[Paper Review] Polarization Modeling and Predictions for DKIST Part 2: Application of the Berreman Calculus to Spectral Polarization Fringes of Beamsplitters and Crystal Retarders

David M. Harrington, Frans Snik+4 more|arXiv (Cornell University)|Jan 1, 2017

Solar and Space Plasma Dynamics57 references17 citations

TL;DR

This paper applies the Berreman calculus—a 4×4 matrix formalism for electromagnetic wave propagation in birefringent media—to model and predict spectral polarization fringes in thick-crystal retarders and beam splitters for the Daniel K. Inouye Solar Telescope (DKIST). The method enables quantitative prediction of fringe periods and amplitudes across optical design parameters, leading to design decisions such as removal of cover windows and optimized anti-reflection coatings, significantly reducing polarization and intensity fringe errors in high-resolution spectropolarimetry.

ABSTRACT

We outline polarization fringe predictions derived from a new application of the Berreman calculus for the Daniel K. Inouye Solar Telescope (DKIST) retarder optics. The DKIST retarder baseline design used 6 crystals, single-layer anti-reflection coatings, thick cover windows and oil between all optical interfaces. This new tool estimates polarization fringes and optic Mueller matrices as functions of all optical design choices. The amplitude and period of polarized fringes under design changes, manufacturing errors, tolerances and several physical factors can now be estimated. This tool compares well with observations of fringes for data collected with the SPINOR spectropolarimeter at the Dunn Solar Telescope using bi-crystalline achromatic retarders as well as laboratory tests. With this new tool, we show impacts of design decisions on polarization fringes as impacted by anti-reflection coatings, oil refractive indices, cover window presence and part thicknesses. This tool helped DKIST decide to remove retarder cover windows and also recommends reconsideration of coating strategies for DKIST. We anticipate this tool to be essential in designing future retarders for mitigation of polarization and intensity fringe errors in other high spectral resolution astronomical systems.

Motivation & Objective

• To develop a predictive framework for spectral polarization fringes in multi-crystal retarders and beam splitters used in high-resolution astronomical instruments.
• To address the dominant error source in high spectral resolution polarimetry: intensity and polarization fringes caused by coherent interference in birefringent optical components.
• To guide DKIST optical design decisions by quantifying the impact of anti-reflection coatings, refractive index matching oils, cover windows, and crystal thicknesses on fringe generation.
• To validate the model against laboratory measurements and observations from the SPINOR spectropolarimeter at the Dunn Solar Telescope.
• To provide a generalizable tool for future astronomical instruments requiring mitigation of fringe-induced polarization errors.

Proposed method

• Adapts the Berreman calculus formalism (4×4 matrix method) to model electromagnetic wave propagation through multiple birefringent layers with arbitrary optical axis orientations.
• Extends the formalism to include isotropic materials such as anti-reflection coatings, cover windows, and index-matching oils, enabling full-stack modeling of complex optical systems.
• Implements a custom Python software package based on the MHW textbook (McCall, Hodgkinson, Wu) to compute Mueller matrices, transmittance, retardance, and polarizance across spectral bands.
• Validates predictions against analytical solutions in the non-interfering limit and against established optical modeling tools (Zemax, TFCalc) for isotropic layers.
• Uses the framework to simulate fringe behavior across varying thicknesses, refractive indices, coating strategies, and thermal conditions.
• Compares model outputs with experimental data from SPINOR at the Dunn Solar Telescope and lab tests on quartz retarders and windows.

Experimental results

Research questions

• RQ1 How do anti-reflection coatings and refractive index matching oils affect the spectral period and amplitude of polarization fringes in multi-crystal retarders?
• RQ2 To what extent do cover windows contribute to fringe generation in DKIST retarders, and can their removal reduce fringe-related errors?
• RQ3 How do variations in crystal thickness and material refractive index influence the spectral period and visibility of polarization fringes?
• RQ4 Can the Berreman calculus accurately predict fringe periods observed in real-world instruments like SPINOR at the Dunn Solar Telescope?
• RQ5 What design trade-offs exist between thermal management, mechanical stability, and fringe suppression in high-precision polarimetric optics?

Key findings

• The Berreman calculus accurately predicts fringe spectral periods for bi-crystalline achromatic retarders, matching observed data from the SPINOR spectropolarimeter at 396.8 nm with a predicted period of 1.25 pm and required spectral resolution R = 638,000.
• For 43 mm thick FIDO beam splitter substrates, the fringe period is 19.7 nm at 1565 nm, requiring R = 159,000 for two-point sampling—indicating that high-resolution instruments are necessary to resolve these fringes.
• The model successfully reproduces fringe behavior in laboratory tests on quartz retarders and simple windows, confirming consistency with known optical physics.
• Model predictions show that cover windows in DKIST retarders significantly contribute to fringe generation, leading to the design decision to remove them to mitigate polarization and intensity fringes.
• The framework quantifies that anti-reflection coatings and oil refractive indices have measurable impacts on fringe amplitude and period, suggesting a need to reconsider coating strategies for DKIST.
• The tool enables systematic trade studies by providing scaling relations between thermal load, optical thickness, and fringe sensitivity, supporting integrated optical and thermal design.