Reflection and transmission of a wave incident on a slab with a time-periodic dielectric function<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>ϵ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>

We present a theoretical description of the response of a dynamic slab, with time-periodic dielectric function $ϵ(t)$, to a normally incident monochromatic plane wave of frequency ${\ensuremath{\omega}}_{o}$. As a consequence of the interaction of this incoming wave with the dynamic slab, the reflected and transmitted waves contain harmonics of the modulating frequency $\ensuremath{\Omega}$, namely, the slab itself becomes a polychromatic source of frequencies ${\ensuremath{\omega}}_{o}\ensuremath{-}n\ensuremath{\Omega}(n=0,\ifmmode\pm\else\textpm\fi{}1,\ifmmode\pm\else\textpm\fi{}2,\dots{})$. We establish a general formalism to quantify the reflected and transmitted fields for any periodic variation in the dielectric function. To achieve this, a description of wave propagation in a dynamic bulk is needed. A theoretical framework to treat such propagation, based on the concept of a temporal photonic crystal, is developed. As a consequence of the Bloch-Floquet theorem, the dispersion relation is a band structure that is periodic with frequency and exhibits forbidden wave vector gaps. The Poynting vectors of the transmitted and reflected fields are analyzed in detail. Our theory is applied to the case in which the dielectric function is modulated sinusoidally. We calculate numerically the magnitudes and phases of the reflection and transmission coefficients for several harmonics ${\ensuremath{\omega}}_{o}\ensuremath{-}n\ensuremath{\Omega}$ and slab thicknesses. Three modulation regimes are considered: weak, moderate, and strong. The response in the weak regime is similar to that of a Fabry-Perot etalon---the strengths of the harmonics are weak. For the strong-modulation regime, the strengths of reflection and transmission coefficients of the harmonics become large; they can even exceed one due to the openness of the system in which an external modulating agent can provide part of the invested energy. Dynamic variation in the dielectric properties of materials can give rise to new effects in wave propagation and to novel optical applications and is readily attainable with present-day technology.