Abstract

A simple model for two-dimensional photonic crystal devices consists of a finite number of possibly different circular cylinders centered on lattice points of a square or triangular lattice and surrounded by a homogeneous or layered background medium. The Dirichlet-to-Neumann (DtN) map method is a special method for analyzing the scattering of an incident wave by such a structure. It is more efficient than existing numerical or semianalytic methods, such as the finite element method and the multipole method, since it takes advantage of the underlying lattice structure and the simple geometry of the unit cells. The DtN map of a unit cell is a relation between a wave field component and its normal derivative on the cell boundary, and it can be used to avoid further computation inside the unit cell. In this paper, an improved DtN map method is developed by constructing special DtN maps for boundary and corner unit cells using the method of fictitious sources.

© 2012 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 University, 1995).
  2. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).
  3. J. M. Jin, The finite Element Method in Electromagnetics, 2nd ed. (Wiley, 2002).
  4. V. Twersky, “Multiple scattering of radiation by an arbitrary configuration of parallel cylinders,” J. Acoust. Soc. Am. 24, 42–46 (1952).
    [CrossRef]
  5. G. O. Olaofe, “Scattering by an arbitrary configuration of parallel circular cylinders,” J. Opt. Soc. Am. 60, 1233–1236 (1970).
    [CrossRef]
  6. D. Felbacq, G. Tayeb, and D. Maystre, “Scattering by a random set of parallel cylinders,” J. Opt. Soc. Am. 11, 2526–2538 (1994).
    [CrossRef]
  7. 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]
  8. J. Yonekura, M. Ikeda, and T. Baba, “Analysis of finite 2-D photonic crystals of columns and lightwave devices using the scattering matrix method,” J. Lightwave Technol. 17, 1500–1508 (1999).
    [CrossRef]
  9. P. A. Martin, Multiple Scattering (Cambridge University, 2006).
  10. P. Pawliuk and M. Yedlin, “Truncating cylindrical wave modes in two-dimensional multiple scattering,” Opt. Lett. 35, 3997–3999 (2010).
    [CrossRef]
  11. R. Borghi, F. Gori, M. Santasiero, F. Frezza, and G. Schettini, “Plane-wave scattering by a set of perfectly conducting circular cylinders in the presence of a plane surface,” J. Opt. Soc. Am. A 13, 2441–2452 (1996).
    [CrossRef]
  12. S. C. Lee and J. A. Grzesik, “Light scattering by closely spaced parallel cylinders embedded in a semi-infinite dielectric medium,” J. Opt. Soc. Am. A 15, 163–173 (1998).
    [CrossRef]
  13. S. C. Lee, “Light scattering by closely spaced parallel cylinders embedded in a finite dielectric slab,” J. Opt. Soc. Am. A 16, 1350–1361 (1999).
    [CrossRef]
  14. A. Coatanhay and J.-M. Conoir, “Scattering near a plane interface using a generalized method of images approach,” J. Comp. Acous. 12, 233–256 (2004).
    [CrossRef]
  15. F. Frezza, L. Pajewski, C. Ponti, and G. Schettini, “Scattering by dielectric circular cylinders in a dielectric slab,” J. Opt. Soc. Am. A 27, 687–695 (2010).
    [CrossRef]
  16. Z. Hu and Y. Y. Lu, “Efficient analysis of photonic crystal devices by Dirichlet-to-Neumann maps,” Opt. Express 16, 17383–17399 (2008).
    [CrossRef]
  17. Z. Hu and Y. Y. Lu, “Improved Dirichlet-to-Neumann map method for modeling extended photonic crystal devices,” Opt. Quantum Electron. 40, 921–932 (2008).
    [CrossRef]
  18. J. Yuan and Y. Y. Lu, “Dirichlet-to-Neumann map method with boundary cells for photonic crystal devices,” Commun. Comput. Phys. 9, 113–128 (2011).
  19. F. Zolla, R. Petit, and M. Cadilhac, “Electromagnetic theory of diffraction by a system of parallel rods: the method of fictitious sources,” J. Opt. Soc. Am. A 11, 1087–1096 (1994).
    [CrossRef]
  20. Y. Huang and Y. Y. Lu, “Scattering from periodic arrays of cylinders by Dirichlet-to-Neumann maps,” J. Lightwave Technol. 24, 3448–3453 (2006).
    [CrossRef]
  21. S. Li and Y. Y. Lu, “Computing photonic crystal defect modes by Dirichlet-to-Neumann maps,” Opt. Express 15, 14454–14466 (2007).
    [CrossRef]
  22. H. Xie and Y. Y. Lu, “Modeling two-dimensional anisotropic photonic crystals by Dirichlet-to-Neumann maps,” J. Opt. Soc. Am. A 26, 1606–1614 (2009).
    [CrossRef]
  23. Y. Wu and Y. Y. Lu, “Dirichlet-to-Neumann map method for analyzing crossed arrays of circular cylinders,” J. Opt. Soc. Am. B 26, 1984–1993 (2009).
    [CrossRef]
  24. S. Li and Y. Y. Lu, “Efficient method for computing leaky modes in two-dimensional photonic crystal waveguides,” J. Lightwave Technol. 28, 978–983 (2010).
    [CrossRef]
  25. L. Yuan and Y. Y. Lu, “Dirichlet-to-Neumann map method for analyzing hole arrays in a slab,” J. Opt. Soc. Am. B 27, 2568–2579 (2010).
    [CrossRef]
  26. J. Yuan and Y. Y. Lu, “Computing photonic band structures by Dirichlet-to-Neumann maps: the triangular lattice,” Opt. Commun. 273, 114–120 (2007).
    [CrossRef]

2011 (1)

J. Yuan and Y. Y. Lu, “Dirichlet-to-Neumann map method with boundary cells for photonic crystal devices,” Commun. Comput. Phys. 9, 113–128 (2011).

2010 (4)

2009 (2)

2008 (2)

Z. Hu and Y. Y. Lu, “Efficient analysis of photonic crystal devices by Dirichlet-to-Neumann maps,” Opt. Express 16, 17383–17399 (2008).
[CrossRef]

Z. Hu and Y. Y. Lu, “Improved Dirichlet-to-Neumann map method for modeling extended photonic crystal devices,” Opt. Quantum Electron. 40, 921–932 (2008).
[CrossRef]

2007 (2)

S. Li and Y. Y. Lu, “Computing photonic crystal defect modes by Dirichlet-to-Neumann maps,” Opt. Express 15, 14454–14466 (2007).
[CrossRef]

J. Yuan and Y. Y. Lu, “Computing photonic band structures by Dirichlet-to-Neumann maps: the triangular lattice,” Opt. Commun. 273, 114–120 (2007).
[CrossRef]

2006 (1)

2004 (1)

A. Coatanhay and J.-M. Conoir, “Scattering near a plane interface using a generalized method of images approach,” J. Comp. Acous. 12, 233–256 (2004).
[CrossRef]

1999 (2)

1998 (1)

1997 (1)

1996 (1)

1994 (2)

1970 (1)

1952 (1)

V. Twersky, “Multiple scattering of radiation by an arbitrary configuration of parallel cylinders,” J. Acoust. Soc. Am. 24, 42–46 (1952).
[CrossRef]

Baba, T.

Borghi, R.

Cadilhac, M.

Coatanhay, A.

A. Coatanhay and J.-M. Conoir, “Scattering near a plane interface using a generalized method of images approach,” J. Comp. Acous. 12, 233–256 (2004).
[CrossRef]

Conoir, J.-M.

A. Coatanhay and J.-M. Conoir, “Scattering near a plane interface using a generalized method of images approach,” J. Comp. Acous. 12, 233–256 (2004).
[CrossRef]

Felbacq, D.

D. Felbacq, G. Tayeb, and D. Maystre, “Scattering by a random set of parallel cylinders,” J. Opt. Soc. Am. 11, 2526–2538 (1994).
[CrossRef]

Frezza, F.

Gori, F.

Grzesik, J. A.

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

Hu, Z.

Z. Hu and Y. Y. Lu, “Improved Dirichlet-to-Neumann map method for modeling extended photonic crystal devices,” Opt. Quantum Electron. 40, 921–932 (2008).
[CrossRef]

Z. Hu and Y. Y. Lu, “Efficient analysis of photonic crystal devices by Dirichlet-to-Neumann maps,” Opt. Express 16, 17383–17399 (2008).
[CrossRef]

Huang, Y.

Ikeda, M.

Jin, J. M.

J. M. Jin, The finite Element Method in Electromagnetics, 2nd ed. (Wiley, 2002).

Joannopoulos, J. D.

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

Lee, S. C.

Li, S.

Lu, Y. Y.

J. Yuan and Y. Y. Lu, “Dirichlet-to-Neumann map method with boundary cells for photonic crystal devices,” Commun. Comput. Phys. 9, 113–128 (2011).

S. Li and Y. Y. Lu, “Efficient method for computing leaky modes in two-dimensional photonic crystal waveguides,” J. Lightwave Technol. 28, 978–983 (2010).
[CrossRef]

L. Yuan and Y. Y. Lu, “Dirichlet-to-Neumann map method for analyzing hole arrays in a slab,” J. Opt. Soc. Am. B 27, 2568–2579 (2010).
[CrossRef]

Y. Wu and Y. Y. Lu, “Dirichlet-to-Neumann map method for analyzing crossed arrays of circular cylinders,” J. Opt. Soc. Am. B 26, 1984–1993 (2009).
[CrossRef]

H. Xie and Y. Y. Lu, “Modeling two-dimensional anisotropic photonic crystals by Dirichlet-to-Neumann maps,” J. Opt. Soc. Am. A 26, 1606–1614 (2009).
[CrossRef]

Z. Hu and Y. Y. Lu, “Efficient analysis of photonic crystal devices by Dirichlet-to-Neumann maps,” Opt. Express 16, 17383–17399 (2008).
[CrossRef]

Z. Hu and Y. Y. Lu, “Improved Dirichlet-to-Neumann map method for modeling extended photonic crystal devices,” Opt. Quantum Electron. 40, 921–932 (2008).
[CrossRef]

J. Yuan and Y. Y. Lu, “Computing photonic band structures by Dirichlet-to-Neumann maps: the triangular lattice,” Opt. Commun. 273, 114–120 (2007).
[CrossRef]

S. Li and Y. Y. Lu, “Computing photonic crystal defect modes by Dirichlet-to-Neumann maps,” Opt. Express 15, 14454–14466 (2007).
[CrossRef]

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

Martin, P. A.

P. A. Martin, Multiple Scattering (Cambridge University, 2006).

Maystre, D.

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]

D. Felbacq, G. Tayeb, and D. Maystre, “Scattering by a random set of parallel cylinders,” J. Opt. Soc. Am. 11, 2526–2538 (1994).
[CrossRef]

Meade, R. D.

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

Olaofe, G. O.

Pajewski, L.

Pawliuk, P.

Petit, R.

Ponti, C.

Santasiero, M.

Schettini, G.

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

Tayeb, G.

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]

D. Felbacq, G. Tayeb, and D. Maystre, “Scattering by a random set of parallel cylinders,” J. Opt. Soc. Am. 11, 2526–2538 (1994).
[CrossRef]

Twersky, V.

V. Twersky, “Multiple scattering of radiation by an arbitrary configuration of parallel cylinders,” J. Acoust. Soc. Am. 24, 42–46 (1952).
[CrossRef]

Winn, J. N.

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

Wu, Y.

Xie, H.

Yedlin, M.

Yonekura, J.

Yuan, J.

J. Yuan and Y. Y. Lu, “Dirichlet-to-Neumann map method with boundary cells for photonic crystal devices,” Commun. Comput. Phys. 9, 113–128 (2011).

J. Yuan and Y. Y. Lu, “Computing photonic band structures by Dirichlet-to-Neumann maps: the triangular lattice,” Opt. Commun. 273, 114–120 (2007).
[CrossRef]

Yuan, L.

Zolla, F.

Commun. Comput. Phys. (1)

J. Yuan and Y. Y. Lu, “Dirichlet-to-Neumann map method with boundary cells for photonic crystal devices,” Commun. Comput. Phys. 9, 113–128 (2011).

J. Acoust. Soc. Am. (1)

V. Twersky, “Multiple scattering of radiation by an arbitrary configuration of parallel cylinders,” J. Acoust. Soc. Am. 24, 42–46 (1952).
[CrossRef]

J. Comp. Acous. (1)

A. Coatanhay and J.-M. Conoir, “Scattering near a plane interface using a generalized method of images approach,” J. Comp. Acous. 12, 233–256 (2004).
[CrossRef]

J. Lightwave Technol. (3)

J. Opt. Soc. Am. (2)

G. O. Olaofe, “Scattering by an arbitrary configuration of parallel circular cylinders,” J. Opt. Soc. Am. 60, 1233–1236 (1970).
[CrossRef]

D. Felbacq, G. Tayeb, and D. Maystre, “Scattering by a random set of parallel cylinders,” J. Opt. Soc. Am. 11, 2526–2538 (1994).
[CrossRef]

J. Opt. Soc. Am. A (7)

J. Opt. Soc. Am. B (2)

Opt. Commun. (1)

J. Yuan and Y. Y. Lu, “Computing photonic band structures by Dirichlet-to-Neumann maps: the triangular lattice,” Opt. Commun. 273, 114–120 (2007).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Opt. Quantum Electron. (1)

Z. Hu and Y. Y. Lu, “Improved Dirichlet-to-Neumann map method for modeling extended photonic crystal devices,” Opt. Quantum Electron. 40, 921–932 (2008).
[CrossRef]

Other (4)

P. A. Martin, Multiple Scattering (Cambridge University, 2006).

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

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

J. M. Jin, The finite Element Method in Electromagnetics, 2nd ed. (Wiley, 2002).

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Figures (4)

Fig. 1.
Fig. 1.

(a) Ten cylinders in a rectangular loop and (b) 13 cylinders in a row.

Fig. 2.
Fig. 2.

Truncated domain and unit cells for Example 1 shown in Fig. 1(a).

Fig. 3.
Fig. 3.

For a boundary unit cell, sources are located on (a) a horizontal line, (b) a U curve, or (c) a V curve. For a corner unit cell, sources are located on (d) a line. Geometric parameters are (a) d=1.3a, w=4.8a; (b) d0=1.6a, d1=a, w=2.25a; (c) d=1.28a, w=1.8a, ϕ=84°; (d) d=2a, w=2a.

Fig. 4.
Fig. 4.

Dielectric slab with 10 holes forming a rectangular loop.

Tables (4)

Tables Icon

Table 1. Maximum Relative Errors of Normal Derivatives of Test Functions on the Edges of a Boundary Unit Cella

Tables Icon

Table 2. Example 1: u(s) at Point A Computed by the Improved and Original DtN Map Methods, the Multipole Method, and the FEM

Tables Icon

Table 3. Example 2: u(s) at Point B Computed by the Improved and Original DtN Map Methods, the Multipole Method, and the FEM

Tables Icon

Table 4. Numerical Solutions for the Dielectric Slab with 10 Holesa

Equations (17)

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

2ux2+2uy2+k02n2(r)u=0,
x[1n2(r)ux]+y[1n2(r)uy]+k02u=0,
u(i)(r)=exp[i(α0x+β0y)]
Λjk[vj1,khj,k1vjkhjk]=[xvj1,kyhj,k1xvjkyhjk],
Λ21[v11(s)v21(s)h21(s)]=[xv11(s)xv21(s)yh21(s)].
Λ11[v11(s)h11(s)]=[xv11(s)yh11(s)].
u(s)(r)j=1p+2p0cjH0(1)(k0n0|rsj|),rΩ(b).
Um(r;t)=Hm(1)(k0n0|rt|)eimθ(rt),
u(s)(r)j=12p0cjH0(1)(k0n0|rsj|),rΩ(c),
(α0,β0)=k0n0(12,12).
u=u*+u(s),
u*(x,y)={u(i)+Rei(α0x+β0y),above the slab,[C1eiβ1y+C2eiβ1y]eiα0x,within the slab,Tei(α0xβ0y),below the slab,
u(s)(r)=l=1Nm=blmHm(1)(k0n0rl)eimθl,
A[b1b2bN]=[IS1T12S1T13S2T21IS2T23S3T31S3T32I][b1b2bN]=[f1f2fN],
(Tlj)mq=Hmq(1)(k0n0rlj)exp[i(qm)θlj],
(Sl)mm=nlJm(ξ)Jm(η)n0Jm(η)Jm(ξ)nlHm(1)(ξ)Jm(η)+n0Jm(η)Hm(1)(ξ),
qlm=imexp[ik0n0rlcos(θlθ(i))imθ(i)]

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