Abstract

A finite-difference frequency-domain method based on the Yee’s cell is utilized to analyze the band diagrams of two-dimensional photonic crystals with square or triangular lattice. The differential operator is replaced by the compact scheme and the index average scheme is introduced to deal with the curved dielectric interfaces in the unit cell. For the triangular lattice, the hexagonal unit cell is converted into a rectangular one for easier mesh generation. The band diagrams for both square and triangular lattices are obtained and the numerical convergence of computed eigen frequencies is examined and compared with other methods.

© 2004 Optical Society of America

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References

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    [CrossRef]

Appl. Numer. Math. (1)

A. Yefet and E. Turkel, �??Fourth order compact implicit method for the Maxwell equations with discontinuous coefficients,�?? Appl. Numer. Math. 33, 125�??134 (2000).
[CrossRef]

Appl. Phys. (1)

M. Qiu, and S. He, �??A nonorthogonal finite-difference time-domain method for computing the band structure of a two-dimaensional photonic crystal with dielectric and metalic inclusions,�?? Appl. Phys. 87, 8268�??8275 (1992).

Appl. Phys. Lett. (1)

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

Comp. Phys. Commun. (1)

L. Shen, S. He, and A. Xiao, "A finite-difference eigenvalue algorithm for calculating the band structure of a photonic crystal,�?? Comp. Phys. Commun. 143, 213�??221 (2002).
[CrossRef]

IEEE Trans. Antenna Propagat. (1)

K. S. Yee, �??Numerical solution of initial boundary value problems involing Maxwell�??s equations on isotropic media,�?? IEEE Trans. Antenna Propagat. 14, 302�??307 (1966).
[CrossRef]

IEEE. Trans. Microwave Theory Tech. (1)

H. Y. D. Yang, �??Finite difference analysis of 2-D photonic crystals,�?? IEEE. Trans. Microwave Theory Tech. 34, 2688�??2695 (1996).
[CrossRef]

Inst. Elec. Eng. Proc. J. (4)

C. L. Xu, W. P. Huang, M. S. Stern, and S. K. Chaudhuri, �??Full-vectorial mode calculations by finite difference method,�?? Inst. Elec. Eng. Proc. J. 141, 281�??286 (1994).
[CrossRef]

M. J. Robertson, S. Ritchie, and P. Dayan, �??Semiconductor waveguides: analysis of optical propagation in single rib structures and directional couplers,�?? Inst. Elec. Eng. Proc. J. 132, 336�??342 (1985).

M. S. Stern, �??Semivectorial polarized finite difference method for optical waveguides with arbitrary index pro-files,�?? Inst. Elec. Eng. Proc. J. 135, 56�??63 (1988).

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

J. Lightwave Technol. (8)

K. Bierwirth, N. Schulz, and F. Arndt, �??Finite-difference analysis of rectangular dielectric waveguides by a new finite difference method,�?? J. Lightwave Technol. 34, 1104�??1113 (1986).

P. Lusse, P. Stuwe, J. Schule, and H.-G. Unger, �??Analysis of vectorial mode fields in optical waveguides by a new finite difference method,�?? J. Lightwave Technol. 12, 487�??494 (1994).
[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 (1994).
[CrossRef]

W. P. Huang, C. L. Xu, S. T. Chu, and S. K. Chaudhuri, �??The finite-difference vector beam propagation method. Analysis ans Assessment,�?? J. Lightwave Technol. 10, 295�??305 (1992).
[CrossRef]

C. L. Xu W. P. Huang, and S. K. Chaudhuri, �??Efficient and accurate vector mode calculations by beam propagation method,�?? J. Lightwave Technol. 11, 1209�??1215 (1993).
[CrossRef]

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]

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]

T. Ando, H. Nakayama, S. Numata, J. Yamauchi, and H. Nakano, �??Eigenmode analysis of optical waveguides by a Yee-mesh-based imaginary-distance propagation method for an arbitrary dielectric interface,�?? J. Lightwave Technol. 20, 1627�??1634 (2002).
[CrossRef]

Opt. Express (2)

Phys. Rev. (1)

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

Phys. Rev. B (2)

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

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulous, and O. L. Alerhand, �??Accurate theoretical analysis of photonic band-gap materials,�?? Phys. Rev. B 48, 8434�??8437 (1993).
[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, �??Existance of photonic gap in periodic dielectric structures,�?? Phys. Rev. Lett. 65, 3152�??3155 (1990).
[CrossRef] [PubMed]

Other (1)

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic crystal: modeling the flow of light (Princeton University Press., Princeton, NJ, 1995)

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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 circles for (a) the square lattice and (b) the triangular lattice.

Fig. 2.
Fig. 2.

Yee’s mesh for the (a) TE and (b) TM modes.

Fig. 3.
Fig. 3.

(a) The unit cell of the PC with triangular lattice and its corresponding PBCs. (b) The modified unit cell.

Fig. 4.
Fig. 4.

Band diagrams for the 2-D PC formed by square-arranged alumina rods with r/a=0.2 and ε=8.9 in the air. (a) TE mode and (b) TM mode.

Fig. 5.
Fig. 5.

The first eigen frequency versus the number of grid points for (a) the TE and (b) the TM modes as k is at the M point. The lines with circles and rectangles are the results obtained by our FDFD method without and with the index average scheme, respectively.

Fig. 6.
Fig. 6.

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. 7.
Fig. 7.

Band diagrams of (a) the TE mode and (b) the TM mode for the 2-D PC formed by triangular-arranged dielectric cylinders with r/a=0.2 and ε=11.4 in the air. Our results (triangles) are compared with the results from the FDTD method (circles) and the PWE method (solid lines).

Fig. 8.
Fig. 8.

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

Equations (42)

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

× E = j ω μ 0 μ H
× H = j ω ε 0 ε E
j ω μ 0 μ H z = E y x E x y
j ω ε 0 ε E x = H z y
j ω ε 0 ε E y = H z x .
j ω μ 0 ( μ z H z ) i + 1 2 , j + 1 2 = ( E y x E x y ) i + 1 2 , j + 1 2
j ω ε 0 ( ε x E x ) i + 1 2 , j = ( H z y ) i + 1 2 , j
j ω ε 0 ( ε y E y ) i , j + 1 2 = ( H z x ) i , j + 1 2
H z x i + 1 , j + 1 2 = H z x i + 1 2 , j + 1 2 + ( Δ x 2 ) 1 ! 2 H z x 2 i + 1 2 , j + 1 2
+ ( Δ x 2 ) 2 2 ! 3 H z x 3 i + 1 2 , j + 1 2 + ( Δ x 2 ) 3 3 ! 4 H z x 4 i + 1 2 , j + 1 2 + H. O . T .
H z x i , j + 1 2 = H z x i + 1 2 , j + 1 2 ( Δ x 2 ) 1 ! 2 H z x 2 i + 1 2 , j + 1 2
+ ( Δ x 2 ) 2 2 ! 3 H z x 3 i + 1 2 , j + 1 2 ( Δ x 2 ) 3 3 ! 4 H z x 4 i + 1 2 , j + 1 2 + H. O . T .
H z x i 1 , j + 1 2 = H z x i + 1 2 , j + 1 2 ( 3 Δ x 2 ) 1 ! 2 H z x 2 i + 1 2 , j + 1 2
+ ( 3 Δ x 2 ) 2 2 ! 3 H z x 3 i + 1 2 , j + 1 2 ( 3 Δ x 2 ) 3 3 ! 4 H z x 4 i + 1 2 , j + 1 2 + H. O . T .
H z i 1 2 , j + 1 2 = H z i + 1 2 , j + 1 2 ( Δ x ) 1 ! H z x i + 1 2 , j + 1 2 + ( Δ x ) 2 2 ! 2 H z x 2 i + 1 2 , j + 1 2
( Δ x ) 3 3 ! 3 H z x 3 i + 1 2 , j + 1 2 + ( Δ x ) 4 4 ! 4 H z x 4 i + 1 2 , j + 1 2 + H. O . T .
H z i + 1 2 , j + 1 2 H z i 1 2 , j + 1 2 =
Δ x 24 ( H z x i + 1 , j + 1 2 + 22 H z x i , j + 1 2 + H z x i 1 , j + 1 2 ) .
H z i + 1 2 , j + 1 2 H z i + 1 2 , j 1 2 =
Δ y 24 ( H z y i + 1 2 , j + 1 + 22 H z y i + 1 2 , j + H z y i + 1 2 , j 1 )
E x i + 1 2 , j + 1 E x i + 1 2 , j =
Δ y 24 ( E x y i + 1 2 , j + 3 2 + 22 E x y i + 1 2 , j + 1 2 + E x y i + 1 2 , j 1 2 )
E y i + 1 , j + 1 2 E y i 1 , j + 1 2 =
Δ x 24 ( E y x i + 3 2 , j + 1 2 + 22 E y x i + 1 2 , j + 1 2 + E y x i 1 2 , j + 1 2 ) .
U · [ H z x ] = V x H z
V · [ H z y ] = V y H z
V · [ E x y ] = U y E x
U · [ E y x ] = U x E y
j ω [ μ 0 μ z 0 0 0 ε 0 ε x 0 0 0 ε 0 ε y ] [ H z E x E y ] =
[ 0 V 1 U y U 1 U x V 1 V y 0 0 U 1 V x 0 0 ] [ H z E x E y ]
k 0 2 H z = μ z 1 { V 1 U y ε x 1 V 1 V y + U 1 U x ε y 1 U 1 V x } H z
j ω ε 0 ε E z = H y x H x y
j ω μ 0 μ H x = E z y
j ω μ 0 μ H y = E z x
j ω [ ε 0 ε z 0 0 0 μ 0 μ x 0 0 0 μ 0 μ y ] [ E z H x H y ] =
[ 0 V 1 V y U 1 V x V 1 U y 0 0 U 1 U x 0 0 ] [ E z H x H y ]
k 0 2 E z = ε z 1 { V 1 V y μ x 1 V 1 U y + U 1 V x μ y 1 U 1 U x } E z .
Ψ ( x + a , y ) = e jk x a Ψ ( x , y )
Ψ ( x , y + a ) = e jk y a Ψ ( x , y )
PBC 1 : Ψ ( x + 3 a 2 , y a 2 ) = e j ( k x 3 a 2 k y a 2 ) Ψ ( x , y )
PBC 2 : Ψ ( x + 3 a 2 , y + a 2 ) = e j ( k x 3 a 2 + k y a 2 ) Ψ ( x , y )
PBC 3 : Ψ ( x , y + a ) = e jk y a Ψ ( x , y ) .

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