In this work, we present the design of a novel mm-wave leaky wave antenna (LWA), which operates at the TM11 high-order mode. While typical LWAs operate at their fundamental mode, e.g., TE10, our proposed TM11 high-order mode LWA takes advantage of its modal distribution, thereby coupling more energy to free-space through appropriately designed slots. Due to the latter, our TM11 mm-wave LWA achieves 1.47 3.24 dB more gain over a bandwidth of 5 GHz (from 35 GHz to-40 GHz) when it is compared to a traditional LWA of the same electrical dimensions that operates at the fundamental TE10-mode.