Abstract

A high-order finite-difference frequency domain method is proposed for the analysis of the band diagrams of two-dimensional photonic crystals. This improved formulation is based on Taylor series expansion, local coordinate transformation, boundary conditions matching, and the generalized Douglas scheme. The nine-point second-order formulas are extended to fourth-order accuracy. This proposed scheme can deal with piecewise homogeneous structures with curved dielectric interfaces.

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  1. E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (1987).
    [CrossRef] [PubMed]
  2. S. John, "Strong localization of photons in certain disordered dielectric superlattices," Phys. Rev. Lett. 58, 2486-2489 (1987).
    [CrossRef] [PubMed]
  3. K. M. Ho, C. T. Chan, and C. M. Soukoulis, "Existence of a photonic gap in periodic dielectric structures," Phys. Rev. Lett. 65, 3152-3155 (1990).
    [CrossRef] [PubMed]
  4. M. Plihal and A. A. Maradudin, "Photonic band structure of two-dimensional systems: The triangular lattice," Phys. Rev. B 44, 8565-8571 (1991).
    [CrossRef]
  5. R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, "Existence of a photonic band gap in two dimensions," Appl. Phys. Lett. 61, 495-497 (1992).
    [CrossRef]
  6. C. T. Chan, Q. L. Yu, and K. M. Ho, "Order-N spectral method for electromagnetic waves," Phys. Rev. B 51, 16635-16642 (1995).
    [CrossRef]
  7. M. Qiu and S. He, "A nonorthogonal finite-difference time-domain method for computing the band structure of a two-dimensional photonic crystal with dielectric and metallic inclusions," Appl. Phys. 87, 8268-8275 (2000).
    [CrossRef]
  8. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd edition, (Artech House, MA USA, 2005).
  9. H. Y. D. Yang, "Finite difference analysis of 2-D photonic crystals," IEEE Trans. Microwave Theory Tech. 44, 2688-2695 (1996).
    [CrossRef]
  10. K. Bierwirth, N. Schulz, and F. Arndt, "Finite-difference analysis of rectangular dielectric waveguide structures," IEEE Trans. Microwave Theory Tech. 34, 1104-1114 (1986).
    [CrossRef]
  11. G. R. Hadley and R. E. Smith, "Full-vector waveguide modeling using an iterative finite-difference method with transparent boundary conditions," J. Lightwave Technol. 13, 465-469 (1995).
    [CrossRef]
  12. W. P. Huang, C. L. Xu, S. T. Chu, and S. K. Chaudhuri, "The finite-difference vector beam propagation method: analysis and assessment," J. Lightwave Technol. 10, 295-305 (1992).
    [CrossRef]
  13. Z. Zhu and T. G. Brown, "Full-vectorial finite-difference analysis of microstructured optical fibers," Opt. Express 10,853-864 (2002), http://www.opticsexpress.org/abstract.cfm?uri=OE-10-17-853.
    [PubMed]
  14. M. S. Stern, "Semivectorial polarized finite difference method for optical waveguides with arbitrary index profiles," Proc. Inst. Elect. Eng. J. 135, 56-63 (1988).
  15. C. Vassallo, "Improvement of finite difference methods for step-index optical waveguides," Proc. Inst. Elect. Eng. J. 139, 137-142 (1992).
  16. Y.-P. Chiou, Y.-C. Chiang, and H.-C. Chang, "Improved three-point formulas considering the interface conditions in the finite-difference analysis of step-index optical devices," J. Lightwave Technol. 18, 243-251 (2000).
    [CrossRef]
  17. J. Xia and J. Yu, "New finite-difference scheme for simulations of step-index waveguides with tilt interfaces," IEEE Photon. Technol. Lett. 15, 1237-1239 (2003).
    [CrossRef]
  18. Y.-C. Chiang, Y.-P. Chiou, and H.-C. Chang, "Improved full-vectorial finite-difference mode solver for optical waveguides with step-index profiles," J. Lightwave Technol. 20, 1609-1618 (2002).
    [CrossRef]
  19. Y.-C. Chiang, Y.-P. Chiou, and H.-C. Chang, "Finite-difference frequency-domain analysis of 2-D photonic crystals with curved dielectric interfaces," J. Lightwave Technol. 26, 971-976 (2008).
    [CrossRef]
  20. C.-P. Yu and H.-C. Chang, "Compact finite-difference frequency-domain method for the analysis of twodimensional photonic crystals," Opt. Express 12,1397-1408 (2004), http://www.opticsexpress.org/abstract.cfm?uri=OE-12-7-1397.
    [CrossRef] [PubMed]
  21. P.-J. Chiang, C.-P. Yu, and H.-C. Chang, "Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method," Phys. Rev. E 75, 026703 (2007).
    [CrossRef]
  22. I. Harari and E. Turkel, "Accurate finite difference methods for time-harmonic wave propagation," J. Comput. Phys. 119, 252-270 (1995).
    [CrossRef]
  23. S. G. Johnson and J. D. Joannopoulos, "The MIT Photonic-Bands Package home page [on line]," http://ab-initio.mit.edu/mpb/.

2008 (1)

2007 (1)

P.-J. Chiang, C.-P. Yu, and H.-C. Chang, "Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method," Phys. Rev. E 75, 026703 (2007).
[CrossRef]

2004 (1)

2003 (1)

J. Xia and J. Yu, "New finite-difference scheme for simulations of step-index waveguides with tilt interfaces," IEEE Photon. Technol. Lett. 15, 1237-1239 (2003).
[CrossRef]

2002 (2)

Y.-C. Chiang, Y.-P. Chiou, and H.-C. Chang, "Improved full-vectorial finite-difference mode solver for optical waveguides with step-index profiles," J. Lightwave Technol. 20, 1609-1618 (2002).
[CrossRef]

Z. Zhu and T. G. Brown, "Full-vectorial finite-difference analysis of microstructured optical fibers," Opt. Express 10,853-864 (2002), http://www.opticsexpress.org/abstract.cfm?uri=OE-10-17-853.
[PubMed]

2000 (2)

Y.-P. Chiou, Y.-C. Chiang, and H.-C. Chang, "Improved three-point formulas considering the interface conditions in the finite-difference analysis of step-index optical devices," J. Lightwave Technol. 18, 243-251 (2000).
[CrossRef]

M. Qiu and S. He, "A nonorthogonal finite-difference time-domain method for computing the band structure of a two-dimensional photonic crystal with dielectric and metallic inclusions," Appl. Phys. 87, 8268-8275 (2000).
[CrossRef]

1996 (1)

H. Y. D. Yang, "Finite difference analysis of 2-D photonic crystals," IEEE Trans. Microwave Theory Tech. 44, 2688-2695 (1996).
[CrossRef]

1995 (3)

C. T. Chan, Q. L. Yu, and K. M. Ho, "Order-N spectral method for electromagnetic waves," Phys. Rev. B 51, 16635-16642 (1995).
[CrossRef]

G. R. Hadley and R. E. Smith, "Full-vector waveguide modeling using an iterative finite-difference method with transparent boundary conditions," J. Lightwave Technol. 13, 465-469 (1995).
[CrossRef]

I. Harari and E. Turkel, "Accurate finite difference methods for time-harmonic wave propagation," J. Comput. Phys. 119, 252-270 (1995).
[CrossRef]

1992 (3)

W. P. Huang, C. L. Xu, S. T. Chu, and S. K. Chaudhuri, "The finite-difference vector beam propagation method: analysis and assessment," J. Lightwave Technol. 10, 295-305 (1992).
[CrossRef]

C. Vassallo, "Improvement of finite difference methods for step-index optical waveguides," Proc. Inst. Elect. Eng. J. 139, 137-142 (1992).

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, "Existence of a photonic band gap in two dimensions," Appl. Phys. Lett. 61, 495-497 (1992).
[CrossRef]

1991 (1)

M. Plihal and A. A. Maradudin, "Photonic band structure of two-dimensional systems: The triangular lattice," Phys. Rev. B 44, 8565-8571 (1991).
[CrossRef]

1990 (1)

K. M. Ho, C. T. Chan, and C. M. Soukoulis, "Existence of a photonic gap in periodic dielectric structures," Phys. Rev. Lett. 65, 3152-3155 (1990).
[CrossRef] [PubMed]

1988 (1)

M. S. Stern, "Semivectorial polarized finite difference method for optical waveguides with arbitrary index profiles," Proc. Inst. Elect. Eng. J. 135, 56-63 (1988).

1987 (2)

E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef] [PubMed]

S. John, "Strong localization of photons in certain disordered dielectric superlattices," Phys. Rev. Lett. 58, 2486-2489 (1987).
[CrossRef] [PubMed]

1986 (1)

K. Bierwirth, N. Schulz, and F. Arndt, "Finite-difference analysis of rectangular dielectric waveguide structures," IEEE Trans. Microwave Theory Tech. 34, 1104-1114 (1986).
[CrossRef]

Arndt, F.

K. Bierwirth, N. Schulz, and F. Arndt, "Finite-difference analysis of rectangular dielectric waveguide structures," IEEE Trans. Microwave Theory Tech. 34, 1104-1114 (1986).
[CrossRef]

Bierwirth, K.

K. Bierwirth, N. Schulz, and F. Arndt, "Finite-difference analysis of rectangular dielectric waveguide structures," IEEE Trans. Microwave Theory Tech. 34, 1104-1114 (1986).
[CrossRef]

Brommer, K. D.

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, "Existence of a photonic band gap in two dimensions," Appl. Phys. Lett. 61, 495-497 (1992).
[CrossRef]

Brown, T. G.

Chan, C. T.

C. T. Chan, Q. L. Yu, and K. M. Ho, "Order-N spectral method for electromagnetic waves," Phys. Rev. B 51, 16635-16642 (1995).
[CrossRef]

K. M. Ho, C. T. Chan, and C. M. Soukoulis, "Existence of a photonic gap in periodic dielectric structures," Phys. Rev. Lett. 65, 3152-3155 (1990).
[CrossRef] [PubMed]

Chang, H.-C.

Chaudhuri, S. K.

W. P. Huang, C. L. Xu, S. T. Chu, and S. K. Chaudhuri, "The finite-difference vector beam propagation method: analysis and assessment," J. Lightwave Technol. 10, 295-305 (1992).
[CrossRef]

Chiang, P.-J.

P.-J. Chiang, C.-P. Yu, and H.-C. Chang, "Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method," Phys. Rev. E 75, 026703 (2007).
[CrossRef]

Chiang, Y.-C.

Chiou, Y.-P.

Chu, S. T.

W. P. Huang, C. L. Xu, S. T. Chu, and S. K. Chaudhuri, "The finite-difference vector beam propagation method: analysis and assessment," J. Lightwave Technol. 10, 295-305 (1992).
[CrossRef]

Hadley, G. R.

G. R. Hadley and R. E. Smith, "Full-vector waveguide modeling using an iterative finite-difference method with transparent boundary conditions," J. Lightwave Technol. 13, 465-469 (1995).
[CrossRef]

Harari, I.

I. Harari and E. Turkel, "Accurate finite difference methods for time-harmonic wave propagation," J. Comput. Phys. 119, 252-270 (1995).
[CrossRef]

He, S.

M. Qiu and S. He, "A nonorthogonal finite-difference time-domain method for computing the band structure of a two-dimensional photonic crystal with dielectric and metallic inclusions," Appl. Phys. 87, 8268-8275 (2000).
[CrossRef]

Ho, K. M.

C. T. Chan, Q. L. Yu, and K. M. Ho, "Order-N spectral method for electromagnetic waves," Phys. Rev. B 51, 16635-16642 (1995).
[CrossRef]

K. M. Ho, C. T. Chan, and C. M. Soukoulis, "Existence of a photonic gap in periodic dielectric structures," Phys. Rev. Lett. 65, 3152-3155 (1990).
[CrossRef] [PubMed]

Huang, W. P.

W. P. Huang, C. L. Xu, S. T. Chu, and S. K. Chaudhuri, "The finite-difference vector beam propagation method: analysis and assessment," J. Lightwave Technol. 10, 295-305 (1992).
[CrossRef]

Joannopoulos, J. D.

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, "Existence of a photonic band gap in two dimensions," Appl. Phys. Lett. 61, 495-497 (1992).
[CrossRef]

John, S.

S. John, "Strong localization of photons in certain disordered dielectric superlattices," Phys. Rev. Lett. 58, 2486-2489 (1987).
[CrossRef] [PubMed]

Maradudin, A. A.

M. Plihal and A. A. Maradudin, "Photonic band structure of two-dimensional systems: The triangular lattice," Phys. Rev. B 44, 8565-8571 (1991).
[CrossRef]

Meade, R. D.

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, "Existence of a photonic band gap in two dimensions," Appl. Phys. Lett. 61, 495-497 (1992).
[CrossRef]

Plihal, M.

M. Plihal and A. A. Maradudin, "Photonic band structure of two-dimensional systems: The triangular lattice," Phys. Rev. B 44, 8565-8571 (1991).
[CrossRef]

Qiu, M.

M. Qiu and S. He, "A nonorthogonal finite-difference time-domain method for computing the band structure of a two-dimensional photonic crystal with dielectric and metallic inclusions," Appl. Phys. 87, 8268-8275 (2000).
[CrossRef]

Rappe, A. M.

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, "Existence of a photonic band gap in two dimensions," Appl. Phys. Lett. 61, 495-497 (1992).
[CrossRef]

Schulz, N.

K. Bierwirth, N. Schulz, and F. Arndt, "Finite-difference analysis of rectangular dielectric waveguide structures," IEEE Trans. Microwave Theory Tech. 34, 1104-1114 (1986).
[CrossRef]

Smith, R. E.

G. R. Hadley and R. E. Smith, "Full-vector waveguide modeling using an iterative finite-difference method with transparent boundary conditions," J. Lightwave Technol. 13, 465-469 (1995).
[CrossRef]

Soukoulis, C. M.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, "Existence of a photonic gap in periodic dielectric structures," Phys. Rev. Lett. 65, 3152-3155 (1990).
[CrossRef] [PubMed]

Stern, M. S.

M. S. Stern, "Semivectorial polarized finite difference method for optical waveguides with arbitrary index profiles," Proc. Inst. Elect. Eng. J. 135, 56-63 (1988).

Turkel, E.

I. Harari and E. Turkel, "Accurate finite difference methods for time-harmonic wave propagation," J. Comput. Phys. 119, 252-270 (1995).
[CrossRef]

Vassallo, C.

C. Vassallo, "Improvement of finite difference methods for step-index optical waveguides," Proc. Inst. Elect. Eng. J. 139, 137-142 (1992).

Xia, J.

J. Xia and J. Yu, "New finite-difference scheme for simulations of step-index waveguides with tilt interfaces," IEEE Photon. Technol. Lett. 15, 1237-1239 (2003).
[CrossRef]

Xu, C. L.

W. P. Huang, C. L. Xu, S. T. Chu, and S. K. Chaudhuri, "The finite-difference vector beam propagation method: analysis and assessment," J. Lightwave Technol. 10, 295-305 (1992).
[CrossRef]

Yablonovitch, E.

E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef] [PubMed]

Yang, H. Y. D.

H. Y. D. Yang, "Finite difference analysis of 2-D photonic crystals," IEEE Trans. Microwave Theory Tech. 44, 2688-2695 (1996).
[CrossRef]

Yu, C.-P.

Yu, J.

J. Xia and J. Yu, "New finite-difference scheme for simulations of step-index waveguides with tilt interfaces," IEEE Photon. Technol. Lett. 15, 1237-1239 (2003).
[CrossRef]

Yu, Q. L.

C. T. Chan, Q. L. Yu, and K. M. Ho, "Order-N spectral method for electromagnetic waves," Phys. Rev. B 51, 16635-16642 (1995).
[CrossRef]

Zhu, Z.

Appl. Phys. (1)

M. Qiu and S. He, "A nonorthogonal finite-difference time-domain method for computing the band structure of a two-dimensional photonic crystal with dielectric and metallic inclusions," Appl. Phys. 87, 8268-8275 (2000).
[CrossRef]

Appl. Phys. Lett. (1)

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, "Existence of a photonic band gap in two dimensions," Appl. Phys. Lett. 61, 495-497 (1992).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

J. Xia and J. Yu, "New finite-difference scheme for simulations of step-index waveguides with tilt interfaces," IEEE Photon. Technol. Lett. 15, 1237-1239 (2003).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (2)

H. Y. D. Yang, "Finite difference analysis of 2-D photonic crystals," IEEE Trans. Microwave Theory Tech. 44, 2688-2695 (1996).
[CrossRef]

K. Bierwirth, N. Schulz, and F. Arndt, "Finite-difference analysis of rectangular dielectric waveguide structures," IEEE Trans. Microwave Theory Tech. 34, 1104-1114 (1986).
[CrossRef]

J. Comput. Phys. (1)

I. Harari and E. Turkel, "Accurate finite difference methods for time-harmonic wave propagation," J. Comput. Phys. 119, 252-270 (1995).
[CrossRef]

J. Lightw. Technol. (1)

Y.-C. Chiang, Y.-P. Chiou, and H.-C. Chang, "Improved full-vectorial finite-difference mode solver for optical waveguides with step-index profiles," J. Lightwave Technol. 20, 1609-1618 (2002).
[CrossRef]

J. Lightwave Technol. (4)

Y.-P. Chiou, Y.-C. Chiang, and H.-C. Chang, "Improved three-point formulas considering the interface conditions in the finite-difference analysis of step-index optical devices," J. Lightwave Technol. 18, 243-251 (2000).
[CrossRef]

G. R. Hadley and R. E. Smith, "Full-vector waveguide modeling using an iterative finite-difference method with transparent boundary conditions," J. Lightwave Technol. 13, 465-469 (1995).
[CrossRef]

W. P. Huang, C. L. Xu, S. T. Chu, and S. K. Chaudhuri, "The finite-difference vector beam propagation method: analysis and assessment," J. Lightwave Technol. 10, 295-305 (1992).
[CrossRef]

Y.-C. Chiang, Y.-P. Chiou, and H.-C. Chang, "Finite-difference frequency-domain analysis of 2-D photonic crystals with curved dielectric interfaces," J. Lightwave Technol. 26, 971-976 (2008).
[CrossRef]

Opt. Express (2)

Phys. Rev. B (2)

C. T. Chan, Q. L. Yu, and K. M. Ho, "Order-N spectral method for electromagnetic waves," Phys. Rev. B 51, 16635-16642 (1995).
[CrossRef]

M. Plihal and A. A. Maradudin, "Photonic band structure of two-dimensional systems: The triangular lattice," Phys. Rev. B 44, 8565-8571 (1991).
[CrossRef]

Phys. Rev. E (1)

P.-J. Chiang, C.-P. Yu, and H.-C. Chang, "Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method," Phys. Rev. E 75, 026703 (2007).
[CrossRef]

Phys. Rev. Lett. (3)

E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef] [PubMed]

S. John, "Strong localization of photons in certain disordered dielectric superlattices," Phys. Rev. Lett. 58, 2486-2489 (1987).
[CrossRef] [PubMed]

K. M. Ho, C. T. Chan, and C. M. Soukoulis, "Existence of a photonic gap in periodic dielectric structures," Phys. Rev. Lett. 65, 3152-3155 (1990).
[CrossRef] [PubMed]

Proc. Inst. Elect. Eng. J. (2)

M. S. Stern, "Semivectorial polarized finite difference method for optical waveguides with arbitrary index profiles," Proc. Inst. Elect. Eng. J. 135, 56-63 (1988).

C. Vassallo, "Improvement of finite difference methods for step-index optical waveguides," Proc. Inst. Elect. Eng. J. 139, 137-142 (1992).

Other (2)

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd edition, (Artech House, MA USA, 2005).

S. G. Johnson and J. D. Joannopoulos, "The MIT Photonic-Bands Package home page [on line]," http://ab-initio.mit.edu/mpb/.

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Figures (8)

Fig. 1.
Fig. 1.

The cross-sectional view of a 2-D PC and its unit cell with a being the lattice distance and r being the radius of the circular rods for (a) the square lattice and (b) the triangular lattice.

Fig. 2.
Fig. 2.

Cross section of general finite-difference mesh with a curved interface.

Fig. 3.
Fig. 3.

The unit cell of the PC with corresponding PBCs for (a) the square lattice case, (b) the triangular lattice case, and (c) the modified unit cell of (b).

Fig. 4.
Fig. 4.

Calculated band diagrams for the 2-D PC formed by square-arranged alumina rods with r/a = 0.2 and ε A1 = 8.9 in the air. The results are compared with those from PWE method [23]. (a) TE mode and (b) TM mode.

Fig. 5.
Fig. 5.

The convergence properties of our method for the 2-D square-lattice PC compared with other methods. (a) TE first band; (b) TE second band; (c) TM first band; (d) TM second band.

Fig. 6.
Fig. 6.

Calculated band diagrams for the 2-D PC formed by triangle-arranged dielectric rods with r/a = 0.2 and ε = 11.4 in the air. The results are compared with those from PWE method [23]. (a) TE mode and (b) TM mode.

Fig. 7.
Fig. 7.

The convergence properties of our method for the 2-D square-lattice PC compared with other methods. (a) TE first band; (b) TE second band; (c) TM first band; (d) TM second band.

Fig. 8.
Fig. 8.

Resulting matrices utilizing this high-order scheme for the calculation of band diagrams of PCs with (a) the rectangular lattice and (b) the triangular lattice.

Equations (24)

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

ϕ ( m + 1 , n + 1 ) TSE ϕ R ; x y LCT ϕ R ; r θ BCM ϕ L ; r θ LCT ϕ L ; x y TSE ϕ ( m , n ) ,
p ϕ r q θ p q R = ϕ ˜ ( r , θ ) L
p + 1 α r q θ p q + 1 R = ϕ ˜ ( r , θ ) θ L
3 E z r 2 θ R = 3 E z r 2 θ L + k 0 2 ( ε L ε R ) E z θ L
3 E z r θ 2 R = 3 E z r θ 2 L
3 E z θ 3 R = 3 E z θ 3 L
3 H z r 2 θ R = 3 H z r 2 θ L + 1 r ( 1 ε R ε L ) H z r L + k 0 2 ( ε L ε R ) H z θ L
3 H z r θ 2 R = ε R ε L 3 H z r θ 2 L
3 H z θ 3 R = 3 H z θ 3 L
3 E z r 3 R = 3 E z r 3 L + k 0 2 ( ε L ε R ) E z r L 1 r k 0 2 ( ε L ε R ) E z L .
3 H z r 3 R = ε R ε L { 3 H z r 3 L + 1 r ( 1 ε L ε R ) 2 H z r 2 L + [ k 0 2 ( ε L ε R ) + 1 r 2 ( 1 ε L ε R ) ] H z r L }
1 r k 0 2 ( ε L ε R ) H z L .
Φ = M · D ( m , n ) + H . O . T . ,
D ( m , n ) M 1 · Φ .
2 x 2 = D xx 1 + c 1 D x + c 2 D xx ,
( D xx + D yy ) E z = 2 E z x 2 + 2 E z y 2 + g 1 3 E z x 3 + g 2 3 E z y 3 ,
( D xx + D yy ) E z + g 1 3 E z x y 2 + g 2 3 E z x 2 y = ( 1 + g 1 x + g 2 y ) ( 2 E z x 2 + 2 E z y 2 ) .
( 2 x 2 + 2 y 2 ) E z = ( D xx + D yy + g 1 D xyy + g 2 D xxy 1 + g 1 D x + g 2 D y ) E z .
PBC-X : ϕ ( x + a , y ) = exp ( j k x a ) ϕ ( x , y )
PBC-Y : ϕ ( x , y + a ) = exp ( j k y a ) ϕ ( x , y )
PBC 1 : ϕ ( x + 3 a 2 , y a 2 ) = exp [ j ( k x 3 a 2 k y a 2 ) ] ϕ ( x , y )
PBC 2 : ϕ ( x + 3 a 2 , y + a 2 ) = exp [ j ( k x 3 a 2 + k y a 2 ) ] ϕ ( x , y )
PBC 3 : ϕ ( x , y + a ) = exp ( j k y a ) ϕ ( x , y ) .
[ A ( ω c ) 2 B ] · Φ ˜ = 0

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