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Optical resonators are used for the realisation of ultra-stable frequencylasers. The use of high reflectivity multi-band coatings allows the frequencylocking of several lasers of different wavelengths to a single cavity. Whilethe noise processes for single wavelength cavities are well known, thecorrelation caused by multi-stack coatings has as yet not been analysedexperimentally. In our work, we stabilise the frequency of a $729\,$nm and a$1069\,$nm laser to one mirror pair and determine the residual-amplitudemodulation (RAM) and photo-thermal noise (PTN). We find correlations in PTNbetween the two lasers and observe coherent cancellation of PTN for the$1069\,$nm coating. We show that the fractional frequency instability of the$729\,$nm laser is limited by RAM at $1\times10^{-14}$. The instability of the$1069\,$nm laser is at $3\times10^{-15}$ close to the thermal noise limit of$1.5\times10^{-15}$.

Coherent photo-thermal noise cancellation in a dual-wavelength optical cavity for narrow-linewidth laser frequency stabilisation