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Abstract
Optical ring lattices containing multiple optical traps in a ring configuration have gradually evolved into indispensable tools in cold atom experiments. However, the conventional methods for generating optical ring lattices require bulky apparatus and work at a limited wavelength range. Recently, metasurfaces have emerged as promising ultra-compact platforms for the implementation of optical ring lattices. Here, we report the generation of ultraviolet optical ring lattices using monolithic AlN metasurfaces. The feasibility of trapping atom and atom-like systems in the optical ring lattices is validated through theoretical calculation of the optical forces using the dipole model. The adoption of AlN metasurfaces can reduce the size and cost of generating optical ring lattices and thus give rise to diverse applications such as transportable atom clocks.
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1. Introduction
The optical ring lattices are structured optical fields that contain multiple optical traps arranged in a ring-configuration. Since their conception, the optical ring lattices have gradually evolved into indispensable tools in the field of cold atoms [1–3] and quantum physics [4,5]. Depending on the purposes and the target particles, optical ring lattices for particle capturing can be either bright or dark. The bright ring lattices is comprised of bright petals arranged in a circular formation, and particles with a refractive index higher than the surrounding medium can be trapped in the intensity maxima. The dark ring lattices appear as dark sites enclosed by bright boundaries, and particles with a refractive index lower than the surrounding medium can be trapped at the intensity minima [6]. Most reported optical ring lattices were generated using programmable optical elements including spatial light modulators [1–4,7–9] and axially symmetric polarizers et al. [10]. These mature methods, though, require bulky apparatuses to implement. In addition, the typical working wavelength is usually limited to the infrared region for optical ring lattices because most liquid-crystal based programmable optical elements are transparent in such wavelength region. Ultraviolet (UV) spectrum features high photon energy, covers a multitude of atomic transition frequencies, and is thus of interest in laser cooling and trapping atoms. By trapping ytterbium (Yb) [11], mercury (Hg) [12], and other atoms [13,14] in a UV optical ring lattice, state-of-the-art applications such as quantum computing [15] and atom clocks [16,17] can be developed. However, it is challenging to generate optical ring lattices in the UV region. Conventional UV optical elements are made of glass with precisely polished surfaces for shaping the wavefront in the designed way. Despite the maturity of the manufacturing process, they have fixed functionality and are unable to generate the optical ring lattices with multiple optical traps. Programmable optical elements work well in the visible and longer wavelength range but they are inconvenient in the UV range due to the severe absorption loss.
Recently, the progress of metasurface technology empowered researchers with novel tools for shaping light beams, revitalizing this field with renewed vigor [18,19]. The metasurfaces can shape light within an ultra-thin film, thus significantly reducing the size and cost of the optical setup compared to conventional methods while maintaining the high degree of tuning flexibility [20,21]. Moreover, the working wavelength range of the optical ring lattices can be expanded due to a broader range of material choices. Previously plasmonic metasurfaces have been reported through Pancharatnam-Berry (PB) phase method in the near-infrared and visible spectrum region [22–24]. Although fabrication of plasmonic metasurfaces is less challenging, the low efficiency due to inherent Joule losses of metals impedes their widespread adoption in the shorter wavelength range [25,26]. Titanium dioxide (TiO$_2$) [27], Gallium Nitride (GaN) [28], and Silicon Nitride (Si$_3$N$_4$) [29] have been utilized for metasurface in the visible range. To push the operation wavelength of metasurface further into UV range, high quality material with wide bandgap is a prerequisite. The niobium pentoxide (Nb$_2$O$_5$) [30] hafnia (HfO$_2$) [31], MgO [32] and Si$_3$N$_4$ [33] have been of interest for metasurfaces in UV range. However, these materials cannot cover the entire UV spectrum down to 200 nm. Aluminum nitride (AlN) has an ultrawide bandgap ($\approx$ 6.2 eV), ensuring high transmission for UV light with wavelength above 200 nm. Its refractive index ($\approx$ 2.2) is also higher than that of other UV transparent materials (Si$_3$N$_4$, HfO$_2$, MgO, MgF$_2$, CaF$_2$, SiO$_2$ et al.). The materials with higher refractive index can confine the optical field more effectively and weaken the undesired optical coupling and resonance, thus increase the tolerance for random fabrication errors [29]. Furthermore, high-quality AlN films with a low dislocation density of $10^7$-$10^8$ cm$^{-2}$ have been reported using metal-organic chemical vapor deposition [34,35] and sputtering and high temperature annealing methods [36]. Therefore, AlN metasurface are excellent choice for versatile beam shaping in the UV range [37,38].
Here, we design monolithic AlN metasurfaces to generate optical ring lattices with 6, 8 and 10 optical traps working at 375 nm. The working wavelength of corresponds to GaN diode lasers. The same method can be applied to trap and cool different atoms by choosing the appropriate atom transition frequencies and UV light sources. The achieved optical ring lattices show high stability along the propagation direction with the optical trap size increasing linearly with the propagation distance. Besides, the AlN metasurfaces are polarization-insensitive, which enable the rotation of light-absorbing particles by freely switching the polarization state of the incident beam when necessary. Finally, the feasibility of capturing particles in the lattice is validated through calculation of optical forces using a point dipole model.
2. Design methods
The AlN metasurfaces are composed of AlN circular pillars on sapphire substrate with varying radii that provide phase modulation covering the entire 0-2$\pi$ range. A schematic illustration of the metasurface and its subwavelength constituents, referred to as meta-atoms, is shown in Fig. 1. To determine the appropriate size of AlN meta-atoms, we conduct parameter sweeps using commercial finite-difference time-domain (FDTD) software from Lumerical Inc. (Vancouver, BC, Canada). A coherent Gaussian light source with a wavelength of $\lambda$ = 375 nm was adopted to emulate incident laser beams in the UV region. The refractive index and extinction coefficient of AlN film were attained from the ellipsometer measurement to construct the material model. Next, AlN meta-atoms were placed in periodic square sites and the period of the square site was set to 360 nm, which was slightly smaller than $\lambda$ to avoid undesired diffractions. Two monitors were respectively placed above the meta-atom top surface in the freespace and below the meta-atom bottom surface in the sapphire substrate to record the transmitted and reflected beams. Perfectly matched layers were implemented on all sides to eliminate boundary effects.
Fig. 1. An illustration of AlN metasurface comprising nanopillars as meta-atoms. Perfectly matched layers were adopted on all-sides of the square lattice containing one.
After the simulation structure for meta-atoms was set up, the height H and the radius R of the meta-atoms were set as variable and optimized subsequently. Parameter sweeps were conducted to obtain meta-atom geometry dependent phase retardation and transmission of the modulated light. Figure 2(a) and Fig. 2(b) show the calculated phase retardation and amplitude transmission of the incident light modulated by AlN meta-atoms with various sizes, respectively. For pillars with a uniform height H = 500 nm, the phase retardation and amplitude transmission of the modulated light as functions of meta-atom radius R were replotted in Fig. 2(c). It can be seen that the phase retardation was continuously modulated in the 0-2$\pi$ range by increasing the radii of meta-atoms from 30 nm to 120 nm while the amplitude transmission coefficient of light maintains above 56%.
Fig. 2. Phase retardation (a) and transmission efficiency (b) of the incident beam as a function of H and R. H = 500 nm is highlighted by white dotted lines on both figures and the obtained results are re-plotted in (c).
Considering that continuous variation of AlN meta-atom radii is challenging for fabrication, it is desirable to discretize the continuous curves in in Fig. 2(c). Here, eight meta-atom radii contributing to phase retardations of k/8$\pi$ (k is odd from 1 to 15) were selected. The lookup table is shown in Table 1.
Table 1. The Lookup Table for AlN Meta-Atom Radius
Then the target optical far field of multiple optical traps in a ring configuration was designed to arrange the meta-atoms properly. Laguerre-Gaussian (LG) modes were chosen because they are shape-invariant during propagation, which is crucial to enhance the stability of optical field. The complex amplitude of LG beams in a cylindrical coordinate $(\rho,\phi,z)$ is described by [7]:
The positions of the optical traps can be revealed by searching the singularities with zero light intensity. Therefore, Eq. (2) is rewritten as the product of non-zero part and the possible singularity-determining $P(\rho, \phi )$.
The optical ring lattices containing 6, 8 and 10 optical traps were optimally obtained by coaxially superposing $LG_{p=0}^{l=8}$ and $LG_{p=0}^{l=2}$, $LG_{p=0}^{l=11}$ and $LG_{p=0}^{l=3}$, $LG_{p=0}^{l=17}$ and $LG_{p=0}^{l=7}$ respectively. The far field intensity and phase profiles are depicted in Fig. 3 in contrast to the local LG beam radius w. The multiple optical traps are uniformly distributed in ring lattices. Each site is accompanied by a helical phase structure centered around the singularity, as highlighted by white dotted line circles. It is worth noting that the intensity around the center of the ring is also zero, but the corresponding phase shift is several times of 2$\pi$. The high-order phase singularities at the center would break down during propagation. Besides, the large size of the central dark site would undermine the accuracy of manipulation. Therefore, the central singularity is typically not employed to manipulate particles.
It was found that the intensity on the outer edge of the optical ring lattices tended to be weaker than the designed pattern, due to the slight difference between a uniform intensity distribution of the incident light assumed in the design process and the actual Gaussian intensity distribution. So when finalizing the hologram pattern designs for generating the optical ring lattices with 6, 8 and 10 optical traps, the weights of the LG modes contributing to the outer edge intensity were increased to 1.5, 1.8 and 2 for $LG_{p=0}^{l=8}$, $LG_{p=0}^{l=11}$ and $LG_{p=0}^{l=17}$ respectively while the weights for LG modes contributing to the inner edge intensity remained 1 for $LG_{p=0}^{l=2}$, $LG_{p=0}^{l=3}$ and $LG_{p=0}^{l=7}$. Since that the meta-atoms are transparent to the incident light, the metasurface is unable to directly modulate the amplitude transmission. A holographic method is used to impose grating carrier phase on the near field phase profile to realize the light intensity redistribution among different diffraction orders. The grating together with the hologram phase pattern must be resolvable by the number of meta-atoms and meta-atoms spacing of the metasurface. The grating period is set to 1.8 $\mu$m. Hence, each grating period is resolved by 5 meta-atoms and the metasurface is resolved by approximately 22 grating periods. Figure 4(a)–4(c) show the calculated near field hologram phase profiles. AlN meta-atoms with different radii were accordingly assembled to construct the metasurfaces. As shown in S1, the radius of each metasurface was set to 20 $\mu$m due to limitation of available computation resources. The beam waist radius of the incident Gaussian source was 40 $\mu$m to ensure nearly homogeneous light intensity on metasurfaces. The in-plane frequency-domain profile and power monitor for recording the near field phase was placed at a distance of 500 nm from the top surfaces of AlN pillars. In Fig. 4(d)–4(f), the phase profiles imparted by metasurfaces present little deviation from the theoretical designs, demonstrating high feasibility of our method.
Fig. 3. Designed optical ring lattices containing 6,8 and 10 optical traps. (a)-(c) Normalized far field intensity distribution, and (d)-(f) far field phase profile.
Fig. 4. The comparison of derived results and metasurface-modulated near field phase profile. (a)-(c) The theoretically designed near field phase profile. (d)-(f) The phase profile at 1 $\mu$m from the surface of the sapphire substrate.
3. Simulation results
Figure 5(a)–5(c) show the far field intensity patterns of optical ring lattices with 6, 8 and 10 optical traps at 1 mm away from the metasurfaces, respectively. The optical traps in the ring lattices have distinct boundaries. The amplitude in Fig. 5(d)–5(f) and the helical phase profile in Fig. 5(g)–5(i) unambiguously demonstrate the characteristic rotating phase profiles of structured vortex beams. Particles in the vicinity of optical traps thus can be captured effectively. The efficiency of the AlN metasurfaces, defined as the ratio of the energy within the pattern of optical ring lattices to the total injected beam energy, is also evaluated. The designed AlN metasurfaces for generating 6, 8 and 10 optical traps have efficiencies of 14.33%, 13.44% and 12.13% respectively. The predominant energy loss is attributed to the discard of unused diffraction orders. These efficiency values are comparable to that of UV optical tweezers generated with spatial light modulators [39], and close to that of silicon-based metasurface working in the infrared region [40]. Nevertheless, our method of shaping UV laser beams with metasurfaces can benefit from a higher damage threshold compared to the traditional methods using spatial light modulators. It was reported that the damage of spatial light modulators occurred when the peak intensity was greater than 3.5 $\times$ 10$^6$ W/cm$^2$ [39]. For sapphire and AlN material, the damage threshold were roughly 10$^{11}$ W/cm$^2$ [41] and 6.7 $\times$ 10$^8$ W/cm$^2$ [42], respectively. One can safely draw the conclusion that by replacing programmable optical elements with AlN metasurfaces, the damage threshold is increased by 2 orders of magnitude. Therefore, decent performance can still be expected from the compact AlN metasurfaces despite modest efficiencies.
Fig. 5. Optical ring lattices containing 6,8 and 10 optical traps at 1mm from the metasurfaces. (a)-(c) Field intensity, (d)-(f) amplitude, and (g)-(i) phase profile of the optical field.
The propagation characteristics of the generated optical field were further investigated. The optical ring lattice with 8 optical traps was taken as an example. As shown in Fig. 6, the size of each site increases linearly with the propagation distance. This linear relationship enables a monolithic AlN metasurface to generate optical traps with controllable areas for various amounts of atoms. Moreover, the optical field pattern becomes sharper with increased propagation distance as the target diffraction order gradually separates from the neighboring order. After the separation is completed, the optical ring lattices remain stable and robust during propagation due to the self-healing property [43].
Fig. 6. The sizes of optical traps as a function of propagation distance. Data obtained from simulations are marked by red crosses. The insets show the field patterns at propagation distances of 1, 5, 10 mm, respectively.
The rotation of particles is a useful degree of freedom in manipulation. There are two mechanisms to send particles into rotation. The first is transferring the orbital angular momentum (OAM) from light field to particles. The optical field with a helical phase structure carries OAM of $m\hslash$ per photon [44], where $m$ is the phase change around the singularity measured in multiples of 2$\pi$. The second is transferring the spin angular momentum (SAM) from light field to particles. The SAM $\sigma _z\hslash$ per photon is determined by the polarization state of the incident light. The OAM and SAM constitute the total angular momentum of $(m + \sigma _z)\hslash$ for trapped particles upon absorption of photons. The OAM value is determined by the design of AlN metasurface. In our design, each site has fixed $m=1$ as depicted in Fig. 3. The SAM can be altered flexibly by inserting polarizers between the Gaussian source and the AlN metasurface. For light with right circular polarization, left circular polarization and linear polarization states, the corresponding $\sigma _z$ values are +1, -1 and 0, respectively. Figure 7 presents the phase retardations imposed by AlN meta-atoms under different polarization states of incident beams. It can be seen that the difference in phase response is negligible, implying the polarization state can be switched to independently change the spinning rate of particles without affecting the helical phase profile.
Fig. 7. Phase retardation under different polarization states. Simulation data obtained under linear polarization states, left and right circular polarization are marked by red triangles, blue circles, and green crosses respectively.
In other words, the rotation of trapped particles can accelerate (same helicity, $m=1$, $\sigma _z = 1$), stop (opposite helicity, $m=1$, $\sigma _z = -1$) or proceed at the normal rate (linear polarization, $m=1$, $\sigma _z = 0$) when applied to light-absorbing particles by changing the polarization of incident beams. This property can transform the optical traps into optical spanners when target particles absorb UV light. For non-absorbing particles, they are confined by the potential wells created by the intensity profile and do not necessarily experience rotation.
Finally, we analyzed the capability of the optical ring lattice with 8 sites to trap particles. As shown in Fig. 8, the target optical ring lattice is set at 5 mm from the metasurface, where it is completely separated from neighboring diffraction orders. The particles were trapped in the off-axis dark sites of the optical ring lattice and simplified as a point dipole neglecting the effect of polarizability and detuning. The Rayleigh scattering model is used to calculate the optical forces because the size of the dipole is much smaller than the wavelength of optical ring lattices (375 nm). The optical forces acting on the dipole can be divided into two parts, namely the scattering force F$_{scat}$ and the gradient force F$_{grad}$. The scattering force arises from the absorption and re-radiation of light by the dipole. The gradient force is caused by the gradual variation of light intensity. The two forces can be given by [6]:
Fig. 8. The gradient force and scattering force analysis of the optical ring lattice with 8 optical traps. (a) Intensity profile with a white dotted line intersecting 2 of the 8 optical traps. (b) From top to bottom, the field in-tensity, the scattering force and the gradient force the dipole experiences as it moves along the white dotted line.
4. Conclusions
In summary, UV optical ring lattices containing 6, 8 and 10 optical traps were successfully generated using monolithic AlN metasurfaces. The corresponding energy efficiencies were 14.33%, 13.44% and 12.13% respectively. The optical ring lattices showed stable propagating properties and the size of the off-axis optical traps increases linearly with propagation distances. Light absorbing particles can be rotated in the off-axis optical traps around the local axis, and the rotation speed can be adjusted by altering the polarization state of the incident beam. Furthermore, particles were verified to be stably held at the dark center of the optical trap in the lattice with the gradient force using a dipole model. AlN metasurfaces make progress towards higher integration and compactness of optical apparatus through flexible wavefront engineering within a single ultra-thin layer, which will facilitate the exploration of exotic phenomena in the realms of transportable optical lattice clocks, cold atoms and quantum physics.
Funding
National Natural Science Foundation of China (62135013, 62234001, 62250071, 62274163); Beijing Nova Program (20230484466); Youth Innovation Promotion Association of the Chinese Academy of Sciences (2022000028, 2023123).
Acknowledgments
The authors acknowledge the financial support provided by National Natural Science Foundation of China (62022080, 62135013, 62274163, 62234001, 62250071); Beijing Nova Program 20230484466; Youth Innovation Promotion Association CAS 2022000028 and 2023123.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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