Carrying out astronomical observations at far-infrared wavelengths is critical in enabling further progress in the fields of cosmology and astrophysics. Such observations will allow additional insight into the birth and evolution of the Universe. To allow progress in these areas, it is necessary to improve the sensitivity and resolution of the instrumentation that is used to carry out these observations. At the high frequencies in question (terahertz), the instruments typically make use of horn antenna fed detector systems. To achieve the required performance, the horn designs must be highly optimised. Full electromagnetic solvers (CST, HFSS, COMSOL etc.) struggle to predict the performance of horn antennas at such high frequencies in a timely manner due to the large electrical size of the structures. It is therefore very challenging to perform the optimisation using such solvers particularly for multi-mode systems where each mode would have to be considered individually. In this paper we outline an alternative technique for modelling multi-mode (partially coherent) horn antennas based on the mode-matching technique, which allows electrically large structures to be modelled in a highly efficient manner. This technique returns a set of scattering matrices which gives a full vector definition of the transmission and re ection characteristics of the resulting design at a given frequency. We demonstrate how this can be used to extract field patterns and other figures of merit that are important for evaluating the electromagnetic performance of a horn design. An efficient genetic algorithm based optimisation technique (using mode-matching) is also presented. The optimisation process is based on a piecewise conical profile horn design and produces a geometry that is optimised with respect to some figure of merit that is of critical importance for the application in question. This allows the instrument to realise the high levels of optical performance that are required for astronomical applications.