Beam quality of Gaussian beams passing through a spherically aberrated annular lens

With power in the bucket(PIB) and β parameter to characterize beam quality,the effect of some parameters,such as spherical aberration coefficient C 4 and obscure ratio ε,on the beam quality of Gaussian beams is analyzed.It is worthwhile to note that PIB is larger at the actual focal plane than that at the geometrical plane due to focal shift,and that β can become less than 1 when a negative spherically aberrated annular lens is properly selected,hence a higher energy density beam can be achieved compared with the case of the annular lens without spherical aberration.