7.3 Reflection of a Plane Wave from a Layer with the Fibrillar Structure
139
Let us briefly consider the problem of reflection of a Gaussian beam with an arbitrary cross section. This problem can be solved by expansion of counter propagating
waves in terms of plane waves in the region of medium 1, their reflection by layer 2,
and reverse transformation with a subsequent Huygens-Fresnel integral transformation to obtain the field in the initial section (see Chap. 4). The laser radiation intensity
is defined by the form (4.69) of Chap. 4.
Substituting the expression (4.67) in (4.69) for the condition that the simulated
layer is fibrillar structure we obtain the dependence of the laser radiation intensity
for electrical parameters of the modeled biological structure.
7.4 Numerical Calculations for a Model Medium
and Conclusions
Let us consider a model medium with the following characteristics: refractive
indices of the layers are equal to n
◦
2 = 1.50, n
◦
3 = 1.40, n
◦
4 = 1.35, n
◦
5 = 1.40, the
characteristic thicknesses of the layers amount to d 2 = 65 · 10
−6 , d 3 = 565 · 10
−6 ,
d 4 = 90 · 10
−6 , n
◦
1 = 1, χ 1 = 0, χ 2 = χ 3 = χ 4 = χ 5 = 10
−5 and the following values of parametersa 1 = −0.0024, b 1 = 0.020, a 2 = 0.021, b 2 = 0.030, a 3 = 0.041,
b 3 = 0.051, c 1 = c 2 = c 3 = 10
−2 .
The values of parameters a1, b1, a2, b2, a3, b3, c1, c2, and c3 are chosen for
the interface of each layer so that the surface shape are as close as possible to the
interface shape of the corresponding layer in the structure of human skin, and the
wavelength is λ=0.63 µm (center of the line of a He–Ne laser).
The calculations were performed for monolayer particle spherulated modeling
red blood cells, while the number of particles in the simulated layer is assumed to
be ten, with the following parameters: the relative refractive index for the first five
spherulated erythrocytes was assumed to be 1.035 + 10
−5 i, for others it was set as
1.033 + 10
−5 i for a particle radius of 4.3 µm, number of cylinders modeling collagen
fibers in the layer was assumed to be nine, the radius cylinder was assumed to be
1 · 10
−10 , 1 · 10
−6 , 2 · 10
−6 , 1 · 10
−6 , 1 · 10
−6 , 3 · 10
−6 , 1 · 10
−6 , 1 · 10
−6 , 2 · 10
−6 .
Note that in the numerical calculations we was considered the normal incidence of the
electromagnetic wave and the reflected field was regarded in the main approximation.
Figure 7.2 shows the cross-section of the scattering field in the case of multiple
scattering on the group, closely spaced cylinders, different radii in the far-field.
To consider scattering in the far field in expression (7.29), we replace of the Hankel
function by its asymptotic representation for kl 1:
H
2
n (kl) ∼
2
π kl
e
−i(kl−(2n+1)π/4)
, l = R cos θ n 0 , γ j p ∼ γ, ∀ j.
139
Let us briefly consider the problem of reflection of a Gaussian beam with an arbitrary cross section. This problem can be solved by expansion of counter propagating
waves in terms of plane waves in the region of medium 1, their reflection by layer 2,
and reverse transformation with a subsequent Huygens-Fresnel integral transformation to obtain the field in the initial section (see Chap. 4). The laser radiation intensity
is defined by the form (4.69) of Chap. 4.
Substituting the expression (4.67) in (4.69) for the condition that the simulated
layer is fibrillar structure we obtain the dependence of the laser radiation intensity
for electrical parameters of the modeled biological structure.
7.4 Numerical Calculations for a Model Medium
and Conclusions
Let us consider a model medium with the following characteristics: refractive
indices of the layers are equal to n
◦
2 = 1.50, n
◦
3 = 1.40, n
◦
4 = 1.35, n
◦
5 = 1.40, the
characteristic thicknesses of the layers amount to d 2 = 65 · 10
−6 , d 3 = 565 · 10
−6 ,
d 4 = 90 · 10
−6 , n
◦
1 = 1, χ 1 = 0, χ 2 = χ 3 = χ 4 = χ 5 = 10
−5 and the following values of parametersa 1 = −0.0024, b 1 = 0.020, a 2 = 0.021, b 2 = 0.030, a 3 = 0.041,
b 3 = 0.051, c 1 = c 2 = c 3 = 10
−2 .
The values of parameters a1, b1, a2, b2, a3, b3, c1, c2, and c3 are chosen for
the interface of each layer so that the surface shape are as close as possible to the
interface shape of the corresponding layer in the structure of human skin, and the
wavelength is λ=0.63 µm (center of the line of a He–Ne laser).
The calculations were performed for monolayer particle spherulated modeling
red blood cells, while the number of particles in the simulated layer is assumed to
be ten, with the following parameters: the relative refractive index for the first five
spherulated erythrocytes was assumed to be 1.035 + 10
−5 i, for others it was set as
1.033 + 10
−5 i for a particle radius of 4.3 µm, number of cylinders modeling collagen
fibers in the layer was assumed to be nine, the radius cylinder was assumed to be
1 · 10
−10 , 1 · 10
−6 , 2 · 10
−6 , 1 · 10
−6 , 1 · 10
−6 , 3 · 10
−6 , 1 · 10
−6 , 1 · 10
−6 , 2 · 10
−6 .
Note that in the numerical calculations we was considered the normal incidence of the
electromagnetic wave and the reflected field was regarded in the main approximation.
Figure 7.2 shows the cross-section of the scattering field in the case of multiple
scattering on the group, closely spaced cylinders, different radii in the far-field.
To consider scattering in the far field in expression (7.29), we replace of the Hankel
function by its asymptotic representation for kl 1:
H
2
n (kl) ∼
2
π kl
e
−i(kl−(2n+1)π/4)
, l = R cos θ n 0 , γ j p ∼ γ, ∀ j.