Abstract

The plane-wave-based transfer matrix method with rational function interpolation and higher-order plane-wave incidence is proposed as an efficient calculation approach to simulate three-dimensional photonic crystal devices. As an example, the dispersion relations and quality factors are calculated for resonant cavity arrays embedded in a woodpile photonic crystal. An interesting ultraslow negative group velocity is observed in this structure.

© 2006 Optical Society of America

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  1. E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).
    [CrossRef] [PubMed]
  2. S. John, Phys. Rev. Lett. 58, 2486 (1987).
    [CrossRef] [PubMed]
  3. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton U. Press, 1995).
  4. E. Özbay, G. Tuttle, M. Sigalas, C. M. Soukoulis, and K. M. Ho, Phys. Rev. B 51, 13961 (1995).
    [CrossRef]
  5. A. Chutinan and S. Noda, Appl. Phys. Lett. 75, 3739 (1999).
    [CrossRef]
  6. M. Okano, A. Chutinan, and S. Noda, Phys. Rev. B 66, 165211 (2002).
    [CrossRef]
  7. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, Opt. Lett. 24, 711 (1999).
    [CrossRef]
  8. M. Bayindir, B. Temelkuran, and E. Özbay, Phys. Rev. Lett. 84, 2140 (2000).
    [CrossRef] [PubMed]
  9. H. Altug and J. Vuckovic, Appl. Phys. Lett. 84, 161 (2004).
    [CrossRef]
  10. H. Altug and J. Vuckovic, Appl. Phys. Lett. 86, 111102 (2005).
    [CrossRef]
  11. Z. Y. Li and L. L. Lin, Phys. Rev. E 67, 046607 (2003).
    [CrossRef]
  12. L. L. Lin, Z. Y. Li, and K. M. Ho, J. Appl. Phys. 94, 811 (2003).
    [CrossRef]
  13. M. Li, Z. Y. Li, K. M. Ho, J. R. Cao, and M. Miyawaki, Opt. Lett. 31, 262 (2006).
    [CrossRef] [PubMed]
  14. K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, Solid State Commun. 89, 413 (1994).
    [CrossRef]
  15. V. Heine, Group Theory in Quantum Mechanics (Pergamom, 1960).
  16. The detail of advantages of TMM over the finite-difference time-domain method can be found in Ref. : (1) full bandgap range rational-function interpolation to obtain continuous spectra, (2) no limitation on the length of the PC along the propagation direction.

2006 (1)

2005 (1)

H. Altug and J. Vuckovic, Appl. Phys. Lett. 86, 111102 (2005).
[CrossRef]

2004 (1)

H. Altug and J. Vuckovic, Appl. Phys. Lett. 84, 161 (2004).
[CrossRef]

2003 (2)

Z. Y. Li and L. L. Lin, Phys. Rev. E 67, 046607 (2003).
[CrossRef]

L. L. Lin, Z. Y. Li, and K. M. Ho, J. Appl. Phys. 94, 811 (2003).
[CrossRef]

2002 (1)

M. Okano, A. Chutinan, and S. Noda, Phys. Rev. B 66, 165211 (2002).
[CrossRef]

2000 (1)

M. Bayindir, B. Temelkuran, and E. Özbay, Phys. Rev. Lett. 84, 2140 (2000).
[CrossRef] [PubMed]

1999 (2)

1995 (1)

E. Özbay, G. Tuttle, M. Sigalas, C. M. Soukoulis, and K. M. Ho, Phys. Rev. B 51, 13961 (1995).
[CrossRef]

1994 (1)

K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, Solid State Commun. 89, 413 (1994).
[CrossRef]

1987 (2)

E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).
[CrossRef] [PubMed]

S. John, Phys. Rev. Lett. 58, 2486 (1987).
[CrossRef] [PubMed]

Altug, H.

H. Altug and J. Vuckovic, Appl. Phys. Lett. 86, 111102 (2005).
[CrossRef]

H. Altug and J. Vuckovic, Appl. Phys. Lett. 84, 161 (2004).
[CrossRef]

Bayindir, M.

M. Bayindir, B. Temelkuran, and E. Özbay, Phys. Rev. Lett. 84, 2140 (2000).
[CrossRef] [PubMed]

Biswas, R.

K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, Solid State Commun. 89, 413 (1994).
[CrossRef]

Cao, J. R.

Chan, C. T.

K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, Solid State Commun. 89, 413 (1994).
[CrossRef]

Chutinan, A.

M. Okano, A. Chutinan, and S. Noda, Phys. Rev. B 66, 165211 (2002).
[CrossRef]

A. Chutinan and S. Noda, Appl. Phys. Lett. 75, 3739 (1999).
[CrossRef]

Heine, V.

V. Heine, Group Theory in Quantum Mechanics (Pergamom, 1960).

Ho, K. M.

M. Li, Z. Y. Li, K. M. Ho, J. R. Cao, and M. Miyawaki, Opt. Lett. 31, 262 (2006).
[CrossRef] [PubMed]

L. L. Lin, Z. Y. Li, and K. M. Ho, J. Appl. Phys. 94, 811 (2003).
[CrossRef]

E. Özbay, G. Tuttle, M. Sigalas, C. M. Soukoulis, and K. M. Ho, Phys. Rev. B 51, 13961 (1995).
[CrossRef]

K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, Solid State Commun. 89, 413 (1994).
[CrossRef]

Joannopoulos, J. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton U. Press, 1995).

John, S.

S. John, Phys. Rev. Lett. 58, 2486 (1987).
[CrossRef] [PubMed]

Lee, R. K.

Li, M.

Li, Z. Y.

M. Li, Z. Y. Li, K. M. Ho, J. R. Cao, and M. Miyawaki, Opt. Lett. 31, 262 (2006).
[CrossRef] [PubMed]

L. L. Lin, Z. Y. Li, and K. M. Ho, J. Appl. Phys. 94, 811 (2003).
[CrossRef]

Z. Y. Li and L. L. Lin, Phys. Rev. E 67, 046607 (2003).
[CrossRef]

Lin, L. L.

Z. Y. Li and L. L. Lin, Phys. Rev. E 67, 046607 (2003).
[CrossRef]

L. L. Lin, Z. Y. Li, and K. M. Ho, J. Appl. Phys. 94, 811 (2003).
[CrossRef]

Meade, R. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton U. Press, 1995).

Miyawaki, M.

Noda, S.

M. Okano, A. Chutinan, and S. Noda, Phys. Rev. B 66, 165211 (2002).
[CrossRef]

A. Chutinan and S. Noda, Appl. Phys. Lett. 75, 3739 (1999).
[CrossRef]

Okano, M.

M. Okano, A. Chutinan, and S. Noda, Phys. Rev. B 66, 165211 (2002).
[CrossRef]

Özbay, E.

M. Bayindir, B. Temelkuran, and E. Özbay, Phys. Rev. Lett. 84, 2140 (2000).
[CrossRef] [PubMed]

E. Özbay, G. Tuttle, M. Sigalas, C. M. Soukoulis, and K. M. Ho, Phys. Rev. B 51, 13961 (1995).
[CrossRef]

Scherer, A.

Sigalas, M.

E. Özbay, G. Tuttle, M. Sigalas, C. M. Soukoulis, and K. M. Ho, Phys. Rev. B 51, 13961 (1995).
[CrossRef]

K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, Solid State Commun. 89, 413 (1994).
[CrossRef]

Soukoulis, C. M.

E. Özbay, G. Tuttle, M. Sigalas, C. M. Soukoulis, and K. M. Ho, Phys. Rev. B 51, 13961 (1995).
[CrossRef]

K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, Solid State Commun. 89, 413 (1994).
[CrossRef]

Temelkuran, B.

M. Bayindir, B. Temelkuran, and E. Özbay, Phys. Rev. Lett. 84, 2140 (2000).
[CrossRef] [PubMed]

Tuttle, G.

E. Özbay, G. Tuttle, M. Sigalas, C. M. Soukoulis, and K. M. Ho, Phys. Rev. B 51, 13961 (1995).
[CrossRef]

Vuckovic, J.

H. Altug and J. Vuckovic, Appl. Phys. Lett. 86, 111102 (2005).
[CrossRef]

H. Altug and J. Vuckovic, Appl. Phys. Lett. 84, 161 (2004).
[CrossRef]

Winn, J. N.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton U. Press, 1995).

Xu, Y.

Yablonovitch, E.

E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).
[CrossRef] [PubMed]

Yariv, A.

Appl. Phys. Lett. (3)

A. Chutinan and S. Noda, Appl. Phys. Lett. 75, 3739 (1999).
[CrossRef]

H. Altug and J. Vuckovic, Appl. Phys. Lett. 84, 161 (2004).
[CrossRef]

H. Altug and J. Vuckovic, Appl. Phys. Lett. 86, 111102 (2005).
[CrossRef]

J. Appl. Phys. (1)

L. L. Lin, Z. Y. Li, and K. M. Ho, J. Appl. Phys. 94, 811 (2003).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. B (2)

M. Okano, A. Chutinan, and S. Noda, Phys. Rev. B 66, 165211 (2002).
[CrossRef]

E. Özbay, G. Tuttle, M. Sigalas, C. M. Soukoulis, and K. M. Ho, Phys. Rev. B 51, 13961 (1995).
[CrossRef]

Phys. Rev. E (1)

Z. Y. Li and L. L. Lin, Phys. Rev. E 67, 046607 (2003).
[CrossRef]

Phys. Rev. Lett. (3)

M. Bayindir, B. Temelkuran, and E. Özbay, Phys. Rev. Lett. 84, 2140 (2000).
[CrossRef] [PubMed]

E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).
[CrossRef] [PubMed]

S. John, Phys. Rev. Lett. 58, 2486 (1987).
[CrossRef] [PubMed]

Solid State Commun. (1)

K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, Solid State Commun. 89, 413 (1994).
[CrossRef]

Other (3)

V. Heine, Group Theory in Quantum Mechanics (Pergamom, 1960).

The detail of advantages of TMM over the finite-difference time-domain method can be found in Ref. : (1) full bandgap range rational-function interpolation to obtain continuous spectra, (2) no limitation on the length of the PC along the propagation direction.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton U. Press, 1995).

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Figures (4)

Fig. 1
Fig. 1

x y -plane periodic array of 3D woodpile PC (with cavity in the 13th layer): (a) 5 × 5 supercell of total 25 layers along the z direction with dielectric cladding of refractive index n cl (cladding is not shown); (b) top view of the 13th layer with the cavity of volume of a 0 × a 0 × 0.35 a 0 and n = 2.4 ; (c) irreducible Brillouin zone of ( q x , q y ) . 3D woodpile PC is composed of dielectric square rods of n = 2.4 and width 0.35 a 0 , where a 0 is the distance between two neighbored rods (i.e., the lattice constant).

Fig. 2
Fig. 2

(a) Incidence of a plane wave with wave vector [ k x = q x + G x , k y = q y + G y , k z = ( k 0 2 k x 2 k y 2 ) 1 2 ] upon the 3D PC cavity array (periodic in the x y -plane); solid (dashed) arrows represent propagating (evanescent) waves. (b) Reciprocal lattice for 5 × 5 supercell PC cavity array; G points inside (outside) the dashed circle [with radius k 0 = n cl ω c and center ( q x , q y ) , shown for ω a ( 2 π c ) = 0.43 , n cl = 1.0 , and q x = q y = 0 ] represent propagating (evanescent) waves. (c) C 2 v group character table for the PC cavity array with E, C 2 , σ x , and σ y symmetry operation.[15] (d) Irreducible representations for the e- and h-polarized incident plane waves of order ( i , j ) , i.e., G x = 2 i π ( 5 a 0 ) and G y = 2 j π ( 5 a 0 ) .

Fig. 3
Fig. 3

Transmission spectra for the plane waves incidence of q x = q y = 0 and order ( i , j ) upon the 3D PC cavity array with e-polarization and h-polarization defined by E i j , x 0 = 1 , E i j , y 0 = 0 and E i j , x 0 = 0 , E i j , y 0 = 1 , respectively. Resonant modes A and B are also labeled. The first bandgap in the z direction [i.e., the (0, 0) incidence] opens from ω a ( 2 π c ) = 0.395 to ω a ( 2 π c ) = 0.515 . There is the third resonant mode at normalized frequency ω a ( 2 π c ) = 0.4635 , which is not shown in this figure.

Fig. 4
Fig. 4

(a) Electric field mode profiles for cavity modes A and B at Γ point ( q x = q y = 0 ) . (b) Dispersion relation of both cavity modes in the 3D PC cavity array. Solid curve, Γ X M ; red dashed curve, Γ X M . Note: Γ X M and Γ X M represent different ( q x , q y ) directions as shown in Fig. 1c.

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