Fig. 1. Schematic diagram of focusing

Fig. 2. Light field distribution of Airy beam under different υ. (a) υ=1; (b) υ=2; (c) υ=3; (d) υ=4; (e) υ=5; (f) υ=6; (g) υ=8; (h) υ=10; (i) υ=15; (j) υ=20

Fig. 3. Relationship between main lobe size of Airy beam and geometric phase modulation parameter

Fig. 4. Light field distribution of Airy beam under different β. (a) β=0.2; (b) β=0.5; (c) β=0.7; (d) β=1.0

Fig. 5. Airy beam energy flow diagram under different υ. (a) υ=1; (b) υ=3; (c) υ=5; (d) υ=8

Fig. 6. Phase-modulated Airy vortex beam focusing mode when l is integer order. (a) l=1; (b) l=2; (c) l=3; (d) l=4; (e) l=5; (f) l=6; (g) l=7; (h) l=8

Fig. 7. Phase distribution of modulated Airy vortex beam. (a) l=1; (b) l=2; (c) l=3; (d) l=4; (e) l=5; (f) l=6; (g) l=7; (h) l=8

Fig. 8. Phase-modulated Airy vortex beam focusing mode when l is fractional order. (a) l=-1.5; (b) l=-2.5; (c) l=-3.5; (d) l=-4.5; (e) l=1.5; (f) l=2.5; (g) l=3.5; (h) l=4.5