# The Potential of Focusing Acoustic Retroreflectors for Architectural Surface Treatment
## 1 Introduction
### Retroreflectors and Focusing Retroreflectors  
Retroreflective array surfaces provide an obvious way of concentrating early reflected sound at the source without knowledge of the source's position. Their effect has been explored in prior studies, albeit in mostly outdoor spaces or in laboratory settings [44; 48; 49; 50; 51; 52]. However, while retroreflective arrays can achieve a substantial boost in sound returned to the source, more may be needed; for this reason, we consider the potential of focusing retroreflectors. Acoustic retroreflection can be achieved through passive acoustic devices that provide space-reversal across their face. The three examples in Figure 2 achieve this over reasonably broad spectral and spatial ranges in different ways, each with distinct advantages and disadvantages. The cat's eye and van Atta reflectors have wider acceptance angles than the cube corner reflector but involve more elaborate construction. For example, a cat's eye lens might be created from a gas balloon [53] (e.g., xenon, which has a refractive index close to 2 with respect to air), from an intricate metamaterial (which can have a graded index of refraction, i.e., a Luneburg lens [54]), or even with a flat lens and mirror [55]. The van Atta reflector [56] is an intricate and bulky construction, and its high-frequency response is inherently limited by its spatial physical discretization. The complexities of these devices can lead to acoustic losses, as well as substantial costs. Hence, the cube corner reflector's simplicity has practical appeal in treating large architectural surfaces. Within its narrower acceptance angle, the cube corner reflector's efficiency varies markedly with incidence angle: common designs are most efficient on-axis, but also have local efficiency peaks for full dihedral and single-face specular reflection angles [57]. Figure 1: Relationship between \(L_{\text{reverb}}\) (Equation (6)) and \(\text{ST}_{V}\) (Equation (8)) of theoretical diffuse rooms, with characteristic room dimension, \(V^{1/3}\), indicated by the colormap. Room volumes range from 27 m\({}^{3}\) to 7500 m\({}^{3}\). A reflective floor in the area around the talker is assumed for theoretical \(\text{ST}_{V}\). Data from real room measurements (using mid-frequency reverberation time and room volume to calculate reverberant level) are shown as magenta points. The right axis gives an example of predicted voice regulation from voice support (from [47]). Each of these devices can be made to focus: the cat's eye reflector can have an increased refractive index lens; the corner reflector can have co-focused parabolic faces (Appendix A); and the van Atta reflector can have pipe lengths increasing towards the center of its face. Alternatively, a converging acoustic lens over the face of any retroreflector can be designed for a focal plane. For a sound source with a spherical wave front, 6 dB is lost from geometric dispersion on the return path from a non-focusing retroreflector to the source, and conceivably more than this 6 dB could be gained from focusing. Focusing retroreflectors are rarely studied because most applications in optics and radar are far-field, and the retroreflection literature in acoustics is underdeveloped. In acoustics, [PERSON] et al. [51] demonstrated an array of four cube corner reflectors (square trihedra), each with two faces optimally warped to enhance retroreflected sound concentration within a few meters. Most of the improved sound concentration of the warped 0.5 m edge-length trihedra was at high frequencies, but a 1 m edge-length version had enhanced effectiveness over a broader frequency band. [PERSON] et al. [58] describe a focusing single-wavelength infra-red electromagnetic retroreflector, exploiting resonance as part of the retroreflective mechanism. Other studies have examined more general focus adjustments of retroreflectors. [PERSON] et al. [59] describe a diverging cube corner optical retroreflector for spacecraft by very slightly curving the trihedral surfaces. Similarly, [PERSON] et al.