Abstract
1. INTRODUCTION
Optical tweezers (OTs), consisting of a tightly focused beam [1], give a contactless way to trap and accurately position small particles. Such ability is attributed to the intensity gradient in the focused field, which induces a restoring force for confining the particles in three dimensions (3D) [2–5]. Nowadays, the OT has become a widely valuable tool in many fields, especially in biophysics [6], and it also provides a typical platform for conducting rigorous and specific phenomena in both classical and quantum light–matter interactions [7–12]. On the other hand, when carrying angular momentum, light can exert torques on the particles, driving them to rotate continuously [13–16]. The tool capable of particle rotation is known as the optical spanner (OS) [17–21], comprising a donut shape with a helical wave-front, and it gives an additional degree of freedom for manipulating objects. In fact, while the OS is most known for its ability in rotating particles, it can also be used to trap particles when the intensity gradient force is repulsive [22,23].
The manipulation of particles with light requires a careful shaping of light itself, and since the OT and OS are based on dramatically different fields, they generally rely on distinct devices for wave-front modulation. To achieve the aforementioned functions with stable trapping and durable spanning, traditionally the beam in the OT is focused by high-numerical-aperture (NA) objectives into a diffraction-limited spot [1,2], while the vortex beam in the OS is usually generated with spatial light modulators [14]. Nevertheless, these relatively bulky devices make it difficult to integrate the OT and OS together, especially in small systems, such as optical fiber and optofluidic platforms.
During the last decade, metasurfaces composed of an array of subwavelength nanostructures have exhibited exceptional strength in the full control over the polarization, amplitude, and phase of light [24–27], and the tremendous booming of applications in nanoscale has been reported [28–33]. A conventional metasurface is primarily dependent on either the geometric phase (i.e., Pancharatnam–Berry phase) or the dynamic phase by means of rotating the orientation angle relative to the reference axis of meta-atoms and the geometry of several specific nanostructures, separately [24,26,27]. Instead, the latest works have revealed that the syntheses of two types of these phases possess the superiority of multiplexing two functionalities into a single metasurface [33–37], which offers an unprecedented opportunity to set up OT and OS with a compact and miniaturized device. Up to now, while the metasurface-based OT or OS has been severally implemented through unitary light field generation [38–43], a structure enabling both the OT and OS is yet to be reported. In this paper, we introduce a versatile metasurface, which integrates the functionality of both OT and OS onto an individual single-layer device. We realize this OT–OS metasurface by using the interplay of the geometric and the dynamic phases, by which the metasurface holds two independent optical responses. As a result, dual focal planes are yielded for the output field, and the focal lengths and topological charges can be tailored at will.
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2. METASURFACE IMPLEMENTATION
Figure 1(a) schematically illustrates the OT–OS metasurface device, excited by a linearly polarized plane wave of wavelength . The output field is featured by sharp intensity gradient and helical phase structures, which are separated coaxially, capable of particle trapping and rotation, respectively. To see how the output field is produced, let us consider an optical element described by a Jones matrix and normally illuminated by a plane wave (defined as the vector ket . When the light field passing through the element, the output field becomes . In this process, two independent phase profiles, and , can be generated in synergy with the polarization of incident field . Assuming that the polarization bases of interest are and with , one may express the spatially varying Jones matrix as
Figure 1.(a) Schematic diagram of the polarization-dependent OT–OS metasurface. (b), (c) Typical unit cell of the OT–OS metasurface with period
The unitary matrix of Eq. (3) can be diagonalized by solving the characteristic equation, and the basis of eigenvalues and eigenvectors of can be obtained (See Appendix A). It is found that an anisotropic meta-atom can be considered as a birefringent waveguide [24] because of the high index contrast between the nanobrick and the air, while the phase shifts (, ) along the two perpendicular symmetry axes and the specific orientation angle along the plane can satisfy the optical element Jones matrix . The calculated relation indicates that and the difference between the phase shifts, , can be analytically expressed by
It is worth noting that Eq. (4) is attributed to the accumulations of the dynamic phase, which needs to build a phase library of nanobricks by varying the geometry of the nanobrick, while Eq. (5) is controlled only by the rotation of the orientation angle . Through the combination of the dynamic phase and the geometric phase, when the linear plane wave passes through a nanobrick, the LCP and RCP states can be transformed into their orthogonal polarization states and hold the same dynamic phase and opposite geometric phase to realize independent optical manipulation.
3. OT–OS METASURFACE DEVICE REALIZATION
Based on the theory mentioned above, the specifically designed unit cell, composed of a fused substrate (refractive index: ), on which () nanobricks are periodically arranged with fixed lattice constant and height , is shown in Figs. 1(b) and 1(c). Finite-difference time domain (FDTD) simulations are performed to calculate transmittance by using the commercial software package “FDTD Solutions” (Lumerical Inc.). Periodic boundary conditions are applied along the and axes, and the perfectly matched layers (PMLs) are applied to the direction. The range of lengths () and widths () of the nanofins covers 50 to 400 nm, for 5 nm increments of each geometric variable, and the results of continuous phase shifts and along the and axes are approximated into twelve discrete phase levels, high transmittances, and polarization conversion efficiencies, as shown in Figs. 1(d) and 1(e). The detailed parameters of each nanobrick can be found in Appendix B.
According to this discussion, the polarization-sensitive metasurface has two independent optical responses and, hence, holds the possibility for integrating the OT and OS. To implement this device, the focused phase formula must be introduced as follows:
Figure 2(a) illustrates the top, zoomed, and angled views of the metasuface configuration. The radius of the entire metasurface is , and there are 918 nanobricks used in the full-metasurface design. Figure 2(b) shows the output field distribution, for a metasurface designed by setting and two focal lengths, and . Two planes of local intensity maxima can be identified at and 9.5 μm, in agreement with the theory. Specially, at , the field is focused into a spot with a waist radius about 0.285 μm [Fig. 2(c)], corresponding to an NA of . On the other hand, the field around constructs the OS, characterized by ring-like intensity and helical phase distributions [Fig. 2(d)].
Figure 2.Simulation for the OT–OS metasurface. (a) The entire OT–OS metasurface configuration with top, zoomed, and angled 3D perspective view. (b) Axial section view of the intensity distribution. The dashed lines indicate the focal plane for the function of OT and OS. (c), (d) Transverse section views of the intensity (left column) and phase (right column) distributions at the focal plane of the OS and OT, respectively. The phase profiles are responsible for the transverse fields. White arrows indicate the polarization state.
For a small particle that can be approximated as an electric dipole, its dynamic behavior in the optical field can be predicted by the dipole model, in which the component () of the optical force is given by [3]
To understand how the particle rotation can be achieved by the OS, one may write the complex electric field as , where and represent the amplitude and phase, respectively. Then, the radiation pressure can be expressed as
To verify the manipulation ability of the OT–OS metasurface, numerical calculations were performed by using a polystyrene (PS) nanosphere (radius: ; ) as a probe. The optical force is computed rigorously by Maxwell stress tensor method [15,44–46]:
Figure 3 shows the calculated force exerted on the particle placed at different positions on the plane. When the particle is close to the tight focusing spot, 3D trapping arises at and , as indicated by Figs. 3(a) and 3(b). The transverse and longitudinal trapping stiffnesses [47] are estimated to be and , respectively. We also notice in Fig. 3(b) the appearance of a small -component force when the particle deviates from the equilibrium position. This should be caused by Belinfante’s spin current due to spin inhomogeneity [44,48–50]. By contrast, when the particle is located around the OS, the zero points of the - and -component forces occur around and , as shown by Figs. 3(c) and 3(d). However, at this position, is shown to be nonzero, and the direction of the force agrees with the phase gradient indicated by Fig. 2(c). These indicate an off-axis trapping, along with an anticlockwise rotation of the particle about the optics axis.
Figure 3.Calculated optical forces on a PS particle located in the vicinity of (a), (c) the focusing spot and (b), (d) the vortex field. Longitudinal force as a function of the
4. DISCUSSION AND CONCLUSION
We have shown the simultaneous creation of the OT and OS, with a longitudinal separation distance of and the topological charge of the OS being . Next, we shall explore the possibility of changing and . Since the polarization-dependent design principle provides a different phase profile with independent control, the focal length of the output field can be adjustable. Figure 4(a), as an example, demonstrates an enhanced OT-to-OS distance, . The OT focal plane remains at . While the position of the OS focal plane is increased to , the helical phase structure is well maintained. The locations of the OT and OS can even be interchanged with , as shown in Fig. 4(b). Except for the focal lengths, the rotating orbit of the spanner can be tuned as well. Figures 4(c) and 4(d) illustrate an integration of the OT with OS of topological charges and 3, respectively. However, it is noted that the local intensity of the OS will decrease with increasing .
Figure 4.Simulated field distributions for (a)
In summary, we have demonstrated the possibility of assembling the OTs and OS onto one identical single-layer metasurface. The mechanism is the coexisting geometric and dynamic phases, which confer two kinds of polarization-dependent optical responses upon the metasurface. In addition, since the two optical responses are independent of each other, the focal planes of the tweezers and spanner are adjustable, and the orbit of the spanner can be flexibly tailored. With the merits of miniaturization, integration, and tunability, the dual phase-controlled metasurface is expected to serve in the next generation of optical manipulation devices. We also anticipate that the emerging field of the metasurface could inspire new ideas for achieving exotic optical forces [51–55].
Acknowledgment
Acknowledgment. The authors are grateful for the stimulating discussions with Professor Shaohui Yan (XIOPM).
APPENDIX A: DERIVATION OF THE JONES MATRIX J(X,Y) AND ITS EIGENVALUES AND EIGENVECTORS
As we discussed in Section?
By solving the characteristic equations of Jones matrix in Eq.?(
Therefore, the Jones matrix can be decomposed into canonical form , where is a diagonal matrix, and is an invertible matrix. We can write the Jones matrix for the realization of two optical responses as
Because can be considered as the rotation matrix of , the phase shift and along two symmetry axes and the rotation angle can be written as the following expression:
Equations?(
APPENDIX B: PARAMETERS OF SELECTED NANOBRICKS
According to the calculation, the transmittances ( and ) and phase shifts ( and ) as a function of the nanobrick size parameters, and , covering 50–400?nm, are perfomed by the FDTD method, as shown in Fig.?
Figure 5.Transmittances (
Detailed Parameters of the Selected Nanobricks Including Length 220 115 0.09 0.61 245 115 0.17 0.64 250 125 0.25 0.78 275 125 0.35 0.80 280 135 0.42 0.89 300 140 0.50 0.97 110 240 0.59 0.12 120 235 0.69 0.17 125 245 0.78 0.23 130 275 0.84 0.38 140 260 0.94 0.38 145 275 0.99 0.47
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