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Journal of Modern Optics

Optical coherence engineering in the polarization and spatial degrees of freedom

K. R. Sethuraj, Bhaskar Kanseri

Published Oct 31, 2019• 6 citations
Publisher

Abstract

We report a method to synthesize the complete range of arbitrary combinations of coherences independently in both the polarization and spatial degrees of freedom (DoFs) by utilizing energy conserving unitary and non-unitary transformations. We further demonstrate that starting from a fully spatially coherent and polarized field, one can achieve two light fields having any degree of mutual coherence, and remixing these fields can result in maximum visibility (unit) interference fringes irrespective of any degree of polarization of the mixed field. We believe that these results provide new insight and may contribute significantly towards the ongoing explorations on optical coherence and its applications.

DOI
10.1080/09500340.2019.1683242

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We report a method to synthesize the complete range of arbitrary combinations of coherences independently in both the polarization and spatial degrees of freedom (DoFs) by utilizing energy conserving unitary and non-unitary transformations. We further demonstrate that starting from a fully spatially coherent and polarized field, one can achieve two light fields having any degree of mutual coherence, and remixing these fields can result in maximum visibility (unit) interference fringes irrespective of any degree of polarization of the mixed field. We believe that these results provide new insight and may contribute significantly towards the ongoing explorations on optical coherence and its applications.
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.
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.
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.
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.