R&D for compact calorimeters for $e^+e^-$ colliders and their application to the LUXE experiment

The FCAL collaboration is preparing large-scale prototypes of special calorimeters to be used in the very forward region at future electron-positron colliders. Two compact calorimeters are foreseen, LumiCal and BeamCal, designed as sandwich calorimeters with very thin sensor planes to keep the Moli\`ere radius small. Silicon sensor prototypes and dedicated FE ASICs have been developed and produced for LumiCal. The ASICs have been designed to cope with the timing and dynamic range requirements. The electromagnetic calorimeters of the LUXE experiment are based on a similar technology to that developed by the FCAL collaboration. LUXE will measure the number of positrons and their energy spectrum in e-laser and $\gamma$-laser interactions or electrons in $\gamma$-laser interactions. Results of recent beam tests will be presented.