396
5 Laser Specklometer, Speckle-Optical Diagnostics and Laser …
S(u, v) =
2
u
1 −
1
2
v
2
2u
2
+ · · ·
−
cos
1
2
u
1
2
u
J o (v) −
v
u
2
J 2 (v) + · · ·
−
sin
1
2
u
1
2
u
v
u
J 1 (v) −
v
u
3
J 3 (v) + · · ·
=
2
u
cos
v
2
2u
−
cos
1
2
u
1
2
u
V 0 (u, v) −
sin
1
2
u
1
2
u
V 1 (u, v),
(5.18)
where the Lommel functions V n (u, v) are
V n (u, v) =
∞
S=0
(−1)
S
v
u
n+2S
J n+2S (v).
The well-known ratio ∂/∂ x
x
−n J n (x)
= −x
−n J n+1 (x) for Bessel functions was
used while deriving (5.17) and (5.18). Substitution of (5.14)–(5.15) and (5.17)–(5.18)
into (5.10) gives the desired expression for the distribution of the amplitude in the
observation point, and the intensity in observation point I = |U(P)|
2 is determined
in the following way:
I (P) =
2
u
2
U
2
1 (u, v) + U
2
2 (u, v)
I 0 ,
u
v
< 1
;
(5.19)
I (P) =
2
u
2
1 + V
2
0 (u, v) + V
2
1 (u, v) − 2V 0 (u, v) cos
1
2
u +
v
2
u
−2V 1 (u, v) sin
1
2
u +
v
2
u
I 0 (|u/v| > 1)
(5.20)
If v = 0, then V 0 (u, 0) = 1, and if V 1 (u, 0) = 0, U 1 (u, 0) = U 2 (u, 0) = 0 then
I (P) ≡ I (u, 0) =
4
u 2 {2 − 2 cos[(1/2)u]}I 0 ≡
sin(u/4)
u/4
2
I 0 ,
(5.21)
where I 0 =
π S
2 A
λd 2
2
; A is the amplitude of the radiation incident on the skin surface.
In order amplitude characteristics of the registered signal correctly reflected alterations of oscillation amplitude of the surface under study, it is necessary that the
value of the amplitudes was within the limits of linear area of the distribution curve
I(P), which is reached at z = 5 μm (the wavelength of the incident radiation λ =
633 nm, r = 4, r = S/d, where d is the distance from the illuminated surface to the
receiver; S is the size of the illuminated spot of the skin surface).
The analysis of plots of muscle oscillation amplitudes shows, for example, in the
contraction mode that the following condition A < z is reached for loads up to
12 kg and time of registration up to 10 s. These dependencies of muscle oscillation
5 Laser Specklometer, Speckle-Optical Diagnostics and Laser …
S(u, v) =
2
u
1 −
1
2
v
2
2u
2
+ · · ·
−
cos
1
2
u
1
2
u
J o (v) −
v
u
2
J 2 (v) + · · ·
−
sin
1
2
u
1
2
u
v
u
J 1 (v) −
v
u
3
J 3 (v) + · · ·
=
2
u
cos
v
2
2u
−
cos
1
2
u
1
2
u
V 0 (u, v) −
sin
1
2
u
1
2
u
V 1 (u, v),
(5.18)
where the Lommel functions V n (u, v) are
V n (u, v) =
∞
S=0
(−1)
S
v
u
n+2S
J n+2S (v).
The well-known ratio ∂/∂ x
x
−n J n (x)
= −x
−n J n+1 (x) for Bessel functions was
used while deriving (5.17) and (5.18). Substitution of (5.14)–(5.15) and (5.17)–(5.18)
into (5.10) gives the desired expression for the distribution of the amplitude in the
observation point, and the intensity in observation point I = |U(P)|
2 is determined
in the following way:
I (P) =
2
u
2
U
2
1 (u, v) + U
2
2 (u, v)
I 0 ,
u
v
< 1
;
(5.19)
I (P) =
2
u
2
1 + V
2
0 (u, v) + V
2
1 (u, v) − 2V 0 (u, v) cos
1
2
u +
v
2
u
−2V 1 (u, v) sin
1
2
u +
v
2
u
I 0 (|u/v| > 1)
(5.20)
If v = 0, then V 0 (u, 0) = 1, and if V 1 (u, 0) = 0, U 1 (u, 0) = U 2 (u, 0) = 0 then
I (P) ≡ I (u, 0) =
4
u 2 {2 − 2 cos[(1/2)u]}I 0 ≡
sin(u/4)
u/4
2
I 0 ,
(5.21)
where I 0 =
π S
2 A
λd 2
2
; A is the amplitude of the radiation incident on the skin surface.
In order amplitude characteristics of the registered signal correctly reflected alterations of oscillation amplitude of the surface under study, it is necessary that the
value of the amplitudes was within the limits of linear area of the distribution curve
I(P), which is reached at z = 5 μm (the wavelength of the incident radiation λ =
633 nm, r = 4, r = S/d, where d is the distance from the illuminated surface to the
receiver; S is the size of the illuminated spot of the skin surface).
The analysis of plots of muscle oscillation amplitudes shows, for example, in the
contraction mode that the following condition A < z is reached for loads up to
12 kg and time of registration up to 10 s. These dependencies of muscle oscillation