The two new kinds of truncated Gaussian beams, known as the half and quarter Gaussian beams, are defined as the product of the fundamental Gaussian beam with the Heaviside unit step function. Using the generalized Collins integral, the exact analytical propagation formulas are derived for the truncated Gaussian beams through paraxial optical systems. Combined with the Gaussian beam decomposition method, the truncated Gaussian beams are used to represent the sharp edges of a field after a hard aperture. The modified Gaussian beam decomposition method presented in this work enables the calculation of the diffraction of a given field from a hard aperture with an arbitrary shape. This solves one of the limitations of the conventional Gaussian beam decomposition method, which is its inability to accurately model the diffraction of fields with sharp edges, especially in the near field. The validity and accuracy of the proposed method are demonstrated using a few exemplary diffraction calculations.
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