# Beam Formation and Vernier Steering Off of a Rough Surface
## 1 Introduction  
Diffuse scattering of light challenges traditional imaging systems, which rely on ballistic light to operate. When light propagates through turbid media, or reflects off an optically rough surface, it scatters diffusely. Since 2007, new imaging techniques have been able to operate under non-ideal conditions with diffuse scattering by incorporating a process called Wavefront Shaping to their imaging processes. Wavefront Shaping is a family of techniques that compensate for diffuse scattering through the use of spatial light modulators (SLMs) or Digital Micromirror Devices (DMDs) to modify the spatial distribution of amplitude or phase of incident wavefronts. Mathematical models for wavefront shaping treat rough surfaces as a complex field that interfere with incident light [1]. Controlling the resultant reflected wavefront can be achieved by modulating the incident light. Properly modulated incident light has been shown to refocus after interfering with scattering surface or turbid media in a process dubbed inverse diffusion. Numerous feedback-based algorithms and approaches such as matrix measurements have been developed that determine the required modulation to refocus light after transmissive and reflective scattering [2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14]. In particular, [PERSON] provides a detailed summary of several different approaches taken [11]. Wavefront Shaping is distinguished from existing free-space adaptive optics due to the use of turbid media or rough surfaces. Depending on the application, the scattering medium may also be static relative to the time required to find an adaptive optics solution. Existing wavefront shaping techniques have been applied to microscopy, tomography, fiber optics, super-resolution imaging, and a growing number of other use cases [15; 16; 17; 18; 19]. Applications of wavefront shaping specifically involving reflection from a rough surface as opposed to transmission through turbid media are not as numerous in the literature, but there is a continuing interest to utilize reflective inverse diffusion to image around corners [9; 12; 13; 14; 20; 21]. There also could exist unexplored use cases for wavefront shaping in LiDAR with certain limitations. The design of LiDAR systems is trending toward miniaturization, which may necessitate integrated photonics or other miniaturized technologies that preclude liquid crystal SLMs or MEMS mirror technology [22], which are uniquely capable of displaying and rapidly updating phase masks required for wavefront shaping. Nevertheless, very large optical phased arrays could potentially perform the function of wavefront shaping. One method that uses wavefront shaping principles to image around corners is to modulate light diffusely reflected from a rough surface to reestablish the light's ballistic path. This method depends on already present illumination in the scene, and it enables imaging as if the reflected light had specularly reflected from a smooth surface. Previous demonstrations of this technique required calibration sources or guide stars in the around-the-corner scene, although techniques are still being investigated that do not require prior knowledge of the calibration source [9; 23]. The scenario when imaging occurs without access or prior knowledge of the scene is sometimes called indirect imaging or non-line-of-sight imaging. A subset of indirect imaging methods require that illumination be actively provided to the scene. These active methods depend on some intermediate surface that has a direct line of sight to the scene and can reflect illumination into it. The intermediate surface functions as either a virtual illumination source, a virtual detector, or both [12; 13; 14; 24; 25]. An indirect imaging system with these imposed limitations must be able to rapidly create and then control a virtual illumination source at any surface, even in the presence of noise. This motivates the need for reflective inverse diffusion to refocus a scattered beam to be used as a virtual illumination source in such a system.