Comparison of density of states and scattering parameters in coaxial photonic crystals: Theory and experiment

Abstract

We present an analytical and experimental study of the scattering parameters in a one dimensional (1D) symmetric photonic crystal and their relation to the density of states (DOS). The 1D photonic crystal is constituted by $N$ alternating wires and loops that are either inserted horizontally or attached vertically between the source and load on a transmission line. The complete knowledge of the scattering matrix coefficients (${S}_{ij}$) allows us to access the DOS and eigenvalues of the finite periodic structure as well as the DOS and dispersion curves of an infinite periodic system. We show the usefulness of the transmission and reflection delay times and highlight their similarities and differences with respect to the DOS, in particular as a function of the absorption strength in the system. For both horizontal and vertical geometries, we show analytically that in a lossless structure, the DOS is proportional to the Friedel phase, namely the derivative of the argument of the determinant of the scattering matrix S. For a low loss system, this proportionality remains still valid with a good approximation and can be used as a practical tool to derive the DOS and therefore the dispersion curves from experimental data. Also, the absorption can be accurately extracted from the measurement of the modulus of the determinant of $S$. However, for increasing strength of dissipation, we show how and why these relationships cease to be valid. Still, the transmission delay time can remain an efficient tool to derive DOS even at relatively high dissipation strength. Additionally, we show that in the vertical geometry the transmission and reflection delay times exhibit negative delta peaks which are related directly to the eigenmodes of the finite system with different boundary conditions on its extremities. Our theoretical results are obtained by means of the Green's function approach, whereas the experimental demonstrations are performed using standard coaxial cables in the radio-frequency domain.