Generalisation of the transfer matrix formulation of the theory of electron and photon conductance

The usual technique for calculating the mesoscopic conductance of materials is based on Pichard's transfer matrix method [J.L. Pichard, Thesis, University of Paris, Orsay, 1984]. The essential ideas of the theory of conductance for electronic systems also extend to electromagnetic systems such as waveguides and photonic devices. In the standard formulation of Pichard's treatment, there is an explicit assumption that the number of channels (or “leads”) in the input and output media is identical. In this paper, we extend the theory to handle the general situation when the input and output media may support different numbers of (propagating) channels, as is needed to study the photon conductance of semi-infinite waveguides. We show that the pivotal conclusions of Pichard's original derivation, including the existence of the key polar decomposition of the transfer matrix, continue to hold, even though the premise on which the original formulation was based (i.e., that the matrices involved are square and invertible) no longer hold. We also demonstrate this generalisation by calculating the conductance ladder for a photonic crystal waveguide terminated in free space.