Abstract

We formulate a degenerate perturbation theory for the vector electromagnetic field of periodic structures and apply it to the problem of the creation of Dirac cones in the Brillouin-zone center by accidental degeneracy of two modes. We derive a necessary condition by which we can easily select candidates of mode combinations that enable the creation of the Dirac cone. We analyze the structure of a matrix that determines the first-order correction to eigen frequencies by examining its transformation by symmetry operations. Thus, we can obtain the analytical solution of dispersion curves in the vicinity of the zone center and can judge the presence of the Dirac cone. All these findings clearly show that the presence or absence of the Dirac cone in the zone center is solely determined by the spatial symmetry of the two modes.

© 2012 OSA

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References

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  1. 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]
  2. S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A78, 033834 (2008).
    [CrossRef]
  3. T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B80, 155103 (2009).
    [CrossRef]
  4. X. Zhang, “Observing zitterbewegung for photons near the Dirac point of a two-dimensional photonic crystal,” Phys. Rev. Lett.100, 113903 (2008).
    [CrossRef] [PubMed]
  5. R. A. Sepkhanov, Y. B. Bazaliy, and C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A75, 063813 (2007).
    [CrossRef]
  6. M. Diem, T. Koschny, and C. M. Soukoulis, “Transmission in the vicinity of the Dirac point in hexagonal photonic crystals,” Physica B405, 2990–2995 (2010).
    [CrossRef]
  7. 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]
  8. K. Sakoda and H-F. Zhou, “Role of structural electromagnetic resonances in a steerable left-handed antenna,” Opt. Express18, 27371–27386 (2010).
    [CrossRef]
  9. K. Sakoda and H.-F. Zhou, “Analytical study of two-dimensional degenerate metamaterial antennas,” Opt. Express19, 13899–13921 (2011).
    [CrossRef] [PubMed]
  10. K. Sakoda, “Dirac cone in two- and three-dimensional metamaterials,” Opt. Express20, 3898–3917 (2012).
    [CrossRef] [PubMed]
  11. K. Sakoda, “Double Dirac cones in triangular-lattice metamaterials,” Opt. Express20, 9925–9939 (2012).
    [CrossRef] [PubMed]
  12. K. Sakoda, “Universality of mode symmetries in creating photonic Dirac cones,” J. Opt. Soc. Am. B29, 2770–2778 (2012).
    [CrossRef]
  13. M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using ε-near-zero materials,” Phys. Rev. Lett.97, 157403 (2006).
    [CrossRef] [PubMed]
  14. A. Alu, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B75, 155410 (2007).
    [CrossRef]
  15. K. Sakoda, Optical Properties of Photonic Crystals, 2nd Ed. (Springer-Verlag, Berlin, 2004).
  16. T. Inui, Y. Tanabe, and Y. Onodera, Group Theory and Its Applications in Physics (Springer, Berlin, 1990).
    [CrossRef]
  17. H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, 5th Edition (World Scientific, Singapore, 2009).
    [CrossRef]

2012 (3)

2011 (2)

K. Sakoda and H.-F. Zhou, “Analytical study of two-dimensional degenerate metamaterial antennas,” Opt. Express19, 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 (2)

M. Diem, T. Koschny, and C. M. Soukoulis, “Transmission in the vicinity of the Dirac point in hexagonal photonic crystals,” Physica B405, 2990–2995 (2010).
[CrossRef]

K. Sakoda and H-F. Zhou, “Role of structural electromagnetic resonances in a steerable left-handed antenna,” Opt. Express18, 27371–27386 (2010).
[CrossRef]

2009 (1)

T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B80, 155103 (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. A78, 033834 (2008).
[CrossRef]

2007 (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. A75, 063813 (2007).
[CrossRef]

A. Alu, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B75, 155410 (2007).
[CrossRef]

2006 (1)

M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using ε-near-zero materials,” Phys. Rev. Lett.97, 157403 (2006).
[CrossRef] [PubMed]

Alu, A.

A. Alu, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B75, 155410 (2007).
[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. A75, 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. A75, 063813 (2007).
[CrossRef]

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 B405, 2990–2995 (2010).
[CrossRef]

Engheta, N.

A. Alu, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B75, 155410 (2007).
[CrossRef]

M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using ε-near-zero materials,” Phys. Rev. Lett.97, 157403 (2006).
[CrossRef] [PubMed]

Haldane, F. D. M.

S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A78, 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]

Haug, H.

H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, 5th Edition (World Scientific, Singapore, 2009).
[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]

Koch, S. W.

H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, 5th Edition (World Scientific, Singapore, 2009).
[CrossRef]

Koschny, T.

M. Diem, T. Koschny, and C. M. Soukoulis, “Transmission in the vicinity of the Dirac point in hexagonal photonic crystals,” Physica B405, 2990–2995 (2010).
[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]

Ochiai, T.

T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B80, 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. B80, 155103 (2009).
[CrossRef]

Onodera, Y.

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

Raghu, S.

S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A78, 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]

Sakoda, K.

Salandrino, A.

A. Alu, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B75, 155410 (2007).
[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. A75, 063813 (2007).
[CrossRef]

Silveirinha, M.

M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using ε-near-zero materials,” Phys. Rev. Lett.97, 157403 (2006).
[CrossRef] [PubMed]

Silveirinha, M. G.

A. Alu, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B75, 155410 (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 B405, 2990–2995 (2010).
[CrossRef]

Tanabe, Y.

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

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.

Zhou, H-F.

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

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 (4)

Phys. Rev. A (2)

S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A78, 033834 (2008).
[CrossRef]

R. A. Sepkhanov, Y. B. Bazaliy, and C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A75, 063813 (2007).
[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. B80, 155103 (2009).
[CrossRef]

A. Alu, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B75, 155410 (2007).
[CrossRef]

Phys. Rev. Lett. (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]

M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using ε-near-zero materials,” Phys. Rev. Lett.97, 157403 (2006).
[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]

Physica B (1)

M. Diem, T. Koschny, and C. M. Soukoulis, “Transmission in the vicinity of the Dirac point in hexagonal photonic crystals,” Physica B405, 2990–2995 (2010).
[CrossRef]

Other (3)

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]

H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, 5th Edition (World Scientific, Singapore, 2009).
[CrossRef]

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Figures (1)

Fig. 1
Fig. 1

Dispersion curves of a photonic crystal composed of the simple-cubic lattice of dielectric spheres with a dielectric constant of 12.6. The vertical axis is the normalized frequency (ωa/2πc), and the horizontal axis is the wave vector in the R (= (π/a, π/a, π/a)), and X (= (π/a, 0, 0)) directions, where R/6, for example, means that the horizontal axis is magnified by six times. The radius of the spheres is 0.160a for (a), 0.210a for (b), and 0.416a for (c). The number associated with each dispersion curve is the multiplicity of the mode.

Tables (4)

Tables Icon

Table 1 Types of dispersion curves generated by accidental degeneracy of two modes (Mode 1 and Mode 2) for one-dimensional lattices of the C2v symmetry. Symmetries of the magnetic field of the two modes on the Γ point are given in the left column. The middle column shows whether { C i j ( k )} vanishes by symmetry (×) or it can be non-zero (○). Shapes of the dispersion curves are given in the right column, where D and Q denote Dirac cone and quadratic dispersion surface, respectively.

Tables Icon

Table 2 Types of dispersion curves generated by accidental degeneracy of two modes (Mode 1 and Mode 2) for square lattices of the C4v symmetry. Symmetries of the magnetic field of the two modes on the Γ point are given in the left column. The middle column shows whether { C i j ( k )} vanishes by symmetry (×) or it can be non-zero (○). Shapes of the dispersion curves are given in the right column, where D and Q denote Dirac cone and quadratic dispersion surface, respectively.

Tables Icon

Table 3 Types of dispersion curves generated by accidental degeneracy of two modes for the triangular lattice of the C6v symmetry. Shapes of the dispersion curves are given in the right column, where D, DD and Q denote Dirac cone, double Dirac cones, and quadratic dispersion surface, respectively.

Tables Icon

Table 4 Types of dispersion curves generated by accidental degeneracy of two modes for the simple-cubic lattice of the Oh symmetry. Symmetries of the magnetic field of the two modes on the Γ point are given in the left column. The middle column shows whether { C i j ( k )} vanishes by symmetry (×) or it can be non-zero (○). Shapes of the dispersion curves are given in the right column, where D, DD, and Q denote Dirac cone, double Dirac cones, and quadratic dispersion surface, respectively.

Equations (53)

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H k n × [ 1 ε ( r ) × H k n ] = λ k n H k n ,
λ k n = ω k n 2 c 2 ,
H k n ( r ) = e i k r u k n ( r ) ,
u k n ( r + a ) = u k n ( r ) ,
H 1 | H 2 1 V V d r H 1 * ( r ) H 2 ( r ) ,
H k n | H k n = δ k k δ n n .
k e i k r e i k r = ( + i k ) × [ 1 ε ( r ) ( + i k ) × ] ,
u k n | u k n 0 1 V 0 V 0 d r u k n * ( r ) u k n ( r ) = δ n n ,
{ u 0 n | n = 1 , 2 , }
Δ k k 0 Δ k ( 1 ) + Δ k ( 2 ) ,
Δ k ( 1 ) = i k × [ 1 ε ( r ) × ] ,
Δ k ( 2 ) = × [ 1 ε ( r ) i k × ] .
C i j ( k ) = u 0 i | Δ k | u 0 j 0 ,
R t = R 1 and detR = ± 1 ,
u 0 i | Δ k ( 1 ) | u 0 j 0 = i k P i j ,
P i j = 1 V 0 V 0 d r u 0 i * ( r ) × [ 1 ε ( r ) × u 0 j ( r ) ] u 0 i | Δ | u 0 j 0 .
V 0 u 0 i | Δ k ( 2 ) | u 0 j 0 V d r { × [ 1 ε ( r ) i k × u 0 j ( r ) ] } u 0 i * ( r ) = S 0 d S { [ 1 ε ( r ) i k × u 0 j ( r ) ] × u 0 i * ( r ) } n + V 0 d r [ 1 ε ( r ) i k × u 0 j ( r ) ] [ × u 0 i * ( r ) ] ,
u 0 i | Δ k ( 2 ) | u 0 j 0 = i k P j i * .
C i j ( k ) = i k ( P i j + P j i * ) .
( × ) 1 = detR ( × ) .
( × ) 1 = detR ( × ) .
k P i j = k V 0 V 0 d r 1 u 0 i * ( r ) 1 × 1 [ 1 ε ( r ) 1 × 1 u 0 j ( r ) ] = ( detR ) 2 V 0 k V 0 d r 1 [ u 0 i * ] ( r ) × { 1 ε ( r ) × [ u 0 j ] ( r ) } = ( R k ) V 0 V 0 d r [ u 0 i * ] ( r ) × { 1 ε ( r ) × [ u 0 j ] ( r ) } = ( R k ) u 0 i | Δ | u 0 j 0 ,
[ u 0 i ] ( r ) R u 0 i ( R 1 r ) .
C k = ( 0 , b k b * k , 0 ) .
b = i e [ u 01 | Δ | u 02 0 + u 02 | Δ | u 01 0 * ] ,
| Δ λ , b k b * k , Δ λ | = 0 .
Δ λ = ± | b | k ,
ω k ω 0 ± | b | c 2 k 2 ω 0 .
C k = ( 0 0 b k x 0 0 b k y b * k x b * k y 0 ) .
| Δ λ , 0 , b k x 0 , Δ λ , b k y b * k x , b * k y , Δ λ , | = 0 ,
Δ λ = 0 , ± | b | k ,
( E , A 2 ) : C k = ( 0 0 b k y 0 0 b k x b * k y b * k x 0 ) , ( E , B 1 ) : C k = ( 0 0 b k x 0 0 b k y b * k x b * k y 0 ) , ( E , B 2 ) : C k = ( 0 0 b k y 0 0 b k x b * k y b * k x 0 ) .
C k = ( 0 0 b k y b k x 0 0 b k x b k y b * k y b * k x 0 0 b * k x b * k y 0 0 ) ,
Δ λ = ± | b | k ( double roots ) .
C k = ( 0 0 b k x 0 0 b k y b * k x b * k y 0 ) .
C k = ( 0 0 b k y 0 0 b k x b * k y b * k x 0 ) .
Δ λ = 0 , ± | b | k .
C k = ( 0 0 0 b k x 0 0 0 b k y 0 0 0 b k z b * k x b * k y b * k z 0 ) .
Δ λ = { 0 ( double roots ) , ± | b | k ,
C k = ( 0 0 0 b k x 3 b k x 0 0 0 b k y 3 b k y 0 0 0 2 b k z 0 b * k x b * k y 2 b * k z 0 0 3 b * k x 3 b * k y 0 0 0 )
Δ λ [ ( Δ λ ) 4 4 | b | 2 k 2 ( Δ λ ) 2 + 12 ( k x 2 k y 2 + k y 2 k z 2 + k z 2 k x 2 ) | b | 4 ] = 0 .
Δ λ = { 0 , ± 2 | b | k 1 ± 1 3 F ( θ , ϕ ) ,
F ( θ , ϕ ) = sin 4 θ sin 2 2 ϕ + sin 2 2 θ 4 .
C k = ( 0 0 0 3 b k x b k x 0 0 0 3 b k y b k y 0 0 0 0 2 b k z 3 b * k x 3 b * k y 0 0 0 b * k x b * k y 2 b * k z 0 0 )
C k = ( 0 0 0 0 b k z b k y 0 0 0 b k z 0 b k x 0 0 0 b k y b k x 0 0 b * k z b * k y 0 0 0 b * k z 0 b * k x 0 0 0 b * k y b * k x 0 0 0 0 )
Δ λ = { 0 ( double roots ) , ± | b | k ( double roots ) .
C k = ( 0 0 0 0 b k z b k y 0 0 0 b k z 0 b k x 0 0 0 b k y b k x 0 0 b * k z b * k y 0 0 0 b * k z 0 b * k x 0 0 0 b * k y b * k x 0 0 0 0 ) ,
( Δ λ ) 3 | b | 2 k 2 Δ λ ± 2 | b | 3 k x k y k z = 0 .
ξ 3 + p ξ + q = 0
ξ l = e 2 π i l / 3 q 2 + ( q 2 ) 2 + ( p 3 ) 3 3 + e 2 π i l / 3 q 2 ( q 2 ) 2 + ( p 3 ) 3 3 ,
Δ λ l ± = | b | k ( e 2 π i l / 3 F 1 ± + e 2 π i l / 3 F 2 ± ) ,
F 1 ± = [ ± sin θ sin 2 θ sin 2 ϕ 4 + sin 2 θ sin 2 2 θ sin 2 2 ϕ 16 1 27 ] 1 3 ,
F 2 ± = [ ± sin θ sin 2 θ sin 2 ϕ 4 sin 2 θ sin 2 2 θ sin 2 2 ϕ 16 1 27 ] 1 3 .

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