Abstract

We develop a numerical algorithm that computes the Green’s function of Maxwell equation for a 2D finite-size photonic crystal, composed of rods of arbitrary shape. The method is based on the boundary integral equation, and a Nyström discretization is used for the numerical solution. To provide an exact solution that validates our code we derive multipole expansions for circular cylinders using our integral equation approach. The numerical method performs very well on the test case. We then apply it to crystals of arbitrary shape and discuss the convergence.

© 2006 Optical Society of America

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  1. A. A. Asatryan, K. Busch, R. C. McPhedran, L.C. Botten, C. Martijn de Sterke, and N. A. Nicorovici, “Twodimensional Green function and local density of states in photonic crystals consisting of a fnite number of cylinders of infnite length,” Phys Rev. E 63046612 (2001)
    [Crossref]
  2. E. Yablonovitch, “Spontaneous Emission in Solid-State Physics and Electronics,” Phys. Rev. Lett. 8, 2059–2062 (1987)
    [Crossref]
  3. G. Tayeb and D. Maystre, “Rigorous theoretical study of finite-size two-dimensional photonic crystals doped by microcavities,” J. Opt. Soc. Am. A,  14, 3323–3332 (1997)
    [Crossref]
  4. H. Ammari, N. Bŕeux, and E. Bonnetier, “Analysis of the radiation properties of a planar antenna on a photonic crystal substrate,” Math. Methods Appl. Sci. 24, 1021–1042 (2001)
    [Crossref]
  5. A. Z. Elsherbeni and A. A. Kishk, “Modeling of cylindrical objects by circular dielectric and conducting cylinders,” IEEE Trans. Antennas Propag 40, 96–99 (1992)
    [Crossref]
  6. D. Colton and R. Kress, Integral Equation Methods in Scattering Theory (John Wiley, New York, 1983)
  7. R. Kress, Linear Integral Equations (New York: Springer-Verlag, 1989).
    [Crossref]
  8. M.A. Haider, S.P. Shipman, and S. Venakides, “Boundary-integral calculations of two-dimensional electromagnetic scattering in infinite photonic crystal slabs: Channel defects and resonances,” SIAM Journal on Applied Mathematics,  62, 2129–2148 (2002)
    [Crossref]
  9. A. Garcia-Martin, D. Hermann, F. Hagmann, K. Busch, and P. Wlfle, “Defect computations in photonic crystals: a solid state theoretical approach,” Nanotechnology 14, 177–183 (2003)
    [Crossref]
  10. D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer Verlag, 1997)
  11. P.A. Martin and P. Ola, “Boundary integral equations for the scattering of electromagnetic waves by a homogeneous dielectric obstacle,” Proc. R. Soc. Edinb., Sect. A 123, 185–208 (1993)
    [Crossref]
  12. R. Kress, “On the numerical solution of a hypersingular integral equation in scattering theory,” J. Comp. Appl. Math. 61, 345–360 (1995)
    [Crossref]
  13. W. Hackbusch, Multi-Grid Methods and Applications (Springer-Verlag, Berlin, 1985)
  14. J.M. Song and W.C. Chew, “FMM and MLFMA in 3D and Fast Illinois Solver Code,” in Fast and Efficient Algorithms in Computational Electromagnetics, Chew, Jin, Michielssen, and Song, eds. (Norwood, MA: Artech House, 2001)
  15. A.A. Asatryan, K. Busch, R.C. McPhedran, L.C. Botten, C.M. de Sterke, and N.A. Nicorovici, “Two-dimensional Green tensor and local density of states in finite-sized two-dimensional photonic crystals,” Waves Random Media 139–25 (2003)
    [Crossref]
  16. W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, 1995)

2003 (2)

A. Garcia-Martin, D. Hermann, F. Hagmann, K. Busch, and P. Wlfle, “Defect computations in photonic crystals: a solid state theoretical approach,” Nanotechnology 14, 177–183 (2003)
[Crossref]

A.A. Asatryan, K. Busch, R.C. McPhedran, L.C. Botten, C.M. de Sterke, and N.A. Nicorovici, “Two-dimensional Green tensor and local density of states in finite-sized two-dimensional photonic crystals,” Waves Random Media 139–25 (2003)
[Crossref]

2002 (1)

M.A. Haider, S.P. Shipman, and S. Venakides, “Boundary-integral calculations of two-dimensional electromagnetic scattering in infinite photonic crystal slabs: Channel defects and resonances,” SIAM Journal on Applied Mathematics,  62, 2129–2148 (2002)
[Crossref]

2001 (2)

A. A. Asatryan, K. Busch, R. C. McPhedran, L.C. Botten, C. Martijn de Sterke, and N. A. Nicorovici, “Twodimensional Green function and local density of states in photonic crystals consisting of a fnite number of cylinders of infnite length,” Phys Rev. E 63046612 (2001)
[Crossref]

H. Ammari, N. Bŕeux, and E. Bonnetier, “Analysis of the radiation properties of a planar antenna on a photonic crystal substrate,” Math. Methods Appl. Sci. 24, 1021–1042 (2001)
[Crossref]

1997 (1)

1995 (1)

R. Kress, “On the numerical solution of a hypersingular integral equation in scattering theory,” J. Comp. Appl. Math. 61, 345–360 (1995)
[Crossref]

1993 (1)

P.A. Martin and P. Ola, “Boundary integral equations for the scattering of electromagnetic waves by a homogeneous dielectric obstacle,” Proc. R. Soc. Edinb., Sect. A 123, 185–208 (1993)
[Crossref]

1992 (1)

A. Z. Elsherbeni and A. A. Kishk, “Modeling of cylindrical objects by circular dielectric and conducting cylinders,” IEEE Trans. Antennas Propag 40, 96–99 (1992)
[Crossref]

1987 (1)

E. Yablonovitch, “Spontaneous Emission in Solid-State Physics and Electronics,” Phys. Rev. Lett. 8, 2059–2062 (1987)
[Crossref]

Ammari, H.

H. Ammari, N. Bŕeux, and E. Bonnetier, “Analysis of the radiation properties of a planar antenna on a photonic crystal substrate,” Math. Methods Appl. Sci. 24, 1021–1042 (2001)
[Crossref]

Asatryan, A. A.

A. A. Asatryan, K. Busch, R. C. McPhedran, L.C. Botten, C. Martijn de Sterke, and N. A. Nicorovici, “Twodimensional Green function and local density of states in photonic crystals consisting of a fnite number of cylinders of infnite length,” Phys Rev. E 63046612 (2001)
[Crossref]

Asatryan, A.A.

A.A. Asatryan, K. Busch, R.C. McPhedran, L.C. Botten, C.M. de Sterke, and N.A. Nicorovici, “Two-dimensional Green tensor and local density of states in finite-sized two-dimensional photonic crystals,” Waves Random Media 139–25 (2003)
[Crossref]

Bonnetier, E.

H. Ammari, N. Bŕeux, and E. Bonnetier, “Analysis of the radiation properties of a planar antenna on a photonic crystal substrate,” Math. Methods Appl. Sci. 24, 1021–1042 (2001)
[Crossref]

Botten, L.C.

A.A. Asatryan, K. Busch, R.C. McPhedran, L.C. Botten, C.M. de Sterke, and N.A. Nicorovici, “Two-dimensional Green tensor and local density of states in finite-sized two-dimensional photonic crystals,” Waves Random Media 139–25 (2003)
[Crossref]

A. A. Asatryan, K. Busch, R. C. McPhedran, L.C. Botten, C. Martijn de Sterke, and N. A. Nicorovici, “Twodimensional Green function and local density of states in photonic crystals consisting of a fnite number of cylinders of infnite length,” Phys Rev. E 63046612 (2001)
[Crossref]

Breux, N.

H. Ammari, N. Bŕeux, and E. Bonnetier, “Analysis of the radiation properties of a planar antenna on a photonic crystal substrate,” Math. Methods Appl. Sci. 24, 1021–1042 (2001)
[Crossref]

Busch, K.

A. Garcia-Martin, D. Hermann, F. Hagmann, K. Busch, and P. Wlfle, “Defect computations in photonic crystals: a solid state theoretical approach,” Nanotechnology 14, 177–183 (2003)
[Crossref]

A.A. Asatryan, K. Busch, R.C. McPhedran, L.C. Botten, C.M. de Sterke, and N.A. Nicorovici, “Two-dimensional Green tensor and local density of states in finite-sized two-dimensional photonic crystals,” Waves Random Media 139–25 (2003)
[Crossref]

A. A. Asatryan, K. Busch, R. C. McPhedran, L.C. Botten, C. Martijn de Sterke, and N. A. Nicorovici, “Twodimensional Green function and local density of states in photonic crystals consisting of a fnite number of cylinders of infnite length,” Phys Rev. E 63046612 (2001)
[Crossref]

Chew, W. C.

W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, 1995)

Chew, W.C.

J.M. Song and W.C. Chew, “FMM and MLFMA in 3D and Fast Illinois Solver Code,” in Fast and Efficient Algorithms in Computational Electromagnetics, Chew, Jin, Michielssen, and Song, eds. (Norwood, MA: Artech House, 2001)

Colton, D.

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer Verlag, 1997)

D. Colton and R. Kress, Integral Equation Methods in Scattering Theory (John Wiley, New York, 1983)

de Sterke, C. Martijn

A. A. Asatryan, K. Busch, R. C. McPhedran, L.C. Botten, C. Martijn de Sterke, and N. A. Nicorovici, “Twodimensional Green function and local density of states in photonic crystals consisting of a fnite number of cylinders of infnite length,” Phys Rev. E 63046612 (2001)
[Crossref]

de Sterke, C.M.

A.A. Asatryan, K. Busch, R.C. McPhedran, L.C. Botten, C.M. de Sterke, and N.A. Nicorovici, “Two-dimensional Green tensor and local density of states in finite-sized two-dimensional photonic crystals,” Waves Random Media 139–25 (2003)
[Crossref]

Elsherbeni, A. Z.

A. Z. Elsherbeni and A. A. Kishk, “Modeling of cylindrical objects by circular dielectric and conducting cylinders,” IEEE Trans. Antennas Propag 40, 96–99 (1992)
[Crossref]

Garcia-Martin, A.

A. Garcia-Martin, D. Hermann, F. Hagmann, K. Busch, and P. Wlfle, “Defect computations in photonic crystals: a solid state theoretical approach,” Nanotechnology 14, 177–183 (2003)
[Crossref]

Hackbusch, W.

W. Hackbusch, Multi-Grid Methods and Applications (Springer-Verlag, Berlin, 1985)

Hagmann, F.

A. Garcia-Martin, D. Hermann, F. Hagmann, K. Busch, and P. Wlfle, “Defect computations in photonic crystals: a solid state theoretical approach,” Nanotechnology 14, 177–183 (2003)
[Crossref]

Haider, M.A.

M.A. Haider, S.P. Shipman, and S. Venakides, “Boundary-integral calculations of two-dimensional electromagnetic scattering in infinite photonic crystal slabs: Channel defects and resonances,” SIAM Journal on Applied Mathematics,  62, 2129–2148 (2002)
[Crossref]

Hermann, D.

A. Garcia-Martin, D. Hermann, F. Hagmann, K. Busch, and P. Wlfle, “Defect computations in photonic crystals: a solid state theoretical approach,” Nanotechnology 14, 177–183 (2003)
[Crossref]

Kishk, A. A.

A. Z. Elsherbeni and A. A. Kishk, “Modeling of cylindrical objects by circular dielectric and conducting cylinders,” IEEE Trans. Antennas Propag 40, 96–99 (1992)
[Crossref]

Kress, R.

R. Kress, “On the numerical solution of a hypersingular integral equation in scattering theory,” J. Comp. Appl. Math. 61, 345–360 (1995)
[Crossref]

D. Colton and R. Kress, Integral Equation Methods in Scattering Theory (John Wiley, New York, 1983)

R. Kress, Linear Integral Equations (New York: Springer-Verlag, 1989).
[Crossref]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer Verlag, 1997)

Martin, P.A.

P.A. Martin and P. Ola, “Boundary integral equations for the scattering of electromagnetic waves by a homogeneous dielectric obstacle,” Proc. R. Soc. Edinb., Sect. A 123, 185–208 (1993)
[Crossref]

Maystre, D.

McPhedran, R. C.

A. A. Asatryan, K. Busch, R. C. McPhedran, L.C. Botten, C. Martijn de Sterke, and N. A. Nicorovici, “Twodimensional Green function and local density of states in photonic crystals consisting of a fnite number of cylinders of infnite length,” Phys Rev. E 63046612 (2001)
[Crossref]

McPhedran, R.C.

A.A. Asatryan, K. Busch, R.C. McPhedran, L.C. Botten, C.M. de Sterke, and N.A. Nicorovici, “Two-dimensional Green tensor and local density of states in finite-sized two-dimensional photonic crystals,” Waves Random Media 139–25 (2003)
[Crossref]

Nicorovici, N. A.

A. A. Asatryan, K. Busch, R. C. McPhedran, L.C. Botten, C. Martijn de Sterke, and N. A. Nicorovici, “Twodimensional Green function and local density of states in photonic crystals consisting of a fnite number of cylinders of infnite length,” Phys Rev. E 63046612 (2001)
[Crossref]

Nicorovici, N.A.

A.A. Asatryan, K. Busch, R.C. McPhedran, L.C. Botten, C.M. de Sterke, and N.A. Nicorovici, “Two-dimensional Green tensor and local density of states in finite-sized two-dimensional photonic crystals,” Waves Random Media 139–25 (2003)
[Crossref]

Ola, P.

P.A. Martin and P. Ola, “Boundary integral equations for the scattering of electromagnetic waves by a homogeneous dielectric obstacle,” Proc. R. Soc. Edinb., Sect. A 123, 185–208 (1993)
[Crossref]

Shipman, S.P.

M.A. Haider, S.P. Shipman, and S. Venakides, “Boundary-integral calculations of two-dimensional electromagnetic scattering in infinite photonic crystal slabs: Channel defects and resonances,” SIAM Journal on Applied Mathematics,  62, 2129–2148 (2002)
[Crossref]

Song, J.M.

J.M. Song and W.C. Chew, “FMM and MLFMA in 3D and Fast Illinois Solver Code,” in Fast and Efficient Algorithms in Computational Electromagnetics, Chew, Jin, Michielssen, and Song, eds. (Norwood, MA: Artech House, 2001)

Tayeb, G.

Venakides, S.

M.A. Haider, S.P. Shipman, and S. Venakides, “Boundary-integral calculations of two-dimensional electromagnetic scattering in infinite photonic crystal slabs: Channel defects and resonances,” SIAM Journal on Applied Mathematics,  62, 2129–2148 (2002)
[Crossref]

Wlfle, P.

A. Garcia-Martin, D. Hermann, F. Hagmann, K. Busch, and P. Wlfle, “Defect computations in photonic crystals: a solid state theoretical approach,” Nanotechnology 14, 177–183 (2003)
[Crossref]

Yablonovitch, E.

E. Yablonovitch, “Spontaneous Emission in Solid-State Physics and Electronics,” Phys. Rev. Lett. 8, 2059–2062 (1987)
[Crossref]

IEEE Trans. Antennas Propag (1)

A. Z. Elsherbeni and A. A. Kishk, “Modeling of cylindrical objects by circular dielectric and conducting cylinders,” IEEE Trans. Antennas Propag 40, 96–99 (1992)
[Crossref]

J. Comp. Appl. Math. (1)

R. Kress, “On the numerical solution of a hypersingular integral equation in scattering theory,” J. Comp. Appl. Math. 61, 345–360 (1995)
[Crossref]

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

Math. Methods Appl. Sci. (1)

H. Ammari, N. Bŕeux, and E. Bonnetier, “Analysis of the radiation properties of a planar antenna on a photonic crystal substrate,” Math. Methods Appl. Sci. 24, 1021–1042 (2001)
[Crossref]

Nanotechnology (1)

A. Garcia-Martin, D. Hermann, F. Hagmann, K. Busch, and P. Wlfle, “Defect computations in photonic crystals: a solid state theoretical approach,” Nanotechnology 14, 177–183 (2003)
[Crossref]

Phys Rev. E (1)

A. A. Asatryan, K. Busch, R. C. McPhedran, L.C. Botten, C. Martijn de Sterke, and N. A. Nicorovici, “Twodimensional Green function and local density of states in photonic crystals consisting of a fnite number of cylinders of infnite length,” Phys Rev. E 63046612 (2001)
[Crossref]

Phys. Rev. Lett. (1)

E. Yablonovitch, “Spontaneous Emission in Solid-State Physics and Electronics,” Phys. Rev. Lett. 8, 2059–2062 (1987)
[Crossref]

Proc. R. Soc. Edinb., Sect. A (1)

P.A. Martin and P. Ola, “Boundary integral equations for the scattering of electromagnetic waves by a homogeneous dielectric obstacle,” Proc. R. Soc. Edinb., Sect. A 123, 185–208 (1993)
[Crossref]

SIAM Journal on Applied Mathematics (1)

M.A. Haider, S.P. Shipman, and S. Venakides, “Boundary-integral calculations of two-dimensional electromagnetic scattering in infinite photonic crystal slabs: Channel defects and resonances,” SIAM Journal on Applied Mathematics,  62, 2129–2148 (2002)
[Crossref]

Waves Random Media (1)

A.A. Asatryan, K. Busch, R.C. McPhedran, L.C. Botten, C.M. de Sterke, and N.A. Nicorovici, “Two-dimensional Green tensor and local density of states in finite-sized two-dimensional photonic crystals,” Waves Random Media 139–25 (2003)
[Crossref]

Other (6)

W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, 1995)

W. Hackbusch, Multi-Grid Methods and Applications (Springer-Verlag, Berlin, 1985)

J.M. Song and W.C. Chew, “FMM and MLFMA in 3D and Fast Illinois Solver Code,” in Fast and Efficient Algorithms in Computational Electromagnetics, Chew, Jin, Michielssen, and Song, eds. (Norwood, MA: Artech House, 2001)

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer Verlag, 1997)

D. Colton and R. Kress, Integral Equation Methods in Scattering Theory (John Wiley, New York, 1983)

R. Kress, Linear Integral Equations (New York: Springer-Verlag, 1989).
[Crossref]

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Figures (8)

Fig. 1.
Fig. 1.

A finite size photonic crystal consisting of 85 circular cylinders

Fig. 2.
Fig. 2.

The absolute value of the Green’s function on the x-coordinate for a finite-size photonic crystal consisting of 85 circular cylinders. Here λ=3.05, using IEM (solid line) and MEM (‘o’).

Fig. 3.
Fig. 3.

The absolute value of the Green’s function on the x-coordinate for a finite-size photonic crystal consisting of 85 circular cylinders. Here λ=6.05 (green), λ=9.05 (blue), λ=11.25 (red) and λ=8.05 (magnum) using IEM.

Fig. 4.
Fig. 4.

The absolute value of the Green’s function on the x- coordinate for a finite-size photonic crystal consisting of 38 cylinders of square shape (blue) and circular shape (red). Here λ=8.05 using IEM.

Fig. 5.
Fig. 5.

The absolute value of the Green’s function on the x- coordinate for a finite-size photonic crystal consisting of 49 cylinders of square shape (blue) and circular shape (red). Here λ=8.05 using IEM.

Fig. 6.
Fig. 6.

The absolute value of the Green’s function on the x-coordinate for a finite-size photonic crystal consisting of 85 square-shaped cylinders. Here λ=2.25 (red) and λ=8.05 (blue) using IEM.

Fig. 7.
Fig. 7.

Non-circular shapes for which analytical methods can not be derived. On the left a non-convex boundary and on the right a square with corners.

Fig. 8.
Fig. 8.

Convergence test. The absolute value of the Green’s function against Nyström points, at the point (-5,5), for a finite-size photonic crystal consisting of 33 non-convex shape cylinders. Here λ=3d, using the integral equation method.

Equations (10)

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

2 G + ω 2 ε G = δ ( x x s ) ,
2 u l + ω 2 ε l u l = 0 in Ω l , l = 1,2 . . . , M ,
2 u 0 + ω 2 ε 0 u 0 = 0 in Ω 0 , ,
u l + ξ l int Φ l = u 0 + ξ ext Φ 0 and N ( u l + ξ l int Φ l ) = N ( u 0 + ξ ext Φ 0 ) on Γ l , l = 1,2 , . . . M ,
lim x x 1 2 ( u 0 x i ω ε 0 u 0 ) = 0 .
2 u l ( x ) = S l l N u l ( x ) D l l u l ( x ) , x Ω l
0 = S l l N u l ( x ) D l l u l ( x ) , x Ω 0 .
u 0 = l = 1 M D 0 l ϕ l .
( ( I D ̂ l l ) ( D ̂ 0 l I ) + S ̂ l l Q ̂ 0 l ) ϕ l + m = 1 , m l M ( ( I D ̂ l l ) D 0 m , l + S ̂ l l Q 0 m , l ) ϕ m = f
f = A ̂ 0 l ϕ l m = 1 , m l M A 0 m , l ϕ m on Γ l , l = 1,2 . . . , M .

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