Temporal metamaterials, created by modulating the refractive index in time, offer powerful means of controlling wave propagation but still lack a systematic design methodology. Here, we develop an analytic inverse-design framework rooted in space-time duality and the established theory of one-dimensional spatial inverse scattering. By prescribing reflection (backward-wave) and transmission (forward-wave) responses in rational-function form, we obtain closed-form refractive-index modulations that are guaranteed to be physically admissible. This approach avoids iterative optimization and provides direct analytic control of the modulation. We illustrate the method with syntheses of mathematical operators, such as derivatives and integrals, as well as Chebyshev- and Butterworth-type filters, and validate the results through finite-difference time-domain simulations. Our findings establish a general route to temporal media with tailored functional and spectral responses, enabling applications in wave-based information processing, programmable filtering, and amplification schemes inspired by photonic time crystals.

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• Inverse problems
• Metamaterials
• Complex materials
• Electromagnetic wave theory
• Finite-difference time-domain method
• First-principles calculations