Abstract

It is shown by tight-binding approximation and group theory that a double Dirac cone, or a pair of two identical Dirac cones, of the electromagnetic dispersion relation can be created in the Brillouin zone center by accidental degeneracy of E1 and E2 modes in triangular-lattice metamaterials of C6v symmetry. The Dirac point thus obtained is equivalent to a zero-index system, so we can expect unique optical propagation phenomena such as constant-phase waveguides and lenses of arbitrary shapes. Zitterbewegung is also expected without disturbance due to an auxiliary quadratic dispersion surface, which is present for other combinations of mode symmetries to materialize the Dirac cones. To the best of the author’s knowledge, this is the first prediction of the presence of a double Dirac cone in metamaterials.

© 2012 Optical Society of America

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References

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  1. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, 1995).
  2. K. Sakoda, Optical Properties of Photonic Crystals, 2nd ed. (Springer-Verlag, Berlin, 2004).
  3. K. Sakoda and J. W. Haus, “Science and engineering of photonic crystals,” Prog. Opt. 54, 271–317 (2010).
    [Crossref]
  4. C. Caloz and T. ItohElectromagnetic Metamaterials: Transmission Line Theory and Microwave Applications (Wiley, 2006).
  5. C. Caloz and T. Ito, “Application of the transmission line theory of left-handed (LH) materials to the realization of a microstrip LH line,” IEEE Antennas and Propagation Society International Symposium 2, 412–415 (2002).
  6. A. Lai, T. Itoh, and C. Caloz, “Composite right/left-handed transmission line metamaterials,” IEEE Microwave Mag.  5, 34–50 (2004).
    [Crossref]
  7. A. Sanada, C. Caloz, and T. Itoh, “Characteristics of the composite right/left-handed transmission lines,” IEEE Microw. Wirel. Compon. Lett. 14, 68–70 (2004).
    [Crossref]
  8. M. Plihal and A. A. Maradudin, “Photonic band structure of a two-dimensional system: The triangular lattice,” Phys. Rev. B 44, 8565–8571 (1991).
    [Crossref]
  9. F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100, 013904 (2008).
    [Crossref] [PubMed]
  10. S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A 78, 033834 (2008).
    [Crossref]
  11. T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B 80, 155103 (2009).
    [Crossref]
  12. X. Zhang, “Observing zitterbewegung for photons near the Dirac point of a two-dimensional photonic crystal,” Phys. Rev. Lett. 100, 113903 (2008).
    [Crossref] [PubMed]
  13. L.-G. Wang, Z.-G. Wang, and S.-Y. Zhu, “Zitterbewegung of optical pulses near the Dirac point inside a negative-zero-positive index metamaterial,” Europhys. Lett. 86, 47008 (2009).
    [Crossref]
  14. R. A. Sepkhanov, Y. B. Bazaliy, and C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A 75, 063813 (2007).
    [Crossref]
  15. M. Diem, T. Koschny, and C. M. Soukoulis, “Transmission in the vicinity of the Dirac point in hexagonal photonic crystals,” Physica B 405, 2990–2995 (2010).
    [Crossref]
  16. L.-G. Wang, Z.-G. Wang, J.-X. Zhang, and S.-Y. Zhu, “Realization of Dirac point with double cones in optics,” Opt. Lett. 34, 1510–1512 (2009).
    [Crossref] [PubMed]
  17. X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nature Mater. 10, 582–586 (2011).
    [Crossref]
  18. K. Sakoda and H.-F. Zhou, “Role of structural electromagnetic resonances in a steerable left-handed antenna,” Opt. Express 18, 27371–27386 (2010).
    [Crossref]
  19. K. Sakoda and H.-F. Zhou, “Analytical study of two-dimensional degenerate metamaterial antennas,” Opt. Express 19, 13899–13921 (2011).
    [Crossref] [PubMed]
  20. K. Sakoda, “Dirac cone in two- and three-dimensional metamaterials,” Opt. Express 20, 3898–3917 (2012).
    [Crossref] [PubMed]
  21. T. Inui, Y. Tanabe, and Y. Onodera, Group Theory and its Applications in Physics (Springer, Berlin, 1990).
    [Crossref]

2012 (1)

2011 (2)

K. Sakoda and H.-F. Zhou, “Analytical study of two-dimensional degenerate metamaterial antennas,” Opt. Express 19, 13899–13921 (2011).
[Crossref] [PubMed]

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nature Mater. 10, 582–586 (2011).
[Crossref]

2010 (3)

K. Sakoda and H.-F. Zhou, “Role of structural electromagnetic resonances in a steerable left-handed antenna,” Opt. Express 18, 27371–27386 (2010).
[Crossref]

M. Diem, T. Koschny, and C. M. Soukoulis, “Transmission in the vicinity of the Dirac point in hexagonal photonic crystals,” Physica B 405, 2990–2995 (2010).
[Crossref]

K. Sakoda and J. W. Haus, “Science and engineering of photonic crystals,” Prog. Opt. 54, 271–317 (2010).
[Crossref]

2009 (3)

T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B 80, 155103 (2009).
[Crossref]

L.-G. Wang, Z.-G. Wang, J.-X. Zhang, and S.-Y. Zhu, “Realization of Dirac point with double cones in optics,” Opt. Lett. 34, 1510–1512 (2009).
[Crossref] [PubMed]

L.-G. Wang, Z.-G. Wang, and S.-Y. Zhu, “Zitterbewegung of optical pulses near the Dirac point inside a negative-zero-positive index metamaterial,” Europhys. Lett. 86, 47008 (2009).
[Crossref]

2008 (3)

X. Zhang, “Observing zitterbewegung for photons near the Dirac point of a two-dimensional photonic crystal,” Phys. Rev. Lett. 100, 113903 (2008).
[Crossref] [PubMed]

F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100, 013904 (2008).
[Crossref] [PubMed]

S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A 78, 033834 (2008).
[Crossref]

2007 (1)

R. A. Sepkhanov, Y. B. Bazaliy, and C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A 75, 063813 (2007).
[Crossref]

2004 (2)

A. Lai, T. Itoh, and C. Caloz, “Composite right/left-handed transmission line metamaterials,” IEEE Microwave Mag.  5, 34–50 (2004).
[Crossref]

A. Sanada, C. Caloz, and T. Itoh, “Characteristics of the composite right/left-handed transmission lines,” IEEE Microw. Wirel. Compon. Lett. 14, 68–70 (2004).
[Crossref]

2002 (1)

C. Caloz and T. Ito, “Application of the transmission line theory of left-handed (LH) materials to the realization of a microstrip LH line,” IEEE Antennas and Propagation Society International Symposium 2, 412–415 (2002).

1991 (1)

M. Plihal and A. A. Maradudin, “Photonic band structure of a two-dimensional system: The triangular lattice,” Phys. Rev. B 44, 8565–8571 (1991).
[Crossref]

Bazaliy, Y. B.

R. A. Sepkhanov, Y. B. Bazaliy, and C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A 75, 063813 (2007).
[Crossref]

Beenakker, C. W. J.

R. A. Sepkhanov, Y. B. Bazaliy, and C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A 75, 063813 (2007).
[Crossref]

Caloz, C.

A. Lai, T. Itoh, and C. Caloz, “Composite right/left-handed transmission line metamaterials,” IEEE Microwave Mag.  5, 34–50 (2004).
[Crossref]

A. Sanada, C. Caloz, and T. Itoh, “Characteristics of the composite right/left-handed transmission lines,” IEEE Microw. Wirel. Compon. Lett. 14, 68–70 (2004).
[Crossref]

C. Caloz and T. Ito, “Application of the transmission line theory of left-handed (LH) materials to the realization of a microstrip LH line,” IEEE Antennas and Propagation Society International Symposium 2, 412–415 (2002).

C. Caloz and T. ItohElectromagnetic Metamaterials: Transmission Line Theory and Microwave Applications (Wiley, 2006).

Chan, C. T.

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nature Mater. 10, 582–586 (2011).
[Crossref]

Diem, M.

M. Diem, T. Koschny, and C. M. Soukoulis, “Transmission in the vicinity of the Dirac point in hexagonal photonic crystals,” Physica B 405, 2990–2995 (2010).
[Crossref]

Haldane, F. D. M.

S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A 78, 033834 (2008).
[Crossref]

F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100, 013904 (2008).
[Crossref] [PubMed]

Hang, Z. H.

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nature Mater. 10, 582–586 (2011).
[Crossref]

Haus, J. W.

K. Sakoda and J. W. Haus, “Science and engineering of photonic crystals,” Prog. Opt. 54, 271–317 (2010).
[Crossref]

Huang, X.

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nature Mater. 10, 582–586 (2011).
[Crossref]

Inui, T.

T. Inui, Y. Tanabe, and Y. Onodera, Group Theory and its Applications in Physics (Springer, Berlin, 1990).
[Crossref]

Ito, T.

C. Caloz and T. Ito, “Application of the transmission line theory of left-handed (LH) materials to the realization of a microstrip LH line,” IEEE Antennas and Propagation Society International Symposium 2, 412–415 (2002).

Itoh, T.

A. Lai, T. Itoh, and C. Caloz, “Composite right/left-handed transmission line metamaterials,” IEEE Microwave Mag.  5, 34–50 (2004).
[Crossref]

A. Sanada, C. Caloz, and T. Itoh, “Characteristics of the composite right/left-handed transmission lines,” IEEE Microw. Wirel. Compon. Lett. 14, 68–70 (2004).
[Crossref]

C. Caloz and T. ItohElectromagnetic Metamaterials: Transmission Line Theory and Microwave Applications (Wiley, 2006).

Joannopoulos, J. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, 1995).

Koschny, T.

M. Diem, T. Koschny, and C. M. Soukoulis, “Transmission in the vicinity of the Dirac point in hexagonal photonic crystals,” Physica B 405, 2990–2995 (2010).
[Crossref]

Lai, A.

A. Lai, T. Itoh, and C. Caloz, “Composite right/left-handed transmission line metamaterials,” IEEE Microwave Mag.  5, 34–50 (2004).
[Crossref]

Lai, Y.

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nature Mater. 10, 582–586 (2011).
[Crossref]

Maradudin, A. A.

M. Plihal and A. A. Maradudin, “Photonic band structure of a two-dimensional system: The triangular lattice,” Phys. Rev. B 44, 8565–8571 (1991).
[Crossref]

Meade, R. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, 1995).

Ochiai, T.

T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B 80, 155103 (2009).
[Crossref]

Onoda, M.

T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B 80, 155103 (2009).
[Crossref]

Onodera, Y.

T. Inui, Y. Tanabe, and Y. Onodera, Group Theory and its Applications in Physics (Springer, Berlin, 1990).
[Crossref]

Plihal, M.

M. Plihal and A. A. Maradudin, “Photonic band structure of a two-dimensional system: The triangular lattice,” Phys. Rev. B 44, 8565–8571 (1991).
[Crossref]

Raghu, S.

F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100, 013904 (2008).
[Crossref] [PubMed]

S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A 78, 033834 (2008).
[Crossref]

Sakoda, K.

Sanada, A.

A. Sanada, C. Caloz, and T. Itoh, “Characteristics of the composite right/left-handed transmission lines,” IEEE Microw. Wirel. Compon. Lett. 14, 68–70 (2004).
[Crossref]

Sepkhanov, R. A.

R. A. Sepkhanov, Y. B. Bazaliy, and C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A 75, 063813 (2007).
[Crossref]

Soukoulis, C. M.

M. Diem, T. Koschny, and C. M. Soukoulis, “Transmission in the vicinity of the Dirac point in hexagonal photonic crystals,” Physica B 405, 2990–2995 (2010).
[Crossref]

Tanabe, Y.

T. Inui, Y. Tanabe, and Y. Onodera, Group Theory and its Applications in Physics (Springer, Berlin, 1990).
[Crossref]

Wang, L.-G.

L.-G. Wang, Z.-G. Wang, J.-X. Zhang, and S.-Y. Zhu, “Realization of Dirac point with double cones in optics,” Opt. Lett. 34, 1510–1512 (2009).
[Crossref] [PubMed]

L.-G. Wang, Z.-G. Wang, and S.-Y. Zhu, “Zitterbewegung of optical pulses near the Dirac point inside a negative-zero-positive index metamaterial,” Europhys. Lett. 86, 47008 (2009).
[Crossref]

Wang, Z.-G.

L.-G. Wang, Z.-G. Wang, and S.-Y. Zhu, “Zitterbewegung of optical pulses near the Dirac point inside a negative-zero-positive index metamaterial,” Europhys. Lett. 86, 47008 (2009).
[Crossref]

L.-G. Wang, Z.-G. Wang, J.-X. Zhang, and S.-Y. Zhu, “Realization of Dirac point with double cones in optics,” Opt. Lett. 34, 1510–1512 (2009).
[Crossref] [PubMed]

Winn, J. N.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, 1995).

Zhang, J.-X.

Zhang, X.

X. Zhang, “Observing zitterbewegung for photons near the Dirac point of a two-dimensional photonic crystal,” Phys. Rev. Lett. 100, 113903 (2008).
[Crossref] [PubMed]

Zheng, H.

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nature Mater. 10, 582–586 (2011).
[Crossref]

Zhou, H.-F.

Zhu, S.-Y.

L.-G. Wang, Z.-G. Wang, and S.-Y. Zhu, “Zitterbewegung of optical pulses near the Dirac point inside a negative-zero-positive index metamaterial,” Europhys. Lett. 86, 47008 (2009).
[Crossref]

L.-G. Wang, Z.-G. Wang, J.-X. Zhang, and S.-Y. Zhu, “Realization of Dirac point with double cones in optics,” Opt. Lett. 34, 1510–1512 (2009).
[Crossref] [PubMed]

Europhys. Lett. (1)

L.-G. Wang, Z.-G. Wang, and S.-Y. Zhu, “Zitterbewegung of optical pulses near the Dirac point inside a negative-zero-positive index metamaterial,” Europhys. Lett. 86, 47008 (2009).
[Crossref]

IEEE Antennas and Propagation Society International Symposium (1)

C. Caloz and T. Ito, “Application of the transmission line theory of left-handed (LH) materials to the realization of a microstrip LH line,” IEEE Antennas and Propagation Society International Symposium 2, 412–415 (2002).

IEEE Microw. Wirel. Compon. Lett. (1)

A. Sanada, C. Caloz, and T. Itoh, “Characteristics of the composite right/left-handed transmission lines,” IEEE Microw. Wirel. Compon. Lett. 14, 68–70 (2004).
[Crossref]

IEEE Microwave Mag (1)

A. Lai, T. Itoh, and C. Caloz, “Composite right/left-handed transmission line metamaterials,” IEEE Microwave Mag.  5, 34–50 (2004).
[Crossref]

Nature Mater. (1)

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nature Mater. 10, 582–586 (2011).
[Crossref]

Opt. Express (3)

Opt. Lett. (1)

Phys. Rev. A (2)

R. A. Sepkhanov, Y. B. Bazaliy, and C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A 75, 063813 (2007).
[Crossref]

S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A 78, 033834 (2008).
[Crossref]

Phys. Rev. B (2)

T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B 80, 155103 (2009).
[Crossref]

M. Plihal and A. A. Maradudin, “Photonic band structure of a two-dimensional system: The triangular lattice,” Phys. Rev. B 44, 8565–8571 (1991).
[Crossref]

Phys. Rev. Lett. (2)

F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100, 013904 (2008).
[Crossref] [PubMed]

X. Zhang, “Observing zitterbewegung for photons near the Dirac point of a two-dimensional photonic crystal,” Phys. Rev. Lett. 100, 113903 (2008).
[Crossref] [PubMed]

Physica B (1)

M. Diem, T. Koschny, and C. M. Soukoulis, “Transmission in the vicinity of the Dirac point in hexagonal photonic crystals,” Physica B 405, 2990–2995 (2010).
[Crossref]

Prog. Opt. (1)

K. Sakoda and J. W. Haus, “Science and engineering of photonic crystals,” Prog. Opt. 54, 271–317 (2010).
[Crossref]

Other (4)

C. Caloz and T. ItohElectromagnetic Metamaterials: Transmission Line Theory and Microwave Applications (Wiley, 2006).

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, 1995).

K. Sakoda, Optical Properties of Photonic Crystals, 2nd ed. (Springer-Verlag, Berlin, 2004).

T. Inui, Y. Tanabe, and Y. Onodera, Group Theory and its Applications in Physics (Springer, Berlin, 1990).
[Crossref]

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Figures (2)
Fig. 1
Fig. 1 (a) Top view of the unit structure with the C6v (regular hexagon) symmetry composed of a thin metallic film. (b) Symmetry operations of the C6v group. There are two sets of three equivalent mirror reflections that are denoted by (σx, σ′x, σ″x) and (σy, σ′y, σ″y). Seven lattice points (the origin and its nearest neighbors) are denoted by integers from 0 to 6. (c) Triangular array of the unit metallic structures on a uniform dielectric-slab waveguide with a back electrode. Localized electromagnetic resonance states are formed between the metallic unit structures and the back electrode [19].

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Fig. 2
Fig. 2 Double Dirac cone realized by accidental degeneracy of E1 and E2 modes of triangular-lattice metamaterials of the C6v symmetry. The origin of the frequency ω is shifted to the degenerate frequency ωΓ. Although the two cones are illustrated as slightly separated from each other to emphasize the presence of two cones, their actual shapes are identical.

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Tables (1)

Tables Icon

Table 1 Parity of resonant states of a single unit structure.

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Equations (96)

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𝒧 H ( r , t ) ≡ ∇ × [ 1 ε ( r ) ∇ × H ( r , t ) ] = − 1 c 2 ∂ 2 ∂ t 2 H ( r , t ) ,
∇ × [ 1 ε s ( r ) ∇ × H ( 1 , 2 ) ( r ) ] = ω 1 2 c 2 H ( 1 , 2 ) ( r ) ,
∇ × [ 1 ε s ( r ) ∇ × H ( 3 , 4 ) ( r ) ] = ω 2 2 c 2 H ( 3 , 4 ) ( r ) ,
∫ V d r H ( i ) * ( r ) ⋅ H ( j ) ( r ) = V δ i j ,
B = ( B 11 B 12 B 13 B 14 B 12 * B 22 B 23 − B 13 B 13 * B 23 * B 33 B 34 B 14 * − B 13 * B 34 * B 44 ) .
B 11 = ω 1 2 c 2 + 2 M 12   cos   k x + 4 M 14   cos   k x a 2   cos   3 k y a 2 ,
B 22 = ω 1 2 c 2 + 2 M 13   cos   k x + 4 M 15   cos   k x a 2   cos   3 k y a 2 ,
B 33 = ω 2 2 c 2 + 2 M 22   cos   k x + 4 M 24   cos   k x a 2   cos   3 k y a 2 ,
B 44 = ω 2 2 c 2 + 2 M 23   cos   k x + 4 M 25   cos   k x a 2   cos   3 k y a 2 ,
B 12 = − 4 M 16   sin   k x a 2   sin   3 k y a 2 ,
B 13 = 4 i M 3   cos   k x a 2   sin   3 k y a 2 ,
B 14 = 2 i M 1   sin   k x a + 4 i M 4   sin   k x a 2   cos   3 k y a 2 ,
B 23 = 2 i M 2   sin   k x a + 4 i M 5   sin   k x a 2   cos   3 k y a 2 ,
B 34 = − 4 M 26   sin   k x a 2   sin   3 k y a 2 ,
ω k ≈ ω Γ ± 3 | M 3 | a c 2 k ω Γ − 3 M a 2 c 2 k 2 16 ω Γ ( double   roots ) .
ω − ω Γ = v k ,
v = 3 | M 3 | a c 2 ω Γ
ω = c n eff k .
n eff = A ( ω )   ( ω − ω Γ ) ,
v = c ω A ( ω ) ≈ c ω Γ A ( ω Γ ) .
n eff = ε eff μ eff ,
ε eff ( ω ) = 1 − ω e p 2 ω 2 ,
μ eff ( ω ) = 1 − ω m p 2 ω 2 .
n eff = 1 − ω Γ 2 ω 2 ,
A ( ω ) = ω + ω Γ ω 2
n eff = ( 1 − ω e p 2 ω 2 )   ( 1 − ω m p 2 ω 2 ) .
ω + ≈ ω e p + ω e p c 2 k 2 2 ( ω e p 2 − ω m p 2 ) ,
ω − ≈ ω m p − ω m p c 2 k 2 2 ( ω e p 2 − ω m p 2 ) ,
ω = ω Γ + a 2 c 2 k 2 4 ω Γ   { ± ( M 1 − M 2 ) 2 + 4 | M 3 | 2   cos 2 2 ϕ − ( M 1 + M 2 ) } ,
L m ( i j ) ≡ 1 V ∫ V d r   H ( i ) * ( r ) ⋅ 𝒧 H ( j )   ( r − r m ) ,
L 0 ( 11 ) = L 0 ( 22 ) ≡ ω 1 2 c 2 + M 11 ,
L 0 ( 12 ) = L 0 ( 21 ) = 0 ,
L 1 ( 11 ) = L 4 ( 11 ) ≡ M 12 ,
L 1 ( 22 ) = L 4 ( 22 ) ≡ M 13 ,
L 1 ( 12 ) = L 4 ( 12 ) = L 1 ( 21 ) = L 4 ( 21 ) = 0 ,
L 2 ( 11 ) = L 3 ( 11 ) = L 5 ( 11 ) = L 6 ( 11 ) ≡ M 14 ,
L 2 ( 22 ) = L 3 ( 22 ) = L 5 ( 22 ) = L 6 ( 22 ) ≡ M 15 ,
L 2 ( 12 ) = − L 3 ( 12 ) = L 5 ( 12 ) = − L 6 ( 12 ) = L 2 ( 21 ) = − L 3 ( 21 ) = L 5 ( 21 ) = − L 6 ( 21 ) ≡ M 16 ,
M 14 = M 12 + 3 M 13 4 ,
M 15 = 3 M 12 + M 13 4 ,
M 16 = 3 ( M 12 − M 13 ) 4 .
L 0 ( 33 ) = L 0 ( 44 ) ≡ ω 2 2 c 2 + M 21 ,
L 0 ( 34 ) = L 0 ( 43 ) = 0 ,
L 1 ( 33 ) = L 4 ( 33 ) ≡ M 22 ,
L 1 ( 44 ) = L 4 ( 44 ) ≡ M 23 ,
L 1 ( 34 ) = L 4 ( 34 ) = L 1 ( 43 ) = L 4 ( 43 ) = 0 ,
L 2 ( 33 ) = L 3 ( 33 ) = L 5 ( 33 ) = L 6 ( 33 ) ≡ M 24 ,
L 2 ( 44 ) = L 3 ( 44 ) = L 5 ( 44 ) = L 6 ( 44 ) ≡ M 25 ,
L 2 ( 34 ) = − L 3 ( 34 ) = L 5 ( 34 ) = − L 6 ( 34 ) = L 2 ( 43 ) = − L 3 ( 43 ) = L 5 ( 43 ) = − L 6 ( 43 ) ≡ M 26 ,
M 24 = M 22 + 3 M 23 4 ,
M 25 = 3 M 22 + M 23 4 ,
M 26 = 3 ( M 22 − M 23 ) 4 ,
L m ( i j ) = 1 V ∫ V d r ′ H ( i ) * ( R − 1 r ′ ) ⋅ [ 𝒧 H ( j ) ( R − 1 r ′ − r m ) ] ,
L m ( i j ) = 1 V ∫ V d r ′ [ R − 1 R H ( i ) * ( R − 1 r ′ ) ] ⋅ [ R − 1 R 𝒧 R − 1 R H ( j ) ( R − 1 r ′ − r m ) ] = 1 V ∫ V d r ′ [ R H ( i ) * ( R − 1 r ′ ) ] ⋅ [ 𝒧 ′ R H ( j ) ( R − 1 r ′ − r m ) ] ,
𝒧 ′ ≡ R 𝒧 R − 1 ,
R H ( i ) ( R − 1 r ) ≡ [ R H ( i ) ]   ( r ) .
L m ( i j ) = 1 V ∫ V d r ′   [ R H ( i ) * ( R − 1 r ′ ) ] ⋅ [ 𝒧 ′ R H ( j ) ( R − 1 { r ′ − R r m } ) ] = 1 V ∫ V d r [ R H ( i ) * ] ( r ) ⋅ 𝒧 [ R H ( j ) ] ( r − R r m )
L 0 ( 13 ) = L 0 ( 14 ) = L 0 ( 23 ) = L 0 ( 24 ) = L 0 ( 31 ) = L 0 ( 41 ) = L 0 ( 32 ) = L 0 ( 42 ) = 0 ,
L 1 ( 13 ) = L 1 ( 24 ) = L 1 ( 31 ) = L 1 ( 42 ) = L 4 ( 13 ) = L 4 ( 24 ) = L 4 ( 31 ) = L 4 ( 42 ) = 0 ,
L 1 ( 14 ) = − L 4 ( 14 ) = − L 1 ( 41 ) * = L 4 ( 41 ) * ≡ M 1 ,
L 1 ( 23 ) = − L 4 ( 23 ) = − L 1 ( 32 ) * = L 4 ( 32 ) * ≡ M 2 ,
L 2 ( 13 ) = L 3 ( 13 ) = − L 5 ( 13 ) = − L 6 ( 13 ) = − L 2 ( 31 ) * = − L 3 ( 31 ) * = L 5 ( 31 ) * = L 6 ( 31 ) * = − L 2 ( 24 )
= − L 3 ( 24 ) = L 5 ( 24 ) = L 6 ( 24 ) = L 2 ( 42 ) * = L 3 ( 42 ) * = − L 5 ( 42 ) * = − L 6 ( 42 ) * ≡ M 3 ,
L 2 ( 14 ) = − L 3 ( 14 ) = − L 5 ( 14 ) = L 6 ( 14 ) = − L 2 ( 41 ) * = L 3 ( 41 ) * = L 5 ( 41 ) * = − L 6 ( 41 ) * ≡ M 4 ,
L 2 ( 23 ) = − L 3 ( 23 ) = − L 5 ( 23 ) = L 6 ( 23 ) = − L 2 ( 32 ) * = L 3 ( 32 ) * = L 5 ( 32 ) * = − L 6 ( 32 ) * ≡ M 5 .
M 3 = 3 ( M 1 + M 2 ) 4 ,
M 4 = − M 1 + 3 M 2 4 ,
M 5 = 3 M 1 − M 2 4 .
L m ( i j ) * = L − m ( j i ) ,
H k   ( r ) = 1 V ∑ m e i k ⋅ r m ∑ i = 1 4 A i H ( i ) ( r − r m ) ,
𝒧 H k ( r ) = ω k 2 c 2 H k   ( r ) ,
| B − ω k 2 c 2 I | = 0 ,
B i j = ∑ m e i k ⋅ r m   L m ( i j ) .
ξ = ω k 2 c 2 , ξ 1 = ω 1 2 c 2 , ξ 2 = ω 2 2 c 2 .
ξ 4 + b 3 ξ 3 + b 2 ξ 2 + b 1 ξ + b 0 = 0 ,
b 3 = − ( B 11 + B 22 + B 33 + B 44 ) ,
b 2 = B 11 B 22 + B 11 B 33 + B 11 B 44 + B 22 B 33 + B 22 B 44 + B 33 B 44 − ( | B 12 | 2 + 2 | B 13 | 2 + | B 14 | 2 + | B 23 | 2 + | B 34 | 2 ) ,
b 1 = − ( B 22 B 33 B 44 + B 11 B 33 B 44 + B 11 B 22 B 44 + B 11 B 22 B 33 ) + | B 12 | 2 ( B 33 + B 44 ) + | B 13 | 2 ( B 11 + B 22 + B 33 + B 44 ) + | B 14 | 2 ( B 22 + B 33 ) + | B 23 | 2 ( B 11 + B 44 ) + | B 34 | 2 ( B 11 + B 22 ) + B 12 B 13 B 14 * + B 12 * B 13 * B 14 − B 12 B 13 * B 23 − B 12 * B 13 B 23 * − B 13 B 14 * B 34 − B 13 * B 14 B 34 * + B 13 B 23 * B 34 * + B 13 * B 23 B 34 ,
b 0 = B 11 B 22 B 33 B 44 + | B 13 | 4 + | B 14 | 2 | B 23 | 2 − | B 12 | 2 B 33 B 44 − | B 13 | 2 ( B 11 B 33 + B 22 B 44 ) − | B 14 | 2 B 22 B 33 − | B 23 | 2 B 11 B 44 − | B 34 | 2 B 11 B 22 − B 11 ( B 13 B 23 * B 34 * + B 13 * B 23 B 34 ) + B 22 ( B 13 B 14 * B 34 + B 13 * B 14 B 34 * ) − B 33 ( B 12 B 13 B 14 * + B 12 * B 13 * B 14 ) + B 44 ( B 12 B 13 * B 23 + B 12 * B 13 B 23 * ) + B 13 2 B 14 * B 23 * + B 13 * 2 B 14 B 23 .
ξ = { ξ 1 + 3 2 ( M 12 + M 13 ) ≡ ξ Γ ( 1 ) ( double   root ) , ξ 2 + 3 2 ( M 22 + M 23 ) ≡ ξ Γ ( 2 ) ( double   root ) ,
η = ξ + b 3 4 .
η 4 + p η 2 + q η + r = 0 ,
p = b 2 − 3 b 3 2 8 ,
q = b 1 − b 2 b 3 2 + b 3 3 8 ,
r = b 0 − b 1 b 3 4 + b 2 b 3 2 16 − 3 b 3 4 256 .
p ≈ − 1 2 ( ξ Γ ( 1 ) − ξ Γ ( 2 ) ) 2 + 2 ( ξ Γ ( 1 ) − ξ Γ ( 0 ) )   ( M 12 + M 13 − M 22 − M 23 )   ( sin 2   k x a 2 + 2   sin 2   k x a 4 + 2   sin 2   3 k y a 4 ) − 4 ( | M 1 | 2 + | M 2 | 2 )   sin 2   k x a − 16   ( | M 4 | 2 + | M 5 | 2 )   sin 2   k x a 2 − 32 | M 3 | 2   sin 2   3 k y a 2 − 8 ( M 1 M 4 * + M 1 * M 4 + M 2 M 5 * + M 2 * M 5 )   sin   k x a   sin   k x a 2 .
q ≈ 0
r ≈ 1 16   ( ξ Γ ( 1 ) − ξ Γ ( 2 ) ) 2                 + 1 2   ( ξ Γ ( 1 ) − ξ Γ ( 2 ) ) 3   ( M 22 + M 23 − M 12 − M 13 )                           × ( sin 2   k x a 2 + 2 sin 2   k x a 4 + 2   sin 2   3 k y a 4 − 4   sin 2   k x a 4   sin 2   3 k y a 4 )                   + ( ξ Γ ( 1 ) − ξ Γ ( 2 ) ) 2                             × [ 8 | M 3 | 2   ( 1 − 4 sin 2   k x a 4 )   sin 2   3 k y a 2                                             + 1 2 { M 12 2 + M 13 2 + M 22 2 + M 23 2 + 10 ( M 12 M 13 + M 22 M 23 )                                                                     − 6 ( M 12 + M 13 )   ( M 22 + M 23 ) } sin 4   k x a 2                                               + 2   { M 14 2 + M 15 2 + M 24 2 + M 25 2 + 10 ( M 14 M 15 + M 24 M 25 )                                                                             − 6 ( M 14 + M 15 ) ( M 24 + M 25 ) } ( sin 2   k x a 4 + sin 2   3 k y a 4 )                                                 + 2   { 5 ( M 12 + M 13 + M 22 + M 23 ) 2 − 16 ( M 12 + M 13 ) ( M 22 + M 23 )                                                                           − 4 ( M 12 M 14 + M 13 M 15 + M 22 M 24 + M 23 M 25 ) }                                                                                                                                                                     × sin 2   k x a 2   ( sin 2   k x a 4 + sin 2   3 k y a 4 )                                                   − 4 ( | M 16 | 2 + | M 26 | 2 ) sin 2   k x a 2 sin 2   3 k y a 2                                                   − 16 ( | M 4 | 2 + | M 5 | 2 )   sin 2   k x a 2 sin 2   3 k y a 4                                                   − ( M 1 M 4 * + M 1 * M 4 + M 2 M 5 * + M 2 * M 5 )   sin   k x a   sin   k x a 2   sin 2   3 k y a 4                                                   + | M 1   sin   k x a + 2 M 4   sin   k x a 2 | 2 + | M 2   sin   k x a + 2 M 5   sin   k x a 2 | 2 ]                               + ( ξ Γ ( 1 ) − ξ Γ ( 2 ) )                                         × [ 16 { M 3 * ( M 16 + M 26 )   ( M 2   sin   k x a + 2 M 5   sin   k x a 2 )                                                               + M 3   ( M 16 * + M 26 * )   ( M 2 *   sin   k x a + 2 M 5 *   sin   k x a 2 )                                                                 − M 3 ( M 16 + M 26 )   ( M 1 *   sin   k x a + 2 M 4 *   sin   k x a 2 )                                                                   − M 3 * ( M 16 * + M 26 * )   ( M 1   sin   k x a + 2 M 4   sin   k x a 2 ) }   sin   k x a 2   sin 2   3 k y a 2                                               + 32 | M 3 | 2   ( M 22 + M 23 − M 12 − M 13 )   sin 2   3 k y a 2                                                                                                                           × ( sin 2   k x a 2 + 2   sin 2   k x a 4 + 2   sin 2   3 k y a 4 )                                                 + 8 { ( M 22 − M 13 )   sin 2   k x a 2 + 2 ( M 24 − M 15 )   ( sin 2   k x a 4 + sin 2   3 k y a 4 ) }                                                                                                                       × | M 1   sin   k x a + 2 M 4   sin   k x a 2 | 2                                                   + 8   { ( M 23 − M 12 )   sin 2   k x a 2 + 2 ( M 25 − M 14 )   ( sin 2   k x a 4 + sin 2   3 k y a 4 ) }                                                                                                                         × | M 2   sin   k x a + 2 M 5   sin   k x a 2 | 2 ]                     + 256 | M 3 | 4   sin 4   3 k y a 2                       + 64 M 3 2   sin 2   3 k y a 2   ( M 2 *   sin   k x a + 2 M 5 *   sin   k x a 2 )   ( M 1 *   sin   k x a + 2 M 4 *   sin   k x a 2 )                       + 64 M 3 * 2   sin 2   3 k y a 2   ( M 2   sin   k x a + 2 M 5   sin   k x a 2 )   ( M 2   sin   k x a + 2 M 4   sin   k x a 2 )                       + 16 | M 1   sin   k x a + 2 M 4   sin   k x a 2 | 2   | M 2 sin k x a + 2 M 5 sin k x a 2 | 2 .
ξ Γ ( 1 ) = ξ Γ ( 2 ) ≡ ξ Γ .
p ≈ − 24 | M 3 | 2 k 2   a 2 ,
q ≈ 0 ,
r ≈ 144 | M 3 | 4 k 4   a 4 ,
η ≈ ± 2 3 | M 3 | k a ( double   roots ) .
b 3 ≈ − 4 ξ Γ + 3 2 M k 2   a 2 ,
ξ ≈ ξ Γ ± 2 3 | M 3 | k a − 3 8 M k 2   a 2 ( double   roots ) .
ω k ≈ ω Γ ± 3 | M 3 | a c 2 k ω Γ − 3 M a 2 c 2 k 2 16 ω Γ ( double   roots ) .

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