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Expanded Gaussian-Laguerre modes provides a more accurate and simple method to analyze the beam-waveguide (BWG) system. Using the propagated mode set and multiplication of the matrices for the BWGs factors, output field patterns with cross-polarized fields can be derived. A Q-band (33-50 GHz) BWG system which has two flat mirrors and two elliptic mirrors for a satellite Earth station antenna is designed. The characteristics of a corrugated conical horn for feeding to the BWG and the output beam patterns of the BWG system are simulated and compared with measured values.