Trapping and self-assembly of particles by photonic chiral surface waves

Trapping and self-assembly of particles by a single photonic chiral surface wave (PCSW) in a waveguide is investigated in this work. Through rigorous calculations, stability analysis, and physical interpretation using response theory of optical force, we show that particles of arbitrary sizes, shapes, and dielectric constants can be stably trapped and assembled in the waveguide which is counterintuitive since a propagating surface wave usually exerts a pushing or pulling force on the particle and transports it. The ability of confinement is due to that the PCSW can navigate the particle and recover to its original state. Because of the unique propagating and scattering properties, there is no interaction among the particles when the particles are far away enough. Thus, the particles are independently trapped during the self-assembly process. This work provides a different approach to manipulate small particles by using the PCSW.

Figure 1

(a), (b) Stereogram schematic and top view of the waveguide that supports a chiral surface mode. The waveguide is formed by an air gap sandwiched between a magneto-optical PC (red) and an ordinary PC (blue). The lattice constants of both PCs are $a$, and the ambient is air. The gray lines denote two thin barriers to prevent the particles from entering into the PCs, the cylinders are infinite along $z$ direction. (c), (d) Band diagrams of the two PCs for TM polarization. The band gap between the third and fourth bands of the ordinary PC and the band gap between the second and third bands of the magneto-optical PC are shaded in yellow, and the Chern numbers of the bands near the band gaps are indicated. (e) Projection bands (blue) of the PCs and the surface band (red) at the interface (the lower is the magneto-optical PC) between the two PCs. The black dot denotes the chiral surface mode that is excited.

Figure 2

The Poynting vectors of a PCSW propagating in the waveguide without (a) and with (b) a particle located in the waveguide. The radius and refractive index of the particle are $r=0.15a$ and $n=4.0$, respectively. The location of the particle is $x=19a$ and $y=0.0973a$. (c), (d) Enlarged view of the rectangle areas in (a) and (b), respectively.

Figure 3

Longitudinal (a) and transversal (b) optical forces acting on the dielectric cylinder as a function of its location ($x,y$) within a rectangular region as marked by the black dotted rectangle in Fig. 1. The radius and refractive index of the cylinder are $r=0.15a$ and $n=2$, respectively. (c), (d) Divergence and curl of the optical force. (e), (f) Zoomed view of the rectangular area in (c) and (d), respectively. The stars denote the equilibrium position.

Figure 4

(a), (b) Optical forces calculated by using the MST (blue solid line) and RTOF (red circles) along an axial segment. The $y$ coordinates of the cylinders in (a) and (b) are fixed at $y=-0.11684a$ and $y=0.0973a$, respectively. The radii and refractive indices of the cylinders are (a) $r=0.12a, n=4$ and (b) $r=0.15a, n=4$, respectively. (c), (d) Phase responses corresponding to (a) and (b), respectively.

Figure 5

Equilibrium positions of the cylinder as functions of its refractive index [(a) and (b)] and its radius [(c) and (d)]. The blue circles denote that the cylinder is trapped at the position $x=Na$, while the orange hexagons represent that the cylinder is trapped at the position $x=\left(N+0.5\right)a$, where $N$ is an integer. The radii in (a) and (b) are $r=0.12a$ and $r=0.15a$, respectively, and the refractive indexes in (c) and (d) are $n=1.5$ and $n=4.0$, respectively. The red stars in (a) and (b) denotes the $y$ coordinates used in Figs. 3 and 3, respectively. The red stars in (b) and (d) are for the same particle. (e), (f) Distribution of electric fields and magnetic fields.

Figure 6

Eigenvalues of the force constant matrices of the equilibrium positions in Fig. 5.

Figure 7

The optical forces acting on two identical cylinders vs their separations. The blue circles (red squares) denote the optical forces acting on the left (right) cylinder. The longitudinal and transversal optical forces are shown in the upper and lower rows, respectively. In each column, the radii, refractive index, and $y$ coordinates of the two cylinders are fixed. From left to right, $r=0.12a, n=1.6, y=-0.1497a$; $r=0.12a, n=4, y=-0.1168a$; $r=0.15a, n=3, =-0.1105a$; and $r=0.15a, n=4.2, y=0.0642a$. (The amplitude of the source was not given).

Figure 8

The optical forces acting on three identical cylinders. (a) and (b) denote $f_x$ and $f_y$, respectively. The first and the third cylinders are fixed with a separation between them of $30a$, while the second cylinder is in the middle of them and moved from left to right with a step of $a$. $\mathrm{\Delta }x$ denotes the distance between the first and the second cylinders. The parameters of the three cylinders are the same as those in Fig. 5.

Figure 9

The stable equilibrium positions of dielectric cylinders with square and triangular shapes as functions of refractive index. The side lengths of the squares and equilateral triangles are both 0.$2a$.

Figure 10

Temperature distribution near the cylinder.

Figure 11

(a) Schematics of the structure with the particle at its original position (gray disk) and the mirror-reflected position (dashed circle). The operations are conducted in two steps: from the original system (b) to its reciprocal pair (c), and to (e) the system after a mirror operation. “+” and “−” signs represent the direction of the biased magnetic field along $z$ direction.