Reversible coherence conversion across optical degrees-of-freedom: a tutorial

The polarization of an optical field corresponds to its electric field vector orientations. Determining the state of polarization is often done by calculating the Stokes parameters, $S^{\mathrm{p}}_{l}$, where l is an integer value from 0 to 3 [1]. To find these parameters experimentally requires measuring the intensity (or power) of an optical field, using a polarization analyzer and a quarter wave plate, for instance, along certain polarization projections; $I^{\mathrm{p}}_0$, $I^{\mathrm{p}}_1$, $I^{\mathrm{p}}_2$, and $I^{\mathrm{p}}_3$ which correspond to the intensity of the horizontal component plus the intensity of the vertical component (the total intensity of the field), the intensity of the horizontal component, the intensity of the (+45^∘)-projection, and the intensity of the right-handed circular projection, respectively. The Stokes parameters are calculated using the equation $S^{\mathrm{p}}_{l} = 2I^{\mathrm{p}}_l - I^{\mathrm{p}}_0$, and the degree of polarization $D_{\mathrm{p}}$ is calculated using the Stokes parameters: $D_\mathrm{p} = (1/S^{\mathrm{p}}_{0})\sqrt{(S^{\mathrm{p}}_{1})^2 + (S^{\mathrm{p}}_{2})^2 + (S^{\mathrm{p}}_{3})^2}$ [21]. The degree of polarization should be interpreted as a ratio of intensities; the polarized part of the field over the total intensity. For an unpolarized field, the normalized Stokes parameters are $[S^{\mathrm{p}}_{0} \; S^{\mathrm{p}}_{1} \; S^{\mathrm{p}}_{2} \; S^{\mathrm{p}}_{3}] = [1 \; 0 \; 0 \; 0]$, and $D_{\mathrm{p}} = 0$. The state of polarization, or polarization coherence, at a point in an optical field may also be described using a coherency matrix which is a product of the Stokes parameters and the Pauli spin matrices ($\hat{\sigma}_{l}$),

$$\begin{align} {\mathbf{G}}_{\mathrm{p}}= \begin{pmatrix} G^\mathrm{HH}&G^\mathrm{HV}\\G^\mathrm{VH}&G^\mathrm{VV}\\ \end{pmatrix} = \frac{1}{2} \sum_{l=0}^{3} S^{\mathrm{p}}_{l} \hat{\sigma}_{l} = \frac{1}{2} \begin{pmatrix} S^{\mathrm{p}}_{0} + S^{\mathrm{p}}_{1}&S^{\mathrm{p}}_{2} - iS^{\mathrm{p}}_{3}\\[4pt] S^{\mathrm{p}}_{2} + iS^{\mathrm{p}}_{3}&S^{\mathrm{p}}_{0} - S^{\mathrm{p}}_{1}\\ \end{pmatrix}, \end{align} \tag{ 1 }$$

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 [1, 23]. The Pauli spin matrices are defined as $\hat{\sigma}_{0} = \begin{pmatrix} 1&0\\0&1\\ \end{pmatrix}, \hat{\sigma}_{1} = \begin{pmatrix} 1&0\\0&-1\\ \end{pmatrix}, \hat{\sigma}_{2} = \begin{pmatrix} 0&1\\1&0\\ \end{pmatrix}, \hat{\sigma}_{3} = \begin{pmatrix} 0&-i\\i&0\\ \end{pmatrix}$, and are used in this context because they underly the geometry of the Poincar$\acute{\mathrm{e}}$ sphere, the 3D geometric representation of all the possible states of polarization, although another complete set of four basis matrices may be used [25]. Here, ${G^\mathrm{HH}}$ and ${G^\mathrm{VV}}$ are the contributions of H and V to the total power of the field, respectively, whereas ${G^\mathrm{HV}}$ and ${G^\mathrm{VH}}$ are correlations between the H and V components. For a horizontally polarized field ${ {\mathbf{G}}_{\mathrm{p}} = \begin{pmatrix} 1&0\\0&0\\ \end{pmatrix}}$, under the normalization $\mathrm{Tr}\{\mathbf{G}_{\mathrm{p}}\} = 1$, where $\mathrm{Tr}\{[\cdot]\}$ is the matrix trace. For a vertically polarized field ${ {\mathbf{G}}_{\mathrm{p}} = \begin{pmatrix} 0&0\\0&1\\ \end{pmatrix}}$, for a 45^∘ polarized field ${ {\mathbf{G}}_{\mathrm{p}} = \frac{1}{2} * \begin{pmatrix} 1&1\\1&1\\ \end{pmatrix}}$, and for an unpolarized field ${ {\mathbf{G}}_{\mathrm{p}} = \frac{1}{2} * \begin{pmatrix} 1&0\\0&1\\ \end{pmatrix}}$.

The coherency matrix $\mathbf{G}_{\mathrm{p}}$ is Hermitian and positive-semidefinite such that it is equal to its conjugate transposed matrices, and has real, non-negative eigenvalues. These eigenvalues determine the entropy of the polarization DoF and the degree of polarization. We define the polarization entropy as $\mathcal{S}_\mathrm{p} = -{\lambda_\mathrm{H}\log_{2}\lambda_\mathrm{H}}-{\lambda_\mathrm{V}\log_{2}\lambda_\mathrm{V}}$, where $\lambda_\mathrm{H}$ and $\lambda_\mathrm{V}$ are the eigenvalues of $\mathbf{G}_{\mathrm{p}}$ [23]. Entropy in this context is a measure of the total disorder in the polarization DoF [26]. We can relate the degree of polarization to the eigenvalues of $\mathbf{G}_{\mathrm{p}}$ by $D_\mathrm{p} = |\lambda_\mathrm{H} - \lambda_\mathrm{V}|$ [1]. By imposing the normalization $\mathrm{Tr}\{\mathbf{G}_{\mathrm{p}}\} = 1$, we have $0\leqslant \mathcal{S}_\mathrm{p}\leqslant 1$ [27] and $0\leqslant D_\mathrm{p}\leqslant 1$. When the field is fully polarized $\mathcal{S}_{\mathrm{p}} = 0$, $D_{\mathrm{p}} = 1$, and when unpolarized $\mathcal{S}_{\mathrm{p}} = 1$, $D_{\mathrm{p}} = 0$. Another relation of the eigenvalues to $D_\mathrm{p}$ is $\lambda_\mathrm{H} = \frac{1 + D_\mathrm{p}}{2}$ and $\lambda_\mathrm{V} = \frac{1 - D_\mathrm{p}}{2}$ when $\lambda_\mathrm{H} - \lambda_\mathrm{V} \gt 0$, and $\lambda_\mathrm{H} = \frac{1 - D_\mathrm{p}}{2}$ and $\lambda_\mathrm{V} = \frac{1 + D_\mathrm{p}}{2}$ when $\lambda_\mathrm{H} - \lambda_\mathrm{V} \gt 0$ [25]. When $D_\mathrm{p} = 0$, $\lambda_j = \frac{1 + D_\mathrm{p}}{2}$. The relationship of the polarization entropy and the degree of polarization is $\mathcal{S}_{\mathrm{p}}(D_{\mathrm{p}}) \equiv -\log_{2}{(w(D_{\mathrm{p}}))}$ where $w(x) = (1/2)(1-x)^{(1-x)/2} (1+x)^{(1+x)/2}$, such that there is a nonlinear dependence of $D_{\mathrm{p}}$ on $\mathcal{S}_{\mathrm{p}}$; as $D_{\mathrm{p}}$ ranges from 0 to 1, $\mathcal{S}_{\mathrm{p}}$ ranges from 1 to 0. This is the red curve in figure 1. Also shown in figure 1 is a portion of the circle defined by $\mathcal{S}_{\mathrm{p}} = \sqrt{1-D_{\mathrm{p}}^2}$, drawn in blue. These two functions correspond to the left y-axis. One can quantitatively compare $\mathcal{S}_{\mathrm{p}} = \sqrt{1-D_{\mathrm{p}}^2}$ to the definition of $\mathcal{S}_{\mathrm{p}}(D_{\mathrm{p}})$ by plotting the absolute values of the first derivatives of each function which are shown as dashed curves in figure 1 and correspond to the right y-axis. These derivatives trend toward $+\infty$ as $D_{\mathrm{p}}$ approaches 1, and highlight that $\mathcal{S}_{\mathrm{p}}(D_{\mathrm{p}})$ does not have the same rate of change as a circular function.

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While the spatial DoF is a continuous variable, it may be reduced to just two points to assess its coherence—evaluating the statistical correlation between said points. Spatial points are said to be correlated if they are within the field's spatial coherence width or coherence area [1]. Spatial coherence for a pair of points in an optical field may also be described in a similar manner as polarization coherence using a $2\times2$ coherency matrix: ${{\mathbf{G}}_\mathrm{s} = \begin{pmatrix} G_{aa}&G_{ab}\\G_{ba}&G_{bb}\\\end{pmatrix}} = (1/2)\sum_{m = 0}^{3} S^\mathrm{s}_{m} \hat{\sigma}_{m}$, where a and b are two points of a scalar field in space, $S^\mathrm{s}_{m}$ are two-point 'spatial Stokes' parameters, $G_{kl} = \langle{E_k(E_l)^*}\rangle$, $\hat{\sigma}_{m}$ are the Pauli spin matrices, and $k,l = a,b$ [21]. Here, ${G_{aa}}$ and ${G_{bb}}$ are the contributions of a and b to the total power of the field, respectively, whereas ${G_{ab}}$ and ${G_{ba}}$ are correlations between the points a and b. The experimental process to calculate the spatial Stokes parameters requires measuring the intensity (or power) of the field along certain spatial projections; $I^{\,\mathrm{s}}_0$, $I^{\,\mathrm{s}}_1$, $I^{\,\mathrm{s}}_2$, and $I^{\,\mathrm{s}}_3$ correspond to the total intensity of the field, the intensity at a or b, the intensity after symmetric mixing of a and b, and the intensity after a $\pi/2$ phase shift is applied to b, respectively, furthermore, the two-point spatial Stokes parameters can be found using the equation $S^{\mathrm{s}}_{m} = 2I^{\,\mathrm{s}}_m - I^{\,\mathrm{s}}_0$ [21].

For a spatially coherent field made up of one point (point a) ${ {\mathbf{G}}_{\mathrm{s}} = \begin{pmatrix} 1&0\\0&0\\ \end{pmatrix}}$, under the normalization $\mathrm{Tr}\{\mathbf{G}_{\mathrm{s}}\} = 1$, where $\mathrm{Tr}\{[\cdot]\}$ is the matrix trace. For a spatially coherent field made up of just point b ${ {\mathbf{G}}_{\mathrm{s}} = \begin{pmatrix} 0&0\\0&1\\ \end{pmatrix}}$, for a spatially coherent field in which two points can be used to assess coherence ${ {\mathbf{G}}_{\mathrm{s}} = \frac{1}{2} * \begin{pmatrix} 1&1\\1&1\\ \end{pmatrix}}$, and for a spatially incoherent field ${ {\mathbf{G}}_{\mathrm{s}} = \frac{1}{2} * \begin{pmatrix} 1&0\\0&1\\ \end{pmatrix}}$. The coherency matrix $\mathbf{G}_{\mathrm{s}}$ is Hermitian and positive-semidefinite. Such is the case with the polarization coherency matrix, the spatial coherency matrix determines the entropy of the spatial DoF and the degree of spatial coherence. We define the spatial entropy as $\mathcal{S}_\mathrm{s} = -{\lambda_a\log_{2}\lambda_a - \lambda_b\log_{2}\lambda_b}$, where λ_a and λ_b are the eigenvalues of $\mathbf{G}_\mathrm{s}$ [24]. We define the degree of spatial coherence as $D_\mathrm{s} = |\lambda_a - \lambda_b|$ [28]. By imposing the normalization $\mathrm{Tr}\{\mathbf{G}_{\mathrm{s}}\} = 1$, we have $0\leqslant \mathcal{S}_\mathrm{s}\leqslant 1$ and $0\leqslant D_\mathrm{s}\leqslant 1$. When the field is spatially coherent $\mathcal{S}_{\mathrm{s}} = 0$, $D_{\mathrm{s}} = 1$, and when spatially incoherent $\mathcal{S}_{\mathrm{s}} = 1$, $D_{\mathrm{s}} = 0$.

As will be outlined in section 3, a key feature of coherence conversion is the correlating and uncorrelating of the spatial DoF and polarization, enabling the sharing of entropy between DoFs and the concentration of entropy in a single DoF [24]. When entropy is shared between two DoFs simultaneously, these DoFs are non-separable. Consider an optical field in which there exists correlations between the spatial and polarization DoFs. For cases in which the two DoFs are not separable (correlated), such is the case for classically entangled cylindrical vector beams [29, 30], ${\mathbf{G}_{\mathrm{p}}}$ and ${\mathbf{G}_{\mathrm{s}}}$ are not sufficient descriptors [23, 28]. Instead the spatial and polarization coherence must be considered jointly. Using a $4\times4$ coherency matrix that describes the full vector-field coherence in terms of discretized space and polarization,

$$\begin{equation} \mathbf{G} = \begin{pmatrix} G^\mathrm{HH}_{aa} & G^\mathrm{HV}_{aa} & G^\mathrm{HH}_{ab} & G^\mathrm{HV}_{ab}\\[3pt] G^\mathrm{VH}_{aa} & G^\mathrm{VV}_{aa} & G^\mathrm{VH}_{ab} & G^\mathrm{VV}_{ab}\\[3pt] G^\mathrm{HH}_{ba} & G^\mathrm{HV}_{ba} & G^\mathrm{HH}_{bb} & G^\mathrm{HV}_{bb}\\[3pt] G^\mathrm{VH}_{ba} & G^\mathrm{VV}_{ba} & G^\mathrm{VH}_{bb} & G^\mathrm{VV}_{bb}\\ \end{pmatrix}, \end{equation} \tag{ 2 }$$

where $G^{ij}_{kl} = \langle{E^i_k(E^j_l)^*}\rangle$, $i,j =$ H,V, and $k,l = a,b$, one can account for correlations between the spatial and polarization DoFs [4, 21–23]. The elements of $\boldsymbol{\mathrm{G}}$ therefore represent normalized correlations between field projections in space and polarization. In cases that exhibit said correlations, $\mathbf{G} \neq {\mathbf{G}_{\mathrm{p}}} \otimes {\mathbf{G}_{\mathrm{s}}}$; this equality is only achieved when the two DoFs are independent of each other (separable). This coherency matrix has proved useful in a variety of contexts, such as evaluating the visibility of Young's double slit interference in the presence of polarization effects [28, 31, 32]. From the Hermitian and non-negative matrix (equation (2)) we can derive the field entropy $\mathcal{S} = -\sum_{m}\lambda_{m}\log_{2}\lambda_{m}$, where λ_m are the real eigenvalues of G: $\{\lambda\} = \{\lambda_{a\mathrm{H}},\lambda_{a\mathrm{V}},\lambda_{b\mathrm{H}},\lambda_{b\mathrm{V}}\}$, and $0\leqslant \mathcal{S}\leqslant 2$ with the normalization $\textrm{Tr}\{\mathbf{G}\} = 1$. Therefore, the field can carry up to a total of 2 bits of entropy across the two DoFs.

2.4.1. Introduction to OCmT.

OCmT is a method used to measure the full 4 × 4 coherence matrix of an optical field in which two DoFs are of interest; it is especially useful when investigating the coherence of DoFs with respect to one another [21, 22]. While this method can be used for a variety of DoFs, this tutorial will only cover OCmT in the context of the spatial and polarization DoFs. The expression for the full coherence matrix represented by equation (2) can be rewritten as

$$\begin{equation} \begin{aligned} \mathbf{G}& = \left(\begin{array}{cccc} S_{00} + S_{01} + S_{10} + S_{11} & S_{02} + S_{12} - i(S_{03} + S_{13}) & S_{20} + S_{21} - i(S_{30} + S_{31}) & S_{22} - S_{33} - i(S_{23} + S_{32}\\ S_{02} + S_{12} + i(S_{03} + S_{13}) & S_{00} - S_{01} + S_{10} - S_{11} & S_{22} - S_{33} + i(S_{23} - S_{32}) & S_{20} - S_{21} - i(S_{30} - S_{31})\\ S_{20} + S_{21} + i(S_{30} + S_{31}) & S_{22} + S_{33} - i(S_{23} - S_{32}) & S_{00} + S_{01} - S_{10} - S_{11} & S_{02} - S_{12} - i(S_{03} - S_{13})\\ S_{22} - S_{33} + i(S_{23} + S_{32}) & S_{20} - S_{21} + i(S_{30} - S_{31}) & S_{02} - S_{12} + i(S_{03} - S_{13}) & S_{00} - S_{01} - S_{10} + S_{11} \\ \end{array}\right)\\ &=\frac{1}{4} \sum_{l,m=0}^{3} S^{l}_{m} \hat{\sigma}^\mathrm{p}_{l} \otimes \hat{\sigma}^\mathrm{s}_{m}, \end{aligned} \end{equation} \tag{ 3 }$$

where the elements are made up of multi-DoF Stokes parameters $(S_{lm} = 4I_{lm} - 2I_{0m} - 2I_{l0} + I_{00}$, $l,m = 0\ldots 3)$ [21]. Here I_lm represents joint intensity projections; to calculate these projections and ultimately the coherence matrix, $\mathrm{\mathbf{G}}$, 16 tomographic projections are needed. The procedure for finding the intensity projections is outlined in figure 2 and table 1.

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Table 1. The tomographic intensity measurements necessary for calculating the joint Stokes parameters, and ultimately G. H: horizontal, V: vertical, D: 45^∘, R: right-handed circular.

		Polarization projection
		H	V	D	R
Spatial projection	a	$I^{\mathrm{H}}_{a}$	$I^{\mathrm{V}}_{a}$	$I^{\mathrm{D}}_{a}$	$I^{\mathrm{R}}_{a}$
	b	$I^{\mathrm{H}}_{b}$	$I^{\mathrm{V}}_{b}$	$I^{\mathrm{D}}_{b}$	$I^{\mathrm{R}}_{b}$
	a + b	$I^{\mathrm{H}}_{(a+b)}$	$I^{\mathrm{V}}_{(a+b)}$	$I^{\mathrm{D}}_{(a+b)}$	$I^{\mathrm{R}}_{(a+b)}$
	a + ib	$I^{\mathrm{H}}_{(a+ib)}$	$I^{\mathrm{V}}_{(a+ib)}$	$I^{\mathrm{D}}_{(a+ib)}$	$I^{\mathrm{R}}_{(a+ib)}$

The 16 tomographic measurements in table 1 can be found using the setup in figure 2 and taking a cross-section of intensity images. The first row of the spatial projection section corresponds to the joint polarization-spatial projections when b is blocked, likewise the second row corresponds to the joint polarization-spatial projections when a is blocked. The third row of the spatial projection section corresponds to the zero-spatial-phase-shift joint spatial-polarization projections when both a and b are superposed, and if coherent: interfere. The fourth row of the spatial projection section corresponds to the joint polarization-spatial projections when b experiences a ($\pi/2$)-spatial-phase-shift with respect to a. The fourth column of the polarization projection section corresponds to the joint polarization-spatial projections when b experiences a ($\pi/2$)-polarization-phase-shift with respect to a. These values are used to calculate the 16 intensity projections (I_lm ) in the following manner: $I_{00} = I^{\mathrm{H}}_{a} + I^{\mathrm{V}}_{a} + I^{\mathrm{H}}_{b} + I^{\mathrm{V}}_{b}$; $I_{01} = I^{\mathrm{H}}_{a} + I^{\mathrm{H}}_{b}$; $I_{02} = I^{\mathrm{D}}_{a} + I^{\mathrm{D}}_{b}$; $I_{03} = I^{\mathrm{R}}_{a} + I^{\mathrm{R}}_{b}$; $I_{10} = I^{\mathrm{H}}_{a} + I^{\mathrm{V}}_{a}$; $I_{11} = I^{\mathrm{H}}_{a}$; $I_{12} = I^{\mathrm{D}}_{a}$; $I_{13} = I^{\mathrm{R}}_{a}$; $I_{20} = I^{\mathrm{H}}_{a}$; $I_{20} = (I^{\mathrm{H}}_{(a+b)} + I^{\mathrm{V}}_{(a+b)})/2$; $I_{21} = I^{\mathrm{H}}_{(a+b)}/2$; $I_{22} = I^{\mathrm{D}}_{(a+b)}/2$; $I_{23} = I^{\mathrm{R}}_{(a+b)}/2$; $I_{30} = (I^{\mathrm{H}}_{(a+ib)} + I^{\mathrm{V}}_{(a+ib)})/2$; $I_{31} = I^{\mathrm{H}}_{(a+ib)}/2$; $I_{32} = I^{\mathrm{D}}_{(a+ib)}/2$; $I_{33} = I^{\mathrm{R}}_{(a+ib)}/2$. Once these intensity projections are obtained, the joint Stokes parameters (S_lm ) can be calculated, and the full optical coherence matrix (equation (3)) can be constructed.

2.4.2. OCmT example 1.

Consider a field that is randomly polarized and spatially coherent. To determine the field's coherence properties, one can apply the OCmT procedure from figure 2. The mutual coherence of the electric field components are summarized in table 2. The field's horizontal and vertical components are equal in intensity due to the randomness of the polarization state. The field should be assessed at two spatial points (a and b) that are equal in intensity; these points can be selected using a Young's double slit with separation less than the spatial coherence width of the field. In this case, $I_a^\mathrm{H} = I_b^\mathrm{H} = I_a^\mathrm{V} = I_b^\mathrm{V} \gt 0$. The interference equation for the vertical projection is $I^{\mathrm{V}}_{(a+b)} = I_a^\mathrm{V} + I_b^\mathrm{V} + 2\sqrt{I^{\mathrm{V}}_{a}I^{\mathrm{V}}_{b}}|g_{V_aV_b}|\cos{\phi} = I_a^\mathrm{V} + I_b^\mathrm{V}$, here the normalized cross-correlation $|g_{V_aV_b}| = 1$, and the cosine of the phase difference (φ) equals 1 because the field is spatially coherent and, for measurement purposes, polarized along the vertical projection. For the uninitiated, note that the wave nature of light is mathematically represented by a phase variable inside a cosine term. Therefore, in this context, φ is the difference in phase between $E_a^\mathrm{V}$ and $E_b^\mathrm{V}$. Similar interference equations are written for the horizontal, 45^∘, and circular projections, all of which are theoretically equal in magnitude due to the random state of polarization. The interference equation for the vertical projection containing a ($\pi/2$)-spatial phase shift is $I^{\mathrm{V}}_{(a+ib)} = I_a^\mathrm{V} + I_b^\mathrm{V} + 2\sqrt{I^{\mathrm{V}}_{a}I^{\mathrm{V}}_{b}}\cos{(\pi/2)} = I_a^\mathrm{V} + I_b^\mathrm{V}$. Similar interference equations with ($\pi/2$)-spatial phase shifts are written for the horizontal and 45^∘-projection, and even for the circular projection which adds a ($\pi/2$)-polarization phase shift between the horizontal and vertical components which already exhibit a random phase relationship and thus the equation is unaffected by polarization phase shifts. These measurements result in the coherence matrix: $\mathbf{G} = \frac{1}{4}\begin{pmatrix} 1&0&1&0\\0&1&0&1\\1&0&1&0\\0&1&0&1\end{pmatrix}$. If this field is restricted to one spatial point (a) such that the field is still unpolarized and spatially coherent, the resulting coherency matrix would be equation (4).

Table 2. Comparing the mutual coherence of electrical field components for the OCmT example from section 2.4.2.

	Mutually coherent?
	$E_b^\mathrm{H}$	$E_b^\mathrm{V}$
$E_a^\mathrm{H}$	Yes	No
$E_a^\mathrm{V}$	No	Yes

2.4.3. OCmT example 2.

Consider a field that is linearly polarized along a vertical projection and made up of two mutually incoherent spatial modes. The mutual coherence of the electric field components are summarized in table 3. It is expected that $I_a^\mathrm{H} = I_b^\mathrm{H} = 0$, and the field is linearly polarized along the vertical projection; $I_a^\mathrm{V} = I_b^\mathrm{V} \gt 0$. Furthermore, $I^{\mathrm{H}}_{(a+b)} = I^{\mathrm{H}}_{(a+ib)} = 0$. The interference equation for the vertical projection: $I^{\mathrm{V}}_{(a+b)} = I_a^\mathrm{V} + I_b^\mathrm{V} + 2\sqrt{I^{\mathrm{V}}_{a}I^{\mathrm{V}}_{b}}|g_{V_aV_b}|\cos{\phi} = I_a^\mathrm{V} + I_b^\mathrm{V}$ does not include a third term because the spatial modes (a and b) are mutually incoherent, therefore, $g_{V_aV_b} = \frac{\langle E_a^\mathrm{V^*}E_b^\mathrm{V} \rangle}{\sqrt{I^{\mathrm{V}}_{a}I^{\mathrm{V}}_{b}}} = 0$. Furthermore, $I^{\mathrm{V}}_{(a+b)} = I^{\mathrm{V}}_{(a+ib)}$, because a spatial phase shift, in this case of $\pi/2$, to one arm (b) does not alter the tomographic measurement if a and b are mutually incoherent, i.e. there exists a random phase relationship between the two points.

Table 3. Comparing the mutual coherence of electrical field components for the OCmT example from section 2.4.3.

	Mutually coherent?
	$E_b^\mathrm{H}$	$E_b^\mathrm{V}$
$E_a^\mathrm{H}$	No	No
$E_a^\mathrm{V}$	No	No

The physical result of superposing a and b at a vertical projection is an incoherent sum—no fringes are present. A 45^∘-projection (D) results in $I_a^\mathrm{D} = I_a^\mathrm{V}/2$, likewise for b, and $I^{\mathrm{D}}_{(a+b)} = I_a^\mathrm{V}/2 + I_b^\mathrm{V}/2$. Similar equations can be written for the circular projection. These measurements result in the coherence matrix shown in equation (5).

Now that we have described the $4\times4$ coherency matrix and a method to measure it, the concept of optical coherence conversion can be discussed. A coherence conversion between the spatial and polarization DoFs relies on the field's ability to transfer entropy between the two DoFs, therefore, the field at the input of the system must be curated with this ability. The field must contain entropy, and thus cannot be fully coherent ($\mathcal{S}_{\mathrm{p}} = 0$, $\mathcal{S}_{\mathrm{s}} = 0$). The field should also not be fully incoherent ($\mathcal{S}_{\mathrm{p}} = 1$, $\mathcal{S}_{\mathrm{s}} = 1$). Two appropriate options, for example, are a polarized ($\mathcal{S}_{\mathrm{p}} = 0$) and spatially incoherent ($\mathcal{S}_{\mathrm{s}} = 1$) field (e.g. a polarized LED), or an unpolarized ($\mathcal{S}_{\mathrm{p}} = 1$) and spatially coherent ($\mathcal{S}_{\mathrm{s}} = 0$) field.

The concept of a reversible coherence converter is outlined in figure 3. In the case of the input field being unpolarized and spatially coherent, i.e. having 1 bit of entropy in the polarization DoF and 0 bits of entropy in the spatial DoF, as the field passes through coherence conversion α, the entropy will move from the polarization DoF to the spatial DoF, thus the output is a polarized and spatially incoherent field. For the case in which the input is polarized and spatially incoherent, as the field passes through coherence conversion β, the entropy will move from the spatial DoF to the polarization DoF, thus the output is unpolarized and spatially coherent. A key feature of this conversion is that it is unitary—no energy is lost during the conversion as is observed in typical filtering procedures to establish coherence in an optical field. Thus, the conversion can continue indefinitely between the DoFs. Also, the total entropy of the field is invariant throughout the coherence conversion.

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The case in figure 3 of the input to the converter being unpolarized and spatially coherent is shown in the experimental setup in figure 4. An example of an appropriate source would be an unpolarized HeNe laser. The first step is to isolate the beam in a single spatial mode using a small aperture on the µm-scale. Because the field encounters a small aperture, it will diffract in space necessitating the use of lenses to relay the beam—maintaining reasonable dimensions as the field encounters optics of finite size; generally on the scale of 25.4 cm. The field after the aperture at the input of coherence conversion α can then be described with the coherence matrix:

$$\begin{equation} \mathbf{G}_{1}=\frac{1}{2}\left(\begin{array}{cccc}1&0&0&0\\0&1&0&0\\0&0&0&0\\0&0&0&0\end{array}\right)=\underbrace{\left(\begin{array}{cc}1&0\\0&0\end{array}\right)}_{\mathbf{G}_{\mathrm{s}}}\otimes\underbrace{\frac{1}{2}\left(\begin{array}{cc}1&0\\0&1\end{array}\right)}_{\mathbf{G}_{\mathrm{p}}}, \end{equation} \tag{ 4 }$$

here the field exists as one point and has no correlations between H and V. As the field encounters the polarization beam splitter (PBS) in coherence conversion α, the random polarization is spatially split into two paths (a and b), each containing a now fully polarized eigenmode; H or V of the original unpolarized state. In doing so, the two DoFs become correlated (non-separable) and entropy is shared between the polarization and spatial DoF, such that $\mathbf{G} \neq {\mathbf{G}_{\mathrm{p}}} \otimes {\mathbf{G}_{\mathrm{s}}}$. With the addition of the half-wave plate (HWP) to rotate the polarization in one arm to be parallel to the polarization in the other arm, the entropy is fully transferred from the polarization DoF to the spatial DoF, and the field is separable once again. The field can be described with the coherence matrix:

$$\begin{equation} \mathbf{G}_{2}=\frac{1}{2}\left(\begin{array}{cccc}0&0&0&0\\0&1&0&0\\0&0&0&0\\0&0&0&1\end{array}\right)=\underbrace{\frac{1}{2}\left(\begin{array}{cc}1&0\\0&1\end{array}\right)}_{\mathbf{G}_{\mathrm{s}}}\otimes\underbrace{\left(\begin{array}{cc}0&0\\0&1\end{array}\right)}_{\mathbf{G}_{\mathrm{p}}}. \end{equation} \tag{ 5 }$$

At this point, the field is linearly polarized and spatially incoherent (there is mutual incoherence between the two arms). In other words, the random phase relationship that exists between H and V prior to PBS is exhibited between a and b after the PBS; the respective elements of $\mathbf{G}_\mathrm{p}$ and $\mathbf{G}_\mathrm{s}$ have completely interchanged in this coherence conversion.

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At the plane of G₂ the field is linearly polarized and there exists mutual incoherence between a and b (equation (5)). Now the field is to pass through coherence converter β. Lenses L₂ and L₃ are used to relay the field from the plane of G₂ to its conjugate plane: the plane of G₃. The field first encounters the HWP and then the PBS. The HWP rotates the polarizations such that they are perpendicular to one another, but a constant phase difference still remains in the polarization DoF. When entering the PBS through orthogonal ports, the orthogonal polarizations overlap in a single spatial mode and the random phase relationship between a and b now exists between H and V, resulting in a spatially coherent and unpolarized field that can be represented by equation (4), here $\mathbf{G}_{3} = \mathbf{G}_{1}$. To benchmark coherence in these contexts, we use OCmT.

Here, we have presented an experimental example in which the input to the coherence converter is spatially coherent and unpolarized. There is also the case in which the input to the system is linearly polarized and spatially incoherent, such as a field represented by equation (5). An example of this would be a linearly polarized LED. If this is the starting field, a reversible coherence converter would be setup by first isolating two mutually spatially incoherent spatial modes from the LED, then passing the field through coherence conversion β, resulting in a spatially coherent and unpolarized field (G₃). This should be followed by coherence conversion α to reconstitute the field's original properties.

In both examples, it is crucial that care is taken in the alignment of the paths' respective relay lenses, such that a and b are the same size. For sources of low temporal coherence length, it is vital that a delay line is implemented in one of the optical paths to minimize the path length difference; without it, any mutual temporal incoherence between the two paths may lead to confounding or misleading results.

Coherence converters have demonstrated unique protective effects against scrambling media. Generally, when coherent light encounters a phase-scrambling medium, its phase in one or more DoFs is randomized or becomes more disordered. To counteract this undesirable disorder, prior to the medium one can transfer the ordered phase in a coherent DoF that is to experience scrambling to an incoherent DoF that is not expected to be scrambled, via a coherence conversion. After the medium, a second coherence conversion can be implemented to transfer the ordered phase back into its original DoF. Thus, coherence conversion can protect specific DoFs from the deleterious effects of scrambling media.

Consider a depolarizing medium that is described by the Mueller matrix $\hat{M} = \mathrm{diag}\{1,0,0,0\}$ and converts any incoming state of polarization to one that is completely random. There are many practical cases in which maintaining a field's initial polarization is desired (e.g. in optical communications) [33–35]. For a field that is completely polarized and encountering this depolarizing medium, e.g. one that is represented by equation (5), its polarization DoF can be protected by a reversible coherence converter. To protect the polarization DoF from randomization, two mutually spatially incoherent points of the field should be isolated and passed through coherence conversion β which transfers the entropy from the spatial DoF to the polarization DoF. The coherence that existed in the polarization DoF is transferred to the spatial DoF. The field can then encounter the depolarizing medium without suffering any deleterious effects. Following the medium with coherence conversion α reconstitutes the field's original coherence properties. Okoro et al demonstrated a proof-of-concept experiment which aimed to protect the polarization DoF in such a case [23]. A linearly polarized and spatially incoherent field was passed through a coherence conversion (organized identical to coherence conversion β) to swap entropy between the DoFs, such that the output is unpolarized and spatially coherent. This was done prior to a depolarizing medium. It was shown that the field's coherence properties are invariant when passed through the depolarizer; this invariance is attributed to the transfer of entropy into the polarization DoF [23]. In this experiment, the reversibility of the coherence conversion was not demonstrated; however the reversibility was successfully demonstrated by Harling et al [24].

In this case, a spatially coherent and unpolarized field was set to encounter a spatially scrambling medium. Protecting a field's spatial properties from scrambling would be beneficial in image transmission, space division multiplexing applications, and other coherence coding schemes [36, 37]. This experiment more closely resembled the setup shown in figure 4. Firstly, the PBS in coherence conversion α was replaced with a regular beam splitter (BS), outputting two mutually spatially coherent points. These points were sent to a simulated spatially scrambling medium using a spatial light modulator (SLM) not shown in figure 4. The simulated medium was a hologram exhibiting the full dynamic phase range of the SLM from 0 to $\gt2\pi$ over columns that extended vertically in the y-direction and were ten pixels wide in the x-direction. The random phase patterns changed randomly at 10 fps for 1 s. Following the SLM, a and b were overlapped to interfere, and images were acquired by a camera that collected multiple frames at 50 fps, resulting in stacks of 50 images. When the SLM was idle, the mutually coherent points demonstrated high visibility, but when the scrambling pattern was present, the visibility decreased by 35% suggesting a large reduction in spatial coherence. These results are shown in figures 5(a) and (b). To avoid this deleterious effect, a field that was spatially coherent and unpolarized, represented by equation (4), was passed through coherence conversion α to protect the spatial DoF by transferring the entropy from the polarization DoF to the spatial DoF. Following the SLM, the field was sent to the SLM and passed through coherence conversion β which reversed the entropy back to the polarization DoF, leaving the spatial DoF coherent; thus reconstituting the field's original coherence properties. Using a BS and mirrors not shown in figure 4, the field was separated and then the two points were overlapped to interfere. These results are shown in figures 5(c) and (d). The visibility remained high when the spatial scrambling medium was imparted to the SLM. Thus, similarly to the previous case of protecting polarization, the field's coherence properties were found to be invariant upon scrambling. We encourage readers to refer to [23, 24] for further experimental details.

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