Recent experiments utilizing resonant cavities have produced remarkable results that limit the level of any possible violation of Lorentz Invariance. However, within the framework of the Standard Model Extension (SME) one parameter κtr remains significantly under-constrained compared to the other parameters. It has been shown an asymmetric optical resonator will be sensitive to violations of LI and specifically much more sensitive to κtr than conventional resonators. In this experiment such an asymmetric resonator is compared to a linear sapphire cavity with the measurement system is based on a 1064 nm YAG laser locked to the resonances of both cavities.