Abstract

We present an open-geometry Fourier modal method based on a new combination of open boundary conditions and an efficient k-space discretization. The open boundary of the computational domain is obtained using basis functions that expand the whole space, and the integrals subsequently appearing due to the continuous nature of the radiation modes are handled using a discretization based on nonuniform sampling of the k space. We apply the method to a variety of photonic structures and demonstrate that our method leads to significantly improved convergence with respect to the number of degrees of freedom, which may pave the way for more accurate and efficient modeling of open nanophotonic structures.

© 2016 Optical Society of America

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References

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  1. K. J. Vahala, “Optical microcavities,” Nature 424, 839–846 (2003).
    [Crossref]
  2. G. Lecamp, P. Lalanne, and J. P. Hugonin, “Very large spontaneous-emission β factors in photonic-crystal waveguides,” Phys. Rev. Lett. 99, 023902 (2007).
    [Crossref]
  3. V. S. C. Manga Rao and S. Hughes, “Single quantum-dot Purcell factor and β factor in a photonic crystal waveguide,” Phys. Rev. B 75, 205437 (2007).
    [Crossref]
  4. S. Strauf and F. Jahnke, “Single quantum dot nanolaser,” Laser Photon. Rev. 5, 607–633 (2011).
    [Crossref]
  5. N. Gregersen, P. Kaer, and J. Mørk, “Modeling and design of high-efficiency single-photon sources,” IEEE J. Sel. Top. Quantum Electron. 19, 1–16 (2013).
    [Crossref]
  6. A. V. Lavrinenko, J. Lægsgaard, N. Gregersen, F. Schmidt, and T. Søndergaard, Numerical Methods in Photonics (CRC Press, 2014), Chap. 7, pp. 197–249.
  7. A. Taflove, Computational Electrodynamics: The Finite-difference Time-domain Method, Antennas and Propagation Library (Artech, 1995).
  8. J. Reddy, An Introduction to the Finite Element Method (McGraw-Hill, 2005).
  9. J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
    [Crossref]
  10. E. Noponen and J. Turunen, “Eigenmode method for electromagnetic synthesis of diffractive elements with three-dimensional profiles,” J. Opt. Soc. Am. A 11, 2494–2502 (1994).
    [Crossref]
  11. M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995).
    [Crossref]
  12. J. P. Hugonin and P. Lalanne, “Perfectly matched layers as nonlinear coordinate transforms: a generalized formalization,” J. Opt. Soc. Am. A 22, 1844–1849 (2005).
    [Crossref]
  13. N. Gregersen, S. Reitzenstein, C. Kistner, M. Strauss, C. Schneider, S. Höfling, L. Worschech, A. Forchel, T. Nielsen, J. Mørk, and J.-M. Gérard, “Numerical and experimental study of the Q factor of high-Q micropillar cavities,” IEEE J. Quantum Electron. 46, 1470–1483 (2010).
    [Crossref]
  14. M. Pisarenco, J. Maubach, I. Setija, and R. Mattheij, “Aperiodic Fourier modal method in contrast-field formulation for simulation of scattering from finite structures,” J. Opt. Soc. Am. A 27, 2423–2431 (2010).
    [Crossref]
  15. P. T. Kristensen, C. V. Vlack, and S. Hughes, “Generalized effective mode volume for leaky optical cavities,” Opt. Lett. 37, 1649–1651 (2012).
    [Crossref]
  16. J. R. de Lasson, “Modeling and simulations of light emission and propagation in open nanophotonic systems,” Ph.D. thesis (Technical University of Denmark, 2015), available at http://orbit.dtu.dk/files/119895633/PhDThesis_Jakobrdl_Oct2015.pdf .
  17. B. Guizal, D. Barchiesi, and D. Felbacq, “Electromagnetic beam diffraction by a finite lamellar structure: an aperiodic coupled-wave method,” J. Opt. Soc. Am. A 20, 2274–2280 (2003).
    [Crossref]
  18. N. Bonod, E. Popov, and M. Nevière, “Differential theory of diffraction by finite cylindrical objects,” J. Opt. Soc. Am. A 22, 481–490 (2005).
    [Crossref]
  19. G. P. Bava, P. Debernardi, and L. Fratta, “Three-dimensional model for vectorial fields in vertical-cavity surface-emitting lasers,” Phys. Rev. A 63, 023816 (2001).
    [Crossref]
  20. M. Dems, I.-S. Chung, P. Nyakas, S. Bischoff, and K. Panajotov, “Numerical methods for modeling photonic-crystal VCSELs,” Opt. Express 18, 16042–16054 (2010).
    [Crossref]
  21. J. Claudon, N. Gregersen, P. Lalanne, and J.-M. Gérard, “Harnessing light with photonic nanowires: fundamentals and applications to quantum optics,” ChemPhysChem 14, 2393–2402 (2013).
    [Crossref]
  22. I. Friedler, P. Lalanne, J. P. Hugonin, J. Claudon, J. M. Gérard, A. Beveratos, and I. Robert-Philip, “Efficient photonic mirrors for semiconductor nanowires,” Opt. Lett. 33, 2635–2637 (2008).
    [Crossref]
  23. L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996).
    [Crossref]
  24. A. V. Lavrinenko, J. Lægsgaard, N. Gregersen, F. Schmidt, and T. Søndergaard, Numerical Methods in Photonics (CRC Press, 2014), Chap. 6, pp. 139–195.
  25. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).
  26. L. Novotny and B. Hecht, Principles of Nano-Optics, 2nd ed. (Cambridge University, 2012), Chap. 8, pp. 224–281.
  27. J. R. de Lasson, T. Christensen, J. Mørk, and N. Gregersen, “Modeling of cavities using the analytic modal method and an open geometry formalism,” J. Opt. Soc. Am. A 29, 1237–1246 (2012).
    [Crossref]
  28. J. P. Boyd, Chebyshev and Fourier Spectral Methods, 2nd ed. (Dover, 2001).
  29. L. Novotny and B. Hecht, Principles of Nano-Optics, 2nd ed. (Cambridge University, 2012), Chap. 10, pp. 313–337.
  30. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
    [Crossref]
  31. E. Popov, M. Nevière, and N. Bonod, “Factorization of products of discontinuous functions applied to Fourier-Bessel basis,” J. Opt. Soc. Am. A 21, 46–52 (2004).
    [Crossref]

2013 (2)

N. Gregersen, P. Kaer, and J. Mørk, “Modeling and design of high-efficiency single-photon sources,” IEEE J. Sel. Top. Quantum Electron. 19, 1–16 (2013).
[Crossref]

J. Claudon, N. Gregersen, P. Lalanne, and J.-M. Gérard, “Harnessing light with photonic nanowires: fundamentals and applications to quantum optics,” ChemPhysChem 14, 2393–2402 (2013).
[Crossref]

2012 (2)

2011 (1)

S. Strauf and F. Jahnke, “Single quantum dot nanolaser,” Laser Photon. Rev. 5, 607–633 (2011).
[Crossref]

2010 (3)

N. Gregersen, S. Reitzenstein, C. Kistner, M. Strauss, C. Schneider, S. Höfling, L. Worschech, A. Forchel, T. Nielsen, J. Mørk, and J.-M. Gérard, “Numerical and experimental study of the Q factor of high-Q micropillar cavities,” IEEE J. Quantum Electron. 46, 1470–1483 (2010).
[Crossref]

M. Pisarenco, J. Maubach, I. Setija, and R. Mattheij, “Aperiodic Fourier modal method in contrast-field formulation for simulation of scattering from finite structures,” J. Opt. Soc. Am. A 27, 2423–2431 (2010).
[Crossref]

M. Dems, I.-S. Chung, P. Nyakas, S. Bischoff, and K. Panajotov, “Numerical methods for modeling photonic-crystal VCSELs,” Opt. Express 18, 16042–16054 (2010).
[Crossref]

2008 (1)

2007 (2)

G. Lecamp, P. Lalanne, and J. P. Hugonin, “Very large spontaneous-emission β factors in photonic-crystal waveguides,” Phys. Rev. Lett. 99, 023902 (2007).
[Crossref]

V. S. C. Manga Rao and S. Hughes, “Single quantum-dot Purcell factor and β factor in a photonic crystal waveguide,” Phys. Rev. B 75, 205437 (2007).
[Crossref]

2005 (2)

2004 (1)

2003 (2)

2001 (1)

G. P. Bava, P. Debernardi, and L. Fratta, “Three-dimensional model for vectorial fields in vertical-cavity surface-emitting lasers,” Phys. Rev. A 63, 023816 (2001).
[Crossref]

1996 (2)

1995 (1)

1994 (2)

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

E. Noponen and J. Turunen, “Eigenmode method for electromagnetic synthesis of diffractive elements with three-dimensional profiles,” J. Opt. Soc. Am. A 11, 2494–2502 (1994).
[Crossref]

Barchiesi, D.

Bava, G. P.

G. P. Bava, P. Debernardi, and L. Fratta, “Three-dimensional model for vectorial fields in vertical-cavity surface-emitting lasers,” Phys. Rev. A 63, 023816 (2001).
[Crossref]

Berenger, J.-P.

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

Beveratos, A.

Bischoff, S.

Bonod, N.

Boyd, J. P.

J. P. Boyd, Chebyshev and Fourier Spectral Methods, 2nd ed. (Dover, 2001).

Christensen, T.

Chung, I.-S.

Claudon, J.

J. Claudon, N. Gregersen, P. Lalanne, and J.-M. Gérard, “Harnessing light with photonic nanowires: fundamentals and applications to quantum optics,” ChemPhysChem 14, 2393–2402 (2013).
[Crossref]

I. Friedler, P. Lalanne, J. P. Hugonin, J. Claudon, J. M. Gérard, A. Beveratos, and I. Robert-Philip, “Efficient photonic mirrors for semiconductor nanowires,” Opt. Lett. 33, 2635–2637 (2008).
[Crossref]

de Lasson, J. R.

J. R. de Lasson, T. Christensen, J. Mørk, and N. Gregersen, “Modeling of cavities using the analytic modal method and an open geometry formalism,” J. Opt. Soc. Am. A 29, 1237–1246 (2012).
[Crossref]

J. R. de Lasson, “Modeling and simulations of light emission and propagation in open nanophotonic systems,” Ph.D. thesis (Technical University of Denmark, 2015), available at http://orbit.dtu.dk/files/119895633/PhDThesis_Jakobrdl_Oct2015.pdf .

Debernardi, P.

G. P. Bava, P. Debernardi, and L. Fratta, “Three-dimensional model for vectorial fields in vertical-cavity surface-emitting lasers,” Phys. Rev. A 63, 023816 (2001).
[Crossref]

Dems, M.

Felbacq, D.

Forchel, A.

N. Gregersen, S. Reitzenstein, C. Kistner, M. Strauss, C. Schneider, S. Höfling, L. Worschech, A. Forchel, T. Nielsen, J. Mørk, and J.-M. Gérard, “Numerical and experimental study of the Q factor of high-Q micropillar cavities,” IEEE J. Quantum Electron. 46, 1470–1483 (2010).
[Crossref]

Fratta, L.

G. P. Bava, P. Debernardi, and L. Fratta, “Three-dimensional model for vectorial fields in vertical-cavity surface-emitting lasers,” Phys. Rev. A 63, 023816 (2001).
[Crossref]

Friedler, I.

Gaylord, T. K.

Gérard, J. M.

Gérard, J.-M.

J. Claudon, N. Gregersen, P. Lalanne, and J.-M. Gérard, “Harnessing light with photonic nanowires: fundamentals and applications to quantum optics,” ChemPhysChem 14, 2393–2402 (2013).
[Crossref]

N. Gregersen, S. Reitzenstein, C. Kistner, M. Strauss, C. Schneider, S. Höfling, L. Worschech, A. Forchel, T. Nielsen, J. Mørk, and J.-M. Gérard, “Numerical and experimental study of the Q factor of high-Q micropillar cavities,” IEEE J. Quantum Electron. 46, 1470–1483 (2010).
[Crossref]

Grann, E. B.

Gregersen, N.

N. Gregersen, P. Kaer, and J. Mørk, “Modeling and design of high-efficiency single-photon sources,” IEEE J. Sel. Top. Quantum Electron. 19, 1–16 (2013).
[Crossref]

J. Claudon, N. Gregersen, P. Lalanne, and J.-M. Gérard, “Harnessing light with photonic nanowires: fundamentals and applications to quantum optics,” ChemPhysChem 14, 2393–2402 (2013).
[Crossref]

J. R. de Lasson, T. Christensen, J. Mørk, and N. Gregersen, “Modeling of cavities using the analytic modal method and an open geometry formalism,” J. Opt. Soc. Am. A 29, 1237–1246 (2012).
[Crossref]

N. Gregersen, S. Reitzenstein, C. Kistner, M. Strauss, C. Schneider, S. Höfling, L. Worschech, A. Forchel, T. Nielsen, J. Mørk, and J.-M. Gérard, “Numerical and experimental study of the Q factor of high-Q micropillar cavities,” IEEE J. Quantum Electron. 46, 1470–1483 (2010).
[Crossref]

A. V. Lavrinenko, J. Lægsgaard, N. Gregersen, F. Schmidt, and T. Søndergaard, Numerical Methods in Photonics (CRC Press, 2014), Chap. 7, pp. 197–249.

A. V. Lavrinenko, J. Lægsgaard, N. Gregersen, F. Schmidt, and T. Søndergaard, Numerical Methods in Photonics (CRC Press, 2014), Chap. 6, pp. 139–195.

Guizal, B.

Hecht, B.

L. Novotny and B. Hecht, Principles of Nano-Optics, 2nd ed. (Cambridge University, 2012), Chap. 8, pp. 224–281.

L. Novotny and B. Hecht, Principles of Nano-Optics, 2nd ed. (Cambridge University, 2012), Chap. 10, pp. 313–337.

Höfling, S.

N. Gregersen, S. Reitzenstein, C. Kistner, M. Strauss, C. Schneider, S. Höfling, L. Worschech, A. Forchel, T. Nielsen, J. Mørk, and J.-M. Gérard, “Numerical and experimental study of the Q factor of high-Q micropillar cavities,” IEEE J. Quantum Electron. 46, 1470–1483 (2010).
[Crossref]

Hughes, S.

P. T. Kristensen, C. V. Vlack, and S. Hughes, “Generalized effective mode volume for leaky optical cavities,” Opt. Lett. 37, 1649–1651 (2012).
[Crossref]

V. S. C. Manga Rao and S. Hughes, “Single quantum-dot Purcell factor and β factor in a photonic crystal waveguide,” Phys. Rev. B 75, 205437 (2007).
[Crossref]

Hugonin, J. P.

Jahnke, F.

S. Strauf and F. Jahnke, “Single quantum dot nanolaser,” Laser Photon. Rev. 5, 607–633 (2011).
[Crossref]

Kaer, P.

N. Gregersen, P. Kaer, and J. Mørk, “Modeling and design of high-efficiency single-photon sources,” IEEE J. Sel. Top. Quantum Electron. 19, 1–16 (2013).
[Crossref]

Kistner, C.

N. Gregersen, S. Reitzenstein, C. Kistner, M. Strauss, C. Schneider, S. Höfling, L. Worschech, A. Forchel, T. Nielsen, J. Mørk, and J.-M. Gérard, “Numerical and experimental study of the Q factor of high-Q micropillar cavities,” IEEE J. Quantum Electron. 46, 1470–1483 (2010).
[Crossref]

Kristensen, P. T.

Lægsgaard, J.

A. V. Lavrinenko, J. Lægsgaard, N. Gregersen, F. Schmidt, and T. Søndergaard, Numerical Methods in Photonics (CRC Press, 2014), Chap. 7, pp. 197–249.

A. V. Lavrinenko, J. Lægsgaard, N. Gregersen, F. Schmidt, and T. Søndergaard, Numerical Methods in Photonics (CRC Press, 2014), Chap. 6, pp. 139–195.

Lalanne, P.

J. Claudon, N. Gregersen, P. Lalanne, and J.-M. Gérard, “Harnessing light with photonic nanowires: fundamentals and applications to quantum optics,” ChemPhysChem 14, 2393–2402 (2013).
[Crossref]

I. Friedler, P. Lalanne, J. P. Hugonin, J. Claudon, J. M. Gérard, A. Beveratos, and I. Robert-Philip, “Efficient photonic mirrors for semiconductor nanowires,” Opt. Lett. 33, 2635–2637 (2008).
[Crossref]

G. Lecamp, P. Lalanne, and J. P. Hugonin, “Very large spontaneous-emission β factors in photonic-crystal waveguides,” Phys. Rev. Lett. 99, 023902 (2007).
[Crossref]

J. P. Hugonin and P. Lalanne, “Perfectly matched layers as nonlinear coordinate transforms: a generalized formalization,” J. Opt. Soc. Am. A 22, 1844–1849 (2005).
[Crossref]

Lavrinenko, A. V.

A. V. Lavrinenko, J. Lægsgaard, N. Gregersen, F. Schmidt, and T. Søndergaard, Numerical Methods in Photonics (CRC Press, 2014), Chap. 7, pp. 197–249.

A. V. Lavrinenko, J. Lægsgaard, N. Gregersen, F. Schmidt, and T. Søndergaard, Numerical Methods in Photonics (CRC Press, 2014), Chap. 6, pp. 139–195.

Lecamp, G.

G. Lecamp, P. Lalanne, and J. P. Hugonin, “Very large spontaneous-emission β factors in photonic-crystal waveguides,” Phys. Rev. Lett. 99, 023902 (2007).
[Crossref]

Li, L.

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

Manga Rao, V. S. C.

V. S. C. Manga Rao and S. Hughes, “Single quantum-dot Purcell factor and β factor in a photonic crystal waveguide,” Phys. Rev. B 75, 205437 (2007).
[Crossref]

Mattheij, R.

Maubach, J.

Moharam, M. G.

Mørk, J.

N. Gregersen, P. Kaer, and J. Mørk, “Modeling and design of high-efficiency single-photon sources,” IEEE J. Sel. Top. Quantum Electron. 19, 1–16 (2013).
[Crossref]

J. R. de Lasson, T. Christensen, J. Mørk, and N. Gregersen, “Modeling of cavities using the analytic modal method and an open geometry formalism,” J. Opt. Soc. Am. A 29, 1237–1246 (2012).
[Crossref]

N. Gregersen, S. Reitzenstein, C. Kistner, M. Strauss, C. Schneider, S. Höfling, L. Worschech, A. Forchel, T. Nielsen, J. Mørk, and J.-M. Gérard, “Numerical and experimental study of the Q factor of high-Q micropillar cavities,” IEEE J. Quantum Electron. 46, 1470–1483 (2010).
[Crossref]

Nevière, M.

Nielsen, T.

N. Gregersen, S. Reitzenstein, C. Kistner, M. Strauss, C. Schneider, S. Höfling, L. Worschech, A. Forchel, T. Nielsen, J. Mørk, and J.-M. Gérard, “Numerical and experimental study of the Q factor of high-Q micropillar cavities,” IEEE J. Quantum Electron. 46, 1470–1483 (2010).
[Crossref]

Noponen, E.

Novotny, L.

L. Novotny and B. Hecht, Principles of Nano-Optics, 2nd ed. (Cambridge University, 2012), Chap. 10, pp. 313–337.

L. Novotny and B. Hecht, Principles of Nano-Optics, 2nd ed. (Cambridge University, 2012), Chap. 8, pp. 224–281.

Nyakas, P.

Panajotov, K.

Pisarenco, M.

Pommet, D. A.

Popov, E.

Reddy, J.

J. Reddy, An Introduction to the Finite Element Method (McGraw-Hill, 2005).

Reitzenstein, S.

N. Gregersen, S. Reitzenstein, C. Kistner, M. Strauss, C. Schneider, S. Höfling, L. Worschech, A. Forchel, T. Nielsen, J. Mørk, and J.-M. Gérard, “Numerical and experimental study of the Q factor of high-Q micropillar cavities,” IEEE J. Quantum Electron. 46, 1470–1483 (2010).
[Crossref]

Robert-Philip, I.

Schmidt, F.

A. V. Lavrinenko, J. Lægsgaard, N. Gregersen, F. Schmidt, and T. Søndergaard, Numerical Methods in Photonics (CRC Press, 2014), Chap. 6, pp. 139–195.

A. V. Lavrinenko, J. Lægsgaard, N. Gregersen, F. Schmidt, and T. Søndergaard, Numerical Methods in Photonics (CRC Press, 2014), Chap. 7, pp. 197–249.

Schneider, C.

N. Gregersen, S. Reitzenstein, C. Kistner, M. Strauss, C. Schneider, S. Höfling, L. Worschech, A. Forchel, T. Nielsen, J. Mørk, and J.-M. Gérard, “Numerical and experimental study of the Q factor of high-Q micropillar cavities,” IEEE J. Quantum Electron. 46, 1470–1483 (2010).
[Crossref]

Setija, I.

Snyder, A. W.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

Søndergaard, T.

A. V. Lavrinenko, J. Lægsgaard, N. Gregersen, F. Schmidt, and T. Søndergaard, Numerical Methods in Photonics (CRC Press, 2014), Chap. 6, pp. 139–195.

A. V. Lavrinenko, J. Lægsgaard, N. Gregersen, F. Schmidt, and T. Søndergaard, Numerical Methods in Photonics (CRC Press, 2014), Chap. 7, pp. 197–249.

Strauf, S.

S. Strauf and F. Jahnke, “Single quantum dot nanolaser,” Laser Photon. Rev. 5, 607–633 (2011).
[Crossref]

Strauss, M.

N. Gregersen, S. Reitzenstein, C. Kistner, M. Strauss, C. Schneider, S. Höfling, L. Worschech, A. Forchel, T. Nielsen, J. Mørk, and J.-M. Gérard, “Numerical and experimental study of the Q factor of high-Q micropillar cavities,” IEEE J. Quantum Electron. 46, 1470–1483 (2010).
[Crossref]

Taflove, A.

A. Taflove, Computational Electrodynamics: The Finite-difference Time-domain Method, Antennas and Propagation Library (Artech, 1995).

Turunen, J.

Vahala, K. J.

K. J. Vahala, “Optical microcavities,” Nature 424, 839–846 (2003).
[Crossref]

Vlack, C. V.

Worschech, L.

N. Gregersen, S. Reitzenstein, C. Kistner, M. Strauss, C. Schneider, S. Höfling, L. Worschech, A. Forchel, T. Nielsen, J. Mørk, and J.-M. Gérard, “Numerical and experimental study of the Q factor of high-Q micropillar cavities,” IEEE J. Quantum Electron. 46, 1470–1483 (2010).
[Crossref]

ChemPhysChem (1)

J. Claudon, N. Gregersen, P. Lalanne, and J.-M. Gérard, “Harnessing light with photonic nanowires: fundamentals and applications to quantum optics,” ChemPhysChem 14, 2393–2402 (2013).
[Crossref]

IEEE J. Quantum Electron. (1)

N. Gregersen, S. Reitzenstein, C. Kistner, M. Strauss, C. Schneider, S. Höfling, L. Worschech, A. Forchel, T. Nielsen, J. Mørk, and J.-M. Gérard, “Numerical and experimental study of the Q factor of high-Q micropillar cavities,” IEEE J. Quantum Electron. 46, 1470–1483 (2010).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

N. Gregersen, P. Kaer, and J. Mørk, “Modeling and design of high-efficiency single-photon sources,” IEEE J. Sel. Top. Quantum Electron. 19, 1–16 (2013).
[Crossref]

J. Comput. Phys. (1)

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

J. Opt. Soc. Am. A (10)

E. Noponen and J. Turunen, “Eigenmode method for electromagnetic synthesis of diffractive elements with three-dimensional profiles,” J. Opt. Soc. Am. A 11, 2494–2502 (1994).
[Crossref]

M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995).
[Crossref]

J. P. Hugonin and P. Lalanne, “Perfectly matched layers as nonlinear coordinate transforms: a generalized formalization,” J. Opt. Soc. Am. A 22, 1844–1849 (2005).
[Crossref]

M. Pisarenco, J. Maubach, I. Setija, and R. Mattheij, “Aperiodic Fourier modal method in contrast-field formulation for simulation of scattering from finite structures,” J. Opt. Soc. Am. A 27, 2423–2431 (2010).
[Crossref]

B. Guizal, D. Barchiesi, and D. Felbacq, “Electromagnetic beam diffraction by a finite lamellar structure: an aperiodic coupled-wave method,” J. Opt. Soc. Am. A 20, 2274–2280 (2003).
[Crossref]

N. Bonod, E. Popov, and M. Nevière, “Differential theory of diffraction by finite cylindrical objects,” J. Opt. Soc. Am. A 22, 481–490 (2005).
[Crossref]

L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996).
[Crossref]

J. R. de Lasson, T. Christensen, J. Mørk, and N. Gregersen, “Modeling of cavities using the analytic modal method and an open geometry formalism,” J. Opt. Soc. Am. A 29, 1237–1246 (2012).
[Crossref]

L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
[Crossref]

E. Popov, M. Nevière, and N. Bonod, “Factorization of products of discontinuous functions applied to Fourier-Bessel basis,” J. Opt. Soc. Am. A 21, 46–52 (2004).
[Crossref]

Laser Photon. Rev. (1)

S. Strauf and F. Jahnke, “Single quantum dot nanolaser,” Laser Photon. Rev. 5, 607–633 (2011).
[Crossref]

Nature (1)

K. J. Vahala, “Optical microcavities,” Nature 424, 839–846 (2003).
[Crossref]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. A (1)

G. P. Bava, P. Debernardi, and L. Fratta, “Three-dimensional model for vectorial fields in vertical-cavity surface-emitting lasers,” Phys. Rev. A 63, 023816 (2001).
[Crossref]

Phys. Rev. B (1)

V. S. C. Manga Rao and S. Hughes, “Single quantum-dot Purcell factor and β factor in a photonic crystal waveguide,” Phys. Rev. B 75, 205437 (2007).
[Crossref]

Phys. Rev. Lett. (1)

G. Lecamp, P. Lalanne, and J. P. Hugonin, “Very large spontaneous-emission β factors in photonic-crystal waveguides,” Phys. Rev. Lett. 99, 023902 (2007).
[Crossref]

Other (9)

A. V. Lavrinenko, J. Lægsgaard, N. Gregersen, F. Schmidt, and T. Søndergaard, Numerical Methods in Photonics (CRC Press, 2014), Chap. 7, pp. 197–249.

A. Taflove, Computational Electrodynamics: The Finite-difference Time-domain Method, Antennas and Propagation Library (Artech, 1995).

J. Reddy, An Introduction to the Finite Element Method (McGraw-Hill, 2005).

J. R. de Lasson, “Modeling and simulations of light emission and propagation in open nanophotonic systems,” Ph.D. thesis (Technical University of Denmark, 2015), available at http://orbit.dtu.dk/files/119895633/PhDThesis_Jakobrdl_Oct2015.pdf .

A. V. Lavrinenko, J. Lægsgaard, N. Gregersen, F. Schmidt, and T. Søndergaard, Numerical Methods in Photonics (CRC Press, 2014), Chap. 6, pp. 139–195.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

L. Novotny and B. Hecht, Principles of Nano-Optics, 2nd ed. (Cambridge University, 2012), Chap. 8, pp. 224–281.

J. P. Boyd, Chebyshev and Fourier Spectral Methods, 2nd ed. (Dover, 2001).

L. Novotny and B. Hecht, Principles of Nano-Optics, 2nd ed. (Cambridge University, 2012), Chap. 10, pp. 313–337.

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Figures (6)

Fig. 1.
Fig. 1. Nonuniform discretization scheme: in a bulk medium, all propagation directions have equal weights. Therefore, the wavevector k is sampled in the ( β , k ) plane using equidistant angles, as shown by θ in the figure. Due to the uniform angle distribution, the k discretization is more dense close to n k 0 .
Fig. 2.
Fig. 2. Fourier components of a point dipole emission defined in Eq. (11). The figures show the calculated radiation and guided mode contributions and the total emission as function of the radial wave number in z -invariant nanowires of varying radii. The nanowire has a refractive index of n w = 3.5 and the wavelength is λ = 950    nm . An equidistant k discretization with 1500 points and k max = 10 k 0 was used.
Fig. 3.
Fig. 3. Example of discretization step sizes Δ k m for a nonuniform discretization with M 1 = M 2 = M 3 = 10 and k cut - off / k 0 = 4 . For comparison, the equidistant discretization is also shown with the corresponding number of modes and cut-off values.
Fig. 4.
Fig. 4. (a) Calculated emission power P num in a bulk medium ( n b = 1 ) normalized with analytical result P ana with a fixed wavelength λ = 950    nm . Both numerical schemes have the wavenumber cut-off value n b k 0 in the bulk medium, and the horizontal axis shows the number of modes included in the calculations. (b) Normalized dipole emission power in air in front of a glass ( ε = 2.25 ) half-space. The dipole is parallel to the interface. (c) Normalized power emitted by point dipole placed in air close to an air–metal interface ( ε = 41 + 2.5 i ). The dipole is perpendicular to the interface. Numerical results in (b) and (c) are calculated using a cut-off value of 2 k 0 and 200 modes. The powers are normalized with the bulk medium value and the distance z 0 from the interface with the wavelength λ = 950    nm .
Fig. 5.
Fig. 5. Emission from a point dipole placed on the axis of an infinitely long rotationally symmetric nanowire of diameter d . (a)  β factor and normalized emission rates to the first and second guided modes HE 11 , E H 11 and radiation modes as functions of d . The nanowire refractive index is n = 3.45 and the wavelength is λ = 950    nm . In both discretization schemes, 1200 modes and cut-off value of 25 k 0 were used. (b) The emission rate to radiation modes calculated with a fixed nanowire diameter 0.3 λ . The horizontal axis shows the number of discretization modes, and the legend shows the cut-off value of the wavenumber in units of k 0 .
Fig. 6.
Fig. 6. Reflection coefficient of the fundamental mode calculated using (a) an equidistant grid and (b) a nonuniform grid with varying number of modes (shown in the legend) and k cut - off = 20 k 0 as a function of the nanowire diameter. The wire and the metal have refractive indices of n w = 3.5 and n Ag = 41 + 2.5 i , respectively, at wavelength λ = 950    nm . (c) The reflection coefficient of the fundamental mode using equidistant (dotted lines) and nonuniform (dashed lines) discretization and varying the cut-off of k m for a nanowire having diameter of 0.22 λ . The values of k m are chosen such that k m is equidistantly/nonuniformly sampled up to value 2 n w k 0 ( n w = 3.5 ), with M shown in the legend. Then extra k m values are added according to the scheme when the cut-off value is increased.

Equations (45)

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× [ × E ( r ) ] = ω 2 μ 0 ε ( r ) E ( r ) ,
E ( r , z ) = j a j E j ( r ) exp ( ± i β j z ) + a ( k ) E ( k , r ) exp ( ± i β ( k ) z ) d k ,
a ( k ) E ( k , r ) exp ( ± i β ( k ) z ) d k l a l E j ( k l , r ) exp ( ± i β l z ) Δ k l ,
Ma = i β a ,
f ( r ) = k c f ( k ) g ( k , r ) d k ,
k c f ( k ) g ( k , r ) d k m = 1 M c f ( k , m ) g ( k , m , r ) Δ k , m ,
E ( r ) = j a j ( r pd , p ) E j ( r ) = j m a j c j , m g m ( r ) Δ k , m e ± i β j ( z z pd ) ,
P j P Bulk = Im { a j E j ( r pd ) } P Bulk = Im { m a j c j , m g m ( r pd ) Δ k , m } P Bulk ,
β = a FM E FM ( r pd ) j a j E j ( r pd ) = a FM m c FM , m g m ( r pd ) Δ k , m j m a j c j , m g m ( r pd ) Δ k , m .
k m = m Δ k = m M + 1 k cut - off ,
E r ( r ) = i m [ j = g.m. a j E j , m ( r ) + j = r.m. a j E j , m ( r ) ] k m Δ k m ,
k m ( 1 ) = k 0 sin ( θ m ) , θ m = π 2 m M 1 + 1 , m = 1 , , M 1 , k m ( 2 ) = k 0 [ 2 sin ( θ m ) ] , θ m = π 2 ( 1 + m M 2 + 1 ) , m = 1 , , M 2 , k m ( 3 ) = k n 2 ( 2 ) + δ 1 m + δ 2 2 m ( m + 1 ) , m = 1 , , M 3 ,
z E φ , n = i n r E z , n i ω μ 0 H r , n ,
z E r , n = r E z , n + i ω μ 0 H φ , n ,
i ω μ 0 H z , n = E φ , n r + E φ , n r i n r E r , n ,
H φ , n z = i n r H z , n + i ω ε ( r ) E r , n ,
H r , n z = H z , n r i ω ε ( r ) E φ , n ,
i ω ε ( r ) E z , n = H φ , n r + H φ , n r i n r H r , n .
Δ E r , n E r , n r 2 2 i n r 2 E φ , n + ω 2 μ 0 ε ( r ) E r , n = 0 ,
Δ E φ , n E φ , n r 2 + 2 i n r 2 E r , n + ω 2 μ 0 ε ( r ) E φ , n = 0 ,
Δ E z + ω 2 μ 0 ε ( r ) E z , n = 0 .
E n ± = E φ , n ± i E r , n .
Δ E n + E n + r 2 + 2 n r 2 E n + + ω 2 μ 0 ε ( r ) E n + = 0 ,
Δ E n E n r 2 2 n r 2 E n + ω 2 μ 0 ε ( r ) E n = 0 .
E n + = E φ , n + i E r , n = k r = 0 2 c n E ( k r , z ) J n 1 ( k r r ) k r d k r ,
E n = E φ , n i E r , n = k r = 0 2 b n E ( k r , z ) J n + 1 ( k r r ) k r d k r ,
E r ( r , φ , z ) = i n = N N m = 1 M k m Δ k m [ b n , m E ( z ) J n + 1 ( k m r ) c n , m E ( z ) J n 1 ( k m r ) ] exp ( i n φ ) ,
E φ ( r , φ , z ) = n = N N m = 1 M k m Δ k m [ b n , m E ( z ) J n + 1 ( k m r ) + c n , m E ( z ) J n 1 ( k m r ) ] exp ( i n φ ) .
i ω μ 0 H z , n = m = 1 M k m 2 Δ k m [ b n , m E c n , m E ] J n ( k m r ) ,
i ω ε ( r ) E z , n = m = 1 M k m 2 Δ k m [ b n , m H c n , m H ] J n ( k m r ) .
E z , n = i ω m , m = 1 M ( [ ε ] n , n ) m , m 1 k m [ b n , m H c n , m H ] J n ( k m r ) ,
d f n ( z ) d z = M n f n ( z ) , n [ N , N ] ,
f n ( z ) = [ b n , m E ( z ) c n , m E ( z ) b n , m H ( z ) c n , m H ( z ) ] M n = [ M n , 11 M n , 12 M n , 21 M n , 22 ] .
β n 2 [ b n , m E c n , m E ] = M n , 12 M n , 21 [ b n , m E c n , m E ] .
d b ˜ n , m E d z = ω μ 0 b ˜ n , m H k m 2 ω m ( [ ε ] m , m n , n ˜ ) 1 k m [ b ˜ n , m H c ˜ n , m H ] ,
d c ˜ n , m E d z = ω μ 0 c ˜ n , m H k m 2 ω m ( [ ε ] m , m n , n ˜ ) 1 k m [ b ˜ n , m H c ˜ n , m H ] ,
d b ˜ n , m H d z = k m 2 2 ω μ 0 ( b ˜ n , m E c ˜ n , m E ) + i 1 2 ω k m r = 0 ε ( r ) E r , n ( r ) J n + 1 ( k m r ) r d r 1 2 ω k m r = 0 ε ( r ) E φ , n ( r ) J n + 1 ( k m r ) r d r ,
d c ˜ n , m H d z = k m 2 2 ω μ 0 ( b ˜ n , m E c ˜ n , m E ) + i 1 2 ω k m r = 0 ε ( r ) E r , n ( r ) J n 1 ( k m r ) r d r + 1 2 ω k m r = 0 ε ( r ) E φ , n ( r ) J n 1 ( k m r ) r d r ,
0 ε ( r ) E φ , n ( r ) J n ± 1 ( k m r ) r d r = m = 1 M k m Δ k m ( [ ε ] m , m n ± 1 , n + 1 b n , m E + [ ε ] m , m n ± 1 , n 1 c n , m E ) ,
E r , n , m ± = 0 E r , n ( r ) J n ± 1 ( k m r ) r d r = m k m Δ k m D r , n , m ± [ 1 ε ] m , m n ± 1 , n ± 1 .
D r , n , m ± = m 1 k m Δ k m ( [ 1 ε ] n ± 1 , n ± 1 ) m , m 1 E r , n , m ± .
i k m D r , n , m + = m = 1 M 1 k m Δ k m ( [ 1 ε ] n + 1 , n + 1 ) m , m 1 k m b n , m E + m , m = 1 M 1 k m Δ k m ( [ 1 ε ] n + 1 , n + 1 ) m , m 1 × k m Δ k m [ Ψ ] m , m n + 1 , n 1 k m c n , m E ,
i k m D r , n , m = m , m = 1 M 1 k m Δ k m ( [ 1 ε ] n 1 , n 1 ) m , m 1 × k m Δ k m [ Ψ ] m , m n 1 , n + 1 k m b n , m E + m = 1 M 1 k m Δ k m ( [ 1 ε ] n 1 , n 1 ) m , m 1 k m c n , m E ,
[ 1 ε ] m , m n , n = r = 0 1 ε ( r ) J n ( k m r ) J n ( k m r ) r d r ,
[ Ψ ] m , m n ± 1 , n 1 = r = 0 J n ± 1 ( k m r ) J n 1 ( k m r ) r d r .

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