Abstract

The lasing spectra and threshold values of material gain for the dipole-type supermodes of an active microdisk concentrically coupled with an external passive microring are investigated. TE polarized modes are treated accurately using the linear electromagnetic formalism of the 2-D lasing eigenvalue problem (LEP) with exact boundary and radiation conditions. The influence of the microring on the lasing frequencies and thresholds is studied numerically, demonstrating threshold reduction opportunities. This is explained through the analysis of the mode near-field patterns and the degree of their overlap with the active region, as suggested by the optical theorem applied to the LEP solutions.

© 2008 Optical Society of America

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References

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  1. A. B. Matsko and V. S. Ilchenko, “Optical resonators with whispering-gallery modes-part 1: basics,” IEEE J. Sel. Top. Quantum Electron. 12, 3-14 (2006).
    [CrossRef]
  2. V. S. Ilchenko and A. B. Matsko, “Optical resonators with whispering-gallery modes-part 2: applications,” IEEE J. Sel. Top. Quantum Electron. 12, 15-32 (2006).
    [CrossRef]
  3. A. I. Nosich, E. I. Smotrova, S. V. Boriskina, T. M. Benson, and P. Sewell, “Trends in microdisk laser research and linear optical modeling,” Opt. Quantum Electron. 39, 1253-1272 (2007).
    [CrossRef]
  4. E. I. Smotrova, A. I. Nosich, T. Benson, and P. Sewell, “Cold-cavity thresholds of microdisks with uniform and non-uniform gain: quasi-3D modeling with accurate 2D analysis,” IEEE J. Sel. Top. Quantum Electron. 11, 1135-1142 (2005).
    [CrossRef]
  5. P. W. Evans and N. Holonyak, Jr., “Room temperature photopump laser operation of native-oxide-defined coupled GaAs-AlAs superlattice microrings,” Appl. Phys. Lett. 69, 2391-2393 (1996).
    [CrossRef]
  6. A. Nakagawa, S. Ishii, and T. Baba, “Photonic molecule laser composed of GaInAsP microdisks,” Appl. Phys. Lett. 86, 041112 (2005).
    [CrossRef]
  7. S. V. Boriskina, “Theoretical prediction of a dramatic Q-factor enhancement and degeneracy removal of WG modes in symmetrical photonic molecules,” Opt. Lett. 31, 338-340 (2006).
    [CrossRef] [PubMed]
  8. E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Optical coupling of WG modes in two identical microdisks and its effect on the lasing spectra and thresholds,” IEEE J. Sel. Top. Quantum Electron. 12, 78-85 (2006).
    [CrossRef]
  9. E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Threshold reduction in a cyclic photonic molecule laser composed of identical microdisks with WG modes,” Opt. Lett. 31, 921-923 (2006).
    [CrossRef] [PubMed]
  10. E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Ultralow lasing thresholds of the pi-type supermodes in cyclic photonic molecules composed of sub-micron disks with monopole and dipole modes,” IEEE Photon. Technol. Lett. 18, 1993-1995 (2006).
    [CrossRef]
  11. A. Jebali, R. F. Mahrt, N. Moll, C. Bauer, G. L. Bona, and W. Bachtold, “Lasing in organic circular grating structures,” J. Appl. Phys. 96, 3043-3049 (2004).
    [CrossRef]
  12. D. Labilloy, H. Benisty, and C. Weisbuch, “High-finesse disk microcavity based on a circular Bragg reflector,” Appl. Phys. Lett. 73, 1314-1316 (1998).
    [CrossRef]
  13. M. Born and E. Wolf, Principles of Optics, 4th ed., (Pergamon, 1968).
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    [CrossRef]
  15. X. Sun and A. Yariv, “Modal properties and modal control in vertically emitting annular Bragg lasers,” Opt. Express 15, 17323-17333 (2007).
    [CrossRef] [PubMed]
  16. J. Scheuer, “Radial Bragg lasers: optimal design for minimal threshold levels and enhanced mode discrimination,” J. Opt. Soc. Am. B 24, 2178-2184 (2007).
    [CrossRef]
  17. H.-J. Moon, G.-W. Park, S.-B. Lee, K. An, and J.-H. Lee, “Laser oscillations of resonance modes in a thin gain-doped ring-type cylindrical microcavity,” Opt. Commun. 235, 401-406 (2004).
    [CrossRef]
  18. Z. Zhang, L. Yang, V. Liu, T. Hong, K. Vahala, and A. Scherer, “Visible submicron microdisk lasers,” Appl. Phys. Lett. 90, 111119 (2007).
    [CrossRef]
  19. G. Hanson and A. Yakovlev, Operator Theory for Electromagnetics (Springer-Verlag, 2002).
  20. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1980).

2007 (5)

2006 (6)

A. B. Matsko and V. S. Ilchenko, “Optical resonators with whispering-gallery modes-part 1: basics,” IEEE J. Sel. Top. Quantum Electron. 12, 3-14 (2006).
[CrossRef]

V. S. Ilchenko and A. B. Matsko, “Optical resonators with whispering-gallery modes-part 2: applications,” IEEE J. Sel. Top. Quantum Electron. 12, 15-32 (2006).
[CrossRef]

S. V. Boriskina, “Theoretical prediction of a dramatic Q-factor enhancement and degeneracy removal of WG modes in symmetrical photonic molecules,” Opt. Lett. 31, 338-340 (2006).
[CrossRef] [PubMed]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Optical coupling of WG modes in two identical microdisks and its effect on the lasing spectra and thresholds,” IEEE J. Sel. Top. Quantum Electron. 12, 78-85 (2006).
[CrossRef]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Threshold reduction in a cyclic photonic molecule laser composed of identical microdisks with WG modes,” Opt. Lett. 31, 921-923 (2006).
[CrossRef] [PubMed]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Ultralow lasing thresholds of the pi-type supermodes in cyclic photonic molecules composed of sub-micron disks with monopole and dipole modes,” IEEE Photon. Technol. Lett. 18, 1993-1995 (2006).
[CrossRef]

2005 (2)

E. I. Smotrova, A. I. Nosich, T. Benson, and P. Sewell, “Cold-cavity thresholds of microdisks with uniform and non-uniform gain: quasi-3D modeling with accurate 2D analysis,” IEEE J. Sel. Top. Quantum Electron. 11, 1135-1142 (2005).
[CrossRef]

A. Nakagawa, S. Ishii, and T. Baba, “Photonic molecule laser composed of GaInAsP microdisks,” Appl. Phys. Lett. 86, 041112 (2005).
[CrossRef]

2004 (2)

H.-J. Moon, G.-W. Park, S.-B. Lee, K. An, and J.-H. Lee, “Laser oscillations of resonance modes in a thin gain-doped ring-type cylindrical microcavity,” Opt. Commun. 235, 401-406 (2004).
[CrossRef]

A. Jebali, R. F. Mahrt, N. Moll, C. Bauer, G. L. Bona, and W. Bachtold, “Lasing in organic circular grating structures,” J. Appl. Phys. 96, 3043-3049 (2004).
[CrossRef]

1998 (1)

D. Labilloy, H. Benisty, and C. Weisbuch, “High-finesse disk microcavity based on a circular Bragg reflector,” Appl. Phys. Lett. 73, 1314-1316 (1998).
[CrossRef]

1996 (1)

P. W. Evans and N. Holonyak, Jr., “Room temperature photopump laser operation of native-oxide-defined coupled GaAs-AlAs superlattice microrings,” Appl. Phys. Lett. 69, 2391-2393 (1996).
[CrossRef]

An, K.

H.-J. Moon, G.-W. Park, S.-B. Lee, K. An, and J.-H. Lee, “Laser oscillations of resonance modes in a thin gain-doped ring-type cylindrical microcavity,” Opt. Commun. 235, 401-406 (2004).
[CrossRef]

Baba, T.

A. Nakagawa, S. Ishii, and T. Baba, “Photonic molecule laser composed of GaInAsP microdisks,” Appl. Phys. Lett. 86, 041112 (2005).
[CrossRef]

Bachtold, W.

Bauer, C.

A. Jebali, R. F. Mahrt, N. Moll, C. Bauer, G. L. Bona, and W. Bachtold, “Lasing in organic circular grating structures,” J. Appl. Phys. 96, 3043-3049 (2004).
[CrossRef]

Benisty, H.

D. Labilloy, H. Benisty, and C. Weisbuch, “High-finesse disk microcavity based on a circular Bragg reflector,” Appl. Phys. Lett. 73, 1314-1316 (1998).
[CrossRef]

Benson, T.

E. I. Smotrova, A. I. Nosich, T. Benson, and P. Sewell, “Cold-cavity thresholds of microdisks with uniform and non-uniform gain: quasi-3D modeling with accurate 2D analysis,” IEEE J. Sel. Top. Quantum Electron. 11, 1135-1142 (2005).
[CrossRef]

Benson, T. M.

A. I. Nosich, E. I. Smotrova, S. V. Boriskina, T. M. Benson, and P. Sewell, “Trends in microdisk laser research and linear optical modeling,” Opt. Quantum Electron. 39, 1253-1272 (2007).
[CrossRef]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Optical coupling of WG modes in two identical microdisks and its effect on the lasing spectra and thresholds,” IEEE J. Sel. Top. Quantum Electron. 12, 78-85 (2006).
[CrossRef]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Threshold reduction in a cyclic photonic molecule laser composed of identical microdisks with WG modes,” Opt. Lett. 31, 921-923 (2006).
[CrossRef] [PubMed]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Ultralow lasing thresholds of the pi-type supermodes in cyclic photonic molecules composed of sub-micron disks with monopole and dipole modes,” IEEE Photon. Technol. Lett. 18, 1993-1995 (2006).
[CrossRef]

Bona, G. L.

A. Jebali, R. F. Mahrt, N. Moll, C. Bauer, G. L. Bona, and W. Bachtold, “Lasing in organic circular grating structures,” J. Appl. Phys. 96, 3043-3049 (2004).
[CrossRef]

Boriskina, S. V.

A. I. Nosich, E. I. Smotrova, S. V. Boriskina, T. M. Benson, and P. Sewell, “Trends in microdisk laser research and linear optical modeling,” Opt. Quantum Electron. 39, 1253-1272 (2007).
[CrossRef]

S. V. Boriskina, “Theoretical prediction of a dramatic Q-factor enhancement and degeneracy removal of WG modes in symmetrical photonic molecules,” Opt. Lett. 31, 338-340 (2006).
[CrossRef] [PubMed]

Born, M.

M. Born and E. Wolf, Principles of Optics, 4th ed., (Pergamon, 1968).

Erni, D.

Evans, P. W.

P. W. Evans and N. Holonyak, Jr., “Room temperature photopump laser operation of native-oxide-defined coupled GaAs-AlAs superlattice microrings,” Appl. Phys. Lett. 69, 2391-2393 (1996).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1980).

Gulde, S.

Hanson, G.

G. Hanson and A. Yakovlev, Operator Theory for Electromagnetics (Springer-Verlag, 2002).

Holonyak, N.

P. W. Evans and N. Holonyak, Jr., “Room temperature photopump laser operation of native-oxide-defined coupled GaAs-AlAs superlattice microrings,” Appl. Phys. Lett. 69, 2391-2393 (1996).
[CrossRef]

Hong, T.

Z. Zhang, L. Yang, V. Liu, T. Hong, K. Vahala, and A. Scherer, “Visible submicron microdisk lasers,” Appl. Phys. Lett. 90, 111119 (2007).
[CrossRef]

Ilchenko, V. S.

A. B. Matsko and V. S. Ilchenko, “Optical resonators with whispering-gallery modes-part 1: basics,” IEEE J. Sel. Top. Quantum Electron. 12, 3-14 (2006).
[CrossRef]

V. S. Ilchenko and A. B. Matsko, “Optical resonators with whispering-gallery modes-part 2: applications,” IEEE J. Sel. Top. Quantum Electron. 12, 15-32 (2006).
[CrossRef]

Ishii, S.

A. Nakagawa, S. Ishii, and T. Baba, “Photonic molecule laser composed of GaInAsP microdisks,” Appl. Phys. Lett. 86, 041112 (2005).
[CrossRef]

Jebali, A.

Labilloy, D.

D. Labilloy, H. Benisty, and C. Weisbuch, “High-finesse disk microcavity based on a circular Bragg reflector,” Appl. Phys. Lett. 73, 1314-1316 (1998).
[CrossRef]

Lee, J.-H.

H.-J. Moon, G.-W. Park, S.-B. Lee, K. An, and J.-H. Lee, “Laser oscillations of resonance modes in a thin gain-doped ring-type cylindrical microcavity,” Opt. Commun. 235, 401-406 (2004).
[CrossRef]

Lee, S.-B.

H.-J. Moon, G.-W. Park, S.-B. Lee, K. An, and J.-H. Lee, “Laser oscillations of resonance modes in a thin gain-doped ring-type cylindrical microcavity,” Opt. Commun. 235, 401-406 (2004).
[CrossRef]

Liu, V.

Z. Zhang, L. Yang, V. Liu, T. Hong, K. Vahala, and A. Scherer, “Visible submicron microdisk lasers,” Appl. Phys. Lett. 90, 111119 (2007).
[CrossRef]

Mahrt, R. F.

Matsko, A. B.

V. S. Ilchenko and A. B. Matsko, “Optical resonators with whispering-gallery modes-part 2: applications,” IEEE J. Sel. Top. Quantum Electron. 12, 15-32 (2006).
[CrossRef]

A. B. Matsko and V. S. Ilchenko, “Optical resonators with whispering-gallery modes-part 1: basics,” IEEE J. Sel. Top. Quantum Electron. 12, 3-14 (2006).
[CrossRef]

Moll, N.

A. Jebali, R. F. Mahrt, N. Moll, C. Bauer, G. L. Bona, and W. Bachtold, “Lasing in organic circular grating structures,” J. Appl. Phys. 96, 3043-3049 (2004).
[CrossRef]

Moon, H.-J.

H.-J. Moon, G.-W. Park, S.-B. Lee, K. An, and J.-H. Lee, “Laser oscillations of resonance modes in a thin gain-doped ring-type cylindrical microcavity,” Opt. Commun. 235, 401-406 (2004).
[CrossRef]

Nakagawa, A.

A. Nakagawa, S. Ishii, and T. Baba, “Photonic molecule laser composed of GaInAsP microdisks,” Appl. Phys. Lett. 86, 041112 (2005).
[CrossRef]

Nosich, A. I.

A. I. Nosich, E. I. Smotrova, S. V. Boriskina, T. M. Benson, and P. Sewell, “Trends in microdisk laser research and linear optical modeling,” Opt. Quantum Electron. 39, 1253-1272 (2007).
[CrossRef]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Optical coupling of WG modes in two identical microdisks and its effect on the lasing spectra and thresholds,” IEEE J. Sel. Top. Quantum Electron. 12, 78-85 (2006).
[CrossRef]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Threshold reduction in a cyclic photonic molecule laser composed of identical microdisks with WG modes,” Opt. Lett. 31, 921-923 (2006).
[CrossRef] [PubMed]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Ultralow lasing thresholds of the pi-type supermodes in cyclic photonic molecules composed of sub-micron disks with monopole and dipole modes,” IEEE Photon. Technol. Lett. 18, 1993-1995 (2006).
[CrossRef]

E. I. Smotrova, A. I. Nosich, T. Benson, and P. Sewell, “Cold-cavity thresholds of microdisks with uniform and non-uniform gain: quasi-3D modeling with accurate 2D analysis,” IEEE J. Sel. Top. Quantum Electron. 11, 1135-1142 (2005).
[CrossRef]

Park, G.-W.

H.-J. Moon, G.-W. Park, S.-B. Lee, K. An, and J.-H. Lee, “Laser oscillations of resonance modes in a thin gain-doped ring-type cylindrical microcavity,” Opt. Commun. 235, 401-406 (2004).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1980).

Scherer, A.

Z. Zhang, L. Yang, V. Liu, T. Hong, K. Vahala, and A. Scherer, “Visible submicron microdisk lasers,” Appl. Phys. Lett. 90, 111119 (2007).
[CrossRef]

Scheuer, J.

Sewell, P.

A. I. Nosich, E. I. Smotrova, S. V. Boriskina, T. M. Benson, and P. Sewell, “Trends in microdisk laser research and linear optical modeling,” Opt. Quantum Electron. 39, 1253-1272 (2007).
[CrossRef]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Optical coupling of WG modes in two identical microdisks and its effect on the lasing spectra and thresholds,” IEEE J. Sel. Top. Quantum Electron. 12, 78-85 (2006).
[CrossRef]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Ultralow lasing thresholds of the pi-type supermodes in cyclic photonic molecules composed of sub-micron disks with monopole and dipole modes,” IEEE Photon. Technol. Lett. 18, 1993-1995 (2006).
[CrossRef]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Threshold reduction in a cyclic photonic molecule laser composed of identical microdisks with WG modes,” Opt. Lett. 31, 921-923 (2006).
[CrossRef] [PubMed]

E. I. Smotrova, A. I. Nosich, T. Benson, and P. Sewell, “Cold-cavity thresholds of microdisks with uniform and non-uniform gain: quasi-3D modeling with accurate 2D analysis,” IEEE J. Sel. Top. Quantum Electron. 11, 1135-1142 (2005).
[CrossRef]

Smotrova, E. I.

A. I. Nosich, E. I. Smotrova, S. V. Boriskina, T. M. Benson, and P. Sewell, “Trends in microdisk laser research and linear optical modeling,” Opt. Quantum Electron. 39, 1253-1272 (2007).
[CrossRef]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Optical coupling of WG modes in two identical microdisks and its effect on the lasing spectra and thresholds,” IEEE J. Sel. Top. Quantum Electron. 12, 78-85 (2006).
[CrossRef]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Ultralow lasing thresholds of the pi-type supermodes in cyclic photonic molecules composed of sub-micron disks with monopole and dipole modes,” IEEE Photon. Technol. Lett. 18, 1993-1995 (2006).
[CrossRef]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Threshold reduction in a cyclic photonic molecule laser composed of identical microdisks with WG modes,” Opt. Lett. 31, 921-923 (2006).
[CrossRef] [PubMed]

E. I. Smotrova, A. I. Nosich, T. Benson, and P. Sewell, “Cold-cavity thresholds of microdisks with uniform and non-uniform gain: quasi-3D modeling with accurate 2D analysis,” IEEE J. Sel. Top. Quantum Electron. 11, 1135-1142 (2005).
[CrossRef]

Sun, X.

Vahala, K.

Z. Zhang, L. Yang, V. Liu, T. Hong, K. Vahala, and A. Scherer, “Visible submicron microdisk lasers,” Appl. Phys. Lett. 90, 111119 (2007).
[CrossRef]

Weisbuch, C.

D. Labilloy, H. Benisty, and C. Weisbuch, “High-finesse disk microcavity based on a circular Bragg reflector,” Appl. Phys. Lett. 73, 1314-1316 (1998).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 4th ed., (Pergamon, 1968).

Yakovlev, A.

G. Hanson and A. Yakovlev, Operator Theory for Electromagnetics (Springer-Verlag, 2002).

Yang, L.

Z. Zhang, L. Yang, V. Liu, T. Hong, K. Vahala, and A. Scherer, “Visible submicron microdisk lasers,” Appl. Phys. Lett. 90, 111119 (2007).
[CrossRef]

Yariv, A.

Zhang, Z.

Z. Zhang, L. Yang, V. Liu, T. Hong, K. Vahala, and A. Scherer, “Visible submicron microdisk lasers,” Appl. Phys. Lett. 90, 111119 (2007).
[CrossRef]

Appl. Phys. Lett. (4)

P. W. Evans and N. Holonyak, Jr., “Room temperature photopump laser operation of native-oxide-defined coupled GaAs-AlAs superlattice microrings,” Appl. Phys. Lett. 69, 2391-2393 (1996).
[CrossRef]

A. Nakagawa, S. Ishii, and T. Baba, “Photonic molecule laser composed of GaInAsP microdisks,” Appl. Phys. Lett. 86, 041112 (2005).
[CrossRef]

D. Labilloy, H. Benisty, and C. Weisbuch, “High-finesse disk microcavity based on a circular Bragg reflector,” Appl. Phys. Lett. 73, 1314-1316 (1998).
[CrossRef]

Z. Zhang, L. Yang, V. Liu, T. Hong, K. Vahala, and A. Scherer, “Visible submicron microdisk lasers,” Appl. Phys. Lett. 90, 111119 (2007).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (4)

A. B. Matsko and V. S. Ilchenko, “Optical resonators with whispering-gallery modes-part 1: basics,” IEEE J. Sel. Top. Quantum Electron. 12, 3-14 (2006).
[CrossRef]

V. S. Ilchenko and A. B. Matsko, “Optical resonators with whispering-gallery modes-part 2: applications,” IEEE J. Sel. Top. Quantum Electron. 12, 15-32 (2006).
[CrossRef]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Optical coupling of WG modes in two identical microdisks and its effect on the lasing spectra and thresholds,” IEEE J. Sel. Top. Quantum Electron. 12, 78-85 (2006).
[CrossRef]

E. I. Smotrova, A. I. Nosich, T. Benson, and P. Sewell, “Cold-cavity thresholds of microdisks with uniform and non-uniform gain: quasi-3D modeling with accurate 2D analysis,” IEEE J. Sel. Top. Quantum Electron. 11, 1135-1142 (2005).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Ultralow lasing thresholds of the pi-type supermodes in cyclic photonic molecules composed of sub-micron disks with monopole and dipole modes,” IEEE Photon. Technol. Lett. 18, 1993-1995 (2006).
[CrossRef]

J. Appl. Phys. (1)

A. Jebali, R. F. Mahrt, N. Moll, C. Bauer, G. L. Bona, and W. Bachtold, “Lasing in organic circular grating structures,” J. Appl. Phys. 96, 3043-3049 (2004).
[CrossRef]

J. Opt. Soc. Am. B (2)

Opt. Commun. (1)

H.-J. Moon, G.-W. Park, S.-B. Lee, K. An, and J.-H. Lee, “Laser oscillations of resonance modes in a thin gain-doped ring-type cylindrical microcavity,” Opt. Commun. 235, 401-406 (2004).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Opt. Quantum Electron. (1)

A. I. Nosich, E. I. Smotrova, S. V. Boriskina, T. M. Benson, and P. Sewell, “Trends in microdisk laser research and linear optical modeling,” Opt. Quantum Electron. 39, 1253-1272 (2007).
[CrossRef]

Other (3)

M. Born and E. Wolf, Principles of Optics, 4th ed., (Pergamon, 1968).

G. Hanson and A. Yakovlev, Operator Theory for Electromagnetics (Springer-Verlag, 2002).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1980).

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Figures (7)

Fig. 1
Fig. 1

In-plane geometry of active circular microcavity concentrically coupled with a passive ring.

Fig. 2
Fig. 2

Normalized lasing frequencies (a) and thresholds (b) of the H 1 , n , q , p supermodes in the active disk loaded with a passive external dielectric ring versus the normalized air-gap separation between the disk and the ring. Ring thickness is w = 0.2 a , refractive indices are α a = α r = 2.63 , α g = 1 .

Fig. 3
Fig. 3

Near-field H z ( r , φ ) patterns for the supermodes H 1 , n , q , p at the points marked in Fig. 2; ring thickness is w = 0.2 a , refractive indices are α a = α r = 2.63 , α g = 1 .  

Fig. 4
Fig. 4

Eigenvalue migration on the plane ( κ , γ ) when the air-gap separation parameter d a varies. Ring thickness is w = 0.2 a , refractive indices are α a = α r = 2.63 , α g = 1 . In-circle numbers correspond to the mode notations of Fig. 2. The open symbols correspond to eigenvalues for supermodes H 1 , n , q , p at the initial value of the parameter d a = 0.01 . The filled color symbols on the curve correspond to eigenvalues of supermodes H 1 , n , q , p at d a = 0.5 , 1.0, 1.5 (from right to left). The off-curve symbols correspond to the eigenvalues for the all-active disk of radius a.

Fig. 5
Fig. 5

Normalized lasing frequencies (a) and thresholds (b) of the H 1 , n , q , p modes in the active disk placed inside a passive ring versus the normalized ring thickness. Air-gap separation is d = 0.5 a , and refractive indices are α a = α r = 2.63 , α g = 1 .

Fig. 6
Fig. 6

Near-field H z ( r , φ ) patterns for the supermodes H 1 , n , q , p at the minima of threshold γ corresponding to the points marked by B to K in Fig. 5. Conversely, case A corresponds to the high threshold. Air-gap separation is d = 0.5 a , refractive indices are α a = α r = 2.63 , α g = 1 .

Fig. 7
Fig. 7

Active-region (a), passive air-gap (b), and passive ring (c) mode overlap factors for the H 1 , n , q , p supermodes in the active disk loaded with a passive dielectric ring versus the normalized air-gap separation between the disk and the ring. Other parameters are the same as in Fig. 2.

Equations (16)

Equations on this page are rendered with MathJax. Learn more.

U ( r , φ ) = m = 0 [ A m s J m ( k v s r ) + B m s H m ( k v s r ) ] cos m φ ,
A m s + 1 J m ( κ v s + 1 ρ s ) + B m s + 1 H m ( 1 ) ( κ v s + 1 ρ s ) = A m s J m ( κ v s ρ s ) + B m s H m ( 1 ) ( κ v s ρ s ) ,
[ A m s + 1 J m ( κ v s + 1 ρ s ) + B m s + 1 H m ( 1 ) ( κ v s + 1 ρ s ) ] v s + 1 = [ A m s J m ( κ v s ρ s ) + B m s H m ( 1 ) ( κ v s ρ s ) ] v s ,
Det [ C ( m ) ( κ , γ ) ] = 0 , m = 0 , 1 , 2 , .
( 1 2 ) S E × H * d s = ( i 2 ) V ( k * ε * Z 0 1 E 2 k μ Z 0 H 2 ) d v ,
( Z 0 2 ) Re S E j × H j * d s = γ j k j α a V a E j ( R , k j , γ j ) 2 d v ,
W j ( k j , γ j ) = V α a 2 E j ( R , k j , γ j ) 2 d v = V a α a 2 E j 2 d v + V g α g 2 E j 2 d v + V r α r 2 E j 2 d v ,
Γ j ( f ) = W j ( f ) W j , W j ( f ) ( k j , γ j ) = V f α f 2 E j ( R , k j , γ j ) 2 d v ,
f = a , g , r .
γ j = ( Z 0 2 ) Re S E j × H j * d s k j α a V a E j ( R , k j , γ j ) 2 d v = α a P j Γ j ( a ) k j W j ,
P j = ( Z 0 2 ) Re S E j × H j * d s
γ j = α a Γ j ( a ) Q ̃ j .
W s = V s α 2 E 2 ρ d ρ d φ = π α 2 Z 0 2 2 ν s ν s * a s a s + 1 [ A m s 2 ( J m 1 J m 1 * + J m + 1 J m + 1 * ) + A m s B m s * ( J m 1 H m 1 ( 1 ) * + J m + 1 H m + 1 ( 1 ) * ) + A m s * B m s ( J m 1 * H m 1 ( 1 ) + J m + 1 * H m + 1 ( 1 ) ) + B m s 2 ( H m 1 ( 1 ) H m 1 ( 1 ) * + H m + 1 ( 1 ) H m + 1 ( 1 ) * ) ] ρ d ρ .
Z m ( κ ν ρ ) T m ( κ ν * ρ ) ρ d ρ = ρ κ [ ν * Z m ( κ ν ρ ) T m 1 * ( κ ν ρ ) ν j Z m 1 * ( κ ν ρ ) T m ( κ ν ρ ) ν 2 ν * 2 ] ,
Z m ( κ α ρ ) T m ( κ α ρ ) ρ d ρ = ρ 2 4 [ 2 Z m ( κ α ρ ) T m ( κ α ρ ) Z m 1 ( κ α ρ ) T m + 1 ( κ α ρ ) Z m + 1 ( κ α ρ ) T m 1 ( κ α ρ ) ] ,
W m n ( a ) = Z 0 2 A 0 2 a 2 2 γ m n κ m n Im [ ν m n J m ( κ ν m n * ) J m 1 ( κ ν m n ) ] .

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