Optical coherency matrix tomography

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.

A fundamental capability that has remained elusive in classical optics is the complete identification of the coherence function for a beam with coupled DoFs. In quantum mechanics, the task of measuring all the elements of a density matrix is known as 'quantum state tomography' 34,35 . The corresponding procedure for multi-DoF beams in classical optics has been studied theoretically 11 , but has not been demonstrated experimentally heretofore. Even in the simplest case of two binary DoFs 6 (e.g., polarization, a bimodal waveguide 36,37 , two coupled single-mode waveguides 38,39 , spatial-parity modes [40][41][42][43][44] , etc.), the associated 4 × 4 coherency matrix G, which is a complete representation of second-order coherence 10,11 , has not been measured in its entirety to date.

Results
In this Article, we present a methodical approach-optical coherency matrix tomography (OCmT)-for measuring the complex elements of 4 × 4 coherency matrices G by appropriating the quantum-state-tomography strategy. To demonstrate the universality of our approach, we implement it with coherent and partially coherent fields having coupled or uncoupled DoFs in three distinct settings involving pairs of points [9][10][11] , spatial-parity modes [40][41][42][43][44] , and orbital angular momentum (OAM) modes 45 -each together with polarization. We identify the minimal set of linearly independent, joint spatial-polarization projective measurements that enable a unique reconstruction of G. Since G is a complete representation of the field, its reconstruction obviates the need to measure directly any coherence descriptors (all of which are scalar functions of the complex elements of G) and, moreover, allows for unambiguous identification of classical entanglement.
The coherence of an optical beam having a single binary DoF is represented by a 2 × 2 Hermitian  (i) the scalar field at two points  r a and  r b , E a and E b ; (ii) the spatial-parity even 'e' and odd 'o' modes of a scalar field E e and E o refs. 40-44; or (iii) a pair of OAM modes, e.g., E 0 and E 1 corresponding to OAM =  0 and 1, respectively 45,47 . When two binary DoFs of the field are relevant, e.g., the first is polarization 'p' and the second is a spatial 's' DoF with modes identified as 'a' and 'b' [ Fig. 1(c)]-the corresponding coherency matrix G is now 4 × 4 refs. 10, 11,  11 . In determining the coherence descriptors of each DoF independently of the other, one first traces over the other DoF to obtain a reduced coherency matrix 10 . The reduced polarization coherency matrix G p , obtained by tracing over the spatial DoF in G, is given by  The elements of the reduced coherency matrices are measured by a system sensitive to one DoF, but not to the other. When the two DoFs are uncoupled, then = ⊗ G G G p s , otherwise the elements of G p and G s lack information about the correlations between the two DoFs that is contained in G. Such correlations are only measurable by a system that is sensitive to both DoFs via joint polarization-spatial measurements.
We pose the following question: what are the necessary and sufficient measurements to reconstruct an arbitrary G for two binary DoFs? This question was solved by Wootters 48 in the context of reconstructing the density matrix ρ for a bipartite quantum system. He showed that the measurements carried out on each subsystem to reconstruct its reduced density matrix are sufficient to reconstruct ρ when carried out jointly-a methodology known as quantum state tomography 34,35 . In our context of a classical optical beam having two binary DoFs, the analogy with the quantum setting allows us to exploit the same strategy. Regardless of the specific form of G, the necessary measurements to carry out OCmT and reconstruct G [ Fig. 1(c)] are those used to reconstruct the reduced coherency matrices [ Fig. 1(a,b)] carried out in cascades of pairs of projections-one for polarization and the other for the spatial DoF. Each measurement yields a real number I lm (projection l for polarization and m for the spatial DoF) corresponding to the projection of a tomographic slice through G. The 16 combinations of . We have performed a series of experiments implementing the OCmT scheme described above using quasi-monochromatic beams having two binary DoFs: polarization and a spatial DoF. We have measured the 4 × 4 coherency matrix G for six different beams corresponding to distinct states of light having the following properties: G 1 : the polarization and spatial DoFs are separable and both are coherent. G 2 : the polarization DoF is coherent while the spatial DoF lacks coherence. G 3 : both the polarization and spatial DoFs lack coherence. G 4 : the polarization and spatial DoFs are classically entangled. G 5 : the polarization and spatial DoFs are classically correlated. G 6 : this beam is a mixture of the separable-coherent beam G 1 and the classically entangled beam G 4 .
We use the sequence of polarization projections described earlier and present below the spatial projections following the H projection (similar spatial projections are carried out following the V, D and R polarization projections).
Polarization with Spatial Position. The first realization of the spatial DoF is the traditional two points, as in the Young's double slit experiment. The polarization and position-coupled beam is prepared in one of six states G 1 through G 6 ; Fig. 2(a) (see Supplementary information for details). OCmT for a such a beam comprises of the polarization analysis followed by the spatial analysis; Fig. 2(b). The spatial analysis may alternatively be carried out by extracting specific intensity points from the far-field intensity patterns for only two values of displacement x on a screen or an array of detectors; Fig. 2 It is important to note that the visibility of fringes is not the parameter sought here to characterize the spatial coherence at 'a' and 'b'; instead the four points identified in Fig. 2(d), together with the set of points obtained for the V, D, and R polarization projections, reveal the complete picture even when polarization and the spatial DoFs are classically entangled.
Polarization with Spatial Parity. The second spatial-DoF realization makes use of one-dimensional even 'e' and odd 'o' spatial-parity modes with respect to x = 0. The polarization and spatial parity-coupled beam is prepared in one of six states G 1 through G 6 ; Fig. 3

(a) (see Supplementary information for details).
OCmT for a such a beam comprises of the polarization analysis followed by the spatial-parity analysis; Fig. 3(b). The four spatial projections are obtained by measuring the power (integrated over the shaded areas in Fig. 3(c)) in the following settings: (1) the total power I 10 = I H of the beam; (2) the power of the even component I 11 = I He obtained from a modified Mach-Zehnder interferometer that separates the beam into the different spatial-parity components 41    beams from the two separable and non-separable classes (see Supplementary information for the complete results). In each experiment, the prepared beam passes first through polarization then spatial-DoF analysis stages (the order may be reversed without changing the outcome). In each of these realizations, permutations of the four polarization projection settings combined with the four spatial projection settings yield 16 measurements for OCmT, which are used to reconstruct G. We make use of a maximal-likelihood algorithm that exploits the constraints set by the trace, hermiticity, and semi-positive-definiteness of G 50 . We portray the real and imaginary components of G using the standard visualization from quantum state tomography. In each plot we provide the coherence descriptor for the polarization D p and spatial DoF D s obtained from their reduced coherency matrices, in addition to the linear entropy , which serves as a measure of the overall beam coherence, where [ Fig. 5(c)]. In all three cases, the separability of the two DoFs is readily detected by visual inspection of G and confirmed by taking the direct product of the reduced coherency matrices.
Beams with non-separable DoFs. We next present two fundamentally distinct classes of beams with non-separable G in Fig. 6(a,b). First, OCmT of a classically entangled beam G 4 is shown in Fig. 6(a), wherein the beam is fully coherent, and yet the measures extracted from reduced coherency matrices indicate complete incoherence. In such a beam, the polarization and spatial modes occur in pairs-e.g., H with 'a' and V with 'b' (but never H with 'b' or V with 'a'). In the traditional view of the double slit experiment, such coupling will produce no interference fringes, and the lack of visibility may be interpreted as the absence of spatial coherence, despite the beam being perfectly spatially coherent. This coupling between the DoFs is in fact encoded in the non-zero off-diagonal elements of G revealed once it is reconstructed through OCmT, but cannot be obtained from G p or G s . Second, a classically correlated beam G 5 is shown in Fig. 6(b) in which the same coupling between polarization and spatial modes occurs as in the previous example, except the different combinations are incoherently mixed and not linearly superposed. The partial global coherence-despite the complete lack of coherence for each DoF-is clear from the fact that not all the diagonal elements of G 5 are equal as is the case in G 3 .
Mixture of beams with separable and non-separable DoFs. Finally, in Fig. 6(c) we depict G 6 corresponding to a beam formed by statistically mixing the separable-coherent beam G 1 and the classically entangled beam G 4 . The measurement of G 6 indicates that part of the apparent incoherence in this beam stems from the intrinsic randomness in the individual DoFs, and part of it from the correlation, or classical entanglement, between the two DoFs.

Discussion
The reconstruction of G allows for the unambiguous and complete mathematical expression of fields that are coherent, partially coherent, or incoherent, in either, or both, DoFs of an optical beam with two binary DoFs. The usefulness of this technique becomes specially apparent in cases where the DoFs are coupled or non-separable, and the traditional scalar measures of coherence provide a conflicting and fallacious account of beam coherence. The apparent absence of coherence in any DoF may be the result of intrinsic randomness due to statistical fluctuations, or due to the coupling or non-separability with another DoF. In the latter case, the measurement of G also provides the way for implementing unitary transformations required to undo such coupling, and restore coherence in the DoFs. The application of our work can be easily seen in the myriad applications of coherence under conditions of coupled DoFs, particularly those involving localized vector beams, sub-diffraction imaging, nanophotonics, and propagation through disordered media. Measurement of G, before and after transmission though a system that couples various DoFs, will help determine the characteristics of the system. This technique may hence find important applications in crystallography, atmospheric optics, and systems involving photonic crystals or anisotropic scatters, etc.
In summary, we have experimentally demonstrated for the first time a methodical, yet versatile, approach to reconstructing the 4 × 4 coherency matrix G of an optical beam having two binary DoFs, which we call optical coherency matrix tomography. We have explored three different physical realizations in which we combine polarization with spatial position, spatial parity, or orbital angular momentum modes. By exploiting the mathematical similarity with quantum state tomography of two photon states, we determine the minimal set of measurements required to reconstruct G. Although we have conducted the experiments for a beam with two binary DoFs, this methodology is equally applicable for a higher number of DoFs with m-ary levels each.