Scattering of Directional Light

Blue Marble the Uniform Earth Image

Scattering of Directional Light

Uri Lachish, Rehovot



Backlight scattering of directional light has been so far considered a diffusive process obeying Lambert's Cosine scattering law. According to it, a sphere image will be maximal at its center and dwindle to zero at its periphery by the cosine function. However, observed images, either celestial, or terrestrial, are all nearly uniform and do not comply with the law. There are no true images that comply with it. Uniform images of the full moon have been discussed as diffusive events. Other bodies have not been discussed and the moon theories, based on its surface properties, are not applicable to them.
A single event, non-diffusive scattering, is suggested to account for all such images. The scattering is coherent and leads to uniform images. It accounts for the "Opposition Effect", enhancement of 180 degrees back-scattering, considered, so far, a separate effect.
Coherent single event scattering may be separated from non-coherent scattering noise by interference methods. Images such as asteroids, may be observed with better resolution or at a longer distance. Bodies within scattering media, as in biological tissues, maybe better observed.

Light, scattering, Lambert, diffuse, coherent, interference

Introduction and Results
Back scattering of directional light, started by Lambert with his cosine scattering law [1], [2]. The law states that the intensity of a back scattered light is proportional to the cosine of the angle θ between a light ray striking on a surface and a line perpendicular to that surface, fig-1. Thus, the scattering intensity is maximal when the surface is perpendicular to the ray at θ = 0 degrees, and it fades by the cosine function to zero when the ray is grazing the surface at θ = 90 degrees.


Fig-1: Back scattering in a multiple event and in a single event. The equivalent scattered light in a multiple event is perpendicular the surface. The maximal scattered light in a single event is aimed back to the source.

In the case of scattering from a sphere the back scattering is maximal at the center of the sphere, and it dwindles to zero by the cosine law when moving toward the sphere periphery [3], [4]. This law seems nearly self-evident, since by looking on the surface through a unit area "a" in the direction of the coming ray. The area on the surface will be proportional to 1/Cos(θ), and the radiation density on the surface will therefore be proportional Cos(θ). The light scattered backwards will be proportional to Cos(θ) since the scattering maximum is perpendicular to the surface, fig-2.


Fig-2: A rendered, simulated, sphere, following Lambert cosine Law [3].

The full moon image, fig-3, is best observed during moonrise [5]. The image corresponds to back scattering of sun light, and it is somewhat surprising therefore, that the full moon image is uniform from the image center toward its periphery, except details of rocks and dry seas. The moon image does not obey Lambert cosine law.

In the last decades images of the full earth, the "blue marble", fig-4, have been taken from space and the uniformity is observed in them [6].


Fig-3: Nasa photo of the full moon [5]     Fig-4: Nasa photo of the full-earth [6]

Similar uniformity or near uniformity is observed in the backward configuration of sun light scattering for images, fig-5, of all the planets and their moons [7].

Fig-5: Full images of the planets [7].

On earth, Fig-6 shows a smooth glass cup filled with milk, Fig-7 shows an egg with fine rough surface, and Fig-8 shows a tennis ball with a coarse rough surface. More photos are in [8]. The photos were taken with a flash in a dark room with a dark background. The photos look similar. There are no images, neither celestial, nor terrestrial, that obey Lambert's cosine law. The only images that do obey the law, fig-2, are rendered, images that are at least partly simulated.


Fig-6: A glass cup with milk        Fig-7: An egg                    Fig-8: A tennis ball

The moon uniformity is discussed in the literature in terms of surface properties; roughness, shading, or retro-reflection [9], [10]. There are no discussions at all of other planet images. The planets and moons surfaces differ significantly in types, structure and morphology, Never the less, all their images are nearly uniform. It is unlikely that the uniformity is an outcome of some specific surface properties, but rather, of a more general and fundamental principle. In particular, the earth full image, fig-5, the "blue marble" [6], contains vast areas of gas phase clouds, liquid phase oceans, and solid phase land, and each of them is nearly uniform separately.

The "opposition effect", the enhanced light scattering for light approaching the backward direction, is observed in celestial and terrestrial bodies as well. It is discussed as an independent effect of the uniformity and is attributed to shading and to coherent light scattering [11], [12], [13], [14], [15]. However, it is not mentioned why the scattering is coherent.

In all these discussions of scattering, it is assumed without exception, that the scattering process is diffusive, that is, the light passes many scattering events before it returns to an observer. However, in such a diffusive process the events are independent and there is no way that the scattering can be coherent. The Lambert cosine scattering law becomes very strong and the scattering process must obey it, which is never the case.

In order to have some insight into these light scattering processes it is useful to consider an electromagnetic light wave travelling within a matter. The wave stimulates dipole oscillations, and each such a dipole becomes a source of a wave itself which adds to the overall light radiation. However, if the material is uniform, the radiation coming from all the dipoles interfere and cancel each other. There will be no light scattering except for the forward direction where the effect is refraction. Light scattering is the outcome of material non-uniformity, so that the interference of light from the dipole sources is not fully destructive [16].

A volume of non-uniform material may be divided into sub-volumes of uniform domains. The intensity of scattered light is the sum of the domain intensities. Domain size determines the material's scattering properties. In a single micron size, or less, there is a wide-angle Rayleigh Scattering that tends to be uniform in all directions. This scattering is typical of gases, liquids and solutions, and is the source of the sky's blue color. In domain size of few tens of microns there is a narrow angle Mie Scattering typical of solids. In both cases the scattering intensity is proportional to the intensity of the stimulating light. The icy rings of Saturn glow in the opposition configuration, fig-9, while the glow from the gaseous surface of the star is by far weaker [17].

Fig-9: Saturn rings glow during opposition [17]

There is a fundamental difference between single event and multiple event scattering. In single event scattering the light is scattered one time before it reaches the observer. Therefore, the dipoles which are stimulated by a single source wave, will all oscillate coherently in one plane perpendicular to the original wave direction. The light scattered by each dipole is maximal in a direction perpendicular to this plane, [16], that is, back to the light source. Therefore, the back scattered light by all of them is also maximal in the back direction to the light source.

In multiple events scattering the light is scattered many times before it reaches the observer. Therefore, any dipole oscillates in a random plane in space and there can't be any correlation or coherence between the radiations of different dipoles. The equivalent radiation will be perpendicular to the surface plane of the scattering material. In this case Lambert's cosine law of light scattering must be obeyed.

Consider a line between a light source and a point on the surface of a scattering sphere defined by the angle θ.

A unit cross section area "a" is perpendicular to this line (fig.-1). The area on the sphere surface observed through the cross-section area "a" will be a / Cos(θ), thus, it is equal to a at the sphere center, and it increases toward the periphery. Similarly, the light density on the sphere is proportional to a * Cos(θ) and it will dwindle to zero at the periphery.

In a single event scattering the scattering intensity back to the light source, Imaxs, is equal to the maximal scattering intensity a:

Imaxs = a * Cos (θ) * 1 / Cos(θ)            (1)

Thus, in the back scattering of a single event the intensity is independent of the surface angle to the source, and a sphere will appear uniform.

In a multiple event scattering the maximal scattering intensity, Imaxm, is perpendicular to the scattering surface, and its component back to the light source is proportional to Cos(θ):

Imaxm = a * Cos(θ) * Cos(θ) * 1 / Cos(θ)           (2)

Thus, in multiple events scattering the intensity will follow Lambert's Cosine scattering law, and the maximum intensity at the sphere center will dwindle by the Cosine function to zero at its periphery.

In a mixture of single and multiple event scattering there will be a sum of constant, Imaxs , and variable, Imaxm , parts.

Why does single event scattering seem to be dominant? In opaque substances the mean free path within the material is too short for many scattering events before the light is absorbed. But also in transparent materials, the probability, that light will return back to the source, will fall with the number of scattering events. A single scattering event seems to have the highest probability.

Summary and conclusions
Back light scattering by a single event provides an account for the nearly uniform full images of all the planets and their moons. There are no objects that comply with Lambert's Cosine scattering law. Single event scattering is automatically coherent since a single electromagnetic source wave stimulates all the scattering dipoles. Thus, the coherence provides an account for the "Opposition Effect", enhancement of 180 degrees back scattering, by constructive light interference. Opposition enhancement has been considered so far, a separate effect from image uniformity.

The coherence of single event scattering may enable separation from non-coherent scattering noise by interferometric methods, so that objects would be observed with better resolution or at a higher distance, for example, in asteroids. Removal of incoherent noise may enable deeper image observation within a scattering medium, such as biological tissues.

Most of the imagery that surround us consists of mainly single event scattering. Illumination engineering may improve by including single scattering in illumination models.


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On the net: December 2020. Revised September 2022

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by the author:
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Scattering of Directional Light