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Polarization Bloch waves in photonic crystals based on vertical cavity surface emitting laser arrays

Dmitri L. Boiko, Gilles Guerrero, and Eli Kapon

Open Access Open Access

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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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).

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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.

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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).

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Fig. 4.
Fig. 4. Structure of the electromagnetic field of the main lasing mode of a VCSEL array (|T 5,x̂〉 photonic mode, ξ = π Λ K z ).

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

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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 ℏ n ln ( r ( x , y ) ) 2 L ] | v m K 〉 = ℏ ω m K | v m K 〉 , m 0 = ℏ 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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