Asymmetric longitudinal optical binding force between two identical dielectric particles with electric and magnetic dipolar responses

In general, the optical binding force between identical particles is thought to be symmetric. However, we demonstrate analytically a counterintuitively asymmetric longitudinal optical binding force between two identical electric and magnetic dipolar dielectric particles. This homodimer is confined in two counterpropagating incoherent plane waves along the dimer's axis. The force consists of the electric dipolar, magnetic dipolar, and electric-magnetic dipolar coupling interactions. The combined effect of these interactions is markedly different than the expected behavior in the Rayleigh approximation. The asymmetric force is a result of the asymmetric forward and backward scattering of the particles due to the dipolar hybridization and coupling interactions. Consequently, it leads to a harmonic driving force on the pair, which decays with the interparticle distance to the first power. We show the rich nonequilibrium dynamics of the dimer and of the two particles impelled by the driving and binding forces and discuss the ranges of particle refractive index and size in which the asymmetric binding force arises. Our results open perspectives for nonequilibrium light-driven multiparticle transport and self-assembly.

Figure 1

Longitudinal optical binding between two identical electric and magnetic dipolar dielectric nanospheres with separation ($R$) immersed in air and illuminated by two incoherent counterpropagating plane waves. (a) Two $p$-polarized plane waves with electric ${\mathbf{E}}_0$ (magnetic ${\mathbf{H}}_0$) fields along the $z$ ($x$) axis propagate along the $y$ axis in Cartesian system (O-xyz) as shown by the wave vector k. $p^{\mathrm{j}}$ ($m^{\mathrm{j}}$) denotes the induced ED (MD) moment in the two particles $A$ and $B$ along the $z$ ($x$) axis. (b) The real parts of the dimensionless $p^{\mathrm{A}}$ (blue solid curve), $p^{\mathrm{B}}$ (blue double dot-dashed curve), $m^{\mathrm{A}}$ (red dashed curve), and $m^{\mathrm{B}}$ (red dotted curve) as functions of $R$ (bottom horizontal axis) and $\mathrm{\Delta }\phi =kR$ (top horizontal axis).

Figure 2

LOBF $F^{\mathrm{B}}$ (black solid curve) on polystyrene ($n_{\mathrm{p}}=1.59$) sphere B (Fig. 1) with radius $a=130\mathrm{nm}$ as well as its electric dipolar $F_{\mathrm{e}}$ (blue dashed curve), magnetic dipolar $F_{\mathrm{m}}$ (red dash-dot-dotted curve), and electric-magnetic dipolar coupling $F_{\mathrm{em}}$ (green dash-dotted curve) components as a function of separation ($R$). $F_{\mathrm{Ray}}$ represents the classical LOBF (magenta short dashed curve) in Rayleigh approximation.

Figure 3

LOBFs $F^{\mathrm{A}}$ (red dashed curve) and $F^{\mathrm{B}}$ (blue short dashed curve) on polystyrene particles $A$ and $B$ with radius $a=130\mathrm{nm}$ (a), 163 nm (b), and 172 nm (c) as well as the driving force $F_{\mathrm{driv}}$ (black solid curve) on the center of the dimer as a function of separation ($R$). For 130-nm particle in subplot (a), the electric and magnetic components $F_{\mathrm{driv}}^{\mathrm{e}}+F_{\mathrm{driv}}^{\mathrm{m}}$ (purple solid curve with square) as well as the coupling component $F_{\mathrm{driv}}^{\mathrm{em}}$ (orange solid curve with triangle) of $F_{\mathrm{driv}}$ are presented. The black solid (dashed) arrows denote the stable (unstable) equilibrium positions of $F_{\mathrm{driv}}$ while the green counterparts denote the stable and unstable separations between the two particles. (d) The four far-field scattering patterns of the homodimer in $x\text{−}y$ (blue solid curves) and $y\text{−}z$ (red dashed curves) planes (see Fig. 1) correspond respectively to the dimer with different radii and separations.

Figure 4

Dependence of the driving force ($F_{\mathrm{driv}}$) on the center of two polystyrene particles vs radius ($a$) and separation ($R$). The color denotes the magnitude of $F_{\mathrm{driv}}$. The black (white) curves represent the stable (unstable) equilibrium positions of $F_{\mathrm{driv}}$. The green (magenta) curves represent the stable (unstable) equilibrium separation of the two particles. The intersections of the white and black lines are located at $a=163\mathrm{nm}$. Inside the dotted rectangle at the bottom of the figure, the black solid circles represent the particles. The thick blue arrows denote the moving direction along the $y$ or $–y$ axis (blue long directional ray) of the center of the dimer. The thin red arrows show the relative motion (attraction or repulsion) between the two particles in areas I–IV.

Figure 5

Dependence of the driving force ($F_{\mathrm{driv}}$) on the center of two silicon ($n_{\mathrm{p}}=3.5$) particles vs radius ($a$) and separation ($R$). The meanings of the black, white, green, and magenta curves as well as the areas I–IV are same as the counterparts in Fig. 4.

Figure 6

LOBFs $F^{\mathrm{A}}$ (red dashed curve) and $F^{\mathrm{B}}$ (blue short dashed curve) on two silicon particles with radius $a=65\mathrm{nm}$ (a), 72 nm (b), 85 nm (c), and 99 nm (d) as well as the driving force $F_{\mathrm{divi}}$ (black solid curve) on the center of the dimer as a function of the separation ($R$). The black solid (dashed) arrows denote the stable (unstable) equilibrium positions of $F_{\mathrm{driv}}$ while the green solid (dashed) arrows denote the stable (unstable) separations between the two particles.

Figure 7

The real parts of the dimensionless $p^{\mathrm{A}}, p^{\mathrm{B}}, m^{\mathrm{A}}$, and $m^{\mathrm{B}}$ in a dielectric homodimer as function of radius $a$ (a) and refractive index $n_{\mathrm{p}}$ (b). The separation is fixed at $R=1.2\lambda$ and the parameters of the incident wave are the same as those in Fig. 2 while $n_{\mathrm{p}}=1.59$ (a) and $a=100\mathrm{nm}$ (b).

Figure 8

The scattering coefficients $Q_{\mathrm{sca}}$ (green dash-dotted curve) including its electric and magnetic multipolar components of single polystyrene (a) and silicon (b) particles as a function of the particle radius ($a$). The thick black solid (dashed) curve denotes the electric (magnetic) dipolar component. The red solid (dashed) denotes the electric (magnetic) quadrupolar component. The thin blue solid (dashed) denotes the electric (magnetic) octupolar component.

Figure 9

Dependence of $F_{\mathrm{Ray}}$ between two polystyrene particles (a) and two silicon particles (b) vs radius ($a$) and separation ($R$). The color denotes the magnitude of the force. The green and magenta curves represent respectively the stable and unstable equilibrium positions of $F_{\mathrm{Ray}}$.

Figure 10

The far-field scattering patterns of a polystyrene homodimer with $a=225\mathrm{nm}$ and $R=1.2\lambda$ including the electric and magnetic quadrupoles (a) and a silicon homodimer with $a=105\mathrm{nm}$ and $R=1.2\lambda$ including a dominant magnetic quadrupole (b) in $x\text{−}y$ (blue solid curves) and $y\text{−}z$ (red dashed curves) planes.