Optimal interplanetary trajectories for Sun-facing ideal diffractive sails

A diffractive sail is a solar sail whose exposed surface is covered by an advanced diffractive metamaterial film with engineered optical properties. This study examines the optimal performance of a diffractive solar sail with a Sun-facing attitude in a typical orbit-to-orbit heliocentric transfer. A Sun-facing attitude, which can be passively maintained through the suitable design of the sail shape, is obtained when the sail nominal plane is perpendicular to the Sun–spacecraft line. Unlike an ideal reflective sail, a Sun-facing diffractive sail generates a large transverse thrust component that can be effectively exploited to change the orbital angular momentum. Using a recent thrust model, this study determines the optimal control law of a Sun-facing ideal diffractive sail and simulates the minimum transfer times for a set of interplanetary mission scenarios. It also quantifies the performance difference between Sun-facing diffractive sail and reflective sail. A case study presents the results of a potential mission to the asteroid 16 Psyche.


Introduction
The interaction between electromagnetic waves and matter is the working principle of photonic propulsion, a propellant-less technology that extracts momentum from solar radiation to generate thrust and navigate the solar system. An example of its effectiveness is provided by conventional solar sails [1,2], which use a thin membrane to reflect the impinging photons coming from the Sun. The mechanism of photon reflection is exploited by other solar sail-based configurations such as heliogyros [3]. Following the success of the IKAROS demonstration mission [4][5][6], the reflective solar sail concept has become the focus of active research, as confirmed by the number of planned missions that will utilize this propulsion technology, including NASA's Near-Earth Asteroid Scout [7] and Solar Cruiser [8].
One of the main disadvantages of reflective solar sails is their limited capability to generate a transverse thrust component, in which the transverse means are perpendicular to the Sun-spacecraft line [9][10][11]. In particular, a Sun-facing reflective solar sail, characterized by a nominal plane normal to the Sun-spacecraft line, generates a purely radial propulsive acceleration; hence, it cannot modify the angular momentum of the osculating orbit. Other optical phenomena affecting the momentum transfer between incoming photons and the spacecraft have been investigated to deal with this problem.
For example, refraction or diffraction effects may be used to extract momentum from the solar radiation pressure and generate a suitable propulsive thrust [12]. In particular, a refractive sail can deflect incoming photons by refracting sunlight across a thin transparent membrane made of polymeric microprisms [13,14], thereby producing a propulsive acceleration. Unlike a reflective solar sail, when microprisms are appropriately designed, a refractive sail can generate a large transverse thrust component even in a nearly Sun-facing orientation. This characteristic makes it easier to modify the specific angular momentum of the osculating orbit with a simple attitude control law. The unique capabilities of the refractive sail have attracted the interest of researchers. Firuzi and Gong [13] first addressed the problem of evaluating the radiation pressure applied to a refractive sail by a ray-tracing method [15] under the assumption that the refractive sail has no wrinkles or billowing effects and is perfectly transmissive. A recent study by Bassetto  [16] proposed a semi-analytical thrust vector model to analyze a set of minimum-time circle-to-circle orbit transfers of a refractive sail-based spacecraft. A typical refractive sail is designed to minimize the light diffraction through the microprism film. This requires the shortest side of the microprisms to be at least ten times greater than the longest wavelength [17]. The diffraction of sunlight can also be exploited to generate thrust. More precisely, the working principle of a diffractive sail involves the use of a metamaterial film to diffract incoming photons, which are deflected from their original path to generate a net propulsive acceleration. In a typical configuration, the diffractive film consists of a polarization grating with a period comparable to the wavelength of the incoming electromagnetic radiation. In principle, many other complex metamaterials may be used with potential advantages in sail performance and thrust-vectoring capability [18].
Recent studies have demonstrated the interest of the scientific community in this innovative propulsive concept. The study by Swartzlander [18] compared the performance of Sun-facing diffractive sails, Littrow diffraction configurations, and conventional reflective sails in the context of Earth-Mars transfers without using an optimal approach. In particular, under the assumption of a constant sail attitude with respect to the Sunspacecraft line, a transparent diffractive sail was found to be superior to a reflective sail because the latter requires more time to reach the target orbit [18]. Another study by Swartzlander [19] showed that a diffraction film with a grating period of 1 µm can convert 83% of the solar blackbody spectrum into spacecraft momentum. Moreover, the non-optimized orbit-raising trajectories of diffractive and reflective sails were compared, and the potential advantages of the former were described. Srivastava and Swartzlander [20] described the optomechanics of a rigid nonspinning light sail that mitigated a catastrophic sail walk-off and tumbling using a flat axicon diffraction grating. Other recent studies by Serak et al. [21], Srivastava et al. [22], Chu et al. [23], and Chu et al. [24] confirmed the potential of this technology in the field of solar sail research.
An interesting feature of the diffractive sail is its ability to provide a large transverse component of thrust, even when its nominal plane is orthogonal to the Sunspacecraft line. This intrinsic characteristic was described in detail by Dubill and Swartzlander [25], which is taken as the starting point of this paper. In particular, Dubill and Swartzlander [25] attempted to maneuver an interplanetary diffractive sail-based light spacecraft to increase the orbital inclination while reducing its distance from the Sun from 1 to 0.32 au with a flight time of approximately 6 years. Such a mission scenario could allow a constellation of diffractive solar sails (or smart dust [26][27][28]) to be placed around the Sun to collect images and other data for space weather monitoring and heliophysics science. The interest in such missions has been confirmed by the research program on solar photonic propulsion recently promoted by the Italian Space Agency [29,30].
Based on the results of Ref. [25], the aim of this study is to develop an analytical expression of the time-optimal steering law [31,32] for a Sun-facing ideal diffractive sail (SFIDS). The thrust control variable was represented by the clock angle, which provides the angular position of the sail body-fixed reference frame with respect to the radial direction. An in-orbit variation of the clock angle may be obtained using a reaction wheel [25] capable of generating a torque along the spacecraft body axis aligned with the Sun-spacecraft line. The optimal steering law was then used to analyze typical heliocentric transfer trajectories in the preliminary mission design phase, including a transfer toward an asteroid [33][34][35].
The remainder of this paper is organized as follows. Section 2 describes the diffractive sail propulsive acceleration model, starting from the literature. Section 3 analyzes the optimal steering law, which is specialized in Section 4 for a set of minimum-time interplanetary transfers. Section 5 deals with the optimal transfers toward the asteroid 16 Psyche, while Section 6 presents some concluding remarks.

Orbital dynamics and sail thrust model description
Consider a spacecraft equipped with a diffractive solar sail as its primary propulsion system. The spacecraft initially covers a heliocentric parking orbit (subscript i) defined by a given set of modified equinoctial orbital elements 36,37]. Recall that the classical orbital elements {a, e, i, Ω, ω} can be calculated from the five MEOEs {p, f, g, h, k} using Eqs. (1)-(5): sin ω = gh − f k, cos ω = f h + gk (4) sin Ω = k, cos Ω = h where a is the semi-major axis, e is the eccentricity, i is the orbital inclination with respect to the ecliptic at epoch J2000.0, Ω is the longitude of the ascending node, and ω is the argument of perihelion. According to Eqs. (1) and (2), the MEOE p coincides with the semilatus rectum of the spacecraft's osculating orbit. The spacecraft true anomaly ν can be expressed as a function of the true longitude L (the last of the six MEOEs [36,37]) by Following Betts [38], spacecraft heliocentric dynamics can be described by introducing the state vector x ∈ R 6×1 defined as whose time derivative is expressed as a function of the spacecraft propulsive acceleration vector a p ∈ R 3×1 aṡ where A ∈ R 6×3 and d ∈ R 6×1 are given by with where µ ⊙ is the Sun's gravitational parameter. The vectorial differential equation (8) is completed using the initial conditions in Eq. (21) defined at the initial time t = t 0 . ≜ 0: where L 0 ≜ L(t 0 ) is the true longitude at time t 0 , which can be calculated by Eq. (6) as a function of the initial spacecraft true anomaly ν 0 ≜ ν(t 0 ) on the heliocentric parking orbit. In Eq. (8), the three components {a p R , a p T , a p N } of the spacecraft propulsive acceleration vector a p are expressed in a radial-tangential-normal reference frame T RTN (C; R, T, N ) of the unit vectors {î R ,î T ,î N }. The origin of T RTN coincides with the spacecraft center-ofmass C. The R-axis lies along the Sun-spacecraft line (i.e., along the direction of the unit vectorî R ). The T -axis belongs to the osculating orbit plane and points toward the direction of the spacecraft (inertial) velocity vector. The N -axis coincides with the direction of the spacecraft angular momentum vector, as shown in Fig. 1.  The propulsive acceleration vector (and its components) can be described as a function of the spacecraft attitude using a suitable diffractive sail thrust model. A mathematical model was first proposed by Swartzlander [18]. This propulsion system representation, which can be considered as the counterpart of the well-known ideal reflective sail (IRS) force model [9,39] for a diffractive sail, was recently adapted by Dubill and Swartzlander [25] to the special case of a Sun-facing sail [40,41]. Note that this particular attitude can be passively maintained by choosing a suitable sail shape [42], that is, by designing a slightly conical surface with the apex directed toward the Sun [43].
Based on the analytical results obtained in Refs. [18,25], the thrust model of an SFIDS can be described by introducing a right-handed body-fixed reference frame T (C; x, y, z) of origin C and unit vectors {î x ,î y ,î z }, as shown in Fig. 2. The (x, y) plane coincides with the sail nominal plane, while the x-axis is in the opposite direction of the grating momentum unit vectorK, that is,î x = −K. Recall that the grating vector of the sail film structure is aligned with the direction of periodicity of the grating [44]. According to Dubill and Swartzlander [25], when the z-axis of the body reference frame is aligned with the Sun-spacecraft line, that is, in a Sun-facing condition with z ≡ R andî R ≡î z , the propulsive acceleration vector a p can be written as where r is the Sun-spacecraft distance, S is the area of the sail reflective surface, c is the speed of light in vacuum, m is the total mass of the spacecraft (assumed to be constant), and I ⊕ is the solar irradiance at a distance r = r ⊕ ≜ 1 au from the Sun. According to Eq. (22), in a Sunfacing condition, the sail propulsive acceleration vector belongs to the plane (x, z) of the body reference frame T , and a p forms 45 deg angle with the Sun-spacecraft line, as shown in Fig. 3, whereâ p ≜ a p / ∥a p ∥ is the propulsive acceleration unit vector. Equation (22) can be rewritten in a more compact form by introducing the characteristic acceleration a c [9]: Analogous to the IRS case, a c is defined as the maximum value of the propulsive acceleration magnitude ∥a p ∥ when r = r ⊕ . Taking Eq. (22) into account, a c is expressed as Such that the propulsive acceleration vector can be rewritten as The three components {a p R , a p T , a p N } of the propulsive acceleration required to complete the vectorial differential equation (8) were obtained by projecting a p onto the radial-tangential-normal reference frame T RTN . To this end, we introduce the sail clock angle δ ∈ [−180, 180) deg, defined as the angle, measured counterclockwise, between the T -axis and x-axis, as shown in Fig. 4, wheren is the unit vector normal to the sail nominal plane in the direction opposite the Sun (i.e., the shadowed side of the sail). From Fig. 4 we obtain and recalling thatî z ≡î R in a Sun-facing condition, Eqs. (24) and (25) yield such that the components {a p R , a p T , a p N } are reduced to According to Eq. (26), for a given value of a c and distance r from the Sun, the SFIDS thrust vector is dependent only on the value of the clock angle δ, which therefore represents the single control variable.
Note that an IRS (when not constrained to maintain a Sun-facing attitude) has two control variables: the clock and cone angle. The latter is defined as the angle between n and the Sun-spacecraft line. The form of the SFIDS propulsive acceleration vector given by Eq. (26) is very different from the general expression used for an IRS without optical degradation [45] and wrinkles [46,47]. Such a difference can be observed in Fig. 5, which shows a typical sail force bubble (i.e., the locus of the propulsive acceleration vector arrow) for an SFIDS and an IRS of equal characteristic acceleration. Figure 5 shows that the SFIDS thrust vector belongs to a conical surface coaxial in the radial (i.e., Sunspacecraft) direction with a half angle of 45 deg, as shown in Fig. 6. In particular, Fig. 5 highlights that an SFIDS provides a single value for the transverse component of the propulsive acceleration vector that is significantly greater  than the maximum transverse value achieved with an IRS of equal characteristic acceleration. Figure 5 also shows that in a "diffractive case", ∥a p ∥ ̸ = 0, that is, a coasting arc cannot be generated in the SFIDS trajectory design. Coasting arcs are often required in the time-optimal transfer trajectory of a solar sail-based spacecraft [48,49]. This raises the question of the performance difference between a common IRS and an SFIDS in a heliocentric minimum-time transfer scenario, which will be discussed in Section 4. Section 3 presents the analysis of the optimal control law and optimal trajectory characteristics of an SFIDS-based mission.

Trajectory optimization
The performance of SFIDS and IRS in a threedimensional heliocentric mission case is compared. To that end, consider an optimal orbit-to-orbit interplanetary transfer, in which the sail thrust vector is steered in such a way to minimize the flight time required to transfer the spacecraft from the Earth's heliocentric orbit to that of the target celestial body. In an orbit-toorbit transfer, the spacecraft angular position remains free at both the beginning and end of the transfer phase to obtain the minimum value of the performance index (i.e., flight time) within the context of an ephemerisfree mission scenario. Accordingly, the values of the spacecraft's true anomalies along the parking and target heliocentric orbits are the outputs of the optimization process. The mathematical model that provides the optimal transfer trajectories of an IRS-based spacecraft was discussed in Ref. [50]. The case of an SFIDS is analyzed in this section.
The heliocentric orbit of a given celestial body is defined by the set {p, f, g, h, k} of MEOEs. Using the SFIDS thrust model discussed in Section 2, the optimization problem requires the determination of the time variation of the sail clock angle δ that maximizes the performance index: with boundary constraints: where the set {p f , f f , g f , h f , k f } denotes the characteristics of the heliocentric target orbit. The optimization problem was solved with an indirect approach [51,52]. Recalling Eq. (8), the Hamiltonian function is written as where λ ∈ R 6×1 is the costate vector defined as and {λ p , λ f , λ g , λ h , λ k , λ L } are six costates. The time derivatives of the costates are given by the Euler-Lagrange equations:λ where x denotes the state vector defined in Eq. (7). The Euler-Lagrange equations were derived from Eq. (35) in analytical form using Eqs. (9)- (20). However, their explicit expressions are omitted here for brevity. The optimal time variation of the clock angle δ = δ ⋆ (t) was obtained by applying Pontryagin's maximum principle, that is, by maximizing at any time instant the portion H ′ of the Hamiltonian function H that is explicitly dependent on the control angle δ, or Using Eqs. (9)- (19) and (27)- (29), the necessary condition ∂H ′ /∂δ = 0 provides the expression of the optimal clock angle as a function of the spacecraft states and costates.
The solution of the optimization problem requires the numerical integration of a set of 12 nonlinear scalar differential equations provided by Eqs. (8) and (35). Ten of the twelve necessary boundary conditions are given by Eqs. (31) and (32). The last two boundary conditions were obtained by enforcing the transversality condition [53].
Bearing in mind Eq. (30), the transversality condition [53] also applies an additional constraint on the final value of the Hamiltonian function, which is useful for calculating the unknown flight time t f .
H(t f ) = 1 (40) The associated two-point boundary value problem (TPBVP) was solved with an absolute error of less than 10 −8 through a hybrid numerical technique that uses gradient-based methods to obtain the optimal flight time and unknown initial value of the costates {λ p , λ f , λ g , λ h , λ k }. It should be noted that the initial value of λ L was obtained from Eq. (40). During the solution of the associated TPBVP, the spacecraft equations of motion and the Euler-Lagrange equations were numerically integrated using a variable order Adams-Bashforth-Moulton solver scheme [54] with absolute and relative errors of 10 −10 .
For a given value of the sail performance parameter a c and a given target celestial body, the previously described optimization problem was solved to obtain the minimum flight time t f . Therefore, it is possible to graphically determine the function t f = t f (a c ) by simply repeating the same procedure for different values of a c . This aspect will be analyzed in Section 4 for a set of potential interplanetary mission scenarios, along with a numerical performance comparison with an IRS-based spacecraft.

Numerical results and SFIDS-IRS comparison
The procedure described in Section 3 was used to evaluate the optimal transfer performance of an SFIDS-based spacecraft in an interplanetary mission to Mercury, Venus, and Mars. The orbital elements of the celestial bodies involved in the numerical simulations were obtained through the JPL Horizons online ephemeris system (all the data were from July 1, 2022). The orbital characteristics, in terms of classical orbital elements and MEOEs, are summarized in Table 1. The table also includes the data of the asteroid 16 Psyche, which will be considered as a case study in Section 5.
In each three-dimensional mission case (Earth-Mercury, Earth-Venus, and Earth-Mars), parametric analysis was performed to evaluate the sensitivity of the minimum flight time t f to the value of the characteristic acceleration, which was assumed to vary within the range The numerical results in terms of the minimum flight time t f as a function of a c for an SFIDS-based spacecraft are summarized in Fig. 7 for the three interplanetary mission cases. The difference in flight times between SFIDS-and IRS-based spacecraft was quantified with the dimensionless parameter: where t f | SFIDS (or t f | IRS ) is the minimum flight time obtained by solving an optimal control problem for an SFIDS-based (or IRS-based) spacecraft. For a given a c , a positive value of D indicates that the performance of the SFIDS is superior to that of the IRS, that is, the SFIDS enables the spacecraft to reach the target planet in a shorter flight time compared with the IRS case. The function D = D(a c ), obtained numerically, is plotted in Fig. 8. Note that this comparison assumes that the SFIDS and IRS have the same characteristic acceleration a c ; however, in principle, the SFIDS should have a smaller area-to-mass ratio than the IRS. A more detailed estimate of the achievable level of performance with an SFIDS is beyond the scope of this study. Figure 8 shows that in each mission scenario, the SFIDS performance is superior to that of the IRS for the entire range of characteristic accelerations considered in the numerical simulations. The percentage difference in flight times was more pronounced when smaller values of the characteristic acceleration were considered. For example, in a three-dimensional Earth-Mars orbit-to-orbit transfer with a medium-high performance sail, the flight time in the IRS-based case was approximately 15%-20% greater than that required by the SFIDS. The percentage difference increased to approximately 70%-90% for a lowperformance sail. The ripples in Fig. 8 were attributed to the slight variations in the flight time when the spacecraft swept angle is an integer multiple of 2π, as discussed in Refs. [3,56] with a simplified (two-dimensional) mission scenario involving a reflective sail. The curves in Figs. 7 and 8 show that the difference in the transverse component of the propulsive acceleration vector between the SFIDS and IRS (highlighted in Fig. 5) has a major effect on the transfer performance, despite the higher maneuverability characteristics of the IRS. Recall that the IRS has two control variables (i.e., two control angles), while the thrust vector of the SFIDS is dependent only on the clock angle. The characteristics of the optimal transfer trajectory of the SFIDS-based spacecraft are analyzed in Section 5 in a more challenging orbit-to-orbit mission scenario.

Case study
In this section, we consider a mission to the asteroid 16 Psyche, a small body located in the belt between Mars and Jupiter. The asteroid is a potential target of exploration in the near future [57] because of its particular metal-rich composition [58][59][60]. In fact NASA and the Jet Propulsion Laboratory planned a robotic   target asteroid. Unfortunately, the launch was indefinitely postponed in June 2022.
The characteristics of the heliocentric orbit of the asteroid are listed in Table 1. These indicate that reaching this small body will be a challenge for a solar sail-based spacecraft because of the distance of the asteroid from the Sun. Indeed, the fact that the solar radiation pressure has an inverse-square variation with the solar distance imposes severe constraints on solar sail-based transfers when the mean radius of the target orbit is larger than the mean Sun-Mars distance. Assuming a characteristic acceleration in the usual range a c ∈ [0.1, 1] mm/s 2 , the minimum orbit-to-orbit flight time of the SFIDS-based spacecraft is shown in Fig. 9. The black circles refer to the cases listed in Table 2. The table also Table 2. in the parking (ν(t 0 )) and target (ν(t f )) heliocentric orbits. The ν(t 0 ) and ν(t f ) values are the outputs of the optimization process. These provide the designer with useful information about the potential launch window at which the minimum transfer time can be achieved. Figure 9 and Table 2 show that the SFIDS-based transfer toward the asteroid 16 Psyche with a flight time comparable to that of NASA's planned mission (i.e., approximately 3.5 years) requires a medium-performance sail with a characteristic acceleration in the range 0.5-0.6 mm/s 2 . However, the SFIDS-based scenario considers a direct transfer, that is, it does not require an intermediate gravity-assisted maneuver. In this case, the diffractive solar sail option is less constrained from the viewpoint of the existing launch windows.
When a low-performance SFIDS is considered for mission optimization, the minimum flight time is in  the order of 15-18 years, and the transfer trajectory shows multiple revolutions around the Sun. For example, Fig. 10 shows the optimal transfer trajectory when a c = 0.1 mm/s 2 , which requires approximately 18 years. However, using a medium-high performance sail in the mission design reduces the flight time to a few years and the complexity of transfer trajectory. For instance, Fig. 11 shows the optimal transfer trajectory when the characteristic acceleration is 1 mm/s 2 . Figure 12 shows the time variation of the optimal clock angle during the transfer. It should be noted that in both low-and high-performance cases, the modulus of the clock angle remained below 20 deg for approximately 80% of the flight time. In the case of a high-performance sail, the ν(t 0 ) and ν(t f ) values listed in Table 2 indicate that the potential optimal launch window is around April 5, 2029.

Conclusions
This study analyzed the performance of a diffractive sail with a Sun-facing attitude in an optimal framework. Starting from the recent ideal thrust model of a diffractive sail and using an indirect approach, we derived the sail optimal control law, that is, the explicit expression of the clock angle (the single control variable of this particular configuration) as a function of the osculating orbit-modified orbital elements and problem costates. The proposed mathematical model was then used to analyze the performance of an SFIDS as a function of the characteristic acceleration in a set of typical orbit-to-orbit interplanetary (three-dimensional) transfer scenarios.
The numerical simulations showed that for missions involving transfers toward both inner and outer planets, the flight times required by an SFIDS were significantly shorter than those required by a more conventional IRS, with the characteristic acceleration being the same. This is an interesting and less obvious result because the IRS has better maneuverability and may exploit the possible presence of coasting arcs along the transfer, which could be obtained by an edge-on flight orientation toward the Sun. The study of an orbit-to-orbit transfer toward the asteroid 16 Psyche also revealed that a mediumperformance Sun-facing diffractive sail may achieve a flight time comparable to that of NASA's planned mission to Psyche.
The potential extensions of this study are twofold. First, the simplified thrust model discussed here can be used to analyze the SFIDS performance when generating and maintaining both artificial equilibrium points and heliocentric non-Keplerian orbits, paralleling the wellknown results obtained with reflective solar sails. Second, the diffractive sail optimal performance can be analyzed by relaxing the Sun-facing attitude constraint, that is, by considering an additional control variable in the sail thrust model. In the latter case, the description of the sail propulsive acceleration vector becomes more complex (compared with the Sun-facing case), and requires a new study on the optimal control law.

Funding note
Open Access funding provided by Università di Pisa within the CRUI-CARE Agreement.
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