Recent advancements in materials and metamaterials with strong, time-varying, nonlinear optical responses have spurred a surge of interest in time-varying photonics. This opens the door to novel optical phenomena including reciprocity breaking, frequency translation, and amplification that can be further optimized by improving the light-matter interaction. Although there has been recent interest in applying topology-based inverse design to this problem, we propose a novel approach in this article. We introduce a method for the inverse design of optical pulse shapes to enhance their interaction with time-varying media. We validate our objective-first approach by maximizing the transmittance of optical pulses of equal intensity through time-varying media. Through this approach, we achieve large, broadband enhancements in pulse energy transmission, including gain, without altering the incident pulse energy. As a final test, we maximize pulse transmission through thin films of indium tin oxide, a time-varying medium when strongly pumped in its ENZ band. Our work presents a new degree of freedom for the exploration, application, and design of time-varying systems and we hope it inspires further research in this direction.
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