  The Floquet-Bloch theorem allows waves in infinite, lossless periodic media
to be expressed as a sum of discrete Floquet-Bloch modes, but its validity is
challenged under the realistic constraints of loss and finite extent. In this
work, we mathematically reveal the existence of Floquet-Bloch modes in the
electromagnetic fields sustained by lossy, finite periodic layered media using
Maxwell's equations alone without invoking the Floquet-Bloch theorem. Starting
with a transfer-matrix representation of the electromagnetic field in a generic
layered medium, we apply Fourier transformation and a series of mathematical
manipulations to isolate a term explicitly dependent on Floquet-Bloch modes.
Fourier-domain representation of the electromagnetic field can be reduced into
a product of the Floquet-Bloch term and two other matrix factors: one governed
by reflections from the medium boundaries and another dependent on layer
composition. Electromagnetic fields in any finite, lossy, layered structure can
now be interpreted in the Fourier-domain by separable factors dependent on
distinct physical features of the structure. The developed theory enables new
methods for analyzing and communicating the electromagnetic properties of
layered metamaterials.
