1. Introduction

Coherence is a fundamental feature of an optical field that determines the possibility of the field interfering with a modified version of itself, whether temporally delayed, spatially displaced, or rotated in polarization [1–6]. As such, optical coherence plays a key role in interferometric-based sensing and imaging techniques, such as label-free optical sectioning in biological media using optical coherence tomography [7] or optical diffraction tomography [8], and in imaging schemes that exploit the coherency function [9]. Generally speaking, coherence can only be increased through filtering procedures; e.g., spatial coherence can be enhanced by use of an aperture, and temporal coherence can be increased through spectral filtering [10]. In such cases, coherence enhancement come at the cost of irreversible energy loss. Conversely, reducing coherence requires a randomizing process that adds entropy to the field; e.g., a rotating ground glass diffuser can decrease the spatial coherence of a field [11], which can help reduce coherent imaging artifacts. Although entropy can be added without loss of energy, the procedure is irreversible.

Experimental studies of coherence usually isolate a particular degree of freedom (DoF) of the field – whether spatial, temporal, or polarization, although the formal definition of optical coherence includes all the DoFs simultaneously [12,13]. The past decade has witnessed new concepts being introduced into the study of optical coherence that originated in quantum mechanics [14–21]. In particular, the mathematical analogy between the density matrix representing the quantum state of a multi-partite system and the coherency matrix for a classical optical field encompassing multiple degrees of freedom (DoFs) allows us to formulate the concept of ‘classical entanglement’ [15,22–28] – based on the initial suggestion by Spreeuw [29,30]. This refers to the non-separability of the joint-DoF coherency matrix into a direct product of conventional single-DoF coherency matrices [15]. Moreover, entropy can be defined for multi-DoF and single-DoF classical fields in a similar fashion to multi-partite and single-particle quantum systems. In addition to their interest for fundamental understanding of optical coherence, these advances have paved the way to applications in particle tracking [31], polarimetry [32–34], and communication channel characterization [35–37].

These developments in the theoretical formalism of optical coherence theory have led to the conceptualization of coherence as a ‘resource’ that may be manipulated across and exchanged between multiple DoFs. This line of thinking informs a strategy for converting coherence from one DoF to another by swapping entropy between these DoFs. For example, we have demonstrated that coherence can be converted unitarily (i.e., without loss of energy) from the polarization to the spatial DoF via entropy swapping, thus converting a linearly polarized field lacking spatial coherence (no fringes observable in a double-slit experiment) to an unpolarized field endowed with spatial coherence (high-visibility double-slit interference fringes) [25]. No entropy is added to or removed from the field, and only lossless reversible optical processes are used in this transformation (see subsequent work in [38,39]). This particular example provides a paradigm for this emerging enterprise of exploiting coherence as a ‘resource’. An open question remains: can lossless and reversible coherence conversion be achieved via entropy swapping in a partially coherent field?

Here, we experimentally validate the reversibility of coherence conversion between two distinct DoFs of an optical field via entropy swapping. The process of coherence conversion is carried out unitarily without loss of energy between the DoFs in both directions. The field is initially prepared so that the polarization DoF carries one bit of entropy (unpolarized), whereas the spatially DoF is fully coherent. That is, the field’s statistical fluctuations are maximized in polarization, but are absent from the spatial DoF. We then perform two successive coherence conversions. In the first, entropy associated with polarization fluctuations are swapped with the spatial DoF; the field becomes spatially incoherent but polarized. A second coherence conversion process then returns the field to its original unpolarized and spatially coherent state. Finally, we exploit this experimental strategy to ‘protect’ the spatial coherence of the field from spatial phase scrambling by first converting the coherence from the spatial DoF to the polarization DoF, and then subsequently converting coherence back to the spatial DoF. At each stage in our experiment, the changes in coherence are evaluated using optical coherence matrix tomography (OCmT) [22,23,40] to measure the full $4\times 4$ coherency matrix encompassing both DoFs.

The paper is organized as follows: First, we outline the theoretical basis for coherence conversion via entropy swapping couched in the formalism of multi-DoF coherency matrices. Second, we describe our experimental approach for reversible coherence conversion and deconversion between the spatial and polarization DoFs. Third, we present our experimental results confirming these predicted field transformations and verifying the ‘protection’ of spatial coherence from a phase-scrambling device.

2. Theoretical formulation of reversible coherence conversion

The polarization coherence at a point in an optical field can be described using a coherency matrix ${ \textbf {G}_{\mathrm {p}}= \begin {pmatrix} G^\mathrm {HH} & G^\mathrm {HV}\\G^\mathrm {VH} & G^\mathrm {VV}\end {pmatrix}}$, where H and V correspond to the horizontal and vertical polarization components, respectively, $G^{ij} = \langle {E^i(E^j)^*}\rangle$, $i,j =$ H,V, and $\langle {\cdot }\rangle$ is an ensemble average [6,25]. Similarly, spatial coherence for a pair of points in a scalar optical field may also be described by a $2\times 2$ coherency matrix: ${\textbf {G}_\mathrm {s}=\begin {pmatrix} G_{aa} & G_{ab}\\G_{ba} & G_{bb}\\\end {pmatrix}}$, where $a$ and $b$ identify two points in space, $G_{kl}=\langle {E_k(E_l)^*}\rangle$, and $k,l=a,b$. Both $\mathbf {G}_{\mathrm {p}}$ and $\mathbf {G}_{\mathrm {s}}$ are Hermitian non-negative matrices, so that their eigenvalues are real and non-negative. We define the polarization entropy as $S_\mathrm {p}=-{\lambda _\mathrm {H}\log _{2}\lambda _\mathrm {H}}-{\lambda _\mathrm {V}\log _{2}\lambda _\mathrm {V}}$, where $\lambda _\mathrm {H}$ and $\lambda _\mathrm {V}$ are the eigenvalues of $\mathbf {G}_{\mathrm {p}}$. We similarly define the spatial entropy as $S_\mathrm {s}=-{\lambda _a\log _{2}\lambda _a - \lambda _b\log _{2}\lambda _b}$, where $\lambda _a$ and $\lambda _b$ are the eigenvalues of $\mathbf {G}_\mathrm {s}$. By imposing the normalization $\mathrm {Tr}\{\mathbf {G}_{\mathrm {p}}\}=1$ and $\mathrm {Tr}\{\mathbf {G}_{\mathrm {s}}\}=1$, where $\mathrm {Tr}\{[\cdot ]\}$ is the matrix trace, we have $0\leq S_\mathrm {p}\leq 1$ and $0\leq S_\mathrm {s}\leq 1$ [41,42]. Each of these field configurations can carry up to one bit of entropy when the field lacks coherence. When the field is fully polarized $S_{\mathrm {p}}=0$, and when unpolarized $S_{\mathrm {p}}=1$; similarly, when the field is spatially coherent $S_{\mathrm {s}}=0$, and when spatially incoherent $S_{\mathrm {s}}=1$. Entropy so-defined can thus be used to quantify the degree of coherence of the particular DoF of the field [43].

Consider now the combined spatial and polarization coherence at a pair of points ($a$ and $b$) in a vector field. Because these two DoFs need not be separable, then $\textbf {G}_\mathrm {s}$ and $\textbf {G}_\mathrm {p}$ are not sufficient descriptors of field coherence [24,25]. Instead, the coherence of such a field must be represented by a $4\times 4$ coherency matrix that describes the full vector-field coherence in terms of discretized space and polarization

Because $\mathbf {G}$, just like $\mathbf {G}_{\mathrm {p}}$ and $\mathbf {G}_{\mathrm {s}}$, is Hermitian and non-negative, its eigenvalues $\{\lambda \} = \{\lambda _{a\mathrm {H}},\lambda _{a\mathrm {V}},\lambda _{b\mathrm {H}},\lambda _{b\mathrm {V}}\}$ are real and non-negative. We define the field entropy as: $S=-\sum _{m}\lambda _{m}\log _{2}\lambda _{m}$, where $\lambda _{m}$ are the eigenvalues of $\mathbf {G}$, and $0\leq S\leq 2$ with the normalization $\mathrm {Tr}\{\mathbf {G}\}=1$. That is, the field can carry up to 2 bits of entropy across both DoFs. It is necessary to use such a coherency matrix to account for the correlations between the DoFs, which may arise from optical transformations that affect the two DoFs jointly [25].

To calculate the entropy for each DoF independently of the other, $S_\mathrm {s}$ and $S_\mathrm {p}$, we define the $2\times 2$ ‘reduced’ spatial and polarization coherency matrices

We have previously shown that this formalism opens up heretofore unexplored opportunities in the study of optical coherence. In particular, this overall approach enables us to exploit and manipulate coherence as a ‘resource’ by reorganizing the field structure to modify the distribution of entropy between the DoFs. The embodiment of the coherence exchange we carry out here is illustrated in Fig. 1. We start with a partially coherent field possessing one bit of entropy $S\!=\!1$. The fluctuations giving rise to this entropy can be completely restricted to one DoF [24,25], whether spatial or polarization – leaving the other DoF fully coherent. Our starting point here is a field that is spatially coherent $S_{\mathrm {s}}\!=\!0$ but unpolarized $S_{\mathrm {p}}\!=\!1$, represented by the coherency matrix

 

Fig. 1. Reversible coherence conversion. Starting with an optical field (represented by the coherency matrix $\mathbf {G}_{1}$) that is spatially coherent ($D_{\mathrm {s}}=1$, $S_{\mathrm {s}}=0$) but unpolarized ($D_{\mathrm {p}}=0$, $S_{\mathrm {p}}=1$), the field traverses a coherence converter that produces a new field ($\mathbf {G}_{2}$) that is spatially incoherent ($D_{\mathrm {s}}=0$, $S_{\mathrm {s}}=1$) yet polarized ($D_{\mathrm {p}}=1$, $S_{\mathrm {p}}=0$). Entropy has been swapped from polarization to the spatial DoF. The new field then traverses a second coherence converter that produces a field ($\mathbf {G}_{3}=\mathbf {G}_{1}$), which is again spatially coherent ($D_{\mathrm {s}}=1$, $S_{\mathrm {s}}=0$) but unpolarized ($D_{\mathrm {p}}=0$, $S_{\mathrm {p}}=1$). Entropy has been swapped back from the spatial DoF to the polarization DoF.

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The coherence conversion $\mathbf {G}_{1}\rightarrow \mathbf {G}_{2}$ results in a field that is now spatially incoherent $S_{\mathrm {s}}\!=\!1$ but polarized $S_{\mathrm {p}}\!=\!0$. The field no longer produces interference fringes in a double-slit interferometer configuration. The total entropy remains fixed at $S\!=\!1$; no entropy has been added or removed from the field as a consequence of the coherence conversion [25]. The field once again, however, has all the entropy confined to one DoF. It is important to note that an intermediary step in this coherence conversion, the two DoFs become correlated (non-separable) and entropy is shared between the two DoFs before the entropy can be fully swapped to and solely compartmentalized in the spatial DoF. Note that the elements $\mathbf {G}_{\mathrm {s}}$ and $\mathbf {G}_{\mathrm {p}}$ have completely interchanged in this coherence conversion. A similar quantum mechanical demonstration has been done in which spatial and polarization qubits of a single photon were unitarily exchanged via a SWAP operation [46]. In the second coherence conversion process $\mathbf {G}_{2}\rightarrow \mathbf {G}_{3}$, entropy swapping is implemented to revert the coherence back from the polarization DoF to the spatial DoF, such that $\mathbf {G}_{3}\!=\!\mathbf {G}_{1}$. The resulting field is spatially coherent $S_{\mathrm {s}}\!=\!0$ but unpolarized $S_{\mathrm {p}}\!=\!1$. High-visibility interference fringes can now be observed in a double slit configuration. The total entropy remains constant $S\!=\!1$, and no energy is lost in the course of the two coherence conversion processes implemented.

Whereas the process of reconstructing the single-DoF coherency matrices $\mathbf {G}_{\mathrm {p}}$ and $\mathbf {G}_{\mathrm {s}}$ from intensity measurements is well-known, the analogous procedure for reconstructing the dual-DoF coherency matrix $\mathbf {G}$ is less familiar. We developed a strategy dubbed OCmT for the process of reconstructing this coherency matrix from intensity measurements by noting the analogy between the dual-DoF coherency matrix for the classical optical field and the density matrix of a two-qubit quantum system. For the latter, the process of quantum state tomography helps reconstruct the density matrix from joint measurements performed on the two-qubits [47–49]; analogously, OCmT can be utilized to reconstruct the coherency matrix $\textbf {G}$ through joint measurements of the two-field DoFs [22,23,40], as we describe in more detail below.

3. Experiment

3.1 Optical arrangement

A schematic of the experimental arrangement used to convert coherence from the spatial DoF to the polarization DoF and back is shown in Fig. 2. The source is a spatially coherent, randomly polarized Helium Neon (HeNe) laser (Thorlabs, HNL100R; 10-mW power at a wavelength of 632.8 nm and a coherence length of $\approx 30$ cm), which is collimated and passed through a 100-$\mu$m-wide rectangular aperture. The coherency matrix is identified here as $\mathbf {G}_{1}$ [Eq. (4)]. The field is then split by a polarizing beam splitter (PBS) into two orthogonally polarized paths identified as $a$ and $b$. At this point, the polarization and spatial paths are correlated, and the field is represented by a coherency matrix that is not separable into coherency matrices for the polarization and spatial DoFs. The entropy is now shared between the DoFs. However, once the linear polarization in $b$ is rotated by a half-wave plate (HWP) to coincide with that in $a$, the initially unpolarized field becomes totally horizontally polarized at $a$ and $b$. Therefore, the operation of the HWP determines whether the field is made up of correlated or separable spatial and polarization coherency matrices; a similar disentangling operation via a HWP was observed in [46]. The polarization entropy at the source has been converted by the PBS to spatial entropy: the two paths $a$ and $b$ (each fully polarized) are mutually spatially incoherent. The field here is represented by the coherency matrix $\mathbf {G}_{2}$ [Eq. (5)]. This constitutes the first coherence conversion $\mathbf {G}_{1}\rightarrow \mathbf {G}_{2}$ (entropy swapped from the polarization DoF to the spatial DoF). To verify this conversion, lenses L$_1$, L$_2$, and L$_3$ relay the field from the $\mathbf {G}_{1}$-plane to the $\mathbf {G}_{2}$-plane via an imaging system. The coherency matrix $\mathbf {G}_{2}$ is then reconstructed via OCmT. Here the field is fully polarized but the paths $a$ and $b$ do not produce interference fringes when superposed spatially.

 

Fig. 2. Experimental setup for reversible coherence conversion. The first coherence conversion transforms the field from $\mathbf {G}_{1}$ to $\mathbf {G}_{2}$. The second coherence conversion transforms $\mathbf {G}_2$ into $\mathbf {G}_3=\mathbf {G}_{1}$. L$_1$–L$_{10}$: Convex spherical lenses; HWP: half-wave plate; PBS: polarizing beam splitter; BS: beam splitter; and SLM: spatial light modulator. The focal length of lenses L$_1$, L$_2$, L$_3$ is 100 mm; that for L$_4$ and L$_5$ is 125 mm; for L$_6$ and L$_7$ is 250 mm; and for L$_8$, L$_9$, and L$_{10}$ is 200 mm. The reflective SLM is shown in transmission mode for simplicity. The phase imparted by the SLM to the incident field (uniform and random) are sketched as an inset.

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The second coherence conversion swaps the field entropy back from the spatial DoF to the polarization DoF. The coherency matrix $\mathbf {G}_{2}$ is thus converted to the coherency matrix $\mathbf {G}_{3}$, where $\mathbf {G}_{3}=\mathbf {G}_{1}$. This conversion $\mathbf {G}_{2}\rightarrow \mathbf {G}_{3}=\mathbf {G}_{1}$ is achieved using the reverse system as that performing the first conversion $\mathbf {G}_{1}\rightarrow \mathbf {G}_{2}$. Lenses L$_4$, L$_5$, L$_6$, and L$_7$ relay the field from the $\mathbf {G}_{2}$ plane to the conjugate plane $\mathbf {G}_3$. A second HWP rotates the linear polarization in $b$ to be orthogonal to that in $a$, such that the two arms can be recombined at a second PBS into a single path. The combined field is now spatially coherent, but unpolarized. The spatial coherence of the field at $\mathbf {G}_{3}$ is further verified by splitting the field into two paths using a non-polarizing beam splitter (BS) after implementing the appropriate unity magnification imaging relay via the lenses L$_8$, L$_9$, and L$_{10}$ to the plane $\mathbf {G}_4$, whereby we reconstruct the coherency matrix via OCmT. Here the field is unpolarized, but the paths $a$ and $b$ produce high-visibility fringes when superposed spatially.

At the plane $\mathbf {G}_{2}$ between the two coherence conversion systems, the field is spatially incoherent but polarized. Besides verifying this statement by reconstructing $\mathbf {G}_{2}$ via OCmT, we also put this statement to the test by introducing a phase-scrambler that reduces the spatial coherence but leaves the polarization unaffected. We confirm that the spatial coherence retrieved in subsequent manipulations has been minimally affected by this phase scrambling procedure. For this purpose, we make use of a phase-only spatial light modulator (SLM; Hamamatsu X15213-07) having $1272\times 1024$ pixels each of pitch 12.5 $\mu$m to digitally scramble the phase. Relay lenses L$_4$ and L$_5$ map the $\mathbf {G}_{2}$-plane to that of the SLM. We make use of this scrambling system only in the final stage of the experiment, and the SLM is idled in the earlier experiments.

3.2 Reconstructing the coherency matrix via OCmT

We carry out OCmT measurements at four planes in our experiments ($\mathbf {G}_{1}$ through $\mathbf {G}_{4}$ in Fig. 2). Such measurements require spatial-polarization projections that are carried out sequentially with regards to the two DoFs. First, the field is projected onto the following polarization bases: (1) H and V, labelled $I_{1}$; (2) linear polarizations at $45^{\circ }$ and $-45^{\circ }$, labelled $I_{2}$; (3) right- and left-hand circular polarizations, labelled $I_{3}$; in addition to measuring the total intensity (no polarization projection, labelled $I_{0}$); see Fig. 3(a). After each of these projections, we carry out spatial projections extracted from the interferograms produced by spatially superposing the fields from the two spatial paths labelled $c$ and $d$ in Fig. 3(a). We extract four measurements from the spatial interferogram; two of these correspond to the intensity at the interferogram center when each one of the slits is blocked in turn [black dots in Fig. 3(a)]. The other two measurements are taken with both of the slits unblocked: the intensity at the interferogram center [blue dot in Fig. 3(a)], and the intensity midway between the center peak and the first interference minimum, corresponding to the result of introducing a $\tfrac {\pi }{2}$ phase shift between the fields at the two slots [green dot in Fig. 3(a)]. From these $4\times 4=16$ real measured quantities we reconstruct the complex $4\times 4$ coherency matrix subject to the restraints of Hermiticity and non-negativeness [22,23].

 

Fig. 3. (a) The polarization-spatial projections required to reconstruct a $4\times 4$ coherency matrix via OCmT. Four polarization projections ($I_{0}$ through $I_{3}$) are cascaded with four spatial projections (identified with the colored dots) obtained from the interferograms produced by spatially overlapping the fields from $c$ and $d$. A total of 16 intensity measurements are acquired to reconstruct the coherency matrix via the process detailed in [50]. (b) Graphical depiction of the elements of the measured and predicted coherency matrix $\mathbf {G}_{1}'$, plotted as a two-dimensional bar diagram, with the height representing the magnitude of the elements of the coherency matrix. (c) Same as (b) for the coherency matrix $\mathbf {G}_{1}$. The fidelity measure for $\mathbf {G}_{1}$ is $F = 0.99$.

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To produce the double-slit interference patterns necessary for reconstructing the coherency matrix, we bring together the fields from paths $a$ and $b$ (via mirrors not shown in Fig. 2). Of course, if the field is confined to one path (e.g., $\mathbf {G}_{1}$ and $\mathbf {G}_{3}$, where $E_{b}\!=\!0$), then no fringes are observed in such an experiment, and only the elements of the $2\times 2$ sub-matrix of $\mathbf {G}$ can have non-zero values. One may further emphasize the spatial coherence by splitting the field at $\mathbf {G}_{3}$ via a beam splitter into paths $a$ and $b$, after which the two fields can be spatially overlapped to confirm the high-visibility (unpolarized) spatial interference fringes. The associated coherency matrix $\mathbf {G}_{4}$ is given by

4. Demonstration of reversible coherence conversion

4.1 Source characterization

To confirm that the source is spatially coherent and unpolarized, we reconstruct $\mathbf {G}_{1}$ via OCmT. The assumption is that the field at paths $a$ and $b$ extracted from the source are each coherent (but may lack mutual coherence). This assumption is established by carrying out an initial measurement of the coherency matrix associated with the source. We perform the OCmT measurements before the initial slit, where the beam diameter is $\approx 4$ mm, by inserting a pair of 100-$\mu$m-wide slits separated by 300 $\mu$m (identified as $c$ and $d$), following the procedure outlined in Fig. 3(a). The experimental and theoretical coherency matrices denoted $\mathbf {G}_{1}'$ are plotted in Fig. 3(b). We find that $\mathbf {G}_{1}'$ indeed corresponds to $\mathbf {G}_{4}$ in Eq. (6); that is, the field is spatially coherent but unpolarized, and the two DoFs are separable. The values of the spatial and polarization entropy extracted from the reconstructed $\mathbf {G}_{1}'$ are $S_\mathrm {s}\approx 5.03\times 10^{-4}$ and $S_\mathrm {p}\approx 1.00$, respectively. The field is thus spatially coherent with transverse coherence width $>300$ $\mu$m. By passing the field through a 100-$\mu$m-wide slit to establish the path $a$, we guarantee full spatial coherence for each subsequent path. From the measured coherency matrix $\mathbf {G}_{1}'$, we can extract the coherency matrix $\mathbf {G}_{1}$ [Fig. 3(c)] in which only path $a$ has non-zero intensity whereas path $b$ is a null. Finally, to assess the fidelity between experimentally reconstructed coherency matrices using OCmT and their respective theoretical coherency matrices, we use $F=\{\mathrm {Tr}[(\sqrt {\mathbf {\Gamma }}\mathrm {\mathbf {G}}\sqrt {\mathbf {\Gamma }})^{1/2}]\}^2$, where $\mathbf {\Gamma }$ is the theoretical coherency matrix and $\mathrm {\mathbf {G}}$ is the measured matrix with the normalization $\mathrm {Tr}[(\sqrt {\mathbf {\Gamma }}\mathrm {\mathbf {G}}\sqrt {\mathbf {\Gamma }})^{1/2}]=1$ [23,51].

4.2 Converting coherence from the spatial DoF to the polarization DoF

The field at $\mathbf {G}_{1}$ is spatially in the form of a rectangle of width 100 $\mu$m along $x$ and is extended along $y$. All the spatial transformations are carried out along $x$. A PBS splits the field in path $a$ into orthogonally polarized field components directed into paths $a$ and $b$. The polarization components are made parallel using a HWP placed in path $b$ [Fig. 2]. This step eliminates the polarization entropy, which is swapped to the spatial DoF: paths $a$ and $b$ are not mutually coherent. The spatial and polarization entropies extracted from the measured coherency matrix $\mathbf {G}_{2}$ are $S_\mathrm {s}\approx 0.80$ and $S_\mathrm {p}\approx 1.2\times 10^{-4}$, respectively. The measured coherency matrix $\mathbf {G}_{2}$ is in good agreement with that expected from Eq. (5), as shown in Fig. 4.

 

Fig. 4. Graphical depiction of the elements of the measured and predicted coherency matrix $\mathbf {G}_{2}$, plotted as a two-dimensional bar diagram, with the height representing the magnitude of the elements of the coherency matrix. The fidelity measure for $\mathbf {G}_{2}$ is $F = 0.95$.

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4.3 Converting coherence back from the polarization DoF to the spatial DoF

Next, we convert coherence back from the polarization DoF to the spatial DoF via entropy swapping. The two paths $a$ and $b$ are combined by a second PBS into a single path after rotating the polarization in path $b$ via a HWP to be orthogonal to the polarization in path $a$. The initially polarized field at $\mathbf {G}_{2}$ is now unpolarized at $\mathbf {G}_{3}$. We confirm this transformation in the state of polarization first by obtaining two Malus curves for the field at the $\mathbf {G}_{2}$ and $\mathbf {G}_{3}$ planes [Fig. 5(a)]. The Malus curve associated with $\mathbf {G}_{2}$ shows a high degree of polarization, whereas that for $\mathbf {G}_{3}$ reveals a large drop in the degree of polarization. A full characterization of $\mathbf {G}_{3}$ via OCmT verifies that the field is indeed unpolarized but spatially coherent [Fig. 5(b)]. The spatial and polarization entropies extracted from $\mathbf {G}_3$ are $S_\mathrm {s}\approx 3.1\times 10^{-3}$ and $S_\mathrm {p}\approx 0.97$, respectively.

 

Fig. 5. (a) Malus curves for the field at planes $\mathbf {G}_{2}$ and $\mathbf {G}_{3}$ obtained by rotating a linear polarizer an angle $\theta$ with respect to H. The solid and dotted lines are theoretical expectations. (b) Graphical depiction of the elements of the measured and predicted coherency matrix $\mathbf {G}_{3}$, plotted as a two-dimensional bar diagram, with the height representing the magnitude of the elements of the coherency matrix. The fidelity measure for $\mathbf {G}_{3}$ is $F = 0.96$.

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At this point, the field has undergone a transfer of entropy from polarization to the spatial DoF, and then from the spatial DoF to polarization, thus reconstituting its original state of coherence. However, $\mathbf {G}_{3}$ corresponds to a field with a null in path $b$, so that the field is spatially coherent by definition. We can verify this spatial coherence by splitting the field once again into two paths $a$ and $b$ (each of which is unpolarized) via a beam splitter. The coherency matrix $\mathbf {G}_{4}$ describing this field is similar to that for $\mathbf {G}_{1}'$. The measured $\mathbf {G}_{4}$ is plotted in Fig. 6, and the extracted spatial and polarization entropies are $S_\mathrm {s}\approx 0.07$ and $S_\mathrm {p}\approx 0.98$, respectively.

 

Fig. 6. Graphical depiction of the elements of the measured and predicted coherency matrix $\mathbf {G}_{4}$, plotted as a two-dimensional bar diagram, with the height representing the magnitude of the elements of the coherency matrix. The fidelity measure for $\mathbf {G}_{4}$ is $F = 0.99$.

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5. ‘Protecting’ the spatial DoF from phase-scrambling

At the plane $\mathbf {G}_{2}$ after the first coherence conversion, the spatial DoF carries one bit of entropy, which is the maximum for a binary DoF, whereas the polarization carries no entropy. If the field undergoes a randomizing transformation that impacts only the spatial DoF, the spatial entropy cannot increase. The second coherence conversion then retrieves the spatial coherence to its original state. We have thus ‘protected’ the spatial coherence of the field from the randomizing process.

In our experiment, we have made use of an SLM to realize a phase-scrambler. The SLM imparts a two-dimensional random phase pattern exhibiting the full dynamic phase range of the SLM from 0 to $>2\pi$ over square superpixels comprising $10\times 10$ pixels [Fig. 2]. The paths $a$ and $b$ each cover one or two superpixels, so that we alter their mutual coherence, but may also reduce the coherence of each path alone (the state of polarization is unaffected). Of course, if the two paths are already mutually incoherent, then the phase scrambling has no impact on the spatial coherence of the field.

We first confirm the phase scrambling implemented by the SLM does indeed reduce spatial coherence. For that purpose, we direct a single spatially coherent circular optical beam with $\approx 4$ mm diameter to the SLM, which thus covers multiple superpixels. The beam reflected from the SLM is then passed through a pair of 100-$\mu$m-wide slits separated by 300 $\mu$m. In absence of the randomizing phase, the interference visibility is $V\!\approx \!92\%$ [Fig. 7(a)], which drops to $V\!\approx \!24\%$ in presence of the phase scrambling [Fig. 7(b)]. This measurement confirms the randomizing effect of the SLM on spatial coherence. Note that the random phase pattern of course deforms the field along both transverse dimensions $x$ and $y$.

 

Fig. 7. (a) Interference pattern produced by an unpolarized circularly symmetric Gaussian beam reflected from the SLM and passed through physical double slits in absence of the scrambling phase; bottom panel is its corresponding interferogram along $x$ after integrating along $y$. (b) Same as (a) after introducing the scrambling phase. (c) Interference pattern resulting from spatially overlapping the paths $a$ and $b$ at $\mathbf {G}_{4}$ in absence of the scrambling phase; bottom panel is its corresponding interferogram along $x$ after integrating along $y$. (d) Same as (c), the interference pattern at $\mathbf {G}_{4}$, after introducing the scrambling phase to the field at $\mathbf {G}_{2}$. (e) Graphical depiction of the elements of the measured and predicted coherency matrix $\mathbf {G}_{4}$ in the presence of the SLM scrambling phase, plotted as a two-dimensional bar diagram, with the height representing the magnitude of the elements of the coherency matrix. The fidelity measure for $\mathbf {G}_{4}$ is $F = 0.98$.

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We next direct the two-point field at plane $\mathbf {G}_{2}$ to the SLM and then through the second conversion. In absence of the randomizing phase, the interferogram obtained at $\mathbf {G}_{4}$ by spatially overlapping paths $a$ and $b$ displays high visibility, $V\!\approx \!95\%$ [Fig. 7(c)]. Once the random phase pattern is activated, two features stand out: the spatial interferogram obtained at $\mathbf {G}_{4}$ is still visible along $x$, but the field structure is randomized along $y$. Because the random phase pattern is not separable along $x$ and $y$, the spatial features can be coupled along these two dimensions. We assess the visibility of the interferogram along $x$ (after integrating the measured intensity along $y$) to be $V\!\approx \!61\%$ [Fig. 7(d)]. This is significantly higher than the visibility in Fig. 7(b) where the spatial DoF was not ‘protected’. The drop in visibility in Fig. 7(d) from that in Fig. 7(c) is attributed to the coupling between the spatial DoFs along $x$ and $y$. Finally, we plot the measured coherency matrix $\mathbf {G}_{4}$ in presence of the SLM scrambling phase. The measurements in Fig. 7(e) have not differed from those in Fig. 6 in absence of the scrambling phase, thus confirming that the spatial coherence has been ‘protected’.

Figure 7(d) demonstrates that the spatial DoF has been ‘protected’ and thus converted along only one spatial dimension. These findings motivated us to use a more appropriate scrambling pattern—one that only varies randomly in the $x$-direction. Also, to introduce more randomness to the field, we varied the pattern in time and collected multiple frames for our analysis. The SLM imparts one-dimensional random phase patterns exhibiting the full dynamic phase range of the SLM from 0 to $>2\pi$ over columns that extend vertically in the $y$-direction and are 10 pixels wide, see the inset of Fig. 2. The random phase patterns on the SLM were .gif file extensions, so they played like movies. The gif contained 10 frames with columns of varying phase shifts and played at 10 fps for 1 s. Interference images were acquired by a camera that collected multiple frames at 50 fps, resulting in stacks of 50 images. The results are shown in Fig. 8. Rather than send a circularly symmetric Gaussian beam to the SLM followed by a pair of double slits (like in Fig. 7), the PBS in the first coherence conversion setup was replaced with a nonpolarizing BS, and the two resulting mutually coherent virtual slits were relayed to the SLM, such that they covered the same area on the SLM as the mutually incoherent slits that were originally outputted from the first coherence conversion. The mutually coherent slits that were incident on the SLM were made to superpose after the SLM, and demonstrated high visibility when the SLM was idle, but exhibited severe phase scrambling and loss of spatial coherence in the presence of the scrambling pattern [Fig. 8(a, b)]. The mutually coherent slits experienced a decrease in visibility of 30% for the mean of the image stack. The mutually incoherent slits from the first coherence conversion (where a PBS is used) were sent to the idle SLM, then through the second conversion where $a$ and $b$ superposed at $\mathbf {G}_{4}$ and were captured by the camera, showing high visibility interference [Fig. 8(c)]. In [Fig. 8(d)], the mutually incoherent slits from the first coherence conversion were sent to the varying 1D phase pattern of the SLM, then through the second conversion where $a$ and $b$ superposed at $\mathbf {G}_{4}$ and were captured by the camera. The mean of the resulting interference patterns demonstrates high visibility interference, experiencing a decrease in visibility of only 1%. The results of Fig. 8 suggest that the first coherence conversion has provided the field with robust protection against random media. However, this level of protection has only been demonstrated for randomness along a single dimension. Future work will be directed to protecting the spatial coherence along both $x$ and $y$ simultaneously to avoid this drawback.

 

Fig. 8. (a) Interference pattern resulting from spatially overlapping two mutually coherent slits, generated by replacing the first PBS of the coherence converter with a BS when the SLM is idle; lower panel is its corresponding interferogram along $x$ after integrating along $y$. (b) The average of 50 interference patterns collected after the SLM while the SLM imparted the 1D scrambling phase pattern on the two mutually coherent slits, generated with a BS; lower panel is its corresponding interferogram along $x$ after integrating along $y$. (c) Interference pattern resulting from spatially overlapping the paths $a$ and $b$ at $\mathbf {G}_{4}$ in absence of the scrambling phase; bottom panel is the corresponding interferogram along $x$ after integrating along $y$. (d) The average of 50 frames collected at $\mathbf {G}_{4}$ while the SLM imparted the 1D scrambling phase pattern to the field at $\mathbf {G}_{2}$; lower panel is its corresponding interferogram along $x$ after integrating along $y$.

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6. Conclusion

We have demonstrated reversible conversion of coherence via entropy swapping between the spatial and polarization DoFs in an optical field. Using only unitary transformations, the coherence is converted back and forth with no loss in energy. Starting with a field that is spatially coherent and randomly polarized, we transferred the statistical fluctuations from the polarization DoF to the spatial DoF, thus engendering spatial incoherence and polarization coherence. We then reversed the entropy transfer to restore the field’s original coherence structure. This process, inspired by the mathematical formalism of quantum mechanics, treats coherence as a resource that can be shared between the DoFs of the field without loss of energy or increase in entropy.

Such entropy engineering can be useful in controlling the interaction of the optical field with randomizing systems, such as turbid or turbulent media. To date, however, coherence conversion and entropy swapping has been demonstrated with simple DoFs; namely binary DoFs that can be described each by a $2\times 2$ coherency matrix, and thus can carry one bit of entropy. This follows in the footsteps of the work done so far in the study of classical entanglement of optical fields [21,52,53] (which is also related to the study of vector beams [54,55]). The next crucial step in this enterprise is to extend the formalism and the experimental techniques to DoFs with large dimensionality, and potentially continuous DoFs. Recently, the first steps have been taken towards this goal by synthesizing coherent [56] and incoherent [57,58] optical fields in which the spatial and temporal DoFs are continuously correlated [20,59]. Implementing entropy swapping in this context offers the potential for powerful technologies that control coherence and protect the information encoded in the field against randomizing media.