Studying orbital angular momentum (OAM) spectra is important for analyzing crosstalk in free-space optical (FSO) communication systems.This work offers a new method of simplifying the expressions for the OAM spectra of Laguerre-Gaussian (LG) beams under both weak/medium and strong atmospheric turbulences.We propose fixing the radius to the extreme point of the intensity distribution,review the expression for the OAM spectrum under weak/medium turbulence,derive the OAM spectrum expression for an LG beam under strong turbulence,and simplify both of them to concise forms.Then,we investigate the accuracy of the simplified expressions through simulations.We find that the simplified expressions permit accurate calculation of the OAM spectrum for large transmitted OAM numbers under any type of turbulence.Finally,we use the simplified expressions to analytically address the broadening of the OAM spectrum caused by atmospheric turbulence.This work should contribute to the concise theoretical derivation of analytical expressions for OAM channel matrices for FSO-OAM communications and the analytical study of the laws goveing OAM spectra.