Fig. 4: Simulated turbulence probing results by using longitudinally structured beams with different center longitudinal wavenumbers Q.

a Simulated beam widths of the longitudinally structured beams with different center longitudinal wavenumbers (i.e., \(Q\) values with wavenumber \(k=2\pi /\lambda\)). For the different \(Q\) values, we choose \(N={N}_{\max }\). b Simulation results for probing “Gaussian-shaped” turbulence distributions with different full widths at half maximum (FWHM). c Relative probing errors for the three different cases of FWHM using longitudinally structured beams with different \(Q\) values. d Calculated aperture diameters required for generating the longitudinally structured beams with different Q values.