By variational approach, we analyze the characteristics of beam propagation through a cubic optical nonlinear medium using a laser beam that has a transverse elliptic Gaussian profile. The analytic solution to the normalized transmittance at the center of the far field as a function of medium position and the beam characteristics is obtained and compared with the numerical simulation, which is realized by a combination of algorithms. We also analyze the peak-valley transmittance difference as a function of medium length, ellipticity, and astigmatism. The relationship between peak-valley normalized transmittance difference of the z-scan trace and aperture size or the slit width are obtained. Meanwhile, the comparison of z-scan characteristics with an elliptic Gaussian beam with those using a circular symmetric Gaussian beam is made.
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