3
Basic conventions
|
Fig. 2 The drawing shows the way in
which the parameters, scattering angle theta, elevation angle phi and
crossing angle alpha are defined. The coordinate system (X, Y, Z) is refered
to as the global coordinate system. |
The drawing in fig. 2 shows the
coordinate system used in the LSA
light scattering program. The scattering angle theta and elevation angle phi,
used in the program, are for a two beams system defined in fig. 2. These angles
are defined in the same way for a one beam system. The two angles theta and phi
are the angles that can be used to describe a direction in a normal spherical
coordinate system. If the crossing angle alpha between the two laser beams is
set to 0°, the beam vectors are parallel to
the global Z-axis, we have the normal definition of the scattering angle and
elevation angle. The coordinate system will then be the same as the coordinate
system normally used to express light scattering theories. Normally the
polarisations of the one or two laser beams are linear. The polarisation
directions for the laser beams are perpendicular to the laser beams and the
polarisation angle for the one or two laser beams is measured relative to the
global coordinate systems Y-axes.
3.1
Light scattering theories
These paragraphs give a short
introduction to the different light scattering theories, that can be performed
with the use of the LSA program. For
a more detailed presentation of the different light scattering theories please
see the reference list in the user manual for the LSA program.
3.2
Geometric optics
|
Fig. 3 The first 3 geometric modes are
showed. The relative index of refraction is 1.334. |
|
Fig. 4 The first 3 geometric modes are
showed. The relative index of refraction is 0.7496. |
The simplest, most intuitive and
oldest light scattering theory is based on geometric optics. The light is in
this theory described with the use of rays. Geometric optics can only be used,
when the particle diameter is must larger than the wavelength of the incoming
light. On the boundaries between the particle and the surrounding medium the
energy in the light is divided between reflected and refracted rays. See the
figures.
The two figures show the traces of
18 rays through a spherical particle with a relative index of refraction of
1.334 and 0.7496 corresponding to water in air and an air bubble in water. The
figures show the first 3 geometric modes corresponding to reflection,
refraction and 2 order refraction. The two figures can be made in the programs
raytracing section.
3.3
Lorenz-Mie theory and Generalizered Lorenz-Mie theory
In the Lorenz-Mie theory it is
assumed that the light scattering particle is placed in a plane wave. The Lorenz-Mie theory gives a correct
description of the scattered light from a particle in a plane wave. Geometric
optics can as we have seen before only be used for particle diameters bigger
than the wavelength and can not be used for certain angles fore example at the
rainbow angles. It can be showed, see van de Hulst [4], that the sum of the
light scattered in all the geometric modes added to the diffracted light will
approach the result from Lorenz-Mie theory for big particles. The Lorenz-Mie
theory (LMT) was proposed independently by the two physicists G. Mie in 1908
and the Dane Lorenz in 1890. Lorenz did not use the Maxwells equations in his
work. He used his own set of equations for the electromagnetic theory. Lorenz
version, of the theory for light scattered from a particle in a plane wave,
gives equations that are very close to the equations that are commonly used
today. Lorenz is also well known for his retarded potentials. The retarded
potentials can be calculated directly from the sources of the fields, and the
electric and magnetic fields E and H can hereafter be calculated directly from the
retarded potentials.
Different research groups have in
resent years generalized the classic Lorenz-Mie theory to include non-plane
waves. Of particular interest is the theory for the light scattered from a
spherical particle placed in a Gaussian beam. The generalized Lorenz-Mie theory
calculations in the LSA program is
based on the Generalized Lorenz-Mie theory (GLMT) and algorithms developed at
the Laboratoire d`Energètique des systèmes et procédés INSA in Rouen in France.
|
Fig. 5 The drawing shows, how the
incoming wave is interpretated in Lorenz-Mie theory and Generalized
Lorenz-Mie theory. |
The main difference between the
Lorenz-Mie theory and the Generalized Lorenz-Mie theory is the introduction of
the g-factors. Different types of incoming waves correspond to different types
of g-factors. The g-factors contain the information on the type of the incoming
wave and the particles position in the wave. The g-factors can be calculated
with the use of 3 different methods. The g-factors in the LSA program are based on the so called locale approximation. The
routines for the g-factors are developed at the Laboratoire d`Energètique des
systèmes et procédés INSA in Rouen in France. The GLMT calculations will in
theory give a complete description of the complex relations in a Phase Doppler
Anemometer. It can be shown that when the locale equiphase fronts in the laser
beam is close to plane wave fronts, the Generalized Lorenz-Mie theory will then
give the same results as the Lorenz-Mie theory. As a general rule of thumb it
can be stated that GLMT and LMT gives the same results, when the diameter of
the laser beam is 10 times bigger than the particle diameter and the particle
is placed in the centre of the laser beam. The figure below shows the
difference between how the incoming wave is interpreted in the Lorenz-Mie
theory and the Generalized Lorenz-Mie theory. The drawings show in the case of
LMT the plane phase fronts and in the GLMT case the Gaussian intensity profile
of the laser beam. The two theories LMT and GLMT will give the same result for
the small particle corresponding to drawing 1 and 2, and a different result for
the big particle corresponding to drawing 3 and 4.
3.4
Diffraction and Rayleigh theory
This paragraph gives a short
description of the scattered light described with the use of diffraction theory
and Rayleigh theory.
The diffracted light from a particle
with a particle diameter d can, in accordance with the princip of Babinet, be
calculated as the diffracted light from a plane wave diffracted from a hole in
a plane with diameter d. The intensity pattern must therefore be the well-known
Airy disk.
Rayleigh theory can be used for
particles that are much smaller than the wavelength of the incoming light. The
theory was first discovered by Lord Rayleigh in 1871. The scattered light from
a Rayleigh scatter is basically the same as the intensity pattern from an
electric dipol. When the particle diameter is must smaller than the wavelength,
then the light scattering result obtained by the use of Lorenz-Mie theory and
Rayleigh theory will be the same. It is the Rayleigh scattering from small
particles, that gives the sky it's blue color. Another example of Rayleigh
scattering is found in optical fibers. Rayleigh scattering is together with
infrared absorption the main sources for the attenuation of light in optical
fibers in the important wavelength range between 1.3 micron and 1.54 micron
used in optical communication.