Invisibility in PT-symmetric complex crystals

TL;DR

This paper provides exact analytical expressions for light scattering in finite PT-symmetric complex crystals, revealing that unidirectional invisibility predicted by earlier theories diminishes as crystal thickness increases.

Contribution

It derives exact formulas for reflection and transmission in finite PT-symmetric crystals, showing the breakdown of unidirectional invisibility with increasing crystal length.

Findings

Unidirectional invisibility occurs only at short crystal lengths.

At intermediate lengths, crystals are reflectionless but not invisible.

Longer crystals lose both reflectionless and invisibility properties.

Abstract

Bragg scattering in sinusoidal PT-symmetric complex crystals of finite thickness is theoretically investigated by the derivation of exact analytical expressions for reflection and transmission coefficients in terms of modified Bessel functions of first kind. The analytical results indicate that unidirectional invisibility, recently predicted for such crystals by coupled-mode theory [Z. Lin et al., Phys. Rev. Lett. 106, 213901 (2011)], breaks down for crystals containing a large number of unit cells. In particular, for a given modulation depth in a shallow sinusoidal potential, three regimes are encountered as the crystal thickness is increased. At short lengths the crystal is reflectionless and invisible when probed from one side (unidirectional invisibility), whereas at intermediate lengths the crystal remains reflectionless but not invisible; for longer crystals both unidirectional reflectionless and invisibility properties are broken.