Abstract

The modal properties, including the resonant vertical radiation, of a type of laser structures based on the annular Bragg resonance (ABR) are studied in detail. The modal threshold gains and the resonance frequencies of such lasers are obtained from the derived governing characteristic equation. Two kinds of ABR lasers, one with a π/2 phase shift in the outer grating and the other without, are analyzed. It is numerically demonstrated that, it’s possible to get a large-area, high-efficiency, single defect mode lasing in ABR lasers if we choose the kind without a π/2 phase shift in the outer grating and also a device size smaller than a critical value.

© 2007 Optical Society of America

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  1. T. Erdogan, and D. G. Hall, "Circularly symmetric distributed feedback semiconductor lasers: An analysis," J. Appl. Phys. 68, 1435-1444 (1990).
    [CrossRef]
  2. C. Wu, M. Svilans, M. Fallahi, T. Makino, J. Glinski, C. Maritan, and C. Blaauw, "Optical Pumped Surface-Emitting DFB GaInAsP/InP Lasers with Circular Grating," Electron. Lett. 27, 1819-1821 (1991).
    [CrossRef]
  3. T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, and M. J. Rooks, "Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum-well semiconductor laser," Appl. Phys. Lett. 60, 1921-1923 (1992).
    [CrossRef]
  4. C. Wu, M. Svilans, M. Fallahi, I. Templeton, T. Makino, J. Glinski, R. Maciejko, S. I. Najafi, C. Maritan, C. Blaauw, and G. Knight, "Room Temperature Operation of Electrically Pumped Surface-Emitting Circular Grating DBR Laser," Electron. Lett. 28, 1037-1039 (1992).
    [CrossRef]
  5. C. Wu, T. Makino, S. I. Najafi, R. Maciejko, M. Svilans, J. Glinski, and M. Fallahi, "Threshold Gain and Threshold Current Analysis of Circular Grating DFB and DBR Lasers," IEEE J. Quantum Electron. 29, 2596-2606 (1993).
    [CrossRef]
  6. A. M. Shams-Zadeh-Amiri, X. Li, and W. P. Huang, "Hankel transform-domain analysis of scattered fields in multilayer planar waveguides and lasers with circular gratings," IEEE J. Quantum Electron. 39, 1086-1098 (2003).
    [CrossRef]
  7. R. H. Jordan, D. G. Hall, O. King, G. W. Wicks, and S. Rishton, "Lasing behavior of circular grating surface-emitting semiconductor lasers," J. Opt. Soc. Am. B 14, 449-453 (1997).
    [CrossRef]
  8. J. Scheuer, and A. Yariv, "Coupled-Waves Approach to the Design and Analysis of Bragg and Photonic Crystal Annual Resonators," IEEE J. Quantum Electron. 39, 1555-1562 (2003).
    [CrossRef]
  9. J. Scheuer, W. M. J. Green, G. DeRose, and A. Yariv, "Low-threshold two-dimensional annular Bragg lasers," Opt. Lett. 29, 2641-2643 (2004).
    [CrossRef] [PubMed]
  10. X. K. Sun, J. Scheuer, and A. Yariv, "Optimal design and reduced threshold in vertically emitting circular Bragg disk resonator lasers," IEEE J. Sel. Top. Quantum Electron. 13, 359-366 (2007).
    [CrossRef]
  11. A. Yariv, Optical Electronics in Modern Communications (Oxford Univ. Press, New York, 1997).
  12. J. Scheuer, W. M. J. Green, G. A. DeRose, and A. Yariv, "InGaAsP Annular Bragg Lasers: Theory, Applications, and Modal Properties," IEEE J. Sel. Top. Quantum Electron. 11, 476-484 (2005).
    [CrossRef]
  13. R. F. Kazarinov, and C. H. Henry, "Second-Order Distributed Feedback Lasers with Mode Selection Provided by First-Order Radiation Losses," IEEE J. Quantum Electron. QE-21, 144-150 (1985).
    [CrossRef]

2007 (1)

X. K. Sun, J. Scheuer, and A. Yariv, "Optimal design and reduced threshold in vertically emitting circular Bragg disk resonator lasers," IEEE J. Sel. Top. Quantum Electron. 13, 359-366 (2007).
[CrossRef]

2005 (1)

J. Scheuer, W. M. J. Green, G. A. DeRose, and A. Yariv, "InGaAsP Annular Bragg Lasers: Theory, Applications, and Modal Properties," IEEE J. Sel. Top. Quantum Electron. 11, 476-484 (2005).
[CrossRef]

2004 (1)

2003 (2)

A. M. Shams-Zadeh-Amiri, X. Li, and W. P. Huang, "Hankel transform-domain analysis of scattered fields in multilayer planar waveguides and lasers with circular gratings," IEEE J. Quantum Electron. 39, 1086-1098 (2003).
[CrossRef]

J. Scheuer, and A. Yariv, "Coupled-Waves Approach to the Design and Analysis of Bragg and Photonic Crystal Annual Resonators," IEEE J. Quantum Electron. 39, 1555-1562 (2003).
[CrossRef]

1997 (1)

1993 (1)

C. Wu, T. Makino, S. I. Najafi, R. Maciejko, M. Svilans, J. Glinski, and M. Fallahi, "Threshold Gain and Threshold Current Analysis of Circular Grating DFB and DBR Lasers," IEEE J. Quantum Electron. 29, 2596-2606 (1993).
[CrossRef]

1992 (2)

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, and M. J. Rooks, "Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum-well semiconductor laser," Appl. Phys. Lett. 60, 1921-1923 (1992).
[CrossRef]

C. Wu, M. Svilans, M. Fallahi, I. Templeton, T. Makino, J. Glinski, R. Maciejko, S. I. Najafi, C. Maritan, C. Blaauw, and G. Knight, "Room Temperature Operation of Electrically Pumped Surface-Emitting Circular Grating DBR Laser," Electron. Lett. 28, 1037-1039 (1992).
[CrossRef]

1991 (1)

C. Wu, M. Svilans, M. Fallahi, T. Makino, J. Glinski, C. Maritan, and C. Blaauw, "Optical Pumped Surface-Emitting DFB GaInAsP/InP Lasers with Circular Grating," Electron. Lett. 27, 1819-1821 (1991).
[CrossRef]

1990 (1)

T. Erdogan, and D. G. Hall, "Circularly symmetric distributed feedback semiconductor lasers: An analysis," J. Appl. Phys. 68, 1435-1444 (1990).
[CrossRef]

1985 (1)

R. F. Kazarinov, and C. H. Henry, "Second-Order Distributed Feedback Lasers with Mode Selection Provided by First-Order Radiation Losses," IEEE J. Quantum Electron. QE-21, 144-150 (1985).
[CrossRef]

Anderson, E. H.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, and M. J. Rooks, "Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum-well semiconductor laser," Appl. Phys. Lett. 60, 1921-1923 (1992).
[CrossRef]

Blaauw, C.

C. Wu, M. Svilans, M. Fallahi, I. Templeton, T. Makino, J. Glinski, R. Maciejko, S. I. Najafi, C. Maritan, C. Blaauw, and G. Knight, "Room Temperature Operation of Electrically Pumped Surface-Emitting Circular Grating DBR Laser," Electron. Lett. 28, 1037-1039 (1992).
[CrossRef]

C. Wu, M. Svilans, M. Fallahi, T. Makino, J. Glinski, C. Maritan, and C. Blaauw, "Optical Pumped Surface-Emitting DFB GaInAsP/InP Lasers with Circular Grating," Electron. Lett. 27, 1819-1821 (1991).
[CrossRef]

DeRose, G.

DeRose, G. A.

J. Scheuer, W. M. J. Green, G. A. DeRose, and A. Yariv, "InGaAsP Annular Bragg Lasers: Theory, Applications, and Modal Properties," IEEE J. Sel. Top. Quantum Electron. 11, 476-484 (2005).
[CrossRef]

Erdogan, T.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, and M. J. Rooks, "Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum-well semiconductor laser," Appl. Phys. Lett. 60, 1921-1923 (1992).
[CrossRef]

T. Erdogan, and D. G. Hall, "Circularly symmetric distributed feedback semiconductor lasers: An analysis," J. Appl. Phys. 68, 1435-1444 (1990).
[CrossRef]

Fallahi, M.

C. Wu, T. Makino, S. I. Najafi, R. Maciejko, M. Svilans, J. Glinski, and M. Fallahi, "Threshold Gain and Threshold Current Analysis of Circular Grating DFB and DBR Lasers," IEEE J. Quantum Electron. 29, 2596-2606 (1993).
[CrossRef]

C. Wu, M. Svilans, M. Fallahi, I. Templeton, T. Makino, J. Glinski, R. Maciejko, S. I. Najafi, C. Maritan, C. Blaauw, and G. Knight, "Room Temperature Operation of Electrically Pumped Surface-Emitting Circular Grating DBR Laser," Electron. Lett. 28, 1037-1039 (1992).
[CrossRef]

C. Wu, M. Svilans, M. Fallahi, T. Makino, J. Glinski, C. Maritan, and C. Blaauw, "Optical Pumped Surface-Emitting DFB GaInAsP/InP Lasers with Circular Grating," Electron. Lett. 27, 1819-1821 (1991).
[CrossRef]

Glinski, J.

C. Wu, T. Makino, S. I. Najafi, R. Maciejko, M. Svilans, J. Glinski, and M. Fallahi, "Threshold Gain and Threshold Current Analysis of Circular Grating DFB and DBR Lasers," IEEE J. Quantum Electron. 29, 2596-2606 (1993).
[CrossRef]

C. Wu, M. Svilans, M. Fallahi, I. Templeton, T. Makino, J. Glinski, R. Maciejko, S. I. Najafi, C. Maritan, C. Blaauw, and G. Knight, "Room Temperature Operation of Electrically Pumped Surface-Emitting Circular Grating DBR Laser," Electron. Lett. 28, 1037-1039 (1992).
[CrossRef]

C. Wu, M. Svilans, M. Fallahi, T. Makino, J. Glinski, C. Maritan, and C. Blaauw, "Optical Pumped Surface-Emitting DFB GaInAsP/InP Lasers with Circular Grating," Electron. Lett. 27, 1819-1821 (1991).
[CrossRef]

Green, W. M. J.

J. Scheuer, W. M. J. Green, G. A. DeRose, and A. Yariv, "InGaAsP Annular Bragg Lasers: Theory, Applications, and Modal Properties," IEEE J. Sel. Top. Quantum Electron. 11, 476-484 (2005).
[CrossRef]

J. Scheuer, W. M. J. Green, G. DeRose, and A. Yariv, "Low-threshold two-dimensional annular Bragg lasers," Opt. Lett. 29, 2641-2643 (2004).
[CrossRef] [PubMed]

Hall, D. G.

R. H. Jordan, D. G. Hall, O. King, G. W. Wicks, and S. Rishton, "Lasing behavior of circular grating surface-emitting semiconductor lasers," J. Opt. Soc. Am. B 14, 449-453 (1997).
[CrossRef]

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, and M. J. Rooks, "Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum-well semiconductor laser," Appl. Phys. Lett. 60, 1921-1923 (1992).
[CrossRef]

T. Erdogan, and D. G. Hall, "Circularly symmetric distributed feedback semiconductor lasers: An analysis," J. Appl. Phys. 68, 1435-1444 (1990).
[CrossRef]

Henry, C. H.

R. F. Kazarinov, and C. H. Henry, "Second-Order Distributed Feedback Lasers with Mode Selection Provided by First-Order Radiation Losses," IEEE J. Quantum Electron. QE-21, 144-150 (1985).
[CrossRef]

Huang, W. P.

A. M. Shams-Zadeh-Amiri, X. Li, and W. P. Huang, "Hankel transform-domain analysis of scattered fields in multilayer planar waveguides and lasers with circular gratings," IEEE J. Quantum Electron. 39, 1086-1098 (2003).
[CrossRef]

Jordan, R. H.

Kazarinov, R. F.

R. F. Kazarinov, and C. H. Henry, "Second-Order Distributed Feedback Lasers with Mode Selection Provided by First-Order Radiation Losses," IEEE J. Quantum Electron. QE-21, 144-150 (1985).
[CrossRef]

King, O.

R. H. Jordan, D. G. Hall, O. King, G. W. Wicks, and S. Rishton, "Lasing behavior of circular grating surface-emitting semiconductor lasers," J. Opt. Soc. Am. B 14, 449-453 (1997).
[CrossRef]

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, and M. J. Rooks, "Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum-well semiconductor laser," Appl. Phys. Lett. 60, 1921-1923 (1992).
[CrossRef]

Knight, G.

C. Wu, M. Svilans, M. Fallahi, I. Templeton, T. Makino, J. Glinski, R. Maciejko, S. I. Najafi, C. Maritan, C. Blaauw, and G. Knight, "Room Temperature Operation of Electrically Pumped Surface-Emitting Circular Grating DBR Laser," Electron. Lett. 28, 1037-1039 (1992).
[CrossRef]

Li, X.

A. M. Shams-Zadeh-Amiri, X. Li, and W. P. Huang, "Hankel transform-domain analysis of scattered fields in multilayer planar waveguides and lasers with circular gratings," IEEE J. Quantum Electron. 39, 1086-1098 (2003).
[CrossRef]

Maciejko, R.

C. Wu, T. Makino, S. I. Najafi, R. Maciejko, M. Svilans, J. Glinski, and M. Fallahi, "Threshold Gain and Threshold Current Analysis of Circular Grating DFB and DBR Lasers," IEEE J. Quantum Electron. 29, 2596-2606 (1993).
[CrossRef]

C. Wu, M. Svilans, M. Fallahi, I. Templeton, T. Makino, J. Glinski, R. Maciejko, S. I. Najafi, C. Maritan, C. Blaauw, and G. Knight, "Room Temperature Operation of Electrically Pumped Surface-Emitting Circular Grating DBR Laser," Electron. Lett. 28, 1037-1039 (1992).
[CrossRef]

Makino, T.

C. Wu, T. Makino, S. I. Najafi, R. Maciejko, M. Svilans, J. Glinski, and M. Fallahi, "Threshold Gain and Threshold Current Analysis of Circular Grating DFB and DBR Lasers," IEEE J. Quantum Electron. 29, 2596-2606 (1993).
[CrossRef]

C. Wu, M. Svilans, M. Fallahi, I. Templeton, T. Makino, J. Glinski, R. Maciejko, S. I. Najafi, C. Maritan, C. Blaauw, and G. Knight, "Room Temperature Operation of Electrically Pumped Surface-Emitting Circular Grating DBR Laser," Electron. Lett. 28, 1037-1039 (1992).
[CrossRef]

C. Wu, M. Svilans, M. Fallahi, T. Makino, J. Glinski, C. Maritan, and C. Blaauw, "Optical 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, I. Templeton, T. Makino, J. Glinski, R. Maciejko, S. I. Najafi, C. Maritan, C. Blaauw, and G. Knight, "Room Temperature Operation of Electrically Pumped Surface-Emitting Circular Grating DBR Laser," Electron. Lett. 28, 1037-1039 (1992).
[CrossRef]

C. Wu, M. Svilans, M. Fallahi, T. Makino, J. Glinski, C. Maritan, and C. Blaauw, "Optical Pumped Surface-Emitting DFB GaInAsP/InP Lasers with Circular Grating," Electron. Lett. 27, 1819-1821 (1991).
[CrossRef]

Najafi, S. I.

C. Wu, T. Makino, S. I. Najafi, R. Maciejko, M. Svilans, J. Glinski, and M. Fallahi, "Threshold Gain and Threshold Current Analysis of Circular Grating DFB and DBR Lasers," IEEE J. Quantum Electron. 29, 2596-2606 (1993).
[CrossRef]

C. Wu, M. Svilans, M. Fallahi, I. Templeton, T. Makino, J. Glinski, R. Maciejko, S. I. Najafi, C. Maritan, C. Blaauw, and G. Knight, "Room Temperature Operation of Electrically Pumped Surface-Emitting Circular Grating DBR Laser," Electron. Lett. 28, 1037-1039 (1992).
[CrossRef]

Rishton, S.

Rooks, M. J.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, and M. J. Rooks, "Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum-well semiconductor laser," Appl. Phys. Lett. 60, 1921-1923 (1992).
[CrossRef]

Scheuer, J.

X. K. Sun, J. Scheuer, and A. Yariv, "Optimal design and reduced threshold in vertically emitting circular Bragg disk resonator lasers," IEEE J. Sel. Top. Quantum Electron. 13, 359-366 (2007).
[CrossRef]

J. Scheuer, W. M. J. Green, G. A. DeRose, and A. Yariv, "InGaAsP Annular Bragg Lasers: Theory, Applications, and Modal Properties," IEEE J. Sel. Top. Quantum Electron. 11, 476-484 (2005).
[CrossRef]

J. Scheuer, W. M. J. Green, G. DeRose, and A. Yariv, "Low-threshold two-dimensional annular Bragg lasers," Opt. Lett. 29, 2641-2643 (2004).
[CrossRef] [PubMed]

J. Scheuer, and A. Yariv, "Coupled-Waves Approach to the Design and Analysis of Bragg and Photonic Crystal Annual Resonators," IEEE J. Quantum Electron. 39, 1555-1562 (2003).
[CrossRef]

Shams-Zadeh-Amiri, A. M.

A. M. Shams-Zadeh-Amiri, X. Li, and W. P. Huang, "Hankel transform-domain analysis of scattered fields in multilayer planar waveguides and lasers with circular gratings," IEEE J. Quantum Electron. 39, 1086-1098 (2003).
[CrossRef]

Sun, X. K.

X. K. Sun, J. Scheuer, and A. Yariv, "Optimal design and reduced threshold in vertically emitting circular Bragg disk resonator lasers," IEEE J. Sel. Top. Quantum Electron. 13, 359-366 (2007).
[CrossRef]

Svilans, M.

C. Wu, T. Makino, S. I. Najafi, R. Maciejko, M. Svilans, J. Glinski, and M. Fallahi, "Threshold Gain and Threshold Current Analysis of Circular Grating DFB and DBR Lasers," IEEE J. Quantum Electron. 29, 2596-2606 (1993).
[CrossRef]

C. Wu, M. Svilans, M. Fallahi, I. Templeton, T. Makino, J. Glinski, R. Maciejko, S. I. Najafi, C. Maritan, C. Blaauw, and G. Knight, "Room Temperature Operation of Electrically Pumped Surface-Emitting Circular Grating DBR Laser," Electron. Lett. 28, 1037-1039 (1992).
[CrossRef]

C. Wu, M. Svilans, M. Fallahi, T. Makino, J. Glinski, C. Maritan, and C. Blaauw, "Optical Pumped Surface-Emitting DFB GaInAsP/InP Lasers with Circular Grating," Electron. Lett. 27, 1819-1821 (1991).
[CrossRef]

Templeton, I.

C. Wu, M. Svilans, M. Fallahi, I. Templeton, T. Makino, J. Glinski, R. Maciejko, S. I. Najafi, C. Maritan, C. Blaauw, and G. Knight, "Room Temperature Operation of Electrically Pumped Surface-Emitting Circular Grating DBR Laser," Electron. Lett. 28, 1037-1039 (1992).
[CrossRef]

Wicks, G. W.

R. H. Jordan, D. G. Hall, O. King, G. W. Wicks, and S. Rishton, "Lasing behavior of circular grating surface-emitting semiconductor lasers," J. Opt. Soc. Am. B 14, 449-453 (1997).
[CrossRef]

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, and M. J. Rooks, "Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum-well semiconductor laser," Appl. Phys. Lett. 60, 1921-1923 (1992).
[CrossRef]

Wu, C.

C. Wu, T. Makino, S. I. Najafi, R. Maciejko, M. Svilans, J. Glinski, and M. Fallahi, "Threshold Gain and Threshold Current Analysis of Circular Grating DFB and DBR Lasers," IEEE J. Quantum Electron. 29, 2596-2606 (1993).
[CrossRef]

C. Wu, M. Svilans, M. Fallahi, I. Templeton, T. Makino, J. Glinski, R. Maciejko, S. I. Najafi, C. Maritan, C. Blaauw, and G. Knight, "Room Temperature Operation of Electrically Pumped Surface-Emitting Circular Grating DBR Laser," Electron. Lett. 28, 1037-1039 (1992).
[CrossRef]

C. Wu, M. Svilans, M. Fallahi, T. Makino, J. Glinski, C. Maritan, and C. Blaauw, "Optical Pumped Surface-Emitting DFB GaInAsP/InP Lasers with Circular Grating," Electron. Lett. 27, 1819-1821 (1991).
[CrossRef]

Yariv, A.

X. K. Sun, J. Scheuer, and A. Yariv, "Optimal design and reduced threshold in vertically emitting circular Bragg disk resonator lasers," IEEE J. Sel. Top. Quantum Electron. 13, 359-366 (2007).
[CrossRef]

J. Scheuer, W. M. J. Green, G. A. DeRose, and A. Yariv, "InGaAsP Annular Bragg Lasers: Theory, Applications, and Modal Properties," IEEE J. Sel. Top. Quantum Electron. 11, 476-484 (2005).
[CrossRef]

J. Scheuer, W. M. J. Green, G. DeRose, and A. Yariv, "Low-threshold two-dimensional annular Bragg lasers," Opt. Lett. 29, 2641-2643 (2004).
[CrossRef] [PubMed]

J. Scheuer, and A. Yariv, "Coupled-Waves Approach to the Design and Analysis of Bragg and Photonic Crystal Annual Resonators," IEEE J. Quantum Electron. 39, 1555-1562 (2003).
[CrossRef]

Appl. Phys. Lett. (1)

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, and M. J. Rooks, "Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum-well semiconductor laser," Appl. Phys. Lett. 60, 1921-1923 (1992).
[CrossRef]

Electron. Lett. (2)

C. Wu, M. Svilans, M. Fallahi, I. Templeton, T. Makino, J. Glinski, R. Maciejko, S. I. Najafi, C. Maritan, C. Blaauw, and G. Knight, "Room Temperature Operation of Electrically Pumped Surface-Emitting Circular Grating DBR Laser," Electron. Lett. 28, 1037-1039 (1992).
[CrossRef]

C. Wu, M. Svilans, M. Fallahi, T. Makino, J. Glinski, C. Maritan, and C. Blaauw, "Optical Pumped Surface-Emitting DFB GaInAsP/InP Lasers with Circular Grating," Electron. Lett. 27, 1819-1821 (1991).
[CrossRef]

IEEE J. Quantum Electron. (4)

C. Wu, T. Makino, S. I. Najafi, R. Maciejko, M. Svilans, J. Glinski, and M. Fallahi, "Threshold Gain and Threshold Current Analysis of Circular Grating DFB and DBR Lasers," IEEE J. Quantum Electron. 29, 2596-2606 (1993).
[CrossRef]

A. M. Shams-Zadeh-Amiri, X. Li, and W. P. Huang, "Hankel transform-domain analysis of scattered fields in multilayer planar waveguides and lasers with circular gratings," IEEE J. Quantum Electron. 39, 1086-1098 (2003).
[CrossRef]

R. F. Kazarinov, and C. H. Henry, "Second-Order Distributed Feedback Lasers with Mode Selection Provided by First-Order Radiation Losses," IEEE J. Quantum Electron. QE-21, 144-150 (1985).
[CrossRef]

J. Scheuer, and A. Yariv, "Coupled-Waves Approach to the Design and Analysis of Bragg and Photonic Crystal Annual Resonators," IEEE J. Quantum Electron. 39, 1555-1562 (2003).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (2)

X. K. Sun, J. Scheuer, and A. Yariv, "Optimal design and reduced threshold in vertically emitting circular Bragg disk resonator lasers," IEEE J. Sel. Top. Quantum Electron. 13, 359-366 (2007).
[CrossRef]

J. Scheuer, W. M. J. Green, G. A. DeRose, and A. Yariv, "InGaAsP Annular Bragg Lasers: Theory, Applications, and Modal Properties," IEEE J. Sel. Top. Quantum Electron. 11, 476-484 (2005).
[CrossRef]

J. Appl. Phys. (1)

T. Erdogan, and D. G. Hall, "Circularly symmetric distributed feedback semiconductor lasers: An analysis," J. Appl. Phys. 68, 1435-1444 (1990).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Lett. (1)

Other (1)

A. Yariv, Optical Electronics in Modern Communications (Oxford Univ. Press, New York, 1997).

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Figures (3)

Fig. 1.
Fig. 1.

Illustration of an annular Bragg laser.

Fig. 2.
Fig. 2.

Emission efficiency η of the defect mode and the first in-band mode, as a function of the normalized exterior boundary radius xb .

Fig. 3.
Fig. 3.

Threshold gain gA and detuning factor δ, of the defect mode and the first in-band mode, as a function of the normalized exterior boundary radius xb .

Tables (2)

Tables Icon

Table 1. Modal threshold gains, detuning factors, and modal field patterns of the ABR lasers (xb =200) which have a π/2 phase shift in the outer grating.

Tables Icon

Table 2. Modal threshold gains, detuning factors, and modal field patterns of the ABR lasers (xb =200) which have the same phase dependence in the inner and outer gratings.

Equations (29)

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[ 1 ρ ρ ( ρ ρ ) + 1 ρ 2 2 φ 2 + k 0 2 n 2 ( ρ , z ) + 2 z 2 ] E z ( ρ , φ , z ) = 0 ,
E z ( ρ , φ , z ) = E z ( m ) ( ρ , z ) exp ( i m φ )
= [ A H m ( 1 ) ( β ρ ) + B H m ( 2 ) ( β ρ ) ] Z ( z ) exp ( i m φ ) ,
( k 0 2 ε r ( z ) + 2 z 2 ) Z ( z ) = β 2 Z ( z ) .
Δ ε ( ρ , z ) = Δ ε 0 l = ± 1 , ± 2 a l ( z ) exp ( i l Φ [ H m ( 1 ) ( β design ρ ) ] )
= Δ ε 0 l = ± 1 , ± 2 a l ( z ) exp ( i l Φ [ H m ( 1 ) ( x ) ] ) exp ( i l δ · x )
= Δ ε 0 ( a 2 ( z ) H m ( 2 ) H m ( 1 ) e 2 i δ · x + a 2 ( z ) H m ( 1 ) H m ( 2 ) e 2 i · x + a 1 ( z ) H m ( 2 ) H m ( 1 ) e i δ · x + a 1 ( z ) H m ( 1 ) H m ( 1 ) e i δ · x ) ,
E z ( m ) ( x , z ) = [ A ( x ) H m ( 1 ) ( x ) + B ( x ) H m ( 2 ) ( x ) ] Z ( z ) + Δ E ( x , z ) .
[ 1 ρ ρ ( ρ ρ ) m 2 ρ 2 ] Δ E = 0 ,
H m ( 1 , 2 ) ( x ) x < < d H m ( 1 , 2 ) ( x ) d x , d n H m ( 1 , 2 ) ( x ) d x n ( ± i ) n H m ( 1 , 2 ) ( x ) ,
2 i Z ( d A d x H m ( 1 ) dB dx H m ( 2 ) ) + i k 0 2 ε i β 2 ( A H m ( 1 ) Z + B H m ( 2 ) Z ) + 1 β 2 ( k 0 2 ε r + i k 0 2 ε i + 2 z 2 ) Δ E
= k 0 2 Δ ε 0 β 2 ( a 2 H m ( 2 ) H m ( 1 ) e 2 i δ · x + a 2 H m ( 1 ) H m ( 2 ) e 2 i δ · x
+ a 1 H m ( 2 ) H m ( 1 ) e i δ · x + a 1 H m ( 1 ) H m ( 1 ) e i δ · x ) ( A H m ( 1 ) Z + B H m ( 2 ) Z + Δ E ) .
{ 2 i dA dx H m ( 1 ) Z + i k 0 2 ε i β 2 A H m ( 1 ) Z = k 0 2 Δ ε 0 β 2 ( a 2 B H m ( 1 ) e 2 i δ · x Z + a 1 Δ E H m ( 1 ) H m ( 1 ) e i δ · x ) ( a ) 2 i dB dx H m ( 2 ) Z + i k 0 2 ε i β 2 B H m ( 2 ) Z = k 0 2 Δ ε 0 β 2 ( a 2 A H m ( 2 ) e 2 i δ · x Z + a 1 Δ E H m ( 1 ) H m ( 2 ) e i δ · x ) ( b ) ( k 0 2 ε r + 2 z 2 ) Δ E = k 0 2 Δ ε 0 ( a 1 A H m ( 1 ) e i δ · x Z + a 1 B H m ( 1 ) e i δ · x Z ) ( c )
Δ E = ( s 1 A e i δ · x + s 1 B e i δ · x ) H m ( 1 ) ,
s l ( z ) = k 0 2 Δ ε 0 + a l ( z ' ) Z ( z ' ) G ( z , z ' ) d z ' ,
{ dA dx = ( g A h 1 , 1 ) A ( h 1 , 1 + i h 2 ) B e 2 i δ · x dB dx = ( g A h 1 , 1 ) B + ( h 1 , 1 + i h 2 ) A e 2 i δ · x ,
{ A ( x ) = [ C 1 exp ( S x ) + C 2 exp ( S x ) ] exp ( i δ · x ) B ( x ) = 1 ν [ C 1 ( S u + i δ ) exp ( S x ) C 2 ( S + u i δ ) exp ( S x ) ] exp ( i δ · x ) ,
{ A ( x ) = A ( 0 ) exp ( g A x ) B ( x ) = B ( 0 ) exp ( g A x ) .
E z ( m ) ( x , z ) = { A 1 ( x ) H m ( 1 ) ( x ) Z ( z ) + B 1 ( x ) H m ( 2 ) ( x ) Z ( z ) + Δ E 1 ( x , z ) , region I : x < x L A 2 e g A x H m ( 1 ) ( x ) Z ( z ) + B 2 e g A x H m ( 2 ) ( x ) Z ( z ) , region II : x L < x < x R A 3 ( x ) H m ( 1 ) ( x ) Z ( z ) + B 3 ( x ) H m ( 2 ) ( x ) Z ( z ) + Δ E 3 ( x , z ) region III : x R < x < x b .
{ E I ( x ) = P C 11 [ e S · x + i δ · x + S u + ν + i δ S + u ν i δ e S · x + i δ · x ] H m ( 1 ) ( x ) P C 11 ν [ ( S u + i δ ) e S · x i δ · x S u + ν + i δ S + u ν i δ ( S + u i δ ) e S · x i δ · x ] H m ( 2 ) ( x ) E II ( x ) = P [ A 2 e g A x H m ( 1 ) ( x ) + B 2 e g A x H m ( 2 ) ( x ) ] E III ( x ) = P C 31 e S · x + i δ · x [ 1 + S u + i δ S + u i δ e 2 S ( x b x ) ] H m ( 1 ) ( x ) P C 31 ( S u + i δ ) ν ' e S · x i δ · x [ 1 e 2 S ( x b x ) ] H m ( 2 ) ( x ) ,
( g A + i ) ( L H S ) I 1 ( g A + i ) ( L H S ) I + 1 . ( g A + i ) ( R H S ) III + 1 ( g A + i ) ( R H S ) III 1 = e 2 g A ( x R x L ) H m ( 1 ) ( x R ) H m ( 2 ) ( x R ) . H m ( 2 ) ( x L ) H m ( 1 ) ( x L ) ,
( L H S ) I =
{ [ e S · x L + i δ · x L + S u + ν + i δ S + u ν i δ ] H m ( 1 ) ( x L ) 1 ν [ ( S u + i δ ) e S · x L i δ · x L S u + ν + i δ S + u ν i δ ( S + u i δ ) e S · x L i δ · x L ] H m ( 2 ) ( x L ) } { [ ( S + i ( δ + 1 ) ) e S · x L + i δ · x L + S u + ν + i δ S + u ν i δ ( S + i ( δ + 1 ) ) e S · x L i δ · x L ] H m ( 1 ) ( x L ) 1 ν [ ( S u + i δ ) ( S i ( δ + 1 ) ) e S · x L i δ · x L + S u + ν + i δ S + u ν i δ ( S + i ( δ + 1 ) ) e S · x L i δ · x L ] H m ( 2 ) ( x L ) }
( R H S ) III = e S · x R + i δ · x R [ 1 + S u + i δ S + u i δ e 2 S ( x b x R ) ] H m 1 ( x R ) ( S u + i δ ) v ' e S · x R i δ · x R [ 1 e 2 S ( x b X R ) ] H m ( 2 ) ( x R ) { ( S + i ( δ + 1 ) ) e S · x R + i δ · x R [ 1 + S u + i δ S + u i δ e 2 S ( x b x R ) ] H m ( 1 ) ( x R ) 2 S S u + i δ S + u i δ e S · x R + i δ · x R e 2 S ( x b x R ) H m ( 1 ) ( x R ) ( S u + i δ ) v ' ( S i ( δ + 1 ) e S · x R + i δ · x R [ 1 e 2 S ( x b x R ) ] H m ( 2 ) ( x R ) 2 S ( S u + i δ ) v ' e S · x R i δ · x R e 2 S ( x b x R ) H m ( 2 ) ( x R ) } .
Θ ( Φ [ H m ( 1 ) ( x ) ] , α ) = arccos α π + 2 π l = 1 sin ( l arccos α ) l cos ( l Φ [ H m ( 1 ) ( x ) ] ) .
r 1 = e i ( δ + i u ) x b ( v e i ( δ + i u ) x b + i δ u ) sinh ( S x b 2 ) + S cosh ( S x b 2 ) ( v e i ( δ + i u ) x b + i δ u ) sinh ( S x b 2 ) + S cosh ( S x b 2 ) ,
r 2 = v sinh ( S x b 2 ) ( i δ u ) sinh ( S x b 2 ) + S cosh ( S x b 2 ) .
exp [ i Φ ( H m ( 2 ) ( x b 2 ) H m ( 1 ) ( x b 2 ) ) ] e 2 i ( x b 2 m π 2 π 4 ) = i e i x b ,

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