Thermal noise is a limiting factor in many high-precision optical experiments. A search is underway for novel optical materials with reduced thermal noise. One such pair of materials, gallium arsenide and aluminum-alloyed gallium arsenide (collectively referred to as AlGaAs), shows promise for its low Brownian noise when compared to conventional materials such as silica and tantala. However, AlGaAs has the potential to produce a high level of thermo-optic noise. We have fabricated a set of AlGaAs crystalline coatings, transferred to fused silica substrates, whose layer structure has been optimized to reduce thermo-optic noise by inducing coherent cancellation of the thermoelastic and thermorefractive effects. By measuring the photothermal transfer function of these mirrors, we find evidence that this optimization has been successful.