3D non-paraxial kernel for two-point correlation modelling in optical and quantum interference at the micro and nano-scales

Abstract

Optical and quantum interference at the micro and nano-scales are of growing interest. Its accurate description bases on the non-paraxial propagation of the spatial correlation, in which the physical observable, determined by the square modulus of the optical and quantum wave functions, is expressed as a modal expansion on a 3D non-paraxial geometric kernel, with the spatial correlation as coefficient. The kernel plays the main role of the model and is deduced from the optical wave equation in free-space as well as from the Schrödinger equation for particle propagation in field-free regions. Two features are analyzed in detail, i.e. the physical implications on the wave and particle interference due to the 3D spatial modulations provided by the local and non-local components of the kernel at the micro and nano-scales, and the decay of the kernel terms with the propagation distance which leads to a novel criterion for the kernel accuracy. The interference modeling is implemented on a matrix algorithm and is illustrated by some examples with nano-structured masks.

Confinement and spatial entanglement: phenomenology of a new interference principle

“…′ ′ r r and (ξ A , ξ D ) (Castañeda et al, 2020) are defined at the S and M planes, respectively, to determine univocally pairs of points on them, which are denoted as The spatial components of both the classical wave equation in free space and the Schrödinger equation for field-free regions are given by the Helmholtz equation…”

“…The exact solution of the Helmholtz equation for any point in the setup can be obtained by applying Green's theorem (Arfken, 1970), which leads to the eigenfunction at the D plane in terms of the expansion (Castañeda et al, 2020)…”

“…For this reason, the integrand of Eq. ( 8) is called the "geometric potential" (Castañeda et al, 2020), whose cross-section at the D plane is Λ(r A ).…”

“…It has been shown that the kernel mode in Eq. ( 15) has a conical shape, and is a geometrical and deterministic function in the SM stage called the correlation cone (Castañeda et al, 2020). Its vertex is placed at the emission point, its basis covers a region of pairs of points (ξ + , ξ -) on the mask and exhibits a Lorentzian cross-section at any distance from the S plane (Figure 5a, b).…”

“…Usually, nonlocality is described as the spatial-correlation attributes of the optical field called spatial coherence (Born & Wolf, 1993;Mandel & Wolf, 1995) or the quantum wave functions of single particles (Feynman & Hibbs, 1965). In contrast, in the context of the confinement principle, it has been shown that nonlocality at the M plane is a geometric condition prepared in the volume of the SM stage in accordance with the stage configuration (Castañeda et al, 2020(Castañeda et al, , 2021. Indeed, by applying the same reasoning as in Eq.…”