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Abstract The near-horizon region of a black hole impacts linear (LP) and circular polarization (CP) through strong lensing of photons, adding large-scale symmetries and anti-symmetries to the polarized image. To probe the signature of lensing in polarimetry, we utilize a geometric model of concentric Gaussian rings of equal radius to investigate the transition in the Fourier plane at which the photon ring signal begins to dominate over the direct image. We find analytic, closed-form expressions for the transition radii in total intensity, LP, and CP, wherein the resultant formulae are composed of ratios of tunable image parameters, with the overall “scale” set primarily by the thickness of the direct image. Using these formulae, we compute the transition radii for time-averaged images of M87* simulations at 230 GHz, studying both magnetically arrested disk (MAD) and standard and normal evolution configurations for various spin and electron heating models. We compare geometric values to radii obtained directly from the simulations through a coherent averaging scheme. We find that nearly all MAD models have a photon ring-dominated CP signal on long baselines shorter than Earth's diameter at 230 GHz. Across favored models for the M87* accretion flow identified by the Event Horizon Telescope (EHT) polarimetric constraints, we quantify the sensitivity and antenna size requirements for the next-generation EHT and the Black Hole Explorer orbiter to detect these features. We find that the stringent requirements for CP favor explorations using long baselines on the ground, while LP remains promising on Earth-space baselines. 

While supermassive black-hole masses have been cataloged across cosmic time, only a few dozen of them have robust spin measurements. By extending and improving the existing Event Horizon Telescope (EHT) array, the next-generation Event Horizon Telescope (ngEHT) will enable multifrequency, polarimetric movies on event-horizon scales, which will place new constraints on the space-time and accretion flow. By combining this information, it is anticipated that the ngEHT may be able to measure tens of supermassive black-hole masses and spins. In this white paper, we discuss existing spin measurements and many proposed techniques with which the ngEHT could potentially measure spins of target supermassive black holes. Spins measured by the ngEHT would represent a completely new sample of sources that, unlike pre-existing samples, would not be biased towards objects with high accretion rates. Such a sample would provide new insights into the accretion, feedback, and cosmic assembly of supermassive black holes.