  We investigate analytically, numerically, and experimentally the modulational
instability in a layered, cubically-nonlinear (Kerr) optical medium that
consists of alternating layers of glass and air. We model this setting using a
nonlinear Schr\"odinger (NLS) equation with a piecewise constant nonlinearity
coefficient and conduct a theoretical analysis of its linear stability,
obtaining a Kronig-Penney equation whose forbidden bands correspond to the
modulationally unstable regimes. We find very good {\it quantitative} agreement
between the theoretical analysis of the Kronig-Penney equation, numerical
simulations of the NLS equation, and the experimental results for the
modulational instability. Because of the periodicity in the evolution variable
arising from the layered medium, we find multiple instability regions rather
than just the one that would occur in uniform media.
