Exploiting extraordinary topological optical forces at bound states in the continuum

Abstract

Polarization singularities and topological vortices in photonic crystal slabs centered at bound states in the continuum (BICs) can be attributed to zero amplitude of polarization vectors. We show that such topological features are also observed in optical forces within the vicinity of BIC, around which the force vectors wind in the momentum space. The topological force carries force topological charge and can be used for trapping and repelling nanoparticles. By tailoring asymmetry of the photonic crystal slab, topological force will contain spinning behavior and shifted force zeros, which can lead to three-dimensional asymmetric trapping. Several off-Γ BICs generate multiple force zeros with various force distribution patterns. Our findings introduce the concepts of topology to optical force around BICs and create opportunities to realize optical force vortices and enhanced reversible forces for manipulating nanoparticles and fluid flow.

Topology-enabled optical forces around the bound states in the continuum are unveiled for unique optical manipulation.

INTRODUCTION

Bound states in the continuum (BICs) in photonic crystal slabs (PhCSs) are known to reveal topological features in the far-field patterns of radiating polarization states (1–5). The polarization vectors will wind around the singular point induced by zero outgoing power, which can be characterized by topological charge of the polarization vortex (1, 6–10). PhCSs supporting BICs have been used to realize ultralow threshold lasing action, and, by taking advantage of their topological nature, optical vortex beams can be generated in momentum space without the need to align preciously to a real-space center (11–15). In view of polarization singularities, breaking the in-plane inversion symmetry of the PhCS will result in splitting of the vortex point at BIC to two circular points and linear polarizations that are able to achieve full coverage of the Poincare sphere (16, 17). Dynamics and evolution of the polarization singularities including spawning circular polarizations and annihilation of topological charge have further been studied by modifying in-plane symmetry and σ_z symmetry of PhCSs (17–19).

Besides the control of polarization state, amplitude, and phase of waves, PhCSs and metasurfaces have also been great candidates for exploiting optical forces, such as optical trapping, manipulation, and meta-vehicles (20–27). By tailoring the scattering direction and structural chirality, optically pumped vehicles, light sails, and motors for transportation of nanoparticles and space exploration have been proposed recently (20, 21, 26, 28). PhCS, particularly the presence of etched holes on the slab, are remarkably suitable for optical trapping (29, 30). Exploiting quasi-BICs to enhance further the electromagnetic field localization markedly increases the optical force perceived by the particles (31, 32).

Here, we introduce the idea of zero optical force by exploiting the analogy between topological polarization singularities and topological zero power radiation, thus defining a force singular point at which the optical force goes to zero. To demonstrate the existence of such force singular point, we studied the behavior of optical forces in the vicinity of BICs, especially the topology of the optical force distribution in the momentum space, and found similar behavior as the one prescribed by the polarization vectors winding around the polarization singular point. We thus exploit the momentum space BIC nonradiative property to produce momentum-dependent vanishing optical forces. By looking in more details on the topological nature of optical force at BIC, we found diverse patterns, including converging and radiating topology for trapping and repelling nanoparticles. We show that the force vectors can also wind around force zero and, thus, can be quantized by similar topological charge of force, which implies the robustness of the topological force against change of the structure’s and targeted objects’ parameters. With the introduction of displacement and twist in the PhCS structure, topological force can reveal various distribution patterns, such as the spinning cyclone, shifted force zero to the off-Γ point, and asymmetric three-dimensional trapping. Last, the topology of the optical force is further manipulated to achieve several off-Γ BIC–induced force zeros.

RESULTS

Design principle of BIC protected topological optical force

Figure 1A schematically presents the PhCS chosen to support controllable BIC mode. It consists of a dual layer metastructure designed to break the z-inversion symmetry. The interface is achieved by etching half the way on top and bottom two arrays of rotated square nanoholes. The lattice period is kept constant at 574 nm and made of silicon. The side length of the square nanoholes is chosen at 330 nm, and the overall slab thickness is fixed at 600 nm. The two holes of the PhCS can be twisted with a rotation of α_top for the top layer and α_bot for the bottom layer. Depending on the rotation angles, α_top and α_bot, two scenarios with untwisted and twisted PhCS are studied. We called the untwisted PhCS regime the condition α_top = α_bot = 0^°, that is, when the structure remains z-inversion symmetric and the twisted PhCS all other asymmetric conditions. For the trapping numerical analysis, we consider a polymeric nanosphere with a radius of 100 nm and a refractive index of 1.6 disposed at 1 μm above the center of the PhCS. The whole structure along with the nanosphere is immersed in water. Because of its topological nature in momentum space, BIC induces a force singular point named force zero (F = 0), around which force distribution will have similar topology as far-field polarization states, hence trapping (repelling) the nanosphere at (off) the Γ point in an untwisted PhCS (Fig. 1B). This can be easily understood as the radiation field goes to zero at BIC, therefore forming a point with force also approaching zero (a force singular point). Whereas for a twisted PhCS, nanoparticles can be trapped in a spinning behavior (Fig. 1C). Figure 1D plots the calculated iso-frequency surfaces of two bands with their quality factors color-mapped for the untwisted PhCS. BIC with infinite quality factor is obtained at Γ point on each band. Note that the lower-band BIC quality factor decreases more slowly, improving the robustness against perturbation in the value of the momentum. The corresponding far-field polarization states are illustrated in Fig. 1 (E and F) with topological distribution, respectively, indicating that the upper- and lower-band BICs have opposite topological charge. In contrast, twisted PhCS can maintain BICs on the two bands as shown in Fig. 1G while generating elliptical polarization states with nearly the same ellipticity (Fig. 1, H and I). It is noteworthy that the solved eigenmodes of PhCS are unaffected by the introduced nanoparticles because the eigenfrequencies corresponding to the eigenmodes of the PhCS and nanoparticle are very different due to the large contrast in the shape, size, and index of the PhCS and nanoparticles. In addition, there is no mutual coupling due to their large enough distance. Therefore, BIC in the PhCS suffers no impact from the introduction and movement of the nanoparticles, paving the base for manipulating nanoparticles in motion. As for practical excitation of the BIC eigenmodes, a Gaussian beam can be used as incident light thanks to its decomposition to different values of in-plane vector k_∥ (3, 11, 13, 14). The different k components of the excitation beam would interact with different k_∥ resonances of the PhCS, spawning the BIC topology and hence topological force.

Fig. 1. BIC-induced topological distribution of optical force for manipulation of nanoparticles.

(A) Schematic of a PhCS supporting BIC with topological optical force for trapping nanospheres in water. Holes in the PhCS can be twisted with a rotation angle of α_top for the top layer and α_bot for the bottom layer. (B) Illustration of topological force for trapping and dispersing nanoparticles on two bands with BICs in untwisted PhCS. (C) Topological force for trapping nanoparticles with spinning pattern in twisted PhCS. Insets in (B) and (C) show the distributions of force vectors in momentum space. With the twisted PhCS, nanoparticles will be trapped to or repelled from the center in a spinning feature like a swirl, which is different from the untwisted case leading to a rectilinear motion. (D) Iso-frequency surface and color-mapped quality factor of the untwisted PhCS for two eigenmodes (α_top = α_bot = 0°). Corresponding distribution of far-field polarization states of (E) the upper band (top) and (F) the lower band (bottom) in (D). (G) Iso-frequency surface and color-mapped quality factor of the twisted PhCS with α_top = 25°, α_bot = 0° for two eigenmodes. Elliptical polarization states induced by the twist PhCS for (H) the upper band (top) and (I) the lower band (bottom) in (G).

To illustrate the BIC-induced topological force in momentum space for manipulating nanoparticles, we are calculating the net optical force on the nanosphere above PhCS as a function of the metastructure geometry. Rigorous calculation of optical forces on arbitrary sized and shaped particles can be obtained by integrating the flow of Maxwell stress tensor (MST) through a closed surface S containing the nanoparticle as follows (21, 24, 33)

$$\mathbf{F}={\oint }_S⟨\overleftrightarrow{\mathbf{T}}⟩\cdot \hat{\mathbf{n}}dS$$ (1)

where $\hat{\mathit{n}}$ is the unit vector normal to the surrounding surface S, and ⟨ · ⟩ is the time-average operator. The MST can be given in the form of electric field E and magnetic field H,

$$⟨T_{ij}⟩=\frac{1}{8\pi }\mathrm{\Re }\left[D_i{E}_j^{\ast }+B_i{H}_j^{\ast }-\frac{1}{2}\left(\mathbf{D}\cdot {\mathbf{E}}^{\ast }+\mathbf{B}\cdot {\mathbf{H}}^{\ast }\right){\delta }_{ij}\right]$$ (2)

Here, δ_ij is the Kronecker delta, and D = εE, B = μH, with ε and μ denoting the permittivity and permeability of the medium, respectively. BIC feature of the PhCS can be seen from the infinite quality factor at Γ point in Fig. 1 (D and G), leading to zero radiated far-field power at this point. This zero radiation is demonstrated in the topological winding electric polarization distribution around Γ point in Fig. 1 (E and F) and Fig. 1 (H and I). Through stress tensor for optical force and winding polarization of BIC singular point, the correlation between the BIC and force zero can be established. The force zero can be achieved either (i) by using BIC that generates near-zero fields or (ii) by designing symmetric distribution of forces that cancel each other with opposite sign. For case (i), BIC results in zero radiation, that is, when both 𝐄 and 𝐇 simultaneously vanish. This is the obvious case with zero stress tensor and zero force integration. For the more subtle case (ii), the force can be arranged to distribute with inversion symmetry around one point, at which forces in opposite direction offset each other and spawning a force zero. Away from the zero-force singular point, we find that the topological winding electric polarization distribution also leads to a winding optical force with diverse manipulating features. To understand the force topology of the two cases, we write the force vector as $\mathbf{F}\left(\mathbf{k}\right)=F_x\left(\mathbf{k}\right)\hat{\mathbf{x}}+F_y\left(\mathbf{k}\right)\hat{\mathbf{y}}$, where $\mathbf{k}=k_x\hat{\mathbf{x}}+k_y\hat{\mathbf{y}}$ is the two-dimensional wave vector, and F_{x, y} is the optical force in the x and y directions, respectively. The force topology is then characterized and quantized with topological charge as (1, 11, 34)

$$q_F=\frac{1}{2\pi }{\oint }_{L}d\mathbf{k}\cdot {\mathrm{\nabla }}_{\mathbf{k}}{\varphi }_F\left(\mathbf{k}\right)$$ (3)

which describes how many times the force vector winds around the force zero. Here, ϕ_F(k) = arg [F_x(k) + iF_y(k)] is the angle of the force vector in the momentum space. Although the proposed topological force is investigated in momentum space, it actually connects to real space based on the relation between the spatial angle and in-plane vector k_∥ (see the Supplementary Materials).

Converging and diverging topological optical force

Figure 2 demonstrates the topological distribution of optical force in momentum space induced by the BIC singular point with untwisted PhCS. Figure 2 (A and B) shows the converging and diverging patterns of the optical force generated by the lower-band and upper-band BICs, respectively. Force converging toward BIC point is represented by red color, and the pattern indicates that when nanoparticles are located within the vicinity of the BIC point, they will perceive a moving and trapping force toward the Γ-BIC. This force is of interest for collecting and accumulating nanoparticles. As for the diverging force pattern, any off-Γ position will have a force that would repel objects away from the center. Insets in Fig. 2 (A and B) show that the angle of the force vector is topologically distributed in momentum space and confirm that the two patterns have an angle difference of 180°. The direction of the converging optical force can be illustrated by calculating the Poynting vector around the object. At a point of k_x = 0, k_y < 0, the Poynting vectors are pointing upward, meaning that the optical force is pushing the nanosphere toward k_y = 0, as shown in Fig. 2C. For k_x < 0, k_y = 0, the Poynting vectors are pointing toward right to bring the nanosphere toward k_x = 0, as shown in Fig. 2D. This behavior is confirmed by looking at the two-dimensional deviation in Fig. 2E (k_x < 0, k_y > 0) and Fig. 2F (k_x > 0, k_y < 0), where the nanoparticle always has Poynting vectors pointing to the BIC point. Figure 2G compares the optical force F_y perceived by a nanosphere alone and after placing a 10 × 10 PhCS excited by a Gaussian beam impinging at normal incidence. Without PhCS, the optical force is exclusively generated by the Gaussian beam. Whereas when there is a finite PhCS, the BIC topological optical force, mainly induced by the field enhancement of the BIC mode, is almost one order of magnitude higher than the one with only the Gaussian beam.

Fig. 2. Topological force distribution around BIC in momentum space with untwisted PhCS.

BIC induces a force singular point (force zero, F = 0) at Γ point with topological distribution of optical force calculated by integrating MST surrounding a sphere 1000 nm above the PhCS for the (A) upper band and (B) lower band. (A) The upper band has optical forces converging toward the center, Γ point. (B) The lower band has optical forces radiating away from the center. Insets plot maps of the angle of the force vector ϕ_F(k) showing topological winding feature of optical force around BIC in momentum space. The converging and radiating force patterns are of rotation symmetry with a degree of 180. Lower-band Poynting vector and electric field profile around the nanosphere with (C) k_x = 0, k_y < 0 (D) k_x < 0, k_y = 0, (E) k_x < 0, k_y > 0, and (F) k_x > 0, k_y < 0. Here, E₀ is equal to 1 × 10⁶V/m. (G) Comparison between magnitude of the converging force with a 10 × 10 PhCS and without PhCS by shining a Gaussian beam with a frequency of 186 THz, a beam width of 287 nm, and a focal plane at z = 850 nm.

Topological optical force with symmetry breaking

To further demonstrate the effect of converging force, Fig. 3A plots the distribution of optical force and force zero with untwisted PhCS when the nanosphere is laterally displaced by about 100 nm in the positive x-axis direction. The force zero has a displacement toward negative k_x, and the topology reveals that the optical force converges toward the shifted off-Γ force zero. When the nanosphere is moving toward Γ point, force zero will approach Γ point on the other side, and the position of nanosphere and force zero will finally coalesce at Γ point, achieving optical trapping condition. Similar phenomenon occurs with force zero shifting along the negative direction of diagonal when displacing nanosphere 141 nm along the positive direction of diagonal (Fig. 3B). Figure 3C shows the force in the x direction for the case in Fig. 3A, where the force zero region bends toward the direction of negative k_x, as indicated by the arrow. Line plot of the force in x direction is shown in Fig. 3D with different displacement of d_x = 50 nm and d_x = 150 nm. Both lines cross zero at positions of k_x ≠ 0, and for larger displacement, the line becomes more tilted toward negative direction. Engineering the PhCS to reveal diverse polarization features originating from the BIC topology and intriguing optical force distribution in momentum space around BIC are expected. To further demonstrate the design versatility offered by this approach, we conceived a PhCS unit cell containing rotated top hole disposed at an angle of 25° with respect to the hole orientation in the lower layer, thus forming a twisted PhCS. The calculated force patterns given in Fig. 3E for the upper band and in Fig. 3F for the lower band also enable a BIC-induced force zero at Γ point. With the twisted PhCS, nanoparticles will be trapped to or repelled from the center with a spinning feature like a swirl, which is different from the untwisted case with the point-to-point force leading to a rectilinear motion. This spinning force feature associated with twisted PhCS at BIC may lead to diverse possibilities in nanoparticle sorting, manipulating, and optofluidics.

Fig. 3. Topological forces with perturbation.

(A) Force distribution around lower-band BIC with untwisted PhCS when the nanosphere is displaced 100 nm along positive x axis. (B) Force distribution around lower-band BIC when the nanosphere is displaced both 100 nm along positive x axis and 100 nm along positive y axis. (C) Map of F_x in momentum space when the nanosphere is displaced 100 nm along positive x axis. The arrow indicates the shift of F_x = 0. (D) F_x as a function of k_x at k_y = 0 with the nanosphere displaced 50 and 100 nm along positive x axis. The optical forces in (C) and (D) are normalized to a value of F₀ = 10⁻¹⁶N. Force distribution above the PhCS around (E) lower-band BIC and (F) upper-band BIC when the PhCS has a rotation angle with α_top = 25° and α_bot = 0°. Insets in (E) show the structure of the perturbed PhCS.

Manipulation with multiple BICs

Modifying the lattice constant of untwisted PhCS to 517 nm, we achieved interesting off-ΓBICs from which we evaluate their topological forces in Fig. 4. We observe that the quality factor presents now four infinite points resulting from four BICs emerging symmetrically at k_x = 0 and k_y = 0. The corresponding force distribution is shown in Fig. 4B. Five force zeros are found, and four of them are directly attributed to the BIC modes observed in Fig. 4A. One exception related to the zero force at Γ point is attributed to the inversion symmetry. The four off-Γ force zeros introduce repelling force outward offset in the opposite direction one with respect to the others to form an additional force zero at the Γ point. The force zero at Γ point can be understood as a converging center, and the BIC points as radiating centers. The distribution of absolute value of optical force is shown in Fig. 4C in log scale, clearly indicating the five force zeros in momentum space. Map of the force angle ϕ_F(k) demonstrates the topological feature with charge of 1 for force zeros at BICs (white arrow) and opposite charge of −1 for force zero induced by symmetry (black arrow) as shown in Fig. 4D. Applying sign function to force in x and y directions results in Fig. 4 (E and F) illustrating the force direction, respectively. BICs generate triangle regions that drag objects to BICs in one direction and pulling them away in the other direction. Another eigenmode has several BICs forming a band that encloses the Γ point (Fig. 4G). In Fig. 4H, the force distribution is radiating outward and goes to zero within a diamond-like region around the Γ point. Absolute value of optical force of this band is given in Fig. 4I. Objects within the force zero region will be nearly stable, and objects outside can be pulling away, a behavior of great interest, for example, to sort and separate nanoparticles.

Fig. 4. Topological features of optical force around off-Γ BIC when the untwisted PhCS lattice constant is reduced to 517 nm.

(A) Quality factor of one eigenmode showing that four off-Γ BICs are obtained. (B) Topological distribution of optical force surrounding four BICs. (C) Absolute value of optical force in log scale shows force zeros occurring at four BICs and one additional at Γ point. (D) Map of the value ϕ_F(k) showing topological features of optical force around force zeros in momentum space. The black and white arrows have opposite topological charge. Map of sign of (E) F_x and (F) F_y in momentum space. (G) Quality factor of another eigenmode shows several BICs enclosing the Γ point. (H) Distribution of optical force around these BICs. Accumulation of these singular points results in nearly zero force within a diamond-like region around the Γ point. (I) Map of absolute value of optical force in log scale around the accumulating BICs.

As for experimental realization, the PhCS can be made of silicon, silicon nitride, titanium dioxide, and other dielectric materials, which have been widely used to fabricate optical PhCSs and observe BIC effects. Here, we use silicon as the structure material and the geometrical parameters of the PhCS that are well compatible with nanofabrication approaches. The proposed PhCS of period larger than 500 nm can be fabricated with typical electron-beam lithography (EBL) and reactive ion etching process. We have also evaluated the trapping performance for different relevant particle types in terms of various refractive indices from 1.4 to 1.8 and various radii from 50 to 200 nm, and we find that the topological distribution of optical forces is stable and robust. We propose that the single layer PhCS can be fabricated by using conventional EBL and etching (5, 15, 35, 36), followed by direct wafer bonding of two PhCS layers (37). Another option could consist in fabricating two layers via EBL, using planarization with spin on glass in between the two lithography steps (38, 39).

DISCUSSION

In conclusion, we unveil the topological feature of optical force distribution in momentum space around BIC. We observe that the force vectors wind around the force singular point similarly as the phase that winds around optical singularities in photonics. We characterized these zero force singular points by a force topological charge and show that converging and diverging force patterns are associated with the same topological charge but with 180° angle difference. Being almost one order of magnitude higher than the Gaussian beam–induced optical forces, these force patterns distributed around force singularities have the capability of trapping and repelling nanoparticles. The topological force can also be engineered to manifest spinning vortices by modifying σ_z asymmetry of the PhCS. Off-Γ BICs generate multiple force zeros and enrich the force patterns for optical manipulation. This study may open up a new avenue for optical trapping and manipulation of nanoparticles using BIC topology and may also inspire the future development of nanosorters and nanorobots in microfluidic systems.

MATERIALS AND METHODS

The simulation results are performed by COMSOL Multiphysics software.

Supplementary Materials

This PDF file includes:

Notes S1 to S8

Figs. S1 to S13

References