Abstract

This Letter presents a split-step (SS) finite-difference time-domain (FDTD) method for the efficient analysis of two-dimensional (2-D) photonic crystals (PhCs) with anisotropic media. The proposed SS FDTD method is formulated with perfectly matched layer boundary conditions and caters for inhomogeneous anisotropic media. Furthermore, the proposed method is derived using the efficient SS1 splitting formulas with simpler right-hand sides that are more efficient and easier to implement. A 2-D PhC cavity with anisotropic media is used as an example to validate the efficiency of the proposed method.

© 2012 Optical Society of America

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  1. A. Figotin and I. Vitebskiy, Phys. Rev. B 67, 165210 (2003).
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  2. A. Figotin and I. Vitebskiy, Phys. Rev. A 76, 053839 (2007).
    [CrossRef]
  3. E. P. Kosmidou, E. E. Kriezis, and T. D. Tsiboukis, IEEE J. Quantum Electron. 41, 657 (2005).
    [CrossRef]
  4. A. S. Vala, B. Rezaei, and M. Kalafi, Physica B 405, 2996 (2010).
    [CrossRef]
  5. M. B. Panah, M. S. Abrishamian, and S. A. Mirtaheri, J. Opt. 13, 015103 (2011).
    [CrossRef]
  6. K.-Y. Jung, B. Donderici, and F. L. Teixeira, Phys. Rev. B 74, 165207 (2006).
    [CrossRef]
  7. J. Shibayama, R. Ando, J. Yamauchi, and H. Nakano, J. Lightwave Technol. 29, 1652 (2011).
    [CrossRef]
  8. K.-Y. Jung, F. L. Teixeira, and R. Lee, IEEE Antennas Wirel. Propag. Lett. 6, 643 (2007).
    [CrossRef]
  9. G. Singh, E. L. Tan, and Z. N. Chen, Opt. Lett. 36, 1494 (2011).
    [CrossRef]
  10. G. Singh, E. L. Tan, and Z. N. Chen, IEEE Trans. Antennas Propag. 58, 3647 (2010).
    [CrossRef]
  11. J.-P. Berenger, J. Comput. Phys. 114, 185 (1994).
    [CrossRef]
  12. F. L. Teixeira and W. C. Chew, IEEE Trans. Antennas Propag. 46, 1386 (1998).
    [CrossRef]
  13. A. P. Zhao and A. V. Raisanen, IEEE Microwave Guided Wave Lett. 8, 15 (1998).
    [CrossRef]
  14. E. L. Tan, IEEE Trans. Antennas Propag. 56, 170 (2008).
    [CrossRef]
  15. G. R. Werner and J. R. Cary, J. Comput. Phys. 226, 1085 (2007).
    [CrossRef]

2011 (3)

2010 (2)

G. Singh, E. L. Tan, and Z. N. Chen, IEEE Trans. Antennas Propag. 58, 3647 (2010).
[CrossRef]

A. S. Vala, B. Rezaei, and M. Kalafi, Physica B 405, 2996 (2010).
[CrossRef]

2008 (1)

E. L. Tan, IEEE Trans. Antennas Propag. 56, 170 (2008).
[CrossRef]

2007 (3)

G. R. Werner and J. R. Cary, J. Comput. Phys. 226, 1085 (2007).
[CrossRef]

A. Figotin and I. Vitebskiy, Phys. Rev. A 76, 053839 (2007).
[CrossRef]

K.-Y. Jung, F. L. Teixeira, and R. Lee, IEEE Antennas Wirel. Propag. Lett. 6, 643 (2007).
[CrossRef]

2006 (1)

K.-Y. Jung, B. Donderici, and F. L. Teixeira, Phys. Rev. B 74, 165207 (2006).
[CrossRef]

2005 (1)

E. P. Kosmidou, E. E. Kriezis, and T. D. Tsiboukis, IEEE J. Quantum Electron. 41, 657 (2005).
[CrossRef]

2003 (1)

A. Figotin and I. Vitebskiy, Phys. Rev. B 67, 165210 (2003).
[CrossRef]

1998 (2)

F. L. Teixeira and W. C. Chew, IEEE Trans. Antennas Propag. 46, 1386 (1998).
[CrossRef]

A. P. Zhao and A. V. Raisanen, IEEE Microwave Guided Wave Lett. 8, 15 (1998).
[CrossRef]

1994 (1)

J.-P. Berenger, J. Comput. Phys. 114, 185 (1994).
[CrossRef]

Abrishamian, M. S.

M. B. Panah, M. S. Abrishamian, and S. A. Mirtaheri, J. Opt. 13, 015103 (2011).
[CrossRef]

Ando, R.

Berenger, J.-P.

J.-P. Berenger, J. Comput. Phys. 114, 185 (1994).
[CrossRef]

Cary, J. R.

G. R. Werner and J. R. Cary, J. Comput. Phys. 226, 1085 (2007).
[CrossRef]

Chen, Z. N.

G. Singh, E. L. Tan, and Z. N. Chen, Opt. Lett. 36, 1494 (2011).
[CrossRef]

G. Singh, E. L. Tan, and Z. N. Chen, IEEE Trans. Antennas Propag. 58, 3647 (2010).
[CrossRef]

Chew, W. C.

F. L. Teixeira and W. C. Chew, IEEE Trans. Antennas Propag. 46, 1386 (1998).
[CrossRef]

Donderici, B.

K.-Y. Jung, B. Donderici, and F. L. Teixeira, Phys. Rev. B 74, 165207 (2006).
[CrossRef]

Figotin, A.

A. Figotin and I. Vitebskiy, Phys. Rev. A 76, 053839 (2007).
[CrossRef]

A. Figotin and I. Vitebskiy, Phys. Rev. B 67, 165210 (2003).
[CrossRef]

Jung, K.-Y.

K.-Y. Jung, F. L. Teixeira, and R. Lee, IEEE Antennas Wirel. Propag. Lett. 6, 643 (2007).
[CrossRef]

K.-Y. Jung, B. Donderici, and F. L. Teixeira, Phys. Rev. B 74, 165207 (2006).
[CrossRef]

Kalafi, M.

A. S. Vala, B. Rezaei, and M. Kalafi, Physica B 405, 2996 (2010).
[CrossRef]

Kosmidou, E. P.

E. P. Kosmidou, E. E. Kriezis, and T. D. Tsiboukis, IEEE J. Quantum Electron. 41, 657 (2005).
[CrossRef]

Kriezis, E. E.

E. P. Kosmidou, E. E. Kriezis, and T. D. Tsiboukis, IEEE J. Quantum Electron. 41, 657 (2005).
[CrossRef]

Lee, R.

K.-Y. Jung, F. L. Teixeira, and R. Lee, IEEE Antennas Wirel. Propag. Lett. 6, 643 (2007).
[CrossRef]

Mirtaheri, S. A.

M. B. Panah, M. S. Abrishamian, and S. A. Mirtaheri, J. Opt. 13, 015103 (2011).
[CrossRef]

Nakano, H.

Panah, M. B.

M. B. Panah, M. S. Abrishamian, and S. A. Mirtaheri, J. Opt. 13, 015103 (2011).
[CrossRef]

Raisanen, A. V.

A. P. Zhao and A. V. Raisanen, IEEE Microwave Guided Wave Lett. 8, 15 (1998).
[CrossRef]

Rezaei, B.

A. S. Vala, B. Rezaei, and M. Kalafi, Physica B 405, 2996 (2010).
[CrossRef]

Shibayama, J.

Singh, G.

G. Singh, E. L. Tan, and Z. N. Chen, Opt. Lett. 36, 1494 (2011).
[CrossRef]

G. Singh, E. L. Tan, and Z. N. Chen, IEEE Trans. Antennas Propag. 58, 3647 (2010).
[CrossRef]

Tan, E. L.

G. Singh, E. L. Tan, and Z. N. Chen, Opt. Lett. 36, 1494 (2011).
[CrossRef]

G. Singh, E. L. Tan, and Z. N. Chen, IEEE Trans. Antennas Propag. 58, 3647 (2010).
[CrossRef]

E. L. Tan, IEEE Trans. Antennas Propag. 56, 170 (2008).
[CrossRef]

Teixeira, F. L.

K.-Y. Jung, F. L. Teixeira, and R. Lee, IEEE Antennas Wirel. Propag. Lett. 6, 643 (2007).
[CrossRef]

K.-Y. Jung, B. Donderici, and F. L. Teixeira, Phys. Rev. B 74, 165207 (2006).
[CrossRef]

F. L. Teixeira and W. C. Chew, IEEE Trans. Antennas Propag. 46, 1386 (1998).
[CrossRef]

Tsiboukis, T. D.

E. P. Kosmidou, E. E. Kriezis, and T. D. Tsiboukis, IEEE J. Quantum Electron. 41, 657 (2005).
[CrossRef]

Vala, A. S.

A. S. Vala, B. Rezaei, and M. Kalafi, Physica B 405, 2996 (2010).
[CrossRef]

Vitebskiy, I.

A. Figotin and I. Vitebskiy, Phys. Rev. A 76, 053839 (2007).
[CrossRef]

A. Figotin and I. Vitebskiy, Phys. Rev. B 67, 165210 (2003).
[CrossRef]

Werner, G. R.

G. R. Werner and J. R. Cary, J. Comput. Phys. 226, 1085 (2007).
[CrossRef]

Yamauchi, J.

Zhao, A. P.

A. P. Zhao and A. V. Raisanen, IEEE Microwave Guided Wave Lett. 8, 15 (1998).
[CrossRef]

IEEE Antennas Wirel. Propag. Lett. (1)

K.-Y. Jung, F. L. Teixeira, and R. Lee, IEEE Antennas Wirel. Propag. Lett. 6, 643 (2007).
[CrossRef]

IEEE J. Quantum Electron. (1)

E. P. Kosmidou, E. E. Kriezis, and T. D. Tsiboukis, IEEE J. Quantum Electron. 41, 657 (2005).
[CrossRef]

IEEE Microwave Guided Wave Lett. (1)

A. P. Zhao and A. V. Raisanen, IEEE Microwave Guided Wave Lett. 8, 15 (1998).
[CrossRef]

IEEE Trans. Antennas Propag. (3)

E. L. Tan, IEEE Trans. Antennas Propag. 56, 170 (2008).
[CrossRef]

G. Singh, E. L. Tan, and Z. N. Chen, IEEE Trans. Antennas Propag. 58, 3647 (2010).
[CrossRef]

F. L. Teixeira and W. C. Chew, IEEE Trans. Antennas Propag. 46, 1386 (1998).
[CrossRef]

J. Comput. Phys. (2)

J.-P. Berenger, J. Comput. Phys. 114, 185 (1994).
[CrossRef]

G. R. Werner and J. R. Cary, J. Comput. Phys. 226, 1085 (2007).
[CrossRef]

J. Lightwave Technol. (1)

J. Opt. (1)

M. B. Panah, M. S. Abrishamian, and S. A. Mirtaheri, J. Opt. 13, 015103 (2011).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (1)

A. Figotin and I. Vitebskiy, Phys. Rev. A 76, 053839 (2007).
[CrossRef]

Phys. Rev. B (2)

A. Figotin and I. Vitebskiy, Phys. Rev. B 67, 165210 (2003).
[CrossRef]

K.-Y. Jung, B. Donderici, and F. L. Teixeira, Phys. Rev. B 74, 165207 (2006).
[CrossRef]

Physica B (1)

A. S. Vala, B. Rezaei, and M. Kalafi, Physica B 405, 2996 (2010).
[CrossRef]

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Figures (2)

Fig. 1.
Fig. 1.

Normalized resonant frequency of PhC cavity versus anisotropy angle ϕ using explicit and SS FDTD method.

Fig. 2.
Fig. 2.

Ez field over PhC for ϕ=30° using SS FDTD method at CFLN=5.

Equations (27)

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

jωϵ0E˜=κ˜×H˜,
jωμ0H˜=ν˜×E˜.
˜=1sxxx^+1syyy^+1szzz^,
jωE˜sζsξ=κϵ0ξξ^×H˜,
Esζt+σξϵ0Esζ=κϵ0ξξ^×H,
Hsζt+σξϵ0Hsζ=νμ0ξξ^×E,
Ut=MU,
U=[Es1,zEs2,zHs1,xHs1,yHs2,xHs2,y]T
Ez=Es1,z+Es2,z,Hx=Hs1,x+Hs2,x,Hy=Hs1,y+Hs2,y.
Input initialization:U^n=D1Un.
Substep 1:(IΔt4A)Vn+12=U^n,
U^n+12=D1Vn+12U^n.
Substep 2:(IΔt4B)Vn+1=U^n+12,
U^n+1=D1Vn+1U^n+12.
Output:Un+1=DU^n+1.
V=[es1,zes2,zhs1,xhs1,yhs2,xhs2,y]T,
A=[0001ϵϵ0x12+σxΔt8ϵ001ϵϵ0x12+σxΔt8ϵ0000000νxyμ0x12+σxΔt8ϵ0νxyμ0x12+σxΔt8ϵ00000νyyμ0x12+σxΔt8ϵ0νyyμ0x12+σxΔt8ϵ00000000000000000],
B=[000000001ϵϵ0y12+σyΔt8ϵ001ϵϵ0y12+σyΔt8ϵ00000000000000νxxμ0y12+σyΔt8ϵ0νxxμ0y12+σyΔt8ϵ00000νyxμ0y12+σyΔt8ϵ0νyxμ0y12+σyΔt8ϵ00000],
D=diag(12+σxΔt8ϵ0,12+σyΔt8ϵ0,12+σxΔt8ϵ0,12+σxΔt8ϵ0,12+σyΔt8ϵ0,12+σyΔt8ϵ0).
βi,jγi12,jyyes1,z|i1,jn+12+(1+βi,jγi12,jyy+βi,jγi+12,jyy)es1,z|i,jn+12βi,jγi+12,jyyes1,z|i+1,jn+12=E^s1,z|i,jn+βi,jγi12,jyyE^s2,z|i1,jn(βi,jγi12,jyy+βi,jγi+12,jyy)E^s2,z|i,jn+βi,jγi+12,jyyE^s2,z|i+1,jn+βi,j(H^s1,y|i+12,jn+H^s2,y|i+12,jnH^s1,y|i12,jnH^s2,y|i12,jn),
βi,j=Δt4ϵ|i,jϵ0Δx12+σx|i,jΔt8ϵ0,
γi+12,jyy=νyy|i+12,jΔt4μ0Δx12+σx|i+12,jΔt8ϵ0.
hs1,y|i+12,jn+12=H^s1,y|i+12,jn+γi+12,jyy(es1,z|i+1,jn+12es1,z|i,jn+12+E^s2,z|i+1,jnE^s2,z|i,jn),
hs1,x|i,j+12n+12=H^s1,x|i,j+12n+{νxyνyy(hs1,yn+12H^s1,yn)}i,j+12,
es2,z|i,jn+12=E^s2,z|i,jn,
hs2,x|i,j+12n+12=H^s2,x|i,j+12n,
hs2,y|i+12,jn+12=H^s2,y|i+12,jn.

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