of 5
Supp
orting
Information for “
Nanophotonic
Heterostructures for Efficient Propulsion and
Radiative Cooling of Relativistic Light Sails”
Ognjen Ilic, Cora M. Went, and Harry A. Atwater
*
Thomas J. Watson Laboratories of Applied Physics
California Institute of Technology
Pasadena, CA 91125
*
haa@caltech.edu
SECTION 1: Parameters
for analyzed hetero
structures
Table S1
.
Designed thin
-
film structures analyzed
in this work.
Reflectance
corresponds to the
averaged value in the
[
$
,
&
]
interval (where
$
=
1
.
2
μ
m
and
&
=
1
.
225
$
, corresponding to
the Doppler
-
shifted laser wavelength at
=
0
.
2
). Note the thinnest allowed layer in our analysis is
5 nm. The refractive index of the gap layer is
gap
=
1
.
Label
Layout: Material(T
hickness
[nm])
g
m
2
g
m
A1
SiO
2
(
206
)
0.
45
3
0.
12
0.1
51
A3
SiO
2
(197)
gap(399
)
SiO
2
(197
)
0.86
7
0.37
0.
068
A5
SiO
2
(180)
gap(421
)
SiO
2
(182)
gap(421
)
SiO
2
(180
)
1.19
0.
58
0
.050
A7
SiO
2
(156)
gap(452
)
SiO
2
(161)
gap(448
)
SiO
2
(161)
gap(452
)
SiO
2
(156
)
1.
39
0.
70
0.
045
A9
SiO
2
(130)
gap(484
)
SiO
2
(140)
gap(473
)
SiO
2
(143)
gap(473
)
SiO
2
(140
)
gap(484
)
SiO
2
(130)
1.51
0.76
0.043
A11
SiO
2
(104)
gap(514
)
SiO
2
(124)
gap(492
)
SiO
2
(131
)
gap(486
)
SiO
2
(131
)
gap(492
)
SiO
2
(124
)
gap(514
)
SiO
2
(104
)
1.58
0.78
0.042
1
B2
Si(61
)
SiO
2
(5)
0.1
54
0.
65
0.
0167
B2’
Si(54
)
SiO
2
(63
)
0.2
64
0.
62
0.02
30
B3
Si(61)
gap(550
)
SiO
2
(5)
0.
154
0.
6
5
0.0165
B3’
Si(51)
gap(619
)
SiO
2
(73
)
0.2
79
0.
64
0.
02
24
B4
Si(34)
gap(506)
Si(33
)
SiO
2
(5)
0.1
66
0.
82
0.
0136
B4’
Si(33)
gap(523)
Si(31
)
SiO
2
(6
)
0.1
6
2
0.
8
1
0
.0137
B4’’
Si(5)
gap(631)
Si(45
)
SiO
2
(65
)
0.
2
58
0.
61
0.
0224
SECTION 2: Expressions for
propulsion and radiative cooling of a laser
-
driven
lightsail
From Eq. (10) in [
1
2
], we have
=
9
:
;
<
=
>
?
@
A
B
@
C
B
, where
is the total mass of the spacecraft (sail +
payload). Since
=
EB
EF
EF
EG
=
EB
EF
푐훽
,
we have
EF
EB
=
;
<
?
>
?
B
9
:
@
C
B
@
A
B
. Substituting
=
$
and
9
=
1
9
A
@
into the previous expression, we arrive at the equation (1) of the main text.
We write the absorbed power by the sail (Eq. 4) as
abs
(
,
)
=
$
(
)
,
, where
$
(
)
,
=
$
(
1
)
/
(
1
+
)
(
,
)
accounts for the Do
ppler
-
shift of the laser photon flux
as seen by the sail
1
,
and
(
,
)
is the absorptivity of the sa
il.
The blackbody spectral intensity is given by
WX
=
9Z
<
=
W
[
@
Exp
^
_
`
a
b
c
A
@
.
The hemispherical
-
spectral
(hs)
emissivity
W
,
in Eq. (5) of the main text is the sum
of the front and the back surface
(hs) e
missivities, namely
W
,
=
W
e
,
+
W
f
,
(
S1
)
where
the hemispherical
-
spectral emissivity is related to the directional
-
spectral emissivity as
W
e
=
1
W
e
,
,
cos
sin
푑휃푑휙
(
S2
)
and
, from Kirc
h
hoff’s law,
W
e
,
,
=
W
e
,
,
=
1
W
e
,
,
W
e
,
,
, where
,
denote the
reflectivity and transmissivity coefficients.
Finally, we can relate the front and the back surface
emissivities as
W
e
=
W
f
.
SECTION 3: Sensitivity of
figure of merit
to
>
gap refractive index
For
multilay
er silica structures from Fig. 3
, we
assume vacuum gap(s) with refractive index
of unity
.
Instead of vacuum, one could envision
gaps formed by low
-
absorption, low
-
density aerogel materials
with
>
1
refractive indices
[25
-
28
].
To characte
rize the RAAD sensitivity to greater
-
than
-
unity
refractive index, we
perform the same
optimization analys
is for silica structures (Fig. 3
), but wi
th
gap
=
1
.
1
(and
gap
~
0
.
1
g cm
A
t
,
to model aerogel density
)
2
. The result is shown in
Fig. S2
,
and the
corresponding structure parameter listed
in Table S2. We note
approximately 25
-
30% increase
in
relative to the case of unity gap refractive index.
Fig S1
.
Spectral reflectance (at normal incidence)
in the laser propulsion
band
for the structures from Table S1.
Fig. S2
.
Comparison of designed silica multilayer stacks with different gap
layer refractive index. Stack parameters are listed in Tables S1 & S2.
Table S2
.
Designed thin
-
film SiO
2
structures when the refractive index of the gap layer is
gap
=
1
.
10
.
The rest of the parameters are the same as in Table S1.
SECTION 4:
Infrared e
xtinction coefficients of silica & silicon
In contrast to vibrational modes in silica, the fundamental vibration in
undoped
silicon has no dipole
moment due to crystal symmetry and is therefore infrared inactive. The interaction of phonons with light
can occur
via higher order atomic displacements
.
Consequently, the
measured
extinction coefficient of
undoped
silicon
(resistivity
>
10
t
Ω
cm
)
in the mid infrared is orders of magnitude smalle
r than that of
silica (Fig. S3)
.
Label
Layout: Material(Thickness
[nm])
g
m
2
g
m
SA1
SiO
2
(206)
0.453
0.12
0.151
SA3
SiO
2
(203)
gap(361)
SiO
2
(203)
0.929
0.30
0.086
SA5
SiO
2
(196)
gap(369)
SiO
2
(193)
gap(369)
SiO
2
(196)
1.36
0.47
0.065
SA7
SiO
2
(186)
gap(382)
SiO
2
(184)
gap(379)
SiO
2
(184)
gap(382)
SiO
2
(186)
1.74
0.61
0.057
References
1.
Landau, L. D. & Lifshitz, E. M.
The Classical Theory of Fields (Fourth Edition)
2,
(Pergamon,
1975).
2.
Bellunato, T., Calvi, M., Matteuzzi, C., Musy, M., Perego, D.
& Storaci, B.
Refractive index of
silica aerogel: Uniformity and dispersion law.
Nucl. Instruments Methods Phys. Res.
Sect. A Accel.
Spectrometers, Detect. Assoc. Equip.
595,
183
186 (2008).
Fig S3
.
(a)
Comparison of
mid
-
IR
extinction coefficients for s
ilica
(Kitamura et al
.
[
18
]) and undoped
silicon (
Ch
andler
-
Horowitz & Amirtharaj [21
])
.
For silicon
,
is
multiplied by 10
3
.
(b) Steady
-
state
temperature for
the
B2 stack (Si + 5nm of SiO
2
,
solid line) and the same stack without SiO
2
(dashed
line). The 5nm SiO
2
film increases
only marginally (
~3%
)
, but
can enable
significantly lower
steady
-
state temperatures.
Here, we assume
$
x
=
100
GW
g
-
1
, and
Si
O
2
=
10
A
€
cm
-
1
.