Rough surface scattering for active-illumination systems

In 1960, Theodore Maiman invented the first working laser,¹ which originally boasted only a few milliwatts of power. Yet, by the 1970s, laser powers reached the megawatt level and the directed-energy (DE) research community came to life.² The technology found in DE applications is inspiring in that it presents game-changing capabilities by offering systems with varying lethality, speed-of-light delivery, and unparalleled precision. With our work, we hope to aid the burgeoning DE research community by investigating the interaction of partially coherent beams with rough surfaces, a topic that plays a key role in active-illumination systems such as those found in DE applications.

Rough surface scattering has been an active area of research for over half a century. Some of the early work dealt with how incoherent light interacts with surfaces—a process modeled using the bidirectional reflectance distribution function (the ratio of scattered radiance over the incident irradiance)—for applications in passive visible and near-infrared remote sensing and computer graphics. For example, in preparation for the NASA Apollo missions, analysis of light scattered from the lunar surface led researchers to conclude that the moon's surface is composed of a particulate material.³ More recently, with the proliferation of laser-based systems, the interaction of coherent laser light with rough surfaces, in particular the statistical behaviors of the resulting speckle patterns, gained considerable interest in fields such as metrology and remote sensing. Goodman's recent text reviews much of the notable research in this area.⁴

Figure 1. In this optical layout, a lens collimates the light from a broadband optical source that is then sent through a Gaussian apodization filter. This creates a Gaussian Schell-model beam that illuminates a rough target. We set the detector two focal lengths (f) from the rough target so that it collects the scattered light in the far field. Narrowband filters are placed in front of the detector to measure the spectrum of the scattered light.

For DE applications, a fully coherent laser beam propagates from the source through the atmosphere resulting in partially coherent beam illumination on the target. Interestingly enough, not much literature exists pertaining to the scattering of partially coherent light from rough surfaces. Most of the current literature deals with the scattering from low-contrast surfaces (i.e., where the index of refraction differs only slightly from unity). These are scattering surfaces in which the Born approximation is valid.⁵

Some of our recent work derived 2D scalar-equivalent expressions for the scattering of partially coherent beams from statistically rough surfaces.^{6, 7} Specifically, the analyses made use of a Gaussian Schell-model (GSM) source in creating the incident-field cross-spectral density function, which allowed us to vary the spatial coherence properties of the incident radiation. In so doing, we developed closed-form solutions for the scattered-field cross-spectral density function to observe the spatial coherence properties of the scattered radiation in the far field. We validated these analytical expressions through computational simulations and showed good agreement between the theoretical predictions and the numerical results.

Missing from this previous analysis was a rigorous experimental verification. Here, we propose an experiment to validate the 2D scattering expressions derived previously: see Figure 1. In our setup, a detector records the spectrum of light scattered from a rough target in the far field for a variety of incident and scattering angles. Note that the light illuminating the rough target is in the form of a GSM beam and that multiple narrowband filters allow us to experimentally record the change in the spectrum. As such, this experimental setup is a novel approach to validating the 2D expressions as well as a 3D theory discussed below.

While the 2D scalar-equivalent theory is a convenient tool for gaining insight into rough surface scattering, a complete understanding of the problem requires the 3D vector solution. Knowledge gained from the 2D analysis reveals that the desired vector solution is, in general, composed of complicated functions of both the source (size and coherence properties) and surface parameters (surface height standard deviation and correlation length).^{6, 7} However, in most scenarios of interest, these complicated expressions should reduce to physically insightful forms.

The proposed vector solution provides all the necessary information about how the coherence and polarization of scattered light is affected by rough surfaces. By formulating the vector solution in a manner consistent with Wolf's unified theory of coherence and polarization,⁸ all physical implications inherent in Wolf's work apply to the desired vector solution. In particular, the vector solution predicts both spectrum and polarization changes upon scattering and allows for the formulation of Stokes parameters and bidirectional reflectance distribution functions. It also permits one to make predictions of speckle behavior.

Based on the novel experiment and the vector solution proposed here, DE applications of this research include studies in adaptive optics systems,⁹ optical phased arrays,¹⁰ and remote sensing using laser radar.^{11, 12} All of these topics serve as future research efforts, which directly relate to our previous and proposed work.

The views expressed in this paper are those of the authors and do not reflect the official policy or position of the US Air Force, the Department of Defense, or the US Government.