Fig. 1. Intensity of AM beams for the case of a fourth-order q=1 and angular phase shift β parameters of (a) π/4, (c) π/2, and (e) 2π/3. A third-order AM beam with q=2 and β parameters of (b) π/4, (d) π/2, and (f) 2π/3.

Fig. 2. Phase distribution of AM beams, where (a), (c), and (e) are fourth order with ellipticity parameter q=1 and angular phase shift given by β parameters of π/4, π/2, and 2π/3, respectively. (b), (d), and (f) correspond to third-order AM beams with q=2 and β parameters of π/4, π/2, and 2π/3, respectively.

Fig. 3. Intensity of AM beams. (a) q=0.01, (b) q=2, (c) q=5, and (d) q=15. In all cases, from top to bottom, α=0, α=0.1, α=0.5, and α=1.

Fig. 4. Intensity of asymmetric Gaussian–Mathieu beams with the following parameters: (a) q=0.01, (b) q=2, (c) q=5, and (d) q=15. In all cases, from top to bottom, α=0, α=0.1, α=0.5, and α=1.

Fig. 5. Experimental setup. From left to right: He–Ne laser at 633 nm, 12 mW; BE, beam expander, 10×; SLM, spatial light modulator, LC2002; L1 and L2, lenses; D, diaphragm; CCD, Thorlabs CCD.

Fig. 6. Experimental fourth-order AM beam obtained with parameters (a) α=1, (b) α=1.67, (c) α=5, (d) α=1.14, (e) α=1.19, and (f) α=1.25. The size of the transverse display area is 5  mm×3  mm. (g) Propagation of the AM beam in the x–z plane.

Fig. 7. Power of asymmetric Gaussian–Mathieu beams as a function of the α parameter at different ellipticities given by the q parameter.

Fig. 8. OAM of asymmetric Gaussian–Mathieu beams as a function of the α parameter at different ellipticities given by the q parameter.