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We studied theoretically and experimentally the propagation of structured Laguerre–Gaussian (sLG) beams through an optical system with general astigmatism based on symplectic ABCD transforms involving geometry of the second-order intensity moments symplectic matrices. The evolution of the coordinate submatrix ellipses accompanying the transformation of intensity patterns at different orientations of the cylindrical lens was studied. It was found that the coordinate submatrix W and the twistedness submatrix M of the symplectic matrix P degenerate in the astigmatic sLG beam with simple astigmatism, which sharply reduces the number of degrees of freedom, while general astigmatism removes the degeneracy. Nevertheless, degeneracy entails a simple relationship between the coordinate element W x y and the twistedness elements M x y and M y x of the submatrix M, which greatly simplifies the measurement of the total orbital angular momentum (OAM), reducing the full cycle of measurements of the Hermite–Gaussian (HG) mode spectrum (amplitudes and phases) of the structured beam to the only measurement of the intensity moment. Moreover, we have shown that Fourier transform by a spherical lens enables us to suppress the astigmatic OAM component and restore the original free-astigmatic sLG beam structure. However, with further propagation, the sLG beam restores its astigmatic structure while maintaining the maximum OAM. [ABSTRACT FROM AUTHOR]

We have demonstrated efficiency of employing the ABCD matrix approach to transform higher-order structured Laguerre–Gaussian (sLG) beams into structurally stable astigmatic sLG (asLG) beams, highlighting their dynamics at propagating. Radical transformations of the beam structure by a cylindrical lens form not only orbital angular momentum (OAM) fast oscillations and bursts, but also make the asLG beams structurally unstable in propagation through cylindrical and spherical lenses when focusing paraxially. But, if the spherical lens performs a Fourier transform of the asLG beam after a cylindrical lens, the symmetric beam emerges at the lens focal plane with a sharp OAM dip; then, the OAM restores its former astigmatism, becoming structurally stable at the far diffraction domain. By investigating the beam structure at the focal area, we have showed that the OAM sharp dip is associated with nothing less than the process of dividing the OAM into the vortex and astigmatic constitutes predicted by Anan'ev and Bekshaev. [ABSTRACT FROM AUTHOR]

We have developed a structured beams theory, computer simulation and experiment that have lost rotational symmetry and propagate in a medium with a general astigmatism. It transpires that the astigmatic medium brings to light such hidden properties of the structured beams as a sharp growing of the orbital angular momentum exceeding the sum of the radial and azimuth numbers, as well as the effect of the beam structure following the cylindrical lens rotation. [ABSTRACT FROM AUTHOR]