Fundamental Limits to Phase and Amplitude Estimation in the High-Strehl Regime

Abstract

Context: Ground-based telescopes are susceptible to seeing, an atmospheric phenomenon that reduces the resolving power of large observatories to that of a home telescope. Compensating these effects is therefore critical to realizing the potential of upcoming extremely large telescopes, a challenging task that requires precise wavefront control. Ultimately, this precision is limited by one's wavefront sensor (WFS) and its capacity to accurately encode phase and amplitude aberrations. Aims: Our attention is on photon noise-limited wavefront sensing in the high-Strehl regime. In particular, we seek fundamental limits to phase and amplitude estimation in addition to a WFS that saturate these bounds. Methods: Information theory is employed for deriving minimum-achievable residual errors, as stipulated by a metric called the Holevo Cramer-Rao bound. Holevo's bound is closely related to another metric called the quantum Cramer-Rao bound, which has already been applied to phase estimation on nearly-corrected wavefronts. Results: We present a WFS that can perfectly extract and phase shift a telescope's piston mode. We show how this phase can be used to tune the apparatus' sensitivity to phase and amplitude, and provide a closed-form expression for the optimal phase shift. For circular apertures, this implementation saturates the fundamental limits, but it can be easily modified to work with arbitrary pupils. Moreover, our proposal uses optics that are manufactureable today and is readily achromatized with geometric phase shifters.

Figures (4)

• Figure 1: Expected long-exposure Strehl ratio against the number of photons collected and modes corrected. Plots are shown for an optimal WFS measuring phase and amplitude with equal weight. We see that adaptive optics has little utility when collecting fewer than four photons for every ten modes. Conversely, Strehl ratios of around of 80% are achievable with about 4.5 photons per mode.
• Figure 2: The piston-adapted WFS (PAWS). Our construction spatially sorts the complex Zernike polynomials in two stages, first using a bi-vortex phase filter and Mach-Zhender interferometer (MZI) to separate $m=\pm n$ modes from the remainder, and then separating piston from high-order $m=\pm n$ polynomials using a quad-vortex mask. We envision an implementation using vector vortex masks, which introduce conjugate phases to left- and right-handed circular polarizations states (hence the $\pm$). Likewise, both apertures inside the MZIs use geometric phase to shift ejected modes. Sensitivity to phase and amplitude is then controlled using another half-wave plate with a hole cut to the aperture size. Rotating this waveplate changes the geometric phase $\varphi$, increasing sensitivity to phase aberrations as $\varphi\rightarrow\pi/2$ or amplitude as $\varphi\rightarrow0$. At $\varphi=\pi/4$ we allocate equal sensitivity between the two aberrations, minimizing the total residual variance. Like the vZWFS, our WFS works on the two orthogonal circular polarizations, imparting conjugate phases to each. Optics right of the figure's center serve to reconstitute the field at the pupil plane and produce two images, one for each circular polarization, via a Wollaston prism. A polarizing cube beamsplitter could also be used. Note that this figure is adapted from trzaska2025zernikemodesortingvortex.
• Figure 3: A circular-symmetric imaging system of diameter $D$ with atmospheric phase overlaid.
• Figure 4: Photon noise-limited high-Strehl sensitivities to the first 54 non-piston (top) phase and (bottom) amplitude Zernike coefficients for various WFS. Sensitivities were calculated using the method found in chambouleyron_modeling_2023, which is equivalent to calculating the Fisher information haffert2023. The QFI referenced in each plot corresponds to coefficient type: (top) phase and (bottom) amplitude.