Abstract

We have developed a theory that unifies the analysis of the modal properties of surface-emitting chirped circular grating lasers. This theory is based on solving the resonance conditions which involve two types of reflectivities of chirped circular gratings. This approach is shown to be in agreement with previous derivations which use the characteristic equations. Utilizing this unified analysis, we obtain the modal properties of circular DFB, disk-, and ring- Bragg resonator lasers. We also compare the threshold gain, single mode range, quality factor, emission efficiency, and modal area of these types of circular grating lasers. It is demonstrated that, under similar conditions, disk Bragg resonator lasers have the highest quality factor, the highest emission efficiency, and the smallest modal area, indicating their suitability in low-threshold, high-efficiency, ultracompact laser design, while ring Bragg resonator lasers have a large single mode range, high emission efficiency, and large modal area, indicating their suitability for high-efficiency, large-area, high-power applications.

© 2008 Optical Society of America

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    [CrossRef] [PubMed]
  2. D. F. Welch, R. Parke, A. Hardy, W. Streifer, and D. R. Scifres, "Low-Threshold Grating-Coupled Surface-Emitting Lasers," Appl. Phys. Lett. 55, 813-815 (1989).
    [CrossRef]
  3. R. Parke, R. Waarts, D. F. Welch, A. Hardy, and W. Streifer, "High-Efficiency, High Uniformity, Grating Coupled Surface Emitting Lasers," Electron. Lett. 26, 125-127 (1990).
    [CrossRef]
  4. T. Kjellberg, M. Hagberg, N. Eriksson, and A. G. Larsson, "Low-Threshold Grating-Coupled Surface-Emitting Lasers with Etch-Stop Layer for Precise Grating Positioning," IEEE Photon. Technol. Lett. 5, 1149-1152 (1993).
    [CrossRef]
  5. F. S. Choa, M. H. Shih, J. Y. Fan, G. J. Simonis, P. L. Liu, T. Tanbunek, R. A. Logan, W. T. Tsang, and A. M. Sergent, "Very Low Threshold 1.55 ?m Grating-Coupled Surface-Emitting Lasers for Optical Signal Processing and Interconnect," Appl. Phys. Lett. 67, 2777-2779 (1995).
    [CrossRef]
  6. R. G. Waarts, "Optical Characterization of Grating Surface Emitting Semiconductor Lasers," Appl. Opt. 29, 2718-2721 (1990).
    [CrossRef] [PubMed]
  7. T. Erdogan and D. G. Hall, "Circularly symmetric distributed feedback semiconductor lasers: An analysis," J. Appl. Phys. 68, 1435-1444 (1990).
    [CrossRef]
  8. 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]
  9. 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]
  10. 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]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  18. P. L. Greene and D. G. Hall, "Effects of Radiation on Circular-Grating DFB Lasers??????Part II: Device and Pump-Beam Parameters," IEEE J. Quantum Electron. 37, 364-371 (2001).
    [CrossRef]
  19. 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]
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    [CrossRef]
  21. W. M. J. Green, J. Scheuer, G. DeRose, and A. Yariv, "Vertically emitting annular Bragg lasers using polymer epitaxial transfer," Appl. Phys. Lett. 85, 3669-3671 (2004).
    [CrossRef]
  22. A. Jebali, R. F. Mahrt, N. Moll, D. Erni, C. Bauer, G.-L. Bona, and W. Bachtold, "Lasing in organic circular grating structures," J. Appl. Phys. 96, 3043-3049 (2004).
    [CrossRef]
  23. 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]
  24. J. Scheuer, W. M. J. Green, G. A. DeRose, and A. Yariv, "Lasing from a circular Bragg nanocavity with an ultrasmall modal volume," Appl. Phys. Lett. 86, 251101 (2005).
    [CrossRef]
  25. G. A. Turnbull, A. Carleton, G. F. Barlow, A. Tahraouhi, T. F. Krauss, K. A. Shore, and I. D. W. Samuel, "Influence of grating characteristics on the operation of circular-grating distributed-feedback polymer lasers," J. Appl. Phys. 98, 023105 (2005).
    [CrossRef]
  26. G. A. Turnbull, A. Carleton, A. Tahraouhi, T. F. Krauss, I. D. W. Samuel, G. F. Barlow, and K. A. Shore, "Effect of gain localization in circular-grating distributed feedback lasers," Appl. Phys. Lett. 87, 201101 (2005).
    [CrossRef]
  27. R. Coccioli, M. Boroditsky, K. W. Kim, Y. Rahmat-Samii, and E. Yablonovitch, "Smallest possible electromagnetic mode volume in a dielectric cavity," IEE Proc.:Optoelectron. 145, 391-397 (1998).
    [CrossRef]
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  29. 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]
  30. J. Scheuer and A. Yariv, "Annular Bragg defect mode resonators," J. Opt. Soc. Am. B 20, 2285-2291 (2003).
    [CrossRef]
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    [CrossRef]
  32. X. K. Sun and A. Yariv, "Modal properties and modal control in vertically emitting annular Bragg lasers," Opt. Express 15, 17323-17333 (2007).
    [CrossRef] [PubMed]
  33. 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]
  34. A. Jebali, D. Erni, S. Gulde, R. F. Mahrt, and W. Bachtold, "Analytical calculation of the Q factor for circular-grating microcavities," J. Opt. Soc. Am. B 24, 906-915 (2007).
    [CrossRef]

2007

2005

J. Scheuer, W. M. J. Green, G. A. DeRose, and A. Yariv, "Lasing from a circular Bragg nanocavity with an ultrasmall modal volume," Appl. Phys. Lett. 86, 251101 (2005).
[CrossRef]

G. A. Turnbull, A. Carleton, G. F. Barlow, A. Tahraouhi, T. F. Krauss, K. A. Shore, and I. D. W. Samuel, "Influence of grating characteristics on the operation of circular-grating distributed-feedback polymer lasers," J. Appl. Phys. 98, 023105 (2005).
[CrossRef]

G. A. Turnbull, A. Carleton, A. Tahraouhi, T. F. Krauss, I. D. W. Samuel, G. F. Barlow, and K. A. Shore, "Effect of gain localization in circular-grating distributed feedback lasers," Appl. Phys. Lett. 87, 201101 (2005).
[CrossRef]

2004

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]

G. F. Barlow, A. Shore, G. A. Turnbull, and I. D. W. Samuel, "Design and analysis of a low-threshold polymer circular-grating distributed-feedback laser," J. Opt. Soc. Am. B 21, 2142-2150 (2004).
[CrossRef]

W. M. J. Green, J. Scheuer, G. DeRose, and A. Yariv, "Vertically emitting annular Bragg lasers using polymer epitaxial transfer," Appl. Phys. Lett. 85, 3669-3671 (2004).
[CrossRef]

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

2003

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]

J. Scheuer and A. Yariv, "Annular Bragg defect mode resonators," J. Opt. Soc. Am. B 20, 2285-2291 (2003).
[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]

2001

C. Bauer, H. Giessen, B. Schnabel, E.-B. Kley, C. Schmitt, and U. Scherf, "A Surface-Emitting Circular Grating Polymer Laser," Adv. Mater. 13, 1161-1164 (2001).
[CrossRef]

P. L. Greene and D. G. Hall, "Effects of Radiation on Circular-Grating DFB Lasers??????Part I: Coupled-Mode Equations," IEEE J. Quantum Electron. 37, 353-364 (2001).
[CrossRef]

P. L. Greene and D. G. Hall, "Effects of Radiation on Circular-Grating DFB Lasers??????Part II: Device and Pump-Beam Parameters," IEEE J. Quantum Electron. 37, 364-371 (2001).
[CrossRef]

2000

A. M. Shams-Zadeh-Amiri, X. Li, and W.-P. Huang, "Above-Threshold Analysis of Second-Order Circular-Grating DFB Lasers," IEEE J. Quantum Electron. 36, 259-267 (2000).
[CrossRef]

1999

1998

C. Olson, P. L. Greene, G. W. Wicks, D. G. Hall, and S. Rishton, "High-order azimuthal spatial modes of concentric-circle-grating surface-emitting semiconductor lasers," Appl. Phys. Lett. 72, 1284-1286 (1998).
[CrossRef]

R. Coccioli, M. Boroditsky, K. W. Kim, Y. Rahmat-Samii, and E. Yablonovitch, "Smallest possible electromagnetic mode volume in a dielectric cavity," IEE Proc.:Optoelectron. 145, 391-397 (1998).
[CrossRef]

1997

1995

F. S. Choa, M. H. Shih, J. Y. Fan, G. J. Simonis, P. L. Liu, T. Tanbunek, R. A. Logan, W. T. Tsang, and A. M. Sergent, "Very Low Threshold 1.55 ?m Grating-Coupled Surface-Emitting Lasers for Optical Signal Processing and Interconnect," Appl. Phys. Lett. 67, 2777-2779 (1995).
[CrossRef]

1993

T. Kjellberg, M. Hagberg, N. Eriksson, and A. G. Larsson, "Low-Threshold Grating-Coupled Surface-Emitting Lasers with Etch-Stop Layer for Precise Grating Positioning," IEEE Photon. Technol. Lett. 5, 1149-1152 (1993).
[CrossRef]

1992

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

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

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

R. Parke, R. Waarts, D. F. Welch, A. Hardy, and W. Streifer, "High-Efficiency, High Uniformity, Grating Coupled Surface Emitting Lasers," Electron. Lett. 26, 125-127 (1990).
[CrossRef]

R. G. Waarts, "Optical Characterization of Grating Surface Emitting Semiconductor Lasers," Appl. Opt. 29, 2718-2721 (1990).
[CrossRef] [PubMed]

1989

S. A. Shakir, T. C. Salvi, and G. C. Dente, "Analysis of Grating-Coupled Surface-Emitting Lasers," Opt. Lett. 14, 937-939 (1989).
[CrossRef] [PubMed]

D. F. Welch, R. Parke, A. Hardy, W. Streifer, and D. R. Scifres, "Low-Threshold Grating-Coupled Surface-Emitting Lasers," Appl. Phys. Lett. 55, 813-815 (1989).
[CrossRef]

1985

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]

1969

E. A. J. Marcatili, "Bends in Optical Dielectric Guides," Bell Syst. Tech. J. 48, 2103-2132 (1969).

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]

Bachtold, W.

A. Jebali, D. Erni, S. Gulde, R. F. Mahrt, and W. Bachtold, "Analytical calculation of the Q factor for circular-grating microcavities," J. Opt. Soc. Am. B 24, 906-915 (2007).
[CrossRef]

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

Barlow, G. F.

G. A. Turnbull, A. Carleton, G. F. Barlow, A. Tahraouhi, T. F. Krauss, K. A. Shore, and I. D. W. Samuel, "Influence of grating characteristics on the operation of circular-grating distributed-feedback polymer lasers," J. Appl. Phys. 98, 023105 (2005).
[CrossRef]

G. A. Turnbull, A. Carleton, A. Tahraouhi, T. F. Krauss, I. D. W. Samuel, G. F. Barlow, and K. A. Shore, "Effect of gain localization in circular-grating distributed feedback lasers," Appl. Phys. Lett. 87, 201101 (2005).
[CrossRef]

G. F. Barlow, A. Shore, G. A. Turnbull, and I. D. W. Samuel, "Design and analysis of a low-threshold polymer circular-grating distributed-feedback laser," J. Opt. Soc. Am. B 21, 2142-2150 (2004).
[CrossRef]

Bauer, C.

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

C. Bauer, H. Giessen, B. Schnabel, E.-B. Kley, C. Schmitt, and U. Scherf, "A Surface-Emitting Circular Grating Polymer Laser," Adv. Mater. 13, 1161-1164 (2001).
[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]

Bona, G.-L.

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

Boroditsky, M.

R. Coccioli, M. Boroditsky, K. W. Kim, Y. Rahmat-Samii, and E. Yablonovitch, "Smallest possible electromagnetic mode volume in a dielectric cavity," IEE Proc.:Optoelectron. 145, 391-397 (1998).
[CrossRef]

Carleton, A.

G. A. Turnbull, A. Carleton, A. Tahraouhi, T. F. Krauss, I. D. W. Samuel, G. F. Barlow, and K. A. Shore, "Effect of gain localization in circular-grating distributed feedback lasers," Appl. Phys. Lett. 87, 201101 (2005).
[CrossRef]

G. A. Turnbull, A. Carleton, G. F. Barlow, A. Tahraouhi, T. F. Krauss, K. A. Shore, and I. D. W. Samuel, "Influence of grating characteristics on the operation of circular-grating distributed-feedback polymer lasers," J. Appl. Phys. 98, 023105 (2005).
[CrossRef]

Choa, F. S.

F. S. Choa, M. H. Shih, J. Y. Fan, G. J. Simonis, P. L. Liu, T. Tanbunek, R. A. Logan, W. T. Tsang, and A. M. Sergent, "Very Low Threshold 1.55 ?m Grating-Coupled Surface-Emitting Lasers for Optical Signal Processing and Interconnect," Appl. Phys. Lett. 67, 2777-2779 (1995).
[CrossRef]

Coccioli, R.

R. Coccioli, M. Boroditsky, K. W. Kim, Y. Rahmat-Samii, and E. Yablonovitch, "Smallest possible electromagnetic mode volume in a dielectric cavity," IEE Proc.:Optoelectron. 145, 391-397 (1998).
[CrossRef]

Dente, G. C.

DeRose, G.

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]

W. M. J. Green, J. Scheuer, G. DeRose, and A. Yariv, "Vertically emitting annular Bragg lasers using polymer epitaxial transfer," Appl. Phys. Lett. 85, 3669-3671 (2004).
[CrossRef]

DeRose, G. A.

J. Scheuer, W. M. J. Green, G. A. DeRose, and A. Yariv, "Lasing from a circular Bragg nanocavity with an ultrasmall modal volume," Appl. Phys. Lett. 86, 251101 (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]

Eriksson, N.

T. Kjellberg, M. Hagberg, N. Eriksson, and A. G. Larsson, "Low-Threshold Grating-Coupled Surface-Emitting Lasers with Etch-Stop Layer for Precise Grating Positioning," IEEE Photon. Technol. Lett. 5, 1149-1152 (1993).
[CrossRef]

Erni, D.

A. Jebali, D. Erni, S. Gulde, R. F. Mahrt, and W. Bachtold, "Analytical calculation of the Q factor for circular-grating microcavities," J. Opt. Soc. Am. B 24, 906-915 (2007).
[CrossRef]

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

Fallahi, M.

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]

Fan, J. Y.

F. S. Choa, M. H. Shih, J. Y. Fan, G. J. Simonis, P. L. Liu, T. Tanbunek, R. A. Logan, W. T. Tsang, and A. M. Sergent, "Very Low Threshold 1.55 ?m Grating-Coupled Surface-Emitting Lasers for Optical Signal Processing and Interconnect," Appl. Phys. Lett. 67, 2777-2779 (1995).
[CrossRef]

Giessen, H.

C. Bauer, H. Giessen, B. Schnabel, E.-B. Kley, C. Schmitt, and U. Scherf, "A Surface-Emitting Circular Grating Polymer Laser," Adv. Mater. 13, 1161-1164 (2001).
[CrossRef]

Glinski, J.

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, "Lasing from a circular Bragg nanocavity with an ultrasmall modal volume," Appl. Phys. Lett. 86, 251101 (2005).
[CrossRef]

W. M. J. Green, J. Scheuer, G. DeRose, and A. Yariv, "Vertically emitting annular Bragg lasers using polymer epitaxial transfer," Appl. Phys. Lett. 85, 3669-3671 (2004).
[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]

Greene, P. L.

P. L. Greene and D. G. Hall, "Effects of Radiation on Circular-Grating DFB Lasers??????Part I: Coupled-Mode Equations," IEEE J. Quantum Electron. 37, 353-364 (2001).
[CrossRef]

P. L. Greene and D. G. Hall, "Effects of Radiation on Circular-Grating DFB Lasers??????Part II: Device and Pump-Beam Parameters," IEEE J. Quantum Electron. 37, 364-371 (2001).
[CrossRef]

C. Olson, P. L. Greene, G. W. Wicks, D. G. Hall, and S. Rishton, "High-order azimuthal spatial modes of concentric-circle-grating surface-emitting semiconductor lasers," Appl. Phys. Lett. 72, 1284-1286 (1998).
[CrossRef]

Gulde, S.

Hagberg, M.

T. Kjellberg, M. Hagberg, N. Eriksson, and A. G. Larsson, "Low-Threshold Grating-Coupled Surface-Emitting Lasers with Etch-Stop Layer for Precise Grating Positioning," IEEE Photon. Technol. Lett. 5, 1149-1152 (1993).
[CrossRef]

Hall, D. G.

P. L. Greene and D. G. Hall, "Effects of Radiation on Circular-Grating DFB Lasers??????Part II: Device and Pump-Beam Parameters," IEEE J. Quantum Electron. 37, 364-371 (2001).
[CrossRef]

P. L. Greene and D. G. Hall, "Effects of Radiation on Circular-Grating DFB Lasers??????Part I: Coupled-Mode Equations," IEEE J. Quantum Electron. 37, 353-364 (2001).
[CrossRef]

C. Olson, P. L. Greene, G. W. Wicks, D. G. Hall, and S. Rishton, "High-order azimuthal spatial modes of concentric-circle-grating surface-emitting semiconductor lasers," Appl. Phys. Lett. 72, 1284-1286 (1998).
[CrossRef]

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]

Hardy, A.

R. Parke, R. Waarts, D. F. Welch, A. Hardy, and W. Streifer, "High-Efficiency, High Uniformity, Grating Coupled Surface Emitting Lasers," Electron. Lett. 26, 125-127 (1990).
[CrossRef]

D. F. Welch, R. Parke, A. Hardy, W. Streifer, and D. R. Scifres, "Low-Threshold Grating-Coupled Surface-Emitting Lasers," Appl. Phys. Lett. 55, 813-815 (1989).
[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]

Huang, W.-P.

A. M. Shams-Zadeh-Amiri, X. Li, and W.-P. Huang, "Above-Threshold Analysis of Second-Order Circular-Grating DFB Lasers," IEEE J. Quantum Electron. 36, 259-267 (2000).
[CrossRef]

Jebali, A.

A. Jebali, D. Erni, S. Gulde, R. F. Mahrt, and W. Bachtold, "Analytical calculation of the Q factor for circular-grating microcavities," J. Opt. Soc. Am. B 24, 906-915 (2007).
[CrossRef]

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

Jordan, R. H.

Kasunic, K. J.

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]

Kim, K. W.

R. Coccioli, M. Boroditsky, K. W. Kim, Y. Rahmat-Samii, and E. Yablonovitch, "Smallest possible electromagnetic mode volume in a dielectric cavity," IEE Proc.:Optoelectron. 145, 391-397 (1998).
[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]

Kjellberg, T.

T. Kjellberg, M. Hagberg, N. Eriksson, and A. G. Larsson, "Low-Threshold Grating-Coupled Surface-Emitting Lasers with Etch-Stop Layer for Precise Grating Positioning," IEEE Photon. Technol. Lett. 5, 1149-1152 (1993).
[CrossRef]

Kley, E.-B.

C. Bauer, H. Giessen, B. Schnabel, E.-B. Kley, C. Schmitt, and U. Scherf, "A Surface-Emitting Circular Grating Polymer Laser," Adv. Mater. 13, 1161-1164 (2001).
[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]

Krauss, T. F.

G. A. Turnbull, A. Carleton, A. Tahraouhi, T. F. Krauss, I. D. W. Samuel, G. F. Barlow, and K. A. Shore, "Effect of gain localization in circular-grating distributed feedback lasers," Appl. Phys. Lett. 87, 201101 (2005).
[CrossRef]

G. A. Turnbull, A. Carleton, G. F. Barlow, A. Tahraouhi, T. F. Krauss, K. A. Shore, and I. D. W. Samuel, "Influence of grating characteristics on the operation of circular-grating distributed-feedback polymer lasers," J. Appl. Phys. 98, 023105 (2005).
[CrossRef]

Larsson, A. G.

T. Kjellberg, M. Hagberg, N. Eriksson, and A. G. Larsson, "Low-Threshold Grating-Coupled Surface-Emitting Lasers with Etch-Stop Layer for Precise Grating Positioning," IEEE Photon. Technol. Lett. 5, 1149-1152 (1993).
[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]

A. M. Shams-Zadeh-Amiri, X. Li, and W.-P. Huang, "Above-Threshold Analysis of Second-Order Circular-Grating DFB Lasers," IEEE J. Quantum Electron. 36, 259-267 (2000).
[CrossRef]

Liu, P. L.

F. S. Choa, M. H. Shih, J. Y. Fan, G. J. Simonis, P. L. Liu, T. Tanbunek, R. A. Logan, W. T. Tsang, and A. M. Sergent, "Very Low Threshold 1.55 ?m Grating-Coupled Surface-Emitting Lasers for Optical Signal Processing and Interconnect," Appl. Phys. Lett. 67, 2777-2779 (1995).
[CrossRef]

Logan, R. A.

F. S. Choa, M. H. Shih, J. Y. Fan, G. J. Simonis, P. L. Liu, T. Tanbunek, R. A. Logan, W. T. Tsang, and A. M. Sergent, "Very Low Threshold 1.55 ?m Grating-Coupled Surface-Emitting Lasers for Optical Signal Processing and Interconnect," Appl. Phys. Lett. 67, 2777-2779 (1995).
[CrossRef]

Maciejko, R.

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]

Mahrt, R. F.

A. Jebali, D. Erni, S. Gulde, R. F. Mahrt, and W. Bachtold, "Analytical calculation of the Q factor for circular-grating microcavities," J. Opt. Soc. Am. B 24, 906-915 (2007).
[CrossRef]

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

Makino, T.

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]

Marcatili, E. A. J.

E. A. J. Marcatili, "Bends in Optical Dielectric Guides," Bell Syst. Tech. J. 48, 2103-2132 (1969).

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]

Moll, N.

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

Najafi, S. 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]

Olson, C.

C. Olson, P. L. Greene, G. W. Wicks, D. G. Hall, and S. Rishton, "High-order azimuthal spatial modes of concentric-circle-grating surface-emitting semiconductor lasers," Appl. Phys. Lett. 72, 1284-1286 (1998).
[CrossRef]

Parke, R.

R. Parke, R. Waarts, D. F. Welch, A. Hardy, and W. Streifer, "High-Efficiency, High Uniformity, Grating Coupled Surface Emitting Lasers," Electron. Lett. 26, 125-127 (1990).
[CrossRef]

D. F. Welch, R. Parke, A. Hardy, W. Streifer, and D. R. Scifres, "Low-Threshold Grating-Coupled Surface-Emitting Lasers," Appl. Phys. Lett. 55, 813-815 (1989).
[CrossRef]

Rahmat-Samii, Y.

R. Coccioli, M. Boroditsky, K. W. Kim, Y. Rahmat-Samii, and E. Yablonovitch, "Smallest possible electromagnetic mode volume in a dielectric cavity," IEE Proc.:Optoelectron. 145, 391-397 (1998).
[CrossRef]

Rishton, S.

C. Olson, P. L. Greene, G. W. Wicks, D. G. Hall, and S. Rishton, "High-order azimuthal spatial modes of concentric-circle-grating surface-emitting semiconductor lasers," Appl. Phys. Lett. 72, 1284-1286 (1998).
[CrossRef]

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]

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]

Salvi, T. C.

Samuel, I. D. W.

G. A. Turnbull, A. Carleton, A. Tahraouhi, T. F. Krauss, I. D. W. Samuel, G. F. Barlow, and K. A. Shore, "Effect of gain localization in circular-grating distributed feedback lasers," Appl. Phys. Lett. 87, 201101 (2005).
[CrossRef]

G. A. Turnbull, A. Carleton, G. F. Barlow, A. Tahraouhi, T. F. Krauss, K. A. Shore, and I. D. W. Samuel, "Influence of grating characteristics on the operation of circular-grating distributed-feedback polymer lasers," J. Appl. Phys. 98, 023105 (2005).
[CrossRef]

G. F. Barlow, A. Shore, G. A. Turnbull, and I. D. W. Samuel, "Design and analysis of a low-threshold polymer circular-grating distributed-feedback laser," J. Opt. Soc. Am. B 21, 2142-2150 (2004).
[CrossRef]

Scherf, U.

C. Bauer, H. Giessen, B. Schnabel, E.-B. Kley, C. Schmitt, and U. Scherf, "A Surface-Emitting Circular Grating Polymer Laser," Adv. Mater. 13, 1161-1164 (2001).
[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, "Lasing from a circular Bragg nanocavity with an ultrasmall modal volume," Appl. Phys. Lett. 86, 251101 (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]

W. M. J. Green, J. Scheuer, G. DeRose, and A. Yariv, "Vertically emitting annular Bragg lasers using polymer epitaxial transfer," Appl. Phys. Lett. 85, 3669-3671 (2004).
[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]

J. Scheuer and A. Yariv, "Annular Bragg defect mode resonators," J. Opt. Soc. Am. B 20, 2285-2291 (2003).
[CrossRef]

Schmitt, C.

C. Bauer, H. Giessen, B. Schnabel, E.-B. Kley, C. Schmitt, and U. Scherf, "A Surface-Emitting Circular Grating Polymer Laser," Adv. Mater. 13, 1161-1164 (2001).
[CrossRef]

Schnabel, B.

C. Bauer, H. Giessen, B. Schnabel, E.-B. Kley, C. Schmitt, and U. Scherf, "A Surface-Emitting Circular Grating Polymer Laser," Adv. Mater. 13, 1161-1164 (2001).
[CrossRef]

Scifres, D. R.

D. F. Welch, R. Parke, A. Hardy, W. Streifer, and D. R. Scifres, "Low-Threshold Grating-Coupled Surface-Emitting Lasers," Appl. Phys. Lett. 55, 813-815 (1989).
[CrossRef]

Sergent, A. M.

F. S. Choa, M. H. Shih, J. Y. Fan, G. J. Simonis, P. L. Liu, T. Tanbunek, R. A. Logan, W. T. Tsang, and A. M. Sergent, "Very Low Threshold 1.55 ?m Grating-Coupled Surface-Emitting Lasers for Optical Signal Processing and Interconnect," Appl. Phys. Lett. 67, 2777-2779 (1995).
[CrossRef]

Shakir, S. A.

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]

A. M. Shams-Zadeh-Amiri, X. Li, and W.-P. Huang, "Above-Threshold Analysis of Second-Order Circular-Grating DFB Lasers," IEEE J. Quantum Electron. 36, 259-267 (2000).
[CrossRef]

Shih, M. H.

F. S. Choa, M. H. Shih, J. Y. Fan, G. J. Simonis, P. L. Liu, T. Tanbunek, R. A. Logan, W. T. Tsang, and A. M. Sergent, "Very Low Threshold 1.55 ?m Grating-Coupled Surface-Emitting Lasers for Optical Signal Processing and Interconnect," Appl. Phys. Lett. 67, 2777-2779 (1995).
[CrossRef]

Shore, A.

Shore, K. A.

G. A. Turnbull, A. Carleton, G. F. Barlow, A. Tahraouhi, T. F. Krauss, K. A. Shore, and I. D. W. Samuel, "Influence of grating characteristics on the operation of circular-grating distributed-feedback polymer lasers," J. Appl. Phys. 98, 023105 (2005).
[CrossRef]

G. A. Turnbull, A. Carleton, A. Tahraouhi, T. F. Krauss, I. D. W. Samuel, G. F. Barlow, and K. A. Shore, "Effect of gain localization in circular-grating distributed feedback lasers," Appl. Phys. Lett. 87, 201101 (2005).
[CrossRef]

Simonis, G. J.

F. S. Choa, M. H. Shih, J. Y. Fan, G. J. Simonis, P. L. Liu, T. Tanbunek, R. A. Logan, W. T. Tsang, and A. M. Sergent, "Very Low Threshold 1.55 ?m Grating-Coupled Surface-Emitting Lasers for Optical Signal Processing and Interconnect," Appl. Phys. Lett. 67, 2777-2779 (1995).
[CrossRef]

Streifer, W.

R. Parke, R. Waarts, D. F. Welch, A. Hardy, and W. Streifer, "High-Efficiency, High Uniformity, Grating Coupled Surface Emitting Lasers," Electron. Lett. 26, 125-127 (1990).
[CrossRef]

D. F. Welch, R. Parke, A. Hardy, W. Streifer, and D. R. Scifres, "Low-Threshold Grating-Coupled Surface-Emitting Lasers," Appl. Phys. Lett. 55, 813-815 (1989).
[CrossRef]

Sun, X. K.

X. K. Sun and A. Yariv, "Modal properties and modal control in vertically emitting annular Bragg lasers," Opt. Express 15, 17323-17333 (2007).
[CrossRef] [PubMed]

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, 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]

Tahraouhi, A.

G. A. Turnbull, A. Carleton, G. F. Barlow, A. Tahraouhi, T. F. Krauss, K. A. Shore, and I. D. W. Samuel, "Influence of grating characteristics on the operation of circular-grating distributed-feedback polymer lasers," J. Appl. Phys. 98, 023105 (2005).
[CrossRef]

G. A. Turnbull, A. Carleton, A. Tahraouhi, T. F. Krauss, I. D. W. Samuel, G. F. Barlow, and K. A. Shore, "Effect of gain localization in circular-grating distributed feedback lasers," Appl. Phys. Lett. 87, 201101 (2005).
[CrossRef]

Tanbunek, T.

F. S. Choa, M. H. Shih, J. Y. Fan, G. J. Simonis, P. L. Liu, T. Tanbunek, R. A. Logan, W. T. Tsang, and A. M. Sergent, "Very Low Threshold 1.55 ?m Grating-Coupled Surface-Emitting Lasers for Optical Signal Processing and Interconnect," Appl. Phys. Lett. 67, 2777-2779 (1995).
[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]

Tsang, W. T.

F. S. Choa, M. H. Shih, J. Y. Fan, G. J. Simonis, P. L. Liu, T. Tanbunek, R. A. Logan, W. T. Tsang, and A. M. Sergent, "Very Low Threshold 1.55 ?m Grating-Coupled Surface-Emitting Lasers for Optical Signal Processing and Interconnect," Appl. Phys. Lett. 67, 2777-2779 (1995).
[CrossRef]

Turnbull, G. A.

G. A. Turnbull, A. Carleton, G. F. Barlow, A. Tahraouhi, T. F. Krauss, K. A. Shore, and I. D. W. Samuel, "Influence of grating characteristics on the operation of circular-grating distributed-feedback polymer lasers," J. Appl. Phys. 98, 023105 (2005).
[CrossRef]

G. A. Turnbull, A. Carleton, A. Tahraouhi, T. F. Krauss, I. D. W. Samuel, G. F. Barlow, and K. A. Shore, "Effect of gain localization in circular-grating distributed feedback lasers," Appl. Phys. Lett. 87, 201101 (2005).
[CrossRef]

G. F. Barlow, A. Shore, G. A. Turnbull, and I. D. W. Samuel, "Design and analysis of a low-threshold polymer circular-grating distributed-feedback laser," J. Opt. Soc. Am. B 21, 2142-2150 (2004).
[CrossRef]

Waarts, R.

R. Parke, R. Waarts, D. F. Welch, A. Hardy, and W. Streifer, "High-Efficiency, High Uniformity, Grating Coupled Surface Emitting Lasers," Electron. Lett. 26, 125-127 (1990).
[CrossRef]

Waarts, R. G.

Welch, D. F.

R. Parke, R. Waarts, D. F. Welch, A. Hardy, and W. Streifer, "High-Efficiency, High Uniformity, Grating Coupled Surface Emitting Lasers," Electron. Lett. 26, 125-127 (1990).
[CrossRef]

D. F. Welch, R. Parke, A. Hardy, W. Streifer, and D. R. Scifres, "Low-Threshold Grating-Coupled Surface-Emitting Lasers," Appl. Phys. Lett. 55, 813-815 (1989).
[CrossRef]

Wicks, G. W.

C. Olson, P. L. Greene, G. W. Wicks, D. G. Hall, and S. Rishton, "High-order azimuthal spatial modes of concentric-circle-grating surface-emitting semiconductor lasers," Appl. Phys. Lett. 72, 1284-1286 (1998).
[CrossRef]

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[CrossRef]

W. M. J. Green, J. Scheuer, G. DeRose, and A. Yariv, "Vertically emitting annular Bragg lasers using polymer epitaxial transfer," Appl. Phys. Lett. 85, 3669-3671 (2004).
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[CrossRef]

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Figures (8)

Fig. 1.
Fig. 1.

Illustration of chirped circular grating lasers: (a) Circular DFB laser; (b) Disk Bragg resonator laser; (c) Ring Bragg resonator laser. Laser radiation is coupled out of the resonators in vertical direction via the gratings.

Fig. 2.
Fig. 2.

Illustration of the two types of boundary conditions for calculating reflectivities. Left: A(0)=B(0), r 1(L)=A(L)/B(L); Right: B(L)=0, r 2(L,x 0)=B(x 0)/A(x 0).

Fig. 3.
Fig. 3.

Reflectivities |r 1(100)| and |r 2(x 0,x 0+100)| for different gain levels gA =0, 5×10-4, 1×10-3, and 4×10-3, in the presence of vertical radiation.

Fig. 4.
Fig. 4.

Evolution of threshold gains of the 5 lowest-order modes of circular DFB, disk-, and ring- Bragg resonator lasers.

Fig. 5.
Fig. 5.

Unnormalized quality factors of circular DFB, disk-, and ring- Bragg resonator lasers.

Fig. 6.
Fig. 6.

Emission efficiencies of circular DFB, disk-, and ring- Bragg resonator lasers.

Fig. 7.
Fig. 7.

Modal areas of circular DFB, disk-, and ring- Bragg resonator lasers. Their top surface area (πx 2 b ) is marked with a black dashed line as a reference.

Fig. 8.
Fig. 8.

Comparison of radial electric field profiles of ring Bragg resonator lasers (xb =90, 113, and 150). For xb =113, the electric field has equal amplitude at the center (x=0) and the defect (x=xb /2).

Tables (1)

Tables Icon

Table 1. Modal threshold gains (gA ), frequency detuning factors (δ), and modal fields of the circular DFB, disk-, and ring- Bragg resonator lasers (xb =200).

Equations (10)

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

{ d A d x = u A v B e 2 i δ · x d B d x = u B + v A e 2 i δ · x
{ d A ~ d x = ( u i δ ) A ~ v B ~ d B ~ d x = ( u i δ ) B ~ + v A ~
{ A ˜ ( x ) = A ˜ ( 0 ) sinh [ S ( x L ) ] + cosh [ S ( x L ) ] sinh [ SL ] + cosh [ SL ] B ˜ ( x ) = A ˜ ( 0 ) v [ ( u i δ ) S ] sinh [ S ( x L ) ] + [ ( u i δ ) S ] cosh [ S ( x L ) ] sinh [ SL ] + cosh [ SL ]
r 1 ( L ) = A ( L ) B ( L ) = A ˜ ( L ) e i δ L B ˜ ( L ) e i δ L = e 2 i δ L ( u v i δ ) sinh [ SL ] + S cosh [ SL ] ( u v i δ ) sinh [ SL ] + S cosh [ SL ] .
r 2 ( x 0 , L ) = B ( x 0 ) A ( x 0 ) = B ˜ ( x 0 ) e i δ x 0 A ˜ ( x 0 ) e i δ x 0 = e 2 i δ x 0 v sinh [ S ( L x 0 ) ] ( u i δ ) sinh [ S ( L x 0 ) ] S cosh [ S ( L x 0 ) ] .
tanh [ S x b ] = S u v i δ .
e 2 ( g A i δ ) x 0 v sinh [ S ( x b x 0 ) ] ( u i δ ) sinh [ S ( x b x 0 ) ] S cosh [ S ( x b x 0 ) ] = 1 .
e 2 ( g A i δ ) ( x R x L ) ( u v i δ ) sinh [ S x L ] + S cosh [ S x L ] ( u v i δ ) sinh [ S x L ] + S cosh [ S x L ] · v sinh [ S ( x b x R ) ] ( u i δ ) sinh [ S ( x b x R ) ] S cosh [ S ( x b x R ) ] = 1 .
Q = 0 D d z 0 2 π d φ 0 ρ b E ( ρ , z ) 2 ρ d ρ grating Δ E ( ρ , z = 0 ) 2 ρ d ρ d φ + 0 D dz 0 2 π E ( ρ = ρ b , z ) 2 ρ b d φ
= 0 D Z 2 ( z ) dz · 0 x b E ( x ) 2 x d x grating Δ E ( x , y = 0 ) 2 x d x + 0 D Z 2 ( z ) d z · E ( x = x b ) 2 β x b

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