Abstract

Conditions for the injection locking of vertical-cavity laser arrays are considered. The injection power required for locking to be achieved depends on various parameters such as the injected-field distribution and frequency as well as the array geometry and pumping strength. The circulating field distribution, driven by the injected field, is controlled mainly by the frequency detuning from the laser resonances as well as by the injected-field distribution.

© 1999 Optical Society of America

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  1. E. Kapon, J. Katz, and A. Yariv, “Supermode analysis of phase-locked arrays of semiconductor lasers,” Opt. Lett. 10, 125–127 (1984).
    [CrossRef]
  2. D. Botez and D. E. Ackley, “Phase-locked arrays of semiconductor diode lasers,” IEEE Circuits Devices Mag. 2, 8–17 (1986).
    [CrossRef]
  3. J. M. Verdiell, R. Frey, and J. P. Huignard, “Analysis of injection-locked gain-guided diode laser arrays,” IEEE J. Quantum Electron. 27, 396–401 (1991).
    [CrossRef]
  4. J. P. Hohimer, A. Owyoung, and G. R. Hadley, “Single-channel injection locking of a diode-laser array with a cw dye laser,” Appl. Phys. Lett. 47, 1244–1246 (1985).
    [CrossRef]
  5. S. MacCormack, J. Feinberg, and M. H. Garrett, “Injection locking a laser-diode array with a phase-conjugate beam,” Opt. Lett. 19, 120–122 (1994).
    [CrossRef] [PubMed]
  6. L. Goldberg, H. F. Taylor, J. F. Weller, and D. R. Scifres, “Injection locking of coupled-stripe diode laser arrays,” Appl. Phys. Lett. 46, 236–240 (1985).
    [CrossRef]
  7. J. L. Jewel, A. Scherer, S. L. McCall, V. H. Lee, S. Walker, J. P. Harbison, and T. Fishman, “Low-threshold electrically pumped vertical cavity surface-emitting microlasers,” Electron. Lett. 25, 1123–1124 (1989).
    [CrossRef]
  8. H. Soda, Y. Motegi, and K. Iga, “Surface-emitting GaInAsP/InP injection laser with short cavity length,” Electron. Lett. 18, 461–463 (1982).
    [CrossRef]
  9. D. G. Deppe, J. P. Van der Ziel, N. Chand, G. L. Zydzik, and S. N. G. Chu, “Phase-coupled two-dimensional AlxGa1−xAs-GaAs vertical-cavity surface-emitting laser array,” Appl. Phys. Lett. 56, 2089–1091 (1990).
    [CrossRef]
  10. H.-J. Yoo, A. Scherer, J. P. Harbison, L. T. Florez, E. G. Paek, B. P. Van der Gaag, J. R. Hayes, A. V. Von Lehmen, E. Kapon, and Y. S. Kwon, “Fabrication of two-dimensional phased array of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 56, 1198–1200 (1990).
    [CrossRef]
  11. P. L. Gourley, M. E. Waren, G. R. Hadley, G. A. Wawter, T. M. Brennan, and B. E. Hammons, “Coherent beams from high efficiency two-dimensional surface-emitting semiconductor laser arrays,” Appl. Phys. Lett. 58, 890–892 (1991).
    [CrossRef]
  12. M. Orenstein, E. Kapon, J. P. Harbison, L. T. Florez, and N. G. Stoffel, “Large two dimensional arrays of phase-locked vertical cavity surface emitting lasers,” Appl. Phys. Lett. 60, 1535–1537 (1992).
    [CrossRef]
  13. R. A. Morgan and K. Kojima, “Optical characteristics of two-dimensionally coherently coupled vertical-cavity surface-emitting laser arrays,” Opt. Lett. 18, 352–354 (1993).
    [CrossRef] [PubMed]
  14. H.-J. Yoo, J. R. Hayes, E. G. Paek, A. Scherer, and Y. S. Kwon, “Array mode analysis of two-dimensional phased array of vertical cavity surface emitting lasers,” IEEE J. Quantum Electron. 26, 1039–1051 (1990).
    [CrossRef]
  15. A. Hardy and E. Kapon, “Coupled mode formulation for parallel laser resonators with application to vertical cavity semiconductor laser array,” IEEE J. Quantum Electron. 32, 966–971 (1996).
    [CrossRef]
  16. T. Fishman, A. Hardy, and E. Kapon, “Formulations for calculating the eigenmodes of vertical cavity laser arrays,” IEEE J. Quantum Electron. 33, 1756–1762 (1997).
    [CrossRef]
  17. H. Li, T. L. Lucas, J. G. McInerney, M. W. Wright, and R. A. Morgan, “Injection locking dynamics of vertical cavity semiconductor lasers under conventional and phase conjugate injection,” IEEE J. Quantum Electron. 32, 227–235 (1996).
    [CrossRef]
  18. J. Y. Law, G. H. M. van Tartwijk, and G. P. Agrawal, “Effects of transverse-mode competition on injection dynamics of vertical-cavity surface-emitting lasers,” Quantum Semiclassic. Opt. 9, 737–747 (1997).
    [CrossRef]
  19. R. Lang, “Injection locking of a semiconductor laser,” IEEE J. Quantum Electron. 18, 976–983 (1982).
    [CrossRef]
  20. A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986).
  21. R. J. Ram, D. I. Babic, R. A. York, and J. E. Bowers, “Spontaneous emission in microcavities with distributed mirrors,” IEEE J. Quantum Electron. 31, 399–410 (1995), and Refs. 10 and 11 therein.
    [CrossRef]
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1997 (2)

T. Fishman, A. Hardy, and E. Kapon, “Formulations for calculating the eigenmodes of vertical cavity laser arrays,” IEEE J. Quantum Electron. 33, 1756–1762 (1997).
[CrossRef]

J. Y. Law, G. H. M. van Tartwijk, and G. P. Agrawal, “Effects of transverse-mode competition on injection dynamics of vertical-cavity surface-emitting lasers,” Quantum Semiclassic. Opt. 9, 737–747 (1997).
[CrossRef]

1996 (2)

H. Li, T. L. Lucas, J. G. McInerney, M. W. Wright, and R. A. Morgan, “Injection locking dynamics of vertical cavity semiconductor lasers under conventional and phase conjugate injection,” IEEE J. Quantum Electron. 32, 227–235 (1996).
[CrossRef]

A. Hardy and E. Kapon, “Coupled mode formulation for parallel laser resonators with application to vertical cavity semiconductor laser array,” IEEE J. Quantum Electron. 32, 966–971 (1996).
[CrossRef]

1995 (1)

R. J. Ram, D. I. Babic, R. A. York, and J. E. Bowers, “Spontaneous emission in microcavities with distributed mirrors,” IEEE J. Quantum Electron. 31, 399–410 (1995), and Refs. 10 and 11 therein.
[CrossRef]

1994 (1)

1993 (1)

1992 (1)

M. Orenstein, E. Kapon, J. P. Harbison, L. T. Florez, and N. G. Stoffel, “Large two dimensional arrays of phase-locked vertical cavity surface emitting lasers,” Appl. Phys. Lett. 60, 1535–1537 (1992).
[CrossRef]

1991 (2)

P. L. Gourley, M. E. Waren, G. R. Hadley, G. A. Wawter, T. M. Brennan, and B. E. Hammons, “Coherent beams from high efficiency two-dimensional surface-emitting semiconductor laser arrays,” Appl. Phys. Lett. 58, 890–892 (1991).
[CrossRef]

J. M. Verdiell, R. Frey, and J. P. Huignard, “Analysis of injection-locked gain-guided diode laser arrays,” IEEE J. Quantum Electron. 27, 396–401 (1991).
[CrossRef]

1990 (3)

H.-J. Yoo, J. R. Hayes, E. G. Paek, A. Scherer, and Y. S. Kwon, “Array mode analysis of two-dimensional phased array of vertical cavity surface emitting lasers,” IEEE J. Quantum Electron. 26, 1039–1051 (1990).
[CrossRef]

D. G. Deppe, J. P. Van der Ziel, N. Chand, G. L. Zydzik, and S. N. G. Chu, “Phase-coupled two-dimensional AlxGa1−xAs-GaAs vertical-cavity surface-emitting laser array,” Appl. Phys. Lett. 56, 2089–1091 (1990).
[CrossRef]

H.-J. Yoo, A. Scherer, J. P. Harbison, L. T. Florez, E. G. Paek, B. P. Van der Gaag, J. R. Hayes, A. V. Von Lehmen, E. Kapon, and Y. S. Kwon, “Fabrication of two-dimensional phased array of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 56, 1198–1200 (1990).
[CrossRef]

1989 (1)

J. L. Jewel, A. Scherer, S. L. McCall, V. H. Lee, S. Walker, J. P. Harbison, and T. Fishman, “Low-threshold electrically pumped vertical cavity surface-emitting microlasers,” Electron. Lett. 25, 1123–1124 (1989).
[CrossRef]

1986 (1)

D. Botez and D. E. Ackley, “Phase-locked arrays of semiconductor diode lasers,” IEEE Circuits Devices Mag. 2, 8–17 (1986).
[CrossRef]

1985 (2)

L. Goldberg, H. F. Taylor, J. F. Weller, and D. R. Scifres, “Injection locking of coupled-stripe diode laser arrays,” Appl. Phys. Lett. 46, 236–240 (1985).
[CrossRef]

J. P. Hohimer, A. Owyoung, and G. R. Hadley, “Single-channel injection locking of a diode-laser array with a cw dye laser,” Appl. Phys. Lett. 47, 1244–1246 (1985).
[CrossRef]

1984 (1)

1982 (2)

R. Lang, “Injection locking of a semiconductor laser,” IEEE J. Quantum Electron. 18, 976–983 (1982).
[CrossRef]

H. Soda, Y. Motegi, and K. Iga, “Surface-emitting GaInAsP/InP injection laser with short cavity length,” Electron. Lett. 18, 461–463 (1982).
[CrossRef]

Ackley, D. E.

D. Botez and D. E. Ackley, “Phase-locked arrays of semiconductor diode lasers,” IEEE Circuits Devices Mag. 2, 8–17 (1986).
[CrossRef]

Agrawal, G. P.

J. Y. Law, G. H. M. van Tartwijk, and G. P. Agrawal, “Effects of transverse-mode competition on injection dynamics of vertical-cavity surface-emitting lasers,” Quantum Semiclassic. Opt. 9, 737–747 (1997).
[CrossRef]

Babic, D. I.

R. J. Ram, D. I. Babic, R. A. York, and J. E. Bowers, “Spontaneous emission in microcavities with distributed mirrors,” IEEE J. Quantum Electron. 31, 399–410 (1995), and Refs. 10 and 11 therein.
[CrossRef]

Botez, D.

D. Botez and D. E. Ackley, “Phase-locked arrays of semiconductor diode lasers,” IEEE Circuits Devices Mag. 2, 8–17 (1986).
[CrossRef]

Bowers, J. E.

R. J. Ram, D. I. Babic, R. A. York, and J. E. Bowers, “Spontaneous emission in microcavities with distributed mirrors,” IEEE J. Quantum Electron. 31, 399–410 (1995), and Refs. 10 and 11 therein.
[CrossRef]

Brennan, T. M.

P. L. Gourley, M. E. Waren, G. R. Hadley, G. A. Wawter, T. M. Brennan, and B. E. Hammons, “Coherent beams from high efficiency two-dimensional surface-emitting semiconductor laser arrays,” Appl. Phys. Lett. 58, 890–892 (1991).
[CrossRef]

Chand, N.

D. G. Deppe, J. P. Van der Ziel, N. Chand, G. L. Zydzik, and S. N. G. Chu, “Phase-coupled two-dimensional AlxGa1−xAs-GaAs vertical-cavity surface-emitting laser array,” Appl. Phys. Lett. 56, 2089–1091 (1990).
[CrossRef]

Chu, S. N. G.

D. G. Deppe, J. P. Van der Ziel, N. Chand, G. L. Zydzik, and S. N. G. Chu, “Phase-coupled two-dimensional AlxGa1−xAs-GaAs vertical-cavity surface-emitting laser array,” Appl. Phys. Lett. 56, 2089–1091 (1990).
[CrossRef]

Deppe, D. G.

D. G. Deppe, J. P. Van der Ziel, N. Chand, G. L. Zydzik, and S. N. G. Chu, “Phase-coupled two-dimensional AlxGa1−xAs-GaAs vertical-cavity surface-emitting laser array,” Appl. Phys. Lett. 56, 2089–1091 (1990).
[CrossRef]

Feinberg, J.

Fishman, T.

T. Fishman, A. Hardy, and E. Kapon, “Formulations for calculating the eigenmodes of vertical cavity laser arrays,” IEEE J. Quantum Electron. 33, 1756–1762 (1997).
[CrossRef]

J. L. Jewel, A. Scherer, S. L. McCall, V. H. Lee, S. Walker, J. P. Harbison, and T. Fishman, “Low-threshold electrically pumped vertical cavity surface-emitting microlasers,” Electron. Lett. 25, 1123–1124 (1989).
[CrossRef]

Florez, L. T.

M. Orenstein, E. Kapon, J. P. Harbison, L. T. Florez, and N. G. Stoffel, “Large two dimensional arrays of phase-locked vertical cavity surface emitting lasers,” Appl. Phys. Lett. 60, 1535–1537 (1992).
[CrossRef]

H.-J. Yoo, A. Scherer, J. P. Harbison, L. T. Florez, E. G. Paek, B. P. Van der Gaag, J. R. Hayes, A. V. Von Lehmen, E. Kapon, and Y. S. Kwon, “Fabrication of two-dimensional phased array of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 56, 1198–1200 (1990).
[CrossRef]

Frey, R.

J. M. Verdiell, R. Frey, and J. P. Huignard, “Analysis of injection-locked gain-guided diode laser arrays,” IEEE J. Quantum Electron. 27, 396–401 (1991).
[CrossRef]

Garrett, M. H.

Goldberg, L.

L. Goldberg, H. F. Taylor, J. F. Weller, and D. R. Scifres, “Injection locking of coupled-stripe diode laser arrays,” Appl. Phys. Lett. 46, 236–240 (1985).
[CrossRef]

Gourley, P. L.

P. L. Gourley, M. E. Waren, G. R. Hadley, G. A. Wawter, T. M. Brennan, and B. E. Hammons, “Coherent beams from high efficiency two-dimensional surface-emitting semiconductor laser arrays,” Appl. Phys. Lett. 58, 890–892 (1991).
[CrossRef]

Hadley, G. R.

P. L. Gourley, M. E. Waren, G. R. Hadley, G. A. Wawter, T. M. Brennan, and B. E. Hammons, “Coherent beams from high efficiency two-dimensional surface-emitting semiconductor laser arrays,” Appl. Phys. Lett. 58, 890–892 (1991).
[CrossRef]

J. P. Hohimer, A. Owyoung, and G. R. Hadley, “Single-channel injection locking of a diode-laser array with a cw dye laser,” Appl. Phys. Lett. 47, 1244–1246 (1985).
[CrossRef]

Hammons, B. E.

P. L. Gourley, M. E. Waren, G. R. Hadley, G. A. Wawter, T. M. Brennan, and B. E. Hammons, “Coherent beams from high efficiency two-dimensional surface-emitting semiconductor laser arrays,” Appl. Phys. Lett. 58, 890–892 (1991).
[CrossRef]

Harbison, J. P.

M. Orenstein, E. Kapon, J. P. Harbison, L. T. Florez, and N. G. Stoffel, “Large two dimensional arrays of phase-locked vertical cavity surface emitting lasers,” Appl. Phys. Lett. 60, 1535–1537 (1992).
[CrossRef]

H.-J. Yoo, A. Scherer, J. P. Harbison, L. T. Florez, E. G. Paek, B. P. Van der Gaag, J. R. Hayes, A. V. Von Lehmen, E. Kapon, and Y. S. Kwon, “Fabrication of two-dimensional phased array of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 56, 1198–1200 (1990).
[CrossRef]

J. L. Jewel, A. Scherer, S. L. McCall, V. H. Lee, S. Walker, J. P. Harbison, and T. Fishman, “Low-threshold electrically pumped vertical cavity surface-emitting microlasers,” Electron. Lett. 25, 1123–1124 (1989).
[CrossRef]

Hardy, A.

T. Fishman, A. Hardy, and E. Kapon, “Formulations for calculating the eigenmodes of vertical cavity laser arrays,” IEEE J. Quantum Electron. 33, 1756–1762 (1997).
[CrossRef]

A. Hardy and E. Kapon, “Coupled mode formulation for parallel laser resonators with application to vertical cavity semiconductor laser array,” IEEE J. Quantum Electron. 32, 966–971 (1996).
[CrossRef]

Hayes, J. R.

H.-J. Yoo, J. R. Hayes, E. G. Paek, A. Scherer, and Y. S. Kwon, “Array mode analysis of two-dimensional phased array of vertical cavity surface emitting lasers,” IEEE J. Quantum Electron. 26, 1039–1051 (1990).
[CrossRef]

H.-J. Yoo, A. Scherer, J. P. Harbison, L. T. Florez, E. G. Paek, B. P. Van der Gaag, J. R. Hayes, A. V. Von Lehmen, E. Kapon, and Y. S. Kwon, “Fabrication of two-dimensional phased array of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 56, 1198–1200 (1990).
[CrossRef]

Hohimer, J. P.

J. P. Hohimer, A. Owyoung, and G. R. Hadley, “Single-channel injection locking of a diode-laser array with a cw dye laser,” Appl. Phys. Lett. 47, 1244–1246 (1985).
[CrossRef]

Huignard, J. P.

J. M. Verdiell, R. Frey, and J. P. Huignard, “Analysis of injection-locked gain-guided diode laser arrays,” IEEE J. Quantum Electron. 27, 396–401 (1991).
[CrossRef]

Iga, K.

H. Soda, Y. Motegi, and K. Iga, “Surface-emitting GaInAsP/InP injection laser with short cavity length,” Electron. Lett. 18, 461–463 (1982).
[CrossRef]

Jewel, J. L.

J. L. Jewel, A. Scherer, S. L. McCall, V. H. Lee, S. Walker, J. P. Harbison, and T. Fishman, “Low-threshold electrically pumped vertical cavity surface-emitting microlasers,” Electron. Lett. 25, 1123–1124 (1989).
[CrossRef]

Kapon, E.

T. Fishman, A. Hardy, and E. Kapon, “Formulations for calculating the eigenmodes of vertical cavity laser arrays,” IEEE J. Quantum Electron. 33, 1756–1762 (1997).
[CrossRef]

A. Hardy and E. Kapon, “Coupled mode formulation for parallel laser resonators with application to vertical cavity semiconductor laser array,” IEEE J. Quantum Electron. 32, 966–971 (1996).
[CrossRef]

M. Orenstein, E. Kapon, J. P. Harbison, L. T. Florez, and N. G. Stoffel, “Large two dimensional arrays of phase-locked vertical cavity surface emitting lasers,” Appl. Phys. Lett. 60, 1535–1537 (1992).
[CrossRef]

H.-J. Yoo, A. Scherer, J. P. Harbison, L. T. Florez, E. G. Paek, B. P. Van der Gaag, J. R. Hayes, A. V. Von Lehmen, E. Kapon, and Y. S. Kwon, “Fabrication of two-dimensional phased array of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 56, 1198–1200 (1990).
[CrossRef]

E. Kapon, J. Katz, and A. Yariv, “Supermode analysis of phase-locked arrays of semiconductor lasers,” Opt. Lett. 10, 125–127 (1984).
[CrossRef]

Katz, J.

Kojima, K.

Kwon, Y. S.

H.-J. Yoo, A. Scherer, J. P. Harbison, L. T. Florez, E. G. Paek, B. P. Van der Gaag, J. R. Hayes, A. V. Von Lehmen, E. Kapon, and Y. S. Kwon, “Fabrication of two-dimensional phased array of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 56, 1198–1200 (1990).
[CrossRef]

H.-J. Yoo, J. R. Hayes, E. G. Paek, A. Scherer, and Y. S. Kwon, “Array mode analysis of two-dimensional phased array of vertical cavity surface emitting lasers,” IEEE J. Quantum Electron. 26, 1039–1051 (1990).
[CrossRef]

Lang, R.

R. Lang, “Injection locking of a semiconductor laser,” IEEE J. Quantum Electron. 18, 976–983 (1982).
[CrossRef]

Law, J. Y.

J. Y. Law, G. H. M. van Tartwijk, and G. P. Agrawal, “Effects of transverse-mode competition on injection dynamics of vertical-cavity surface-emitting lasers,” Quantum Semiclassic. Opt. 9, 737–747 (1997).
[CrossRef]

Lee, V. H.

J. L. Jewel, A. Scherer, S. L. McCall, V. H. Lee, S. Walker, J. P. Harbison, and T. Fishman, “Low-threshold electrically pumped vertical cavity surface-emitting microlasers,” Electron. Lett. 25, 1123–1124 (1989).
[CrossRef]

Li, H.

H. Li, T. L. Lucas, J. G. McInerney, M. W. Wright, and R. A. Morgan, “Injection locking dynamics of vertical cavity semiconductor lasers under conventional and phase conjugate injection,” IEEE J. Quantum Electron. 32, 227–235 (1996).
[CrossRef]

Lucas, T. L.

H. Li, T. L. Lucas, J. G. McInerney, M. W. Wright, and R. A. Morgan, “Injection locking dynamics of vertical cavity semiconductor lasers under conventional and phase conjugate injection,” IEEE J. Quantum Electron. 32, 227–235 (1996).
[CrossRef]

MacCormack, S.

McCall, S. L.

J. L. Jewel, A. Scherer, S. L. McCall, V. H. Lee, S. Walker, J. P. Harbison, and T. Fishman, “Low-threshold electrically pumped vertical cavity surface-emitting microlasers,” Electron. Lett. 25, 1123–1124 (1989).
[CrossRef]

McInerney, J. G.

H. Li, T. L. Lucas, J. G. McInerney, M. W. Wright, and R. A. Morgan, “Injection locking dynamics of vertical cavity semiconductor lasers under conventional and phase conjugate injection,” IEEE J. Quantum Electron. 32, 227–235 (1996).
[CrossRef]

Morgan, R. A.

H. Li, T. L. Lucas, J. G. McInerney, M. W. Wright, and R. A. Morgan, “Injection locking dynamics of vertical cavity semiconductor lasers under conventional and phase conjugate injection,” IEEE J. Quantum Electron. 32, 227–235 (1996).
[CrossRef]

R. A. Morgan and K. Kojima, “Optical characteristics of two-dimensionally coherently coupled vertical-cavity surface-emitting laser arrays,” Opt. Lett. 18, 352–354 (1993).
[CrossRef] [PubMed]

Motegi, Y.

H. Soda, Y. Motegi, and K. Iga, “Surface-emitting GaInAsP/InP injection laser with short cavity length,” Electron. Lett. 18, 461–463 (1982).
[CrossRef]

Orenstein, M.

M. Orenstein, E. Kapon, J. P. Harbison, L. T. Florez, and N. G. Stoffel, “Large two dimensional arrays of phase-locked vertical cavity surface emitting lasers,” Appl. Phys. Lett. 60, 1535–1537 (1992).
[CrossRef]

Owyoung, A.

J. P. Hohimer, A. Owyoung, and G. R. Hadley, “Single-channel injection locking of a diode-laser array with a cw dye laser,” Appl. Phys. Lett. 47, 1244–1246 (1985).
[CrossRef]

Paek, E. G.

H.-J. Yoo, J. R. Hayes, E. G. Paek, A. Scherer, and Y. S. Kwon, “Array mode analysis of two-dimensional phased array of vertical cavity surface emitting lasers,” IEEE J. Quantum Electron. 26, 1039–1051 (1990).
[CrossRef]

H.-J. Yoo, A. Scherer, J. P. Harbison, L. T. Florez, E. G. Paek, B. P. Van der Gaag, J. R. Hayes, A. V. Von Lehmen, E. Kapon, and Y. S. Kwon, “Fabrication of two-dimensional phased array of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 56, 1198–1200 (1990).
[CrossRef]

Ram, R. J.

R. J. Ram, D. I. Babic, R. A. York, and J. E. Bowers, “Spontaneous emission in microcavities with distributed mirrors,” IEEE J. Quantum Electron. 31, 399–410 (1995), and Refs. 10 and 11 therein.
[CrossRef]

Scherer, A.

H.-J. Yoo, A. Scherer, J. P. Harbison, L. T. Florez, E. G. Paek, B. P. Van der Gaag, J. R. Hayes, A. V. Von Lehmen, E. Kapon, and Y. S. Kwon, “Fabrication of two-dimensional phased array of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 56, 1198–1200 (1990).
[CrossRef]

H.-J. Yoo, J. R. Hayes, E. G. Paek, A. Scherer, and Y. S. Kwon, “Array mode analysis of two-dimensional phased array of vertical cavity surface emitting lasers,” IEEE J. Quantum Electron. 26, 1039–1051 (1990).
[CrossRef]

J. L. Jewel, A. Scherer, S. L. McCall, V. H. Lee, S. Walker, J. P. Harbison, and T. Fishman, “Low-threshold electrically pumped vertical cavity surface-emitting microlasers,” Electron. Lett. 25, 1123–1124 (1989).
[CrossRef]

Scifres, D. R.

L. Goldberg, H. F. Taylor, J. F. Weller, and D. R. Scifres, “Injection locking of coupled-stripe diode laser arrays,” Appl. Phys. Lett. 46, 236–240 (1985).
[CrossRef]

Soda, H.

H. Soda, Y. Motegi, and K. Iga, “Surface-emitting GaInAsP/InP injection laser with short cavity length,” Electron. Lett. 18, 461–463 (1982).
[CrossRef]

Stoffel, N. G.

M. Orenstein, E. Kapon, J. P. Harbison, L. T. Florez, and N. G. Stoffel, “Large two dimensional arrays of phase-locked vertical cavity surface emitting lasers,” Appl. Phys. Lett. 60, 1535–1537 (1992).
[CrossRef]

Taylor, H. F.

L. Goldberg, H. F. Taylor, J. F. Weller, and D. R. Scifres, “Injection locking of coupled-stripe diode laser arrays,” Appl. Phys. Lett. 46, 236–240 (1985).
[CrossRef]

Van der Gaag, B. P.

H.-J. Yoo, A. Scherer, J. P. Harbison, L. T. Florez, E. G. Paek, B. P. Van der Gaag, J. R. Hayes, A. V. Von Lehmen, E. Kapon, and Y. S. Kwon, “Fabrication of two-dimensional phased array of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 56, 1198–1200 (1990).
[CrossRef]

Van der Ziel, J. P.

D. G. Deppe, J. P. Van der Ziel, N. Chand, G. L. Zydzik, and S. N. G. Chu, “Phase-coupled two-dimensional AlxGa1−xAs-GaAs vertical-cavity surface-emitting laser array,” Appl. Phys. Lett. 56, 2089–1091 (1990).
[CrossRef]

van Tartwijk, G. H. M.

J. Y. Law, G. H. M. van Tartwijk, and G. P. Agrawal, “Effects of transverse-mode competition on injection dynamics of vertical-cavity surface-emitting lasers,” Quantum Semiclassic. Opt. 9, 737–747 (1997).
[CrossRef]

Verdiell, J. M.

J. M. Verdiell, R. Frey, and J. P. Huignard, “Analysis of injection-locked gain-guided diode laser arrays,” IEEE J. Quantum Electron. 27, 396–401 (1991).
[CrossRef]

Von Lehmen, A. V.

H.-J. Yoo, A. Scherer, J. P. Harbison, L. T. Florez, E. G. Paek, B. P. Van der Gaag, J. R. Hayes, A. V. Von Lehmen, E. Kapon, and Y. S. Kwon, “Fabrication of two-dimensional phased array of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 56, 1198–1200 (1990).
[CrossRef]

Walker, S.

J. L. Jewel, A. Scherer, S. L. McCall, V. H. Lee, S. Walker, J. P. Harbison, and T. Fishman, “Low-threshold electrically pumped vertical cavity surface-emitting microlasers,” Electron. Lett. 25, 1123–1124 (1989).
[CrossRef]

Waren, M. E.

P. L. Gourley, M. E. Waren, G. R. Hadley, G. A. Wawter, T. M. Brennan, and B. E. Hammons, “Coherent beams from high efficiency two-dimensional surface-emitting semiconductor laser arrays,” Appl. Phys. Lett. 58, 890–892 (1991).
[CrossRef]

Wawter, G. A.

P. L. Gourley, M. E. Waren, G. R. Hadley, G. A. Wawter, T. M. Brennan, and B. E. Hammons, “Coherent beams from high efficiency two-dimensional surface-emitting semiconductor laser arrays,” Appl. Phys. Lett. 58, 890–892 (1991).
[CrossRef]

Weller, J. F.

L. Goldberg, H. F. Taylor, J. F. Weller, and D. R. Scifres, “Injection locking of coupled-stripe diode laser arrays,” Appl. Phys. Lett. 46, 236–240 (1985).
[CrossRef]

Wright, M. W.

H. Li, T. L. Lucas, J. G. McInerney, M. W. Wright, and R. A. Morgan, “Injection locking dynamics of vertical cavity semiconductor lasers under conventional and phase conjugate injection,” IEEE J. Quantum Electron. 32, 227–235 (1996).
[CrossRef]

Yariv, A.

Yoo, H.-J.

H.-J. Yoo, J. R. Hayes, E. G. Paek, A. Scherer, and Y. S. Kwon, “Array mode analysis of two-dimensional phased array of vertical cavity surface emitting lasers,” IEEE J. Quantum Electron. 26, 1039–1051 (1990).
[CrossRef]

H.-J. Yoo, A. Scherer, J. P. Harbison, L. T. Florez, E. G. Paek, B. P. Van der Gaag, J. R. Hayes, A. V. Von Lehmen, E. Kapon, and Y. S. Kwon, “Fabrication of two-dimensional phased array of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 56, 1198–1200 (1990).
[CrossRef]

York, R. A.

R. J. Ram, D. I. Babic, R. A. York, and J. E. Bowers, “Spontaneous emission in microcavities with distributed mirrors,” IEEE J. Quantum Electron. 31, 399–410 (1995), and Refs. 10 and 11 therein.
[CrossRef]

Zydzik, G. L.

D. G. Deppe, J. P. Van der Ziel, N. Chand, G. L. Zydzik, and S. N. G. Chu, “Phase-coupled two-dimensional AlxGa1−xAs-GaAs vertical-cavity surface-emitting laser array,” Appl. Phys. Lett. 56, 2089–1091 (1990).
[CrossRef]

Appl. Phys. Lett. (6)

D. G. Deppe, J. P. Van der Ziel, N. Chand, G. L. Zydzik, and S. N. G. Chu, “Phase-coupled two-dimensional AlxGa1−xAs-GaAs vertical-cavity surface-emitting laser array,” Appl. Phys. Lett. 56, 2089–1091 (1990).
[CrossRef]

H.-J. Yoo, A. Scherer, J. P. Harbison, L. T. Florez, E. G. Paek, B. P. Van der Gaag, J. R. Hayes, A. V. Von Lehmen, E. Kapon, and Y. S. Kwon, “Fabrication of two-dimensional phased array of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 56, 1198–1200 (1990).
[CrossRef]

P. L. Gourley, M. E. Waren, G. R. Hadley, G. A. Wawter, T. M. Brennan, and B. E. Hammons, “Coherent beams from high efficiency two-dimensional surface-emitting semiconductor laser arrays,” Appl. Phys. Lett. 58, 890–892 (1991).
[CrossRef]

M. Orenstein, E. Kapon, J. P. Harbison, L. T. Florez, and N. G. Stoffel, “Large two dimensional arrays of phase-locked vertical cavity surface emitting lasers,” Appl. Phys. Lett. 60, 1535–1537 (1992).
[CrossRef]

L. Goldberg, H. F. Taylor, J. F. Weller, and D. R. Scifres, “Injection locking of coupled-stripe diode laser arrays,” Appl. Phys. Lett. 46, 236–240 (1985).
[CrossRef]

J. P. Hohimer, A. Owyoung, and G. R. Hadley, “Single-channel injection locking of a diode-laser array with a cw dye laser,” Appl. Phys. Lett. 47, 1244–1246 (1985).
[CrossRef]

Electron. Lett. (2)

J. L. Jewel, A. Scherer, S. L. McCall, V. H. Lee, S. Walker, J. P. Harbison, and T. Fishman, “Low-threshold electrically pumped vertical cavity surface-emitting microlasers,” Electron. Lett. 25, 1123–1124 (1989).
[CrossRef]

H. Soda, Y. Motegi, and K. Iga, “Surface-emitting GaInAsP/InP injection laser with short cavity length,” Electron. Lett. 18, 461–463 (1982).
[CrossRef]

IEEE Circuits Devices Mag. (1)

D. Botez and D. E. Ackley, “Phase-locked arrays of semiconductor diode lasers,” IEEE Circuits Devices Mag. 2, 8–17 (1986).
[CrossRef]

IEEE J. Quantum Electron. (7)

J. M. Verdiell, R. Frey, and J. P. Huignard, “Analysis of injection-locked gain-guided diode laser arrays,” IEEE J. Quantum Electron. 27, 396–401 (1991).
[CrossRef]

R. J. Ram, D. I. Babic, R. A. York, and J. E. Bowers, “Spontaneous emission in microcavities with distributed mirrors,” IEEE J. Quantum Electron. 31, 399–410 (1995), and Refs. 10 and 11 therein.
[CrossRef]

R. Lang, “Injection locking of a semiconductor laser,” IEEE J. Quantum Electron. 18, 976–983 (1982).
[CrossRef]

H.-J. Yoo, J. R. Hayes, E. G. Paek, A. Scherer, and Y. S. Kwon, “Array mode analysis of two-dimensional phased array of vertical cavity surface emitting lasers,” IEEE J. Quantum Electron. 26, 1039–1051 (1990).
[CrossRef]

A. Hardy and E. Kapon, “Coupled mode formulation for parallel laser resonators with application to vertical cavity semiconductor laser array,” IEEE J. Quantum Electron. 32, 966–971 (1996).
[CrossRef]

T. Fishman, A. Hardy, and E. Kapon, “Formulations for calculating the eigenmodes of vertical cavity laser arrays,” IEEE J. Quantum Electron. 33, 1756–1762 (1997).
[CrossRef]

H. Li, T. L. Lucas, J. G. McInerney, M. W. Wright, and R. A. Morgan, “Injection locking dynamics of vertical cavity semiconductor lasers under conventional and phase conjugate injection,” IEEE J. Quantum Electron. 32, 227–235 (1996).
[CrossRef]

Opt. Lett. (3)

Quantum Semiclassic. Opt. (1)

J. Y. Law, G. H. M. van Tartwijk, and G. P. Agrawal, “Effects of transverse-mode competition on injection dynamics of vertical-cavity surface-emitting lasers,” Quantum Semiclassic. Opt. 9, 737–747 (1997).
[CrossRef]

Other (3)

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

F. B. Hildebrand, Methods of Applied Mathematics, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1965).

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Figures (8)

Fig. 1
Fig. 1

Schematic 1D representation of a phased-locked VCSEL array. Vertical cavities, which are formed by the higher-reflectivity zones ρ=ρe, are coupled by means of diffraction. Light is emitted from the uniform-reflectivity mirror at the bottom of the array.

Fig. 2
Fig. 2

(a) Schematic illustration of the phasor field components that describe injection locking in VCSEL arrays. Both the driven field Ec and the injected field Einj are oscillating at ω=ωinj, whereas the free-running field Ef (not shown) is oscillating at ω=ωf. At steady state, the driven field Ec(y) is a phasor sum of Einj(y) and the round-trip propagated field Ec(x) [see Eq. (1)]. (b) Schematic of the unfolded laser cavity.

Fig. 3
Fig. 3

Contour diagram of Pc/Pinj for a single-element (1D) VCSEL with d=8 µm and D=12 µm. The ordinate is the gain |γ| (the dashed horizontal line is at the threshold γth), and the abscissa is the normalized injection frequency detuning Δωinj/ω0. The injected field is a Gaussian with a spot size of σ=1 µm centered at x0=0.5 µm. The number at each contour corresponds to log(Pc/Pinj).

Fig. 4
Fig. 4

Contour diagram of Pc/Pinj for a 1D two-element VCSEL array with d=4 µm, s=5 µm, and D=13 µm. The ordinate is the gain |γ|, and the abscissa is the normalized injection frequency detuning Δωinj/ω0. The injected field is a Gaussian with a spot size of σ=1 µm centered at x0=2.5 µm. The number at each contour corresponds to log(Pc/Pinj). The horizontal dashed line is at the threshold |γ|=γth. Points A–F correspond to some typical injection parameters for which the field distributions are presented in Fig. 5.

Fig. 5
Fig. 5

Field distributions Ec(x) (solid curves) that correspond to points A–F in Fig. 4: (a) field distribution at point F; (b) field distribution at point B, which is quite similar to the field distributions at points C and D; (c) field distribution at point E; (d) field distribution at point F. As a reference, the mirror reflectivity ρ(x) (solid curves) and the injected Gaussian field (dashed curves) are also plotted. For comparison, the symmetric u2(x) and the antisymmetric u1(x) eigenmodes are also given (dotted curves).

Fig. 6
Fig. 6

Normalized locking regimes of the two-element array (d=4 µm, s=5 µm, and D=13 µm) for various injection-field distributions Einj(x): u1(x), solid curve; u2(x), dashed curve; 10u2(x)+u1(x), circles; and a Gaussian beam, squares.

Fig. 7
Fig. 7

Power and gain for the two-element array of Fig. 6(a). Normalized driven power Pc/APsat and normalized free-running power Pf/APsat as a function of pump level γ0. (b) Saturable gain |γ| as a function of pump level γ0. The normalized injected power is Pinj/APsat=4.4×10-6, and the normalized injected frequency detuning is Δωinj/ω0=3.18×10-4. For γth<γ0<γ0(l) the VCSEL is locked (|γ|<γth, Pf=0). For γ0<γth the VCSEL is operated as an amplifier for the injected signal. For γ0>γ0(l), |γ|=γth, and the VCSEL is unlocked (Pf>0).

Fig. 8
Fig. 8

A 2D contour diagrams of the driven field |Ec(x)| of a 2×2 element array with d1=4, s=5 µm, D=13 µm, and a background reflectivity of ρb=0.7. The four lowest-order resonance frequencies are at Δω1/ω0=9.18×10-4, Δω2/ω0=Δω3/ω0=9.63×10-4, and Δω4/ω0=1.005×10-3. The gain is |γ|=γth=1.015, and the injected frequency is ωinj=ω4. (a) Symmetrically injected Gaussian beam with x0=(0, 0) and σ=4 µm. (b) Asymmetrically injected Gaussian beam with x0=(2.5, 2.5) and σ=1 µm. The array elements are marked with dashed lines.

Equations (19)

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Ec(y)=Einj(y)+γρ(y)Kω0(y, x)Ec(x)dx,
Kω0(y, x)=1jλexp(jkr)rcos(nˆ, r),
Kω0(y, x)=1jλexp(j2kL)2L×expjk4L[(y1-x1)2+(y2-x2)2].
um(y)=γmρ(y)Kω0(y, x)um(x)dx,
arrayaperture1ρ(x)ul(x)um(x)dx=δlm.
Phase{γm}2πN-(ωm-ω0)(2L/v),
Phase{γ}2πN-(ωinj-ω0)(2L/v).
Ec(x)=m=1cmum(x)m=1Mcmum(x),
Einj(x)=m=1amum(x)m=1Mamum(x),
cm=am1-γ/γm=am1-|γ/γm|exp[i(ωm-ωinj)(2L/v)].
am=arrayaperture1ρ(x)um(x)Einj(x)dx.
Ef(y)=γfρ(y)Kω0(y, x)Ef(x)dx.
|γf|=|γ|=exp[2d(g-α)/2],
g=g01+(Pc+Pf)/APsat.
gth=α+(1/d)ln|γ1|.
Pf=APsat(g0/gth-1)Pf(0).
PinjlockPf(0)=1-γthγm2+γthγm(2L/v)(ωinj-ωm)2,
PinjlockPf(0)=(2L/v)|ωinj-ω1|=2πN|ωinj-ω1|/ω0.
Einj(x)=Ei exp-(x-x0)2σ2,

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