Highlights Structural Stability of a Lightsail for Laser-Driven Interstellar Flight Dan-Cornelius Savu Andrew J. Higgins

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Highlights

Structural Stability of a Lightsail for Laser-Driven Interstellar Flight

Dan-Cornelius Savu, Andrew J. Higgins

•The shape of a ﬂat lightsail was perturbed to test its resistance to deformation.

•Quasi-static analytical expressions were developed for the critical stability point.

•Dynamics of the perturbed lightsail was studied using numerical models.

•Analytical expressions predict the stability boundary seen in numerical simulations.

•Sail tensioning, not increasing material stiﬀness, is a method to obtain stability.

arXiv:2210.14399v1 [physics.space-ph] 26 Oct 2022

Structural Stability of a Lightsail for Laser-Driven Interstellar Flight

Dan-Cornelius Savua,1, Andrew J. Higginsa,2

aDepartment of Mechanical Engineering, McGill University, 817 Sherbrooke St. W., Montreal, Quebec, H3A 0C3, Canada

Abstract

The structural stability of a lightsail under the intense laser ﬂux necessary for interstellar ﬂight is studied analytically and numer-

ically. A sinusoidal perturbation is introduced into a two-dimensional thin-ﬁlm sail to determine if the sail remains stable or if

the perturbations grow in amplitude. A perfectly reﬂective sail material that gives specular reﬂection of the laser illumination is

assumed in determining the resulting loading on the sail, although other reﬂection models can be incorporated as well. The quasi-

static solution of the critical point between shape stability and instability is found by equating the bending moments induced on

the sail due to radiation pressure with the restoring moments caused by the strength of the sail material and the tension applied at

the edges of the sail. From this quasi-static solution, analytical expressions for the critical value of elastic modulus and boundary

tension magnitude are found as a function of sail properties (e.g., thickness) and the amplitude and wave number of the initial

sinusoidal perturbation. These same expressions are also derived from a more formal variational energy (virtual work) approach.

A numerical model of the complete lightsail dynamics is developed by discretizing the lightsail into rectangular ﬁnite elements.

By introducing torsional and rectilinear springs between the elements into the numerical model, a hierarchy of models is produced

that can incorporate the eﬀects of bending and applied tension. The numerical models permit the transient dynamics of a perturbed

lightsail to be compared to the analytic results of the quasi-static analysis, visualized as stability maps that show the rate of pertur-

bation growth as a function of sail thickness, elastic modulus, and applied tension. The analytic theory is able to correctly predict

the stability boundary found in the numerical simulations. The stiﬀness required to make a thin lightsail stable against uncontrolled

perturbation growth appears to be unfeasible for known materials, however, a relatively modest tensioning of the sail (e.g., via an

inﬂatable structure or spinning of the sail) is able to maintain the sail shape under all wavelengths and amplitudes of perturbations.

Keywords: lightsail, structural stability, laser-driven propulsion, interstellar ﬂight

Nomenclature

Latin Symbols

aabsorption coeﬃcient

ampmax maximum lightsail perturbation amplitude

Adiscrete element in-plane area

Accross-sectional area

Bbnon-Lambertian coeﬃcient of the lightsail back (non-

reﬂecting) surface

Bfnon-Lambertian coeﬃcient of the lightsail front (re-

ﬂecting) surface

cspeed of light in vacuum

ddiameter

Ematerial Young’s modulus

Ecr critical Young’s modulus predicted by the direct ap-

proach

1Undergraduate Research Assistant, Department of Mechanical Engineer-

ing, 817 Sherbrooke St. W. Email address: dan-cornelius.savu@mail.mcgill.ca

2Professor, Department of Mechanical Engineering, 817 Sherbrooke St. W.

Email address: andrew.higgins@mcgill.ca

Ecrmax maximum critical Young’s modulus value predicted

by the direct approach

EcrδWcritical Young’s modulus predicted by the energy

approach

EcrmaxδWmaximum critical Young’s modulus value predicted

by the energy approach

f0ﬂat element radiation force magnitude

fiith element radiation force

fnradiation force component directed along the ele-

ment normal direction

ftradiation force component directed along the ele-

ment tangential direction

fforcing vector

fmodiﬁed forcing vector

g0non-inertial D’Alembert vertical acceleration

hthickness

Isecond moment of area

IGmoment of inertia about the center of mass

I0laser beam intensity

ksrectilinear spring constant

kttorsional spring constant

Preprint submitted to Acta Astronautica October 27, 2022

lelement length

lsrectilinear spring elongation at rest (always set to 0)

Lworking length

Lthe Lagrangian

mdiscrete element mass

mgeneral radiated element force direction

Mbulk mass

Mmoment/torque

Mmass matrix

Mmodiﬁed mass matrix

nnumber of rigid sail elements

nnormal vector

prradiation pressure

qvector of generalised coordinates with components

rreﬂection coeﬃcient

rposition vector

sfraction of light that is specularly reﬂected

ttime

tﬁnal ﬁnal simulation runtime

Tboundary tension magnitude

Tcr critical boundary tension magnitude

Tcrmax maximum critical boundary tension value

Tboundary tension vector

winitial perturbations vertical displacement

Wwidth

Wwork done

xix-position of the ith element

xmodiﬁed vector of generalised coordinates

yiy-position of the ith element

zielongation of the ith element

Greek Symbols

εbemissivity of the lightsail back (non-reﬂecting) sur-

face

εfemissivity of the lightsail front (reﬂecting) surface

θiangular position of the ith element with respect to

the positive x-axis

κcurvature

νmode number of initial perturbations

ρdensity

τtime to doubling of initial perturbation amplitude

˜τtransmission coeﬃcient

1. Introduction

Concentrating laser-light energy onto a reﬂective foil to per-

mit fast transportation within the solar system and beyond has

been actively considered since the 1980s [1, 2] with serious

academic discussions dating as far back as the 1960s [3, 4].

Because of the exponential rate of development of ﬁber-optic-

based lasers within the telecommunication and laser machining

industries, the laser-driven spacecraft is now steadily turning

from concept to a present-day reality. Arbitrarily large laser

beams can now be made by constructing phased arrays of lasers

using inexpensive optical components [5, 6]. With modern ini-

tiatives such as the Breakthrough Starshot project, which in-

tends to use a ∼1 gram lightsail laser-accelerated to 20% the

speed of light to reach the nearest star Proxima Centauri within

20 human years, interstellar ﬂight in the 21st century may be-

come an actuality, but for this to happen various interdisci-

plinary scientiﬁc and engineering challenges still need to be

overcome [5–7]. One such challenge that needs to be solved,

given the large laser intensities involved (∼10 GW/m2) and the

ideally low sail inertia, is the problem of the dynamic and struc-

tural stability of the lightsail.

The directional or beam-riding stability of laser-driven light-

sails has already been studied for a variety of rigid sail shapes.

The beam-riding stability of conical and spherically curved light-

sails attached to the spacecraft via a rigid boom have been in-

vestigated [8, 9], and Srinivansan et al. have shown that a hyper-

boloid shaped lightsail impinged upon its convex surface by the

laser beam has passive directional stability [10]. Manchester

and Loeb also demonstrated that a spherically shaped lightsail—

a lightsail akin to a balloon—is inherently directionally sta-

ble provided that the laser beam intensity proﬁle was Gaus-

sian multi-modal or donut-shaped [11]. The inﬂuence of dy-

namic dampening upon directional lightsail stability has also

been considered recently [12, 13]. The inclusion of dissipa-

tion eﬀects appears to eﬀectively dampen the lightsail motions

lateral to the beam axis thereby further increasing directional

stability. Shirin et al. have numerically shown, for example,

that dynamically dampened conical-shaped lightsails become

exponentially stable—as opposed to their marginally stable un-

dampened counterparts [12].

Directional stability has also been shown to be possible with-

out recourse to curved sail shapes by means of engineered op-

tical lightsail surfaces and materials. This has been achieved

through the addition of nanoscale (dielectric) structures to the

surface of a ﬂat lightsail, a procedure enabled through the recent

advances in optical design and nanofabrication. These so-called

photonic metasurfaces allow for control over the magnitude and

direction of the reﬂected and transmitted incident laser light

while adding relatively little mass to the lightsail because of

their nanoscopic nature. Swartzlander et al., in a series of the-

oretical, computational, and experimental studies have shown

that a ﬂat lightsail whose reﬂecting surface is equipped with

diﬀractive gratings is directionally stable [14–18]. Myilswamy

et al. have also shown that a nonlinear photonic crystal lightsail

can help minimize the dynamic asymmetry caused by the at-

mospherically distorted laser beam [19]. Lightsails employing

(a)

(b)

Figure 1: Three-dimensional analytical (plate) model of the lightsail; (a) the lightsail, ﬂat; (b) the lightsail, smoothly perturbed with an incident uniform laser beam.

The problem here considered is whether the perturbations will grow in amplitude or not under the large laser loads.

Bloch-wave type scatterers and other engineered optical meta-

surfaces have also been reported to be directionally stable [20–

22]. Santi et al. have also suggested the use of thin-ﬁlm multi-

layered optical structures (Bragg mirrors) for the actualization

of curved lightsails with enhanced passive beam-riding stability

[23].

The studies mentioned so far have assumed the lightsail to

be rigid and either ideally ﬂat or perfectly smooth, that is, ab-

sent of deformation of the ideal sail shape. Deformation of

the lightsail has been built into several other models in an at-

tempt to quantify their inﬂuence upon the orientation and tra-

jectory of solar-driven sails [24–26]. Huang et al., for exam-

ple, inquired into the deviation of the resultant solar radiation

pressure force due to sail deformation eﬀects caused by wrin-

kling and billowing via the use of point cloud and triangular

mesh methods [26]. Structural analysis of the sail, considering

beams/booms and membranes, has also been undertaken [27–

31]. Liu et al. studied the attitude dynamics and the vibrations

of a square solar sail supported by four beams, the presence

of which allowed them to neglect the detailed sail membrane

vibrations and wrinkle eﬀects [28, 29]. Wong and Pellegrino

inquired theoretically, numerically, and experimentally into the

visible membrane wrinkling amplitude and wavelength growth

when tension is gradually applied to the corners of an initially

ﬂat, square solar sail membrane [27]. Other studies concerning

the problem of sail structural response have also been under-

taken for particular sail shapes [32, 33]. Cassenti and Cassenti

have also proposed the tensioning of the lightsail via the use of a

boundary ring—much akin to a drumskin—as a potential solu-

tion to the structural vibration problem caused by the presence

of a non-uniform laser beam [34]. A more recent study inquired

into the thermal and mechanical stresses that the lightsail expe-

riences during laser-driven acceleration and has concluded that

spherically curved lightsails of appreciable curvature are better

suited to sustain the large laser loads [35]. Of note are also the

experimental eﬀorts of Myrabo et al. who conducted investi-

gations on the problem of sail stability by subjecting lightsail

prototypes to laser loads in vacuum [36].

Recently, the application of the radiation pressure regime

for laser-driven acceleration of thin foils under extremely in-

tense ﬂuxes (exceeding 1021 W/cm2) has been considered as

a technique for heavy ion acceleration [37–39]. Such radia-

tion pressure acceleration technology could overcome the ve-

locity limitations of more traditional pulsed-laser acceleration

technologies, such as ablation, and has potential application to

fast ignition in inertial conﬁnement fusion. Under such intense

ﬂuxes, the structural stability of the thin-foil lightsail under ra-

diation pressure can be treated using the Rayleigh-Taylor for-

malism of interface instability [40]. The regime of laser ﬂux in

this application is ﬁfteen orders of magnitude greater than that

proposed for the laser-driven interstellar lightsail and thus is not

likely of direct relevance to the problem under consideration in

the present study.

Altogether, while some studies have inquired into the struc-

tural stresses and strains supported by solar sails, it should be

noted that solar-driven sails are generally not designed to toler-

ate high photon pressures and resultingly most solar sails envi-

sioned so far—like the IKAROS, NanoSail-D2, and LightSail

2 solar sails [41–44]—sustain accelerations orders of magni-

tude less than the accelerations that, for example, the Break-

through Starshot laser-driven lightsail would need to withstand

[6, 7]. Consequently, the structural analysis of solar-driven sails

has been chieﬂy concerned with the inﬂuence of lightsail de-

formations upon the spacecraft trajectory whereas, by contrast,

the comparatively large accelerations sustained by laser-driven

lightsails could cause a deformed lightsail to crumple beyond

functional use. Thus far, no inquiry into the detailed vibra-

tions and deformations of a lightsail under high photon radi-

ation loading has been completed. In particular, no study in-

quired into whether an ideally thin lightsail is capable of sus-

taining large laser-driven accelerations despite the inevitable

presence of perturbations (see Fig. 1). These perturbations—

arising from multiple possible sources such as atmospheric dis-

turbances, beam non-uniformities, shape distortion of the sail,

etc.—prevent perfectly uniform loading of the sail, and this

complication of the laser-lightsail dynamics may cause the dis-

tortions in sail shape to grow in amplitude. Even if the per-

turbations are, through careful lightsail deployment, kept small

in magnitude, the following question remains: Under the large

photonic pressure loads required for feasible interstellar ﬂight,

will the structural perturbations of the lightsail grow out of

bound or will the lightsail remain ﬂat?

The present study addresses the question of lightsail shape

stability with perturbations by considering a ﬁrst-principles ap-

proach to the problem. First, a continuous model of a sinu-

soidally perturbed lightsail under radiation load allows an ana-

lytical, quasi-static analysis of the critical point between light-

sail shape stability and instability in Section 2. A Lagrangian-

based ﬁnite element (FE) numerical model of the perturbed

lightsail is then constructed using rigid rectangular slices to

simulate the full dynamics of the lightsail while undergoing ac-

celeration in Section 3. The rigid-element numerical model was

further generalized to include torsion and rectilinear springs

to investigate the inﬂuence upon lightsail structural stability of

material bending stiﬀness and applied tension, respectively. In

Section 4, the quasi-static derived analytical expressions are

compared to the numerical results, which consisted of multi-

ple lightsail dynamics simulations with each simulation vary-

ing lightsail geometry, material modulus, and/or applied ten-

sion magnitude. The engineering implications of the analysis

are then explored.

2. Theoretical Considerations

As a preliminary consideration of the problem of a lightsail

under radiation loading, an L×W×hcontinuous elastic plate

model was ﬁrst constructed that would allow for a static anal-

ysis of the criticality between a structurally stable and unstable

lightsail (see Fig. 1). The 3-dimensional plate model was then

simpliﬁed to a 2-dimensional beam model by setting the light-

sail width, W, equal to unit length (in meters). The continuous

elastic beam model is shown in Fig. 2. The following analysis

considers introducing a sinusoidal perturbation into the lightsail

shape. The restoring bending moment caused by this deforma-

tion is then compared to the moment induced by the incident ra-

diation interacting with the curve surface of the lightsail. If the

radiation-induced moment exceeds the bending moment asso-

ciated with the imposed perturbation and acts in the same direc-

tion as the perturbation, then presumably the lightsail continues

to further deform. If the radiation-induced moment is less than

the restoring bending moment of the perturbation, then the sail

would be expected to return toward its original, ﬂat conﬁgura-

tion. By equating the bending moment of the sinusoidal pertur-

bation with the radiation-induced moment, a critical condition

for lightsail stability can be deﬁned in terms of the radiation

intensity, the elastic modulus and dimensions of the lightsail,

and the amplitude and wave number of the perturbation. The

formalism of this approach follows here.

The beam model analysis is here conducted in a non-inertial

reference frame accelerating at g0, the lightsail’s vertical ac-

celeration, with a body force term ρh W g0acting in the y-

direction on the mass elements of the lightsail in accordance

with D’Alembert’s principle, thereby reducing the dynamic prob-

lem to a quasi-static problem.3The presence of radiation pres-

sure is modeled by the distributed loading, pr. The expression

of prcan be derived by ﬁrst considering the kinematics and

dynamics of an inﬁnitesimal ﬂat sail element of length dsas

shown in Fig. 3 for reference. For a lightsail element made of

a material with reﬂection coeﬃcient r, absorption coeﬃcient a,

and transmission coeﬃcient ˜τ, the total force imparted by ra-

diation of oblique incidence angle θcan be resolved into the

normal-tangential components

fn=I0

c"(1 +rs) cos2θ+Bf(1 −s)rcos θ

+(1 −r−˜τ)εfBf+εbBb

εf+εb

cos θ#n,

(1)

ft=I0

c(1−rs)cos θsin θt,(2)

where sstands for the fraction of light that is specularly re-

ﬂected; εfand εbstand for the emissivity of the front (reﬂect-

ing) and back (non-reﬂecting) surfaces of the sail, respectively;

and Bfand Bbare the coeﬃcients accounting for the poten-

tially non-Lambertian nature of the sail element surfaces. To-

gether, these two forces generate a resultant whose direction,

m, is skewed away from the element’s normal:

f=qf2

n+f2

tm.(3)

This optical lightsail model was ﬁrst proposed by Forward [45]

and then was further discussed by Wright and McInnes [46,

47] now termed the Forward-Wright-McInnes or FWM model.4

For the purposes of this paper, the lightsail element will be as-

sumed to be perfectly reﬂective (r=1=⇒a=˜τ=0) with all

beam reﬂections being specular (s=1). The resulting force per

element then becomes

f=2I0

cAcos2θn,(4)

a force that is entirely normal to the sail element. Normaliz-

ing (4) with respect to the element area generates the radiation

pressure:

pr=2I0

ccos θ. (5)

3Given the large magnitude of the vertical accelerations, it is here assumed

that g0=¨y¨x,¨

θ(s), and thus the horizontal and rotational inertia of the light-

sail were deemed negligible and their d’Alembert equivalent was not included

into the lightsail beam model.

4The authors are aware that more accurate lightsail optical models using

vector theories such as the Rayleigh-Rice theory have been proposed [48, 49],

but the FWM model was used here to ensure a straightforward simpliﬁcation to

the case of an ideal reﬂector. The implementation of physically more accurate

optical models can be incorporated into future studies.

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HighlightsStructuralStabilityofaLightsailforLaser-DrivenInterstellarFlightDan-CorneliusSavu,AndrewJ.HigginsTheshapeofaatlightsailwasperturbedtotestitsresistancetodeformation.Quasi-staticanalyticalexpressionsweredevelopedforthecriticalstabilitypoint.Dynamicsoftheperturbedlightsailwasstudiedusingn...

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Highlights Structural Stability of a Lightsail for Laser-Driven Interstellar Flight Dan-Cornelius Savu Andrew J. Higgins

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