
MagicBall
Scattering of Directional Light
Blue Marble the Uniform Earth Image
particlesinabox
Scattering of Directional Light
Uri Lachish, Rehovot
Orcid:
https://orcid.org/0000000192360254
Abstract
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, nondiffusive 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 backscattering, considered, so far, a separate effect.
Coherent single event scattering may be separated from noncoherent 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.
Keywords
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, fig1. 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.
Fig1: 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
selfevident, 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, fig2.
Fig2: A rendered, simulated, sphere, following Lambert cosine Law [3].
The full moon image, fig3, 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", fig4, have been taken from space
and the uniformity is observed in them [6].
Fig3: Nasa photo of the full moon [5] Fig4: Nasa photo of the fullearth [6]
Similar uniformity or near uniformity is observed in the backward
configuration of sun light scattering for images, fig5, of all the planets and their moons [7].
Fig5: Full images of the planets [7].
On earth, Fig6 shows a smooth glass cup filled with milk, Fig7 shows an egg with fine rough
surface, and Fig8 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, fig2, are rendered, images
that are at least partly simulated.
Fig6: A glass cup with milk
Fig7: An egg
Fig8: A tennis ball
Discussion
The moon uniformity is discussed in the literature in terms of surface
properties; roughness, shading, or retroreflection [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, fig5, 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 nonuniformity, so
that the interference of light from the dipole sources is not fully destructive [16].
A volume of nonuniform material may be divided into subvolumes 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 wideangle 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, fig9, while the glow from the gaseous surface of the star is by far weaker [17].
Fig9: 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 crosssection 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, I_{max}^{s}, is equal
to the maximal scattering intensity a:
I_{max}^{s} = 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, I_{max}^{m}, is
perpendicular to the scattering surface, and its component back to the light source is
proportional to Cos(θ):
I_{max}^{m} = 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, I_{max}^{s} , and
variable, I_{max}^{m} , 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 noncoherent 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.
References
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https://owlcation.com/stem/TrueColorPhotosofAllthePlanets
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On the net: December 2020. Revised September 2022
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