PROCESO DE INFECCIÓN

In this section we describe an automatic polarization recognition algorithm that can directly identify the existence of circular polarization in the time-frequency domain.

Consider a general single frequency signal in 3D

B(t) = ˆξB0cosωt+ ˆηB1sinωt, (4.44)

where ˆξ and ˆη are two unit vectors in 3D space and are perpendicular to each other. With

B0 = 0 orB1= 0, the equation can describe a linear polarized wave. With non-zeroB0 and

B1, the equation describes an elliptically polarized wave. WithB0 =B1 6= 0, the equation describes a circularly polarized wave.

The time derivative of B is

˙

B=−ωξBˆ 0sinωt+ωηBˆ 1cosωt. (4.45)

The cross product and inner product ofB and B˙ are

B×B˙ = ω( ˆξ×ηˆ)B0B1, (4.46)

B·B˙ = ω(B₁2−B₀2) sinωtcosωt= (ω/2)(B₁2−B₀2) sin 2ωt. (4.47)

The time average of |B·B˙|within any π/ω period is

<|B·B˙|>π/ω= (ω/2)|B02−B12| Rπ/4 0 cosθdθ π/4 = (2ω/π)|B 2 0−B12|, (4.48)

For linearly polarized signal, B0 = 0 or B1 = 0. Without loss of generality, assume

B1 = 0 and B0 6= 0. There are

For circularly polarized signal, B0 =B1. There are

B×B˙ =ωkBˆ ₀2 <|B·B˙|>π/ω= 0, (4.50)

where ˆk≡ξˆ×ηˆis the unit wavevector perpendicular to both ˆξ and ˆη.

Therefore|B×B˙|is proportional to the amplitude of the circularly polarized component of a 3D signal and<|B·B˙|>π/ω is proportional to the amplitude of the linearly polarized

component of the signal. In general cases, we define the polarization factor α ≡ B1/B0. Without loss of generality, we assume|B1|<|B0|so |α|<1. There are

|B×B˙| <|B·B˙|>_π/ω = π 2 α 1−α2 (4.51) and α=− 1 2β + r 1 + 1 4β2, β = 2|B×B˙| π <|B·B˙|>_π/ω. (4.52) α is essentially the ellipticity of the polarization. Note that Eq.4.52 guarantees 0≤α≤1.

α = 1 (or β → +∞) means perfectly circular polarization and α = 0 (or β → 0) means linear polarization.

α is a function of time and frequency. Therefore to identify circular polarization we simply need to computeα in a given time-frequency domain and look forα close to unity. The merit of this polarization recognition algorithm is that it is coordinate invariant because only cross product and inner product of 3D vectors are involved.

The direct output of the whistler probe is B˙(t). After integration with time we obtain B(t). We also need to use digital bandpass filter to select single-frequency modes in order to compute|B(t, f)×B˙(t, f)|and |B(t, f)·B˙(t, f)|.

We compute α(t, f) for shot 16940, 17012 and 17014. In shot 17014 the probe was placed at the same location as shot 17012. Figure 4.12 shows the calculation results for valid measurements (i.e., |B˙(t, f)| is larger than the sensitivity of the probe). It is seen that the distribution of α(t, f) of shot 16940 is different from 17012 or 17014, especially in the 7−11 MHz range. In shot 17012 and 17014, α(t, f) ≥ 0.7 is valid at most times and frequencies, indicating that circularly polarized magnetic fluctuations are ubiquitous in these two shots.

Figure 4.12: Polarization factor α in time-frequency domains for shot 16940 (top row), 17012 (middle row) and 17014 (bottom row). In each row the two colormap plots are the same except the color indexing range. We only computed α(t, f) for ˙B(t, f) >300 T/sec, i.e., at least three times as large as the probe sensitivity.

Figure 4.13: Top left, top right, bottom left panels: Three components of the normalized wavevector ˆk of shot 17012 computed by Eq. 4.53 in the time-frequency domain. Bottom right panel: inner product of ˆk and the direction of the background magnetic field ˆB0. Only the valid measurements are used (i.e., ˙B(t, f) at least three times larger than the probe sensitivity).

Figure 4.14: Inner product of the normalized wavevector ˆk and the direction of the back- ground magnetic field ˆB0 of shot 17014 and 16940.

Equation 4.46 shows that the wavevector direction ˆk≡ξˆ×ηˆcan be given by

ˆ

k= B×B˙

|B×B˙|. (4.53)

Note that Eq. 4.53 has assumed that B is right-hand polarized because the phase of B˙ is behind B by π/2 for right-hand polarized wave. Figure 4.13 shows the three components of the normalized wavevector ˆk·rˆ, ˆk·θˆ, and ˆk·zˆ, and the inner product of ˆk and ˆB0 computed using Eq. 4.53. Here ˆB0 is the direction of the background magnetic field. It is seen that most waves propagate with angles smaller than 60◦ (i.e., ˆk·Bˆ0>0.5). Figure4.14 gives the calculation results for shot 16940 and 17014. Measurements of shot 17014 also reveals ˆk·Bˆ0 > 0.5 at most times and frequencies similar to shot 17012 because the two measurements are performed in the same configuration (probe at r = 6 cm). However, when the probe is immersed in the central dense plasma jet, shot 16940 gives a much more randomly distributed ˆk·Bˆ0. This is because the wavelengths of the whistler waves in the central dense jet is not significantly larger than the size of the whistler probe. Hence the wave polarization inferred from shot 16940 is not very reliable.