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In ref. [1] we analyzed the properties of a Degenerate Optical ParametricOscillator (DOPO) tuned to the first transverse mode family at the signalfrequency. Above threshold, a Hermite-Gauss mode with an arbitrary orientationin the transverse plane is emitted, and thus the rotational invariance of thesystem is broken. When quantum effects were taken into account, it was found onthe one hand, that quantum noise is able to induce a random rotation on thisclassically emitted mode. On the other hand, the analysis of a balancedhomodyne detection in which the local oscillator (LO) was orthogonal to theexcited mode at any time, showed that squeezing in the quadrature selected bythe LO was found for every phase of this one, squeezing being perfect for api/2 phase. This last fact revealed an apparent paradox: If all quadratures arebelow shot noise level, the uncertainty principle seems to be violated. In [1]we stated that the explanation behind this paradox is that the quadratures ofthe rotating orthogonal mode do not form a canonical pair, and the extra noiseis transferred to the diffusing orientation. Thes notes are devoted to provethis claim.

On canonical pairs in 2-transverse-mode DOPOs