## Abstract

Studying the coherence of an optical field is typically compartmentalized with respect to its different physical 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 regard 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.

© 2017 Optical Society of America

## Corrections

30 August 2017: A typographical correction was made to the author affiliations.

## 1. INTRODUCTION

Optical coherence is evaluated by assessing the correlations between field fluctuations at different points in space and time [1]. When multiple degrees of freedom (DoFs) of an optical field—spatial, temporal, and polarization—are relevant, the coherence of each DoF is typically studied separately. For example, spatial coherence is evaluated via double-slit interference [2], temporal coherence through two-path (e.g., Michelson) interference [3], and polarization coherence by measuring the Stokes parameters [4]. Although this traditional approach succeeds when the DoFs are uncoupled, it fails at capturing key features of the field coherence if they are correlated [5,6].

Here we show that coherence can be viewed as a “resource”
that can be reversibly converted from one DoF of the field to another. We
demonstrate experimentally the reversible and energy-conserving (unitary)
conversion of coherence between the spatial and polarization DoFs of an
optical field. Starting from a linearly polarized field having *no
spatial coherence* (a complete lack of double-slit
interference visibility), we convert the field without filtering or loss
of energy into one that displays spatial coherence (high-visibility
interference fringes) but is *unpolarized*. The optical
arrangement we describe engenders an internal reorganization of the field
energy that leads to a migration of the entropy associated with the
statistical fluctuations from one DoF (spatial) to another (polarization).
This coherency conversion is confirmed by measuring the full
$4\times 4$ coherency matrix that provides a complete
description of two-point vector-field coherence via “optical
coherency matrix tomography” (OCmT) [7,8]. The tomographic
measurement of coherence is carried out at different stages in the
experimental setup to confirm the transformations involved in the
coherence conversion process. As an application to highlight the
usefulness of reallocating the entropy from one DoF of the field to
another, we show that the field can be reconfigured to be invariant under
the impact of a depolarizer or polarization scrambler that transforms any
input polarization to unpolarized light. By transferring all the field
entropy into polarization, the polarization scrambler cannot further
increase the polarization entropy, which thus emerges unchanged. Since the
coherence conversion procedure is reversible and no energy is lost, the
field may be reversed to its original fully polarized configuration after
traversing the polarization scrambler.

The paper first briefly reviews the matrix approach to quantifying optical coherence for a single DoF and multiple DoFs. Second, we describe the concept of an optical-coherence converter, a system that reversibly transforms coherence—viewed as a resource—from one DoF of a field to another without loss of energy. Starting from a field with one coherent DoF and another incoherent DoF, we reversibly convert the field such that the former DoF becomes incoherent and the latter coherent. Third, we present the experimental arrangement used in confirming these predictions and the measurement scheme to identify multi-DoF beam coherence. Finally, we demonstrate the invariance of a field with respect to a polarization scrambler before presenting our conclusions.

## 2. MULTI-DoF COHERENCE

#### A. Polarization Coherence and Spatial Coherence

Partial polarization is described by a polarization coherency matrix
${\mathit{G}}_{\mathrm{p}}=(\genfrac{}{}{0ex}{}{{G}^{\mathrm{HH}}}{{G}^{\mathrm{VH}}}\genfrac{}{}{0ex}{}{{G}^{\mathrm{HV}}}{{G}^{\mathrm{VV}}})$, where $\mathrm{H}$ and $\mathrm{V}$ identify the horizontal and vertical
polarization components, respectively, ${G}^{ij}=\u27e8{E}^{i}{({E}^{j})}^{*}\u27e9$, $i,j=\mathrm{H},\mathrm{V}$, and $\u27e8\xb7\u27e9$ is an ensemble average [9]. Here,
${\mathit{G}}_{\mathrm{p}}$ is Hermitian
(${\mathit{G}}_{\mathrm{p}}^{\u2020}={\mathit{G}}_{\mathrm{p}}$), semi-positive, normalized such that
${G}^{\mathrm{HH}}+{G}^{\mathrm{VV}}=1$, ${G}^{\mathrm{HH}}$ and ${G}^{\mathrm{VV}}$ are the contributions of H and V to
the total power, respectively, and ${G}^{\mathrm{HV}}$ is their normalized correlation. The
degree of polarization is ${D}_{\mathrm{p}}=|{\lambda}_{\mathrm{H}}-{\lambda}_{\mathrm{V}}|$, where ${\lambda}_{\mathrm{H}}$ and ${\lambda}_{\mathrm{V}}$ are the eigenvalues of
${\mathit{G}}_{\mathrm{p}}$ [10]. Spatial coherence can be similarly described via a
spatial coherency matrix for points $\overrightarrow{a}$ and $\overrightarrow{b}$, ${\mathit{G}}_{\mathrm{s}}=(\genfrac{}{}{0ex}{}{{G}_{aa}}{{G}_{ba}}\genfrac{}{}{0ex}{}{{G}_{ab}}{{G}_{bb}})$, where ${G}_{kl}=\u27e8{E}_{k}{E}_{l}^{*}\u27e9$, $k,l=a,b$. The properties of
${\mathit{G}}_{\mathrm{s}}$ are similar to those of
${\mathit{G}}_{\mathrm{p}}$. The visibility of the interference
fringes observed by superposing the fields from
$\overrightarrow{a}$ and $\overrightarrow{b}$ is $V=2|{G}_{ab}|$. Alternatively, the degree of spatial
coherence ${D}_{\mathrm{s}}=|{\lambda}_{a}-{\lambda}_{b}|$ represents the
*maximum* visibility obtained after equalizing the
amplitudes at $\overrightarrow{a}$ and $\overrightarrow{b}$, where ${\lambda}_{a}$ and ${\lambda}_{b}$ are the eigenvalues of
${\mathit{G}}_{\mathrm{s}}$ [2,11].

A DoF represented by a $2\times 2$ coherency matrix carries up to
1 bit of entropy; e.g., the polarization entropy is
${S}_{\mathrm{p}}=-{\lambda}_{\mathrm{H}}{\mathrm{log}}_{2}{\lambda}_{\mathrm{H}}-{\lambda}_{\mathrm{V}}{\mathrm{log}}_{2}{\lambda}_{\mathrm{V}}$, where $0\le {S}_{\mathrm{p}}\le 1$. The zero-entropy state
${S}_{\mathrm{p}}=0$ corresponds to a fully polarized
field (no statistical fluctuations), whereas the maximal-entropy state
${S}_{\mathrm{p}}=1$ corresponds to an unpolarized field
(maximal fluctuations) [12];
similarly for the spatial DoF based on ${\mathit{G}}_{\mathrm{s}}$. Entropy so defined is a
*unitary invariant* of the field DoF: it cannot be
changed by applying lossless deterministic optical
transformations.

#### B. Joint Polarization and Spatial Coherence Formalism

Evaluating ${\mathit{G}}_{\mathrm{p}}$ and ${\mathit{G}}_{\mathrm{s}}$ is *not* sufficient to
completely identify the coherence of a vector field in which the
polarization and spatial DoFS are potentially correlated. A
$4\times 4$ coherency matrix
$\mathit{G}$ is necessary to capture the full
vector-field coherence [6,13],

*not a unitary invariant*of the field [15,16], and reversible optical transformations that span the spatial and polarization DoFs can increase $V$ [17,18].

Each *physically independent* DoF (spatial and
polarization) carries 1 bit of entropy, so the vector field now
carries 2 bits of entropy: $S=-{\lambda}_{a\mathrm{H}}{\mathrm{log}}_{2}{\lambda}_{a\mathrm{H}}-{\lambda}_{a\mathrm{V}}{\mathrm{log}}_{2}{\lambda}_{a\mathrm{V}}-{\lambda}_{b\mathrm{H}}{\mathrm{log}}_{2}{\lambda}_{b\mathrm{H}}-{\lambda}_{b\mathrm{V}}{\mathrm{log}}_{2}{\lambda}_{b\mathrm{V}}$, where $0\le S\le 2$ and $\{\lambda \}=\{{\lambda}_{a\mathrm{H}},{\lambda}_{a\mathrm{V}},{\lambda}_{b\mathrm{H}},{\lambda}_{b\mathrm{V}}\}$ are the real positive eigenvalues of
$\mathit{G}$. The zero-entropy state
$S=0$ corresponds to a fully polarized
*and* spatially coherent field (no statistical
fluctuations in either DoF and $\{\lambda \}=\{\mathrm{1,0},\mathrm{0,0}\}$), whereas the maximal-entropy state
$S=2$ corresponds to an unpolarized
spatially incoherent field (maximal fluctuations in both DoFs and
$\{\lambda \}=\frac{1}{4}\{\mathrm{1,1},\mathrm{1,1}\}$).

## 3. CONCEPT OF OPTICAL COHERENCY CONVERSION

In general, $S\le {S}_{\mathrm{p}}+{S}_{\mathrm{s}}$, with equality achieved only when the two DoFs are independent, in which case $\mathit{G}$ can be written in separable form $\mathit{G}={\mathit{G}}_{\mathrm{s}}\otimes {\mathit{G}}_{\mathrm{p}}$. In general, ${S}_{\mathrm{s}}$ and ${S}_{\mathrm{p}}$ are obtained from the $2\times 2$ “reduced” spatial and polarization coherency matrices

The concept of an optical-coherence converter is illustrated in
Fig. 1(a). Consider the
case when the field carries 1 bit of entropy
($S=1$) and the DoFs are independent
($S={S}_{\mathrm{p}}+{S}_{\mathrm{s}}$), in which case a single DoF can
accommodate this entropy. The field may be maximally
*incoherent* but polarized (${S}_{\mathrm{s}}=1$ and ${S}_{\mathrm{p}}=0$), whereupon no interference fringes can
be observed [Fig. 1(b)].
Alternatively, the field may be spatially coherent but *un*
polarized (${S}_{\mathrm{s}}=0$ and ${S}_{\mathrm{p}}=1$), in which case full-visibility fringes
can be observed [Fig. 1(c)].
We demonstrate here that an optical field can be
*reversibly* transformed from the former configuration
to the latter without loss of energy, thus *converting*
coherence from one DoF (polarization) to the other (spatial). Throughout
the procedure, $S$ remains constant; that is, no uncertainty
is added or removed from the field, only an internal reorganization of the
field engendered by a unitary transformation confines the statistical
fluctuations to one DoF while freeing the other from uncertainty. We call
such a system a “coherence converter.”

The optical arrangement we propose to convert coherency between the spatial
and polarization DoFs is depicted in Fig. 2. We start from two points
${\overrightarrow{a}}^{\prime}$ and ${\overrightarrow{b}}^{\prime}$ of equal intensity in a spatially
*in* coherent H-polarized field (the fields are
mutually incoherent or statistically independent), which thus produce no
interference fringes. The polarization at ${\overrightarrow{b}}^{\prime}$ is rotated to become orthogonal to that
at ${\overrightarrow{a}}^{\prime}$
$(\mathrm{H}\to \mathrm{V})$ before combining the fields at a
polarizing beam splitter (PBS), which yields an *un*
polarized field. We then split the field into two points
$\overrightarrow{a}$ and $\overrightarrow{b}$ using a *non* polarizing
beam splitter, which creates two copies of the field that can demonstrate
high-visibility interference fringes. We proceed now to present the
measurements at each step of this coherency-conversion process.

## 4. SOURCE CHARACTERIZATION

The optical field we study is extracted from a broadband, unpolarized LED (center wavelength 850 nm, 30 nm FWHM bandwidth; Thorlabs M850L3 IR). The field is spectrally filtered (10 nm FWHM), polarized along H, and spatially filtered through a 100 μm wide slit placed at a distance of 180 mm from the source. The “input” plane that includes the points ${\overrightarrow{a}}^{\prime}$ and ${\overrightarrow{b}}^{\prime}$ (each defined by a 100 μm wide slit) is located 420 mm away from the slit [the source in Fig. 2(a)]. We first confirm that the field is spatially coherent within ${\overrightarrow{a}}^{\prime}$ and ${\overrightarrow{b}}^{\prime}$ separately (i.e., the spatial coherence width of the field, estimated to be $\sim 1\text{\hspace{0.17em}}\mathrm{mm}$, is larger than the slit width). This is accomplished using a narrow pair of slits (50 μm wide separated by $\mathrm{\Delta}=150\text{\hspace{0.17em}}\mathrm{\mu m}$) at either ${\overrightarrow{a}}^{\prime}$ or ${\overrightarrow{b}}^{\prime}$ and observing the double-slit interference on a CCD camera (Hamamatsu 1394) at a distance of $d=200\text{\hspace{0.17em}}\mathrm{mm}$ away. High-visibility fringes ($V=0.98$) are observed separated by $\lambda d/\mathrm{\Delta}\approx 1170\text{\hspace{0.17em}}\mathrm{\mu m}$. Next, we superpose the fields from ${\overrightarrow{a}}^{\prime}$ and ${\overrightarrow{b}}^{\prime}$ [“measurement” in Fig. 2(a)] and observe no interference fringes [Fig. 3(a)], confirming that the two points are separated by more than the field coherence width. We have thus confirmed the relationship between the two length scales involved: the sizes of the locations at ${\overrightarrow{a}}^{\prime}$ and ${\overrightarrow{b}}^{\prime}$ (100 μm) are smaller than the coherence width, and the separation between them (10 mm) is larger. The field that we start from is linearly polarized (scalar) but spatially incoherent, thus $\mathit{G}$ has the form

To fully characterize the field coherence across the spatial and polarization DoFs, we measure ${\mathit{G}}_{1}$ via OCmT [7,8], which requires 16 measurements to reconstruct ${\mathit{G}}_{1}$. Since four polarization projections are required to identify ${\mathit{G}}_{\mathrm{p}}$ and four spatial projections are required to determine ${\mathit{G}}_{\mathrm{s}}$ for a scalar field, $4\times 4$ linearly independent combinations of these spatial and polarization projections are necessary to reconstruct $\mathit{G}$ subject to the constraints of Hermiticity, semi-positiveness, and unity trace. These measurements are in one-to-one correspondence to those required to reconstruct a two-qubit density matrix in quantum mechanics, a process known as “quantum state tomography” [20–22]. Carrying out these optical measurements (see [8] for details), ${\mathit{G}}_{1}$ is reconstructed [Fig. 3(b)] and found to be in good agreement with the theoretical expectation [Fig. 3(c)], with the remaining slight deviations attributable to unequal powers at ${\overrightarrow{a}}^{\prime}$ and ${\overrightarrow{b}}^{\prime}$.

The measured coherency matrix ${\mathit{G}}_{1}$ in the $({\overrightarrow{a}}^{\prime},{\overrightarrow{b}}^{\prime})$ plane yields $S=1.001$, and the reduced spatial and polarization
coherency matrices ${\mathit{G}}_{\mathrm{s}}^{(\mathrm{r})}$ and ${\mathit{G}}_{\mathrm{p}}^{(\mathrm{r})}$ obtained from $\mathit{G}$ yield ${S}_{\mathrm{s}}=0.991$ and ${S}_{\mathrm{p}}=0.037$, respectively. The field entropy is thus
associated with the spatial DoF and *not* polarization,
resulting in an absence of interference fringes [Fig. 3(a)]. The lack of interference
fringes is consistent with the fact that all measures of spatial coherence
or double-slit interference fringes for a *vector field*
rely on the cross-correlation matrix [14] or beam-coherence matrix [23], ${\mathit{G}}_{ab}=\left(\genfrac{}{}{0ex}{}{{G}_{ab}^{\mathrm{HH}}}{{G}_{ab}^{\mathrm{VH}}}\genfrac{}{}{0ex}{}{{G}_{ab}^{\mathrm{HV}}}{{G}_{ab}^{\mathrm{VV}}}\right)$, which is the top-right
$2\times 2$ block of the coherency matrix
$\mathit{G}$. For example, the degree of coherence
proposed by Wolf [14], the degree
of electromagnetic coherence [15,16], the complex
degree of mutual polarization [24],
the visibility predicted through a generalized form of the
Fresnel–Arago law [25], or
the maximal visibility obtainable through local unitary transformations
[17,18] all predict that the observed visibility will be
zero if all the entries in ${\mathit{G}}_{ab}$ are zero. Because the measured and
theoretically expected ${\mathit{G}}_{1}$ has all zero entries in the
${\mathit{G}}_{ab}$ block, it follows that interference
fringes do not form. We proceed to show that interference fringes
nevertheless appear once the coherence has been *reversibly
converted* from polarization to the spatial DoF.

## 5. CONVERTING COHERENCE FROM POLARIZATION TO SPACE

Converting the coherence reversibly from polarization to the spatial DoF
entails maximizing the polarization entropy ${S}_{\mathrm{p}}$ and minimizing the spatial entropy
${S}_{\mathrm{s}}$ at *fixed* total entropy
$S$. A half-wave plate (HWP) placed after
${\overrightarrow{b}}^{\prime}$ rotates the polarization
$\mathrm{H}\to \mathrm{V}$, resulting in a new coherency matrix
$\mathit{G}=\frac{1}{2}\mathrm{diag}\{\mathrm{1,0},\mathrm{0,1}\}$. The fields from
${\overrightarrow{a}}^{\prime}$ and ${\overrightarrow{b}}^{\prime}$ are then directed by mirrors to the two
input ports of a PBS (Thorlabs CM1-PBS252), where the V component is
transmitted and H is reflected [Fig. 4(a)]. Consequently, the H and V components overlap
in the same output port at ${\overrightarrow{a}}^{\prime \prime}$ (minimal power in the other port
${\overrightarrow{b}}^{\prime \prime}$). Note that a PBS is a reversible device:
when the two output fields reverse their direction and return to the PBS,
the input fields are reconstituted. The field is now unpolarized, which we
confirm by registering a flat Malus curve and comparing it to the
sinusoidal Malus curve produced by the linearly polarized input fields
[Fig. 4(b)]. That is,
each incident field on the PBS is linearly polarized, whereas their
superposition at the output—with the initial optical power now
concentrated in a single path—is randomly polarized. The coherency
matrix ${\mathit{G}}_{2}$ at ${\overrightarrow{a}}^{\prime \prime}$ and ${\overrightarrow{b}}^{\prime \prime}$ is reconstructed via OCmT
[Fig. 4(c)] and is found
to be in good agreement with the expected form [Fig. 4(d)]:

It is important to note that the two spatial modes derived from ${\overrightarrow{a}}^{\prime}$ and ${\overrightarrow{b}}^{\prime}$ completely overlap in a single mode after the PBS. This is confirmed by placing double slits separated by 150 μm right after the PBS and observing high-visibility interference fringes (thus eliminating the possibility of incoherence due to reduced spatial overlap [26]).

From the reconstructed ${\mathit{G}}_{2}$ in the $({\overrightarrow{a}}^{\prime \prime},{\overrightarrow{b}}^{\prime \prime})$ plane, the total entropy is
$S=0.997$, but the new values of entropy for the
spatial and polarization DoFs are ${S}_{\mathrm{s}}=0.001$ and ${S}_{\mathrm{p}}=0.996$, respectively. *The entropy has
now been converted between the two DoFs*. To observe
interference fringes, the randomly polarized field at
${\overrightarrow{a}}^{\prime \prime}$ is split symmetrically into two halves by
a 50:50 *non*-polarizing beam splitter to points
$\overrightarrow{a}$ and $\overrightarrow{b}$ [Fig. 5(a)], which can then be overlapped to produce
high-visibility fringes. This step does not change the values of
$S$, ${S}_{\mathrm{s}}$, or ${S}_{\mathrm{p}}$. The $(\overrightarrow{a},\overrightarrow{b})$ plane is the image plane relayed from the
$({\overrightarrow{a}}^{\prime},{\overrightarrow{b}}^{\prime})$ plane by a lens [Fig. 2(c)]. The coherency matrix
${\mathit{G}}_{3}$ in this plane is

The measured ${\mathit{G}}_{3}$ reconstructed via OCmT [Fig. 5(b)] is in good agreement with the theoretical expectation [Fig. 5(c)].

Now that we have ${G}_{ab}^{\mathrm{HH}}+{G}_{ab}^{\mathrm{VV}}=\frac{1}{2}$, the predicted visibility is
$V=1$ [14]. Furthermore, because the two DoFs are independent
*and* the field is unpolarized [as is clear from the
separable form of ${\mathit{G}}_{3}$ in Eq. (5)], projecting the
polarization on any direction will not affect the high visibility
[Fig. 5(d)]. Note that
the coherence time of the field is determined by its spectral bandwidth,
and observing the fringes [Fig. 5(d)] requires introducing a relative delay in the path of the
fields from $\overrightarrow{a}$ or $\overrightarrow{b}$ before overlapping them [delay line 2 in
Fig. 2(c)]. The
variation in the measured visibility with introduced relative time delay
corresponds to the expected field coherence time $\approx 100\text{\hspace{0.17em}}\mathrm{fs}$ (coherence length
$\approx 30\text{\hspace{0.17em}}\mathrm{\mu m}$) [Fig. 5(e)].

## 6. SURVIVING RANDOMIZATION THROUGH ENTROPY REALLOCATION

The ability to redistribute or reallocate the field entropy
$S$ between the DoFs can be exploited in
protecting a DoF from the deleterious impact of a randomizing medium.
Consider a depolarizing medium represented by a Mueller matrix
$\widehat{M}=\mathrm{diag}\{\mathrm{1,0},\mathrm{0,0}\}$ that converts *any* state
of polarization into a completely unpolarized state. The initial field
${\mathit{G}}_{1}$ [Figs. 3(b) and 3(c)] would be converted into the incoherent unpolarized field
${\mathit{G}}_{1}^{\prime}=\frac{1}{4}\mathrm{diag}\{\mathrm{1,1},\mathrm{1,1}\}$ upon traversing this medium with
${S}_{\mathrm{p}}\to 1$, such that $S\to 2$. If, however, coherence is first
reversibly converted from the spatial DoF to polarization
(${S}_{\mathrm{p}}\to 1$ and ${S}_{\mathrm{s}}\to 0$), then traversing a depolarizing medium
*cannot increase*
${S}_{\mathrm{p}}$, and the field is thus left unchanged.
Subsequently, the coherence conversion can be reversed and a polarized
field is retrieved after emerging from the depolarizing medium without
loss of energy.

We have carried out the proof-of-concept experiment depicted in Fig. 6(a) where we place a depolarizer or polarization scrambler in the path of the field ${\mathit{G}}_{2}$ in the $({\overrightarrow{a}}^{\prime \prime},{\overrightarrow{b}}^{\prime \prime})$ plane. The polarization scrambler is implemented by rotating a HWP. The CCD camera exposure time is increased to 10 s, corresponding to the rotation time of the wave plate, to capture the averaged interference pattern in a single shot. The Mueller matrix associated with a polarization scrambler is $\widehat{M}=\mathrm{diag}\{\mathrm{1,0},\mathrm{0,0}\}$. The visibility observed after the $(\overrightarrow{a},\overrightarrow{b})$ plane remains high. This result can be modeled theoretically by first noting that a unitary transformation $\widehat{U}$ transforms the field according to ${\mathit{G}}_{2}\to {\mathit{G}}_{2}^{\prime}=\widehat{U}{\mathit{G}}_{2}{\widehat{U}}^{\u2020}$, where $\widehat{U}$ is a $4\times 4$ unitary transformation that spans the spatial and polarization DoFs. If the device in question impacts the two DoFs independently, then $\widehat{U}={\widehat{U}}_{\mathrm{s}}\otimes {\widehat{U}}_{\mathrm{p}}$, where ${\widehat{U}}_{\mathrm{s}}$ and ${\widehat{U}}_{\mathrm{p}}$ are $2\times 2$ unitary transformations for the spatial and polarization DoFs, respectively. The impact of a randomizing but energy-conserving transformation can be modeled as a statistical ensemble of unitary transformations [27–29]. The transformation of ${\mathit{G}}_{2}$ upon traversing a polarization scrambler can be expressed as

*invariant with regard to any polarization randomization*.

If the polarization scrambler is position dependent, the coherency matrix ${\mathit{G}}_{2}$ will no longer be invariant under randomization (because the spatial and polarization DoFs become coupled). In the experiment illustrated in Fig. 6(b), the polarization scrambler is placed at $\overrightarrow{a}$ in the plane of ${\mathit{G}}_{3}$. The spatial-polarization transformation of ${\mathit{G}}_{3}$ takes the form

## 7. DISCUSSION AND CONCLUSION

To facilitate the analysis of the coherence of optical fields encompassing multiple DoFs, it is becoming increasingly clear that the mathematical formalism of multipartite quantum mechanical states is most useful [5,6,30]. The underlying foundation for this utility is the analogy between the mathematical descriptions used in these domains. The Hilbert space of a multipartite system is the tensor product of the Hilbert spaces of the single-particle subsystems. In classical optics, the multiple DoFs of a beam are described in a linear vector space having formally the structure of a tensor product of the linear vector spaces of the individual DoFs. In the quantum setting, pure multipartite states that cannot be separated into products of single-particle states are said to be entangled, whereas in the classical setting, fields in which the DoFs cannot be separated are now being called “classically entangled” [6]; coherence can be viewed as a “resource” that may be reversibly converted from one DoF of the beam to another, just as entanglement is a resource shared among multiple quantum particles. There is a critical difference though between the quantum and classical settings. Entanglement between initially separable particles requires nonlocal operations of particle–particle interactions (which are particularly challenging for photons [31]); on the other hand, entangling operations that couple different DoFs of a beam are readily available in classical optics. Adopting this information-driven standpoint has led to a host of novel opportunities and applications. For example, Bell’s measure, originally developed for ascertaining nonlocality, becomes useful in quantifying the coherence of a multi-DoF beam and assessing the resources required to synthesize a beam of given characteristics [6]; beams in which spatial modes and polarization are classically entangled have been shown to enable fast particle tracking [32], local teleportation [33,34], and full Mueller characterization of a sample [35,36]; and introducing spatiotemporal spectral correlations leads to propagation-invariant pulsed optical beams [37,38].

We have demonstrated—for the first time to the best of our knowledge—the “conversion” or transformation of coherence from one DoF of an optical field to another, namely, from polarization to the spatial DoF. Starting from a field that is spatially incoherent but polarized, we redirect the statistical fluctuations from space toward polarization, resulting in an unpolarized field that is spatially coherent. Specifically, the 1 bit of entropy characterizing the spatial DoF was removed and added instead to the initially zero-entropy polarization. This is confirmed by multiple tests: (i) the field is found to be spatially coherent by reconstructing the $4\times 4$ coherency matrix and observing the high-visibility interference fringes after overlapping the two field paths; (ii) the field is unpolarized as confirmed by the invariance of the outcome under different polarization projections; and (iii) the polarization and spatial DoFs are separable, as confirmed by repeating the spatial interference measurements and observing the invariance of the visibilities for all polarization projections. Entropy engineering of partially coherent optical fields can open the path to a variety of possible future extensions and applications with regard to optimizing the interaction of optical fields with disordered media.

## Funding

Office of Naval Research (ONR) (N00014-14-1-0260, N00014-17-1-2458).

## Acknowledgment

The authors thank A. Dogariu and D. N. Christodoulides for useful discussions.

## REFERENCES

**1. **L. Mandel and E. Wolf, *Optical Coherence and Quantum
Optics* (Cambridge
University, 1995).

**2. **F. Zernike, “The concept of
degree of coherence and its application to optical
problems,” Physica **5**, 785–795
(1938). [CrossRef]

**3. **M. Born and E. Wolf, *Principles of Optics*,
7th ed. (Cambridge
University, 1999).

**4. **C. Brosseau, *Fundamentals of Polarized
Light* (Wiley,
1998).

**5. **X.-F. Qian and J. H. Eberly, “Entanglement and
classical polarization states,”
Opt. Lett. **36**, 4110–4112
(2011). [CrossRef]

**6. **K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s
measure in classical optical coherence,”
Nat. Photonics **7**, 72–78
(2013). [CrossRef]

**7. **A. F. Abouraddy, K. H. Kagalwala, and B. E. A. Saleh, “Two-point optical
coherency matrix tomography,” Opt.
Lett. **39**, 2411–2414
(2014). [CrossRef]

**8. **K. H. Kagalwala, H. E. Kondakci, A. F. Abouraddy, and B. E. A. Saleh, “Optical coherency
matrix tomography,” Sci.
Rep. **5**, 15333 (2015). [CrossRef]

**9. **E. Wolf, *Introduction to the Theory of
Coherence and Polarization of Light*
(Cambridge University,
2007).

**10. **A. Al-Qasimi, O. Korotkova, D. James, and E. Wolf, “Definitions of the
degree of polarization of a light beam,”
Opt. Lett. **32**, 1015–1016
(2007). [CrossRef]

**11. **A. F. Abouraddy, “What is the maximum
attainable visibility by a partially coherent electromagnetic
field in Young’s double-slit
interference?” Opt. Express **25**, 18320–18331
(2017). [CrossRef]

**12. **C. Brosseau and A. Dogariu, “Symmetry properties
and polarization descriptors for an arbitrary electromagnetic
wavefield,” in *Progress in
Optics*, E. Wolf, ed.
(Elsevier, 2006),
Vol. 49, Chap. 4,
pp. 315–380.

**13. **F. Gori, M. Santarsiero, and R. Borghi, “Vector mode
analysis of a Young interferometer,”
Opt. Lett. **31**, 858–860
(2006). [CrossRef]

**14. **E. Wolf, “Unified theory of
coherence and polarization of random electromagnetic
beams,” Phys. Lett. A **312**, 263–267
(2003). [CrossRef]

**15. **J. Tervo, T. Setälä, and A. T. Friberg, “Degree of coherence
of electromagnetic fields,” Opt.
Express **11**, 1137–1143
(2003). [CrossRef]

**16. **T. Setälä, J. Tervo, and A. T. Friberg, “Complete
electromagnetic coherence in the space-frequency
domain,” Opt. Lett. **29**, 328–330
(2004). [CrossRef]

**17. **F. Gori, M. Santarsiero, and R. Borghi, “Maximizing
Young’s fringe visibility through reversible optical
transformations,” Opt.
Lett. **32**, 588–590
(2007). [CrossRef]

**18. **R. Martínez-Herrero and P. M. Mejías, “Maximum visibility
under unitary transformations in two-pinhole interference for
electromagnetic fields,” Opt.
Lett. **32**, 1471–1473
(2007). [CrossRef]

**19. **A. Peres, *Quantum Theory: Concepts and
Methods* (Kluwer
Academic, 1995).

**20. **W. K. Wootters, “Local accessibility
of quantum states,” in *Complexity,
Entropy, and the Physics of Information*, W. H. Zurek, ed., Vol. VIII of
SFI Studies in the Sciences of Complexity
(Addison-Wesley,
1990),
pp. 39–46.

**21. **D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of
qubits,” Phys. Rev. A **64**, 052312 (2001). [CrossRef]

**22. **A. F. Abouraddy, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Quantum
entanglement and the two-photon Stokes
parameters,” Opt. Commun. **201**, 93–98
(2002). [CrossRef]

**23. **F. Gori, “Matrix treatment
for partially polarized, partially coherent
beams,” Opt. Lett. **23**, 241–243
(1998). [CrossRef]

**24. **J. Ellis and A. Dogariu, “Complex degree of
mutual polarization,” Opt.
Lett. **29**, 536–538
(2004). [CrossRef]

**25. **M. Mujat, A. Dogariu, and E. Wolf, “A law of
interference of electromagnetic beams of any state of coherence
and polarization and the Fresnel–Arago interference
laws,” J. Opt. Soc. Am. A **21**, 2414–2417
(2004). [CrossRef]

**26. **A. Aiello, F. Toppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum-like
nonseparable structures in optical beams,”
New J. Phys. **17**, 043024 (2015). [CrossRef]

**27. **K. Kim, L. Mandel, and E. Wolf, “Relationship
between Jones and Mueller matrices for random
media,” J. Opt. Soc. Am. A **4**, 433–437
(1987). [CrossRef]

**28. **J. J. Gil, “Characteristic
properties of Mueller matrices,” J.
Opt. Soc. Am. A **17**, 328–334
(2000). [CrossRef]

**29. **O. Gamel and D. F. V. James, “Causality and
complete positivity of classical polarization
maps,” Opt. Lett. **36**, 2821–2823
(2011). [CrossRef]

**30. **B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum
entanglement resolves a basic issue in polarization
optics,” Phys. Rev. Lett. **104**, 023901
(2010). [CrossRef]

**31. **V. Venkataraman, K. Saha, and A. L. Gaeta, “Phase modulation at
the few-photon level for weak-nonlinearity-based quantum
computing,” Nat. Photonics **7**, 138–141
(2013). [CrossRef]

**32. **S. Berg-Johansen, F. Töppel, B. Stiller, P. Banzer, M. Ornigotti, E. Giacobino, G. Leuchs, A. Aiello, and C. Marquardt, “Classically
entangled optical beams for high-speed kinematic
sensing,” Optica **2**, 864–868
(2015). [CrossRef]

**33. **S. M. Hashemi Rafsanjani, M. Mirhosseini, O. S. Magana-Loaiza, and R. W. Boyd, “State transfer
based on classical nonseparability,”
Phys. Rev. A **92**, 023827 (2015). [CrossRef]

**34. **D. Guzman-Silva, R. Brüning, F. Zimmermann, C. Vetter, M. Grafe, M. Heinrich, S. Nolte, M. Duparre, A. Aiello, M. Ornigotti, and A. Szameit, “Demonstration of
local teleportation using classical
entanglement,” Laser Photon.
Rev. **10**, 317–321
(2016). [CrossRef]

**35. **F. Töppel, A. Aiello, C. Marquardt, E. Giacobino, and G. Leuchs, “Classical
entanglement in polarization metrology,”
New J. Phys. **16**, 073019 (2014). [CrossRef]

**36. **S. Tripathi and K. C. Toussaint, “Rapid Mueller
matrix polarimetry based on parallelized polarization state
generation and detection,” Opt.
Express **17**, 21396–21407
(2009). [CrossRef]

**37. **H. E. Kondakci and A. F. Abouraddy, “Diffraction-free
pulsed optical beams via space-time
correlations,” Opt. Express **24**, 28659–28668
(2016). [CrossRef]

**38. **K. J. Parker and M. A. Alonso, “Longitudinal
ISO-phase condition and needle pulses,”
Opt. Express **24**, 28669–28677
(2016). [CrossRef]