Abstract

We report numerical results for gain saturation in circular-grating distributed-feedback semiconductor lasers. Our model includes a radially varying nonlinear gain and a uniformly distributed grating loss in the solution of the coupled-mode equations. Our results show that lossy, high-power operation results in an optimum coupling strength for efficient conversion of pump power into useful output power. Our results also show a multimode spectrum for large-coupling strengths, a consequence of mode competition induced by a spatially varying gain distribution. Single-mode selection entails operating at approximately the optimum coupling coefficient determined for efficient pumping.

© 1997 Optical Society of America

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References

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  1. C. Wu, T. Makino, S. 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. QE-29, 2596–2606 (1993).
  2. T. Erdogan and D. G. Hall, “Circularly symmetric distributed feedback semiconductor laser: an analysis,” J. Appl. Phys. 68, 1435–1444 (1990).
    [Crossref]
  3. T. Erdogan and D. G. Hall, “Circularly symmetric distributed feedback laser: coupled mode treatment of TE vector fields,” IEEE J. Quantum Electron. QE-28, 612–623 (1992).
    [Crossref]
  4. C. Wu, M. Svilans, M. Fallahi, I. M. 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. M. Fallahi, F. Chatenoud, M. Dion, I. Templeton, R. Barber, and J. Thompson, “Circular-grating surface-emitting distributed Bragg reflector lasers on an InGaAs-GaAs structure for 0.98 µm applications,” IEEE J. Sel. Topics Quantum Electron. 1, 382–386 (1995).
    [Crossref]
  6. M. Fallahi, N. Peyghambarian, K. Kasunic, M. Dion, and Z. Wasilewski, “Circular-grating surface-emitting DBR laser array for free-space applications,” Electron. Lett. 32, 1583–1585 (1996).
    [Crossref]
  7. 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, Al-GaAs/GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
    [Crossref]
  8. W. W. Rigrod, “Gain saturation and output power of optical masers,” J. Appl. Phys. 34, 2602–2609 (1963).
    [Crossref]
  9. W. W. Rigrod, “Saturation effects in high-gain lasers,” J. Appl. Phys. 36, 2487–2490 (1965).
    [Crossref]
  10. K. O. Hill and A. Watanabe, “Envelope gain saturation in distributed-feedback lasers,” Appl. Opt. 14, 950–961 (1975).
    [Crossref] [PubMed]
  11. W. Press, B. Flannery, S. Teukolsky, and W. Vetterling, Numerical Recipes (Cambridge U. Press, New York, 1986).
  12. H. A. Haus, “Gain saturation in distributed feedback lasers,” Appl. Opt. 14, 2650–2652 (1975).
    [Crossref] [PubMed]
  13. R. F. Harrinton, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961), Appendix D.
  14. H. Soda, Y. Kotaki, H. Sudo, H. Ishikawa, S. Yamakoshi, and H. Imai, “Stability in single longitudinal mode operation in GaInAsP/InP phase-adjusted DFB lasers,” IEEE J. Quantum Electron. QE-23, 804–814 (1987).
    [Crossref]

1996 (1)

M. Fallahi, N. Peyghambarian, K. Kasunic, M. Dion, and Z. Wasilewski, “Circular-grating surface-emitting DBR laser array for free-space applications,” Electron. Lett. 32, 1583–1585 (1996).
[Crossref]

1995 (1)

M. Fallahi, F. Chatenoud, M. Dion, I. Templeton, R. Barber, and J. Thompson, “Circular-grating surface-emitting distributed Bragg reflector lasers on an InGaAs-GaAs structure for 0.98 µm applications,” IEEE J. Sel. Topics Quantum Electron. 1, 382–386 (1995).
[Crossref]

1993 (1)

C. Wu, T. Makino, S. 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. QE-29, 2596–2606 (1993).

1992 (3)

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

C. Wu, M. Svilans, M. Fallahi, I. M. 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]

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, Al-GaAs/GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[Crossref]

1990 (1)

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

1987 (1)

H. Soda, Y. Kotaki, H. Sudo, H. Ishikawa, S. Yamakoshi, and H. Imai, “Stability in single longitudinal mode operation in GaInAsP/InP phase-adjusted DFB lasers,” IEEE J. Quantum Electron. QE-23, 804–814 (1987).
[Crossref]

1975 (2)

1965 (1)

W. W. Rigrod, “Saturation effects in high-gain lasers,” J. Appl. Phys. 36, 2487–2490 (1965).
[Crossref]

1963 (1)

W. W. Rigrod, “Gain saturation and output power of optical masers,” J. Appl. Phys. 34, 2602–2609 (1963).
[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, Al-GaAs/GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[Crossref]

Barber, R.

M. Fallahi, F. Chatenoud, M. Dion, I. Templeton, R. Barber, and J. Thompson, “Circular-grating surface-emitting distributed Bragg reflector lasers on an InGaAs-GaAs structure for 0.98 µm applications,” IEEE J. Sel. Topics Quantum Electron. 1, 382–386 (1995).
[Crossref]

Blaauw, C.

C. Wu, M. Svilans, M. Fallahi, I. M. 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]

Chatenoud, F.

M. Fallahi, F. Chatenoud, M. Dion, I. Templeton, R. Barber, and J. Thompson, “Circular-grating surface-emitting distributed Bragg reflector lasers on an InGaAs-GaAs structure for 0.98 µm applications,” IEEE J. Sel. Topics Quantum Electron. 1, 382–386 (1995).
[Crossref]

Dion, M.

M. Fallahi, N. Peyghambarian, K. Kasunic, M. Dion, and Z. Wasilewski, “Circular-grating surface-emitting DBR laser array for free-space applications,” Electron. Lett. 32, 1583–1585 (1996).
[Crossref]

M. Fallahi, F. Chatenoud, M. Dion, I. Templeton, R. Barber, and J. Thompson, “Circular-grating surface-emitting distributed Bragg reflector lasers on an InGaAs-GaAs structure for 0.98 µm applications,” IEEE J. Sel. Topics Quantum Electron. 1, 382–386 (1995).
[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, Al-GaAs/GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[Crossref]

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

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

Fallahi, M.

M. Fallahi, N. Peyghambarian, K. Kasunic, M. Dion, and Z. Wasilewski, “Circular-grating surface-emitting DBR laser array for free-space applications,” Electron. Lett. 32, 1583–1585 (1996).
[Crossref]

M. Fallahi, F. Chatenoud, M. Dion, I. Templeton, R. Barber, and J. Thompson, “Circular-grating surface-emitting distributed Bragg reflector lasers on an InGaAs-GaAs structure for 0.98 µm applications,” IEEE J. Sel. Topics Quantum Electron. 1, 382–386 (1995).
[Crossref]

C. Wu, T. Makino, S. 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. QE-29, 2596–2606 (1993).

C. Wu, M. Svilans, M. Fallahi, I. M. 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]

Flannery, B.

W. Press, B. Flannery, S. Teukolsky, and W. Vetterling, Numerical Recipes (Cambridge U. Press, New York, 1986).

Glinski, J.

C. Wu, T. Makino, S. 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. QE-29, 2596–2606 (1993).

C. Wu, M. Svilans, M. Fallahi, I. M. 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]

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. QE-28, 612–623 (1992).
[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, Al-GaAs/GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[Crossref]

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

Harrinton, R. F.

R. F. Harrinton, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961), Appendix D.

Haus, H. A.

Hill, K. O.

Imai, H.

H. Soda, Y. Kotaki, H. Sudo, H. Ishikawa, S. Yamakoshi, and H. Imai, “Stability in single longitudinal mode operation in GaInAsP/InP phase-adjusted DFB lasers,” IEEE J. Quantum Electron. QE-23, 804–814 (1987).
[Crossref]

Ishikawa, H.

H. Soda, Y. Kotaki, H. Sudo, H. Ishikawa, S. Yamakoshi, and H. Imai, “Stability in single longitudinal mode operation in GaInAsP/InP phase-adjusted DFB lasers,” IEEE J. Quantum Electron. QE-23, 804–814 (1987).
[Crossref]

Kasunic, K.

M. Fallahi, N. Peyghambarian, K. Kasunic, M. Dion, and Z. Wasilewski, “Circular-grating surface-emitting DBR laser array for free-space applications,” Electron. Lett. 32, 1583–1585 (1996).
[Crossref]

King, O.

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, Al-GaAs/GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[Crossref]

Knight, G.

C. Wu, M. Svilans, M. Fallahi, I. M. 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]

Kotaki, Y.

H. Soda, Y. Kotaki, H. Sudo, H. Ishikawa, S. Yamakoshi, and H. Imai, “Stability in single longitudinal mode operation in GaInAsP/InP phase-adjusted DFB lasers,” IEEE J. Quantum Electron. QE-23, 804–814 (1987).
[Crossref]

Maciejko, R.

C. Wu, T. Makino, S. 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. QE-29, 2596–2606 (1993).

C. Wu, M. Svilans, M. Fallahi, I. M. 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. 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. QE-29, 2596–2606 (1993).

C. Wu, M. Svilans, M. Fallahi, I. M. 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]

Maritan, C.

C. Wu, M. Svilans, M. Fallahi, I. M. 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]

Najafi, S.

C. Wu, T. Makino, S. 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. QE-29, 2596–2606 (1993).

Najafi, S. I.

C. Wu, M. Svilans, M. Fallahi, I. M. 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]

Peyghambarian, N.

M. Fallahi, N. Peyghambarian, K. Kasunic, M. Dion, and Z. Wasilewski, “Circular-grating surface-emitting DBR laser array for free-space applications,” Electron. Lett. 32, 1583–1585 (1996).
[Crossref]

Press, W.

W. Press, B. Flannery, S. Teukolsky, and W. Vetterling, Numerical Recipes (Cambridge U. Press, New York, 1986).

Rigrod, W. W.

W. W. Rigrod, “Saturation effects in high-gain lasers,” J. Appl. Phys. 36, 2487–2490 (1965).
[Crossref]

W. W. Rigrod, “Gain saturation and output power of optical masers,” J. Appl. Phys. 34, 2602–2609 (1963).
[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, Al-GaAs/GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[Crossref]

Soda, H.

H. Soda, Y. Kotaki, H. Sudo, H. Ishikawa, S. Yamakoshi, and H. Imai, “Stability in single longitudinal mode operation in GaInAsP/InP phase-adjusted DFB lasers,” IEEE J. Quantum Electron. QE-23, 804–814 (1987).
[Crossref]

Sudo, H.

H. Soda, Y. Kotaki, H. Sudo, H. Ishikawa, S. Yamakoshi, and H. Imai, “Stability in single longitudinal mode operation in GaInAsP/InP phase-adjusted DFB lasers,” IEEE J. Quantum Electron. QE-23, 804–814 (1987).
[Crossref]

Svilans, M.

C. Wu, T. Makino, S. 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. QE-29, 2596–2606 (1993).

C. Wu, M. Svilans, M. Fallahi, I. M. 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]

Templeton, I.

M. Fallahi, F. Chatenoud, M. Dion, I. Templeton, R. Barber, and J. Thompson, “Circular-grating surface-emitting distributed Bragg reflector lasers on an InGaAs-GaAs structure for 0.98 µm applications,” IEEE J. Sel. Topics Quantum Electron. 1, 382–386 (1995).
[Crossref]

Templeton, I. M.

C. Wu, M. Svilans, M. Fallahi, I. M. 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]

Teukolsky, S.

W. Press, B. Flannery, S. Teukolsky, and W. Vetterling, Numerical Recipes (Cambridge U. Press, New York, 1986).

Thompson, J.

M. Fallahi, F. Chatenoud, M. Dion, I. Templeton, R. Barber, and J. Thompson, “Circular-grating surface-emitting distributed Bragg reflector lasers on an InGaAs-GaAs structure for 0.98 µm applications,” IEEE J. Sel. Topics Quantum Electron. 1, 382–386 (1995).
[Crossref]

Vetterling, W.

W. Press, B. Flannery, S. Teukolsky, and W. Vetterling, Numerical Recipes (Cambridge U. Press, New York, 1986).

Wasilewski, Z.

M. Fallahi, N. Peyghambarian, K. Kasunic, M. Dion, and Z. Wasilewski, “Circular-grating surface-emitting DBR laser array for free-space applications,” Electron. Lett. 32, 1583–1585 (1996).
[Crossref]

Watanabe, A.

Wicks, G. W.

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, Al-GaAs/GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[Crossref]

Wu, C.

C. Wu, T. Makino, S. 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. QE-29, 2596–2606 (1993).

C. Wu, M. Svilans, M. Fallahi, I. M. 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]

Yamakoshi, S.

H. Soda, Y. Kotaki, H. Sudo, H. Ishikawa, S. Yamakoshi, and H. Imai, “Stability in single longitudinal mode operation in GaInAsP/InP phase-adjusted DFB lasers,” IEEE J. Quantum Electron. QE-23, 804–814 (1987).
[Crossref]

Appl. Opt. (2)

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, Al-GaAs/GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[Crossref]

Electron. Lett. (2)

M. Fallahi, N. Peyghambarian, K. Kasunic, M. Dion, and Z. Wasilewski, “Circular-grating surface-emitting DBR laser array for free-space applications,” Electron. Lett. 32, 1583–1585 (1996).
[Crossref]

C. Wu, M. Svilans, M. Fallahi, I. M. 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]

IEEE J. Quantum Electron. (3)

C. Wu, T. Makino, S. 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. QE-29, 2596–2606 (1993).

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

H. Soda, Y. Kotaki, H. Sudo, H. Ishikawa, S. Yamakoshi, and H. Imai, “Stability in single longitudinal mode operation in GaInAsP/InP phase-adjusted DFB lasers,” IEEE J. Quantum Electron. QE-23, 804–814 (1987).
[Crossref]

IEEE J. Sel. Topics Quantum Electron. (1)

M. Fallahi, F. Chatenoud, M. Dion, I. Templeton, R. Barber, and J. Thompson, “Circular-grating surface-emitting distributed Bragg reflector lasers on an InGaAs-GaAs structure for 0.98 µm applications,” IEEE J. Sel. Topics Quantum Electron. 1, 382–386 (1995).
[Crossref]

J. Appl. Phys. (3)

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

W. W. Rigrod, “Gain saturation and output power of optical masers,” J. Appl. Phys. 34, 2602–2609 (1963).
[Crossref]

W. W. Rigrod, “Saturation effects in high-gain lasers,” J. Appl. Phys. 36, 2487–2490 (1965).
[Crossref]

Other (2)

R. F. Harrinton, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961), Appendix D.

W. Press, B. Flannery, S. Teukolsky, and W. Vetterling, Numerical Recipes (Cambridge U. Press, New York, 1986).

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Figures (9)

Fig. 1
Fig. 1

Schematic of CGDFB semiconductor laser. The top view shows radius R and electric-field polarization for circularly symmetric (m=0) radial mode; the cross section shows output power Po.

Fig. 2
Fig. 2

Unsaturated gain versus detuning for Po=10-5 and 10; αg=0.0; κR=0.1, 0.5, and 2.5. Solid circles indicate even radial modes, for which Ω=0 and m=0; hollow circles indicate odd radial modes (Ω=π and m=0, which is equivalent to Ω=0 and m=1). Continuous curves are obtained by varying Ω from 0 to 2π for m=0.

Fig. 3
Fig. 3

Unsaturated gain versus detuning for (a) Po=1 and (b) Po=10. Grating losses αgR are included in both plots; real coupling is κR=2.5; the mode is the central radial mode (Ω=0, m=0) at δR=0.

Fig. 4
Fig. 4

Unsaturated gain versus coupling for central radial modes. Included are effects of grating losses. Output power Po=10-5.

Fig. 5
Fig. 5

Unsaturated gain versus coupling for central radial modes (m=0 at δR=0) and first side modes (m=1 at δR±π). Grating losses are included; the output power is above threshold (Po=10).

Fig. 6
Fig. 6

Power (in units of saturation power) versus CGDFB radius for various couplings and grating losses. The mode is the central radial mode; Po=10. For κR=0.1 the power distributions are essentially identical for both lossless and lossy gratings.

Fig. 7
Fig. 7

Saturated gain versus CGDFB radius for conditions of Fig. 6. For simplicity, data are not shown for κR=0.5.

Fig. 8
Fig. 8

Comparison of saturated gain versus CGDFB radius for the central radial mode (m=0) and the first side modes (m=1). Output power Po=10; coupling κR=2.5; grating losses αgR=0.0 and 0.5.

Fig. 9
Fig. 9

Comparison of power distributions for conditions of Fig. 8.

Equations (9)

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

dAmdr=iπβr2Km(r)Hm(2)(βr)[Am(r)Hm(1)(βr)+Bm(r)Hm(2)(βr)],
dBmdr=-iπβr2Km(r)Hm(1)(βr)[Am(r)Hm(1)(βr)+Bm(r)Hm(2)(βr)],
Km(r)=i[αg-αm(r)]+2κ cos(2β0r-Ω),
αm(r)=αom1+P(r)/C(r)Isat,
αm(r)=αom1+[|Am(r)|2+|Bm(r)|2]/r.
|B(R)|=0.
|A(R)|2=Po,
ddr[|A(r)|2-|B(r)|2]=2[αm(r)-αg]×[|A(r)|2+|B(r)|2].
|A(R)|2=20R[αm(r)-αg][|A(r)|2+|B(r)|2]dr.

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