Fig. 1. Schematic of a focusing optical system. a is the radius of the particle, np is the refractive index of the particle, and nm is the refractive index of the ambient.

Fig. 2. (a) Intensity distribution for different values of M at the focal plane. (b) Two-dimensional intensity distribution for M=30.

Fig. 3. Effect of the beam index M on (a)–(c) Fscat (the scattering force), (d)–(f) Fgrad,t (the transverse gradient force). Fgrad,t=(Fgrad,x2+Fgrad,y2)1/2 while z1<0; Fgrad,t=−(Fgrad,x2+Fgrad,y2)1/2 while z1>0. (g) and (h) The effect of the beam index on Fgrad,z (the longitudinal gradient force). The parameters are chosen as y=0 and δ=0.1  mm.

Fig. 4. Effect of the coherence width δ on (a)–(c) Fscat, (d)–(f) Fgrad,t, and (g) and (h) Fgrad,z. The parameters are chosen as y=0, M=30.

Fig. 5. Effect of the elliptical ratio γ on (a)–(c) Fscat, (d)–(f) Fgrad,t, and (g) and (h) Fgrad,z. The parameters are chosen as y=0, M=30, δ=0.1  mm, and wy=0.1  mm.

Fig. 6. Two-dimensional force distribution of the REGBs near the focus point. (a1)–(a5) are the scattering force distributions. (b1)–(b5) are the transverse gradient force distributions. (c1)–(c5) are the longitudinal gradient forces distributions at positions z1=−0.3, −0.1, 0, 0.1, and 0.3 mm.

Fig. 7. When z1=0.1  μm, the radiation forces Fscatm, Fgrad,tm, Fgrad,zm produced by partially coherent multi-REGBs depend on (a) the initial beam index M with δ=0.1  mm and γ=6, (b) the coherence width δ with M=30 and γ=6, (c) the elliptical ratio γ with M=30 and δ=0.1  mm, and (d) the particle radius a with M=30, δ=0.1  mm, and γ=6. Fb is the Brownian force.