Abstract

Injection locking of vertical-cavity surface-emitting laser arrays is analyzed at steady state, including the effect of spatial hole burning. A free-running laser array (i.e., without injection), that operates well-above threshold, can exhibit multimode oscillations. Consequently, each of the free-running modes (at different frequencies) needs a different locking injection power. For low pump levels, just above threshold, the array is single mode and, as expected, the results bear a close resemblance to those of the average gain analysis, which ignores spatial hole burning.

© 2000 Optical Society of America

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  1. E. Kapon, J. Katz, A. Yariv, “Supermode analysis of phase-locked arrays of semiconductor lasers,” Opt. Lett. 10, 125–127 (1984).
    [CrossRef]
  2. D. Botez, D. E. Ackley, “Phase-locked arrays of semiconductor diode lasers,” IEEE Circuits Devices Mag. 2, 8–17 (1986).
    [CrossRef]
  3. H. Pier, E. Kapon, “Photon localization in lattices of coupled vertical-cavity surface-emitting lasers between one and two dimensionalities,” Opt. Lett. 22, 546–548 (1997).
    [CrossRef] [PubMed]
  4. T. Fishman, E. Kapon, H. Pier, A. Hardy, “Modal expansion analysis of strained photonic lattices based on vertical cavity surface emitting laser arrays,” Appl. Phys. Lett. 74, 3595–3597 (1999), and references therein.
  5. M. Orenstein, E. Kapon, J. P. Harbison, L. T. Florez, N. G. Stoffel, “Large two dimensional arrays of phase-locked vertical cavity surface emitting lasers,” Appl. Phys. Lett. 60, 1535–1537 (1992).
    [CrossRef]
  6. 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, Y. S. Kwon, “Fabrication of a two-dimensional phased array of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 56, 1198–1200 (1990).
    [CrossRef]
  7. T. Fishman, A. Hardy, E. Kapon, “Formulations for calculating the eigenmodes of vertical cavity laser arrays,” IEEE J. Quantum Electron. 33, 1756–1762 (1997).
    [CrossRef]
  8. J. M. Verdiell, R. Frey, J. P. Huignard, “Analysis of injection-locked gain-guided diode laser arrays,” IEEE J. Quantum Electron. 27, 396–401 (1991).
    [CrossRef]
  9. J. P. Hohimer, A. Owyoung, 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]
  10. S. MacCormack, J. Feinberg, M. H. Garrett, “Injection locking a laser-diode array with a phase-conjugate beam,” Opt. Lett. 19, 120–122 (1994).
    [CrossRef] [PubMed]
  11. L. Golberg, H. F. Taylor, J. F. Weller, “Injection locking of coupled-stripe diode laser arrays,” Appl. Phys. Lett. 46, 236–240 (1985).
    [CrossRef]
  12. T. Fishman, A. Hardy, E. Kapon, H. Pier, “Injection locking of shear-strain photonic lattices based on VCSEL arrays,” in Advanced Semiconductor Lasers and Their Applications, L. Hollberg, R. J. Lang, eds., Vol. 31 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2000).
  13. H. Li, T. L. Lucas, J. G. McInerney, M. W. Wright, 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]
  14. J. Y. Law, G. H. M. van Tartwijk, 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]
  15. T. Fishman, A. Hardy, “Injection-locking analysis of vertical-cavity laser arrays,” J. Opt. Soc. Am. B 16, 38–45 (1999).
    [CrossRef]
  16. G. P. Agrawal, N. K. Dutta, Long-Wavelength Semiconductor Lasers (Van Nostrand-Reinhold, New York, 1986).
    [CrossRef]
  17. S.-S. Wang, H. G. Winful, “Propagation model for the dynamics of gain-guided semiconductor laser arrays,” J. Appl. Phys. 73, 462–464 (1993).
    [CrossRef]
  18. R. Lang, “Injection locking of a semiconductor laser,” IEEE J. Quantum Electron. QE-18, 976–983 (1982).
    [CrossRef]
  19. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
  20. R. J. Ram, D. I. Babic, R. A. York, J. E. Bowers, “Spontaneous emission in microcavities with distributed mirrors,” IEEE J. Quantum Electron. 31, 399–410 (1995), and Refs. 10 and 11 therein.
  21. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).
  22. M. Shimizu, F. Koyama, K. Iga, “Transverse mode analysis for surface emitting laser using beam propagation method,” Trans. Inst. Electron. Inf. Commun. Eng. Part E 74, 3334–3340 (1991).
  23. F. B. Hildebrand, Methods of Applied Mathematics, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1965).
  24. A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
    [CrossRef]

1999 (2)

T. Fishman, E. Kapon, H. Pier, A. Hardy, “Modal expansion analysis of strained photonic lattices based on vertical cavity surface emitting laser arrays,” Appl. Phys. Lett. 74, 3595–3597 (1999), and references therein.

T. Fishman, A. Hardy, “Injection-locking analysis of vertical-cavity laser arrays,” J. Opt. Soc. Am. B 16, 38–45 (1999).
[CrossRef]

1997 (3)

H. Pier, E. Kapon, “Photon localization in lattices of coupled vertical-cavity surface-emitting lasers between one and two dimensionalities,” Opt. Lett. 22, 546–548 (1997).
[CrossRef] [PubMed]

J. Y. Law, G. H. M. van Tartwijk, 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]

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

1996 (1)

H. Li, T. L. Lucas, J. G. McInerney, M. W. Wright, 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]

1995 (1)

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

1994 (1)

1993 (1)

S.-S. Wang, H. G. Winful, “Propagation model for the dynamics of gain-guided semiconductor laser arrays,” J. Appl. Phys. 73, 462–464 (1993).
[CrossRef]

1992 (1)

M. Orenstein, E. Kapon, J. P. Harbison, L. T. Florez, 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)

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

M. Shimizu, F. Koyama, K. Iga, “Transverse mode analysis for surface emitting laser using beam propagation method,” Trans. Inst. Electron. Inf. Commun. Eng. Part E 74, 3334–3340 (1991).

1990 (1)

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, Y. S. Kwon, “Fabrication of a two-dimensional phased array of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 56, 1198–1200 (1990).
[CrossRef]

1986 (1)

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

1985 (2)

J. P. Hohimer, A. Owyoung, 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]

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

1984 (1)

1982 (1)

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

1961 (1)

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
[CrossRef]

Ackley, D. E.

D. Botez, 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, 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]

G. P. Agrawal, N. K. Dutta, Long-Wavelength Semiconductor Lasers (Van Nostrand-Reinhold, New York, 1986).
[CrossRef]

Babic, D. I.

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

Botez, D.

D. Botez, 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, J. E. Bowers, “Spontaneous emission in microcavities with distributed mirrors,” IEEE J. Quantum Electron. 31, 399–410 (1995), and Refs. 10 and 11 therein.

Dutta, N. K.

G. P. Agrawal, N. K. Dutta, Long-Wavelength Semiconductor Lasers (Van Nostrand-Reinhold, New York, 1986).
[CrossRef]

Feinberg, J.

Fishman, T.

T. Fishman, A. Hardy, “Injection-locking analysis of vertical-cavity laser arrays,” J. Opt. Soc. Am. B 16, 38–45 (1999).
[CrossRef]

T. Fishman, E. Kapon, H. Pier, A. Hardy, “Modal expansion analysis of strained photonic lattices based on vertical cavity surface emitting laser arrays,” Appl. Phys. Lett. 74, 3595–3597 (1999), and references therein.

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

T. Fishman, A. Hardy, E. Kapon, H. Pier, “Injection locking of shear-strain photonic lattices based on VCSEL arrays,” in Advanced Semiconductor Lasers and Their Applications, L. Hollberg, R. J. Lang, eds., Vol. 31 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2000).

Florez, L. T.

M. Orenstein, E. Kapon, J. P. Harbison, L. T. Florez, 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, Y. S. Kwon, “Fabrication of a two-dimensional phased array of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 56, 1198–1200 (1990).
[CrossRef]

Fox, A. G.

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
[CrossRef]

Frey, R.

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

Garrett, M. H.

Golberg, L.

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

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).

Hadley, G. R.

J. P. Hohimer, A. Owyoung, 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]

Harbison, J. P.

M. Orenstein, E. Kapon, J. P. Harbison, L. T. Florez, 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, Y. S. Kwon, “Fabrication of a two-dimensional phased array of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 56, 1198–1200 (1990).
[CrossRef]

Hardy, A.

T. Fishman, E. Kapon, H. Pier, A. Hardy, “Modal expansion analysis of strained photonic lattices based on vertical cavity surface emitting laser arrays,” Appl. Phys. Lett. 74, 3595–3597 (1999), and references therein.

T. Fishman, A. Hardy, “Injection-locking analysis of vertical-cavity laser arrays,” J. Opt. Soc. Am. B 16, 38–45 (1999).
[CrossRef]

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

T. Fishman, A. Hardy, E. Kapon, H. Pier, “Injection locking of shear-strain photonic lattices based on VCSEL arrays,” in Advanced Semiconductor Lasers and Their Applications, L. Hollberg, R. J. Lang, eds., Vol. 31 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2000).

Hayes, J. R.

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, Y. S. Kwon, “Fabrication of a two-dimensional phased array of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 56, 1198–1200 (1990).
[CrossRef]

Hildebrand, F. B.

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

Hohimer, J. P.

J. P. Hohimer, A. Owyoung, 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, J. P. Huignard, “Analysis of injection-locked gain-guided diode laser arrays,” IEEE J. Quantum Electron. 27, 396–401 (1991).
[CrossRef]

Iga, K.

M. Shimizu, F. Koyama, K. Iga, “Transverse mode analysis for surface emitting laser using beam propagation method,” Trans. Inst. Electron. Inf. Commun. Eng. Part E 74, 3334–3340 (1991).

Kapon, E.

T. Fishman, E. Kapon, H. Pier, A. Hardy, “Modal expansion analysis of strained photonic lattices based on vertical cavity surface emitting laser arrays,” Appl. Phys. Lett. 74, 3595–3597 (1999), and references therein.

H. Pier, E. Kapon, “Photon localization in lattices of coupled vertical-cavity surface-emitting lasers between one and two dimensionalities,” Opt. Lett. 22, 546–548 (1997).
[CrossRef] [PubMed]

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

M. Orenstein, E. Kapon, J. P. Harbison, L. T. Florez, 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, Y. S. Kwon, “Fabrication of a two-dimensional phased array of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 56, 1198–1200 (1990).
[CrossRef]

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

T. Fishman, A. Hardy, E. Kapon, H. Pier, “Injection locking of shear-strain photonic lattices based on VCSEL arrays,” in Advanced Semiconductor Lasers and Their Applications, L. Hollberg, R. J. Lang, eds., Vol. 31 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2000).

Katz, J.

Koyama, F.

M. Shimizu, F. Koyama, K. Iga, “Transverse mode analysis for surface emitting laser using beam propagation method,” Trans. Inst. Electron. Inf. Commun. Eng. Part E 74, 3334–3340 (1991).

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, Y. S. Kwon, “Fabrication of a two-dimensional phased array of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 56, 1198–1200 (1990).
[CrossRef]

Lang, R.

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

Law, J. Y.

J. Y. Law, G. H. M. van Tartwijk, 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]

Li, H.

H. Li, T. L. Lucas, J. G. McInerney, M. W. Wright, 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]

Li, T.

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
[CrossRef]

Lucas, T. L.

H. Li, T. L. Lucas, J. G. McInerney, M. W. Wright, 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.

McInerney, J. G.

H. Li, T. L. Lucas, J. G. McInerney, M. W. Wright, 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, 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]

Orenstein, M.

M. Orenstein, E. Kapon, J. P. Harbison, L. T. Florez, 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, 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, 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, Y. S. Kwon, “Fabrication of a two-dimensional phased array of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 56, 1198–1200 (1990).
[CrossRef]

Pier, H.

T. Fishman, E. Kapon, H. Pier, A. Hardy, “Modal expansion analysis of strained photonic lattices based on vertical cavity surface emitting laser arrays,” Appl. Phys. Lett. 74, 3595–3597 (1999), and references therein.

H. Pier, E. Kapon, “Photon localization in lattices of coupled vertical-cavity surface-emitting lasers between one and two dimensionalities,” Opt. Lett. 22, 546–548 (1997).
[CrossRef] [PubMed]

T. Fishman, A. Hardy, E. Kapon, H. Pier, “Injection locking of shear-strain photonic lattices based on VCSEL arrays,” in Advanced Semiconductor Lasers and Their Applications, L. Hollberg, R. J. Lang, eds., Vol. 31 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2000).

Ram, R. J.

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

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, Y. S. Kwon, “Fabrication of a two-dimensional phased array of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 56, 1198–1200 (1990).
[CrossRef]

Shimizu, M.

M. Shimizu, F. Koyama, K. Iga, “Transverse mode analysis for surface emitting laser using beam propagation method,” Trans. Inst. Electron. Inf. Commun. Eng. Part E 74, 3334–3340 (1991).

Siegman, A. E.

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

Stoffel, N. G.

M. Orenstein, E. Kapon, J. P. Harbison, L. T. Florez, 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. Golberg, H. F. Taylor, J. F. Weller, “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, Y. S. Kwon, “Fabrication of a two-dimensional phased array of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 56, 1198–1200 (1990).
[CrossRef]

van Tartwijk, G. H. M.

J. Y. Law, G. H. M. van Tartwijk, 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, 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, Y. S. Kwon, “Fabrication of a two-dimensional phased array of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 56, 1198–1200 (1990).
[CrossRef]

Wang, S.-S.

S.-S. Wang, H. G. Winful, “Propagation model for the dynamics of gain-guided semiconductor laser arrays,” J. Appl. Phys. 73, 462–464 (1993).
[CrossRef]

Weller, J. F.

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

Winful, H. G.

S.-S. Wang, H. G. Winful, “Propagation model for the dynamics of gain-guided semiconductor laser arrays,” J. Appl. Phys. 73, 462–464 (1993).
[CrossRef]

Wright, M. W.

H. Li, T. L. Lucas, J. G. McInerney, M. W. Wright, 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, 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, Y. S. Kwon, “Fabrication of a 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, J. E. Bowers, “Spontaneous emission in microcavities with distributed mirrors,” IEEE J. Quantum Electron. 31, 399–410 (1995), and Refs. 10 and 11 therein.

Appl. Phys. Lett. (5)

T. Fishman, E. Kapon, H. Pier, A. Hardy, “Modal expansion analysis of strained photonic lattices based on vertical cavity surface emitting laser arrays,” Appl. Phys. Lett. 74, 3595–3597 (1999), and references therein.

M. Orenstein, E. Kapon, J. P. Harbison, L. T. Florez, 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, Y. S. Kwon, “Fabrication of a two-dimensional phased array of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 56, 1198–1200 (1990).
[CrossRef]

J. P. Hohimer, A. Owyoung, 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]

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

Bell Syst. Tech. J. (1)

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Schematic representation of a VCSEL array.

Fig. 2
Fig. 2

Various fields that propagate in the cavity and are represented in Eq. (1).

Fig. 3
Fig. 3

Normalized free-running power P˜fi/P sat of the two lowest-order (the in-phase and the antiphase) modes as a function of the normalized pump level g p /gth for a two-element array with ρ b = 0.98, d = 4 µm, s = 5 µm, and D = 13 µm (see Fig. 1). The triangles, squares, and circles are, respectively, for L d = 2.4, L d = 0.7, and L d = 0 µm, whereas the dashed line represents the average gain analysis result.

Fig. 4
Fig. 4

Locking regimes of the two-element array of Fig. 3 with (a) β c = 0 and L d = 0.7 µm. For g p = 1.1gth (dashed curve) the laser is single mode. Therefore, only a single line is needed to describe the locking regime. For g p = 2g th (solid curve) both the antiphase u 1(y) and the in-phase u 2(y) modes are above threshold (see Fig. 3). Thus two lines are needed, as explained in the text. (b) Locking regimes of the same array with β c = 2 at g p = 2gth.

Equations (29)

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Ecy=Einjy+γyρy  Kω0y, xEcxdx.
Efiy=γfiyρy  Kω0y, xEfixdx,
Kω0y, x=1jλexpjk2L2Lexpj k4Ly1-x12+y2-x22,
γy|γy|expjϕinj-jϕβy,
γfiy|γfiy|expjϕfi-jϕβiy,
|γfiy|=|γy|=exp2dagy-α)/2,
Ld2T2+1+Pcy+i PfiyPsatgy=gpy.
gy=gpy1+Pcy+iPfiyPsat.
ϕinj2πN+ωinj-ω02L/vp,
ϕfi2πN+ωfi-ω02L/vp.
Ecy=m=1 cmumym=1Mf cmumy,
Einjy=m=1 amumym=1Mf amumy,
Efiy=m=1 bmiumym=1Mf bmiumy,
umy=γmρy  Kω0y, xumxdx,
arrayapertureulyumyρydy=δlm.
C=I-expjϕinjMG-1A
Bi=expjϕfiMGBi,
mlm=arrayapertureulyumyρy |γy|exp-jϕβydy.
am=arrayapertureumyEinjyρydy.
gth=α+1/daln|γ1|.
P˜inj=1S  |Einjy|2dy,
P˜fi=1S  Pfiydy0.
P˜c=1S  Pcydy
Einjy=Ei exp-y-y02/σ2,
Ecny=Einjy+γn-1yρy× Kω0y, xEcn-1xdx,
Efny=γn-1yρy  Kω0y, xEfn-1xdx,
Efn-1y=iEfn-1iy=i qin-1ueffiy.
μeffiueffiy=ρeffy  Kω0y, xueffixdx,
qin-1= Efn-1yueffiyρeffydy.

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