In recent years, a growing number of studies have reported thermal emission of light with engineered properties, illustrating the fact that mid-infrared radiation is not being dictated exclusively by Planck’s law.1^–4 Thermal metasurfaces have demonstrated control of polarization state, directly at the source level without requiring further external optical components.5 These achievements match the growing interest for both sources and detectors of infrared circularly polarized light, as a specific signature in some phenomenon such as light emitted by interstellar objects,6 or its importance in infrared spectroscopy,7^,8 sensing, and detection in unfavorable environments.9 Therefore, a number of papers have reported thermal emission of light with a significant degree of circular polarization.10^–12 This paper focuses on the accurate characterization and analysis of the polarization properties of this type of source, where the total emitted spectrum remains most often broadband.

The Stokes vector, the components of which describe the polarization state of incoming electromagnetic radiation, can be characterized by performing several intensity measurements, consisting of projections of the incoming light polarization state onto different states of a polarization state analyzer (PSA). The simplest PSA consists of two elements: a retarder and a polarizer. A PSA state then corresponds to different orientations of the retarder and the polarizer. Several works such as Refs. 13–15 have proposed polarimeters based on a fixed polarizer and a rotatable retarder. They show that for such a PSA, there is a set of four states providing the least error–sensitive measurement framework for one given wavelength. However, because of the high dependence of the waveplate retardance with respect to the wavelength, the quality of the corresponding PSA configuration quickly deteriorates when moving away from this optimal wavelength. When the total spectral window explored remains small compared with the central wavelength of analysis, the spectral behavior of simple retarding components can be modeled and taken into account without affecting the complete Stokes vector characterization.10 Nevertheless, situations of low or zero retardance occur in highly broadband scenarios, making Stokes measurements impossible at some wavelengths or at least generating significant noise and uncertainties on Stokes vector components.

The issue of performing accurate Stokes parameters measurements is tied to the more general challenge of manipulating polarization on a broad range of wavelengths. Notable efforts have been pursued to design broadband polarization components. Some investigations are based on the combinations of several waveplates to achieve achromatic operation of a retarder16^–18 or on exploiting topological phase shifts in noncoplanar reflections by metallic mirrors.19 Several recent demonstrations of broadband polarization state generation rely on nanostructures. Graphene-based metamaterials20^,21 or electric tuning of a few layers of 2D materials22 have been recently proposed to control the polarization state in the mid-wave infrared (MWIR) in the terahertz or in the telecommunication wavelength range. Plasmonic23^,24 and other resonant metallic structures25^,26 have also been demonstrated. Some functionalities, such as polarization conversion or rotation, can also be addressed by the use of liquid crystals.27^,28 Such demonstrations may eventually lead to more widespread use but remain so far in the research labs.

We now comment on more commercially available approaches to analyze the polarization state of highly polychromatic light components. A first approach is to perform multiple measurements over smaller spectral windows with an analyzer, the properties of which are systematically adjusted to the spectral window: several retarders, e.g., plates of various materials or thicknesses, or tunable retarders. In that spirit, a PSA based on a liquid crystal variable retarder (LCVR) can be used so that the optimal retardance is reached for several wavelengths sequentially. The Stokes vector spectrum can then be reconstructed with optimal error bars at the end of the measurement process. However, performing sequences of measurements with either several sets of waveplates and filters or with an LCVR has a considerable time cost, each measurement at a given wavelength involving four analysis states of the PSA. In addition, working with smaller spectral windows also means much less signal to collect when working with thermal emission. When interested in spectral measurements, this well-known problem is solved using a Fourier transform infrared spectrometer (FTIR) spectrometer. The multiplexing, of Fellgett’s advantage, refers to the improvement in signal-to-noise ratio in spectral measurements obtained thanks to measurements collecting the entire infrared spectrum at once, instead of performing a series of measurements on smaller, low-intensity spectral windows. In the case of LCVR, one can also note their rare availability in the MWIR.29^,30 LCVRs are also sensitive to local environmental conditions, yielding a retardance that may derive on an hour timescale, resulting in large uncertainties as reported by Sharp et al.31 An LCVR-based PSA would need systematic calibration before use, and a controlled knowledge of the PSA cannot be guaranteed over all measurements. Although in principle this LCVR-based PSA yields optimal results on the Stokes vector, it has considerable drawbacks when stability and durability of the PSA are required.

An alternative to these solutions involves the use of an achromatic retarder such as a Fresnel rhomb retarder, which allows for constant retardance over a large spectral range. Such elements are broadly available in the visible and near-infrared range but not in the MWIR, where transparent materials are limited. Although efficient and spectrally robust infrared polarimeters based on Fresnel rhombs have been demonstrated,32^,33 one intrinsic drawback of such retarders is that they are difficult to align to prevent beam wandering when the retarder is rotated, which adds to the complexity of the setup when using an FTIR spectrometer. Fresnel rhombs may be replaced by a retarder cut out of birefringent material with limited dispersion, therefore transparent in the MWIR. Such birefringent and transparent materials include calomel, ${\mathrm{MgF}}_2$, ZnS, CdS, CdSe, ${\mathrm{TiO}}_2$, quartz, and sapphire, which display birefringence in the MWIR. An interesting approach proposed by Ref. 34 consists of the combination of a CdS and a CdSe waveplate to obtain a quasi-achromatic retarder over the infrared range by significantly reducing the spectral dependence of the retardance. However, the majority of commercial solutions in the MWIR use ${\mathrm{MgF}}_2$ because it is transparent between $\sim 0.12$ and $7\mu \mathrm{m}$ although it shows a nonnegligible dispersion in retardance. A PSA based on an ${\mathrm{MgF}}_2$ waveplate and a polarizer with a broadband operating range in the MWIR would then offer a simple and robust solution.

This paper is motivated by the challenges of polarization state analysis of thermal emission. In this paper, we present a minimal PSA optimized for the MWIR, consisting of a commercial quarter-waveplate and a linear polarizer. We discuss PSA configurations that are (1) easily built with a simple slab of a birefringent material (${\mathrm{MgF}}_2$) and a polarizer, (2) spectrally robust over a broad spectral range from 2000 to $4000{\mathrm{cm}}^{-1}$, i.e., between 2.5 and $5\mu \mathrm{m}$, while (3) minimizing the number of analysis states needed to fully determine the Stokes vector of a radiation beam. In particular, we explore three configurations consisting of five analysis states, and we compare them with a well-known configuration of four analysis states. We then report the calibration and operation of such a PSA in combination with an FTIR spectrometer, allowing for the reconstruction of the Stokes over a range of wavelengths covering a significant part of the mid-wave infrared range by acquiring five complete spectra.

The Stokes vector fully characterizes the polarization state of a partially coherent radiation. To measure its components, a Stokes polarimeter allows for performing linear operations on the unknown four-component Stokes vector $\mathbf{S}$. Such a PSA usually consists of a combination of linear polarizers and retarders. The retardance and relative orientations of these elements form a set of free parameters describing a state of the PSA. For state labeled $j$, once the light to be analyzed goes through the PSA, it is collected by a detector and a power ${\mathcal{P}}_j$ is measured. The measurement process can therefore be seen as a projection of the unknown Stokes vector, reading ${\mathcal{P}}_j={\mathbf{W}}_j^T\mathbf{S}$, where ${\mathbf{W}}_j$ is a four-component vector describing the state of the PSA onto which $\mathbf{S}$ is projected. To recover the Stokes vector, different projections can be performed onto $M\ge 4$ different states of the PSA so that $j$ runs from 1 to $M$. The PSA states can then be conveniently represented by an $M\times 4$ matrix $\mathbf{W}$, the line of which $j$ equals ${\mathbf{W}}_j^T$. $\mathbf{W}$ will be called the polarization analysis matrix thereafter. In the same way, the $M$ intensities measured with the PSA can be stored in a single column vector $\mathcal{P}$ so that the system reads in a compact manner $$\mathcal{P}=\mathbf{WS}.$$(1)

For a Stokes polarimeter consisting of a rotatable retarder followed by a rotatable polarizer, there are three degrees of freedom for a given wavelength: the angles formed by the principal axes of the retarder ${\theta }_{\mathrm{C}}$ and of the polarizer ${\theta }_{\mathrm{P}}$ with respect to a common fixed reference and the retardance $\delta$ of the retarder. The scope of our discussion will be limited thereafter to this type of two-element polarimeter proposed to use a fixed polarizer and a quarter-waveplate polarizer (QWP) at a fixed wavelength,15 therefore leaving the orientation of the retarder as the unique parameter, which could be used to optimize the polarimeter. A variant was proposed by Ref. 13, where the polarizer is fixed when the rotation of the polarizer and its retardance are left as free parameters. Each set of values for the free parameters $\{{\theta }_{\mathrm{P}},{\theta }_{\mathrm{C}},\delta \}$ describes a PSA state labeled $j$, which can be described by vector ${\mathbf{W}}_j$$${\mathbf{W}}_j=(\begin{array}{cccc}1& 0& 0& 0\end{array}){\mathbf{M}}_{\mathrm{PSA},\mathrm{j}},$$(2)where ${\mathbf{M}}_{\mathrm{PSA},j}=\mathbf{R}(-{\theta }_{\mathrm{P},j})\mathbf{PR}({\theta }_{\mathrm{P},j})\mathbf{R}(-{\theta }_{\mathrm{C},j})\mathbf{C}({\delta }_j)\mathbf{R}({\theta }_{\mathrm{C},j})$, $\mathbf{P}$ and $\mathbf{C}(\delta )$ are the Mueller matrices of the PSA linear polarizer and QWP, respectively, and $\mathbf{R}(\theta )$ is the rotation matrix of angle $\theta$. The latter implicitly assumes that the detector is polarization insensitive.

When $M=4$, the solution is straightforward and reads $\mathbf{S}={\mathbf{W}}^{-1}\mathbf{P}$. Note that $\mathbf{W}$ should be nonsingular so that the problem is not underdetermined. If a higher number of projection states is needed, Eq. (1) generally does not have a solution, but it can be solved as a linear least-squares problem, where the solution $\mathbf{S}$ minimizes $\Vert \mathbf{WS}-\mathcal{P}\Vert$, $\parallel ·\parallel$ denoting the Euclidean norm. We will keep this norm with its subordinate matrix norm in the rest of this paper. The minimum point of the least-squares problem, which has the smallest Euclidean norm, is given by the pseudo-inverse estimator $$\widehat{\mathbf{S}}={\mathbf{W}}^{+}\mathcal{P},$$(3)where ${\mathbf{W}}^{+}=({\mathbf{W}}^T\mathbf{W}{)}^{-1}{\mathbf{W}}^T$ is the Moore–Penrose pseudo-inverse of the polarization analysis matrix.35

However, the question remains as to how the $M$ states of the PSA must be chosen so that the solution $\widehat{\mathbf{S}}$ accurately describes the Stokes vector under investigation, in particular, in real experimental conditions bound to uncertainties on measured intensities $\mathrm{\Delta }\mathcal{P}$ or on the polarization analysis matrix itself $\mathrm{\Delta }\mathbf{W}$ due to misalignment of the optical elements or errors on the correct knowledge of each element’s optical properties. The key element is the condition number (CN) of the polarization analysis matrix, $\mathrm{cond}(\mathbf{W})\ge 1$,13^,15 which controls how sensitive the instrument is to errors. $\mathrm{cond}(\mathbf{W})$ should be as close to 1 as possible to ensure that Eq. (1) represents a “well-posed” problem as called in the numerical analysis literature.36^,37

In an ideal error and fluctuation-free framework, the pseudo-inverse estimator $\widehat{\mathbf{S}}={\mathbf{W}}^{+}\mathcal{P}$ is the solution to the linear least-squares formulation of the problem. However, to account for experimental sources of error such as uncertainties on measured intensities $\mathrm{\Delta }\mathcal{P}$ or on the polarization analysis matrix itself $\mathrm{\Delta }\mathbf{W}$ and quantify their impact on the estimator $\widehat{\mathbf{S}}$, the solution to Eq. (1) changes into $$\widehat{\mathbf{S}}+\mathrm{\Delta }\mathbf{S}=(\mathbf{W}+\mathrm{\Delta }\mathbf{W}{)}^{+}(\mathcal{P}+\mathrm{\Delta }\mathcal{P}).$$(4)

In the case where $\mathbf{W}$ is perfectly known $\mathrm{\Delta }\mathbf{W}=0$, the variance of each component of the Stokes vector is given by the diagonal elements of the covariance matrix ${\mathbf{K}}_{\widehat{\mathbf{S}}}$35$${\mathbf{K}}_{\widehat{\mathbf{S}}}={\sigma }^2({\mathbf{W}}^T\mathbf{W}{)}^{-1},$$(5)where ${\sigma }^2$ is the variance on each measured intensity ${\mathcal{P}}_j$ assumed to be common to all $j=1,\dots ,M$. This provides a lower bound for the uncertainty affecting each component of the Stokes estimate as $\mathrm{\Delta }\mathbf{W}=0$ is a strong assumption. It can be shown37 that, provided $\parallel \mathbf{W}\widehat{\mathbf{S}}-\mathcal{P}\parallel /\parallel \mathbf{W}\widehat{\mathbf{S}}\parallel \ll 1$, the uncertainty affecting the determination of the norm of the Stokes vector $\parallel \mathrm{\Delta }\mathbf{S}\parallel$ is bounded by $$\frac{\parallel \mathrm{\Delta }\mathbf{S}\parallel }{\parallel \widehat{\mathbf{S}}\parallel }\le \kappa \{\frac{\parallel \mathrm{\Delta }\mathcal{P}\parallel }{\parallel \mathcal{P}\parallel }+\frac{\parallel \mathrm{\Delta }\mathbf{W}\parallel }{\parallel \mathbf{W}\parallel }[1+\kappa \frac{\parallel \mathbf{W}\widehat{\mathbf{S}}-\mathcal{P}\parallel }{\parallel \mathbf{W}\widehat{\mathbf{S}}\parallel }\left]\right\},$$(6)where $\kappa =\mathrm{cond}(\mathbf{W})={\sigma }_{\mathrm{Max}}(\mathbf{W})/{\sigma }_{\min }(\mathbf{W})\ge 1$ is the CN of the measurement matrix. It appears that $\kappa$ is a key quantity assessing the PSA quality as it determines the relevance of the Stokes vector estimate resulting from the inversion process. In an optimized polarimeter, $\kappa$ should be minimal, therefore as close as 1 as possible. The latter may be achieved by carefully choosing the parameters $\{{\theta }_{\mathrm{C},j},{\theta }_{\mathrm{P},j},{\delta }_j\},j=1,\dots ,M$ that determine the configuration of the polarimeter. Note that, contrary to Eq. (5), Eq. (6) gives a large upper bound for the relative error on the Stokes estimate as it only guarantees an upper bound on the Euclidean norm of the deviation on the Stokes vector, therefore ignoring how each component of the Stokes vector may be individually affected. It is therefore interesting to consider both quantities when analyzing the data.

Several figures of merit have been considered to estimate the quality of a PSA, two of which are the CN of the associated measurement matrix $\kappa$ and the equally weighted variance (EWV) first proposed by Ref. 13 and defined as $$\mathrm{EWV}=\mathrm{Tr}({\mathbf{K}}_{\widehat{\mathbf{S}}}).$$(7)

The EWV equals the trace of the covariance matrix of the Stokes vector estimate, as defined in Eq. (5), characterizing the pseudo-inverse estimate $\widehat{\mathbf{S}}$, assuming the variance on all measured intensities is equal. From the definition of both quantities, optimizing the PSA is equivalent to minimizing the CN of $\mathbf{W}$ or minimizing the EWV. Reference 35 shows the equivalence of both these metrics. In this particular case, a handling of the error bars specific to each component of the Stokes vector has been proposed by Refs. 35 and 38.

The representation of the PSA states can be done by placing them on a Poincaré sphere as proposed by Ref. 15. Indeed, from Eq. (2), ${\mathbf{W}}_j$ is a four-element vector with $\sum _{i=1}^3{w}_{j,i}^2\le w_{j,0}^2$. If the PSA does not induce depolarization, the equality is reached, and the states of the PSA can be placed on the unit sphere and form the vertices of a polyhedron.

Reference 14 shows that to minimize the EWV or the CN of $\mathbf{W}$, the PSA states must be chosen so as to form a spherical two-design on this sphere. If this condition is verified, the PSA is optimal for these metrics, which guarantees the best possible error bars on the estimate $\widehat{\mathbf{S}}$. For a four-state PSA, a spherical two-design is obtained by setting the retardance of the retarder to 132 deg, yielding a regular tetrahedron as shown in Fig. 1(a).

In the following, we consider a PSA consisting of a rotatable retarder followed by a rotatable linear polarizer. The retarder used in this work is made of ${\mathrm{MgF}}_2$, and we choose it to have a retardance of $\delta =90\mathrm{deg}$ at $2000{\mathrm{cm}}^{-1}$ ($5\mu \mathrm{m}$). The corresponding plot of the retardance over the entire available wavelength range is displayed in Fig. 2.

For such crystalline waveplates, the retardance can only be chosen at one specific wavelength by setting the thickness of the plate $\ell$. Taking into account the material spectral dispersion impacting the birefringence $\mathrm{\Delta }n(\lambda )$, one can define the full retardance spectral dependence $\delta (\lambda )=\frac{\mathrm{\Delta }n(\lambda )\ell }{\lambda }\times 360$ as set. Therefore, $\delta (\lambda )$ is considered a fixed parameter for all PSAs described later on. Each configuration of the PSA is then defined by a set of $M$ pairs of angles $({\theta }_C,{\theta }_P)$, respectively, denoting the orientation of the retarder and polarizer relative to a common reference axis. The corresponding PSA analysis states are wavelength dependent and can be computed using $\delta (\lambda )$. Therefore, each analysis state of our PSA is defined by a pair of angles $({\theta }_C,{\theta }_P)$ denoting their orientation relative to a common reference axis.

A straightforward first approach to analyze the polarization state of an incoming radiation with the best estimator is to follow the approach briefly described in the previous section: we can first design a four-state PSA, the states of which form a spherical two-design on the Poincaré sphere. This is only possible when the retardance is equal to 132 deg. As $\delta$ is wavelength dependent, this means that this optimal configuration can only be reached at a specific optimal wavelength. In our case, such an optimal wavelength is $3.5\mu \mathrm{m}$ ($2850{\mathrm{cm}}^{-1}$).

The configuration corresponding to such a four-state PSA at the optimal wavelength (and optimal retardance) forms a regular tetrahedron and is represented in Fig. 1(a). This configuration reaches a CN of $\sqrt{3}$ and an EWV of 10, the optimal values of the blue curves plotted in Figs. 1(a) and 1(b), respectively. One should note, however, on the same plots that these metrics quickly degrade when moving away from the optimal wavelength, i.e., when the retardance differs from 132 deg.

As we will discuss later, the optimal hallmark of this four-state PSA is to some extent connected to the uniform spread of the polarization states over the Poincaré sphere at the optimal wavelength. Similarly, the lack of spectral robustness of this configuration is related to the modification of the spreading of the states when changing the wavelength. This is illustrated in Fig. 4(a), where the tetrahedron representing the four states is depicted for various values of the retardance and displays dramatic changes from a close-to-optimal spread with a quasiregular tetrahedron at $\delta =125\mathrm{deg}$ to a situation where the four states are close to collapse in a linear state at low retardance values.

To mitigate the issue of spectral robustness of the PSA, we investigate solutions consisting of five analysis states, which is hardly more complex. Although Ref. 35 has shown that spherical two-designs do not exist for the specific case of five-state PSAs, we decided to explore the possibilities of such configurations. Compared with other well-known spherical two-designs such as the octahedron $(M=6)$, the cube $(M=8)$, or the icosahedron $(M=12)$, it only implies a minimal increase of the number of measurements required for a full determination of a Stokes vector.

We emphasize that the CN and the EWV optimization metrics are equivalent only for a fixed number of analysis states so that a direct comparison between these metrics for a four-state and a five-state PSA should be avoided. As formalized by Ref. 35, the EWV has a stronger physical meaning as it directly takes into account the total number of analysis states. As expected, the larger amount of analysis states, the lesser the variance on the Stokes estimate. The CN is, however, enough for the purpose of this paper, which aims at finding a PSA with few projection states that enable spectrally robust analysis of the Stokes vector of an incoming radiation.

In the following, we present three configurations consisting of five analysis states each and compare their performance not only in terms of optimal CN and EWV but also in terms of spectral robustness of these two quantities. All these PSA designs are based on a set of orientations of the retarder and polarizer, which, for a given retardance of the retarder, hence, for a given wavelength, form the polyhedra depicted in Fig. 1 on the Poincaré sphere. A classical set consists of only four canonical states: three linear states (horizontal, vertical, and tilted to 45 deg) and a circular state. Although such a configuration has been extensively used in the past, it produces a nonoptimal PSA with a CN of 0.31 related to the nonuniform spread of the corresponding polyhedron vertices on the Poincaré sphere. The first solution that we discuss here is a straightforward generalization of such classical set to five analysis states that will be called the extended classic set thereafter. It consists of three linear states of the classical set and two circular states (at the wavelength for which the waveplate is QWP) with opposite handedness. The polyhedron representing the extended classic set is sketched in Fig. 1(b) at $2000{\mathrm{cm}}^{-1}$. The second and third configurations are variations where we tried to improve the spread of the states over the surface of the Poincaré sphere. The second configuration consists of four linearly polarized states (horizontal, vertical, tilted by 45 deg and $-45\mathrm{deg}$) and one circularly polarized state at the wavelength for which the waveplate is QWP. The choice of the handedness of the circular state is arbitrary as it does not have any influence on the performance of the PSA in terms of CN or EWV. The polyhedron associated with this second configuration takes the shape of a regular pyramid with a fixed square base at $2000{\mathrm{cm}}^{-1}$ where the retardance reaches 90 deg. This square pyramid is depicted in Fig. 1(c). The third configuration consists of three linearly polarized analysis states equally separated with respect to each other and obtained by orienting the polarizer at 0, 60, and 120 deg, in addition to two circularly polarized states with opposite handedness at the wavelength for which the waveplate is QWP. It corresponds to an equilateral triangular bipyramid when optimal, as shown in Fig. 1(d).

Figure 3 shows the spectral evolution of both the EWV and CN of the polarization analysis matrix for the four aforementioned configurations. Without surprise, the CN and the EWV associated with the extended classical five-state set [orange lines on Figs. 3(a) and 3(b)] are relatively far from the optimal (the maximal inverse CN reaches 0.38), but interestingly, they do not show a strong spectral dependence between 1500 and $2500{\mathrm{cm}}^{-1}$, thus showing a promising spectral robustness. By contrast, the CN and the EWV associated with the fixed-base square pyramid configuration show similar optimal values but are more spectrally dependent. Therefore, this latter configuration is of limited interest for spectroscopic applications. Finally, the spectral dependence of the CN and the EWV of the triangular bipyramid configuration is similar to that of the classical five-state set in terms of spectral stability but has the remarkable advantage of offering a close to optimal PSA, with an EWV up to 97% of the optimal value reached by the square pyramid set. Such behavior indicates that the triangular bipyramid set allows for reconstruction of the Stokes vector on a large spectral range, where the retardance of the waveplate takes values between $\delta =50$ and 140 deg at least, with low sensitivity to noise, and by making five measurements only. When the retardance is close to 180 deg, the waveplate is close to the half waveplate condition and produces quasi linearly polarized states. In these conditions, the volume of the related polyhedrons in the Poincaré sphere is quite small, implying that both the inverse CN and the inverse EWV tend to 0. The use of exclusively linearly polarized analysis states does not allow for the proper measurement of the third component of the Stokes vector $S_3$ as such linearly polarized analysis states are not appropriate to measure the ellipticity of the polarization. Therefore, one common feature to all sets is the divergence of both the CN and EWV figures of merit when the retardance of the waveplate is close to $180\mathrm{deg}$.

To understand the origins of the spectral dependence of the CN and EWV metrics for the five-state PSA, it is informative to look at the evolution of the PSA states on the Poincaré sphere as a function of the wavelength, as previously done for the four-state tetrahedron PSA. In Fig. 3(a), we show the evolution of the CN, and in Figs. 4(b)–4(d), we show the shape of some polyhedra for different retardances $\delta =20$, 55, 90, 125, and 160 deg. From these plots, a rule of thumb can be derived to allow for minimization of $\kappa$: the larger the volume underpinned by vertices representing the projection states, the lesser the CN. Note, however, as pointed out by Ref. 14, that maximizing the volume of the polyhedron representing the states of a PSA does not necessarily yield the least noise-sensitive PSA. However, close-to-vanishing volumes result in degraded EWV or CN on the polarization analysis matrix.

From Figs. 3 and 4, the spectral stability may be partially explained by the fact that the location of several vertices of the polyhedron is spectrally independent. Configurations with a high number of linear states, such as the fixed-base square pyramid, triangular bipyramid, and extended classic set, allow for symmetry in both the CN and EWV figures of merit around the optimal retardance. This can be understood by looking at the evolution of the polyhedron representing each set of five states as a function of retardance, where volumes and shapes evolve smoothly, in contrast to shapes such as the tetrahedron for which the lack of spectrally stable points implies strong shape and size variations with wavelength.

From there, a first hint can be drawn: to design a spectrally robust PSA, the $M$ states must remain close to the optimal positions for all wavelengths in the spectral range of interest. For that purpose, a maximum number of linearly polarized projection states must be privileged. Other aspects, such as the shape of the polyhedron, its symmetry, and its ability to uniformly span the volume of the Poincaré sphere, play a role that determines the spectral response of the PSA. An exhaustive analysis of the geometrical aspects of the polyhedra leading to optimal configurations in terms of spectral robustness is out of the scope of the present study.

In this section on the design of a broadband polarimeter, we have presented several configurations of five-state PSA displaying different levels of performance both in terms of optimal CN or EWV and in terms of spectral robustness. The best candidate we identified to ensure high-performance broadband operation is the triangular bipyramid configuration, which is able to provide a close-to-optimal operation in the 1500 to $2500{\mathrm{cm}}^{-1}$ range. In what follows, we describe the experimental implementation of such a PSA with a linear polarizer and a QWP at $2000{\mathrm{cm}}^{-1}$, the calibration procedure of the instrument, and results from tests run on different polarization states.

Calibration of the triangular bipyramid PSA based on a retarder and a linear polarizer means experimentally measuring the polarization analysis matrix $\mathbf{W}$. In this part, we report two sets of results providing a measurement of the analysis matrix, the first one providing a reference measurement, and the second one being less accurate but more easily performed in a lab, the results of which can be compared with the reference we provide.

We first discuss the reference measurement. A realistic estimate of the polarization analysis matrix components can be obtained by measuring the Mueller matrices $\mathbf{C}$ and $\mathbf{P}$ of the QWP and polarizer, respectively, using a well-calibrated Mueller polarimeter. However, this method requires access to a Mueller polarimetry setup, which is uncommon in the MWIR. We have used the facility described in Ref. 32 to characterize each optical component of our PSA. Quite importantly, this setup is based on the use of Fresnel rhomb retarders, making the whole instrument quasi-achromatic in the MWIR. Using Eq. (2), we are able to construct the polarization analysis matrix from the characterization of all individual components of our PSA. The results of this procedure are displayed as blue solid lines in the polarization analysis matrix frame of Fig. 5. In Fig. 5(a), we plot in solid blue lines the degree of polarization (DOP) as obtained from the polarization analysis matrix components extracted from the Mueller matrix procedure. The use of $\mathrm{ZnSe}$ Fresnel rhomb with flat retardance over the entire probed spectral range leads to spectrally featureless measurements.32 The results of this first method can be compared with a theoretical polarization analysis matrix obtained from Eq. (2) but using the theoretical expressions of the Mueller matrices of an ideal linear polarizer and a retarder. For the Mueller matrix of the QWP, the only quantity that needs to be determined is the retardance $\delta$ of the QWP. This can be experimentally measured using a standard setup (see Ref. 40 for instance). The retarder is inserted between two parallel linear polarizers and oriented at $45\mathrm{deg}$ to the polarizer axes, and the measurement is repeated with crossed polarizers. From these measurements, the cosine of the retardance can be easily extracted by taking the ratio between the difference and the sum of the first and second intensity measurements. The retardance of our commercial QWP is available from the manufacturer (and displayed in Fig. 2). Comparing the polarization analysis matrices obtained from this theoretical estimation using ideal optical components to the reference measurement, we get $\parallel \mathrm{\Delta }\mathbf{W}\parallel /\parallel \mathbf{W}\parallel \le 0.03$ in average over the spectral range studied—which enables us to evaluate the right-hand side of Eq. (6). This confirms the validity of the measurement performed on the Mueller polarimeter, which we will now use as a reference for a second calibration procedure that does not require access to highly specific polarimetric instrumentation.

The polarization analysis matrix can be probed using $N$ well-known input polarization states and recording the intensity resulting from their projection onto each state of the PSA as proposed by Ref. 41. We perform the second calibration procedure by generating these input states with a polarization state generator (PSG) consisting of the same elements as the PSA but reversed, i.e., a linear polarizer followed by a waveplate that we choose to be a QWP at $2000{\mathrm{cm}}^{-1}$. An unpolarized blackbody is placed in front of the PSG so that the generated polarization state is perfectly known and fixed by the relative angles of the polarizer ${\theta }_{\mathrm{P}}$ and waveplate ${\theta }_{\mathrm{C}}$ principal axes with respect to one fixed common reference. The transmissivity of each element is not relevant for a Stokes polarimetry experiment as the Stokes vector normalized to the total intensity $S_0$ is the quantity of interest. Using these well-characterized elements, we can easily produce our desired polarization states to probe $\mathbf{W}$. Stacking the input states in a matrix called $\mathbf{G}$ and the results of the projection in matrix $\mathbf{T}$, the equations read $$\mathbf{WG}=\mathbf{T}.$$(8)

The polarization analysis matrix can be recovered by computing $\tilde{\mathbf{W}}={\mathbf{TG}}^{+}$, where $\tilde{\mathbf{W}}=\mathbf{WA}$, $\mathbf{A}=\mathbf{G} {\mathbf{G}}^{+}$ is the orthogonal projector onto the range of $\mathbf{G}$. As long as $\mathbf{W}$ is evaluated on a vector in the range of $\mathbf{G}$, $\tilde{\mathbf{W}}=\mathbf{W}$. To probe the measurements matrix, a minimum of four distinct input states must be chosen. In what follows, we choose to use the same five input states of the bipyramid configuration: $({\theta }_{\mathrm{P}},{\theta }_{\mathrm{C}})=\{(0^{°},0^{°}),({60}^{°},{60}^{°}),({120}^{°},{120}^{°}),(0^{°},{45}^{°}),(0^{°},-{45}^{°})\}$, in other words, $\mathbf{G}(\delta )={\mathbf{W}}^T(-\delta )$.

An important aspect of this procedure is that the intensity of each measurement is recorded on an FTIR spectrometer that collects all wavelengths at once. The complete spectral dependence of the polarization analysis matrix components is recovered from the intensity spectra acquired for each input state probed by the system. This prevents the use of filters in front of the blackbody. Doing so would drastically reduce the incoming light flux on the detectors, therefore significantly affecting the measurement accuracy. We benefit here from the multiplexing advantage of the FTIR spectrometer to perform quicker and still precise measurements.

The results of the second procedure are plotted as solid red and orange lines in Figs. 5(a) and 5(b) so that the whole figure compares the matrix obtained with the Mueller polarimeter procedure and the matrix obtained from the second calibration procedure. The DOP is obtained from both procedures. Agreement between both methods is overall excellent, with a $<3\%$ discrepancy on most components of the polarization analysis matrix. Noisy data around $3350{\mathrm{cm}}^{-1}$ from the second calibration procedure is due to the degraded CN of matrix $\mathbf{G}$. This can be explained by the fact that the QWP used to generate $\mathbf{G}$ reaches a 180 deg retardance in this spectral range. It follows that the fourth column of the measurement matrix, which represents the retardance-sensitive part, is not accurately probed, which results in the noisy behavior in this spectral range.

The quality of the method we used to experimentally characterize $\mathbf{W}$ is visible in the high degree of polarization computed for each of the PSA states. We obtain good agreement with theory to a relative uncertainty of $<10\%$ on each component of the polarization analysis matrix over a large spectral range from 2000 to $4000{\mathrm{cm}}^{-1}$.

The role of a standard PSA is to measure the polarization state of a monochromatic incident light beam via intensity measurements. Here, we note that the calibration procedure gives us access to all components of the polarization analysis matrix over a broad spectral range. This means that our PSA is now able to provide measurements of the polarization state of broadband light via the measurement of spectra. By combining our PSA and an FTIR, we are able to reconstruct the polarization state of each monochromatic component of a given beam by measuring five different spectra, without requiring any spectral filtering.

To check the performances of our triangular bipyramid PSA, we generate well-known polarization states with the same combination of a blackbody and a PSG as used in the previous section to characterize the components of $\mathbf{W}$ (see Fig. 6). The light is analyzed by the PSA and focused on the entrance slit of the FTIR spectrometer. The mounts of the ${\mathrm{MgF}}_2$ waveplate and the polarizer are manually rotated to place the PSA in the five different configurations of the five-state bipyramid.

We generate three polarization states with the PSG defined by tilting angles $({\theta }_{\mathrm{P}},{\theta }_{\mathrm{C}})=\{(0^{°},0^{°})$, $({45}^{°},{45}^{°})$, $(0^{°},-{45}^{°})\}$ of the PSG polarizer and waveplate, respectively. Such polarization states are theoretically represented by the following normalized Stokes vector $(\begin{array}{c}\mathrm{1,0},0\end{array})$, $(\begin{array}{c}\mathrm{0,1},0\end{array})$, and $(\begin{array}{c}\cos \{\delta \},0,\sin \{\delta \}\end{array})$. Applying Eq. (3) allows access to the Stokes vector estimate, i.e., the maximum likelihood estimate of the Stokes vector under analysis. Figures 7(a)–7(c) show the evaluation of the normalized components of the Stokes vector of the three polarization states used for the test, where $\mathbf{W}$ was evaluated with Eq. (2) from the experimental measurement of the Mueller matrices of each component of the PSA. Confidence bars on the obtained values are computed from Eq. (5) (darker shades) and Eq. (6) (paler shades), respectively. The latter were computed using $\parallel \mathrm{\Delta }\mathbf{W}\parallel /\parallel \mathbf{W}\parallel =0.03$, which is the estimated relative uncertainty on the polarization analysis matrix.

Good agreement of the Stokes estimate with the expected values is obtained. Figures 7(a) and 7(b) show two vanishing normalized Stokes components and one component close to 1 as expected. Figure 7(c) shows a purely circular polarization state at $2000{\mathrm{cm}}^{-1}$ as expected with a QWP at this wavelength. The polarization state evolves into an elliptical one with continuous variation of the ellipticity until the retardance of the waveplate reaches 180 deg at $3350{\mathrm{cm}}^{-1}$. As such, the waveplate behaves as a half-waveplate, and the generated polarization state becomes linear. However, large fluctuations around this wavelength are visible. They are due to the degraded CN of the polarization analysis matrix $\mathbf{W}$. This limitation is intrinsic to the choice of the waveplate and the material it is made of. A possible solution to overcome this defect is the use of a second retarder with a different spectral retardance, particularly around 2900 to $3500{\mathrm{cm}}^{-1}$, to cover the whole spectral range with higher quality. The drawback of this solution is that it would be necessary to add at least two states to the PSA and two states to the calibration set using the second retarder. As a result, the improved PSA would have seven instead of five states, which will increase measurement time and instrumental complexity. In spite of this, we can see that the estimated normalized Stokes vector still gives a reasonable estimation, although noisy, as they are highly sensitive to any noise affecting the intensity measurements. The use of a nonoptimum configuration $(M=4)$ states affects the uncertainties of the measurement in the proportion displayed in Fig. 3, in particular with an EWV degraded by a factor between 1.2 and 3 for wavenumbers around $2500{\mathrm{cm}}^{-1}$ and below. These results confirm the validity of our method and the robustness of our PSA on a large spectral range in the MWIR.

In summary, we reported in this paper the design and characterization of a PSA able to analyze an unknown polarization state on a large spectral range, spanning from 2000 to $4000{\mathrm{cm}}^{-1}$. The deduced Stokes vector accuracy is estimated to be at most 10% on the operating range, outside of the spectral span where the retardance of the waveplate is an integer multiple of 180 deg. This PSA is built from a waveplate made of ${\mathrm{MgF}}_2$, a quarter waveplate at $2000{\mathrm{cm}}^{-1}$, and a linear polarizer. It was designed to provide five states optimizing the conditioning of the procedure of polarization analysis over a broad range of wavelengths. Combining such a PSA and an FTIR allows to perform the analysis of the polarization state of all monochromatic components of an incoming light-beam in parallel: in other words, no spectral filtering is required in our procedure. The PSA we propose can therefore be used to perform accurate characterization of polarized sources of thermal emission, where narrowband operation often means working with low levels of signal.

Biographies of the authors are not available.