Based on quadratic approximation and δ expansion in the first order, the analytical expressions for multi-Gaussian beams in a turbulent atmosphere have been derived. By comparing the two approaches with numerical calculations, the relative errors of average intensity are investigated. As special cases, the relative errors for Gaussian beams, flattened Gaussian beams, and annular beams are investigated. The investigation shows that the method of δ expansion in the first order agrees well with numerical calculations, no matter what the effect of turbulence. If the effect of turbulence is large enough, the relative error of on-axis intensity trends to a constant. The maximum of relative error is about 2.8%. However, quadratic approximation does not give satisfying results under some circumstance even when the effect of turbulence is small. The relative error of on-axis intensity reaches 9.2% when the effect of turbulence is large enough.
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