2D Relative Phase Reconstruction in Plasma Diagnostics

Abstract

Interferometric analysis methods for measuring plasma properties are presented with emphasis on emerging trends in 2D phase reconstruction. Using recent imagery from exploded-wire experiments the relative phase profiles from independent interferograms are reconstructed. The well-known Fourier Transform Method is presented and discussed. Then, the electron and atom densities are recovered from the phase by solving a linear system of equations in the form of line-integrated density profiles. The mathematical models of the line density and phase function are described and elucidate why interferograms of plasma suffer from low contrast, high signal-to-noise ratio and poorly defined fringes. Although these effects pose challenges for phase reconstruction, the interferometric diagnostic continues to advance the plasma science.

1. Introduction

This chapter presents an engineering perspective on recent analysis methods used to measure plasma properties from interferograms [1, 2]. The electron and atomic densities [3, 4], volume distribution, expansion velocity, and atomic polarizability [5] affect the amount of fringe line shift. To recover these properties, the basic two-step approach is to recover the relative phase difference represented by the shifted fringe lines, and to invert the relationship between the phase difference and the desired properties.

However, as described in Section 2 and unlike profilometry applications [6] where the interference pattern is stable over a lengthy observation period, the plasma medium changes rapidly and continually. For example, laser ablation methods produce an abruptly expanding plasma lobe. Whereas the electrically exploded wire produces an expanding cylindrical volume that lasts for a few hundred nanoseconds. In each case, the plasma presents an inhomogeneous, lossy and dispersive medium to the probing laser. These medium properties cause radiometric variation in the fringe pattern such as low contrast and poorly defined fringe lines.

Before addressing the fringe line analysis, Section 3 reviews the mathematical model of the phase function for light wave propagation through the plasma volume. The electromagnetic phase accrual through the inhomogeneous medium, and the plasma’s refractive index are represented with line integrals of the electron and atom densities. Thus, there are two integral operations to invert during a density measurement. The reader is referred to classic references like [1, 2, 7] for additional factors to consider when the experiment includes controlling magnetic and electric fields. Recent works with electrically exploded wires [8, 9] and dual wavelength interferometry are discussed as a means to recover both density profiles.

Then, Section 4 presents a summary of different fringe analysis methods with emphasis on the Fourier Transform Method (FTM) developed by Takeda, Ina and Kobayashi [10] and continuously improved since [11, 12]. FTM is likely the most well-known method for extracting 2D phase information from the interferogram in surface profilometry and 3D shape measurement. As eloquently described by [13] these applications also give insight into the time–space analog and the time-frequency duality represented by different interferometry experiments.

Section 4 also discusses improvements to FTM which are generally based on iterative filtering [14] or pre-filtering [15]. The latter is presented with examples from exploded wire experiments. Section 5 briefly highlights different works on plasma interferometry and recent advances in the field of plasma science from interferometry. The chapter concludes with some thoughts on how the 2D phase analysis provides a rich understanding of the plasma.

2. Plasma diagnostic setup

2.1 Plasma creation and probing laser setup

Some of the common approaches to create a laboratory plasma include laser ablation, the spark gap, and exploded wires as illustrated in Figure 1a. Each drawing shows a probing laser that traverses the plasma medium. Although not shown, the probing laser passes through an interferometer before the charge-coupled-device (CCD) camera captures the interference pattern. Figure 1a shows how a driving laser irradiates a sample of very high purity (99.99% or better). The driving laser’s power density can range from ${10}^9\text{to}{10}^{12}\mathrm{W}\cdot {\mathrm{cm}}^{-2}$ for short bursts ($\approx 100\mathrm{ps}$) to create a plasma plume or jet [16, 17]. The driving laser’s pulse duration and pulse power, and the geometric shape of the sample control the plasma jet’s direction and size. The spark gap of [18] as represented in Figure 1b uses a pulsed high-voltage to cause the air between the electrodes to break down and form a spark plasma. The low-collisional spark plasma produces a current in air of $\approx 2.7\mathrm{kA}$ and lasts for $\approx 4.2\mathrm{ms}$. An alternative approach uses electrically-exploded wires shown in Figure 1c. The thin wire samples also of very high purity have very small diameters of a few microns and are driven from the electrodes with a high-current pulse ($\approx 2.5\mathrm{kA}$). The pulse has a short rise time of a few nano-seconds and the sudden burst of energy causes the wire to vaporize as a cylindrically expanding plasma [4].

Figure 1.

Illustration of common setup to create plasma and image with a single camera. (a) Driving laser ablates material. (b) Electrical breakdown of air produces electron stream. (c) High driving current vaporizes wire.

Several recent works use exploding wires to study how the plasma forms and behaves under varying conditions [4, 8, 19]. In particular, the setup of [8] uses two interferograms at 1064 and 532 nm to measure electron and atom densities (illustrated in Figure 2a). Also, the addition of a second wire as shown in Figure 1 allows study of colliding plasma flows [20]. In Figure 2, beam splitters and mirrors are understood to change the optical path of probe laser 2.

Figure 2.

Illustration of recent methods with electrically exploded wires [8, 20]. (a) Single and (b) double wire setups.

2.2 Plasma diagnostic methods

Plasma presents electrical, optical and mechanical behaviors that are observable from the electromagnetic (EM) wave emissions in the visible through x-ray regimes and the EM wave propagation through the plasma. Schlieren, shadowgraph and interferometric images reveal temporal and spatial variation of the plasma’s index of refraction. Of interest are the interferometer methods such as [8] that measured electron and atom densities using the classical Mach-Zehnder and also the shearing air wedge interferometer [21].

The basic optical setup of [8] as developed from the experiences of [4, 5] is shown in Figure 3a. The 1064-nm probing laser is passed through a harmonic doubler to produce a co-linear probing laser at 532 nm. Both beams are adjusted to illuminate and traverse the windowed vacuum chamber. The interferometer is adjusted to provide a regular fringe pattern before wire explosion. Figure 3b shows the resulting interference patterns collected at each wavelength (left: 1064 nm, right: 532 nm) at different time intervals before and after wire explosion. The shadowed regions centered near the top and bottom of each image are the electrodes, and the wire’s shadow appears in the $t=0\mathrm{ns}$ imagery. The CCDs’ reference frames are defined with $x$ along the optical path and $y$ − $z$ in the CCD focal planes. Lastly, the setup excludes static electric and magnetic fields, and the laser frequencies are well above the plasma cutoff frequency.

Figure 3.

(a) Optical path for dual-wavelength interferometry, and (b) time-sequenced interferograms. $\copyright$ 2020 IEEE. Reprinted, with permission, from [8].

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.

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.

In the discussion below, the pixel coordinates $\left(y,z\right)$ are described as $\mathbf{r}=\widehat{\mathbf{y}}y+\widehat{\mathbf{z}}z$, and before wire vaporization ($t=0\mathit{ns}$) the total electric field at $\mathbf{r}$ is

$$E\left(\mathbf{r}\right)=E_0{e}^{{\mathit{ik}}_{0}L}\left[1+e^{{\mathit{ik}}_0\mathrm{\ell }}{e}^{i\mathrm{\Delta }\mathbf{k}\cdot \mathbf{r}}\right],$$E1

where the time convention is $e^{-\mathit{i\omega t}}$ 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 $\mathrm{\ell }$, the wave amplitude is $E_0$, and the free-space wavenumber is $k_0=2\pi /\lambda$ at wavelength $\lambda$. In (1), the additional phase term $\mathrm{\Delta }\mathbf{k}\cdot \mathbf{r}$ represents the effect of the interferometer which is adjusted to produce straight fringe lines with spatial frequencies ${\kappa }_y=\mathrm{\Delta }\mathbf{k}\cdot \widehat{\mathbf{y}}$ and ${\kappa }_z=\mathrm{\Delta }\mathbf{k}\cdot \widehat{\mathbf{z}}$. 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 ($t=0\mathrm{ns}$) is

$$I_{\mathrm{ref}}\left(\mathbf{r}\right)=a_r\left(\mathbf{r}\right)+b_r\left(\mathbf{r}\right)cos\left(\mathrm{\Delta }\mathbf{k}\cdot \mathbf{r}+k_0\mathrm{\ell }\right),$$E2

where the background $a_r$ and contrast $b_r$ are proportional to ${\left|E_0\right|}^2$. 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 $t=0\mathrm{ns}$ the plasma has wavenumber $k=k_{0}n$ and the propagation phase through the plasma is $k_0{\int }_{\mathrm{\ell }}n\mathrm{d}x$ [24]. The test image has intensity

$$I_{\mathrm{tst}}\left(\mathbf{r}\right)=a_t\left(\mathbf{r}\right)+b_t\left(\mathbf{r}\right)cos\left[\mathrm{\Delta }\mathbf{k}\cdot \mathbf{r}+k_0{\int }_{\mathrm{\ell }}n\left(x,\mathbf{r}\right)\mathrm{d}x\right],$$E3

where the index of refraction $n$ is inhomogeneous and the background and contrast are proportional to ${\left|E_0\right|}^2/{\int }_{\mathrm{\ell }}n\mathrm{d}x$.

The index of refraction, assuming complete ionization is expressed with the atomic and electron densities $N_a$ and $N_e$, respectively, as

$$n\left(x,\mathbf{r}\right)=1+\alpha N_a\left(x,\mathbf{r}\right)-\beta N_e\left(x,\mathbf{r}\right).$$E4

where $\alpha$ is the dynamic polarizability of the material, $\beta ={\beta }_0{\lambda }^2$, ${\beta }_0=e^2/\left(16{\pi }^3{\epsilon }_0{m}_e{c}_0^2\right)$, ${\epsilon }_0$ is the free-space permittivity, $m_e$ is the electron mass, $c_0$ 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 $\mathrm{\Delta }\mathbf{k}\cdot \mathbf{r}$ and the amount of line shift is determined by the line-integrated densities

$${\chi }_a\left(\mathbf{r}\right)={\int }_{\mathrm{\ell }}{N}_a\left(x,\mathbf{r}\right)\mathrm{d}x,$$E5

$${\chi }_e\left(\mathbf{r}\right)={\int }_{\mathrm{\ell }}{N}_e\left(x,\mathbf{r}\right)\mathrm{d}x.$$E6

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 $\mathcal{A}\left\{\cdot \right\}$ as [25].

$${\chi }_a\left(y,z\right)=\mathcal{A}\left\{N_a\right\}=2{\int }_y^{\infty }\frac{N_a\left(r,z\right)y}{\sqrt{r^2-y^2}}\mathrm{d}r,$$E7

$${\chi }_e\left(y,z\right)=\mathcal{A}\left\{N_e\right\}=2{\int }_y^{\infty }\frac{N_e\left(r,z\right)y}{\sqrt{r^2-y^2}}\mathrm{d}r.$$E8

where $r^2=x^2+y^2$. 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

$$\mathrm{\Phi }\left(y,z\right)={\kappa }_{y}y+{\kappa }_{z}z+k_0\mathrm{\ell }+k_0{\mathit{\alpha \chi }}_a\left(y,z\right)-k_0{\mathit{\beta \chi }}_e\left(y,z\right),$$E9

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

$$\mathrm{\Delta }\varphi \left(y,z\right)=k_0{\mathit{\alpha \chi }}_a\left(y,z\right)-k_0{\mathit{\beta \chi }}_e\left(y,z\right).$$E10

In cases where $N_a\ll N_e$, $\mathrm{\Delta }\varphi \approx -k_0{\mathit{\beta \chi }}_e$ and $N_e$ is measured using the inverse Abel transform as

$$N_e\left(r,z\right)={\mathcal{A}}^{-1}\left\{\frac{-\mathrm{\Delta }\varphi \left(y,z\right)}{k_0\beta }\right\}=-\frac{1}{\pi }{\int }_r^{\infty }\frac{\partial {\chi }_e\left(y,z\right)}{\partial y}\frac{1}{\sqrt{y^2-r^2}}\mathrm{d}y.$$E11

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 $\delta =\mathrm{\Delta }\varphi /\left(2\pi \right)$. Therefore, $\delta$ has implicit units of lines. When the interferogram has high contrast or high signal-to-noise ratio (SNR) it is straightforward to visually calculate $\delta$ 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 contour-fitting algorithm was also proposed to calculate $\delta \left(y\right)$, but the 2D calculation of $\delta \left(y,z\right)$ with FTM is presented in Section 4.3.

Figure 5.

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

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 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 $\mathrm{\Delta }{D}_y$ 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 $D_y$ from each image in the region where $\delta >0.1$ [26]. With $D_{y1}$ and $D_{y2}$ equal to the spans at times $t_1$ and $t_2$, the expansion velocity is $\left(D_{y2}-D_{y1}\right)/\left(t_2-t_1\right)$ 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 $N_e/N_a$ 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 cross-sectional slice of the wire with the corresponding integrated phase inferred by the interferogram. With the linear density defined as $N_{\mathrm{lin}}=\iint N_a\mathrm{d}x\mathrm{d}y$ the dynamic polarizability is measured as

$$\alpha \left(\lambda \right)=\frac{k_0}{N_{\mathrm{lin}}}{\int }_{D_y}{\chi }_a\left(y\right)\mathrm{d}y,$$E12

where the $z$ dependency is suppressed, and $D_y$ 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.

4. Relative phase reconstruction with Fourier transform method (FTM)

4.1 Relative and absolute phase reconstruction

Before addressing the analysis methods, it is helpful to note how the measured phase is relative to a known reference. Unlike radio frequency methods, the phase reference is generally unknown in optical interferometry. Thus, the term phase reconstruction implicitly means relative phase reconstruction. The phase difference developed in Section 3.1 is the subject of the phase reconstruction methods and referred to as the phase function and phase profile in this section.

4.2 1D fringe analysis

When the interference pattern has a uniform behavior along one dimension such as ${\kappa }_y=0$ then each fringe line can be visually analyzed to measure the amount of line shift. In the case of a uniformly tilted line, i.e. ${\kappa }_y\ne 0$ and ${\kappa }_z\ne 0$ then the image can be rotated such that ${\kappa }_y=0$. Also, the principle component analysis can be performed on the fringe line to measure the shift, but firstly, requires a method to extract the fringe line. The approach in [8] treated the intensity image as a surface and extracted the contours for each fringe line in the image. After averaging, normalization and fitting of the lines to a Gaussian curve for each wavelength, the line-integrated densities were recovered and inverted with the Abel transform to calculate the electron and atom densities. The 1D approach is unlikely to work well for closed loop fringe lines that occur in surface profilometry and collisional plasma interferometry.

4.3 2D fringe analysis

The 2D fringe analysis is applicable to temporal and spatial analysis, 1D and 2D interference patterns, and straight lines and open and closed curves. Approaches to reconstruct $\varphi \left(y,z\right)$ include principal component analysis [27, 28], phase shifting [29], wavelet analysis [30] and Fourier analysis [10, 15]. The latter has become popular in profilometry, 3D shape reconstruction and more recently measurement of nano-scale and femtosecond observables. In much of the literature the original work of [10] has become known as the Fourier Transform Method (FTM).

FTM is well-known for 2D phase reconstruction in optical metrology of surface flatness [31], surface height, strain [32] and defect [33] profilometry, surface motion [34], and 3D shape measurement. However, the spatial-carrier fringe behavior has also been used to observe extreme physical phenomenon [35] such as magnetic fields, electron waves and ultra-violet lithography, as well as beam propagation [36], plasma property measurement [37], refractive index studies of polymeric substrates [38], nano-scale surface metrology [39], corneal topography [40] and biological tissue characterization [41].

The essence of the approach is to analyze the 2D Fourier spectrum of the fringe pattern and to extract the phase function using a digital homodyne receiver process. In treating the interferogram as a 2D signal with spatial dimensions and spatial frequencies ${\kappa }_y$ and ${\kappa }_z$, the perturbation is a modulating signal to be recovered as in a communication system. FTM first demodulates the image by the spatial frequency and then filters the baseband content. The solution is succinctly and elegantly described as a two-step algorithm [6, 12, 13, 35]. The listing in Algorithm 1 includes two additional steps to estimate the spatial frequencies and to unwrap the phase function.

Algorithm 1. Fourier Transform Method [12].

Given input image $I\left(y,z\right)=a\left(y,z\right)+b\left(y,z\right)cos\left[{\kappa }_{y}y+{\kappa }_{z}z+\varphi \left(y,z\right)\right]$.

Measure spatial frequencies ${\kappa }_y$ and ${\kappa }_z$.

Demodulate $I$ to baseband and low-pass filter using the Fourier Transform as

$$c\left(y,z\right)=h\left(y,z\right)\ast \left[I\left(y,z\right)e^{i{\kappa }_{y}y}{e}^{i{\kappa }_{z}z}\right],$$E13

Recover the wrapped phase $\varphi$ as

$$\varphi \left(y,z\right)=arctan\left(\frac{\mathrm{ℐ}m\left\{c\left(y,z\right)\right\}}{\mathrm{ℛ}e\left\{c\left(y,z\right)\right\}}\right).$$E14

Unwrap phase: $\tilde{\varphi }=\mathit{\text{unwrap}}\left(\varphi \right)$.

return unwrapped phase $\tilde{\varphi }\left(y,z\right)$.

In (13), $h$ is an ideal 2D rectangular window function with size $Q_y\times Q_z$, and $Q_y=⌈2\pi /{\kappa }_y⌉$ and $Q_z=⌈2\pi /{\kappa }_z⌉$ where $⌈\cdot ⌉$ denotes the ceiling function. The last step to unwrap the phase may be accomplished with a variety of techniques. To demonstrate the FTM approach, an interference image is simulated as $I\left(y,z\right)=a\left(y,z\right)+b\left(y,z\right)cos\left[{\kappa }_{y}y+{\kappa }_{z}z+\varphi \left(y,z\right)\right]$ with spatial frequencies ${\kappa }_y=\pi /5\mathrm{rad}\cdot {\mathrm{m}}^{-1}$ and ${\kappa }_z=\pi \mathrm{rad}\cdot {\mathrm{m}}^{-1}$. The phase function $\varphi$ is defined as

$$\varphi \left(y,z\right)=\pi exp\left[-0.1{\left(z/10+y-0.1\right)}^2\right],$$E15

and the background and contrast functions are specified as

$$a\left(y,z\right)=0.6+\mathit{\gamma \delta }\left(y,z\right),$$E16

$$b\left(y,z\right)=1.$$E17

In (16), $\gamma \sim \mathcal{N}\left(0,{\sigma }^2\right)$ is a random variable from the standard normal distribution with zero mean and variance ${\sigma }^2$. The variance is set to produce phase noise with 10-dB signal-to-noise ratio.

The simulated phase and interferogram are shown in Figure 6a and b. The phase profile causes fringe lines similar to what are observed with a single exploded wire. However, the radiometric effects are ignored in the example. The results of the different stages of Algorithm 1 are also shown in Figure 6c–f. The spectrum of $g$ has three components: a direct-current (DC)-like term, the spatial carrier, and the carrier’s conjugate.

Figure 6.

Example of FTM using simulated noisy 532-nm interferogram. (a) phase, (b) image, (c) signal spectrum, (d), filter spectrum, (e) filtered signal spectrum, (f) recovered phase.

The DC component centered at ${\kappa }_y={\kappa }_z=0$ is mainly due to the background $a$ but often includes residual energy from the carrier and its conjugate. The region of interest is centered at the spatial carrier frequency $\left({\kappa }_y,{\kappa }_z\right)$ and has a spectral shape determined by $c={\mathit{be}}^{\mathit{i\varphi }}$. Therefore, the phase recovery is sensitive to the design of the filter $h$. Figure 6f shows the result when the ideal rectangular impulse response is used per [12] and has a noticeable ripple and effect of Gibbs phenomenon. The root mean square error between $\varphi$ and the reconstructed and unwrapped phase $\tilde{\varphi }$ is $0.67$ radians.

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 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.

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 $f_y={\kappa }_y/\left(2\pi \right)$ and $f_z={\kappa }_z/\left(2\pi \right)$. Examples with (a) low spatial frequency, (b) low SNR, and (c) low contrast and low SNR.

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, 43, 44, 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 $I\left(y,z\right)=a\left(y,z\right)+b\left(y,z\right)cos\left({\kappa }_{y}y+{\kappa }_{z}z+{\varphi }_S\left(y,z\right)\right)$. The most prominent is the iterative model-based approach of [14] that uses Zernike polynomials to model the phase profile ${\varphi }_S$ and then iteratively improve the model using narrowing filters. In context of Figure 7, the algorithm of [14] first finds the initial model ${\varphi }_{\mathrm{Mdl}}$ 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 ${\varphi }_S$ 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 ${\varphi }_{S,\mathrm{Mdl}}$, the original demodulated and DC-rejected image is iteratively filtered to remove the conjugate phase. The extracted phase error $\mathrm{\Delta }\varphi ={\varphi }_S-{\varphi }_{S,\mathrm{Mdl}}$ is used to update the model until the phase ${\varphi }_S$ 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 $b\left(y,z\right)=1$. 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 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.

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.

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 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 ${\delta }_1=\mathrm{\Delta }{\varphi }_1/\left(2\pi \right)$ (1064 nm) and ${\delta }_2={\mathrm{\Delta }}_2/\left(2\pi \right)$ (532 nm), the line-integrated densities ${\chi }_a$ and ${\chi }_e$ are determined from the linear system of equations

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⁻³).

$$\mathit{\delta }=\mathrm{A}\mathit{\chi },$$E18

where $\mathit{\delta }={\left[{\delta }_1,{\delta }_2\right]}^T$, $\mathit{\chi }={\left[{\chi }_a,{\chi }_e\right]}^T$, the superscript $T$ denotes the transpose operator, and the matrix

$$\mathbf{A}=\left[\begin{array}{cc}{\alpha }_1/{\lambda }_1& -{\beta }_0{\lambda }_1\\ {\alpha }_2/{\lambda }_2& -{\beta }_0{\lambda }_2\end{array}\right],$$E19

is easily inverted. In (19), the dynamic atomic polarizabilities at 1064 and 532 nm are ${\alpha }_1=8.7\times {10}^{-24}$ and ${\alpha }_2=10.8\times {10}^{-24}{\mathrm{cm}}^3$, respectively [5]. The explicit expressions for the line-integrated densities are

$${\chi }_a=\mathrm{\Delta }\left(-{\beta }_0{\lambda }_2{\delta }_1+{\beta }_0{\lambda }_1{\delta }_2\right),$$E20

$${\chi }_e=\mathrm{\Delta }\left(-{\alpha }_2{\delta }_1/{\lambda }_2+{\alpha }_1{\delta }_2/{\lambda }_1\right),$$E21

where $\mathrm{\Delta }={\lambda }_1{\lambda }_2/\left({\alpha }_2{\beta }_0{\lambda }_1^2-{\alpha }_1{\beta }_0{\lambda }_2^2\right)$. The line-integrated densities are transformed into the volumetric densities (Figure 9c) by applying the inverse Abel Transform as

$$\left\{\begin{array}{l}{N}_a\left(r,z\right)\\ N_e\left(r,z\right)\end{array}\right\}={\mathcal{A}}^{-1}\left\{\begin{array}{l}{\chi }_a\left(y,z\right)\\ {\chi }_e\left(y,z\right)\end{array}\right\},$$E22

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 $\pm 1$ pixel. The relative error is $\pm 100/N_m$% where $N_m$ 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 $\pi /10\mathrm{rad}$ for general purposes. The error analysis of [5], also showed the phase-shift reconstruction accuracy should be on the order of $2\pi /20$ which is equivalent to 0.05 lines. For a maximum line shift ${\delta }_M$, the accuracy is $\pm 100/\left(20{\delta }_M\right)$%. For an interferogram with up to 3 or 4 lines of shift, the accuracy is estimated as $\pm 7$%. 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 peak-to-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:

$$\left[\begin{array}{l}{\widehat{N}}_a\\ {\widehat{N}}_e\end{array}\right]={\mathcal{A}}^{-1}\left\{{\mathbf{A}}^{-1}\mathcal{F}\left\{\left[\begin{array}{l}a+bcos\left(\mathrm{\Delta }{\mathbf{k}}_1\cdot \mathbf{r}+g\left(\mathcal{A}\left\{N_a\right\},\mathcal{A}\left\{N_e\right\}\right)\right)\\ a+bcos\left(\mathrm{\Delta }{\mathbf{k}}_2\cdot \mathbf{r}+g\left(\mathcal{A}\left\{N_a\right\},\mathcal{A}\left\{N_e\right\}\right)\right)\end{array}\right]\right\}\right\},$$E23

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 ${\widehat{N}}_a$ and ${\widehat{N}}_e$, and $\mathcal{F}$ 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.

5. Advances in plasma diagnostics with interferometry

There are numerous examples of plasma diagnostics with optical and x-ray interferometry. This section highlights a few to illustrate the specific advances in plasma science. Much of the interferometry science with plasma seeks to reconstruct the signal phase and refractive index of the plasma as a means to measure the electron density. However, the various types of plasmas and interferometers have shown additional benefits of the interferometric diagnostic.

For example, Feister et al. [47] developed a temporally multi-scale interferometer to investigate the target pre-ablation by ultra intense pulses ($>{10}^{18}\mathrm{W}\cdot {\mathrm{cm}}^{-2}$). The setup acquired three-phase interference images using a laterally sheared Michelson setup. The combination of phase reconstruction from the interferogram and the ultra-fast shadowgraphy confirmed how self-emission that corrupted the electron density measurement was caused by pre-ablation of the sample. The femtosecond time scale confirmed that the leading edge of the intense pulse caused the pre-ablation and led to images showing nanosecond formation of pre-plasma, femtosecond interaction of the ultra-intense main pulse, and picosecond hydrodynamic expansion. The physical insights gained by the different time-sequenced views were applied to modeling and simulation and subsequent hardware design. In the x-ray approach of Nilsen and Johnson [48], a 14.7-nm Pd laser x-ray interferometry confirmed how bound electrons caused anomalous fringe behavior in Al plasma that resulted in a refractive index less than one.

Since then, other works [9, 20] investigated physical behavior of expanding plasma volumes when created from two electrically exploded wires. In [9], aluminum (Al) and polymide-coated tungsten (W) plasma from a 1-kA current was observed with dual-wavelength (532 and 1064 nm) interferometry. They found that the atomization of Al expanded with a constant velocity of several km/s before stagnating in the middle region of the two wires. They concluded the Al plasma comprised mostly atoms as observed by the density calculations and comparison to the linear wire density. They also discussed how the W wires were more difficult to transition to vapor due to the higher melting point and concluded that a dual-pulse generator would be needed in future experiments. A similar study of mixing flow of two expanding plasma was reported in [20] where they also used two electrically exploded wires to study the hot plasma coronas as they collided. The modeling and simulation study showed that a thin layer between the expanding coronas can be sustained in high Mach number flows resulting from hydrodynamic mixing. It also showed how the dense core material enhances a thin shell instability but has yet to be mapped to observation.

In the recent work of [23], the spontaneous magnetic fields that arise during laser ablation were studied using simultaneous measurement of the polarization plane rotation and plasma electron density. A ${10}^{16}\mathrm{W}\cdot {\mathrm{m}}^{-2}$ iodine laser irradiated planar and thick Cu targets to generate a plasma in air. However, unlike other types of plasma studies, the phase $\varphi$ and amplitude $b$ are necessary to measure the polarization state. Thus, the paper presents a detailed summary of FTM and intermediate steps to recover the amplitude. The introduction of [23] includes a historical perspective of the polaro-interferometry methods for SMF measurement. Ultimately, the work reported a new multi-frame complex interferometry system that can measure the distributions of the magnetic field and electron density from the same interferogram.

6. Conclusions

Advances in plasma science greatly benefit from optical and x-ray interferometry. The first part of the chapter summarized the plasma experiment setup and the models for the plasma refractivity and the interferogram’s 2D signal phase. The Abel inversion technique was described for cases where the electron density greatly exceeds the atom density. However, the optical interferometer can cause significant radiometric variation in the image, low contrast, and low signal-to-noise. These artifacts require special attention for accurate phase reconstruction. Of the various techniques to recover the phase from a single image (e.g, principle component analysis, contour fitting of the fringe lines, and Fourier analysis), the Fourier Transform Method (FTM) is presented with detailed algorithmic steps. Additionally, recent iterative improvements to FTM and a simpler smoothing and leveling pre-filtering algorithm are highlighted. While it is less capable than the iterative method, the pre-filtering approach is demonstrated using dual-wavelength interferometry of exploded Aluminum wires.

However the plasma is formed, the interference patterns are well-suited for 2D Fourier analysis and several plasma experiments were highlighted. The experiments have confirmed previously theorized observations about the effects of bound electrons on refractive index and how precisely the plasma forms during ablation, and how it expands, emits, and eventually recombines. As bandpass and polarimetric filters and beamsplitters improve, and cameras increase in acquisition speed and sensitivity, the interferometric diagnostic will also improve to provide new and greater understanding of the plasma.

Acknowledgments

The author sincerely thanks Andrew Hamilton and Dr. Vladamir Sotnikov for support of previous work and helpful discussions, and Michael Gruesbeck and Javonne Baker for testing the smoothing and leveling algorithm. Lastly, this work was greatly improved by the review of Dr. Gennady Sarkisov whose vast experience with exploding wire experiments and plasma diagnostics motivated the initial work.

Conflict of interest

The author declares no conflict of interest.

Thanks

Thank you Xiomy for your steadfast love and faith as we live to honor Him.