Higher-order complex source for elegant Laguerre–Gaussian waves

We introduce a higher-order complex source that generates elegant Laguerre-Gaussian waves with radial mode number n and angular mode number m. We derive the integral and differential representations for the elegant Laguerre-Gaussian wave that in the appropriate limit yields the corresponding elegant Laguerre-Gaussian beam. From the spectral representation of the elegant Lauguerre-Gaussian wave we determine the first three orders of nonparaxial corrections for the corresponding paraxial elegant Laguerre-Gaussian beam.