Abstract

Many photonic crystal (PhC) devices are nonperiodic structures due to the introduced defects in an otherwise perfectly periodic PhC, and they are often connected by PhC waveguides that serve as input and output ports. Numerical simulation of a PhC device requires boundary conditions to terminate PhC waveguides that extend to infinity. The rigorous boundary condition for terminating a PhC waveguide is a nonlocal condition that connects the wave field on the entire surface (or line in two-dimensional problems) transverse to the waveguide axis, and it is relatively difficult to use, especially for realistic devices, such as those in PhC slabs. In this paper, a simple approximate boundary condition involving a few points in the waveguide axis direction is introduced. The new boundary condition is used with the Dirichlet-to-Neumann map method to take advantage of the lattice structures and identical unit cells in PhC devices. Comparisons with the rigorous nonlocal boundary condition indicate that the simple boundary condition gives accurate solutions if the computational domain is enlarged by a few lattice constants in each direction.

© 2012 Optical Society of America

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  1. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed.(Princeton University, 2008).
  2. D. W. Prather, S. Shi, A. Sharkawy, and G. J. Schneider, Photonic Crystals: Theory, Applications, and Fabrication (Wiley, 2009).
  3. A. Taflove and S. C. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method, 2nd ed. (Artech House, 2000).
  4. J. Jin, The Finite Element Method in Electromagnetics, 2nd ed. (Wiley, 2002).
  5. J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
    [CrossRef]
  6. W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretching coordinates,” Microw. Opt. Technol. Lett. 7, 599–604 (1994).
    [CrossRef]
  7. G. O. Olaofe, “Scattering by an arbitrary configuration of parallel circular cylinders,” J. Opt. Soc. Am. 60, 1233–1236 (1970).
    [CrossRef]
  8. 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]
  9. 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]
  10. 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]
  11. P. A. Martin, Multiple Scattering (Cambridge University, 2006).
  12. Z. Hu and Y. Y. Lu, “Efficient analysis of photonic crystal devices by Dirichlet-to-Neumann maps,” Opt. Express 16, 17383–17399 (2008).
  13. 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]
  14. 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]
  15. J. Yuan, Y. Y. Lu, and X. Antoine, “Modeling photonic crystals by boundary integral equations and Dirichlet-to-Neumann maps,” J. Comput. Phys. 227, 4617–4629 (2008).
    [CrossRef]
  16. P. Joly, J.-R. Li, and S. Fliss, “Exact boundary conditions for periodic waveguides containing a local perturbation,” Commun. Comput. Phys. 1, 945–973 (2006).
  17. M. Ehrhardt, J. G. Sun, and C. Zheng, “Evaluation of scattering operators for semi-infinite periodic arrays,” Commun. Math. Sci. 7, 347–364 (2009).
  18. Y. Huang, Y. Y. Lu, and S. Li, “Analyzing photonic crystal waveguides by Dirichlet-to-Neumann maps,” J. Opt. Soc. Am. B 24, 2860–2867 (2007).
    [CrossRef]
  19. 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]
  20. L. Yuan and Y. Y. Lu, “An efficient numerical method for analyzing photonic crystal slab waveguides,” J. Opt. Soc. Am. B 28, 2265–2270 (2011).
    [CrossRef]
  21. A. Mekis, J. C. Chen, I. Kurland, S. H. 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]
  22. M. Koshiba, Y. Tsuji, and M. Hikari, “Time-domain beam propagation method and its application to photonic crystal circuits,” J. Lightwave Technol. 18, 102–110 (2000).
    [CrossRef]
  23. J. Smajic, C. Hafner, and D. Erni, “Design and optimization of an achromatic photonic crystal bend,” Opt. Express 11, 1378–1384 (2003).
    [CrossRef]
  24. C. P. Yu and H. C. Chang, “Applications of the finite difference mode solution method to photonic crystal structures,” Opt. Quantum Electron. 36, 145–163 (2004).
    [CrossRef]
  25. S. H. Fan, S. G. Johnson, J. D. Joannopoulos, C. Manolatou, and H. A. Haus, “Waveguide branches in photonic crystals,” J. Opt. Soc. Am. B 18, 162–165 (2001).
    [CrossRef]
  26. J. Nocedal and S. J. Wright, Numerical Optimization, 2nd ed. (Springer-Verlag, 2006).
  27. 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]
  28. M. Ehrhardt, ed., Wave Propagation in Periodic Media—Analysis, Numerical Techniques and Practical Applications, Progress in Computational Physics, Vol. 1 (Bentham Science, 2010).

2011

2010

2009

M. Ehrhardt, J. G. Sun, and C. Zheng, “Evaluation of scattering operators for semi-infinite periodic arrays,” Commun. Math. Sci. 7, 347–364 (2009).

2008

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, Y. Y. Lu, and X. Antoine, “Modeling photonic crystals by boundary integral equations and Dirichlet-to-Neumann maps,” J. Comput. Phys. 227, 4617–4629 (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).

2007

2006

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]

P. Joly, J.-R. Li, and S. Fliss, “Exact boundary conditions for periodic waveguides containing a local perturbation,” Commun. Comput. Phys. 1, 945–973 (2006).

2004

C. P. Yu and H. C. Chang, “Applications of the finite difference mode solution method to photonic crystal structures,” Opt. Quantum Electron. 36, 145–163 (2004).
[CrossRef]

2003

2001

2000

1999

1997

1996

A. Mekis, J. C. Chen, I. Kurland, S. H. 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]

1994

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]

J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretching coordinates,” Microw. Opt. Technol. Lett. 7, 599–604 (1994).
[CrossRef]

1970

Antoine, X.

J. Yuan, Y. Y. Lu, and X. Antoine, “Modeling photonic crystals by boundary integral equations and Dirichlet-to-Neumann maps,” J. Comput. Phys. 227, 4617–4629 (2008).
[CrossRef]

Baba, T.

Berenger, J.

J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

Chang, H. C.

C. P. Yu and H. C. Chang, “Applications of the finite difference mode solution method to photonic crystal structures,” Opt. Quantum Electron. 36, 145–163 (2004).
[CrossRef]

Chen, J. C.

A. Mekis, J. C. Chen, I. Kurland, S. H. 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]

Chew, W. C.

W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretching coordinates,” Microw. Opt. Technol. Lett. 7, 599–604 (1994).
[CrossRef]

Ehrhardt, M.

M. Ehrhardt, J. G. Sun, and C. Zheng, “Evaluation of scattering operators for semi-infinite periodic arrays,” Commun. Math. Sci. 7, 347–364 (2009).

Erni, D.

Fan, S. H.

S. H. Fan, S. G. Johnson, J. D. Joannopoulos, C. Manolatou, and H. A. Haus, “Waveguide branches in photonic crystals,” J. Opt. Soc. Am. B 18, 162–165 (2001).
[CrossRef]

A. Mekis, J. C. Chen, I. Kurland, S. H. 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]

Felbacq, D.

Fliss, S.

P. Joly, J.-R. Li, and S. Fliss, “Exact boundary conditions for periodic waveguides containing a local perturbation,” Commun. Comput. Phys. 1, 945–973 (2006).

Hafner, C.

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method, 2nd ed. (Artech House, 2000).

Haus, H. A.

Hikari, M.

Hu, Z.

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

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]

Huang, Y.

Ikeda, M.

Jin, J.

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

Joannopoulos, J. D.

S. H. Fan, S. G. Johnson, J. D. Joannopoulos, C. Manolatou, and H. A. Haus, “Waveguide branches in photonic crystals,” J. Opt. Soc. Am. B 18, 162–165 (2001).
[CrossRef]

A. Mekis, J. C. Chen, I. Kurland, S. H. 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]

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed.(Princeton University, 2008).

Johnson, S. G.

S. H. Fan, S. G. Johnson, J. D. Joannopoulos, C. Manolatou, and H. A. Haus, “Waveguide branches in photonic crystals,” J. Opt. Soc. Am. B 18, 162–165 (2001).
[CrossRef]

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed.(Princeton University, 2008).

Joly, P.

P. Joly, J.-R. Li, and S. Fliss, “Exact boundary conditions for periodic waveguides containing a local perturbation,” Commun. Comput. Phys. 1, 945–973 (2006).

Koshiba, M.

Kurland, I.

A. Mekis, J. C. Chen, I. Kurland, S. H. 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]

Li, J.-R.

P. Joly, J.-R. Li, and S. Fliss, “Exact boundary conditions for periodic waveguides containing a local perturbation,” Commun. Comput. Phys. 1, 945–973 (2006).

Li, S.

Lu, Y. Y.

Manolatou, C.

Martin, P. A.

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

Maystre, D.

Meade, R. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed.(Princeton University, 2008).

Mekis, A.

A. Mekis, J. C. Chen, I. Kurland, S. H. 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]

Nocedal, J.

J. Nocedal and S. J. Wright, Numerical Optimization, 2nd ed. (Springer-Verlag, 2006).

Olaofe, G. O.

Prather, D. W.

D. W. Prather, S. Shi, A. Sharkawy, and G. J. Schneider, Photonic Crystals: Theory, Applications, and Fabrication (Wiley, 2009).

Schneider, G. J.

D. W. Prather, S. Shi, A. Sharkawy, and G. J. Schneider, Photonic Crystals: Theory, Applications, and Fabrication (Wiley, 2009).

Sharkawy, A.

D. W. Prather, S. Shi, A. Sharkawy, and G. J. Schneider, Photonic Crystals: Theory, Applications, and Fabrication (Wiley, 2009).

Shi, S.

D. W. Prather, S. Shi, A. Sharkawy, and G. J. Schneider, Photonic Crystals: Theory, Applications, and Fabrication (Wiley, 2009).

Smajic, J.

Sun, J. G.

M. Ehrhardt, J. G. Sun, and C. Zheng, “Evaluation of scattering operators for semi-infinite periodic arrays,” Commun. Math. Sci. 7, 347–364 (2009).

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method, 2nd ed. (Artech House, 2000).

Tayeb, G.

Tsuji, Y.

Villeneuve, P. R.

A. Mekis, J. C. Chen, I. Kurland, S. H. 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]

Weedon, W. H.

W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretching coordinates,” Microw. Opt. Technol. Lett. 7, 599–604 (1994).
[CrossRef]

Winn, J. N.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed.(Princeton University, 2008).

Wright, S. J.

J. Nocedal and S. J. Wright, Numerical Optimization, 2nd ed. (Springer-Verlag, 2006).

Yonekura, J.

Yu, C. P.

C. P. Yu and H. C. Chang, “Applications of the finite difference mode solution method to photonic crystal structures,” Opt. Quantum Electron. 36, 145–163 (2004).
[CrossRef]

Yuan, J.

J. Yuan, Y. Y. Lu, and X. Antoine, “Modeling photonic crystals by boundary integral equations and Dirichlet-to-Neumann maps,” J. Comput. Phys. 227, 4617–4629 (2008).
[CrossRef]

Yuan, L.

Zheng, C.

M. Ehrhardt, J. G. Sun, and C. Zheng, “Evaluation of scattering operators for semi-infinite periodic arrays,” Commun. Math. Sci. 7, 347–364 (2009).

Commun. Comput. Phys.

P. Joly, J.-R. Li, and S. Fliss, “Exact boundary conditions for periodic waveguides containing a local perturbation,” Commun. Comput. Phys. 1, 945–973 (2006).

Commun. Math. Sci.

M. Ehrhardt, J. G. Sun, and C. Zheng, “Evaluation of scattering operators for semi-infinite periodic arrays,” Commun. Math. Sci. 7, 347–364 (2009).

J. Comput. Phys.

J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

J. Yuan, Y. Y. Lu, and X. Antoine, “Modeling photonic crystals by boundary integral equations and Dirichlet-to-Neumann maps,” J. Comput. Phys. 227, 4617–4629 (2008).
[CrossRef]

J. Lightwave Technol.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Microw. Opt. Technol. Lett.

W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretching coordinates,” Microw. Opt. Technol. Lett. 7, 599–604 (1994).
[CrossRef]

Opt. Express

Opt. Quantum Electron.

C. P. Yu and H. C. Chang, “Applications of the finite difference mode solution method to photonic crystal structures,” Opt. Quantum Electron. 36, 145–163 (2004).
[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]

Phys. Rev. Lett.

A. Mekis, J. C. Chen, I. Kurland, S. H. 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]

Other

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed.(Princeton University, 2008).

D. W. Prather, S. Shi, A. Sharkawy, and G. J. Schneider, Photonic Crystals: Theory, Applications, and Fabrication (Wiley, 2009).

A. Taflove and S. C. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method, 2nd ed. (Artech House, 2000).

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

J. Nocedal and S. J. Wright, Numerical Optimization, 2nd ed. (Springer-Verlag, 2006).

M. Ehrhardt, ed., Wave Propagation in Periodic Media—Analysis, Numerical Techniques and Practical Applications, Progress in Computational Physics, Vol. 1 (Bentham Science, 2010).

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

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Figures (4)

Fig. 1.
Fig. 1.

90° bend in a 2D PhC and truncated domain with 11×11 unit cells.

Fig. 2.
Fig. 2.

T-junctions in a PhC with a square lattice of rods. (a) a T-junction of Fan et al. [25], (b) an optimized T-junction for high transmission in a large frequency range.

Fig. 3.
Fig. 3.

(a) Transmission spectra of the first optimized T-junction (Topt) and the T-junction in [25] with rt=0.07a. (b) Transmission spectra of the second optimized T-junction (Topt) and the T-junction in [25] with rt=0.03a.

Fig. 4.
Fig. 4.

Electric field patterns for the second optimized T-junction. Panels (a), (b), (c), and (d) correspond to ωa/(2πc)=0.33, 0.35, 0.38, and 0.41, respectively.

Equations (10)

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

x2u+y2u+k02n2(x,y)u=0,
Λ(jk)[vj1,khj,k1vjkhjk]=[xvj1,kyhj,k1xvjkyhjk],
Λ31(11)v01+Λ32(11)h10+Λ33(11)v11+Λ34(11)h11=Λ11(21)v11+Λ12(21)h20+Λ13(21)v21+Λ13(21)h21,
xu=Lu+(L+L)u(inc),x=x0,
uu(inc)+l=1l*blϕl(x,y)eiβlx,
l=1l*(1ξeiβla)=1+B1ξ+B2ξ2++Bl*ξl*.
wj=w(xj,y)=eiβljaw(x0,y)=(eiβla)jw0.
w0+B1w1+B2w2++Bl*wl*=0.
u0+B1u1++Bl*ul*=u0(inc)+B1u1(inc)++Bl*ul*(inc),
u0eiβ1au1=u0(inc)eiβ1au1(inc).

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