This result agrees with the simulation shown in figure3.4. Therefore, MIR fails to make the clear image in the strong fluctuations.
3.4 Limit of phase error
0 0.5 1 1.5 2 2.5 3
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
ch3 ch2
ch1
Amp(a.u)
t(s) (a)
0 0.5 1 1.5 2 2.5 3
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 ch3 ch2 ch1
Amp(a.u.)
t(s) (b)
Figure 3.11: The wave propagation in the (a) horizontal and (b) vertical directions by a rotation reflector. The propagation of the wave is indicated by the arrows.
3.4 Limit of phase error
I(a.u.) I(a.u.)
Q(a.u.)Q(a.u.)
Figure 3.12: The wave propagation in the vertical direction by a rotation reflector in the case of strong fluctuation (d = 0.5 cm, k = 6 cm−1). (a) the phase, (b) amplitude and (c) Lissajous’ plots
3.4 Limit of phase error
constant. It can be used to calibrate the optics system and explain the experimental signals. In the present imaging system, the incident beam is perpendicular to the cutoff surface if the optical system is well designed and arranged. Therefore, the refractive effect is not so serious as expected in the plasma. The approximation in this work might be used for the O-mode plasma. However, it is not true under the most conditions. The full wave equation simulation should be used for the actual plasma test because of the strong diffractive effect. Several similar works, based on synthetic imaging technique and finite-difference time-domain (FDTD) methods, have been carried out [12–14]. Nevertheless, the phase distortion is still a crucial problem in MIR experiments.
2S/kĵ
Cutoff surface
x z
D D n
L
D/2
Lens d
Figure 3.13: Schematic illustration of the beam diffraction
The diffraction effect causes the obscure image with low brightness (not zero) and with mismatch of the wavefront and the fluctuation. It is mainly decided by the size of the aperture optics, the displacement in radial direction and perpendicular wavenumber.
The phase distortion can be estimated from the optical arrangement in MIR system.
Figure3.13shows the schematic illustration of the beam diffraction in the MIR system.
3.4 Limit of phase error
Where, n is a surface normal vector and α1 is the cutting angle of the cutoff surface, Dis the diameter of the optical lens, Lis the distance between the optical lens and the reflector surface, k⊥ is the perpendicular wavenumber, d is the displacement in the x direction (fluctuation amplitude). The parallel launching beam is deflected by the angle 2α1 with the reflector surface. The phase error increases when the lens can’t collect the main reflected beam. If we assume the modulated wave is sinusoidal and L ≫ D, the relationship betweend and k is given as
4k⊥dL
D <1 (3.7)
for the in-focus imaging. The error of the detected phase becomes significant when the radial displacement of the cutoff surface is larger than D/(4k⊥L). As shown in figure 3.9(b), the perpendicular wavenumberk⊥ of the modulated wave is estimated about 0.2 cm−1. Therefore, the radial displacement should be smaller than 0.16 cm for the phase fluctuation without distortion. In the case of small fluctuation, the radial displacement is about 0.1 cm, so the circular IQ plots are obtained. In the case of medial and large fluctuations, the radial displacement are 0.2 cm and 0.3 cm, respectively. So the IQ plots are distorted.
0 0.2 0.4 0.6 0.8 1
0 0.5 1 1.5 2 2.5
G I S
4kdL/D
kd>D/4L kd<D/4L
Figure 3.14: The error of the phase fluctuation as a function of 4k⊥dL/D
Figure3.14 shows the error of the phase fluctuation as a function of 4k⊥dL/D. The error of the measured phase is smaller than 0.1π in the case of 4k⊥dL/D < 1. This
3.4 Limit of phase error
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0 0.5 1 1.5 2 2.5
distribution
4kdL/D
TPE-RX, L=1m, D=0.4m
Figure 3.15: The distribution of the fluctuation measured by MIR in TPE-RX
suggests that a clear image of the cutoff surface may be made by MIR. The phase error is increased as 4k⊥dL/D is increased in the case of 4k⊥dL/D > 1. Therefore, MIR can’t make a clear image of the strong fluctuations withk⊥d > D/4Ldue to the strong diffraction effect of the reflected wave.
The equation 3.7 clarifies the relation between the optical parameters of MIR and the fluctuation parameters of the cutoff surface. It can be used to estimate the sensi-tivity of the MIR optical system to the turbulence. It can be also used to estimate the optical parameters if we know the fluctuation parameters in plasma. The perpendicular wavenumber and the radial displacement of the fluctuations are coupled. In general, high k⊥ fluctuations have small radial displacement in the experiment. The measured k⊥d is determined by the geometrical parameter of the optical system. In the case of TPE-RX, the distance between the plasma and the main mirror is about 100 cm and the diameter of the main mirror is about 40 cm. Therefore, MIR can measure the fluctuation with k⊥d <0.1 in TPE-RX plasma.
Figure 3.15 shows the distribution of the fluctuations as a function of 4k⊥dL/D in TPE-RX. The distribution decreases as 4k⊥dL/D increases. The tail of the distribution in the highk⊥drange may be caused by the strong highkfluctuation or the intermittent burst of the turbulence (see chapter 6), which is often observed in the reversed-field pinch
3.4 Limit of phase error
plasma [41]. Nevertheless, the fluctuations mainly distribute in the range of 4k⊥dL/D <
0.8 which suggests present MIR optical systems in TPE-RX can make a clear image of the cutoff surface in plasma.
3.4 Limit of phase error
Chapter 4
Development of the Spectral Analysis Techniques
4.1 Introduction
The analysis method is important in the experimental study of turbulence. The real experimental signals are often submerged in the strong background noises such as elec-tronic noise and thermal noise, especially when the signal is very weak. The turbulence has a large number of modes and different ranges of correlations. It is similar to a se-ries of wave packets, which contains many different scales of fluctuations. On the other hand, the turbulence is transient and it always rapidly changes in spatial and temporal domains. The spectrum of the turbulence is broad due to many active modes in the wave packets. The transient turbulent structures may cause the distortion of the spectrum.
Sometimes the turbulence is similar to the random noise. Therefore, it is hard to see something from the signal even in the frequency domain. Proper selection of analysis methods can give the direct evidence of the underlying physics of turbulence. On the contrary, miss selection of the analysis methods may lead fake results.
Many numerical noise reduction techniques have been developed in previous studies [61–63]. These techniques use the statistical feature of the random noises, whose power spectral density is similar in the whole frequency band. The expected error of the averaging in Fourier space (or real space and time) decreases monotonically as a function