Optimal polarization modulation and calibration schemes

TL;DR

This work generalizes polarization modulation and calibration efficiency to account for state-dependent throughputs and nonuniform photon-noise, enabling accurate demodulation and optimization for spatial modulators such as SIMPol MPS. The authors derive a generalized demodulation matrix $\mathbf{D}$ and show how modulation efficiencies relate to $\mathbf{D}$ and the throughput matrix $\mathbf{T}$, recovering the classical results when $\mathbf{T}=\mathbb{1}$. They extend the framework to calibration, deriving optimal calibration-inversion $\mathbf{E}$ and a calibration-efficiency metric, and demonstrate practical gains through SIMPol and DKIST case studies, including sequence optimization and retarder choices. The approach provides a unified, implementable method to maximize polarimetric accuracy and minimize calibration time in real instruments with nonuniform channel throughputs. This has direct implications for solar polarimetry where high photon budgets and rapid, precise calibrations are essential.

Abstract

We review the algebraic definition of the efficiency of a polarization modulation scheme, which is commonly adopted for solar and stellar spectro-polarimetry applications, and generalize it to allow distinct states of the modulation cycle to have arbitrary throughput and different photon-noise statistics for each state. Such a generalization becomes necessary to model and optimize the polarimetric efficiency of instruments implementing spatial polarization modulation schemes, where different optical paths are assigned to different polarization analysis states, which may be characterized by different throughput values. The proposed algebraic extension also proves essential for introducing a workable concept of the efficiency of a polarization calibration scheme, which can then be used to create a merit function for the optimization of calibration sequences, which take into account the specific characteristics of the polarimetric instrument and of its calibration optics.

Figures (3)

• Figure 1: Polarization analysis ellipses of the first five orders of the SIMPol MPS, as derived from laboratory calibrations, along with their percentage throughput. The MPS design optimized the throughput and polarization efficiency of the four 1st-order channels. The total throughput of the five channels shown here is ${\sim}\,67$%, and the diattenuation values are all above ${\sim}\,97$%. The (0,0) order dominates the efficiency losses of the MPS grating, along with all the other diffracted channels of order ${>}\, 1$ (not shown). The orders shown in the figure follow the same order as in the modulation matrix of Eq. (\ref{['eq:modSIMPol']}), moving CW from the (0,+1) order at the top, and ending on the (0,0) order at the center. The reference direction for linear polarization corresponds to the horizontal axis of the plots.
• Figure 2: Calibration efficiency plots between 370 and 1700 nm for the DKIST optically contacted SiO$_2$ calibration retarder, assuming an optimally efficient and balanced 4-state polarization modulator. Colored curves show $S_1$ (black), $S_2$ (blue), $S_3$ (red), and $S_4$ (green). The yellow dot-dashed curve corresponds to the RSS of the $S_{2,3,4}$ efficiencies. Left: Using one of the calibration sequences currently implemented at the DKIST (left panel of Table \ref{['tab:DKIST']}). Right: After optimization of the DKIST sequence, resulting in the calibration sequence in the center panel of Table \ref{['tab:DKIST']} (see caption of Table \ref{['tab:DKIST']} for details); we note that the curves for the linear polarization parameters $S_2$ and $S_3$ are overlapping, and the RSS of the $S_{2,3,4}$ efficiencies is essentially identical to the efficiency of $S_1$, demonstrating the optimal performance of the calibration sequence for this retarder.
• Figure 3: Calibration efficiency plots between 370 and 5000 nm for the DKIST compound MgF$_2$ elliptical calibration retarder (EliCal), assuming an optimally efficient and balanced 4-state polarization modulator. Colored curves are as in Fig. \ref{['fig:DKIST']}. The optimized calibration sequence uses fixed angular steps for both calibration polarizer and retarder, as shown in the right panel of Table \ref{['tab:DKIST']}.