46
K. D. Dahm and D. J. Dahm
• θ is the scattering angle. While the intensity of the scattered light is a function of
angle, the consequence of the cos
2 functionality is that the scattering pattern is
symmetrical, and essentially the same amount of light gets scattered “forward”
as “backward,” relative to the incident beam.
• λ is the wavelength, which is raised to the −4 power. This means that scatter is
essentially zero at very high wavelengths.
• η is the refractive index. It is possible to incorporate absorption into the calculation
by expressing the refractive index as a complex number, with the real component
representing scatter and the imaginary component representing absorption.
For larger particles, modeling the particle as a single scattering center becomes
unrealistic-one must distinguish between scatter from different locations on the
particle. Given enough time and computing power, for a specific particle size and
shape, it is theoretically possible to solve wave equations for scatter emanating from
every point on the particle, sum these, and quantify the intensity of light emanating
from the particle in any direction. In practice, the shape that has been studied the
most is the sphere. Building a theoretical model of a sphere is simplified by the fact
that a sphere presents the same dimensions (e.g., cross-sectional area, depth) to the
beam regardless of the orientation of the beam. Mie scattering theory can be applied
to spheres of any size, but is especially useful when the wavelength of the light and
the size of the particle are of comparable magnitude, or when the particle is larger
than the wavelength.
While Mie computations are complex, some outcomes will here be discussed
qualitatively. Mie predicts vanishingly small scatter for very small particles and/or
very large wavelengths, the latter result being consistent with the (1/λ
4 ) functionality
of the Rayleigh equation. The intensity of scatter increases as particle size becomes
larger relative to wavelength and reaches a maximum when the circumference of
the sphere is equal to the wavelength, as shown in Fig. 3.6. Scattering intensity
oscillates as the ratio of circumference to wavelength increases further. Another
result predicted by the Mie equations is that scatter from spheres is not isotropic.
According to Mie scattering theory, the majority of the scattered light continues in a
generally “forward” direction, as illustrated in Fig. 3.7. In a limiting case, the largest
spheres will approximate the scattering pattern of a planar surface. A crucial point
is that Mie’s equations were derived specifically for spheres and cannot be applied
to other shapes. Bass et al. have noted frequent misuse of the Mie theory, stating “in
defiance of logic and history every particle under the sun has been dubbed a ‘Mie
scatterer,’ and Mie scattering has been promoted from a particular theory of limited
applicability to the unearned rank of general scattering process… Using Mie theory
for particles other than spheres is risky, especially for computing scattering toward
the backward direction.” [5].
K. D. Dahm and D. J. Dahm
• θ is the scattering angle. While the intensity of the scattered light is a function of
angle, the consequence of the cos
2 functionality is that the scattering pattern is
symmetrical, and essentially the same amount of light gets scattered “forward”
as “backward,” relative to the incident beam.
• λ is the wavelength, which is raised to the −4 power. This means that scatter is
essentially zero at very high wavelengths.
• η is the refractive index. It is possible to incorporate absorption into the calculation
by expressing the refractive index as a complex number, with the real component
representing scatter and the imaginary component representing absorption.
For larger particles, modeling the particle as a single scattering center becomes
unrealistic-one must distinguish between scatter from different locations on the
particle. Given enough time and computing power, for a specific particle size and
shape, it is theoretically possible to solve wave equations for scatter emanating from
every point on the particle, sum these, and quantify the intensity of light emanating
from the particle in any direction. In practice, the shape that has been studied the
most is the sphere. Building a theoretical model of a sphere is simplified by the fact
that a sphere presents the same dimensions (e.g., cross-sectional area, depth) to the
beam regardless of the orientation of the beam. Mie scattering theory can be applied
to spheres of any size, but is especially useful when the wavelength of the light and
the size of the particle are of comparable magnitude, or when the particle is larger
than the wavelength.
While Mie computations are complex, some outcomes will here be discussed
qualitatively. Mie predicts vanishingly small scatter for very small particles and/or
very large wavelengths, the latter result being consistent with the (1/λ
4 ) functionality
of the Rayleigh equation. The intensity of scatter increases as particle size becomes
larger relative to wavelength and reaches a maximum when the circumference of
the sphere is equal to the wavelength, as shown in Fig. 3.6. Scattering intensity
oscillates as the ratio of circumference to wavelength increases further. Another
result predicted by the Mie equations is that scatter from spheres is not isotropic.
According to Mie scattering theory, the majority of the scattered light continues in a
generally “forward” direction, as illustrated in Fig. 3.7. In a limiting case, the largest
spheres will approximate the scattering pattern of a planar surface. A crucial point
is that Mie’s equations were derived specifically for spheres and cannot be applied
to other shapes. Bass et al. have noted frequent misuse of the Mie theory, stating “in
defiance of logic and history every particle under the sun has been dubbed a ‘Mie
scatterer,’ and Mie scattering has been promoted from a particular theory of limited
applicability to the unearned rank of general scattering process… Using Mie theory
for particles other than spheres is risky, especially for computing scattering toward
the backward direction.” [5].