Self-Induced Backaction Optical Pulling Force

We achieve long-range and continuous optical pulling in a periodic photonic crystal background, which supports a unique Bloch mode with the self-collimation effect. Most interestingly, the pulling force reported here is mainly contributed by the intensity gradient force originating from the self-induced backaction of the object to the self-collimation mode. This force is sharply distinguished from the widely held conception of optical tractor beams based on the scattering force. Also, this pulling force is insensitive to the angle of incidence and can pull multiple objects simultaneously.

Figure 1

(a) Schematic of the optical pulling force enabled by the SC mode in a PC. The upper inset shows the intensity envelopes across the SC mode axis without (gray dashed line) and with (red solid line) the object. The green ellipse shows the object. (b) Detailed structure of the square lattice (lattice constant $a$) PC and the object. The parameters are $n_a=3.4$, $n_b=1.33$, $n_o=1.5$, $r=0.1a$, and semiaxes of $r_x$ and $r_y$. (c) The second band in the reduced Brillion zone for the TM mode (electric field perpendicular to the $xy$ plane). For $\omega =0.5(2\pi c/a)$, the equifrequency contour is square, which is the distinguishing characteristic of the SC mode. (d) SC mode propagation analysis when a light beam is incident on the PC. Here, ${\mathbf{k}}_{\text{in},r,t}$ (${\mathbf{v}}_{\text{in},r,t}$, ${\theta }_{\text{in},r,t}$) are the wave vectors (group velocities, angles) of the incident, reflected, and transmitted beams, respectively. (e) Distribution of $E_z(x,y)$ for the SC mode when excited by a Gaussian beam at normal incidence. The small circles show the position of the PC rods.

Figure 2

Illustrating how the total optical force $F_{x,\text{total}}$ changes with the shape of the object. $r_y=0.1\mu \mathrm{m}$ is fixed, and $r_x$ varies from 0.1 to $2.4\mu \mathrm{m}$. (a) Changes of $F_{x,\text{total}}$ vs $(x_o,r_x)$. The regions with positive forces are in white. (b) Optical force for $r_x=0.1\mu \mathrm{m}$ (black dashed curve) and $r_x=0.2\mu \mathrm{m}$ (blue curve). (c)–(e) show the same as (b) except for different $r_x$.

Figure 3

(a) The solid (blue) curve is the total optical force calculated by Eq. (1), and the dashed (red) curve shows the gradient force calculated by Eq. (3), at normal incidence. The parameters are $r_y=0.1\mu \mathrm{m}$ and $r_x=1.9\mu \mathrm{m}$, and $x_{o1}$ and $x_{o2}$ are two typical positions of the object center that are further analyzed in Fig. 4. (b) Variation of $F_x$ vs the incidence angles of ${\theta }_{\text{in}}=0°$, 6°, and 9°.

Figure 4

(a) Self-induced backaction of the object on the SC mode at normal incidence. The inset shows an enlarged image of the object, where the integration path (black and dashed lines) for the force calculation is also shown. The object is centered at $x_{o2}=1.9a$ [cf. Fig. 3]. All the other parameters are the same as those in Fig. 3. (b) Section of (a) along the SC beam axis $y=0$. The blue curve (solid) shows the SC intensity envelope. Since the object locates in the negative gradient region, it will be pulled by the negative gradient force towards the source. (c) Enlargement of the negative intensity profile of (b) around the object. (d) The same as (c) except for the object centered at $x_{o2}=2.4a$ [cf. Fig. 3].