An orthonormal family of super Lorentz–Gauss (SLG) modes is proposed to describe the highly divergent higher-order modes. The first-order and the second-order SLG modes ${\mathrm{SLG}}_{01}$ and ${\mathrm{SLG}}_{11}$ are illustrated as examples. Analytical propagation formulas of the ${\mathrm{SLG}}_{01}$ and ${\mathrm{SLG}}_{11}$ modes through a paraxial $ABCD$ optical system are derived, and analytical beam propagation factors of the ${\mathrm{SLG}}_{01}$ and ${\mathrm{SLG}}_{11}$ modes are presented. The paraxial propagation properties of the ${\mathrm{SLG}}_{01}$ and ${\mathrm{SLG}}_{11}$ modes in free space are also compared with those of the corresponding Hermite–Gaussian (HG) ${\mathrm{HG}}_{01}$ and ${\mathrm{HG}}_{11}$ modes, respectively. This research indicates that SLG modes are more appropriate than HG modes to describe the highly divergent higher-order modes.