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arXiv:2601.18797

On-chip control of the coherence matrix of four-mode partially coherent light: rank, entropy, and modal Stokes parameters

Amin Hashemi, Abbas Shiri, Bahaa E. A. Saleh, Andrea Blanco‐Redondo, Ayman F. Abouraddy

Published Jan 26, 2026• 2 citations
arXivHTMLPDF

Abstract

Partially coherent light offers salutary capabilities in optical information processing that cannot be matched by coherent light. To date, this `coherence advantage' has been confirmed in proof-of-principle optical communications protocols using bulk optics. Taking full advantage of such opportunities necessitates processing multimode partially coherent light in integrated photonics platforms that alone provide the requisite stability for cascaded operations on a large scale. Here we demonstrate on-chip manipulation of four-mode partially coherent light described by a $4\times4$ Hermitian coherence matrix. Starting with generic maximally incoherent light, we utilize an on-chip hexagonal mesh of Mach-Zehnder interferometers to perform all the unitary and non-unitary tasks that are critical for realizing structured coherence: controlling the coherence rank (the number of non-zero eigenvalues of the coherence matrix); tuning the field entropy; molding the structure of the coherence matrix via $4\times4$ unitary transformations constructed out of sequences of $2\times2$ unitaries acting on pairs of modes; and tomographic reconstruction of the coherence matrix by measuring the modal Stokes parameters associated with Kronecker-Pauli matrices. These results confirm the scalability of utilizing $2\times2$ on-chip building blocks for the synthesis and reconstruction of high-dimensional coherence matrices, and provide a decisive step towards large-scale on-chip manipulation of massively moded partially coherent light for applications in optical information processing.

DOI
10.48550/arxiv.2601.18797

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Partially coherent light offers salutary capabilities in optical information processing that cannot be matched by coherent light. To date, this `coherence advantage' has been confirmed in proof-of-principle optical communications protocols using bulk optics. Taking full advantage of such opportunities necessitates processing multimode partially coherent light in integrated photonics platforms that alone provide the requisite stability for cascaded operations on a large scale. Here we demonstrate on-chip manipulation of four-mode partially coherent light described by a 4×44\times44×4 Hermitian coherence matrix. Starting with generic maximally incoherent light, we utilize an on-chip hexagonal mesh of Mach-Zehnder interferometers to perform all the unitary and non-unitary tasks that are critical for realizing structured coherence: controlling the coherence rank (the number of non-zero eigenvalues of the coherence matrix); tuning the field entropy; molding the structure of the coherence matrix via 4×44\times44×4 unitary transformations constructed out of sequences of 2×22\times22×2 unitaries acting on pairs of modes; and tomographic reconstruction of the coherence matrix by measuring the modal Stokes parameters associated with Kronecker-Pauli matrices. These results confirm the scalability of utilizing 2×22\times22×2 on-chip building blocks for the synthesis and reconstruction of high-dimensional coherence matrices, and provide a decisive step towards large-scale on-chip manipulation of massively moded partially coherent light for applications in optical information processing.
Partially coherent light is typically studied in the context of freely propagating continuous fields. Recent developments have indicated the existence of a `coherence advantage' in multimode optical communications, where partially coherent light outperforms coherent light. However, exploiting partial coherence in such applications requires manipulating multimode field coherence in programmable on-chip platforms. We present here the first example of on-chip synthesis and characterization of two-mode optical fields in an integrated on-chip hexagonal mesh of Mach-Zehnder interferometers. Starting with incoherent two-mode light, we adjust the degree of coherence on the chip with non-unitary transformations, construct 2×22\times22×2 unitary transformations to synthesize prescribed coherence matrices, and reconstruct the coherence matrices via measurements of the spatial Stokes parameters. These results indicate the possibility of deploying programmable photonics for producing large-dimensional structured partially coherent light for applications in communications, cryptography, sensing, and spectroscopy.
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Recent advances in optical imaging and communication increasingly involve high-dimensional, partially coherent light, creating a growing need for scalable tools to measure and manipulate coherence. Here, we demonstrate the automatic separation of spatially partially coherent light into "coherence modes" -- its orthogonal and mutually incoherent components. To make this separation possible, we exploit variational processing in layered self-configuring interferometer architectures in a silicon photonic circuit. This process formally finds and measures the eigenvectors and eigenvalues of the coherency matrix, hence measuring the partially coherent state, while leaving it intact and separated after optimization. Furthermore, we show that mutually incoherent beams, if spatially orthogonal, can be automatically separated even if they are completely overlapped, hence separating unknown laser beams based only on their mutual incoherence. Our experiment finds and separates the two strongest coherence modes starting from a nine-mode sampling of the partially or fully overlapping fields from two independent lasers. The method requires a number of physical components that scales linearly with the rank rrr of the coherency matrix and operates through a sequence of rrr in situ gradient-based optimizations enabled by electronic drive frequency multiplexing of interferometer phase shifters. We benchmark its performance against a mixture-based tomographic method, also implemented on chip. These results establish a scalable framework for programmable coherence analysis and control in imaging, communication, and photonic information processing.
For partially coherent optical fields in which a single binary degree of freedom (DoF) is relevant, such as polarization, entropy uniquely identifies the class of optical fields that can be converted into each other via unitary transformations. However, when multiple DoFs are taken into consideration, entropy no longer serves this purpose. We investigate the structure of the family of iso-entropy partially coherent optical fields defined by two binary DoFs (polarization and two spatial modes) and described by a 4x4 coherence matrix G. We find that the rank of G (the number of its non-zero eigenvalues) plays a critical role in this context: whereby any pair of iso-entropy rank-2 fields can be converted into each other unitarily, this is not necessarily the case for a pair of rank-3 or rank-4 fields. Furthermore, unitary transformations between iso-entropy fields of different ranks are strictly forbidden. Instead, such conversions require entropy-maintaining non-unitary transformations that potentially combine filtering projections and randomizing operations. We experimentally synthesize partially coherent iso-entropy optical fields of all ranks, and tomographically reconstruct their coherence matrices. Moreover, we steer the coherence matrix over iso-entropy trajectories that maintain a fixed rank (intra-rank conversion) or that involve changes in the rank (inter-rank conversion). These findings offer a new perspective for the potential utility of partially coherent light in optical communications and sensing.
For partially coherent optical fields in which a single binary degree of freedom (DOF) is relevant, such as polarization, entropy uniquely identifies the class of optical fields that can be converted into each other via unitary transformations. However, when multiple DOFs are taken into consideration, entropy no longer serves this purpose. We investigate the structure of the family of isoentropy partially coherent optical fields defined by two binary DOFs (polarization and two spatial modes) and described by a $4\ifmmode\times\else\texttimes\fi{}4$ coherence matrix G\mathbf{G}G. We find that the rank of G\mathbf{G}G (the number of its nonzero eigenvalues) plays a critical role in this context: whereby any pair of isoentropy rank-2 fields can be converted into each other unitarily, this is not necessarily the case for a pair of rank-3 or rank-4 fields. Furthermore, unitary transformations between isoentropy fields of different ranks are strictly forbidden. Instead, such conversions require entropy-maintaining nonunitary transformations that potentially combine filtering projections and randomizing operations. We experimentally synthesize partially coherent isoentropy optical fields of all ranks and tomographically reconstruct their coherence matrices. Moreover, we steer the coherence matrix over isoentropy trajectories that maintain a fixed rank (intrarank conversion) or that involve changes in the rank (interrank conversion). These findings offer a different perspective for the potential utility of partially coherent light in optical communications and sensing.
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.