On the lowest-frequency bandgap of 1D phononic crystals

This manuscript puts forward and verifies a novel analytical approach to design phononic crystals that feature a bandgap at the lowest possible frequencies. This new approach is verified against numerical optimization. It rests on the exact form of the trace of the cumulative transfer matrix. This matrix arises from the product of $N$ elementary transfer matrices, where $N$ represents the number of layers in the unit cell of the crystal. The paper presents first a new proof of the harmonic decomposition of the trace (which, unlike the original derivation, does not resort to group-theoretical concepts), and then goes to demonstrate that the long-wavelength asymptotics of the function that governs the dispersion relation can be described in closed form for any layering, plus that it possesses a simple and explicit form that facilitates its study. Using this asymptotic result for low frequencies, we then propose an analytical curvature-minimization approach to devise layerings that yield an opening of the first frequency bandgap at the lowest possible frequency value. Finally, we also show that minimizing the frequency at which the first gap opens can be at odds with obtaining broadband attenuation, i.e., maximizing the width of the first frequency bandgap.