## Abstract

Stokes polarimetry is a mature topic in optics, most commonly performed to extract the polarization structure of optical fields for a range of diverse applications. For historical reasons, most Stokes polarimetry approaches are based on static optical polarization components that must be manually adjusted, prohibiting automated, real-time analysis of fast changing fields. Here we provide a tutorial on performing Stokes polarimetry in an all-digital approach, exploiting a modern optical toolkit based on liquid-crystal-on-silicon spatial light modulators and digital micromirror devices. We explain in a tutorial fashion how to implement two digital approaches, based on these two devices, for extracting Stokes parameters in a fast, cheap, and dynamic manner. After outlining the core concepts, we demonstrate their applicability to the modern topic of structured light, and highlight some common experimental issues. In particular, we illustrate how digital Stokes polarimetry can be used to measure key optical parameters such as the state of polarization, degree of vectorness, and intra-modal phase of complex light fields.

© 2020 Optical Society of America

## 1. INTRODUCTION

Understanding the state of polarization (SoP) plays an important role in many areas, such as imaging, metrology, fiber-based communication, microscopy, radar, and quantum optics [1,2]. Therefore, its measurement and assessment are extremely important, resulting in a plethora of research dedicated to the topic. The most common theoretical approach to describe the SoP of light is based on the polarization ellipse [3,4]. Although this approach allows one to describe various SoPs by a single equation, it has some drawbacks [5]. First, it is not possible to physically observe, via any measurement technique, the polarization ellipse, due to the fact that light’s vector traces out an ellipse in an extremely short time period (on the order of ${10^{- 5}}\,\,{\rm s}$). The second drawback is that this description is applicable only to completely polarized light. Therefore, the polarization ellipse is not an ideal means to describe light’s SoP. In 1852, Sir George Gabriel Stokes solved this problem by demonstrating that one can describe the SoP of light with just four parameters, now known as the Stokes parameters [5,6]. The first parameter, ${S_0}$, describes the total intensity of the optical field, while the remaining three parameters, ${S_1},{S_2}$, and ${S_3}$, its polarization state. This can be applied to completely polarized, partially polarized, and also non-polarized light. The realization of these observables removed the need to measure the polarization ellipse and its associated limitations.

Since Stokes’ formulation of these four parameters, many measurement techniques have been dedicated to their physical acquisition. The most commonly used techniques are based on measuring all six orthogonally polarized intensity profiles, namely, ${I_H},{I_V},{I_D},{I_A},{I_R}$, and ${I_L}$. Here, the subscripts for the intensities, $I$, denote: $H$, horizontal; $V$, vertical; $D$, diagonal; $A$, anti-diagonal; $R$, right-circular; and $L$, left-circular. For a more detailed discussion of these techniques (based on polarizers and phase retarders), the reader is directed to [7–10]. The Stokes parameters have also been described in terms of the density matrix in order to formulate an analogy between photon polarization and elementary particles [11], as well as being applied to investigate light–matter interactions [12]. An easier, more frequently implemented approach consists of measuring only four intensity profiles, rather than the standard, over-complete set of six. This approach also makes use of the two previous mentioned optical components, namely, a polarizer and a phase retarder. Here the angle of the polarizer and phase retarder are sequentially altered in order to extract the associated intensity profile [13]. Although these techniques have been around for many years, their simple arrangement and ease in experimental implementation mean they are still popular in many optics laboratories.

Adaptations and improvements of these long-standing measurement techniques have included the implementation of wavefront [14] and amplitude division [15,16], as well as the use of photopolarimeters [17,18], to negate the need for sequential measurements. Automation approaches incorporating a monochromator and a single photon counting unit [19] have also been developed. Polarimetry techniques utilizing liquid crystal polymer or sub-wavelength metasurfaces have also been shown to allow for simultaneous measurements, some of which (namely, the division of focal plane polarization cameras) provide full polarimetry imaging [20–24]. A real-time technique based on three-way polarization-preserving beam splitters (BSs) has also been proposed [25].

Today there exists a modern, digital toolkit comprising spatial light modulators (SLMs) [26–28] and digital micromirror devices (DMDs) [29–31] that has fueled the creation and detection of structured light [32–36], particularly that with exotic polarization structures [37–49]. Naturally, measuring the polarization structure is just as important as creating it. Can one implement a digital toolkit for polarimetry? Yes, these digital devices have shown to be ideal tools not only in creating unique polarization structures, but also in performing Stokes polarimetry for the detection of structured polarization states. For example, liquid-crystal-on-silicon SLMs (LCoS-SLMs) have been used for an all-digital approach to polarization analysis and used to extract wavefronts [50] and vector quality factors [51,52], infer quantum states [53,54], and determine the motion of objects [55], with the benefit of efficient single shot analysis with no moving parts. More recently, DMDs have been used for Stokes analysis [56,57], concurrence determination [58,59], wavefront analysis [60,61], and phase imaging [62], ushering in a faster, and cheaper alternative (on the order of a few hundred dollars), but at a lower efficiency [63].

Here we provide a tutorial-style explanation of the theoretical and experimental requirements for measuring the Stokes parameters via two real-time, digital approaches, one of which makes use of a LCoS-SLM, while the other uses a DMD. These two digital methods are demonstrated on various vector vortex beams, which have shown growing interest in the area of structured light. We provide all the necessary code required for the reader to easily reproduce the results contained within this tutorial-style manuscript. Finally, we show how these measured Stokes parameters can be used to determine various optical properties, such as the SoP and intra-modal phase, as well as determining the degree of non-separability of vector beams.

## 2. STOKES PARAMETERS

In this section, we will provide a detailed overview of the formalism of the Stokes parameters and their conventional measurement. Thereafter, we will extend the concept to using modern digital devices.

Consider a pair of orthogonal monochromatic plane waves at $z = 0$, which can be represented as

with amplitudes ${E_{0x}}$ and ${E_{0y}}$, angular frequency $\omega$, time $t$, and phase factors ${\delta _x}$ and ${\delta _y}$. We can obtain an equation describing the polarization ellipse by removing the term “$\omega t$” between Eqs. (1) and (2):Making use of the following simple relationships:

Each term in parentheses in the above equation is one of the non-normalized Stokes parameters for the plane wave, defined as

Since we are concerned with intensities (leading to perfect squares), at every spatial position on a transverse wave, we have that

Here the first Stokes parameter ${S_0}$ is equal to the total intensity of the light, while the remaining parameters describe its polarization state. ${S_1}$ represents the amount of linear polarization (horizontal or vertical), ${S_2}$ represents the amount of diagonal polarization (45° or 135°), and finally, ${S_3}$ represents the amount of circular polarization (right or left) present in the beam. Using Schwarz’s inequality, we can show that for any possible state of light, the Stokes parameters will satisfy the following inequality:

We can rewrite the above in complex notation to find

Here ${E_x}$ and ${E_y}$ are complex amplitudes. Repeating the same calculation leads us to the more general definition of the Stokes parameters:

It is possible to write the Stokes parameters above based on the six polarization intensities, namely, ${I_H},{I_V},{I_D},{I_A},{I_R}$, and ${I_L}$ [10]:

These intensities, listed in Eq. (15), can be measured in the laboratory by making use of two optical components, namely, a polarizer and a retarder, shown in Fig. 1(a).

A polarizer transmits the polarization components of an optical field along the polarizer’s transmission axis. Therefore, in the case of a polarizer with its transmission axis set at an angle of $\theta$, the total electric field emerging from the polarizer is

and the intensity of this field is determined asTherefore, all four linear polarization intensities $({I_H},{I_V},{I_D},{I_A})$ can be measured via a polarizer in accordance with Eqs. (18) and (16):

A phase retarder is a phase shifting element, which set at an angle of $\phi$ will add a phase of $\frac{\phi}{2}$ to the $x$ component of the electric field (${E_x}$), and a phase of ${-}\frac{\phi}{2}$ to the $y$ component of the electric field (${E_y}$); such an example is a quarter-wave plate (QWP), giving a phase shift of $\frac{\pi}{2}$.

In the case of having both a polarizer and a retarder, the total electric field emerging from these two optics will be

with its intensity calculated asBy adding a QWP, we can measure circular intensities $({I_R},{I_L})$, in accordance with Eqs. (21) and (16):

Measuring these six intensities is sufficient to find all the Stokes parameters—known as *full Stokes polarimetry*. These six measurements are over-complete, and in fact, it has been shown that only four suitable measurements is sufficient [10]. In order to determine these four intensity measurements, let us rewrite Eq. (21) in terms of the following relationships:

By implementing the definition of the Stokes parameters [Eq. (9)] and simplifying, we can rewrite Eq. (21) as

By setting the QWP at 90°, and placing it before a polarizer (set at 45° and 135°, respectively) the intensity profiles, ${I_R}$ and ${I_L}$, can be acquired [as seen in Fig. 1(a-1)]. The two linear intensities, ${I_D}$ and ${I_H}$, are recorded by removing the QWP and setting the polarizer to angular orientations of 0° and 45°, respectively [depicted in Fig. 1(a-2)]. Solving for the Stokes parameters, we obtain

*reduced Stokes polarimetry*.

## 3. DIGITAL STOKES POLARIMETRY

To illustrate the concept of digital Stokes polarimetry techniques, we will construct analogies from the standard, conventional method, outlined in the section above. Here the essential components of the two digital approaches are amplitude and phase modulating devices, the first of which is a LCoS-SLM, which uses electrically controlled birefringence for optical phase modulation, while the second approach makes use of a DMD, consisting of an array of individually actuating micromirrors for amplitude modulation [64–67]. The main difference between the DMD and the LCoS-SLM, which significantly affects their versatility, is the fact that DMDs are polarization invariant, while the majority of LCoS-SLMs are not. In the case of the DMD, orthogonal polarization components are equally modulated by encoded holograms, diffracting the modulated beam into its associated first diffraction order, whereas, LCoS-SLMs exhibit diffraction inefficiency in one of the orthogonal polarization components (resulting in its remaining in the zeroth order), with only the complimentary component being diffracted into the first order [50]. We discuss the concept behind each approach in detail below.

#### A. Digital LCoS-SLM Method

Just as the conventional Stokes polarimetry requires the projection of a beam of interest into the various polarization states, the digital approaches follow the same requirement. It is well known that an optical field $U(\vec r)$ can be decomposed into orthogonal polarization bases, such as $|H\rangle$ and $|V\rangle$, with the use of a polarizing BS (PBS). Here $| \rangle$ denotes ket-notation for the representation of vector space. However, a lesser-known optic that is capable of mapping the right- and left-circular polarization bases ($|R\rangle$ and $|L\rangle$) to independent optical paths (let us call these paths $A$ and $B$) is a polarization-dependent diffraction grating (PG) [68–71]. One should note that when using a PG, the input field may have a spatially inhomogeneous or homogeneous polarization structure (i.e., a vector or scalar beam, respectively) but the components in the polarization basis will always be scalar [37]. We will use a PG for the experimental demonstrations in this tutorial, but we stress that a PBS (or other polarization-sensitive components) for the linear states would work just as well.

First, the incident field is split into two fields ${U_A}(\vec r)$ and ${U_B}(\vec r)$ representing the orthogonally polarized components, $|R\rangle$ and $|L\rangle$, as illustrated in Fig. 1(b-1) and mathematically described as

Next, to measure two additional intensity profiles for this particular digital method, namely, ${I_D}$ and ${I_A}$, usually a QWP with its fast axis set at 45° would need to be introduced before the PG to project ${I_D}$ and ${I_A}$ into the two detectable paths, $A$ (${I_R}$) and $B$ (${I_L}$), respectively. However, since we know that a QWP induces a quarter-wavelength phase shift on an incident beam, converting linearly polarized light to circular and vice versa, we can achieve this digitally by merely encoding the LCoS-SLM with an additional $\pi /2$ phase term, as illustrated in Fig. 1(b-2). The advantage with this technique [50] is that the optics (namely, PG and LCoS-SLM) remain static, while only a phase-change (affecting only the incident vertical component) is encoded on the LCoS-SLM, to mimic the transformation of the QWP [Fig. 1(b-2)].

In implementing this technique, we would like to point out that the intensity profiles that can be extracted allow for the reconstruction of only three of the four Stokes parameters (namely, ${S_0}$, ${S_2}$, and ${S_3}$) evident from Eq. (25). We would like to point out to the reader that he/she could approximate ${S_1}$ from Eq. (10). Since the LCoS-SLM is polarization sensitive, acquisition of one of the two remaining polarization states necessary for the reconstruction of the Stokes parameter, ${S_1}$, (namely, ${I_H}$) for this conceptual configuration is not possible. Even though this approach does not offer reconstruction of all four Stokes parameters, it is still possible to investigate interesting optical properties from only two of the Stokes parameters (namely, ${S_2}$ and ${S_3}$), such as the intra-modal phase, which is discussed in Section 5.

#### B. Digital DMD Method

For the second digital approach [57] we make use of the previously discussed PG together with a DMD. Here the intensity profiles ${I_R}$ and ${I_L}$ are again mapped to independent paths $A$ and $B$, respectively, with the implementation of the PG [Fig. 1(c)]. Each path is directed to one half of a DMD screen, which can be independently encoded with a binary amplitude and phase modulating hologram (labeled as ${H_A}$ and ${H_B}$) according to the following functions (based on the algorithm presented by Lee [72]):

Here ${G_x}$ and ${G_y}$ are the directional spatial carrier (grating) frequencies, while ${\Phi _{A/B}}(\vec r) = \frac{{\arg ({T_{A/B}}(\vec r))}}{\pi}$ and ${A_{A/B}}(\vec r) = \frac{{\mathop {\sin}\nolimits^{- 1} (abs({T_{A/B}}(\vec r)))}}{\pi}$ are the respective phase and amplitude modulation terms, with ${T_{A/B}}(\vec r)$ being the transmission function intended for the modulation of each path. For simple spatially constant (i.e., $\forall$ $\vec r$) phase modulation, the transmission functions required can be represented as

where ${c_{A/B}}$ are merely independent constants. The first diffraction order of each path reflected off the DMD, due to the encoded diffraction grating, carries the desired modulation given by ${U_{A/B}}{T_{A/B}}$. The modulated paths are combined by coupling them into orthogonal ports of a 50:50 BS, after which the necessary intensity profiles are acquired.The field $U^\prime (\vec r)$, after the BS, now represents a superposition of the independently modulated orthogonally polarized components of the initial field $U(\vec r)$:

With this digital approach, it is easy to see how the ${I_R}$ and ${I_L}$ measurements can be obtained naturally from the PG by allowing ${T_{A/B}} = 0$ (the DMD is in the off-state) and ${T_{B/A}} = {e^{i{c_{B/A}}}}$, where $U^\prime = {U_{B/A}}{e^{i{\varphi _{B/A}}}}{e^{i{c_{B/A}}}}$ (i.e., either only the $|L\rangle /|R\rangle$ component with some spatially constant phase). In order to acquire the ${I_H}$ measurement, one can make use of the relationship $|H\rangle = \frac{1}{{\sqrt 2}}(|R\rangle + |L\rangle)$, which implies that a phase difference of $\Delta \varphi = {\varphi _A} - {\varphi _B} = 0$ is required. Subsequently, from the relationship $|D\rangle = |R\rangle + {e^{i\frac{\pi}{2}}}|L\rangle$, an addition of $\frac{\pi}{2}$ to one of the constants, ${c_{A/B}}$, will allow for the measurement of ${I_D}$. These phase modulation steps can be achieved by simply adjusting the hologram transmission functions ${T_{A/B}}$, which are substituted into the holograms ${H_{A/B}}$ through ${A_{A/B}}$ and ${\Phi _{A/B}}$.

To summarize, the entire procedure can be compactly described via the Jones matrix representation of the transformations associated with each optical element (under the assumption ${\varphi _A} = {\varphi _B} = 0$), given by

Here $A,B \in \{0,1\}$ and must be set according to Table 1, which describes the system transformations that the reader will need to use in order to acquire the four intensity measurements. The reader must also take note that only one of the BS outputs needs to be monitored. We recommend monitoring the ${\rm BS}_{1,1}^{{\rm out}}$ matrix element, as this corresponds to the $|H\rangle$ component as shown by Eq. (30), while $BS_{2,1}^{{\rm out}}$ corresponds to $i|V\rangle$.

Having described the underlying principles of the two digital measurement techniques, we will now outline their respective experimental setups, highlighting any nuances and issues to be considered when implementing these techniques in the lab.

## 4. EXPERIMENTAL REALIZATIONS AND RESULTS

#### A. Conventional Method

For completeness, we will start with and consider the experimental realization of the standard, conventional Stokes polarimetry method, depicted in Fig. 2(a). To demonstrate the approach, we apply it to a radially polarized beam, which we must first generate. To do so, we used a $\lambda = 633$ nm He–Ne laser, producing a horizontally polarized Gaussian beam, and passed it through a half-wave plate (HWP, with its fast axis orientated at 45° to convert the incident polarization to vertical), and a q-plate [73] ($q = 1/2$, which imparts an azimuthally varying phase of ${e^{{il\phi}}}$ via an anisotropic medium, with $\ell = 1$) in order to generate a radially polarized (TE) vector vortex beam, represented by

Note that q-plates, sometimes called S-plates, are commercially available today. Here ${\rm LG}_p^l(\vec r)$ refers to a Laguerre–Gaussian mode, given in polar coordinates by [74]

In reproducing this experiment in the laboratory, the reader should expect to capture intensity profiles similar to those depicted in Fig. 3, together with their corresponding Stokes parameters, calculated via Eq. (25). Here simulated predictions of the four intensity profiles with their corresponding Stokes parameters are given as insets to help aid the reader with what to expect. The code StokesParametersExperimental.py [75] can be used by the reader to calculate the Stokes parameters, based on their measured intensity profiles, as well as making use of the code StokesParametersTheoretical.py [75] to calculate the expected, theoretical Stokes parameters.

#### B. Digital LCoS-SLM Method

The experimental procedure for digitally extracting Stokes parameters with the use of a LCoS-SLM is outlined in Fig. 2(b). Here a He–Ne laser was expanded and collimated to illuminate the screen of a reflective LCoS-SLM (Pluto Holoeye, $1920 \times 1080$ pixels, 8 µm pixel pitch, $60\% {-} 80\%$ diffraction efficiency) preceded by a polarizer to generate linear polarization [namely, diagonal (45°)]. For this particular case, we chose the screen of the LCoS-SLM to be encoded with a phase-only azimuthal hologram [inset in Fig. 2(b)]. The LCoS-SLM diffracts only the vertically polarized component into the off-axis, diffraction orders that possess the encoded phase profile, while the horizontal component remains unaffected in the on-axis, non-diffracted (zero) order. We exploited this polarization-variant property to generate the following structured field, consisting of two orthogonal components:

Here each component consists of a common Gaussian field, while the vertical component has an additional phase term of $\exp (i\delta (r,\phi))$, which for this particular case is $\exp (i\ell \phi$), where $\ell = 1$. The generated field (denoted in the first block in Fig. 4) was imaged from the plane of the LCoS-SLM screen to the CCD camera (preceded by a PG), where the various intensity profiles were recorded. The hologram required to create our beam of interest [first inset in Fig. 2(b)] was encoded onto the LCoS-SLM, followed by the same hologram encoded with an additional $\pi /2$ phase term [producing the second inset in Fig. 2(b)].

For each of the two holograms, the corresponding intensity profiles (${I_D}$, ${I_A}$ and ${I_R}$, ${I_L}$) (middle block in Fig. 4) were captured, respectively. Their corresponding Stokes parameters (${S_0}$, ${S_2}$, and ${S_3}$), calculated via Eq. (25), were determined and are presented in the right-most block in Fig. 4. Here we would like to point out to the reader that since the LCoS-SLM is polarization dependent, it will not be possible (with the current presented configuration) to perform a phase-dependent transformation in order to extract the horizontal and vertical intensity profiles, (${I_H}$ and ${I_V}$) needed for the calculation of the Stokes parameter ${S_1}$. However, simulated predictions of the four intensity profiles together with their corresponding Stokes parameters are given as insets, so that readers can verify their measurements on this particular field [described by a superposition of the two equations in Eq. (34)] if they so acquire the Stokes measurements via another method. As in the conventional approach, readers can make use of the two codes StokesParametersExperimental.py and StokesParametersTheoretical.py [75] to calculate their experimental and theoretically expected Stokes parameters.

#### C. Digital DMD Method

The experimental procedure for obtaining the four Stokes intensity measurements, via the digital DMD approach [depicted in Fig. 2(c)], was conducted on a radially polarized vector vortex beam, as described in Eq. (32), and generated in the same manner as that described for the conventional case [Fig. 2(a)]. The vector beam was propagated through a PG that mapped the left- and right-circularly polarized components to two independent paths, $A$ and $B$, respectively. The two paths were then directed to their associated halves of the DMD screen (TI-DLP650NE), which were addressed separately to facilitate the independent modulation of the left- and right-circularly polarized components. Ideally, DMDs have an inherent diffraction efficiency of 60%; additional details regarding their diffraction efficiencies can be found in [31] At the DMD screen, binary holographic gratings, generated using the technique presented by Lee [72] [Eqs. (27) and (28)], were displayed. An expanded representation of the holograms encoded onto the DMD screen are provided as insets in Fig. 2(c). The gratings used were generated in order to effect a constant (in the transverse plane) phase modulation. By encoding the grating on either DMD half $A/B$ and encoding zeros (off-states) in the other, the right and left components were independently imaged by the CCD [shown by the top two holograms in Fig. 2(c)]. By encoding gratings onto both halves and adjusting the constant phase modulation of each half independently, relative phases of $\delta = \{0,\frac{\pi}{2}\}$ were induced into the interfering beams, resulting in the imaging of the horizontal and diagonal components by the CCD [as shown by the bottom two holograms in Fig. 2(c)]. The first diffraction orders (which carry the desired modulation) from each half of the DMD were isolated (from the zeroth and higher diffraction orders) with the use of an aperture and coupled into orthogonal ports of a 50:50 BS, through which the paths were interfered. The resulting beam (from one of the BS output ports) was then imaged by a CCD camera. One important aspect that the reader needs to consider when implementing this in the laboratory is that the paths exiting the BS must be completely co-propagating. The resulting interference pattern—as captured by the CCD—for misaligned beam paths will be characterized by fringes (vertical for $x$ misalignment and horizontal for $y$ misalignment). The reader will notice that as good alignment is approached, the spatial frequency of the fringes will decrease until the lobes (for the example case presented) are visible. For the benefit of the reader, this procedure is simulated in Fig. 5.

Another important aspect to consider is that practically in constructing this experiment, one cannot guarantee equal path lengths of the orthogonally polarized components to within the wavelength of visible light. The path length discrepancy will manifest as some non-zero phase difference between the orthogonally polarized components $\Delta \varphi \ne 0$. This can be compensated for by encoding the transmission functions ${T_{A/B}} = {e^{i{c_{A/B}}}}$ onto both of the relevant sides of the DMD, where one of the constants is held fixed (e.g., ${c_A} = 0$) and the other is varied until the image captured by the CCD correlates well with a theoretically predicted ${I_H}$. For the benefit of the reader, a simulated example of this procedure is shown in Fig. 6, where orthogonal components ${U_A} = LG_0^1$ and ${U_B} = LG_0^{- 1}$ are interfered with $\Delta \varphi = \frac{\pi}{2}$; ${c_A} = 0$ is held fixed, while ${c_B} = \{0,\frac{\pi}{8},\frac{\pi}{6},\frac{\pi}{4},\frac{\pi}{2}\}$ is varied. Expanded images of the holographic gratings as well as the behavior of the intensity patterns illustrates how the final grating with ${c_B} = \frac{\pi}{2}$ compensates for the phase difference $\Delta \varphi$ and results in the expected ${I_H}$ intensity pattern for the given components.

Reproducing this digital measurement, the reader should obtain intensity profiles ${I_R},{I_L},{I_H}$, and ${I_D}$, as well as the Stokes parameters calculated according to Eq. (25), similar to those depicted in Fig. 7. Simulated predictions are given as insets to help aid the reader with what to expect. Again the codes StokesParametersExperimental.py and StokesParametersTheoretical.py [75] can be used to calculate the reader’s experimental and theoretical Stokes parameters.

We would like to highlight to the reader that if a vector vortex beam propagates by some non-negligible factor of the Rayliegh length ${z_R}$ (which is the distance over which the beam remains well collimated), there will be some curvature observed in the intensity lobes of the ${I_H}$ and ${I_D}$ components. This curvature (or “spiraling” of the lobes) will manifest in the Stokes parameters ${S_1}$ and ${S_2}$ as illustrated with exemplar results obtained using the digital DMD method in Fig. 8. The effect of this on the SoP and intra-modal phase, $\delta$, will be discussed in Section 5. This presents us with a possible improvement on the measurement device in the form of the requirement of a $4f$ imaging system in order to image the desired measurement plane onto the DMD screen directly. Another notable feature is that the ${S_3}$ parameter was approximated to zero; discrepancies in this parameter may arise from the modulated components reaching the BS with different spot sizes, indicating another possible improvement in the form of an appropriate imaging system between the DMD and the BS. The latter improvement would be of large consequence to calculating the handedness of circular polarization, and this will become apparent as the complete reconstruction of polarization structure from the Stokes parameters is discussed in Section 5.

## 5. APPLICATIONS

In this section, we outline various applications of Stokes polarimetry that the reader can implement when digitally measuring Stokes parameters. The examples provided here make use of measurements extracted via the two digital Stokes polarimetry methods.

#### A. Polarization Reconstruction

The Stokes parameters have been shown to describe a polarization state by relating the amplitudes and relative phases of the $x$ and $y$ electric field components according to Eq. (9). They can, however, be mapped to the more convenient description of a given SoP through expression of the latter three parameters as the Cartesian co-ordinates of a point on the Poincaré sphere according to

Here $2\psi \in [0,2\pi]$ is the azimuthal spherical co-ordinate, and $2\chi \in [0,\pi]$ is the radial spherical co-ordinate; these also happen to be twice the angles describing the orientation and ellipticity angle of the polarization ellipse [10]. The relationship between the Poincaré sphere co-ordinates and the polarization ellipse properties is shown in Fig. 9. The linear polarization states ($H, V, D$, and $A$) lie on the equator of the sphere, while the right and left circular polarization states on the north and south poles, respectively. Elliptical states are located on the remaining surface of the sphere.

This relationship to the polarization ellipse, which can be used to easily describe the SoP at each point $(x,y)$ in the transverse plane of an optical field, allows for the complete reconstruction of the SoP at this plane from the four parameters ${S_0}(x,y),{S_1}(x,y),{S_2}(x,y)$, and ${S_3}(x,y)$. First let us consider rearranging and equating Eqs. (36) and (37) in the following manner:

Next, let us consider taking the sum in the quadrature of Eqs. (36) and (37) and equating to Eq. (38) to obtain

Finally, if we consider the expression ${S_3} = {I_R} - {I_L}$, we can see the handedness ${H_{{\rm CP}}}$ of any dominant circular polarization (if ${S_3} = 0$, there is only linear polarization) will be

Using these relationships, the orientation $\psi (x,y)$, ellipticity $\chi (x,y)$, and handedness ${H_{{\rm CP}}}(x,y)$ of the polarization ellipse at every point in a given field can be determined from the Stokes parameters within the resolution of the CCD used to image the Stokes intensity profiles. Reconstructions of the SoP for the conventional and DMD digital approaches can be seen in the top row in Fig. (10), illustrating good agreement with their theoretical predictions (given as insets). By implementing the ${4f}$-imaging system discussed in Section 4.C, one can reduce the curvature that manifests in the SoP [as seen in the right-most image of the top row in Fig. (10)]. The code SOPCodeExperimental.py [75] can be used by readers to reconstruct the SoP from their experimental images, while the code SOPCodeTheoretical.py [75] can be used to obtain predictions of the SoP.

We would like to point out to the reader that the curvature described in the experimental realization of the digital DMD method (Section 4.C) will be evident in the SoP of the beam, as illustrated by simulated examples shown in the top row in Fig. 11. Therefore, should the reader obtain such “spiraling” structures in the SoPs, it is evidence that the field of interest is no longer collimated.

#### B. Intra-Modal Phase Reconstruction

Another optical property that the reader can extract from the Stokes measurements is the phase difference $\delta (r,\phi)$ between the orthogonal polarization components (in this case, horizontal and vertical). Here we consider that a vector light field, which is described completely by the ellipticity of the polarization ellipse, is given by [10]

Equation (42) illustrates that the intra-modal phase difference is described by the Stokes parameters ${S_3}$ and ${S_2}$, which are easily measured via four specific intensity measurements. Note that ${S_1}$ is not needed. The intra-modal phases reconstructed via the conventional and digital DMD approaches, for the vector vortex mode described by Eq. (32), are depicted in the bottom row in Fig. 10, which consist of quadrants varying by $\pi$. If the reader wishes to extract the intra-modal phase for the field described by Eq. (34), he/she should expect to obtain a phase structure similar to the first image in Fig. 12. Since the generated mode consists of a superposition of two Gaussian fields of orthogonal polarization, where the vertical component has an additional azimuthal phase of $\exp (i1\phi)$, the reconstructed phase profile is an azimuthal phase variation of $\ell = 1$. We would like to highlight to the reader that these measurements can also identify the handedness of azimuthal phase profiles [50].

Another aspect that the reader should be aware of is that this approach can be extended to investigate the intra-modal phase profile of propagating optical fields. This can be executed either by moving the Stokes polarimetry device (conventional, LCoS-SLM, or DMD approaches) along the beam’s propagation axis or via digitally simulating free-space propagation [76] on either the LCoS-SLM or DMD. Simulating the propagation requires performing the Fourier transform of the generated beam via a physical lens, while simultaneously encoding the phase term $\exp (i{k_z}z)$ on the digital component (LCoS-SLM or DMD) and subsequently taking the inverse Fourier transform with a physical lens to the plane of the detector, ${{\cal F}^{- 1}}({\cal F}[U(r)]\exp (i{k_z}z))$. We would like to draw the attention of the reader to additional details pertaining to this technique, provided in [50,76].

The propagation of the intra-modal phase for a vector vortex mode (of $\ell = 1$), generated with the experimental setup outlined in the LCoS-SLM approach, is depicted in Fig. 12. It is evident that as the beam propagates, the phase distribution spirals around the propagation axis in accordance with the encoded phase profile [$\exp (i{k_z}z)$]. Similarly, the intra-modal phase for the radial vector vortex beam [Eq. (32)] used in the conventional and digital DMD approaches, as a function of increasing curvature (or propagation distance), can be seen in the bottom row in Fig. 11, also illustrating the spiraling nature of the phase quadrants around a singularity.

#### C. Vectorness Measure

Stokes polarimetry can also be used to quantify the amount of non-separability of a classical optical field. In order to explain this concept, we will consider the quantum regime.

In quantum mechanics, non-separability refers to quantum correlations between two particles where it is not possible to describe the quantum state of each particle independently from the state of the other, even when they are separated by a large distance. In the classical regime (considered here), non-separability refers to correlations between two degrees of freedom, such as the spatial profile and polarization of an optical field [52], as in the case of vector vortex beams:

Here $|{\bar u_R}\rangle$ and $|{\bar u_L}\rangle$ stand for the unnormalized arbitrary complex spatial states on a transversal plane, and $|R\rangle$ and $|L\rangle$ represent the basis for right- and left-circular polarization.

We can write the density matrix corresponding to this vector vortex beam [Eq. (43)] with respect to the polarization bases $|R\rangle$ and $|L\rangle$ as

To measure the vector nature (or “vectorness”) of the beam described in Eq. (43), we will consider the measure of concurrence [51,58,77]. Other measures (not considered here), such as entanglement entropy, can also be implemented [52,77].

Concurrence is an entanglement measure for pairs of two-level systems, defined by

In the case of vector vortex beams [Eq. (43)], spatial states $|{\bar u_R}\rangle$ and $|{\bar u_L}\rangle$ can be expressed in terms of the transverse position $|x,y\rangle$ and parametrized in terms of their amplitude and phase:

In order to make use of Eq. (48) to formulate a measure for the vectorness of an optical field, we need to apply an orthogonalization process (such as Gram–Schmidt) to make $|{\bar u_R}\rangle$ and $|{\bar u_L}\rangle$ orthogonal. We define normalized states $|{u_i}\rangle = |{\bar u_i}\rangle /\sqrt {|\langle {{\bar u}_i}|{{\bar u}_i}\rangle |}$ for $i \in R,L$, as well as select the global phase of $|{u_R}\rangle$ and $|{u_L}\rangle$, such that $\langle {u_R}|{u_L}\rangle = {\langle {u_L}|{u_R}\rangle ^*}$. This allows us to compute the concurrence of a generalized vector beam Eq. (43) with the use of Eq. (48), as follows:

By computing the global Stokes parameters ($\vec S = \int {\rm d}{\vec r_ \bot}\vec S({{{\vec r}_ \bot}})$), we can express the concurrence (or vectorness) as a function of the measured Stokes parameters

We would like to highlight to the reader that by implementing Eq. (51), which requires a simple substitution of the measured Stokes parameters, one can obtain a measure of the incident beam’s degree of non-separability, an example of which is depicted in Fig. 13, where the weighting of the orthogonal polarization states ($R$ and $L$) in Eq. (43) are controlled with the use of a QWP placed before the $q$-plate, as in Figs. 2(a) and 2(c).

## 6. CONCLUSION

Here we have provided the necessary details in order to implement two digital, real-time methods for measuring Stokes parameters and consequently, the SoP, intra-modal phase, and degree of non-separability of optical fields (demonstrated here with various structured, vector beams). We demonstrate how one can make use of amplitude and phase modulating devices, namely, a polarization-sensitive LCoS-SLM and a polarization-insensitive DMD, to act as phase retarders, allowing one to perform all-digital, real-time polarimetry experiments. We hope this tutorial will prove useful in teaching digital Stokes polarimetry techniques, as well as measuring unique optical parameters in undergraduate and postgraduate laboratories.

## Funding

Department of Science and Technology, Republic of South Africa (DSI–CSIR Interbursary Support Programme).

## Acknowledgment

K. S. acknowledges funding from the DSI–CSIR Interbursary Support Programme.

## Disclosures

The authors declare no conflicts of interest.

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