Sensing ELT petal modes with a discrete beam combiner: an end-to-end theoretical model
The European Southern Observatory's Extremely Large Telescope (ELT), the 39-m telescope under construction in Chile, forms its entrance pupil from 798 hexagonal segments that the six 0.5-m-wide spider arms fragment into six "petals". The low-wind / island effect, adaptive-optics residuals, and mechanical drift imprint an independent piston on each petal. These differential-piston (petal) modes remain largely undetected by the ELT's baseline continuous-pupil wavefront sensors, a major bottleneck for diffraction-limited, high-contrast imaging. This paper presents a complete theoretical model of a single-mode photonic discrete beam combiner (DBC) configured to sense the six petal pistons. The model constructs the coupled-mode description of a two-dimensional evanescently coupled waveguide lattice with six inputs and forty-five outputs and derives its visibility-to-pixel matrix (V2PM); treats the pupil remapping that routes one sub-aperture per petal into the six inputs; develops the five-dimensional petal-mode basis of the fragmented pupil; and propagates the petal modes through the DBC to quantify the sensitivity. Optimising the input-port spacing and interaction length by minimising the V2PM condition number yields a well-conditioned device (CN = 10.4). It recovers petal pistons with unit linear gain over the full +/-lambda/2 (+/-800 nm) unambiguous range and separates the five modes with negligible cross-talk (< 10^-3). For 1% photometric noise, the reconstructed-piston error is 4.7 nm rms at lambda_0 = 1.6 micron. Tested against petal amplitudes from end-to-end ELT simulations (~40-300 nm rms atmospheric residual; ~0.3-1 micron low-wind events), the DBC recovers each piston to a few nm rms at H band, with a two-wavelength extension capturing the larger events. The DBC is therefore a compact, chromatically robust petalometer for the fragmented ELT pupil.