Abstract

Partially coherent, transversely coupled laser arrays are investigated within a stochastic coupled mode formalism. Predictions of the coherence or correlation functions in both the spectral and time domains are made. It is demonstrated that the coherence properties of the system in both domains are strongly dependent on the number and intensity of coupled modes. The theory can be useful for the study of semiconductor laser arrays, particularly vertical-cavity surface-emitting laser arrays.

© 2010 Optical Society of America

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  1. H. Yoo, A. Scherer, J. Harbison, L. Florez, E. Paek, B. van der Gaag, J. Hayes, A. von Lehmen, E. Kapon, and Y. Kwon, “Fabrication of a two-dimensional phased array of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 56, 1198-1200 (1990).
    [CrossRef]
  2. R. Morgan, K. Kojima, T. Mullally, G. Guth, M. Focht, R. Leibenguth, and M. Asom, “High-power coherently coupled 8×8 vertical cavity surface emitting laser array,” Appl. Phys. Lett. 61, 1160-1162 (1992).
    [CrossRef]
  3. M. Orenstein, E. Kapon, N. Stoffel, J. Harbison, L. Florez, and J. Wullert, “Two-dimensional phase locked arrays of vertical-cavity semiconductor lasers by mirror reflectivity modulation,” Appl. Phys. Lett. 58, 804-806 (1991).
    [CrossRef]
  4. L. Bao, N. Kim, L. Mawst, N. Elkin, V. Troshchieva, D. Vysotsky, and A. Napartovich, “Near-diffraction-limited coherent emission from large aperture antiguided vertical-cavity surface-emitting laser arrays,” Appl. Phys. Lett. 84, 320-322 (2004).
    [CrossRef]
  5. A. Lehman, D. Siriani, and K. Choquette, “Two-dimensional electronic beam-steering with implant-defined coherent vcsel arrays,” Electron. Lett. 43, 1202-1203 (2007).
    [CrossRef]
  6. P. Gourley, M. Warren, G. Hadley, G. Vawter, T. Brennan, and B. Hammons, “Coherent beams from high efficiency two-dimensional surface-emitting semiconductor laser arrays,” Appl. Phys. Lett. 58, 890-892 (1991).
    [CrossRef]
  7. F. Monti di Sopra, M. Brunner, H. Gauggel, H. Zappe, M. Moser, R. Hövel, and E. Kapon, “Continuous-wave operation of phase-coupled vertical-cavity surface-emitting laser arrays,” Appl. Phys. Lett. 77, 2283-2285 (2000).
    [CrossRef]
  8. A. Danner, J. Lee, J. Raftery, Jr, N. Yokouchi, and K. Choquette, “Coupled-defect photonic crystal vertical cavity surface emitting lasers,” Electron. Lett. 39, 1323-1324 (2003).
    [CrossRef]
  9. D. Serkland, K. Choquette, G. Hadley, K. Geib, and A. Allerman, “Two-element phased array of antiguided vertical-cavity lasers,” Appl. Phys. Lett. 75, 3754-3756 (1999).
    [CrossRef]
  10. D. Zhou and L. Mawst, “Two-dimensional phase-locked antiguided vertical-cavity surface-emitting laser arrays,”Appl. Phys. Lett. 77, 2307-2309 (2000).
    [CrossRef]
  11. K. Ebeling and L. Coldren, “Analysis of multielement semiconductor lasers,” J. Appl. Phys. 54, 2962-2969 (1983).
    [CrossRef]
  12. G. Hadley, “Modes of a two-dimensional phase-locked array of vertical-cavity surface-emitting lasers,” Opt. Lett. 15, 1215-1217 (1990).
    [CrossRef] [PubMed]
  13. 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]
  14. J. Butler, D. Ackley, and D. Botez, “Coupled-mode analysis of phase-locked injection laser arrays,” Appl. Phys. Lett. 44, 293-295 (1984).
    [CrossRef]
  15. E. Kapon, J. Katz, and A. Yariv, “Supermode analysis of phase-locked arrays of semiconductor lasers,” Opt. Lett. 9, 125-127 (1984).
    [CrossRef] [PubMed]
  16. A. Lehman, J. Raftery, P. Carney, and K. Choquette, “Coherence of photonic crystal vertical-cavity surface-emitting laser arrays,” IEEE J. Quantum Electron. 43, 25-30 (2007).
    [CrossRef]
  17. A. C. Lehman Harren, K. D. Choquette, and P. S. Carney, “Partial coherence in coupled photonic crystal vertical cavity laser arrays,” Opt. Lett. 34, 905-907 (2009).
    [CrossRef] [PubMed]
  18. L. A. Coldren and E. R. Hegblom, Vertical-Cavity Surface-Emitting Lasers: Design, Fabrication, and Applications (Cambridge Univ. Press, 1999), Chap. 2, pp. 32-67.
  19. S. L. Chuang, Physics of Optoelectronic Devices (Wiley, 1995).
  20. R. Syms, “Simple approximate theory for twin-guide Fabry-Perot laser amplifier switches,” J. Mod. Opt. 38, 1167-1180 (1991).
    [CrossRef]
  21. A. Einstein, “Method for the determination of the statistical values of observations concerning quantities subject to irregular fluctuations,” Arch. Sci. Phys. Nat. 37, 254-256 (1914).
  22. N. Wiener, “Generalized harmonic analysis,” Acta Math. 55, 117-258 (1930).
    [CrossRef]
  23. A. Y. Khintchine, “Correlation theory of stationary stochastic processes,” Math. Ann. 109, 604-615 (1934).
    [CrossRef]
  24. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995).

2009

2007

A. Lehman, D. Siriani, and K. Choquette, “Two-dimensional electronic beam-steering with implant-defined coherent vcsel arrays,” Electron. Lett. 43, 1202-1203 (2007).
[CrossRef]

A. Lehman, J. Raftery, P. Carney, and K. Choquette, “Coherence of photonic crystal vertical-cavity surface-emitting laser arrays,” IEEE J. Quantum Electron. 43, 25-30 (2007).
[CrossRef]

2004

L. Bao, N. Kim, L. Mawst, N. Elkin, V. Troshchieva, D. Vysotsky, and A. Napartovich, “Near-diffraction-limited coherent emission from large aperture antiguided vertical-cavity surface-emitting laser arrays,” Appl. Phys. Lett. 84, 320-322 (2004).
[CrossRef]

2003

A. Danner, J. Lee, J. Raftery, Jr, N. Yokouchi, and K. Choquette, “Coupled-defect photonic crystal vertical cavity surface emitting lasers,” Electron. Lett. 39, 1323-1324 (2003).
[CrossRef]

2000

F. Monti di Sopra, M. Brunner, H. Gauggel, H. Zappe, M. Moser, R. Hövel, and E. Kapon, “Continuous-wave operation of phase-coupled vertical-cavity surface-emitting laser arrays,” Appl. Phys. Lett. 77, 2283-2285 (2000).
[CrossRef]

D. Zhou and L. Mawst, “Two-dimensional phase-locked antiguided vertical-cavity surface-emitting laser arrays,”Appl. Phys. Lett. 77, 2307-2309 (2000).
[CrossRef]

1999

D. Serkland, K. Choquette, G. Hadley, K. Geib, and A. Allerman, “Two-element phased array of antiguided vertical-cavity lasers,” Appl. Phys. Lett. 75, 3754-3756 (1999).
[CrossRef]

1997

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]

1992

R. Morgan, K. Kojima, T. Mullally, G. Guth, M. Focht, R. Leibenguth, and M. Asom, “High-power coherently coupled 8×8 vertical cavity surface emitting laser array,” Appl. Phys. Lett. 61, 1160-1162 (1992).
[CrossRef]

1991

M. Orenstein, E. Kapon, N. Stoffel, J. Harbison, L. Florez, and J. Wullert, “Two-dimensional phase locked arrays of vertical-cavity semiconductor lasers by mirror reflectivity modulation,” Appl. Phys. Lett. 58, 804-806 (1991).
[CrossRef]

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

R. Syms, “Simple approximate theory for twin-guide Fabry-Perot laser amplifier switches,” J. Mod. Opt. 38, 1167-1180 (1991).
[CrossRef]

1990

H. Yoo, A. Scherer, J. Harbison, L. Florez, E. Paek, B. van der Gaag, J. Hayes, A. von Lehmen, E. Kapon, and Y. Kwon, “Fabrication of a two-dimensional phased array of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 56, 1198-1200 (1990).
[CrossRef]

G. Hadley, “Modes of a two-dimensional phase-locked array of vertical-cavity surface-emitting lasers,” Opt. Lett. 15, 1215-1217 (1990).
[CrossRef] [PubMed]

1984

E. Kapon, J. Katz, and A. Yariv, “Supermode analysis of phase-locked arrays of semiconductor lasers,” Opt. Lett. 9, 125-127 (1984).
[CrossRef] [PubMed]

J. Butler, D. Ackley, and D. Botez, “Coupled-mode analysis of phase-locked injection laser arrays,” Appl. Phys. Lett. 44, 293-295 (1984).
[CrossRef]

1983

K. Ebeling and L. Coldren, “Analysis of multielement semiconductor lasers,” J. Appl. Phys. 54, 2962-2969 (1983).
[CrossRef]

1934

A. Y. Khintchine, “Correlation theory of stationary stochastic processes,” Math. Ann. 109, 604-615 (1934).
[CrossRef]

1930

N. Wiener, “Generalized harmonic analysis,” Acta Math. 55, 117-258 (1930).
[CrossRef]

1914

A. Einstein, “Method for the determination of the statistical values of observations concerning quantities subject to irregular fluctuations,” Arch. Sci. Phys. Nat. 37, 254-256 (1914).

Ackley, D.

J. Butler, D. Ackley, and D. Botez, “Coupled-mode analysis of phase-locked injection laser arrays,” Appl. Phys. Lett. 44, 293-295 (1984).
[CrossRef]

Allerman, A.

D. Serkland, K. Choquette, G. Hadley, K. Geib, and A. Allerman, “Two-element phased array of antiguided vertical-cavity lasers,” Appl. Phys. Lett. 75, 3754-3756 (1999).
[CrossRef]

Asom, M.

R. Morgan, K. Kojima, T. Mullally, G. Guth, M. Focht, R. Leibenguth, and M. Asom, “High-power coherently coupled 8×8 vertical cavity surface emitting laser array,” Appl. Phys. Lett. 61, 1160-1162 (1992).
[CrossRef]

Bao, L.

L. Bao, N. Kim, L. Mawst, N. Elkin, V. Troshchieva, D. Vysotsky, and A. Napartovich, “Near-diffraction-limited coherent emission from large aperture antiguided vertical-cavity surface-emitting laser arrays,” Appl. Phys. Lett. 84, 320-322 (2004).
[CrossRef]

Botez, D.

J. Butler, D. Ackley, and D. Botez, “Coupled-mode analysis of phase-locked injection laser arrays,” Appl. Phys. Lett. 44, 293-295 (1984).
[CrossRef]

Brennan, T.

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

Brunner, M.

F. Monti di Sopra, M. Brunner, H. Gauggel, H. Zappe, M. Moser, R. Hövel, and E. Kapon, “Continuous-wave operation of phase-coupled vertical-cavity surface-emitting laser arrays,” Appl. Phys. Lett. 77, 2283-2285 (2000).
[CrossRef]

Butler, J.

J. Butler, D. Ackley, and D. Botez, “Coupled-mode analysis of phase-locked injection laser arrays,” Appl. Phys. Lett. 44, 293-295 (1984).
[CrossRef]

Carney, P.

A. Lehman, J. Raftery, P. Carney, and K. Choquette, “Coherence of photonic crystal vertical-cavity surface-emitting laser arrays,” IEEE J. Quantum Electron. 43, 25-30 (2007).
[CrossRef]

Carney, P. S.

Choquette, K.

A. Lehman, J. Raftery, P. Carney, and K. Choquette, “Coherence of photonic crystal vertical-cavity surface-emitting laser arrays,” IEEE J. Quantum Electron. 43, 25-30 (2007).
[CrossRef]

A. Lehman, D. Siriani, and K. Choquette, “Two-dimensional electronic beam-steering with implant-defined coherent vcsel arrays,” Electron. Lett. 43, 1202-1203 (2007).
[CrossRef]

A. Danner, J. Lee, J. Raftery, Jr, N. Yokouchi, and K. Choquette, “Coupled-defect photonic crystal vertical cavity surface emitting lasers,” Electron. Lett. 39, 1323-1324 (2003).
[CrossRef]

D. Serkland, K. Choquette, G. Hadley, K. Geib, and A. Allerman, “Two-element phased array of antiguided vertical-cavity lasers,” Appl. Phys. Lett. 75, 3754-3756 (1999).
[CrossRef]

Choquette, K. D.

Chuang, S. L.

S. L. Chuang, Physics of Optoelectronic Devices (Wiley, 1995).

Coldren, L.

K. Ebeling and L. Coldren, “Analysis of multielement semiconductor lasers,” J. Appl. Phys. 54, 2962-2969 (1983).
[CrossRef]

Coldren, L. A.

L. A. Coldren and E. R. Hegblom, Vertical-Cavity Surface-Emitting Lasers: Design, Fabrication, and Applications (Cambridge Univ. Press, 1999), Chap. 2, pp. 32-67.

Danner, A.

A. Danner, J. Lee, J. Raftery, Jr, N. Yokouchi, and K. Choquette, “Coupled-defect photonic crystal vertical cavity surface emitting lasers,” Electron. Lett. 39, 1323-1324 (2003).
[CrossRef]

Ebeling, K.

K. Ebeling and L. Coldren, “Analysis of multielement semiconductor lasers,” J. Appl. Phys. 54, 2962-2969 (1983).
[CrossRef]

Einstein, A.

A. Einstein, “Method for the determination of the statistical values of observations concerning quantities subject to irregular fluctuations,” Arch. Sci. Phys. Nat. 37, 254-256 (1914).

Elkin, N.

L. Bao, N. Kim, L. Mawst, N. Elkin, V. Troshchieva, D. Vysotsky, and A. Napartovich, “Near-diffraction-limited coherent emission from large aperture antiguided vertical-cavity surface-emitting laser arrays,” Appl. Phys. Lett. 84, 320-322 (2004).
[CrossRef]

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]

Florez, L.

M. Orenstein, E. Kapon, N. Stoffel, J. Harbison, L. Florez, and J. Wullert, “Two-dimensional phase locked arrays of vertical-cavity semiconductor lasers by mirror reflectivity modulation,” Appl. Phys. Lett. 58, 804-806 (1991).
[CrossRef]

H. Yoo, A. Scherer, J. Harbison, L. Florez, E. Paek, B. van der Gaag, J. Hayes, A. von Lehmen, E. Kapon, and Y. Kwon, “Fabrication of a two-dimensional phased array of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 56, 1198-1200 (1990).
[CrossRef]

Focht, M.

R. Morgan, K. Kojima, T. Mullally, G. Guth, M. Focht, R. Leibenguth, and M. Asom, “High-power coherently coupled 8×8 vertical cavity surface emitting laser array,” Appl. Phys. Lett. 61, 1160-1162 (1992).
[CrossRef]

Gauggel, H.

F. Monti di Sopra, M. Brunner, H. Gauggel, H. Zappe, M. Moser, R. Hövel, and E. Kapon, “Continuous-wave operation of phase-coupled vertical-cavity surface-emitting laser arrays,” Appl. Phys. Lett. 77, 2283-2285 (2000).
[CrossRef]

Geib, K.

D. Serkland, K. Choquette, G. Hadley, K. Geib, and A. Allerman, “Two-element phased array of antiguided vertical-cavity lasers,” Appl. Phys. Lett. 75, 3754-3756 (1999).
[CrossRef]

Gourley, P.

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

Guth, G.

R. Morgan, K. Kojima, T. Mullally, G. Guth, M. Focht, R. Leibenguth, and M. Asom, “High-power coherently coupled 8×8 vertical cavity surface emitting laser array,” Appl. Phys. Lett. 61, 1160-1162 (1992).
[CrossRef]

Hadley, G.

D. Serkland, K. Choquette, G. Hadley, K. Geib, and A. Allerman, “Two-element phased array of antiguided vertical-cavity lasers,” Appl. Phys. Lett. 75, 3754-3756 (1999).
[CrossRef]

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

G. Hadley, “Modes of a two-dimensional phase-locked array of vertical-cavity surface-emitting lasers,” Opt. Lett. 15, 1215-1217 (1990).
[CrossRef] [PubMed]

Hammons, B.

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

Harbison, J.

M. Orenstein, E. Kapon, N. Stoffel, J. Harbison, L. Florez, and J. Wullert, “Two-dimensional phase locked arrays of vertical-cavity semiconductor lasers by mirror reflectivity modulation,” Appl. Phys. Lett. 58, 804-806 (1991).
[CrossRef]

H. Yoo, A. Scherer, J. Harbison, L. Florez, E. Paek, B. van der Gaag, J. Hayes, A. von Lehmen, E. Kapon, and Y. 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, A. Hardy, and E. Kapon, “Formulations for calculating the eigenmodes of vertical-cavity laser arrays,” IEEE J. Quantum Electron. 33, 1756-1762 (1997).
[CrossRef]

Hayes, J.

H. Yoo, A. Scherer, J. Harbison, L. Florez, E. Paek, B. van der Gaag, J. Hayes, A. von Lehmen, E. Kapon, and Y. Kwon, “Fabrication of a two-dimensional phased array of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 56, 1198-1200 (1990).
[CrossRef]

Hegblom, E. R.

L. A. Coldren and E. R. Hegblom, Vertical-Cavity Surface-Emitting Lasers: Design, Fabrication, and Applications (Cambridge Univ. Press, 1999), Chap. 2, pp. 32-67.

Hövel, R.

F. Monti di Sopra, M. Brunner, H. Gauggel, H. Zappe, M. Moser, R. Hövel, and E. Kapon, “Continuous-wave operation of phase-coupled vertical-cavity surface-emitting laser arrays,” Appl. Phys. Lett. 77, 2283-2285 (2000).
[CrossRef]

Kapon, E.

F. Monti di Sopra, M. Brunner, H. Gauggel, H. Zappe, M. Moser, R. Hövel, and E. Kapon, “Continuous-wave operation of phase-coupled vertical-cavity surface-emitting laser arrays,” Appl. Phys. Lett. 77, 2283-2285 (2000).
[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]

M. Orenstein, E. Kapon, N. Stoffel, J. Harbison, L. Florez, and J. Wullert, “Two-dimensional phase locked arrays of vertical-cavity semiconductor lasers by mirror reflectivity modulation,” Appl. Phys. Lett. 58, 804-806 (1991).
[CrossRef]

H. Yoo, A. Scherer, J. Harbison, L. Florez, E. Paek, B. van der Gaag, J. Hayes, A. von Lehmen, E. Kapon, and Y. 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, and A. Yariv, “Supermode analysis of phase-locked arrays of semiconductor lasers,” Opt. Lett. 9, 125-127 (1984).
[CrossRef] [PubMed]

Katz, J.

Khintchine, A. Y.

A. Y. Khintchine, “Correlation theory of stationary stochastic processes,” Math. Ann. 109, 604-615 (1934).
[CrossRef]

Kim, N.

L. Bao, N. Kim, L. Mawst, N. Elkin, V. Troshchieva, D. Vysotsky, and A. Napartovich, “Near-diffraction-limited coherent emission from large aperture antiguided vertical-cavity surface-emitting laser arrays,” Appl. Phys. Lett. 84, 320-322 (2004).
[CrossRef]

Kojima, K.

R. Morgan, K. Kojima, T. Mullally, G. Guth, M. Focht, R. Leibenguth, and M. Asom, “High-power coherently coupled 8×8 vertical cavity surface emitting laser array,” Appl. Phys. Lett. 61, 1160-1162 (1992).
[CrossRef]

Kwon, Y.

H. Yoo, A. Scherer, J. Harbison, L. Florez, E. Paek, B. van der Gaag, J. Hayes, A. von Lehmen, E. Kapon, and Y. Kwon, “Fabrication of a two-dimensional phased array of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 56, 1198-1200 (1990).
[CrossRef]

Lee, J.

A. Danner, J. Lee, J. Raftery, Jr, N. Yokouchi, and K. Choquette, “Coupled-defect photonic crystal vertical cavity surface emitting lasers,” Electron. Lett. 39, 1323-1324 (2003).
[CrossRef]

Lehman, A.

A. Lehman, D. Siriani, and K. Choquette, “Two-dimensional electronic beam-steering with implant-defined coherent vcsel arrays,” Electron. Lett. 43, 1202-1203 (2007).
[CrossRef]

A. Lehman, J. Raftery, P. Carney, and K. Choquette, “Coherence of photonic crystal vertical-cavity surface-emitting laser arrays,” IEEE J. Quantum Electron. 43, 25-30 (2007).
[CrossRef]

Lehman Harren, A. C.

Leibenguth, R.

R. Morgan, K. Kojima, T. Mullally, G. Guth, M. Focht, R. Leibenguth, and M. Asom, “High-power coherently coupled 8×8 vertical cavity surface emitting laser array,” Appl. Phys. Lett. 61, 1160-1162 (1992).
[CrossRef]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995).

Mawst, L.

L. Bao, N. Kim, L. Mawst, N. Elkin, V. Troshchieva, D. Vysotsky, and A. Napartovich, “Near-diffraction-limited coherent emission from large aperture antiguided vertical-cavity surface-emitting laser arrays,” Appl. Phys. Lett. 84, 320-322 (2004).
[CrossRef]

D. Zhou and L. Mawst, “Two-dimensional phase-locked antiguided vertical-cavity surface-emitting laser arrays,”Appl. Phys. Lett. 77, 2307-2309 (2000).
[CrossRef]

Monti di Sopra, F.

F. Monti di Sopra, M. Brunner, H. Gauggel, H. Zappe, M. Moser, R. Hövel, and E. Kapon, “Continuous-wave operation of phase-coupled vertical-cavity surface-emitting laser arrays,” Appl. Phys. Lett. 77, 2283-2285 (2000).
[CrossRef]

Morgan, R.

R. Morgan, K. Kojima, T. Mullally, G. Guth, M. Focht, R. Leibenguth, and M. Asom, “High-power coherently coupled 8×8 vertical cavity surface emitting laser array,” Appl. Phys. Lett. 61, 1160-1162 (1992).
[CrossRef]

Moser, M.

F. Monti di Sopra, M. Brunner, H. Gauggel, H. Zappe, M. Moser, R. Hövel, and E. Kapon, “Continuous-wave operation of phase-coupled vertical-cavity surface-emitting laser arrays,” Appl. Phys. Lett. 77, 2283-2285 (2000).
[CrossRef]

Mullally, T.

R. Morgan, K. Kojima, T. Mullally, G. Guth, M. Focht, R. Leibenguth, and M. Asom, “High-power coherently coupled 8×8 vertical cavity surface emitting laser array,” Appl. Phys. Lett. 61, 1160-1162 (1992).
[CrossRef]

Napartovich, A.

L. Bao, N. Kim, L. Mawst, N. Elkin, V. Troshchieva, D. Vysotsky, and A. Napartovich, “Near-diffraction-limited coherent emission from large aperture antiguided vertical-cavity surface-emitting laser arrays,” Appl. Phys. Lett. 84, 320-322 (2004).
[CrossRef]

Orenstein, M.

M. Orenstein, E. Kapon, N. Stoffel, J. Harbison, L. Florez, and J. Wullert, “Two-dimensional phase locked arrays of vertical-cavity semiconductor lasers by mirror reflectivity modulation,” Appl. Phys. Lett. 58, 804-806 (1991).
[CrossRef]

Paek, E.

H. Yoo, A. Scherer, J. Harbison, L. Florez, E. Paek, B. van der Gaag, J. Hayes, A. von Lehmen, E. Kapon, and Y. Kwon, “Fabrication of a two-dimensional phased array of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 56, 1198-1200 (1990).
[CrossRef]

Raftery, J.

A. Lehman, J. Raftery, P. Carney, and K. Choquette, “Coherence of photonic crystal vertical-cavity surface-emitting laser arrays,” IEEE J. Quantum Electron. 43, 25-30 (2007).
[CrossRef]

A. Danner, J. Lee, J. Raftery, Jr, N. Yokouchi, and K. Choquette, “Coupled-defect photonic crystal vertical cavity surface emitting lasers,” Electron. Lett. 39, 1323-1324 (2003).
[CrossRef]

Scherer, A.

H. Yoo, A. Scherer, J. Harbison, L. Florez, E. Paek, B. van der Gaag, J. Hayes, A. von Lehmen, E. Kapon, and Y. Kwon, “Fabrication of a two-dimensional phased array of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 56, 1198-1200 (1990).
[CrossRef]

Serkland, D.

D. Serkland, K. Choquette, G. Hadley, K. Geib, and A. Allerman, “Two-element phased array of antiguided vertical-cavity lasers,” Appl. Phys. Lett. 75, 3754-3756 (1999).
[CrossRef]

Siriani, D.

A. Lehman, D. Siriani, and K. Choquette, “Two-dimensional electronic beam-steering with implant-defined coherent vcsel arrays,” Electron. Lett. 43, 1202-1203 (2007).
[CrossRef]

Stoffel, N.

M. Orenstein, E. Kapon, N. Stoffel, J. Harbison, L. Florez, and J. Wullert, “Two-dimensional phase locked arrays of vertical-cavity semiconductor lasers by mirror reflectivity modulation,” Appl. Phys. Lett. 58, 804-806 (1991).
[CrossRef]

Syms, R.

R. Syms, “Simple approximate theory for twin-guide Fabry-Perot laser amplifier switches,” J. Mod. Opt. 38, 1167-1180 (1991).
[CrossRef]

Troshchieva, V.

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[CrossRef]

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P. Gourley, M. Warren, G. Hadley, G. Vawter, T. Brennan, and B. Hammons, “Coherent beams from high efficiency two-dimensional surface-emitting semiconductor laser arrays,” Appl. Phys. Lett. 58, 890-892 (1991).
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H. Yoo, A. Scherer, J. Harbison, L. Florez, E. Paek, B. van der Gaag, J. Hayes, A. von Lehmen, E. Kapon, and Y. Kwon, “Fabrication of a two-dimensional phased array of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 56, 1198-1200 (1990).
[CrossRef]

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F. Monti di Sopra, M. Brunner, H. Gauggel, H. Zappe, M. Moser, R. Hövel, and E. Kapon, “Continuous-wave operation of phase-coupled vertical-cavity surface-emitting laser arrays,” Appl. Phys. Lett. 77, 2283-2285 (2000).
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D. Zhou and L. Mawst, “Two-dimensional phase-locked antiguided vertical-cavity surface-emitting laser arrays,”Appl. Phys. Lett. 77, 2307-2309 (2000).
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[CrossRef]

L. Bao, N. Kim, L. Mawst, N. Elkin, V. Troshchieva, D. Vysotsky, and A. Napartovich, “Near-diffraction-limited coherent emission from large aperture antiguided vertical-cavity surface-emitting laser arrays,” Appl. Phys. Lett. 84, 320-322 (2004).
[CrossRef]

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

F. Monti di Sopra, M. Brunner, H. Gauggel, H. Zappe, M. Moser, R. Hövel, and E. Kapon, “Continuous-wave operation of phase-coupled vertical-cavity surface-emitting laser arrays,” Appl. Phys. Lett. 77, 2283-2285 (2000).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Sketch of the laser array system under investigation.

Fig. 2
Fig. 2

Unperturbed power spectra for no detuning between the propagation constants of guides a and b. The spectra are identical.

Fig. 3
Fig. 3

(a) Deterministic and (b) stochastic coupled power spectra for no detuning between the propagation constants of guides a and b. The spectra from the two guides exactly overlap.

Fig. 4
Fig. 4

Maximum complex degree of coherence plotted as a function of the coupling strength κ for random boundary conditions and no detuning.

Fig. 5
Fig. 5

Unperturbed power spectra for a small detuning between the propagation constants of guides a and b. The spectrum from a is shown with a solid curve, and the spectrum from b is shown with a dashed curve.

Fig. 6
Fig. 6

(a) Deterministic and (b) stochastic coupled power spectra for a small detuning between the propagation constants of guides a and b and κ = 0.005 . The spectrum from a is shown with a solid curve, and the spectrum from b is shown with a dashed curve.

Fig. 7
Fig. 7

Degree of coherence for symmetric coupled waveguides as a function of the ratio of the − Mode spectral weighting to that of the + mode.

Fig. 8
Fig. 8

(a) Degree of coherence and (b) visibility for detuning Δ n = 0.005 as a function of the ratio of the − mode spectral weighting to that of the + mode. The three curves are for coupling strengths κ = 0.01 (solid), κ = 0.001 (dashed), and κ = 0.0001 (dotted).

Tables (3)

Tables Icon

Table 1 Definitions of Terms (units, μ m 1 )

Tables Icon

Table 2 Definitions of Unitless Terms

Tables Icon

Table 3 Parameters Used for Symmetric Calculation

Equations (43)

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U ( x , y , z ) = a ( z ) U ( a ) ( x , y ) + b ( z ) U ( b ) ( x , y ) ,
[ a ( z 0 ) b ( z 0 ) ] = V F V 1 [ a ( 0 ) b ( 0 ) ] ,
V = [ K K Δ β + ψ Δ β ψ ] ,
F = [ ξ + ( ω ) 0 0 ξ ( ω ) ] .
ξ + ( ω ) = σ + ( 1 R 2 ) e i β + L 1 R 2 e i 2 β + L ,
ξ ( ω ) = σ ( 1 R 2 ) e i β L 1 R 2 e i 2 β L .
β p = ω c n p ,
K = ω c κ ,
V = [ κ κ Δ n + η Δ n η ] ,
ξ + ( ω ) = σ + α α + i ( ω ω + ) ,
ξ ( ω ) = σ α α + i ( ω ω ) ,
[ a ( z 0 ) b ( z 0 ) ] = 1 2 η [ A a a a ( 0 ) + A a b b ( 0 ) A b a a ( 0 ) + A b b b ( 0 ) ] ,
A a a = ξ + ( ω ) ( η Δ n ) + ξ ( ω ) ( η + Δ n ) ,
A a b = ξ + ( ω ) κ ξ ( ω ) κ ,
A b a = ξ + ( ω ) κ * ξ ( ω ) κ * ,
A b b = ξ + ( ω ) ( η + Δ n ) + ξ ( ω ) ( η Δ n ) .
W = [ a * ( z 0 ) a ( z 0 ) a * ( z 0 ) b ( z 0 ) b * ( z 0 ) a ( z 0 ) b * ( z 0 ) b ( z 0 ) ] = 1 | 2 η | 2 [ W a a W a b W b a W b b ] .
W a b = [ W a a W b b ] 1 2 e i ϕ ,
| μ ( ω ) | = | W a b [ W a a W b b ] 1 2 | ,
W [ V a V b ] = λ [ V a V b ] .
[ V a V b ] = 1 2 η * [ W a a 1 2 e i ϕ W b b 1 2 ] = 1 2 η * [ A a a a ( 0 ) + A a b b ( 0 ) A b a a ( 0 ) + A b b b ( 0 ) ] * ,
a * ( 0 ) a ( 0 ) = S a ( 0 ) ,
b * ( 0 ) b ( 0 ) = S b ( 0 ) ,
a * ( 0 ) b ( 0 ) = 0 .
W = [ a * ( z 0 ) a ( z 0 ) a * ( z 0 ) b ( z 0 ) b * ( z 0 ) a ( z 0 ) b * ( z 0 ) b ( z 0 ) ] = 1 | 2 η | 2 [ W a a W a b W b a W b b ] .
W a a = | A a a | 2 S a ( 0 ) + | A a b | 2 S b ( 0 ) ,
W b b = | A b a | 2 S a ( 0 ) + | A b b | 2 S b ( 0 ) ,
W a b = A a a * A b a S a ( 0 ) + A a b * A b b S b ( 0 ) .
[ V a V b ] = [ 1 0 ] ,
[ V a V b ] = [ 0 1 ] ,
[ V a V b ] = [ κ Δ n ± η ] * ,
Γ = 1 | 2 η | 2 α ( π 2 ) 1 2 [ Γ a a Γ a b Γ b a Γ b b ] .
I FF = Γ a a ( 0 ) + Γ b b ( 0 ) + 2 R { Γ a b ( τ ) } ,
I FF = I a + + I b + + I a + I b + 2 [ I a + I b + ] 1 2 e α | τ | cos ( ω + τ + ϕ ) 2 [ I a I b ] 1 2 e α | τ | cos ( ω τ + ϕ ) ,
I a + = σ + 2 S ( 0 ) [ ( η Δ n ) 2 + | κ | 2 ] ,
I a = σ 2 S ( 0 ) [ ( η + Δ n ) 2 + | κ | 2 ] ,
I b + = σ + 2 S ( 0 ) [ ( η + Δ n ) 2 + | κ | 2 ] ,
I b = σ 2 S ( 0 ) [ ( η Δ n ) 2 + | κ | 2 ] .
ω ¯ = ω + ω + 2 ,
Δ ω = ω ω + 2 ,
I FF I a + + I b + + I a + I b + 2 ( [ I a + I b + ] 1 2 [ I a I b ] 1 2 ) cos ( ω ¯ τ + ϕ ) .
| γ | = | [ I a + I b + ] 1 2 [ I a I b ] 1 2 [ ( I a + + I a ) ( I b + + I b ) ] 1 2 | .
V = 2 [ I a + I b + ] 1 2 [ I a I b ] 1 2 I a + + I b + + I a + I b .

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