Article
https://doi.org/10.1038/s41467-024-47476-1
Dynamically stable radiation pressure
propulsion of
fl
exible lightsails for
interstellar exploration
Ramon Gao
1,2
, Michael D. Kelzenberg
1,2
&HarryA.Atwater
1
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 explorat
ion within and beyond the solar system,
posing substantial challenges for materials science and engineering. We ana-
lyze the structural and photonic design of
fl
exible lightsails by developing a
mesh-based multiphysics si
mulator based on linear el
astic theory. We observe
spin-stabilized
fl
exible lightsail shapes and designs that are immune to shape
collapse during acceleration and exhibit
beam-riding stabili
ty despite defor-
mations caused by photon pressure and thermal expansion. Excitingly,
nanophotonic lightsails based on planar silicon nitride membranes patterned
with suitable optical metagratings exhi
bit both mechanically and dynamically
stable propulsion along the pump lase
r axis. These advances suggest that
laser-driven acceleration of
membrane-like lightsails to the relativistic speeds
needed to access interstellar distances i
s conceptually feasible, and that their
fabrication could be achieved by scaling up modern microfabrication
technology.
Centuriesofastronomicalobservationsanddecadesofroboticspace
exploration have been dedicated to the study of our own solar system.
Exoplanets orbiting other sun-like stars were
fi
rst conclusively detec-
ted in the 1990s
1
, posing the question of whether life exists elsewhere
in the universe
2
. However, exoplanets are far too distant to be directly
imaged by telescopes, nor could conventional space probes reach
them within the timescale of human civilization. Among the three
space probes that have reached interstellar space, Voyager 1 has tra-
veled the farthest, being presently 0.0025 light years away from our
sun. The nearest star to our own is Proxima Centauri, 4.2 light years
away, and hosts at least two exoplanets, with Proxima Centauri b lying
in the habitable zone
3
. Exploration of such exoplanets will require
signi
fi
cant advances in propulsion technology.
Lightsails utilize radiation pressure rather than reaction mass for
spacecraft propulsion, potentially allowing space probes to reach far
greater distances within a human lifetime. The concept dates to at least
400 years ago when Kepler observed that comet tails point away from
the sun as if blown by a solar wind
4
,
5
. Solar-powered lightsails have
been demonstrated through the recent JAXA IKAROS
6
,NASANanoSail-
D
7
, and the Planetary Society LightSail missions
8
,andhavebeenpro-
posed to enable a mission to the solar gravitational lens (SGL), nearly
0.01 light years from the sun, from which exoplanets could be studied
with far greater resolution than with any conceivable telescope
9
,
10
.
Whereas sunlight provides a relatively weak force for accelerating
spacecraft in Earth
’
s vicinity (~10
μ
Nm
−
2
for a perfect re
fl
ector at 1 AU),
far greater accelerating forces can be produced if a high-power density
laser is focused onto a lightsail. Laser-propelled lightsails can, in
principle, be accelerated to relativistic velocities, offering a promising
pathway for interstellar exploration using ultralight space probes for
direct
fl
yby missions
11
–
13
. Due in part to the announcement of the
Breakthrough Starshot Initiative in 2016, which seeks to enable this
capability within the next generation
14
,
15
, recent investigations have
explored the viability of laser-driven lightsails as a basis for interstellar
spacecraft propulsion
13
,
16
–
18
. A major challenge for such lightsails is the
Received: 1 March 2023
Accepted: 2 April 2024
Check for updates
1
Thomas J. Watson Laboratory of Applied Physics, California Institute of Technology, Pasadena, CA 91125, USA.
2
These authors contributed equally: Ramon
Gao, Michael D. Kelzenberg.
e-mail:
haa@caltech.edu
Nature Communications
| (2024) 15:4203
1
1234567890():,;
1234567890():,;
need to maximize re
fl
ectance while minimizing weight and limiting
optical absorption to extremely low values, prompting multilayer or
nanophotonic designs
19
–
23
. The lightsails must also be designed to be
structurally and dynamically stable during acceleration, passively fol-
lowing the optical axis of propulsion
24
–
38
without collapsing or tearing.
Many designs for rigid-body beam-riding lightsails have been pro-
posed, and membrane deformation was modeled for gas-
fi
lled sphe-
rical lightsails
39
, but to date, no study has considered the mechanical
fl
exibility of meter-scale, fully unsupported membranes and its effects
on acceleration stability. Notably, to achieve relativistic velocities, the
Starshot mission calls for a ~10 m
2
, ~1 g lightsail, which therefore, must
be on the order of 100 atomic layers thick on average, including all
framing or stiffening, suggesting that mechanical
fl
exibility cannot be
neglected in lightsail design.
Here, we examine the selection of materials, the structural and
photonic design, and dynamic mechanical stability of
fl
exible lightsail
membranes to investigate whether interstellar lightsail spacecraft can
be realized with real materials of
fi
nite stiffness and strength. We
identify key material properties required for relativistic
fl
exible light-
sails, then develop a multiphysics numerical simulation approach to
explore the deformation and passively stabilized acceleration of
spinning
fl
exible lightsails with either specular scattering concave
shapes or
fl
at membranes with embedded metagrating nanophotonic
elements.
Results
Materials considerations
The Breakthrough Starshot Initiative
14
has challenged a global com-
munity of scientists and engineers to design a ~1-g interstellar probe
that will travel 4.2 light years to reach Proxima Centauri b, the nearest
known habitable-zone exoplanet, within ~20 years of launch, as well as
the necessary propulsion, communication, and instrumentation sys-
tems for such a mission. To accelerate the spacecraft to ~0.2
c
, a ~10 m
2
lightsail weighing ~1 g would be propelled from low-earth orbit by an
earth-based laser at incident power densities approaching ~10 GW m
−
2
,
experiencing ~10,000 Gs of acceleration for ~1000 s
13
,
15
. A lightsail
suitable for this mission must address immense engineering obstacles,
challenging the limits of materials science and engineering.
One challenge is that the lightsail must have reasonably high
optical re
fl
ectance to produce thrust from the accelerating beam yet
must exhibit near-zero optical absorption (
≲
1 ppm) and high thermal
emissivity to prevent overheating. A handful of dielectric and semi-
conductor materials have been identi
fi
ed as potentially viable
candidates
16
–
18
. Optimized nanophotonic metamaterials comprising
combinations of such materials can offer favorable combinations of
enhanced re
fl
ectance, low absorption, and high emissivity
19
–
22
,
40
.
Lightsail materials and designs must also offer adequate mechanical
strength and stiffness to endure the acceleration conditions neces-
sary for interstellar propulsion. Table
1
shows key room-temperature
mechanical properties and performance metrics for several candi-
date lightsail materials, with more details listed in Supplementary
Table 1.
Bulk crystalline dielectrics and semiconductors such as Si, quartz
(SiO
2
), and diamond 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 each of these materials can achieve remarkable
degrees of purity and scale of manufacture, present-day technology
has yet to produce pure, defect-free, submicron-thick membranes over
10 m
2
areas.
Among two-dimensional crystals, MoS
2
appears particularly
promising for lightsail applications owing to its high strength and
refractive index
20
. The reported tensile strength for micron-scale
suspended membranes of mono- or bi-layer MoS
2
is nearly three
times higher
41
than that of any other material listed in Supplementary
Table 1. Understanding the achievable strength and optical trans-
parency of MoS
2
fi
lms fabricated over large or nonplanar surfaces at
relevant layer thicknesses and elevated temperatures is of con-
siderable interest.
Another interesting class of materials includes amorphous or
nanocrystalline deposited thin
fi
lms. Promisingly, submicron-thick
silicon nitride membranes have been fabricated at wafer scale and
further patterned with photonic crystal designs for near-unity
re
fl
ectance
42
,
43
. Widely employed in MEMS and cavity opto-
mechanics applications
44
,
45
, high-stress silicon nitride (Si
3
N
4
)isa
particularly promising candidate material for lightsail development
due to its ultralow extinction coef
fi
cients on the order of 10
−
6
at near-
infrared wavelengths and large modulus and tensile strength.
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 to endure the forces and optical intensities of
the propulsion laser beam.
Table 1 | Figures of merit for mechanical strength, including the stationary burst diameter
D
SB
and maximum spin frequency
f
max
, of candidate lightsail materials
67 Pa
D
SB
f
max
Material
Young
’
smodulus,
E
(GPa)
Tensile strength,
σ
T
(GPa)
Density,
ρ
(g cm
−
3
)
Thickness,
t
for
0.1 g m
−
2
(nm)
D
SB
burst for
67 Pa (m)
f
max
spin for
10 m
2
(Hz)
Silicon (111 surf.)
67
169
2.1
2.33
43
1.09
133
Diamond (PECVD nano)
68
750
Up to 7.5
3.27
31
1.35
179
SiO
2
(tempered glass)
69
,
70
77
Up to 1.0
2.42
42
0.52
91
Si
3
N
4
(LPCVD
fi
lm)
44
,
71
270
6.4
2.7
37
3.96
215
MoS
2
(multilayer)
41
,
72
200
–
330
21
5.02
20
19.9
285
Aluminum
72
0.50
2.80
36
0.16
58
Polyimide
73
2.5
0.069
1.42
70
0.09
30
This table is not intended to suggest that lightsails should be constructed from uniformly thick, continuous membranes or to impose an upper limit for
thickness, but rather to facilitate
fi
rst-order
comparison of the limiting structural capabilities of the candidate materials. More detailed properties are provided in Supplementary Table 1.
Article
https://doi.org/10.1038/s41467-024-47476-1
Nature Communications
| (2024) 15:4203
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Stability considerations
Our work addresses two key challenges for stable lightsail acceleration
and potential solutions to them (Fig.
1
):
beam-riding stability
, the ability
of the lightsail to follow along the beam axis without external gui-
dance, and
structural stability
, the ability of the lightsail to survive the
acceleration sequence without collapsing, rupturing or excessively
deforming.
Passive beam-riding stability is necessary for relativistic lightsails
because the large acceleration distance precludes closed-loop beam
control to provide trajectory corrections. When the lightsail becomes
misaligned with the beam, its design must produce corrective optical
forces based on spatial power density variations on the lightsail. In
practice, the laser system would be constructed no larger nor more
powerful than necessary to achieve the desired
fi
nal velocity, such that
the system would operate at or near the diffraction limit during the
fi
nal acceleration phase. As depicted in Fig.
1
a, a
fl
at specularly
re
fl
ective disk does not offer beam-riding stability and will tilt and veer
away from the beam, whereas certain geometrically concave re
fl
ector
shapes, including cones
26
–
28
, hyperboloids
24
, paraboloids, and other
parametric shapes
37
can achieve stable beam-riding behavior. Convex
shapes such as spheres can exhibit stability using more complex
higher-order beam pro
fi
les
26
. Nonspecular surfaces can be employed
to produce restoring forces and torques, even for
fl
at lightsails
23
,
29
–
33
,
38
,
and have been shown to enhance lateral and rotational maneuver-
ability of solar lightsails
46
.
Our study addresses marginal (undamped) beam-riding stability
during acceleration, where the lightsail exhibits bounded, oscillatory
displacement and tilting about the beam axis in response to an initial
beam-lightsail misalignment. Con
tinuous perturbations to the beam-
lightsail alignment could cause the oscillatory motion to grow in
amplitude, eventually ejecting marginally stable lightsails from the
beam. Furthermore, for nonrigid structures such as
fl
exible
membranes
47
, gradual energy buildup in vibrational or acoustic modes
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 structural vibrations to achieve
asymptotically stable propulsion. Passive damping could be achieved
with damped internal degrees of freedom
36
, nonlinear optical
materials
35
, or materials with highly varying temperature-dependent
optical properties to enable hysteresis of the restoring forces.
Regarding structural stability, the lightsail must survive the
acceleration forces without experiencing mechanical failure or defor-
mations that would disrupt beam-riding. Optimized membrane
designs could incorporate multiple materials
21
, complex geometries,
and intricate spatial patterning, e.g., perforations
20
,
22
,
35
,
43
or optical
resonators
23
,
29
,
30
,
38
,
40
to maximize re
fl
ectance, emissivity, and tensile
strength. Rigid shell theory has been applied to study stress distribu-
tions in parametrically shaped lightsails
48
, and 2D analytic and
fi
nite-
element models of deformation instabilities have been reported for
uniformly illuminated lightsails
49
, but the behavior of unsupported or
loosely supported
fl
exible membranes subject to nonuniform forces is
generally complex
47
.
Thin unsupported membranes will generally collapse and crum-
ple upon themselves when subject to focused laser propulsion (Fig.
1
b,
left). A curved surface offers greater structural rigidity than a
fl
at
membrane while also conferring the bene
fi
ts of improved stress dis-
tribution 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 (Fig.
1
b, center left). When curved
lightsails become slightly deformed, their elongated regions present
larger cross-sectional areas and smaller incidence angles, resulting in
an increased effective photon pressure, whereas the narrowing regions
similarly experience decreasing effective photon pressure, furthering
the distortion and leading to collapse. Adding structural reinforce-
ment or framing to resist distortion comes with a mass penalty.
Potential structural reinforcement approaches include microlattices
50
,
gas-
fi
lled envelopes
39
,
51
, annular tensioning, fractal supports
52
,ten-
segrity structures
53
, or lamination with low-density or corrugated
backing layer(s). Ultimately, given mass and material constraints, even
a structurally rigidi
fi
ed lightsail will likely deform during acceleration,
potentially changing the membrane
’
sstressdistributionoralteringits
beam-riding properties. The proposed lightsail membranes are gen-
erally partially transparent; thus, any frame or backing materials may
still be exposed to high laser intensities even if placed behind the
lightsail surface, further limiting materials selection.
Alternatively, spin-stabilization can prevent shape collapse by
effectively rigidifying the lightsail via inertial tensioning and gyrosco-
pically stabilizing the lightsail to resist tilting, all while avoiding the
added mass and complexity of structural reinforcement. However,
spin-stabilization greatly complicates the dynamics of the lightsail,
particularly for
fl
exible membranes prone to complex instabilities
47
,
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 dyna-
mically stable under nonspinning (rigid-body) conditions, particularly
in the case of angular misalignment between the beam and spin
axes
26
,
27
. Thus, the use of spin-stabilization for ultrathin
fl
exible light-
sails can be a challenging design objective.
Table
1
introduces two
fi
gures of merit to facilitate
fi
rst-order
comparison of the limiting structural capabilities and to provide
insights into the general viability of constructing large-area
Flat
reflector
Concave
reflector
Diffractive
metasurface
Convex +
hollow beam
Beam-riding stability
Structural stability
Flat
membrane
Stable
Unstable
(a)
(b)
Curved
membrane
collapse
Framing
Stiffening
Spin stabilizing
Fig. 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 arrows depict the accelerating beam position, the orange arrows indicate
the direction of re
fl
ected light, and the blue arrows indicate the force of radiation
pressure.
Article
https://doi.org/10.1038/s41467-024-47476-1
Nature Communications
| (2024) 15:4203
3
structurally stable lightsails from the candidate materials. The
sta-
tionary burst diameter D
SB
is the maximal diameter at which a
fl
at
circular membrane (rather than a plate or shell) of areal density
0.1 g m
−
2
, rigidly clamped at its perimeter, can sustain a pressure of
67 Pa (the effective photon pressure of 10 GW m
−
2
illumination for
unity re
fl
ectance) applied to one side without rupturing
54
.Thisper-
tains to the construction of a perimeter-supported lightsail, e.g.,
spanning a ring-shaped support frame, but rather than making speci
fi
c
assumptions about the mass, rigidity, or pretensioning of such a sup-
port structure, we consider the simpler and more conservative case in
which the perimeter is stationary and rigidly clamped without pre-
tensioning. Precluding free
fl
ight of the membrane,
D
SB
should thus be
interpreted as a comparative
fi
gure of merit rather than a practical size
limit for perimeter-supported interstellar lightsails. Realistically, the
support structure must have
fi
nite (preferably small) mass so that it
could be accelerated with the lightsail, and the beam would necessarily
taper off at the lightsail edge, enabling larger lightsails to be con-
structed than indicated by
D
SB
(see Supplementary Note 2 for example
cases). The design of a practical lightsail spacecraft must address its
speci
fi
c support structure(s) and payload(s) and must also consider
optical and mechanical properties of the membrane throughout the
range of illumination conditions and operating temperatures experi-
enced during acceleration
—
none of which are captured by
D
SB
,
although we address some of these issues in greater detail in our
numerical simulations below. But interestingly, some candidate
membrane materials (Si
3
N
4
,MoS
2
)are,inprinciple,strongenoughto
span 10 m
2
areas (
D
SB
> 3.6 m) with perimeter support
—
even in the
stationary case. On the other hand, conventional solar lightsail mate-
rials such as aluminum and polyimide are considerably weaker and
likely unable to span meter-scale areas between structural supports in
interstellar lightsail applications, noting that they can more decisively
be ruled out on the basis of their optical absorption alone
16
.
The second
fi
gure of merit is the
maximum spin frequency f
max
at
which a
fl
at, 10 m
2
circular membrane (
D
= 3.6 m) could be spin-
stabilized without rupturing due to tensile failure
55
.Forthedesigns
considered here, relatively high spin frequencies are required to pro-
duce both shape stability and beam-riding stability, with resulting
stress approaching a signi
fi
cant fraction of the materials
’
tensile limits.
The viability of spin stabilization depends on the spin frequency, the
acceleration conditions, and the speci
fi
c design of the lightsail.
Mesh-based multiphysics modeling of
fl
exible lightsails
To simulate
fl
exible lightsail membranes of various shapes and optical
designs, a triangular surface mesh is constructed (Fig.
2
a) to represent
themembraneasamass-springdynamicalmodelin
fi
nite-difference
time-domain simulations. Light
–
matter interactions are calculated
over the enclosed triangular mesh elements: Incident light produces
photon pressure forces, optical absorption heats the lightsail, and
thermal radiation cools the lightsail (Fig.
2
b). In the simplest type of
optical interaction, the photon pressure force results from specular
re
fl
ection (Fig.
2
c), with the resulting force directed normal to the
surface. We
fi
rst assume
fi
xed values for re
fl
ectance and absorptance
to model curved and
fl
at specular lightsails. Then, we improve the
specular surface model to include angle-dependent re
fl
ectance and
absorptance based on Fresnel coef
fi
cients and consider the effects of
multiple re
fl
ections of light within concave curved lightsails using
simpli
fi
ed ray tracing. Finally, we present simulations of nonspecularly
re
fl
ecting surfaces, demonstrating that diffractive metagratings
(Fig.
2
d) allow
fl
at lightsails to achieve beam-riding stability. With
future work, lightsails made from optical metasurfaces (Fig.
2
e) with
various optical behavior could be studied using this simulation
approach.
Our simulations have studied only the
fi
rst few seconds of accel-
eration, with the lightsail being misaligned to the beam at
t
=0s, to
observe its shape evolution and whether its motion exhibits marginal
stability over many periods of beam-riding oscillation, to determine
temperature and stress distributions and to identify thermal or
mechanical membrane failure. Future efforts could also consider the
effects of local temperature and strain on optical and mechanical
properties, study damping or active control surfaces
56
, include beam
pro
fi
les varying in time or distance from the source, or address rela-
tivistic effects necessary to model the full acceleration duration to
interstellar velocities.
Dynamics of
fl
exible curved lightsails
The simulated behavior of
fl
at versus curved (paraboloid) lightsails and
theeffectsofspinstabilizationaredepictedinFig.
3
, using optical and
mechanical properties corresponding to a ~43 nm-thick Si membrane
(0.1 g m
−
2
). We assume
fi
xed values for specular re
fl
ectance (0.45),
absorption (1.4 × 10
−
7
), and emissivity (0.1), with the latter two values
being signi
fi
cantly higher than expected for the Si membrane alone.
This is done to approximate a radiative cooling surface of the
A
k
m
a
I
I
R
R
−1
R
+1
R
0
R
I
I
bc
de
T
T
−1
T
+1
T
0
Fig. 2 | Modeling
fl
exible 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
I
. Modeled behaviors include
b
absorption of light
and thermal emission, heating and cooling the structure, driving heat
fl
ow, thermal
expansion, and changes in material properties;
c
specular re
fl
ection
R
and trans-
mission
T
of light, producing photon pressure, and in some cases, causing re
fl
ected
light to impinge other triangles;
d
optical diffraction with exemplary re
fl
ected
orders
R
±1
,
R
0
and transmitted orders
T
±1
,
T
0
from periodic wavelength-scale sur-
face patterning, producing transverse directional forces from photon pressure, and
e
optical wavefront shaping such as re
fl
ective beam steering with subwavelength
optical metasurfaces.
Article
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Nature Communications
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