Speckled speckled speckle

Speckle is the spatial fluctuation of irradiance seen when coherent light is reflected from a rough surface. It is due to light reflected from the surface's many nooks and crannies accumulating vastly-discrepant time delays, spanning much more than an optical period, en route to an observation point. Although speckle with continuous-wave (cw) illumination is well understood, the emerging interest in non-line-of-sight (NLoS) imaging using coherent light has created the need to understand the higher-order speckle that results from multiple rough-surface reflections, viz., speckled speckle and speckled speckled speckle. Moreover, the recent introduction of phasor-field ($\mathcal{P}$-field) NLoS imaging---which relies on amplitude-modulated coherent illumination---requires pushing beyond cw scenarios for speckle and higher-order speckle. In this paper, we take first steps in addressing the foregoing needs using a three-diffuser transmissive geometry that is a proxy for three-bounce NLoS imaging. In the small-diffusers limit, we show that the irradiance variances of cw and modulated $n$th-order speckle coincide and are $(2^n-1)$-times those of ordinary (first-order) speckle. The more important case for NLoS imaging, however, involves extended diffuse reflectors. For our transmissive geometry with extended diffusers, we treat third-order cw speckle and first-order modulated speckle. Our results there imply that speckle is unlikely to impede successful operation of coherent-illumination cw imagers, and they suggest that the same might be true for $\mathcal{P}$-field imagers.