Abstract

A fast full-wave technique based on a nonoverlapping domain decomposition implementation of the finite-element method is developed for the simulation of large-scale three-dimensional (3D) photonic crystal devices modeled with millions of unknowns. The technique is highly efficient because it fully exploits the geometrical redundancy found in photonic crystal problems. It solves for the electric field everywhere in the problem domain using higher-order vector basis functions that accurately model the property of the electric field. As a 3D full-wave technique, the proposed method can easily take into account the effect of radiation loss on the device parameters such as quality factor, transmission, and reflection coefficients. The numerical results of various photonic crystal devices are presented to demonstrate the application, efficiency, and capability of this method.

© 2007 Optical Society of America

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References

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  1. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).
  2. S. Johnson and J. D. Joannopoulos, Photonic Crystals: the Road from Theory to Practice (Kluwer Academic, 2002).
  3. A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).
    [CrossRef] [PubMed]
  4. S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, "Channel drop filters in photonic crystals," Opt. Express 3, 4-11 (1998).
    [CrossRef] [PubMed]
  5. S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. J. Khan, C. Manolatou, and H. A. Haus, "Theoretical analysis of channel drop tunneling processes," Phys. Rev. B 59, 15882-15892 (1999).
    [CrossRef]
  6. R. D. Meade, A. M. Rappe, K. M. Brommer, J. D. Joannopoulos, and O. L. Alerhand, "Accurate theoretical analysis of photonic bandgap materials," Phys. Rev. B 48, 8434-8437 (1993).
    [CrossRef]
  7. S. G. Johnson and J. D. Joannopoulos, "Block-iterative frequency-domain methods for Maxwell's equations in a plane wave basis," Opt. Express 8, 173-180 (2001).
    [CrossRef] [PubMed]
  8. P. M. Bell, J. B. Pendry, L. M. Moreno, and A. J. Ward, "A program for calculating photonic band structures and transmission coefficients of complex structures," Comput. Phys. Commun. 85, 306-322 (1995).
    [CrossRef]
  9. M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, "Theoretical investigation of off-plane propagation of electromagnetic waves in two-dimensional photonic crystals," Phys. Rev. B 58, 6791-6794 (1998).
    [CrossRef]
  10. T. Baba, A. Motegi, T. Iwai, N. Fukaya, Y. Watanabe, and A. Sakai, "Light propagation characteristics of straight single-line-defect waveguides in photonic crystal slabs fabricated into a silicon-on-insulator substrate," IEEE J. Quantum Electron. 38, 743-752 (2002).
    [CrossRef]
  11. M. Qiu and S. L. 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).
    [CrossRef]
  12. L. Wu and S. L. He, "Revised finite-difference time-domain algorithm in a nonorthogonal coordinate system and its application to the computation of the band structure of a photonic crystal," J. Appl. Phys. 91, 6499-6506 (2002).
    [CrossRef]
  13. M. N. Vouvakis, Z. Cendes, and J.-F. Lee, "A FEM domain decomposition method for photonic and electromagnetic band gap structures," IEEE Trans. Antennas Propag. 54, 721-733 (2006).
    [CrossRef]
  14. Y. Li and J. M. Jin, "A vector dual-primal finite element tearing and interconnecting method for solving 3-D large-scale electromagnetic problems," IEEE Trans. Antennas Propag. 54, 3000-3009 (2006).
    [CrossRef]
  15. C. Farhat, P. Avery, R. Tezaur, and J. Li, "FETI-DPH: a dual-primal domain decomposition method for acoustic scattering," J. Comput. Acoust. 13, 499-524 (2005).
    [CrossRef]
  16. D. Rixen and C. Farhat, "A simple and efficient extension of a class of substructure based preconditioners to heterogeneous structural mechanics problems," Int. J. Numer. Methods Eng. 44, 489-516 (1999).
    [CrossRef]
  17. Z. Lou, "Time-domain finite-element simulation of large antennas and antenna arrays," Ph.D. (University of Illinois at Urbana-Champaign, 2006).
  18. H. Mosallaei and Y. Rahmat-Samii, "Periodic bandgap and effective dielectric materials in electromagnetics: characterization and applications in nanocavities and waveguides," IEEE Trans. Antennas Propag. 51, 549-563 (2003).
    [CrossRef]
  19. A. Sharkawy, S. Shi, and D. W. Prather, "Implementations of optical vias in high-density photonic crystal optical networks," J. Microlith., Microfab., Microsyst. 2, 300-308 (2003).
    [CrossRef]
  20. S. G. Johnson, C. Manolatou, S. Fan, P. Villeneuve, J. D. Joannopoulos, and H. A. Haus, "Elimination of cross talk in waveguide intersections," Opt. Lett. 23, 1855-1857 (1998).
    [CrossRef]
  21. S. Lan and H. Ishikawa, "Broadband waveguide intersections with low cross talk in photonic crystal circuits," Opt. Lett. 27, 1567-1569 (2002).
    [CrossRef]
  22. Y.-G. Roh, S. Yoon, H. Jeon, S.-H. Han, and Q.-H. Park, "Experimental verification of cross talk reduction in photonic crystal waveguide crossings," Appl. Phys. Lett. 85, 3351-3353 (2004).
    [CrossRef]

2006

M. N. Vouvakis, Z. Cendes, and J.-F. Lee, "A FEM domain decomposition method for photonic and electromagnetic band gap structures," IEEE Trans. Antennas Propag. 54, 721-733 (2006).
[CrossRef]

Y. Li and J. M. Jin, "A vector dual-primal finite element tearing and interconnecting method for solving 3-D large-scale electromagnetic problems," IEEE Trans. Antennas Propag. 54, 3000-3009 (2006).
[CrossRef]

2005

C. Farhat, P. Avery, R. Tezaur, and J. Li, "FETI-DPH: a dual-primal domain decomposition method for acoustic scattering," J. Comput. Acoust. 13, 499-524 (2005).
[CrossRef]

2004

Y.-G. Roh, S. Yoon, H. Jeon, S.-H. Han, and Q.-H. Park, "Experimental verification of cross talk reduction in photonic crystal waveguide crossings," Appl. Phys. Lett. 85, 3351-3353 (2004).
[CrossRef]

2003

H. Mosallaei and Y. Rahmat-Samii, "Periodic bandgap and effective dielectric materials in electromagnetics: characterization and applications in nanocavities and waveguides," IEEE Trans. Antennas Propag. 51, 549-563 (2003).
[CrossRef]

A. Sharkawy, S. Shi, and D. W. Prather, "Implementations of optical vias in high-density photonic crystal optical networks," J. Microlith., Microfab., Microsyst. 2, 300-308 (2003).
[CrossRef]

2002

L. Wu and S. L. He, "Revised finite-difference time-domain algorithm in a nonorthogonal coordinate system and its application to the computation of the band structure of a photonic crystal," J. Appl. Phys. 91, 6499-6506 (2002).
[CrossRef]

T. Baba, A. Motegi, T. Iwai, N. Fukaya, Y. Watanabe, and A. Sakai, "Light propagation characteristics of straight single-line-defect waveguides in photonic crystal slabs fabricated into a silicon-on-insulator substrate," IEEE J. Quantum Electron. 38, 743-752 (2002).
[CrossRef]

S. Lan and H. Ishikawa, "Broadband waveguide intersections with low cross talk in photonic crystal circuits," Opt. Lett. 27, 1567-1569 (2002).
[CrossRef]

2001

2000

M. Qiu and S. L. 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).
[CrossRef]

1999

D. Rixen and C. Farhat, "A simple and efficient extension of a class of substructure based preconditioners to heterogeneous structural mechanics problems," Int. J. Numer. Methods Eng. 44, 489-516 (1999).
[CrossRef]

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. J. Khan, C. Manolatou, and H. A. Haus, "Theoretical analysis of channel drop tunneling processes," Phys. Rev. B 59, 15882-15892 (1999).
[CrossRef]

1998

1996

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

1995

P. M. Bell, J. B. Pendry, L. M. Moreno, and A. J. Ward, "A program for calculating photonic band structures and transmission coefficients of complex structures," Comput. Phys. Commun. 85, 306-322 (1995).
[CrossRef]

1993

R. D. Meade, A. M. Rappe, K. M. Brommer, J. D. Joannopoulos, and O. L. Alerhand, "Accurate theoretical analysis of photonic bandgap materials," Phys. Rev. B 48, 8434-8437 (1993).
[CrossRef]

Appl. Phys. Lett.

Y.-G. Roh, S. Yoon, H. Jeon, S.-H. Han, and Q.-H. Park, "Experimental verification of cross talk reduction in photonic crystal waveguide crossings," Appl. Phys. Lett. 85, 3351-3353 (2004).
[CrossRef]

Comput. Phys. Commun.

P. M. Bell, J. B. Pendry, L. M. Moreno, and A. J. Ward, "A program for calculating photonic band structures and transmission coefficients of complex structures," Comput. Phys. Commun. 85, 306-322 (1995).
[CrossRef]

IEEE J. Quantum Electron.

T. Baba, A. Motegi, T. Iwai, N. Fukaya, Y. Watanabe, and A. Sakai, "Light propagation characteristics of straight single-line-defect waveguides in photonic crystal slabs fabricated into a silicon-on-insulator substrate," IEEE J. Quantum Electron. 38, 743-752 (2002).
[CrossRef]

IEEE Trans. Antennas Propag.

M. N. Vouvakis, Z. Cendes, and J.-F. Lee, "A FEM domain decomposition method for photonic and electromagnetic band gap structures," IEEE Trans. Antennas Propag. 54, 721-733 (2006).
[CrossRef]

Y. Li and J. M. Jin, "A vector dual-primal finite element tearing and interconnecting method for solving 3-D large-scale electromagnetic problems," IEEE Trans. Antennas Propag. 54, 3000-3009 (2006).
[CrossRef]

H. Mosallaei and Y. Rahmat-Samii, "Periodic bandgap and effective dielectric materials in electromagnetics: characterization and applications in nanocavities and waveguides," IEEE Trans. Antennas Propag. 51, 549-563 (2003).
[CrossRef]

Int. J. Numer. Methods Eng.

D. Rixen and C. Farhat, "A simple and efficient extension of a class of substructure based preconditioners to heterogeneous structural mechanics problems," Int. J. Numer. Methods Eng. 44, 489-516 (1999).
[CrossRef]

J. Appl. Phys.

M. Qiu and S. L. 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).
[CrossRef]

L. Wu and S. L. He, "Revised finite-difference time-domain algorithm in a nonorthogonal coordinate system and its application to the computation of the band structure of a photonic crystal," J. Appl. Phys. 91, 6499-6506 (2002).
[CrossRef]

J. Comput. Acoust.

C. Farhat, P. Avery, R. Tezaur, and J. Li, "FETI-DPH: a dual-primal domain decomposition method for acoustic scattering," J. Comput. Acoust. 13, 499-524 (2005).
[CrossRef]

J. Microlith., Microfab., Microsyst.

A. Sharkawy, S. Shi, and D. W. Prather, "Implementations of optical vias in high-density photonic crystal optical networks," J. Microlith., Microfab., Microsyst. 2, 300-308 (2003).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. B

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. J. Khan, C. Manolatou, and H. A. Haus, "Theoretical analysis of channel drop tunneling processes," Phys. Rev. B 59, 15882-15892 (1999).
[CrossRef]

R. D. Meade, A. M. Rappe, K. M. Brommer, J. D. Joannopoulos, and O. L. Alerhand, "Accurate theoretical analysis of photonic bandgap materials," Phys. Rev. B 48, 8434-8437 (1993).
[CrossRef]

M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, "Theoretical investigation of off-plane propagation of electromagnetic waves in two-dimensional photonic crystals," Phys. Rev. B 58, 6791-6794 (1998).
[CrossRef]

Phys. Rev. Lett.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

Other

Z. Lou, "Time-domain finite-element simulation of large antennas and antenna arrays," Ph.D. (University of Illinois at Urbana-Champaign, 2006).

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

S. Johnson and J. D. Joannopoulos, Photonic Crystals: the Road from Theory to Practice (Kluwer Academic, 2002).

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Figures (18)

Fig. 1
Fig. 1

PhC structure decomposed into 19 subdomains with each subdomain containing one unit cell. The small black circles represent the locations of the corner edges.

Fig. 2
Fig. 2

Geometry of an array of dielectric cylinders on a conducting plane with the center cylinder replaced by a monopole.

Fig. 3
Fig. 3

Magnitude of the reflection at the monopole port as a function of frequency.

Fig. 4
Fig. 4

Three-layer PhC nanocavity. (a) Geometry. (b) Normalized magnitude of the electric field at the midplane.

Fig. 5
Fig. 5

Seven-layer PhC nanocavity. (a) Geometry. (b) Normalized magnitude of the electric field at the midplane.

Fig. 6
Fig. 6

Fourteen-layer PhC nanocavity. (a) Geometry. (b) Normalized magnitude of the electric field at the midplane.

Fig. 7
Fig. 7

Normalized magnitude of the electric field in the x z plane through the center of the cavity.

Fig. 8
Fig. 8

Quality factors for three photonic crystal nanocavities.

Fig. 9
Fig. 9

Convergence history for solving three different PhC nanocavity problems.

Fig. 10
Fig. 10

PhC optical circuit. (a) Top view. (b) Normalized real part of the electric field at the midplane.

Fig. 11
Fig. 11

Original PCW intersection with cross talk. (a) Geometry. (b) Normalized real part of the electric field at the midplane.

Fig. 12
Fig. 12

Modified PCW intersection without cross talk. (a) Geometry. The shaded alumina rod in the center has a diameter of d = 1.33 a . (b) Normalized real part of the electric field at the midplane.

Fig. 13
Fig. 13

Systematic illustration of observation planes used in the calculation of power flow.

Fig. 14
Fig. 14

Normalized power transmission and reflection coefficients. (a) Original PCW intersection with cross talk. (b) Modified PCW intersection without cross talk.

Fig. 15
Fig. 15

Original 60° PCW bend. (a) Geometry. (b) Normalized magnitude of the electric field at the midplane.

Fig. 16
Fig. 16

Modified 60° PCW bend. (a) Geometry. The small air hole in the black cell has a radius of r = 0.22 a . (b) Normalized magnitude of the electric field at the midplane.

Fig. 17
Fig. 17

Systematic illustration of observation planes used in the calculation of power flow.

Fig. 18
Fig. 18

Normalized power in the 60° PCW. (a) Original PCW. (b) Modified PCW.

Tables (1)

Tables Icon

Table 1 Computational Statistics of the FETI-DPEM Method for Three Photonic Crystal Nanocavity Simulations

Equations (24)

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× 1 μ r × E s k 0 2 ϵ r E s = j k 0 Z 0 J imp s , in Ω s R 3 ,
n ̂ s × ( × E s ) + j k 0 n ̂ s × ( n ̂ s × E s ) = U s , on Ω ABC s ,
n ̂ s × 1 μ r × × E s = Λ s , on Γ s .
K s E s = f s λ s .
K s = Ω s [ 1 μ r ( × N s ) ( × N s ) T k 0 2 ϵ r N s N s T ] d V + j k 0 Ω ABC s ( n ̂ s × N s ) ( n ̂ s × N s ) T d S ,
f s = j k 0 Z 0 Ω s N s J imp s d V Ω ABC s N s U s d S ,
λ s = Γ s N s ( n ̂ s × 1 μ r × × E s ) d S = Γ s N s Λ s d S ,
E s = [ E V s E I s E c s ] T = [ E r s E c s ] T ,
E = [ E r 1 E r N s E c ] T .
K s = [ K r r s K r c s K r c s T K c c s ] , f s = [ f r s f c s ] .
[ K r r s K r c s K r c s T K c c s ] [ E r s E c s ] = [ f r s λ s f c s ] .
B r s E r s = B r s K r r s 1 ( f r s B r s T λ K r c s E c s ) ,
( K c c s K r c s T K r r s 1 K r c s ) B c s E c = f c s K r c s T K r r s 1 f r s + K r c s T K r r s 1 B r s T λ ,
s = 1 N s B r s E r s = s = 1 N s B r s K r r s 1 ( f r s B r s T λ K r c s B c s E c ) = 0 .
s = 1 N s B c s T ( K c c s K r c s T K r r s 1 K r c s ) B c s E c = s = 1 N s B c s T ( f c s K r c s T K r r s 1 f r s + K r c s T K r r s 1 B r s T λ ) .
[ F r r + F r c K ̃ c c 1 F r c T ] λ = d r F r c K ̃ c c 1 f ̃ c ,
K ̃ c c = s = 1 N s K ̃ c c s = s = 1 N s [ B c s T K c c s B c s ( K r c s B c s ) T K r r s 1 ( K r c s B c s ) ] ,
F r r = s = 1 N s F r r s = s = 1 N s B r s K r r s 1 B r s T ,
F r c = s = 1 N s F r c s = s = 1 N s B r s K r r s 1 K r c s B c s ,
d r = s = 1 N s d r s = s = 1 N s B r s K r r s 1 f r s ,
f ̃ c = s = 1 N s f ̃ c s = s = 1 N s ( B c s T f c s B c s T K r c s T K r r s 1 f r s ) .
( F r r D ) 1 = s = 1 N s B r s [ 0 0 0 S I I s ] B r s T ,
S I I s = K I I s K I V s K V V s 1 K I V s T ,
Q = ω 0 U P ,

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