4.4 Application of Spectrum of Dynamic Speckles Intensity …
339
But analytical solution of (4.63) is not possible because of difficulties with calculations. Therefore, the assigned task was solved through numerical simulations on
PC and presenting graphically dependences of ratio of amplitudes of the first two
harmonics b 2 /b 1 on the amplitude of oscillations of the object h for different parameters determining conditions of illumination of the diffuser and observation of the
speckle-field, which are the width and the length of the waist of a focusing lens,
wavelength of laser radiation λ and oscillation frequency of the diffuser ω [112, 113].
Fourier transformation b n can be presented in the form [107]:
|b n | =
b n b ∗
n .
(4.64)
Having calculated the module |b n |, the amplitude value of n-th harmonic can be
obtained
b n =
ω
2π
2
⎡
⎢
⎢
⎣
⎛
⎜
⎝
2π
ω
¨
0
ω sin[ω(t 1 + t 2 )]dt 1 dt 2
A 2 [sin(ωt 1 ) − sin(ωt 2 )]
2
+ 1
⎞
⎟
⎠
2
+
⎛
⎜
⎝
2π
ω
¨
0
sin[nω(t 1 + t 2 )]dt 1 dt 2
A 2 [sin(ωt 1 ) − sin(ωt 2 )]
2
+ 1
⎞
⎟
⎠
2
⎤
⎥
⎥
⎦
1
2
(4.65)
where A = h/2a.
The calculations were performed for wavelength λ = 0.6328 μm and oscillation
frequency ω = 50 Hz, as well as for values of waist width of the focusing lens W 0,
which lie in the range from 10 μm to 10 mm, as further increase is unreasonable for
these selected values W 0 because in this case b 2 /b 1 → 1.
The results of numeric simulations of dependence of amplitude harmonics b 2 /b 1
on alteration of waist width of lens W 0 for different oscillation amplitudes of diffuser
are presented in Fig. 4.13. The analysis of the obtained data has showed that increase
W 0 of the focusing lens W 0 leads to narrowing of spectrum of intensity fluctuations
of the speckle-field and decrease of the ratio of spectral harmonic amplitudes b 2 /b 1
for this value of diffuser oscillation amplitude. Ratio b 2 /b 1 changes more slowly at
the increase of oscillation amplitude if the value of waist W 0 of the focusing lens is
larger.
Decorrelation of speckles is determined by the change of wave phases from each
elementary diffuser and will be changing the quicker, the narrower the beam waist
is. Decrease of waist width leads to the increase of ratio b 2 /b 1 for the given value of
amplitude of diffuser oscillations h.
Thus, the conducted calculations gave the possibility to develop a method for
determining oscillation amplitude of objects through changing the ratio of spectral
harmonics amplitudes b 2 /b 1 .
339
But analytical solution of (4.63) is not possible because of difficulties with calculations. Therefore, the assigned task was solved through numerical simulations on
PC and presenting graphically dependences of ratio of amplitudes of the first two
harmonics b 2 /b 1 on the amplitude of oscillations of the object h for different parameters determining conditions of illumination of the diffuser and observation of the
speckle-field, which are the width and the length of the waist of a focusing lens,
wavelength of laser radiation λ and oscillation frequency of the diffuser ω [112, 113].
Fourier transformation b n can be presented in the form [107]:
|b n | =
b n b ∗
n .
(4.64)
Having calculated the module |b n |, the amplitude value of n-th harmonic can be
obtained
b n =
ω
2π
2
⎡
⎢
⎢
⎣
⎛
⎜
⎝
2π
ω
¨
0
ω sin[ω(t 1 + t 2 )]dt 1 dt 2
A 2 [sin(ωt 1 ) − sin(ωt 2 )]
2
+ 1
⎞
⎟
⎠
2
+
⎛
⎜
⎝
2π
ω
¨
0
sin[nω(t 1 + t 2 )]dt 1 dt 2
A 2 [sin(ωt 1 ) − sin(ωt 2 )]
2
+ 1
⎞
⎟
⎠
2
⎤
⎥
⎥
⎦
1
2
(4.65)
where A = h/2a.
The calculations were performed for wavelength λ = 0.6328 μm and oscillation
frequency ω = 50 Hz, as well as for values of waist width of the focusing lens W 0,
which lie in the range from 10 μm to 10 mm, as further increase is unreasonable for
these selected values W 0 because in this case b 2 /b 1 → 1.
The results of numeric simulations of dependence of amplitude harmonics b 2 /b 1
on alteration of waist width of lens W 0 for different oscillation amplitudes of diffuser
are presented in Fig. 4.13. The analysis of the obtained data has showed that increase
W 0 of the focusing lens W 0 leads to narrowing of spectrum of intensity fluctuations
of the speckle-field and decrease of the ratio of spectral harmonic amplitudes b 2 /b 1
for this value of diffuser oscillation amplitude. Ratio b 2 /b 1 changes more slowly at
the increase of oscillation amplitude if the value of waist W 0 of the focusing lens is
larger.
Decorrelation of speckles is determined by the change of wave phases from each
elementary diffuser and will be changing the quicker, the narrower the beam waist
is. Decrease of waist width leads to the increase of ratio b 2 /b 1 for the given value of
amplitude of diffuser oscillations h.
Thus, the conducted calculations gave the possibility to develop a method for
determining oscillation amplitude of objects through changing the ratio of spectral
harmonics amplitudes b 2 /b 1 .