Electromagnetic momentum inside a material is notion quite subtle to define, related to the Abraham-Minkowski debate. With new class of metamaterials emerging, allowing for extreme electromagnetic parameters such as near-zero refractive index materials or time-varying materials, those subtilties should treated with great care. Here, we revise fundamental radiative processes, momentum transfer experiments, diffraction, Doppler shift, Heisenberg inequality and microscopy applications inside near-zero refractive index. Furthermore, we demonstrate that the Minkowski momentum -related to spatial translation - is a conserved quantity inside time-varying media by three independent approaches. However, we stress how the Abraham momentum – related to energy transport – is not a conserved quantity in time-varying media.
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