Abstract

The optimization of PhC waveguides is a key issue for successfully designing PhC devices. Since this design task is computationally expensive, efficient methods are demanded. The available codes for computing photonic bands are also applied to PhC waveguides. They are reliable but not very efficient, which is even more pronounced for dispersive material. We present a method based on higher order finite elements with curved cells, which allows to solve for the band structure taking directly into account the dispersiveness of the materials. This is accomplished by reformulating the wave equations as a linear eigenproblem in the complex wave-vectors k. For this method, we demonstrate the high efficiency for the computation of guided PhC waveguide modes by a convergence analysis.

© 2010 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
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  5. J. Li, T. P. White, L. O’Faolain, A. Gomez-Iglesias, and T. F. Krauss, "Systematic design of flat band slow light in photonic crystal waveguides," Opt. Express 16(9), 6227-6232 (2008). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-16-9-6227.
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  7. N. Kono and M. Koshiba, "Three-dimensional finite element analysis of nonreciprocal phase shifts in magneto-photonic crystal waveguides," Opt. Express 13(23), 9155-9166 (2005). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-13-23-9155.
    [CrossRef] [PubMed]
  8. K. Busch, "Photonic band structure theory: assessment and perspectives," C. R. Physique 3(53), 53-66 (2002).
    [CrossRef]
  9. D. C. Dobson, J. Gopalakrishnan, and J. E. Pasciak, "An Efficient Method for Band Structure Calculations in 3D Photonic Crystals," J. Comput. Phys. 161(2), 668-679 (2000).
    [CrossRef]
  10. D. Boffi, M. Conforti, and L. Gastaldi, "Modified edge finite elements for photonic crystals," Numer. Math. 105(2), 249-266 (2006).
    [CrossRef]
  11. K. Schmidt and P. Kauf, "Computation of the band structure of two-dimensional Photonic Crystals with hp Finite Elements," Comp. Meth. App. Mech. Engr. 198, 1249-1259 (2009).
    [CrossRef]
  12. J. Smajic, C. Hafner, K. Rauscher, and D. Erni, "Computation of Radiation Leakage in Photonic Crystal Waveguides," in Proc. Progr. Electromagn. Res. Symp. 2004, Pisa, Italy, pp. 21-24 (2004).
  13. K. Sakoda, Optical Properties of Photonic Crystals, Springer Series in Optical Sciences, 2nd ed. (Springer, Berlin, 2005).
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    [CrossRef]
  15. P. Kuchment and B. Ong, "On guided waves in photonic crystal waveguides," in Waves in Periodic and Random Media, vol. 339 of Contemp. Math., pp. 105-115 (AMS, Providence, USA, 2004).
  16. H. Ammari and F. Santosa, "Guided Waves in a Photonic Bandgap Structure with a Line Defect," SIAM J. Appl. Math. 64(6), 2018-2033 (2004). URL http://link.aip.org/link/?SMM/64/2018/1.
    [CrossRef]
  17. S. G. Johnson and J. D. Joannopoulos, "Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis," Opt. Express 8(3), 173-190 (2001). URL http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173.
    [CrossRef] [PubMed]
  18. A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, London, 1995).
  19. G. Stark, M. Mishrikey, F. Robin, H. Jäckel, C. Hafner, R. Vahdieck, and D. Erni, "Position dependence of FDTD mode detection in photonic crystal systems," Int. J. Numer. Model. 29, 201-218 (2009).
    [CrossRef]
  20. I. Babuška and B. Q. Guo, "The h, p and h-p version of the finite-element method - basic theory and applications," Adv. Eng. Software 15, 159-174 (1992).
    [CrossRef]
  21. C. Schwab, p- and hp- Finite Element Methods - Theory and Applications in Solid and Fluid Mechanics (Oxford Science Publications, 1998).
  22. A. von Rhein, S. Greulich-Weber, and R. B. Wehrspohn, "Berechnung von photonischen Kristallen mit Hilfe des Comsol-Elektromagnetikmoduls," in Proc. of the COMSOL Users Conf. 2006, Frankfurt, Germany, pp. 91-96 (2006).
  23. W. Jiang, R. Chen, and X. Lu, "Theory of light refraction at the surface of a photonic crystal," Phys. Rev. B 71, 245115 (2005).
    [CrossRef]
  24. C. Engström and M. Richter, "On the Spectrum of an Operator Pencil with Applications to Wave Propagation in Periodic and Frequency Dependent Materials," SIAM J. Appl. Math. 70(1), 231-247 (2009). URL http: //link.aip.org/link/?SMM/70/231/1.
    [CrossRef]
  25. C. Engström, C. Hafner, and K. Schmidt, "Computations of lossy Bloch waves in two-dimensional photonic crystals," J. Comput. Theory Nanosci. 6, 775-783 (2009).
    [CrossRef]
  26. P. Frauenfelder and C. Lage, "Concepts - An Object-Oriented Software Package for Partial Differential Equations," Math. Model. Numer. Anal. 36(5), 937-951 (2002). URL http://www.edpsciences.org/ articles/m2an/abs/2002/05/m2anns12/m2anns12.html.
    [CrossRef]
  27. K. Busch, G. Schneider, L. Tkeshelashvili, and H. Uecker, "Justification of the nonlinear Schrödinger equation in spatially periodic media," Z. Angew. Math. Phys. 57(6), 905-939 (2006).
    [CrossRef]
  28. P. Kuchment, Floquet Theory for Partial Differential Equations (Birkhäuser Verlag, Basel, 1993).
    [CrossRef]
  29. P. Kuchment, "The mathematics of photonic crystals," in Mathematical modeling in optical science, vol. 22 of Frontiers Appl. Math., pp. 207-272 (SIAM, Philadelphia, USA, 2001).
    [CrossRef]
  30. F. Tisseur and K. Meerbergen, "The Quadratic Eigenvalue Problem," SIAM Review 43(2), 235-286 (2001).
    [CrossRef]
  31. G. E. Karniadakis and S. J. Sherwin, Spectral/hp Element Methods for Computational Fluid Dynamics (Oxford University Press, Oxford, 2005).
  32. Concepts Development Team, Webpage of Numerical C++ Library Concepts 2 (2009). URL http://www. concepts.math.ethz.ch.
  33. S. Adachi, "Optical properties of In1−xGaxAsyP1−y alloys," Phys. Rev. B 39(17), 12,612-12,621 (1989).
    [CrossRef]
  34. P. Strasser, R. Flückiger, R. Wüest, F. Robin, and H. Jäckel, "InP-based compact photonic crystal directional coupler with large operation range," Opt. Express 15(13), 8472-8478 (2007). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-15-13-8472.
    [CrossRef] [PubMed]

2009

K. Schmidt and P. Kauf, "Computation of the band structure of two-dimensional Photonic Crystals with hp Finite Elements," Comp. Meth. App. Mech. Engr. 198, 1249-1259 (2009).
[CrossRef]

G. Stark, M. Mishrikey, F. Robin, H. Jäckel, C. Hafner, R. Vahdieck, and D. Erni, "Position dependence of FDTD mode detection in photonic crystal systems," Int. J. Numer. Model. 29, 201-218 (2009).
[CrossRef]

C. Engström, C. Hafner, and K. Schmidt, "Computations of lossy Bloch waves in two-dimensional photonic crystals," J. Comput. Theory Nanosci. 6, 775-783 (2009).
[CrossRef]

2006

K. Busch, G. Schneider, L. Tkeshelashvili, and H. Uecker, "Justification of the nonlinear Schrödinger equation in spatially periodic media," Z. Angew. Math. Phys. 57(6), 905-939 (2006).
[CrossRef]

D. Boffi, M. Conforti, and L. Gastaldi, "Modified edge finite elements for photonic crystals," Numer. Math. 105(2), 249-266 (2006).
[CrossRef]

2005

S. Soussi, "Convergence of the supercell method for defect modes calculations in photonic crystals," SIAM J. Numer. Anal. 43, 1175-1201 (2005).
[CrossRef]

W. Jiang, R. Chen, and X. Lu, "Theory of light refraction at the surface of a photonic crystal," Phys. Rev. B 71, 245115 (2005).
[CrossRef]

2002

K. Busch, "Photonic band structure theory: assessment and perspectives," C. R. Physique 3(53), 53-66 (2002).
[CrossRef]

2001

M. Agio and C. M. Soukoulis, "Ministop bands in single-defect photonic crystal waveguides," Phys. Rev. E 64(5), 055603 (2001).
[CrossRef]

F. Tisseur and K. Meerbergen, "The Quadratic Eigenvalue Problem," SIAM Review 43(2), 235-286 (2001).
[CrossRef]

2000

D. C. Dobson, J. Gopalakrishnan, and J. E. Pasciak, "An Efficient Method for Band Structure Calculations in 3D Photonic Crystals," J. Comput. Phys. 161(2), 668-679 (2000).
[CrossRef]

1992

I. Babuška and B. Q. Guo, "The h, p and h-p version of the finite-element method - basic theory and applications," Adv. Eng. Software 15, 159-174 (1992).
[CrossRef]

1989

S. Adachi, "Optical properties of In1−xGaxAsyP1−y alloys," Phys. Rev. B 39(17), 12,612-12,621 (1989).
[CrossRef]

Adachi, S.

S. Adachi, "Optical properties of In1−xGaxAsyP1−y alloys," Phys. Rev. B 39(17), 12,612-12,621 (1989).
[CrossRef]

Agio, M.

M. Agio and C. M. Soukoulis, "Ministop bands in single-defect photonic crystal waveguides," Phys. Rev. E 64(5), 055603 (2001).
[CrossRef]

Babuška, I.

I. Babuška and B. Q. Guo, "The h, p and h-p version of the finite-element method - basic theory and applications," Adv. Eng. Software 15, 159-174 (1992).
[CrossRef]

Boffi, D.

D. Boffi, M. Conforti, and L. Gastaldi, "Modified edge finite elements for photonic crystals," Numer. Math. 105(2), 249-266 (2006).
[CrossRef]

Busch, K.

K. Busch, G. Schneider, L. Tkeshelashvili, and H. Uecker, "Justification of the nonlinear Schrödinger equation in spatially periodic media," Z. Angew. Math. Phys. 57(6), 905-939 (2006).
[CrossRef]

K. Busch, "Photonic band structure theory: assessment and perspectives," C. R. Physique 3(53), 53-66 (2002).
[CrossRef]

Chen, R.

W. Jiang, R. Chen, and X. Lu, "Theory of light refraction at the surface of a photonic crystal," Phys. Rev. B 71, 245115 (2005).
[CrossRef]

Conforti, M.

D. Boffi, M. Conforti, and L. Gastaldi, "Modified edge finite elements for photonic crystals," Numer. Math. 105(2), 249-266 (2006).
[CrossRef]

Dobson, D. C.

D. C. Dobson, J. Gopalakrishnan, and J. E. Pasciak, "An Efficient Method for Band Structure Calculations in 3D Photonic Crystals," J. Comput. Phys. 161(2), 668-679 (2000).
[CrossRef]

Engström, C.

C. Engström, C. Hafner, and K. Schmidt, "Computations of lossy Bloch waves in two-dimensional photonic crystals," J. Comput. Theory Nanosci. 6, 775-783 (2009).
[CrossRef]

Erni, D.

G. Stark, M. Mishrikey, F. Robin, H. Jäckel, C. Hafner, R. Vahdieck, and D. Erni, "Position dependence of FDTD mode detection in photonic crystal systems," Int. J. Numer. Model. 29, 201-218 (2009).
[CrossRef]

Gastaldi, L.

D. Boffi, M. Conforti, and L. Gastaldi, "Modified edge finite elements for photonic crystals," Numer. Math. 105(2), 249-266 (2006).
[CrossRef]

Gopalakrishnan, J.

D. C. Dobson, J. Gopalakrishnan, and J. E. Pasciak, "An Efficient Method for Band Structure Calculations in 3D Photonic Crystals," J. Comput. Phys. 161(2), 668-679 (2000).
[CrossRef]

Guo, B. Q.

I. Babuška and B. Q. Guo, "The h, p and h-p version of the finite-element method - basic theory and applications," Adv. Eng. Software 15, 159-174 (1992).
[CrossRef]

Hafner, C.

C. Engström, C. Hafner, and K. Schmidt, "Computations of lossy Bloch waves in two-dimensional photonic crystals," J. Comput. Theory Nanosci. 6, 775-783 (2009).
[CrossRef]

G. Stark, M. Mishrikey, F. Robin, H. Jäckel, C. Hafner, R. Vahdieck, and D. Erni, "Position dependence of FDTD mode detection in photonic crystal systems," Int. J. Numer. Model. 29, 201-218 (2009).
[CrossRef]

Jäckel, H.

G. Stark, M. Mishrikey, F. Robin, H. Jäckel, C. Hafner, R. Vahdieck, and D. Erni, "Position dependence of FDTD mode detection in photonic crystal systems," Int. J. Numer. Model. 29, 201-218 (2009).
[CrossRef]

Jiang, W.

W. Jiang, R. Chen, and X. Lu, "Theory of light refraction at the surface of a photonic crystal," Phys. Rev. B 71, 245115 (2005).
[CrossRef]

Kauf, P.

K. Schmidt and P. Kauf, "Computation of the band structure of two-dimensional Photonic Crystals with hp Finite Elements," Comp. Meth. App. Mech. Engr. 198, 1249-1259 (2009).
[CrossRef]

Lu, X.

W. Jiang, R. Chen, and X. Lu, "Theory of light refraction at the surface of a photonic crystal," Phys. Rev. B 71, 245115 (2005).
[CrossRef]

Meerbergen, K.

F. Tisseur and K. Meerbergen, "The Quadratic Eigenvalue Problem," SIAM Review 43(2), 235-286 (2001).
[CrossRef]

Mishrikey, M.

G. Stark, M. Mishrikey, F. Robin, H. Jäckel, C. Hafner, R. Vahdieck, and D. Erni, "Position dependence of FDTD mode detection in photonic crystal systems," Int. J. Numer. Model. 29, 201-218 (2009).
[CrossRef]

Pasciak, J. E.

D. C. Dobson, J. Gopalakrishnan, and J. E. Pasciak, "An Efficient Method for Band Structure Calculations in 3D Photonic Crystals," J. Comput. Phys. 161(2), 668-679 (2000).
[CrossRef]

Robin, F.

G. Stark, M. Mishrikey, F. Robin, H. Jäckel, C. Hafner, R. Vahdieck, and D. Erni, "Position dependence of FDTD mode detection in photonic crystal systems," Int. J. Numer. Model. 29, 201-218 (2009).
[CrossRef]

Schmidt, K.

K. Schmidt and P. Kauf, "Computation of the band structure of two-dimensional Photonic Crystals with hp Finite Elements," Comp. Meth. App. Mech. Engr. 198, 1249-1259 (2009).
[CrossRef]

C. Engström, C. Hafner, and K. Schmidt, "Computations of lossy Bloch waves in two-dimensional photonic crystals," J. Comput. Theory Nanosci. 6, 775-783 (2009).
[CrossRef]

Schneider, G.

K. Busch, G. Schneider, L. Tkeshelashvili, and H. Uecker, "Justification of the nonlinear Schrödinger equation in spatially periodic media," Z. Angew. Math. Phys. 57(6), 905-939 (2006).
[CrossRef]

Soukoulis, C. M.

M. Agio and C. M. Soukoulis, "Ministop bands in single-defect photonic crystal waveguides," Phys. Rev. E 64(5), 055603 (2001).
[CrossRef]

Soussi, S.

S. Soussi, "Convergence of the supercell method for defect modes calculations in photonic crystals," SIAM J. Numer. Anal. 43, 1175-1201 (2005).
[CrossRef]

Stark, G.

G. Stark, M. Mishrikey, F. Robin, H. Jäckel, C. Hafner, R. Vahdieck, and D. Erni, "Position dependence of FDTD mode detection in photonic crystal systems," Int. J. Numer. Model. 29, 201-218 (2009).
[CrossRef]

Tisseur, F.

F. Tisseur and K. Meerbergen, "The Quadratic Eigenvalue Problem," SIAM Review 43(2), 235-286 (2001).
[CrossRef]

Tkeshelashvili, L.

K. Busch, G. Schneider, L. Tkeshelashvili, and H. Uecker, "Justification of the nonlinear Schrödinger equation in spatially periodic media," Z. Angew. Math. Phys. 57(6), 905-939 (2006).
[CrossRef]

Uecker, H.

K. Busch, G. Schneider, L. Tkeshelashvili, and H. Uecker, "Justification of the nonlinear Schrödinger equation in spatially periodic media," Z. Angew. Math. Phys. 57(6), 905-939 (2006).
[CrossRef]

Vahdieck, R.

G. Stark, M. Mishrikey, F. Robin, H. Jäckel, C. Hafner, R. Vahdieck, and D. Erni, "Position dependence of FDTD mode detection in photonic crystal systems," Int. J. Numer. Model. 29, 201-218 (2009).
[CrossRef]

Adv. Eng. Software

I. Babuška and B. Q. Guo, "The h, p and h-p version of the finite-element method - basic theory and applications," Adv. Eng. Software 15, 159-174 (1992).
[CrossRef]

C. R. Physique

K. Busch, "Photonic band structure theory: assessment and perspectives," C. R. Physique 3(53), 53-66 (2002).
[CrossRef]

Comp. Meth. App. Mech. Engr.

K. Schmidt and P. Kauf, "Computation of the band structure of two-dimensional Photonic Crystals with hp Finite Elements," Comp. Meth. App. Mech. Engr. 198, 1249-1259 (2009).
[CrossRef]

Int. J. Numer. Model.

G. Stark, M. Mishrikey, F. Robin, H. Jäckel, C. Hafner, R. Vahdieck, and D. Erni, "Position dependence of FDTD mode detection in photonic crystal systems," Int. J. Numer. Model. 29, 201-218 (2009).
[CrossRef]

J. Comput. Phys.

D. C. Dobson, J. Gopalakrishnan, and J. E. Pasciak, "An Efficient Method for Band Structure Calculations in 3D Photonic Crystals," J. Comput. Phys. 161(2), 668-679 (2000).
[CrossRef]

J. Comput. Theory Nanosci.

C. Engström, C. Hafner, and K. Schmidt, "Computations of lossy Bloch waves in two-dimensional photonic crystals," J. Comput. Theory Nanosci. 6, 775-783 (2009).
[CrossRef]

Numer. Math.

D. Boffi, M. Conforti, and L. Gastaldi, "Modified edge finite elements for photonic crystals," Numer. Math. 105(2), 249-266 (2006).
[CrossRef]

Phys. Rev. B

W. Jiang, R. Chen, and X. Lu, "Theory of light refraction at the surface of a photonic crystal," Phys. Rev. B 71, 245115 (2005).
[CrossRef]

S. Adachi, "Optical properties of In1−xGaxAsyP1−y alloys," Phys. Rev. B 39(17), 12,612-12,621 (1989).
[CrossRef]

Phys. Rev. E

M. Agio and C. M. Soukoulis, "Ministop bands in single-defect photonic crystal waveguides," Phys. Rev. E 64(5), 055603 (2001).
[CrossRef]

SIAM J. Numer. Anal.

S. Soussi, "Convergence of the supercell method for defect modes calculations in photonic crystals," SIAM J. Numer. Anal. 43, 1175-1201 (2005).
[CrossRef]

SIAM Review

F. Tisseur and K. Meerbergen, "The Quadratic Eigenvalue Problem," SIAM Review 43(2), 235-286 (2001).
[CrossRef]

Z. Angew. Math. Phys.

K. Busch, G. Schneider, L. Tkeshelashvili, and H. Uecker, "Justification of the nonlinear Schrödinger equation in spatially periodic media," Z. Angew. Math. Phys. 57(6), 905-939 (2006).
[CrossRef]

Other

P. Kuchment, Floquet Theory for Partial Differential Equations (Birkhäuser Verlag, Basel, 1993).
[CrossRef]

P. Kuchment, "The mathematics of photonic crystals," in Mathematical modeling in optical science, vol. 22 of Frontiers Appl. Math., pp. 207-272 (SIAM, Philadelphia, USA, 2001).
[CrossRef]

G. E. Karniadakis and S. J. Sherwin, Spectral/hp Element Methods for Computational Fluid Dynamics (Oxford University Press, Oxford, 2005).

Concepts Development Team, Webpage of Numerical C++ Library Concepts 2 (2009). URL http://www. concepts.math.ethz.ch.

C. Engström and M. Richter, "On the Spectrum of an Operator Pencil with Applications to Wave Propagation in Periodic and Frequency Dependent Materials," SIAM J. Appl. Math. 70(1), 231-247 (2009). URL http: //link.aip.org/link/?SMM/70/231/1.
[CrossRef]

P. Frauenfelder and C. Lage, "Concepts - An Object-Oriented Software Package for Partial Differential Equations," Math. Model. Numer. Anal. 36(5), 937-951 (2002). URL http://www.edpsciences.org/ articles/m2an/abs/2002/05/m2anns12/m2anns12.html.
[CrossRef]

P. Strasser, R. Flückiger, R. Wüest, F. Robin, and H. Jäckel, "InP-based compact photonic crystal directional coupler with large operation range," Opt. Express 15(13), 8472-8478 (2007). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-15-13-8472.
[CrossRef] [PubMed]

P. Kuchment and B. Ong, "On guided waves in photonic crystal waveguides," in Waves in Periodic and Random Media, vol. 339 of Contemp. Math., pp. 105-115 (AMS, Providence, USA, 2004).

H. Ammari and F. Santosa, "Guided Waves in a Photonic Bandgap Structure with a Line Defect," SIAM J. Appl. Math. 64(6), 2018-2033 (2004). URL http://link.aip.org/link/?SMM/64/2018/1.
[CrossRef]

S. G. Johnson and J. D. Joannopoulos, "Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis," Opt. Express 8(3), 173-190 (2001). URL http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173.
[CrossRef] [PubMed]

A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, London, 1995).

J. D. Joannopoulos and S. G. Johnson, Photonic Crystals, 2nd ed. (Princeton University Press, 2008).

C. Schwab, p- and hp- Finite Element Methods - Theory and Applications in Solid and Fluid Mechanics (Oxford Science Publications, 1998).

A. von Rhein, S. Greulich-Weber, and R. B. Wehrspohn, "Berechnung von photonischen Kristallen mit Hilfe des Comsol-Elektromagnetikmoduls," in Proc. of the COMSOL Users Conf. 2006, Frankfurt, Germany, pp. 91-96 (2006).

M. Soljacic and J. D. Joannopoulos, "Enhancement of nonlinear effects using photonic crystals," Nat. Mater. 3(4), 211-219 (2004). URL http://dx.doi.org/10.1038/nmat1097.
[CrossRef] [PubMed]

T. F. Krauss, "Why do we need slow light," Nat. Photon. 2(8), 448-450 (2008). URL http://www.nature.com/nphoton/journal/v2/n8/full/nphoton.2008.139.html.
[CrossRef]

J. Li, T. P. White, L. O’Faolain, A. Gomez-Iglesias, and T. F. Krauss, "Systematic design of flat band slow light in photonic crystal waveguides," Opt. Express 16(9), 6227-6232 (2008). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-16-9-6227.
[CrossRef] [PubMed]

S. Kubo, D. Mori, and T. Baba, "Low-group-velocity and low-dispersion slow light in photonic crystal waveguides," Opt. Lett. 32(20), 2981-2983 (2007). URL http://ol.osa.org/abstract.cfm?URI=ol-32-20-2981.
[CrossRef] [PubMed]

N. Kono and M. Koshiba, "Three-dimensional finite element analysis of nonreciprocal phase shifts in magneto-photonic crystal waveguides," Opt. Express 13(23), 9155-9166 (2005). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-13-23-9155.
[CrossRef] [PubMed]

J. Smajic, C. Hafner, K. Rauscher, and D. Erni, "Computation of Radiation Leakage in Photonic Crystal Waveguides," in Proc. Progr. Electromagn. Res. Symp. 2004, Pisa, Italy, pp. 21-24 (2004).

K. Sakoda, Optical Properties of Photonic Crystals, Springer Series in Optical Sciences, 2nd ed. (Springer, Berlin, 2005).

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Figures (8)

Fig. 1.
Fig. 1.

(a) Scanning electron microscope (SEM) picture showing the top view of a PhC W1 waveguide fabricated by etching deep holes in a InP/InGaAsP/InP layer structure with finite PhC-width. (b) 3D sketch of a supercell including the vertical layer structure.

Fig. 2.
Fig. 2.

(a) Illustration of a PhC W1 waveguide in 2D based on a hexagonal lattice with three rows of holes on both sides of the channel and a possible super-cell Ω. (b) Analogous illustration of PhC W1 waveguide based on a square lattice with a super-cell Ω.

Fig. 3.
Fig. 3.

In the sub-figures (a)–(c) the real part of the magnetic field component for three typical guided PhC waveguides modes at different frequencies are shown: (a) an even mode where the slope of the dispersion is relatively steep, (b) another even mode where the slope is flat and (c) a typical odd mode. In sub-figure (d) the mesh of the super-cell is illustrated. The configuration is that of Sec. 3.3.

Fig. 4.
Fig. 4.

The same band diagram, once computed with the ω-formulation and uniformly distributed wave-vectors ki shown in (a), and once – see (b) – with the k-formulation and uniformly distributed frequencies ωi . The PhC configuration is that introduced in Sec. 3.3.

Fig. 5.
Fig. 5.

A comparison of the convergence for various codes, each using an ω-formulation. We plot the average error of all eigenfrequency values, which are part of the dispersion of the guided PhC modes versus the degree of freedom (left) and computing time (right). MPB (red) shows a algebraic convergence, COMSOL (blue) and Concepts with h-refinement (green) converge quadratically. The best convergence is achieved with Concepts and refinement of the polynomial degree of the basis functions (black curve). The configuration is that of the test PhC waveguide of Sec. 3.3.

Fig. 6.
Fig. 6.

This figure shows the convergence rate of p-FEM and h-FEM based on the k-formulation. The test configuration is that of Sec. 3.3. The shown error is the averaged error of the wave-vector eigenvalues of the guided modes for frequency values between ω a 2 π c = 0.23 and ω a 2 π c = 0.28 which is shown in dependence on the degrees of freedom (left) and the computational time (right). The curve for h-refinement and uniform polynomial degree p = 2 (dashed line) reveals coarsely the behavior of a potential COMSOL implementation of the k-formulation. The p-FEM curves were fitted both to an algebraic convergence (convergence rate of 7.2) as well as to an exponential law.

Fig. 7.
Fig. 7.

On the left side, a comparison of the band diagrams for the case of a constant dielectric constant (blue) and for the case of a dispersive permittivity (red) are shown. Both have been computed using the k-formulation. Despite their non-scalability we labelled the frequency and the wave-vector for comparison purpose in dimensionless units. The absolute wave-length is labelled on the right axis. On the right side the real part of the permittivity model of Adachi [33] for InP is plotted.

Fig. 8.
Fig. 8.

The error ∆k in the wave-vector k for a given frequency ω 0 ( ω 0 a 2 π c = 0.26 and a = 400nm) as a function of the deviation ∆ε of the permittivity ε(ω 0) = 10.007. The frequency is kept to ω 0. For the k-formulation for each frequency the right permittivity value can be used directly (corresponding to ∆ε = 0) whereas with the ω-formulation an iteration procedure has to be used.

Equations (22)

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curl E ( x ) = i ω μ 0 H ( x ) , curl H ( x ) = i ω ε 0 ε ( x , ω ) E ( x ) ,
u ˜ ( k , x ) = ( 𝓣 u ) ( k , x ) = e i k x 1 ( 𝓕 u ) ( k , x ) : = a 2 π e i k x 1 Σ n Z u ( x n a e 1 ) e inka ,
𝓣 ( curl E ( x ) ) = i ω μ 0 𝓣 ( H ( x ) ) , 𝓣 ( curl H ( x ) ) = i ω ε 0 ε ( x , ω ) 𝓣 ( E ( x ) ) ,
curl k u ̃ ( k , x ) : = curl u ̃ ( k , x ) + i k e 1 × u ̃ ( k , x ) = ( u ̃ 3 x 2 u ̃ 3 x 1 u ̃ 2 x 1 u ̃ 1 x 2 ) + i k ( 0 u ̃ 3 u ̃ 2 ) ,
curl k E ̃ ( k , x ) = i ω μ 0 H ̃ ( k , x ) , curl k H ̃ ( k , x ) = i ω ε 0 ε ( x , ω ) E ̃ ( k , x ) .
( + i k e 1 ) · ( + i k e 1 ) and ( + i k e 1 ) ε 1 ( x , ω ) · ( + i k e 1 ) .
( + i k e 1 ) · ( + i k e 1 ) E ̃ ( k , x ) = ω 2 c 2 ε ( x , ω ) E ̃ ( k , x ) ,
E ̃ ( k , x + a e 1 ) = E ̃ ( k , x ) ,
x 1 E ̃ ( k , x + a e 1 ) = x 1 E ̃ ( k , x ) ,
( + i k e 1 ) · 1 ε ( x , ω ) ( + i k e 1 ) H ̃ ( k , x ) = ω 2 c 2 H ̃ ( k , x ) ,
H ̃ ( k , x + a e 1 ) = H ̃ ( k , x ) ,
1 ε ( x + a e 1 , ω ) x 1 H ̃ ( k , x + a e 1 ) = 1 ε ( x , ω ) x 1 H ̃ ( k , x ) ,
Ω ( + i k e 1 ) E ̃ ( k , x ) · ( i k e 1 ) v ( x ) ¯ d x = ( ω c ) 2 Ω ε ( x , ω ) E ̃ ( k , x ) v ( x ) ¯ d x ,
Ω 1 ε ( x , ω ) ( + i k e 1 ) H ̃ ( k , x ) · ( i k e 1 ) v ( x ) ¯ d x = ( ω c ) 2 Ω H ̃ ( k , x ) v ( x ) ¯ d x .
Ω ε ( x , ω ) u ( x ) · v ( x ) ¯ d x a ( u , v ) ( ω c ) 2 Ω ε + 1 ( x , ω ) u ( x ) · v ( x ) ¯ d x b + 1 ( u , v )
+ k i Ω ε ( u ( x ) . v ( x ) ¯ x 1 u ( x ) x 1 . v ( x ) ¯ ) d x c ( u , v ) + k 2 Ω ε u ( x ) . v ( x ) d x b ( u , v ) = 0 .
k 2 b ( u , v , ω ) + k c ( u , v , ω ) + a ( u , v , ω ) ( ω c ) 2 b + 1 ( u , v , ω ) = 0 .
( k 2 M 3 ( ω ) + k M 2 ( ω ) + M 1 ( ω ) ) u = 0 .
M 1 ( ω ) : = ( a ( ϕ j , ϕ i ) ( ω c ) 2 b + 1 ( ϕ j , ϕ i ) ) i , j = 1 N
M 2 ( ω ) : = ( c ( ϕ j , ϕ i ) ) i , j = 1 N , M 3 ( ω ) : = ( b ( ϕ j , ϕ i ) ) i , j = 1 N .
( M 2 ( ω ) M 1 ( ω ) I 0 ) = : A ( ω ) x = k ( M 3 ( ω ) 0 0 I ) = : M ( ω ) x
ε ( ω ) = 1 ω 2 Σ i = 0 1 [ α i ( 2 Σ ± 1 ± ħ ω E 0 + δ i = 1 Δ 0 ) β i ln ( 1 ħ ω E 1 + δ i = 1 Δ 1 ) ] + C ( 1 ( ħ ω E 0 ) 2 ) ( 1 ( ħ ω E 0 ) 2 ) 2 + ( ħ ω E 0 ) 2 γ 2 ,

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