Open Access
Add to BibSonomy
Abstract
The p-polarization component of radiation pressure force from an unpolarized blackbody light source is predicted by the use of a Maxwell equation solver for a right triangular prism grating of period 2 μm and refractive index 3.5. The transmitted and reflected angular scattering distributions are found to qualitatively agree with diffraction theory: At relatively short wavelengths the transmitted light is concentrated near the refraction angle, and reflected light is concentrated near the reflection angle. Owing to diffraction and multiple internal reflections, however, the spectral irradiance of transmitted and reflected light was found to significantly vary with wavelength. We found that the high value of the refractive index produced a large fraction of reflected light, thereby reducing the net transverse component of radiation pressure force. These results suggest that low index transmission gratings, anti-reflection coatings, optimized metasurface films, or reflection gratings should be explored for future solar sailing missions.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Since the use of radiation pressure for in-space propulsion was first proposed in the early 1920s and 1930s by Tsiolkovsky [1] and Tsander [2], the attraction of solar sailing in outer space to otherwise unreachable destinations has never faded. Solar sails are large, flexible, and traditionally, reflective surfaces, using the momentum imparted upon them by photons from the Sun as the primary means of propulsion [3]. Unlike chemical and ionic propulsion engines, which have limitations due to the requirement of transporting the required propellant (see for example Ref. [4]) a solar sailing spacecraft or “sailcraft” ideally carries no propellant, therefore lowering the total mass and potentially increasing the acceleration. A sailcraft is instead propelled by the unlimited supply of radiation pressure from the sun. Since the first solar sailing science mission was proposed, designed to study the 1986 visit of Halley’s Comet [5,6], the technical readiness level of solar sails has steadily improved – especially within the last two decades. With the recent revival and increased interest in new solar sailing missions by the Planetary Society, NASA, JAXA and other international agencies, in-space solar sailing demonstration missions have become feasible, e.g., IKAROS (JAXA) [7,8], NanoSail-D (NASA) [9,10], LightSail (Planetary Society) [11], Sun–Earth sub-L1 Lagrange point halo orbit mission [12], near-Earth asteroid rendezvous (NASA) [13], the upcoming Near-Earth Asteroid Scout mission (NASA) [14,15], European Space Agency sail deployment mission [16,17], and the NASA Solar Cruiser mission concept to support the heliophysics community [18].
The objective of this report is to explore, by use of a numerical Maxwell equation solver, the transfer of optical momentum to mechanical momentum for a light sail comprised of a linear array of prismatic refractive elements. The following formalisms may be heuristically applied to other periodic structures (e.g., metamaterials, geometric phase elements, or freeform surfaces) and light sources (e.g., lasers). In past years a flat solar sail comprised of a reflective metal-coated polyimide or polyester thin film has been assumed for the missions listed above. Such a sail has several disadvantages, including a lack of attitude control about the roll axis due to negligible radiation pressure torque normal to the sail. Faceted reflective sails [19], liquid crystal-based devices [8,20–22] and anomalously reflecting metasurfaces were proposed for attitude control [23]. What is more, a reflective sail is unable to reach to theoretical maximum efficiency of $100\%$ owing to the tilt angle requirement for spiral orbit trajectories [24]. Optical diffraction provides an alternative mechanism for solar sailing [24–26]. The veracity of this concept was first demonstrated in the laboratory by use of a vacuum torsion oscillator and laser [27,28]. Since then various diffractive sail designs have been explored for both solar and laser-driven sails [29–37].
Solar sails designed to follow inward or outward spiral trajectories require a significant transverse component of force, perpendicular to the sun line. However, radiation pressure force on an ideal reflective sail is parallel to the surface normal, and thus the sail must be tilted with respect to the sun line to achieve a transverse component of force. The amount of solar power projected onto the sail decreases with tilt angle, thereby diminishing the force. In contrast, a non-tilted (sun facing) diffractive sail can, in principle, experience a significant transverse component of radiation pressure force, thereby allowing navigation without sacrificing the amount of solar power projected onto the sail. A Fourier series analysis of an idealized right triangular diffraction grating found that at wavelengths much smaller than the grating period the radiation pressure is dictated by Snell’s law for a transmission grating (and the law of reflection for a reflection grating), whereas at wavelengths shorter than, but on the order of the grating period, diffraction may weaken the radiation pressure force [26].
To better understand the latter phenomenon and other non-idealized light scattering affects on radiation pressure we numerically solved Maxwell’s equations for a right triangular prism grating of base period $\Lambda = 2 \ [\mu m]$ and refractive index $n = 3.5$, with a variable prism apex angle $\alpha$. A schematic is illustrated in Fig. 1. The angle of a deviated refracted ray through the back surface, $\psi$, is governed by Snell’s law, $\sin \psi = n \sin \theta ''$, where $\theta '' = \alpha -\theta '$ is the incident angle at the back surface, and $n \sin \theta ' = \sin \alpha$ describes refraction through the front surface. For example, if $\alpha = 21^\circ$ and $n=3.5$ then $\psi =66^\circ$ represents the deviation angle from the sun line (the $z$-axis). For an infinitely periodic structure and spatially coherent illumination, the far-field transmitted light is expected to deviate from the sun line at discrete angles, $\psi _m$, described by the grating equation: $\sin \psi _m = m \lambda /\Lambda$ where $\lambda$ is the optical wavelength and $m$ is a signed integer. We adopt the angle sign convention where $\psi$ is postive as illustrated in Fig. 1. Likewise $\psi _m$ is positive if $m > 0$. Front surface reflection deviates light from the sun line at the angle $\psi _r = -\pi + 2\alpha$, and reflected diffraction orders at angles $\psi _{r,m}$ are also expected. A so-called “blaze” condition occurs when $\psi _m$ coincides with $\psi$ (or $\psi _{r,m}$ coincides with $\psi _r$) at particular wavelengths.
Fig. 1. Light incident upon a sun-facing diffractive sail comprised of an array of right prisms of base length $\Lambda$, apex angle $\alpha$, and refractive index $n$. Transmitted deviation angle, $\psi$. Reflected deviation angle, $\psi _r = -\pi + 2\alpha$. Net force, $\vec {F}$. Internal ray angles $\theta '$ and $\theta ''$.
2. Theory
The solar radiation pressure force on a sail of area $A$, located $1 \; [\text {AU}]$ from the sun, may be expressed
where $I_{sun} = 1.36 \; [kW/m^2]$ is the solar constant, and $c$ is the speed of light. The momentum transfer efficiency vector, $\vec {\eta } = \eta _x \hat {x} + \eta _z \hat {z}$, where $\hat {z}$ and $\hat {x}$ are respectively unit vectors parallel and perpendicular to the sun line, is determined from the scattered distribution of light. These values may be estimated from diffraction theory if multiple internal reflections and shadowing near edges can be ignored [26]. Accurate values that make fewer assumptions may be obtained by use of a finite-difference time-domain (FDTD) Maxwell equation solver like Lumerical.This paper considers a spatially coherent transverse magnetic (p-polarized) incident field having frequency components:
where $A_{\nu }$ is the spectral field amplitude, $\nu$ is the optical frequency, and $\lambda = c/\nu$ is the optical wavelength. The incident spectral irradiance is related to the squared amplitude, which may be expressed $I_{\nu }' = A_{\nu }^2$. Radiation pressure from the highly transmissive p-polarization component of light is of particular interest for future comparisons with other polarization-dependent scattering elements that may be used as spacecraft attitude control devices.When the sail is illuminated with sunlight, $I_{\nu }'$ may be represented by the blackbody spectrum at the temperature $T = 5770 \;[K]$:
The integrated spectral irradiance (i.e., the irradiance) for any distribution $I_\nu '$ may be expressed
Inserting Eq. (3) into Eq. (4) one obtains the solar constant: $I_{bb} = I_{sun} = 1.36 \; [kW/m^2]$.
Lacking a blackbody option in Lumerical, we instead specified a “broadband” source option having a quasi-Gaussian spectral irradiance distribution function $I_{qg,\nu }'$. Data corresponding to the spectral irradiance for free space propagation was found to have a peak value at $\nu _{peak}=1596 \; [THz]$ and FWHM of $1900 \; [THz]$. Scattering from the grating was determined at $N = 500$ equally spaced intervals spanning the range $[\nu _{0}, \nu _{max}]$, where $\nu _0 = c/\Lambda =$ $150 \; [THz]$ is the grating cut-off frequency (corresponding to $\lambda _{max} = \Lambda$), and $\nu _{max} = 3000 \; [THz]$ (corresponding to $\lambda _{min} = 0.1 \; [\mu m]$). Note that the numerically determined value of the irradiance for the quasi-Gaussian source is not equal to the solar constant. We also note that $6\%$ of the solar blackbody irradiance is cut off from diffraction owing to the relatively small value of $\Lambda$ (i.e., large value of $\nu _0$). The values of $I'_{bb,\nu } /I_{sun}$ and $I'_{qg,\nu } /I_{qg}$ are plotted in Fig. 2.
Fig. 2. Normalized spectral irradiance distribution functions. Solar blackbody at $T = 5770 \;[K]$ (solid line). Quasi-Gaussian (dashed line). The cutoff frequency for a $\Lambda = 2 \; [\mu m]$ grating period occurs at $\nu _0 = 150 \; [THz]$.
The two-dimensional layout of the numerical computation space is illustrated in Fig. 3 for a single unit cell of the infinitely periodic prism grating. The electric field is polarized in the plane of incidence. The computational grid is comprised of rectangular elements of size $\delta x \times \delta z$, where $\delta x = 2.85 [nm]$ and $\delta z = 10 [nm]$. Mode expansion monitors are placed before and after the prism to respectively capture spectral modal reflection data, $R_{\nu,m} = I_{\nu,m}'^R / I_{qg,\nu }'$ and spectral modal transmission data, $T_{\nu,m} = I_{\nu,m}'^T / I_{qg,\nu }'$ for each allowed diffraction order, $m$, where $I_{\nu,m}'^R$ and $I_{\nu,m}'^T$ are respective the numerically determined spectral modal irradiance distributions at each diffraction order.
Fig. 3. Numerical computation space layout. Right triangular prism (turquoise) of apex angle $\alpha$, grating period $\Lambda = 2 \; [\mu m]$ and refractive index $n=3.5$. Incident wave vector $\vec {k}_i$ and transverse magnetic field $\vec {E}_i$. Numerical grid elements: $\delta x =2.85[nm], \delta z = 10 \; [nm]$. Spectral data is obtained over 500 data equally spaced frequency values between $150$ and $3000 \; [THz]$.
The transverse component of the momentum transfer efficiency vector for the quasi-Gaussian light source may be numerically calculated from the generalized equations [26],
To determine the transverse efficiency for the solar blackbody spectrum we make use of the same numerically determined values of $R_{\nu,m}$ and $T_{\nu,m}$, and replace Eq. (5) with
Note that we have opted to normalized the efficiency by the entire solar irradiance, rather than the half value corresponding to a single polarization. The latter approach doubles the alternative efficiency value.
3. Results and analysis
The numerically determined modal spectral transmission and reflection values, $T_{\nu,m}$ and $R_{\nu,m}$ are shown in Fig. 4 for the case $\alpha = 16^\circ$, $n=3.5$, $\psi = 44^\circ$. Frequencies in the range from $0$ to $\nu _0 = 150 \; [THz]$ are cut-off from scattering into all modes $m \ne 0$. In Fig. 4(A) the dominate diffraction orders $\tilde {m}$ correspond to the blaze condition where $\psi _{\tilde {m}} = \sin ^{-1} (\tilde {m} \nu _0 / \nu ) \approx \psi$. In Fig. 4(B) the dominate diffraction orders $\tilde {m}$ correspond to the blaze angle where $\sin \psi _{r,\tilde {m}} \approx \sin \psi _r = \sin (-\pi + 2\alpha ) = - \sin (2\alpha )$. An examination of the color bars in these figures suggests that, on average, the dominant transmitted modes are significantly more intense than the dominant reflected modes, by a factor of roughly $5/2$. Whereas transmitted light is primarily scattered at positive angles, the reflected light scatters at primarily negative angles. Considering these two opposing scattering angles along with the higher values of irradiance in the transmitted case, we expect, after summing over all modes and integrating over the optical frequency according to Eq. (5) or (6), that radiation pressure of the transmitted light will dominate, resulting in a negative component of force $F_x$ as illustrated in Fig. 1.
Fig. 4. (A) Spectral modal transmission distribution $T_{\nu,m}$ diffracted from a right triangular prism of apex angle $16^\circ$, grating period $\Lambda = 2 \; [\mu m]$ and refractive index $n=3.5$. (B) Similarly modal spectral irradiance distribution of reflected light.
The values of $T_{\nu,m}$ and $R_{\nu,m}$ are plotted in Fig. 5 for the lower order diffraction modes. These plots reveal several important features that are targets for optimization in future work. (1) Owing to the large refractive index the Fresnel transmission coefficients for p-polarized light at the front and back surfaces are respectively $71.6\%$ and $96.3\%$, resulting in a net transmission of $68.9\%$. It is evident in Fig. 5 that $T_{\nu,m}$ rarely exceeds this value, with peak values generally closer to $60\%$. We attribute the $\sim 9\%$ loss to shadowing effects, whereupon a refracted ray intersects the prism edge rather than the back surface. Roughly $9.4 \%$ of the incident light refracts into the prism edge [26]. We note that a smaller refractive index will decrease both the losses from Fresnel reflections and shadowing. (2) Negative order transmission values are only significant at $m=-1$, and are attributed to the $\text {sinc}^2$ variation of $T_{\nu,m}$ as a function of $m$ [26]. A longer period grating may be used to shift this phenomenon to smaller frequencies where the solar spectrum has negligible values (e.g., in the far infrared) [26]. A longer period grating would also decrease the cut off frequency $\nu _0 = c / \Lambda$ (e.g., see Fig. 2), thereby allowing the sail to make fuller use of the entire solar spectrum. (3) Fine scale spectral variations may be attributed to multiple internal reflections which are known to rapidly vary with frequency in thin films. (4) The effect of the front surface Fresnel reflection coefficient, $28.4\%$, is evident in the values of the negative diffraction orders of $R_{\nu,m}$ in Fig. 5. Enhancements at low negative orders may be attributed to thin film interference effects and shadowing consequences. (5) Surprisingly a few spikes of the zero order reflection values $R_{\nu,0}$ are found at low frequencies. We attributed such retro-reflections to thin film interference.
Fig. 5. Spectral modal transmission and reflection values, $T_{\nu,m}$ and $R_{\nu,m}$ for a prism grating of period $\Lambda = 2 \; [\mu m]$, angle $\alpha = 21^\circ$, and refractive index $n=3.5$ for diffraction orders $m$. Negative diffraction orders have been multiplied by $-1$ to aid the eye. Transverse momentum transfer efficiency $\eta _{x,\nu }$ for an equivalent ideal grating (dashed line) [26].
One may naturally expect the values of $\eta _x$ to change for different values of apex angle, grating period, and refractive index. To illustrate this we varied the value of $\alpha$ for the case described above with $n=3.5$, $\Lambda = 2\; [\mu m]$, p-polarized light, and a quasi-Gaussian incident spectral irradiance from $150$ to $3000 \; [THz]$. The transmitted, reflected, and summed values of $\eta _x^T$ and $\eta _x^R$ are plotted in Fig. 6. From a qualitative point of view $\eta$ may be related to an effective scattering angle, $\psi _{\text {eff}}$: $\sin \psi _{\text {eff}} \equiv -\eta$. Therefore the plotted values of $\eta$ have been multiplied by $-1$ to indicate the scattering direction consistent with Fig. 1. The value of $-\eta _x^T$ is found to increase until $\alpha = 21^\circ$, after which it decreases. This decrease is attributed to the back-surface total internal reflection angle, occurring when $\alpha \ge 23^\circ$ as predicted from Snell’s law. and an optical thickness of $nH = 2.69 \;[\mu m]$.
Fig. 6. Transmitted (red), reflected (green), and net (black) transverse momentum transfer efficiencies as a function of prism apex angle $\alpha$ for period $\Lambda = 2 \; [\mu m]$, refractive index $n=3.5$, and p-polarized light. Geometrical optics estimation (dashed red line). A uniform (normalized from quasi-Gaussian) spectral irradiance ranging from $150$ to $3000 \; [THz]$ (or wavelengths $0.1$ to $2.0 \; [\mu m]$) is assumed.
It is not surprising that the maximum value of $-\eta _x^T$ appears when the transmitted refraction angle is almost $90^\circ$. However, one must also account for Fresnel transmission through the two surfaces. In the case of p-polarized light the net transmission is a maximum when $\alpha = 22^\circ$, reaching a value of $T_{net,p} = 72\%$. (We note that net Fresnel transmission for s-polarized light is $T_{net,s} =20\%$ at this apex angle.) From a geometrical optics point of view the net efficiency may be estimated from the product $T_{net} \sin \psi$, where $\psi$ is the transmitted deviation angle (see Fig. 1). For p-polarized light, $n=3.5$, and $\alpha = 22^\circ$ we calculate $T_{net,p} \sin \psi = 0.69$. In comparison, the numerically generated data provides $-\eta _x = 0.44$. The discrepancy between these two values is attributed to diffraction, multiple reflections, and shadowing from edges. Owing to the opposing scattering angles of reflected light (see discussion of Fig. 4), the net value $-(\eta _x^T + \eta _x^R)$ is typically reduced further in Fig. 6.
Further work is required to optimize the system parameters $n$, $\alpha$, and $\Lambda$ to achieve values of $\eta _x^T + \eta _x^R \approx -\sin \psi$. For example, $\eta _x^R$ has negligible values in Fig. 6 when $\alpha < 9^\circ$, i.e., when the average optical thickness of the prism $(n \Lambda /2) \tan \alpha < 0.55 \; [\mu m]$, therefore suggesting that relatively thin metasurface structures may provide a favorable alternative to relatively thick right prism gratings. On the other hand, a purely reflective grating structure would obviate the problem of opposing scattering angles, and may therefore reach greater efficiency magnitudes, although shadowing remains a problem. Finally, an examination of Fig. 1 indicates that the right prism structure considered here is unsupported by a substrate. In future work the addition of a thin substrate that functions as an anti-reflection coating may reduce the value of $|\eta _x^R|$. Our preliminary exploration indicates that a $0.14 \; [\mu m]$ thick film on both surfaces of refractive index $\sim 2.25$ increases $|\eta _x^T|$ from a value of $0.167$ to $0.59$.
4. Conclusions
Transmitted and reflected light scattering from a right triangular transmission grating of period $\Lambda = 2 \; [\mu m]$ and index $n=3.5$ was numerically calculated by means of FDTD software for an incident p-polarized spatially coherent source having a quasi-Gaussian spectral irradiance distribution ranging from $150$ to $3000 \; [THz]$ (or wavelengths from $0.1$ to $2.0 \; [\mu m]$). Normalized spectral modal transmission and reflection values were determined, from which the transverse momentum transfer efficiency was calculated for the corresponding p-polarized component of the solar blackbody spectrum. Our findings suggest approaches to optimize the transfer efficiency in future work. Radiation pressure from front surface reflection at the high index surface was found to compete against the pressure from transmitted light, thereby reducing the net transfer efficiency. This suggests the exploration of lower index materials or anti-reflection surfaces. A periodic array of other shaped elements [38,39] may also be expected to provide radiation pressure. The small grating period assumed in this study had undesirable consequences, cutting off $6\%$ of the solar spectrum and scattering low frequency transmitted light into the wrong direction. An optimization study that includes both polarization components of light and that varies the refractive index, grating period, and transmission angle may be expected to significantly improvement in the value of the transfer efficiency, which in principle may approach $100\%$. Here we report an efficiency of roughly $31\%$ when scaled against the entire blackbody irradiance, or $62\%$ when scaled against the p-polarized blackbody irradiance. To overcome the competition between transmitted and reflected scattering directions, reflection (rather than transmission) gratings may be considered. Future work should also explore the transfer efficiency associated with natural sunlight which is partially spatially coherent – particularly when the sailcraft approaches the sun. The foregoing theoretical and numerical formalisms may be heuristically applied to determine the radiation pressure force on a broad class of periodic structures by use of finite difference time domain methods.
Funding
NASA Space Technology Mission Directorate (80NSSC19K0975).
Acknowledgments
GS is grateful to Prateek R. Srivastava and Ryan M. Crum (Rochester Institute of Technology, Rochester, NY) for verifying some of the numerical calculations using the FDTD program, MEEP, and to Rajesh Menon (University of Utah, Salt Lake City, UT) for discussions of radiation pressure on diffractive structures. GS and SMK are grateful to Les Johnson and Andy Heaton (NASA Marshall Space Flight Center, Huntsville, AL) for discussions related to solar sailing.
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.
References
1. K. E. Tsiolkovsky, “Extension of Man into Outer Space,” In Proceedings of the Symposium on Jet Propulsion2, United Scientific and Technical Presses (NIT), 1936 (1921).
2. K. Tsander, From a Scientific Heritage, NASA Technical Translation TTF-541 (1967), a translation of Iz Nauchnogo Naslediya, Nauca Press, Moscow (1924).
3. C. R. McInnes, Solar Sailing: Technology, Dynamics and Mission Applications (Springer, 2004).
4. D. Pettit, “The Tyranny of the Rocket Equation,” https://www.nasa.gov/mission_pages/station/expeditions/expedition30/tryanny.html (2012).
5. L. Friedman, Starsailing: Solar Sails and Interstellar Travel (John Wiley and Sons, 1988).
6. P. Gilster, Centauri Dreams: Imagining and Planning Interstellar Exploration (Copernicus Books, 2004).
7. O. Mori, H. Sawada, R. Furnase, M. Morimoto, T. Endo, T. Yamamoto, Y. Tsudu, Y. Kawakatsu, J. Kawaguchi, Y. Miyazaki, and Y. Shirasawa, “First Solar Power Sail Demonstration by IKAROS,” Transactions of the Japan Society for Aeronautical and Space Science, Aerospace Technology Japan (2010).
8. Y. Shirasawa, O. Mori, N. Okuizumi, Y. Satou, A. Yamasaki, H. Furuya, T. Nishizawa, H. Sakamoto, and G. Ono, “Evaluation of Sail Mechanics of IKAROS on its Slow-Spin and Reverse-Spin Operation,” Advances in Solar Sailing (Springer, 2014), pp. 57–74.
9. L. Johnson, R. Young, E. Montgomery, and D. Alhorn, “Status of solar sail technology within NASA,” Adv. Space Res. 48(11), 1687–1694 (2011). [CrossRef]
10. A. Heaton, B. F. Faller, and C. K. Katan, “NanoSail:D Orbital and Attitude Dynamics,” Advances in Solar Sailing (Springer, 2014), pp. 95–113.
11. R. Ridenoure, R. Munakata, S. Wong, A. Diaz, D. Spencer, D. A. Stetson, B. Betts, B. Plante, J. Foley, and J. Bellardo, “Testing the lightsail program: Advancing solar sailing technology using a cubesat platform,” J. Small. Satell. 5, 531–550 (2016).
12. J. Heiligers and C. Mclnnes, “Novel solar sail mission concepts for space weather forecasting,” 24th AAS/AIAA Space Flight Mechanics Meeting (2014).
13. A. Peloni, M. Ceriotti, and B. Dachwald, “Solar-sail trajectory design for a multiple near-earth-asteroid rendezvous mission,” J. Guid. Control Dyn. 39(12), 2712–2724 (2016). [CrossRef]
14. A. Heaton, K. Miller, and N. Ahmad, “Near earth asteroid scout solar sail thrust and torque model,” International Symposium on Solar Sailing (2017).
15. E. Mahoney, “NEA Scout,” (2018). https://www.nasa.gov/content/nea-scout.
16. M. Leipold, M. Eiden, C. E. Garner, L. Herbeck, D. Kassing, T. Niederstadt, T. Krüger, G. Pagel, M. Rezazad, H. Rozemeijer, W. Seboldt, C. Schöppinger, C. Sickinger, and W. Unckenbold, “Solar sail technology development and demonstration,” Acta Astronaut. 52(2-6), 317–326 (2003). [CrossRef]
17. S. Gong and M. Macdonald, “Review on solar sail technology,” Astrodyn. 3(2), 93–125 (2019). [CrossRef]
18. J. B Pezent, R. Sood, A. Heaton, K. Miller, and L. Johnson, “Preliminary trajectory design for NASA’s solar cruiser: A technology demonstration mission,” Acta Astronaut. 183, 134–140 (2021). [CrossRef]
19. J. A. Atchison and M. A. Peck, “A passive, sun-pointing, millimeter-scale solar sail,” Acta Astronaut. 67(1-2), 108–121 (2010). [CrossRef]
20. D. Ma, J. Murray, and J. N. Munday, “Controllable propulsion by light:steering a solar sail via tunable radiation pressure,” Adv. Opt. Mater. 5(4), 1600668 (2017). [CrossRef]
21. A. L. Dubill, “Attitude Control for Circumnavigating the Sun with Diffractive Solar Sails,” Thesis (Rochester Institute of Technology, 2020).
22. Y. Chu, S. Firuzi, and S. Gong, “Controllable liquid crystal diffractive sail and its potential applications,” Acta Astronaut. 182, 37–45 (2021). [CrossRef]
23. D. Ullery, S. Soleymani, A. Heaton, J. Orphee, L. Johnson, R. Sood, P. Kung, and S. M. Kim, “Strong solar radiation forces from anomalously reflecting metasurfaces for solar sail attitude control,” Sci. Rep. 8(1), 10026 (2018). [CrossRef]
24. G. A. Swartzlander Jr., “Radiation pressure on a diffractive sailcraft,” J. Opt. Soc. Am. B 34(6), C25–C30 (2017). [CrossRef]
25. A. L. Dubill and G. A. Swartzlander Jr., “Circumnavigating the sun with diffractive solar sails,” Acta Astronaut. 187, 190–195 (2021). [CrossRef]
26. G. A. Swartzlander Jr., “Theory of radiation pressure on a diffractive solar sail,” J. Opt. Soc. Am. B 39(9), 2556–2563 (2022). [CrossRef]
27. Y.-J. L. Chu, E. Jansson, and G. A. Swartzlander Jr., “Measurements of radiation pressure owing to the grating momentum,” Phys. Rev. Lett. 121(6), 063903 (2018). [CrossRef]
28. Y.-J. L. Chu, N. V. Tabiryan, and G. A. Swartzlander Jr., “Experimental verification of a bigrating beam rider,” Phys. Rev. Lett. 123(24), 244302 (2019). [CrossRef]
29. S. Firuzi and S. Gong, “Refractive sail and its applications in solar sailing,” Aerosp. Sci. Technol. 77, 362–372 (2018). [CrossRef]
30. P. R. Srivastava, Y.-J. L. Chu, and G. A. Swartzlander Jr., “Stable diffractive beam rider,” Opt. Lett. 44(12), 3082–3085 (2019). [CrossRef]
31. O. Ilic and H. A. Atwater, “Self-stabilizing photonic levitation and propulsion of nanostructured macroscopic objects,” Nat. Photonics 13(4), 289–295 (2019). [CrossRef]
32. P. R. Srivastava and G. A. Swartzlander Jr., “Optomechanics of a stable diffractive axicon light sail,” Eur. Phys. J. Plus 135(7), 570 (2020). [CrossRef]
33. A. R. Davoyan, J. N. Munday, N. Tabiryan, G. A. Swartzlander Jr., and L. Johnson, “Photonic materials for interstellar solar sailing,” Optica 8(5), 722–734 (2021). [CrossRef]
34. S. Firuzi, Y. Song, and S. Gong, “Gradient-index solar sail and its optimal orbital control,” Aerosp. Sci. Technol. 119, 107103 (2021). [CrossRef]
35. S.-F. Shi, B. Zeng, H.-L. Han, X. Hong, H.-Z. Tsai, H. S. Jung, A. Zett, M. F. Crommie, and F. Wang, “Optimizing broadband terahertz modulation with hybrid graphene/metasurface structures,” Nano Lett. 15(1), 372–377 (2015). [CrossRef]
36. C. Wang, W. Liu, Z. Li, H. Cheng, Z. Li, S. Chen, and J. Tian, “Dynamically tunable deep subwavelength high-order anomalous reflection using graphene metasurfaces,” Adv. Opt. Mater. 6(3), 1701047 (2018). [CrossRef]
37. Q. Li, X. Cai, T. Liu, M. Jia, Q. Wu, H. Zhou, H. Liu, Q. Wang, X. Ling, C. Chen, F. Ding, Q. He, Y. Zhang, S. Xiao, and L. Zhou, “Gate-tuned graphene meta-devices for dynamically controlling terahertz wavefronts,” Nanophotonics 11(9), 2085–2096 (2022). [CrossRef]
38. G. A. Swartzlander, T. J. Peterson, A. B. Artusio-Glimpse, and A. D. Raisanen, “Stable optical lift,” Nat. Photonics 5(1), 48–51 (2011). [CrossRef]
39. S. H. Simpson, S. Hanna, T. J. Peterson, and G. A. Swartzlander Jr., “Optical lift from dielectric semicylinders,” Opt. Lett. 37(19), 4038–4040 (2012). [CrossRef]
