Abstract

We investigate the spontaneous emission modifications when ensembles of quantum dots (QDs) with differing emission frequencies and finite Lorentzian linewidths are coupled to a microcavity. Using contour integrals we develop a general expression for the rate enhancement when neither the emitter nor the cavity resonance can be treated as a delta function. We show that the ensemble cavity-coupled luminescence lifetimes are generally suppressed in the case of spherical cavities and that the spontaneous emission dynamics of the cavity coupled component becomes increasingly stretched as the coupling factor increases. The Q-factor measured from the luminescence spectrum can be much lower than the intrinsic cavity Q-factor, and is in many practical situations limited by the QD spectral width. The mode spectrum observed in the photoluminescence (PL) spectrum can be largely determined by the QD emission linewidth, permitting this parameter to be extracted without requiring single-particle spectroscopy. In the case of Si-QDs, the linewidth cannot be significantly greater than 10 meV in order to observe spherical cavity resonances in the PL spectrum.

© 2010 OSA

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  4. T. Tawara, H. Kamada, S. Hughes, H. Okamoto, M. Notomi, and T. Sogawa, “Cavity mode emission in weakly coupled quantum dot--cavity systems,” Opt. Express 17(8), 6643–6654 (2009).
    [CrossRef] [PubMed]
  5. J.-Y. Marzin, J.-M. Gérard, A. Izraël, D. Barrier, and G. Bastard, “Photoluminescence of single InAs quantum dots obtained by self-organized growth on GaAs,” Phys. Rev. Lett. 73(5), 716–719 (1994).
    [CrossRef] [PubMed]
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  7. K. Leosson, D. Birkedal, I. Magnusdottir, W. Langbein, and J. M. Hvam, “Homogeneous linewidth of self-assembled III–V quantum dots observed in single-dot photoluminescence,” Physica E 17, 1–6 (2003).
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  27. V. Belyakov, V. Burdov, R. Lockwood, and A. Meldrum, “Silicon Nanocrystals: Fundamental Theory and Implications for Stimulated Emission,” Adv. Opt. Technol. 2008, 279502 (2008).
  28. M. Dovrat, Y. Goshen, J. Jedrzejewski, I. Balberg, and A. Sa’ar, “Radiative versus nonradiative decay processes in silicon nanocrystals probed by time-resolved photoluminescence spectroscopy,” Phys. Rev. B 69(15), 155311 (2004).
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  31. K. Ujihara, “Spontaneous emission and the concept of effective area in a very short optical cavity with plane-parallel dielectric mirrors,” Jpn. J. Appl. Phys. 30(Part 2, No. 5B5b), L901–L903 (1991).
    [CrossRef]
  32. V. Vinciguerra, G. Franzo, F. Priolo, F. Iacona, and C. Spinella, “Quantum confinement and recombination dynamics in silicon nanocrystals embedded in Si/SiO2 superlattices,” J. Appl. Phys. 87(11), 8165–8173 (2000).
    [CrossRef]
  33. J. Valenta, R. Juhasz, and J. Linnros, “Photoluminescence spectroscopy of single silicon quantum dots,” Appl. Phys. Lett. 80(6), 1070 (2002).
    [CrossRef]
  34. A. Beltaos and A. Meldrum, “Whispering gallery modes in silicon-nanocrystal-coated silica microspheres,” J. Lumin. 126(2), 607–613 (2007).
    [CrossRef]
  35. A. Pitanti, M. Ghulinyan, D. Navarro-Urrios, G. Pucker, and L. Pavesi, “Probing the spontaneous emission dynamics in Si-nanocrystals-based microdisk resonators,” Phys. Rev. Lett. 104(10), 103901 (2010).
    [CrossRef] [PubMed]

2010 (1)

A. Pitanti, M. Ghulinyan, D. Navarro-Urrios, G. Pucker, and L. Pavesi, “Probing the spontaneous emission dynamics in Si-nanocrystals-based microdisk resonators,” Phys. Rev. Lett. 104(10), 103901 (2010).
[CrossRef] [PubMed]

2009 (4)

T. Tawara, H. Kamada, S. Hughes, H. Okamoto, M. Notomi, and T. Sogawa, “Cavity mode emission in weakly coupled quantum dot--cavity systems,” Opt. Express 17(8), 6643–6654 (2009).
[CrossRef] [PubMed]

P. Bianucci, J. R. Rodríguez, C. M. Clements, J. G. C. Veinot, and A. Meldrum, “Silicon nanocrystal luminescence coupled to whispering gallery modes in optical fibers,” J. Appl. Phys. 105(2), 023108 (2009).
[CrossRef]

A. Francois and M. Himmelhaus, “Whispering gallery mode biosensor operated in the stimulated emission regime,” Appl. Phys. Lett. 94(3), 031101 (2009).
[CrossRef]

P. Bianucci, J. R. Rodriguez, F. C. Lenz, J. C. G. Veinot, and A. Meldrum, “Mode structure in the luminescence of Si-nc in cylindrical microcavities,” Physica E 41(6), 1107–1110 (2009).
[CrossRef]

2008 (6)

R. Kekatpure and M. L. Brongersma, “Fundamental photophysics and optical loss processes in Si-nanocrystal-doped microdisk resonators,” Phys. Rev. A 78(2), 023829 (2008).
[CrossRef]

B. Gayral and J. M. Gérard, “Photoluminescence experiment on quantum dots embedded in a large Purcell-factor microcavity,” Phys. Rev. B 78(23), 235306 (2008).
[CrossRef]

L. Pavesi, “Silicon-Based light sources for silicon integrated circuits,” Adv. Opt. Technol. 2008, 416926 (2008).

M. Ghulinyan, D. Navarro-Urrios, A. Pitanti, A. Lui, G. Pucker, and L. Pavesi, “Whispering-gallery modes and light emission from a Si-nanocrystal-based single microdisk resonator,” Opt. Express 16(17), 13218–13224 (2008).
[CrossRef] [PubMed]

M. Kaniber, A. Laucht, A. Neumann, M. Villas-Bôas, M. Bichler, M.-C. Amann, and J. J. Finley, “Investigation of the nonresonant dot-cavity coupling in two-dimensional photonic crystal nanocavities,” Phys. Rev. B 77(16), 161303 (2008).
[CrossRef]

V. Belyakov, V. Burdov, R. Lockwood, and A. Meldrum, “Silicon Nanocrystals: Fundamental Theory and Implications for Stimulated Emission,” Adv. Opt. Technol. 2008, 279502 (2008).

2007 (3)

A. Beltaos and A. Meldrum, “Whispering gallery modes in silicon-nanocrystal-coated silica microspheres,” J. Lumin. 126(2), 607–613 (2007).
[CrossRef]

A. Belarouci and F. Gourbilleau, “Microcavity enhanced spontaneous emission from silicon nanocrystals,” J. Appl. Phys. 101(7), 073108 (2007).
[CrossRef]

J. Wang, X. F. Wang, Q. Li, A. Hryciw, and A. Meldrum, “The microstructure of SiO thin films: from nanoclusters to nanocrystals,” Philos. Mag. 87(1), 11–27 (2007).
[CrossRef]

2006 (2)

Y. Yamamoto, “Quantum Communication and Information Processing with Quantum Dots,” Quantum Inf. Process. 5(5), 299–311 (2006).
[CrossRef]

N. Le Thomas, U. Woggon, O. Schöps, M. V. Artemyev, M. Kazes, and U. Banin, “Cavity QED with semiconductor nanocrystals,” Nano Lett. 6(3), 557–561 (2006).
[CrossRef] [PubMed]

2005 (3)

D. E. Gómez, I. Pastoriza-Santos, and P. Mulvaney, “Tunable whispering gallery mode emission from quantum-dot-doped microspheres,” Small 1(2), 238–241 (2005).
[CrossRef]

I. Sychugov, R. Juhasz, J. Valenta, and J. Linnros, “Narrow luminescence linewidth of a silicon quantum dot,” Phys. Rev. Lett. 94(8), 087405 (2005).
[CrossRef] [PubMed]

M. N. Berberan-Santos, E. N. Bodunov, and B. Valeur, “Mathematical functions for the analysis of luminescence decays with underlying distributions 1. Kohlrausch decay function (stretched exponential),” Chem. Phys. 315(1-2), 171–182 (2005).
[CrossRef]

2004 (1)

M. Dovrat, Y. Goshen, J. Jedrzejewski, I. Balberg, and A. Sa’ar, “Radiative versus nonradiative decay processes in silicon nanocrystals probed by time-resolved photoluminescence spectroscopy,” Phys. Rev. B 69(15), 155311 (2004).
[CrossRef]

2003 (1)

K. Leosson, D. Birkedal, I. Magnusdottir, W. Langbein, and J. M. Hvam, “Homogeneous linewidth of self-assembled III–V quantum dots observed in single-dot photoluminescence,” Physica E 17, 1–6 (2003).
[CrossRef]

2002 (1)

J. Valenta, R. Juhasz, and J. Linnros, “Photoluminescence spectroscopy of single silicon quantum dots,” Appl. Phys. Lett. 80(6), 1070 (2002).
[CrossRef]

2001 (1)

S. Chan, Y. Li, R. J. Rothberg, B. L. Miller, and P. M. Fauchet, “Nanoscale silicon microcavities for biosensing,” Mater. Sci. Eng. C 15(1-2), 277–282 (2001).
[CrossRef]

2000 (2)

V. Vinciguerra, G. Franzo, F. Priolo, F. Iacona, and C. Spinella, “Quantum confinement and recombination dynamics in silicon nanocrystals embedded in Si/SiO2 superlattices,” J. Appl. Phys. 87(11), 8165–8173 (2000).
[CrossRef]

X. Fan, P. Palinginis, S. Lacey, H. Wang, and M. C. Lonergan, “Coupling semiconductor nanocrystals to a fused-silica microsphere: a quantum-dot microcavity with extremely high Q factors,” Opt. Lett. 25(21), 1600–1602 (2000).
[CrossRef]

1998 (1)

J. Gérard, B. Sermage, B. Gayral, B. Legrand, E. Costard, and V. Thierry-Mieg, “Enhanced spontaneous emission by quantum boxes in a monolithic optical microcavity,” Phys. Rev. Lett. 81(5), 1110–1113 (1998).
[CrossRef]

1994 (1)

J.-Y. Marzin, J.-M. Gérard, A. Izraël, D. Barrier, and G. Bastard, “Photoluminescence of single InAs quantum dots obtained by self-organized growth on GaAs,” Phys. Rev. Lett. 73(5), 716–719 (1994).
[CrossRef] [PubMed]

1991 (1)

K. Ujihara, “Spontaneous emission and the concept of effective area in a very short optical cavity with plane-parallel dielectric mirrors,” Jpn. J. Appl. Phys. 30(Part 2, No. 5B5b), L901–L903 (1991).
[CrossRef]

1946 (2)

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

H. Pollard, “The representation of $e^{ - x^\lambda }$ as a Laplace integral,” Bull. Am. Math. Soc. 52(10), 908–911 (1946).
[CrossRef]

1854 (1)

R. Kohlrausch, “Ueber das Dellmann'sche Elektrometer,” Ann. Phys. 91, 353–405 (1854).

Amann, M.-C.

M. Kaniber, A. Laucht, A. Neumann, M. Villas-Bôas, M. Bichler, M.-C. Amann, and J. J. Finley, “Investigation of the nonresonant dot-cavity coupling in two-dimensional photonic crystal nanocavities,” Phys. Rev. B 77(16), 161303 (2008).
[CrossRef]

Artemyev, M. V.

N. Le Thomas, U. Woggon, O. Schöps, M. V. Artemyev, M. Kazes, and U. Banin, “Cavity QED with semiconductor nanocrystals,” Nano Lett. 6(3), 557–561 (2006).
[CrossRef] [PubMed]

Balberg, I.

M. Dovrat, Y. Goshen, J. Jedrzejewski, I. Balberg, and A. Sa’ar, “Radiative versus nonradiative decay processes in silicon nanocrystals probed by time-resolved photoluminescence spectroscopy,” Phys. Rev. B 69(15), 155311 (2004).
[CrossRef]

Banin, U.

N. Le Thomas, U. Woggon, O. Schöps, M. V. Artemyev, M. Kazes, and U. Banin, “Cavity QED with semiconductor nanocrystals,” Nano Lett. 6(3), 557–561 (2006).
[CrossRef] [PubMed]

Barrier, D.

J.-Y. Marzin, J.-M. Gérard, A. Izraël, D. Barrier, and G. Bastard, “Photoluminescence of single InAs quantum dots obtained by self-organized growth on GaAs,” Phys. Rev. Lett. 73(5), 716–719 (1994).
[CrossRef] [PubMed]

Bastard, G.

J.-Y. Marzin, J.-M. Gérard, A. Izraël, D. Barrier, and G. Bastard, “Photoluminescence of single InAs quantum dots obtained by self-organized growth on GaAs,” Phys. Rev. Lett. 73(5), 716–719 (1994).
[CrossRef] [PubMed]

Belarouci, A.

A. Belarouci and F. Gourbilleau, “Microcavity enhanced spontaneous emission from silicon nanocrystals,” J. Appl. Phys. 101(7), 073108 (2007).
[CrossRef]

Beltaos, A.

A. Beltaos and A. Meldrum, “Whispering gallery modes in silicon-nanocrystal-coated silica microspheres,” J. Lumin. 126(2), 607–613 (2007).
[CrossRef]

Belyakov, V.

V. Belyakov, V. Burdov, R. Lockwood, and A. Meldrum, “Silicon Nanocrystals: Fundamental Theory and Implications for Stimulated Emission,” Adv. Opt. Technol. 2008, 279502 (2008).

Berberan-Santos, M. N.

M. N. Berberan-Santos, E. N. Bodunov, and B. Valeur, “Mathematical functions for the analysis of luminescence decays with underlying distributions 1. Kohlrausch decay function (stretched exponential),” Chem. Phys. 315(1-2), 171–182 (2005).
[CrossRef]

Bianucci, P.

P. Bianucci, J. R. Rodriguez, F. C. Lenz, J. C. G. Veinot, and A. Meldrum, “Mode structure in the luminescence of Si-nc in cylindrical microcavities,” Physica E 41(6), 1107–1110 (2009).
[CrossRef]

P. Bianucci, J. R. Rodríguez, C. M. Clements, J. G. C. Veinot, and A. Meldrum, “Silicon nanocrystal luminescence coupled to whispering gallery modes in optical fibers,” J. Appl. Phys. 105(2), 023108 (2009).
[CrossRef]

Bichler, M.

M. Kaniber, A. Laucht, A. Neumann, M. Villas-Bôas, M. Bichler, M.-C. Amann, and J. J. Finley, “Investigation of the nonresonant dot-cavity coupling in two-dimensional photonic crystal nanocavities,” Phys. Rev. B 77(16), 161303 (2008).
[CrossRef]

Birkedal, D.

K. Leosson, D. Birkedal, I. Magnusdottir, W. Langbein, and J. M. Hvam, “Homogeneous linewidth of self-assembled III–V quantum dots observed in single-dot photoluminescence,” Physica E 17, 1–6 (2003).
[CrossRef]

Bodunov, E. N.

M. N. Berberan-Santos, E. N. Bodunov, and B. Valeur, “Mathematical functions for the analysis of luminescence decays with underlying distributions 1. Kohlrausch decay function (stretched exponential),” Chem. Phys. 315(1-2), 171–182 (2005).
[CrossRef]

Brongersma, M. L.

R. Kekatpure and M. L. Brongersma, “Fundamental photophysics and optical loss processes in Si-nanocrystal-doped microdisk resonators,” Phys. Rev. A 78(2), 023829 (2008).
[CrossRef]

Burdov, V.

V. Belyakov, V. Burdov, R. Lockwood, and A. Meldrum, “Silicon Nanocrystals: Fundamental Theory and Implications for Stimulated Emission,” Adv. Opt. Technol. 2008, 279502 (2008).

Chan, S.

S. Chan, Y. Li, R. J. Rothberg, B. L. Miller, and P. M. Fauchet, “Nanoscale silicon microcavities for biosensing,” Mater. Sci. Eng. C 15(1-2), 277–282 (2001).
[CrossRef]

Clements, C. M.

P. Bianucci, J. R. Rodríguez, C. M. Clements, J. G. C. Veinot, and A. Meldrum, “Silicon nanocrystal luminescence coupled to whispering gallery modes in optical fibers,” J. Appl. Phys. 105(2), 023108 (2009).
[CrossRef]

Costard, E.

J. Gérard, B. Sermage, B. Gayral, B. Legrand, E. Costard, and V. Thierry-Mieg, “Enhanced spontaneous emission by quantum boxes in a monolithic optical microcavity,” Phys. Rev. Lett. 81(5), 1110–1113 (1998).
[CrossRef]

Dovrat, M.

M. Dovrat, Y. Goshen, J. Jedrzejewski, I. Balberg, and A. Sa’ar, “Radiative versus nonradiative decay processes in silicon nanocrystals probed by time-resolved photoluminescence spectroscopy,” Phys. Rev. B 69(15), 155311 (2004).
[CrossRef]

Fan, X.

Fauchet, P. M.

S. Chan, Y. Li, R. J. Rothberg, B. L. Miller, and P. M. Fauchet, “Nanoscale silicon microcavities for biosensing,” Mater. Sci. Eng. C 15(1-2), 277–282 (2001).
[CrossRef]

Finley, J. J.

M. Kaniber, A. Laucht, A. Neumann, M. Villas-Bôas, M. Bichler, M.-C. Amann, and J. J. Finley, “Investigation of the nonresonant dot-cavity coupling in two-dimensional photonic crystal nanocavities,” Phys. Rev. B 77(16), 161303 (2008).
[CrossRef]

Francois, A.

A. Francois and M. Himmelhaus, “Whispering gallery mode biosensor operated in the stimulated emission regime,” Appl. Phys. Lett. 94(3), 031101 (2009).
[CrossRef]

Franzo, G.

V. Vinciguerra, G. Franzo, F. Priolo, F. Iacona, and C. Spinella, “Quantum confinement and recombination dynamics in silicon nanocrystals embedded in Si/SiO2 superlattices,” J. Appl. Phys. 87(11), 8165–8173 (2000).
[CrossRef]

Gayral, B.

B. Gayral and J. M. Gérard, “Photoluminescence experiment on quantum dots embedded in a large Purcell-factor microcavity,” Phys. Rev. B 78(23), 235306 (2008).
[CrossRef]

J. Gérard, B. Sermage, B. Gayral, B. Legrand, E. Costard, and V. Thierry-Mieg, “Enhanced spontaneous emission by quantum boxes in a monolithic optical microcavity,” Phys. Rev. Lett. 81(5), 1110–1113 (1998).
[CrossRef]

Gérard, J.

J. Gérard, B. Sermage, B. Gayral, B. Legrand, E. Costard, and V. Thierry-Mieg, “Enhanced spontaneous emission by quantum boxes in a monolithic optical microcavity,” Phys. Rev. Lett. 81(5), 1110–1113 (1998).
[CrossRef]

Gérard, J. M.

B. Gayral and J. M. Gérard, “Photoluminescence experiment on quantum dots embedded in a large Purcell-factor microcavity,” Phys. Rev. B 78(23), 235306 (2008).
[CrossRef]

Gérard, J.-M.

J.-Y. Marzin, J.-M. Gérard, A. Izraël, D. Barrier, and G. Bastard, “Photoluminescence of single InAs quantum dots obtained by self-organized growth on GaAs,” Phys. Rev. Lett. 73(5), 716–719 (1994).
[CrossRef] [PubMed]

Ghulinyan, M.

A. Pitanti, M. Ghulinyan, D. Navarro-Urrios, G. Pucker, and L. Pavesi, “Probing the spontaneous emission dynamics in Si-nanocrystals-based microdisk resonators,” Phys. Rev. Lett. 104(10), 103901 (2010).
[CrossRef] [PubMed]

M. Ghulinyan, D. Navarro-Urrios, A. Pitanti, A. Lui, G. Pucker, and L. Pavesi, “Whispering-gallery modes and light emission from a Si-nanocrystal-based single microdisk resonator,” Opt. Express 16(17), 13218–13224 (2008).
[CrossRef] [PubMed]

Gómez, D. E.

D. E. Gómez, I. Pastoriza-Santos, and P. Mulvaney, “Tunable whispering gallery mode emission from quantum-dot-doped microspheres,” Small 1(2), 238–241 (2005).
[CrossRef]

Goshen, Y.

M. Dovrat, Y. Goshen, J. Jedrzejewski, I. Balberg, and A. Sa’ar, “Radiative versus nonradiative decay processes in silicon nanocrystals probed by time-resolved photoluminescence spectroscopy,” Phys. Rev. B 69(15), 155311 (2004).
[CrossRef]

Gourbilleau, F.

A. Belarouci and F. Gourbilleau, “Microcavity enhanced spontaneous emission from silicon nanocrystals,” J. Appl. Phys. 101(7), 073108 (2007).
[CrossRef]

Himmelhaus, M.

A. Francois and M. Himmelhaus, “Whispering gallery mode biosensor operated in the stimulated emission regime,” Appl. Phys. Lett. 94(3), 031101 (2009).
[CrossRef]

Hryciw, A.

J. Wang, X. F. Wang, Q. Li, A. Hryciw, and A. Meldrum, “The microstructure of SiO thin films: from nanoclusters to nanocrystals,” Philos. Mag. 87(1), 11–27 (2007).
[CrossRef]

Hughes, S.

Hvam, J. M.

K. Leosson, D. Birkedal, I. Magnusdottir, W. Langbein, and J. M. Hvam, “Homogeneous linewidth of self-assembled III–V quantum dots observed in single-dot photoluminescence,” Physica E 17, 1–6 (2003).
[CrossRef]

Iacona, F.

V. Vinciguerra, G. Franzo, F. Priolo, F. Iacona, and C. Spinella, “Quantum confinement and recombination dynamics in silicon nanocrystals embedded in Si/SiO2 superlattices,” J. Appl. Phys. 87(11), 8165–8173 (2000).
[CrossRef]

Izraël, A.

J.-Y. Marzin, J.-M. Gérard, A. Izraël, D. Barrier, and G. Bastard, “Photoluminescence of single InAs quantum dots obtained by self-organized growth on GaAs,” Phys. Rev. Lett. 73(5), 716–719 (1994).
[CrossRef] [PubMed]

Jedrzejewski, J.

M. Dovrat, Y. Goshen, J. Jedrzejewski, I. Balberg, and A. Sa’ar, “Radiative versus nonradiative decay processes in silicon nanocrystals probed by time-resolved photoluminescence spectroscopy,” Phys. Rev. B 69(15), 155311 (2004).
[CrossRef]

Juhasz, R.

I. Sychugov, R. Juhasz, J. Valenta, and J. Linnros, “Narrow luminescence linewidth of a silicon quantum dot,” Phys. Rev. Lett. 94(8), 087405 (2005).
[CrossRef] [PubMed]

J. Valenta, R. Juhasz, and J. Linnros, “Photoluminescence spectroscopy of single silicon quantum dots,” Appl. Phys. Lett. 80(6), 1070 (2002).
[CrossRef]

Kamada, H.

Kaniber, M.

M. Kaniber, A. Laucht, A. Neumann, M. Villas-Bôas, M. Bichler, M.-C. Amann, and J. J. Finley, “Investigation of the nonresonant dot-cavity coupling in two-dimensional photonic crystal nanocavities,” Phys. Rev. B 77(16), 161303 (2008).
[CrossRef]

Kazes, M.

N. Le Thomas, U. Woggon, O. Schöps, M. V. Artemyev, M. Kazes, and U. Banin, “Cavity QED with semiconductor nanocrystals,” Nano Lett. 6(3), 557–561 (2006).
[CrossRef] [PubMed]

Kekatpure, R.

R. Kekatpure and M. L. Brongersma, “Fundamental photophysics and optical loss processes in Si-nanocrystal-doped microdisk resonators,” Phys. Rev. A 78(2), 023829 (2008).
[CrossRef]

Kohlrausch, R.

R. Kohlrausch, “Ueber das Dellmann'sche Elektrometer,” Ann. Phys. 91, 353–405 (1854).

Lacey, S.

Langbein, W.

K. Leosson, D. Birkedal, I. Magnusdottir, W. Langbein, and J. M. Hvam, “Homogeneous linewidth of self-assembled III–V quantum dots observed in single-dot photoluminescence,” Physica E 17, 1–6 (2003).
[CrossRef]

Laucht, A.

M. Kaniber, A. Laucht, A. Neumann, M. Villas-Bôas, M. Bichler, M.-C. Amann, and J. J. Finley, “Investigation of the nonresonant dot-cavity coupling in two-dimensional photonic crystal nanocavities,” Phys. Rev. B 77(16), 161303 (2008).
[CrossRef]

Le Thomas, N.

N. Le Thomas, U. Woggon, O. Schöps, M. V. Artemyev, M. Kazes, and U. Banin, “Cavity QED with semiconductor nanocrystals,” Nano Lett. 6(3), 557–561 (2006).
[CrossRef] [PubMed]

Legrand, B.

J. Gérard, B. Sermage, B. Gayral, B. Legrand, E. Costard, and V. Thierry-Mieg, “Enhanced spontaneous emission by quantum boxes in a monolithic optical microcavity,” Phys. Rev. Lett. 81(5), 1110–1113 (1998).
[CrossRef]

Lenz, F. C.

P. Bianucci, J. R. Rodriguez, F. C. Lenz, J. C. G. Veinot, and A. Meldrum, “Mode structure in the luminescence of Si-nc in cylindrical microcavities,” Physica E 41(6), 1107–1110 (2009).
[CrossRef]

Leosson, K.

K. Leosson, D. Birkedal, I. Magnusdottir, W. Langbein, and J. M. Hvam, “Homogeneous linewidth of self-assembled III–V quantum dots observed in single-dot photoluminescence,” Physica E 17, 1–6 (2003).
[CrossRef]

Li, Q.

J. Wang, X. F. Wang, Q. Li, A. Hryciw, and A. Meldrum, “The microstructure of SiO thin films: from nanoclusters to nanocrystals,” Philos. Mag. 87(1), 11–27 (2007).
[CrossRef]

Li, Y.

S. Chan, Y. Li, R. J. Rothberg, B. L. Miller, and P. M. Fauchet, “Nanoscale silicon microcavities for biosensing,” Mater. Sci. Eng. C 15(1-2), 277–282 (2001).
[CrossRef]

Linnros, J.

I. Sychugov, R. Juhasz, J. Valenta, and J. Linnros, “Narrow luminescence linewidth of a silicon quantum dot,” Phys. Rev. Lett. 94(8), 087405 (2005).
[CrossRef] [PubMed]

J. Valenta, R. Juhasz, and J. Linnros, “Photoluminescence spectroscopy of single silicon quantum dots,” Appl. Phys. Lett. 80(6), 1070 (2002).
[CrossRef]

Lockwood, R.

V. Belyakov, V. Burdov, R. Lockwood, and A. Meldrum, “Silicon Nanocrystals: Fundamental Theory and Implications for Stimulated Emission,” Adv. Opt. Technol. 2008, 279502 (2008).

Lonergan, M. C.

Lui, A.

Magnusdottir, I.

K. Leosson, D. Birkedal, I. Magnusdottir, W. Langbein, and J. M. Hvam, “Homogeneous linewidth of self-assembled III–V quantum dots observed in single-dot photoluminescence,” Physica E 17, 1–6 (2003).
[CrossRef]

Marzin, J.-Y.

J.-Y. Marzin, J.-M. Gérard, A. Izraël, D. Barrier, and G. Bastard, “Photoluminescence of single InAs quantum dots obtained by self-organized growth on GaAs,” Phys. Rev. Lett. 73(5), 716–719 (1994).
[CrossRef] [PubMed]

Meldrum, A.

P. Bianucci, J. R. Rodríguez, C. M. Clements, J. G. C. Veinot, and A. Meldrum, “Silicon nanocrystal luminescence coupled to whispering gallery modes in optical fibers,” J. Appl. Phys. 105(2), 023108 (2009).
[CrossRef]

P. Bianucci, J. R. Rodriguez, F. C. Lenz, J. C. G. Veinot, and A. Meldrum, “Mode structure in the luminescence of Si-nc in cylindrical microcavities,” Physica E 41(6), 1107–1110 (2009).
[CrossRef]

V. Belyakov, V. Burdov, R. Lockwood, and A. Meldrum, “Silicon Nanocrystals: Fundamental Theory and Implications for Stimulated Emission,” Adv. Opt. Technol. 2008, 279502 (2008).

J. Wang, X. F. Wang, Q. Li, A. Hryciw, and A. Meldrum, “The microstructure of SiO thin films: from nanoclusters to nanocrystals,” Philos. Mag. 87(1), 11–27 (2007).
[CrossRef]

A. Beltaos and A. Meldrum, “Whispering gallery modes in silicon-nanocrystal-coated silica microspheres,” J. Lumin. 126(2), 607–613 (2007).
[CrossRef]

Miller, B. L.

S. Chan, Y. Li, R. J. Rothberg, B. L. Miller, and P. M. Fauchet, “Nanoscale silicon microcavities for biosensing,” Mater. Sci. Eng. C 15(1-2), 277–282 (2001).
[CrossRef]

Mulvaney, P.

D. E. Gómez, I. Pastoriza-Santos, and P. Mulvaney, “Tunable whispering gallery mode emission from quantum-dot-doped microspheres,” Small 1(2), 238–241 (2005).
[CrossRef]

Navarro-Urrios, D.

A. Pitanti, M. Ghulinyan, D. Navarro-Urrios, G. Pucker, and L. Pavesi, “Probing the spontaneous emission dynamics in Si-nanocrystals-based microdisk resonators,” Phys. Rev. Lett. 104(10), 103901 (2010).
[CrossRef] [PubMed]

M. Ghulinyan, D. Navarro-Urrios, A. Pitanti, A. Lui, G. Pucker, and L. Pavesi, “Whispering-gallery modes and light emission from a Si-nanocrystal-based single microdisk resonator,” Opt. Express 16(17), 13218–13224 (2008).
[CrossRef] [PubMed]

Neumann, A.

M. Kaniber, A. Laucht, A. Neumann, M. Villas-Bôas, M. Bichler, M.-C. Amann, and J. J. Finley, “Investigation of the nonresonant dot-cavity coupling in two-dimensional photonic crystal nanocavities,” Phys. Rev. B 77(16), 161303 (2008).
[CrossRef]

Notomi, M.

Okamoto, H.

Palinginis, P.

Pastoriza-Santos, I.

D. E. Gómez, I. Pastoriza-Santos, and P. Mulvaney, “Tunable whispering gallery mode emission from quantum-dot-doped microspheres,” Small 1(2), 238–241 (2005).
[CrossRef]

Pavesi, L.

A. Pitanti, M. Ghulinyan, D. Navarro-Urrios, G. Pucker, and L. Pavesi, “Probing the spontaneous emission dynamics in Si-nanocrystals-based microdisk resonators,” Phys. Rev. Lett. 104(10), 103901 (2010).
[CrossRef] [PubMed]

M. Ghulinyan, D. Navarro-Urrios, A. Pitanti, A. Lui, G. Pucker, and L. Pavesi, “Whispering-gallery modes and light emission from a Si-nanocrystal-based single microdisk resonator,” Opt. Express 16(17), 13218–13224 (2008).
[CrossRef] [PubMed]

L. Pavesi, “Silicon-Based light sources for silicon integrated circuits,” Adv. Opt. Technol. 2008, 416926 (2008).

Pitanti, A.

A. Pitanti, M. Ghulinyan, D. Navarro-Urrios, G. Pucker, and L. Pavesi, “Probing the spontaneous emission dynamics in Si-nanocrystals-based microdisk resonators,” Phys. Rev. Lett. 104(10), 103901 (2010).
[CrossRef] [PubMed]

M. Ghulinyan, D. Navarro-Urrios, A. Pitanti, A. Lui, G. Pucker, and L. Pavesi, “Whispering-gallery modes and light emission from a Si-nanocrystal-based single microdisk resonator,” Opt. Express 16(17), 13218–13224 (2008).
[CrossRef] [PubMed]

Pollard, H.

H. Pollard, “The representation of $e^{ - x^\lambda }$ as a Laplace integral,” Bull. Am. Math. Soc. 52(10), 908–911 (1946).
[CrossRef]

Priolo, F.

V. Vinciguerra, G. Franzo, F. Priolo, F. Iacona, and C. Spinella, “Quantum confinement and recombination dynamics in silicon nanocrystals embedded in Si/SiO2 superlattices,” J. Appl. Phys. 87(11), 8165–8173 (2000).
[CrossRef]

Pucker, G.

A. Pitanti, M. Ghulinyan, D. Navarro-Urrios, G. Pucker, and L. Pavesi, “Probing the spontaneous emission dynamics in Si-nanocrystals-based microdisk resonators,” Phys. Rev. Lett. 104(10), 103901 (2010).
[CrossRef] [PubMed]

M. Ghulinyan, D. Navarro-Urrios, A. Pitanti, A. Lui, G. Pucker, and L. Pavesi, “Whispering-gallery modes and light emission from a Si-nanocrystal-based single microdisk resonator,” Opt. Express 16(17), 13218–13224 (2008).
[CrossRef] [PubMed]

Purcell, E. M.

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

Rodriguez, J. R.

P. Bianucci, J. R. Rodriguez, F. C. Lenz, J. C. G. Veinot, and A. Meldrum, “Mode structure in the luminescence of Si-nc in cylindrical microcavities,” Physica E 41(6), 1107–1110 (2009).
[CrossRef]

Rodríguez, J. R.

P. Bianucci, J. R. Rodríguez, C. M. Clements, J. G. C. Veinot, and A. Meldrum, “Silicon nanocrystal luminescence coupled to whispering gallery modes in optical fibers,” J. Appl. Phys. 105(2), 023108 (2009).
[CrossRef]

Rothberg, R. J.

S. Chan, Y. Li, R. J. Rothberg, B. L. Miller, and P. M. Fauchet, “Nanoscale silicon microcavities for biosensing,” Mater. Sci. Eng. C 15(1-2), 277–282 (2001).
[CrossRef]

Sa’ar, A.

M. Dovrat, Y. Goshen, J. Jedrzejewski, I. Balberg, and A. Sa’ar, “Radiative versus nonradiative decay processes in silicon nanocrystals probed by time-resolved photoluminescence spectroscopy,” Phys. Rev. B 69(15), 155311 (2004).
[CrossRef]

Schöps, O.

N. Le Thomas, U. Woggon, O. Schöps, M. V. Artemyev, M. Kazes, and U. Banin, “Cavity QED with semiconductor nanocrystals,” Nano Lett. 6(3), 557–561 (2006).
[CrossRef] [PubMed]

Sermage, B.

J. Gérard, B. Sermage, B. Gayral, B. Legrand, E. Costard, and V. Thierry-Mieg, “Enhanced spontaneous emission by quantum boxes in a monolithic optical microcavity,” Phys. Rev. Lett. 81(5), 1110–1113 (1998).
[CrossRef]

Sogawa, T.

Spinella, C.

V. Vinciguerra, G. Franzo, F. Priolo, F. Iacona, and C. Spinella, “Quantum confinement and recombination dynamics in silicon nanocrystals embedded in Si/SiO2 superlattices,” J. Appl. Phys. 87(11), 8165–8173 (2000).
[CrossRef]

Sychugov, I.

I. Sychugov, R. Juhasz, J. Valenta, and J. Linnros, “Narrow luminescence linewidth of a silicon quantum dot,” Phys. Rev. Lett. 94(8), 087405 (2005).
[CrossRef] [PubMed]

Tawara, T.

Thierry-Mieg, V.

J. Gérard, B. Sermage, B. Gayral, B. Legrand, E. Costard, and V. Thierry-Mieg, “Enhanced spontaneous emission by quantum boxes in a monolithic optical microcavity,” Phys. Rev. Lett. 81(5), 1110–1113 (1998).
[CrossRef]

Ujihara, K.

K. Ujihara, “Spontaneous emission and the concept of effective area in a very short optical cavity with plane-parallel dielectric mirrors,” Jpn. J. Appl. Phys. 30(Part 2, No. 5B5b), L901–L903 (1991).
[CrossRef]

Valenta, J.

I. Sychugov, R. Juhasz, J. Valenta, and J. Linnros, “Narrow luminescence linewidth of a silicon quantum dot,” Phys. Rev. Lett. 94(8), 087405 (2005).
[CrossRef] [PubMed]

J. Valenta, R. Juhasz, and J. Linnros, “Photoluminescence spectroscopy of single silicon quantum dots,” Appl. Phys. Lett. 80(6), 1070 (2002).
[CrossRef]

Valeur, B.

M. N. Berberan-Santos, E. N. Bodunov, and B. Valeur, “Mathematical functions for the analysis of luminescence decays with underlying distributions 1. Kohlrausch decay function (stretched exponential),” Chem. Phys. 315(1-2), 171–182 (2005).
[CrossRef]

Veinot, J. C. G.

P. Bianucci, J. R. Rodriguez, F. C. Lenz, J. C. G. Veinot, and A. Meldrum, “Mode structure in the luminescence of Si-nc in cylindrical microcavities,” Physica E 41(6), 1107–1110 (2009).
[CrossRef]

Veinot, J. G. C.

P. Bianucci, J. R. Rodríguez, C. M. Clements, J. G. C. Veinot, and A. Meldrum, “Silicon nanocrystal luminescence coupled to whispering gallery modes in optical fibers,” J. Appl. Phys. 105(2), 023108 (2009).
[CrossRef]

Villas-Bôas, M.

M. Kaniber, A. Laucht, A. Neumann, M. Villas-Bôas, M. Bichler, M.-C. Amann, and J. J. Finley, “Investigation of the nonresonant dot-cavity coupling in two-dimensional photonic crystal nanocavities,” Phys. Rev. B 77(16), 161303 (2008).
[CrossRef]

Vinciguerra, V.

V. Vinciguerra, G. Franzo, F. Priolo, F. Iacona, and C. Spinella, “Quantum confinement and recombination dynamics in silicon nanocrystals embedded in Si/SiO2 superlattices,” J. Appl. Phys. 87(11), 8165–8173 (2000).
[CrossRef]

Wang, H.

Wang, J.

J. Wang, X. F. Wang, Q. Li, A. Hryciw, and A. Meldrum, “The microstructure of SiO thin films: from nanoclusters to nanocrystals,” Philos. Mag. 87(1), 11–27 (2007).
[CrossRef]

Wang, X. F.

J. Wang, X. F. Wang, Q. Li, A. Hryciw, and A. Meldrum, “The microstructure of SiO thin films: from nanoclusters to nanocrystals,” Philos. Mag. 87(1), 11–27 (2007).
[CrossRef]

Woggon, U.

N. Le Thomas, U. Woggon, O. Schöps, M. V. Artemyev, M. Kazes, and U. Banin, “Cavity QED with semiconductor nanocrystals,” Nano Lett. 6(3), 557–561 (2006).
[CrossRef] [PubMed]

Yamamoto, Y.

Y. Yamamoto, “Quantum Communication and Information Processing with Quantum Dots,” Quantum Inf. Process. 5(5), 299–311 (2006).
[CrossRef]

Adv. Opt. Technol. (2)

L. Pavesi, “Silicon-Based light sources for silicon integrated circuits,” Adv. Opt. Technol. 2008, 416926 (2008).

V. Belyakov, V. Burdov, R. Lockwood, and A. Meldrum, “Silicon Nanocrystals: Fundamental Theory and Implications for Stimulated Emission,” Adv. Opt. Technol. 2008, 279502 (2008).

Ann. Phys. (1)

R. Kohlrausch, “Ueber das Dellmann'sche Elektrometer,” Ann. Phys. 91, 353–405 (1854).

Appl. Phys. Lett. (2)

A. Francois and M. Himmelhaus, “Whispering gallery mode biosensor operated in the stimulated emission regime,” Appl. Phys. Lett. 94(3), 031101 (2009).
[CrossRef]

J. Valenta, R. Juhasz, and J. Linnros, “Photoluminescence spectroscopy of single silicon quantum dots,” Appl. Phys. Lett. 80(6), 1070 (2002).
[CrossRef]

Bull. Am. Math. Soc. (1)

H. Pollard, “The representation of $e^{ - x^\lambda }$ as a Laplace integral,” Bull. Am. Math. Soc. 52(10), 908–911 (1946).
[CrossRef]

Chem. Phys. (1)

M. N. Berberan-Santos, E. N. Bodunov, and B. Valeur, “Mathematical functions for the analysis of luminescence decays with underlying distributions 1. Kohlrausch decay function (stretched exponential),” Chem. Phys. 315(1-2), 171–182 (2005).
[CrossRef]

J. Appl. Phys. (3)

P. Bianucci, J. R. Rodríguez, C. M. Clements, J. G. C. Veinot, and A. Meldrum, “Silicon nanocrystal luminescence coupled to whispering gallery modes in optical fibers,” J. Appl. Phys. 105(2), 023108 (2009).
[CrossRef]

A. Belarouci and F. Gourbilleau, “Microcavity enhanced spontaneous emission from silicon nanocrystals,” J. Appl. Phys. 101(7), 073108 (2007).
[CrossRef]

V. Vinciguerra, G. Franzo, F. Priolo, F. Iacona, and C. Spinella, “Quantum confinement and recombination dynamics in silicon nanocrystals embedded in Si/SiO2 superlattices,” J. Appl. Phys. 87(11), 8165–8173 (2000).
[CrossRef]

J. Lumin. (1)

A. Beltaos and A. Meldrum, “Whispering gallery modes in silicon-nanocrystal-coated silica microspheres,” J. Lumin. 126(2), 607–613 (2007).
[CrossRef]

Jpn. J. Appl. Phys. (1)

K. Ujihara, “Spontaneous emission and the concept of effective area in a very short optical cavity with plane-parallel dielectric mirrors,” Jpn. J. Appl. Phys. 30(Part 2, No. 5B5b), L901–L903 (1991).
[CrossRef]

Mater. Sci. Eng. C (1)

S. Chan, Y. Li, R. J. Rothberg, B. L. Miller, and P. M. Fauchet, “Nanoscale silicon microcavities for biosensing,” Mater. Sci. Eng. C 15(1-2), 277–282 (2001).
[CrossRef]

Nano Lett. (1)

N. Le Thomas, U. Woggon, O. Schöps, M. V. Artemyev, M. Kazes, and U. Banin, “Cavity QED with semiconductor nanocrystals,” Nano Lett. 6(3), 557–561 (2006).
[CrossRef] [PubMed]

Opt. Express (2)

Opt. Lett. (1)

Philos. Mag. (1)

J. Wang, X. F. Wang, Q. Li, A. Hryciw, and A. Meldrum, “The microstructure of SiO thin films: from nanoclusters to nanocrystals,” Philos. Mag. 87(1), 11–27 (2007).
[CrossRef]

Phys. Rev. (1)

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

Phys. Rev. A (1)

R. Kekatpure and M. L. Brongersma, “Fundamental photophysics and optical loss processes in Si-nanocrystal-doped microdisk resonators,” Phys. Rev. A 78(2), 023829 (2008).
[CrossRef]

Phys. Rev. B (3)

B. Gayral and J. M. Gérard, “Photoluminescence experiment on quantum dots embedded in a large Purcell-factor microcavity,” Phys. Rev. B 78(23), 235306 (2008).
[CrossRef]

M. Kaniber, A. Laucht, A. Neumann, M. Villas-Bôas, M. Bichler, M.-C. Amann, and J. J. Finley, “Investigation of the nonresonant dot-cavity coupling in two-dimensional photonic crystal nanocavities,” Phys. Rev. B 77(16), 161303 (2008).
[CrossRef]

M. Dovrat, Y. Goshen, J. Jedrzejewski, I. Balberg, and A. Sa’ar, “Radiative versus nonradiative decay processes in silicon nanocrystals probed by time-resolved photoluminescence spectroscopy,” Phys. Rev. B 69(15), 155311 (2004).
[CrossRef]

Phys. Rev. Lett. (4)

J. Gérard, B. Sermage, B. Gayral, B. Legrand, E. Costard, and V. Thierry-Mieg, “Enhanced spontaneous emission by quantum boxes in a monolithic optical microcavity,” Phys. Rev. Lett. 81(5), 1110–1113 (1998).
[CrossRef]

A. Pitanti, M. Ghulinyan, D. Navarro-Urrios, G. Pucker, and L. Pavesi, “Probing the spontaneous emission dynamics in Si-nanocrystals-based microdisk resonators,” Phys. Rev. Lett. 104(10), 103901 (2010).
[CrossRef] [PubMed]

J.-Y. Marzin, J.-M. Gérard, A. Izraël, D. Barrier, and G. Bastard, “Photoluminescence of single InAs quantum dots obtained by self-organized growth on GaAs,” Phys. Rev. Lett. 73(5), 716–719 (1994).
[CrossRef] [PubMed]

I. Sychugov, R. Juhasz, J. Valenta, and J. Linnros, “Narrow luminescence linewidth of a silicon quantum dot,” Phys. Rev. Lett. 94(8), 087405 (2005).
[CrossRef] [PubMed]

Physica E (2)

P. Bianucci, J. R. Rodriguez, F. C. Lenz, J. C. G. Veinot, and A. Meldrum, “Mode structure in the luminescence of Si-nc in cylindrical microcavities,” Physica E 41(6), 1107–1110 (2009).
[CrossRef]

K. Leosson, D. Birkedal, I. Magnusdottir, W. Langbein, and J. M. Hvam, “Homogeneous linewidth of self-assembled III–V quantum dots observed in single-dot photoluminescence,” Physica E 17, 1–6 (2003).
[CrossRef]

Quantum Inf. Process. (1)

Y. Yamamoto, “Quantum Communication and Information Processing with Quantum Dots,” Quantum Inf. Process. 5(5), 299–311 (2006).
[CrossRef]

Small (1)

D. E. Gómez, I. Pastoriza-Santos, and P. Mulvaney, “Tunable whispering gallery mode emission from quantum-dot-doped microspheres,” Small 1(2), 238–241 (2005).
[CrossRef]

Other (4)

A. M. Fox, Quantum Optics: An Intruduction, Oxford University Press, Oxford, 2006.

H. Yokoyama, Y. Nambu, and T. Kawakami, “Controlling spontaneous emission and optical microcavities,” in Confined Electrons and Photons, edited by E. Burstein and C. Weisbuch, Plenum Press, NY, 1995, pp. 427–466.

L. A. Weinstein, Open Resonators and Open Waveguides. The Golem Press, Boulder, CO, 1969.

S. M. Spillane, Fiber-coupled Ultra-high-Q Microresonators for Nonlinear and Quantum Optics, California Institute of Technology PhD Thesis, 2004; see also V.B. Braginsky, M.L. Gorodetsky, “Quality-factor and nonlinear properties of optical whispering-gallery modes,” Phys. Lett. A 137, 393–397 (1989).

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Figures (8)

Fig. 1
Fig. 1

Illustration of the emissions into cavity, guided, and “free-medium” modes for (a) planar, (b) cylindrical, and (c) spherical cavities. The spherical cavity is, in many respects, the simplest one because of the lack of guided modes and overall fairly simple mode structure without too many radial modes. QDs emit into the different modes at rates given by Fermi’s golden rule for each geometry

Fig. 2
Fig. 2

The PL spectrum calculated from I(λ) = P(λ)W(λ), for the Si-QD size distribution shown in the inset. The points in wavelength space are clearly visible, and are more widely separated at shorter wavelengths because of the non-linear relationship between QD radius and energy gap. The filtered spectrum is obtained by setting a bandpass filter and decreasing the data spacing – in this case the region between 790 and 850 nm is “expanded” to include 500 discrete points; as many as 10,000 can be used for a 12-hour simulation on a standard PC.

Fig. 3
Fig. 3

(a) A sample cavity spectrum (blue lines; FSR = 12.6 GHz corresponding to the TE modes of a ~5-μm-diameter sphere, Δfcav = 0.5 GHz, Q ≈700) compared with that of single QDs (ΔEQD = 3 meV; red curves), illustrating the relatives widths and overlap of the spectra. For clarity, only 40 QD Lorentzians are shown for 40 central wavelengths; the actual simulations used 500 to 2000 QD center frequencies uniformly distributed in wavelength space. Panels (b) and (c) show the variation of Fp and βcpl as a function of wavelength (effectively, the range of QD central wavelengths) for the cavity-QD spectra shown in (a). In this case, Fp reaches a maximum near 0.2 on resonance, compared to an idealized Purcell factor of ~1. The coupling factor varies between 0.01 and 0.2, depending on the resonance overlap. The slight increase in the peak values with wavelength arises from the frequency dependence of the rate enhancement in Eq. (12). Note that the resonances in Fp and β are much wider than the intrinsic cavity resonances, due to the 3-meV QD spectral width.

Fig. 4
Fig. 4

Simulated PL spectra for a microsphere with a diameter of 10 μm, for several different ratios of cavity:free-space collection efficiencies. The Si-QDs lognormal distribution parameters corresponded to a mean radius and deviation of 2.5 and 0.5 nm, respectively, and the linewidth was 5 meV.

Fig. 5
Fig. 5

(a) Simulated PL spectra for a microsphere with a d = 10 μm, and the corresponding background corrected emission spectrum. (b) Background-corrected spectra for a set of different microspheres with diameters ranging from 3 to 20 μm. The inset shows the “ideal” (cavity intrinsic) Q factors, in which there are only radiation losses. The observed PL Q-factors are much lower than these values (green fit) due to the 5 meV bandwidth of the QDs that limits the PL Q-factor to approximately 300, despite the much higher intrinsic cavity Q. The orange line in the inset shows the QPL calculated from the QD and cavity linewidths, and are quite close to those obtained by background subtraction and peak fitting.

Fig. 6
Fig. 6

Radiative rate as a function of Si-QD radius: green → “infinite” medium; red → 10-μm-diameter microsphere with ΔEQD = 20 meV; blue → the same microsphere with ΔEQD = 1 meV. The cavity rates are entirely suppressed compared to free space, and will only become enhanced at the resonance positions for still smaller values of ΔEQD . The inset shows the corresponding ensemble lifetimes via Laplace transform of the data in the main panel. We see already hints of the general trend of β decreasing and τ increasing as the PL resonances become better defined (β appears somewhat higher for the red curve than the green, suggesting that the rate distribution could be more compressed). Although the general trends can be seen in diagrams like these, the exact values of β and τ obtained by fitting also depend sensitively on the sampling in time and frequency space and on the width of the filter.

Fig. 7
Fig. 7

Radiative rate for Si-QDs for a cavity with a small mode volume. In (a) the black curve shows the “infinite medium” rate, while the colors correspond to cavity-coupled emission with different values of ΔEQD : red → 0.5 meV; orange → 1 meV; green → 5 meV; blue → 10 meV; navy → 20 meV. The corresponding cavity-only and global lifetimes are shown in (b) and (c), respectively, with the same color coding. In (c), the black line represents the free-medium (no-cavity) decay. The inset to (c) shows the corresponding values of τ (red) and β (blue) for the global decays on separate vertical axes. When ΔEQD ≤ 0.5 meV, the trends of increasing β and decreasing τ with QD bandwidth can no longer be followed, since such narrow resonances cannot be well-sampled in frequency space. Here, the decrease in β with ΔEQD is monotonic, suggesting that the difference observed between the red and green curves in Fig. 6 may be a data-fitting artifact.

Fig. 8
Fig. 8

(a) Simulated TE-polarized PL for a 20-μm-diameter sphere with Si-QDs of different bandwidths ΔEQD . The curves shown are, in order: blue (ΔEQD = 1 meV), turquoise (ΔEQD = 5 meV), green (ΔEQD = 10 meV), orange (ΔEQD = 20 meV), red (ΔEQD = 100 meV). The mode structure becomes increasingly difficult to observe as the QD bandwidth increases. Panel (b) shows a comparison of simulated PL (red) with experimental PL data (blue) from a Si-QD-coated microsphere with a diameter of 20 μm (see PL image in the inset, in which the spectrometer slit can be observed on the right edge of the luminescent sphere) [34]. The simulation used ΔEQD = 3 meV to provide a close fit to the experimental data. To maximize the visibility of the mode spectrum, we used a greater collection efficiency by a factor 108 for mode PL compared with background PL, which would correspond to the energy trapped in a sphere with a Q factor near 109, close to the maximum effectively possible for a silica sphere. Additional structure in the experimental spectra (especially some shoulders in the PL resonances) are due to the presence of the opposite polarization and/or higher-order radial modes [34].

Equations (17)

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Wcav=2π20|f|H|i|2ρ(ω)Λ(ω)dω
ρ(ω)=2πΔωcav4(ωωcav)2+Δωcav2
M2=|f|H|i|2=ξ2ω0μ22εV
Wcav(ω0)=2π2ξ2ω0μ22εV2πΔωcavΔωcav24(ω0ωcav)2+Δωcav2
Fp=2π2ξ2ω0μ22εV2πΔωcav[Δωcav24(ω0ωcav)2+Δωcav2]3πε0c3ω03μ2
Fp=ξ23Q(λ/n)34π2V
Wcav(ωcav)=2π2ξ2ω0μ22εV2πΔω0Δω024(ωcavω0)2+Δω02
Wcav(ω0,ωcav)=A01π2δ0δ02+(ωω0)2ω0δcavδcav2+(ωωcav)2dω
Wcav(ω0,ωcav)Aδ04π2i[(1+ω0iδ0)(1ωω0iδ0)(1ωωcaviδcav1ωωcav+iδcav)+(1ω0iδ0)(1ωω0+iδ0)(1ωωcaviδcav1ωωcav+iδcav)]dω
Wcav(ω0,ωcav)=A2πi[ω0+iδ0ωcavω0iδ0iδcavω0iδ0ωcavω0+iδ0+iδcav]
Wcav(ω0,ωcav)=Aπωcavδ0+ω0δcav(ωcavω0)2+(δ0+δcav)2
Wcav(ω0,ωcav)=2ξ2μ2εVω0Δωcav+ωcavΔω04(ω0ωcav)2+(Δωcav+Δω0)2
Wcav(ω0,ωcav)=Aπωcavδ0+ω0δcav(ωcavω0)2+(δ0+δcav)2+12π2[ω0(ω0ωcav)+δ0(δ0δcav)(ω0ωcav)2+(δ0δcav)2logω02+δ02ωcav2+δcav2ω0(ω0ωcav)+δ0(δ0+δcav)(ω0ωcav)2+(δ0+δcav)2logω02+δ02ωcav2+δcav2+ωcavδ0ω0δcav(ωcavω0)2+(δ0δcav)2(tan1δ0ω0+tan1δcavωcav)ωcavδ0+ω0δcav(ωcavω0)2+(δ0+δcav)2(tan1δ0ω0tan1δcavωcav)]
Wcav=A2π(ω0δ0+1π[1ω0δ0tan1δ0ω0])(ωcav=ω0,δcav=δ0)
I(t)=0H(W)exp(Wt)dW
εg(QD)(R)=εg2+D1/R2
Wcav(ω0,ωcav)=ABΔωcav+Δω04(ω0ωcav)2+(Δωcav+Δω0)2

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