Role of intertwined Hamiltonian in two dimensional classical optics

Intertwined Hamiltonian formalism originally has its roots in quantum field theory and non-relativistic quantum mechanics. In this work, we develop the non-relativistic two dimensional intertwined Hamiltonian formalism in classical optics. We obtain the properties of the intertwined media in detail and show that the differential part of intertwining operator is a series in Euclidean algebra generators. Also, we investigate quadratic gradient-index medium as an example of this structure, and obtain the intertwining operator and intertwined medium refractive index. Moreover, we study the preservation of quantum properties in the intertwined medium. For this, we consider superposition preservation as the most important property of quantum characters. We show that when a Schrödinger cat state is generated in gradient-index medium, we can construct another Schrödinger cat state in the intertwined one.