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Floquet–Bloch eigenwaves and bandgaps in a di-periodic potential

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Abstract

We present general solutions to the Schrödinger equation for a di-periodic potential composed of two frequencies, thus generalizing the standard sinusoidal potential. The Schrödinger equation with a di-periodic potential becomes a three-term Whittaker–Hill equation that is solved by two different approaches. The first is applying the central equation formalism, which allows determining the Floquet–Bloch eigenwaves and band structure as a function of the potential parameters. In the second approach, we transform the Whittaker–Hill equation into an Ince equation with a suitable change of variable. In this case, we get a complete set of orthogonal solutions described by Ince functions. The well-known Mathieu functions obtained with purely sinusoidal potentials are a special case of Ince functions.

© 2021 Optical Society of America

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Supplementary Material (1)

NameDescription
Visualization 1       Eigenvalue surfaces in the parametric space of the Whittaker-Hill equation.

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Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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