**3. Measuring plasma properties with interferometry**

#### **3.1 Electron and atom density**

In **Figure 4a**, the probe laser passes through the plasma column as a ray and signifies negligible refraction through the inhomogeneous medium. The interferometer (not shown between the plasma and the CCD) establishes the baseline interference pattern and the CCD camera measures the disturbed fringe pattern as caused by the plasma. The shaded discs in the column represent different cross-sectional views of the plasma. Also, the dark to light coloring of **Figure 4b** shows how each particle density has a radial and negative gradient. A more complete model for the density is given in [9]. Referring to **Figure 2**, the Cartesian reference is defined at the junction between the wire and the lower electrode.

In the discussion below, the pixel coordinates ð Þ *y*, *z* are described as **r** ¼ **y**^*y* þ **z**^*z*, and before wire vaporization (*t* ¼ 0 *ns*) the total electric field at **r** is

$$E(\mathbf{r}) = E\_0 e^{ik\_0 L} \left[ \mathbf{1} + e^{ik\_0 \ell} e^{i \Delta \mathbf{k} \cdot \mathbf{r}} \right],\tag{1}$$

where the time convention is *e*�*iω<sup>t</sup>* and the model assumes ideal lossless and dispersionless optics for convenience. The common path length before the plasma is *L*, the length through the plasma region is ℓ, the wave amplitude is *E*0, and the

*2D Relative Phase Reconstruction in Plasma Diagnostics DOI: http://dx.doi.org/10.5772/intechopen.104748*

**Figure 4.**

*Simplified illustration (without refraction) of probe laser passing through plasma of radial density gradient. (a) 3D and (b) 2D vantage of single ray through plasma.*

free-space wavenumber is *k*<sup>0</sup> ¼ 2*π=λ* at wavelength *λ*. In (1), the additional phase term Δ**k** � **r** represents the effect of the interferometer which is adjusted to produce straight fringe lines with spatial frequencies *κ<sup>y</sup>* ¼ Δ**k** � **y**^ and *κ<sup>z</sup>* ¼ Δ**k** � **z**^. As seen in **Figure 3b** there may be curvature of the fringe lines but those effects are ignored in (1).

Therefore, the intensity pattern in the reference image (*t* ¼ 0 ns) is

$$I\_{\rm ref}(\mathbf{r}) = a\_r(\mathbf{r}) + b\_r(\mathbf{r}) \cos(\Delta \mathbf{k} \cdot \mathbf{r} + k\_0 e'),\tag{2}$$

where the background *ar* and contrast *br* are proportional to j j *E*<sup>0</sup> 2 . The non-ideal effects of the optical system cause spatially varying background and contrast [22] that can be measured following the techniques of [23].

After time *t* ¼ 0 ns the plasma has wavenumber *k* ¼ *k*0*n* and the propagation phase through the plasma is *k*<sup>0</sup> Ð <sup>ℓ</sup>*n*d*x* [24]. The test image has intensity

$$I\_{\rm txt}(\mathbf{r}) = a\_t(\mathbf{r}) + b\_t(\mathbf{r}) \cos \left[ \Delta \mathbf{k} \cdot \mathbf{r} + k\_0 \int\_{\ell} n(\mathbf{x}, \mathbf{r}) d\mathbf{x} \right], \tag{3}$$

where the index of refraction *n* is inhomogeneous and the background and contrast are proportional to j j *E*<sup>0</sup> 2 *=* Ð <sup>ℓ</sup>*n*d*x*.

The index of refraction, assuming complete ionization is expressed with the atomic and electron densities *Na* and *Ne*, respectively, as

$$n(\mathbf{x}, \mathbf{r}) = \mathbf{1} + a\mathbf{N}\_a(\mathbf{x}, \mathbf{r}) - \beta \mathbf{N}\_e(\mathbf{x}, \mathbf{r}).\tag{4}$$

where *<sup>α</sup>* is the dynamic polarizability of the material, *<sup>β</sup>* <sup>¼</sup> *<sup>β</sup>*0*λ*<sup>2</sup> , *<sup>β</sup>*<sup>0</sup> <sup>¼</sup> *<sup>e</sup>*<sup>2</sup>*<sup>=</sup>* <sup>16</sup>*π*<sup>3</sup>*ε*0*mec*<sup>2</sup> 0 � �, *ε*<sup>0</sup> is the free-space permittivity, *me* is the electron mass, *c*<sup>0</sup> is speed of light in vacuum, and *e* is unit charge.

Upon substituting (4) into (3) it is clear how the fringe spacing in the interferogram is determined by Δ**k** � **r** and the amount of line shift is determined by the line-integrated densities

$$\chi\_a(\mathbf{r}) = \int\_{\ell} N\_a(\mathbf{x}, \mathbf{r}) \, \mathrm{d}x,\tag{5}$$

$$\chi\_{\epsilon}(\mathbf{r}) = \int\_{\ell} N\_{\epsilon}(\mathbf{x}, \mathbf{r}) \, \mathbf{dx}. \tag{6}$$

When the plasma has axial symmetry and radial density profiles as illustrated in **Figure 4**, Eqs (5) and (6) can be expressed in terms of the forward Abel transform Af g� as [25].

$$\chi\_a(\mathbf{y}, \mathbf{z}) = \mathcal{A}\{\mathbf{N}\_a\} = 2 \int\_{\mathcal{Y}}^{\infty} \frac{\mathbf{N}\_a(\mathbf{r}, \mathbf{z}) \mathbf{y}}{\sqrt{\mathbf{r}^2 - \mathbf{y}^2}} \,\mathrm{d}\mathbf{r},\tag{7}$$

$$\chi\_{\varepsilon}(\boldsymbol{y},\boldsymbol{z}) = \mathcal{A}\{\boldsymbol{N}\_{\varepsilon}\} = 2\int\_{\mathcal{Y}} \frac{\boldsymbol{N}\_{\varepsilon}(\boldsymbol{r},\boldsymbol{z})\boldsymbol{y}}{\sqrt{\boldsymbol{r}^{2} - \boldsymbol{y}^{2}}} \,\mathrm{d}\boldsymbol{r}.\tag{8}$$

where *<sup>r</sup>*<sup>2</sup> <sup>¼</sup> *<sup>x</sup>*<sup>2</sup> <sup>þ</sup> *<sup>y</sup>*<sup>2</sup> . The upper limit of the integration is usually restricted to the radius *R* of the plasma column.

Upon expanding the phase argument of (3) with (4)–(6) as

$$
\Phi(y, z) = \kappa\_{\mathcal{Y}} y + \kappa\_{\mathcal{z}} z + k\_0 \ell' + k\_0 a \chi\_a(y, z) - k\_0 \beta \chi\_\epsilon(y, z), \tag{9}
$$

and comparing with the phase argument of (2) the phase difference caused by the plasma is

$$
\Delta\phi(\mathbf{y},\mathbf{z}) = k\_0 a \chi\_a(\mathbf{y},\mathbf{z}) - k\_0 \beta \chi\_\epsilon(\mathbf{y},\mathbf{z}).\tag{10}
$$

In cases where *Na* ≪ *Ne*, Δ*ϕ*≈ � *k*0*βχ<sup>e</sup>* and *Ne* is measured using the inverse Abel transform as

$$N\_{\epsilon}(r,z) = \mathcal{A}^{-1}\left\{\frac{-\Delta\phi(\mathbf{y},z)}{k\_0\beta}\right\} = -\frac{1}{\pi}\int\_{r}^{\infty} \frac{\partial\chi\_{\epsilon}(\mathbf{y},z)}{\partial\mathbf{y}} \frac{1}{\sqrt{\mathbf{y}^2 - r^2}} \,\mathrm{d}\mathbf{y}.\tag{11}$$

Otherwise, two independent wavelength measurements may be used to solve for the line-integrated densities. Then, each radial density profile is determined with Abel inversion of the respective line-integrated density [4]. Also, when the amount of line shift is normalized by the spacing between undisturbed fringe lines the resulting fringe ratio is equivalent to the normalized phase difference *δ* ¼ Δ*ϕ=*ð Þ 2*π* . Therefore, *δ* has implicit units of lines. When the interferogram has high contrast or high signal-tonoise ratio (SNR) it is straightforward to visually calculate *δ* and then apply the inverse Abel transform.

The time-sequenced, 1D fringe ratios of [8] are shown in **Figure 5** and were calculated by visual inspection of the average fringe shift in each image. A contourfitting algorithm was also proposed to calculate *δ*ð Þ*y* , but the 2D calculation of *δ*ð Þ *y*, *z* with FTM is presented in Section 4.3.

#### **3.2 Expansion velocity**

The expansion velocity is calculated from the change in the plasma's radial dimension as observed over time. Schlieren imagery and shadowograms are better suited to

*2D Relative Phase Reconstruction in Plasma Diagnostics DOI: http://dx.doi.org/10.5772/intechopen.104748*

#### **Figure 5.**

*Time-sequenced fringe ratio from applying a contour-fitting algorithm to Cu interferograms.* © *2020 IEEE. Reprinted, with permission, from [8].*

visually observe the precise change of the plasma radius over time, but the changes are observable in the interferogram. The process is to measure the span Δ*Dy* where the fringe line begins to shift away from the reference line and where it just returns to the reference line. A common approach is to determine *Dy* from each image in the region where *δ*>0*:*1 [26]. With *Dy*<sup>1</sup> and *Dy*<sup>2</sup> equal to the spans at times *t*<sup>1</sup> and *t*2, the expansion velocity is *Dy*<sup>2</sup> � *Dy*<sup>1</sup> � �*=*ð Þ *<sup>t</sup>*<sup>2</sup> � *<sup>t</sup>*<sup>1</sup> as shown by the shaded regions of **Figure 4**. The accuracy is limited by the pixel size and thickness and clarity of the fringe line. Also, the expansion speed is highly dependent on the method used to produce the plasma. Speeds of a few km/s are commonly observed in wire experiments but simulation of colliding flows shows speeds at high orders of Mach [20].

#### **3.3 Ionization ratio**

The ionization ratio *Ne=Na* is highly dependent on the experimental conditions. For example, excess electrons from the excitation current must be considered. Also, some plasmas will have greater concentration of different types of ions. Therefore, the refractivity model should include terms for the higher order ions as in [1]. It has been noted that it is unlikely one can measure individual ion density from a single interferogram. In particular, the work of [8] should be considered as an effort to measure electron and atom densities rather than electron and ion densities.

#### **3.4 Polarizability**

Atomic polarizability is measurable from interferometry as described in [5]. The authors of [5] use the integrated-phase technique that equates the density of a crosssectional slice of the wire with the corresponding integrated phase inferred by the interferogram. With the linear density defined as *<sup>N</sup>*lin <sup>¼</sup> ÐÐ *Na*d*x*d*<sup>y</sup>* the dynamic polarizability is measured as

$$a(\lambda) = \frac{k\_0}{N\_{\rm lin}} \int\_{D\_y} \chi\_a(y) \, \mathrm{d}y,\tag{12}$$

where the *z* dependency is suppressed, and *Dy* is the span of plasma according to the where the fringe shifts occur. The authors of [5] also carefully note the assumptions of total vaporization, negligible free-electron refraction, and that the shifted fringe lines represent the region of the plasma. Under the conditions they report dynamic polarizability with an accuracy of 10% for Mg, Ag, Al, Cu, and Au samples at laser wavelengths of 1064 and 532 nm. The measured values of Al as given in [5] are used in the example of Section 4.6.
