Fig. 1. Wavefront intensity distributions of 2D butterfly-shaped beams. (a) Bu(0,0,Z,W); (b) Bu(X,0,Z,0)

Fig. 2. Caustics of 3D butterfly-shaped beams. (a) Bu(-5<a1<5,0,Z,W); (b) Bu(X,0,Z,-5<a4<5)

Fig. 3. Initial light field distributions of butterfly-shaped beams. (a1)-(a3) Bu(X,Y,a3,a4) when a4=0 and a3=-3, 0, 3; (b1)-(b3) Bu(X,a2,Z,a4) when a4=0 and a2=-4, 0, 4

Fig. 4. 3D caustic surfaces of butterfly-shaped beams. (a) 3D caustic surface of Bu(X,Y,a3,a4) when a4=0 and -3<a3<3; (b) 3D caustic surface of Bu(X,a2,Z,a4) when a4=0 and -4<a2<4

Fig. 5. Experimental setup

Fig. 6. Light field intensity distributions of Bu(X,Y,a3=0,a4=0,ξ). (a1)-(c1) Simulated butterfly-shaped light field intensity distributions for planes of ξ=0,2,4; (a2)-(c2) experimental intensity distributions for planes with ξb of 0 nm, 400 nm, and 800 nm

Fig. 7. Caustics of Bu(X,Y,a3,a4,ξ) during propagation

Fig. 8. Spectrum distributions. (a1)-(c1) Numerical spectrum intensity distributions of Bu(X,Y,a3,a4), Bu(X,a2,a3,W), and Bu(a1,Y,Z,a4); (a2)-(c2) numerical spectrum phase distributions corresponding to Figs. 8(a1)-8(c1); (a3)-(c3) experimental results corresponding to Figs. 8(a1)-8(c1)