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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References

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  1. John D. Joannopoulos, R. D. Meade, 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).
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  3. M. Orenstein, E. Kapon, 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]
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    [CrossRef]
  5. P.L. Gourley, M.E. Warren, 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, 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.
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    [CrossRef]
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    [CrossRef]
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    [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]
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  19. D. L. Boiko and E. Kapon �??Theory of vertical cavity photonic crystals, �?? (manuscript in preparation).
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  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).
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  25. L.D. Landau, E.M. Lifshitz, Mechanics (�??Nauka�??, Moscow, 1974)

Appl. Phys. Lett (1)

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]

Appl. Phys. Lett. (5)

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.-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]

M. Orenstein, E. Kapon, 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, 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).

CLEO 2000 (1)

C.-A. Berseth, G. Guerrero, 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.

ICPS 2002 (1)

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, <a href=" http://www.icps2002.org">http://www.icps2002.org</a>

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)

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]

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]

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 (1)

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

Opt. Lett. (2)

Phys. Rev. B (1)

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

Phys. Rev. E (1)

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

Science (1)

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

Zh. Eksp. Teor. Fiz (Sov. Phys. JETP) (1)

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

Other (3)

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

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

L.D. Landau, E.M. Lifshitz, Mechanics (�??Nauka�??, Moscow, 1974)

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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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