Topological features of Bloch impedance

Abstract

The bulk-boundary correspondence (b-bc) principle states that the presence and number of evanescent bandgap modes at an interface between two periodic media depend on the topological invariants (Chern numbers in 2D or Zak phases in 1D) of propagating modes at completely different frequencies in all Bloch bands below that bandgap. The objective of this letter is to explain, on physical grounds, this connection between modes with completely different characteristics. We assume periodic lossless 1D structures and lattice cells with mirror symmetry; in this case the Zak phase is unambiguously defined. The letter presents a systematic study of the behavior of electromagnetic Bloch impedance, defined as the ratio of electrical and magnetic fields in a Bloch wave at the boundary of a lattice cell. The impedance-centric view confers transparent physical meaning on the bulk-boundary correspondence principle. Borrowing from the semiconductor terminology, we classify the bandgaps as p- and n-type at the Γ and X points, depending on whether the Bloch impedance has a pole (p) or a null (n) at the bottom of that gap. An interface mode exists only for pn-junctions per our definition. We expect these ideas to be extendable to problems in higher dimensions, with a variety of emerging applications.