Abstract

The vectorial model of two-dimensional photonic crystals based on coherently coupled arrays of Vertical Cavity Surface - Emitting Lasers (VCSELs) is proposed in non-Hermitian Hamiltonian eigenproblem formulation. The polarization modes of square-symmetry photonic lattices are investigated theoretically. Rich mode structure with complimentary patterns of intensity for orthogonal polarizations of electromagnetic Bloch wave is predicted. The predicted near-field patterns of the polarization modes are confirmed in measurements of InGaAs/AlGaAs VCSEL arrays emitting at 965nm wavelength.

© 2004 Optical Society of America

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  1. John D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic crystals: molding the flow of light, (Princeton University Press, Princeton, 1995)
  2. P. Russell, “Photonic Crystal Fibers,” Science 299, 358 (2003).
    [CrossRef] [PubMed]
  3. M. Orenstein, E. Kapon, and N. G. Stoffel et al., “Two-dimensional phase-locked arrays of vertical-cavity semiconductor lasers by mirror reflectivity modulation,” Appl. Phys. Lett. 58, 804 (1991).
    [CrossRef]
  4. 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 (1992).
    [CrossRef]
  5. P.L. Gourley, M.E. Warren, and G.R. Hadley et al., “Coherent beams from high efficiency two-dimensional surface-emitting semiconductor laser arrays,” Appl. Phys. Lett. 58, 890 (1991).
    [CrossRef]
  6. H. Pier and E. Kapon, “Photon localization in lattices of coupled vertical-cavity surface-emitting lasers with dimensionalities between one and two,” Opt. Lett. 22, 546 (1997).
    [CrossRef] [PubMed]
  7. A. Golshani, H. Pier, E. Kapon, and M. Moser, “Photon mode localization in disordered arrays of vertical cavity surface emitting lasers,” J. Appl. Phys. 85, 2454 (1999)
    [CrossRef]
  8. H. Pier, E. Kapon, and M. Moser, “Strain effects and phase transitions in photonic resonator crystals,” Nature (London) 407, 880–883 (2000).
    [CrossRef]
  9. C.-A. Berseth, G. Guerrero, and E. Kapon, and et al., “Mode confinement in VCSEL-based photonic heterostructures,” in Conference on Lasers and Electro-Optics, CLEO 2000, OSA Technical Digest (Optical Society of America, Washington, D.C., 2000), pp 171–172, CtuA48.
  10. H.-J. Yoo, J. R. Hayes, E. G. Paek, A. Scherer, and Y. -S. Kwon, “Array mode analysis of two-dimension phased arrays of vertical cavity surface emitting lasers,” IEEE J. Quantum Electron. 26, 1039 (1990).
    [CrossRef]
  11. A. Hardy and E. Kapon, “Coupled-mode formulations for parallel-laser resonators with application to vertical-cavity semiconductor-laser arrays,” IEEE J. Quantum Electron. 32, 966 (1996).
    [CrossRef]
  12. T. Fishman, E. Kapon, H. Pier, and A. Hardy, “Modal expansion analysis of strained photonic lattices based on vertical cavity surface emitting laser arrays,” Appl. Phys. Lett. 74, 3595 (1999).
    [CrossRef]
  13. L.J. Mawst, ““Anti” up the aperture [antiguided VCSEL structures],” IEEE Circuits and Devices Magazine,  19, 34 (2003)
    [CrossRef]
  14. D.-S. Song, Y.-J Lee, H.-W. Choi, and Y.-H. Lee, “Polarization-controlled, single-transverse-mode, photonic-crystal, vertical-cavity, surface-emitting lasers” Appl. Phys. Lett. 82, 3182 (2003)
    [CrossRef]
  15. P. Debernardi, G. P. Bava, F. Monti di Sopra, and M. B. Willemsen, “Features of vectorial modes in phase-coupled VCSEL arrays: experiments and theory,” IEEE J. Quantum Electron. 39, 109–119 (2003).
    [CrossRef]
  16. K. M. Leung and Y. F. Liu, “Photon band structures: The plane-wave method,” Phys. Rev. B 41, 10188–10190 (1990).
    [CrossRef]
  17. A. M. Khromykh, “Ring Generator in a Rotating Reference System,” Zh. Eksp. Teor. Fiz (Sov. Phys. JETP) 50, 281 (1966).
  18. D. L. Boiko, G. Guerrero, and E. Kapon, “Bloch wave states in photonic crystals based on VCSEL arrays,”in Proceedings of the 26-th International Conference on the Physics of Semiconductors 2002 (ICPS 2002), A.R. Long and J.H. Davies, ed. (The Institute of Physics, Conference Series Number, 171, London, 2003), pp P278-7, http://www.icps2002.org
  19. D. L. Boiko and E. Kapon “Theory of vertical cavity photonic crystals,” (manuscript in preparation).
  20. G. Guerrero, D.L. Boiko, and E. Kapon, “Dynamics of polarization modes in photonic crystals based on arrays of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett., (to be published).
  21. W.L. Erikson and S. Singh, “Polarization properties of Maxwell-Gaussian laser beams,” Phys. Rev. E 49, 5778 (1994).
    [CrossRef]
  22. E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. 48, 2071 (1969).
  23. J. E. Goell, “A circular-harmonic computer analysis of rectangular dielectric waveguides,” Bell Syst. Tech. J. 48, 2133 (1969).
  24. E. Kapon, J. Katz, and A. Yariv, “Supermode analysis of phase-locked arrays of semiconductor lasers,” Opt. Lett.9, 125 (1984); E. Kapon, “Supermode analysis of phase-locked arrays of semiconductor lasers,” Opt. Lett.9, 318 (1984).
    [CrossRef] [PubMed]
  25. L.D. Landau and E.M. Lifshitz, Mechanics (“Nauka”, Moscow, 1974)

2003 (4)

L.J. Mawst, ““Anti” up the aperture [antiguided VCSEL structures],” IEEE Circuits and Devices Magazine,  19, 34 (2003)
[CrossRef]

D.-S. Song, Y.-J Lee, H.-W. Choi, and Y.-H. Lee, “Polarization-controlled, single-transverse-mode, photonic-crystal, vertical-cavity, surface-emitting lasers” Appl. Phys. Lett. 82, 3182 (2003)
[CrossRef]

P. Debernardi, G. P. Bava, F. Monti di Sopra, and M. B. Willemsen, “Features of vectorial modes in phase-coupled VCSEL arrays: experiments and theory,” IEEE J. Quantum Electron. 39, 109–119 (2003).
[CrossRef]

P. Russell, “Photonic Crystal Fibers,” Science 299, 358 (2003).
[CrossRef] [PubMed]

2000 (1)

H. Pier, E. Kapon, and M. Moser, “Strain effects and phase transitions in photonic resonator crystals,” Nature (London) 407, 880–883 (2000).
[CrossRef]

1999 (2)

A. Golshani, H. Pier, E. Kapon, and M. Moser, “Photon mode localization in disordered arrays of vertical cavity surface emitting lasers,” J. Appl. Phys. 85, 2454 (1999)
[CrossRef]

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

1997 (1)

1996 (1)

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

1994 (1)

W.L. Erikson and S. Singh, “Polarization properties of Maxwell-Gaussian laser beams,” Phys. Rev. E 49, 5778 (1994).
[CrossRef]

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 (1992).
[CrossRef]

1991 (2)

P.L. Gourley, M.E. Warren, and G.R. Hadley et al., “Coherent beams from high efficiency two-dimensional surface-emitting semiconductor laser arrays,” Appl. Phys. Lett. 58, 890 (1991).
[CrossRef]

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

1990 (2)

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

K. M. Leung and Y. F. Liu, “Photon band structures: The plane-wave method,” Phys. Rev. B 41, 10188–10190 (1990).
[CrossRef]

1969 (2)

E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. 48, 2071 (1969).

J. E. Goell, “A circular-harmonic computer analysis of rectangular dielectric waveguides,” Bell Syst. Tech. J. 48, 2133 (1969).

1966 (1)

A. M. Khromykh, “Ring Generator in a Rotating Reference System,” Zh. Eksp. Teor. Fiz (Sov. Phys. JETP) 50, 281 (1966).

Bava, G. P.

P. Debernardi, G. P. Bava, F. Monti di Sopra, and M. B. Willemsen, “Features of vectorial modes in phase-coupled VCSEL arrays: experiments and theory,” IEEE J. Quantum Electron. 39, 109–119 (2003).
[CrossRef]

Berseth, C.-A.

C.-A. Berseth, G. Guerrero, and E. Kapon, and et al., “Mode confinement in VCSEL-based photonic heterostructures,” in Conference on Lasers and Electro-Optics, CLEO 2000, OSA Technical Digest (Optical Society of America, Washington, D.C., 2000), pp 171–172, CtuA48.

Boiko, D. L.

D. L. Boiko and E. Kapon “Theory of vertical cavity photonic crystals,” (manuscript in preparation).

D. L. Boiko, G. Guerrero, and E. Kapon, “Bloch wave states in photonic crystals based on VCSEL arrays,”in Proceedings of the 26-th International Conference on the Physics of Semiconductors 2002 (ICPS 2002), A.R. Long and J.H. Davies, ed. (The Institute of Physics, Conference Series Number, 171, London, 2003), pp P278-7, http://www.icps2002.org

Boiko, D.L.

G. Guerrero, D.L. Boiko, and E. Kapon, “Dynamics of polarization modes in photonic crystals based on arrays of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett., (to be published).

Choi, H.-W.

D.-S. Song, Y.-J Lee, H.-W. Choi, and Y.-H. Lee, “Polarization-controlled, single-transverse-mode, photonic-crystal, vertical-cavity, surface-emitting lasers” Appl. Phys. Lett. 82, 3182 (2003)
[CrossRef]

Debernardi, P.

P. Debernardi, G. P. Bava, F. Monti di Sopra, and M. B. Willemsen, “Features of vectorial modes in phase-coupled VCSEL arrays: experiments and theory,” IEEE J. Quantum Electron. 39, 109–119 (2003).
[CrossRef]

Erikson, W.L.

W.L. Erikson and S. Singh, “Polarization properties of Maxwell-Gaussian laser beams,” Phys. Rev. E 49, 5778 (1994).
[CrossRef]

Fishman, T.

T. Fishman, E. Kapon, H. Pier, and A. Hardy, “Modal expansion analysis of strained photonic lattices based on vertical cavity surface emitting laser arrays,” Appl. Phys. Lett. 74, 3595 (1999).
[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 (1992).
[CrossRef]

Goell, J. E.

J. E. Goell, “A circular-harmonic computer analysis of rectangular dielectric waveguides,” Bell Syst. Tech. J. 48, 2133 (1969).

Golshani, A.

A. Golshani, H. Pier, E. Kapon, and M. Moser, “Photon mode localization in disordered arrays of vertical cavity surface emitting lasers,” J. Appl. Phys. 85, 2454 (1999)
[CrossRef]

Gourley, P.L.

P.L. Gourley, M.E. Warren, and G.R. Hadley et al., “Coherent beams from high efficiency two-dimensional surface-emitting semiconductor laser arrays,” Appl. Phys. Lett. 58, 890 (1991).
[CrossRef]

Guerrero, G.

C.-A. Berseth, G. Guerrero, and E. Kapon, and et al., “Mode confinement in VCSEL-based photonic heterostructures,” in Conference on Lasers and Electro-Optics, CLEO 2000, OSA Technical Digest (Optical Society of America, Washington, D.C., 2000), pp 171–172, CtuA48.

G. Guerrero, D.L. Boiko, and E. Kapon, “Dynamics of polarization modes in photonic crystals based on arrays of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett., (to be published).

D. L. Boiko, G. Guerrero, and E. Kapon, “Bloch wave states in photonic crystals based on VCSEL arrays,”in Proceedings of the 26-th International Conference on the Physics of Semiconductors 2002 (ICPS 2002), A.R. Long and J.H. Davies, ed. (The Institute of Physics, Conference Series Number, 171, London, 2003), pp P278-7, http://www.icps2002.org

Hadley, G.R.

P.L. Gourley, M.E. Warren, and G.R. Hadley et al., “Coherent beams from high efficiency two-dimensional surface-emitting semiconductor laser arrays,” Appl. Phys. Lett. 58, 890 (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 (1992).
[CrossRef]

Hardy, A.

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

A. Hardy and E. Kapon, “Coupled-mode formulations for parallel-laser resonators with application to vertical-cavity semiconductor-laser arrays,” IEEE J. Quantum Electron. 32, 966 (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-dimension phased arrays of vertical cavity surface emitting lasers,” IEEE J. Quantum Electron. 26, 1039 (1990).
[CrossRef]

Joannopoulos, John D.

John D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic crystals: molding the flow of light, (Princeton University Press, Princeton, 1995)

Kapon, E.

H. Pier, E. Kapon, and M. Moser, “Strain effects and phase transitions in photonic resonator crystals,” Nature (London) 407, 880–883 (2000).
[CrossRef]

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

A. Golshani, H. Pier, E. Kapon, and M. Moser, “Photon mode localization in disordered arrays of vertical cavity surface emitting lasers,” J. Appl. Phys. 85, 2454 (1999)
[CrossRef]

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

A. Hardy and E. Kapon, “Coupled-mode formulations for parallel-laser resonators with application to vertical-cavity semiconductor-laser arrays,” IEEE J. Quantum Electron. 32, 966 (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 (1992).
[CrossRef]

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

G. Guerrero, D.L. Boiko, and E. Kapon, “Dynamics of polarization modes in photonic crystals based on arrays of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett., (to be published).

E. Kapon, J. Katz, and A. Yariv, “Supermode analysis of phase-locked arrays of semiconductor lasers,” Opt. Lett.9, 125 (1984); E. Kapon, “Supermode analysis of phase-locked arrays of semiconductor lasers,” Opt. Lett.9, 318 (1984).
[CrossRef] [PubMed]

E. Kapon, J. Katz, and A. Yariv, “Supermode analysis of phase-locked arrays of semiconductor lasers,” Opt. Lett.9, 125 (1984); E. Kapon, “Supermode analysis of phase-locked arrays of semiconductor lasers,” Opt. Lett.9, 318 (1984).
[CrossRef] [PubMed]

C.-A. Berseth, G. Guerrero, and E. Kapon, and et al., “Mode confinement in VCSEL-based photonic heterostructures,” in Conference on Lasers and Electro-Optics, CLEO 2000, OSA Technical Digest (Optical Society of America, Washington, D.C., 2000), pp 171–172, CtuA48.

D. L. Boiko, G. Guerrero, and E. Kapon, “Bloch wave states in photonic crystals based on VCSEL arrays,”in Proceedings of the 26-th International Conference on the Physics of Semiconductors 2002 (ICPS 2002), A.R. Long and J.H. Davies, ed. (The Institute of Physics, Conference Series Number, 171, London, 2003), pp P278-7, http://www.icps2002.org

D. L. Boiko and E. Kapon “Theory of vertical cavity photonic crystals,” (manuscript in preparation).

Katz, J.

E. Kapon, J. Katz, and A. Yariv, “Supermode analysis of phase-locked arrays of semiconductor lasers,” Opt. Lett.9, 125 (1984); E. Kapon, “Supermode analysis of phase-locked arrays of semiconductor lasers,” Opt. Lett.9, 318 (1984).
[CrossRef] [PubMed]

Khromykh, A. M.

A. M. Khromykh, “Ring Generator in a Rotating Reference System,” Zh. Eksp. Teor. Fiz (Sov. Phys. JETP) 50, 281 (1966).

Kwon, Y. -S.

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

Landau, L.D.

L.D. Landau and E.M. Lifshitz, Mechanics (“Nauka”, Moscow, 1974)

Lee, Y.-H.

D.-S. Song, Y.-J Lee, H.-W. Choi, and Y.-H. Lee, “Polarization-controlled, single-transverse-mode, photonic-crystal, vertical-cavity, surface-emitting lasers” Appl. Phys. Lett. 82, 3182 (2003)
[CrossRef]

Lee, Y.-J

D.-S. Song, Y.-J Lee, H.-W. Choi, and Y.-H. Lee, “Polarization-controlled, single-transverse-mode, photonic-crystal, vertical-cavity, surface-emitting lasers” Appl. Phys. Lett. 82, 3182 (2003)
[CrossRef]

Leung, K. M.

K. M. Leung and Y. F. Liu, “Photon band structures: The plane-wave method,” Phys. Rev. B 41, 10188–10190 (1990).
[CrossRef]

Lifshitz, E.M.

L.D. Landau and E.M. Lifshitz, Mechanics (“Nauka”, Moscow, 1974)

Liu, Y. F.

K. M. Leung and Y. F. Liu, “Photon band structures: The plane-wave method,” Phys. Rev. B 41, 10188–10190 (1990).
[CrossRef]

Marcatili, E. A. J.

E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. 48, 2071 (1969).

Mawst, L.J.

L.J. Mawst, ““Anti” up the aperture [antiguided VCSEL structures],” IEEE Circuits and Devices Magazine,  19, 34 (2003)
[CrossRef]

Meade, R. D.

John D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic crystals: molding the flow of light, (Princeton University Press, Princeton, 1995)

Monti di Sopra, F.

P. Debernardi, G. P. Bava, F. Monti di Sopra, and M. B. Willemsen, “Features of vectorial modes in phase-coupled VCSEL arrays: experiments and theory,” IEEE J. Quantum Electron. 39, 109–119 (2003).
[CrossRef]

Moser, M.

H. Pier, E. Kapon, and M. Moser, “Strain effects and phase transitions in photonic resonator crystals,” Nature (London) 407, 880–883 (2000).
[CrossRef]

A. Golshani, H. Pier, E. Kapon, and M. Moser, “Photon mode localization in disordered arrays of vertical cavity surface emitting lasers,” J. Appl. Phys. 85, 2454 (1999)
[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 (1992).
[CrossRef]

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

Paek, E. G.

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

Pier, H.

H. Pier, E. Kapon, and M. Moser, “Strain effects and phase transitions in photonic resonator crystals,” Nature (London) 407, 880–883 (2000).
[CrossRef]

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

A. Golshani, H. Pier, E. Kapon, and M. Moser, “Photon mode localization in disordered arrays of vertical cavity surface emitting lasers,” J. Appl. Phys. 85, 2454 (1999)
[CrossRef]

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

Russell, P.

P. Russell, “Photonic Crystal Fibers,” Science 299, 358 (2003).
[CrossRef] [PubMed]

Scherer, A.

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

Singh, S.

W.L. Erikson and S. Singh, “Polarization properties of Maxwell-Gaussian laser beams,” Phys. Rev. E 49, 5778 (1994).
[CrossRef]

Song, D.-S.

D.-S. Song, Y.-J Lee, H.-W. Choi, and Y.-H. Lee, “Polarization-controlled, single-transverse-mode, photonic-crystal, vertical-cavity, surface-emitting lasers” Appl. Phys. Lett. 82, 3182 (2003)
[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 (1992).
[CrossRef]

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

Warren, M.E.

P.L. Gourley, M.E. Warren, and G.R. Hadley et al., “Coherent beams from high efficiency two-dimensional surface-emitting semiconductor laser arrays,” Appl. Phys. Lett. 58, 890 (1991).
[CrossRef]

Willemsen, M. B.

P. Debernardi, G. P. Bava, F. Monti di Sopra, and M. B. Willemsen, “Features of vectorial modes in phase-coupled VCSEL arrays: experiments and theory,” IEEE J. Quantum Electron. 39, 109–119 (2003).
[CrossRef]

Winn, J. N.

John D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic crystals: molding the flow of light, (Princeton University Press, Princeton, 1995)

Yariv, A.

E. Kapon, J. Katz, and A. Yariv, “Supermode analysis of phase-locked arrays of semiconductor lasers,” Opt. Lett.9, 125 (1984); E. Kapon, “Supermode analysis of phase-locked arrays of semiconductor lasers,” Opt. Lett.9, 318 (1984).
[CrossRef] [PubMed]

Yoo, H.-J.

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

Appl. Phys. Lett. (5)

D.-S. Song, Y.-J Lee, H.-W. Choi, and Y.-H. Lee, “Polarization-controlled, single-transverse-mode, photonic-crystal, vertical-cavity, surface-emitting lasers” Appl. Phys. Lett. 82, 3182 (2003)
[CrossRef]

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

M. Orenstein, E. Kapon, and N. G. Stoffel et al., “Two-dimensional phase-locked arrays of vertical-cavity semiconductor lasers by mirror reflectivity modulation,” Appl. Phys. Lett. 58, 804 (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 (1992).
[CrossRef]

P.L. Gourley, M.E. Warren, and G.R. Hadley et al., “Coherent beams from high efficiency two-dimensional surface-emitting semiconductor laser arrays,” Appl. Phys. Lett. 58, 890 (1991).
[CrossRef]

Bell Syst. Tech. J. (2)

E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. 48, 2071 (1969).

J. E. Goell, “A circular-harmonic computer analysis of rectangular dielectric waveguides,” Bell Syst. Tech. J. 48, 2133 (1969).

IEEE Circuits and Devices Magazine (1)

L.J. Mawst, ““Anti” up the aperture [antiguided VCSEL structures],” IEEE Circuits and Devices Magazine,  19, 34 (2003)
[CrossRef]

IEEE J. Quantum Electron. (3)

P. Debernardi, G. P. Bava, F. Monti di Sopra, and M. B. Willemsen, “Features of vectorial modes in phase-coupled VCSEL arrays: experiments and theory,” IEEE J. Quantum Electron. 39, 109–119 (2003).
[CrossRef]

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

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

J. Appl. Phys. (1)

A. Golshani, H. Pier, E. Kapon, and M. Moser, “Photon mode localization in disordered arrays of vertical cavity surface emitting lasers,” J. Appl. Phys. 85, 2454 (1999)
[CrossRef]

Nature (London) (1)

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Figures (5)

Fig.1. .
Fig.1. .

odel of the VCSEL-based photonic crystal (a), Brillouin zone of the equivalent 3D photonic crystal (b), empty lattice test (dashed lines) and simplified diagram of the energy bands Ω m K = ( n Re ω m K c ) 2 K z 2 π Λ (solid lines) (c).

Fig. 2.
Fig. 2.

Calculated intensity patterns of the main polarization components at the Δ, Z, and T points of the Brillouin zone ; arrows show the polarization direction.

Fig. 3.
Fig. 3.

Losses of different modes as a function of the pattern fill factor FF (ratio of the areas of the high-reflectivity pixel and of the unit cell).

Fig. 4.
Fig. 4.

Structure of the electromagnetic field of the main lasing mode of a VCSEL array (|T 5,〉 photonic mode, ξ = π Λ K z ).

Fig. 5.
Fig. 5.

Measured polarization-resolved NF intensity pattern of the |T 5,〉 state of a continuous - wave lasing 4×4 VCSEL array.

Equations (6)

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D = ε E + [ H × g ] , B = μ H + [ g × E ] , g = z ̂ i c ω ln r ( x , y ) N δ ( z 2 N L )
[ m o ( c n ) 2 + p ̂ 2 2 m o + i c n ln ( r ( x , y ) ) 2 L ] | v m K = ω m K | v m K , m 0 = K z n c
E ̂ | T 5 , x ̂ x ̂ cos ( π Λ x ) cos ( π Λ y ) z ̂ i π Λ K z sin ( π Λ x ) cos ( π Λ y ) + y ̂ π 2 2 Λ 2 K z 2 sin ( π Λ x ) sin ( π Λ y )
E ̂ T 5 , y ̂ y ̂ cos ( π Λ x ) cos ( π Λ y ) z ̂ i π Λ K z cos ( π Λ x ) sin ( π Λ y ) + x ̂ π 2 2 Λ 2 K z 2 sin ( π Λ x ) sin ( π Λ y )
E ( 2 ) = y ̂ π 2 2 Λ 2 K z 2 f ( x , y ) sin ( π Λ x ) sin ( π Λ y ) + y ̂ π 2 Λ K z 2 { f y sin ( π Λ x ) cos ( π Λ y ) + f x ( π Λ x ) sin ( π Λ y ) }
+ y ̂ 1 2 K z 2 cos ( π Λ x ) cos ( π Λ y ) 2 f ( x , y ) x y

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