## 3 Simulation Results  
The optimization procedure described in the preceding section has been used to evaluate the diffractive sail transfer performance. A set of heliocentric canonical units [64] has been introduced to reduce the numerical sensitivity of the two-point boundary value problem, and in all of the numerical simulations, the differential Equations (5)-(8) and (13)-(16) have been integrated in double precision using a variable order Adams-Bashforth-Moulton solver [65] scheme with absolute and relative errors of \(10^{-12}\). The diffractive sail circle-to-circle transfer performance has been analyzed by selecting different possible combinations of \(a_{c}\), \(r_{0}\), and \(r_{f}\). In the remaining part of this section, the numerical results correspond to the important case when \(r_{0}=r_{\oplus}=1\,\)au and \(a_{c}=1\,\)mm\(/\)s\({}^{2}\). In fact, \(r_{0}=1\,\)au describes the case of a solar sail deployment along a parabolic escape orbit relative to the Earth and a heliocentric model in which the eccentricity of the Earth's orbit is neglected. Moreover, \(a_{c}=1\,\mathrm{mm}/\mathrm{s}^{2}\) corresponds to when the solar sail characteristic acceleration assumes a sort of canonical value, as indicated by [PERSON] for a reflective sail (2018). With the aim of obtaining a parametric study of the diffractive sail performance in heliocentric transfer trajectories, the target orbit radius has been selected in the range \(r_{f}\in[0.3,\,0.95]\,\mathrm{au}\) (or \(r_{f}\in[1.05,\,5.2]\,\mathrm{au}\)) for an orbit lowering (or an orbit raising). The minimum flight time as a function of the target radius is shown in Figure 9, which also reports, for comparative purposes, the optimal transfer time obtained with a reflective solar sail assuming an ideal force model (2018). In the latter case, the trajectory optimization has been studied by adapting the procedure described in Ref. (2018). The interesting aspect that emerges from Figure 9 is that the minimum flight time of a diffractive sail is better (i.e., smaller) than that of a reflective sail of equal characteristic acceleration, provided the target radius is greater (for an orbit raising) or lower (for an orbit lowering) than a critical value, which depends on the selected \(a_{c}\). In particular, when \(a_{c}=1\,\mathrm{mm}/\mathrm{s}^{2}\), Figure 9 shows that a reflective sail outperforms a diffractive sail with EOCPs when \(r_{f}\in[0.9,1.12]\,\mathrm{au}\). On the other hand, if \(r_{f}<0.9\,\mathrm{au}\) or \(r_{f}>1.12\,\mathrm{au}\), that is, in a wide range of circle-to-circle mission cases, the flight times required by a diffractive sail with EOCPs are smaller than the reflective counterpart. Such an interesting result is related to the complex interaction between the capability of a Sun-facing diffractive sail to generate a transverse component of the propulsive acceleration greater than that obtained through an unconstrained reflective sail and the higher maneuverability of a reflective sail, which can steer the thrust vector in the half-plane \(\mathbf{a}\cdot\mathbf{\hat{i}}_{r}>0\). The latter consideration, which can be extended to the more general case of a three-dimensional heliocentric mission scenario, has been thoroughly analyzed in Ref. (2018), where a comparison between the force bubble of the reflective sail with that of a diffractive sail is presented; see the scheme of Figure 10 which reports the force bubble comparison in the three-dimensional case. Figure 9: Minimum flight time as a function of the target orbit radius when \(a_{c}=1\,\mathrm{mm}/\mathrm{s}^{2}\): comparison between an ideal and unconstrained reflective sail (red dashed line) and a Sun-facing diffractive sail with EOCPs (solid black line). Figure 11 shows the final value of the spacecraft polar angle \(\theta(t_{f})\) (calculated with respect to the initial Sun-spacecraft line) as a function of the target orbit radius \(r_{f}\) for the selected values of \(\{a_{c},r_{0}\}\).