Non-orthogonal Transformations of Structured Light Using Ellipticity-Dependent Ince-Gaussian Modes

The Ince-Gaussian modes form a complete set of solutions to the paraxial wave equation parametrized by an ellipticity parameter {\epsilon}, enabling a continuous transition between Laguerre-Gaussian and Hermite-Gaussian modes While each fixed {\epsilon} defines an orthogonal basis, modes associated with different ellipticities are not mutually orthogonal, and no explicit transformation between such bases has been reported. Here, we derive the first explicit finite analytical expression to transformation between Ince-Gaussian bases of arbitrary ellipticity, enabling direct and experimentally accessible mapping between non-orthogonal structured-light representations. We further demonstrate an experimental implementation using spatial light modulators to perform ellipticity-resolved modal decomposition. This framework introduces ellipticity as a controllable degree of freedom for structured light engineering, enabling new strategiesfor mode conversion, encoding, and high-dimensional optical information processing.