We theoretically describe the quantum properties of a large Fresnel numberdegenerate optical parametric oscillator with spherical mirrors that is pumpedby a Gaussian beam. The resonator is tuned so that the resonance frequency of agiven transverse mode family coincides with the down-converted frequency. Afterdemonstrating that only the lower orbital angular momentum (OAM) Laguerre-Gaussmodes are amplified above threshold, we focus on the quantum properties of therest of (classically empty) modes. We find that combinations of opposite OAM(Hybrid Laguerre-Gauss modes) can exhibit arbitrary large quadrature squeezingfor the lower OAM non amplified modes.
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