## 1 Introduction
This solar sail, which uses a diffractive (instead of reflective) film [49], is able to generate an in-plane thrust component even if the incoming photon direction is perpendicular to the sail nominal plane [50, 51, 52]; see Figure 1.  
Exploiting the characteristics of an ideal diffractive sail, the authors [53] have recently analyzed the performance of a solar sail-based spacecraft with a Sun-facing attitude in an interplanetary mission scenario, assuming that the sail nominal plane may rotate around the Sun-spacecraft line. The trajectory design model discussed in [53] extends the preliminary results obtained by [PERSON] and [PERSON] [54], and it is based on the assumption that the diffractive sail-induced thrust vector belongs to a conical surface with a fixed half angle and coaxial with the Sun-spacecraft line; see Figure 2. The thrust vector orientation with respect to an orbital reference frame (that is, the azimuthal position of the thrust vector direction along the virtual cone depicted in Figure 2) can be chosen by rotating the sail nominal plane of a suitable angle around the Sun-spacecraft line. Figure 1: Conceptual scheme of the thrust vector direction in a Sun-facing solar sail with a reflective or a diffractive film. Using such a simplified model, Ref. [53] analyzes three-dimensional orbit-to-orbit transfers within an optimal framework, where the performance index to be minimized is the total flight time necessary for the diffractive sail to complete an assigned heliocentric transfer between two assigned Keplerian orbits. The aim of this paper is to simplify the diffractive sail guidance scheme considered in Ref. [53], assuming a sail nominal plane with a fixed orientation relative to an orbital reference frame. The in-plane component of the diffractive sail-induced thrust vector may be changed by means of active (switchable) diffractive elements, that is, using a sail film designed with a controlled binary arrayed grating [46]. In fact, according to the conceptual propulsion system design described in Ref. [46], a diffractive sail potentially offers some interesting features that are difficult (or even impossible) to achieve with a conventional reflective solar sail, that is, with a metalized high-reflective (typically aluminum-based) thin film [55; 56]. In principle, as pointed out by [PERSON] [46], the diffractive sail concept gives the opportunity: (i) to recycle photons to generate solar-electric power, using a bi-layer (or, more in general, a multi-layer) thin film containing photo-voltaic cells; and (ii) to employ electro-optically controlled binary metamaterial (either for the entire sail or for a subset of panels) to change the direction of the in-plane thrust vector component. Actually, the use of active electro-optically controlled panels (EOCPs) is a well-known idea, which has already been exploited in solar sail design [57] since the construction of IKAROS spacecraft, which represents a milestone for solar sail technology. Indeed, during its interplanetary flight, IKAROS spacecraft was able to both generate electric power through very thin solar cells attached to the sail film and to execute attitude control maneuvers by means of advanced electro-controlled LCD panels installed near the edges of its membrane [2]; see the scheme of Figure 3.  
The use of a sail film with electro-controlled and variable reflectivity panels has also been proposed as a means to (slightly) change the magnitude of the sail-induced thrust vector in order to perform missions that require a propulsive acceleration vector with a fixed (orbital) direction and a magnitude variable within a narrow range. Possible examples are the maintenance of a collinear artificial equilibrium point [58] in the Sun-planet elliptic-restricted three-body problem [59] or the guidance of a smart dust in a scientific mission aiming at an in situ study of Earth's magnetotail [60; 61]. Figure 2: Single degree-of-freedom diffractive sail considered in the trajectory analysis discussed in Ref. [53].