Abstract

A mathematical approach is presented that is based on coupled-mode theory (CMT); it is extended to infinite perfect photonic structures and combined with the supercell method for analysis of infinite photonic crystals with introduced point (0D) and linear (1D) defects. This approach shows a strong advantage over most existing techniques in regards to time and consumption of resources, and thus allows one to quickly analyze operational characteristics of different photonic devices [photonic crystal fibers (PCFs), phased arrays of VCSELs, etc.] over a wide range of physical parameters.

© 2009 Optical Society of America

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References

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  1. A. Snyder, “Coupled-mode theory for optical fibers,” J. Opt. Soc. Am. 62, 1267-1277 (1972).
    [CrossRef]
  2. V. Shteeman, D. Boiko, E. Kapon, and A. A. Hardy, “Extension of coupled mode analysis to periodic large arrays of identical waveguides for photonic crystals applications,” IEEE J. Quantum Electron. 43, 215-224 (2007).
    [CrossRef]
  3. V. Shteeman, I. Nusinsly, E. Kapon, and A. A. Hardy, “Extension of coupled mode analysis to infinite photonic superlattices,” IEEE J. Quantum Electron. 44, 826-833 (2008).
    [CrossRef]
  4. A. A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. LT-3, 1135-1146 (1985).
    [CrossRef]
  5. A. A. Hardy and W. Streifer. “Coupled modes of multiwaveguide systems and phased arrays,” J. Lightwave Technol. LT-4, 90-99 (1986).
    [CrossRef]
  6. A. A. Hardy and E. Kapon, “Coupled-mode formulations for parallel-laser resonators with application to vertical-cavity semiconductor-laser arrays,” IEEE J. Quantum Electron. 32, 996-971 (1996).
    [CrossRef]
  7. M. Bayer, T. Gutbrod, A. Forchel, T. Reinecke, P. Knipp, R. Werner, and J. Reithmaier, “Optical demonstration of a crystal band structure formation,” Phys. Rev. Lett. 83, 5374-5377 (1999).
    [CrossRef]
  8. L. Lundeberg, D. L. Boiko, and E. Kapon, “Coupled islands of photonic crystal heterostructures implemented with vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 87, 241120 (2005).
    [CrossRef]
  9. F. Luan, A. George, T. Hedley, G. Pearce, D. Bird, J.C. Knight, and P. St. J. Russell, “All-solid photonic bandgap fiber,” Opt. Lett. 29, 2369-2371 (2004).
    [CrossRef] [PubMed]
  10. J. C.Knight, J. Broeng, T. Birks, and P. St. J. Russell, “Photonic band gap guidance in optical fibers,” Science 282, 1476-1478 (1998).
    [CrossRef] [PubMed]
  11. A. Millo, L. Lobachinsky, and A. Katzir, “Single-mode octagonal photonic crystal fibers for the middle infrared,” Appl. Phys. Lett. 92, 021112 (2008).
    [CrossRef]
  12. K. Sakoda, Optical Properties of Photonic Crystals, (Springer, 2001), pp. 123-142.
  13. J. Joannopoulos, R. Meade, and J. Winn, Photonic Crystals. Molding the Flow of Light (Princeton U. Press, 1995), pp. 54-76.
  14. J. C. Slater and G. F. Koster, “Simplified LCAO method for the periodic potential problem,” Phys. Rev. 94, 1498-1524 (1954).
    [CrossRef]
  15. M. Punkkinen, T. Korhonen, K. Kokko, and I. Väyrynen, “The electronic band structure of Si/SiO2-superlattices: A first-principles study,” Phys. Status Solidi B 214, r17-r18 (1999).
    [CrossRef]
  16. R. Grimes, A. Catlow, and A. Shluger, Quantum Mechanical Cluster Calculations in Solid State Studies (World Scientific, 1992), pp. 168-171.
  17. R. Evarestov, Quantum Chemistry of Solids (Springer, 2007), pp. 105-140.
  18. P. Hertel, Lectures on Theoretical Physics. Dielectric Waveguides (U. Osnabrueck, 2005), pp. 16-19.
  19. L. Landau and E. Lifschitz, Quantum Mechanics: Non-relativistic Theory (Butterworth Heinemann, 2003), pp. 177-182.

2008

V. Shteeman, I. Nusinsly, E. Kapon, and A. A. Hardy, “Extension of coupled mode analysis to infinite photonic superlattices,” IEEE J. Quantum Electron. 44, 826-833 (2008).
[CrossRef]

A. Millo, L. Lobachinsky, and A. Katzir, “Single-mode octagonal photonic crystal fibers for the middle infrared,” Appl. Phys. Lett. 92, 021112 (2008).
[CrossRef]

2007

V. Shteeman, D. Boiko, E. Kapon, and A. A. Hardy, “Extension of coupled mode analysis to periodic large arrays of identical waveguides for photonic crystals applications,” IEEE J. Quantum Electron. 43, 215-224 (2007).
[CrossRef]

2005

L. Lundeberg, D. L. Boiko, and E. Kapon, “Coupled islands of photonic crystal heterostructures implemented with vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 87, 241120 (2005).
[CrossRef]

2004

1999

M. Punkkinen, T. Korhonen, K. Kokko, and I. Väyrynen, “The electronic band structure of Si/SiO2-superlattices: A first-principles study,” Phys. Status Solidi B 214, r17-r18 (1999).
[CrossRef]

M. Bayer, T. Gutbrod, A. Forchel, T. Reinecke, P. Knipp, R. Werner, and J. Reithmaier, “Optical demonstration of a crystal band structure formation,” Phys. Rev. Lett. 83, 5374-5377 (1999).
[CrossRef]

1998

J. C.Knight, J. Broeng, T. Birks, and P. St. J. Russell, “Photonic band gap guidance in optical fibers,” Science 282, 1476-1478 (1998).
[CrossRef] [PubMed]

1996

A. A. Hardy and E. Kapon, “Coupled-mode formulations for parallel-laser resonators with application to vertical-cavity semiconductor-laser arrays,” IEEE J. Quantum Electron. 32, 996-971 (1996).
[CrossRef]

1986

A. A. Hardy and W. Streifer. “Coupled modes of multiwaveguide systems and phased arrays,” J. Lightwave Technol. LT-4, 90-99 (1986).
[CrossRef]

1985

A. A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. LT-3, 1135-1146 (1985).
[CrossRef]

1972

1954

J. C. Slater and G. F. Koster, “Simplified LCAO method for the periodic potential problem,” Phys. Rev. 94, 1498-1524 (1954).
[CrossRef]

Bayer, M.

M. Bayer, T. Gutbrod, A. Forchel, T. Reinecke, P. Knipp, R. Werner, and J. Reithmaier, “Optical demonstration of a crystal band structure formation,” Phys. Rev. Lett. 83, 5374-5377 (1999).
[CrossRef]

Bird, D.

Birks, T.

J. C.Knight, J. Broeng, T. Birks, and P. St. J. Russell, “Photonic band gap guidance in optical fibers,” Science 282, 1476-1478 (1998).
[CrossRef] [PubMed]

Boiko, D.

V. Shteeman, D. Boiko, E. Kapon, and A. A. Hardy, “Extension of coupled mode analysis to periodic large arrays of identical waveguides for photonic crystals applications,” IEEE J. Quantum Electron. 43, 215-224 (2007).
[CrossRef]

Boiko, D. L.

L. Lundeberg, D. L. Boiko, and E. Kapon, “Coupled islands of photonic crystal heterostructures implemented with vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 87, 241120 (2005).
[CrossRef]

Broeng, J.

J. C.Knight, J. Broeng, T. Birks, and P. St. J. Russell, “Photonic band gap guidance in optical fibers,” Science 282, 1476-1478 (1998).
[CrossRef] [PubMed]

Catlow, A.

R. Grimes, A. Catlow, and A. Shluger, Quantum Mechanical Cluster Calculations in Solid State Studies (World Scientific, 1992), pp. 168-171.

Evarestov, R.

R. Evarestov, Quantum Chemistry of Solids (Springer, 2007), pp. 105-140.

Forchel, A.

M. Bayer, T. Gutbrod, A. Forchel, T. Reinecke, P. Knipp, R. Werner, and J. Reithmaier, “Optical demonstration of a crystal band structure formation,” Phys. Rev. Lett. 83, 5374-5377 (1999).
[CrossRef]

George, A.

Grimes, R.

R. Grimes, A. Catlow, and A. Shluger, Quantum Mechanical Cluster Calculations in Solid State Studies (World Scientific, 1992), pp. 168-171.

Gutbrod, T.

M. Bayer, T. Gutbrod, A. Forchel, T. Reinecke, P. Knipp, R. Werner, and J. Reithmaier, “Optical demonstration of a crystal band structure formation,” Phys. Rev. Lett. 83, 5374-5377 (1999).
[CrossRef]

Hardy, A. A.

V. Shteeman, I. Nusinsly, E. Kapon, and A. A. Hardy, “Extension of coupled mode analysis to infinite photonic superlattices,” IEEE J. Quantum Electron. 44, 826-833 (2008).
[CrossRef]

V. Shteeman, D. Boiko, E. Kapon, and A. A. Hardy, “Extension of coupled mode analysis to periodic large arrays of identical waveguides for photonic crystals applications,” IEEE J. Quantum Electron. 43, 215-224 (2007).
[CrossRef]

A. A. Hardy and E. Kapon, “Coupled-mode formulations for parallel-laser resonators with application to vertical-cavity semiconductor-laser arrays,” IEEE J. Quantum Electron. 32, 996-971 (1996).
[CrossRef]

A. A. Hardy and W. Streifer. “Coupled modes of multiwaveguide systems and phased arrays,” J. Lightwave Technol. LT-4, 90-99 (1986).
[CrossRef]

A. A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. LT-3, 1135-1146 (1985).
[CrossRef]

Hedley, T.

Hertel, P.

P. Hertel, Lectures on Theoretical Physics. Dielectric Waveguides (U. Osnabrueck, 2005), pp. 16-19.

Joannopoulos, J.

J. Joannopoulos, R. Meade, and J. Winn, Photonic Crystals. Molding the Flow of Light (Princeton U. Press, 1995), pp. 54-76.

Kapon, E.

V. Shteeman, I. Nusinsly, E. Kapon, and A. A. Hardy, “Extension of coupled mode analysis to infinite photonic superlattices,” IEEE J. Quantum Electron. 44, 826-833 (2008).
[CrossRef]

V. Shteeman, D. Boiko, E. Kapon, and A. A. Hardy, “Extension of coupled mode analysis to periodic large arrays of identical waveguides for photonic crystals applications,” IEEE J. Quantum Electron. 43, 215-224 (2007).
[CrossRef]

L. Lundeberg, D. L. Boiko, and E. Kapon, “Coupled islands of photonic crystal heterostructures implemented with vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 87, 241120 (2005).
[CrossRef]

A. A. Hardy and E. Kapon, “Coupled-mode formulations for parallel-laser resonators with application to vertical-cavity semiconductor-laser arrays,” IEEE J. Quantum Electron. 32, 996-971 (1996).
[CrossRef]

Katzir, A.

A. Millo, L. Lobachinsky, and A. Katzir, “Single-mode octagonal photonic crystal fibers for the middle infrared,” Appl. Phys. Lett. 92, 021112 (2008).
[CrossRef]

Knight, J. C.

F. Luan, A. George, T. Hedley, G. Pearce, D. Bird, J.C. Knight, and P. St. J. Russell, “All-solid photonic bandgap fiber,” Opt. Lett. 29, 2369-2371 (2004).
[CrossRef] [PubMed]

J. C.Knight, J. Broeng, T. Birks, and P. St. J. Russell, “Photonic band gap guidance in optical fibers,” Science 282, 1476-1478 (1998).
[CrossRef] [PubMed]

Knipp, P.

M. Bayer, T. Gutbrod, A. Forchel, T. Reinecke, P. Knipp, R. Werner, and J. Reithmaier, “Optical demonstration of a crystal band structure formation,” Phys. Rev. Lett. 83, 5374-5377 (1999).
[CrossRef]

Kokko, K.

M. Punkkinen, T. Korhonen, K. Kokko, and I. Väyrynen, “The electronic band structure of Si/SiO2-superlattices: A first-principles study,” Phys. Status Solidi B 214, r17-r18 (1999).
[CrossRef]

Korhonen, T.

M. Punkkinen, T. Korhonen, K. Kokko, and I. Väyrynen, “The electronic band structure of Si/SiO2-superlattices: A first-principles study,” Phys. Status Solidi B 214, r17-r18 (1999).
[CrossRef]

Koster, G. F.

J. C. Slater and G. F. Koster, “Simplified LCAO method for the periodic potential problem,” Phys. Rev. 94, 1498-1524 (1954).
[CrossRef]

Landau, L.

L. Landau and E. Lifschitz, Quantum Mechanics: Non-relativistic Theory (Butterworth Heinemann, 2003), pp. 177-182.

Lifschitz, E.

L. Landau and E. Lifschitz, Quantum Mechanics: Non-relativistic Theory (Butterworth Heinemann, 2003), pp. 177-182.

Lobachinsky, L.

A. Millo, L. Lobachinsky, and A. Katzir, “Single-mode octagonal photonic crystal fibers for the middle infrared,” Appl. Phys. Lett. 92, 021112 (2008).
[CrossRef]

Luan, F.

Lundeberg, L.

L. Lundeberg, D. L. Boiko, and E. Kapon, “Coupled islands of photonic crystal heterostructures implemented with vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 87, 241120 (2005).
[CrossRef]

Meade, R.

J. Joannopoulos, R. Meade, and J. Winn, Photonic Crystals. Molding the Flow of Light (Princeton U. Press, 1995), pp. 54-76.

Millo, A.

A. Millo, L. Lobachinsky, and A. Katzir, “Single-mode octagonal photonic crystal fibers for the middle infrared,” Appl. Phys. Lett. 92, 021112 (2008).
[CrossRef]

Nusinsly, I.

V. Shteeman, I. Nusinsly, E. Kapon, and A. A. Hardy, “Extension of coupled mode analysis to infinite photonic superlattices,” IEEE J. Quantum Electron. 44, 826-833 (2008).
[CrossRef]

Pearce, G.

Punkkinen, M.

M. Punkkinen, T. Korhonen, K. Kokko, and I. Väyrynen, “The electronic band structure of Si/SiO2-superlattices: A first-principles study,” Phys. Status Solidi B 214, r17-r18 (1999).
[CrossRef]

Reinecke, T.

M. Bayer, T. Gutbrod, A. Forchel, T. Reinecke, P. Knipp, R. Werner, and J. Reithmaier, “Optical demonstration of a crystal band structure formation,” Phys. Rev. Lett. 83, 5374-5377 (1999).
[CrossRef]

Reithmaier, J.

M. Bayer, T. Gutbrod, A. Forchel, T. Reinecke, P. Knipp, R. Werner, and J. Reithmaier, “Optical demonstration of a crystal band structure formation,” Phys. Rev. Lett. 83, 5374-5377 (1999).
[CrossRef]

Sakoda, K.

K. Sakoda, Optical Properties of Photonic Crystals, (Springer, 2001), pp. 123-142.

Shluger, A.

R. Grimes, A. Catlow, and A. Shluger, Quantum Mechanical Cluster Calculations in Solid State Studies (World Scientific, 1992), pp. 168-171.

Shteeman, V.

V. Shteeman, I. Nusinsly, E. Kapon, and A. A. Hardy, “Extension of coupled mode analysis to infinite photonic superlattices,” IEEE J. Quantum Electron. 44, 826-833 (2008).
[CrossRef]

V. Shteeman, D. Boiko, E. Kapon, and A. A. Hardy, “Extension of coupled mode analysis to periodic large arrays of identical waveguides for photonic crystals applications,” IEEE J. Quantum Electron. 43, 215-224 (2007).
[CrossRef]

Slater, J. C.

J. C. Slater and G. F. Koster, “Simplified LCAO method for the periodic potential problem,” Phys. Rev. 94, 1498-1524 (1954).
[CrossRef]

Snyder, A.

St. J. Russell, P.

F. Luan, A. George, T. Hedley, G. Pearce, D. Bird, J.C. Knight, and P. St. J. Russell, “All-solid photonic bandgap fiber,” Opt. Lett. 29, 2369-2371 (2004).
[CrossRef] [PubMed]

J. C.Knight, J. Broeng, T. Birks, and P. St. J. Russell, “Photonic band gap guidance in optical fibers,” Science 282, 1476-1478 (1998).
[CrossRef] [PubMed]

Streifer, W.

A. A. Hardy and W. Streifer. “Coupled modes of multiwaveguide systems and phased arrays,” J. Lightwave Technol. LT-4, 90-99 (1986).
[CrossRef]

A. A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. LT-3, 1135-1146 (1985).
[CrossRef]

Väyrynen, I.

M. Punkkinen, T. Korhonen, K. Kokko, and I. Väyrynen, “The electronic band structure of Si/SiO2-superlattices: A first-principles study,” Phys. Status Solidi B 214, r17-r18 (1999).
[CrossRef]

Werner, R.

M. Bayer, T. Gutbrod, A. Forchel, T. Reinecke, P. Knipp, R. Werner, and J. Reithmaier, “Optical demonstration of a crystal band structure formation,” Phys. Rev. Lett. 83, 5374-5377 (1999).
[CrossRef]

Winn, J.

J. Joannopoulos, R. Meade, and J. Winn, Photonic Crystals. Molding the Flow of Light (Princeton U. Press, 1995), pp. 54-76.

Appl. Phys. Lett.

L. Lundeberg, D. L. Boiko, and E. Kapon, “Coupled islands of photonic crystal heterostructures implemented with vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 87, 241120 (2005).
[CrossRef]

A. Millo, L. Lobachinsky, and A. Katzir, “Single-mode octagonal photonic crystal fibers for the middle infrared,” Appl. Phys. Lett. 92, 021112 (2008).
[CrossRef]

IEEE J. Quantum Electron.

A. A. Hardy and E. Kapon, “Coupled-mode formulations for parallel-laser resonators with application to vertical-cavity semiconductor-laser arrays,” IEEE J. Quantum Electron. 32, 996-971 (1996).
[CrossRef]

V. Shteeman, D. Boiko, E. Kapon, and A. A. Hardy, “Extension of coupled mode analysis to periodic large arrays of identical waveguides for photonic crystals applications,” IEEE J. Quantum Electron. 43, 215-224 (2007).
[CrossRef]

V. Shteeman, I. Nusinsly, E. Kapon, and A. A. Hardy, “Extension of coupled mode analysis to infinite photonic superlattices,” IEEE J. Quantum Electron. 44, 826-833 (2008).
[CrossRef]

J. Lightwave Technol.

A. A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. LT-3, 1135-1146 (1985).
[CrossRef]

A. A. Hardy and W. Streifer. “Coupled modes of multiwaveguide systems and phased arrays,” J. Lightwave Technol. LT-4, 90-99 (1986).
[CrossRef]

J. Opt. Soc. Am.

Opt. Lett.

Phys. Rev.

J. C. Slater and G. F. Koster, “Simplified LCAO method for the periodic potential problem,” Phys. Rev. 94, 1498-1524 (1954).
[CrossRef]

Phys. Rev. Lett.

M. Bayer, T. Gutbrod, A. Forchel, T. Reinecke, P. Knipp, R. Werner, and J. Reithmaier, “Optical demonstration of a crystal band structure formation,” Phys. Rev. Lett. 83, 5374-5377 (1999).
[CrossRef]

Phys. Status Solidi B

M. Punkkinen, T. Korhonen, K. Kokko, and I. Väyrynen, “The electronic band structure of Si/SiO2-superlattices: A first-principles study,” Phys. Status Solidi B 214, r17-r18 (1999).
[CrossRef]

Science

J. C.Knight, J. Broeng, T. Birks, and P. St. J. Russell, “Photonic band gap guidance in optical fibers,” Science 282, 1476-1478 (1998).
[CrossRef] [PubMed]

Other

R. Grimes, A. Catlow, and A. Shluger, Quantum Mechanical Cluster Calculations in Solid State Studies (World Scientific, 1992), pp. 168-171.

R. Evarestov, Quantum Chemistry of Solids (Springer, 2007), pp. 105-140.

P. Hertel, Lectures on Theoretical Physics. Dielectric Waveguides (U. Osnabrueck, 2005), pp. 16-19.

L. Landau and E. Lifschitz, Quantum Mechanics: Non-relativistic Theory (Butterworth Heinemann, 2003), pp. 177-182.

K. Sakoda, Optical Properties of Photonic Crystals, (Springer, 2001), pp. 123-142.

J. Joannopoulos, R. Meade, and J. Winn, Photonic Crystals. Molding the Flow of Light (Princeton U. Press, 1995), pp. 54-76.

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Figures (6)

Fig. 1
Fig. 1

PCF with point defect. Milk-white circles represent regular waveguides. Gray rings represent defect waveguides.

Fig. 2
Fig. 2

Explanation for the defect bands selection.

Fig. 3
Fig. 3

Propagation constants as a function of λ 0 for PCF, depicted in Fig. 1. Solid and dashed lines represent photonic bands and gap states of infinite crystal, computed with extended CMT [3]. Circles represent photonic bands and gap states of finite PCF (eight waveguides on each outer side of the crystal), computed with conventional CMT [4]. Plus markers (+) are gap states of finite PCF, computed with Helmholtz equation (finite differences solution).

Fig. 4
Fig. 4

2D photonic crystal with two linear defects. Milk-white circles represent regular waveguides. Black dotted circles and checkered circles represent defect waveguides.

Fig. 5
Fig. 5

Band structure of photonic array, depicted in Fig. 4 (for λ 0 = 0.9 μ m ). Solid lines represent 2D and 1D bands of infinite crystal, computed with extended CMT [3]. Circles represent 2D and 1D bands of finite 30 × 30 photonic crystal, computed with conventional CMT [4].

Fig. 6
Fig. 6

Propagation constants as a function of λ 0 for photonic array, depicted in Fig. 4. Black dashed curves indicate mean values of 1D defect bands, computed with extended CMT [3]. Gray solid curves indicate top and bottom of 2D bulk bands of infinite array. Small black circles and small gray circles represent 2D and 1D bands of finite 30 × 30 photonic crystal, computed with conventional CMT [4].

Tables (6)

Tables Icon

Table 1 Parameters of Solitary Waveguides of Photonic Crystal Depicted in Fig. 1

Tables Icon

Table 2 Parameters of Photonic Lattice and Superlattice Used in Computation of Band Structure of the Infinite PCF Depicted in Fig. 1

Tables Icon

Table 3 Relative Error in Estimation of Propagation Constants of the First Four Gap States, Computed by Different Methods

Tables Icon

Table 4 Parameters of Solitary Waveguides of Photonic Crystal Depicted in Fig. 4

Tables Icon

Table 5 Parameters of Photonic Lattice and Superlattice Used in Computation of Band Structure of the Infinite Array Depicted in Fig. 4

Tables Icon

Table 6 Total CPU Time Required for Computation of Eigenmodes, Eigenvalues, and Band Structure of the Photonic Arrays Discussed in Sections 2, 3

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

( E t ( x , y , z ) H t ( x , y , z ) ) = e i σ z [ p = 1 S y q = 1 S x m = 1 G { p , q } A k x , k y , p , q , m ( E k x , k y , p , q , m 2 D Bloch ( x , y ) H k x , k y , p , q , m 2 D Bloch ( x , y ) ) + d = 1 N PD μ = 1 G PD { d } B μ , d ( E μ , d PD ( x , y ) H μ , d PD ( x , y ) ) ] ,
( E t ( x , y , z ) H t ( x , y , z ) ) = e i σ z [ p = 1 S y q = 1 S x m = 1 G { p , q } A k x , k y , p , q , m ( E k x , k y , p , q , m 2 D Bloch ( x , y ) H k x , k y , p , q , m 2 D Bloch ( x , y ) ) + α = x , y d α = 1 N α LD s α = 1 G LD { d α } C k α , s α , d α ( E k α , s α , d α 1 D Bloch ( x , y ) H k α , s α , d α 1 D Bloch ( x , y ) ) ] ,
( E t ( x , y , z ) H t ( x , y , z ) ) = e i σ z i = 1 j = 1 p = 1 S y q = 1 S x m = 1 G { p , q } u m , 0 { i , j p , q } ( E t m { i , j p , q } ( x , y ) H t m { i , j p , q } ( x , y ) ) ,
( M ͇ ̂ σ ( k x , k y ) I ̂ ) U ͇ ̂ 0 = 0 ,
M ͇ ̂ = P ͇ ̂ ( k x , k y ) 1 [ B ͇ ̂ P ͇ ̂ ( k x , k y ) + K ͇ ̃ ̂ ( k x , k y ) ] .
( E t ( x , y , z ) H t ( x , y , z ) ) = e i σ z j = 1 p = 1 S y q = 1 S x m = 1 G { p , q } u m , 0 { 1 , j p , q } ( E t m { 1 , j p , q } ( x , y ) H t m { 1 , j p , q } ( x , y ) ) .

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