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Abstract
The radiation pressure force and torque on a one-dimensional bi-grating composed of a Si-SiO2 high contrast binary metagrating is analyzed for the purpose of stable beam riding whereupon a high power laser having an expanding Gaussian irradiance distribution propels the grating in outer space, free from gravitational forces. The binary metagrating structure has been simultaneously optimized to afford high forward thrust, and corrective restoring forces and torques in the event of small linear and angular disturbances. We demonstrate that stability may be enhanced at the expense of forward thrust. The validity of our metamaterial findings is reinforced owing to good agreements between finite-difference time-domain and finite element numerical methods. To reduce mass and enhance forward acceleration this laser-driven sail was designed to be free of a stabilizing boom.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Optical tweezers or single-beam optical traps are one of the most powerful means of contactless manipulation of nanoscopic/microscopic objects with a very diverse set of applications ranging from biology [1] to quantum optomechanics [2]. But the promise of optical tweezers to manipulate macroscopic objects of meter-scale has been held in check by the need for a tightly-focused beam that creates a gradient force in a very limited volume and effective distance. By engineering the scattering properties of the objects and the shape of the beam, the dynamics of macroscopic objects may be controlled and used for successful trapping, manipulation, levitation, and even propulsion without the need for a tightly focused beam. Most recently, NASA and Breakthrough Starshot Initiative aim to leverage these radiation pressure forces to propel a meter-scale light sail to the nearest stars with relativistic speeds. Nearly all the stable geometric designs proposed so far in the literature are subprime in three senses: (a) the ability of a conical/spherical/concave/convex sail to maintain its shape is questionable (b) a flat diffractive/nanophotonic design demands an undesirable mast to offset the center of mass away from the sail to achieve stability [3] and (c) all the designs sacrifice on thrust force for levitation/propulsion to enable a restoring force for stability.
In this work, we demonstrate the stability of a bi-grating lightsail comprised of subwavelength unit-cells of Si/$\mathrm {SiO_2}$. The unit-cell geometry is optimized to engineer diffraction efficiencies such that the sail is stable against linear and rotational perturbation without any offset between the center of mass and the sail while simultaneously achieving maximum forward thrust. This work is different from [4] in the sense: (1) forward thrust is also an objective of optimization (2) a stability basin is compared between two different forward thrusts that quantify the trade-off between forward thrust and stable initial conditions. What’s more, the forces and torque are validated using FDTD (MEEP) and FEM (COMSOL) methods. The two-dimensional dynamical analysis and electromagnetic response presented in this work may be extended into three dimensions using the scheme described in [5,6].
2. Theory
Consider the structure shown in Fig. 1 composed of two different panels $L$ (left) and $R$ (right) of length $\ell$. Each panel is comprised of an artificially designed sub-wavelength periodic lattice whose properties primarily arise from the design and distribution of meta-atoms or unit cells. Inspired by the principles of form birefringence and effective medium theory, the unit-cell of choice in this work is a ridge-width-modulated high contrast grating with $\mathrm {Si}$ nano-pillars on a low index $\mathrm {SiO_2}$ substrate. As shown in Fig. 1 (inset) is a multi-layer subwavlength binary unit-cell of period $\Lambda$. High-index Si nanopillars of height $h$ are deposited on low-index $\mathrm {SiO_2}$ substrate of thickness $t$. The width and position of the nanopillar is $w_{1,2}$ and $p_{1,2}$ respectively. There exist multiple benefits to using this subwavelength binary design: (a) this design has been shown to offer stable levitation [7], (b) optimal thermal management via radiative cooling [8], and (c) the feature sizes may be easily realized with existing e-beam lithography.
The bi-grating may levitate or propel when illuminated owing to radiation pressure forces. To model these forces, it is convenient to consider two different frames of reference: $(\hat {x}, \hat {z})$ and $(\hat {p}, \hat {n})$ attached to the laser and bi-grating respectively. The bi-grating has a total mass $M$ (including a payload) and the center of the bi-grating coincides with the center of mass of the system at $x_{cm}$ such that there is no offset between the two. The two coordinate spaces are related to each other as $p = (x - x_{cm})/\cos \Theta$ where $\Theta = -\theta _i$ is the attitude of the sail rotated about $\hat {y}$ and $\theta _{i}$ is the angle of the beam incident on the bi-grating.
Fig. 1. High contrast metasurface bi-grating configuration propelled by a Gaussian beam. The bi-grating is comprised of unit-cells with geometric parameters shown in the inset. $p_{1,2}$ is measured from the center of the unit-cell.
The meta-atoms offer exceptional control of wavefront and scattering response near their resonance, which may be engineered for desired force and torque on the bi-grating structure. Assuming a non-relativistic non-spinning bi-grating, the incident and scattered wavelengths are equal in the reference frame of the structure, and thus, the respective wave vectors may be expressed $\vec {k}_{i}= k\hat {n}$ and $\vec {k}_{m}^{L,R}= k (\cos \theta _m^{L,R} \; \hat {n} + \sin \theta _m^{L,R} \; \hat {p})$, where $k = 2\pi /\lambda$. The diffraction angles are governed by the grating equation,
where and $\mathbb {K}^{L,R}$ is the grating momentum such that $\mathbb {K}^{L}=-\mathbb {K}^R$ and $(\theta ^{L,R}_{t_{m}}$ , $\theta ^{L,R}_{r_{m}})$ are the diffraction angles for the $m^{\mathrm {th}}$ (transmitted, reflected) orders. The change in photon momentum $\Delta \vec {k}^{L,R}=(\vec {k}_i-\vec {k}_m^{L,R})/k_i$ in the reference frame of unit-cell may now be expressedThe bi-grating may experience force and torque when illuminated by a laser beam of peak power $P_0$, wavelength $\lambda <<L$, characteristic beam-width $2w_0$, and a Gaussian irradiance distribution
The non-spinning two-dimensional system described above entails 3 degrees of freedom: translation along $\hat {x}$ and rotation about $\hat {y}$ while it is propelled along $\hat {z}$. We define a state vector $\mathbf {x} = [x, \theta ]^T$ to analyze the stable transverse dynamics of the bi-grating system. In a close analogy of a oscillating spring system, the transverse dynamics of the bi-grating may be linearized and expressed as a set of ordinary differential equations (ODE) $\ddot {\mathbf {x}} = -K\mathbf {x}$, where K is a Jacobian with stiffness coefficients
We may conclude from Eq. (2) and Eq. (6) that it is possible to engineer a unit-cell geometry that simultaneously high forward thrust and stability against small perturbation. However, the forces depends on $\eta$ and analytically characterizing the diffraction efficiencies is very difficult except for simpler geometries. For the aforementioned unit-cell, rigorous diffraction theory must be applied. We make use of an FDTD based package in Python called MEEP [9] to solve for Maxwells Equation. The boundary conditions are assumed to be PML (Bloch periodic) along $\hat {n}$($\hat {p}$) . To account for fabrication constraints and numerical dispersion problem in FDTD, we limit our resolution to a step size of 20 nm or 50 pixels per micron. When an in-the plane polarizatied light of wavelength $\lambda$ is incident on the structure, the simulation is ran until the fields have decayed to $10^{-6}$ of their peak value. A similar FEA simulation is performed in COMSOL to validate the results from MEEP. The problem may now be formulated in terms of a multi-objective optimization problem i.e., for the set of variables $(\Lambda, h, p_{1,2}, w_{1,2})$ along with $\mathrm {Si (n=3.5220) /SiO_2 (n=1.4582)}$, we are seeking a sail design with with two figures of merit (FOM): (a) $\mathrm {FOM_1} = \mathrm {Im}(\mathbf {eig}(K))$ is minimized until it is 0 and (b) the forward thrust $\mathrm {F_z/F_0}$ is maximized, where $F_0 = 2P_0/c$. The optimization is performed using a genetic algorithm called NSGA-II [10] for the bi-grating-beam system for the following parameters: $L=1$, $w_0=0.5L$, $\lambda =1.2 \mu \mathrm {m}$, $M=1 \mathrm {gm}$, and $P_0=10$ kW.
3. Result and analysis
Since, this is a multi-objective optimization problem, there is no single solution. Instead, there exist many solutions that are Pareto-optimal i.e., lie at the optimal trade-off between two competing objectives. From the multiple Pareto-optimal solutions that were achieved, selected some fabrication-friendly designs shown in Table 1. The diffraction efficiency of both the unit-cell geometries comprising the left panel is evaluated using MEEP(FDTD) and COMSOL(FEA) and is shown in Fig. 2(a) and (b) as the function of incident angle. An excellent agreement is evident between the two methods. The change in photon momentum $\delta \vec {K}(\theta )$ is evaluated using both diffraction efficiency and Maxwell stress tensor and an excellent agreement between the two may be seen in Fig. 2(c) and (d). Clearly both the structure have very high reflectance that enables high forward thrust and hence $\Delta k^L_z$, whereas a non-zero $\Delta k^L_x$ enables restoring stable force for non-equilibrium positions, as shown in Fig. 2(c) and (d).
Fig. 2. The diffraction efficiencies of each orders for a sail tilted by $\Theta$. (A) and (B) correspond to the two structures with $F_z/(P_0/c)=1.26$ (Unit-cell I) and $F_z/(P_0/c)=1.70$ (Unit-cell II) respectively. Straight line and circles correspond to MEEP (FDTD) and COMSOL (FEM) results respectively.
Table 1. Optimized Geometries [in $\mu$m] (t=0.5 $\mu$m, $\lambda =1.2$$\mu$m)
The total force and torque on the bi-grating, however, is the result of the individual response of each panel with respect to the beam center. Thus, the force and torque are the functions of both angle and displacement from the equilibrium point. Shown in Fig. 3 is the force and torque on the bi-grating for both the structures assuming an expanded filling ($w_0=\ell$) and non-expanded underfilling ($w_0=0.5\ell$) beam. It is evident from the figure the restoring force and torque are linear for small perturbations from the equilibrium point and the slope of these curves correspond to Jacobians as described in Eq. (8) for linear stability analysis purposes.
Fig. 3. Force and Torque as a function of displacement and rotational perturbation for the two sail unit-cell designs:(a) Unit-cell I ($w_0=\ell /2, F_z/F_0=1.20$) (b) Unit-cell II($w_0=\ell /2, F_z/F_0=1.70$) (c) Unit-cell I($w_0=\ell, F_z/F_0=0.60$) (d) Unit-cell II($w_0=\ell, F_z/F_0=0.45$).
Linear stability analysis however provides only a partial picture of stability i.e., the stability of the system for very small perturbations near the equilibrium point. And hence the equations of motion must be solved using numerical methods like Runge-Kutta to fully gain an understanding of system stability. For an initial perturbation of $(\delta x/\ell = 0.01, \delta \theta = 0.01 \mathrm {rad}, \delta \dot {x}/\ell = 0, \delta \dot {\theta } = 0)$, the equations of motion are solved for the two geometries and two different beam width $w_0=0.5\ell$ and $w_0=\ell$. The results are shown in Fig. 4. The phase maps for all the cases are closed, suggesting a stable system. Please note, these simulations were performed for $P_0=10$kW and for a gigawatt class laser as proposed in the Breakthrough Starshot system will remain closed and bounded albeit the frequency of oscillation will be scaled accordingly.
Fig. 4. Runge-Kutta solution of equation of motion for the two sail designs: (A) Unit-cell I($w_0=\ell /2, F_z/F_0=1.20$) (B) Unit-cell II($w_0=\ell /2, F_z/F_0=1.70$) (C) Unit-cell I($w_0=\ell, F_z/F_0=0.60$) (D) Unit-cell II($w_0=\ell, F_z/F_0=0.45$).
The above simulation is repeated for a range of initial conditions to test the limits of stability and is shown in Fig. 5. It is evident from Fig. 5(A) and (B), an increase in forward thrust from 120% to 170% leads to 3.4% shrinkage in stable basin for an underfilling beam-width. Similar behavior may be observed for an expanding beam Fig. 5(C) and (D) i.e., high forward thrust implies sacrifice on permissible stable conditions. However, the expanded beam imparts lower forward thrust than the non-expanded underfilling beam.
Fig. 5. Light Sail with stable trajectory for initital perturbation and rotation (A) Unit-cell I($w_0=\ell /2, F_z/F_0=1.20$) (B) Unit-cell II($w_0=\ell /2, F_z/F_0=1.70$) (C) Unit-cell I($w_0=\ell, F_z/F_0=0.60$) (D) Unit-cell II($w_0=\ell, F_z/F_0=0.45$)
4. Conclusion
In summary, this study presents significant advancements in the development and utilization of rigid bi-grating light sails. Innovative $\mathrm {Si-SiO_2}$ bi-grating metasurface configurations have been engineered to ride Gaussian beams at a 1.2-micron wavelength, showcasing stability even in the face of expanding beams, rendering them highly applicable for practical uses. Two distinct geometries are introduced, each delivering substantial forward thrust of 120% and 170% (normalized by $P_0/c$).
The precision and dependability of this research are evident through the cross-validation of diffraction modes, employing both finite-difference time-domain (FDTD) simulations with MEEP and finite element method (FEM) simulations with COMSOL. Forces and torque are assessed using diffraction modes and the Maxwell stress tensor, demonstrating excellent agreement. Additionally, a basin of stability has been calculated, considering the impact of initial perturbations in position and tilt, providing valuable insights into the trade-off between forward thrust and stability against initial perturbations. Our findings underscore the remarkable stability of both proposed structures, even under a 1% perturbation in position and tilt.
In conclusion, the $\mathrm {Si-SiO_2}$ bi-grating metasurface geometries proposed in this study exhibit promising strides in light sail technology, demonstrating both stability and impressive thrust capabilities in the presence of expanding beams. However, practical challenges, such as sensitivity to wavelength and non-rigidity, warrant consideration. The potential integration of tunable phased-array lasers could enhance the practicality of our designs, addressing challenges related to wavelength sensitivity. Furthermore, the non-rigidity of the light sail should be acknowledged and may be effectively modeled using Lagrangian mechanics [11]. Additionally, optimizing the local deformation of the all-dielectric thin non-rigid sail can be achieved by applying the principles of conformal metasurfaces [12].
Funding
Breakthrough Prize Foundation (7dBPF Starshot LLC); National Aeronautics and Space Administration (80NSSC19K0975).
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.
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