Changing the microstructure properties of a space-time metamaterial while a wave is propagating through it, in general, requires addition or removal of energy, which can be of an exponential form depending on the type of modulation. This limits the realization and application of space-time metamaterials. We resolve this issue by introducing a mechanism of conserving energy at temporal metasurfaces in a non-linear setting. The idea is first demonstrated by considering a wave-packet propagating in a discrete medium of a one-dimensional (1D) chain of springs and masses, where using our energy conserving mechanism, we show that the spring stiffness can be incremented at several time interfaces and the energy will still be conserved. We then consider an interesting application of time-reversed imaging in 1D and two-dimensional (2D) spring-mass systems with a wave packet traveling in the homogenized regime. Our numerical simulations show that, in 1D, when the wave packet hits the time-interface, two sets of waves are generated, one traveling forward in time and the other traveling backward. The time-reversed waves re-converge at the location of the source, and we observe its regeneration. In 2D, we use more complicated initial shapes and, even then, we observe regeneration of the original image or source. Thus, we achieve time-reversed imaging with conservation of energy in a non-linear system. The energy conserving mechanism can be easily extended to continuum media. Some possible ideas and concerns in experimental realization of space-time media are highlighted in conclusion and in the supplementary material.
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