# Broadband Diffractive Solar Sail

Prateek R. Srivastava  1 Ryan M. Crum  1 and Grover A. Swartzlander  Jr.1
1Chester F. Carlson Center for Imaging Science, Rochester Institute of
Technology, Rochester, NY 14623, USA<EMAIL_ADDRESS>
††journal: opticajournal††articletype: Research Article

The transverse radiation pressure force and acceleration is compared for two
parametrically optimized designs: prismatic and two-pillar metasurface
gratings. The numerical results were cross-verified with both Maxwell stress
tensor and modal analysis. Solar blackbody irradiance was assumed for
wavelengths ranging from 0.33 $\mu$m to the grating cutoff at 1.5 $\mu$m,
encompassing 83% of the solar constant. This multi-objective optimizer study
found that neither design comprised of $\mathrm{Si_{3}N_{4}}$ performed as
well as those corresponding to a low refractive index, low mass density
material. The predicted transverse acceleration of the optimized low-index
metasurface grating is compared to that of a state-of-the-art reflective solar
sail.

##  1 Introduction

The in-space propulsion of sailcraft via solar radiation pressure was
originally pioneered by in the 1920s by Tsander and Tsiolkvosy [1, 2]. In
contrast to rockets which both transport significant amounts of fuel mass and
make discrete orbit-changing burns, solar sails can attain extraordinarily
high velocities given a low mass and continuous acceleration. Space
organizations such as NASA, JAXA, and the Planetary Society, have improved the
technical readiness level of solar sails in recent years, culminating in an
assortment of proposed space science missions [3]. The advent of solar sailing
has stimulated advanced concepts that consider the mission objectives as part
of the sail design. For example, missions having a spiral trajectory toward or
away from the sun benefit from a sail having an optimal “lift" force
perpendicular to the sun line. To achieve lift a traditional reflective sail
must be tilted away from the sun; consequently the maximum lift cannot be
achieved owing to the reduced illumination projected area. In contrast,
optical scattering mechanisms like diffraction provide alternative means of
transferring photon momentum to the sail in a preferred sun-facing orientation
[4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. The maximum transverse force
on the sail occurs when sunlight is uniformly scattered at 90∘ with respect to
the surface normal of a sun-facing sail.

Figure 1: Schematic diagram of a solar sail with constituent (A) prism and (B)
subwavelength pillar elements of period $\Lambda$ and. The sail diffracts
incident light $\vec{k}_{i}$ by $\theta_{m}$ into $\vec{k}_{m}$ owing to
$\mathbb{\vec{K}}$, resulting in net radiation pressure force $\vec{F}$.

##  2 Theory

To advance the understanding of diffractive sails we explore two designs: a
triangular prismatic grating and a metasurface grating comprised of two
pillars. Two material strategies are analyzed for each design. First we
consider an arbitrary non-dispersive dielectric material having a refractive
index $n_{1}$ placed on a thin substrate of index $n_{2}=1.5$. Finite
difference time domain (FDTD) methods are used to account for internal and
external reflections of both polarization component of light, and moreover,
the angular scattering distribution across a broad band of optical
frequencies. Likewise, we determine the angular scattering distribution when
the grating and thin substrate are made with $\mathrm{Si_{3}N_{4}}$. The
schematic illustration shown in Fig. 1 depicts a portion of a flat rigid
infinitely periodic grating with period $\Lambda$ in the $x,z$-plane of
incidence for a sun-facing configuration, comprised of either (A) prism
elements or (B) pillars on a thin substrate. Structural flexing and non-normal
incidence angle are beyond the scope of this baseline study. The grating
period $\Lambda=1.5\;[\mu\text{m}]$, or equivalently the grating frequency
$\tilde{\nu}=c/\Lambda=200[\text{THz}]$ was selected from a consideration of
the spectral cut-off condition, the prism mass, and diffraction effects. The
fraction of blackbody irradiance cut off from diffraction decreases with
increasing value of $\Lambda$, whereas the mass of a prism varies as
$\Lambda^{2}$. A large value of the transverse acceleration generally requires
negligible spectral cut off and low mass, which combined with a diffraction
analysis, provides a value of roughly $\Lambda=1.5\;[\mu\text{m}]$.

Light is transmitted or reflected light into discreet diffraction angles
$\theta_{m}$ measured with respect to the back surface normal as depicted in
Fig. 1. In the reference frame of the sail, the incident and scattered
wavelengths are equal, and thus, the respective wave vectors may be expressed
$\vec{k}_{i}=k\hat{z}$ and
$\vec{k}_{m}=k(\cos\theta_{m}\;\hat{z}+\sin\theta_{m}\;\hat{x})$, where
$k=2\pi/\lambda$. The diffraction angles are governed by the grating equation:
$\sin\theta_{m}=m\lambda/\Lambda$ assuming normal incidence. We note that
$\cos\theta_{m}=\pm\sqrt{1-\sin^{2}\theta_{m}}$, where $+(-)$ corresponds to
transmitted (reflected) light. The $m^{\text{th}}$ order photon momentum
transfer efficiency imparted to the sail at the optical frequency
$\nu=c/\lambda$ may be expressed
$\vec{\eta}_{\nu,m}=(\vec{k}_{i}-\vec{k}_{m})/k$
$=(1-\cos\theta_{m})\;\hat{z}-\sin\theta_{m}\;\hat{x}$, where $c$ is the speed
of light, and normal incidence is assumed. For a light source having a
spectral irradiance distribution $\tilde{I}(\nu)$ the net momentum transfer
efficiency $\vec{\eta}$ may be found by integrating over all frequencies and
summing over all allowed diffraction orders for both polarization modes [12].
For an unpolarized source like the sun, we assume the spectral irradiance is
equally divided into $s$ and $p$ polarization states.

Figure 2: Fraction of integrated solar black body spectral irradiance for the
range $\nu_{min}=c/\Lambda$ to $\nu_{max}$, where
$I_{sun}=1360\;\mathrm{[W/m^{2}]}$. Insert: Black body spectral irradiance
with range $\nu_{min}=200\;[\text{THz}]$ $(\Lambda=1.5\;[\mu\text{m}])$ and
$\nu_{max}=900\;[\text{THz}]$ ($0.83I_{sun}$). Arrows: Range of maximum mode
number $M$.

The net radiation pressure force on the sail may be expressed
$\vec{F}=F_{0}\vec{\eta}$, where $F_{0}=I_{0}A/c$ where $A$ is the sail area
and $I_{0}$ is the irradiance. For example the solar blackbody irradiance
between $\nu_{\text{min}}$ and $\nu_{\text{max}}$ of a band-limited blackbody
source a distance $r$ from the sun may be expressed

$I_{0}=\frac{R_{S}^{2}}{r^{2}}\int_{\nu_{\text{min}}}^{\nu_{\text{max}}}\tilde{I}(\nu)d\nu=\frac{R_{S}^{2}}{r^{2}}\frac{2\pi
h}{c^{2}}\int_{\nu_{\text{min}}}^{\nu_{\text{max}}}\frac{\nu^{3}\;d\nu}{\exp(h\nu/k_{B}T)-1}$
(1)

where $R_{S}=6.957\times 10^{8}$ [m] is the solar radius, $h=6.626\times
10^{-34}\mathrm{[J\cdot s]}$ is the Planck constant, $k_{B}=1.381\times
10^{-23}[\mathrm{J/K}]$ is the Boltzmann constant, and we assign $T=5770.2$ as
the effective absolute temperature of the sun. Below we assume $r$ corresponds
to 1 [AU]. The case $\nu_{min,max}=0,\infty$ corresponds to the so-called
solar-constant, $I_{sun}=1360\;[\text{W/m}^{2}]$. Values of $I_{0}$ are
plotted in Fig. 2 as a function of the grating period for
$\nu_{min}=\tilde{\nu}=c/\Lambda$ and two different values of $\nu_{max}$:
$\infty$ (blue line) and $900\;[\text{THz}]$ (red line). The case used for our
FDTD model, $\lambda_{min}=0.333\;[\mu\text{m}]$ and
$\lambda_{max}=\Lambda=1.5\;[\mu\text{m}]$ ($\nu_{min}=200\;[\text{THz}]$, and
$\nu_{max}=900\;[\text{THz}]$) includes up to four diffraction orders and
spans 83% of the solar spectrum. Although wider bandwidths are of interest,
FDTD run times become prohibitively long.

Following Ref [12] the net radiation pressure force on the sail owing to a
band-limited source may be expressed

$\begin{split}\vec{F}^{s,p}=\frac{A}{c}\int_{\nu_{\text{min}}}^{\nu_{\text{max}}}\sum_{m=M_{\nu}^{-}}^{M_{\nu}^{+}}\tilde{I}^{s,p}_{m}(\nu)\left((1-\cos\theta_{m})\;\hat{z}-\sin\theta_{m}\;\hat{x}\right)\mathrm{d}\nu\end{split}$
(2)

Figure 3:  FDTD Schematic: Unit cell of period $\Lambda$ of (A) prism and (B)
meta gratings with plane wave source (red line), field monitors (blue lines),
and perfectly absorbing boundary layers (green areas).

where $\tilde{I}^{s}_{m}(\nu)$ and $\tilde{I}^{p}_{m}(\nu)$ respectively
correspond to the value of the spectral irradiance scattered into the
$m^{\text{th}}$ diffraction order for the $s$ and $p$ polarization states, and
where $\theta_{m}$ depends on frequency owing to the grating equation which
may be expressed, $\sin\theta_{m}=mc/\nu\Lambda$. The frequency-dependent cut-
off mode numbers at the normal incident are given by
$M_{\nu}^{\pm}=\pm\mathrm{INT}[\nu/\tilde{\nu}]$ (or equivalently
$\pm\mathrm{INT}[\Lambda/\lambda]$)  where $\mathrm{INT}$ represents the
integer value of the argument rounded toward zero. In a lossless system having
no guided surface waves that extend to infinity, we expect

$\tilde{I}(\nu)=\sum_{m=M_{\nu}^{-}}^{M_{\nu}^{+}}\left(\tilde{I}^{s}_{m}(\nu)+\tilde{I}^{p}_{m}(\nu)\right)$
(3)

In general $\tilde{I}^{s}_{m}(\nu)\neq\tilde{I}^{p}_{m}(\nu)$ owing to
polarization-dependent scattering.

The Maxwell stress tensor $\overline{\overline{T}}_{\nu}$ may be evaluated at
each frequency as an alternative method to evaluate the net force $\vec{F}$:

$\vec{F}^{s,p}=\int_{\nu_{\text{min}}}^{\nu_{\text{max}}}\vec{F}_{\nu}^{s,p}d\nu=\int_{\nu_{\text{min}}}^{\nu_{\text{max}}}\left(\oint_{S}\overline{\overline{T}}^{s,p}_{\nu,ij}\cdot
d\vec{S}\ \right)d\nu$ (4)

where $S$ is an arbitrary surface enclosing the sail, $\text{d}\vec{S}$ is the
elemental area vector and

$\overline{\overline{T}}_{\nu,ij}^{s,p}=\epsilon_{0}(E^{s,p}_{\nu,i}E^{s,p}_{\nu,j}-\frac{1}{2}|E^{s,p}_{\nu}|^{2}\delta_{ij})+\frac{1}{\mu_{0}}(B^{s,p}_{\nu,i}B^{s,p}_{\nu,j}-\frac{1}{2}|B^{s,p}_{\nu}|^{2}\delta_{ij})$
(5)

where $\epsilon_{0}$ and $\mu_{0}$ are respectively the vacuum permittivity
and permeability, $E$ and $B$ are respectively electric and magnetic field
amplitudes, and $\delta_{ij}$ is the Kronecker delta function. For a structure
that is periodic in the plane of incidence as depicted in Fig. 3 and extended
over a distance $L_{y}$ out of the plane, the only elemental areas that
contribute to (4) are $d\vec{S}_{z=\pm z_{0}}=\pm dx\;dy\;\hat{x}\pm
dx\;dy\;\hat{z}$. The force exerted across the area $L_{y}\times\Lambda$ of an
infinitely period grating may therefore be expressed

$\begin{split}\vec{F}^{s,p}&=\int_{\nu_{\text{min}}}^{\nu_{\text{max}}}\left(\int_{\Lambda
L_{y}}\left((\overline{\overline{T}}^{s,p}_{\nu,ij}\cdot
d\vec{S})_{z=-z_{0}}+(\overline{\overline{T}}^{s,p}_{\nu,ij}\cdot
d\vec{S})_{z=+z_{0}}\right)\right)d\nu\\\
&=L_{y}\int_{0}^{\Lambda}\big{(}(-T_{xx}-T_{zz})_{z=-z_{0}}+(T_{xx}+T_{zz})_{z=+z_{0}}\big{)}dx\end{split}$
(6)

where $z_{0}$ is an arbitrary distance from the grating, and the final
integral includes the frequency-integrated stress tensor components $T_{xx}$
and $T_{zz}$.

##  3 Numerical Methods

We used the open source FDTD numerical solver MEEP [17] to solve Eq.s (4) -
(6), making use of fast built-in “methods" like ForceSpectra to calculate
forces in a specified ForceRegion. To cross-validate the force values we
randomly compared them to values obtained using Eq.(2), this time using
diffraction mode options in MEEP. In both cases, Bloch periodic boundary
conditions were employed. The power spectrum of a broadband source in MEEP is
defined as the distribution function GaussianSource(fcen, fwidth) where fcen
and fwidth are respectively the center and width of the Gaussian distribution.
Force calculations are made in the frequency domain and we scaled them to
correspond to the solar blackbody spectral irradiance. The red line in Fig. 3
depicts a planar light source propagating in the $\hat{z}$ direction. The blue
lines represent so-called monitors where the electromagnetic fields
$\vec{E}_{\nu}^{s,p}$ and $\vec{B}_{\nu}^{s,p}$ are evaluated for the
determination of the Maxwell stress tensor, and where alternatively the
spectral irradiance $\tilde{I}_{m}^{s,p}(\nu)$ may be determined to evaluate
Eq. (2). The green lines in Fig. 3 represent perfectly matched layers. The
square numerical grid elements were set to $\delta x$ = $\delta z$ = 20 [nm].
The simulation ran until either $E_{z}$ or $H_{z}$ decayed to $10^{-6}$ of the
peak value.

Table 1: Multi-Objective Optimization Scheme: Nine variables, three objectives, and four constraints.  $\mathbf{x\in}$ | $[x_{1,2},w_{1,2},h_{1,2},n_{1,2},t]$
---|---
$\mathbf{max}:$ | $F_{x}^{s}(\mathbf{x}),\;F_{x}^{p}(\mathbf{x})$
$\mathbf{min}:$ | $\mathrm{mass}(\mathbf{x})$
$\mathbf{such\ that:}$ | $1.5\leq n_{1,2}\leq 3.5$
$\mathbf{such\ that:}$ | $-\Lambda/2\leq x_{1,2}\leq\Lambda/2$
$\mathbf{such\ that:}$ | $0\leq w_{1,2},h_{1,2}\leq\Lambda$
$\mathbf{such\ that:}$ | $0.1\mu m\leq t\leq 0.5\mu m$

The focus of this study was to determine optimized parameters of the two
structures depicted in Fig. 3, both having the same period
$\Lambda=1.5\;[\mu\text{m}]$: (A) a prismatic grating and substrate having
four optimization parameters $n_{1}$, $n_{2}$, $h_{1}$, and $t$; and (B) a
metasurface comprised of two pillars and a substrate having nine optimization
parameters $n_{1}$, $n_{2}$, $h_{1}$, $h_{2}$, $w_{1}$, $w_{2}$, $x_{1}$,
$x_{2}$, and $t$. We employed a multi-objective optimizer NSGA-II (with 40
agents, 40 offspring, 150 generations) [18] with the range of parameter values
listed in Table 1. The objectives are to achieve the largest values of
transverse force for both polarizations and to minimize the mass. A
representative set of 40 solutions (called Pareto-optimal) were obtained. The
same procedure was followed for silicon nitride $(n_{\mathrm{Si_{3}N_{4}}})$
structures, but in this case $n_{1}=n_{2}$ and $h_{1}=h_{2}$. Silicon nitride
is relatively stable in a space environment, its optical properties are well
characterized, and its lithographic fabrication techniques are mature. The
refractive index $n_{\mathrm{Si_{3}N_{4}}}$ varies from $\sim 2.00$ at $200$
[THz] to $\sim 2.15$ at $900$ [THz] [19]:

$n_{\mathrm{Si_{3}N_{4}}}^{2}-1=\frac{3.0249\lambda^{2}}{\lambda^{2}-0.1353406^{2}}+\frac{40314\lambda^{2}}{\lambda^{2}-1239.842^{2}}$
(7)

A solar sail is typically used to achieve a spiral trajectory toward or away
from the sun. In this case, the flight time may be minimized when the
transverse (lift) component of acceleration $F_{x}/M_{\text{sc}}$ is a
maximum, where $M_{sc}=m_{\text{sail}}+m_{\text{pl}}$ is the total mass of the
sailcraft, $m_{\text{sail}}$ is the mass of the diffractive sail material,
$m_{\text{pl}}$ is the mass of the payload and structural support mechanisms,
and $F_{x}=F_{x}^{s}+F_{x}^{p}$. The transverse acceleration is optimized when
both $F_{x}^{s}$ and $F_{x}^{p}$ are maximized and $m_{\text{sail}}$ is
minimized. The sail mass of our two designs may be expressed

$m_{\text{sail}}^{\text{prism}}=\left(\frac{1}{2}\rho_{1}h+\rho_{2}t\right)N_{x}^{2}\Lambda^{2}=\left(\frac{1}{2}\rho_{1}h+\rho_{2}t\right)A\\\
$ (8a)
$\begin{split}m_{\text{sail}}^{\text{meta}}&=\rho_{1}(N_{x}w_{1}h_{1}+N_{x}w_{2}h_{2})N_{x}\Lambda+\rho_{2}N_{x}^{2}\Lambda^{2}t\\\
&=\left(\rho_{1}w_{1}h_{1}/\Lambda+\rho_{1}w_{2}h_{2}/\Lambda+\rho_{2}t\right)A\end{split}$
(8b)

where $N_{x}$ is the number of grating periods across the sail, and $A$ is the
area of a square sail. Ignoring the payload mass ($m_{pl}=0$) and writing the
transverse component of force $F_{x}=I_{0}A\eta_{x}/c=m_{\text{sail}}a_{x}$ we
obtain the transverse acceleration for our unladen structures:

$a_{x}^{\text{prism}}=\frac{I_{0}}{\alpha
c}\;\frac{\eta_{x}}{\frac{1}{2}n_{1}h+n_{2}t}$ (9a)
$a_{x}^{\text{meta}}=\frac{I_{0}}{\alpha
c}\;\frac{\eta_{x}}{n_{1}(w_{1}\mathsf{f}_{1}+w_{2}\mathsf{f}_{2})+n_{2}t}$
(9b)

where $\mathsf{f}_{1,2}=h_{1,2}/\Lambda$ is the fill factor, and for
convenience we associate the refractive index and mass density with a
proportionality factor $\alpha$: $\rho_{1,2}\equiv\alpha n_{1,2}$. Using the
space qualified polyimide material CP1 [20] as an example, with a specific
gravity $\text{s.g.}=1.54$ and a mean refractive index of 1.57 we obtain
$\alpha=0.98\times 10^{3}\;[\text{kg/m}^{3}]$. For our silicon nitride
structures we instead combine its specific gravity, $\text{s.g.}=3.17$ [21]
with the mean index, $2.02$, to obtain $\alpha=1.57\times
10^{3}\;[\text{kg/m}^{3}]$. As seen in Eq. 9b the transverse acceleration is
independent of the sail area and is implicitly dependent on the grating period
$\Lambda$ via the efficiency factor $\eta_{x}$ (which is found by numerically
determining the transverse force $F_{x}$ value).

Table 2: Optimized parameters and cost function values for (A) prism and (B) meta gratings of arbitrary dispersionless materials, and (C) prism and (D) meta gratings for $\mathrm{Si_{3}N_{4}}$, each with period $\Lambda=1.5[\mu\text{m}]$, $L_{y}=1$ [m], $L_{x}=N\Lambda=1$ [m], $A=L_{x}L_{y}$.  Parameters | A | B | C | D
---|---|---|---|---
$h_{1}[\mu$m] | 0.76 | 1.12 | 1.02 | 0.62
$h_{2}[\mu$m] | - | 1.26 | - | $h_{1}$
$w_{1}[\mu$m] | - | 0.32 | - | 0.16
$w_{2}[\mu$m] | - | 0.16 | - | 0.24
$x_{1}[\mu$m] | - | 0.06 | - | 0.38
$x_{2}[\mu$m] | - | 0.44 | - | 0.1
Prism Angle |  26.9∘ | - |  34.2∘ | -
$n_{1}$ | 2.43 | 1.55 | $\text{Si}_{3}\text{N}_{4}$ | $\text{Si}_{3}\text{N}_{4}$
$n_{2}$ | 1.5 | 1.5 | $\text{Si}_{3}\text{N}_{4}$ | $\text{Si}_{3}\text{N}_{4}$
$t[\mu$m] | 0.1 | 0.1 | 0.1 | 0.11
Force [nN] | 785 | 787 | 722 | 416
mass [$\times 10^{-3}$ kg] | 1.07 | 0.73 | 1.93 | 0.84
$a_{x}$ [$\mu$m/$\mathrm{s}^{2}$] | 731 | 1080 | 373 | 494

##  4 Results & Analysis

Forty representative Pareto-optimal solutions are plotted in Fig.4 for the two
gratings having nine arbitrary parameters (A) and (B), and for the two
gratings comprised of $\mathrm{Si_{3}N_{4}}$ (C) and (D). The net transverse
radiation pressure force $F_{x}$ is plotted against the total mass of the
sail, $m_{\text{sail}}$. In all cases a trend in the data appears: Higher mass
sails provide higher forces. To select the most optimal design for each
structure we use the greatest value of the transverse acceleration
$a_{x}=F_{x}/m_{\text{sail}}$ as the deciding factor (see straight line in
Fig.4). The parameters for the Pareto-optimal solution that intersects this
line are tabulated in Table 2 for the four different cases.

We find that both the prismatic and metasurface structures having arbitrary
refractive indexes are able to produce large values of $F_{x}$, as is evident
in Fig.4 for Case A and Case B. However, owing to the lower mass of the
metasurface structure, its optimal acceleration
$a_{x}=1080\;[\mu\text{m}/s^{2}]$ is $48\%$ greater than that of the prism
grating. The ${\mathrm{Si_{3}N_{4}}}$ structures, Case C and Case D, depict
significantly less values of optimized acceleration.  These values may be
compared with a conventional aluminized polyimide sail [22] which is roughly
$3\;[\mu\text{m}]$ thick and achieves a momentum transfer efficiency of
roughly $90\%$ of the ideal value of $0.77$ :
$a_{x}=680\;[\mu\text{m}/s^{2}]$. This comparison suggests that an optimized
metasurface sail is a competitive alternative to a conventional reflective
sail. However, amongst the many unknown fabrication, packaging, unfurling, and
space weathering issues is whether a large robust metasurface grating can be
fabricated on a thin $(<1\;[\mu\text{m}])$substrate [14].

To better understand the spectral force characteristics of the four sails
examined in this study, we plot the transverse spectral force distribution
$F_{\nu,x}=F_{\nu,x}^{s}+F_{\nu,x}^{p}$ in Fig. 5. The blue line represents
the FDTD-obtained values corresponding to the Maxwell stress tensor
calculations, whereas the circles represent the values corresponding to our
FDTD modal analysis. The excellent agreement between these two approaches
provides a level of cross-validation of the methods. Fluctuations of the value
of $F_{\nu,x}$ are indicative of pronounced diffractive variations of the
transmitted and reflected light at different optical frequencies, as expected
for a small period grating [4]. Also plotted in Fig. 5 are theoretical values
of force for the ideal limit $\eta_{x}=1$ (black line) and the ideal
reflective sail $\eta_{x}=0.77$ (red line):
$F_{x,\nu}=\eta_{x}\tilde{I}{\nu}A/c$.  These results suggest that the
diffractive sails explored in this study may equal or exceed the acceleration
of a reflective sail only if there is a small-mass advantage of the former.
The prism and pillar designs suffer from the effects of external and internal
reflections which can scatter light that opposes the desired transverse
scattering direction. For example front surface reflections from the prism in
Fig. 3 (A) have positive values of $k_{x}$ which oppose the transmitted
(refracted) rays. Those reflected rays carry 17% of incident beam power owing
to Fresnel reflections. Less than two thirds of the incident radiation is
refracted out the back surface owing to internal reflections and shadowing
effects from the steep side facets. It is yet unknown whether the added mass
of anti-reflection coatings would provide increased the transverse
acceleration. Other unknowns that are beyond the scope of this paper include
the practical limits of assumptions about the rigidity of the sail, the
coherence properties of the incident sunlight, and whether the sail can be
packaged and unfurled without changing its optical properties.

Figure 4: Pareto optimal solutions for (A) prismatic and (B) metasurface
gratings having arbitrary refractive indexes, and for (C) prismatic and (D)
metasurface gratings comprise of silicon nitride. A sun-facing square sail of
area $1[\text{m}^{2}]$ illuminated with a band-limited solar black body is
assumed. The optimal transverse acceleration $a_{x}$ for each case is
determined from the slope of the straight line, and the corresponding design
parameter values for the intersecting points are given in Table 1.  Figure 5:
Spectral transverse force distribution $F_{x,\nu}$ determined from Maxwell
stress tensor (blue line) and modal analysis (open circles) for the four cases
described in the text, and for an idealized reflective sail (red line) and the
upper theoretical bound (black line). An area of $1\;\mathrm{m}^{2}$ is
assumed.

##  5 Conclusions

We performed FDTD simulations coupled with a NSGA-II multi-objective optimizer
to determine design parameters for four different grating structures, each
having a period of $1.5\;[\mu\text{m}]$ and a sail area of
$1\;[\text{m}^{2}]$. The small grating period was selected to satisfy a small
desired mass and a marginal cutoff wavelength of the solar blackbody spectrum.
Our optimization study included 3 objectives and up to 9 variables, as well as
both s and p polarization. The transverse component of radiation pressure
force was determined for a truncated solar black body radiator (200-900 [THz]
or equivalently, 0.33 to $1.5[\mu\text{m}]$) at 1 [AU] for the purpose of two-
orbit changing maneuvers in space. An optimized metasurface grating comprised
of two pillars per period was found to provide $48\%$ more transverse
acceleration than an optimized prism grating owing to the small mass of the
former grating. We found that Silicon Nitride did not perform well for either
the prism or two-pillar metasurface design. Although none of the structures
provided radiation pressure force values exceeding those of an ideal flat
reflective sail, the diffractive sail may nevertheless provide an acceleration
advantage if the proposed sun-facing diffractive sail spacecraft has a total
lower mass than a reflective sailcraft. The design of alternatives to flat
reflective sails is an emerging area of research and we therefore believe
continued exploration of diffractive designs such as hybrid
reflective/transmissive structures will provide more efficient solar sails in
the future.

FundingNational Aeronautics and Space Administration(NASA) (80NSSC19K0975,
80MSFC22F0165), Johns Hopkins University Applied Physics Lab (177864).

AcknowledgmentWe are grateful to Charles (Les) Johnson and Andrew F. Heaton
(NASA George C. Marshall Space Flight Center, Huntsville, AL), and to Amber L.
Dubill (Johns Hopkins Applied Physics Laboratory) for discussions related to
solar sailing. We also thank Rajesh Menon and Apratim Majumder (U. Utah, Salt
Lake City, UT) for meta-material and FDTD modeling discussions.

Disclosures The authors declare no conflicts of interest.

Data availability Data underlying the results presented in this paper are not
publicly available at this time but may be obtained from the authors upon
reasonable request.

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