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The aim of the Cryogenic wide-Area Light Detectors with Excellent Resolution (CALDER) project is the development of light detectors with active area of 5 $$\,\times \,$$ 5 cm $$^{2}$$ and noise energy resolution smaller than 20 eV RMS, implementing phonon-mediated kinetic inductance detectors. The detectors are developed to improve the background suppression in large-mass bolometric experiments such as CUORE, via the double read-out of the light and the heat released by particles interacting in the bolometers. In this work, we present the characterization of the first light detectors developed by CALDER. We describe the analysis tools to evaluate the resonator parameters (resonant frequency and quality factors) taking into account simultaneously all the resonance distortions introduced by the read-out chain (as the feed-line impedance and its mismatch) and by the power stored in the resonator itself. We detail the method for the selection of the optimal point for the detector operation (maximizing the signal-to-noise ratio). Finally, we present the response of the detector to optical pulses in the energy range of $$0{-}30$$ keV.