Hybrid Hermite-Laguerre-Gaussian vector modes

Vector modes are well-defined field distributions with spatially varying polarization states, rendering them irreducible to the product of a single spatial mode and a single polarization state. Traditionally, the spatial degree of freedom of vector modes is constructed using two orthogonal modes from the same family. Here, we introduce a novel class of vector modes whose spatial degree of freedom is encoded by combining modes from both the Hermite- and Laguerre-Gaussian families, ensuring that the modes are shape-invariant upon propagation. This superposition is not arbitrary, and we provide a detailed explanation of the methodology employed to achieve it. This new class of vector modes, which we term hybrid Hermite-Laguerre-Gaussian (HHLG) vector modes, gives rise to subsets of modes exhibiting polarization dependence on propagation due to the difference in mode orders between the constituent modes, while remaining eigenmodes of free space. To the best of our knowledge, this is the first demonstration of vector modes composed of two scalar modes originating from different families.