## 1 Introduction  
Risley prisms are optomechanical devices that have, in their most common configuration, a pair of rotational optical wedges [1, 2]. They are mostly utilized for laser scanning [3, 4, 5, 6, 7], but also for optical pointing, by positioning for example the FOV of a camera on an area of interest for machine vision applications [8, 9]. Other Risley prism devices may consist of a pair of tilting [10, 11], rotational and tilting [12], or translational prisms [13, 14]. Rotational doublets have also been developed to minimize optical dispersion [15]. Other solutions include devices with two rotational gratings [16], with two pairs of prisms to increase the FOV [17], or with three rotational prisms [18], the latter for a supplemental degree of freedom in the design of the generated scan patterns.  
Risley prisms are utilized in a large range of applications, including not only laser scanning [3, 4, 5, 6, 7, 10, 11, 12, 15, 16, 17, 18] and optical pointing [8, 9], but also holography [19], interferometry [13, 20], polarimetry [21], light attenuation [14], optical metrology [22], and imagingthe latter for confocal microscopy [23]. The dimensions of the prisms vary from hundreds of millimeters in diameter, for example for devices utilized in satellite positioning [24], to millimeters in diameter for endoscopy [25].  
Regardless of their type, dimensions, and applications, the characteristic of rotational Risley prisms is that they generate complicated scan patterns. This major disadvantage makes such scanning systems much less utilized than the most common GSs [26; 27; 28]. The advantage of 2D GSs, as well as of their alternatives, polygon mirror (PM) plus GS systems [29; 30; 31] or MEMS with oscillatory mirrors [32; 33; 34], is that they provide the (uniform/constant speed) raster scanning necessary, for example, for laser manufacturing or for imaging [35; 36; 37; 31]. The latter includes hot topics such as Optical Coherence Tomography (OCT) [38; 39; 40; 41; 42] for biomedical investigations [32; 43; 44; 45; 46; 47; 48; 49; 50; 51; 52; 53; 54; 55; 56; 57; 58; 59; 60; 61; 62; 63; 64; 65; 66; 67;  
The second aim is to obtain with the proposed graphical method the simulated scan patterns of a pair of Risley prisms and to fully characterize them by performing their multi-parameter analysis, considering all the constructive and functional parameters of the device. Both the shapes and the dimensions of the patterns are targeted by the analysis, the latter using the theoretically deduced angular and linear deviations.  
The third aim is to extract rules-of-thumb to design such optomechanical scanners in order to obtain certain shapes and dimensions of their scan patterns. Examples of experimental ascertainments of the patterns, to demonstrate how theory and simulations can be validated, complete the study.  
The remaining of this paper includes, in Section 2, the configurations and equations (including for angular deviations) of scanners with a pair of rotational Risley prisms. Linear deviations (maximum and minimum, with exact and approximate expressions) are deduced in Section 3. In Section 4, the graphical method we have developed is presented, as well as examples of its results, i.e., simulated scan patterns. A selection of relevant examples is made with regard to all system parameters to allow for discussion of the shape of patterns. A multi-parameter analysis, of both angular and linear deviations, is performed in Section 5. It is correlated with the simulated scan patterns, extracting rules-of-thumb to be able to optimally design such scanners. A brief experimental validation is made in Section 7. Conclusions and directions of future work are provided in Section 8.