Abstract: The elegant Laguerre-Gaussian(L-G) beams and their propagation properties are studied in detail using the propagation equation derived from the generalized Huygens-Fresnel diffraction integral. The beam propagation factor(M2 factor) and power in the bucket(PIB) are used to characterize beam quality,a comparison of the beam quality of elegant and standard L-G beams is made and the results are analyzed.