisibility devices,” New J. Phys. 8, 118 (2006).
[Crossref]

U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys. 8, 247 (2006).
[Crossref]

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
[Crossref] [PubMed]

U. Leonhardt and P. Piwnicki, “Relativistic effects of light in moving media with extremely low group velocity,” Phys. Rev. Lett. 84, 822–825 (2000).
[Crossref] [PubMed]

Li, M.

Lu, X.

X. Lu, F.-W. Yang, and Y. Xie, “Strong gravitational field time delay for photons coupled toweyl tensor in a schwarzschild black hole,” Eur. Phys. J. C 357, 76 (2016).

Miao, R.-X.

Milton, G. W.

G. W. Milton and N.-A. P. Nicorovici, “On the cloaking effects associated with anomalous localized resonance,” Proc. Roy. Soc. Lond. A: Math. Phys. Eng. Sci. 462, 3027–3059 (2006).
[Crossref]

Misner, C. W.

C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (W. H. Freeman and Company, 1973).

Momeni, D.

D. Momeni and M. R. Setare, “A note on holographic superconductors with weyl corrections,” Mod. Phys. Lett. A 26, 2889 (2011).
[Crossref]

Mutlu, M.

M. Mutlu, A. E. Akosman, A. E. Serebryannikov, and E. Ozbay, “Diodelike asymmetric transmission of linearly polarized waves using magnetoelectric coupling and electromagnetic wave tunneling,” Phys. Rev. Lett. 108, 213905 (2012).
[Crossref] [PubMed]

M. Mutlu, A. E. Akosman, A. E. Serebryannikov, and E. Ozbay, “Asymmetric transmission of linearly polarized waves and polarization angle dependent wave rotation using a chiral metamaterial,” Opt. Express 19, 14290–14299 (2011).
[Crossref] [PubMed]

Nicorovici, N.-A. P.

G. W. Milton and N.-A. P. Nicorovici, “On the cloaking effects associated with anomalous localized resonance,” Proc. Roy. Soc. Lond. A: Math. Phys. Eng. Sci. 462, 3027–3059 (2006).
[Crossref]

Ozbay, E.

M. Mutlu, A. E. Akosman, A. E. Serebryannikov, and E. Ozbay, “Diodelike asymmetric transmission of linearly polarized waves using magnetoelectric coupling and electromagnetic wave tunneling,” Phys. Rev. Lett. 108, 213905 (2012).
[Crossref] [PubMed]

M. Mutlu, A. E. Akosman, A. E. Serebryannikov, and E. Ozbay, “Asymmetric transmission of linearly polarized waves and polarization angle dependent wave rotation using a chiral metamaterial,” Opt. Express 19, 14290–14299 (2011).
[Crossref] [PubMed]

Pan, Q.

J. Jing, S. Chen, Q. Pan, and J. Wang, “Detect black holes using photons for coupling model of electromagnetic and gravitational fields,” arXiv p. 1704.08794 (2017).

Pendry, J. B.

Petrov, A. Z.

A. Z. Petrov, “The Classification of spaces defining gravitational fields,” Gen. Rel. Grav. 32, 1661–1663 (2000).
[Crossref]

Philbin, T. G.

U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys. 8, 247 (2006).
[Crossref]

Piwnicki, P.

U. Leonhardt and P. Piwnicki, “Relativistic effects of light in moving media with extremely low group velocity,” Phys. Rev. Lett. 84, 822–825 (2000).
[Crossref] [PubMed]

Post, E. J.

E. J. Post, Formal Structure of Electromagnetics (Wiley, 1962).

Rosquist, K.

K. Rosquist, “Letter: A moving medium simulation of schwarzschild black hole optics,” Gen. Rel. Grav. 36, 1977–1982 (2004).
[Crossref]

Schurig, D.

Serebryannikov, A. E.

M. Mutlu, A. E. Akosman, A. E. Serebryannikov, and E. Ozbay, “Diodelike asymmetric transmission of linearly polarized waves using magnetoelectric coupling and electromagnetic wave tunneling,” Phys. Rev. Lett. 108, 213905 (2012).
[Crossref] [PubMed]

M. Mutlu, A. E. Akosman, A. E. Serebryannikov, and E. Ozbay, “Asymmetric transmission of linearly polarized waves and polarization angle dependent wave rotation using a chiral metamaterial,” Opt. Express 19, 14290–14299 (2011).
[Crossref] [PubMed]

Setare, M. R.

D. Momeni and M. R. Setare, “A note on holographic superconductors with weyl corrections,” Mod. Phys. Lett. A 26, 2889 (2011).
[Crossref]

Sheng, P.

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nature Materials 9, 387 (2010).
[Crossref] [PubMed]

Smith, D. R.

Stephani, H.

H. Stephani, Relativity: An Introduction to Special and General Relativity, E-libro (Cambridge University, 2004).
[Crossref]

Thorne, K. S.

C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (W. H. Freeman and Company, 1973).

Uhlmann, G.

A. Greenleaf, M. Lassas, and G. Uhlmann, “Anisotropic conductivities that cannot be detected by eit,” Physiol. Meas. 24, 413 (2003).
[Crossref] [PubMed]

Wang, J.

J. Jing, S. Chen, Q. Pan, and J. Wang, “Detect black holes using photons for coupling model of electromagnetic and gravitational fields,” arXiv p. 1704.08794 (2017).

Wheeler, J. A.

C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (W. H. Freeman and Company, 1973).

Xie, Y.

X. Lu, F.-W. Yang, and Y. Xie, “Strong gravitational field time delay for photons coupled toweyl tensor in a schwarzschild black hole,” Eur. Phys. J. C 357, 76 (2016).

Yang, F.-W.

X. Lu, F.-W. Yang, and Y. Xie, “Strong gravitational field time delay for photons coupled toweyl tensor in a schwarzschild black hole,” Eur. Phys. J. C 357, 76 (2016).

Eur. Phys. J. C (1)

X. Lu, F.-W. Yang, and Y. Xie, “Strong gravitational field time delay for photons coupled toweyl tensor in a schwarzschild black hole,” Eur. Phys. J. C 357, 76 (2016).

Gen. Rel. Grav. (2)

K. Rosquist, “Letter: A moving medium simulation of schwarzschild black hole optics,” Gen. Rel. Grav. 36, 1977–1982 (2004).
[Crossref]

A. Z. Petrov, “The Classification of spaces defining gravitational fields,” Gen. Rel. Grav. 32, 1661–1663 (2000).
[Crossref]

Mod. Phys. Lett. A (1)

D. Momeni and M. R. Setare, “A note on holographic superconductors with weyl corrections,” Mod. Phys. Lett. A 26, 2889 (2011).
[Crossref]

Nature Materials (1)

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nature Materials 9, 387 (2010).
[Crossref] [PubMed]

New J. Phys. (2)

U. Leonhardt, “Notes on conformal invisibility devices,” New J. Phys. 8, 118 (2006).
[Crossref]

U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys. 8, 247 (2006).
[Crossref]

Opt. Express (4)

Phys. Rev. (1)

I. T. Drummond, “Bimetric QED,” Phys. Rev. D95, 025006 (2017).

Phys. Rev. E (1)

A. Alù and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E 72, 016623 (2005).
[Crossref]

Phys. Rev. Lett. (2)

U. Leonhardt and P. Piwnicki, “Relativistic effects of light in moving media with extremely low group velocity,” Phys. Rev. Lett. 84, 822–825 (2000).
[Crossref] [PubMed]

M. Mutlu, A. E. Akosman, A. E. Serebryannikov, and E. Ozbay, “Diodelike asymmetric transmission of linearly polarized waves using magnetoelectric coupling and electromagnetic wave tunneling,” Phys. Rev. Lett. 108, 213905 (2012).
[Crossref] [PubMed]

Physiol. Meas. (1)

A. Greenleaf, M. Lassas, and G. Uhlmann, “Anisotropic conductivities that cannot be detected by eit,” Physiol. Meas. 24, 413 (2003).
[Crossref] [PubMed]

Proc. Roy. Soc. Lond. A: Math. Phys. Eng. Sci. (1)

G. W. Milton and N.-A. P. Nicorovici, “On the cloaking effects associated with anomalous localized resonance,” Proc. Roy. Soc. Lond. A: Math. Phys. Eng. Sci. 462, 3027–3059 (2006).
[Crossref]

Science (2)

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
[Crossref] [PubMed]

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

Other (6)

C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (W. H. Freeman and Company, 1973).

J. Jing, S. Chen, Q. Pan, and J. Wang, “Detect black holes using photons for coupling model of electromagnetic and gravitational fields,” arXiv p. 1704.08794 (2017).

E. J. Post, Formal Structure of Electromagnetics (Wiley, 1962).

C. Erices, “Riemannian geometry as a tool for optical metamaterial design,” Master’s thesis, Pontificia Universidad Católica de Chile (2012).

G. Gbur, Chapter 5 - Nonradiating Sources and Other “Invisible” Objects (Elsevier, 2003), pp. 273–315.

H. Stephani, Relativity: An Introduction to Special and General Relativity, E-libro (Cambridge University, 2004).
[Crossref]

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Equations (62)

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μ i j = i j ,
I = 1 4 d 4 x χ α β γ δ F α β F γ δ ,
χ α β γ δ = χ β α γ δ = χ β α δ γ , χ β α γ δ = χ γ δ β α .
F μ ν , λ + F ν λ , μ + F λ μ , ν = 0 , H μ μ , ν = 0 ,
H μ ν = 1 2 χ μ ν α β F α β
χ μ ν α β = 1 2 g ( g μ α g ν β g μ β g ν α ) .
χ α β γ δ = 1 2 g Φ ( g α γ g β δ g α δ g β γ ) + 1 2 g W α β γ δ .
W α β γ δ g β δ = 0 ,
χ α β γ δ g β δ = 3 2 g Φ g α γ .
Φ = 1 6 g χ α β γ δ g β δ g α γ .
( χ α β γ δ δ σ τ 1 4 χ α β τ δ δ σ γ ) g β δ g α τ = 0 .
D = ε ˜ E + γ ˜ B ,
H = μ 1 B γ ˜ t E ,
ε ˜ i j = g Φ ( g 0 i g 0 j g 00 g i j ) g W 0 i 0 j ,
γ ˜ i j = g Φ l k j g 0 l g i k + 1 2 g l k j W 0 i l k ,
μ i j 1 = 1 2 Φ g i l k j m n g l m g k n + 1 4 g i m n j l k W m n l k .
D = ε E + γ H ,
B = μ H + γ t E ,
ε i j = ε ˜ i j + γ ˜ i k μ k m γ ˜ j m , γ i j = γ ˜ i k μ k j .
I = 1 4 d 4 x g Φ F α β F α β 1 8 d 4 x g W α β γ δ F α β F γ δ .
D α ( Φ F α β + 1 2 W γ δ α β F γ δ ) = 0 ,
A μ = Re { a μ e i S / }
k μ = μ S .
Φ k β k β a α k β k μ W α β μ ν a ν = 0 .
k μ a μ = 0 .
W α β μ ν k P β k P ν = A k P α k P μ ,
W α β γ δ = Re [ 4 ψ V α β V γ δ ] = 2 ( ψ V α β V γ δ + ψ ¯ V ¯ α β V ¯ γ δ ) ,
V α β = n α m β n β m α .
n μ = 1 2 ( 1 0 0 1 ) , l μ = 1 2 ( 1 0 0 1 ) , m μ = 1 2 ( 0 1 i 0 ) , m ¯ μ = 1 2 ( 0 1 i 0 ) ,
x ν = ( t x y z ) .
W 1313 = W 1310 = W 0101 = W 2332 = W 2302 = W 2002 = α , W 1323 = W 1320 = W 1023 = W 0102 = β .
ε = ( 1 1 + α 0 0 0 1 1 α 0 0 0 1 ) , μ = ( 1 1 α 0 0 0 1 1 + α 0 0 0 1 ) , γ = ( 0 α 1 + α 0 α 1 α 0 0 0 0 0 ) .
b μ = k ( u 0 0 1 ) = k ( u + 1 ) 2 n μ + k ( u 1 ) 2 l μ ,
0 = Φ b β b β a α b β b μ W α β μ ν a ν = k 2 ( Φ ( u 2 1 ) a α 2 ( u 1 ) 2 α Re ( m α m ν ) a ν ) .
a μ = 1 2 ( e i α m μ + e i θ m ¯ μ ) ,
u + 1 u 1 = α Φ e 2 i θ ,
u x = 1 + α / Φ 1 α / Φ , a x μ = ( 0 1 0 0 ) , u y = 1 α / Φ 1 + α / Φ , a y μ = ( 0 0 1 0 ) .
T μ ν = H ¯ μ α F α ν 1 4 η μ ν H ¯ α β F α β .
ρ = T 00 = k 2 u ,
S = u 2 k 2 z ^ ,
W α β γ δ = Re [ 2 ψ D ( V α β U γ β + U α β V γ δ + M α β M γ δ ) ] ,
U α β = l α m ¯ β + l β m ¯ α , M α β = m α m ¯ β m β m ¯ α n α l β + n β l α .
ε = ( ( 1 + α D ) 2 + β D 2 1 + α D 0 0 0 ( 1 + α D ) 2 + β D 2 1 + α D 0 0 0 ( 1 2 α D ) 2 + 4 β D 2 1 2 α D ) , μ = ( 1 1 + α D 0 0 0 1 1 + α D 0 0 0 1 1 2 α D ) ,
γ = β D ( 1 1 + α D 0 0 0 1 1 + α D 0 0 0 2 1 2 α D ) .
W α β γ δ = Re [ 4 ψ V α β V γ δ + 2 ψ 3 ( V α β M γ δ + M α β V γ δ ) ] ,
u x = 1 + α 2 + 4 α ( α 3 2 β 3 2 ) + 2 ( α + 2 ( α 3 2 β 3 2 ) ) 2 + 16 α 3 2 β 3 2 ( α 2 1 + 4 α ( α 3 2 β 3 2 ) + 4 ( α 3 2 + β 3 2 ) ) ,
u y = 1 + α 2 + 4 α ( α 3 2 β 3 2 ) 2 ( α + 2 ( α 3 2 β 3 2 ) ) 2 + 16 α 3 2 β 3 2 ( α 2 1 + 4 α ( α 3 2 β 3 2 ) + 4 ( α 3 2 + β 3 2 ) )
u x = 1 + σ 2 1 σ 2 , u y = 1 ,
ε = ( 1 0 α 3 0 1 1 α 3 2 0 α 3 0 1 + α 3 2 ) , μ = ( 1 1 α 3 2 0 α 3 1 α 3 2 0 1 0 α 3 1 α 3 2 0 1 1 α 3 2 ) ,
γ = ( 0 0 0 α 3 2 1 α 3 2 0 α 3 1 α 3 2 0 α 3 0 ) .
u = β 3 2 β D 2 + 1 β 3 2 + β D 2 1 .
ε = ( 1 + β D 2 β 3 2 β D 2 1 β 3 2 β 3 2 β D 1 β 3 2 ( 1 + β 3 2 ) β 3 β D 1 β 3 2 β 3 2 β D 1 β 3 2 1 β 3 2 + β D 2 1 β 3 2 β 3 ( 1 β 3 2 + β D 2 ) 1 β 3 2 ( 1 + β 3 2 ) β 3 β D 1 β 3 2 β 3 ( 1 β 3 2 + β D 2 ) 1 β 3 2 1 β 3 2 ( β 3 2 1 ) 1 β 3 2 ) ,
μ = ( 1 0 0 0 1 1 β 3 2 β 3 1 β 3 2 0 β 3 1 β 3 2 1 1 β 3 2 ) ,
γ = ( β D β 3 2 1 β 3 2 β 3 1 β 3 2 0 β D 1 β 3 2 β 3 β D 1 β 3 2 β 3 β 3 β D 1 β 3 2 β D 1 β 3 2 ) .
W α β γ δ = 2 ( ψ 1 V α β V γ δ + ψ 2 V ¯ α β V ¯ γ δ ) ,
0 = Φ ( ω 2 k 2 ) a α ( ω k ) 2 ψ 1 m α m ν a ν ( ω k ) 2 ψ 2 m ¯ α m ¯ ν a ν ,
ω + k = 1 Φ ( α 2 + i β 2 ) ( ω k ) e 2 i θ ,
ω + k = α 1 Φ ( ω k ) e 2 i θ .
ω + k = ± ( z 1 + i z 2 ) ( ω k ) ,
u ± = ω R k = 1 z 1 2 z 2 2 ( 1 z 1 ) 2 + z 2 2 , η ± = 2 k z 2 ( 1 z 1 ) 2 + z 2 2 .
u ± 1 z 1 2 ( 1 z 1 ) 2 , η ± 2 k z 2 ( 1 z 1 ) 2 ,
u + 2 , 2 2 3 / 2 > η + k > 2 2 3 / 2 , u 2 , 2 > η k > 2 .

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