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Within the expansive domain of optical sciences, achieving the precise characterization of light beams stands as a fundamental pursuit, pivotal for various applications, including telecommunications and imaging technologies. This study introduces an innovative methodology aimed at reconstructing the Wigner distribution function of optical signals; a crucial tool in comprehending the time-frequency behavior exhibited by these signals. The proposed approach integrates two robust mathematical tools: the discrete realization of fractional Fourier transform, and the propagator of the quantum harmonic oscillator in waveguide arrays. This integration offers a direct and efficient method for characterizing optical signals by reconstructing their Wigner distribution function in the scope of integrated optics. We provide evidence of how having knowledge of the signal propagation amid the phase-space reconstruction, has desirable advantages in respect to only knowing the signal state., Comment: 10 pages, 7 figures

The effect of a non-unitary transformation on an initial Hermitian operator is studied. The initial (Hermitian) optical system is a Glauber-Fock optical lattice. The resulting non-Hermitian Hamiltonian models an anisotropic (Glauber-Fock) waveguide array of the Hatano-Nelson-type. Several cases are analyzed and exact analytical solutions for both the Hermitian and non-Hermitian Schr\"odinger problems are given, as they are simply connected. Indeed, such transformation can be regarded as a non-unitary Supersymmetric (SUSY) transformation and the resulting non-Hermitian Hamiltonian can be considered as representing an open system that interchanges energy with the environment.

A non-unitary transformation leading to a Hatano-Nelson problem is performed on an array of equally-spaced optical waveguides. Such transformation produces a non-reciprocal system of waveguides, as the corresponding Hamiltonian becomes non-Hermitian. This may be achieved by judiciously choosing an attenuation (amplification) of the injected (or exciting) field. The non-Hermitian transport induced by such transformation is studied for several cases and closed analytical solutions are obtained. The corresponding non-Hermitian Hamiltonian may represent an open system that interacts with the environment, either loosing to or being provided with energy from the exterior., Comment: arXiv admin note: substantial text overlap with arXiv:2307.06425

A medium with specific anisotropic refractive indices can induce a supersymmetric behavior in the propagation of polarized electromagnetic waves, in an analogue fashion to a quantum mechanical system. The polarizations of the wave are the ones which behave as superpartners from each other. For this to happen, the anisotropy of the medium must be transverse to the direction of propagation of the wave, with different refractive indices along the direction of each polarization. These refractive indices must follow a very specific relation in order to trigger the supersymetric response of the electromagnetic wave, each of them with spatial dependence on the longitudinal (propagation) direction of the wave. In this form, in these materials, different polarized light can be used to test supersymmetry in an optical fashion.

By adapting the Madelung-Bohm formalism to paraxial wave propagation we show, by using Ermakov-Lewis techniques, that the Gouy phase is related to the form of the phase chosen in order to produce a Gaussian function as a propagated field. For this, we introduce a quantum mechanical invariant, that it is explicitly time dependent despite the fact that the Hamiltonian is itself time-independent. We finally show that the effective Bohm {\it index of refraction} generates a GRIN medium that produces the focusing needed for the Gouy phase.

We present an integral of diffraction based on particular eigenfunctions of the Laplacian in two dimensions. We show how to propagate some fields, in particular a Bessel field, a superposition of Airy beams, both over the square root of the radial coordinate, and show how to construct a field that reproduces itself periodically in propagation, i.e., a field that renders the Talbot effect. Additionally, it is shown that the superposition of Airy beams produces self-focusing., Comment: 5 pages, 4 figures

It is shown that field propagation in linear and quadratic gradient-index (GRIN) media obeys the same rules of free propagation in the sense that a field propagating in free space has a (mathematical) form that may be {\it exported} to those particular GRIN media. The Bohm potential is introduced in order to explain the reason of such behavior: it changes the dynamics by modifying the original potential. The concrete cases of two different initials conditions for each potential are analyzed.

This contribution has two main purposes. First, we show using classical optics how to model two coupled quantum harmonic oscillators and two interacting quantized fields. Second, we use quantum mechanical techniques to solve, exactly, the propagation of light through a particular type of graded index medium. In passing, we show that the system presents phase transitions., Comment: 11 pages, 3 figures

We analyse the paraxial field propagation in the realm of classical optics, showing that it can be written as the action of the fractional Fourier transform, followed by the squeeze operator applied to the initial field. Secondly, we show that a wavelet transform may be viewed as the application of a displacement and squeeze operator onto the mother wavelet function., Comment: 10 pages, no figures

We demonstrate that superpositions of coherent and displaced Fock states, also referred to as generalized Schr\"odinger cats cats, can be created by application of a nonlinear displacement operator which is a deformed version of the Glauber displacement operator. Consequently, such generalized cat states can be formally considered as nonlinear coherent states. We then show that Glauber-Fock photonic lattices endowed with alternating positive and negative coupling coefficients give rise to classical analogs of such cat states. In addition, it is pointed out that the analytic propagator of these deformed Glauber-Fock arrays explicitly contains the Wigner operator opening the possibility to observe Wigner functions of the quantum harmonic oscillator in the classical domain., Comment: 17 pages, 5 figures

We propose a set of photonic crystals that realize a nonlinear quantum Rabi model equivalent to a two-level system driven by the phase of a quantized electromagnetic field. The crystals are exactly soluble in the weak-coupling regime and their dispersion relation is discrete. The system is diagonalized by normal modes equivalent to a dressed state basis. In the strong-coupling regime, we use perturbation theory and find that the dispersion relation is continuous and give the normal modes of the crystal in terms of continued fractions. We show that these photonic crystals allow state reconstruction in the form of coherent oscillations in the weak-coupling regime., Comment: 15 pages, 4 figures

Coherent states and their generalisations, displaced Fock states, are of fundamental importance to quantum optics. Here we present a direct observation of a classical analogue for the emergence of these states from the eigenstates of the harmonic oscillator. To this end, the light propagation in a Glauber-Fock waveguide lattice serves as equivalent for the displacement of Fock states in phase space. Theoretical calculations and analogue classical experiments show that the square-root distribution of the coupling parameter in such lattices supports a new family of intriguing quantum correlations not encountered in uniform arrays. Due to the broken shift-invariance of the lattice, these correlations strongly depend on the transverse position. Consequently, quantum random walks with this extra degree of freedom may be realised in Glauber-Fock lattices., Comment: 5 pages, 4 figures