Abstract

The creation of photonic Dirac cones by accidental degeneracy in the Brillouin-zone center was recently reported for both metamaterials with localized electromagnetic resonant states and dielectric photonic crystals without well-defined resonance. Based on the anticipation that there should be a common physical origin in this phenomenon, we systematically examined the relation between mode symmetries and shapes of dispersion curves for both systems. The result strongly suggests the presence of universality of mode symmetries that enable the creation of photonic Dirac cones irrespective of the details of the sample structure.

© 2012 Optical Society of America

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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]
  2. S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A 78, 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. B 80, 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]
  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. A 75, 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 B 405, 2990–2995 (2010).
    [CrossRef]
  7. S. H. Nam, A. J. Taylor, and A. Efimov, “Diabolical point and conical-like diffraction in periodic plasmonic nanostructures,” Opt. Express 18, 10120–10126 (2010).
    [CrossRef]
  8. 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,” Nat. Mater. 10, 582–586 (2011).
    [CrossRef]
  9. 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]
  10. A. Alu, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B 75, 155410 (2007).
    [CrossRef]
  11. K. Sakoda, “Dirac cone in two- and three-dimensional metamaterials,” Opt. Express 20, 3898–3917 (2012).
    [CrossRef]
  12. K. Sakoda, “Double Dirac cones in triangular-lattice metamaterials,” Opt. Express 20, 9925–9939 (2012).
    [CrossRef]
  13. K. Sakoda, Optical Properties of Photonic Crystals, 2nd ed. (Springer-Verlag, 2004).
  14. T. Inui, Y. Tanabe, and Y. Onodera, Group Theory and Its Applications in Physics (Springer, 1990).

2012 (2)

2011 (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,” Nat. 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 B 405, 2990–2995 (2010).
[CrossRef]

S. H. Nam, A. J. Taylor, and A. Efimov, “Diabolical point and conical-like diffraction in periodic plasmonic nanostructures,” Opt. Express 18, 10120–10126 (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. B 80, 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]

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]

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 (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]

A. Alu, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B 75, 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]

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. B 75, 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. 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]

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,” Nat. 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]

Efimov, A.

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. B 75, 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]

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]

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,” Nat. Mater. 10, 582–586 (2011).
[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,” Nat. Mater. 10, 582–586 (2011).
[CrossRef]

Inui, T.

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

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, 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,” Nat. Mater. 10, 582–586 (2011).
[CrossRef]

Nam, S. H.

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, 1990).

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]

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.

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. B 75, 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. A 75, 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]

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. B 75, 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 B 405, 2990–2995 (2010).
[CrossRef]

Tanabe, Y.

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

Taylor, A. J.

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]

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,” Nat. Mater. 10, 582–586 (2011).
[CrossRef]

Nat. 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,” Nat. Mater. 10, 582–586 (2011).
[CrossRef]

Opt. Express (3)

Phys. Rev. A (2)

S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A 78, 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. A 75, 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. B 80, 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. B 75, 155410 (2007).
[CrossRef]

Phys. Rev. Lett. (3)

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]

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

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]

Other (2)

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

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

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Figures (7)

Fig. 1.
Fig. 1.

Normalized eigenfrequency (ωa/2πc) for the TE polarization on the Γ point of the photonic crystal composed of a regular triangular array of circular air cylinders in a medium with a dielectric constant of 12.6 (GaAs). The horizontal axis is the normalized radius of the air cylinders (ρ/a). The degeneracy points denoted by a to e correspond to the five panels in Fig. 2.

Fig. 2.
Fig. 2.

Dispersion curves of the triangular-lattice photonic crystal for TE polarization with a normalized radius of (a) 0.407, (b) 0.433, (c) 0.459, (d) 0.474, and (e) 0.492. (f) Brillouin zone of the triangular lattice. The frequencies of accidental degeneracy are circled in (a)–(e). M/10 and K/10 in (e) mean that the horizontal axis is magnified by 10 times.

Fig. 3.
Fig. 3.

The second to fifth lowest normalized eigenfrequency (ωa/2πc) for the TM polarization on the Γ point of the photonic crystal composed of a regular triangular array of circular air cylinders in a dielectric medium with ε=12.6 (GaAs). The horizontal axis is the normalized radius of the air cylinders (ρ/a). The degeneracy points denoted by a and b correspond to the two panels in Fig. 4.

Fig. 4.
Fig. 4.

Dispersion curves of the triangular-lattice photonic crystal for TM polarization with a normalized radius of (a) 0.445, (b) 0.483. The frequencies of accidental degeneracy are circled.

Fig. 5.
Fig. 5.

(a) Brillouin zone of the square lattice of C4v symmetry. Dispersion curves of the square-lattice photonic crystal (b) for TE polarization with a normalized radius of 0.299 and for TM polarization with a normalized radius of (c) 0.421 and (d) 0.464. The frequencies of accidental degeneracy are circled. M/2 and X/2 in (c) mean that the horizontal axis is magnified by two times.

Fig. 6.
Fig. 6.

Numbering of the lattice points of the triangular lattice and symmetry operations of the C6v point group.

Fig. 7.
Fig. 7.

Numbering of the lattice points of the square lattice and symmetry operations of the C4v point group.

Tables (2)

Tables Icon

Table 1. Types of Dispersion Curves Generated by Accidental Degeneracy of Two Modes (Mode 1 and Mode 2) for Triangular-Lattice Metamaterials of C6v Symmetrya

Tables Icon

Table 2. Types of Dispersion Curves Generated by Accidental Degeneracy of Two Modes (Mode 1 and Mode 2) for Square-Lattice Metamaterials of C4v Symmetrya

Equations (71)

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Lm(ij)1VVdrH(i)*(r)·LH(j)(rrm),
LHk×[1ε(r)×Hk]=ωk2c2Hk,
χ(H)(R)=(detR)χ(E)(R).
LH×[1ε(r)×H]=1c22t2H,
LsH(i)(r)×[1εs×H(i)(r)]=ωi2c2H(i)(r),
VdrH(i)*(r)·H(j)(r)=Vδij,
[RH](r)RH(R1r),
Lm(ij)=1VVdr[RH(i)*](r)·L[RH(j)](rRrm),
LHk(r)=ωk2c2Hk(r),
|Bωk2c2I|=0,
Bij=meik·rmLm(ij).
L0(11)M10,L0(22)M20,
L0(12)=L0(21)=0,
L1(11)=L2(11)=L3(11)=L4(11)=L5(11)=L6(11),M11,
L1(22)=L2(22)=L3(22)=L4(22)=L5(22)=L6(22),M21,
L1(12)=L3(12)=L5(12)=L2(12)=L4(12)=L6(12)M1.
Lm(ij)=Lm(ji)*,
B11=ξΓ(1)+M11[2(coskxa1)+4(coskxa2cos3kya21)],
B22=ξΓ(2)+M21[2(coskxa1)+4(coskxa2cos3kya21)],
B12=2iM1(sinkxa2sinkxa2cos3kya2),
ξΓ(1)M10+6M11,
ξΓ(2)M20+6M21,
ω{ωΓ3a2c2k2M112ωΓ,ωΓ3a2c2k2M212ωΓ,
Lm(12)=0(m=0,1,,6)
L0(00)M01,L0(11)L0(22)=M11,
L1(00)=L2(00)=L3(00)=L4(00)=L5(00)=L6(00)M02,
L1(11)=L4(11)M12,L1(22)=L4(22)M13,
L2(11)=L3(11)=L5(11)=L6(11)M14,
L2(22)=L3(22)=L5(22)=L6(22)M15,
L0(12)=L1(12)=L4(12)=0,
L2(12)=L3(12)=L5(12)=L6(12)M16,
L0(01)=L0(02)=L1(02)=L4(02)=0,
L1(01)=L4(01)M1,
L2(01)=L3(01)=L5(01)=L6(01)M2,
L2(02)=L3(02)=L5(02)=L6(02)M3.
M14=M12+3M134,
M15=3M12+M134,
M16=3(M12M13)4,
M1=2M2,M3=3M2.
ω{ωΓ±3akc2|M2|ωΓa2k2M4ωΓ,ωΓa2k2M4ωΓ,
M3=3M2.
L0(00)M01,
L1(00)=L2(00)=L3(00)=L4(00)=L5(00)=L6(00)M02,
L0(01)=L0(02)=L1(01)=L4(01)=0,
L2(01)=L3(01)=L5(01)=L6(01)M1,
L1(02)=L4(02)M2,
L2(02)=L3(02)=L5(02)=L6(02)M3.
M1=3M3,M2=2M3.
ω{ωΓ[M±3(|M1|2+|M16|2sin22ϕ)]a2c2k24ωΓ,ωΓa2c2k2M4ωΓ,
M1=3M3.
L0(11)M10,L0(22)M20,L0(12)=0,
L1(11)=L2(11)=L3(11)=L4(11)M11,
L1(22)=L2(22)=L3(22)=L4(22)M22,
L1(12)=L2(12)=L3(12)=L4(12)M1.
ωωΓa2c2k24ωΓ[M11+M21±(M11M21)2+4|M1|2cosϕ],
L0(12)=L1(12)=L2(12)=L3(12)=L4(12)=0.
L0(00)M01,L0(11)L0(22)=M11,
L1(00)=L2(00)=L3(00)=L4(00)M02,
L1(11)=L3(11)=L2(22)=L4(22)M12,
L2(11)=L4(11)=L1(22)=L3(22)M13,
L0(12)=L1(12)=L2(12)=L3(12)=L4(12)=0,
L0(01)=L0(02)=0,
L1(01)=L3(01)=L2(02)=L4(02)M1,
L1(02)=L3(02)=L2(01)=L4(01)=0.
B00=M01+2M02(coskxa+coskya),
B11=M11+2M12coskxa+2M13coskya,
B22=M11+2M13coskxa+2M12coskya,
B01=2iM1sinkxa,
B02=2iM1sinkya,
B12=0.
L1(01)=L3(01)=L2(02)=L4(02)M1.

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