Dynamically stable radiation pressure propulsion of flexible lightsails for interstellar exploration

Meter-scale, submicron-thick lightsail spacecraft, propelled to relativistic velocities via photon pressure using high-power density laser radiation, offer a potentially new route to space exploration within and beyond the solar system, posing substantial challenges for materials science and engineering. We analyze the structural and photonic design of flexible lightsails by developing a mesh-based multiphysics simulator based on linear elastic theory. We observe spin-stabilized flexible lightsail shapes and designs that are immune to shape collapse during acceleration and exhibit beam-riding stability despite deformations caused by photon pressure and thermal expansion. Excitingly, nanophotonic lightsails based on planar silicon nitride membranes patterned with suitable optical metagratings exhibit both mechanically and dynamically stable propulsion along the pump laser axis. These advances suggest that laser-driven acceleration of membrane-like lightsails to the relativistic speeds needed to access interstellar distances is conceptually feasible, and that their fabrication could be achieved by scaling up modern microfabrication technology.


Introduction
The concept of harvesting radiation pressure to propel spacecraft dates to at least some 400 years ago, when Kepler observed that the gas tails of comets point away from the sun as if blown by a solar wind (1).The physics of radiation pressure became known when Maxwell published his theory of electromagnetism in the 19 th century, giving rise to formal development of the concept of solar lightsails by Tsiolkovsky, Tsander, and others in the early 20 th century (2).Efforts to field solar lightsail spacecraft have led to recent successes including the JAXA IKAROS (3), NASA NanoSail-D (4), and the Planetary Society LightSail missions (5).
Whereas sunlight provides a relatively weak force for accelerating spacecraft in Earth's vicinity (~10 μN/m 2 for a perfect reflector at 1 AU), far greater accelerating forces can be produced if a high power density laser is focused onto a lightsail.Simple analysis suggests that laserpropelled lightsails can in principle be accelerated to relativistic velocities, offering a promising pathway for interstellar exploration using ultralight space probes (6)(7)(8).Due in part to the announcement of the Breakthrough Starshot Initiative in 2016, which seeks to enable this capability within the next generation (9,10), recent investigations have explored the viability of laser-driven lightsails as a basis for interstellar spacecraft propulsion (8,(11)(12)(13).A major challenge for such lightsails is the need to maximize reflectance while minimizing weight and limiting optical absorption to extremely low values, prompting multilayer or nanophotonic designs (14)(15)(16)(17)(18).Given the extreme rates of acceleration and the distances over which this acceleration will occur, it is necessary that such lightsails are designed to be structurally and dynamically stable, such that they can be propelled along the pump laser beam optical axis (19)(20)(21)(22)(23)(24)(25)(26)(27)(28)(29)(30)(31)(32)(33) with shapestable configuration.Several designs for rigid-or constrained-body beam-riding lightsails have been proposed, but to date no studies have considered the mechanical and beam-riding stability of meter-scale unsupported flexible membranes for interstellar propulsion.Notably, to achieve the target velocity of ~0.2c, the Starshot mission concept calls for ~1 g lightsail that is several meters in diameter; thus the membrane must be on the order of 100 atomic layers thick on average, including all framing or stiffening, suggesting that the flexibility of the lightsail must be taken into account in its design.
Here we consider the selection of materials, the structural and photonic design, and dynamic mechanical stability of flexible lightsail membranes, to investigate whether interstellar lightsail spacecraft can be realized with real materials, considering their finite stiffness and strength.We identify key material properties required for relativistic flexible lightsails, then develop a multiphysics simulation approach to explore the deformation and passively stabilized acceleration of spinning flexible lightsails with either specular scattering concave shapes or flat membranes with embedded metagrating nanophotonic elements.

Materials considerations
The Breakthrough Starshot Initiative (9) has challenged a global community of scientists and engineers to design a ~1-gram interstellar probe that will travel 4.2 light-years to reach Proxima Centauri B, the nearest known habitablezone exoplanet, within ~20 years of launch, as well as the necessary propulsion, communication, and instrumentation systems for such a mission.To accelerate the spacecraft to the required speed of ~0.2c, a ~10 m 2 lightsail weighing ~1 g would be propelled by an earthbased laser at incident power densities approaching ~10 GW/m 2 , experiencing ~10,000 Gs of acceleration for ~1000 seconds (8,10).A lightsail suitable for this mission must address immense engineering obstacles that will challenge the limits of materials science and engineering.One such challenge is that the lightsail must have reasonably high optical reflectance to produce thrust from the accelerating beam, yet must exhibit near-zero optical absorption (~1 ppm or less) and high thermal emissivity to prevent overheating.Recent studies have identified a handful of dielectric and semiconductor materials as potentially viable candidates (11)(12)(13), and nanophotonic designs made from these materials (or their combinations), have been reported to offer favorable reflectance, low absorption, and high emissivity (14)(15)(16)(17)34).In addition to achieving suitable optical properties over a wide temperature range, lightsail materials and designs must offer adequate mechanical strength and stiffness to endure the acceleration conditions necessary for interstellar propulsion.Table 1 shows key room-temperature mechanical properties and structural performance metrics for several of the candidate lightsail materials identified by previous studies, as well as for two common solar sail materials: aluminum and polyimide.More detailed properties are references provided in Table S1.
Among the candidates are several bulk crystalline dielectrics and semiconductors, including Si, quartz (SiO2), and diamond, which are hard, brittle, and have among the highest moduli and theoretical strengths of known bulk materials.Despite this such materials are rarely used in bulk structural applications, and are notorious for brittle failure in tension due to cracks initiated at surface defects.In practice, attainable specimen strength is limited almost entirely by the ability to fabricate device structures with defect-free surfaces.Although many decades of materials science and engineering have enabled each of these materials to achieve remarkable degrees of purity and scale of manufacture, present-day technology has yet to produce pure, defectfree, submicron-thick membranes over 10 m 2 areas.Two-dimensional crystals represent another class of candidate materials, among which MoS2 appears particularly promising for lightsail applications owing to its high strength and refractive index (15).The reported tensile strength for micron-scale suspended membranes of mono-or bi-layer MoS2 is nearly three times higher than that of any other material listed in Table S1.(35) Understanding the achievable strength and optical transparency of MoS2 films fabricated over large or nonplanar surfaces, at relevant layer thicknesses, and at elevated temperatures, is of considerable interest.
Another interesting class of materials for lightsail development is that of amorphous or nanocrystalline deposited thin films, including silicon nitride.Such thinfilm materials are widely used for modern MEMS.Promisingly, sub-micron thickness silicon nitride membranes have been fabricated at wafer scale, and further patterned with photonic crystal designs for nearunity reflectance (36,37).Ultralow extinction coefficients on the order of 10 -6 at near-infrared wavelengths can be achieved with high-stress stoichiometric silicon nitride (Si3N4), which is commonly employed in MEMS and cavity optomechanics applications (38,39).With favorable mechanical properties including high modulus and tensile strength, and potential research synergies between the fields of cavity optomechanics and optical levitation, Si3N4 is a particularly promising candidate material for lightsail development.
Ultimately, considerable effort will be required to develop any suitable materials system(s) to the scale of manufacture required for the interstellar lightsails proposed by the Starshot initiative, and careful consideration must be paid to the resulting mechanical and optical properties of the lightsail materials over a wide range of operating temperatures.

Stability considerations
In addition to possessing adequate optical and mechanical properties to endure the forces and optical intensities of the propulsion laser beam, the overall lightsail design must provide for adequate stability during acceleration.
Our work addresses two key aspects of stability: beamriding stability, the ability of the lightsail to follow along the beam axis without external guidance, and structural stability, the ability of the lightsail to survive the acceleration sequence without collapse, disruptive deformation of its shape, or tensile failure of its constitutive materials.These challenges and potential solutions are depicted schematically in Figure 1.
It is tempting to assume that the lightsail should be propelled by a beam of uniform laser intensity, to minimize thermal gradients and force nonuniformities that could distort the lightsail shape.This is the operating regime for solar sails, which navigate via active local control of solar reflectance or other attitude control mechanisms (1, 2).However, uniform plane-wave illumination is impractical for laser-propelled interstellar lightsails, as it would require a laser source of inconceivable power and aperture area to overcome diffraction of the beam over the extreme distance of acceleration.Assuming the propulsion laser would be constructed no larger than necessary to achieve the target mission velocity, the system must operate at or near the diffraction limit during the final phase of acceleration.We therefore restrict our study of beam-riding stability to static, weakly-focused low-order Gaussian beam intensity profiles.Other beam profiles such as higher-order Gaussian beams or doughnut beams (21,25) may be useful at earlier stages of acceleration when the propulsion system is not limited by diffraction.
Passive beam-riding stability is necessary for relativistic lightsail acceleration, because it is not feasible to provide closed-loop propulsive corrections by modulating or adjusting the propulsion beam in response to observations of the lightsail, owing to the large acceleration distances and final lightsail velocity.Limited by the speed of light, the round-trip delay between lightsail observation and the arrival of corrective modulation from the laser source in an active feedback loop would range up to several minutes at the end of the acceleration phase, whereas non-beamriding lightsails can veer off course on a timescale of milliseconds.Additionally, atmospheric turbulence and practical technological limitations will cause at least some perturbation to the desired position and profile of the beam (40).Thus, although some initial prescriptive corrective actions may be feasible from the laser source, the lightsail itself must ultimately be capable of aligning its acceleration trajectory to the beam axis without groundbased intervention, based solely on the local beam gradient.The challenge of steering the spacecraft then becomes primarily that of correctly pointing and slewing the direction of the ground-based laser source during acceleration.
A simple lightsail structure such as a flat specularly reflective disk is dynamically unstable and will eventually tilt and veer away from the beam.Several approaches to achieving beam-riding stability are depicted in Fig. 1A.Certain geometrically concave reflector shapes, including cones (21)(22)(23), hyperboloids (19), paraboloids, and other parametric shapes (32) have been predicted to offer stable beam-riding behavior, while other normally unstable convex shapes such as spheres can follow a stable trajectory by using more complex higher-order beam profiles (21).In addition to shaped specular lightsails, non-specular surfaces can be employed to produce restoring forces and torques, even for flat lightsails, by tailoring asymmetric optical properties to effect transverse forces (18,(24)(25)(26)(27)(28)33).Non-specular surfaces have been developed for solar lightsails to achieve greater maneuverability including enhanced lateral and rotational forces (41).
Our present study addresses only marginal (undamped) beam-riding stability, in which the lightsail exhibits bounded, oscillatory displacement and tilting about the beam axis in response to a finite beam-lightsail misalignment, during acceleration.Continuous perturbations to the beam-lightsail alignment during propulsion, e.g., due to atmospheric turbulence, can cause the oscillatory motion of the lightsail spacecraft to grow in magnitude, which could eventually cause marginally stable lightsails to escape the beam.Furthermore, for nonrigid structures, and flexible membranes in particular (42), the energy buildup in acoustic modes (shape distortions) could also destabilize or overstress the lightsail.Therefore, interstellar lightsails will likely require either active or passive means of damping their beam-riding oscillations and shape vibrations to achieve asymptotically stable propulsion along the desired cruise trajectory.Passive damping approaches might include the use of structures with damped internal degrees of freedom (31), employing nonlinear optical materials (30), or utilizing materials with highly varying temperaturedependent optical properties to enable hysteresis of the restoring forces.Active optical control surfaces for improved beam-riding stability have been demonstrated for solar lightsails (43), but developing such control surfaces to operate under the extreme beam intensities and low mass budget proposed for interstellar lightsail propulsion remains an unsolved challenge.
Turning our attention to structural stability, the interstellar lightsail must be capable of surviving the acceleration forces without collapsing upon itself or experiencing mechanical failure.This is a substantial challenge for the Starshot concept, which calls for meter-scale lightsail membranes of average thickness below 100 nm.Table 1 shows the allowable average thickness for each membrane type.This is not intended to suggest that lightsails should be constructed from uniformly thick continuous membranes, or to impose an upper limit for structural thickness.Optimized lightsail designs will likely incorporate multiple materials (16) and complex spatial patterning, e.g., perforations (15,17,30,37) or optical resonators (18,24,25,33,34), so as to maximize reflectance, emissivity, and tensile strength.However, with such limited mass budget, the finite strength and structural rigidity of the lightsail must be considered.
A prior study addressed tensile strength requirements by treating the lightsail as a rigid parametrically shaped shell, finding that certain surface curvature ranges minimize stress (44).Another recent study presented 2D analytic and finite-element models of deformation instabilities in uniformly illuminated lightsail membranes (45).In the absence of external constraints, the behavior of unsupported or loosely supported flexible membranes subject to nonuniform forces is considerably complex (42).
In general, thin unsupported membranes will collapse and crumple upon themselves when subject to focused laser propulsion, as depicted in Fig. 1B.A curved surface offers greater structural rigidity than a flat membrane, while also conferring the benefits of improved stress distribution that make thin curved shells useful in structural applications.However, open concave shapes such as cones and paraboloids are still prone to collapsing by elongation, an intrinsic instability for such shapes.Structural reinforcement such as framing could be added, but only at the cost of reducing the membrane mass.Potential approaches for structural reinforcement include microlattices (46), gas-filled envelopes (47,48), annular tensioning, fractal supports (49), tensegrity structures (50), or lamination with low-density or corrugated backing layer(s).Ultimately, given mass and material constraints, even a structurally rigidified lightsail will likely deform during acceleration, potentially changing the distribution of stress within the membrane or altering its beam-riding properties.An additional challenge for any structural materials is that the proposed lightsail membranes are generally partially transparent.Thus, even if placed behind the lightsail surface, the frame or backing materials may still be exposed to a high laser intensity, limiting materials selection.
As an alternative to structural support, spin-stabilization may be employed to prevent shape collapse.This effectively rigidifies the lightsail via inertial tensioning, and also gyroscopically stabilizes the lightsail to resist tilting, all while avoiding the added mass and complexity of structural reinforcement.For this reason, our work to date has focused on spin stabilized lightsails.However, spin-stabilization greatly complicates the dynamics of the lightsail, particularly for flexible membranes which are prone to complex instabilities (42), and is not necessarily effective for all structures under all conditions.Perhaps most counterintuitively, gyroscopic effects can disrupt the beam-riding behavior of certain lightsail designs that would be dynamically stable under non-spinning (rigidbody) conditions, particularly in the case of angular misalignment between the beam axis and the spin axis (21,22).Thus, the use of spin-stabilization to prevent shape collapse in ultrathin flexible lightsails can be a challenging design objective.
To provide first-order insights into the general viability of constructing large-area structurally stable lightsails from the candidate materials, we have defined two figures of merit in Table 1.The first is the stationary burst diameter (Dmax), which is the maximal diameter at which a flat circular membrane of areal density 0.1 g/m 2 , rigidly clamped at its perimeter, can sustain a pressure of 67 Pa applied to one side without rupturing (51).This is the effective photon pressure of 10 GW/m 2 illumination, assuming unity reflectance.Practical lightsail designs may have lower reflectance; may incorporate multiple materials or inhomogeneous patterning, owing to the need to optimize tradeoffs between reflectance, thermal properties, and strength; and would need to operate at substantially elevated temperatures.Dmax is thus only intended to serve as an order-of-magnitude indicator of the viability of large-area perimeter-supported membranes for this application.The assumed 'stationary' perimeter constraint provides an overestimate of the required membrane tensile strength, since any viable perimeter structure would not be stationary, but instead must have an extremely small mass that would accelerate along with the lightsail.But interestingly, even this simplified calculation suggests that while conventional solar lightsail materials such as aluminum and polyimide (Kapton) are far too weak to span meter-scale areas between structural supports, some candidate membrane materials (Si3N4 and MoS2) are in principle strong enough to span 10 m 2 areas (Dmax > 3.6 m) with perimeter support only -even in the stationary case.This is an encouraging conclusion for the development of structurally supported lightsails.
The second figure of merit in Table 1 is the maximum spin speed (fmax) at which a flat, 10 m 2 circular membrane could be spun without rupturing due to tensile failure.This is relevant because, for the designs considered here, relatively high spin speeds are required to produce both shape stability and beam-riding stability, often approaching the materials' tensile limits.The viability of spin stabilization depends on the spin speed, the acceleration conditions, and the specific design of the lightsail.For this reason, we have developed multiphysics numerical simulation methods to investigate the dynamic stability of flexible lightsails.

Mesh-based simulator for flexible lightsails
To model realistic flexible lightsail membranes of various shapes and optical designs, a triangular surface mesh is constructed (Fig. 2A).Each vertex is assigned a mass based on the local membrane thickness, the area of the adjoining triangles, and the material density.Elastic behavior of the membrane is captured by the edges, each of which is assigned a linear elastic coefficient (i.e., spring constant) based on the mesh geometry and local material properties.This approach omits the negligible bending stiffness and the specific shear modulus of the material, but provides reasonable first-order insights into the behavior of ultrathin membranes under tensile loading, which is the predominant type of loading in lightsail applications.In future efforts, non-isotropic material properties and the full elastic behavior of the lightsail material(s) could be considered.
Light-matter interactions are evaluated over each enclosed triangular mesh element, where incident light produces photon pressure forces, optical absorption heats the lightsail, and thermal radiation cools the lightsail (Fig. 2B).The heating, cooling, and optical forces calculated at each triangular element are distributed to the adjoining nodes, which represent the temperature distribution, momentum, and shape of the structure.Thermal conduction is calculated along the mesh edges based on the local material properties and mesh geometry, whereas temperature is calculated at each node based on its mass and the specific heat of the material.As the temperature distribution is known throughout the structure, we also include the effects of linear thermal expansion, which contributes to thermal strain.
In the simplest type of optical interaction, the force of photon pressure acting on a triangular element is governed by the effect of specular reflection from the surface (Fig. 2C), with the resulting force occurring normal to the surface.The photon pressure is calculated based on the local beam intensity, the relative polarization, the incidence angle to the surface, and the local membrane properties.Future efforts could also consider the optical effects of local temperature, strain, or the time-varying state of active control surfaces, as well as beam profiles that vary in time or distance from the source.Our present work has studied only the first seconds (up to 10 s) of acceleration following an initial beam-lightsail misalignment, which is adequate for observing marginally stable behavior over many periods of oscillation, determining steady-state temperature distributions, and identifying many types of instabilities.The present model does not address relativistic effects necessary to model the full acceleration duration to interstellar mission velocities.
In the next section, we first assume constant values for the reflectance, transmittance, and correspondingly absorptance to model the basic behavior of curved and flat specular lightsails.Then, we will introduce improvements to the optical calculations, including angle-dependent reflectance and absorption based on Fresnel coefficients, and considering the effects of multiple reflections of light within concave curved lightsail.Finally, we present simulations of non-specularly reflecting surfaces such as diffractive metagratings (Fig. 2D), which allow flat lightsails to achieve beam-riding stability.With future work, this basic simulation approach could be adapted to study lightsails made from optical metasurfaces (Fig. 2E) with a wide range of optical behaviors.
To simulate acceleration of the lightsail, and to assess its apparent stability, we implement a finite-difference timedomain approach wherein we calculate the forces and heat flow acting at each mesh vertex, then evaluate the resulting changes in position, velocity, and temperature over a time step Δt.With sufficiently small Δt, we can simulate the propagation of membrane vibrational modes, and can obtain reasonable predictions of the lightsail dynamic behavior during the initial acceleration phase.Thermal and mechanical membrane failures can be detected when a nodal temperature exceeds a threshold value or when the strain in an edge exceeds the tensile limit of the material.

Dynamics of flexible curved lightsails
Fig. 3 depicts the simulated behavior of flat versus curved (paraboloid) lightsails and the effects of spin stabilization, using optical and mechanical properties roughly corresponding to a 43 nm thick Si membrane (0.1 g/m 2 ) whose properties are parameterized at room temperature.We first consider a 1-meter diameter flat lightsail, illuminated by a λ = 1550 nm Gaussian beam profile, with 4 GW/m 2 peak intensity and a 0.5-meter beam waist, offset by 80 mm from the center of the lightsail.This lightsail size was chosen as a compromise between computational cost and the desire to simulate large macroscopic structures with reasonable mesh accuracy.Because these lightsails are smaller than 10 m 2 in area as proposed for Starshot, they can be spun faster than the values shown in Table 1; the unloaded maximum spin speed for the flat membrane in this case is ~470 Hz.Without spin stabilization, the flat lightsail membrane is structurally unstable and collapses upon itself as expected.Spin stabilization (fspin = 135 Hz) prevents the lightsail from collapsing, but lacking any means for beam-riding stability, the lightsail quickly veers away from the beam axis.A paraboloid shape can offer beam-riding stability according to rigid-body calculations, but in the flexible mesh simulation, the membrane quickly becomes elongated and collapses upon itself.With inadequate spin stabilization (fspin = 90 Hz), the shape collapse is delayed but not prevented, in this case leading to tensile failure.With adequate spin stabilization (fspin = 135 Hz), the shape remains stable, and beam-riding stability is achieved throughout the 1 s duration of the simulation.Animations of all five cases are available in Supplementary Video 1.
To facilitate comparison, all lightsails in Fig. 3 have the same surface area and thus the same total mass.As a result, the paraboloid lightsails are smaller in diameter, and thus accelerate more slowly than the flat lightsails due to their smaller aperture area.Therefore, a drawback of deeply curved shapes is that they tend to be heavier than flat lightsails of the same aperture area and thickness.Also, the sloped peripheral surfaces of the paraboloids do not propel the lightsail along the z direction as efficiently, since some of the photon pressure is directed radially.Light reflected from these edge areas might in fact impinge somewhere on the opposite side of the lightsail, thus imparting additional photon pressure there, potentially affecting the acceleration and stability of the lightsail.We thus improved our simulation code by considering multiple reflections within the lightsail using a simplified raytracing approach, and by calculating reflectance and absorption based on Fresnel coefficients, thus better modelling the angle dependence of light interaction (Fig. 4).Animations of these and other raytracing-based simulations are shown in Supplementary Video 2. Fig. 4A compares the shape and temperature behavior of a 1-meter diameter spin-stabilized paraboloid lightsail representing a 43-nm thick Si membrane, with and without the effects of multiple reflections within the lightsail, with simulation conditions being otherwise the same as for the stabilized paraboloid shown in Fig. 3. Due to the modest reflectivity of silicon (0.45 for λ = 1550 nm at normal incidence), the effects of reflected light can substantially disrupt the lightsail stability.Considering only the effects of the incident light beam, the lightsail trajectory appears stable, similar to that shown in Fig. 3; but upon introducing the effects of secondary reflections, the lightsail shape and trajectory become unstable.While the secondary reflections do increase the total photon pressure on the lightsail, resulting in faster acceleration, reflected light striking the opposite side of the lightsail counteracts the restoring forces and torques produced by the first reflection, thus destabilizing the lightsail.Also evident from the temperature profiles is the localized heating caused by the focusing of reflected light, with the peak temperature increasing from ~700 K to ~1000 K.
Increased temperatures are problematic for lightsails because materials generally weaken or decompose at elevated temperatures.An upper temperature limit may be imposed by material sublimation or decomposition, as even small amounts of material loss could substantially weaken or alter such thin lightsails (15).If we limit the mass loss to 1%, for a 1 g, 10 m 2 lightsail for 1000 s, literature values predict a limiting temperature of ~1300 K for crystalline Si (52), suggesting that the projected temperatures above are acceptable.However, for semiconductor materials, free-carrier absorption increases dramatically with temperature as the bandgap narrows, which may lead to a thermal runaway situation at a much lower threshold temperature.Furthermore, two-photon absorption may trigger thermal runaway above certain laser intensities, regardless of initial temperature.A recent analysis of an optimized Si-based nanophotonic lightsail estimated the threshold temperature for thermal runaway to be only 400-500 K, and placed an upper limit on beam intensity at ~5 GW/m 2 (53).Thus, our simulations predict unsurvivable temperatures for Si membranes, and unsurvivable light intensities in the regions of focused secondary reflections.Nonetheless, we can conclude that spin-stabilization can prevent shape collapse of flexible curved lightsails.While multiple reflections within deeply curved lightsails can increase the acceleration rate, they also increase the risk of localized hotspots from focused light and can also reduce or disrupt beam-riding stability.However, this only affects curved shapes which are deep enough to encounter multiple reflections over the range of tilt angles and shape deformations experienced during acceleration.Another challenge is that curved lightsail shapes would likely be more difficult to fabricate at the meter scale, and for crystalline materials, would introduce weaknesses at joints, grain boundaries, or wherever weaker crystal planes are exposed.We are thus motivated to investigate flat membranes as an alternative to curved shapes, owing to likely easier fabrication and scale-up, and to the lack of internal secondary reflections within the lightsail.However, we note that not all curved shapes are destabilized by internal reflections, and that shallower spin-stabilized curved shapes can achieve stability without encountering conditions that produce secondary internal reflections (21) (Supplementary Video 2).
Since Si exhibits thermal runaway at relatively low threshold temperatures and beam intensities (53), we turned our attention to a different material for the flat lightsails.Even if radiative cooling of Si-based lightsails could be improved, using any material with such a low runaway threshold temperature appears problematic, since any local defect, contamination, or brief localized focusing of light exceeding the two-photon absorption threshold, could initiate catastrophic thermal runaway spreading across the entire lightsail.Si3N4 is used extensively in other high-temperature applications, and its larger optical bandgap (~5 eV) and lower free-carrier absorption are attractive.Furthermore, amorphous Si3N4 films of excellent optical quality can be deposited using LPCVD, suggesting an easier route for fabrication over large or complex surfaces (36,37).A drawback to Si3N4 is its relatively low refractive index (n ~ 2), resulting in lower reflectance and less efficient diffraction.
It is difficult to estimate the practical limiting temperature for Si3N4 lightsails based on its properties reported in literature, owing to the diversity of its applications, the varying stoichiometry, density, and stress produced by chemical vapor deposition methods, and the relative complexity of the N-Si system at high temperatures.As an upper limit, we estimate the temperature at which vacuum decomposition would occur (again choosing a threshold of 1% decomposition over 1000 s) to be ~1600 K, based on decomposition rates for crystalline powders of Si3N4 (54).Practical thermal limits would likely be much lower, as the decomposition evolves nitrogen, leaving elemental silicon at the material surface, which could dramatically increase optical absorption and lead to thermal runaway.Other high-temperature risks include weakening, changes to stress distribution, activation of traps or defects, or crystallization of the material.Further experimental measurements are necessary to accurately determine the limiting temperatures and power densities for Si3N4 lightsails.

Optical design for passive stabilization of flat lightsails
Passive stabilization of lightsail dynamics requires the presence of restoring forces and torques.The previously discussed concave curved shapes achieve this via their shape alone, but flat specular lightsails cannot achieve beam-riding stability because specular reflection only produces forces normal to the surface.One approach to obtain beam-riding designs for flat lightsails is to make use of engineered optical anisotropy.In diffractive gratings with symmetric unit cells, such optical anisotropy can be achieved with nematic liquid crystals (55).Alternatively, optical anisotropy can be created by designing asymmetric diffractive metagratings, e.g., with the unit cells comprising two resonators of dissimilar widths (24,33).In such structures, anisotropic scattering of incident light into the grating diffraction orders manifests in optical forces transverse to the membrane.Moreover, optical metasurfaces comprising subwavelength scatters in the form of disks (18), blocks (25), or spheres (29) can be used to shape the wavefronts of scattered light, redirecting incident photon momentum in anomalous ways to produce beam-riding stability.
We describe stable designs for flat lightsails by designing asymmetric diffractive metagratings, patterned from Si3N4 as shown in Fig. 5.A specifically designed pair of mirror-symmetrically arranged metagratings can passively stabilize translations and rotations along one axis (24,33).Consequently, we employ two distinct and perpendicularly arranged metagrating designs to enable stabilization of translations along both x and y, and rotations θ about yBF (pitch) and ϕ about xBF (roll).As shown in Fig. 5A, a circular lightsail is partitioned into four sectors, forming two orthogonal pairs of symmetrically opposed wedges.We assume a linearly polarized incident beam, with its electric field aligned with the body-frame y-axis yBF.Thus the blue sectors (1/6 of the lightsail area) experience transverse-electric (TE) polarization, and the brown sectors (1/3 of the lightsail area) experience transverse-magnetic (TM) polarization, and the specific asymmetric metagratings for each sector (Fig. 5B) provide stabilizing forces and torques for their respective design planes and polarization.For spinstabilized lightsails, we assume that the beam polarization rotates synchronously with the spinning lightsail.Electromagnetic simulations were performed to determine the optical response of metagrating unit cells.
For a laser propulsion wavelength of λ = 1064 nm, we identified self-stabilizing metagrating designs using linearized stability analysis.While non-spinning designs are marginally stable if the eigenvalues of the Jacobian matrix derived from the lightsail equations of motion are purely imaginary, for spinning lightsails as linear-time periodic systems, we must employ Floquet theory to assess stability of the designs (56,57).Specifically, our chosen unit cell designs for lightsails spinning at 120 Hz produce absolute values of eigenvalues equal to 1, i.e., , which is a sufficient and necessary condition for marginal stability.We study the initial acceleration of our marginally stable lightsail designs, subject to an initial alignment error, which allows us to verify the beam-riding stability predicted by linearized rigid-body Floquet analysis, and importantly, to investigate whether these spinning sails retain their beam-riding stability when the assumption of rigidity is removed.
The two metagrating designs each support m = ±1 diffraction orders in addition to the specular order in reflection and transmission.Asymmetry in the intensities of the diffracted orders provides the mechanism for lateral restoring forces, while asymmetry in the angular dependence of optical thrust provides the mechanism for restoring torques.Assuming a Gaussian beam with a width equal to 40% of the lightsail diameter, i.e., w = 0.4D, we calculated the normalized optical forces and torques induced on a rigid lightsail of the proposed design, over a range of incidence angles (θ, ϕ) and translational offsets (x, y).These induced forces do not depend on acceleration distance z and yaw tilt ψ because we neglect beam divergence and assume synchronous rotation of the polarization.Stabilizing behavior is evident from the negative slopes of Fx and Fy versus x and y, respectively, with zero crossings (equilibrium positions, indicated by gray isolines) present near the beam center (x, y = 0) over the full ±10° range of plotted tilt angles θ and over a ~ ±5° range of roll angles ϕ, respectively (Fig. 5C, 5D).The relative insensitivity of lateral equilibrium position to tilt angle appears to be beneficial for improving stability in the spinning case.
Restoring torques limit angular rotation relative to the optical axis, although the situation is less straightforward for the spinning case.Beam-center optical torques about x and y are shown in Fig. 5E, exhibiting stabilizing polarity and derivative over a ±6.5° range of pitch and roll.While the TE metagrating provides a larger torque about y, the TM metagrating yields slightly stronger optical forces along y.We note that τx(ϕ) is markedly nonlinear beyond ~ ±1.5°, which restricts conclusions drawn from linear stability analysis to this angular range.Rotations beyond ±1.5° will give rise to nonlinear dynamics, resulting in possible coupling to and between distinct frequency components.Our time-domain numerical simulations allow this behavior to be studied by considering the full angle-resolved optical response of the metagratings.

Dynamics of metagrating-based lightsail
To verify our predictions about dynamical stability of rigid lightsails patterned with the composite metagrating design reported here, we numerically solved the equations of motion.The dynamics of flexible lightsails with the same metagrating motif were also simulated using our mesh-based modeling approach.The lightsail diameter is D = 1 m, for which the chosen composite metagrating design yields a total mass of m = 0.867 g.A Gaussian propulsion beam with a peak intensity of I0 = 1 GW/m 2 and a width of 0.4D = 40 cm was assumed.
We present here an exemplary case of passive stabilization of a flexible metagrating lightsail, in which an initial translational offset of x = y = 5 cm in the lightsail position relative to the beam optical axis and an initial (pitch and roll) tilt of θ = ϕ = −2° was assumed (Fig. 6).In the Supplementary Information, we also present results for passive stabilization of a flexible metagrating lightsail being only initially displaced (Fig. S3), but not tilted relative to the beam optical axis.Snapshots of the flexible lightsail position, orientation and shape every 0.5 s are shown in Fig. 6A; an animation of the simulation is available as Supplementary Video 3.For the studied duration of t = 5 s, the lightsail oscillates about the beam axis while remaining relatively flat and level, with no visibly apparent shape distortion thanks to the sufficiently large tensioning forces arising from spin-stabilization.Due to the finite absorptivity of Si3N4, the center region of the lightsail reaches a maximum temperature of 959 K.In contrast, the peripheral area remains significantly cooler (Fig. S4a), heating up to a maximum temperature 489 K.The slower heat up process on the edge of the lightsail can be attributed to limited heat transport from the hot center of the lightsail, owing to the low thermal conductance of the silicon nitride membrane.Thermal conduction dominates over direct absorption as a source of heating in the peripheral areas of the lightsail, due to the underfilling laser beam.The peak temperature appears sufficiently below the vacuum decomposition temperature of Si3N4 (54), although this temperature is likely too hot for most payloads.Increasing the assumed hemispherical emissivity of 0.1 for thin Si3N4 membranes would be desirable, for example with additional metasurface designs for selective thermal radiation in the mid-infrared regime, or addition of other material layers (15,16,58).Our simulation predicts a maximum strain of 0.091% in the Si3N4 membrane (Fig. S5A).With a Young's modulus of 270 GPa, such strain translates to a tensile stress of approximately 246 MPa, which is > 40 times lower than the reported 6.4 GPa tensile limit of Si3N4.(Table 1).Therefore, a meter-sized flexible lightsail is expected to exhibit mechanical stability in its propulsion phase despite being subject to large thermal gradients, spin tensioning, and nonuniform beam intensity.
Examining the trajectory of flexible and rigid lightsails indicates that their motion is bounded and thus the dynamics appear to be marginally stable as expected (Fig. 6B).During the entire 5 s duration of simulated propulsion, the lightsails remain within 180 cm of the beam center, as they traverse triangle-like trajectories in the x-y plane.Comparing the trajectory of the flexible lightsail to that of the identically patterned rigid version, both exhibit similar behavior consistent with marginal stability.Plotting the oscillatory displacement of the lightsail centers-of-mass along x, y and the radial distance r versus time (Fig. 6C) better reveals the slight deviations in trajectory.In the beginning, both flexible and rigid lightsails follow almost indiscernible trajectories.After 0.8 seconds, differences in x and y become more visible, but do not grow continuously over the studied time duration, ruling out the accumulation of numerical errors due to insufficiently small time stepping as a possible reason.Instead, we attribute the small differences in position to the role of shape distortions in flexible lightsails and the effect of thermal expansion.
To elucidate the influence of temperature and thermal strain in the flexible lightsail simulations, we simulated propulsion under conditions of zero absorptivity and emissivity to keep the lightsail temperature constant at 300 K (Fig. S6A).The resulting trajectory is again very similar that of the flexible and the rigid lightsail but does not match either perfectly.However, a closer look reveals a closer resemblance in dynamics between the thermally inactive flexible lightsail and the rigid lightsail, which suggests that thermal effects play a bigger role than shape distortions, both of which exist due to the non-uniformity of the laser and optical pressure as well as the resulting non-uniform temperature distribution and thermal strain.We also observe oscillatory motions with multiple frequency components for both translations and rotations.Examining the lightsail tilt angles θ and ϕ versus time for the rigid lightsail (Fig. 6D, 6E), we observe a fastoscillating component at 240 Hz for both tilt angles associated with the assumed 120 Hz spin speed due to its two-fold cyclic symmetry (see insets) superimposed upon multiple slower nutation/precession frequencies.Throughout the simulation, although the pitch and roll angles grow larger than the initial tilt offset, both θ and ϕ remain bounded between ±7°.For the flexible lightsail tilt, we present the distribution of pitch and roll angles for all mesh triangles across the lightsail surface as normalized time-domain histograms in Fig. 6F and Fig. 6G.We can observe overwhelmingly similar and bounded rotation dynamics for the flexible lightsail, proving again the effectiveness of spin stabilization.At closer look at shorter time scales reveals subtle differences in the time evolution of pitch and roll angles, as indicated by an angular spread of tilt angles of ~1°.
The simulations provide a high-fidelity numerical approximation of the initial lightsail trajectory, stress distribution, and shape evolution, which is sufficient to characterize the general beam-riding and structural behavior of stable lightsail designs, and to definitively identify unstable designs.The specific design presented here appears marginally stable for the chosen initial conditions throughout the 5 s duration of acceleration.However, substantial deviations from this design and set of chosen parameters can produce unstable behavior.Decreasing the spin frequency from 120 Hz to 80 Hz, increasing the beam diameter from 0.4D to 0.5D, or increasing the gap between resonators by 20% for both TE and TM unit cells all result in unstable dynamics (Fig. S7), which highlights the importance of judiciously choosing the beam width, spin frequency and optical design for passive stabilization.

Conclusions
We have presented time-domain multiphysics simulations of flexible lightsail membranes undergoing the initial stages of acceleration toward relativistic velocities due to radiation pressure propulsion.In this work we have explored both the lightsail beam-riding stability and dynamic structural stability.Specifically, we have shown proof-of-concept examples of flexible, meter-scale lightsails, spin-stabilized to tension the lightsail, that exhibit a stable shape without any stiffening elements.We have observed that certain concave specularly reflecting lightsail shapes such as paraboloids can enable both beamriding stability and shape stability, and have also demonstrated passively stabilized flat lightsail designs based on Si3N4 metagratings.The latter is of particular interest for experimental lightsail development, owing to the favorable mechanical strength and low optical absorption of Si3N4, and its ability to be fabricated in planar thin-film form at the wafer scale.Specifically, we have demonstrated that high-speed spin stabilization at 120 Hz is largely effective in rigidifying a flexible metagrating-based lightsail to exhibit similar dynamics compared to its rigid counterpart, while the subtle differences between flexible and rigid metagrating lightsails can be explained by both structural deformations and thermal effects.
We note that the size and average illumination intensity for the designs reported here fall below the nominal design targets proposed by the Breakthrough Starshot program for interstellar missions.Furthermore, the dimensions our metagratings causes them to be heavier than the nominal target of ~0.1 g/m 2 .Further optimization of the metagratings and lightsail structure, potentially including the addition of other materials, will be necessary to produce a full-scale Starshot lightsail design.Our present design represents an important first step towards this goal, and the simulation tools reported here will likely be useful in achieving this goal.Future work should be directed towards modelling the temperature dependence of optical reflectivity, absorptivity, and emissivity, in order to better understand the upper limits of achievable acceleration--a key factor in determining the viability of interstellar exploration via laser-propelled lightsails.For many materials, experimental efforts may be needed to probe high temperature properties.Other second-order effects may also be worthy of investigation, such as the effects of strain on optical properties.Our simulation approach may also be useful in addressing other challenges for interstellar lightsail development, such as payload integration and codesign of the propulsive laser system.Despite numerous simplifications, we have addressed the most relevant physics for flexible lightsail acceleration and flight, including first-order linear elastic behavior, heat flow, and optical scattering.We have presented timedomain simulations of stable lightsail structures undergoing up to five seconds of acceleration.Future work may allow longer simulation durations, but regardless of the chosen simulation duration, it is difficult to infer absolute stability from time-domain simulations of marginally stable lightsails, so a more useful future application of our approach might be the improvement and optimization of lightsail designs.Our present lightsail patterning was selected based on parametric optimization under rigid-body Floquet theory, but the complexity of flexible lightsail dynamics suggests that a more advanced optimization approach based on numerical time-domain simulations may yield more favorable designs, particularly as increasingly complex building blocks and physical behaviors are modelled.Future refinements, such as implementing temperature-dependent optical properties, or improving numerical time-stepping with implicit and higher-order methods, may allow for studies of acceleration over a longer period.Nevertheless, study of the initial seconds of lightsail acceleration provides considerable insight into flexible lightsail design.In connection with the work reported here, we have published an open-source version of our simulation code (59) to further expand effort by the lightsail community to develop new and improved designs for interstellar propulsion, optical levitation, and long-range optical manipulation of macroscopic objects.

Materials and Methods
Electromagnetic response of the TE and TM metagrating designs were simulated in COMSOL Multiphysics assuming periodic Floquet boundary conditions.For highstress stoichiometric silicon nitride, we assumed a refractive index of Re(n) = 2 and an extinction coefficient of Im(n) = 2 × 10 -6 at λ = 1064 nm (59).The TE and TM metagrating unit cells shown in Fig. 5B are defined by w1 TE = 600 nm, w1 TM = 520 nm, w2 TE/TM = 200 nm, d TE = 1600 nm, d TM = 1350 nm and an gap of 190 nm and 200 nm between resonators for the TE and TM unit cells, respectively.The resonators' height and substrate thickness were chosen to be 400 nm and 200 nm, respectively.The process of identifying these selfstabilizing unit cell designs, which was based on Floquet theory, i.e., evaluation of the absolute values of the eigenvalues of the monodromy (state transition) matrix, is described in more detail in the Supplementary Information.Except for the resonator height and substrate thickness, all geometrical parameters were varied systematically to select and compare suitable metagrating designs.By sweeping the incidence angle between ±25° for both pitch (θ) and roll (ϕ) tilt, angle-dependent optical pressures can be obtained via integration of the Maxwell Stress tensor around the respective unit cell.We used the exported look-up-tables of optical pressures as inputs to our rigid and flexible membrane dynamics simulations.In the former case, optically induced forces and torques can be derived assuming a Gaussian beam characterized by its peak intensity I0 and beam width w.For a given set of initial conditions (position, velocity, angular orientation, and angular frequency), the coupled equations of motion were evolved numerically using MATLAB's ode45 solver to obtain the trajectory and time-dependent displacement and tilt of propelled rigid lightsails described by their centers-of-mass.Normalized relevant quantities can be converted to real-life values by specifying I0, the lightsail diameter D and calculating the normalized time constant t0 = (mc/I0) 1/2 , where m is the total mass of the lightsail.
For more detailed description of the modeling and dynamical simulation of flexible curved and flat lightsails, we refer to the Supplementary Information.The MATLAB code has been made available on GitHub.

Tabulation of Material Properties
We have collected a number of candidate material property values from the literature for the purpose of simulating the structural dynamics of lightsails.These appear as table S1 below.This is not intended as an exhaustive list or ranking of candidate materials for the interstellar lightsail, and importantly, it should be noted that the published properties of these materials can vary greatly depending on the method of fabrication, as well as the test geometry and method of measurement.Furthermore, most properties are reported based on room-temperature measurements, whereas during acceleration, lightsails will operate at elevated temperatures where material properties have been less comprehensively studied.We did not attempt to model temperature-dependent mechanical properties in the present study, although it would be straightforward to add this capability in the future.
Ultimately, further characterization of the lightsail material(s), as fabricated and over their intended operating temperature range, will be required to draw conclusions about the viability of any specific lightsail design.
Aluminum and polyimide are typical materials used for solar sails; we include them as a point of comparison.Note that the stringent requirements of ultralow optical absorption preclude the use of even the most reflective of metals for the interstellar lightsail application.It also seems unlikely that polymers could be used structurally in this application, owing to their low strength and limited temperature range.Other materials offering exceptional mechanical strength such as graphene and carbon nanotubes can also likely be ruled out owing to their high optical absorption.However, there are likely a wide range of dielectrics and wide-bandgap semiconductors which may prove useful in lightsail applications, in addition to those shown in Table S1.
For materials such as crystalline silicon, SiO2 and diamond, the highest recorded strengths have been achieved by small (< 50 μm diameter) filaments of high-purity materials with pristine surfaces, tested in bending over a small mandrel to further limit the stressed surface area and thus the chances of encountering a surface defect.It is uncertain if such high strengths could be achieved in a membrane geometry.Furthermore, crystalline materials, whether bulk or 2D, may exhibit reduced strength if used to fabricate arbitrarily curved lightsail surfaces such as spheres, cones, or paraboloids, owing to relative weakness of certain crystal planes, or the inability to perfectly join crystal surfaces at domain boundaries.

Mesh-based simulator for flexible lightsails
We have developed a time-domain simulation code for studying the dynamic behavior of lightsails under acceleration.This is facilitated by modelling the lightsails as a discrete mesh, wherein the nodes represent mass, inertia, temperature, and shape; the edges represent the stiffness and thermal conductivity of the material; and enclosed triangles represent the surface area through which light interacts with the lightsail.Nodes of the mesh are assigned positions along the desired surface, with their spacing chosen to yield approximately uniform edge length and aspect ratio among the triangles.An example simulation mesh for a paraboloid lightsail is plotted in Fig. S1.This code has been open-sourced at: https://github.com/Starshot-LightsailSimulations begin with generation of the mesh.Currently, supported meshes must have two-dimensional topology, but can represent any three-dimensional surface so long the surface is not self-shading at any time.The provided mesh generator script provides parametric options to generate round, square, or hexagonal lightsails, with either flat, spherical, parabolic, or cone/pyramid vertical profiles.Non-round footprints can specify either smooth or faceted vertical profiling.Region and texture mapping is also performed in the mesh generator.This assigns varying mechanical and optical properties to various regions of the lightsail.Regions can be defined in either Cartesian or polar mapping schemes.
The simulation process is outlined in Fig. S2.Briefly, evolution of the shape and position of the sail is calculated iteratively in the time domain, using a fixed time step calculated to be substantially smaller than any vibrational modes of the mesh (typically, 1/20 th to 1/10 th of the period of the highest resonant frequency).Modeled physics include: Mechanical response based on linear elastic theory, tensile failure detection, radiative cooling, thermal conduction, thermal expansion, ray-tracing for specular surfaces, and calculation of optical forces and absorption using either fixed values of reflectance and absorption, a 1D look-up table (LUT) to represent calculations of specular behavior using the transfer matrix method, or a 2D LUT representing the angle-dependent response of nanophotonic or metagrating surfaces.
Upon mechanical or failure of the membrane, the simulation can then be terminated, or allowed to proceed to determine the margin by which the chosen conditions will exceed the material capabilities.Alternately, to enable cursory depictions of the progression of such failures, we can delete the affected elements from the ongoing simulation mesh at the moment of failure, but this is not intended to accurately model the dynamics of tensile or thermal failures.Specifically, our simulator does not model collisions between the collapsed lightsail elements, neglects beam occlusion effects for inverted shapes, and treats tensile failure simplistically; thus the fully collapsed and tattered shapes are not simulated accurately.The images of mechanical failure are included to better show the general progression of the shape instabilities.In Fig. 3 of the main text, we chose a hexagonal perimeter shape for the flat membranes to better illustrate the collapse.
The present approach cannot be used to study the scenario of polarization mismatch, which is necessary to evaluate whether our spinning lightsails are stable in non-rotating beams, or whether lightsails can self-synchronize their rotation to that of the beam during acceleration.Moreover, due the accumulation of numerical errors introduced by explicit time stepping in our code, simulations cannot be performed over indefinite timescales to definitively prove marginal stability.Where we adopted the following notation for partial derivatives, & .
In our case with only pitch-and roll-restoring behavior and translational stability, many of the matrix elements are either zero, very small and thus approximately zero, or can be calculated analytically, leaving us with a Jacobian matrix of full rank that has a reduced dimension given by By numerically evaluating the remaining nonzero matrix elements of the Jacobian matrix, the presence of real parts in any of its eigenvalues indicates exponential growth of the respective solution to the equations of motion and thus instability of the laser-propelled system.Due to the lack of damping terms in the system's equations of motion, eigenvalues with real parts will always come in pairs of positive and negative real part.
The case of spinning rigid lightsails requires a more careful stability analysis, where the absolute values of the complex eigenvalues of the monodromy matrix, which can be obtained from numerical integration involving the system's Jacobian matrix, determine whether spinning lightsails are stable or not.Importantly, is no longer assumed to be zero (or close to zero), but takes on a finite value, i.e., times our desired spinning frequency, instead.Similarly, the yaw angle will vary between and during a period of and thus be time-dependent.To underline these differences, we evaluate with constant and to be To further simplify, we remind ourselves that can be linearly expanded around the "equilibrium" as Noting that Which means that is not a true equilibrium.Nevertheless, evaluating the second term on the right-hand side of the Taylor-expanded equation above yields From this, it follows that constant As for the first case, we observe multiple frequency components within the simulated trajectories and tilt angles (Fig. S3D-2G), with the most noticeable one being the again slow frequency component at 240 Hz superimposed upon slower frequencies of approximately 2.5 Hz and 0.6 Hz.The observation of displacement along x and y being more tightly confined can also be made for the pitch and roll angles of the rigid lightsail, as they remain bounded within ±1.3° during the simulated timespan, suggesting a lesser degree of deformation and vibration in the membrane.The temporal evolution of pitch and roll angle distributions of the flexible lightsail again follows closely θ and ϕ of the rigid lightsail, confirming that spin stabilization at 120 Hz is sufficiently fast enough to treat our flexible lightsail as quasi-rigid.Nevertheless, we note that a finite angular spread of pitch and roll angles of ~1° can be observed for all mesh elements constituting the flexible lightsail.Finally, due to the discretized surface of the flexible lightsail, signs of mesh elements on the perimeter experiencing larger rotations remain visible in the insets of Fig. S3F and S3G despite truncating histogram bins with only few elements (less than 10 within bins of width 0.05°).

Temperature & strain analysis of passively stabilized flexible metagrating-based lightsails
As mentioned in the main text, our flexible lightsail simulator stores several variables of interest for postprocessing and analysis, including the peak and average temperature of the lightsail during propulsion and the maximum strain on the lightsail due to mechanical forces and thermal expansion, downsampled by a factor of 8 for memory management.Due to the underfilling beam width of w = 0.4D, regardless of whether the lightsail is initially only translated or also tilted, the difference between the peak, average and minimum temperature of points on the lightsail can be several hundreds of Kelvin (Fig. S4).While the center of the lightsail heats up to a peak temperature of just below 1000 K during propulsion, its perimeter or edge points experience a temperature rise of less than 200 K, the difference of which results in an average temperature in between these two extremes.Including an initial tilt to the simulated trajectories induces more variation in especially both peak (center) and minimum (edge) temperatures of the accelerated lightsail.

Figure 1 .
Figure 1.Conceptual illustrations of design approaches.Designs for achieving (a) beam-riding stability, and (b) structural stability, in lightsail membranes.In panel (a), the red arrow depicts the accelerating beam position, the orange arrows indicate the direction of reflected light, and the blue arrows indicate the force of radiation pressure.

Figure 2 .
Figure 2. Modeling flexible lightsails and light-matter interaction with a mesh-based time-domain simulator.(A) Ultrathin and meter-scale lightsails and their deformations can be modeled by a mesh comprising masses m (nodes) connected by springs with stiffnesses k (edges), enclosing triangles of area A. Light-matter interactions are calculated for each mesh triangle based on discretization of the incident light as localized beam I0.Modeled behaviors include (B) absorption of light and thermal emission, which heat and cool the structure, driving heat flow, thermal expansion, and changes in material properties; (C) specular reflection and transmission of light, producing photon pressure, and in some cases, causing reflected light to impinge other triangles; (D) optical diffraction from periodic wavelength-scale surface patterning, producing transverse directional forces from photon pressure, and (E) optical wavefront shaping such as beam steering with subwavelength optical metasurfaces.

Figure 3 .
Figure 3. Simulation results for flat versus curved specular lightsails, with and without spin stabilization.Illumination is in the +z direction starting shortly after t = 0, with a Gaussian profile (I0 = 4 GW/m 2 , Rwaist = 0.5 m, λ = 1.55 μm), offset by 80 mm from the initial lightsail centers.Left: Surface renderings show temperature, shape, and lateral position of each lightsail at the indicated times during simulation.Surface shading was applied to enhance depiction of shape.The vertical magenta lines show the beam centerlines.All lightsail images appear at the same scale; however, their vertical positions have been shifted for presentation.Right plots: The distance between the lightsail center of mass and the beam centerline (above), and the lightsail z velocity (below), plotted versus time.Animations of all five simulations are available as Supplementary Video 1.

Figure 4 .
Figure 4. Effects of multiple internal light reflections within spin-stabilized flexible paraboloid lightsail.Simulated shape (a), acceleration (b), peak temperature (c), and trajectory (d) of a 1-m diameter paraboloid lightsail, with and without the effects of internal light reflection within the lightsail.Paraboloid lightsails and acceleration conditions are similar to those in Fig. 3. Animations of these and other raytracing-based simulations are available as Supplementary Video 2.

Figure 6 .
Figure 6.Acceleration dynamics of a flexible and a rigid spinning lightsail based on the same composite metagrating pattern.Lightsails are initially offset by x = y =50 mm from the beam center and rotated by θ = ϕ = −2°.(A) Snapshots of the beamriding flexible lightsail's position, angular orientation, temperature and shape at different times.(B) Lightsail trajectory throughout the 5 s simulation duration.(C) Lightsail xand y-position and radial distance r from the beam center versus time, exhibiting bounded and oscillation around the equilibrium at x, y = 0. (D), (E) Evolution of pitch θ and roll ϕ, respectively, of the rigid lightsail versus time, showing multi-frequency oscillation around the equilibrium at θ, ϕ = 0°.(F), (G) Distribution of θ and ϕ angles, respectively, of all mesh elements comprising the flexible lightsail versus time, showing both bounded oscillations and limited angular spread, with minor shape distortion observed through the range of surface tilt angles at any given time.For (D) -(G), insets show fast-frequency oscillations within a reduced time window (0.1 s).An animation of this simulation is available as Supplementary Video 3

Fig. S1 .
Fig. S1.Three-dimensional surfaces can be constructed using Delaunay triangulation to model the simulation mesh of a paraboloid lightsail.

Table 1 .
Figures of merit for mechanical strength of candidate lightsail materials.

Table S1 .
Summary of published mechanical properties of candidate lightsail materials Supplementary InformationDynamically Stable Radiation Pressure Propulsion of Flexible Lightsails for Interstellar Exploration SI-9