Fig. 1. (a) Electric field Ey of the incident pulse in the x–y plane at z=0 at t=14T. (b) Transverse intensity distribution of the incident beam averaged for the entire pulse at x=5λ [the black dotted line in (a)] at t=14T. (c) Electron density distribution at t=22T when the pulse arrives at the target surface. The black lines in (b) and (c) are the outlines of the 1/e maximum intensity of the incident beam and the hole of the target. (d) Frequency spectrum of the laser field after interaction between the input pulse and target. The field signal corresponds to y=2.5λ and z=0 at t=26T.

Fig. 2. Reflected transverse electric field distributions of the (a1) first, (b1) third, (c1) fifth, and (d1) seventh harmonics in the z–y plane at x=5λ and t=26T. Corresponding transverse intensity distributions of harmonics averaged on entire pulse are shown in (a2)–(d2).

Fig. 3. Relation between the mode αn of the nth harmonic and its harmonic order n for the fundamental frequency beam with α0=0.7. The gray dotted line is the scaling relation between nα0 and the harmonic order n.

Fig. 4. Superposition of (a) first, (b) third, (c) fifth, and (d) seventh harmonics for different integer modes.

Fig. 5. Fraunhofer diffraction patterns of first, third, fifth, and seventh order harmonics of fractional (first row) and integer (second row) vortex beams under different conditions. The illustrations in the bottom left of the first column are corresponding holes. (a1)–(d1) Fundamental fractional vortex beam with α0=0.7 diffracted by the hole shape of α0=0.7. (a2)–(d2) Fundamental integer vortex beam LG01 diffracted by the hole shape of LG01.