Abstract

An improved finite-difference frequency-domain method based on Taylor series expansion, local coordinate transformation, and boundary condition matching is developed for analyzing the band diagrams of 2-D photonic crystals. This newly proposed scheme can deal with piecewise homogeneous structures with curved dielectric interfaces. It also shows much better performance in modeling structures of highly index contrast ratio. We have verified this improved scheme with frequently-used plane-wave expansion method through the calculation of the band diagrams of a 2-D photonic crystal with square lattice. We also compare the scheme with the staircase approximation. We further improve the convergence behavior by adopting high-order terms. The result of this adopting is also demonstrated.

© 2008 IEEE

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  1. E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (1987).
  2. S. John, "Strong localization of photons in certain disordered dielectric superlattices," Phys. Rev. Lett. 58, 2486-2489 (1987).
  3. K. M. Ho, C. T. Chan, C. M. Soukoulis, "Existence of a photonic gap in periodic dielectric structures," Phys. Rev. Lett. 65, 3152-3155 (1990).
  4. M. Plihal, A. A. Maradudin, "Photonic band structure of two-dimensional systems: The triangular lattice," Phys. Rev. B 44, 8565-8571 (1991).
  5. R. D. Meade, K. D. Brommer, A. M. Rappe, J. D. Joannopoulos, "Existence of a photonic band gap in two dimensions," Appl. Phys. Lett. 61, 495-497 (1992).
  6. C. T. Chan, Q. L. Yu, K. M. Ho, "Order-N spectral method for electromagnetic waves," Phys. Rev. B 51, 16635-16642 (1995).
  7. M. Qiu, 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," J. Appl. Phys. 87, 8268-8275 (2000).
  8. H. Y. D. Yang, "Finite difference analysis of 2-D photonic crystals," IEEE Trans. Microw. Theory Technol. 44, 2688-2695 (1996).
  9. L. Shen, S. He, S. Xiao, "A finite-difference eigenvalue algorithm for calculating the band structure of a photonic crystal," Comput. Phys. Commun. 143, 213-221 (2002).
  10. K. Bierwirth, N. Schulz, F. Arndt, "Finite-difference analysis of rectangular dielectric waveguide structures," IEEE Trans. Microw. Theory Technol. 34, 1104-1114 (1986).
  11. G. R. Hadley, R. E. Smith, "Full-vector waveguide modeling using an iterative finite-difference method with transparent boundary conditions," J. Lightw. Technol. 13, 465-469 (1995).
  12. W. P. Huang, C. L. Xu, S. T. Chu, S. K. Chaudhuri, "The finite-difference vector beam propagation method: Analysis and assessment," IEEE J. Lightw. Technol. 10, 295-305 (1992).
  13. Z. Zhu, T. G. Brown, "Full-vectorial finite-difference analysis of microstructured optical fibers," Opt. Express 10, 853-864 (2002).
  14. M. S. Stern, "Semivectorial polarized finite difference method for optical waveguides with arbitrary index profiles," Inst. Elect. Eng. Proc.–J. Optoelectron. 135, 56-63 (1988).
  15. C. Vassallo, "Improvement of finite difference methods for step-index optical waveguides," Inst. Elec. Eng. Proc.–J. 139, 137-142 (1992).
  16. Y.-P. Chiou, Y.-C. Chiang, H.-C. Chang, "Improved three-point formulas considering the interface conditions in the finite-difference analysis of step-index optical devices," IEEE J. Lightw. Technol. 18, 243-251 (2000).
  17. J. Xia, J. Yu, "New finite-difference scheme for simulations of step-index waveguides with tilt interfaces," IEEE Photon. Technol. Lett. 15, 1237-1239 (2003).
  18. Y.-C. Chiang, Y.-P. Chiou, H.-C. Chang, "Improved full-vectorial finite-difference mode solver for optical waveguides with step-index profiles," IEEE J. Lightw. Technol. 20, 1609-1618 (2002).
  19. S. G. Johnson, J. D. Joannopoulos, “The MIT photonic-bands package home page,” (2007) http://ab-initio.mit.edu/mpb/.

2003 (1)

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

2002 (3)

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

L. Shen, S. He, S. Xiao, "A finite-difference eigenvalue algorithm for calculating the band structure of a photonic crystal," Comput. Phys. Commun. 143, 213-221 (2002).

Z. Zhu, T. G. Brown, "Full-vectorial finite-difference analysis of microstructured optical fibers," Opt. Express 10, 853-864 (2002).

2000 (2)

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

M. Qiu, 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," J. Appl. Phys. 87, 8268-8275 (2000).

1996 (1)

H. Y. D. Yang, "Finite difference analysis of 2-D photonic crystals," IEEE Trans. Microw. Theory Technol. 44, 2688-2695 (1996).

1995 (2)

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

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

1992 (3)

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

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

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

1991 (1)

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

1990 (1)

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

1988 (1)

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

1987 (2)

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

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

1986 (1)

K. Bierwirth, N. Schulz, F. Arndt, "Finite-difference analysis of rectangular dielectric waveguide structures," IEEE Trans. Microw. Theory Technol. 34, 1104-1114 (1986).

Appl. Phys. Lett. (1)

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

Comput. Phys. Commun. (1)

L. Shen, S. He, S. Xiao, "A finite-difference eigenvalue algorithm for calculating the band structure of a photonic crystal," Comput. Phys. Commun. 143, 213-221 (2002).

IEEE J. Lightw. Technol. (1)

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

IEEE J. Lightw. Technol. (2)

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

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

IEEE Photon. Technol. Lett. (1)

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

IEEE Trans. Microw. Theory Technol. (2)

K. Bierwirth, N. Schulz, F. Arndt, "Finite-difference analysis of rectangular dielectric waveguide structures," IEEE Trans. Microw. Theory Technol. 34, 1104-1114 (1986).

H. Y. D. Yang, "Finite difference analysis of 2-D photonic crystals," IEEE Trans. Microw. Theory Technol. 44, 2688-2695 (1996).

Inst. Elect. Eng. Proc.–J. Optoelectron. (1)

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

Inst. Elec. Eng. Proc.–J. (1)

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

J. Lightw. Technol. (1)

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

J. Appl. Phys. (1)

M. Qiu, 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," J. Appl. Phys. 87, 8268-8275 (2000).

Opt. Express (1)

Phys. Rev. B (1)

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

Phys. Rev. B (1)

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

Phys. Rev. Lett. (3)

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

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

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

Other (1)

S. G. Johnson, J. D. Joannopoulos, “The MIT photonic-bands package home page,” (2007) http://ab-initio.mit.edu/mpb/.

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