Basis-neutral Hilbert-space analyzers

TL;DR

This paper introduces a novel method to analyze optical beams in arbitrary spatial-mode bases by extending the concept of temporal delay to the spatial domain, enabling advanced spatial-mode analysis for quantum communication.

Contribution

It presents a generalized spatial delay concept and a reconfigurable interferometer for basis-neutral spatial-mode analysis, expanding beyond traditional temporal interferometry principles.

Findings

Implemented a stable, reconfigurable spatial-light-modulator-based interferometer.

Demonstrated analysis of beams in Hermite-Gaussian and Laguerre-Gaussian modes.

Enabled projection onto any spatial modal basis.

Abstract

Interferometry is one of the central organizing principles of optics. Key to interferometry is the concept of optical delay, which facilitates spectral analysis in terms of time-harmonics. In contrast, when analyzing a beam in a Hilbert space spanned by spatial modes -- a critical task for spatial-mode multiplexing and quantum communication -- basis-specific principles are invoked that are altogether distinct from that of `delay.' Here, we extend the traditional concept of temporal delay to the spatial domain, thereby enabling the analysis of a beam in an arbitrary spatial-mode basis -- exemplified using Hermite-Gaussian and radial Laguerre-Gaussian modes. Such generalized delays correspond to optical implementations of fractional transforms; for example, the fractional Hankel transform is the generalized delay associated with the space of Laguerre-Gaussian modes, and an interferometer incorporating such a `delay' obtains modal weights in the associated Hilbert space. By implementing an inherently stable, reconfigurable spatial-light-modulator-based polarization- interferometer, we have constructed a `Hilbert-space analyzer' capable of projecting optical beams onto any modal basis.