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.

© 2009 Optical Society of America

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  1. E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (1987).
    [CrossRef]
  2. S. John, "Strong localization of photons in certain disordered dielectric superlattices," Phys. Rev. Lett. 58, 2486-2489 (1987).
    [CrossRef]
  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]
  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.
  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]
  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.

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]

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

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]

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]

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

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]

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]

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]

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]

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]

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.

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]

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]

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]

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

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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