The authors study partially coherent optical fields, for which polarization and the spatial degrees of freedom (DoF) are relevant, by exploring the rank of the associated coherence matrix. They show that rank-2 fields can always be rendered separable and can concentrate entropy entirely into one DoF, whereas all rank-3 fields are nonseparable and always have some locked entropy in one DoF.
The authors study partially coherent optical fields, for which polarization and the spatial degrees of freedom (DoF) are relevant, by exploring the rank of the associated coherence matrix. They show that rank-2 fields can always be rendered separable and can concentrate entropy entirely into one DoF, whereas all rank-3 fields are nonseparable and always have some locked entropy in one DoF.
We introduce a taxonomy for partially coherent optical fields spanning multiple degrees of freedom (DoFs) based on the rank of the associated coherence matrix (the number of non-zero eigenvalues). When DoFs comprise two spatial modes and polarization, a fourfold classification emerges, with rank-1 fields corresponding to fully coherent fields. We demonstrate theoretically and confirm experimentally that these classes have heretofore unrecognized different properties. Specifically, whereas rank-2 fields can always be rendered separable with respect to its DoFs via a unitary transformation, rank-3 fields are always non-separable. Consequently, the entropy for a rank-2 field can always be concentrated into a single DoF (thus ridding the other DoF of statistical fluctuations), whereas some entropy is always 'locked' in one DoF of a rank-3 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.
In optical communications, logical bits are encoded in physical degrees-of-freedom (DoFs) of the electromagnetic field. Consequently, optical scattering in a communications channel compromises the information transfer. In the worst-case-scenario, bit-to-bit stochastically varying scattering that couples the DoFs to each other—including even unused DoFs—can decouple the transmitter and receiver when relying on conventional physical encoding schemes and preclude the utilization of adaptive techniques as a counter-measure. Here, we show that partially coherent optical fields help circumvent the worst-case-scenario of rapidly varying, strong optical scattering, even when the channel is rendered informationally opaque for conventional approaches. Using a channel in which the spatial and polarization DoFs are relevant, we encode the logical bits in the unitarily invariant coherence rank (the number of non-zero eigenvalues of the field coherence matrix) and prepare our partially coherent fields with maximal entropies of 0, 1, 1.585, and 2 bits for rank-1, rank-2, rank-3, and rank-4 coherence matrices, respectively. This scheme demonstrates scattering-immune optical communications with 100 % fidelity. These results unveil an unexpected utility for partially coherent light in optical communications through challenging environments.
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. We find that the rank of 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.
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.
In optical communications, logical bits (0 and 1) encoded in light’s degrees-of-freedom (DoFs) are confounded in scattering channels. This scattering may affect a single DoF, e.g., a rotation in polarization, or DoFs may be coupled, e.g., polarization may be coupled to spatial modes, leading to a bit error in both scenarios. Adaptive techniques that probe the channel may be used to prescribe an optical field with properties that counteract the scattering. However, these techniques are ineffective if the channel changes stochastically bit-to-bit. In this work, we utilize partially coherent light to circumvent the deleterious effects of a channel that rapidly and randomly scatters one or two DoFs, including operations that couple DoFs. Partially coherent fields in which two DoFs are relevant can be described by a 4×4 coherence matrix, the rank of which (the coherence-rank) is invariant to unitary channels. In our experiments, we use channels that stochastically change bit-to-bit, unitarily scattering the polarization and spatial DoFs leading to scrambled information transfer when bits are encoded in just one of these DoFs. The scattering from these channels is subverted when data is encoded in the coherence-rank. These results offer a new method of using partially coherent light for optical communications through scrambling channels.