Tunable transverse optical force via nonreciprocity

The transverse optical force adds an extra degree of freedom for optical manipulation. In this paper, based on the full wave simulation, we find that an achiral sphere can experience a transverse optical force in the field of a circularly polarized or linearly diagonally polarized electric dipole source in the vicinity of a doped InSb substrate. When the dipole is circularly polarized, the transverse force results from the asymmetrical distribution of the excited surface waves. However, when the dipole is linearly diagonally polarized, the transverse force stems from the combined effect of the surface waves and asymmetrical propagating waves. More importantly, the magnitude and direction of the transverse force can be adjusted by applying a magnetic field parallel to the InSb substrate, due to the generated additional nonreciprocity. Our paper expands the scope of transverse optical force by nonreciprocity.

Figure 1

(a) Schematic plot illustrating a manipulated polystyrene sphere in the optical field of an electric dipole source near a doped InSb [26, 27] substrate. (b) The real part of InSb's dielectric tensor components vs the wavelength at a magnetic field of $B=1\text{T}$. Refer to the inset for the specific form of the dielectric tensor [28]. The shaded area indicates the approximate locations of SPhPs.

Figure 2

The absolute values of the angular spectrum for the $p$-polarized component along $k_x$ ($k_y=0$) are displayed for an electric dipole source with circular polarization (a) and diagonal linear polarization (b). The insets illustrate the amplitude of the electric field for the dipole source radiating in free space. (c, d) Electric field induced by an electric dipole source in the presence of a doped InSb substrate with the permittivity described by the Drude model [26, 28]. The dipole source is positioned at $z_0=0.5\mu \mathrm{m}$ above the substrate surface. The source displays circular and diagonal linear polarization in (c) and (d), respectively. The color map and white arrows signify the amplitude and direction of the electric field. The dipole sources used in (c) and (d) and the insets in (a) and (b) have a radiation wavelength of $50\mu \mathrm{m}$, with a dipole strength of $2.544\times {10}^{-14}\mathrm{C}\mathrm{m}$. The color bars for (c) and (d) and for the insets in (a) and (b) are unified, as shown in (c).

Figure 3

Transverse optical force exerted on a polystyrene sphere [see the system depicted in Fig. 1]. (a), (b) Electric field, induced by an electric dipole source in the presence of a doped InSb substrate, plotted on the $xy$ plane with $z=15\mu \mathrm{m}$. The dipole source is positioned at $z_0=0.5\mu \mathrm{m}$ above the substrate surface, with a radiation wavelength of $50\mu \mathrm{m}$ and a strength of $2.544\times {10}^{-14}\mathrm{C}\mathrm{m}$. The source carries circular and diagonal linear polarization in (a) and (b), respectively. The color map denotes the electric field's amplitude. The white arrows in (b) represent the $x$ component of the complex Poynting vector's imaginary part within the considered region on the $y$ axis ($x=0$, $z=15\mu \mathrm{m}$). (c), (d) Instantaneous distribution of the real electric field propagating along the $y$ axis ($x=0$) in the case of an electric dipole $\mathbf{p}=(1,0,1)$ without and with a doped InSb substrate. The purple arrows represent the projection of the electric field on the $xz$ plane. (e), (f) Transverse optical force when a polystyrene sphere is located on the $y$ axis ($x=0$, $z=15\mu \mathrm{m}$) in the optical fields of (a) and (b). The polystyrene sphere has a permittivity of 2.52 and a radius of $10\mu \mathrm{m}$.

Figure 4

(a) The dispersion of SPhPs vs $k_x$ with $k_y=0$ supported by the doped InSb substrate. The solid orange line depicts the symmetric dispersion of SPhPs with zero magnetic field, while the dashed red and dotted blue lines show the asymmetric dispersion of SPhPs with magnetic fields of $B=1$ and $-1\text{T}$, respectively. The horizontal gray dashed line corresponds to the frequency used in this paper. (b, c) The electric field generated by an electric dipole source in the presence of an InSb substrate plotted on the $xy$ plane at $z=15\mu \mathrm{m}$ when a magnetic field of various values is applied along the $y$ axis (the Voigt configuration). The dipole source is situated $0.5\mu \mathrm{m}$ above the substrate surface, with a radiation wavelength of $50\mu \mathrm{m}$ and a strength of $2.544\times {10}^{-14}\mathrm{C}\mathrm{m}$. The electric dipole source carries circular and diagonal linear polarization in (b) and (c), respectively. The color map represents the amplitude of the electric field. (d), (e) Transverse optical forces vs the magnetic field. The polystyrene sphere with a permittivity of 2.52 and a radius of $10\mu \mathrm{m}$ is located at $(x/\lambda ,y/\lambda ,z/\lambda )=(0,2.7,0.3)$, as shown by the white dashed circles in (b) and (c).

Figure 5

(a) The ratio of the transverse force to the longitudinal force ($F_x/F_y$) vs the polystyrene sphere's position on the $y$ axis ($x=0,z=15\mu \mathrm{m}$) for both circularly polarized and linearly diagonally polarized dipole source cases. (b) The longitudinal force $F_y$ and (c) the optical force $F_z$ perpendicular to the substrate, exerted on a polystyrene sphere vs the spheres' position on the $y$ axis ($x=0,z=15\mu \mathrm{m}$). (d) The transverse optical force exerted on a polystyrene sphere located at $(x/\lambda ,y/\lambda ,z/\lambda )=(0,2.7,0.3)$ vs the distance between the dipole source and substrate surface. (e, f) $F_x$ and $F_z$ exerted on a polystyrene sphere located at $(x/\lambda ,y/\lambda )=(0,2.7)$ vs the spheres' radius. The polystyrene sphere in (e) and (f) is positioned $5\mu \mathrm{m}$ away from the $z=0$ plane at their base. The polystyrene sphere (with a permittivity of 2.52) has a radius of $10\mu \mathrm{m}$ in (a)–(d). The dipole source, with a radiation wavelength of $50\mu \mathrm{m}$, is positioned at $z_0=0.5\mu \mathrm{m}$ above the substrate surface except in (d). In (a), (d), and (e), the results in the cases of circularly polarized and linearly diagonally polarized dipole sources are presented in the left and right vertical axes, respectively.

Figure 6

(a) Transverse optical forces exerted on a silicon sphere (with a permittivity of 12.25 and a radius of $10\mu \mathrm{m}$) located at $(x/\lambda ,y/\lambda ,z/\lambda )=(0,2.7,0.3)$ vs the applied magnetic field. (b) The $z$ component of the optical force exerted on a silicon sphere located at $(x/\lambda ,y/\lambda )=(0,2.7)$ vs the spheres' radius. The dipole source, with a radiation wavelength of $50\mu \mathrm{m}$, is positioned at $z_0=0.5\mu \mathrm{m}$ above the substrate surface in (a) and (b). (c) Transverse optical forces exerted on either a silicon or polystyrene sphere vs the sphere's radius. (d) The $z$ component of optical forces (pure gradient force) exerted on either a silicon or polystyrene sphere vs the sphere's radius. The incident light in (c) and (d) is a linearly diagonally polarized evanescent wave, which propagates along the $y$ direction and decays in the $z>0$ half space. The evanescent wave's exponential decay rate is set to $0.6k$, with a wavelength of $50\mu \mathrm{m}$, and the intensity is $1\text{mW}/\mu {\mathrm{m}}^2$ at the $z=0$ plane. Both the silicon sphere and the polystyrene sphere in (b), (c), and (d) are situated $5\mu \mathrm{m}$ away from the $z=0$ plane at their base.