Material-Independent and Size-Independent Tractor Beams for Dipole Objects

A Bessel beam without an axial gradient can exert a pulling force on an object [A. Novitsky, C. W. Qiu, and H. Wang, Phys. Rev. Lett. 107, 203601 (2011)]. However, it cannot be called a “tractor beam” per se, as long as the light pulling effect is ultrasensitive to the object’s material and size, a perturbation of which will make the optical traction go away. In this Letter, we investigate and report on the universality for a Bessel beam to be either a material-independent or size-independent optical tractor beam within the dipolar regime. Moreover, a general condition for a nonparaxial laser to be simultaneously a material- and size-independent tractor beam is proposed. These universal pulling effects and conditions are discussed in association with insight on modified far-field scattering, scattering resonances, and induced polarizabilities. Interestingly, we find that the acoustic pulling force exhibits only size independence, owing to the acoustic scattering theory in contrast to the light scattering counterpart. The findings pave the way for the realistic engineering and application of universal tractor beams pulling a wide variety of objects.

Figure 1

Mie coefficients (a) $a_1$ and (b) $b_1$ for the nonmagnetic ($\mu =1$) spherical particle. On the right-hand side of the figures, the region of invalid dipole approximation ($|a_2|>0.2|a_1|$) is shaded. (c) Pulling force by a nonparaxial beam ($\alpha$ is large) versus pushing force for paraxial beam ($\alpha$ is small).

Figure 2

Density plots demonstrate the dipole force $⟨f_z⟩=⟨F_z⟩k^2/|c_1{|}^2<0$ calculated according to Eq. (3) for (a) transparent and (c) absorptive [$\epsilon =\mathrm{Re}(\epsilon )+0.1i$] nonmagnetic spherical beads. Region beyond the dipole approximation is shaded. Black curves in (a) and (c) depict exact the optical forceop $⟨f_z⟩$ at $\epsilon =6$ and $\epsilon =6+0.1i$, respectively, when higher moments have been taken into account. Numbers 1 and 2 indicate scattered far field for pushing ($k_{0}R=0.7$) and pulling ($k_{0}R=1.15$) optical forces, respectively. Black arrows show the forward scattering limited by the cone angle $\alpha =70°$. (b) Absolute value of the difference of the Mie coefficients $|a_1-b_1|$ demonstrates that the pulling force (shaded region) appears near ${\alpha }_e={\alpha }_m$. The nonparaxial Bessel beam has characteristics $m=1$, $\alpha =70°$, $c_2/c_1=i$.

Figure 3

The optical pulling force $⟨f_z⟩=⟨F_z⟩k^2/|c_1{|}^2$ for magnetic particles ($\mu =1.2$). Parameters: $m=1$, $\alpha =70°$, $c_2/c_1=i$. In the inset, the force $⟨f_z⟩$ vs $k_{0}R$ (beyond the dipole approximation) at $\epsilon =1.2$ is shown.

Figure 4

Dimensionless pulling acoustic force $Y_P<0$ in the dipole approximation versus density ratio $\lambda$ and dimensionless sphere radius $k_{0}R$ for fluid sound speed ratio $\sigma =0.861$ and Bessel beam’s cone angle $\alpha =70°$. Black curve in the inset corresponds to $Y_P$ for a benzene sphere in water ($\lambda =0.874$) [8]. Shaded region indicates a strong enough quadrupole term using $|d_2|/\sqrt{|d_0{d}_1|}>0.2$.