## Abstract

Recent research has studied the feasibility of using laser radiation pressure to propel lightweight spacecraft, such as sails, at relativistic speeds. One major challenge is the effect of laser beam distortion on sail stability. We propose and investigate the use of lightsails based on Kerr nonlinear photonic crystals as a passive method for increasing sail stability. The key concept is to flatten the dependence of reflected intensity on incident intensity at the laser wavelength, using a specially designed, guided-resonance mode of the nonlinear photonic crystal. We use coupled-mode theory to analyze the resonance characteristics that yield the flattest curve. We then design a silicon nitride photonic crystal that supports a resonance with the desired properties. We show that our design simultaneously provides both high stability and high thrust on the sail, unlike designs based on linear materials.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The proposed use of radiation pressure to propel objects in space has long been a subject of fascination [1–4]. Both solar and laser illumination have been considered as potential light sources [1–12]. Most recently, the NASA Starlight and Breakthrough Starshot Initiative have proposed to use laser beams to propel spacecrafts to relativistic speeds [13,14]. Some of the major challenges and criteria for the laser sail design have previously been summarized in Ref. [15]. Here, we focus on the crucial issue of beam distortion. For systems that use ground-based lasers as light sources, the, the incident light on the laser sail will be distorted by travel through the atmosphere [16–22]. Such beam distortions may dramatically impact sail stability, causing unwanted torque and deflecting the sail trajectory. In this paper, we propose and analyze the use of a nonlinear optical material for the sail to help passively stabilize against beam distortions.

The effect of beam distortion on sail stability is illustrated in Fig. 1. Suppose the ideal beam intensity is symmetric around the sail center [black line in Fig. 1(a)]. Beam distortions will disturb the intensity, potentially resulting in asymmetric profiles (e.g. red, dashed line in Fig. 1(a)]. For a linear sail material, the reflected intensity is proportional to the incident intensity [red, dashed line in Fig. 1(b)]. As a result, the radiation pressure will also be asymmetric with respect to the sail center [red, dashed line in Fig. 1(c)]. This can create a torque on the sail and result in spinning (indicated schematically by the arrows). Spinning may derail the sail trajectory, a problem that is made particularly acute by the lack of restoring forces in the vacuum of outer space.

Beam distortion is a common problem in astronomy, where adaptive optics solutions are used to correct images for atmospheric perturbations [23–27]. Given real-time information about the effect of the atmosphere on the beam spot, the wave fronts from the ground-based laser might in principle be pre-compensated (e.g. deliberately distorted) so as to yield a uniform, undistorted spot on the sail. However, given the extremely high target speeds to be attained by the sail (0.2*c* at the end of acceleration) and the fundamental limits on the response time of adaptive optics introduced by the travel time of light between the atmosphere and the ground (∼ milliseconds), adaptive optics alone may not be sufficient to ensure stability. It is thus of interest to consider how the sail itself might be designed to provide stabilization against residual beam distortions.

To reduce the effects of beam distortion on sail stability, we introduce a nonlinear material for the sail. In particular, we propose to use a nonlinear photonic crystal (PhC) to provide a relatively constant reflected intensity over a range of incident intensities. This property is depicted schematically in Fig. 1(b) (blue, dashed line). For the nonlinear PhC, the radiation pressure across the sail will also be flattened, as shown in Fig. 1(c). This reduces asymmetry, and hence unwanted torque. In Section 2 below, we analyze the optimal characteristics for the PhC design within a semi-analytical approach. We then present a specific PhC design implementation in Section 3 and analyze its performance. We show that the nonlinear PhC achieves high stability while maintaining high overall thrust, in contrast to comparison linear systems. Lastly, we discuss the implications of this approach on system design, particularly in light of the large Doppler shifts present at relativistic speeds.

## 2. Effect of nonlinear resonance on stability and thrust

For a non-absorptive sail, the local radiation pressure on the sail due to normally incident light is given by [28]:

where ${I_{ref}}({x,y} )$ is the local reflected intensity (also known as irradiance; defined as the energy reflected per unit area per unit time from the surface).We consider a photonic-crystal (PhC) slab structure, consisting of a high-index, finite thickness layer patterned with a periodic array of holes, as a reflector. Photonic crystals support electromagnetic modes known as guided resonances [29,30] which produce characteristic resonant, Fano line shapes in their reflection spectra.

We assume the length scale of intensity perturbations in the beam is orders of magnitude larger than the unit cell size of the photonic crystal; the unit cell is typically on the order of microns. Moreover, we assume that the total size of the sail (meters in length) is orders of magnitude larger than both the wavelength and the lattice constant. In this limit, we assume that the local radiation pressure on the sail is proportional to the local, reflected intensity, which is implicitly averaged over the unit cell.

From temporal coupled-mode theory, a mathematical formula for the ratio of reflected to incident intensity is [31],

*I*and

_{ref}*I*are the reflected and input intensities, averaged over the unit cell. They may vary with position (

_{in}*x,y*) on a scale much larger than the unit cell, due to variation in intensity of the incident beam. Here, $r$ and

*t*are real-valued parameters representing the reflectivity and transmissivity of the unpatterned slab, and

*τ*is the cavity lifetime.

*ω*is the resonance frequency, which may also vary in position due to nonlinear effects. For a Kerr nonlinear system [32], the resonance frequency shifts with increasing input intensity as

_{0}(x,y)*I*is a characteristic intensity, given by [33] where

_{0}*n*is the Kerr coefficient, $Q = {\omega _{res}}\tau /2$ is the quality factor of the resonant cavity, and

_{2}*k*is the dimensionless nonlinear feedback factor that defines the degree of the spatial confinement of the mode in PhC slab,

From Eqs. (2)–(5), we see that as the intensity of light incident on the slab increases, the resonance frequency shifts as well. The resulting shift in the reflection spectrum results in a nonlinear dependence of reflected intensity on incident intensity. Below, we study how the nonlinear dependence can be controlled by adjusting the initial wavelength, quality factor, and line-shape of the resonance.

For concreteness, we start by considering a quality factor *Q *= 5000, and *r* = −0.18. We assume that $t = 1 - {r^2}$ (no intrinsic material absorption). We define the detuning $\Delta = {\lambda _{in}} - {\lambda _{res}}\; $ as the difference between the laser wavelength $({\lambda _{in}} = 1064\textrm{nm})$ and the wavelength of the resonance in the absence of nonlinearity, ${\lambda _{res}} = 2\pi c/{\omega _{res}}$.

The reflection spectrum is shown in Fig. 2(a) for $\Delta = 0$. The red curve plots the resonance in the low-intensity limit. The reflection is strongly peaked as a function of wavelength, as is characteristic of a resonance mode. The resonance is initially centered at the input laser wavelength $({{\lambda_{in}} = {\lambda_{res}}} )$. As the input intensity increases, the spectrum shifts to the right due to the modification of the resonance wavelength by the Kerr nonlinearity. This results in a change in reflectivity at the laser wavelength, as indicated by the circles in Fig. 2(a).

The shift in the reflection spectrum is nonlinear with laser intensity and strongly depends on the initial detuning of the laser wavelength from the resonance, as shown in Fig. 2(b). For the zero-detuning case, the shift is monotonic with laser intensity (yellow line). The plot also shows several cases of non-zero detuning. For sufficiently large, positive detuning (e.g. Δ = 0.2nm), the curve becomes multivalued; there is a region of I* _{in}*/I

*for which more than one solution for Δλ*

_{o}_{res}exists. However, for sufficiently large values of I

*/I*

_{in}*(i.e. > 2 in the figure), all curves shown are single valued and monotonically increasing, and the shift increases with increasingly positive detuning.*

_{o}The reflectivity at the laser wavelength is shown in Fig. 2(c). Starting with the zero-detuning case (yellow line), the reflectivity decreases with increasing laser intensity, as expected from the symbols in Fig. 2(a). For negative detuning, the reflectivity also decreases monotonically. For positive detuning, the reflectivity first increases, and then decreases, with increasing intensity. The nonmonotonic behavior results from the shifting of the resonance peak through the laser wavelength. For all curves shown, sufficiently large values of I* _{in}*/I

*yield decreasing reflectivity with increasing intensity.*

_{o}The reflected intensity as a function of input intensity is shown in Fig. 2(d). We will refer to this function as the *input-reflected* intensity *curve*. The reflected intensity increases and then flattens as a function of input intensity for all values of detuning shown. We will focus on a range of input intensities for which all curves are relatively flat, indicated by the grey, shaded box (3 < I* _{in}*/I

*< 4). In this region, increase in input intensity is largely compensated by the reduction of reflectivity shown in Fig. 2(c). As a result, the reflected intensity is nearly constant. Figure 2(d) thus illustrates the key feature of our approach: by designing the sail to support a guided resonance mode and a Kerr nonlinearity, we achieve a relatively constant reflected intensity over a range of input intensities. This effect will mitigate the type of instability illustrated in Fig. 1, which results from a linear increase in reflected intensity with input intensity. We note that Fig. 2(d) also shows that the reflected intensity increases with positive detuning, increasing the thrust on the sail.*

_{o}Figure 3 examines how the flatness and amplitude of the reflected intensity depend on the quality factor of the resonance, for various values of detuning. To quantify flatness, we calculate the average value of *dI _{ref}/dI_{in}* over an input intensity range from

*3I*to

_{0}*4I*[e.g. the average slope of the input-reflected intensity curve in Fig. 2(d)]. The results are shown in Fig. 3(a). We exclude the region of parameter space where the reflected intensity curve is multivalued between

_{0}*3I*to

_{0}*4I*, indicated by the grey region in the plot. An average slope of zero is obtained for values of detuning and quality factor lying along the dashed, green line. This line thus indicates the optimal condition for sail stability. As

_{0}*Q*increases, the optimal detuning decreases as well. Figure 3(b) shows the averaged reflected intensity. For a given value of

*Q*, the average reflected intensity increases with detuning. Operating along the maximum flatness curve (green, dashed line) yields a relatively high value of reflected intensity, though not maximum. Since the thrust on the sail is proportional to the reflected intensity, the region of parameter space lying above the dashed line and below the grey region represents a trade-off between maximum stability and maximum thrust.

The choice of *Q* will determine the required input intensity for operation. Since *I _{o}* is proportional to 1/

*Q*

^{2}from Eq. (4), using a higher-

*Q*structure will require less input intensity from the laser to shift the system into the nonlinear regime shown in Fig. 2(d), e.g. to ensure that

*I*> 3

_{in}*I*.

_{o}The shape of the resonance used [e.g. the red curve in Fig. 2(a)] affects the nonlinear behavior of the reflected intensity. Figure 4(a) shows the reflection spectrum in the limit of low intensity for different *r* values. The quality factor is fixed at 5000, and *t* is equal to *1 - r ^{2}*. The spectrum is completely symmetric for

*r =*0. Nonzero values of

*r*make the spectrum asymmetric. Focusing on wavelengths below the resonance peak (λ < 1064nm), reflectivity decreases with decreasing

*r*. This tends to make the reflectivity fall off faster with input intensity, as the Kerr nonlinearity shifts the resonance peak to higher wavelengths. Figure 4(b) shows the input-reflected intensity curve for zero detuning. As expected, the flattest response is obtained for

*r*= −0.2. Increasing

*r*thus reduces sail stability. There is a trade-off between stability and thrust; decreasing

*r*reduces the value of

*I*

_{ref}/I_{o}.

## 3. Design implementation of lightsail

To implement our approach, we have designed a photonic-crystal slab that supports a nonlinear guided-mode resonance with the required characteristics. We consider a silicon nitride slab patterned with a square lattice of air holes. Silicon nitride was selected due to its combination of significant Kerr nonlinearity and negligible absorption at wavelengths close to 1064nm. The hole radius *r = *0.22*a*, thickness *t = *0.76*a*, and lattice constant *a = *1µm were chosen to place a guided-mode resonance close to 1064nm. We simulated the reflection spectrum for normally-incident light using Lumerical FDTD solutions, using the wavelength-dependent refractive index data found in Ref. [34]. In particular, we obtained the reflection spectrum in the low-intensity limit by setting the Kerr coefficient to zero in the FDTD simulation.

The low-intensity reflection spectrum is shown in Fig. 5(b). From the figure, we see that the slab supports a Fano resonance with a quality factor *Q = *5217 at 1064.1nm and a peak reflectivity close to 1. Fitting the spectrum to the Fano line shape gives *r = *−0.15 and *t = *0.96.

To calculate the nonlinear response of the photonic crystal, we use Eq. (2) along with the fitted values of the resonance parameters. The result is shown in Fig. 5(c). It is evident from the figure that the photonic crystal demonstrates a flattened input-reflected intensity curve for intensity levels above 3*I _{0}*. To find the physical value of the characteristic intensity

*I*, we compute the nonlinear feedback factor using Eq. (5) and the electric field obtained from the FDTD simulation. Taking

_{o}*n*= 10

_{2}^{−18}m

^{2}/W [35], we find that

*k*≈ 0.014. This yields an estimate of 4

*I*0.3W/m

_{0 }=^{2}for unit cell, or approximately 300GW/m

^{2}for a lattice constant of 1µm. Operation at a target value of 100GW/m

^{2}[13] would require a further increase in

*Q*to ∼9100.

To provide insight into how our nonlinear approach benefits sail performance, we compare our design to several comparison cases. For linear systems, we find that increased stability necessarily comes at the cost of decreased thrust. The nonlinear system breaks this constraint, providing both high stability and high thrust simultaneously.

We first consider a perfect mirror (*R *= 1). Figure 6(a) shows the average slope of the input-reflected intensity curve as a function of detuning from resonance. Low average slopes correspond to high stability. Figure 6(b) shows the normalized, average reflected intensity. High values correspond to high thrust density. Both quantities are constant and equal to 1 (green, diamonds). The perfect mirror thus provides low stability and high thrust. This behavior results from the input-reflected intensity curve shown in Fig. 6(c), which is linear with slope 1.

The case of a linear, dielectric slab with the same thickness and linear refractive index as our design is shown by the yellow stars in Figs. 6(a) and 6(b). The average slope is equal to the average reflectivity, as is true for any linear system. Both quantities are low relative to the perfect mirror, corresponding to high stability and low thrust density. In Fig. 6(c), this case corresponds to a much lower-lying and flatter line than for the perfect mirror.

Patterning the linear slab with holes, in the same pattern as our design, creates a guided-resonance mode. Since the Kerr coefficient is set to zero, the resonance does not shift with input intensity. The dependence of average slope and normalized, average reflected intensity on detuning [shown by blue squares in Figs. 6(a) and 6(b)] is determined entirely by the line shape shown in Fig. 5(b). Both quantities are identical and decrease with detuning. Depending on the value of detuning chosen, this structure can provide either low stability and high thrust density, or high stability and low thrust. However, high stability and high thrust density cannot be obtained simultaneously. Visually, this trade-off is illustrated by Fig. 6(c). Changing the detuning shifts the input-reflected intensity curve between that of the mirror and that of the linear slab. Increased stability (flatter line) comes at the cost of decreased thrust density (lower reflected intensity).

The nonlinear system breaks the fundamental constraints of the linear system to simultaneously provide high stability and high thrust density. As shown by the red dots in Fig. 6, the average slope [Fig. 6(a)] approaches zero at a detuning of approximately 0.25nm. The normalized reflected intensity meanwhile increases with detuning as shown in Fig. 6(b). Both high stability (small average slope) and high thrust density (large reflected intensity) are obtained by using a detuning close to 0.25. With this choice, the overall performance of the nonlinear guided-resonance mode design is superior to all the comparison cases. The input-reflected intensity curve is shown in Fig. 6(c). This choice of detuning provides a flat line at high reflected intensity.

## 4. Discussion

In practice, the variation in incident intensity with position across the sail will depend on the beam shape chosen (such as a Gaussian), as well as any beam distortions. Provided that the incident intensity lies between 3*I _{0}* and 4

*I*across the whole sail, the reflected intensity will be nearly constant in our design. One approach is to use a beam that is much bigger than the sail, so that the portion of the beam intersecting the sail lies entirely in this range. However, this approach sacrifices power. Alternatively, one could design the sail such that the characteristic intensity

_{0}*I*varies across the sail with the same profile as the beam shape. Since

_{0}*I*is proportional to

_{0}*1/Q*from Eq. (4), one might potentially vary the photonic crystal pattern adiabatically to change the

^{2}*Q*. Previous work in literature has studied the systematic tuning of

*Q*by varying the size of features in the photonic-crystal lattice [29,36,37].

An important complexity of any resonant-based approach, including the one presented above, arise from the Doppler shift [15]. For a laser wavelength of 1064nm and a velocity of 0.2*c*, the sail will see a Doppler-shifted wavelength of approximately 1298nm. This shift is much larger than the resonance width used above. Two approaches are possible in principle. First, an on-board tuning mechanism might be used to align the resonance wavelength to the Doppler-shifted laser. While electro-optic or thermo-optic effects are likely too small for this purpose, designs based on optomechanical effects [38] might potentially provide large shifts and self-tuning capabilities. In this case, care must be taken to choose a photonic crystal material with low absorption over the entire Doppler-shifted range, in order to detrimental heating. Alternatively, the wavelength of the ground-based laser could be tuned to compensate for the Doppler shift. Recent advances in the commercialization of robust, easily tunable CW Ti:Sapphire lasers suggest this approach may also prove attractive.

We note that the goal of the approach used in this paper is to reduce torque and tilt of the sail. In the event that the sail does tilt, the reflection will in general depend on angle. Future work may need to consider advance strategies that involve either minimizing the angle dependence.[39–42], or deliberately harnessing angle-dependence to generate a corrective torque [43].

Lastly, we note that use of a micropatterned sail will introduce more stringent fabrication requirements than unpatterned sails, particularly over large sail areas. However, we expect that the relevant fabrication technology will benefit from continued advances in display technology, particularly in the area of roll-to-roll nanoimprint lithography [44].

## 5. Conclusion

In conclusion, we have proposed a passive solution for increasing the stability of a laser-propelled sail in the presence of intensity distortions in the incident beam. The key concept of our approach is to design the sail to exhibit a flattened input-reflected intensity curve at the laser wavelength. In particular, we achieve this goal using a guided-resonance mode in a Kerr nonlinear photonic crystal. We used semi-analytic coupled mode theory to analyze the characteristics of the mode that yield the flattest curve. We presented a concrete design based on a silicon nitride photonic crystal that supports a mode with the desired characteristics. We compared the performance of our nonlinear design with designs based on linear materials. For linear systems, the linear dependence of reflected intensity on incident intensity imposes a trade-off between stability and thrust on the sail. We show that our nonlinear design lifts this constraint, simultaneously providing both high stability and high thrust at fixed detuning of the laser from the resonance.

## Acknowledgments

Computation for work described in this paper was supported by the University of Southern California Center for High-Performance Computing and Communications.

## Disclosures

The authors declare no conflicts of interest.

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