In this thesis, I have developed a range of theoretical and numerical techniques for modelling the behaviour of partially coherent optical systems and multi-mode detectors. The numerical simulations were carried out for the ultra-low-noise Transition Edge Sensors (TESs) being proposed for use on the SAFARI instrument on the cooled aperture infrared space telescope SPICA (34 - 210 μm). The optical behaviour of the SAFARI system is described in terms of the optical modes of the telescope, as distinct from the optical modes of the detector. The performance of the TESs were assessed in terms of signal power, background power and photon noise. To establish a method for precisely characterising and calibrating ultra-low-noise TESs, a cryogenic test system was designed and engineered to measure the optical efficiencies of the SAFARI TESs. The multi-mode, partially coherent illumination conditions of the measurement system were engineered to be precisely the same as those of the telescope. A major difference between the test system and the telescope’s optics is that the telescope will have focusing elements, but the test system was designed to avoid focusing elements in order to keep the optical path as clean as possible. The theoretical formalism and numerical models were adapted accordingly to address this difference. The numerical simulations show that the test system could provide near identical optical performance as that of the telescope system even though the focusing elements were absent. I also performed experimental measurements to investigate the optical efficiencies of the multi-mode TESs. The detectors worked exceedingly well in all respects with satisfactory optical efficiencies. In addition, it has been shown that the optical model provides a good description of the optical behaviour of the test system and detectors. Further modal analysis was developed to study losses in the multi-mode horns. The optical behaviour of the waveguide-mounted thin absorbing films in the far-infrared was modelled using a mode-matching method.