Partially-coherent light transport in a scene with diffractive materials. The insets visualise the shape of the coherence area (see Fig. 4) of light that is sourced from a light source or scattered by matter. A spherical source gives rise to light with (a) highly isotropic spatial coherence, while a cylindrical source produces light that is (b) significantly more coherent in one transverse direction than in the other (a phenomenon we term coherence anisotropy). Light then propagates away from the source and interacts with matter. These physical processes—the coherence of light and light-matter interaction—are mutually-dependant processes: Coherence drives the optical response of the interaction of light with matter, and, conversely, the properties of matter alter the coherence properties of the scattered radiation. (c) Thin coating over the wings of a silver scarab induces interference. The distinct colours on the left and right wings arise solely due to the difference in the spectral composition and coherence of the incident light. The surface is smooth and the scattered light retains the coherence shape of the incident light. (d,e) On the other hand, scatter by rough surfaces induces coherence properties and anisotropies that are dictated by the surface parameters. (f) Diffraction grating by (unrecorded) DVD disks. Note that the secondary diffraction lobes diminish due to the limited spatial coherence of light. One of the primary theoretical conclusions of this paper is that it is the ensemble-averaged reflectivity of the matter that drives the coherence shape of the scattered light. This can be seen in (c) and (f), where the induced interference does not influence the scattered radiation’s coherence properties.
Abstract
Our purpose in this paper is two-fold: introduce a computationally-tractable decomposition of the coherence properties of light; and, present a general-purpose light-matter interaction framework for partially-coherent light. In a recent publication, Steinberg and Yan 2021 introduced a framework that generalises the classical radiometry-based light transport to physical optics. This facilitates a qualitative increase in the scope of optical phenomena that can be rendered, however with the additional expressibility comes greater analytic difficulty: This coherence of light, which is the core quantity of physical light transport, depends initially on the characteristics of the light source, and mutates on interaction with matter and propagation. Furthermore, current tools that aim to quantify the interaction of partially-coherent light with matter remain limited to specific materials and are computationally intensive. To practically represent a wide class of coherence functions, we decompose their modal content in Hermite-Gauss space and derive a set of light-matter interaction formulae, which quantify how matter scatters light and affects its coherence properties. Then, we model matter as a locally-stationary random process, generalizing the prevalent deterministic and stationary stochastic descriptions. This gives rise to a framework that is able to formulate the interaction of arbitrary partially-coherent light with a wide class of matter. Indeed, we will show that our presented formalism unifies a few of the state-of-the-art scatter and diffraction formulae into one cohesive theory. This formulae include the sourcing of partially-coherent light, scatter by rough surfaces and microgeometry, diffraction grating and interference by a layered structure.
(top) A golden scarab with rough wings, that are coated with a moderately-thick dielectric layer, illuminated by a light source of varying size (left to right).
Rendered using our physical light transport formalism.
Greater light source admit less coherent radiation, hence interference patterns diminish.
On the right a mirror is placed that reflects light from the source back to the golden scarab.
The greater distance travelled by the reflected light means that this light is more coherent and stronger interference patterns are visible on the right wing, compared with the left wing.
(bottom) Same scene rendered with classical radiometric light transport, where the golden scarab's wings are rendered using Belcour and Barla [2017].
The effects of partial coherence are reproduced by sampling many individual points on the source, i.e. a superposition of many uncorrelated plane waves that constitute the wave ensemble.
However, a great many such samples are needed to faithfully reconstruct the ensemble's statistics, which our method captures with a single ray.
The required sample count also grows exponentially with path depth: light that is scattered and diffused by the rough surface of the body of the smaller copper scarab reaches the bottom parts of golden scarab wings.
That light is very poorly coherent, however the radiometric light transport renderings showcase erroneous interference patterns.
Our method correctly captures the poor coherence of the incident light.
A primary contribution made in this paper is a proposed formal model that is used to describe matter (physical objects, surfaces and media) and quantify its response to incident radiation.
To accurately solve the problem of diffraction and scattering of light by matter, matter features need to be accurately described at a sub-wavelength resolution.
A purely deterministic model that aims to describe that matter (for example, a high-resolution heightmap of a surface’s microstructure) would typically be practical only for tiny or the simplest of objects.
In-place of a purely deterministic description, it is common to use statistical models.
For example, common surface scatter theories in applied optics describe a surface via its spatial frequencies.
Statistical models are also prevalent in graphics and quantify the surface facet distribution, the correlation between different type of scattering particles in a medium, and so on.
Statistical models are computationally convenient, as far less data is required to represent the sub-micron features of the matter.
Such data (e.g, the power spectral density of a surface) can often also be measured directly from real-world objects.
However, purely statistical models lack the capacity to represent interesting features and irregularities (such as scratches), and it is these features that create interesting optical responses.
We introduce a hybrid matter model as a locally-stationary stochastic process (a generalization of wide-sense stationary processes).
The autocorrelation of the scattering function for such matter can be written as
where $\sigma$ is the scattering function and $\vec{r}_{1,2}$ are a pair of spatial points in the matter.
$\langle\cdot\rangle$ denotes ensemble-averaging and $|\cdot|$ the complex magnitude.
The function $R$ is the (normalized) stationary autocorrelation that simply quantifies the statistical similarity of the matter scattering properties at a spatial distance of $\vec{r}_1-\vec{r}_2$.
Then, the autocorrelation of the locally-stationary scattering, $\Sigma$, as described above, can be understood as augmenting the typical statistical representation of matter (e.g., the PSD—which is the spectral decomposition of the stationary autocorrelation $R$—that is used to model rough surfaces) with a spatially-varying, deterministic description of scattering features.
The small micro-scale features are described statistically via $R$—the autocorrelation between the scattering behaviour of these micro features—while the larger matter features (scratches in a surface, suspended particles in paint, etc.) are described by the ensemble-averaged scattering intensity $\langle|\sigma|^2\rangle$.
That is, a hybrid model, partially statistical, and partially explicit.
In the paper we derive a theoretical conclusion that can be formulated as a strong version of the Van Cittert–Zernike theorem, implying that:
Under locally-stationary matter, one aspect of the matter description—the ensemble-averaged scattering behaviour (the first term in the equation above)—fully drives the diffraction process and the coherence of the scattered beam; on the other hand, another aspect—the statistical matter correlation $R$—fully dictates the process of interference.
The consequence of the above can be restated as the interesting conclusion: It is the correlation of scattering features (quantified by $R$) that gives rise to observable interference, while matter features that can not be captured by $\langle|\sigma|^2\rangle$ (i.e., disappear on ensemble-averaging) have no affect on the coherence of the scattered beam.
Computational Aspects
This work builds upon the foundations laid out by our A Generic Framework for Physical Light Transport (Steinberg and Yan 2021).
In physical light transport, the quantity that describes a light beam is the cross-spectral density (CSD) function of the wave ensemble (the CSD replaces the classical radiance).
In conjunction with the optical contributions outlined above, this work also aims to solve a couple of computational challenges that arise with physical light transport:
The classical radiance is easy to represent in a computer program: it is a simple numeric value.
In contrast, the CSD is a function, it mutates on propagation, changes when scattered by matter or diffused by media, and so forth.
How do we represent a large family of CSD functions, non-symbolically (which is slow and cumbersome)?
Under classical, radiometric theory, light-matter interaction reduces to the (seemingly) simple product of the B(S)DF with the radiance.
The problem of interaction of light with matter in physical light transport—that is, computing the CSD of the scattered beam—can be formulated as a Fresnel- or Fraunhofer-region diffraction problem, which under typical Fourier optics treatment reduces to a (fractional) Fourier transform (again, see Steinberg and Yan [2021] for details).
While this (linear) integral relation is analytically simple, computing a (double-spatial) FFT for each scattering event is impractical: it is very slow and introduces errors with each interaction (due to limited resolution and numeric reasons).
How do we formulate general, but practical, physical-optics light-matter interaction formulae?
The idea is to work in Hermite-Gauss space by expanding the CSD functions under the multivariate anisotropic Hermite-Gauss (AHG) functional basis.
The AHG functions are defined with respect to a matrix $\mathbf{\Theta}$—which we term the shape matrix—which manifests important physics: it can be understood as the coherence volume of the light beam.
Capturing this information in this fashion means that a variety of CSD functions can be represented with a good degree of accuracy even with little AHG modes.
The AHG functions also possess additional analytic properties that make them an excellent choice for our purposes:
For example, they are the eigenfunctions of the (fractional) Fourier transform.
Formally, if $\Psi_{\mathbf{\nu}}^{\mathbf{\Theta}}$ is an $n$-dimensional AHG function with degree $\mathbf{\nu}\in\mathbb{N}^n$ and shape matrix $\mathbf{\Theta}$, then
where $\mathcal{F}$ is the Fourier transform operator.
This means that a beam, with its CSD represented using AHG modes, remains such an AHG beam after Fourier transforming: i.e., the coefficients change but the modal representation stays the same.
Computationally, this is very convenient.
Recalling that simple diffraction is nothing more than a (fractional) Fourier transform, the above suggest that working in AHG basis may serve to develop simpler formulae.
In the paper we show that this is indeed the case and derive coherence-aware physical optics formulae for a variety of light-matter interactions (see figure below).
Under our theory, all of these light-matter interactions are unified into one, cohesive framework.
Examples of light-matter interaction that we discuss in this paper. (a) Sourcing of partially-coherent light: Charges and current in a light source give rise to electromagnetic radiation. The amount of charge together with its spatial correlation fully describe the emitted radiation and its coherence properties. (b) Scatter by a statistical surface: A surface (brown) is decomposed into its spatial frequencies (illustrated in gray). The power of each frequency is given by the power spectral density (PSD) function $S_\text{surf}$, which plays an important role in describing the scattered radiation and is the spectral decomposition of the stationary correlation $R$. (c) Scatter by explicitly-defined microgeometry: A surface patch is described by the (explicit) height perturbations $h(\vec{\xi})$, where $\vec{\xi}$ is a point on the surface plane (gray dashed line) and $\langle|\sigma(\vec{\xi})|^2\rangle$ is the (ensemble-averaged) scattered intensity. (d) Layered structure: A dielectric layer over a substrate, where the surface of each is described statistically via the stationary autocorrelation functions $R_1$ and $R_2$, respectively. The surfaces are assumed to be statistically independent. (e) Diffraction grating: A two-dimensional grid of small and identical, but otherwise arbitrary, optical elements produces a diffraction grating. The elements are regularly spaced and the position of the center of each element is denoted $\vec{\alpha}_{pq}$.
