Efficient computation of photonic crystal waveguide modes with dispersive material
Optics Express, Vol. 18, Issue 7, pp. 7307-7322 (2010)
http://dx.doi.org/10.1364/OE.18.007307
Acrobat PDF (4202 KB)
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
1. Introduction
2. M. Agio and C. M. Soukoulis, “Ministop bands in single-defect photonic crystal waveguides,” Phys. Rev. E 64(5), 055603 (2001). [CrossRef]
3. 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]
4. 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]
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]
6. 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]
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]
- the formulation of the mathematical eigenvalue problem for k_{i} of a given ω_{i},
- the modelling of guided PhC modes in an infinite geometry, and
- the discretization of the problem.
8. K. Busch, “Photonic band structure theory: assessment and perspectives,” C. R. Physique 3(53), 53–66 (2002). [CrossRef]
14. S. Soussi, “Convergence of the supercell method for defect modes calculations in photonic crystals,” SIAM J. Numer. Anal. 43, 1175–1201 (2005). [CrossRef]
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]
19. G. Stark, M. Mishrikey, F. Robin, H. Jackel, 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]
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]
- Formulation: Quadratic eigenvalue problem in k for each ω_{i}.Using the Bloch transform we obtain a formulation on the unit-cell with periodic boundary conditions, which exhibits a quadratic dependence on k and a linear dependence on in ω ^{2}. This formulation can be interpreted as ω- as well as k-formulation. In the following, we focus on the k-formulation. The quadratic equation in k can be linearized, such that efficient and well understood algorithms for solving eigenproblems with linear sparse matrices can be applied. The same approach was recently presented for the band structure calculation of TM modes in infinite crystals in [23Fig. 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., 24
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]
] and in [2524. C. Engstrom 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/1SMM/70/231/1. [CrossRef]
] for frequency-dependent materials which we applied in this project to PhC waveguides.25. C. Engstrom, C. Hafner, and K. Schmidt, “Computations of lossy Bloch waves in two-dimensional photonic crystals,” J. Comput. Theor. Nanosci. 6, 775–783 (2009). [CrossRef]
- Modelling in an infinite geometry: Super-cell approach.For comparison we pursue the most often used approach – the super-cell approach – with periodic boundary conditions at all boundaries. This approach does not influence the formulation to be quadratic in wave-vector k. But it introduces a small modelling error which will be studied in a forthcoming article in comparison with two other approaches which also lead to quadratic eigenproblems in k.
- Discretization: High-order FEM on a coarse mesh with curved cells.To be able to approximate the waveguide modes to a high precision with moderate computational costs we use the p-version of the FEM on a coarse quadrilateral mesh with curved cells [11, 25
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]
]. For this system we implemented both, the k-formulation and the ω-formulation for strictly periodic boundary conditions using the numerical C++ library Concepts [2625. C. Engstrom, C. Hafner, and K. Schmidt, “Computations of lossy Bloch waves in two-dimensional photonic crystals,” J. Comput. Theor. Nanosci. 6, 775–783 (2009). [CrossRef]
]. For practical geometries with smooth interfaces we will show the expected exponential convergence of the method and will compare it to the results of MPB and the FEM software COMSOL.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/m2annsl2/m2annsl2.html. [CrossRef]
2. The eigenvalue problem in the super-cell
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]
3. The numerical solution of the eigenvalue problem
3.1. The general eigenvalue problem
3.2. Computational frameworks for the calculation of the modes
- The search for the eigenfrequencies ω for particular values of k_{i} which we call the ω-formulation.
- The search for the eigen wave-vector k for particular frequencies ω_{i} which we call the k-formulation.
3.3. The matrix eigenvalue problem
30. F. Tisseur and K. Meerbergen, “The Quadratic Eigenvalue Problem,” SIAM Review 43(2), 235–286 (2001). [CrossRef]
3.4. Discretization by the p-version of the FEM
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]
32. Webpage of Numerical C++ Library Concepts 2 (2009). URL http://www.concepts.math.ethz.ch.
4. Numerical results
4.1. Comparison of the p-FEM with other methods for the ω-formulation
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]
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]
4.2. Convergence analysis for the k-formulation
4.3. Computing Band Diagrams including Dispersive Material Models
33. S. Adachi, “Optical properties of In_{1-x}Ga_{x}As_{y}P_{1-y} alloys,” Phys. Rev. B 39(17), 12,612–12,621 (1989). [CrossRef]
33. S. Adachi, “Optical properties of In_{1-x}Ga_{x}As_{y}P_{1-y} alloys,” Phys. Rev. B 39(17), 12,612–12,621 (1989). [CrossRef]
5. Conclusion
34. P. Strasser, R. Flückiger, R. Wüest, F. Robin, and H. Jackel, “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]
Acknowledgements
References and links
1. | J. D. Joannopoulos and S. G. Johnson, Photonic Crystals, 2nd ed. (Princeton University Press, 2008). |
2. | M. Agio and C. M. Soukoulis, “Ministop bands in single-defect photonic crystal waveguides,” Phys. Rev. E 64(5), 055603 (2001). [CrossRef] |
3. | 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] |
4. | 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] |
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] |
6. | 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] |
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 (200). [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). |
14. | S. Soussi, “Convergence of the supercell method for defect modes calculations in photonic crystals,” SIAM J. Numer. Anal. 43, 1175–1201 (2005). [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. Jackel, 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 (Oxfold 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. Engstrom 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/1SMM/70/231/1. [CrossRef] |
25. | C. Engstrom, C. Hafner, and K. Schmidt, “Computations of lossy Bloch waves in two-dimensional photonic crystals,” J. Comput. Theor. 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/m2annsl2/m2annsl2.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. | Webpage of Numerical C++ Library Concepts 2 (2009). URL http://www.concepts.math.ethz.ch. |
33. | S. Adachi, “Optical properties of In_{1-x}Ga_{x}As_{y}P_{1-y} alloys,” Phys. Rev. B 39(17), 12,612–12,621 (1989). [CrossRef] |
34. | P. Strasser, R. Flückiger, R. Wüest, F. Robin, and H. Jackel, “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] |
OCIS Codes
(000.4430) General : Numerical approximation and analysis
(130.5296) Integrated optics : Photonic crystal waveguides
ToC Category:
Photonic Crystals
History
Original Manuscript: January 12, 2010
Revised Manuscript: February 18, 2010
Manuscript Accepted: February 21, 2010
Published: March 24, 2010
Citation
Kersten Schmidt and Roman Kappeler, "Efficient computation of photonic crystal waveguide modes with dispersive material," Opt. Express 18, 7307-7322 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-7-7307
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References
- J. D. Joannopoulos and S. G. Johnson, Photonic Crystals, 2nd ed. (Princeton University Press, 2008).
- M. Agio and C. M. Soukoulis, "Ministop bands in single-defect photonic crystal waveguides," Phys. Rev. E 64(5), 055603 (2001). [CrossRef]
- 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]
- K. Busch, "Photonic band structure theory: assessment and perspectives," C. R. Physique 3(53), 53-66 (2002). [CrossRef]
- 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]
- D. Boffi, M. Conforti, and L. Gastaldi, "Modified edge finite elements for photonic crystals," Numer. Math. 105(2), 249-266 (2006). [CrossRef]
- 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]
- 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).
- S. Soussi, "Convergence of the supercell method for defect modes calculations in photonic crystals," SIAM J. Numer. Anal. 43, 1175-1201 (2005). [CrossRef]
- 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).
- 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]
- 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. 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).
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- F. Tisseur and K. Meerbergen, "The Quadratic Eigenvalue Problem," SIAM Review 43(2), 235-286 (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.
- S. Adachi, "Optical properties of In1−xGaxAsyP1−y alloys," Phys. Rev. B 39(17), 12,612-12,621 (1989). [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]
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