Abstract

We propose and analyze a new type of a resonator in an annular geometry that is based on a single defect surrounded by radial Bragg reflectors on both sides. We show that the conditions for efficient mode confinement are different from those of the conventional Bragg waveguiding in a rectangular geometry. A simple and intuitive approach to the design of optimal radial Bragg reflectors is proposed and employed, yielding chirped gratings. Small bending radii and strong control over the resonator dispersion are possible by the Bragg confinement. A design compromise between large free-spectral-range requirements and fabrication tolerances is suggested.

© 2003 Optical Society of America

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References

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  1. A. Yariv, “Critical coupling and its control in optical waveguide-ring resonator systems,” IEEE Photon. Technol. Lett. 14, 483–485 (2002).
    [CrossRef]
  2. B. E. Little, “Ultracompact Si-SiO2 microring resonator optical dropping filter,” Opt. Lett. 23, 1570–1572 (1998).
    [CrossRef]
  3. C. K. Madsen and J. H. Zhao, Optical Filter Design and Analysis: a Signal-Processing Approach (Wiley, New York, 1999).
  4. M. Toda, “Single-mode behavior of a circular grating for potential disk-shaped DFB lasers,” IEEE J. Quantum Electron. 26, 473–481 (1990).
    [CrossRef]
  5. X. H. Zheng and S. Lacroix, “Mode coupling in circular-cylindrical system and its application to fingerprint resonators,” J. Lightwave Technol. 8, 1509–1516 (1990).
    [CrossRef]
  6. T. Erdogan and D. G. Hall, “Circularly symmetric distributed feedback laser: an analysis,” J. Appl. Phys. 68, 1435–1444 (1990).
    [CrossRef]
  7. T. Erdogan and D. G. Hall, “Circularly symmetric distributed feedback laser: coupled mode treatment of TE vector fields,” IEEE J. Quantum Electron. 28, 612–623 (1992).
    [CrossRef]
  8. M. A. Kaliteevski, R. A. Abram, V. V. Nikolaev, and G. S. Sokolovski, “Bragg reflectors for cylindrical waves,” J. Mod. Opt. 46, 875–890 (1999).
    [CrossRef]
  9. C. Wu, M. Svilans, M. Fallahi, T. Makino, J. Glinski, C. Maritan, and C. Blaauw, “Optically pumped surface-emitting DFB GaInAsP/InP lasers with circular grating,” Electron. Lett. 27, 1819–1821 (1991).
    [CrossRef]
  10. D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, C. J. M. Smith, R. Hourdré, and U. Oesterle, “High-finesse disk microcavity based on a circular Bragg reflector,” Appl. Phys. Lett. 73, 1314–1316 (1998).
    [CrossRef]
  11. A. Shaw, B. Roycroft, J. Hegarty, D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, C. J. M. Smith, R. Stanely, R. Hourdré, and U. Oesterle, “Lasing properties of disk microcavity based on circular Bragg reflector,” Appl. Phys. Lett. 75, 3051–3053 (1999).
    [CrossRef]
  12. D. Ochoa, R. Hourdré, M. Ilegems, H. Benisty, T. F. Krauss, and C. J. M. Smith, “Diffraction of cylindrical Bragg reflectors surrounding an in-place semiconductor microcavity,” Phys. Rev. B 61, 4806–4812 (2000).
    [CrossRef]
  13. J. Scheuer and A. Yariv, “Two-dimensional optical ring resonators based on radial Bragg resonance,” Opt. Lett. 28, 1528–1530 (2003).
    [CrossRef] [PubMed]
  14. S. Kim, H. Ryu, H. Park, G. Kim, Y. Choi, Y. Lee, and J. Kim, “Two-dimensional photonic crystal hexagonal waveguide ring laser,” Appl. Phys. Lett. 81, 2499–2501 (2002).
    [CrossRef]
  15. See for example, A. Yariv, Optical Electronics in Modern Communications, 5th ed. (Oxford University, New York, 1997).
  16. P. Yeh, A. Yariv, and E. Marom, “Theory of Bragg fiber,” J. Opt. Soc. Am. 68, 1196–1201 (1978).
    [CrossRef]
  17. S. G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T. D. Engeness, M. Soljačić, S. A. Jacobs, J. D. Joannopoulos, and Y. Fink, “Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers,” Opt. Express 9, 748–779 (2001).
    [CrossRef] [PubMed]
  18. M. Heiblum and J. H. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. QE-11, 75–83 (1975).
    [CrossRef]
  19. L. Djaloshinski and M. Orenstein, “Disk and ring microcavity lasers and their concentric coupling,” IEEE J. Quantum Electron. 35, 737–744 (1999).
    [CrossRef]
  20. G. Meltz, W. W. Morey, and W. H. Glenn, “Formation of Bragg gratings in optical fibers by a transverse holographic method,” Opt. Lett. 14, 823–825 (1989).
    [CrossRef] [PubMed]

2003 (1)

2002 (2)

A. Yariv, “Critical coupling and its control in optical waveguide-ring resonator systems,” IEEE Photon. Technol. Lett. 14, 483–485 (2002).
[CrossRef]

S. Kim, H. Ryu, H. Park, G. Kim, Y. Choi, Y. Lee, and J. Kim, “Two-dimensional photonic crystal hexagonal waveguide ring laser,” Appl. Phys. Lett. 81, 2499–2501 (2002).
[CrossRef]

2001 (1)

2000 (1)

D. Ochoa, R. Hourdré, M. Ilegems, H. Benisty, T. F. Krauss, and C. J. M. Smith, “Diffraction of cylindrical Bragg reflectors surrounding an in-place semiconductor microcavity,” Phys. Rev. B 61, 4806–4812 (2000).
[CrossRef]

1999 (3)

A. Shaw, B. Roycroft, J. Hegarty, D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, C. J. M. Smith, R. Stanely, R. Hourdré, and U. Oesterle, “Lasing properties of disk microcavity based on circular Bragg reflector,” Appl. Phys. Lett. 75, 3051–3053 (1999).
[CrossRef]

M. A. Kaliteevski, R. A. Abram, V. V. Nikolaev, and G. S. Sokolovski, “Bragg reflectors for cylindrical waves,” J. Mod. Opt. 46, 875–890 (1999).
[CrossRef]

L. Djaloshinski and M. Orenstein, “Disk and ring microcavity lasers and their concentric coupling,” IEEE J. Quantum Electron. 35, 737–744 (1999).
[CrossRef]

1998 (2)

B. E. Little, “Ultracompact Si-SiO2 microring resonator optical dropping filter,” Opt. Lett. 23, 1570–1572 (1998).
[CrossRef]

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, C. J. M. Smith, R. Hourdré, and U. Oesterle, “High-finesse disk microcavity based on a circular Bragg reflector,” Appl. Phys. Lett. 73, 1314–1316 (1998).
[CrossRef]

1992 (1)

T. Erdogan and D. G. Hall, “Circularly symmetric distributed feedback laser: coupled mode treatment of TE vector fields,” IEEE J. Quantum Electron. 28, 612–623 (1992).
[CrossRef]

1991 (1)

C. Wu, M. Svilans, M. Fallahi, T. Makino, J. Glinski, C. Maritan, and C. Blaauw, “Optically pumped surface-emitting DFB GaInAsP/InP lasers with circular grating,” Electron. Lett. 27, 1819–1821 (1991).
[CrossRef]

1990 (3)

M. Toda, “Single-mode behavior of a circular grating for potential disk-shaped DFB lasers,” IEEE J. Quantum Electron. 26, 473–481 (1990).
[CrossRef]

X. H. Zheng and S. Lacroix, “Mode coupling in circular-cylindrical system and its application to fingerprint resonators,” J. Lightwave Technol. 8, 1509–1516 (1990).
[CrossRef]

T. Erdogan and D. G. Hall, “Circularly symmetric distributed feedback laser: an analysis,” J. Appl. Phys. 68, 1435–1444 (1990).
[CrossRef]

1989 (1)

1978 (1)

1975 (1)

M. Heiblum and J. H. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. QE-11, 75–83 (1975).
[CrossRef]

Abram, R. A.

M. A. Kaliteevski, R. A. Abram, V. V. Nikolaev, and G. S. Sokolovski, “Bragg reflectors for cylindrical waves,” J. Mod. Opt. 46, 875–890 (1999).
[CrossRef]

Benisty, H.

D. Ochoa, R. Hourdré, M. Ilegems, H. Benisty, T. F. Krauss, and C. J. M. Smith, “Diffraction of cylindrical Bragg reflectors surrounding an in-place semiconductor microcavity,” Phys. Rev. B 61, 4806–4812 (2000).
[CrossRef]

A. Shaw, B. Roycroft, J. Hegarty, D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, C. J. M. Smith, R. Stanely, R. Hourdré, and U. Oesterle, “Lasing properties of disk microcavity based on circular Bragg reflector,” Appl. Phys. Lett. 75, 3051–3053 (1999).
[CrossRef]

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, C. J. M. Smith, R. Hourdré, and U. Oesterle, “High-finesse disk microcavity based on a circular Bragg reflector,” Appl. Phys. Lett. 73, 1314–1316 (1998).
[CrossRef]

Blaauw, C.

C. Wu, M. Svilans, M. Fallahi, T. Makino, J. Glinski, C. Maritan, and C. Blaauw, “Optically pumped surface-emitting DFB GaInAsP/InP lasers with circular grating,” Electron. Lett. 27, 1819–1821 (1991).
[CrossRef]

Choi, Y.

S. Kim, H. Ryu, H. Park, G. Kim, Y. Choi, Y. Lee, and J. Kim, “Two-dimensional photonic crystal hexagonal waveguide ring laser,” Appl. Phys. Lett. 81, 2499–2501 (2002).
[CrossRef]

Djaloshinski, L.

L. Djaloshinski and M. Orenstein, “Disk and ring microcavity lasers and their concentric coupling,” IEEE J. Quantum Electron. 35, 737–744 (1999).
[CrossRef]

Engeness, T. D.

Erdogan, T.

T. Erdogan and D. G. Hall, “Circularly symmetric distributed feedback laser: coupled mode treatment of TE vector fields,” IEEE J. Quantum Electron. 28, 612–623 (1992).
[CrossRef]

T. Erdogan and D. G. Hall, “Circularly symmetric distributed feedback laser: an analysis,” J. Appl. Phys. 68, 1435–1444 (1990).
[CrossRef]

Fallahi, M.

C. Wu, M. Svilans, M. Fallahi, T. Makino, J. Glinski, C. Maritan, and C. Blaauw, “Optically pumped surface-emitting DFB GaInAsP/InP lasers with circular grating,” Electron. Lett. 27, 1819–1821 (1991).
[CrossRef]

Fink, Y.

Glenn, W. H.

Glinski, J.

C. Wu, M. Svilans, M. Fallahi, T. Makino, J. Glinski, C. Maritan, and C. Blaauw, “Optically pumped surface-emitting DFB GaInAsP/InP lasers with circular grating,” Electron. Lett. 27, 1819–1821 (1991).
[CrossRef]

Hall, D. G.

T. Erdogan and D. G. Hall, “Circularly symmetric distributed feedback laser: coupled mode treatment of TE vector fields,” IEEE J. Quantum Electron. 28, 612–623 (1992).
[CrossRef]

T. Erdogan and D. G. Hall, “Circularly symmetric distributed feedback laser: an analysis,” J. Appl. Phys. 68, 1435–1444 (1990).
[CrossRef]

Harris, J. H.

M. Heiblum and J. H. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. QE-11, 75–83 (1975).
[CrossRef]

Hegarty, J.

A. Shaw, B. Roycroft, J. Hegarty, D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, C. J. M. Smith, R. Stanely, R. Hourdré, and U. Oesterle, “Lasing properties of disk microcavity based on circular Bragg reflector,” Appl. Phys. Lett. 75, 3051–3053 (1999).
[CrossRef]

Heiblum, M.

M. Heiblum and J. H. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. QE-11, 75–83 (1975).
[CrossRef]

Hourdré, R.

D. Ochoa, R. Hourdré, M. Ilegems, H. Benisty, T. F. Krauss, and C. J. M. Smith, “Diffraction of cylindrical Bragg reflectors surrounding an in-place semiconductor microcavity,” Phys. Rev. B 61, 4806–4812 (2000).
[CrossRef]

A. Shaw, B. Roycroft, J. Hegarty, D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, C. J. M. Smith, R. Stanely, R. Hourdré, and U. Oesterle, “Lasing properties of disk microcavity based on circular Bragg reflector,” Appl. Phys. Lett. 75, 3051–3053 (1999).
[CrossRef]

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, C. J. M. Smith, R. Hourdré, and U. Oesterle, “High-finesse disk microcavity based on a circular Bragg reflector,” Appl. Phys. Lett. 73, 1314–1316 (1998).
[CrossRef]

Ibanescu, M.

Ilegems, M.

D. Ochoa, R. Hourdré, M. Ilegems, H. Benisty, T. F. Krauss, and C. J. M. Smith, “Diffraction of cylindrical Bragg reflectors surrounding an in-place semiconductor microcavity,” Phys. Rev. B 61, 4806–4812 (2000).
[CrossRef]

Jacobs, S. A.

Joannopoulos, J. D.

Johnson, S. G.

Kaliteevski, M. A.

M. A. Kaliteevski, R. A. Abram, V. V. Nikolaev, and G. S. Sokolovski, “Bragg reflectors for cylindrical waves,” J. Mod. Opt. 46, 875–890 (1999).
[CrossRef]

Kim, G.

S. Kim, H. Ryu, H. Park, G. Kim, Y. Choi, Y. Lee, and J. Kim, “Two-dimensional photonic crystal hexagonal waveguide ring laser,” Appl. Phys. Lett. 81, 2499–2501 (2002).
[CrossRef]

Kim, J.

S. Kim, H. Ryu, H. Park, G. Kim, Y. Choi, Y. Lee, and J. Kim, “Two-dimensional photonic crystal hexagonal waveguide ring laser,” Appl. Phys. Lett. 81, 2499–2501 (2002).
[CrossRef]

Kim, S.

S. Kim, H. Ryu, H. Park, G. Kim, Y. Choi, Y. Lee, and J. Kim, “Two-dimensional photonic crystal hexagonal waveguide ring laser,” Appl. Phys. Lett. 81, 2499–2501 (2002).
[CrossRef]

Krauss, T. F.

D. Ochoa, R. Hourdré, M. Ilegems, H. Benisty, T. F. Krauss, and C. J. M. Smith, “Diffraction of cylindrical Bragg reflectors surrounding an in-place semiconductor microcavity,” Phys. Rev. B 61, 4806–4812 (2000).
[CrossRef]

A. Shaw, B. Roycroft, J. Hegarty, D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, C. J. M. Smith, R. Stanely, R. Hourdré, and U. Oesterle, “Lasing properties of disk microcavity based on circular Bragg reflector,” Appl. Phys. Lett. 75, 3051–3053 (1999).
[CrossRef]

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, C. J. M. Smith, R. Hourdré, and U. Oesterle, “High-finesse disk microcavity based on a circular Bragg reflector,” Appl. Phys. Lett. 73, 1314–1316 (1998).
[CrossRef]

Labilloy, D.

A. Shaw, B. Roycroft, J. Hegarty, D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, C. J. M. Smith, R. Stanely, R. Hourdré, and U. Oesterle, “Lasing properties of disk microcavity based on circular Bragg reflector,” Appl. Phys. Lett. 75, 3051–3053 (1999).
[CrossRef]

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, C. J. M. Smith, R. Hourdré, and U. Oesterle, “High-finesse disk microcavity based on a circular Bragg reflector,” Appl. Phys. Lett. 73, 1314–1316 (1998).
[CrossRef]

Lacroix, S.

X. H. Zheng and S. Lacroix, “Mode coupling in circular-cylindrical system and its application to fingerprint resonators,” J. Lightwave Technol. 8, 1509–1516 (1990).
[CrossRef]

Lee, Y.

S. Kim, H. Ryu, H. Park, G. Kim, Y. Choi, Y. Lee, and J. Kim, “Two-dimensional photonic crystal hexagonal waveguide ring laser,” Appl. Phys. Lett. 81, 2499–2501 (2002).
[CrossRef]

Little, B. E.

Makino, T.

C. Wu, M. Svilans, M. Fallahi, T. Makino, J. Glinski, C. Maritan, and C. Blaauw, “Optically pumped surface-emitting DFB GaInAsP/InP lasers with circular grating,” Electron. Lett. 27, 1819–1821 (1991).
[CrossRef]

Maritan, C.

C. Wu, M. Svilans, M. Fallahi, T. Makino, J. Glinski, C. Maritan, and C. Blaauw, “Optically pumped surface-emitting DFB GaInAsP/InP lasers with circular grating,” Electron. Lett. 27, 1819–1821 (1991).
[CrossRef]

Marom, E.

Meltz, G.

Morey, W. W.

Nikolaev, V. V.

M. A. Kaliteevski, R. A. Abram, V. V. Nikolaev, and G. S. Sokolovski, “Bragg reflectors for cylindrical waves,” J. Mod. Opt. 46, 875–890 (1999).
[CrossRef]

Ochoa, D.

D. Ochoa, R. Hourdré, M. Ilegems, H. Benisty, T. F. Krauss, and C. J. M. Smith, “Diffraction of cylindrical Bragg reflectors surrounding an in-place semiconductor microcavity,” Phys. Rev. B 61, 4806–4812 (2000).
[CrossRef]

Oesterle, U.

A. Shaw, B. Roycroft, J. Hegarty, D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, C. J. M. Smith, R. Stanely, R. Hourdré, and U. Oesterle, “Lasing properties of disk microcavity based on circular Bragg reflector,” Appl. Phys. Lett. 75, 3051–3053 (1999).
[CrossRef]

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, C. J. M. Smith, R. Hourdré, and U. Oesterle, “High-finesse disk microcavity based on a circular Bragg reflector,” Appl. Phys. Lett. 73, 1314–1316 (1998).
[CrossRef]

Orenstein, M.

L. Djaloshinski and M. Orenstein, “Disk and ring microcavity lasers and their concentric coupling,” IEEE J. Quantum Electron. 35, 737–744 (1999).
[CrossRef]

Park, H.

S. Kim, H. Ryu, H. Park, G. Kim, Y. Choi, Y. Lee, and J. Kim, “Two-dimensional photonic crystal hexagonal waveguide ring laser,” Appl. Phys. Lett. 81, 2499–2501 (2002).
[CrossRef]

Roycroft, B.

A. Shaw, B. Roycroft, J. Hegarty, D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, C. J. M. Smith, R. Stanely, R. Hourdré, and U. Oesterle, “Lasing properties of disk microcavity based on circular Bragg reflector,” Appl. Phys. Lett. 75, 3051–3053 (1999).
[CrossRef]

Ryu, H.

S. Kim, H. Ryu, H. Park, G. Kim, Y. Choi, Y. Lee, and J. Kim, “Two-dimensional photonic crystal hexagonal waveguide ring laser,” Appl. Phys. Lett. 81, 2499–2501 (2002).
[CrossRef]

Scheuer, J.

Shaw, A.

A. Shaw, B. Roycroft, J. Hegarty, D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, C. J. M. Smith, R. Stanely, R. Hourdré, and U. Oesterle, “Lasing properties of disk microcavity based on circular Bragg reflector,” Appl. Phys. Lett. 75, 3051–3053 (1999).
[CrossRef]

Skorobogatiy, M.

Smith, C. J. M.

D. Ochoa, R. Hourdré, M. Ilegems, H. Benisty, T. F. Krauss, and C. J. M. Smith, “Diffraction of cylindrical Bragg reflectors surrounding an in-place semiconductor microcavity,” Phys. Rev. B 61, 4806–4812 (2000).
[CrossRef]

A. Shaw, B. Roycroft, J. Hegarty, D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, C. J. M. Smith, R. Stanely, R. Hourdré, and U. Oesterle, “Lasing properties of disk microcavity based on circular Bragg reflector,” Appl. Phys. Lett. 75, 3051–3053 (1999).
[CrossRef]

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, C. J. M. Smith, R. Hourdré, and U. Oesterle, “High-finesse disk microcavity based on a circular Bragg reflector,” Appl. Phys. Lett. 73, 1314–1316 (1998).
[CrossRef]

Sokolovski, G. S.

M. A. Kaliteevski, R. A. Abram, V. V. Nikolaev, and G. S. Sokolovski, “Bragg reflectors for cylindrical waves,” J. Mod. Opt. 46, 875–890 (1999).
[CrossRef]

Soljacic, M.

Stanely, R.

A. Shaw, B. Roycroft, J. Hegarty, D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, C. J. M. Smith, R. Stanely, R. Hourdré, and U. Oesterle, “Lasing properties of disk microcavity based on circular Bragg reflector,” Appl. Phys. Lett. 75, 3051–3053 (1999).
[CrossRef]

Svilans, M.

C. Wu, M. Svilans, M. Fallahi, T. Makino, J. Glinski, C. Maritan, and C. Blaauw, “Optically pumped surface-emitting DFB GaInAsP/InP lasers with circular grating,” Electron. Lett. 27, 1819–1821 (1991).
[CrossRef]

Toda, M.

M. Toda, “Single-mode behavior of a circular grating for potential disk-shaped DFB lasers,” IEEE J. Quantum Electron. 26, 473–481 (1990).
[CrossRef]

Weisberg, O.

Weisbuch, C.

A. Shaw, B. Roycroft, J. Hegarty, D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, C. J. M. Smith, R. Stanely, R. Hourdré, and U. Oesterle, “Lasing properties of disk microcavity based on circular Bragg reflector,” Appl. Phys. Lett. 75, 3051–3053 (1999).
[CrossRef]

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, C. J. M. Smith, R. Hourdré, and U. Oesterle, “High-finesse disk microcavity based on a circular Bragg reflector,” Appl. Phys. Lett. 73, 1314–1316 (1998).
[CrossRef]

Wu, C.

C. Wu, M. Svilans, M. Fallahi, T. Makino, J. Glinski, C. Maritan, and C. Blaauw, “Optically pumped surface-emitting DFB GaInAsP/InP lasers with circular grating,” Electron. Lett. 27, 1819–1821 (1991).
[CrossRef]

Yariv, A.

Yeh, P.

Zheng, X. H.

X. H. Zheng and S. Lacroix, “Mode coupling in circular-cylindrical system and its application to fingerprint resonators,” J. Lightwave Technol. 8, 1509–1516 (1990).
[CrossRef]

Appl. Phys. Lett. (3)

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, C. J. M. Smith, R. Hourdré, and U. Oesterle, “High-finesse disk microcavity based on a circular Bragg reflector,” Appl. Phys. Lett. 73, 1314–1316 (1998).
[CrossRef]

A. Shaw, B. Roycroft, J. Hegarty, D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, C. J. M. Smith, R. Stanely, R. Hourdré, and U. Oesterle, “Lasing properties of disk microcavity based on circular Bragg reflector,” Appl. Phys. Lett. 75, 3051–3053 (1999).
[CrossRef]

S. Kim, H. Ryu, H. Park, G. Kim, Y. Choi, Y. Lee, and J. Kim, “Two-dimensional photonic crystal hexagonal waveguide ring laser,” Appl. Phys. Lett. 81, 2499–2501 (2002).
[CrossRef]

Electron. Lett. (1)

C. Wu, M. Svilans, M. Fallahi, T. Makino, J. Glinski, C. Maritan, and C. Blaauw, “Optically pumped surface-emitting DFB GaInAsP/InP lasers with circular grating,” Electron. Lett. 27, 1819–1821 (1991).
[CrossRef]

IEEE J. Quantum Electron. (4)

M. Heiblum and J. H. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. QE-11, 75–83 (1975).
[CrossRef]

L. Djaloshinski and M. Orenstein, “Disk and ring microcavity lasers and their concentric coupling,” IEEE J. Quantum Electron. 35, 737–744 (1999).
[CrossRef]

T. Erdogan and D. G. Hall, “Circularly symmetric distributed feedback laser: coupled mode treatment of TE vector fields,” IEEE J. Quantum Electron. 28, 612–623 (1992).
[CrossRef]

M. Toda, “Single-mode behavior of a circular grating for potential disk-shaped DFB lasers,” IEEE J. Quantum Electron. 26, 473–481 (1990).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

A. Yariv, “Critical coupling and its control in optical waveguide-ring resonator systems,” IEEE Photon. Technol. Lett. 14, 483–485 (2002).
[CrossRef]

J. Appl. Phys. (1)

T. Erdogan and D. G. Hall, “Circularly symmetric distributed feedback laser: an analysis,” J. Appl. Phys. 68, 1435–1444 (1990).
[CrossRef]

J. Lightwave Technol. (1)

X. H. Zheng and S. Lacroix, “Mode coupling in circular-cylindrical system and its application to fingerprint resonators,” J. Lightwave Technol. 8, 1509–1516 (1990).
[CrossRef]

J. Mod. Opt. (1)

M. A. Kaliteevski, R. A. Abram, V. V. Nikolaev, and G. S. Sokolovski, “Bragg reflectors for cylindrical waves,” J. Mod. Opt. 46, 875–890 (1999).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Express (1)

Opt. Lett. (3)

Phys. Rev. B (1)

D. Ochoa, R. Hourdré, M. Ilegems, H. Benisty, T. F. Krauss, and C. J. M. Smith, “Diffraction of cylindrical Bragg reflectors surrounding an in-place semiconductor microcavity,” Phys. Rev. B 61, 4806–4812 (2000).
[CrossRef]

Other (2)

C. K. Madsen and J. H. Zhao, Optical Filter Design and Analysis: a Signal-Processing Approach (Wiley, New York, 1999).

See for example, A. Yariv, Optical Electronics in Modern Communications, 5th ed. (Oxford University, New York, 1997).

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Figures (12)

Fig. 1
Fig. 1

Illustration of the annular-defect-mode resonator structure. Dark rings are of refractive index n1, narrow light rings n2, wide light ring ndef, center ncore.

Fig. 2
Fig. 2

Radial refractive index profile (A) and the equivalent index profile (B) of an annular defect surrounded by Bragg reflectors. The maximal and minimal refractive indices are 1.5 and 1, respectively, and the grating period is ∼1 μm.

Fig. 3
Fig. 3

Illustration of the design rule used to realize a highly efficient, radial Bragg reflector.

Fig. 4
Fig. 4

Radial index profile (A) and electrical field distribution (B) of an annular-defect-mode resonator.

Fig. 5
Fig. 5

High-index (stars) and low-index (circles) layer widths of the resonator shown in Fig. 4.

Fig. 6
Fig. 6

Resonance wavelengths (circles) and quadratic fit (solid curve) of the resonator shown in Fig. 4.

Fig. 7
Fig. 7

Modal field profiles for m=6 (dotted curve), 7 (solid curve) and 10 (dashed-dotted curve) of the resonator shown in Fig. 4.

Fig. 8
Fig. 8

Resonance wavelengths (circles) and quadratic fit (solid curve) of a resonator based on lower-contrast Bragg reflectors.

Fig. 9
Fig. 9

Modal field profile of a resonator based on lower-contrast Bragg reflectors.

Fig. 10
Fig. 10

Comparison of the modal field profile shown in Fig. 4(A) and the modal field of a resonator based on second-order Bragg reflectors with similar parameters (B).

Fig. 11
Fig. 11

Resonance wavelengths (circles) and quadratic fit (solid curve) of a second-order-Bragg-reflector-based resonator.

Fig. 12
Fig. 12

Resonance wavelengths (circles) and quadratic fit (solid curve) of a resonator based on composite Bragg reflectors with parameters similar to those of the resonator of Fig. 4. The low-index layers are quarter-wavelength in width and the high-index layers are three-quarter-wavelength in width.

Equations (24)

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1ρρρ ρ+1ρ22θ2+k02n2(ρ)+2z2EzHz=0,
Ez=R(ρ)exp[i(mθ±βz)]misaninteger,
ρ22Rρ2+ρ Rρ+{[k2(ρ)-β2]ρ2-m2}R=0,
Rm(ρ)=AJm(kj2-β2ρ)+BYm(kj2-β2ρ),
Ez=[AJm(kj2-β2ρ)+BYm(kj2-β2ρ)]×cos(βz+ϕ)exp(imθ),
Hz=[CJm(kj2-β2ρ)+DYm(kj2-β2ρ)]×sin(βz+ϕ)exp(imθ).
Eθ=iγj2ωμ Hzρ-mρEzz,
Eρ=1γj22Ezzρ-mωμρ Hz,
Hθ=-iγj2ω Ezρ+mρHzz,
Hρ=1γj22Hzzρ+mωρ Ez,
EzHθHzEθ=J(γjρ)Y(γjρ)00nj2γj J(γjρ)nj2γj Y(γjρ)mβρω0γj2 J(γjρ)mβρω0γj2 Y(γjρ)00J(γjρ)Y(γjρ)mβρωμγj2 J(γjρ)mβρωμγj2 Y(γjρ)1γj J(γjρ)1γj Y(γjρ)AjBjCjDjM¯¯j(ρ)AjBjCjDj,
Aj+1Bj+1Cj+1Dj+1=M¯¯j+1-1(ρj)M¯¯j(ρj)AjBjCjDj.
Mj=J(γjρ)Y(γjρ)nj2γj J(γjρ)nj2γj Y(γjρ).
ρ=R exp(U/R),
θ=V/R,
U=R ln(ρ/R),
V=θR,
n(ρ)=neq(ρ)R/ρ,
2EU2+2EV2+k02neq2(U)E=0,
π/2=kdU=k02neq2-βV2dU,
βV=m/R.
lπ=kdU=k02neq2-βV2dU,l=1, 2, 3,
(2l+1)π/2=kdU=k02neq2-βV2dU,
l=1, 2, 3 .

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