Surface figure metrology for reflective membrane mirrors based on phase-measuring deflectometry

TL;DR

This work tackles the challenge of metrology for lightweight, non-rigid membrane mirrors intended for next-generation telescopes by implementing a phase-measuring deflectometry (PMD) system augmented with a rigorous ray-tracing model and iterative gradient-to-height reconstruction. The approach delivers a robust pipeline for extracting surface gradients from fringe distortions, converting them into a high-fidelity surface figure via a 36-term Zernike expansion, and assessing dynamic stability with Monte Carlo-supported uncertainty quantification. Key contributions include a Gray-code based absolute phase unwrapping that resolves $2\pi$ ambiguities, a closed-form phase-to-gradient relation with projection correction, and a robust, initialization-insensitive reconstruction algorithm that can handle large deformations. The results demonstrate micrometer-scale stability and a surface figure accuracy sufficient for many infrared wavelengths, validating PMD as a practical metrology tool for membrane-based telescope systems and guiding future automation and real-time capabilities.

Abstract

Reflective membrane mirrors provide a lightweight, low-cost alternative to traditional optics for next-generation large-aperture telescopes, but their non-rigid, thin structure poses challenges for surface metrology. We present a phase-measuring deflectometry (PMD) system enhanced with tailored ray-tracing and iterative reconstruction to enable non-contact measurement of large membrane optics. The system successfully characterizes the surface figure and evaluates the dynamic stability of a 1-meter Hencky-type membrane mirror. Our results demonstrate the effectiveness of PMD as a practical metrology tool for future membrane-based telescope systems.

Figures (14)

• Figure 1: Left: The 1-meter aperture membrane mirror prototype featuring an aluminum-coated PET membrane ($50\ \mu$m thickness) stressed by Nitrogen gas pressure in a Hencky configuration. Middle: Rear view showing the high-precision laser displacement sensor ($\pm 1\ \mu$m repeatability) that provides real-time feedback to the PID control system for active surface stabilization. Right: Experiment setup comprising an LCD fringe projector and a CMOS industrial camera with a 16 mm F/2.8 lens.
• Figure 2: Workflow diagram of our PMD surface reconstruction process for membrane mirrors, showing the sequence from experiment to surface-figure output.
• Figure 3: Parallel processing architecture of PMD-SFM for high-throughput PMD measurements. The Python-based system implements a producer-consumer model where: the main thread acquires images from the camera and pushes them into a queue buffer, while a dedicated I/O thread asynchronously writes the queued images from RAM to SSD. This parallel pipeline eliminates storage bottlenecks, significantly reducing total measurement latency.
• Figure 4: Structured light patterns displayed on the LCD for absolute phase measurement: (top) Phase-shifted sinusoidal fringes for wrapped phase acquisition and binary Gray codes for phase unwrapping (bottom). Patterns shown are for $x$-direction measurement, with $y$-direction equivalents generated by 90 degree counterclockwise rotation.
• Figure 5: Camera-captured images of the displayed fringe patterns (see Fig. \ref{['fig:stripe']}), showing the reflected sinusoidal-fringe (top) and Gray-code (bottom) patterns used for phase measurement.