The stack is a sphere of radius \(r_1\) surrounded by \(N\) dielectric shells, each with a constant refractive index. The authors consider the TM solution to the Helmholtz equation and they construct a matrix that connects the tangential fields \((H_\varphi, E_\theta)\) at two consecutive shell boundaries. A reflection coefficient \(R\) is obtained from the (complex) amplitudes of forward and backward travelling waves, and the impedance \(Z\) equals the value of \(E_\theta/H_\varphi\) at the boundary \(r= r_1\). These quantities will depend on a number of variables, including the frequency \(f\); and results of practical interest could be obtained only by numerical treatment.
For the shells the authors choose materials typical in optical applications; and they assume, furthermore, that all shells are quarter-wavelength layers and that only the lowest TM mode is present. They calculate \(|R|\) and \(|Z|\) as functions of \(f\), first for \(N =20\), in which case it is found that two stop bands are formed. Other values of \(N\) are also considered; and they investigate, moreover, the influence of small losses and random variations in shell thicknesses, as well as the variation with mode number.
Finally, the cavity resonator case is discussed, and for the lowest modes, they calculate eigenfrequency and quality factor as functions of \(N\). Again, they consider the influence of losses and random variations in thicknesses.