Abstract

We propose a topological description for gaps of one-dimensional symmetric all-dielectric photonic crystals (PCs). It is shown that, in the propagating direction, the effective electromagnetic parameters of PCs can be derived from one unit cell with mirror symmetry. Besides, at the frequencies of gaps, these symmetric PCs can be described as photonic insulators with effective negative permittivity or negative permeability. Moreover, based on the mapping of Maxwell's equations to the Dirac equation and the band inversion achieved by tuning the material and structural parameters, we demonstrate that the gaps of PCs with effective negative permittivity or negative permeability possess different topological orders. Lastly, we show that a bound state is robust against the disorder under a zero-average-effective-mass condition in a heterostructure made of two PCs with different topological orders.

© 2016 Optical Society of America

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References

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2015 (1)

C. Poli, M. Bellec, U. Kuhl, F. Mortessagne, and H. Schomerus, “Selective enhancement of topologically induced interface states in a dielectric resonator chain,” Nat. Commun. 6, 6710 (2015).
[Crossref] [PubMed]

2014 (4)

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8(11), 821–829 (2014).
[Crossref]

W. Tan, Y. Sun, H. Chen, and S. Q. Shen, “Photonic simulation of topological excitations in metamaterials,” Sci. Rep. 4, 3842 (2014).
[Crossref] [PubMed]

M. Xiao, Z. Q. Zhang, and C. T. Chan, “Surface Impedance and Bulk Band Geometric Phases in One-Dimensional Systems,” Phys. Rev. X 4(2), 021017 (2014).
[Crossref]

M. Xiao, G. C. Ma, Z. Y. Yang, P. Shen, Z. Q. Zhang, and C. T. Chan, “Geometric phase and band inversion in periodic acoustic systems,” Nat. Phys. 4(2), 021017 (2014).

2013 (1)

M. Atala, M. Aidelsburger, J. T. Barreiro, D. Abanin, T. Kitagawa, E. Demler, and I. Bloch, “Direct measurement of the Zak phase in topological Bloch bands,” Nat. Phys. 9(12), 795–800 (2013).
[Crossref]

2011 (1)

X. L. Qi and S. C. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys. 83(4), 1057–1110 (2011).
[Crossref]

2010 (1)

M. Z. Hasan and C. L. Kane, “Colloquium: Topological insulators,” Rev. Mod. Phys. 82(4), 3045–3067 (2010).
[Crossref]

2009 (1)

D. A. Genov, S. Zhang, and X. Zhang, “Mimicking celestial mechanics in metamaterials,” Nat. Phys. 5(9), 687–692 (2009).
[Crossref]

2008 (2)

J. Y. Guo, Y. Sun, H. Q. Li, Y. W. Zhang, and H. Chen, “Optical tamm states in dielectric photonic crystal heterostructure,” Chin. Phys. Lett. 25(6), 2093–2096 (2008).
[Crossref]

J. Guo, Y. Sun, Y. Zhang, H. Li, H. Jiang, and H. Chen, “Experimental investigation of interface states in photonic crystal heterostructures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78(2), 026607 (2008).
[Crossref] [PubMed]

2007 (1)

2006 (2)

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[Crossref] [PubMed]

U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006).
[Crossref] [PubMed]

2004 (1)

H. Jiang, H. Chen, H. Li, Y. Zhang, J. Zi, and S. Zhu, “Properties of one-dimensional photonic crystals containing single-negative materials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(6), 066607 (2004).
[Crossref] [PubMed]

2003 (1)

A. Alù and N. Engheta, “Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency,” IEEE. T. Antenn. Propag. 51(10), 2558–2571 (2003).
[Crossref]

2002 (1)

D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65(19), 195104 (2002).
[Crossref]

2001 (1)

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental Verification of a Negative Index of Refraction,” Science 292(5514), 77–79 (2001).
[Crossref] [PubMed]

2000 (2)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
[Crossref] [PubMed]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
[Crossref] [PubMed]

Abanin, D.

M. Atala, M. Aidelsburger, J. T. Barreiro, D. Abanin, T. Kitagawa, E. Demler, and I. Bloch, “Direct measurement of the Zak phase in topological Bloch bands,” Nat. Phys. 9(12), 795–800 (2013).
[Crossref]

Aidelsburger, M.

M. Atala, M. Aidelsburger, J. T. Barreiro, D. Abanin, T. Kitagawa, E. Demler, and I. Bloch, “Direct measurement of the Zak phase in topological Bloch bands,” Nat. Phys. 9(12), 795–800 (2013).
[Crossref]

Alù, A.

A. Alù and N. Engheta, “Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency,” IEEE. T. Antenn. Propag. 51(10), 2558–2571 (2003).
[Crossref]

Atala, M.

M. Atala, M. Aidelsburger, J. T. Barreiro, D. Abanin, T. Kitagawa, E. Demler, and I. Bloch, “Direct measurement of the Zak phase in topological Bloch bands,” Nat. Phys. 9(12), 795–800 (2013).
[Crossref]

Barreiro, J. T.

M. Atala, M. Aidelsburger, J. T. Barreiro, D. Abanin, T. Kitagawa, E. Demler, and I. Bloch, “Direct measurement of the Zak phase in topological Bloch bands,” Nat. Phys. 9(12), 795–800 (2013).
[Crossref]

Bellec, M.

C. Poli, M. Bellec, U. Kuhl, F. Mortessagne, and H. Schomerus, “Selective enhancement of topologically induced interface states in a dielectric resonator chain,” Nat. Commun. 6, 6710 (2015).
[Crossref] [PubMed]

Bloch, I.

M. Atala, M. Aidelsburger, J. T. Barreiro, D. Abanin, T. Kitagawa, E. Demler, and I. Bloch, “Direct measurement of the Zak phase in topological Bloch bands,” Nat. Phys. 9(12), 795–800 (2013).
[Crossref]

Chan, C. T.

M. Xiao, Z. Q. Zhang, and C. T. Chan, “Surface Impedance and Bulk Band Geometric Phases in One-Dimensional Systems,” Phys. Rev. X 4(2), 021017 (2014).
[Crossref]

M. Xiao, G. C. Ma, Z. Y. Yang, P. Shen, Z. Q. Zhang, and C. T. Chan, “Geometric phase and band inversion in periodic acoustic systems,” Nat. Phys. 4(2), 021017 (2014).

Chen, H.

W. Tan, Y. Sun, H. Chen, and S. Q. Shen, “Photonic simulation of topological excitations in metamaterials,” Sci. Rep. 4, 3842 (2014).
[Crossref] [PubMed]

J. Guo, Y. Sun, Y. Zhang, H. Li, H. Jiang, and H. Chen, “Experimental investigation of interface states in photonic crystal heterostructures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78(2), 026607 (2008).
[Crossref] [PubMed]

J. Y. Guo, Y. Sun, H. Q. Li, Y. W. Zhang, and H. Chen, “Optical tamm states in dielectric photonic crystal heterostructure,” Chin. Phys. Lett. 25(6), 2093–2096 (2008).
[Crossref]

H. T. Jiang, H. Chen, and S. Y. Zhu, “Rabi splitting with excitons in effective (near) zero-index media,” Opt. Lett. 32(14), 1980–1982 (2007).
[Crossref] [PubMed]

H. Jiang, H. Chen, H. Li, Y. Zhang, J. Zi, and S. Zhu, “Properties of one-dimensional photonic crystals containing single-negative materials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(6), 066607 (2004).
[Crossref] [PubMed]

Demler, E.

M. Atala, M. Aidelsburger, J. T. Barreiro, D. Abanin, T. Kitagawa, E. Demler, and I. Bloch, “Direct measurement of the Zak phase in topological Bloch bands,” Nat. Phys. 9(12), 795–800 (2013).
[Crossref]

Engheta, N.

A. Alù and N. Engheta, “Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency,” IEEE. T. Antenn. Propag. 51(10), 2558–2571 (2003).
[Crossref]

Genov, D. A.

D. A. Genov, S. Zhang, and X. Zhang, “Mimicking celestial mechanics in metamaterials,” Nat. Phys. 5(9), 687–692 (2009).
[Crossref]

Guo, J.

J. Guo, Y. Sun, Y. Zhang, H. Li, H. Jiang, and H. Chen, “Experimental investigation of interface states in photonic crystal heterostructures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78(2), 026607 (2008).
[Crossref] [PubMed]

Guo, J. Y.

J. Y. Guo, Y. Sun, H. Q. Li, Y. W. Zhang, and H. Chen, “Optical tamm states in dielectric photonic crystal heterostructure,” Chin. Phys. Lett. 25(6), 2093–2096 (2008).
[Crossref]

Hasan, M. Z.

M. Z. Hasan and C. L. Kane, “Colloquium: Topological insulators,” Rev. Mod. Phys. 82(4), 3045–3067 (2010).
[Crossref]

Jiang, H.

J. Guo, Y. Sun, Y. Zhang, H. Li, H. Jiang, and H. Chen, “Experimental investigation of interface states in photonic crystal heterostructures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78(2), 026607 (2008).
[Crossref] [PubMed]

H. Jiang, H. Chen, H. Li, Y. Zhang, J. Zi, and S. Zhu, “Properties of one-dimensional photonic crystals containing single-negative materials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(6), 066607 (2004).
[Crossref] [PubMed]

Jiang, H. T.

Joannopoulos, J. D.

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8(11), 821–829 (2014).
[Crossref]

Kane, C. L.

M. Z. Hasan and C. L. Kane, “Colloquium: Topological insulators,” Rev. Mod. Phys. 82(4), 3045–3067 (2010).
[Crossref]

Kitagawa, T.

M. Atala, M. Aidelsburger, J. T. Barreiro, D. Abanin, T. Kitagawa, E. Demler, and I. Bloch, “Direct measurement of the Zak phase in topological Bloch bands,” Nat. Phys. 9(12), 795–800 (2013).
[Crossref]

Kuhl, U.

C. Poli, M. Bellec, U. Kuhl, F. Mortessagne, and H. Schomerus, “Selective enhancement of topologically induced interface states in a dielectric resonator chain,” Nat. Commun. 6, 6710 (2015).
[Crossref] [PubMed]

Leonhardt, U.

U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006).
[Crossref] [PubMed]

Li, H.

J. Guo, Y. Sun, Y. Zhang, H. Li, H. Jiang, and H. Chen, “Experimental investigation of interface states in photonic crystal heterostructures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78(2), 026607 (2008).
[Crossref] [PubMed]

H. Jiang, H. Chen, H. Li, Y. Zhang, J. Zi, and S. Zhu, “Properties of one-dimensional photonic crystals containing single-negative materials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(6), 066607 (2004).
[Crossref] [PubMed]

Li, H. Q.

J. Y. Guo, Y. Sun, H. Q. Li, Y. W. Zhang, and H. Chen, “Optical tamm states in dielectric photonic crystal heterostructure,” Chin. Phys. Lett. 25(6), 2093–2096 (2008).
[Crossref]

Lu, L.

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8(11), 821–829 (2014).
[Crossref]

Ma, G. C.

M. Xiao, G. C. Ma, Z. Y. Yang, P. Shen, Z. Q. Zhang, and C. T. Chan, “Geometric phase and band inversion in periodic acoustic systems,” Nat. Phys. 4(2), 021017 (2014).

Markoš, P.

D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65(19), 195104 (2002).
[Crossref]

Mortessagne, F.

C. Poli, M. Bellec, U. Kuhl, F. Mortessagne, and H. Schomerus, “Selective enhancement of topologically induced interface states in a dielectric resonator chain,” Nat. Commun. 6, 6710 (2015).
[Crossref] [PubMed]

Nemat-Nasser, S. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
[Crossref] [PubMed]

Padilla, W. J.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
[Crossref] [PubMed]

Pendry, J. B.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[Crossref] [PubMed]

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
[Crossref] [PubMed]

Poli, C.

C. Poli, M. Bellec, U. Kuhl, F. Mortessagne, and H. Schomerus, “Selective enhancement of topologically induced interface states in a dielectric resonator chain,” Nat. Commun. 6, 6710 (2015).
[Crossref] [PubMed]

Qi, X. L.

X. L. Qi and S. C. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys. 83(4), 1057–1110 (2011).
[Crossref]

Schomerus, H.

C. Poli, M. Bellec, U. Kuhl, F. Mortessagne, and H. Schomerus, “Selective enhancement of topologically induced interface states in a dielectric resonator chain,” Nat. Commun. 6, 6710 (2015).
[Crossref] [PubMed]

Schultz, S.

D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65(19), 195104 (2002).
[Crossref]

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental Verification of a Negative Index of Refraction,” Science 292(5514), 77–79 (2001).
[Crossref] [PubMed]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
[Crossref] [PubMed]

Schurig, D.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[Crossref] [PubMed]

Shelby, R. A.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental Verification of a Negative Index of Refraction,” Science 292(5514), 77–79 (2001).
[Crossref] [PubMed]

Shen, P.

M. Xiao, G. C. Ma, Z. Y. Yang, P. Shen, Z. Q. Zhang, and C. T. Chan, “Geometric phase and band inversion in periodic acoustic systems,” Nat. Phys. 4(2), 021017 (2014).

Shen, S. Q.

W. Tan, Y. Sun, H. Chen, and S. Q. Shen, “Photonic simulation of topological excitations in metamaterials,” Sci. Rep. 4, 3842 (2014).
[Crossref] [PubMed]

Smith, D. R.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[Crossref] [PubMed]

D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65(19), 195104 (2002).
[Crossref]

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental Verification of a Negative Index of Refraction,” Science 292(5514), 77–79 (2001).
[Crossref] [PubMed]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
[Crossref] [PubMed]

Soljacic, M.

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8(11), 821–829 (2014).
[Crossref]

Soukoulis, C. M.

D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65(19), 195104 (2002).
[Crossref]

Sun, Y.

W. Tan, Y. Sun, H. Chen, and S. Q. Shen, “Photonic simulation of topological excitations in metamaterials,” Sci. Rep. 4, 3842 (2014).
[Crossref] [PubMed]

J. Y. Guo, Y. Sun, H. Q. Li, Y. W. Zhang, and H. Chen, “Optical tamm states in dielectric photonic crystal heterostructure,” Chin. Phys. Lett. 25(6), 2093–2096 (2008).
[Crossref]

J. Guo, Y. Sun, Y. Zhang, H. Li, H. Jiang, and H. Chen, “Experimental investigation of interface states in photonic crystal heterostructures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78(2), 026607 (2008).
[Crossref] [PubMed]

Tan, W.

W. Tan, Y. Sun, H. Chen, and S. Q. Shen, “Photonic simulation of topological excitations in metamaterials,” Sci. Rep. 4, 3842 (2014).
[Crossref] [PubMed]

Vier, D. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
[Crossref] [PubMed]

Xiao, M.

M. Xiao, Z. Q. Zhang, and C. T. Chan, “Surface Impedance and Bulk Band Geometric Phases in One-Dimensional Systems,” Phys. Rev. X 4(2), 021017 (2014).
[Crossref]

M. Xiao, G. C. Ma, Z. Y. Yang, P. Shen, Z. Q. Zhang, and C. T. Chan, “Geometric phase and band inversion in periodic acoustic systems,” Nat. Phys. 4(2), 021017 (2014).

Yang, Z. Y.

M. Xiao, G. C. Ma, Z. Y. Yang, P. Shen, Z. Q. Zhang, and C. T. Chan, “Geometric phase and band inversion in periodic acoustic systems,” Nat. Phys. 4(2), 021017 (2014).

Zhang, S.

D. A. Genov, S. Zhang, and X. Zhang, “Mimicking celestial mechanics in metamaterials,” Nat. Phys. 5(9), 687–692 (2009).
[Crossref]

Zhang, S. C.

X. L. Qi and S. C. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys. 83(4), 1057–1110 (2011).
[Crossref]

Zhang, X.

D. A. Genov, S. Zhang, and X. Zhang, “Mimicking celestial mechanics in metamaterials,” Nat. Phys. 5(9), 687–692 (2009).
[Crossref]

Zhang, Y.

J. Guo, Y. Sun, Y. Zhang, H. Li, H. Jiang, and H. Chen, “Experimental investigation of interface states in photonic crystal heterostructures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78(2), 026607 (2008).
[Crossref] [PubMed]

H. Jiang, H. Chen, H. Li, Y. Zhang, J. Zi, and S. Zhu, “Properties of one-dimensional photonic crystals containing single-negative materials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(6), 066607 (2004).
[Crossref] [PubMed]

Zhang, Y. W.

J. Y. Guo, Y. Sun, H. Q. Li, Y. W. Zhang, and H. Chen, “Optical tamm states in dielectric photonic crystal heterostructure,” Chin. Phys. Lett. 25(6), 2093–2096 (2008).
[Crossref]

Zhang, Z. Q.

M. Xiao, G. C. Ma, Z. Y. Yang, P. Shen, Z. Q. Zhang, and C. T. Chan, “Geometric phase and band inversion in periodic acoustic systems,” Nat. Phys. 4(2), 021017 (2014).

M. Xiao, Z. Q. Zhang, and C. T. Chan, “Surface Impedance and Bulk Band Geometric Phases in One-Dimensional Systems,” Phys. Rev. X 4(2), 021017 (2014).
[Crossref]

Zhu, S.

H. Jiang, H. Chen, H. Li, Y. Zhang, J. Zi, and S. Zhu, “Properties of one-dimensional photonic crystals containing single-negative materials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(6), 066607 (2004).
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Figures (6)

Fig. 1
Fig. 1 Schematic of a symmetric unit cell A B B A . The black dashed line marks the center of the structure.
Fig. 2
Fig. 2 (a) The band structure of the infinite-periodic PC ( A B B A ) N in which ε a = 1.8 , ε b = 1 , μ a = μ b = 1 , d a = 0.4 Λ and d b = 0.6 Λ . Γ is a real number and denotes the Bloch phase. The gray regions indicate the open gaps. One gap is closed at 2.5. The serial numbers of gaps are listed at the right label. (b) ε e f f and μ e f f of the structure after homogenization for frequencies within the gaps. (c) Reflection phase spectra of the finite-periodic PC ( A B B A ) 8 and the homogenized material with same thicknesses, respectively. (d) Trace( M A B B A ) versus frequency.
Fig. 3
Fig. 3 The electric field distributions in the HMM and in the PC ( A B B A ) 8 . The interfaces of symmetric unit cells are denoted by the white lines. The normalized frequencies are (a) f = 0.32 within the 1st gap and (b) f = 2.16 within the 6th gap, respectively.
Fig. 4
Fig. 4 (a) The ε e f f and μ e f f as a function of the permittivity of dielectric A and frequency. The parameters of PC are given in Fig. 2. (b) The ε e f f and μ e f f as a function of the filling fraction of dielectric A and frequency. Pale green plane is drawn as the reference plane. The black curves in plane correspond to the band-edges. When the permittivity (the filling fraction) of dielectric A is 4 or (0.4), the gap is closed. The closed point at the normalized frequency 2.5 is a topological transition point.
Fig. 5
Fig. 5 (a) and (b) represent the band structures (black curve) of two types of PCs. The cyan strip represents the MNG gap and the magenta strip represents the ENG gap. (c) ε e f f μ e f f of A B B A and C D D C within the gaps.
Fig. 6
Fig. 6 (a) A kind of randomness configuration introduced in the MNG PI. The m ¯ = 0 condition is satisfied for the heterostructure. (b) Transmittance spectra of the heterostructures with disorder under the m ¯ = 0 condition (the red solid line) and without disorder (the scattered open circles), respectively. (c) Transmittance spectra of the heterostructures with disorders under the m ¯ = 0 condition (the red solid line) and under the m ¯ 0 condition (the short dashed line), respectively. A magnification around 2.028 is given for a better view.

Equations (12)

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M = [ cos δ j i Z j sin δ j i Z j sin δ j cos δ j ] ,
M A B B A = M A M B M B M A = [ M 11 M 12 M 21 M 22 ] ,
M 1 1 = M 2 2 = cos δ a cos δ b 1 2 p A B + sin δ a sin δ b ,
M 12 = i Z a [ sin δ a cos δ b + 1 2 p A B + cos δ a sin δ b + 1 2 p A B sin δ b ] ,
M 21 = i Z a [ sin δ a cos δ b + 1 2 p A B + cos δ a sin δ b 1 2 p A B sin δ b ] ,
M A B B A = d e f [ cos Γ i Z e f f sin Γ i Z e f f sin Γ cos Γ ] ,
Trace( M A B B A ) = 2 cos Γ .
M A B B A = ( 1 ) m [ cos i ξ i Z e f f sin i ξ i Z e f f sin i ξ cos i ξ ] = [ M 11 M 12 M 21 M 22 ] .
M homog = [ cos i ξ i Z e f f sin i ξ i Z e f f sin i ξ cos i ξ ] = ( 1 ) m [ M 11 M 12 M 21 M 22 ] .
r = M 11 M 12 + M 21 M 22 M 11 M 12 M 21 + M 22 .
x E z = i ω μ 0 μ r ( x ) H y , x H y = i ω ε 0 ε r ( x ) E z .
[ i σ x x + m ( x ) σ z + V ( x ) ] φ = E φ ,

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