Abstract

An approximate perturbation based method for fast calculation and investigation of complex-shaped two-dimensional photonic crystals is presented. Both E- and H-polarizations are analyzed. Useful analytical formulas for calculating the dispersion relations are developed. The accuracy of the approximations is examined against numerical calculations, showing good match for a wide range of photonic crystal parameters. The present approach can be useful for the investigation of various physical effects inside photonic crystal structures as well as for the design of new photonic crystal devices.

© 2010 Optical Society of America

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References

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  1. K. Sakoda, Optical Properties of Photonic Crystals, 2nd ed. (Springer-Verlag, 2005).
  2. R. Wang, X.-H. Wang, B.-Y. Gu, and G.-Z. Yang, “Effects of shapes and orientations of scatterers and lattice symmetries on the photonic band gap in two-dimensional photonic crystals,” J. Appl. Phys. 90, 4307–4313 (2001).
    [CrossRef]
  3. 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]
  4. I. Ponomarev, “Separation of variables in the computation of spectra in 2-D photonic crystals,” SIAM J. Appl. Math. 61, 1202–1218 (2000).
    [CrossRef]
  5. T. J. Shepherd, P. J. Roberts, and R. Loudon, “Soluble two-dimensional photonic crystal model,” Phys. Rev. E 55, 6024–6038 (1997).
    [CrossRef]
  6. I. Nusinsky and A. A. Hardy, “Approximate analysis of two-dimensional photonic crystals with rectangular geometry. I. E-polarization,” J. Opt. Soc. Am. B 25, 1135–1143 (2008).
    [CrossRef]
  7. I. Nusinsky and A. A. Hardy, “Approximate analysis of two-dimensional photonic crystals with rectangular geometry. Part II: H-polarization,” J. Opt. Soc. Am. B 26, 1497–1505 (2009).
    [CrossRef]
  8. M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).
  9. H. W. Kuhn and A. W. Tucker, “Nonlinear programming,” in Proceedings of 2nd Berkeley Symposium (University of California Press, 1951), pp. 481–492.
  10. G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, 1985), pp. 945–950.
  11. D. J. Griffiths, Introduction to Quantum Mechanics (Prentice-Hall, 1995).
  12. C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics (Wiley, 1977), Vol. 1, pp. 406–411.

2009 (1)

2008 (1)

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]

2005 (1)

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

2001 (1)

R. Wang, X.-H. Wang, B.-Y. Gu, and G.-Z. Yang, “Effects of shapes and orientations of scatterers and lattice symmetries on the photonic band gap in two-dimensional photonic crystals,” J. Appl. Phys. 90, 4307–4313 (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]

1999 (1)

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

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]

1995 (1)

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

1985 (1)

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, 1985), pp. 945–950.

1977 (1)

C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics (Wiley, 1977), Vol. 1, pp. 406–411.

1951 (1)

H. W. Kuhn and A. W. Tucker, “Nonlinear programming,” in Proceedings of 2nd Berkeley Symposium (University of California Press, 1951), pp. 481–492.

Arfken, G.

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, 1985), pp. 945–950.

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]

Cohen-Tannoudji, C.

C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics (Wiley, 1977), Vol. 1, pp. 406–411.

Diu, B.

C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics (Wiley, 1977), Vol. 1, pp. 406–411.

Griffiths, D. J.

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

Gu, B. -Y.

R. Wang, X.-H. Wang, B.-Y. Gu, and G.-Z. Yang, “Effects of shapes and orientations of scatterers and lattice symmetries on the photonic band gap in two-dimensional photonic crystals,” J. Appl. Phys. 90, 4307–4313 (2001).
[CrossRef]

Hardy, A. A.

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]

Kuhn, H. W.

H. W. Kuhn and A. W. Tucker, “Nonlinear programming,” in Proceedings of 2nd Berkeley Symposium (University of California Press, 1951), pp. 481–492.

Laloe, F.

C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics (Wiley, 1977), Vol. 1, pp. 406–411.

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]

Nusinsky, I.

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]

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.

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

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]

Tucker, A. W.

H. W. Kuhn and A. W. Tucker, “Nonlinear programming,” in Proceedings of 2nd Berkeley Symposium (University of California Press, 1951), pp. 481–492.

Wang, R.

R. Wang, X.-H. Wang, B.-Y. Gu, and G.-Z. Yang, “Effects of shapes and orientations of scatterers and lattice symmetries on the photonic band gap in two-dimensional photonic crystals,” J. Appl. Phys. 90, 4307–4313 (2001).
[CrossRef]

Wang, X. -H.

R. Wang, X.-H. Wang, B.-Y. Gu, and G.-Z. Yang, “Effects of shapes and orientations of scatterers and lattice symmetries on the photonic band gap in two-dimensional photonic crystals,” J. Appl. Phys. 90, 4307–4313 (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]

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]

Yang, G. -Z.

R. Wang, X.-H. Wang, B.-Y. Gu, and G.-Z. Yang, “Effects of shapes and orientations of scatterers and lattice symmetries on the photonic band gap in two-dimensional photonic crystals,” J. Appl. Phys. 90, 4307–4313 (2001).
[CrossRef]

J. Appl. Phys. (2)

R. Wang, X.-H. Wang, B.-Y. Gu, and G.-Z. Yang, “Effects of shapes and orientations of scatterers and lattice symmetries on the photonic band gap in two-dimensional photonic crystals,” J. Appl. Phys. 90, 4307–4313 (2001).
[CrossRef]

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]

J. Opt. Soc. Am. B (2)

Phys. Rev. E (1)

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

SIAM J. Appl. Math. (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]

Other (6)

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

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

H. W. Kuhn and A. W. Tucker, “Nonlinear programming,” in Proceedings of 2nd Berkeley Symposium (University of California Press, 1951), pp. 481–492.

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, 1985), pp. 945–950.

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

C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics (Wiley, 1977), Vol. 1, pp. 406–411.

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Figures (6)

Fig. 1
Fig. 1

Distribution of the dielectric function ε ( x , y ) of two-dimensional photonic crystals with marked subcells (dotted lines). The dashed lines are for unit cell: (a) rectangular-shaped rods and (b) circular rods.

Fig. 2
Fig. 2

Distribution of the dielectric function ε ( x , y ) of various complex-shaped structures. The dotted lines show possible subcells.

Fig. 3
Fig. 3

Distribution of the approximate zero order function in the elementary cell: (a) two-dimensional separable function ε ̃ ( x , y ) ; (b),(c) one-dimensional components ε ̃ x ( x ) , ε ̃ y ( y ) .

Fig. 4
Fig. 4

Same as Fig. 3, for ε ̃ ̃ ( x , y ) , ε ̃ ̃ x ( x ) , and ε ̃ ̃ y ( y ) respectively.

Fig. 5
Fig. 5

Band structure of photonic crystal shown in Fig. 1a, with following parameters: ε 1 = 1 , ε 2 = 9 , a 2 = 0.2 L , a 1 = a 3 = 0.3 L , and a 4 = 0.2 L . (a) E-polarization; (b) H-polarization. The solid lines correspond to numerical solutions calculated through the plane-wave expansion method. The dashed lines are for zero order approximation, whereas dot marks are for the first order approximation.

Fig. 6
Fig. 6

Same as Fig. 5, for structure presented in Fig. 1b with ε 1 = 1 , ε 2 = 6.25 , and r = 0.3 L .

Equations (34)

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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 ) ,
x 1 ε ( x , y ) H z ( x , y ) x + y 1 ε ( x , y ) H z ( x , y ) y = ω 2 c 2 H z ( x , y ) ,
2 E ̃ z ( x , y ) x 2 + 2 E ̃ z ( x , y ) y 2 = ε ̃ ( x , y ) ω ̃ E 2 c 2 E ̃ z ( x , y ) ,
x 1 ε ̃ ̃ ( x , y ) H ̃ z ( x , y ) x + y 1 ε ̃ ̃ ( x , y ) H ̃ z ( x , y ) y = ω ̃ H 2 c 2 ε ̃ ( x , y ) ε ̃ ̃ ( x , y ) H ̃ z ( x , y ) .
ε ̃ ( x , y ) = ε ̃ x ( x ) + ε ̃ y ( y ) ,
ε ̃ ̃ ( x , y ) = ε ̃ ̃ x ( x ) ε ̃ ̃ y ( y ) .
R e 0 L x 0 L y [ ε ( x , y ) ε ̃ ( x , y ) ] 2 d x d y = min .
R u x 0 L x 0 L y [   ln   ε ̃ ̃ ( x , y ) x   ln | ε ̃ ( x , y ) | x ] 2 d x d y = min ,
R u y 0 L x 0 L y [   ln   ε ̃ ̃ ( x , y ) y   ln | ε ̃ ( x , y ) | y ] 2 d x d y = min .
[ e 1 e 4 ] = Q 1 A ,
Q = [ L + a 1 a 2 a 3 a 4 a 1 L + a 2 a 3 a 4 a 1 a 2 L + a 3 a 4 a 1 a 2 a 3 L + a 4 ] ,
A = [ 2 ε 1 ( L a 4 ) + 2 ε 2 a 4 2 ε 1 ( a 1 + a 3 ) + 2 ε 2 ( a 2 + a 4 ) 2 ε 1 ( L a 4 ) + 2 ε 2 a 4 2 ε 2 L ] ,
[ ln u 2 u 1 ln u 3 u 2 ln u 4 u 3 ] = 2 L [ ln | e 1 + e 2 2 e 1 | ln | 2 e 2 e 1 + e 2 | ln | e 2 + e 3 e 1 + e 3 | ln | e 2 + e 4 e 1 + e 4 | ln | e 1 + e 3 e 1 + e 2 | ln | e 2 + e 3 2 e 2 | ln | 2 e 3 e 2 + e 3 | ln | e 3 + e 4 e 2 + e 4 | ln | e 1 + e 4 e 1 + e 3 | ln | e 2 + e 4 e 2 + e 3 | ln | e 3 + e 4 2 e 3 | ln | 2 e 4 e 3 + e 4 | ] [ a 1 a 2 a 3 a 4 ] ,
cos ( K x B L ) = i = 1 4 cos ( k i x a i ) + ( k 1 x k 3 x k 2 x k 4 x γ 12 γ 34 + k 2 x k 4 x k 1 x k 3 x γ 12 γ 34 ) i = 1 4 sin ( k i x a i ) 1 2 i , j = 1 i j 4 ( k i x γ i j k j x + γ i j k j x k i x ) sin ( k i x a i ) sin ( k j x a j ) cos ( k p x a p ) cos ( k q x a q ) ,
cos ( K y B L ) = i = 1 4 cos ( k i y a i ) + ( k 1 y k 3 y k 2 y k 4 y γ 12 γ 34 + k 2 y k 4 y k 1 y k 3 y γ 12 γ 34 ) i = 1 4 sin ( k i y a i ) 1 2 i , j = 1 i j 4 ( k i y γ i j k j y + γ i j k j y k i y ) sin ( k i y a i ) sin ( k j y a j ) cos ( k p y a p ) cos ( k q y a q ) ,
A = [ 2 ε 1 S 1 + 2 ε 2 ( L S 1 ) 2 ε 1 ( 2 S 2 + a 2 ) + 2 ε 2 ( L 2 S 2 a 2 ) 2 ε 1 S 3 + 2 ε 2 ( L S 3 ) 2 ε 2 L ] ,
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 ) ,
E z n ( x , y ) = E ̃ z n ( x , y ) + E z n ( 1 ) ( x , y ) ,
H z n ( x , y ) = H ̃ z n ( x , y ) + H z n ( 1 ) ( x , y ) ,
ω n , E ( 1 ) ( K x B , K y B ) = ω ̃ n , E ( K x B , K y B ) 2 0 L x 0 L y Δ ε ( x , y ) | E ̃ z n ( x , y ) | 2 d x d y 0 L x 0 L y ε ̃ ( x , y ) | E ̃ z n ( x , y ) | 2 d x d y ,
ω n , H ( 1 ) ( K x B , K y B ) = ω ̃ n , H ( K x B , K y B ) [ 0 L x 0 L y Δ ε ( x , y ) ε ̃ ̃ ( x , y ) | H ̃ z n | 2 + c 2 ω ̃ n , H 2 ( K x B , K y B ) 0 L x 0 L y H ̃ z n D H ̃ z n ] d x d y 2 0 L x 0 L y ε ̃ ( x , y ) ε ̃ ̃ ( x , y ) | H ̃ z n | 2 d x d y ,
D = [ x ( ln ε ̃ ̃ ε ) 1 ε ̃ ̃ x + y ( ln ε ̃ ̃ ε ) 1 ε ̃ ̃ y ] .
T = ( ω ̃ m 1 0 0 0 0 0 0 ω ̃ m M ) c 2 2 ( ω ̃ m 1 N m 1 m 1 0 0 ω ̃ m M N m M m M ) 1 ( P m 1 m 1 P m 1 m M P m M m 1 P m M m M ) ,
P i j = ω ̃ i , E 2 c 2 0 L x 0 L y Δ ε ( x , y ) E ̃ z i E ̃ z j d x d y ,
N i j = 0 L x 0 L y ε ̃ ( x , y ) E ̃ z i E ̃ z j d x d y ,
P i j = ω ̃ i , H 2 c 2 0 L x 0 L y Δ ε ( x , y ) ε ̃ ̃ ( x , y ) H ̃ z i H z j d x d y + 0 L x 0 L y H ̃ z i D H ̃ z j d x d y ,
N i j = 0 L x 0 L y ε ̃ ( x , y ) ε ̃ ̃ ( x , y ) H ̃ z i H z j d x d y .
R e 0 L x 0 L y [ ε ( x , y ) ε ̃ ( x , y ) ] 2 d x d y = min ,
e i 1 ,     i = 1 , , I .
Λ ( e 1 , e 2 , , e I , μ 1 , , μ I ) = 0 L x 0 L y [ ε ( x , y ) ε ̃ ( x , y ) ] 2 d x d y i = 1 I μ i ( e i 1 ) ,
Λ / e i = 0 ,
e i 1 ,
μ i 0 ,
μ i ( e i 1 ) = 0.

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