Partial coherence in coupled photonic crystal vertical cavity lasers

Abstract

A model for the emergence of partial coherence in arrays of VCSELs is presented. The spectra of the uncoupled lasers determine the coherence properties of the coupled system. Predictions the model is verified experimentally.Recent observations of partially coherent behavior in arrays of semiconductor lasers [1] are not addressed or predicted theoretically in the literature. In this work, a two-by-one array of lasers is modeled as classical, linear, stochastic, coupled-oscillators. The results are verified experimentally.The VCSEL array is produced by fabricating two-dimensional photonic crystal (PhC) patterns into the top facet of a layered system of semiconductor materials with a thin planar active region optically confined between layers of alternating high and low index material (Bragg reflectors). Defects in the PhC define transversely coupled laser arrays [1][2][3]. The VCSELs are electrically pumped and may be addressed independently. The VCSELs exhibit a far-field intensity pattern such as might be observed in a Young's type experiment with polychromatic, partially-coherent illumination of the pinholes [4]. The intensity of the field on a far-zone up to propagation factors, is given by the expression [4]where τ is the time delay associated with the path length difference from each of the laser facets to the point of observation andΓ ij is the correlation between the field at the ith and jth facet. It will prove convenient to work not withΓ ij (τ ) but with the cross-spectral density, W ij (ω) the Fourier transform with respect to τ .In each realization of the ensemble of random fields, the output of each VCSEL is modeled as an oscillator with amplitude U j (ω) satisfying the frequency-domain, driven harmonic oscillator equation f j (ω)U j (ω) = S