Optical Model of Medium for the Numerical Imitation of Wave Surface
Forming High-Intensity Reflected Radiant Energy
Olga V. Shefer1,a, Vitaliy V. Loskutov1,b, Olga V. Rozhneva1,c
1
National Research Tomsk Polytechnic University, Tomsk, Lenin Av., 30, Russia, 634050
a
[email protected], [email protected], [email protected]
Keywords: anomalous backscattering, plate crystals, orientation, radiant energy.
Abstract. An optical model of medium employing a system of plane particles is presented to
study numerically the high-intensity reflected radiant energy. The characteristics of anomalous
backscattering of optical radiation are received by the method of physical optics. The calculation
results for spectral dependence of backscattering coefficient on microphysical and optical
properties of particles are displayed in this work. The informative property of characteristics of
specularly reflected radiation was determined to learn the nature of particle matter and their
microphysical parameters.
Introduction
During the observation of atmospheric crystal formations and water surface, a bright reflected
light can often be noticed. A number of optical effects occurring in nature, such as sunlight
columns, false sun, hotspots or bright flares at water surface, can be explained by the interaction
of sunlight (or moonlight) with the flat surface of water or crystals [1–3]. However, researchers
often encounter difficulties with feedback signal during the laser sensing of crystal clouds. The
reflected signal of extreme intensity can cause illumination of photo-detector or even its
complete failure. The high-amplitude lidar returns research is prosperous due to the fact that
reflective property is primarily found in oriented particles with flat faces as well as unperturbed
water surface. Additionally, during the remote laser sensing a high-amplitude signal can occur
due to reflection from water surface which has large radius of curvature. Defining the parameters
of medium component that provide an anomalous backscattering can be used as a tracer to
evaluate the microphysical and macro physical properties of the test object.
The research part
To calculate the anomalous backscattering coefficient we are using the following equation
[4]:
   S (a)  N(a) da ,
(1)
a
where βπ is backscattering coefficient, Sπ is backscattering cross section for a particle, N(a) is
function of the particle size distribution. The following formula displays that backscattering
cross section is proportional to the scattered light intensity.
S   I ,
(2)
where the proportionality factor ω includes all the values that make a lidar equation as
multipliers. Using the optical physics method while considering the incident light polarization,
the derived equation can be used to calculate the backscattering cross section [5]:
S  W  (M11 
I
I2
I
M12  3 M13  4 M14 ) ,
I1
I1
I1
(3)
where Ii (i=1, 2, 3, 4) are Stokes vector parameters for incident light, Mij is Mueller matrix
elements. The W multiplier is determined by wavenumber and angle function that is Fraunhofer
diffraction equation.
Since the particle distribution can be adequately described by a modified gamma-function [6],
we use the following equation:

   a 
1
1  a 
N(a)  C 
     exp 
.
G(  1) am  am 
 am 
(4)
Formula (4) includes the following parameters: С is plate concentration, am is plate radius
corresponding to the N(a) maximum,  is dimensionless parameter, characterizing the steepness
of the function maximum, G (+1) is gamma-function. As a separate scatterer, let’s take a round
plate with а radius and d thickness. Additionally, the dependency between a and d is taken in
account as following [6]:
d=2.020(2a) 0.449.
With the normal plate crystal position relative to ray direction (β=0°) and considering (4), we
simplify (1) as following:
2
 (  0)  (n  1) (n  1) k 2   N(a) a 4 da ,
(5)
a
where k is wavenumber (k=2π/λ, λ is wavelength of incident radiation), n  n  i   is a complex
refraction index (n is refraction index, χ is absorption index) of plate crystal.
To analyze the experimental data, the mean plate radius a is used. This parameter can be
determined from the following formula:


0
0
a  N (a)da   a  N (a )da .
(6)
Then, substituting expression (4) for N(a) into Eq. (6), the parameter a can be determined as
 1
a  am 1   .
 
(7)
We have done a numerical research of anomalous backscattering coefficient for the light
reflected from the system of plate crystals using (5). For the calculations, the following input
parameters were used as particle geometrical dimensions (a, d), complex refraction index values
(n, χ), parameters for particle size distribution (C, μ, a ), wavelength (λ). A non-polarized
incident radiation has been used to calculate the results.
The Fig. 1 displays the calculation results for anomalous backscattering coefficient depending
on incident radiation wavelength at the different values of refraction index. It is clear that
material optical properties have significant influence on the reflected light intensity. Changing
the real part of refraction index by a fraction of a unit leads to increase in β π by an order of
magnitude.
Fig. 1. Dependence of the backscattering coefficient on the wavelength (βπ(λ)) at a =100 μm, C=1 l-1, μ=5, χ=10-4
for different n: n=1.21 (curve 1); n=1.31 (curve 2); n=1.71 (curve 3); n=1.91 (curve 4).
Fig. 2 displays spectral dependence of βπ(λ) for ice material, which was calculated based
upon n = n(λ) data [7] with different microphysical medium parameters as a and μ. At Fig. 2 it
is shown that βπ(λ) curves have features similar to n(λ). Particle size distribution parameters have
a significant influence on the βπ. Clearly, an increase in the dimensional parameter ( a /λ) leads to
increase in the reflected light amplitude. Despite the unified change tendencies of βπ(λ) change
with the varying distribution parameters ( a , С, μ), they have different influence during the data
analysis with fixed wavelength.
Fig. 2. Dependence of the backscattering coefficient on the wavelength (βπ(λ)) at C=1 l-1, n = n(λ) for ice [7] for
different a and μ: a =100 μm, μ=5 (curve 1); a =200 μm, μ=5 (curve 2); a =100 μm, μ=1 (curve 3); a =100 μm,
μ=10 (curve 4).
The Fig. 3 displays the results of anomalous backscattering coefficient calculation depending
on average particle size with different μ parameter values. The curves of Fig. 3 represent the
exponential dependence βπ( a ). At small values of the parameter μ, i.e., for a substantial
dispersion of particle sizes, we can observe higher values of anomalous backscattering
coefficient.
Fig. 3. Dependence of the backscattering coefficient on average plate radius (βπ( a )) at n=1.4, χ=10-4, C=1 l-1,
λ=0.694 μm for different μ: μ=1 (curve 1); μ=5 (curve 2); μ=10 (curve 3); μ=20 (curve 4).
Currently, the experiment employs only relatively low βπ values corresponding to small-sized
plates with small concentration in the scattering volume. For this experiment in particular, for
λ=0.694 μm, С=0.8 l-1 and a =37 μm the calculated anomalously high values of backscattering
coefficient reached 17 km-1 [1, 2]. Comparing the experimental βπ values and corresponding
numerical βπ values that take in account the concentration, linearly included in (4), we can easily
verify their quality and quantitative agreement.
Conclusions
An uncertainty during the anomalous backscattering evaluation can be eliminated by taking a
priori information in account as well as considering the normalized characteristics. It should be
noted that researching the reflected light properties while considering a possibility of small-angle
scanning and using bistatic sensing scheme can allow to receive informative data for a highprobability calculation of optical, microphysical and dynamic medium properties despite having
an stable or wavy surface. Our general research direction is related to development of ways to
use optical methods to determine the physical parameters of the medium. Currently, the
following task becomes relevant for the ranged surveillance of water surface layer and
atmosphere crystal formations to find solutions for ecological problems, to study the light
transfer through layered media containing primarily oriented plate particles.
This work was supported in part by the Ministry of Education and Science of the Russian
Federation (State Assignment for 2014–2016, R&D work No. 645)
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