Abstract

We present an eigenmode expansion technique for calculating the properties of a dipole emitter inside a micropillar. We consider a solution domain of infinite extent, implying no outer boundary conditions for the electric field, and expand the field on analytic eigenmodes. In contrast to finite-sized simulation domains, this avoids the issue of parasitic reflections from artificial boundaries. We compute the Purcell factor in a two-dimensional micropillar and explore two discretization techniques for the continuous radiation modes. Specifically, an equidistant and a nonequidistant discretization are employed, and while both converge, only the nonequidistant discretization exhibits uniform convergence. These results demonstrate that the method leads to more accurate results than existing simulation techniques and constitutes a promising basis for further work.

© 2012 Optical Society of America

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  1. K. Vahala, “Optical microcavities,” Nature 424, 839–846 (2003).
    [CrossRef]
  2. E. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).
  3. J. Claudon, J. Bleuse, N. S. Malik, M. Bazin, P. Jaffrennou, N. Gregersen, C. Sauvan, P. Lalanne, and J. M. Gérard, “A highly efficient single-photon source based on a quantum dot in a photonic nanowire,” Nat. Photonics 4, 174–177 (2010).
    [CrossRef]
  4. A. Taflove and S. Hagness, Computational Electromagnetics: The Finite-Difference Time-Domain Method (Artech House, 2005).
  5. J. Reddy, An Introduction to the Finite Element Method (McGraw-Hill Science/Engineering/Math, 2005).
  6. E. Silberstein, P. Lalanne, J. P. Hugonin, and Q. Cao, “Use of grating theories in integrated optics,” J. Opt. Soc. Am. A 18, 2865–2875 (2001).
    [CrossRef]
  7. P. Bienstman and R. Baets, “Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers,” Opt. Quantum Electron. 33, 327–341(2001).
    [CrossRef]
  8. P. Bienstman, “Rigorous and efficient modelling of wavelength scale photonic components,” Ph.D. thesis (University of Gent, Department of Information Technology, 2001).
  9. J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
    [CrossRef]
  10. N. Gregersen and J. Mørk, “An improved perfectly matched layer for the eigenmode expansion technique,” Opt. Quantum Electron. 40, 957–966 (2008).
    [CrossRef]
  11. 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]
  12. N. Gregersen, S. Reitzenstein, C. Kistner, M. Strauss, C. Schneider, S. Höfling, L. Worschech, A. Forchel, T. R. 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]
  13. N. Bonod, E. Popov, and M. Neviére, “Differential theory of diffraction by finite cylindrical objects,” J. Opt. Soc. Am. A 22, 481–491 (2005).
    [CrossRef]
  14. I. Tigelis and A. Manenkov, “Scattering from an abruptly terminated asymmetrical slab waveguide,” J. Opt. Soc. Am. A 16, 523–532 (1999).
    [CrossRef]
  15. I. Tigelis and A. Manenkov, “Analysis of mode scattering from an abruptly ended dielectric slab waveguide by an accelerated iteration technique,” J. Opt. Soc. Am. A 17, 2249–2259(2000).
    [CrossRef]
  16. P. Kristensen, P. Lodahl, and J. Mørk, “Light propagation in finite-sized photonic crystals: multiple scattering using an electric field integral equation,” J. Opt. Soc. Am. B 27, 228–237(2010).
    [CrossRef]
  17. J. M. Gérard, “Solid-state cavity-quantum electrodynamics with self-assembled quantum dots,” in Single Quantum Dots: Physics and Applications, P. Michler, ed. (Springer-Verlag, 2003), pp. 269–315.
  18. L. Novotny and B. Hecht, Principles of Nano-Optics, 1st ed. (Cambridge University, 2006), Chap. 8, pp. 250–303.
  19. A. Snyder and J. Love, Optical Waveguide Theory, 1st ed. (Chapman and Hall, 1983), Chap. 31, pp. 601–622.
  20. 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]
  21. 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]
  22. N. K. Uzunoglu, C. N. Capsalis, and I. Tigelis, “Scattering from and abruptly terminated single-mode-fiber waveguide,” J. Opt. Soc. Am. A 4, 2150–2157 (1987).
    [CrossRef]
  23. P. Kristensen, “Light–matter interaction in nanostructured materials,” Ph.D. thesis (Technical University of Denmark, Department of Photonics Engineering, 2009).
  24. P. Lalanne, J. P. Hugonin, and J. M. Gérard, “Electromagnetic study of the quality factor of pillar microcavities in the small diameter limit,” Appl. Phys. Lett. 84, 4726–4728 (2004).
    [CrossRef]
  25. S. Reitzenstein and A. Forchel, “Quantum dot micropillars,” J. Phys. D 43, 033001 (2010).
    [CrossRef]
  26. M. Lermer, N. Gregersen, F. Dunzer, S. Reitzenstein, S. Höfling, J. Mørk, L. Worschech, M. Kamp, and A. Forchel, “Bloch-wave engineering of quantum dot micropillars for cavity quantum electrodynamics experiments,” Phys. Rev. Lett. 108, 057402 (2012).
    [CrossRef]
  27. L. W. Li, H. G. Wee, and M. S. Leong, “Dyadic Green’s functions inside/outside a dielectric elliptical cylinder: theory and application,” IEEE Trans. Antennas Propag. 51, 564–574 (2003).
    [CrossRef]

2012 (1)

M. Lermer, N. Gregersen, F. Dunzer, S. Reitzenstein, S. Höfling, J. Mørk, L. Worschech, M. Kamp, and A. Forchel, “Bloch-wave engineering of quantum dot micropillars for cavity quantum electrodynamics experiments,” Phys. Rev. Lett. 108, 057402 (2012).
[CrossRef]

2010 (4)

J. Claudon, J. Bleuse, N. S. Malik, M. Bazin, P. Jaffrennou, N. Gregersen, C. Sauvan, P. Lalanne, and J. M. Gérard, “A highly efficient single-photon source based on a quantum dot in a photonic nanowire,” Nat. Photonics 4, 174–177 (2010).
[CrossRef]

N. Gregersen, S. Reitzenstein, C. Kistner, M. Strauss, C. Schneider, S. Höfling, L. Worschech, A. Forchel, T. R. 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]

S. Reitzenstein and A. Forchel, “Quantum dot micropillars,” J. Phys. D 43, 033001 (2010).
[CrossRef]

P. Kristensen, P. Lodahl, and J. Mørk, “Light propagation in finite-sized photonic crystals: multiple scattering using an electric field integral equation,” J. Opt. Soc. Am. B 27, 228–237(2010).
[CrossRef]

2008 (1)

N. Gregersen and J. Mørk, “An improved perfectly matched layer for the eigenmode expansion technique,” Opt. Quantum Electron. 40, 957–966 (2008).
[CrossRef]

2005 (2)

2004 (1)

P. Lalanne, J. P. Hugonin, and J. M. Gérard, “Electromagnetic study of the quality factor of pillar microcavities in the small diameter limit,” Appl. Phys. Lett. 84, 4726–4728 (2004).
[CrossRef]

2003 (2)

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

L. W. Li, H. G. Wee, and M. S. Leong, “Dyadic Green’s functions inside/outside a dielectric elliptical cylinder: theory and application,” IEEE Trans. Antennas Propag. 51, 564–574 (2003).
[CrossRef]

2001 (2)

E. Silberstein, P. Lalanne, J. P. Hugonin, and Q. Cao, “Use of grating theories in integrated optics,” J. Opt. Soc. Am. A 18, 2865–2875 (2001).
[CrossRef]

P. Bienstman and R. Baets, “Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers,” Opt. Quantum Electron. 33, 327–341(2001).
[CrossRef]

2000 (1)

1999 (1)

1996 (1)

1995 (1)

1994 (1)

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

1987 (1)

1946 (1)

E. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).

Baets, R.

P. Bienstman and R. Baets, “Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers,” Opt. Quantum Electron. 33, 327–341(2001).
[CrossRef]

Bazin, M.

J. Claudon, J. Bleuse, N. S. Malik, M. Bazin, P. Jaffrennou, N. Gregersen, C. Sauvan, P. Lalanne, and J. M. Gérard, “A highly efficient single-photon source based on a quantum dot in a photonic nanowire,” Nat. Photonics 4, 174–177 (2010).
[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]

Bienstman, P.

P. Bienstman and R. Baets, “Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers,” Opt. Quantum Electron. 33, 327–341(2001).
[CrossRef]

P. Bienstman, “Rigorous and efficient modelling of wavelength scale photonic components,” Ph.D. thesis (University of Gent, Department of Information Technology, 2001).

Bleuse, J.

J. Claudon, J. Bleuse, N. S. Malik, M. Bazin, P. Jaffrennou, N. Gregersen, C. Sauvan, P. Lalanne, and J. M. Gérard, “A highly efficient single-photon source based on a quantum dot in a photonic nanowire,” Nat. Photonics 4, 174–177 (2010).
[CrossRef]

Bonod, N.

Cao, Q.

Capsalis, C. N.

Claudon, J.

J. Claudon, J. Bleuse, N. S. Malik, M. Bazin, P. Jaffrennou, N. Gregersen, C. Sauvan, P. Lalanne, and J. M. Gérard, “A highly efficient single-photon source based on a quantum dot in a photonic nanowire,” Nat. Photonics 4, 174–177 (2010).
[CrossRef]

Dunzer, F.

M. Lermer, N. Gregersen, F. Dunzer, S. Reitzenstein, S. Höfling, J. Mørk, L. Worschech, M. Kamp, and A. Forchel, “Bloch-wave engineering of quantum dot micropillars for cavity quantum electrodynamics experiments,” Phys. Rev. Lett. 108, 057402 (2012).
[CrossRef]

Forchel, A.

M. Lermer, N. Gregersen, F. Dunzer, S. Reitzenstein, S. Höfling, J. Mørk, L. Worschech, M. Kamp, and A. Forchel, “Bloch-wave engineering of quantum dot micropillars for cavity quantum electrodynamics experiments,” Phys. Rev. Lett. 108, 057402 (2012).
[CrossRef]

N. Gregersen, S. Reitzenstein, C. Kistner, M. Strauss, C. Schneider, S. Höfling, L. Worschech, A. Forchel, T. R. 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]

S. Reitzenstein and A. Forchel, “Quantum dot micropillars,” J. Phys. D 43, 033001 (2010).
[CrossRef]

Gaylord, T. K.

Gérard, J. M.

J. Claudon, J. Bleuse, N. S. Malik, M. Bazin, P. Jaffrennou, N. Gregersen, C. Sauvan, P. Lalanne, and J. M. Gérard, “A highly efficient single-photon source based on a quantum dot in a photonic nanowire,” Nat. Photonics 4, 174–177 (2010).
[CrossRef]

N. Gregersen, S. Reitzenstein, C. Kistner, M. Strauss, C. Schneider, S. Höfling, L. Worschech, A. Forchel, T. R. 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]

P. Lalanne, J. P. Hugonin, and J. M. Gérard, “Electromagnetic study of the quality factor of pillar microcavities in the small diameter limit,” Appl. Phys. Lett. 84, 4726–4728 (2004).
[CrossRef]

J. M. Gérard, “Solid-state cavity-quantum electrodynamics with self-assembled quantum dots,” in Single Quantum Dots: Physics and Applications, P. Michler, ed. (Springer-Verlag, 2003), pp. 269–315.

Grann, E. B.

Gregersen, N.

M. Lermer, N. Gregersen, F. Dunzer, S. Reitzenstein, S. Höfling, J. Mørk, L. Worschech, M. Kamp, and A. Forchel, “Bloch-wave engineering of quantum dot micropillars for cavity quantum electrodynamics experiments,” Phys. Rev. Lett. 108, 057402 (2012).
[CrossRef]

N. Gregersen, S. Reitzenstein, C. Kistner, M. Strauss, C. Schneider, S. Höfling, L. Worschech, A. Forchel, T. R. 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]

J. Claudon, J. Bleuse, N. S. Malik, M. Bazin, P. Jaffrennou, N. Gregersen, C. Sauvan, P. Lalanne, and J. M. Gérard, “A highly efficient single-photon source based on a quantum dot in a photonic nanowire,” Nat. Photonics 4, 174–177 (2010).
[CrossRef]

N. Gregersen and J. Mørk, “An improved perfectly matched layer for the eigenmode expansion technique,” Opt. Quantum Electron. 40, 957–966 (2008).
[CrossRef]

Hagness, S.

A. Taflove and S. Hagness, Computational Electromagnetics: The Finite-Difference Time-Domain Method (Artech House, 2005).

Hecht, B.

L. Novotny and B. Hecht, Principles of Nano-Optics, 1st ed. (Cambridge University, 2006), Chap. 8, pp. 250–303.

Höfling, S.

M. Lermer, N. Gregersen, F. Dunzer, S. Reitzenstein, S. Höfling, J. Mørk, L. Worschech, M. Kamp, and A. Forchel, “Bloch-wave engineering of quantum dot micropillars for cavity quantum electrodynamics experiments,” Phys. Rev. Lett. 108, 057402 (2012).
[CrossRef]

N. Gregersen, S. Reitzenstein, C. Kistner, M. Strauss, C. Schneider, S. Höfling, L. Worschech, A. Forchel, T. R. 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]

Hugonin, J. P.

Jaffrennou, P.

J. Claudon, J. Bleuse, N. S. Malik, M. Bazin, P. Jaffrennou, N. Gregersen, C. Sauvan, P. Lalanne, and J. M. Gérard, “A highly efficient single-photon source based on a quantum dot in a photonic nanowire,” Nat. Photonics 4, 174–177 (2010).
[CrossRef]

Kamp, M.

M. Lermer, N. Gregersen, F. Dunzer, S. Reitzenstein, S. Höfling, J. Mørk, L. Worschech, M. Kamp, and A. Forchel, “Bloch-wave engineering of quantum dot micropillars for cavity quantum electrodynamics experiments,” Phys. Rev. Lett. 108, 057402 (2012).
[CrossRef]

Kistner, C.

N. Gregersen, S. Reitzenstein, C. Kistner, M. Strauss, C. Schneider, S. Höfling, L. Worschech, A. Forchel, T. R. 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.

P. Kristensen, P. Lodahl, and J. Mørk, “Light propagation in finite-sized photonic crystals: multiple scattering using an electric field integral equation,” J. Opt. Soc. Am. B 27, 228–237(2010).
[CrossRef]

P. Kristensen, “Light–matter interaction in nanostructured materials,” Ph.D. thesis (Technical University of Denmark, Department of Photonics Engineering, 2009).

Lalanne, P.

J. Claudon, J. Bleuse, N. S. Malik, M. Bazin, P. Jaffrennou, N. Gregersen, C. Sauvan, P. Lalanne, and J. M. Gérard, “A highly efficient single-photon source based on a quantum dot in a photonic nanowire,” Nat. Photonics 4, 174–177 (2010).
[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]

P. Lalanne, J. P. Hugonin, and J. M. Gérard, “Electromagnetic study of the quality factor of pillar microcavities in the small diameter limit,” Appl. Phys. Lett. 84, 4726–4728 (2004).
[CrossRef]

E. Silberstein, P. Lalanne, J. P. Hugonin, and Q. Cao, “Use of grating theories in integrated optics,” J. Opt. Soc. Am. A 18, 2865–2875 (2001).
[CrossRef]

Leong, M. S.

L. W. Li, H. G. Wee, and M. S. Leong, “Dyadic Green’s functions inside/outside a dielectric elliptical cylinder: theory and application,” IEEE Trans. Antennas Propag. 51, 564–574 (2003).
[CrossRef]

Lermer, M.

M. Lermer, N. Gregersen, F. Dunzer, S. Reitzenstein, S. Höfling, J. Mørk, L. Worschech, M. Kamp, and A. Forchel, “Bloch-wave engineering of quantum dot micropillars for cavity quantum electrodynamics experiments,” Phys. Rev. Lett. 108, 057402 (2012).
[CrossRef]

Li, L.

Li, L. W.

L. W. Li, H. G. Wee, and M. S. Leong, “Dyadic Green’s functions inside/outside a dielectric elliptical cylinder: theory and application,” IEEE Trans. Antennas Propag. 51, 564–574 (2003).
[CrossRef]

Lodahl, P.

Love, J.

A. Snyder and J. Love, Optical Waveguide Theory, 1st ed. (Chapman and Hall, 1983), Chap. 31, pp. 601–622.

Malik, N. S.

J. Claudon, J. Bleuse, N. S. Malik, M. Bazin, P. Jaffrennou, N. Gregersen, C. Sauvan, P. Lalanne, and J. M. Gérard, “A highly efficient single-photon source based on a quantum dot in a photonic nanowire,” Nat. Photonics 4, 174–177 (2010).
[CrossRef]

Manenkov, A.

Moharam, M. G.

Mørk, J.

M. Lermer, N. Gregersen, F. Dunzer, S. Reitzenstein, S. Höfling, J. Mørk, L. Worschech, M. Kamp, and A. Forchel, “Bloch-wave engineering of quantum dot micropillars for cavity quantum electrodynamics experiments,” Phys. Rev. Lett. 108, 057402 (2012).
[CrossRef]

P. Kristensen, P. Lodahl, and J. Mørk, “Light propagation in finite-sized photonic crystals: multiple scattering using an electric field integral equation,” J. Opt. Soc. Am. B 27, 228–237(2010).
[CrossRef]

N. Gregersen, S. Reitzenstein, C. Kistner, M. Strauss, C. Schneider, S. Höfling, L. Worschech, A. Forchel, T. R. 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]

N. Gregersen and J. Mørk, “An improved perfectly matched layer for the eigenmode expansion technique,” Opt. Quantum Electron. 40, 957–966 (2008).
[CrossRef]

Neviére, M.

Nielsen, T. R.

N. Gregersen, S. Reitzenstein, C. Kistner, M. Strauss, C. Schneider, S. Höfling, L. Worschech, A. Forchel, T. R. 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]

Novotny, L.

L. Novotny and B. Hecht, Principles of Nano-Optics, 1st ed. (Cambridge University, 2006), Chap. 8, pp. 250–303.

Pommet, D. A.

Popov, E.

Purcell, E.

E. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).

Reddy, J.

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

Reitzenstein, S.

M. Lermer, N. Gregersen, F. Dunzer, S. Reitzenstein, S. Höfling, J. Mørk, L. Worschech, M. Kamp, and A. Forchel, “Bloch-wave engineering of quantum dot micropillars for cavity quantum electrodynamics experiments,” Phys. Rev. Lett. 108, 057402 (2012).
[CrossRef]

N. Gregersen, S. Reitzenstein, C. Kistner, M. Strauss, C. Schneider, S. Höfling, L. Worschech, A. Forchel, T. R. 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]

S. Reitzenstein and A. Forchel, “Quantum dot micropillars,” J. Phys. D 43, 033001 (2010).
[CrossRef]

Sauvan, C.

J. Claudon, J. Bleuse, N. S. Malik, M. Bazin, P. Jaffrennou, N. Gregersen, C. Sauvan, P. Lalanne, and J. M. Gérard, “A highly efficient single-photon source based on a quantum dot in a photonic nanowire,” Nat. Photonics 4, 174–177 (2010).
[CrossRef]

Schneider, C.

N. Gregersen, S. Reitzenstein, C. Kistner, M. Strauss, C. Schneider, S. Höfling, L. Worschech, A. Forchel, T. R. 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]

Silberstein, E.

Snyder, A.

A. Snyder and J. Love, Optical Waveguide Theory, 1st ed. (Chapman and Hall, 1983), Chap. 31, pp. 601–622.

Strauss, M.

N. Gregersen, S. Reitzenstein, C. Kistner, M. Strauss, C. Schneider, S. Höfling, L. Worschech, A. Forchel, T. R. 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 and S. Hagness, Computational Electromagnetics: The Finite-Difference Time-Domain Method (Artech House, 2005).

Tigelis, I.

Uzunoglu, N. K.

Vahala, K.

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

Wee, H. G.

L. W. Li, H. G. Wee, and M. S. Leong, “Dyadic Green’s functions inside/outside a dielectric elliptical cylinder: theory and application,” IEEE Trans. Antennas Propag. 51, 564–574 (2003).
[CrossRef]

Worschech, L.

M. Lermer, N. Gregersen, F. Dunzer, S. Reitzenstein, S. Höfling, J. Mørk, L. Worschech, M. Kamp, and A. Forchel, “Bloch-wave engineering of quantum dot micropillars for cavity quantum electrodynamics experiments,” Phys. Rev. Lett. 108, 057402 (2012).
[CrossRef]

N. Gregersen, S. Reitzenstein, C. Kistner, M. Strauss, C. Schneider, S. Höfling, L. Worschech, A. Forchel, T. R. 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]

Appl. Phys. Lett. (1)

P. Lalanne, J. P. Hugonin, and J. M. Gérard, “Electromagnetic study of the quality factor of pillar microcavities in the small diameter limit,” Appl. Phys. Lett. 84, 4726–4728 (2004).
[CrossRef]

IEEE J. Quantum Electron. (1)

N. Gregersen, S. Reitzenstein, C. Kistner, M. Strauss, C. Schneider, S. Höfling, L. Worschech, A. Forchel, T. R. 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 Trans. Antennas Propag. (1)

L. W. Li, H. G. Wee, and M. S. Leong, “Dyadic Green’s functions inside/outside a dielectric elliptical cylinder: theory and application,” IEEE Trans. Antennas Propag. 51, 564–574 (2003).
[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 (8)

J. Opt. Soc. Am. B (1)

J. Phys. D (1)

S. Reitzenstein and A. Forchel, “Quantum dot micropillars,” J. Phys. D 43, 033001 (2010).
[CrossRef]

Nat. Photonics (1)

J. Claudon, J. Bleuse, N. S. Malik, M. Bazin, P. Jaffrennou, N. Gregersen, C. Sauvan, P. Lalanne, and J. M. Gérard, “A highly efficient single-photon source based on a quantum dot in a photonic nanowire,” Nat. Photonics 4, 174–177 (2010).
[CrossRef]

Nature (1)

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

Opt. Quantum Electron. (2)

N. Gregersen and J. Mørk, “An improved perfectly matched layer for the eigenmode expansion technique,” Opt. Quantum Electron. 40, 957–966 (2008).
[CrossRef]

P. Bienstman and R. Baets, “Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers,” Opt. Quantum Electron. 33, 327–341(2001).
[CrossRef]

Phys. Rev. (1)

E. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).

Phys. Rev. Lett. (1)

M. Lermer, N. Gregersen, F. Dunzer, S. Reitzenstein, S. Höfling, J. Mørk, L. Worschech, M. Kamp, and A. Forchel, “Bloch-wave engineering of quantum dot micropillars for cavity quantum electrodynamics experiments,” Phys. Rev. Lett. 108, 057402 (2012).
[CrossRef]

Other (7)

J. M. Gérard, “Solid-state cavity-quantum electrodynamics with self-assembled quantum dots,” in Single Quantum Dots: Physics and Applications, P. Michler, ed. (Springer-Verlag, 2003), pp. 269–315.

L. Novotny and B. Hecht, Principles of Nano-Optics, 1st ed. (Cambridge University, 2006), Chap. 8, pp. 250–303.

A. Snyder and J. Love, Optical Waveguide Theory, 1st ed. (Chapman and Hall, 1983), Chap. 31, pp. 601–622.

P. Kristensen, “Light–matter interaction in nanostructured materials,” Ph.D. thesis (Technical University of Denmark, Department of Photonics Engineering, 2009).

A. Taflove and S. Hagness, Computational Electromagnetics: The Finite-Difference Time-Domain Method (Artech House, 2005).

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

P. Bienstman, “Rigorous and efficient modelling of wavelength scale photonic components,” Ph.D. thesis (University of Gent, Department of Information Technology, 2001).

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Figures (4)

Fig. 1.
Fig. 1.

(Left) Emitter (red dot) inside a micropillar in the open geometry, illustrated in an xz plane. Different colors represent different refractive indices. The micropillar is surrounded by vacuum. (Right) Calculated intensity profile of the micropillar design introduced in Section 3. Gray dashed lines outline the boundary of the micropillar.

Fig. 2.
Fig. 2.

Two adjacent waveguide layers in the open geometry, illustrated in an xz plane. The vertical arrow illustrates the illumination from layer 1, while the dashed arrows indicate the scattering of the field at the interface.

Fig. 3.
Fig. 3.

Illustration of the relation nk0=ρ2+β2 in the (ρ,β) plane, where the quarter circle represents the propagation constant nk0. Solid lines indicate the equidistant angular spacing used in the θ discretization.

Fig. 4.
Fig. 4.

Simulation results for the Purcell factor. Results are presented as a function of the number of included modes for both discretization methods.

Equations (60)

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Fp=γγ0,
Fp=QV3λ34π2n3,
Fp=PP0,
P=J02Re(Ey(r0)),
E{1}(x,z)=Uj^{1}(x)eiβj^z+j1=1N1Rj1,j^{1}eiβj1{1}zUj1{1}(x)+m1=120Rm1,j^{1}(ρ1)eiβ{1}(ρ1)zψm1{1}(x,ρ1)dρ1,
E{2}(x,z)=j2=1N2Tj2,j^{2}eiβj2{2}zUj2{2}(x)+m2=120Tm2,j^{2}(ρ2)eiβ{2}(ρ2)zψm2{2}(x,ρ2)dρ2.
E{1}(x,z)=ψm^{1}(x,ρ^)eiβ{1}(ρ^)z+j1=1N1Rj1,m^{1}(ρ^)eiβj1{1}zUj1{1}(x)+m1=120Rm1,m^{1}(ρ1,ρ^)eiβ{1}(ρ1)zψm1{1}(x,ρ1)dρ1,
E{2}(x,z)=j2=1N2Tj2,m^{2}(ρ^)eiβj2{2}zUj2{2}(x)+m2=120Tm2,m^{2}(ρ2,ρ^)eiβ{2}(ρ2)zψm2{2}(x,ρ2)dρ2,
Φj^(x)E{1}(x,0)=E{2}(x,0),
Θm^(x,ρ^)E{1}(x,0)=E{2}(x,0).
Φj^(x)=Φ0,j^(x)+ΛK2*(x)|Φj^,
Θm^(x,ρ^)=Θ0,m^(x,ρ^)+ΛK2*(x)|Θm^(ρ^),
Φj^(x)=j1=1N1cj1Uj1{1}(x)+m1=120cm1(ρ)ψm1{1}(x,ρ)dρ,
Φ0,j^(x)=j1=1N1cj10Uj1{1}(x)+m1=120cm10(ρ)ψm1{1}(x,ρ)dρ.
cj1=cj10+ΛK2(x,x)×{j1=1N1cj1Uj1{1}(x)+m1=120cm1(ρ)ψm1{1}(x,ρ)dρ}×(Uj1{1}(x))*dxdx,
cm1(ρ)=cm10(ρ)+ΛK2(x,x)×{j1=1N1cj1Uj1{1}(x)+m1=120cm1(ρ)ψm1{1}(x,ρ)dρ}×(ψm1{1}(x,ρ))*dxdx.
dk=dk0+k=1N1+2LKk,kdk,
dk={ck1kN1c1(ρkN1)N1+1kN1+Lc2(ρkN1L)N1+L+1kN1+2L,
dk0={ck01kN1c10(ρkN1)N1+1kN1+Lc20(ρkN1L)N1+L+1kN1+2L.
d=d0+Kd,
d=(IK)1d0.
d0=ζdin,
ζkk=2Λ{βk{1}1kN1β1(ρkN1)N1+1kN1+Lβ1(ρkN1L)N1+L+1kN1+2L.
d=(IK)1ζdin.
d=din+dr.
dr=Rdin,
R=(IK)1ζI.
ρ=nk0cos(θ),
V=QFp3λ34π2n3=0.36μm3.
Uj{q}(x)=1Nj{q}{aj{q},clexp(ihj{q}(x+D2)),<xD2,aj{q},coexp(ihj{q},co(xD2))+bj{q},coexp(ihj{q},co(x+D2)),D2xD2,bj{q},clexp(ihj{q}(xD2)),D2x<,
ψm{q}(x,ρ)=1Nm{q}(ρ){am{q},1(ρ)exp(iρx)+bm{q},1(ρ)exp(iρx),<xD2,am{q},2(ρ)exp(iρco{q}x)+bm{q},2(ρ)exp(iρco{q}x),D2xD2,am{q},3(ρ)exp(iρx)+bm{q},3(ρ)exp(iρx),D2x<,
g|f=(f|g)*f(x)g*(x)dx.
Uk{q}|Uj{q}=δjk,
ψm{q}(ρ)|Uj{q}=0,
ψm{q}(ρ)|ψm{q}(ρ)=δmmδ(ρρ),
δ(xx)=j=1NqUj(x)(Uj{q}(x))*+m=120ψm{q}(x,ρ)(ψm{q}(x,ρ))*dρ.
Rj1,j^{1}=Uj1{1}|Φδj1j^,
Rj1,m^{1}(ρ^)=Uj1{1}|Θm^(ρ^),
Rm1,j^{1}(ρ1)=ψm1{1}(ρ1)|Φ,
Rm1,m^{1}(ρ1,ρ^)=ψm1{1}(x,ρ1)|Θm^(ρ^)δm1m^δ(ρ1ρ^),
Tj2,j^{2}=Uj2{2}|Φ,
Tj2,m^{2}(ρ^)=Uj2{2}|Θm^(ρ^),
Tm2,j^{2}(ρ2)=ψm2{2}(ρ2)|Φ,
Rj1,m^{1}(ρ^)=ψm2{2}(ρ2)|Θm^(ρ^).
Λ=1k0(ncl{1}+ncl{2}),
Φ0,j^(x)=2βj^{1}k0(ncl{1}+ncl{2})Uj^{1}(x),
Θ0,m^(x,ρ^)=2β1(ρ^)k0(ncl{1}+ncl{2})ψm^{1}(x,ρ^),
K2(x,x)=j1=1N1(βj1{1}k0ncl{1})Uj1{1}(x)(Uj1{1}(x))*+m1=120(β1(ρ1)k0ncl{1})ψm1{1}(x,ρ1)×(ψm1{1}(x,ρ1))*dρ1+j2=1N2(βj2{2}k0ncl{2})Uj2{2}(x)(Uj2{2}(x))*+m2=120(β2(ρ2)k0ncl{2})ψm2{2}(x,ρ2)×(ψm2{2}(x,ρ2))*dρ2.
K=[KGGKRGKGRKRR],
[Kk,kGG,Kk,kRG,1kN1,1kN1,1kN1,N1+1kN1+2L,Kk,kGR,Kk,kRR,N1+1kN1+2L,N1+1kN1+2L,1kN1,N1+1kN1+2L,].
Kk,kGG=Λ[(βk{1}k0ncl{1})δkk+j2=1N2(βj2{2}k0ncl{2})×Uj2{2}|Uk{1}Uk{1}|Uj2{2}+m2=120(β2(ρ2)k0ncl{2})ψm2{2}(ρ2)|Uk{1}×Uk{1}|ψm2{2}(ρ2)dρ2],
Kk,k˜RG=ΛΔρkN1Lk[j2=1N2(βj2{2}k0ncl{2})Uj2{2}|ψηk{1}(ρkN1Lk)Uk{1}|Uj2{2}+m2=120(β2(ρ2)k0ncl{2})ψm2{2}(ρ2)|ψηk{1}(ρkN1Lk)×Uk{1}|ψm2{2}(ρ2)dρ2],
Kk˜,kGR=Λ[j2=1N2(βj2{2}k0ncl{2})Uj2{2}|Uk{1}ψηk{1}(ρkN1Lk)|Uj2{2}+m2=120(β2(ρ2)k0ncl{2})ψηk{1}(ρkN1Lk)|ψm2{2}(ρ2)×ψm2{2}(ρ2)|Uk{1}dρ2],
Kk˜,k˜RR=Λ[Kηk,ηkND(ρkN1Lk,ρkN1Lk)ΔρkN1Lk+Kηk,ηkD(ρkN1Lk)δkLk,kLk],
Km1,m1ND(ρ,ρ)=j2=1N2[βj2{2}k0ncl{2}]ψm1{1}(ρ)|Uj2{2}Uj2{2}|ψm1{1}(ρ)+m2=12{0[β2(ρ2)k0ncl{2}]×Hm2,m1{2,1}(ρ2,ρ)Hm1,m2{1,2}(ρ,ρ2)dρ2+[β2(ρ)k0ncl{2}]Gm2,m1{2,1}(ρ)Hm1,m2{1,2}(ρ,ρ)+[β2(ρ)k0ncl{2}]Hm2,m1{2,1}(ρ,ρ)Gm1,m2{1,2}(ρ)},
Km1,m1D(ρ)=δm1,m1[β1(ρ)k0ncl{1}]+m2=12Gm2,m1{2,1}(ρ)Gm1,m2{1,2}(ρ)[β2(ρ)k0ncl{2}].
ηk={1N1<kN1+L2N1+L<kN1+2L,
Lk={0N1<kN1+LLN1+L<kN1+2L,
k˜=kN1Lk,
ψm{q}(ρ)|ψm{q}(ρ)=δ(ρρ)Gm,m{q,q}(ρ)+Hm,m{q,q}(ρ,ρ).

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