Abstract

Using analytical techniques the dispersion relations for two-dimensional photonic crystals with rectangular geometry are analyzed. In this part of the work E polarization is presented. By comparing with accurate numerical calculations, we show that our analysis provides a good description of the physical properties for this type of photonic crystal. Besides the significantly shorter calculation time, the analytical treatment provides an important insight into the photonic bands’ formation and their properties. The presented approach and derived analytical expressions can be useful for the investigation of photonic band structures as well as for the design of novel photonic crystal devices.

© 2008 Optical Society of America

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References

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  1. K. Inoue and K. Ohtaka, eds., Photonic Crystals, Physics, Fabrication and Applications (Springer, 2004).
  2. M. Plihal, A. Shambrook, and A. A. Maradudin, “Two-dimensional photonic band structures,” Opt. Commun. 80, 199-204 (1991).
    [CrossRef]
  3. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).
  4. J. B. Pendry, “Calculating photonic band structure,” J. Phys.: Condens. Matter 8, 1085-1108 (1996).
    [CrossRef]
  5. S. N. Kawakami, “Analytically solvable model of photonic crystal structures and novel phenomena,” J. Lightwave Technol. 20, 1644-1650 (2002).
    [CrossRef]
  6. I. Ponomarev, “Separation of variables in the computation of spectra in 2-D photonic crystals,” SIAM J. Appl. Math. 61, 1202-1218 (2000).
    [CrossRef]
  7. K. Samokhvalova, C. Chen, and B.-L. Qian, “Analytical and numerical calculations of the dispersion characteristics of two-dimensional dielectric photonic band gap structures,” J. Appl. Phys. 99, 63104 (2006).
    [CrossRef]
  8. T. J. Shepherd, P. J. Roberts, and R. Loudon, “Soluble two-dimensional photonic crystal model,” Phys. Rev. E 55, 6024-6038 (1997).
    [CrossRef]
  9. A. Figotin and P. Kuchment, “Band gap structure of spectra of periodic dielectric and acoustic media. II. Two-dimensional photonic crystals,” SIAM J. Appl. Math. 56, 1561-1620 (1996).
    [CrossRef]
  10. L. Chang, C.-C. Ho, H.-S. Wei, and G. Y. Wu, “Effective medium theory with dimensionality reduction for band structures of photonic crystals,” J. Appl. Phys. 101, 053109 (2007).
    [CrossRef]
  11. M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).
  12. K. Sakoda, Optical Properties of Photonic Crystals, 2nd ed. (Springer-Verlag, 2005).
  13. D. J. Griffiths, Introduction to Quantum Mechanics (Prentice-Hall, 1995).
  14. S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell's equations with shifting materials boundaries,” Phys. Rev. E 65, 066611 (2002).
    [CrossRef]
  15. W. Magnus and S. Winkler, Hill's Equation (Wiley, 1966).
  16. I. Nusinsky and A. A. Hardy, “Band gap-analysis of one-dimensional photonic crystals and conditions for gap closing,” Phys. Rev. B 73, 125104 (2006).
    [CrossRef]
  17. A. H. Nayfen, Introduction to Perturbation Techniques (Wiley, 1993).
  18. P. E. Barclay, K. Srinivasan, and O. Painter, “Design of photonic crystal waveguides for evanescent coupling to optical fiber tapers and integration with high-Q cavities,” J. Opt. Soc. Am. B 20, 2274-2284 (2003).
    [CrossRef]
  19. H. Kitahara, N. Tsumura, H. Kondo, M. W. Takeda, J. W. Haus, Z. Yuan, N. Kawai, K. Sakoda, and K. Inoue, “Terahertz wave dispersion in two-dimensional photonic crystals,” Phys. Rev. B 64, 045202 (2001).
    [CrossRef]

2007 (1)

L. Chang, C.-C. Ho, H.-S. Wei, and G. Y. Wu, “Effective medium theory with dimensionality reduction for band structures of photonic crystals,” J. Appl. Phys. 101, 053109 (2007).
[CrossRef]

2006 (2)

K. Samokhvalova, C. Chen, and B.-L. Qian, “Analytical and numerical calculations of the dispersion characteristics of two-dimensional dielectric photonic band gap structures,” J. Appl. Phys. 99, 63104 (2006).
[CrossRef]

I. Nusinsky and A. A. Hardy, “Band gap-analysis of one-dimensional photonic crystals and conditions for gap closing,” Phys. Rev. B 73, 125104 (2006).
[CrossRef]

2003 (1)

2002 (2)

S. N. Kawakami, “Analytically solvable model of photonic crystal structures and novel phenomena,” J. Lightwave Technol. 20, 1644-1650 (2002).
[CrossRef]

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell's equations with shifting materials boundaries,” Phys. Rev. E 65, 066611 (2002).
[CrossRef]

2001 (1)

H. Kitahara, N. Tsumura, H. Kondo, M. W. Takeda, J. W. Haus, Z. Yuan, N. Kawai, K. Sakoda, and K. Inoue, “Terahertz wave dispersion in two-dimensional photonic crystals,” Phys. Rev. B 64, 045202 (2001).
[CrossRef]

2000 (1)

I. Ponomarev, “Separation of variables in the computation of spectra in 2-D photonic crystals,” SIAM J. Appl. Math. 61, 1202-1218 (2000).
[CrossRef]

1997 (1)

T. J. Shepherd, P. J. Roberts, and R. Loudon, “Soluble two-dimensional photonic crystal model,” Phys. Rev. E 55, 6024-6038 (1997).
[CrossRef]

1996 (2)

A. Figotin and P. Kuchment, “Band gap structure of spectra of periodic dielectric and acoustic media. II. Two-dimensional photonic crystals,” SIAM J. Appl. Math. 56, 1561-1620 (1996).
[CrossRef]

J. B. Pendry, “Calculating photonic band structure,” J. Phys.: Condens. Matter 8, 1085-1108 (1996).
[CrossRef]

1991 (1)

M. Plihal, A. Shambrook, and A. A. Maradudin, “Two-dimensional photonic band structures,” Opt. Commun. 80, 199-204 (1991).
[CrossRef]

Barclay, P. E.

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).

Chang, L.

L. Chang, C.-C. Ho, H.-S. Wei, and G. Y. Wu, “Effective medium theory with dimensionality reduction for band structures of photonic crystals,” J. Appl. Phys. 101, 053109 (2007).
[CrossRef]

Chen, C.

K. Samokhvalova, C. Chen, and B.-L. Qian, “Analytical and numerical calculations of the dispersion characteristics of two-dimensional dielectric photonic band gap structures,” J. Appl. Phys. 99, 63104 (2006).
[CrossRef]

Figotin, A.

A. Figotin and P. Kuchment, “Band gap structure of spectra of periodic dielectric and acoustic media. II. Two-dimensional photonic crystals,” SIAM J. Appl. Math. 56, 1561-1620 (1996).
[CrossRef]

Fink, Y.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell's equations with shifting materials boundaries,” Phys. Rev. E 65, 066611 (2002).
[CrossRef]

Griffiths, D. J.

D. J. Griffiths, Introduction to Quantum Mechanics (Prentice-Hall, 1995).

Hagness, S. C.

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

Hardy, A. A.

I. Nusinsky and A. A. Hardy, “Band gap-analysis of one-dimensional photonic crystals and conditions for gap closing,” Phys. Rev. B 73, 125104 (2006).
[CrossRef]

Haus, J. W.

H. Kitahara, N. Tsumura, H. Kondo, M. W. Takeda, J. W. Haus, Z. Yuan, N. Kawai, K. Sakoda, and K. Inoue, “Terahertz wave dispersion in two-dimensional photonic crystals,” Phys. Rev. B 64, 045202 (2001).
[CrossRef]

Ho, C.-C.

L. Chang, C.-C. Ho, H.-S. Wei, and G. Y. Wu, “Effective medium theory with dimensionality reduction for band structures of photonic crystals,” J. Appl. Phys. 101, 053109 (2007).
[CrossRef]

Ibanescu, M.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell's equations with shifting materials boundaries,” Phys. Rev. E 65, 066611 (2002).
[CrossRef]

Inoue, K.

H. Kitahara, N. Tsumura, H. Kondo, M. W. Takeda, J. W. Haus, Z. Yuan, N. Kawai, K. Sakoda, and K. Inoue, “Terahertz wave dispersion in two-dimensional photonic crystals,” Phys. Rev. B 64, 045202 (2001).
[CrossRef]

K. Inoue and K. Ohtaka, eds., Photonic Crystals, Physics, Fabrication and Applications (Springer, 2004).

Joannopoulos, J. D.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell's equations with shifting materials boundaries,” Phys. Rev. E 65, 066611 (2002).
[CrossRef]

Johnson, S. G.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell's equations with shifting materials boundaries,” Phys. Rev. E 65, 066611 (2002).
[CrossRef]

Kawai, N.

H. Kitahara, N. Tsumura, H. Kondo, M. W. Takeda, J. W. Haus, Z. Yuan, N. Kawai, K. Sakoda, and K. Inoue, “Terahertz wave dispersion in two-dimensional photonic crystals,” Phys. Rev. B 64, 045202 (2001).
[CrossRef]

Kawakami, S. N.

Kitahara, H.

H. Kitahara, N. Tsumura, H. Kondo, M. W. Takeda, J. W. Haus, Z. Yuan, N. Kawai, K. Sakoda, and K. Inoue, “Terahertz wave dispersion in two-dimensional photonic crystals,” Phys. Rev. B 64, 045202 (2001).
[CrossRef]

Kondo, H.

H. Kitahara, N. Tsumura, H. Kondo, M. W. Takeda, J. W. Haus, Z. Yuan, N. Kawai, K. Sakoda, and K. Inoue, “Terahertz wave dispersion in two-dimensional photonic crystals,” Phys. Rev. B 64, 045202 (2001).
[CrossRef]

Kuchment, P.

A. Figotin and P. Kuchment, “Band gap structure of spectra of periodic dielectric and acoustic media. II. Two-dimensional photonic crystals,” SIAM J. Appl. Math. 56, 1561-1620 (1996).
[CrossRef]

Loudon, R.

T. J. Shepherd, P. J. Roberts, and R. Loudon, “Soluble two-dimensional photonic crystal model,” Phys. Rev. E 55, 6024-6038 (1997).
[CrossRef]

Magnus, W.

W. Magnus and S. Winkler, Hill's Equation (Wiley, 1966).

Maradudin, A. A.

M. Plihal, A. Shambrook, and A. A. Maradudin, “Two-dimensional photonic band structures,” Opt. Commun. 80, 199-204 (1991).
[CrossRef]

Nayfen, A. H.

A. H. Nayfen, Introduction to Perturbation Techniques (Wiley, 1993).

Nusinsky, I.

I. Nusinsky and A. A. Hardy, “Band gap-analysis of one-dimensional photonic crystals and conditions for gap closing,” Phys. Rev. B 73, 125104 (2006).
[CrossRef]

Ohtaka, K.

K. Inoue and K. Ohtaka, eds., Photonic Crystals, Physics, Fabrication and Applications (Springer, 2004).

Painter, O.

Pendry, J. B.

J. B. Pendry, “Calculating photonic band structure,” J. Phys.: Condens. Matter 8, 1085-1108 (1996).
[CrossRef]

Plihal, M.

M. Plihal, A. Shambrook, and A. A. Maradudin, “Two-dimensional photonic band structures,” Opt. Commun. 80, 199-204 (1991).
[CrossRef]

Ponomarev, I.

I. Ponomarev, “Separation of variables in the computation of spectra in 2-D photonic crystals,” SIAM J. Appl. Math. 61, 1202-1218 (2000).
[CrossRef]

Qian, B.-L.

K. Samokhvalova, C. Chen, and B.-L. Qian, “Analytical and numerical calculations of the dispersion characteristics of two-dimensional dielectric photonic band gap structures,” J. Appl. Phys. 99, 63104 (2006).
[CrossRef]

Roberts, P. J.

T. J. Shepherd, P. J. Roberts, and R. Loudon, “Soluble two-dimensional photonic crystal model,” Phys. Rev. E 55, 6024-6038 (1997).
[CrossRef]

Sakoda, K.

H. Kitahara, N. Tsumura, H. Kondo, M. W. Takeda, J. W. Haus, Z. Yuan, N. Kawai, K. Sakoda, and K. Inoue, “Terahertz wave dispersion in two-dimensional photonic crystals,” Phys. Rev. B 64, 045202 (2001).
[CrossRef]

K. Sakoda, Optical Properties of Photonic Crystals, 2nd ed. (Springer-Verlag, 2005).

Samokhvalova, K.

K. Samokhvalova, C. Chen, and B.-L. Qian, “Analytical and numerical calculations of the dispersion characteristics of two-dimensional dielectric photonic band gap structures,” J. Appl. Phys. 99, 63104 (2006).
[CrossRef]

Shambrook, A.

M. Plihal, A. Shambrook, and A. A. Maradudin, “Two-dimensional photonic band structures,” Opt. Commun. 80, 199-204 (1991).
[CrossRef]

Shepherd, T. J.

T. J. Shepherd, P. J. Roberts, and R. Loudon, “Soluble two-dimensional photonic crystal model,” Phys. Rev. E 55, 6024-6038 (1997).
[CrossRef]

Skorobogatiy, M. A.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell's equations with shifting materials boundaries,” Phys. Rev. E 65, 066611 (2002).
[CrossRef]

Srinivasan, K.

Taflove, A.

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

Takeda, M. W.

H. Kitahara, N. Tsumura, H. Kondo, M. W. Takeda, J. W. Haus, Z. Yuan, N. Kawai, K. Sakoda, and K. Inoue, “Terahertz wave dispersion in two-dimensional photonic crystals,” Phys. Rev. B 64, 045202 (2001).
[CrossRef]

Tsumura, N.

H. Kitahara, N. Tsumura, H. Kondo, M. W. Takeda, J. W. Haus, Z. Yuan, N. Kawai, K. Sakoda, and K. Inoue, “Terahertz wave dispersion in two-dimensional photonic crystals,” Phys. Rev. B 64, 045202 (2001).
[CrossRef]

Wei, H.-S.

L. Chang, C.-C. Ho, H.-S. Wei, and G. Y. Wu, “Effective medium theory with dimensionality reduction for band structures of photonic crystals,” J. Appl. Phys. 101, 053109 (2007).
[CrossRef]

Weisberg, O.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell's equations with shifting materials boundaries,” Phys. Rev. E 65, 066611 (2002).
[CrossRef]

Winkler, S.

W. Magnus and S. Winkler, Hill's Equation (Wiley, 1966).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).

Wu, G. Y.

L. Chang, C.-C. Ho, H.-S. Wei, and G. Y. Wu, “Effective medium theory with dimensionality reduction for band structures of photonic crystals,” J. Appl. Phys. 101, 053109 (2007).
[CrossRef]

Yuan, Z.

H. Kitahara, N. Tsumura, H. Kondo, M. W. Takeda, J. W. Haus, Z. Yuan, N. Kawai, K. Sakoda, and K. Inoue, “Terahertz wave dispersion in two-dimensional photonic crystals,” Phys. Rev. B 64, 045202 (2001).
[CrossRef]

J. Appl. Phys. (2)

L. Chang, C.-C. Ho, H.-S. Wei, and G. Y. Wu, “Effective medium theory with dimensionality reduction for band structures of photonic crystals,” J. Appl. Phys. 101, 053109 (2007).
[CrossRef]

K. Samokhvalova, C. Chen, and B.-L. Qian, “Analytical and numerical calculations of the dispersion characteristics of two-dimensional dielectric photonic band gap structures,” J. Appl. Phys. 99, 63104 (2006).
[CrossRef]

J. Lightwave Technol. (1)

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

J. Phys.: Condens. Matter (1)

J. B. Pendry, “Calculating photonic band structure,” J. Phys.: Condens. Matter 8, 1085-1108 (1996).
[CrossRef]

Opt. Commun. (1)

M. Plihal, A. Shambrook, and A. A. Maradudin, “Two-dimensional photonic band structures,” Opt. Commun. 80, 199-204 (1991).
[CrossRef]

Phys. Rev. B (2)

I. Nusinsky and A. A. Hardy, “Band gap-analysis of one-dimensional photonic crystals and conditions for gap closing,” Phys. Rev. B 73, 125104 (2006).
[CrossRef]

H. Kitahara, N. Tsumura, H. Kondo, M. W. Takeda, J. W. Haus, Z. Yuan, N. Kawai, K. Sakoda, and K. Inoue, “Terahertz wave dispersion in two-dimensional photonic crystals,” Phys. Rev. B 64, 045202 (2001).
[CrossRef]

Phys. Rev. E (2)

T. J. Shepherd, P. J. Roberts, and R. Loudon, “Soluble two-dimensional photonic crystal model,” Phys. Rev. E 55, 6024-6038 (1997).
[CrossRef]

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell's equations with shifting materials boundaries,” Phys. Rev. E 65, 066611 (2002).
[CrossRef]

SIAM J. Appl. Math. (2)

A. Figotin and P. Kuchment, “Band gap structure of spectra of periodic dielectric and acoustic media. II. Two-dimensional photonic crystals,” SIAM J. Appl. Math. 56, 1561-1620 (1996).
[CrossRef]

I. Ponomarev, “Separation of variables in the computation of spectra in 2-D photonic crystals,” SIAM J. Appl. Math. 61, 1202-1218 (2000).
[CrossRef]

Other (7)

W. Magnus and S. Winkler, Hill's Equation (Wiley, 1966).

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).

K. Sakoda, Optical Properties of Photonic Crystals, 2nd ed. (Springer-Verlag, 2005).

D. J. Griffiths, Introduction to Quantum Mechanics (Prentice-Hall, 1995).

A. H. Nayfen, Introduction to Perturbation Techniques (Wiley, 1993).

K. Inoue and K. Ohtaka, eds., Photonic Crystals, Physics, Fabrication and Applications (Springer, 2004).

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

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Figures (5)

Fig. 1
Fig. 1

Distribution of the dielectric function ε ( x , y ) of a two-dimensional rectangular photonic crystal (top view).

Fig. 2
Fig. 2

Distribution of the dielectric function in the elementary cell of two-dimensional rectangular photonic crystals: (a) The main problem, ε ( x , y ) ; (b) the modified exactly solvable structure, ε ̃ ( x , y ) .

Fig. 3
Fig. 3

Band structure of a photonic crystal with the following parameters: ε 1 = 1 , ε 2 = n 2 2 = 3 . 4 2 = 11.56 , and a = a x = a y = 0.7 L x = 0.7 L y . The solid curves present a numerical solution calculated by the plane wave expansion method. (a) Dashed lines are for the zero-order approximation (that is, ω ̃ ). The filled circles are for the first-order approximation, Δ ω n ( 1 ) . (b) The open circles are for the second-order approximation, Δ ω n ( 2 ) .

Fig. 4
Fig. 4

Same as Fig. 3a for ε 1 = 11.56 , ε 2 = 1 , and a = a x = a y = 0.7 L x = 0.7 L y .

Fig. 5
Fig. 5

Maximum relative error (in percentages) for the first three bands as a function of the standard deviation, σ, between ε ( x , y ) and ε ̃ ( x , y ) for the square structures with 1 = ε 1 < ε 2 .

Equations (45)

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2 E z ( x , y ) x 2 + 2 E z ( x , y ) y 2 = ε ( x , y ) ω 2 c 2 E z ( x , y ) ,
ε ̃ ( x , y ) = ε ̃ x ( x ) + ε ̃ y ( y ) .
ε ̃ i ( i ) = { 0.5 ε 1 0 < i < a i e 0.5 ε 1 a i < i < L i , i = x , y .
e = ( ε 1 + ε 2 ) b x b y + ε 2 ( a x b y + a y b x ) b x L y + b y L x .
cos ( K x B L x ) = cos ( k 1 x a x ) cos ( k 2 x b x ) 1 2 ( k 1 x k 2 x + k 2 x k 1 x ) sin ( k 1 x a x ) sin ( k 2 x b x ) ,
cos ( K y B L y ) = cos ( k 1 y a y ) cos ( k 2 y b y ) 1 2 ( k 1 y k 2 y + k 2 y k 1 y ) sin ( k 1 y a y ) sin ( k 2 y b y ) ,
k 1 x 2 = ω ̃ 2 c 2 ( 0.5 ε 1 β 2 ) ,
k 2 x 2 = ω ̃ 2 c 2 ( e 0.5 ε 1 β 2 ) ,
k 1 y 2 = ω ̃ 2 c 2 ( 0.5 ε 1 + β 2 ) ,
k 2 y 2 = ω ̃ 2 c 2 ( e 0.5 ε 1 + β 2 ) ,
ω n ( K x B , K y B ) = ω ̃ n ( K x B , K y B ) + ω n ( 1 ) ( K x B , K y B ) + ω n ( 2 ) ( K x B , K y B ) ,
E z n ( x , y ) = E ̃ z n ( x , y ) + E z n ( 1 ) ( x , y ) + E z n ( 2 ) ( x , y ) ,
ω n ( 1 ) ( K x B , K y B ) = ω ̃ n ( K x B , K y B ) 2 Δ ε ( x , y ) E ̃ z n ( x , y ) 2 d x d y ε ̃ ( x , y ) E ̃ z n ( x , y ) 2 d x d y ,
E ̃ z n ( x + L x , y + L y ) = E ̃ z n ( x , y ) exp ( i K x B L x + i K y B L y ) .
ω n ( 1 ) ( K x B , K y B ) = ω ̃ n ( K x B , K y B ) 2 ( 0 a x a y L y + a x L x 0 a y + a x L x a y L y ) Δ ε ( x , y ) E ̃ z n 2 d x d y 0 L x 0 L y ε ̃ ( x , y ) E ̃ z n 2 d x d y .
ω n ( 1 ) ( K x B , K y B ) = 0.5 ω ̃ n ( K x B , K y B ) ( ε 2 e ) ( f a b n n + f b a n n ) + ( ε 1 + ε 2 2 e ) f b b n n ε 1 f a a n n + e ( f a b n n + f b a n n ) + ( 2 e ε 1 ) f b b n n ,
T = 0.5 ω m i ( 0 ) ( N m 1 m 1 N m 1 m M N m M m 1 N m M m M ) 1 ( W m 1 m 1 W m 1 m M W m M m 1 W m M m M ) ,
W m i m j = 0 L x 0 L y Δ ε ( x , y ) E ̃ ¯ z m i E ̃ z m j d x d y = ( ε 2 e ) ( f a b m i m j + f b a m i m j ) + ( ε 1 + ε 2 2 e ) f b b m i m j ,
N m i m j = 0 L x 0 L y ε ̃ ( x , y ) E ̃ ¯ z m i E ̃ z m j d x d y = ε 1 f a a m i m j + e ( f b a m i m j + f a b m i m j ) + ( 2 e ε 1 ) f b b m i m j ,
ω ( 1 ) = ω ̃ n 4 ( W n n N n n + W m m N m m ± ( W n n N n n W m m N m m ) 2 + 4 W m n 2 N n n N m m ) .
ω ( 1 ) = ω ̃ n m 4 ( W n n N n n + W m m N m m ± ( W n n N n n W m m N m m ) 2 + 4 W m n 2 N n n N m m + 4 δ n m ω ̃ n m ( W n n N n n W m m N m m ) ) ± δ n m 2 ,
ω n ( 2 ) = ω ̃ n 0 L x 0 L y Δ ε E ̃ ¯ z n E z n ( 1 ) d x d y + 2 ω n ( 1 ) ( 0 L x 0 L y ε ̃ E ̃ ¯ z n E z n ( 1 ) d x d y + 0 L x 0 L y Δ ε E ̃ z n 2 d x d y ) 2 0 L x 0 L y ε ̃ ( x , y ) E ̃ z n 2 d x d y ( ω n ( 1 ) ) 2 2 ω ̃ n ,
E z n ( 1 ) ( x , y ) = m n C m n E ̃ z m ( x , y ) ,
E z n ( 1 ) ( x , y ) = ( ω ̃ n ) 2 m n E ̃ z m ( x , y ) ω ̃ m 2 ω ̃ n 2 0 L x 0 L y Δ ε ( x , y ) E ̃ ¯ z m ( x , y ) E ̃ z n ( x , y ) d x d y 0 L x 0 L y ε ̃ ( x , y ) E ̃ z m ( x , y ) 2 d x d y .
ω n ( 2 ) = ( ω ̃ n ) 3 2 m n 0 L x 0 L y Δ ε ( x , y ) E ̃ ¯ z m ( x , y ) E ̃ z n ( x , y ) d x d y 2 ( ω ̃ m 2 ω ̃ n 2 ) 0 L x 0 L y ε ̃ ( x , y ) E ̃ z n ( x , y ) 2 d x d y 0 L x 0 L y ε ̃ ( x , y ) E ̃ z m ( x , y ) 2 d x d y + 3 ω n ( 1 ) 2 2 ω ̃ n .
ω n ( 2 ) = 0.5 ω ̃ n 3 N n n m n W m n 2 ( ω ̃ m 2 ω ̃ n 2 ) N m m + 3 ω n ( 1 ) 2 2 ω ̃ n .
E ̃ z ( x , y ) = ( X 1 ( x ) + A x X 2 ( x ) ) ( Y 1 ( y ) + A y Y 2 ( y ) ) .
X 1 ( x ) = { cos ( k 1 x x ) 0 x a x cos ( k 1 x a x ) cos ( k 2 x ( x a x ) ) k 1 x k 2 x sin ( k 1 x a x ) sin ( k 2 x ( x a x ) ) a x x L x ,
Y 1 ( y ) = { cos ( k 1 y y ) 0 y a y cos ( k 1 y a y ) cos ( k 2 y ( y a y ) ) k 1 y k 2 y sin ( k 1 y a y ) sin ( k 2 y ( x a y ) ) a y y L y ,
X 2 ( x ) = { sin ( k 1 x x ) k 1 x 0 x a x sin ( k 1 x a x ) k 1 x cos ( k 2 x ( x a x ) ) + cos ( k 1 x a x ) k 2 x sin ( k 2 x ( x a x ) ) a x x L x ,
Y 2 ( y ) = { sin ( k 1 y y ) k 1 y 0 y a y sin ( k 1 y a y ) k 1 y cos ( k 2 y ( y a y ) ) + cos ( k 1 y a y ) k 2 y sin ( k 2 y ( y a y ) ) a y y L y ,
A x = X 1 ( L x ) exp ( i K x B L x ) X 2 ( L x ) ,
A y = Y 1 ( L y ) exp ( i K y B L y ) Y 2 ( L y ) ,
f a a m n = 0 a x 0 a y E ̃ ¯ z m E ̃ z n d x d y = g a x m n g a y m n ,
f b a m n = a x L x 0 a y E ̃ ¯ z m E ̃ z n d x d y = g b x m n g a y m n ,
f a b m n = 0 a x a y L y E ̃ ¯ z m E ̃ z n d x d y = g a x m n g b y m n ,
f b b m n = a x L x a y L y E ̃ ¯ z m E ̃ z n d x d y = g b x m n g b y m n .
g a i m n = a i 2 ( 1 + A ¯ i m A i n k 1 i m k 1 i n ) sinc ( a i ( k 1 i m k 1 i n ) ) + a i 2 ( 1 A ¯ i m A i n k 1 i m k 1 i n ) sinc ( a i ( k 1 i m + k 1 i n ) ) + ( A i n k 1 i n A ¯ i m k 1 i m ) cos ( a i ( k 1 i m k 1 i n ) ) 1 2 ( k 1 i m k 1 i n ) ( A i n k 1 i n + A ¯ i m k 1 i m ) cos ( a i ( k 1 i m + k 1 i n ) ) 1 2 ( k 1 i m + k 1 i n ) ,
g b i m n = b i 2 ( B ¯ i m B i n + C ¯ i m C i n k 2 i m k 2 i n ) sinc ( b i ( k 2 i m k 2 i n ) ) + b i 2 ( B ¯ i m B i n C ¯ i m C i n k 2 i m k 2 i n ) sinc ( b i ( k 2 i m + k 2 i n ) ) + ( B ¯ i m C i n k 2 i n C ¯ i m B i n k 2 i m ) cos ( b i ( k 2 i m k 2 i n ) ) 1 2 ( k 2 i m k 2 i n ) ( B ¯ i m C i n k 2 i n + C ¯ i m B i n k 2 i m ) cos ( b i ( k 2 i m + k 2 i n ) ) 1 2 ( k 2 i m + k 2 i n )
i = x , y
B i n = cos ( k 1 i n a i ) + A i n sin ( k 1 i n a i ) k 1 i n , i = x , y ,
C i n = k 1 i n sin ( k 1 i n a i ) + A i n cos ( k 1 i n a i ) , i = x , y ,
g a i n n = a i 2 ( 1 + A i 2 k 1 i 2 ) + sinc ( 2 k 1 i a i ) a i 2 ( 1 A i 2 k 1 i 2 ) + A i + A ¯ i 2 a i 2 sinc 2 ( 2 k 1 i a i ) ,
g b i n n = b i 2 ( B i 2 + C i 2 k 2 i 2 ) + sinc ( 2 k 2 i b i ) b i 2 ( B i 2 C i 2 k 2 i 2 ) + B ¯ i C i + C ¯ i B i 2 b i 2 sinc 2 ( 2 k 2 i b i ) .
ω ̃ n ε ̃ E ̃ ¯ z m E ̃ z n d x d y = ω ̃ ¯ m ε ̃ E ̃ ¯ z m E ̃ z n d x d y ,

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