Abstract

The periodicity of photonic crystals can be utilized to develop efficient numerical methods for analyzing light waves propagating in these structures. The Dirichlet-to-Neumann (DtN) operator of a unit cell maps the wave field on the boundary of the unit cell to its normal derivative, and it can be used to reduce the computation to the edges of the unit cells. For two-dimensional photonic crystals with complex unit cells, each containing a number of possibly different circular cylinders, we develop an efficient multipole method for constructing the DtN maps. The DtN maps are used to calculate the transmission and reflection spectra for finite photonic crystals with complex unit cells.

© 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. J. Cox and D. C. Dobson, "Maximizing band gaps in two-dimensional photonic crystals," SIAM J. Appl. Math. 59, 2108-2120 (1999).
    [CrossRef]
  3. H. Altug and J. Vuckovic, "Two-dimensional coupled photonic crystal resonator arrays," Appl. Phys. Lett. 84, 161-163 (2004).
    [CrossRef]
  4. K. Amemiya and K. Ohtaka, "Calculation of transmittance of light for an array of dielectric rods using vector cylindrical waves: complex unit cells," J. Phys. Soc. Jpn. 72, 1244-1253 (2003).
    [CrossRef]
  5. P. Lalanne and G. M. Morris, "Highly improved convergence of the coupled-wave method for TM polarization," J. Opt. Soc. Am. A 13, 779-784 (1996).
    [CrossRef]
  6. L. Li, "Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings," J. Opt. Soc. Am. A 13, 1024-1035 (1996).
    [CrossRef]
  7. G. Bao, Z. M. Chen, and H. J. Wu, "Adaptive finite-element method for diffraction gratings," J. Opt. Soc. Am. A 22, 1106-1114 (2005).
    [CrossRef]
  8. R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, and C. M. de Sterke, "Calculation of electromagnetic properties of regular and random arrays of metallic and dielectric cylinders," Phys. Rev. E 60, 7614-7617 (1999).
    [CrossRef]
  9. T. Kushta and K. Yasumoto, "Electromagnetic scattering from periodic arrays of two circular cylinders per unit cell," Prog. Electromagn. Res. 29, 69-85 (2000).
    [CrossRef]
  10. S. Venakides, M. A. Haider, and V. Papanicolaou, "Boundary integral calculations of two-dimensional electromagnetic scattering by photonic crystal Fabry-Perot structures," SIAM J. Appl. Math. 60, 1686-1706 (2000).
    [CrossRef]
  11. Y. X. Huang and Y. Y. Lu, "Scattering from periodic arrays of cylinders by Dirichlet-to-Neumann maps," J. Lightwave Technol. 24, 3448-3453 (2006).
    [CrossRef]
  12. D. Felbacq, G. Tayeb, and D. Maystre, "Scattering by a random set of parallel cylinders," J. Opt. Soc. Am. A 11, 2526-2538 (1994).
    [CrossRef]
  13. J. H. Yuan and Y. Y. Lu, "Photonic bandgap calculations using Dirichlet-to-Neumann maps," J. Opt. Soc. Am. A 23, 3217-3222 (2006).
    [CrossRef]
  14. J. H. Yuan, Y. Y. Lu, and X. Antoine, "Modeling photonic crystals by boundary integral equations and Dirichlet-to-Neumann maps," submitted to J. Comput. Phys.
  15. Y. X. Huang and Y. Y. Lu, "Modeling photonic crystals with complex unit cells by Dirichlet-to-Neumann maps," J. Comput. Math. 25, 227-349 (2007).
  16. K. Sakoda, "Numerical analysis of the interference patterns in the optical transmission spectra of a square photonic lattice," J. Opt. Soc. Am. B 14, 1961-1966 (1997).
    [CrossRef]

2007 (1)

Y. X. Huang and Y. Y. Lu, "Modeling photonic crystals with complex unit cells by Dirichlet-to-Neumann maps," J. Comput. Math. 25, 227-349 (2007).

2006 (2)

2005 (1)

2004 (1)

H. Altug and J. Vuckovic, "Two-dimensional coupled photonic crystal resonator arrays," Appl. Phys. Lett. 84, 161-163 (2004).
[CrossRef]

2003 (1)

K. Amemiya and K. Ohtaka, "Calculation of transmittance of light for an array of dielectric rods using vector cylindrical waves: complex unit cells," J. Phys. Soc. Jpn. 72, 1244-1253 (2003).
[CrossRef]

2000 (2)

T. Kushta and K. Yasumoto, "Electromagnetic scattering from periodic arrays of two circular cylinders per unit cell," Prog. Electromagn. Res. 29, 69-85 (2000).
[CrossRef]

S. Venakides, M. A. Haider, and V. Papanicolaou, "Boundary integral calculations of two-dimensional electromagnetic scattering by photonic crystal Fabry-Perot structures," SIAM J. Appl. Math. 60, 1686-1706 (2000).
[CrossRef]

1999 (2)

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, and C. M. de Sterke, "Calculation of electromagnetic properties of regular and random arrays of metallic and dielectric cylinders," Phys. Rev. E 60, 7614-7617 (1999).
[CrossRef]

S. J. Cox and D. C. Dobson, "Maximizing band gaps in two-dimensional photonic crystals," SIAM J. Appl. Math. 59, 2108-2120 (1999).
[CrossRef]

1997 (1)

1996 (2)

1994 (1)

Altug, H.

H. Altug and J. Vuckovic, "Two-dimensional coupled photonic crystal resonator arrays," Appl. Phys. Lett. 84, 161-163 (2004).
[CrossRef]

Amemiya, K.

K. Amemiya and K. Ohtaka, "Calculation of transmittance of light for an array of dielectric rods using vector cylindrical waves: complex unit cells," J. Phys. Soc. Jpn. 72, 1244-1253 (2003).
[CrossRef]

Antoine, X.

J. H. Yuan, Y. Y. Lu, and X. Antoine, "Modeling photonic crystals by boundary integral equations and Dirichlet-to-Neumann maps," submitted to J. Comput. Phys.

Asatryan, A. A.

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, and C. M. de Sterke, "Calculation of electromagnetic properties of regular and random arrays of metallic and dielectric cylinders," Phys. Rev. E 60, 7614-7617 (1999).
[CrossRef]

Bao, G.

Botten, L. C.

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, and C. M. de Sterke, "Calculation of electromagnetic properties of regular and random arrays of metallic and dielectric cylinders," Phys. Rev. E 60, 7614-7617 (1999).
[CrossRef]

Chen, Z. M.

Cox, S. J.

S. J. Cox and D. C. Dobson, "Maximizing band gaps in two-dimensional photonic crystals," SIAM J. Appl. Math. 59, 2108-2120 (1999).
[CrossRef]

de Sterke, C. M.

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, and C. M. de Sterke, "Calculation of electromagnetic properties of regular and random arrays of metallic and dielectric cylinders," Phys. Rev. E 60, 7614-7617 (1999).
[CrossRef]

Dobson, D. C.

S. J. Cox and D. C. Dobson, "Maximizing band gaps in two-dimensional photonic crystals," SIAM J. Appl. Math. 59, 2108-2120 (1999).
[CrossRef]

Felbacq, D.

Haider, M. A.

S. Venakides, M. A. Haider, and V. Papanicolaou, "Boundary integral calculations of two-dimensional electromagnetic scattering by photonic crystal Fabry-Perot structures," SIAM J. Appl. Math. 60, 1686-1706 (2000).
[CrossRef]

Huang, Y. X.

Y. X. Huang and Y. Y. Lu, "Modeling photonic crystals with complex unit cells by Dirichlet-to-Neumann maps," J. Comput. Math. 25, 227-349 (2007).

Y. X. Huang and Y. Y. Lu, "Scattering from periodic arrays of cylinders by Dirichlet-to-Neumann maps," J. Lightwave Technol. 24, 3448-3453 (2006).
[CrossRef]

Joannopoulos, J. D.

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

Kushta, T.

T. Kushta and K. Yasumoto, "Electromagnetic scattering from periodic arrays of two circular cylinders per unit cell," Prog. Electromagn. Res. 29, 69-85 (2000).
[CrossRef]

Lalanne, P.

Li, L.

Lu, Y. Y.

Y. X. Huang and Y. Y. Lu, "Modeling photonic crystals with complex unit cells by Dirichlet-to-Neumann maps," J. Comput. Math. 25, 227-349 (2007).

J. H. Yuan and Y. Y. Lu, "Photonic bandgap calculations using Dirichlet-to-Neumann maps," J. Opt. Soc. Am. A 23, 3217-3222 (2006).
[CrossRef]

Y. X. Huang and Y. Y. Lu, "Scattering from periodic arrays of cylinders by Dirichlet-to-Neumann maps," J. Lightwave Technol. 24, 3448-3453 (2006).
[CrossRef]

J. H. Yuan, Y. Y. Lu, and X. Antoine, "Modeling photonic crystals by boundary integral equations and Dirichlet-to-Neumann maps," submitted to J. Comput. Phys.

Maystre, D.

McPhedran, R. C.

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, and C. M. de Sterke, "Calculation of electromagnetic properties of regular and random arrays of metallic and dielectric cylinders," Phys. Rev. E 60, 7614-7617 (1999).
[CrossRef]

Meade, R. D.

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

Morris, G. M.

Nicorovici, N. A.

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, and C. M. de Sterke, "Calculation of electromagnetic properties of regular and random arrays of metallic and dielectric cylinders," Phys. Rev. E 60, 7614-7617 (1999).
[CrossRef]

Ohtaka, K.

K. Amemiya and K. Ohtaka, "Calculation of transmittance of light for an array of dielectric rods using vector cylindrical waves: complex unit cells," J. Phys. Soc. Jpn. 72, 1244-1253 (2003).
[CrossRef]

Papanicolaou, V.

S. Venakides, M. A. Haider, and V. Papanicolaou, "Boundary integral calculations of two-dimensional electromagnetic scattering by photonic crystal Fabry-Perot structures," SIAM J. Appl. Math. 60, 1686-1706 (2000).
[CrossRef]

Robinson, P. A.

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, and C. M. de Sterke, "Calculation of electromagnetic properties of regular and random arrays of metallic and dielectric cylinders," Phys. Rev. E 60, 7614-7617 (1999).
[CrossRef]

Sakoda, K.

Tayeb, G.

Venakides, S.

S. Venakides, M. A. Haider, and V. Papanicolaou, "Boundary integral calculations of two-dimensional electromagnetic scattering by photonic crystal Fabry-Perot structures," SIAM J. Appl. Math. 60, 1686-1706 (2000).
[CrossRef]

Vuckovic, J.

H. Altug and J. Vuckovic, "Two-dimensional coupled photonic crystal resonator arrays," Appl. Phys. Lett. 84, 161-163 (2004).
[CrossRef]

Winn, J. N.

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

Wu, H. J.

Yasumoto, K.

T. Kushta and K. Yasumoto, "Electromagnetic scattering from periodic arrays of two circular cylinders per unit cell," Prog. Electromagn. Res. 29, 69-85 (2000).
[CrossRef]

Yuan, J. H.

J. H. Yuan and Y. Y. Lu, "Photonic bandgap calculations using Dirichlet-to-Neumann maps," J. Opt. Soc. Am. A 23, 3217-3222 (2006).
[CrossRef]

J. H. Yuan, Y. Y. Lu, and X. Antoine, "Modeling photonic crystals by boundary integral equations and Dirichlet-to-Neumann maps," submitted to J. Comput. Phys.

Appl. Phys. Lett. (1)

H. Altug and J. Vuckovic, "Two-dimensional coupled photonic crystal resonator arrays," Appl. Phys. Lett. 84, 161-163 (2004).
[CrossRef]

J. Comput. Math. (1)

Y. X. Huang and Y. Y. Lu, "Modeling photonic crystals with complex unit cells by Dirichlet-to-Neumann maps," J. Comput. Math. 25, 227-349 (2007).

J. Lightwave Technol. (1)

J. Opt. Soc. Am. A (5)

J. Opt. Soc. Am. B (1)

J. Phys. Soc. Jpn. (1)

K. Amemiya and K. Ohtaka, "Calculation of transmittance of light for an array of dielectric rods using vector cylindrical waves: complex unit cells," J. Phys. Soc. Jpn. 72, 1244-1253 (2003).
[CrossRef]

Phys. Rev. E (1)

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, and C. M. de Sterke, "Calculation of electromagnetic properties of regular and random arrays of metallic and dielectric cylinders," Phys. Rev. E 60, 7614-7617 (1999).
[CrossRef]

Prog. Electromagn. Res. (1)

T. Kushta and K. Yasumoto, "Electromagnetic scattering from periodic arrays of two circular cylinders per unit cell," Prog. Electromagn. Res. 29, 69-85 (2000).
[CrossRef]

SIAM J. Appl. Math. (2)

S. Venakides, M. A. Haider, and V. Papanicolaou, "Boundary integral calculations of two-dimensional electromagnetic scattering by photonic crystal Fabry-Perot structures," SIAM J. Appl. Math. 60, 1686-1706 (2000).
[CrossRef]

S. J. Cox and D. C. Dobson, "Maximizing band gaps in two-dimensional photonic crystals," SIAM J. Appl. Math. 59, 2108-2120 (1999).
[CrossRef]

Other (2)

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

J. H. Yuan, Y. Y. Lu, and X. Antoine, "Modeling photonic crystals by boundary integral equations and Dirichlet-to-Neumann maps," submitted to J. Comput. Phys.

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Figures (6)

Fig. 1
Fig. 1

Complex unit cell containing a few circular cylinders.

Fig. 2
Fig. 2

Array of two different dielectric cylinders.

Fig. 3
Fig. 3

Reflection spectrum of the array of two different dielectric cylinders.

Fig. 4
Fig. 4

Complex unit cells of a finite photonic crystal structure (air holes in a dielectric medium). Case 1, all four air holes are moved closer to the center; case 2, two lower air holes are moved closer horizontally.

Fig. 5
Fig. 5

Transmission spectra of a finite photonic crystal (air-holes in a dielectric medium). Case 1: four air-holes are moved closer to the center.

Fig. 6
Fig. 6

Transmission spectra of a finite photonic crystal (air holes in a dielectric medium). Case 2, two lower air holes are moved closer together.

Equations (16)

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x 2 u + y 2 u + k 0 2 n 2 u = 0 ,
ϕ k ( i ) ( x , y ) = exp [ i k 0 n 0 ( x cos ρ k + y sin ρ k ) ] ,
ϕ k ( s ) = l = 1 N m = b l m ( k ) Φ l m for Φ l m ( x , y ) = H m ( 1 ) ( ζ ) e i m θ l ,
[ I S 1 T 12 S 1 T 13 S 2 T 21 I S 2 T 23 S 3 T 31 S 3 T 32 I ] [ b 1 ( k ) b 2 ( k ) b 3 ( k ) ] = [ S 1 g 1 ( k ) S 2 g 2 ( k ) S 3 g 3 ( k ) ] ,
( g l ( k ) ) m = i m exp [ i k 0 n 0 d l cos ( ρ k φ l ) i m ρ k ] ,
( T l j ) m q = H m q ( 1 ) ( k 0 n 0 r l j ) e i ( q m ) θ l j ,
( S l ) m m = n l J m ( ξ ) J m ( η ) n 0 J m ( η ) J m ( ξ ) n l H m ( 1 ) ( ξ ) J m ( η ) + n 0 J m ( η ) H m ( 1 ) ( ξ ) ,
ϕ k ( i ) ( x , y ) = i k 0 n 0 exp [ i k 0 n 0 ( x cos ρ k + y sin ρ k ) ] [ cos ρ k sin ρ k ] ,
Φ l m ( x , y ) = k 0 n 0 e i m θ l H m ( 1 ) ( ζ ) [ cos θ l sin θ l ] + i m r l e i m θ l H m ( 1 ) ( ζ ) [ sin θ l cos θ l ] .
u ( i ) ( x , y ) = e i [ α 0 x β 0 ( y D ) ] , y > D ,
u ( r ) ( x , y ) = j = R j e i [ α j x + β j ( y D ) ] , y > D ,
u ( t ) ( x , y ) = j = T j e i ( α j x γ j y ) , y < 0 ,
α j = α 0 + 2 π j L , β j = k 0 2 n t o p 2 α j 2 , γ j = k 0 2 n bot 2 α j 2 ,
Q ( y ) u ( x , y ) = y u ( x , y ) , Y ( y ) u ( x , y ) = u ( x , 0 ) .
M [ u j u j + 1 ] = [ M 11 M 12 M 21 M 22 ] [ u j u j + 1 ] = y [ u j u j + 1 ] ,
Z = [ Q ( y j ) M 11 ] 1 M 12 , Q ( y j + 1 ) = M 22 + M 21 Z , Y ( y j + 1 ) = Y ( y j ) Z .

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