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. ThierryMieg, 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]
  7. Y. Yamamoto, S. Machida, and G. Bjork, “Microcavity Semiconductor-Laser with Enhanced Spontaneous Emission,” Phys. Rev. A 44, 657 (1991).
    [Crossref] [PubMed]
  8. 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]
  9. V. Sandoghdar, F. Treussart, J. Hare, V. LefevreSeguin, 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]
  13. 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. 89 (2002).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  16. 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]
  17. R. R. Chance, Prock R., and R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 37, 1 (1978).
    [Crossref]
  18. H. Chew, “Radiation and Lifetimes of Atoms inside Dielectric Particles,” Phys. Rev. A. 38, 3410 (1988).
    [Crossref] [PubMed]
  19. W. Lukosz and R. E. Kunz, “Light-Emission by Magnetic and Electric Dipoles Close to a Plane Interface .1. Total Radiated Power,” J. Opt. Soc. Am. 67, 1607 (1977).
    [Crossref]
  20. W. K. H. Panofsky and M. Phillips, Classical electricity and Magnetism, (Addison-Weskley, MA, 1956).
  21. H. Chew, “Transition Rates of Atoms near Spherical Surfaces,” J. Chem. Phys. 87, 1355 (1987).
    [Crossref]
  22. M. P. vanExter, G. Nienhuis, and J. P. Woerdman, “Two simple expressions for the spontaneous emission factor beta,” Phys. Rev. A. 54, 3553 (1996).
    [Crossref]
  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. 66 (2002).
    [Crossref]
  25. M. Bayer, T. L. Reinecke, F. Weidner, A. Larionov, A. McDonald, and A. Forchel, “Inhibition and nhancement of the spontaneous emission of quantum dots in structured microresonators,” Phys. Rev. Lett. 86, 3168 (2001).
    [Crossref] [PubMed]
  26. P. Das and H. Metiu, “Enhancement of Molecular Fluorescence and Photochemistry by Small Metal Particles,” J. Phys. Chem. 89, 4680 (1985).
    [Crossref]
  27. W. C. Chew, Waves and Fields in inhomogeneous Media (Van Nostrand Reinhold, New York, 1990).
  28. Jackson, Classical Electrodynamics (John Wiley & Sons, Inc., New York, 1999).

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. 89 (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. 66 (2002).
[Crossref]

2001 (1)

M. Bayer, T. L. Reinecke, F. Weidner, A. Larionov, A. McDonald, and A. Forchel, “Inhibition and nhancement 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 (3)

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]

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

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)

V. Sandoghdar, F. Treussart, J. Hare, V. LefevreSeguin, 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. ThierryMieg, and T. Rivera, “Quantum boxes as active probes for photonic microstructures: The pillar microcavity case,” Appl. Phys. Lett. 69, 449 (1996).
[Crossref]

M. P. vanExter, G. Nienhuis, and J. P. Woerdman, “Two simple expressions for the spontaneous emission factor beta,” Phys. Rev. A. 54, 3553 (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)

E. Yablonovitch, “Inhibited Spontaneous Emission in Solid-State Physics and Electronics,” Phys. Rev. Lett. 58, 2059 (1987).
[Crossref] [PubMed]

H. Chew, “Transition Rates of Atoms near Spherical Surfaces,” J. Chem. Phys. 87, 1355 (1987).
[Crossref]

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, Prock R., 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. ThierryMieg, 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 nhancement 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, Prock R., 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]

Chew, W. C.

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

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. ThierryMieg, 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 nhancement of the spontaneous emission of quantum dots in structured microresonators,” Phys. Rev. Lett. 86, 3168 (2001).
[Crossref] [PubMed]

Gayral, B.

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]

Gerard, J. M.

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]

J. M. Gerard, D. Barrier, J. Y. Marzin, R. Kuszelewicz, L. Manin, E. Costard, V. ThierryMieg, 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. LefevreSeguin, 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. LefevreSeguin, 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]

Jackson,

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

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. ThierryMieg, 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 nhancement of the spontaneous emission of quantum dots in structured microresonators,” Phys. Rev. Lett. 86, 3168 (2001).
[Crossref] [PubMed]

LefevreSeguin, V.

V. Sandoghdar, F. Treussart, J. Hare, V. LefevreSeguin, 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. ThierryMieg, 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. ThierryMieg, 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 nhancement 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. vanExter, 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]

Panofsky, W. K. H.

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

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.

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. 89 (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. 66 (2002).
[Crossref]

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]

Phillips, M.

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

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. 89 (2002).
[Crossref] [PubMed]

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]

R., Prock

R. R. Chance, Prock R., and R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 37, 1 (1978).
[Crossref]

Raimond, J. M.

V. Sandoghdar, F. Treussart, J. Hare, V. LefevreSeguin, 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 nhancement 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. ThierryMieg, 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. LefevreSeguin, 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. 89 (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. 66 (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, Prock R., 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. 89 (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]

ThierryMieg, V.

J. M. Gerard, D. Barrier, J. Y. Marzin, R. Kuszelewicz, L. Manin, E. Costard, V. ThierryMieg, 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. LefevreSeguin, 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]

vanExter, M. P.

M. P. vanExter, 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. 66 (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. 89 (2002).
[Crossref] [PubMed]

Weidner, F.

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

Woerdman, J. P.

M. P. vanExter, 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.

J. Vuckovic, M. Pelton, A. Scherer, and Y. Yamamoto, “Optimization of three-dimensional micropost microcavities for cavity quantum electrodynamics,” Phys. Rev. A. 66 (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. 89 (2002).
[Crossref] [PubMed]

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. 89 (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, Prock R., 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. ThierryMieg, 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)

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]

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. LefevreSeguin, J. M. Raimond, and S. Haroche, “Very low threshold whispering-gallery-mode microsphere laser,” Phys. Rev. A. 54, R1777 (1996).
[Crossref] [PubMed]

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]

J. Vuckovic, M. Pelton, A. Scherer, and Y. Yamamoto, “Optimization of three-dimensional micropost microcavities for cavity quantum electrodynamics,” Phys. Rev. A. 66 (2002).
[Crossref]

M. P. vanExter, G. Nienhuis, and J. P. Woerdman, “Two simple expressions for the spontaneous emission factor beta,” Phys. Rev. A. 54, 3553 (1996).
[Crossref]

Phys. Rev. Lett. (4)

E. Yablonovitch, “Inhibited Spontaneous Emission in Solid-State Physics and Electronics,” Phys. Rev. Lett. 58, 2059 (1987).
[Crossref] [PubMed]

M. Bayer, T. L. Reinecke, F. Weidner, A. Larionov, A. McDonald, and A. Forchel, “Inhibition and nhancement of the spontaneous emission of quantum dots in structured microresonators,” Phys. Rev. Lett. 86, 3168 (2001).
[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. 89 (2002).
[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).

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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