Abstract

We present a numerical method for the accurate and efficient simulation of strongly localized light sources, such as quantum dots, embedded in dielectric micro-optical structures. We apply the method in order to optimize the photon extraction efficiency of a single-photon emitter consisting of a quantum dot embedded into a multi-layer stack with further lateral structures. Furthermore, we present methods to study the robustness of the extraction efficiency with respect to fabrication errors and defects.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  26. S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91, 043902 (2003).
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  28. L. Zschiedrich, H. Greiner, S. Burger, and F. Schmidt, “Numerical analysis of nanostructures for enhanced light extraction from OLEDs,” Proc. SPIE 8641, 86410B (2013).
    [Crossref]
  29. M. Gschrey, R. Schmidt, J.-H. Schulze, A. Strittmatter, S. Rodt, and S. Reitzenstein, “Resolution and alignment accuracy of low-temperature in situ electron beam lithography for nanophotonic device fabrication,” Journal of Vacuum Science & Technology B, Nanotechnology and Microelectronics: Materials, Processing, Measurement, and Phenomena 33, 021603 (2015).
  30. P.-I. Schneider, X. Garcia Santiago, C. Rockstuhl, and S. Burger, “Global optimization of complex optical structures using Bayesian optimization based on Gaussian processes,” Proc. SPIE 10335, 103350O (2017).
    [Crossref]
  31. N. Garcia and E. Stoll, “Monte carlo calculation for electromagnetic-wave scattering from random rough surfaces,” Phys. Rev. Lett. 52, 1798–1801 (1984).
    [Crossref]
  32. M. Andersen, S. Stobbe, A. Sørensen, and P. Lodahl, “Strongly modified plasmon-matter inetraction with mesoscopic quantum emitters,” Nature Physics 7, 215–218 (2010).
    [Crossref]
  33. P. Tighineanu, A. S. Sørensen, S. Stobbe, and P. Lodahl, “Unraveling the mesoscopic character of quantum dots in nanophotonics,” Phys. Rev. Lett. 114, 247401 (2015).
    [Crossref] [PubMed]

2018 (1)

A. Kaganskiy, S. Fischbach, A. Strittmatter, S. Rodt, T. Heindel, and S. Reitzenstein, “Enhancing the photon-extraction efficiency of site-controlled quantum dots by deterministically fabricated microlenses,” Optics Communications 413, 162–166 (2018).
[Crossref]

2017 (4)

P.-I. Schneider, X. Garcia Santiago, C. Rockstuhl, and S. Burger, “Global optimization of complex optical structures using Bayesian optimization based on Gaussian processes,” Proc. SPIE 10335, 103350O (2017).
[Crossref]

R. S. Daveau, K. C. Balram, T. Pregnolato, J. Liu, E. H. Lee, J. D. Song, V. Verma, R. Mirin, S. W. Nam, L. Midolo, S. Stobbe, K. Srinivasan, and P. Lodahl, “Efficient fiber-coupled single-photon source based on quantum dots in a photonic-crystal waveguide,” Optica 4, 178–184 (2017).
[Crossref] [PubMed]

Y.-M. He, J. Liu, S. Maier, M. Emmerling, S. Gerhardt, M. Davanço, K. Srinivasan, C. Schneider, and S. Höfling, “Deterministic implementation of a bright, on-demand single-photon source with near-unity indistinguishability via quantum dot imaging,” Optica 4, 802–808 (2017).
[Crossref] [PubMed]

M. Calic, C. Jarlov, P. Gallo, B. Dwir, A. Rudra, and E. Kapon, “Deterministic radiative coupling of two semiconductor quantum dots to the optical mode of a photonic crystal nanocavity,” Sci. Rep. 7, 4100 (2017).
[Crossref] [PubMed]

2016 (4)

J. Yang, J.-P. Hugonin, and P. Lalanne, “Near-to-far field transformations for radiative and guided waves,” ACS Photonics 3, 395–402 (2016).
[Crossref]

N. Somaschi, V. Giesz, L. De Santis, J. C. Loredo, M. P. Almeida, G. Hornecker, S. L. Portalupi, T. Grange, C. Antón, J. Demory, C. Gómez, I. Sagnes, N. D. Lanzillotti-Kimura, A. Lemaítre, A. Auffeves, A. G. White, L. Lanco, and P. Senellart, “Near-optimal single-photon sources in the solid state,” Nat. Photonics 10, 340 (2016).
[Crossref]

I. Aharonovich, D. Englund, and M. Toth, “Solid-state single-photon emitters,” Nat. Photonics 10, 631 (2016).
[Crossref]

D. Cadeddu, J. Teissier, F. R. Braakman, N. Gregersen, P. Stepanov, J.-M. Gérard, J. Claudon, R. J. Warburton, M. Poggio, and M. Munsch, “A fiber-coupled quantum-dot on a photonic tip,” Applied Physics Letters 108, 011112 (2016).
[Crossref]

2015 (6)

C.-M. Lee, H.-J. Lim, C. Schneider, S. Maier, S. Höfling, M. Kamp, and Y.-H. Lee, “Efficient single photon source based on m-fibre-coupled tunable microcavity,” Scientific Reports 5, 14309 (2015).
[Crossref]

L. Sapienza, M. Davanço, A. Badolato, and K. Srinivasan, “Nanoscale optical positioning of single quantum dots for bright and pure single-photon emission,” Nat. Communications 6, 7833 (2015).
[Crossref]

M. Gschrey, A. Thoma, P. Schnauber, M. Seifried, R. Schmidt, B. Wohlfeil, L. Krüger, J.-H. Schulze, T. Heindel, S. Burger, F. Schmidt, A. Strittmatter, S. Rodt, and S. Reitzenstein, “Highly indistinguishable photons from deterministic quantum-dot microlenses utilizing three-dimensional in situ electron-beam lithography,” Nat. Communications 6, 7662 (2015).
[Crossref]

M. Gschrey, R. Schmidt, J.-H. Schulze, A. Strittmatter, S. Rodt, and S. Reitzenstein, “Resolution and alignment accuracy of low-temperature in situ electron beam lithography for nanophotonic device fabrication,” Journal of Vacuum Science & Technology B, Nanotechnology and Microelectronics: Materials, Processing, Measurement, and Phenomena 33, 021603 (2015).

S. Burger, L. Zschiedrich, J. Pomplun, S. Herrmann, and F. Schmidt, “Hp-finite element method for simulating light scattering from complex 3D structures,” Proc. SPIE 9424, 94240Z (2015).
[Crossref]

P. Tighineanu, A. S. Sørensen, S. Stobbe, and P. Lodahl, “Unraveling the mesoscopic character of quantum dots in nanophotonics,” Phys. Rev. Lett. 114, 247401 (2015).
[Crossref] [PubMed]

2014 (1)

G. Bulgarini, M. E. Reimer, M. Bouwes Bavinck, K. D. Jöns, D. Dalacu, P. J. Poole, E. P. A. M. Bakkers, and V. Zwiller, “Nanowire waveguides launching single photons in a gaussian mode for ideal fiber coupling,” Nano Letters 14, 4102–4106 (2014). PMID: .
[Crossref] [PubMed]

2013 (3)

L. Zschiedrich, H. Greiner, S. Burger, and F. Schmidt, “Numerical analysis of nanostructures for enhanced light extraction from OLEDs,” Proc. SPIE 8641, 86410B (2013).
[Crossref]

B. Maes, J. Petráček, S. Burger, P. Kwiecien, J. Luksch, and I. Richter, “Simulations of high-Q optical nanocavities with a gradual 1D bandgap,” Opt. Express 21, 6794 (2013).
[Crossref] [PubMed]

M. Bergot and M. Duruflé, “High-order optimal edge elements for pyramids, prisms and hexahedra,” Journal of Computational Physics 232, 189–213 (2013).
[Crossref]

2011 (1)

M. Davanço, M. T. Rakher, W. Wegscheider, D. Schuh, A. Badolato, and K. Srinivasan, “Efficient quantum dot single photon extraction into an optical fiber using a nanophotonic directional coupler,” Applied Physics Letters 99, 121101 (2011).
[Crossref]

2010 (3)

F. Haupt, S. S. R. Oemrawsingh, S. M. Thon, H. Kim, D. Kleckner, D. Ding, D. J. Suntrup, P. M. Petroff, and D. Bouwmeester, “Fiber-connectorized micropillar cavities,” Applied Physics Letters 97, 131113 (2010).
[Crossref]

Y. Chen, T. R. Nielsen, N. Gregersen, P. Lodahl, and J. Mørk, “Finite-element modeling of spontaneous emission of a quantum emitter at nanoscale proximity to plasmonic waveguides,” Phys. Rev. B 81, 125431 (2010).
[Crossref]

M. Andersen, S. Stobbe, A. Sørensen, and P. Lodahl, “Strongly modified plasmon-matter inetraction with mesoscopic quantum emitters,” Nature Physics 7, 215–218 (2010).
[Crossref]

2009 (1)

A. Muller, E. B. Flagg, M. Metcalfe, J. Lawall, and G. S. Solomon, “Coupling an epitaxial quantum dot to a fiber-based external-mirror microcavity,” Applied Physics Letters 95, 173101 (2009).
[Crossref]

2005 (1)

A. Badolato, K. Hennessy, M. Atatüre, J. Dreiser, E. Hu, P. M. Petroff, and A. Imamoğlu, “Deterministic coupling of single quantum dots to single nanocavity modes,” Science 308, 1158–1161 (2005).
[Crossref] [PubMed]

2003 (1)

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91, 043902 (2003).
[Crossref] [PubMed]

2002 (1)

W. Barnes, G. Björk, J. Gérard, P. Jonsson, J. Wasey, P. Worthing, and V. Zwiller, “Solid-state single photon sources: light collection strategies,” Eur. Phys. J. D 18, 197–210 (2002).
[Crossref]

1984 (1)

N. Garcia and E. Stoll, “Monte carlo calculation for electromagnetic-wave scattering from random rough surfaces,” Phys. Rev. Lett. 52, 1798–1801 (1984).
[Crossref]

1946 (1)

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Proceedings of the American Physical Society 69, 681 (1946).

Aharonovich, I.

I. Aharonovich, D. Englund, and M. Toth, “Solid-state single-photon emitters,” Nat. Photonics 10, 631 (2016).
[Crossref]

Almeida, M. P.

N. Somaschi, V. Giesz, L. De Santis, J. C. Loredo, M. P. Almeida, G. Hornecker, S. L. Portalupi, T. Grange, C. Antón, J. Demory, C. Gómez, I. Sagnes, N. D. Lanzillotti-Kimura, A. Lemaítre, A. Auffeves, A. G. White, L. Lanco, and P. Senellart, “Near-optimal single-photon sources in the solid state,” Nat. Photonics 10, 340 (2016).
[Crossref]

Andersen, M.

M. Andersen, S. Stobbe, A. Sørensen, and P. Lodahl, “Strongly modified plasmon-matter inetraction with mesoscopic quantum emitters,” Nature Physics 7, 215–218 (2010).
[Crossref]

Antón, C.

N. Somaschi, V. Giesz, L. De Santis, J. C. Loredo, M. P. Almeida, G. Hornecker, S. L. Portalupi, T. Grange, C. Antón, J. Demory, C. Gómez, I. Sagnes, N. D. Lanzillotti-Kimura, A. Lemaítre, A. Auffeves, A. G. White, L. Lanco, and P. Senellart, “Near-optimal single-photon sources in the solid state,” Nat. Photonics 10, 340 (2016).
[Crossref]

Atatüre, M.

A. Badolato, K. Hennessy, M. Atatüre, J. Dreiser, E. Hu, P. M. Petroff, and A. Imamoğlu, “Deterministic coupling of single quantum dots to single nanocavity modes,” Science 308, 1158–1161 (2005).
[Crossref] [PubMed]

Auffeves, A.

N. Somaschi, V. Giesz, L. De Santis, J. C. Loredo, M. P. Almeida, G. Hornecker, S. L. Portalupi, T. Grange, C. Antón, J. Demory, C. Gómez, I. Sagnes, N. D. Lanzillotti-Kimura, A. Lemaítre, A. Auffeves, A. G. White, L. Lanco, and P. Senellart, “Near-optimal single-photon sources in the solid state,” Nat. Photonics 10, 340 (2016).
[Crossref]

Badolato, A.

L. Sapienza, M. Davanço, A. Badolato, and K. Srinivasan, “Nanoscale optical positioning of single quantum dots for bright and pure single-photon emission,” Nat. Communications 6, 7833 (2015).
[Crossref]

M. Davanço, M. T. Rakher, W. Wegscheider, D. Schuh, A. Badolato, and K. Srinivasan, “Efficient quantum dot single photon extraction into an optical fiber using a nanophotonic directional coupler,” Applied Physics Letters 99, 121101 (2011).
[Crossref]

A. Badolato, K. Hennessy, M. Atatüre, J. Dreiser, E. Hu, P. M. Petroff, and A. Imamoğlu, “Deterministic coupling of single quantum dots to single nanocavity modes,” Science 308, 1158–1161 (2005).
[Crossref] [PubMed]

Bakkers, E. P. A. M.

G. Bulgarini, M. E. Reimer, M. Bouwes Bavinck, K. D. Jöns, D. Dalacu, P. J. Poole, E. P. A. M. Bakkers, and V. Zwiller, “Nanowire waveguides launching single photons in a gaussian mode for ideal fiber coupling,” Nano Letters 14, 4102–4106 (2014). PMID: .
[Crossref] [PubMed]

Balram, K. C.

Barnes, W.

W. Barnes, G. Björk, J. Gérard, P. Jonsson, J. Wasey, P. Worthing, and V. Zwiller, “Solid-state single photon sources: light collection strategies,” Eur. Phys. J. D 18, 197–210 (2002).
[Crossref]

Bergot, M.

M. Bergot and M. Duruflé, “High-order optimal edge elements for pyramids, prisms and hexahedra,” Journal of Computational Physics 232, 189–213 (2013).
[Crossref]

Björk, G.

W. Barnes, G. Björk, J. Gérard, P. Jonsson, J. Wasey, P. Worthing, and V. Zwiller, “Solid-state single photon sources: light collection strategies,” Eur. Phys. J. D 18, 197–210 (2002).
[Crossref]

Bouwes Bavinck, M.

G. Bulgarini, M. E. Reimer, M. Bouwes Bavinck, K. D. Jöns, D. Dalacu, P. J. Poole, E. P. A. M. Bakkers, and V. Zwiller, “Nanowire waveguides launching single photons in a gaussian mode for ideal fiber coupling,” Nano Letters 14, 4102–4106 (2014). PMID: .
[Crossref] [PubMed]

Bouwmeester, D.

F. Haupt, S. S. R. Oemrawsingh, S. M. Thon, H. Kim, D. Kleckner, D. Ding, D. J. Suntrup, P. M. Petroff, and D. Bouwmeester, “Fiber-connectorized micropillar cavities,” Applied Physics Letters 97, 131113 (2010).
[Crossref]

Braakman, F. R.

D. Cadeddu, J. Teissier, F. R. Braakman, N. Gregersen, P. Stepanov, J.-M. Gérard, J. Claudon, R. J. Warburton, M. Poggio, and M. Munsch, “A fiber-coupled quantum-dot on a photonic tip,” Applied Physics Letters 108, 011112 (2016).
[Crossref]

Bulgarini, G.

G. Bulgarini, M. E. Reimer, M. Bouwes Bavinck, K. D. Jöns, D. Dalacu, P. J. Poole, E. P. A. M. Bakkers, and V. Zwiller, “Nanowire waveguides launching single photons in a gaussian mode for ideal fiber coupling,” Nano Letters 14, 4102–4106 (2014). PMID: .
[Crossref] [PubMed]

Burger, S.

P.-I. Schneider, X. Garcia Santiago, C. Rockstuhl, and S. Burger, “Global optimization of complex optical structures using Bayesian optimization based on Gaussian processes,” Proc. SPIE 10335, 103350O (2017).
[Crossref]

S. Burger, L. Zschiedrich, J. Pomplun, S. Herrmann, and F. Schmidt, “Hp-finite element method for simulating light scattering from complex 3D structures,” Proc. SPIE 9424, 94240Z (2015).
[Crossref]

M. Gschrey, A. Thoma, P. Schnauber, M. Seifried, R. Schmidt, B. Wohlfeil, L. Krüger, J.-H. Schulze, T. Heindel, S. Burger, F. Schmidt, A. Strittmatter, S. Rodt, and S. Reitzenstein, “Highly indistinguishable photons from deterministic quantum-dot microlenses utilizing three-dimensional in situ electron-beam lithography,” Nat. Communications 6, 7662 (2015).
[Crossref]

B. Maes, J. Petráček, S. Burger, P. Kwiecien, J. Luksch, and I. Richter, “Simulations of high-Q optical nanocavities with a gradual 1D bandgap,” Opt. Express 21, 6794 (2013).
[Crossref] [PubMed]

L. Zschiedrich, H. Greiner, S. Burger, and F. Schmidt, “Numerical analysis of nanostructures for enhanced light extraction from OLEDs,” Proc. SPIE 8641, 86410B (2013).
[Crossref]

Cadeddu, D.

D. Cadeddu, J. Teissier, F. R. Braakman, N. Gregersen, P. Stepanov, J.-M. Gérard, J. Claudon, R. J. Warburton, M. Poggio, and M. Munsch, “A fiber-coupled quantum-dot on a photonic tip,” Applied Physics Letters 108, 011112 (2016).
[Crossref]

Calic, M.

M. Calic, C. Jarlov, P. Gallo, B. Dwir, A. Rudra, and E. Kapon, “Deterministic radiative coupling of two semiconductor quantum dots to the optical mode of a photonic crystal nanocavity,” Sci. Rep. 7, 4100 (2017).
[Crossref] [PubMed]

Chen, Y.

Y. Chen, T. R. Nielsen, N. Gregersen, P. Lodahl, and J. Mørk, “Finite-element modeling of spontaneous emission of a quantum emitter at nanoscale proximity to plasmonic waveguides,” Phys. Rev. B 81, 125431 (2010).
[Crossref]

Claudon, J.

D. Cadeddu, J. Teissier, F. R. Braakman, N. Gregersen, P. Stepanov, J.-M. Gérard, J. Claudon, R. J. Warburton, M. Poggio, and M. Munsch, “A fiber-coupled quantum-dot on a photonic tip,” Applied Physics Letters 108, 011112 (2016).
[Crossref]

Dalacu, D.

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A. Kaganskiy, S. Fischbach, A. Strittmatter, S. Rodt, T. Heindel, and S. Reitzenstein, “Enhancing the photon-extraction efficiency of site-controlled quantum dots by deterministically fabricated microlenses,” Optics Communications 413, 162–166 (2018).
[Crossref]

M. Gschrey, R. Schmidt, J.-H. Schulze, A. Strittmatter, S. Rodt, and S. Reitzenstein, “Resolution and alignment accuracy of low-temperature in situ electron beam lithography for nanophotonic device fabrication,” Journal of Vacuum Science & Technology B, Nanotechnology and Microelectronics: Materials, Processing, Measurement, and Phenomena 33, 021603 (2015).

M. Gschrey, A. Thoma, P. Schnauber, M. Seifried, R. Schmidt, B. Wohlfeil, L. Krüger, J.-H. Schulze, T. Heindel, S. Burger, F. Schmidt, A. Strittmatter, S. Rodt, and S. Reitzenstein, “Highly indistinguishable photons from deterministic quantum-dot microlenses utilizing three-dimensional in situ electron-beam lithography,” Nat. Communications 6, 7662 (2015).
[Crossref]

A. Schlehahn, S. Fischbach, R. Schmidt, A. Kaganskiy, A. Strittmatter, S. Rodt, T. Heindel, and S. Reitzenstein, “A stand-alone fiber-coupled single-photon source,” arXiv preprint arXiv:1703.10536 (2017).

Richter, I.

Rockstuhl, C.

P.-I. Schneider, X. Garcia Santiago, C. Rockstuhl, and S. Burger, “Global optimization of complex optical structures using Bayesian optimization based on Gaussian processes,” Proc. SPIE 10335, 103350O (2017).
[Crossref]

Rodt, S.

A. Kaganskiy, S. Fischbach, A. Strittmatter, S. Rodt, T. Heindel, and S. Reitzenstein, “Enhancing the photon-extraction efficiency of site-controlled quantum dots by deterministically fabricated microlenses,” Optics Communications 413, 162–166 (2018).
[Crossref]

M. Gschrey, R. Schmidt, J.-H. Schulze, A. Strittmatter, S. Rodt, and S. Reitzenstein, “Resolution and alignment accuracy of low-temperature in situ electron beam lithography for nanophotonic device fabrication,” Journal of Vacuum Science & Technology B, Nanotechnology and Microelectronics: Materials, Processing, Measurement, and Phenomena 33, 021603 (2015).

M. Gschrey, A. Thoma, P. Schnauber, M. Seifried, R. Schmidt, B. Wohlfeil, L. Krüger, J.-H. Schulze, T. Heindel, S. Burger, F. Schmidt, A. Strittmatter, S. Rodt, and S. Reitzenstein, “Highly indistinguishable photons from deterministic quantum-dot microlenses utilizing three-dimensional in situ electron-beam lithography,” Nat. Communications 6, 7662 (2015).
[Crossref]

A. Schlehahn, S. Fischbach, R. Schmidt, A. Kaganskiy, A. Strittmatter, S. Rodt, T. Heindel, and S. Reitzenstein, “A stand-alone fiber-coupled single-photon source,” arXiv preprint arXiv:1703.10536 (2017).

Rudra, A.

M. Calic, C. Jarlov, P. Gallo, B. Dwir, A. Rudra, and E. Kapon, “Deterministic radiative coupling of two semiconductor quantum dots to the optical mode of a photonic crystal nanocavity,” Sci. Rep. 7, 4100 (2017).
[Crossref] [PubMed]

Sagnes, I.

N. Somaschi, V. Giesz, L. De Santis, J. C. Loredo, M. P. Almeida, G. Hornecker, S. L. Portalupi, T. Grange, C. Antón, J. Demory, C. Gómez, I. Sagnes, N. D. Lanzillotti-Kimura, A. Lemaítre, A. Auffeves, A. G. White, L. Lanco, and P. Senellart, “Near-optimal single-photon sources in the solid state,” Nat. Photonics 10, 340 (2016).
[Crossref]

Sapienza, L.

L. Sapienza, M. Davanço, A. Badolato, and K. Srinivasan, “Nanoscale optical positioning of single quantum dots for bright and pure single-photon emission,” Nat. Communications 6, 7833 (2015).
[Crossref]

Schlehahn, A.

A. Schlehahn, S. Fischbach, R. Schmidt, A. Kaganskiy, A. Strittmatter, S. Rodt, T. Heindel, and S. Reitzenstein, “A stand-alone fiber-coupled single-photon source,” arXiv preprint arXiv:1703.10536 (2017).

Schmidt, F.

S. Burger, L. Zschiedrich, J. Pomplun, S. Herrmann, and F. Schmidt, “Hp-finite element method for simulating light scattering from complex 3D structures,” Proc. SPIE 9424, 94240Z (2015).
[Crossref]

M. Gschrey, A. Thoma, P. Schnauber, M. Seifried, R. Schmidt, B. Wohlfeil, L. Krüger, J.-H. Schulze, T. Heindel, S. Burger, F. Schmidt, A. Strittmatter, S. Rodt, and S. Reitzenstein, “Highly indistinguishable photons from deterministic quantum-dot microlenses utilizing three-dimensional in situ electron-beam lithography,” Nat. Communications 6, 7662 (2015).
[Crossref]

L. Zschiedrich, H. Greiner, S. Burger, and F. Schmidt, “Numerical analysis of nanostructures for enhanced light extraction from OLEDs,” Proc. SPIE 8641, 86410B (2013).
[Crossref]

A. Lavrinenko, J. Lægsgaard, N. Gregersen, F. Schmidt, and T. Søndergaard,. Numerical Methods in Photonics, Optical Sciences and Applications of Light (CRC, 2014).

Schmidt, R.

M. Gschrey, R. Schmidt, J.-H. Schulze, A. Strittmatter, S. Rodt, and S. Reitzenstein, “Resolution and alignment accuracy of low-temperature in situ electron beam lithography for nanophotonic device fabrication,” Journal of Vacuum Science & Technology B, Nanotechnology and Microelectronics: Materials, Processing, Measurement, and Phenomena 33, 021603 (2015).

M. Gschrey, A. Thoma, P. Schnauber, M. Seifried, R. Schmidt, B. Wohlfeil, L. Krüger, J.-H. Schulze, T. Heindel, S. Burger, F. Schmidt, A. Strittmatter, S. Rodt, and S. Reitzenstein, “Highly indistinguishable photons from deterministic quantum-dot microlenses utilizing three-dimensional in situ electron-beam lithography,” Nat. Communications 6, 7662 (2015).
[Crossref]

A. Schlehahn, S. Fischbach, R. Schmidt, A. Kaganskiy, A. Strittmatter, S. Rodt, T. Heindel, and S. Reitzenstein, “A stand-alone fiber-coupled single-photon source,” arXiv preprint arXiv:1703.10536 (2017).

Schnauber, P.

M. Gschrey, A. Thoma, P. Schnauber, M. Seifried, R. Schmidt, B. Wohlfeil, L. Krüger, J.-H. Schulze, T. Heindel, S. Burger, F. Schmidt, A. Strittmatter, S. Rodt, and S. Reitzenstein, “Highly indistinguishable photons from deterministic quantum-dot microlenses utilizing three-dimensional in situ electron-beam lithography,” Nat. Communications 6, 7662 (2015).
[Crossref]

Schneider, C.

Schneider, P.-I.

P.-I. Schneider, X. Garcia Santiago, C. Rockstuhl, and S. Burger, “Global optimization of complex optical structures using Bayesian optimization based on Gaussian processes,” Proc. SPIE 10335, 103350O (2017).
[Crossref]

Schuh, D.

M. Davanço, M. T. Rakher, W. Wegscheider, D. Schuh, A. Badolato, and K. Srinivasan, “Efficient quantum dot single photon extraction into an optical fiber using a nanophotonic directional coupler,” Applied Physics Letters 99, 121101 (2011).
[Crossref]

Schulze, J.-H.

M. Gschrey, A. Thoma, P. Schnauber, M. Seifried, R. Schmidt, B. Wohlfeil, L. Krüger, J.-H. Schulze, T. Heindel, S. Burger, F. Schmidt, A. Strittmatter, S. Rodt, and S. Reitzenstein, “Highly indistinguishable photons from deterministic quantum-dot microlenses utilizing three-dimensional in situ electron-beam lithography,” Nat. Communications 6, 7662 (2015).
[Crossref]

M. Gschrey, R. Schmidt, J.-H. Schulze, A. Strittmatter, S. Rodt, and S. Reitzenstein, “Resolution and alignment accuracy of low-temperature in situ electron beam lithography for nanophotonic device fabrication,” Journal of Vacuum Science & Technology B, Nanotechnology and Microelectronics: Materials, Processing, Measurement, and Phenomena 33, 021603 (2015).

Seifried, M.

M. Gschrey, A. Thoma, P. Schnauber, M. Seifried, R. Schmidt, B. Wohlfeil, L. Krüger, J.-H. Schulze, T. Heindel, S. Burger, F. Schmidt, A. Strittmatter, S. Rodt, and S. Reitzenstein, “Highly indistinguishable photons from deterministic quantum-dot microlenses utilizing three-dimensional in situ electron-beam lithography,” Nat. Communications 6, 7662 (2015).
[Crossref]

Senellart, P.

N. Somaschi, V. Giesz, L. De Santis, J. C. Loredo, M. P. Almeida, G. Hornecker, S. L. Portalupi, T. Grange, C. Antón, J. Demory, C. Gómez, I. Sagnes, N. D. Lanzillotti-Kimura, A. Lemaítre, A. Auffeves, A. G. White, L. Lanco, and P. Senellart, “Near-optimal single-photon sources in the solid state,” Nat. Photonics 10, 340 (2016).
[Crossref]

Solomon, G. S.

A. Muller, E. B. Flagg, M. Metcalfe, J. Lawall, and G. S. Solomon, “Coupling an epitaxial quantum dot to a fiber-based external-mirror microcavity,” Applied Physics Letters 95, 173101 (2009).
[Crossref]

Somaschi, N.

N. Somaschi, V. Giesz, L. De Santis, J. C. Loredo, M. P. Almeida, G. Hornecker, S. L. Portalupi, T. Grange, C. Antón, J. Demory, C. Gómez, I. Sagnes, N. D. Lanzillotti-Kimura, A. Lemaítre, A. Auffeves, A. G. White, L. Lanco, and P. Senellart, “Near-optimal single-photon sources in the solid state,” Nat. Photonics 10, 340 (2016).
[Crossref]

Søndergaard, T.

A. Lavrinenko, J. Lægsgaard, N. Gregersen, F. Schmidt, and T. Søndergaard,. Numerical Methods in Photonics, Optical Sciences and Applications of Light (CRC, 2014).

Song, J. D.

Sørensen, A.

M. Andersen, S. Stobbe, A. Sørensen, and P. Lodahl, “Strongly modified plasmon-matter inetraction with mesoscopic quantum emitters,” Nature Physics 7, 215–218 (2010).
[Crossref]

Sørensen, A. S.

P. Tighineanu, A. S. Sørensen, S. Stobbe, and P. Lodahl, “Unraveling the mesoscopic character of quantum dots in nanophotonics,” Phys. Rev. Lett. 114, 247401 (2015).
[Crossref] [PubMed]

Spillane, S. M.

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91, 043902 (2003).
[Crossref] [PubMed]

Srinivasan, K.

R. S. Daveau, K. C. Balram, T. Pregnolato, J. Liu, E. H. Lee, J. D. Song, V. Verma, R. Mirin, S. W. Nam, L. Midolo, S. Stobbe, K. Srinivasan, and P. Lodahl, “Efficient fiber-coupled single-photon source based on quantum dots in a photonic-crystal waveguide,” Optica 4, 178–184 (2017).
[Crossref] [PubMed]

Y.-M. He, J. Liu, S. Maier, M. Emmerling, S. Gerhardt, M. Davanço, K. Srinivasan, C. Schneider, and S. Höfling, “Deterministic implementation of a bright, on-demand single-photon source with near-unity indistinguishability via quantum dot imaging,” Optica 4, 802–808 (2017).
[Crossref] [PubMed]

L. Sapienza, M. Davanço, A. Badolato, and K. Srinivasan, “Nanoscale optical positioning of single quantum dots for bright and pure single-photon emission,” Nat. Communications 6, 7833 (2015).
[Crossref]

M. Davanço, M. T. Rakher, W. Wegscheider, D. Schuh, A. Badolato, and K. Srinivasan, “Efficient quantum dot single photon extraction into an optical fiber using a nanophotonic directional coupler,” Applied Physics Letters 99, 121101 (2011).
[Crossref]

Stepanov, P.

D. Cadeddu, J. Teissier, F. R. Braakman, N. Gregersen, P. Stepanov, J.-M. Gérard, J. Claudon, R. J. Warburton, M. Poggio, and M. Munsch, “A fiber-coupled quantum-dot on a photonic tip,” Applied Physics Letters 108, 011112 (2016).
[Crossref]

Stobbe, S.

R. S. Daveau, K. C. Balram, T. Pregnolato, J. Liu, E. H. Lee, J. D. Song, V. Verma, R. Mirin, S. W. Nam, L. Midolo, S. Stobbe, K. Srinivasan, and P. Lodahl, “Efficient fiber-coupled single-photon source based on quantum dots in a photonic-crystal waveguide,” Optica 4, 178–184 (2017).
[Crossref] [PubMed]

P. Tighineanu, A. S. Sørensen, S. Stobbe, and P. Lodahl, “Unraveling the mesoscopic character of quantum dots in nanophotonics,” Phys. Rev. Lett. 114, 247401 (2015).
[Crossref] [PubMed]

M. Andersen, S. Stobbe, A. Sørensen, and P. Lodahl, “Strongly modified plasmon-matter inetraction with mesoscopic quantum emitters,” Nature Physics 7, 215–218 (2010).
[Crossref]

Stoll, E.

N. Garcia and E. Stoll, “Monte carlo calculation for electromagnetic-wave scattering from random rough surfaces,” Phys. Rev. Lett. 52, 1798–1801 (1984).
[Crossref]

Strittmatter, A.

A. Kaganskiy, S. Fischbach, A. Strittmatter, S. Rodt, T. Heindel, and S. Reitzenstein, “Enhancing the photon-extraction efficiency of site-controlled quantum dots by deterministically fabricated microlenses,” Optics Communications 413, 162–166 (2018).
[Crossref]

M. Gschrey, R. Schmidt, J.-H. Schulze, A. Strittmatter, S. Rodt, and S. Reitzenstein, “Resolution and alignment accuracy of low-temperature in situ electron beam lithography for nanophotonic device fabrication,” Journal of Vacuum Science & Technology B, Nanotechnology and Microelectronics: Materials, Processing, Measurement, and Phenomena 33, 021603 (2015).

M. Gschrey, A. Thoma, P. Schnauber, M. Seifried, R. Schmidt, B. Wohlfeil, L. Krüger, J.-H. Schulze, T. Heindel, S. Burger, F. Schmidt, A. Strittmatter, S. Rodt, and S. Reitzenstein, “Highly indistinguishable photons from deterministic quantum-dot microlenses utilizing three-dimensional in situ electron-beam lithography,” Nat. Communications 6, 7662 (2015).
[Crossref]

A. Schlehahn, S. Fischbach, R. Schmidt, A. Kaganskiy, A. Strittmatter, S. Rodt, T. Heindel, and S. Reitzenstein, “A stand-alone fiber-coupled single-photon source,” arXiv preprint arXiv:1703.10536 (2017).

Suntrup, D. J.

F. Haupt, S. S. R. Oemrawsingh, S. M. Thon, H. Kim, D. Kleckner, D. Ding, D. J. Suntrup, P. M. Petroff, and D. Bouwmeester, “Fiber-connectorized micropillar cavities,” Applied Physics Letters 97, 131113 (2010).
[Crossref]

Teissier, J.

D. Cadeddu, J. Teissier, F. R. Braakman, N. Gregersen, P. Stepanov, J.-M. Gérard, J. Claudon, R. J. Warburton, M. Poggio, and M. Munsch, “A fiber-coupled quantum-dot on a photonic tip,” Applied Physics Letters 108, 011112 (2016).
[Crossref]

Thoma, A.

M. Gschrey, A. Thoma, P. Schnauber, M. Seifried, R. Schmidt, B. Wohlfeil, L. Krüger, J.-H. Schulze, T. Heindel, S. Burger, F. Schmidt, A. Strittmatter, S. Rodt, and S. Reitzenstein, “Highly indistinguishable photons from deterministic quantum-dot microlenses utilizing three-dimensional in situ electron-beam lithography,” Nat. Communications 6, 7662 (2015).
[Crossref]

Thon, S. M.

F. Haupt, S. S. R. Oemrawsingh, S. M. Thon, H. Kim, D. Kleckner, D. Ding, D. J. Suntrup, P. M. Petroff, and D. Bouwmeester, “Fiber-connectorized micropillar cavities,” Applied Physics Letters 97, 131113 (2010).
[Crossref]

Tighineanu, P.

P. Tighineanu, A. S. Sørensen, S. Stobbe, and P. Lodahl, “Unraveling the mesoscopic character of quantum dots in nanophotonics,” Phys. Rev. Lett. 114, 247401 (2015).
[Crossref] [PubMed]

Toth, M.

I. Aharonovich, D. Englund, and M. Toth, “Solid-state single-photon emitters,” Nat. Photonics 10, 631 (2016).
[Crossref]

Vahala, K. J.

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91, 043902 (2003).
[Crossref] [PubMed]

Verma, V.

Warburton, R. J.

D. Cadeddu, J. Teissier, F. R. Braakman, N. Gregersen, P. Stepanov, J.-M. Gérard, J. Claudon, R. J. Warburton, M. Poggio, and M. Munsch, “A fiber-coupled quantum-dot on a photonic tip,” Applied Physics Letters 108, 011112 (2016).
[Crossref]

Wasey, J.

W. Barnes, G. Björk, J. Gérard, P. Jonsson, J. Wasey, P. Worthing, and V. Zwiller, “Solid-state single photon sources: light collection strategies,” Eur. Phys. J. D 18, 197–210 (2002).
[Crossref]

Wegscheider, W.

M. Davanço, M. T. Rakher, W. Wegscheider, D. Schuh, A. Badolato, and K. Srinivasan, “Efficient quantum dot single photon extraction into an optical fiber using a nanophotonic directional coupler,” Applied Physics Letters 99, 121101 (2011).
[Crossref]

White, A. G.

N. Somaschi, V. Giesz, L. De Santis, J. C. Loredo, M. P. Almeida, G. Hornecker, S. L. Portalupi, T. Grange, C. Antón, J. Demory, C. Gómez, I. Sagnes, N. D. Lanzillotti-Kimura, A. Lemaítre, A. Auffeves, A. G. White, L. Lanco, and P. Senellart, “Near-optimal single-photon sources in the solid state,” Nat. Photonics 10, 340 (2016).
[Crossref]

Wohlfeil, B.

M. Gschrey, A. Thoma, P. Schnauber, M. Seifried, R. Schmidt, B. Wohlfeil, L. Krüger, J.-H. Schulze, T. Heindel, S. Burger, F. Schmidt, A. Strittmatter, S. Rodt, and S. Reitzenstein, “Highly indistinguishable photons from deterministic quantum-dot microlenses utilizing three-dimensional in situ electron-beam lithography,” Nat. Communications 6, 7662 (2015).
[Crossref]

Worthing, P.

W. Barnes, G. Björk, J. Gérard, P. Jonsson, J. Wasey, P. Worthing, and V. Zwiller, “Solid-state single photon sources: light collection strategies,” Eur. Phys. J. D 18, 197–210 (2002).
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Yang, J.

J. Yang, J.-P. Hugonin, and P. Lalanne, “Near-to-far field transformations for radiative and guided waves,” ACS Photonics 3, 395–402 (2016).
[Crossref]

Zschiedrich, L.

S. Burger, L. Zschiedrich, J. Pomplun, S. Herrmann, and F. Schmidt, “Hp-finite element method for simulating light scattering from complex 3D structures,” Proc. SPIE 9424, 94240Z (2015).
[Crossref]

L. Zschiedrich, H. Greiner, S. Burger, and F. Schmidt, “Numerical analysis of nanostructures for enhanced light extraction from OLEDs,” Proc. SPIE 8641, 86410B (2013).
[Crossref]

Zwiller, V.

G. Bulgarini, M. E. Reimer, M. Bouwes Bavinck, K. D. Jöns, D. Dalacu, P. J. Poole, E. P. A. M. Bakkers, and V. Zwiller, “Nanowire waveguides launching single photons in a gaussian mode for ideal fiber coupling,” Nano Letters 14, 4102–4106 (2014). PMID: .
[Crossref] [PubMed]

W. Barnes, G. Björk, J. Gérard, P. Jonsson, J. Wasey, P. Worthing, and V. Zwiller, “Solid-state single photon sources: light collection strategies,” Eur. Phys. J. D 18, 197–210 (2002).
[Crossref]

ACS Photonics (1)

J. Yang, J.-P. Hugonin, and P. Lalanne, “Near-to-far field transformations for radiative and guided waves,” ACS Photonics 3, 395–402 (2016).
[Crossref]

Applied Physics Letters (4)

A. Muller, E. B. Flagg, M. Metcalfe, J. Lawall, and G. S. Solomon, “Coupling an epitaxial quantum dot to a fiber-based external-mirror microcavity,” Applied Physics Letters 95, 173101 (2009).
[Crossref]

F. Haupt, S. S. R. Oemrawsingh, S. M. Thon, H. Kim, D. Kleckner, D. Ding, D. J. Suntrup, P. M. Petroff, and D. Bouwmeester, “Fiber-connectorized micropillar cavities,” Applied Physics Letters 97, 131113 (2010).
[Crossref]

M. Davanço, M. T. Rakher, W. Wegscheider, D. Schuh, A. Badolato, and K. Srinivasan, “Efficient quantum dot single photon extraction into an optical fiber using a nanophotonic directional coupler,” Applied Physics Letters 99, 121101 (2011).
[Crossref]

D. Cadeddu, J. Teissier, F. R. Braakman, N. Gregersen, P. Stepanov, J.-M. Gérard, J. Claudon, R. J. Warburton, M. Poggio, and M. Munsch, “A fiber-coupled quantum-dot on a photonic tip,” Applied Physics Letters 108, 011112 (2016).
[Crossref]

Eur. Phys. J. D (1)

W. Barnes, G. Björk, J. Gérard, P. Jonsson, J. Wasey, P. Worthing, and V. Zwiller, “Solid-state single photon sources: light collection strategies,” Eur. Phys. J. D 18, 197–210 (2002).
[Crossref]

Journal of Computational Physics (1)

M. Bergot and M. Duruflé, “High-order optimal edge elements for pyramids, prisms and hexahedra,” Journal of Computational Physics 232, 189–213 (2013).
[Crossref]

Journal of Vacuum Science & Technology B, Nanotechnology and Microelectronics: Materials, Processing, Measurement, and Phenomena (1)

M. Gschrey, R. Schmidt, J.-H. Schulze, A. Strittmatter, S. Rodt, and S. Reitzenstein, “Resolution and alignment accuracy of low-temperature in situ electron beam lithography for nanophotonic device fabrication,” Journal of Vacuum Science & Technology B, Nanotechnology and Microelectronics: Materials, Processing, Measurement, and Phenomena 33, 021603 (2015).

Nano Letters (1)

G. Bulgarini, M. E. Reimer, M. Bouwes Bavinck, K. D. Jöns, D. Dalacu, P. J. Poole, E. P. A. M. Bakkers, and V. Zwiller, “Nanowire waveguides launching single photons in a gaussian mode for ideal fiber coupling,” Nano Letters 14, 4102–4106 (2014). PMID: .
[Crossref] [PubMed]

Nat. Communications (2)

M. Gschrey, A. Thoma, P. Schnauber, M. Seifried, R. Schmidt, B. Wohlfeil, L. Krüger, J.-H. Schulze, T. Heindel, S. Burger, F. Schmidt, A. Strittmatter, S. Rodt, and S. Reitzenstein, “Highly indistinguishable photons from deterministic quantum-dot microlenses utilizing three-dimensional in situ electron-beam lithography,” Nat. Communications 6, 7662 (2015).
[Crossref]

L. Sapienza, M. Davanço, A. Badolato, and K. Srinivasan, “Nanoscale optical positioning of single quantum dots for bright and pure single-photon emission,” Nat. Communications 6, 7833 (2015).
[Crossref]

Nat. Photonics (2)

N. Somaschi, V. Giesz, L. De Santis, J. C. Loredo, M. P. Almeida, G. Hornecker, S. L. Portalupi, T. Grange, C. Antón, J. Demory, C. Gómez, I. Sagnes, N. D. Lanzillotti-Kimura, A. Lemaítre, A. Auffeves, A. G. White, L. Lanco, and P. Senellart, “Near-optimal single-photon sources in the solid state,” Nat. Photonics 10, 340 (2016).
[Crossref]

I. Aharonovich, D. Englund, and M. Toth, “Solid-state single-photon emitters,” Nat. Photonics 10, 631 (2016).
[Crossref]

Nature Physics (1)

M. Andersen, S. Stobbe, A. Sørensen, and P. Lodahl, “Strongly modified plasmon-matter inetraction with mesoscopic quantum emitters,” Nature Physics 7, 215–218 (2010).
[Crossref]

Opt. Express (1)

Optica (2)

Optics Communications (1)

A. Kaganskiy, S. Fischbach, A. Strittmatter, S. Rodt, T. Heindel, and S. Reitzenstein, “Enhancing the photon-extraction efficiency of site-controlled quantum dots by deterministically fabricated microlenses,” Optics Communications 413, 162–166 (2018).
[Crossref]

Phys. Rev. B (1)

Y. Chen, T. R. Nielsen, N. Gregersen, P. Lodahl, and J. Mørk, “Finite-element modeling of spontaneous emission of a quantum emitter at nanoscale proximity to plasmonic waveguides,” Phys. Rev. B 81, 125431 (2010).
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Phys. Rev. Lett. (3)

P. Tighineanu, A. S. Sørensen, S. Stobbe, and P. Lodahl, “Unraveling the mesoscopic character of quantum dots in nanophotonics,” Phys. Rev. Lett. 114, 247401 (2015).
[Crossref] [PubMed]

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91, 043902 (2003).
[Crossref] [PubMed]

N. Garcia and E. Stoll, “Monte carlo calculation for electromagnetic-wave scattering from random rough surfaces,” Phys. Rev. Lett. 52, 1798–1801 (1984).
[Crossref]

Proc. SPIE (3)

L. Zschiedrich, H. Greiner, S. Burger, and F. Schmidt, “Numerical analysis of nanostructures for enhanced light extraction from OLEDs,” Proc. SPIE 8641, 86410B (2013).
[Crossref]

P.-I. Schneider, X. Garcia Santiago, C. Rockstuhl, and S. Burger, “Global optimization of complex optical structures using Bayesian optimization based on Gaussian processes,” Proc. SPIE 10335, 103350O (2017).
[Crossref]

S. Burger, L. Zschiedrich, J. Pomplun, S. Herrmann, and F. Schmidt, “Hp-finite element method for simulating light scattering from complex 3D structures,” Proc. SPIE 9424, 94240Z (2015).
[Crossref]

Proceedings of the American Physical Society (1)

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Proceedings of the American Physical Society 69, 681 (1946).

Sci. Rep. (1)

M. Calic, C. Jarlov, P. Gallo, B. Dwir, A. Rudra, and E. Kapon, “Deterministic radiative coupling of two semiconductor quantum dots to the optical mode of a photonic crystal nanocavity,” Sci. Rep. 7, 4100 (2017).
[Crossref] [PubMed]

Science (1)

A. Badolato, K. Hennessy, M. Atatüre, J. Dreiser, E. Hu, P. M. Petroff, and A. Imamoğlu, “Deterministic coupling of single quantum dots to single nanocavity modes,” Science 308, 1158–1161 (2005).
[Crossref] [PubMed]

Scientific Reports (1)

C.-M. Lee, H.-J. Lim, C. Schneider, S. Maier, S. Höfling, M. Kamp, and Y.-H. Lee, “Efficient single photon source based on m-fibre-coupled tunable microcavity,” Scientific Reports 5, 14309 (2015).
[Crossref]

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A. Schlehahn, S. Fischbach, R. Schmidt, A. Kaganskiy, A. Strittmatter, S. Rodt, T. Heindel, and S. Reitzenstein, “A stand-alone fiber-coupled single-photon source,” arXiv preprint arXiv:1703.10536 (2017).

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[Crossref]

A. Lavrinenko, J. Lægsgaard, N. Gregersen, F. Schmidt, and T. Søndergaard,. Numerical Methods in Photonics, Optical Sciences and Applications of Light (CRC, 2014).

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006).
[Crossref]

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Figures (7)

Fig. 1
Fig. 1 The considered system consists of a QD dipole source (red point) embedded into a diffractive structure (GaAs, blue), a Bragg reflector (alternating layers of GaAs (blue) and Al0.9Ga0.1As (gray)), and an optical fiber with homogeneous fiber core (orange) and fiber cladding (yellow). The Bragg reflector is grown on a substrate made of GaAs (blue) and has a GaAs top layer (blue, thickness = 195 nm). The system is parametrized by 5 length scales, the diameter of the fiber core (dcore), the width and height of the lens or mesa (wlens, hlens), the elevation of the dipole (hdip), and the distance between lens and fiber (slf). Left: Spherical lens setup. Right: Mesa setup.
Fig. 2
Fig. 2 Left: Convergence of the absolute error of the coupling efficiency as a function of the polynomial degree p of the finite-element ansatz functions (finite-element degree). The 3D calculations (blue) converge significantly slower than the 2D calculations, which exploit the cylindrical symmetry. Right: Computational time of the 2D and 3D calculations as a function of the polynomial degree of the finite elements. The converged 2D calculations are about 3 orders of magnitude faster than the 3D calculations.
Fig. 3
Fig. 3 Maximum coupling efficiency η into a single-mode fiber as defined in Eq. (14) for different widths and heights of a spherical lens (left) and a mesa (right) at a wavelength of 1,300 nm. The maximum is taken with respect to a scan of the dipole elevation within the range of 0 to 50 nm in steps of 10 nm. The values of the fiber core diameter and the lens-fiber distance are dcore = 1, 500 nm and slf = 100 nm, respectively. The best coupling efficiencies obtained in this optimization step are η = 26.2% for the spherical lens setup and η = 21.0% for the mesa setup
Fig. 4
Fig. 4 Coupling efficiency η into a single-mode fiber as defined in Eq. (14) for different fiber core diameters and lens-fiber distances to the spherical lens (left) and mesa (right) at a wavelength of 1, 300 nm. The best coupling efficiencies obtained are η = 29.9% for the spherical lens setup and η = 23.2% for the mesa setup. The other system parameters are set to the values of the first optimization step (see Fig. 3). The optimized parameters are summarized in Table 1
Fig. 5
Fig. 5 Visualization of the energy density on a logarithmic scale of the light field emitted by the QD embedded in the optimized spherical lens (left) and mesa (right). A cut through the optimized geometry is shown in front of the energy-density plots with the fiber core in purple, the fiber cladding in red, the spherical lens or mesa structure in green and the layers of the Bragg reflector in light and dark blue. The optimal mesa has a smaller coupling efficiency although less radiation is propagating into the Bragg reflector. This indicates that the specific intensity and phase profile of the light field entering the fiber is of importance.
Fig. 6
Fig. 6 Sensitivity of the coupling efficiency η into a single-mode fiber with respect to deviations of the geometry parameters from their optimized values (see table 1) for the spherical lens setup (left) and the mesa setup (right).
Fig. 7
Fig. 7 Left: Mean and variance of coupling efficiency η into a single-mode fiber for surface roughness of different amplitudes added to the surface of the optimized spherical lens. Right: Visualization of the 3D mesh of the spherical lens for different roughness amplitudes.

Tables (1)

Tables Icon

Table 1 Optimized parameters and corresponding coupling efficiency η for spherical lens and mesa obtained by a two-step optimization.

Equations (26)

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J ( r , t ) = Re { i ω p δ ( r r QD ) e i ω t } .
× μ 1 × E ( r , ω ) ϵ ω 2 E ( r , ω ) = i ω J ( r , ω ) .
× μ d 1 × E s ( r ) ϵ d ω 2 E s ( r ) = i ω J ( r )
× μ 1 × E c ( r ) ϵ ω 2 E c ( r ) = × ( μ 1 μ d 1 ) × E s ( r ) + ( ϵ ϵ d ) ω 2 E s ( r ) .
E c ( r ) = n = E n ( r , z ) e i n ϕ .
f ( r ) = × ( μ 1 μ d 1 ) × E s ( r ) + ( ϵ ϵ d ) ω 2 E s ( r ) ,
f ( r ) = n = f n ( r , z ) e i n ϕ with f n ( r , z ) = 1 2 π 0 2 π d ϕ f ( r ) e i n ϕ .
1 2 π 0 2 π d ϕ e i m ϕ n = ( × μ 1 × E n ( r , z ) ϵ ω 2 E n ( r , z ) f n ( r , z ) ) e i n ϕ = n = ( ˜ × μ 1 ˜ × E n ( r , z ) ϵ ω 2 E n ( r , z ) f n ( r , z ) ) 1 2 π 0 2 π d ϕ e i ( n m ) ϕ = δ n m = ˜ × μ 1 ˜ × E m ( r , z ) ϵ ω 2 E m ( r , z ) f m ( r , z ) = 0 ,
× μ 1 × E n ( r ) ϵ ω 2 E n ( r ) = 0 with E n ( r ) = E n ( r , ϕ ) e i k n z .
E scatt = n E n , E scatt E n .
E 1 , E 2 = 1 2 i ω d n ( E 1 × μ 1 E 2 ) = 1 2 d n ( E 1 E 2 ) ,
P = 1 2 Re { n ( E scatt × H scatt * ) } = Re { E scatt , E scatt * } = Re { n m E n , E scatt E m , E scatt * E n , E m } = Re { n E n , E scatt E n , E scatt * } .
P = n | E n , E scatt | 2 .
η = P 1 + P 2 P tot
H ^ int ( r ) = A ^ ( r ) n q n m n i r n ,
p = i n q n m n Ψ g | r n | Ψ e .
α = | p | ω 2 ϵ 0 V ,
0 = z = δ dA e z ( E ˜ m × μ 1 × E n ) z = 0 dA e z ( E ˜ m × μ 1 × E n ) 2 × [ 0 , δ ] dV μ 1 ( × E ˜ m ) ( × E n ) ϵ ω 2 E ˜ m E n = z = δ dA e z { E ˜ m × μ 1 × E n + ( μ 1 × E ˜ m ) × E n } z = 0 dA e z { E ˜ m × μ 1 × E n + ( μ 1 × E ˜ m ) × E n } 2 × [ 0 , δ ] dV ( × μ 1 × E ˜ m ) E n ϵ ω 2 E ˜ m E n . = 0
E 1 , E 2 = 1 2 i ω d n ( E 1 × μ 1 × E 2 ) = 1 2 d n ( E 1 × H 2 ) ,
× μ 1 × E n ( r ) ϵ ω 2 E n ( r ) = 0 .
E ˜ n ( x , y , z ) = ( E x ( x , y , z ) E y ( x , y , z ) E z ( x , y , z ) )
2 × [ 0 , δ ] dV E ˜ m ( × μ 1 × E n ϵ ω 2 E n ) = 0
0 = z = δ dA e z ( E ˜ m × μ 1 × E n ) z = 0 dA e z ( E ˜ m × μ 1 × E n ) 2 × [ 0 , δ ] dV μ 1 ( × E ˜ m ) ( × E n ) ϵ ω 2 E ˜ m E n = z = δ dA e z { E ˜ m × μ 1 × E n + ( μ 1 × E ˜ m ) × E n } 2 × [ 0 , δ ] dV ( × μ 1 × E ˜ m ) E n ϵ ω 2 E ˜ m E n . = 0
E n ( x , y , z ) = E n ( x , y , 0 ) e i k n z .
dA e z ( E ˜ m × μ 1 × E n ) = dA e z ( μ 1 × E ˜ m ) × E n .
0 = ( e i ( k n k m ) δ 1 ) z = 0 dA e z ( E ˜ m × μ 1 × E ˜ n ) .

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