1. INTRODUCTION

Solar radiation pressure has scientific origins dating back to 1619 when Kepler proposed an explanation for comet tails [1,2]. Centuries later, this phenomenon was found to have the profound effect of arresting the gravitational collapse of high mass stars [3]. Applications to space propulsion were first described in the early 1900s by Tsiolkovsky [4] and Tsander [5], whereby a solar sail gains momentum owing to reflected sunlight from a metallic sheet. Navigating the heavens by means of solar radiation pressure provides two advantages not afforded by rocket propulsion: continuous acceleration and an inexhaustible energy source [6–9]. High delta-V missions such as solar polar orbiters [7,10,11], as well as low thrust sub-L1 halo orbits [12,13] are two examples of heliophysics science missions that are enabled with solar sails. Perhaps counter to one’s initial intuition, the component of radiation pressure force perpendicular to the sun line, rather than the parallel component, is of paramount importance for many near-term missions [6,14,15]. Whereas the latter component elongates the orbital ellipticity of a sailcraft, the former enables spiral trajectories that are useful for rendezvous missions to the inner or outer planets. For example, navigation toward the Sun from a quasi-circular orbit at 1 AU (and at a position beyond the influence of Earth’s gravity) is achieved by tangentially decelerating the sailcraft along its orbit, resulting in an inward spiral trajectory. (Space missions requiring a hyperbolic or highly elliptical orbit, on the other hand, make use of the component of radiation pressure force that is parallel to the sun line, but such missions are not practical unless the magnitude of light pressure is comparable to the opposing pressure exerted by solar gravity.) For a sail based on the law of reflection, this requires tilting the sail normal away from the sun line, which has the disadvantage of reducing the solar power projected onto the sail surface. Nevertheless, solar sails benefit from a seemingly limitless supply of externally supplied photons, unlike rockets, which carry a limited volume of propellant. Although rockets must be used to loft a solar sail into space, once there, the change of velocity afforded by a sail can greatly exceed that of a rocket [6,16,17]. This advantage affords opportunities to deliver science instruments to orbits that cannot be reached by rockets alone. While the technical readiness level of solar sails started to improve in the 1970s [8,12,18], only recently have demonstration light sail missions been tested in space [13,19,20]. As of this writing, the Planetary Society’s Lightsail-2 mission has been circling the Earth for three years [21]. If the history of aviation is a guide, one may expect many innovations in the design and functionality of solar sails in the coming decades. Diffractive solar sails were proposed in 2017 as an alternative to reflective sails [15]. A potential advantage of a diffractive sail is the generation of a significant transverse radiation pressure force while in a Sun-facing orientation. A flat reflective sail must be tilted with respect to the sun line to produce a transverse force. The first experimental verification of the transverse force on a diffraction grating was reported in 2018 by use of a vacuum torsion oscillator and a laser [22]. Since then, various diffractive sail designs have been explored for both solar- and laser-driven sails [23–28]. This paper describes a method to calculate the spectrally averaged radiation pressure force, momentum transfer efficiency, and acceleration on a Sun-facing diffractive sail.

2. RADIATION PRESSURE FORCE

In 1873, Maxwell determined that electromagnetic radiation is associated with a pressure that is proportional to the irradiance and inversely proportional to the speed of light [29]. The birth of quantum mechanics allowed light of wavelength $\lambda$ to be described by particles (photons) of momentum $\hbar \vec k$, where $\hbar$ is the Planck constant, and $\vec k = k\hat k$ is the wave vector of magnitude $k = 2\pi /\lambda$ and unit vector $\hat k$ in the direction of propagation. The radiation pressure force on a light sail may then be described with Newton’s third law, which describes conservation of momentum, i.e., momentum imparted to a sail is equal and opposite to the net momentum change experienced by all scattered photons. For example, if the incident and deviated wave vectors of a photon deviated by the angle ${\theta _d}$ are respectively expressed as ${\vec k_i} = k\hat z$ and ${\vec k_d} = k(\cos {\theta _d}\hat z + \sin {\theta _d}\hat x)$, then the momentum imparted to the sail is given by $\Delta {\vec p_s} = - \hbar \Delta \vec k = \hbar k[(1 - \cos {\theta _d})\hat z - \sin {\theta _d}\hat x]$. The sail experiences a positive impulse along the direction of the sun line, $\hat z$ ranging from zero to $2\hbar k$, and a transverse component ranging between ${\pm}\hbar k$. The irradiance associated with an incident flux of $N$ photons per unit area over a time increment $\Delta t$ is given by $I = N\hbar ck/\Delta t$, and thus the force on a sail of area $A$ and projection angle $\psi$ may be expressed as

In general, the direction of scattered light may depend on multiple factors besides incidence angle and wavelength. However, to gain a baseline understanding of the radiation pressure on a diffraction grating, only the normal incidence case is considered below, as illustrated in Fig. 1, ignoring coherence, polarization, internal and external reflections, diffuse scattering, absorption, and Doppler shifts. The net solar radiation pressure force is found by integrating Eq. (1) across the solar spectrum, replacing $I$, ${\eta _x}$, and ${\eta _z}$ with spectral equivalents, ${I_\nu}$, ${\eta _{x,\nu}}$, and ${\eta _{z,\nu}}$, where $\nu = c/\lambda$ is the optical frequency.

 

Fig. 1. Sunlight incident upon a Sun-facing diffractive sail (incidence angle $\psi = 0$) of period $\Lambda$. Different wavelengths scatter at different angles ${\theta _s}$. Each photon of wavelength $\lambda$ is associated with incident wave vector ${\vec k_i}$ directed along the sun line $\hat z$ and scattered wave vector(s) ${\vec k_s}$. The sum of all scattering events produces a radiation pressure force $\vec F$.

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The gravitational attraction of the Sun produces an additional force on the sail:

3. REFRACTION FROM A RIGHT PRISM

A sawtooth grating may be described as a periodic array of refractive or reflective right triangles. For the case of the transmission grating described below, it is instructive to describe the refractive properties of a right prism having refractive index $n$, base length $\Lambda$, height $H$, and prism angle $\alpha = {\tan}^{- 1} (H/\Lambda)$. This structure is illustrated in Fig. 2 and serves as the unit cell of a periodic grating in Section 5. The configuration in Fig. 2(A) is illuminated from the “rough side,” whereas Fig. 2(B) is said to depict “smooth-side” illumination. Both structures are Sun-facing $(\psi = 0)$, with the sun line parallel to the $\hat z$ direction. In both cases, light is scattered in the ${-}\hat x$ direction, resulting in a transverse force ${F_x} \gt 0$ (i.e., ${\eta _x} \gt 0$). For rough-side illumination, the incidence and scattering angles are respectively ${\theta _1} = \alpha$ and ${\theta _d} = - {\theta _4}$, where Snell’s law and geometry provide

 

Fig. 2. Single element of an array of right prisms of base $\Lambda$, apex angle $\alpha$, and height $H = \Lambda \,\tan \,\alpha$ having refractive index $n$ surrounded by vacuum. Insets depict periodic gratings. (A) Rough side incidence with deviation angle ${\theta _d} = - {\theta _4}$. First surface angle of incidence: ${\theta _1} = \alpha$. Refraction angles ${\theta _{2,4}}$. Second surface angle of incidence ${\theta _3} = \alpha - {\theta _2}$. Points A, B, C, D: $(x,z) = (x + \Delta x,\; - H)$, $(x + \Delta x,\; - {h_2})$, $(x,0)$, $(x + \Delta x,0)$, where ${h_2} = H - (x + \Delta x)\tan \alpha$, $\Delta x = (H - x\tan \alpha)/(\cot {\theta _3} + \tan \alpha)$. Shadow region width: $x_{\textit{sh}}^\prime = H/(\cot {\theta _3} + \tan \alpha)$. (B) Smooth side incidence with deviation angle ${\theta _d} = - ({\theta ^{{\prime \prime}}} - \alpha)$, second surface incidence angle ${\theta ^\prime} = \alpha$, and refraction angle ${\theta ^{{\prime \prime}}}$.

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For smooth-side illumination, the transmitted refraction angle $\theta ^{\prime \prime} $ is related to the prism angle by Snell’s law: $n\sin \alpha = \sin \theta ^{\prime \prime} $, and the deviation angle is given by ${\theta _d} = - (\theta ^{\prime \prime} - \alpha)$. Comparisons of the deviation angles for rough- and smooth-side illumination are plotted in Fig. 3 as a function of the prism angle $\alpha$ for different values of the refractive index $n$. Rough-side illumination clearly produces deviation angles as large as 90°, whereas the smooth-side cases do not. For the latter case, the critical angle condition ${\alpha _{\textit{cr}}} = {\sin}^{- 1} (1/n)$ limits the deviation angle to a maximum value of ${90^ \circ} - \alpha$. Owing to this limitation, only rough-side illumination is considered below.

 

Fig. 3. Deviation angle versus apex angle for different values of refractive index, $n$. (A) Rough-side illumination. (B) Smooth-side illumination. A deviation of $|{\theta _d}| = 60^ \circ$ (red line) may be possible in both cases, depending on the refractive index.

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For rough-side illumination, a “shadow region” of width $x_{\textit{sh}}^\prime = \Lambda \tan \alpha /(\cot {\theta _3} + \tan \alpha)$ produces rays that are not described by the discussion above. The relative extent of $x_{\textit{sh}}^\prime /\Lambda$ decreases with increasing values of the refractive index. For example, $x_{\textit{sh}}^\prime $ is 10% of $\Lambda$ when $n = 3.5$ and $|{\theta _d}| = 70^ \circ$. Like internally and externally reflected rays, the scattering of rays associated with the shadow region is ignored below and left as a topic of future study. For example, alternative design approaches based on metasurfaces may allow small surface height modulations that suppress shadowing effects.

If the refractive index varies with optical frequency, the deviation angle will also vary. Consequently, the net momentum transfer efficiency is found by integrating across the spectral irradiance distribution ${I_\nu}$:

4. DIFFRACTION FROM A SINGLE RIGHT PRISM

If the base length is comparable to the wavelength of light, then one must account for diffraction from the single prism element. At a given optical frequency, the transmitted electric field at the output face of the prism may be expressed as

Examples of the far-field irradiance ${\tilde I_\nu}({k_x}) = |{\tilde E_\nu}({k_x}{)|^2}$ are plotted in Fig. 4 at $\lambda = 0.5\;[{\unicode{x00B5}{\rm m}}]$ (blue line) and at $2.5\;[{\unicode{x00B5}{\rm m}}]$ (red line) for ${I_\nu} = 1\;[{{\rm W/m}^2}/{\rm Hz}]$. Nearly half the irradiance distribution is cut off in the long wavelength case. For this example, a deviation angle ${\theta _d}={ 71.4^ \circ}$ was assumed, corresponding to $n = 3.5$, $\Lambda = 10\;[{\unicode{x00B5}{\rm m}}]$, and $H = 4\;[{\unicode{x00B5}{\rm m}}]$. Unlike the above geometric optics descriptions, the deviation is not single-valued except in the case of $\lambda \ll \Lambda$. Consequently, the net momentum imparted to the sail at a particular value of frequency must be obtained by integration, resulting in a net spectral efficiency vector with components ${\eta _{x,\nu}} = - \langle {k_{x,\nu}} \rangle /k$ and ${\eta _{z,\nu}} = 1 - \langle {k_{z,\nu}} \rangle /k$, where the effective wave vector components may be expressed as

 

Fig. 4. Diffraction from a right prism of base $\Lambda = 10\;[{\unicode{x00B5}{\rm m}}]$, height $H = 4\;[{\unicode{x00B5}{\rm m}}]$, and refractive index $n = 3.5$ at two wavelengths: $\lambda = 0.5$ (blue) and $2.5\;[{\unicode{x00B5}{\rm m}}]$ (red). A positive deviation angle is assumed: ${\theta _d}={ 71.4^ \circ}$. Erect arrows: transverse efficiencies ${\eta _x}$ relative to the incident power. Discrete points correspond to diffraction orders of a right triangular grating of period $\Lambda = 10\;[{\unicode{x00B5}{\rm m}}]$. Inverted arrows: grating transverse efficiencies.

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5. DIFFRACTION FROM A RIGHT PRISM GRATING

A diffraction grating comprising an infinite array for right prisms, coherently illuminated from the rough side as illustrated in the inset of Fig. 2(A), is expected to produce discrete diffraction orders ${k_m}$ confined to the envelope function described by the modulus of Eq. (7). Similar to the field in Eq. (6), the piece-wise continuous electric field at the transmitting interface may be expressed as

The diffraction orders of the propagating field corresponding to an infinite array of right triangular prisms of period $\Lambda = 10\;[{\unicode{x00B5}{\rm m}}]$ are depicted in Fig. 4 for $\lambda = 0.5$ and $2.5\;[{\unicode{x00B5}{\rm m}}]$. The short (long) wavelength case supports mode numbers ranging from ${\pm}20$ (${\pm}4$). The irradiance values shown in Fig. 4 have been scaled by $1/{N^2}$ to aid the eye. The centroid values for these discrete cases are ${\eta _{x,\nu}} = \langle {k_{x,\nu}} \rangle /k = 0.950$ for $\lambda = 0.5\;[{\unicode{x00B5}{\rm m}}]$ and ${\eta _{x,\nu}} = \langle {k_{x,\nu}} \rangle /k = 0.982$ for $\lambda = 2.5\;[{\unicode{x00B5}{\rm m}}]$, respectively corresponding to effective deviation angles ${\theta _{{\rm eff}}}={ 71.8^ \circ}$ and 79.1°. These values are greater than those in the single-grating cases described at the end of Section 4 because little light has been cut off.

Solutions of Eq. (11b) are depicted in Fig. 5 for the wavelength range $\lambda :(0.2\;[{\unicode{x00B5}{\rm m}}],2.0\;[{\unicode{x00B5}{\rm m}}])$, without the inclusion of the cutoff modes, for the case of $\Lambda = 2\;[{\unicode{x00B5}{\rm m}}]$ and ${\theta _d} = - {60^ \circ}$. Salient features for this case, such as the wavelength of peak diffraction orders and cutoff mode numbers, are listed in Table 1. As expected for a negative deviation angle, the dominate diffraction orders correspond to negative values of $m$.

 

Fig. 5. Spectral diffraction amplitudes $|{\tilde E_{m,\nu}}/{A_\nu}|$ for a right prism grating of period $\Lambda = 2\;[{\unicode{x00B5}{\rm m}}]$ and refractive deviation angle ${\theta _d} = - {60^ \circ}$. Maxima occur at $\lambda = (\Lambda /m)\sin {\theta _d}$. Cutoff modes are not shown.

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Table 1. Dominant Band, ${m^\prime}$, with Corresponding Peak Wavelengths ${\lambda _{{\rm peak}}}$, Where $|\tilde E_m^\prime |/{A_\lambda} = 1$^a

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The diffraction spectrum exhibits cutoff wavelengths at the wavelength values $\Lambda$, $\Lambda /2$, $\Lambda /3\ldots$, or equivalently at the frequencies ${\nu _0}$, $2{\nu _0}$, $3{\nu _0},\ldots$, where ${\nu _0} = c/\Lambda$ is the fundamental frequency associated with the grating. These boundaries are marked with vertical dashed lines in Fig. 5 and listed as $\lambda _{{m^\prime}}^{{\prime \prime}}$ in Table 1. The wavelength at which $|{\tilde E_{{m^\prime}}}|$ is maximum, i.e., where $|{\tilde E_{{m^\prime}}}| = {A_\nu}\;{\delta _{m,{m^\prime}}}$ (where ${\delta _{m,{m^\prime}}}$ is the Kronecker delta function), is found by setting $\Delta {k_{{m^\prime}}}\Lambda /2 = 0$, i.e., at ${\lambda _{{\rm peak}}} = (1/{m^\prime})\Lambda \sin {\theta _d}$ (see Eq. (12)). The range between $\lambda _{{m^\prime}}^\prime = \Lambda \sin {\theta _d}/({m^\prime} - 1)$ and $\lambda _{{m^\prime}}^{{\prime \prime}} = - \Lambda /{m^\prime}$ is called the ${m^{\prime }}$th dominant band. For example, the ${m^\prime} = - 2$ dominant band ranges from $\lambda _{{m^\prime}}^\prime = 0.577\;[\unicode{x00B5}{\rm m}]$ to $\lambda _{{m^\prime}}^{{\prime \prime}} = 1.0\;[\unicode{x00B5}{\rm m}]$. It is evident in Table 1 that as the magnitude of the dominant mode number $|{m^\prime}|$ increases, the magnitude of $|{\theta _{{m^\prime}}}|$ at $\lambda _{{m^\prime}}^\prime $ increases. For example, ${\theta _{{m^\prime} = - 2}} = - {35.3^ \circ}$, whereas ${\theta _{{m^\prime} = - 8}} = - {50.3^ \circ}$. This suggests that on average, short wavelength light diffracts at higher angles, approaching the refraction angle ${\theta _d}$, compared to long wavelength light (as expected from the geometric optics approximation that assumes $\lambda \to 0$.) The importance of this for solar sailing will become evident in the discussion below.

6. MOMENTUM TRANSFER EFFICIENCY OF A SUNLIT DIFFRACTIVE SAIL

The net force and efficiency may be determined by integrating over all scattered light. Here the illumination is represented by the solar blackbody spectrum with plane waves normally incident upon the rough side of the grating. The small angular extent of the Sun (e.g., $\Delta {\theta _{\rm{sun}}}={ 0.5^ \circ}$ at 1 AU) is ignored. The spatial coherence length ${L_c} = 2\lambda /\pi \Delta {\theta _{\rm{sun}}}$ is also assumed to be large enough compared to the grating period to afford narrowing of the diffraction peaks, e.g., ${L_c}/\Lambda \gt 4$. This condition is satisfied for $\Lambda = 10\;[{\unicode{x00B5}{\rm m}}]$ if $\lambda \gt 0.55;[{\unicode{x00B5}{\rm m}}]$ (i.e., $\nu = c/\lambda \lt 550\;{\rm THz} $), which includes more than half the power emitted by the Sun. In comparison, a grating period of $\Lambda = 2\;[{\unicode{x00B5}{\rm m}}]$ requires $\lambda \gt 0.11;[{\unicode{x00B5}{\rm m}}]$ (i.e., $\nu \lt 2740\;{\rm THz} $), which includes nearly the entire solar blackbody spectrum. Hence, for grating periods $\Lambda \le 10\;[{\unicode{x00B5}{\rm m}}]$, the results in Section 5 may be used to determine the radiation pressure force. A full coherence analysis is beyond the scope of this paper.

Numerical integration across all diffraction orders and a broad band of wavelengths is required to determine the net momentum transfer efficiency. As illustrated in Fig. 5, the interval between cutoff frequencies is not regular. On the other hand, the interval between corresponding frequencies is regular, with cutoff frequencies separated by $\Delta\nu = {\nu _0} = c/\Lambda$. Numerical integration across optical frequencies is therefore made to ensure regular sampling $\delta \nu = {\nu _0}/{N_\nu}$ when employing the trapezoid rule, where ${N_\nu} \gt 100$ samples was used.

The blackbody spectral irradiance distribution in frequency space differs from that in wavelength space, providing a peak value (Wien’s displacement law) at ${\nu _{{\rm peak}}} = (5.879 \times {10^{10}}\;[{\rm Hz/K}])\;T$. At solar temperature $T = 5770\;[{\rm K}]$, ${\nu _{{\rm peak}}} = 339.2\;[{\rm THz}]$ (which corresponds to the wavelength $c/{\nu _{{\rm peak}}} = 884.4\;[{\rm nm}]$). The blackbody spectral irradiance a distance $r$ from the sun may be expressed in units of $[{{\rm W/m}^2}/{\rm Hz}]$:

Owing to orthogonality between diffraction orders, the net force $\vec F$ at $1\;[{\rm AU}]$ from the Sun may be decomposed into force components ${\vec F_{m,\nu}}$ attributed to each $m$th order diffraction mode for a given frequency $\nu$:

The momentum transfer efficiency is defined as the ratio of the net force, Eq. (16a), and the normalizing force parameter ${P_{{\rm in}}}/c$, where ${P_{{\rm in}}}$ is the net power projected on a Sun-facing sail:

The momentum transfer efficiency for two cases, $\Lambda = 2\;{\rm and}\;10\;[{\unicode{x00B5}{\rm m}}]$, are depicted in Fig. 6 for a grating of index $n = 3.5$, apex angle $\alpha ={ 20^ \circ}$, and refractive deviation angle ${\theta _d} = \pm {60.5^ \circ}$. The value ${\nu _0} = c/\Lambda$ also corresponds to the low frequency (long wavelength) cutoff, below which ${\eta _{x,\nu}} = 0$. At frequencies greater than but close to the cutoff (with corresponding wavelength less than but on the same order as the grating period), strong diffraction effects are evidenced by pronounced variations of the efficiency values. The transverse spectral efficiency ${\eta _{x,\nu}}$ varies with frequency, reaching an extremum of roughly $|{\eta _{x,\nu}}| = 0.94$ at $\nu = \Delta \nu$, falling to almost zero at $\nu = 2\Delta \nu$ in Fig. 6(A). On the other hand, for large frequency values (wavelengths much smaller than the grating period), diffractive modulations are less pronounced, with the transverse efficiency converging toward the value for pure refraction: $|{\eta _{x,\nu}}| \approx |\sin {\theta _d}| = 0.87$. As suggested in the discussion of Table 1, the average magnitude of efficiency indeed increases with increasing frequency (decreasing wavelength). Comparing the cutoff frequency for the two cases in Fig. 6(A), it is evident that the longer period grating provides fuller coverage of the solar spectrum, thereby providing a stronger radiation pressure force. Similar phenomena are seen for ${\eta _{z,\nu}}$ in Fig. 6(B), with the magnitude tending toward $\cos {\theta _d} = 0.49$ at high frequencies. The reader is reminded that these calculations do not account for reflected light, such as the backscatter of light in the cutoff frequency band.

 

Fig. 6. Spectral momentum transfer efficiency components, ${\eta _{x,\nu}}$ (top) and ${\eta _{z,\nu}}$ (bottom) for grating periods $\Lambda = 2$ and $10\;[{\unicode{x00B5}{\rm m}}]$ and refractive deviation angle ${\theta _d} = - {60.5^ \circ}$. Cutoff frequency and modal bands $\Delta \nu = c/\Lambda$.

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Integrating the calculated values of spectral efficiency across the optical frequency band provides the net efficiency components ${\eta _x}$ and ${\eta _z}$. Examples of these components are plotted as a function of the grating period $\Lambda$ in Fig. 7 for the case of $n = 3.5$, $\alpha ={ 20^ \circ}$, ${\theta _d} = \pm {60.5^ \circ}$, and solar blackbody illumination across the wavelength band $0.1\;[{\unicode{x00B5}{\rm m}}]$ to the cutoff wavelength $\Lambda$. The magnitude of the transverse efficiency increases with the grating period, with a knee or vertex at $\Lambda \approx 5\;[{\unicode{x00B5}{\rm m}}]$ and an asymptotic value $|{\eta _x}| = |\sin {\theta _d}| = 0.87$. The effective deflection angle of sunlight may be defined by the relation ${\eta _x} = - \sin {\theta _{{\rm eff}}}$. Values of ${\theta _{{\rm eff}}}$ are plotted in Fig. 7, illustrating the long period asymptotic values approaching the refractive deflection angle.

 

Fig. 7. Momentum transfer efficiency components $|{\eta _x}|$ and ${\eta _z}$ of a transmission grating of apex angle $\alpha ={ 20^ \circ}$, refractive index $n = 3.5$, and refractive deviation angle $|{\theta _d}| = 60.5^ \circ$ for solar blackbody illumination. The relation $|{\eta _x}| = \sin {\theta _{{\rm eff}}}$ defines the effective deviation angle ${\theta _{{\rm eff}}}$ (blue line). For comparison, an ideal mirror provides $|{\eta _x}| = 0.77$ (green line). Ratio $|{\eta _x}|/\Lambda$ (gray line).

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The above values may be compared with the momentum transfer efficiency of an ideal reflective sail: $\vec \eta = 2{\cos}^2 {\theta _i}(\cos {\theta _i}\hat z - \sin {\theta _i}\hat x)$ [6]. At the optimum incidence angle ${\theta _i} = \pm {35.3^ \circ}$, $|{\eta _x}| = 0.77$, thereby providing an effective deflection angle $|{\theta _{{\rm eff}}}| = 50.3^ \circ$. The foregoing grating analysis based on a series of prisms, each having a refractive deflection angle of ${\theta _d} = \pm {60.5^ \circ}$, demonstrates $|{\eta _x}| \gt 0.77$ for grating periods $\Lambda \gt 4.8\;[{\unicode{x00B5}{\rm m}}]$. Designing the prisms to achieve larger values of $|{\theta _d}|$ is also expected to support high values of $|{\eta _x}|$.

For the longitudinal efficiency along the sun line, ${\eta _z} = 1.09$ for an ideal mirror, which is significantly greater than the corresponding value for the analyzed prism grating. For orbit changing maneuvers, a large value of ${\eta _z}$ increases the orbital eccentricity, which is typically not desirable. However, as stated in Section 1, the component of radiation pressure force along the sun line is typically negligible compared to solar gravity, and thus the value of ${\eta _z}$ is not a concern. Note that a fuller grating analysis that includes reflected light from the grating is expected to increase the value of ${\eta _z}$ above those shown in Fig. 7.

7. ACCELERATION

The foregoing analysis suggests that a long period grating has a greater transverse momentum transfer efficiency, and therefore a greater force, compared to a short period grating. The paramount concern for orbit changing maneuvers in space, however, is the transverse acceleration:

Setting the mass ratio $\mu= {m^\prime}/{m_s}$ as a dimensionless design parameter, the acceleration may be expressed as

Comparing ${({\eta _x}/\bar H)_{{\rm diff}}}$ to ${({\eta _x}/H)_{{\rm refl}}}$, where ${\bar H_{{\rm diff}}} = (1/2)\Lambda \tan \alpha$ is the average thickness of the grating, and ${H_{{\rm refl}}}$ is the thickness of a reflecting film, the condition ${({\eta _x}/\bar H)_{{\rm diff}}} \gt ({\eta _x}/H{)_{{\rm refl}}}$ suggests greater acceleration for a diffraction grating if ${\bar H_{{\rm diff}}}/{H_{{\rm refl}}} \lt ({\eta _x}{)_{{\rm diff}}}/({\eta _x}{)_{{\rm refl}}}$. For the ideal case, if ${({\eta _x})_{{\rm refl}}} = 0.77$ and ${({\eta _x})_{{\rm diff}}} = 1.0$, this condition provides ${\bar H_{{\rm diff}}}/{H_{{\rm refl}}} \lt 1.3$, thereby requiring a prism height $\Lambda \tan \alpha \lt 2.6{H_{{\rm refl}}}$. For example, if ${H_{{\rm refl}}} = 3\;[\unicode{x00B5}{\rm m}]$, then $\Lambda \tan \alpha \lt 7.8\;[{\unicode{x00B5}{\rm m}}]$. What is more, if this condition is expressed $\Lambda \tan \alpha /{H_{{\rm refl}}} \lt 2({\eta _x}{)_{{\rm diff}}}/({\eta _x}{)_{{\rm refl}}}$, then equality between the prism height and film thickness $(\Lambda \tan \alpha = {H_{{\rm refl}}})$ suggests that ${({\eta _x})_{{\rm diff}}}$ may be as small as $(1/2)({\eta _x}{)_{{\rm refl}}}$. This analysis assumes a massless substrate supporting the array of prisms. Nevertheless, there is a clear acceleration benefit of a lower mass prism grating having the same height as the thickness of a reflecting film.

8. CONCLUSION

To navigate within the neighborhood of a few AUs from the Sun via spiral trajectories, a spacecraft must experience thrust perpendicular to the rays of the Sun. Such a force may be created by means of radiation pressure whereupon sunlight is deviated by an angle approaching 90°. Using the law of reflection to deflect light is counterproductive since the fraction of solar power projected onto the sail decreases with tilt angle. In contrast, a diffraction grating may achieve large deviation angles in a Sun-facing orientation. To demonstrate this, the radiation pressure on an idealized transmission grating comprising right prisms was determined using Fourier series analysis for wavelengths spanning the solar spectrum. A similar analysis may be made for a reflection grating. This paper serves as a baseline study with numerous simplifying assumptions. The primary outcomes are: (1) at wavelengths much smaller than the grating period (where geometric optics is valid), the light deviation angle approaches that predicted by Snell’s law; (2) as the wavelength approaches the grating period, the deviation angle exhibits pronounced wavelength-dependent modulation and consequently a smaller deviation angle; (3) the transverse component of the radiation pressure force for a Sun-facing diffractive sail exceeds that of an ideal flat reflective sail when the spectrally averaged deviation angle exceeds 50.3° (or equivalently, when the transverse momentum transfer efficiency exceeds $\sin {50.3^ \circ} = 0.77$). For example, this study examined a case where the effective angle reaches as high as 60.5°, corresponding to an efficiency of $\sin {60.5^ \circ} = 0.87$. (4) While the efficiency increases asymptotically with the grating period, the more important acceleration increases inversely with the period.

Future work in this area may include an optimization analysis of sailcraft acceleration that includes internal and external reflections, material absorption and dispersion, polarization, spatial coherence, and particularly, alternative beam deviation mechanisms such as reflective or transmissive metasurfaces [30–40] and highly birefringent thin geometric phase films such as cycloidal diffractive wave plates [41–43]. Optimization approaches must include both the momentum transfer efficiency of the sail and also the impact on the total mass of the sailcraft. For example, advanced materials may afford added functionality that allows space flight hardware such as heat radiators, photovoltaics, antennas, or attitude control devices to be replaced with lower mass elements that are integrated into the sail.