  Nonlocality is a fundamental concept in photonics. For instance, nonlocal
wave-matter interactions in spatially modulated metamaterials enable novel
effects, such as giant electromagnetic chirality, artificial magnetism, and
negative refraction. Here, we investigate the effects induced by spatial
nonlocality in {\em temporal} metamaterials, i.e., media with a dielectric
permittivity rapidly modulated in time. Via a rigorous multiscale approach, we
introduce a general and compact formalism for the nonlocal effective medium
theory of temporally periodic metamaterials. In particular, we study two
scenarios: {\em i)} a periodic temporal modulation, and {{\em ii)}} a temporal
boundary where the permittivity is abruptly changed in time and subject to
periodic modulation. We show that these configurations can give rise to
peculiar nonlocal effects, and we highlight the similarities and differences
with respect to the spatial-metamaterial counterparts. Interestingly, by
tailoring the effective boundary wave-matter interactions, we also identify an
intriguing configuration for which a temporal metamaterial can perform the
first-order derivative of an incident wavepacket. Our theoretical results,
backed by full-wave numerical simulations, introduce key physical ingredients
that may pave the way for novel applications. By fully exploiting the
time-reversal symmetry breaking, nonlocal temporal metamaterials promise a
great potential for efficient, tunable optical computing devices.

Optics;Materials Science