In this paper, we first present an experimental demonstration of terahertz radiation pulse generation with energy up to 5 pJ under the electron emission during ultrafast optical discharge of a vacuum photodiode. We use a femtosecond optical excitation of metallic copper photocathode for the generation of ultrashort electron bunch and up to 45 kV/cm external electric field for the photo-emitted electron acceleration. Measurements of terahertz pulses energy as a function of emitted charge density, incidence angle of optical radiation and applied electric field have been provided. Spectral and polarization characteristics of generated terahertz pulses have also been studied. The proposed semi-analytical model and simulations in COMSOL Multiphysics prove the experimental data and allow for the optimization of experimental conditions aimed at flexible control of radiation parameters.