The electromagnetic waves can be controlled either by space, as it happens in the case of photonic crystals,
or by similar variations in time-dependent parameters of the medium. For a spatially homogenous medium,
if there is a sudden change in the permittivity of the medium with time, it gives rise to the reflection and
transmission of incident waves[1]. Earlier studies had proposed that the reflected wave travels backward
from the temporal boundary while the transmitted wave follows a normal way. It is expected that a periodic
variation of permittivity, with respect to time, might give rise to several interesting phenomena, like frequency
band gap and generation of new frequencies[2, 3]. The medium with periodic variation of the permittivity is
called “Temporal Photonic Crystal (TPC)”. In the frequency domain, the amplitude of the electromagnetic
wave propagation through TPC may increase or decrease depending on the increase or decrease of the
permittivity of the medium[4, 5].
As the permittivity obeys causality and Kramer-Kronig relations, the incident wave convolves with timevarying permittivity. Thus, the displacement vector is the convolution of the permittivity and electric field.
Using the finite-difference time-domain method on a one-dimensional spatio-temporal slab, it is possible to
study the reflection and transmission coefficients. Our preliminary studies indicate that there is an apparent
violation of the conservation of energy that can be explained in terms of the work done by the medium on
the wave [6].
Keywords— electromagnetic waves, photonic crystal, temporal photonic crystal, time-dependent permittivity,
Kramer-Kronig relations, convolution
References
[1] F.R. Morgenthaler. Velocity modulation of electromagnetic waves. IRE Transactions on Microwave Theory and
Techniques, 6(2):167–172, 1958.
[2] Jorge R Zurita-S´anchez, P Halevi, and Juan C Cervantes-Gonz´alez. Reflection and transmission of a wave incident
on a slab with a time-periodic dielectric function ϵ(t). Physical Review A, 79(5):053821, 2009.
[3] JR Reyes-Ayona and P Halevi. Observation of genuine wave vector (k or β) gap in a dynamic transmission line
and temporal photonic crystals. Applied Physics Letters, 107(7):074101, 2015.
[4] JT Mendon¸ca and PK Shukla. Time refraction and time reflection: two basic concepts. Physica Scripta, 65(2):160,
2002.
[5] Armen G Hayrapetyan, J¨org B G¨otte, Karen K Grigoryan, Stephan Fritzsche, and Rubik G Petrosyan. Electromagnetic wave propagation in spatially homogeneous yet smoothly time-varying dielectric media. Journal of
Quantitative Spectroscopy and Radiative Transfer, 178:158–166, 2016.
[6] Kumar S Subramanian V, Balasubramanian S. Reflection and transmission from a temporal boundary. presented
at ANZCOP summer meeting, Canberra, Australia, 2023.