Beam test, simulation, and performance evaluation of PbF$_2$ and PWO-UF crystals with SiPM readout for a semi-homogeneous calorimeter prototype with longitudinal segmentation

Crilin (Crystal Calorimeter with Longitudinal Information) is a semi-homogeneous, longitudinally segmented electromagnetic calorimeter based on high-$Z$, ultra-fast crystals with UV-extended SiPM readout. The Crilin design has been proposed as a candidate solution for both a future Muon Collider barrel ECAL and for the Small Angle Calorimeter of the HIKE experiment. As a part of the Crilin development program, we have carried out beam tests of small ($10\times10\times40$~mm$^3$) lead fluoride (PbF$_2$) and ultra-fast lead tungstate (PbWO$_4$, PWO) crystals with 120~GeV electrons at the CERN SPS to study the light yield, timing response, and systematics of light collection with a proposed readout scheme. For a single crystal of PbF$_2$, corresponding to a single Crilin cell, a time resolution of better than 25~ps is obtained for $>$3 GeV of deposited energy. For a single cell of \pwo, a time resolution of better than 45~ps is obtained for the same range of deposited energy. This timing performance fully satisfies the design requirements for the Muon Collider and HIKE experiments. Further optimizations of the readout scheme and crystal surface preparation are expected to bring further improvements.