Closed loop and stubs

Closed loops with finite open loops attached to the robust zeros of one of their two twin states are the systems investigated in this chapter. Such attached finite loops are called stubs. For simplicity, here only the cases where all the stubs have the same length are considered. This length is chosen to be commensurable with the closed loop length. So the interface points of such systems remain robust zeros for many eigenfunctions of the final systems. As a consequence, the corresponding eigenwavelengths are robust. All the states of such systems are obtained with particular attention to the path ones. An eigenfunction zero is robust because its eigenstate cannot be activated by any action applied on it. The path states are robust because they are confined within one-dimensional paths, and when they are degenerate they can turn around defects. Multiplexer and wide reflection band applications of such systems are presented. It is shown that some path states can be transferred from one infinite line to another without any perturbation. Such single wavelength transfers are of infinite quality. It is possible to transform them into finite-width resonances just by tuning the stub lengths.