⇦Chinese Physics Letters, 2023, Vol. 40, No. 11, Article code 114201 Mechanical Characterization of Broadband Achromatic Optical Vortex Metalens Zhechun Lu (陆哲淳)1, Yuehua Deng (邓越华)1, Yang Yu (于洋)1*, Chengzhi Huang (黄承志)2, and Junbo Yang (杨俊波)1 Affiliations 1College of Science, National University of Defense Technology, Changsha 410073, China 2Key Laboratory of Luminescence Analysisand Molecular Sensing (Ministry of Education), College of Pharmaceutical Sciences, Southwest University, Chongqing 400715, China Received 21 August 2023; accepted manuscript online 20 September 2023; published online 16 October 2023 *Corresponding author. Email: yuyang08a@nudt.edu.cn Citation Text: Lu Z C, Deng Y H, Yu Y et al. 2023 Chin. Phys. Lett. 40 114201    Abstract Metalenses, which may effectively manipulate the wavefront of incident light, have been proposed and extensively utilized in the development of various planar optical devices for specialized purposes. However, similar to traditional lenses, the metalens suffers from chromatic aberration problems due to the significant phase dispersion in each unit structure and the limited operational bandwidth. To mitigate the impact of chromatic aberration, we integrate a phase compensation approach with a novel utilization of a phase shift function to define the adjusted phase criterion satisfied by each $\alpha$-Si resonance unit. This approach may lead to development of an innovative optical tweezer known as an achromatic optical vortex metalens (AOVM), offering reliable focusing capabilities across the $1300$ nm and $1600$ nm incident light range. Numerical simulations are conducted to investigate the optical properties of $200$ nm diameter SiO$_{2}$ particles at the focal plane of the AOVM. The trapping ability of the AOVM is successfully validated, exhibiting favorable characteristics including constant optical force, stable kinematic state of trapped particles, and consistent capture positions, surpassing those of the optical vortex metalens.
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 DOI:10.1088/0256-307X/40/11/114201 © 2023 Chinese Physics Society Article Text Optical forces have been widely used to catch and manipulate particles since they were first discovered,[1] and this method of optical capture is known as optical tweezers.[2] Traditional optical tweezers have undergone significant advancements in optical capture and manipulation after decades of development.[3-5] However, traditional optical tweezers rely on high numerical aperture (NA) objectives and other complex optical components for beam expansion and steering. These requirements make classical optics less flexible in terms of manipulation and control, limiting their integration with modern optical devices. To address this challenge, it is imperative to seek a simple yet efficient capture tool that allows seamless integration. The emergence of metalens technology offers promising prospects for the development of compact optical systems. Metalenses utilize subwavelength array structures to introduce phase shifts, enabling spatial surface optical responses that can modify the polarization, phase, and amplitude of light in both reflection and transmission.[6] Consequently, metalenses have found extensive applications in the design and fabrication of optical elements, outperforming conventional diffractive optical elements.[7-9] Furthermore, their capacity to break the diffraction limit due to their subwavelength resolution propels them to previously unheard-of heights in light field manipulation.[10-12] The use of miniaturized on-chip optical tweezers has gained prominence in capturing and controlling nanoscale particles and even smaller biomolecules. These optical tweezers can generate far-field foci with strong gradient forces, enabling the creation of monofocal, multifocal, and vortex beams.[13-16] Moreover, they facilitate multidimensional manipulation.[17-20] Optical vortex (OV) beams, characterized by a helical phase factor exp$(il\theta)$ and orbital angular momentum (OAM), are particularly effective in trapping particles and transferring angular momentum to them. This results in periodic or non-periodic orbital motion of the particles.[21-23] Such rotational movement is crucial for various microsystems[24] that capture asymmetric objects[25] or particles with birefringent effects.[26] Additionally, optical vortices have proven to be efficient in capturing particles with low refractive indices.[27] The combination of vortex light with metalenses is widely employed in the fields of physical detection, optical manipulation, and communication. This allows functional integration and miniaturization of devices.[28-30] Indeed, the influence of wavelength and bandwidth on particle capture stability has been relatively overlooked in the context of metalenses and photodynamics.[31-33] Metalenses exhibit chromatic aberration similar to conventional lenses due to the structural dispersion and phase response modes inherent to their optical confinement.[34-36] As a result, wavelengths within a specific bandwidth cannot be focused precisely at the same location. This leads to non-uniform effects on the magnitude of the optical force exerted on particles and the position of particle capture. In the field of biology, maintaining a consistent optical force across different wavelengths is beneficial for reducing photodamage to biological specimens.[37] To address this, we propose a broadband achromatic optical vortex metalens (AOVM) optical tweezer system based on a phase-compensated approach in this study. This system is capable of generating a focal-stable OV in the incident light range of 1300–1600 nm, while also maintaining a high numerical aperture for improved focusing and capture efficiency. We conduct numerical simulations to evaluate the performance of this system. The results demonstrate that the proposed AOVM optical tweezers can function as an optical spanner, preserving the optical force, kinematic state, and particle capture position for SiO$_2$ spheres with a radius of $200$ nm. These findings highlight the effectiveness of the broadband approach, improving the impact of wavelength bandwidth and expanding the utility of the AOVM system for photodynamic applications.
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Fig. 1. (a) Normal metalenses have chromatic aberration, which means that the longer the wavelength, the shorter the focal length. Achromatic metalenses make the focal length the same at all wavelengths by modulating the cell structure. (b) The geometric phase formed by the unit structure's angle $\theta$ of rotation acts as the fundamental phase at the maximum wavelength $\lambda_{\max}$, while the resonant phase generated by the structure's size is used to calculate the phase difference $\Delta \phi$ between each band and the fundamental phase.
The focal length of a conventional lens is determined by the curvature radius and the lens material's refractive index. When the radius is fixed, the focal length changes with the refractive index. In other words, for the same material, shorter wavelengths correspond to higher refractive indices and shorter focal lengths. This disparity in imaging colors is known as positional chromatic aberration or axial chromatic aberration. Traditional lens systems cannot completely eliminate chromatic aberration using a single lens. To fully correct chromatic aberration, multiple lenses made of different materials are required. This necessitates strict material selection and poses challenges in lens manufacturing processes. In contrast, metalens technology offers a solution that does not involve cascading numerous lenses. Instead, the unit structure of the lens is modified to adapt to changing wavelengths, enabling correction of chromatic aberration. This design approach has been previously documented[38,39] and is depicted in Fig. 1(a). The OV typically refers to a type of beam called a phase vortex beam. It is characterized by having a spiral wavefront and a phase singularity at the center of the beam. As a result, the OV exhibits a zero-intensity region at its center, and its intensity distribution takes on a toroidal shape. One commonly encountered type of phase vortex beam is the Laguerre–Gaussian beam, which can be expressed as a complex amplitude by adding the phase term exp$(il\theta)$, and the typical phase distribution is as follows: \begin{align} \phi_{\scriptscriptstyle{\rm V}}(r,\lambda)=-\frac{2\pi}\lambda{(\sqrt{r^2+f^2}-f)}+l\varphi, \tag {1} \end{align} where $r=\sqrt{x^2+y^2}$ indicates the distance from the positional coordinates $(x,y)$ of each unit structure to the center of the metalens, $\lambda$ is the incident wavelength, $f$ is the planned focal length, $l$ is the vortex light's topological charge, and $\varphi=\arctan(y/x)$ is the azimuthal angle. Varying phase distributions are created as a result of different incident wavelengths. Therefore, the process can be stated as follows: \begin{align} \phi(r,\lambda)=\,&\phi_{V}(r,\lambda_{\max})+\Delta\phi(r,\lambda), \tag {2}\\ \Delta\phi(r,\lambda)=\,&-\Big[2\pi(\sqrt{r^2+f^2}-f)\Big]\Big(\frac{1}{\lambda} -\frac{1}{\lambda_{\max}}\Big), \tag {3} \end{align} where $\phi(r,\lambda_{\max})$ indicates the phase distribution satisfied by the maximum operating wavelength and can be regarded as the fundamental phase, which is exclusively related to $\lambda_{\max}$, and thus the geometrical phase can be used to modulate the phase of each structuring unit. The geometric phase, also known as the Pancharatnam–Berry phase, is generated by rotating the individual structural units of a metalens that are impacted by circular polarization.[40] This rotation leads to a phase modulation that depends solely on the orientation of the scattering elements. The phase difference, denoted as $\Delta\phi(r,\lambda)$, is the phase difference between the incident wavelengths and $\lambda_{\max}$, and it is inversely proportional to the wavelength $\lambda$. By appropriately designing the resonant phase of each unit structure, this phase difference can be achieved. The resonant phase differs from the geometrical phase, allowing them to be combined without interfering with each other. This combination is demonstrated in Fig. 1(b). Due to the waveguide-like cavity resonance properties of the nanopillars, the induced light field becomes highly concentrated within the dielectric structure, which possesses a significantly higher refractive index compared to its surroundings. Consequently, there is weak and negligible optical coupling with neighboring structures.[41] The wavelength-independent geometrical phase serves as the foundational phase required for the proper functioning of the metalens surface device, and the wavelength-dependent resonance phase serves as a compensating phase to correct chromatic aberration between different wavelengths. In order to statistically quantify this phase compensation, we use $\phi_{\rm s}$ to build cell structures that meet the requirements. The phase difference formed at each place $(x,y)$ is clearly seen in Fig. 1(b), and the phase difference increases as the wavelength decreases. As a result, the resonant phase of each unit structure must be satisfied during the fundamental phase of $\lambda_{\max}$: \begin{align} \phi_{\mathrm{s}}(r,\lambda)=\Delta\phi(r,\lambda_{\min}) \Big(\frac{\alpha}{\lambda}+\beta\Big), \tag {4} \end{align} where $\alpha=\frac{\lambda_{\max}\lambda_{\min}}{\lambda_{\max}-\lambda_{\min}}$, $\beta=\frac{\lambda_{\min}}{\lambda_{\min}-\lambda_{\max}}$, and $\Delta\phi(r,\lambda_{\min})$ signifies the greatest phase difference at distinct places across the full working band. The operational wavelengths we picked for testing the theory are $\lambda_{\min}=1300$ nm and $\lambda_{\max}=1600$ nm in the regularly used communication region.
Table 1. Phase (rad) of the designed structure along the $x$-axis at different wavelengths.
$W$ (nm) $L$ (nm) $1300$ nm $1400$ nm $1500$ nm $1600$ nm
$305$ $195$ $-2.95$ 2.14 1.30 0.36
$335$ $200$ $-2.33$ 2.51 1.55 0.60
$240$ $460$ $-0.97$ $-2.30$ 0.84 0.14
$280$ $385$ $-0.18$ $-2.91$ 1.91 $-0.62$
$445$ $65$ 0.83 0.32 $-0.13$ $-0.82$
$355$ $115$ 1.29 0.72 0.20 $-0.52$
$315$ $155$ 1.94 1.20 0.58 $-0.20$
$305$ $180$ 2.66 1.73 0.96 0.12
cpl-40-11-114201-fig2.png
Fig. 2. (a) Electric field distribution in the axial direction for the AOVM at different wavelengths with focal lengths essentially around $7.1$ µm. (b) The distribution of the electric field in the axial direction for the optical vortex metalens (OVM) at different wavelengths, where the focal length rises with decreasing wavelength, demonstrating the anomalous dispersion of the metalens.
The topological charge $l$ of the OV is 2, the overall metalens size is $25\,µ {\rm m}\times25\,µ $m, the focal length is $f=8$ µm, and NA = 0.84. The substrate is built by the fused SiO$_2$ material, the unit structure is formed by $\alpha$-Si nanofins with a unit period of $P_x=P_y=0.55$ µm and a height of $H=1.4$ µm, and the unit structure's length and width are scaled from $80$ nm to $460$ nm, respectively, with a step of $5$ nm. According to the created database, $\phi_{\rm s}$ is used to choose the nanostructures that satisfy the phase compensation, and at the same time, in order to ensure the high polarization conversion efficiency,[42] try to satisfy the phase difference between the long and short axes of the nanofins being $\pi$. As indicated in Table 1, we discretize the phase into 8 cells, each of which is chosen based on $\phi_{\rm s}$ for the linear connection satisfied at different wavelengths. We select the optimum cell placement based on the varying phase compensation requirements at different sites. The geometrical phase of the cell is used to satisfy the focusing vortex effect while rotating the cell direction, as shown in Fig. 1(c). However, the table shows that each cell does not strictly satisfy the linear connection with the wavelength, and only the best cell structure can be chosen. In Fig. 2, a comparison is made between the focusing images obtained using the AOVM and those produced by OVM. Figure 2(a) shows the distribution of the $z$-axis electric field strength of the AOVM at different wavelengths. The results demonstrate that the focal point remains consistent at around $7.1$ µm for all wavelengths, with NA = $0.87$. This confirms the effectiveness of the achromatic approach employed in the AOVM, where the focal length does not vary with wavelength. On the other hand, Fig. 2(b) displays the $z$-axis electric field strength distribution of the OVM at different wavelengths. It can be observed that the metalens exhibits anomalous dispersion, meaning that as the wavelength becomes shorter, the focal length becomes longer. As a consequence, the conventional metalens design struggles to meet the requirements of practical usage, as it demands an extremely narrow range of incident wavelengths. In contrast, the achromatic metalens can be constructed to exhibit strong broadband characteristics, accommodating a wide range of wavelengths as required. This makes it well-suited for practical applications where a broad range of incident wavelengths needs to be focused effectively. To verify the improved stability of the optical spanner performance in the AOVM, we conduct calculations to determine the optical forces exerted by the vortex light field on particles in both the transverse and axial directions. Considering that the size of the sphere is comparable to the wavelength, we choose to model it under the Lorentz–Mie mechanism and use the Maxwell stress tensor (MST) method for the calculation. Light is an electromagnetic wave and the interaction of light with matter can be viewed as an electromagnetic scattering problem. By solving Maxwell's equations, we find that the Lorentz force formula can describe the force of an electromagnetic field on a charge. Charged particles can obtain energy and velocity from the electromagnetic field because the electromagnetic field exerts a force on the charge that makes it possible for them to move in it. This clarifies the existence of energy and momentum in the electromagnetic field. When a charged particle moves through an electromagnetic field, the electromagnetic field and the charged particle exchange energy and momentum. The force-momentum conservation equation in electromagnetic fields is \begin{align} \frac{\partial}{\partial t}\int_{V}(\boldsymbol{g}_{\rm f}+\boldsymbol{g}_{\rm p})dV=\int_{V}\nabla\cdot\boldsymbol{T}dV, \tag {5} \end{align} where $\boldsymbol{g}_{\rm f}$ is the momentum density of the charge system and $\boldsymbol{g}_{\rm p}$ is the momentum density of the electromagnetic field in a vacuum. The total momentum of the electromagnetic field and the charge system in volume $V$ is represented by the rate of change in time on the left side of the equation. The entire force exerted on volume $V$ from outside should be represented by the volume fraction on the right side of the equation. The total force of the electromagnetic field acting into the volume $V$ is \begin{align} \boldsymbol{F}=\int\limits_{V}\nabla\cdot\boldsymbol{T}=\oint\limits_{S}\boldsymbol{T}\cdot\boldsymbol{n}ds. \tag {6} \end{align} Here, the volumetric force acting within a volume $V$ can be thought of as a tension on a closed surface $s$ enclosing the volume $V$, and ${\boldsymbol T}$ is the tension tensor of the electromagnetic field acting per unit area. Since Maxwell was the one who initially introduced the electromagnetic field tensor, it is often referred to as the MST. The time average of the electromagnetic field's stress tensor can be represented in terms of the field's complex amplitude for a time-harmonic field as follows: \begin{align} \langle T\rangle=\frac{1}{2}{\rm Re}\Big[\varepsilon_{0}EE^{*}+\mu_{0}HH^{*}-\frac{1}{2}(\varepsilon_{0}E^{*}+\mu_{0}H^{*})I\Big]. \tag {7} \end{align} The force exerted by the incident beam on dielectric particles can be decomposed into two components: scattering force and gradient force. The gradient force, which is proportional to the gradient of transverse light intensity, acts towards the region with the highest light intensity. It is a crucial factor in the formation of an optical trap. On the other hand, the scattering force is directed along the Poynting vector of the beam propagation and pushes the dielectric particles. To determine the total optical force on the particle, we integrate the MST across the surface of the particle. This can be expressed as[31] \begin{align} \langle F\rangle=\,&\int\Big\{\frac{\varepsilon}{2}{\rm Re}[(E\cdot n)E^*]-\frac{\varepsilon}{4}(E\cdot E^*)n\notag\\ &+\frac{\mu}{2}{\rm Re}[\mu(H\cdot n)H^*]-\frac{\mu}{4}(H\cdot H^*)n\Big\}ds, \tag {8} \end{align} where $\varepsilon$ and $\mu$ denote the relative permittivity and relative permeability of the medium surrounding the particle, respectively, and $n$ is the unit normal perpendicular to the integration plane. To test the stability of this optical trapping, we calculate the trapping potential by integrating the optical forces, i.e.,[43] \begin{align} U(r_0)=-\int_{\infty}^{r_0}F(r)\mathrm{d}r, \tag {9} \end{align} where $U(r_0)$ is the amount of energy required to move a particle from infinity to position $r_0$. To overcome the interference of thermal effects and create a robust optical trap, a trapping potential depth larger than 10 $k_{\rm B}T$ is usually required.[44,45] Here $k_{\rm B}$ is the Boltzmann constant, and $T$ is the temperature, which is assumed to be $300$ K in this study.
cpl-40-11-114201-fig3.png
Fig. 3. The magnitude of the optical force exerted on the SiO$_2$ sphere along the $x$-axis into the AOVM (a) and the OVM (b), as well as the calculated potential depth.
Figure 3 shows an examination of the transverse optical forces generated by the OVM and the AOVM. Since the OV is fully symmetric, we only investigate the optical force along the $x$-axis of a SiO$_2$ sphere with a diameter of $200$ nm. Figure 3(a) displays the optical force generated by the SiO$_2$ sphere at different positions along the $x$-axis of the AOVM. It can be observed that the magnitude of the optical force generated is relatively consistent across different wavelengths, except at 1600 nm where there is a slight variation of about 0.4 pN. This could be due to the lower frequency and energy per photon of the incident wavelength, leading to a smaller final optical force compared to other wavelengths. Additionally, the potential depth of the potential barriers formed in each band is more stable, and can reach over 110 $k_{\rm B}T$ in all bands except at $1600$ nm. In contrast, Fig. 3(b) illustrates the optical force created in the $x$-axis direction by the OVM operating at a wavelength of $1600$ nm. There is a significant difference in the optical force generated when light of different wavelengths is incident, which is largely due to the material's varying intrinsic dispersion at different frequencies. Achromatic aberration is used to compensate for this dispersion by introducing an extra phase. Another important factor is the focusing efficiency at different wavelengths, as shown in Fig. 4(c). The OVM has a large efficiency gap, achieving $43.9\%$ at the design wavelength but only $16.8\%$ at other wavelengths, leading to unstable focusing and an unstable optical force gap. On the other hand, although the AOVM has a lower average efficiency of $25.5\%$, it is more stable. Overall, the AOVM can generate stable transverse trapping pressures and potential barriers for trapping particles over a wide range of wavelengths, immobilizing them at the highest intensity of the ring light.
cpl-40-11-114201-fig4.png
Fig. 4. (a) The value of optical forces and potential depth at various wavelengths produced by a SiO$_2$ sphere traveling along the $z$-axis of the AOVM. (b) The strength of the centripetal force and torque produced by the SiO$_2$ sphere as it rotates around the vortex beam. (c) Efficiency of the AOVM and the OVM focusing at various wavelengths.
Figure 4(a) shows the amplitude of the optical force on the sphere in the $z$-axis direction at various wavelengths. It can be observed that the capture potential is significantly larger than 10 $k_{\rm B}T$, indicating that the particle can be stably captured in the axial direction. The equilibrium position is approximately 7.1 µm, with a maximum distance difference of 0.8 µm. This suggests that strict achromatic aberration has not been achieved, and the focal length has not been stabilized at a constant value. In comparison to the OVM shown in Fig. 2(b), it can be observed that the particles trapped at different wavelengths exhibit considerably smaller displacements in the $z$-axis direction, and no significant movement is observed. This suggests that the SiO$_2$ spheres can be stably trapped within a wide range around the maximum intensity of the focused OV beam's ring, based on the transverse and longitudinal trapping barriers. To further confirm the stability of the trapping mechanism, the moments of the SiO$_2$ spheres traveling around the vortex ring in the axial plane are calculated. This analysis confirms that the AOVM possesses a highly stable “optical spanner” characteristic. Initially, the optical forces acting in the $x$ and $y$ directions on particles positioned on the main ring at different azimuthal angles are determined. The centripetal force exerted by the particle around the ring can be estimated using the relationships of $F_x$ and $F_y$ to the centripetal force $F_{\varphi}$ as follows: $F_x=\sin\phi\cdot F_{\varphi}$ and $F_y=-\cos\phi\cdot F_{\varphi}$. In Fig. 4(b), the blue line represents the magnitude of the centripetal force at different wavelengths. It can be observed that as the wavelength increases, the centripetal force decreases. The average centripetal force across the wavelength band is found to be 0.53 pN. To determine the total torque for motion around the ring at each wavelength we need to consider both the tangential force and the radius of the OV. The diameters of the OV at 1300 nm, 1400 nm, 1500 nm, and 1600 nm are 0.65 µm, 0.85 µm, 0.95 µm, and 0.95 µm, respectively. The moments at each wavelength remain relatively stable, with an average moment of $0.44\times10^{-22}$ N/m, as indicated by the red line in Fig. 4(b). This suggests that the AOVM can generate reasonably stable moments at different wavelengths, allowing the “optical spanner” to function consistently across a wide range of wavelengths. Figure 4(c) displays the focusing efficiency of the AOVM and the OVM at different wavelengths. The lower focusing efficiency of the AOVM may be attributed to the coupling between the unit structures that arises from the simultaneous combination of geometrical and resonant phases, which leads to an overall reduction in efficiency. However, when compared to the OVM, the efficiency of the AOVM is more consistent, demonstrating its stability across a wide wavelength spectrum. To summarize, we have proposed an innovative phase shift function based on the concept of phase compensation to specify the magnitude of the compensated phase that needs to be supplied by each cell of the metalens. However, it is impossible to design a structure that completely satisfies the compensated phase, resulting in some variance in the focus lengths at different wavelengths, with a maximum difference of 0.8 µm. We use the MST method to calculate the optical forces generated along the transverse and longitudinal axes of the SiO$_2$ sphere, confirming that this AOVM can achieve a stable optical spanner capability in a longer wavelength band than a conventional OVM. Based on numerical simulations, the AOVM described in this study can provide stable transverse and longitudinal trapping optical forces in a wider wavelength range than an OVM at a single wavelength. Moreover, we have determined the amount of torque on the SiO$_2$ spheres when they are trapped by the OV and move around the ring. Although the radius of the OV beam varies slightly over different wavelengths, the torque remains generally steady. We anticipate that if each cell structure can be made to rigorously satisfy the phase compensation criteria in future research, the performance of this AOVM can be further stabilized and improved, leading to more practical applications. Acknowledgement. This work was supported by the National Natural Science Foundation of China (Grant Nos. 62275269, 60907003, 61805278, 61875168, and 22134005), and the Chongqing Talents Program for Outstanding Scientists (Grant No. cstc2021ycjh-bgzxm0178). 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