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Journal of Optics

Reversible coherence conversion across optical degrees-of-freedom: a tutorial

Mitchell Harling, Varun A. Kelkar, Ayman F. Abouraddy, Kimani C. Toussaint

Published Mar 24, 2023• 7 citations
Publisher

Abstract

Abstract This work introduces the concept and experimentation of reversible optical coherence conversion, and outlines optical coherency matrix tomography as a method for measuring the full coherence of a field. To date, coherence conversion has only been demonstrated between the spatial and polarization degrees-of-freedom (DoFs). Coherence conversion offers a new control over an optical field’s DoFs—protecting against deleterious scrambling effects.

DOI
10.1088/2040-8986/acc74e

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Abstract This work introduces the concept and experimentation of reversible optical coherence conversion, and outlines optical coherency matrix tomography as a method for measuring the full coherence of a field. To date, coherence conversion has only been demonstrated between the spatial and polarization degrees-of-freedom (DoFs). Coherence conversion offers a new control over an optical field’s DoFs—protecting against deleterious scrambling effects.
The entropy associated with an optical field quantifies the field fluctuations and thus its coherence. Any binary optical degree-of-freedom (DoF) - such as polarization or the field at a pair of points in space - can each carry up to one bit of entropy. We demonstrate here that entropy can be reversibly swapped between different DoFs, such that coherence is converted back and forth between them without loss of energy. Specifically, starting with a spatially coherent but unpolarized field carrying one bit of entropy, we unitarily convert the coherence from the spatial DoF to polarization to produce a spatially incoherent but polarized field by swapping the entropy between the two DoFs. Next, we implement the inverse unitary operator, thus converting the coherence back to yield once again a spatially coherent yet unpolarized field. We exploit the intermediate stage between the two coherence conversions - where the spatial coherence has been converted to the polarization DoF - to verify that the field has become immune to the deleterious impact of spatial phase scrambling. Maximizing the spatial entropy protects the spatial DoF by preventing it from taking on any additional fluctuations. After the second coherence conversion, spatial coherence is readily retrieved, and the effect of spatial phase scrambling circumvented.
The coherence of an optical beam having multiple degrees of freedom (DoFs) is described by a coherency matrix G spanning these DoFs. This optical coherency matrix has not been measured in its entirety to date--even in the simplest case of two binary DoFs where G is a 4 × 4 matrix. We establish a methodical yet versatile approach--optical coherency matrix tomography--for reconstructing G that exploits the analogy between this problem in classical optics and that of tomographically reconstructing the density matrix associated with multipartite quantum states in quantum information science. Here G is reconstructed from a minimal set of linearly independent measurements, each a cascade of projective measurements for each DoF. We report the first experimental measurements of the 4 × 4 coherency matrix G associated with an electromagnetic beam in which polarization and a spatial DoF are relevant, ranging from the traditional two-point Young's double slit to spatial parity and orbital angular momentum modes.
Studying the coherence of an optical field is typically compartmentalized with respect to its different optical degrees of freedom (DoFs) -- spatial, temporal, and polarization. Although this traditional approach succeeds when the DoFs are uncoupled, it fails at capturing key features of the field's coherence if the DOFs are indeed correlated -- a situation that arises often. By viewing coherence as a `resource' that can be shared among the DoFs, it becomes possible to convert the entropy associated with the fluctuations in one DoF to another DoF that is initially fluctuation-free. Here, we verify experimentally that coherence can indeed be reversibly exchanged -- without loss of energy -- between polarization and the spatial DoF of a partially coherent field. Starting from a linearly polarized spatially incoherent field -- one that produces no spatial interference fringes -- we obtain a spatially coherent field that is unpolarized. By reallocating the entropy to polarization, the field becomes invariant with regards to the action of a polarization scrambler, thus suggesting a strategy for avoiding the deleterious effects of a randomizing system on a DoF of the optical field.
Optical coherence is a well-established branch of physical optics in which the statistical properties of fluctuating optical fields are described in terms of correlation functions over continuous spatial and temporal degrees of freedom (DoFs). Nevertheless, in any practical setting, only discrete DoFs are ever accessible. In these scenarios, the modes are fixed, stable, and deterministic, and partial coherence arises from random relative complex amplitudes, a configuration we refer to as 'structured coherence'. Advances in structured coherence have recently unveiled new conceptual ground in optical communications and information processing in which partial coherence may be preferable to full coherence, which we call a 'coherence advantage'. We formulate structured coherence in terms of coherence matrices to investigate these recent theoretical and experimental breakthroughs. We first review optical fields characterized by a binary DoF via 2x2 Hermitian coherence matrices and introduce key concepts that take on new significance for larger-dimensional modal sets. Next, we examine the structured coherence of two binary DoFs, which can be described by 4x4 coherence matrices. We introduce the concept of coherence rank, entropy swapping, and optical cross-purity. In the perspective outlined here, coherence is viewed as a 'resource', which can be exchanged between DoFs, concentrated into a DoF or into particular modes, or spread over the DoFs. We then examine larger-dimensional modal sets, which allow for more versatile applications in optical information processing. The formulation presented here lends itself particularly to the manipulation of partial optical coherence in integrated photonic platforms, thereby opening myriad avenues for novel fundamental investigations of structured coherence and exploiting the coherence advantage in optical communications and information processing.
We explore the concepts of coherence and entanglement as they apply to both the classical and quantum natures of light. In the classical domain, we take inspiration from the tools and concepts developed in foundational quantum mechanics and quantum information science to gain a better understanding of classical coherence theory of light with multiple degrees of freedom (DoFs). First, we use polarization and spatial parity DoFs to demonstrate the notion of classical entanglement, and show that Bell's measure can serve as a useful tool in distinguishing between classical optical coherence theory. Second, we establish a methodical yet versatile approach called 'optical coherency matrix tomography' for reconstructing the coherency matrix of an electromagnetic beam with multiple DoFs. This technique exploits the analogy between this problem in classical optics and that of tomographically reconstructing the density matrix associated with multipartite quantum states in quantum information science. Third, we report the first experimental measurements of the 4 x 4 coherency matrix associated with an electromagnetic beam in which polarization and a spatial DoF are relevant, ranging from the traditional two-point Young's double slit to spatial parity and orbital angular momentum modes. In the quantum domain, we use the modal structure of classical fields to develop qubits and structure Hilbert spaces for use in quantum information processing. Advancing to three-qubit logic gates is an important step towards the success of optical schemes for quantum computing. We experimentally implement a variety of two- and three- qubit, linear and deterministic, single-photon, controlled, quantum logic gates using polarization and spatial parity qubits. Lastly, we demonstrate the implementation of two-qubit single-photon logic using polarization and orbital angular momentum qubits.