Abstract

We formulated an analytical model and analyzed the modification of spontaneous emission in Bragg onion resonators. We consider both the case of a single light emitter and a uniformly distributed ensemble of light emitters within the resonator. We obtain an expression for the average radiation rate of the light emitters ensemble and discuss the modification of the average radiation rate as a function of cavity parameters such as the core radius, the number of Bragg cladding layers, the index contrast of the Bragg cladding, and the refractive index of surrounding medium. We also consider the possibility of non-exponential decay of the light emitter ensemble due to the strong dependence of spontaneous emission on the location and polarization of individual light emitter. We conclude that Bragg onion resonators can both enhance and inhibit spontaneous emission by several orders of magnitude. This property can have significant impact in the field of cavity quantum electrodynamics (QED).

© 2006 Optical Society of America

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  1. J. M. Gerard, D. Barrier, J. Y. Marzin, R. Kuszelewicz, L. Manin, E. Costard, V. Thierry Mieg and T. Rivera, "Quantum boxes as active probes for photonic microstructures: The pillar microcavity case," Appl. Phys. Lett. 69, 449 (1996).
    [CrossRef]
  2. J. M. Gerard, and B. Gayral, "Strong Purcell effect for InAs quantum boxes in three-dimensional solid-state microcavities," J. Lightwave Technol. 17, 2089 (1999).
    [CrossRef]
  3. B. Gayral, J. M. Gerard, A. Lemaitre, C. Dupuis, L. Manin and J. L. Pelouard, "High-Q wet-etched GaAs microdisks containing InAs quantum boxes," Appl. Phys. Lett. 75, 1908 (1999).
    [CrossRef]
  4. D. K. Armani, T. J. Kippenberg, S. M. Spillane and K. J. Vahala, "Ultra-high-Q toroid microcavity on a chip," Nature 421, 925 (2003).
    [CrossRef] [PubMed]
  5. K. Srinivasan, P. E. Barclay, O. Painter, J. X. Chen, A. Y. Cho and C. Gmachl, "Experimental demonstration of a high quality factor photonic crystal microcavity," Appl. Phys. Lett. 83, 1915 (2003).
    [CrossRef]
  6. K. J. Vahala, "Optical microcavities," Nature 424, 839 (2003).
    [CrossRef] [PubMed]
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    [CrossRef]
  9. V. Sandoghdar, F. Treussart, J. Hare, V. Lefevre-Seguin, J. M. Raimond and S. Haroche, "Very low threshold whispering-gallery-mode microsphere laser," Phys. Rev. A. 54, R1777 (1996).
    [CrossRef] [PubMed]
  10. D. L. Huffaker, L. A. Graham, H. Deng and D. G. Deppe, "Sub-40 mu A continuous-wave lasing in an oxidized vertical-cavity surface-emitting laser with dielectric mirrors," IEEE Photon. Technol. Lett. 8, 974 (1996).
    [CrossRef]
  11. P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. D. Zhang, E. Hu and A. Imamoglu, "A quantum dot single-photon turnstile device," Science 290, 2282 (2000).
    [CrossRef] [PubMed]
  12. J. M. Gerard, 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, 1110 (1998).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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  23. E. Yablonovitch, "Inhibited spontaneous emission in Solid-State Physics and Electronics," Phys. Rev. Lett. 58, 2059 (1987).
    [CrossRef] [PubMed]
  24. J. Vuckovic, M. Pelton, A. Scherer and Y. Yamamoto, "Optimization of three-dimensional micropost microcavities for cavity quantum electrodynamics," Phys. Rev. A. 66023808 (2002).
    [CrossRef]
  25. M. Bayer, T. L. Reinecke, F. Weidner, A. Larionov, A. McDonald and A. Forchel, "Inhibition and enhancement of the spontaneous emission of quantum dots in structured microresonators," Phys. Rev. Lett. 86, 3168 (2001).
    [CrossRef] [PubMed]
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    [CrossRef]
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2004 (1)

2003 (4)

Y. Xu, W. Liang,A. Yariv,J. G. Fleming and S. Y. Lin, "High-quality-factor Bragg onion resonators with omnidirectional reflector cladding," Opt. Lett. 28, 2144 (2003).
[CrossRef] [PubMed]

D. K. Armani, T. J. Kippenberg, S. M. Spillane and K. J. Vahala, "Ultra-high-Q toroid microcavity on a chip," Nature 421, 925 (2003).
[CrossRef] [PubMed]

K. Srinivasan, P. E. Barclay, O. Painter, J. X. Chen, A. Y. Cho and C. Gmachl, "Experimental demonstration of a high quality factor photonic crystal microcavity," Appl. Phys. Lett. 83, 1915 (2003).
[CrossRef]

K. J. Vahala, "Optical microcavities," Nature 424, 839 (2003).
[CrossRef] [PubMed]

2002 (2)

M. Pelton, C. Santori, J. Vuckovic, B. Y. Zhang, G. S. Solomon, J. Plant and Y. Yamamoto, "Efficient source of single photons: A single quantum dot in a micropost microcavity," Phys. Rev. Lett. 89299602 (2002).
[CrossRef] [PubMed]

J. Vuckovic, M. Pelton, A. Scherer and Y. Yamamoto, "Optimization of three-dimensional micropost microcavities for cavity quantum electrodynamics," Phys. Rev. A. 66023808 (2002).
[CrossRef]

2001 (1)

M. Bayer, T. L. Reinecke, F. Weidner, A. Larionov, A. McDonald and A. Forchel, "Inhibition and enhancement of the spontaneous emission of quantum dots in structured microresonators," Phys. Rev. Lett. 86, 3168 (2001).
[CrossRef] [PubMed]

2000 (1)

P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. D. Zhang, E. Hu and A. Imamoglu, "A quantum dot single-photon turnstile device," Science 290, 2282 (2000).
[CrossRef] [PubMed]

1999 (2)

B. Gayral, J. M. Gerard, A. Lemaitre, C. Dupuis, L. Manin and J. L. Pelouard, "High-Q wet-etched GaAs microdisks containing InAs quantum boxes," Appl. Phys. Lett. 75, 1908 (1999).
[CrossRef]

J. M. Gerard, and B. Gayral, "Strong Purcell effect for InAs quantum boxes in three-dimensional solid-state microcavities," J. Lightwave Technol. 17, 2089 (1999).
[CrossRef]

1998 (1)

J. M. Gerard, 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, 1110 (1998).
[CrossRef]

1996 (4)

M. P. van Exter, G. Nienhuis and J. P. Woerdman, "Two simple expressions for the spontaneous emission factor beta," Phys. Rev. A. 54, 3553 (1996).
[CrossRef]

V. Sandoghdar, F. Treussart, J. Hare, V. Lefevre-Seguin, J. M. Raimond and S. Haroche, "Very low threshold whispering-gallery-mode microsphere laser," Phys. Rev. A. 54, R1777 (1996).
[CrossRef] [PubMed]

D. L. Huffaker, L. A. Graham, H. Deng and D. G. Deppe, "Sub-40 mu A continuous-wave lasing in an oxidized vertical-cavity surface-emitting laser with dielectric mirrors," IEEE Photon. Technol. Lett. 8, 974 (1996).
[CrossRef]

J. M. Gerard, D. Barrier, J. Y. Marzin, R. Kuszelewicz, L. Manin, E. Costard, V. Thierry Mieg and T. Rivera, "Quantum boxes as active probes for photonic microstructures: The pillar microcavity case," Appl. Phys. Lett. 69, 449 (1996).
[CrossRef]

1995 (1)

M. H. Macdougal, P. D. Dapkus,V. Pudikov, H. M. Zhao and G. M. Yang, "Ultralow threshold current vertical-cavity surface-emitting lasers with alas Oxide-Gaas distributed Bragg reflectors," IEEE Photon. Technol. Lett. 7, 229 (1995).
[CrossRef]

1994 (1)

K. G. Sullivan and D. G. Hall, "Radiation in spherically symmetrical structures.2. Enhancement and inhibition of Dipole Radiation in a Spherical Bragg Cavity," Phys. Rev. A. 50, 2708 (1994).
[CrossRef] [PubMed]

1991 (1)

Y. Yamamoto, S. Machida and G. Bjork, "Microcavity semiconductor-laser with enhanced spontaneous emission," Phys. Rev. A 44, 657 (1991).
[CrossRef] [PubMed]

1988 (1)

H. Chew, "Radiation and lifetimes of Atoms inside dielectric particles," Phys. Rev. A. 38, 3410 (1988).
[CrossRef] [PubMed]

1987 (2)

H. Chew, "Transition rates of Atoms near Spherical Surfaces," J. Chem. Phys. 87, 1355 (1987).
[CrossRef]

E. Yablonovitch, "Inhibited spontaneous emission in Solid-State Physics and Electronics," Phys. Rev. Lett. 58, 2059 (1987).
[CrossRef] [PubMed]

1985 (1)

P. Das, and H. Metiu, "Enhancement of molecular fluorescence and photochemistry by Small Metal Particles," J. Phys. Chem. 89, 4680 (1985).
[CrossRef]

1978 (1)

R. R. Chance, A. Prock and R. Silbey, "Molecular fluorescence and energy transfer near interfaces," Adv. Chem. Phys. 37, 1 (1978).
[CrossRef]

1977 (1)

Armani, D. K.

D. K. Armani, T. J. Kippenberg, S. M. Spillane and K. J. Vahala, "Ultra-high-Q toroid microcavity on a chip," Nature 421, 925 (2003).
[CrossRef] [PubMed]

Barclay, P. E.

K. Srinivasan, P. E. Barclay, O. Painter, J. X. Chen, A. Y. Cho and C. Gmachl, "Experimental demonstration of a high quality factor photonic crystal microcavity," Appl. Phys. Lett. 83, 1915 (2003).
[CrossRef]

Barrier, D.

J. M. Gerard, D. Barrier, J. Y. Marzin, R. Kuszelewicz, L. Manin, E. Costard, V. Thierry Mieg and T. Rivera, "Quantum boxes as active probes for photonic microstructures: The pillar microcavity case," Appl. Phys. Lett. 69, 449 (1996).
[CrossRef]

Bayer, M.

M. Bayer, T. L. Reinecke, F. Weidner, A. Larionov, A. McDonald and A. Forchel, "Inhibition and enhancement of the spontaneous emission of quantum dots in structured microresonators," Phys. Rev. Lett. 86, 3168 (2001).
[CrossRef] [PubMed]

Becher, C.

P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. D. Zhang, E. Hu and A. Imamoglu, "A quantum dot single-photon turnstile device," Science 290, 2282 (2000).
[CrossRef] [PubMed]

Bjork, G.

Y. Yamamoto, S. Machida and G. Bjork, "Microcavity semiconductor-laser with enhanced spontaneous emission," Phys. Rev. A 44, 657 (1991).
[CrossRef] [PubMed]

Chance, R. R.

R. R. Chance, A. Prock and R. Silbey, "Molecular fluorescence and energy transfer near interfaces," Adv. Chem. Phys. 37, 1 (1978).
[CrossRef]

Chen, J. X.

K. Srinivasan, P. E. Barclay, O. Painter, J. X. Chen, A. Y. Cho and C. Gmachl, "Experimental demonstration of a high quality factor photonic crystal microcavity," Appl. Phys. Lett. 83, 1915 (2003).
[CrossRef]

Chew, H.

H. Chew, "Radiation and lifetimes of Atoms inside dielectric particles," Phys. Rev. A. 38, 3410 (1988).
[CrossRef] [PubMed]

H. Chew, "Transition rates of Atoms near Spherical Surfaces," J. Chem. Phys. 87, 1355 (1987).
[CrossRef]

Cho, A. Y.

K. Srinivasan, P. E. Barclay, O. Painter, J. X. Chen, A. Y. Cho and C. Gmachl, "Experimental demonstration of a high quality factor photonic crystal microcavity," Appl. Phys. Lett. 83, 1915 (2003).
[CrossRef]

Costard, E.

J. M. Gerard, 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, 1110 (1998).
[CrossRef]

J. M. Gerard, D. Barrier, J. Y. Marzin, R. Kuszelewicz, L. Manin, E. Costard, V. Thierry Mieg and T. Rivera, "Quantum boxes as active probes for photonic microstructures: The pillar microcavity case," Appl. Phys. Lett. 69, 449 (1996).
[CrossRef]

Dapkus, P. D.

M. H. Macdougal, P. D. Dapkus,V. Pudikov, H. M. Zhao and G. M. Yang, "Ultralow threshold current vertical-cavity surface-emitting lasers with alas Oxide-Gaas distributed Bragg reflectors," IEEE Photon. Technol. Lett. 7, 229 (1995).
[CrossRef]

Das, P.

P. Das, and H. Metiu, "Enhancement of molecular fluorescence and photochemistry by Small Metal Particles," J. Phys. Chem. 89, 4680 (1985).
[CrossRef]

Deng, H.

D. L. Huffaker, L. A. Graham, H. Deng and D. G. Deppe, "Sub-40 mu A continuous-wave lasing in an oxidized vertical-cavity surface-emitting laser with dielectric mirrors," IEEE Photon. Technol. Lett. 8, 974 (1996).
[CrossRef]

Deppe, D. G.

D. L. Huffaker, L. A. Graham, H. Deng and D. G. Deppe, "Sub-40 mu A continuous-wave lasing in an oxidized vertical-cavity surface-emitting laser with dielectric mirrors," IEEE Photon. Technol. Lett. 8, 974 (1996).
[CrossRef]

Dupuis, C.

B. Gayral, J. M. Gerard, A. Lemaitre, C. Dupuis, L. Manin and J. L. Pelouard, "High-Q wet-etched GaAs microdisks containing InAs quantum boxes," Appl. Phys. Lett. 75, 1908 (1999).
[CrossRef]

Fleming, J. G.

Forchel, A.

M. Bayer, T. L. Reinecke, F. Weidner, A. Larionov, A. McDonald and A. Forchel, "Inhibition and enhancement of the spontaneous emission of quantum dots in structured microresonators," Phys. Rev. Lett. 86, 3168 (2001).
[CrossRef] [PubMed]

Gayral, B.

B. Gayral, J. M. Gerard, A. Lemaitre, C. Dupuis, L. Manin and J. L. Pelouard, "High-Q wet-etched GaAs microdisks containing InAs quantum boxes," Appl. Phys. Lett. 75, 1908 (1999).
[CrossRef]

J. M. Gerard, and B. Gayral, "Strong Purcell effect for InAs quantum boxes in three-dimensional solid-state microcavities," J. Lightwave Technol. 17, 2089 (1999).
[CrossRef]

J. M. Gerard, 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, 1110 (1998).
[CrossRef]

Gerard, J. M.

J. M. Gerard, and B. Gayral, "Strong Purcell effect for InAs quantum boxes in three-dimensional solid-state microcavities," J. Lightwave Technol. 17, 2089 (1999).
[CrossRef]

B. Gayral, J. M. Gerard, A. Lemaitre, C. Dupuis, L. Manin and J. L. Pelouard, "High-Q wet-etched GaAs microdisks containing InAs quantum boxes," Appl. Phys. Lett. 75, 1908 (1999).
[CrossRef]

J. M. Gerard, 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, 1110 (1998).
[CrossRef]

J. M. Gerard, D. Barrier, J. Y. Marzin, R. Kuszelewicz, L. Manin, E. Costard, V. Thierry Mieg and T. Rivera, "Quantum boxes as active probes for photonic microstructures: The pillar microcavity case," Appl. Phys. Lett. 69, 449 (1996).
[CrossRef]

Gmachl, C.

K. Srinivasan, P. E. Barclay, O. Painter, J. X. Chen, A. Y. Cho and C. Gmachl, "Experimental demonstration of a high quality factor photonic crystal microcavity," Appl. Phys. Lett. 83, 1915 (2003).
[CrossRef]

Graham, L. A.

D. L. Huffaker, L. A. Graham, H. Deng and D. G. Deppe, "Sub-40 mu A continuous-wave lasing in an oxidized vertical-cavity surface-emitting laser with dielectric mirrors," IEEE Photon. Technol. Lett. 8, 974 (1996).
[CrossRef]

Hall, D. G.

K. G. Sullivan and D. G. Hall, "Radiation in spherically symmetrical structures.2. Enhancement and inhibition of Dipole Radiation in a Spherical Bragg Cavity," Phys. Rev. A. 50, 2708 (1994).
[CrossRef] [PubMed]

Hare, J.

V. Sandoghdar, F. Treussart, J. Hare, V. Lefevre-Seguin, J. M. Raimond and S. Haroche, "Very low threshold whispering-gallery-mode microsphere laser," Phys. Rev. A. 54, R1777 (1996).
[CrossRef] [PubMed]

Haroche, S.

V. Sandoghdar, F. Treussart, J. Hare, V. Lefevre-Seguin, J. M. Raimond and S. Haroche, "Very low threshold whispering-gallery-mode microsphere laser," Phys. Rev. A. 54, R1777 (1996).
[CrossRef] [PubMed]

Hu, E.

P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. D. Zhang, E. Hu and A. Imamoglu, "A quantum dot single-photon turnstile device," Science 290, 2282 (2000).
[CrossRef] [PubMed]

Huffaker, D. L.

D. L. Huffaker, L. A. Graham, H. Deng and D. G. Deppe, "Sub-40 mu A continuous-wave lasing in an oxidized vertical-cavity surface-emitting laser with dielectric mirrors," IEEE Photon. Technol. Lett. 8, 974 (1996).
[CrossRef]

Imamoglu, A.

P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. D. Zhang, E. Hu and A. Imamoglu, "A quantum dot single-photon turnstile device," Science 290, 2282 (2000).
[CrossRef] [PubMed]

Kippenberg, T. J.

D. K. Armani, T. J. Kippenberg, S. M. Spillane and K. J. Vahala, "Ultra-high-Q toroid microcavity on a chip," Nature 421, 925 (2003).
[CrossRef] [PubMed]

Kiraz, A.

P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. D. Zhang, E. Hu and A. Imamoglu, "A quantum dot single-photon turnstile device," Science 290, 2282 (2000).
[CrossRef] [PubMed]

Kunz, R. E.

Kuszelewicz, R.

J. M. Gerard, D. Barrier, J. Y. Marzin, R. Kuszelewicz, L. Manin, E. Costard, V. Thierry Mieg and T. Rivera, "Quantum boxes as active probes for photonic microstructures: The pillar microcavity case," Appl. Phys. Lett. 69, 449 (1996).
[CrossRef]

Larionov, A.

M. Bayer, T. L. Reinecke, F. Weidner, A. Larionov, A. McDonald and A. Forchel, "Inhibition and enhancement of the spontaneous emission of quantum dots in structured microresonators," Phys. Rev. Lett. 86, 3168 (2001).
[CrossRef] [PubMed]

Lefevre-Seguin, V.

V. Sandoghdar, F. Treussart, J. Hare, V. Lefevre-Seguin, J. M. Raimond and S. Haroche, "Very low threshold whispering-gallery-mode microsphere laser," Phys. Rev. A. 54, R1777 (1996).
[CrossRef] [PubMed]

Legrand, B.

J. M. Gerard, 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, 1110 (1998).
[CrossRef]

Lemaitre, A.

B. Gayral, J. M. Gerard, A. Lemaitre, C. Dupuis, L. Manin and J. L. Pelouard, "High-Q wet-etched GaAs microdisks containing InAs quantum boxes," Appl. Phys. Lett. 75, 1908 (1999).
[CrossRef]

Liang, W.

Lin, S. Y.

Lukosz, W.

Macdougal, M. H.

M. H. Macdougal, P. D. Dapkus,V. Pudikov, H. M. Zhao and G. M. Yang, "Ultralow threshold current vertical-cavity surface-emitting lasers with alas Oxide-Gaas distributed Bragg reflectors," IEEE Photon. Technol. Lett. 7, 229 (1995).
[CrossRef]

Machida, S.

Y. Yamamoto, S. Machida and G. Bjork, "Microcavity semiconductor-laser with enhanced spontaneous emission," Phys. Rev. A 44, 657 (1991).
[CrossRef] [PubMed]

Manin, L.

B. Gayral, J. M. Gerard, A. Lemaitre, C. Dupuis, L. Manin and J. L. Pelouard, "High-Q wet-etched GaAs microdisks containing InAs quantum boxes," Appl. Phys. Lett. 75, 1908 (1999).
[CrossRef]

J. M. Gerard, D. Barrier, J. Y. Marzin, R. Kuszelewicz, L. Manin, E. Costard, V. Thierry Mieg and T. Rivera, "Quantum boxes as active probes for photonic microstructures: The pillar microcavity case," Appl. Phys. Lett. 69, 449 (1996).
[CrossRef]

Marzin, J. Y.

J. M. Gerard, D. Barrier, J. Y. Marzin, R. Kuszelewicz, L. Manin, E. Costard, V. Thierry Mieg and T. Rivera, "Quantum boxes as active probes for photonic microstructures: The pillar microcavity case," Appl. Phys. Lett. 69, 449 (1996).
[CrossRef]

McDonald, A.

M. Bayer, T. L. Reinecke, F. Weidner, A. Larionov, A. McDonald and A. Forchel, "Inhibition and enhancement of the spontaneous emission of quantum dots in structured microresonators," Phys. Rev. Lett. 86, 3168 (2001).
[CrossRef] [PubMed]

Metiu, H.

P. Das, and H. Metiu, "Enhancement of molecular fluorescence and photochemistry by Small Metal Particles," J. Phys. Chem. 89, 4680 (1985).
[CrossRef]

Michler, P.

P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. D. Zhang, E. Hu and A. Imamoglu, "A quantum dot single-photon turnstile device," Science 290, 2282 (2000).
[CrossRef] [PubMed]

Nienhuis, G.

M. P. van Exter, G. Nienhuis and J. P. Woerdman, "Two simple expressions for the spontaneous emission factor beta," Phys. Rev. A. 54, 3553 (1996).
[CrossRef]

Painter, O.

K. Srinivasan, P. E. Barclay, O. Painter, J. X. Chen, A. Y. Cho and C. Gmachl, "Experimental demonstration of a high quality factor photonic crystal microcavity," Appl. Phys. Lett. 83, 1915 (2003).
[CrossRef]

Pelouard, J. L.

B. Gayral, J. M. Gerard, A. Lemaitre, C. Dupuis, L. Manin and J. L. Pelouard, "High-Q wet-etched GaAs microdisks containing InAs quantum boxes," Appl. Phys. Lett. 75, 1908 (1999).
[CrossRef]

Pelton, M.

J. Vuckovic, M. Pelton, A. Scherer and Y. Yamamoto, "Optimization of three-dimensional micropost microcavities for cavity quantum electrodynamics," Phys. Rev. A. 66023808 (2002).
[CrossRef]

M. Pelton, C. Santori, J. Vuckovic, B. Y. Zhang, G. S. Solomon, J. Plant and Y. Yamamoto, "Efficient source of single photons: A single quantum dot in a micropost microcavity," Phys. Rev. Lett. 89299602 (2002).
[CrossRef] [PubMed]

Petroff, P. M.

P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. D. Zhang, E. Hu and A. Imamoglu, "A quantum dot single-photon turnstile device," Science 290, 2282 (2000).
[CrossRef] [PubMed]

Plant, J.

M. Pelton, C. Santori, J. Vuckovic, B. Y. Zhang, G. S. Solomon, J. Plant and Y. Yamamoto, "Efficient source of single photons: A single quantum dot in a micropost microcavity," Phys. Rev. Lett. 89299602 (2002).
[CrossRef] [PubMed]

Prock, A.

R. R. Chance, A. Prock and R. Silbey, "Molecular fluorescence and energy transfer near interfaces," Adv. Chem. Phys. 37, 1 (1978).
[CrossRef]

Pudikov, V.

M. H. Macdougal, P. D. Dapkus,V. Pudikov, H. M. Zhao and G. M. Yang, "Ultralow threshold current vertical-cavity surface-emitting lasers with alas Oxide-Gaas distributed Bragg reflectors," IEEE Photon. Technol. Lett. 7, 229 (1995).
[CrossRef]

Raimond, J. M.

V. Sandoghdar, F. Treussart, J. Hare, V. Lefevre-Seguin, J. M. Raimond and S. Haroche, "Very low threshold whispering-gallery-mode microsphere laser," Phys. Rev. A. 54, R1777 (1996).
[CrossRef] [PubMed]

Reinecke, T. L.

M. Bayer, T. L. Reinecke, F. Weidner, A. Larionov, A. McDonald and A. Forchel, "Inhibition and enhancement of the spontaneous emission of quantum dots in structured microresonators," Phys. Rev. Lett. 86, 3168 (2001).
[CrossRef] [PubMed]

Rivera, T.

J. M. Gerard, D. Barrier, J. Y. Marzin, R. Kuszelewicz, L. Manin, E. Costard, V. Thierry Mieg and T. Rivera, "Quantum boxes as active probes for photonic microstructures: The pillar microcavity case," Appl. Phys. Lett. 69, 449 (1996).
[CrossRef]

Sandoghdar, V.

V. Sandoghdar, F. Treussart, J. Hare, V. Lefevre-Seguin, J. M. Raimond and S. Haroche, "Very low threshold whispering-gallery-mode microsphere laser," Phys. Rev. A. 54, R1777 (1996).
[CrossRef] [PubMed]

Santori, C.

M. Pelton, C. Santori, J. Vuckovic, B. Y. Zhang, G. S. Solomon, J. Plant and Y. Yamamoto, "Efficient source of single photons: A single quantum dot in a micropost microcavity," Phys. Rev. Lett. 89299602 (2002).
[CrossRef] [PubMed]

Scherer, A.

J. Vuckovic, M. Pelton, A. Scherer and Y. Yamamoto, "Optimization of three-dimensional micropost microcavities for cavity quantum electrodynamics," Phys. Rev. A. 66023808 (2002).
[CrossRef]

Schoenfeld, W. V.

P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. D. Zhang, E. Hu and A. Imamoglu, "A quantum dot single-photon turnstile device," Science 290, 2282 (2000).
[CrossRef] [PubMed]

Sermage, B.

J. M. Gerard, 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, 1110 (1998).
[CrossRef]

Silbey, R.

R. R. Chance, A. Prock and R. Silbey, "Molecular fluorescence and energy transfer near interfaces," Adv. Chem. Phys. 37, 1 (1978).
[CrossRef]

Solomon, G. S.

M. Pelton, C. Santori, J. Vuckovic, B. Y. Zhang, G. S. Solomon, J. Plant and Y. Yamamoto, "Efficient source of single photons: A single quantum dot in a micropost microcavity," Phys. Rev. Lett. 89299602 (2002).
[CrossRef] [PubMed]

Spillane, S. M.

D. K. Armani, T. J. Kippenberg, S. M. Spillane and K. J. Vahala, "Ultra-high-Q toroid microcavity on a chip," Nature 421, 925 (2003).
[CrossRef] [PubMed]

Srinivasan, K.

K. Srinivasan, P. E. Barclay, O. Painter, J. X. Chen, A. Y. Cho and C. Gmachl, "Experimental demonstration of a high quality factor photonic crystal microcavity," Appl. Phys. Lett. 83, 1915 (2003).
[CrossRef]

Sullivan, K. G.

K. G. Sullivan and D. G. Hall, "Radiation in spherically symmetrical structures.2. Enhancement and inhibition of Dipole Radiation in a Spherical Bragg Cavity," Phys. Rev. A. 50, 2708 (1994).
[CrossRef] [PubMed]

Thierry Mieg, V.

J. M. Gerard, D. Barrier, J. Y. Marzin, R. Kuszelewicz, L. Manin, E. Costard, V. Thierry Mieg and T. Rivera, "Quantum boxes as active probes for photonic microstructures: The pillar microcavity case," Appl. Phys. Lett. 69, 449 (1996).
[CrossRef]

Thierry-Mieg, V.

J. M. Gerard, 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, 1110 (1998).
[CrossRef]

Treussart, F.

V. Sandoghdar, F. Treussart, J. Hare, V. Lefevre-Seguin, J. M. Raimond and S. Haroche, "Very low threshold whispering-gallery-mode microsphere laser," Phys. Rev. A. 54, R1777 (1996).
[CrossRef] [PubMed]

Vahala, K. J.

D. K. Armani, T. J. Kippenberg, S. M. Spillane and K. J. Vahala, "Ultra-high-Q toroid microcavity on a chip," Nature 421, 925 (2003).
[CrossRef] [PubMed]

K. J. Vahala, "Optical microcavities," Nature 424, 839 (2003).
[CrossRef] [PubMed]

van Exter, M. P.

M. P. van Exter, G. Nienhuis and J. P. Woerdman, "Two simple expressions for the spontaneous emission factor beta," Phys. Rev. A. 54, 3553 (1996).
[CrossRef]

Vuckovic, J.

J. Vuckovic, M. Pelton, A. Scherer and Y. Yamamoto, "Optimization of three-dimensional micropost microcavities for cavity quantum electrodynamics," Phys. Rev. A. 66023808 (2002).
[CrossRef]

M. Pelton, C. Santori, J. Vuckovic, B. Y. Zhang, G. S. Solomon, J. Plant and Y. Yamamoto, "Efficient source of single photons: A single quantum dot in a micropost microcavity," Phys. Rev. Lett. 89299602 (2002).
[CrossRef] [PubMed]

Weidner, F.

M. Bayer, T. L. Reinecke, F. Weidner, A. Larionov, A. McDonald and A. Forchel, "Inhibition and enhancement of the spontaneous emission of quantum dots in structured microresonators," Phys. Rev. Lett. 86, 3168 (2001).
[CrossRef] [PubMed]

Woerdman, J. P.

M. P. van Exter, G. Nienhuis and J. P. Woerdman, "Two simple expressions for the spontaneous emission factor beta," Phys. Rev. A. 54, 3553 (1996).
[CrossRef]

Xu, Y.

Yablonovitch, E.

E. Yablonovitch, "Inhibited spontaneous emission in Solid-State Physics and Electronics," Phys. Rev. Lett. 58, 2059 (1987).
[CrossRef] [PubMed]

Yamamoto, Y.

M. Pelton, C. Santori, J. Vuckovic, B. Y. Zhang, G. S. Solomon, J. Plant and Y. Yamamoto, "Efficient source of single photons: A single quantum dot in a micropost microcavity," Phys. Rev. Lett. 89299602 (2002).
[CrossRef] [PubMed]

J. Vuckovic, M. Pelton, A. Scherer and Y. Yamamoto, "Optimization of three-dimensional micropost microcavities for cavity quantum electrodynamics," Phys. Rev. A. 66023808 (2002).
[CrossRef]

Y. Yamamoto, S. Machida and G. Bjork, "Microcavity semiconductor-laser with enhanced spontaneous emission," Phys. Rev. A 44, 657 (1991).
[CrossRef] [PubMed]

Yang, G. M.

M. H. Macdougal, P. D. Dapkus,V. Pudikov, H. M. Zhao and G. M. Yang, "Ultralow threshold current vertical-cavity surface-emitting lasers with alas Oxide-Gaas distributed Bragg reflectors," IEEE Photon. Technol. Lett. 7, 229 (1995).
[CrossRef]

Yariv, A.

Zhang, B. Y.

M. Pelton, C. Santori, J. Vuckovic, B. Y. Zhang, G. S. Solomon, J. Plant and Y. Yamamoto, "Efficient source of single photons: A single quantum dot in a micropost microcavity," Phys. Rev. Lett. 89299602 (2002).
[CrossRef] [PubMed]

Zhang, L. D.

P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. D. Zhang, E. Hu and A. Imamoglu, "A quantum dot single-photon turnstile device," Science 290, 2282 (2000).
[CrossRef] [PubMed]

Zhao, H. M.

M. H. Macdougal, P. D. Dapkus,V. Pudikov, H. M. Zhao and G. M. Yang, "Ultralow threshold current vertical-cavity surface-emitting lasers with alas Oxide-Gaas distributed Bragg reflectors," IEEE Photon. Technol. Lett. 7, 229 (1995).
[CrossRef]

Adv. Chem. Phys. (1)

R. R. Chance, A. Prock and R. Silbey, "Molecular fluorescence and energy transfer near interfaces," Adv. Chem. Phys. 37, 1 (1978).
[CrossRef]

Appl. Phys. Lett. (3)

J. M. Gerard, D. Barrier, J. Y. Marzin, R. Kuszelewicz, L. Manin, E. Costard, V. Thierry Mieg and T. Rivera, "Quantum boxes as active probes for photonic microstructures: The pillar microcavity case," Appl. Phys. Lett. 69, 449 (1996).
[CrossRef]

B. Gayral, J. M. Gerard, A. Lemaitre, C. Dupuis, L. Manin and J. L. Pelouard, "High-Q wet-etched GaAs microdisks containing InAs quantum boxes," Appl. Phys. Lett. 75, 1908 (1999).
[CrossRef]

K. Srinivasan, P. E. Barclay, O. Painter, J. X. Chen, A. Y. Cho and C. Gmachl, "Experimental demonstration of a high quality factor photonic crystal microcavity," Appl. Phys. Lett. 83, 1915 (2003).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

M. H. Macdougal, P. D. Dapkus,V. Pudikov, H. M. Zhao and G. M. Yang, "Ultralow threshold current vertical-cavity surface-emitting lasers with alas Oxide-Gaas distributed Bragg reflectors," IEEE Photon. Technol. Lett. 7, 229 (1995).
[CrossRef]

D. L. Huffaker, L. A. Graham, H. Deng and D. G. Deppe, "Sub-40 mu A continuous-wave lasing in an oxidized vertical-cavity surface-emitting laser with dielectric mirrors," IEEE Photon. Technol. Lett. 8, 974 (1996).
[CrossRef]

J. Chem. Phys. (1)

H. Chew, "Transition rates of Atoms near Spherical Surfaces," J. Chem. Phys. 87, 1355 (1987).
[CrossRef]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. (1)

J. Phys. Chem. (1)

P. Das, and H. Metiu, "Enhancement of molecular fluorescence and photochemistry by Small Metal Particles," J. Phys. Chem. 89, 4680 (1985).
[CrossRef]

Nature (2)

K. J. Vahala, "Optical microcavities," Nature 424, 839 (2003).
[CrossRef] [PubMed]

D. K. Armani, T. J. Kippenberg, S. M. Spillane and K. J. Vahala, "Ultra-high-Q toroid microcavity on a chip," Nature 421, 925 (2003).
[CrossRef] [PubMed]

Opt. Lett. (2)

Phys. Rev. A (1)

Y. Yamamoto, S. Machida and G. Bjork, "Microcavity semiconductor-laser with enhanced spontaneous emission," Phys. Rev. A 44, 657 (1991).
[CrossRef] [PubMed]

Phys. Rev. A. (5)

V. Sandoghdar, F. Treussart, J. Hare, V. Lefevre-Seguin, J. M. Raimond and S. Haroche, "Very low threshold whispering-gallery-mode microsphere laser," Phys. Rev. A. 54, R1777 (1996).
[CrossRef] [PubMed]

J. Vuckovic, M. Pelton, A. Scherer and Y. Yamamoto, "Optimization of three-dimensional micropost microcavities for cavity quantum electrodynamics," Phys. Rev. A. 66023808 (2002).
[CrossRef]

M. P. van Exter, G. Nienhuis and J. P. Woerdman, "Two simple expressions for the spontaneous emission factor beta," Phys. Rev. A. 54, 3553 (1996).
[CrossRef]

K. G. Sullivan and D. G. Hall, "Radiation in spherically symmetrical structures.2. Enhancement and inhibition of Dipole Radiation in a Spherical Bragg Cavity," Phys. Rev. A. 50, 2708 (1994).
[CrossRef] [PubMed]

H. Chew, "Radiation and lifetimes of Atoms inside dielectric particles," Phys. Rev. A. 38, 3410 (1988).
[CrossRef] [PubMed]

Phys. Rev. Lett. (4)

E. Yablonovitch, "Inhibited spontaneous emission in Solid-State Physics and Electronics," Phys. Rev. Lett. 58, 2059 (1987).
[CrossRef] [PubMed]

J. M. Gerard, 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, 1110 (1998).
[CrossRef]

M. Pelton, C. Santori, J. Vuckovic, B. Y. Zhang, G. S. Solomon, J. Plant and Y. Yamamoto, "Efficient source of single photons: A single quantum dot in a micropost microcavity," Phys. Rev. Lett. 89299602 (2002).
[CrossRef] [PubMed]

M. Bayer, T. L. Reinecke, F. Weidner, A. Larionov, A. McDonald and A. Forchel, "Inhibition and enhancement of the spontaneous emission of quantum dots in structured microresonators," Phys. Rev. Lett. 86, 3168 (2001).
[CrossRef] [PubMed]

Science (1)

P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. D. Zhang, E. Hu and A. Imamoglu, "A quantum dot single-photon turnstile device," Science 290, 2282 (2000).
[CrossRef] [PubMed]

Other (3)

W. C. Chew, Waves and Fields in inhomogeneous Media (Van Nostrand Reinhold, New York, 1990).

J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, Inc., New York, 1999).

W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism, (Addison-Weskley, MA, 1956).

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Figures (11)

Fig. 1.
Fig. 1.

SEM image of a sliced Bragg onion resonator.

Fig. 2.
Fig. 2.

(a). Enhancement of the spontaneous emission. (b). Inhibition of the spontaneous emission. (c). Spectral dependence of the normalized frequency shift. Here the dipole is fixed at the center of the Bragg onion sphere. The dotted, dash-dotted, dashed and solid lines correspond to a rising Bragg layer number of NBragg = 3, 4, 5 and 6.

Fig. 3.
Fig. 3.

(a). Radial dependence of the electric field of TE1, TM2 and TE24 eigenmodes. TE1 and TE24 modes only have a transverse electric component while TM2 mode has both the radial and the transverse electric components. (b)–(d). Radial dependence of the normalized radial damping rate b /b 0 (red dashed line) and transverse b ///b 0(blue solid line). Results in (b)–(d) are calculated at the wavelength λ =1.556445μm, 1.559715μm and 1.541255μm, which are the eigen-wavelength of TE1, TM2 and TE24 modes respectively. Ngragg = 6 is used here.

Fig. 4.
Fig. 4.

The radial dependence of the magnetic field of the cladding modes. The fields of both modes are evanescent in the core. The field of TM32 mode decays quickly in the cladding due to the Bragg reflection. While the field of TM44 mode is propagating in the cladding layer and is confined by TIR at the outer surface. Here we use rco = 7μm and NBragg =7.

Fig. 5.
Fig. 5.

The partial averaged spontaneous emission rate as a function of the modal number L (i.e. the lth term in Eq. (22)).

Fig. 6.
Fig. 6.

(a) Spectrum of the onion resonator eigenmodes with modal number l ≤ 24 . (b). Spectral dependence of the ensemble averaged damping rate. rco = 7μn and NBragg = 4 are used in the calculation.

Fig. 7.
Fig. 7.

Enhancement of the ensemble averaged spontaneous emission decaying rate as a function of the cladding layer number. The “plus”, “star” and “circle” are values of the peaks in Fig. 6 corresponding to TE1, TM2 and TE24 resonance modes respectively.

Fig. 8.
Fig. 8.

Suppression of the damping rate for different core radii of 7μm and 4.65 μm respectively. In both (a) and (b), the green circles and the red stars correspond to Bragg layer number NBragg = 6 and 7 respectively.

Fig. 9.
Fig. 9.

(a) The partial averaged damping rate as a function of the modal number l(the lth term in Eq. (22)). (b) and (c). Averaged damping rate for different cladding layer index contrast. We use core radius of 7 μm in (a) and (b), and 4.65μm in (c). The data is calculated at λ = 1.543μm in (a). In all figures, the red “asterisk” is for n 1/n 2 = 2.1/1.5 and NBrag , = 15, the green “circle” is for n 1/n 2 = 3.5/1.5 and NBragg = 6.

Fig. 10.
Fig. 10.

(a) The partial averaged damping rate as a function of the modal number l (the lth term in Eq. (22)). (b) and (c). The averaged damping rate for the Bragg onion resonator immersed in different media. We assume the core area and surrounding area are filled with the light emitting media whose index is n 0. We use core radius of 7 μm in (a) and (b), and 4.65μm in (c). The data is calculated at λ = 1.548μm in (a)-(c). In all figures, we use NBragg = 7, n 0 =1 for the “asterisk” and n 0 =1.33 for the “circle”.

Fig. 11.
Fig. 11.

Normalized radiation power (the left column) and the derived averaged decaying rate b(t) (the right column) as a function of time. The red solid line stands for the exponential decaying, the blue dashed line stands for the non-exponential decaying. The exponential decaying rate is calculated with Eq. (22). We use parameters rco = 7 μm, NBragg = 4 in the calculation and choose two wavelengths: 1.55972μm (eigenwavelength of TM2 mode) and 1.548μm (off resonance).

Equations (66)

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p ̈ + ω 0 2 p = ( q 2 m ) E R ( t ) b 0 p ˙
m q 2 = n 0 ω 0 2 ( 6 π ε 0 b 0 c 0 3 )
p = p 0 exp [ ( + b 2 ) t ]
E R ( t ) = E 0 exp [ ( + b 2 ) t ]
b b 0 = 1 + Re ( E 0 E S )
( ω ω 0 ) b 0 = 1 2 Im ( E 0 E S )
E S = i μ 0 ω 0 2 p 0 k 6 π
E 0 = ε 1 lim r r o { α ̂ . [ D R TE ( r ) + D R TM ( r ) ] }
E 0 E s = 12 π l = 1 m = l l [ ρ l TE h l 1 ( k r co ) h l 2 ( k r co ) ρ l TE h l 1 ( k r co ) α ̂ E lm TE ( r 0 ) 2 + ρ l TM h l 1 ( k r co ) h l 2 ( k r co ) ρ l TM h l 1 ( k r co ) α ̂ E lm TM ( r 0 ) 2 ]
E lm TM = × j l ( kr ) χ lm k
E lm TE = j l ( kr ) L ̂ Y lm θ φ l ( l + 1 ) = j l ( kr ) X lm
[ H E ] = [ h l 1 ( k n r ) X l , m h l 2 ( k r r ) X l , m Z n i k n × h l 1 ( k r r ) X l , m Z n i k n × h l 2 ( k n r ) X l , m ] [ A n B n ] ( TM )
[ E H ] = [ Z n h l 1 ( k n r ) X l , m Z n h l 2 ( k n r ) X l , m i k n × h l 1 ( k n r ) X l , m i k n × h l 2 ( k n r ) X l , m ] [ C n D n ] ( TE )
ρ l TM = B co h l 2 ( k r co ) A co h l 1 ( k r co ) & ρ l TE = D co h l 2 ( k r co ) C co h l 1 ( k r co )
[ A co B co ] = M l TM [ A out B out ] ( TM mode ) & [ C co D co ] = M l TE [ C out D out ] ( TM mode )
B co A co = ( M l TM ) 2,1 ( M l TM ) 1,1
D co C co = ( M l TE ) 2,1 ( M l TE ) 1,1
ρ l TM = ( M l TM ) 2,1 h l 2 ( k r co ) ( M l TM ) 1,1 h l 1 ( k r co ) & ρ l TE = ( M l TE ) 2,1 h l 2 ( k r co ) ( M l TE ) 1,1 h l 1 ( k r co )
b b 0 = 1 + 12 π l = 1 m = l l Re ( ( M l TM , TE ) 2,1 ( M l TM , TE ) 1,1 ( M l TM , TE ) 2,1 ) α . E lm TM , TE ( r 0 ) 2
Δ ω b 0 = 6 π l = 1 m = l l Im ( ( M l TM , TE ) 2,1 ( M l TM , TE ) 1,1 ( M l TM , TE ) 2,1 ) α . E lm TM , TE ( r 0 ) 2
b b 0 = P cav P bulk = 12 π l = 1 m = l l 1 2 1 ( M l TM , TE ) 1,1 ( M l TM , TE ) 2,1 2 α . E lm TM , TE ( r 0 ) 2
b b 0 = 1 + 3 l = 1 Re ( ( M l TM ) 2,1 ( M l TM ) 1,1 ( M l TM ) 2,1 ) l(l+1)(2l+1) j i 2 (kr) 2
b // b 0 = 1 + 3 2 l = 1 ( 2 l + 1 ) Re { ( M l TM ) 2,1 ( M l TM ) 1,1 ( M l TM ) 2,1 [ d ( kr j l ) kr . d ( kr ) ] 2 + ( M l TE ) 2,1 ( M l TE ) 1,1 ( M l TE ) 2,1 j l 2 }
b ( r ) b 0 dir = b r Ω / b 0 d Ω / 4 π
α ̂ . E lm TM , TE 2 / 4 π = ( ( E lm TM , TE ) r 2 + ( E lm TM , TE ) θ 2 + ( E lm TM , TE ) φ 2 / 3
b ( r ) b 0 dir = ( b / b 0 + 2 b / / / b 0 ) 3
b b 0 vol = 0 r co b ( r ) b 0 dir d 3 r ( 4 3 π r co 3 )
b b 0 vol = 1 + 3 r co 3 l = 1 ( 2 l + 1 ) Re { ( M l TE ) 2,1 ( M l TE ) 1,1 ( M l TE ) 2,1 P l + ( M l TE ) 2,1 ( M l TE ) 1,1 ( M l TE ) 2,1 Q l }
Q l = 1 2 ( k r co ) 3 [ j l 2 ( k r co ) j l + 1 ( k r co ) j l 1 ( k r co ) ] & P l = Q l 1 l k r co j l 2 ( k r co )
b b 0 vol = 2 ( k r co ) 3 l = 1 ( 2 l + 1 ) 2 [ 1 ( M l TM ) 1,1 ( M l TM ) 2,1 2 P l + 1 ( M l TE ) 1,1 ( M l TE ) 2,1 2 Q l ]
β = b b 0 vol = Q cav λ 3 ( 4 π 2 V cav eff )
N ( t ) = exp ( b r Ω t ) n r Ω d 3 r d Ω 4 π
N ( t ) = n 0 exp ( b ( r ) b 0 dir t ) d 3 r
P ( t ) = ħυ . dN ( t ) dt = ħυ n 0 exp ( b ( r ) b 0 dir t ) b ( r ) b 0 dir d 3 r
b ( t ) = 1 P ( t ) . ( dP ( t ) dt )
× × D s ( r ) ω 0 2 μ 0 ε D s ( r ) = i μ 0 ω 0 ε J ( r )
[ r D s ( r ) ] TM = i J 0 ω 0 r 0 × 0 × r 0 g r r 0 , r r 0
[ r B s ( r ) ] TE = μ 0 J 0 α ̂ 0 × r 0 g r r 0 , r r 0
g r r 0 = ik l = 0 m = l l j l ( k r < ) h l 1 ( k r > ) Y lm θ φ Y lm * θ 0 φ 0
D R TM ( r ) = r × [ O S TM ( r 0 ) l = 0 m = l l 1 l ( l + 1 ) 2 ( A lm ) TM j l ( kr ) L ̂ Y lm θ φ ]
D R TE ( r ) = ε ω 0 O S TE ( r 0 ) l = 0 m = l l 1 l ( l + 1 ) 2 ( A lm ) TE j l ( kr ) L ̂ Y lm θ φ
O S TM ( r 0 ) = k J 0 ω 0 α ̂ 0 × 0 × r 0
O S TE ( r 0 ) = i μ 0 k J 0 α ̂ 0 × r 0
( A lm ) TE , TM = ρ l TE , TM h l 1 ( k r co ) j l ( k r 0 ) Y lm * θ 0 φ 0 h l 2 ( k r co ) ρ l TE , TM h l 1 ( k r co )
E R E S = 12 π l = 1 m = l l [ ρ l TE h l 1 ( k r co ) h l 2 ( k r co ) ρ l TE h l 1 ( k r co ) α ̂ E lm TE ( r 0 ) 2 + ρ l TM h l 1 ( k r co ) h l 2 ( k r co ) ρ l TM h l 1 ( k r co ) α ̂ E lm TE ( r 0 ) 2 ]
E lm TM = × j l ( kr ) X lm k
E lm TE = j l ( kr ) L ̂ Y lm θ φ = j l ( kr ) X lm l ( l + 1 )
E dip = Z co l , m [ i k co a E l m × h l 1 ( k co r ) X lm + a M l m h l 1 ( k co r ) X lm ]
a E l m = i . k co 2 l ( l + 1 ) Y lm * { d dr [ r j l ( k co r ) ] + i k co ( r J ) j l ( k co r ) } d r
a E l m = i k co 2 l ( l + 1 ) Y lm * ( r × J ) j l ( k co r ) d r
a E l m = i k co J 0 l ( l + 1 ) α ̂ × × r j l ( k co r ) Y lm * θ φ | r = r 0
a M l m = i k co 2 J 0 l ( l + 1 ) α ̂ × × r j l ( k co r ) Y lm * θ φ | r = r 0
E core = E dip + E ref
E ref = Z co l , m [ i k co b E ( l , m ) × j l ( k co r ) X lm + b M ( l , m ) j l ( k cot r ) X lm ]
E rad = Z out l , m [ i k out c E ( l , m ) × h l 1 ( k out r ) X lm + c M ( l , m ) h l 1 ( k out r ) X lm ]
A co l = a E ( l ) + b E ( l ) 2 , B co l = b E ( l ) 2 , A out l = c E ( l ) , B out l = 0
C co l = a M ( l ) + b M ( l ) 2 , D co l = b M ( l ) 2 , C out l = c M ( l ) , D out l = 0
c E = a E ( M l TM ) 1,1 ( M l TM ) 2,1 & b M = a M ( M l TM ) 1,1 ( M l TM ) 2,1
P cav = Z out ( 2 k out 2 ) l , m c E l m 2 + c M l m 2
b b 0 = P cav P bulk = ( ε co ε out ) 3 2 12 π lm 1 2 1 ( M l TM , TE ) 1,1 ( M 2,1 TM , TE ) 2.1 2 α E lm TM , TE ( k co r 0 ) 2
P Z n 2 k n 2 ( A n 2 B n 2 ) ( TM mode ) & P = Z n 2 k n 2 ( C n 2 C n 2 ) ( TE mode )
( A co 2 B co 2 ) = ( A out 2 B out 2 ) & ( C co 2 D co 2 ) = ( C out 2 D out 2 )
( M l ) 1,1 2 ( M l ) 2,1 2 = 1
1 + 2 Re ( ( M l ) 2,1 ( M l ) 1,1 ( M l ) 2,1 ) = 1 ( M l ) 1,1 ( M l ) 2,1 2
12 π lm α E lm TM , TE ( k co r 0 ) 2 2 = 1
b b 0 = P cav P bulk = 1 + 12 π lm Re ( ( M l ) 2,1 TM , TE ( M l ) 1,1 TM , TE ( M l ) 2,1 TM , TE ) α E lm TM , TE ( k co r 0 ) 2

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