Fig. 1. Schematics of the method for generating (a1)–(c1) even and (a2)–(c2) helical Mathieu beams. (a1) and (a2) Generation of ring-shaped Fourier spectra through Fourier transformation of a Gaussian beam imposed with an MPP. (b1) and (b2) Generation of a Mathieu beam through Fourier transformation of Fourier spectra after the filter. (c1) and (c2) Simulation of intensity distribution along the propagation axis.

Fig. 2. (a) and (b) Femtosecond laser platform for TPP. (c) Phase distribution of designed even MPP with m=2 and q=12. (d) Picture of fabricated MPPs on the glass. (e) SEM image of the same MPP. Height profiles of (f) designed MPP and (g) fabricated MPP.

Fig. 3. Helical MPP with m=3 and q=2. (a) Designed phase distribution, (b) SEM image, and (c) height profile of the fabricated phase plate.

Fig. 4. (a) Phase distribution of even MPP with phase modulation error. (b) Phase distribution of helical MPP with discretized phase values of 31 levels. The overlap integral and efficiency of (c) even and (d) helical Mathieu beams at different phase modulations. Marks denote the value when the phase value is discretized into 31 levels. The bars on the marks show the deviation of the value caused by levels of phase value discretization.

Fig. 5. Experimental setup for the generation and characterization of MG beams. BE, beam expander; L1 and L2, lenses; RS, ring slit; M, mirror; and RAP, right-angle prism.

Fig. 6. Measured Fourier spectra of MG beams with different orders m and ellipticity parameters q.

Fig. 7. Measured intensity distribution of even MG beams with different orders m and ellipticity parameters q.

Fig. 8. Experimental results of even MG beams with m=2 and q=12. (a) Intensity distribution at different propagation distances. (b) Axial intensity distribution along the propagation.

Fig. 9. Measured intensity distribution of helical MG beams with different m and q.

Fig. 10. Experimental results of helical MG beams with m=3 and q=2. (a) Intensity distribution at different propagation distances. (b) Axial intensity distribution along the propagation.