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The gaussian beam (more exactly in this case "the embedded gaussian beam") propagates along the Z axis. The plane perpendicular to Z is defined by the two orthogonal axis X and Y. The beam is transformed by a cylindrical lens working in the XZ plane into an elliptical beam which small/large axis are coinciding with the X and Y axis. The page calculates the half width and the position of the emerging beam waist in the XZ plane and the corresponding half width in the YZ plane as functions of the waist radius and position of the incident beam, the wavelength, the focal length of the cylindrical lens and the M2 factor.