**4.4 FTM with plasma interferometry**

**Figure 7** shows examples of measured plasma interferograms (left column). The images exhibit significant intensity variation and regions of very low contrast. The top and middle images have 10–15 dB SNR and the bottom image has 5 dB SNR. The spatial frequency spectra are shown in the middle column with a bounding box drawn

#### **Figure 7.**

*Interferograms (left column), spatial frequency spectrum (middle column) with filter domain shown as a black box, and reconstructed fringe order (right column). The spatial frequencies are f <sup>y</sup>* ¼ *κy=*ð Þ 2*π and f <sup>z</sup>* ¼ *κz=*ð Þ 2*π* . *Examples with (a) low spatial frequency, (b) low SNR, and (c) low contrast and low SNR.*

about the positive carrier frequency. The box has dimensions of an ideal 10-dB bandwidth rectangular convolution filter. In the top image, the energy located about the upper carrier frequency is spread and overlaps with the DC content in the center of the image. Lastly, the fringe lines in some of the images show high frequency ripple caused by the interferometer.

From the frequency images in the middle column of **Figure 7** it is easy to see how the FTM filter *h* requires careful design [42–45]. However, there are various tradeoffs between accuracy, computation time, and filter design. Such details are often left out of the discussion on FTM's role in phase reconstruction. The effects in the imagery affect the accuracy of the phase reconstruction and ultimately the electron and atom density measurements. The right column of **Figure 7** shows the fringe ratio as recovered with FTM. In the top row, the vertical sidelobes of the DC energy bleeds through the bandpass filter at the carrier causing ripple in the fringe order. Note: the DC content is shifted to the carrier in FTM, i.e. Eq. (13) of Algorithm 1. The intensity variation of the middle row is due to intensity variation of the probing laser and the inhomogeneous plasma volume. The spatial dependencies of *a* and *b* cause additional frequency spreading about the carrier. Likewise, the bottom interference pattern has very low contrast, i.e. low SNR and the energy about the carrier is very weak. Thus, the energy spectra are distorted and the fringe ratios are coarsely recovered as shown in **Figure 7b** and **c**.

#### **4.5 Improvements to FTM**

Given the various image artifacts that hinder phase reconstruction, several FTM improvements have been developed to improve the phase accuracy of interference patterns with high density fringe lines. The different approaches can be considered as pre-filtering or iterative filtering of *I y*ð Þ¼ , *<sup>z</sup> a y*ð Þþ , *<sup>z</sup> b y*ð Þ , *<sup>z</sup>* cos *<sup>κ</sup>yy* <sup>þ</sup> *<sup>κ</sup>zz* <sup>þ</sup> *<sup>ϕ</sup>S*ð Þ *<sup>y</sup>*, *<sup>z</sup>* . The most prominent is the iterative model-based approach of [14] that uses Zernike polynomials to model the phase profile *ϕ<sup>S</sup>* and then iteratively improve the model using narrowing filters. In context of **Figure 7**, the algorithm of [14] first finds the initial model *ϕ*Mdl using conventional FTM and removes the background *a* with a DC rejection filter. However, some residual energy from the DC region remains causing the ripple observed in reconstructed phase *ϕ<sup>S</sup>* and as observed in the fringe order of **Figure 7**. The phase function is fitted with a suitable 2D smooth polynomial which effectively excludes the rippling. Using the modeled phase function *ϕS*,Mdl, the original demodulated and DC-rejected image is iteratively filtered to remove the conjugate phase. The extracted phase error Δ*ϕ* ¼ *ϕ<sup>S</sup>* � *ϕ<sup>S</sup>*,Mdl is used to update the model until the phase *ϕ<sup>S</sup>* converges. The authors of [14] report only three iterations were needed in their simulations and the phase accuracy improves by a factor of ten.

The pre-filtering approach proposed in [8] filters the background function *a* and artificially equalizes the envelope function as *b y*ð Þ¼ , *z* 1. The equalization algorithm is given in [46] and the approach is shown by simulation to improve phase reconstruction by a factor of five. While the accuracy is less than the iterative method, it is much simpler to implement. However, it is limited to open fringe lines and is unknown to perform as well as the iterative method for high-density fringe lines. **Figure 8** shows examples of the smoothing and leveling algorithm of [46] with the measured interferograms of **Figure 7**. The fringe lines are noticeably improved and easy to analyze by visual inspection. Also, the spectra show a significant reduction of DC content. The resulting fringe ratio plots have less ripple and the density profiles shown in **Figure 8b** *2D Relative Phase Reconstruction in Plasma Diagnostics DOI: http://dx.doi.org/10.5772/intechopen.104748*

#### **Figure 8.**

*Smoothed and leveled interferograms of Figure 6 (left column), spatial frequency spectrum (middle column) with filter domain shown as a black box, and FTM reconstructed fringe order (right column). Recovered from image with (a) low spatial frequency, (b) low SNR, and (c) low contrast and low SNR.*

and **c** are improved over **Figure 7b** and **c**. In context of the iterative FTM algorithm [14], the phase accuracy using the pre-filtering approach is limited because the residual energy of the signal conjugate can still bleed into the final baseband signal. However, the pre-filtering approach could serve as the initial phase estimate.

#### **4.6 Example of electron and atomic density measurement**

To demonstrate the density measurement the procedure of [8] is adapted for 2D electron and atomic density measurements. The pre-filtered interference images are

**Figure 9.**

*2D phase reconstruction and density measurement of Al plasma following set up of [8]. (a) 1064-nm (top) and 532-nm (bottom) interference patterns. (b) 1064-nm (top) and 532-nm (bottom) fringe ratios, and (c) atomic (top) and electron (bottom) volumetric densities (cm*�*<sup>3</sup> ).*

shown in **Figure 9a** and the measurement setup and wire dimensions are given in [8]. Each phase profile is reconstructed with FTM and presented as fringe ratio (**Figure 9b**). From the two independent line shift measurements *δ*<sup>1</sup> ¼ Δ*ϕ*1*=*ð Þ 2*π* (1064 nm) and *δ*<sup>2</sup> ¼ Δ2*=*ð Þ 2*π* (532 nm), the line-integrated densities *χ<sup>a</sup>* and *χ<sup>e</sup>* are determined from the linear system of equations

$$
\delta = \mathbf{A}\boldsymbol{\chi},\tag{18}
$$

where *δ* ¼ ½ � *δ*1, *δ*<sup>2</sup> *<sup>T</sup>*, *<sup>χ</sup>* <sup>¼</sup> *<sup>χ</sup>a*, *<sup>χ</sup><sup>e</sup>* ½ �*<sup>T</sup>*, the superscript *<sup>T</sup>* denotes the transpose operator, and the matrix

$$\mathbf{A} = \begin{bmatrix} \alpha\_1/\lambda\_1 & -\beta\_0\lambda\_1 \\ \alpha\_2/\lambda\_2 & -\beta\_0\lambda\_2 \end{bmatrix},\tag{19}$$

is easily inverted. In (19), the dynamic atomic polarizabilities at 1064 and 532 nm are *<sup>α</sup>*<sup>1</sup> <sup>¼</sup> <sup>8</sup>*:*<sup>7</sup> � <sup>10</sup>�<sup>24</sup> and *<sup>α</sup>*<sup>2</sup> <sup>¼</sup> <sup>10</sup>*:*<sup>8</sup> � <sup>10</sup>�<sup>24</sup> cm3, respectively [5]. The explicit expressions for the line-integrated densities are

$$\chi\_a = \Delta(-\beta\_0\lambda\_2\delta\_1 + \beta\_0\lambda\_1\delta\_2),\tag{20}$$

$$
\chi\_{\varepsilon} = \Delta(-a\_2 \delta\_1 / \lambda\_2 + a\_1 \delta\_2 / \lambda\_1),
\tag{21}
$$

where <sup>Δ</sup> <sup>¼</sup> *<sup>λ</sup>*1*λ*2*<sup>=</sup> <sup>α</sup>*2*β*0*λ*<sup>2</sup> <sup>1</sup> � *<sup>α</sup>*1*β*0*λ*<sup>2</sup> 2 . The line-integrated densities are transformed into the volumetric densities (**Figure 9c**) by applying the inverse Abel Transform as

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

$$\begin{Bmatrix} N\_a(r, z) \\ N\_\epsilon(r, z) \end{Bmatrix} = \mathcal{A}^{-1} \begin{Bmatrix} \chi\_a(\mathbf{y}, z) \\ \chi\_\epsilon(\mathbf{y}, z) \end{Bmatrix},\tag{22}$$

#### **4.7 Measurement accuracy**

The accuracy of the density measurement depends on the accuracies of (1) the line shift measurement, and (2) the numerical accuracy of the Abel inversion. The numerical accuracy of the transform is determined by the specific implementation. Therefore, the direct integration methods [25] depend on the pixel size but can easily achieve double precision accuracy. Thus, the primary source of measurement error is due to the relative phase shift (i.e., line shift). A visual analysis of the fringe pattern has the best accuracy because the line shift can be measured to �1 pixel. The relative error is �100*=Nm*% where *Nm* is the number of pixels at the maximum line shift. However, for the generic 2D pattern or when the image has noise and speckle, the accuracy depends on the complete phase reconstruction method. According to [15], the overall phase measurement algorithm includes pre-filtering to reduce noise and speckle, relative phase recovery such as with FTM, and phase unwrapping, and the phase evaluation should be better than *π=*10 rad for general purposes. The error analysis of [5], also showed the phase-shift reconstruction accuracy should be on the order of 2*π=*20 which is equivalent to 0.05 lines. For a maximum line shift *δM*, the accuracy is �100*=*ð Þ 20*δ<sup>M</sup>* %. For an interferogram with up to 3 or 4 lines of shift, the accuracy is estimated as �7%. Other approaches to show improvement of the phase measurement first determine the phase difference with the phase-shifting method [27], and then compare it to each of the errors from the iterative FTM [14] and the conventional FTM. The error bound is given by the peakto-valley measurement and the average error is given as root-mean-square error. The improved FTM with iterative filter narrowing is shown to improve accuracy by a factor of ten.

An open problem is to perform a rigorous sensitivity analysis. It can be completed with modeling and simulation using the following model:

$$\begin{Bmatrix} \hat{N}\_a \\ \hat{N}\_\epsilon \end{Bmatrix} = \mathcal{A}^{-1} \left\{ \mathbf{A}^{-1} \mathcal{F} \left\{ \begin{bmatrix} a+b\cos\left(\Delta \mathbf{k}\_1 \cdot \mathbf{r} + \mathbf{g}(\mathcal{A}\{N\_a\}, \mathcal{A}\{N\_\epsilon\})\right) \\ a+b\cos\left(\Delta \mathbf{k}\_2 \cdot \mathbf{r} + \mathbf{g}(\mathcal{A}\{N\_a\}, \mathcal{A}\{N\_\epsilon\})\right) \end{bmatrix} \right\} \right\},\tag{23}$$

where *g* is the mapping of the simulated line-integrated densities to the phase function described in (9) of Section 3. The measured densities are denoted as *N*^ *<sup>a</sup>* and *N*^ *<sup>e</sup>*, and F denotes the FTM or other phase reconstruction (line shift measurement) method.

By appropriate choice of noise terms affecting *a* and *b* (e.g., multiplicative phase noise, additive Gaussian noise, fringe line distortions, radiometric variation or other optical aberrations), (23) can determine the effects on accuracy due to different parameters in the phase recovery algorithm and measurement setup.
