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Currently, astronomical wavefront sensor (WFS) are optimized for two types of sources: natural guide stars (NGS) using an unresolved star as the reference or laser guide stars (LGS) with a large resolved beacon. In the former case, the coherent approximation is valid, while in the latter case, the incoherent approximation can be applied; we examine cases where neither approximation is valid or the partially coherent case. To date, there has been limited research on optimizing a WFS design for a partially coherent beacon, though there are at least two important applications: uplink corrected laser beacons and artificially induced partial coherence in modulated pyramid designs. Using Hopkins' formula1 we have derived an approximate expression for the Fisher information matrix (FIM) for any wavefront sensor design using Fourier plane filtering2 with a particular emphasis on the pyramid and Zernike WFS. From this expression we show that degradation in WFS sensitivity is primarily due to the combining of incoherent pupil point pairs. The expression also gives rise to an alternative method for modeling WFS in the partially coherent regime using the magnitude of the complex coherence factor and the WFS's coherent impulse response (CIR) to approximate the detector output. We explore two methods for optimizing WFS sensitivity with a partially coherent beacon: pupil segmentation and impulse response engineering. In the former case, the pupil is subdivided via to match the coherence area of the source and a separate coherent wavefront sensing technique is used to retrieve higher order modes in each subaperture. The latter approach optimizes the CIR of a WFS to limit the spread of light to the coherence area of the source. The effectiveness of these methods is explored through wave-optic simulation, using both the newly developed CIR method and the more standard fast Fourier transform (FFT) approach. The simulation results are also compared to measurements from a bench-level adaptive optics (AO) system using a novel software-defined WFS with two spatial light modulators (SLM) at the pupil plane and Fourier plane respectively to allow for rapid configuration of multiple WFS designs without the need for hardware changes. Bench-level measurements of the WFS sensitivity to photon noise are presented to further validate the theoretical and simulation predictions for optimal wavefront sensing with partially coherent beacons.
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