Fig. 1. Polyhedra inscribed in the Poincaré sphere representing the four-state and five-state PSAs discussed in this paper for their respective optimal retardance: (a) four-state regular tetrahedron, (b) five-state extended classic set, (c) five-state square pyramid, and (d) five-state triangular bipyramid.

Fig. 2. Spectral retardance of the ${\mathrm{MgF}}_2$ used in the study (Edmund Optics #85–112), with a quarter-waveplate behavior at $5\mu \mathrm{m}$.

Fig. 3. Comparison of measurement matrix for a PSA consisting of an ${\mathrm{MgF}}_2$ waveplate, a QWP at $2000{\mathrm{cm}}^{-1}$, followed by a linear polarizer according to (a) the inverse CN and (b) the inverse EWV optimization metrics. In both panels (a) and (b), we compare a tetrahedron (four states, blue), a classic set of five states (orange), a triangular bipyramid (five states, yellow), and a square pyramid with fixed base (five states, green). In panel (a), the dashed line represents the optimal CN that can be reached according to Ref. 39. In panel (b), the dashed lines represent the optimal EWV that can be achieved for an $M=4$ state PSA and an $M=5$ state PSA.

Fig. 4. Evolution of the PSA states on the Poincaré sphere according to the waveplate retardance for (a) four state tetrahedron, (b) extended five state classic, (c) five-state square-based pyramid, and (d) five-state triangular bipyramid.

Fig. 5. (a, pink frame) DOP and (b, orange frame) components of the five-state triangular bipyramid polarization analysis matrix $\mathbf{W}$ as a function of wavenumber. (Blue) DOP (in the pink frame) and (light blue) $\mathbf{W}$ (in the orange frame) computed using Eq. (2) and Mueller matrices of the optical components experimentally measured with a Mueller polarimeter. (Red) DOP and (orange) $\mathbf{W}$ computed according to the calibration method described by Eq. (8). Horizontal axis runs from 2000 to $4000{\mathrm{cm}}^{-1}$ for all panels.

Fig. 6. Scheme of the experimental setup for the characterization of the PSA. The combination of a blackbody and a PSG is used to generate arbitrary states of polarization, analyzed by the PSA. The entire light flux is collected by an FTIR spectrometer. For a given incoming polarization state, five spectra are recorded, corresponding to the five configurations of the PSA.

Fig. 7. Analysis with our proposed PSA using a PSG to generate three states: (a) normalized Stokes vector $(\begin{array}{c}1,0,0\end{array})$, (b) normalized Stokes vector $(\begin{array}{c}\mathrm{0,1},0\end{array})$, and (c) normalized Stokes vector $(\begin{array}{c}\cos \{\delta \},0,\sin \{\delta \}\end{array})$. The spectral evolution of the normalized Stokes vector components is plotted. The thick lines represent the normalized Stokes vector components estimating $\widehat{S}$. Darker shadowed areas represent $1\sigma$ error bars extracted from the covariance matrix. Lighter shadowed areas represent $1\sigma$ error bars computed from Eq. (6) with $\parallel \mathrm{\Delta }\mathbf{W}\parallel /\parallel \mathbf{W}\parallel =0.03$ which is the estimated relative uncertainty on the knowledge of the PSA employed.