## 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.

### Keywords

- Abel transform
- density measurement
- Fourier transform method
- optical interferometry
- plasma properties

## 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

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.

### 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

## 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

where the time convention is

Therefore, the intensity pattern in the reference image (

where the background

After time

where the index of refraction

The index of refraction, assuming complete ionization is expressed with the atomic and electron densities

where

Upon substituting (4) into (3) it is clear how the fringe spacing in the interferogram is determined by

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

where

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

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

In cases where

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

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

### 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

### 3.3 Ionization ratio

The ionization ratio

### 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

where the

## 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

### 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

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

** Algorithm 1.**Fourier Transform Method [12].

** Given**input image

Measure spatial frequencies

Demodulate

Recover the wrapped phase

Unwrap phase:

** return**unwrapped phase

In (13),

and the background and contrast functions are specified as

In (16),

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

The DC component centered at

### 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.

From the frequency images in the middle column of Figure 7 it is easy to see how the FTM filter

### 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

The pre-filtering approach proposed in [8] filters the background function

### 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

where

is easily inverted. In (19), the dynamic atomic polarizabilities at 1064 and 532 nm are

where

### 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

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

where

By appropriate choice of noise terms affecting

## 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 (

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

## 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.

## References

- 1.
Huddlestone R, Leonard S, editors. Plasma Diagnostic Techniques. New York, San Francisco and London: Academic Press; 1965 - 2.
Lovberg RH, Griem HR, editors. Plasma Physics Part B (Methods of Experimental Physics). New York and London: Academic Press; 1971 - 3.
Pikuz SA, Romanova VM, Baryshnikov NV, Hu M, Kusse BR, Sinars DB, et al. A simple air wedge shearing interferometer for studying exploding wires. Review of Scientific Instruments. 2011; 72 (1):1098-1100 - 4.
Sarkisov GS, Rosenthal SE, Cochrane KR, Struve KW, Deeny C, McDaniel DH. Nanosecond electrical explosion of thin aluminum wires in a vacuum: Experimental and computational investigations. Physical Review E. 2005; 71 (046404):1-21 - 5.
Sarkisov G, Beigman I, Shevelko V, Struve K. Interferometric measurements of dynamic polarizabilities for metal atoms using electrically exploding wires in a vacuum. Physical Review A. 2006; 73 :042501 - 6.
Takeda M, Yamamoto H. Fourier-transform speckle profilometry: Three-dimensional shape measurements of diffuse objects with large height steps and/or spatially isolated surfaces. Applied Optics. 1994; 33 (34):7829-7837 - 7.
Chen FF. Introduction to Plasma Physics and Controlled Fusion. New York and London: Plenum Press; 1984 - 8.
Hamilton A, Saville MA, Sotnikov V. Dual-wavelength interferometric measurement of electrically exploded aluminum and copper wires in low-pressure air for electron and ion density calculation. IEEE Transactions on Plasma Science. 2020; 48 (9):3144-3151 - 9.
Wu J, Li X, Lu Y, Lebedev SV, Yang Z, Jia S, et al. Atomization and merging of two Al and W wires driven by a 1 kA, 10 ns current pulse. Physics of Plasmas. 2016; 23 (112703) - 10.
Takeda M, Ina H, Kobayashi S. Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry. Journal of the Optical Society of America. 1982; 72 (1):156-160 - 11.
Takeda M. Temporal versus spatial carrier techniques for heterodyne interferometery. In: Arsenault HH, editor. Optics and the Information Age, International Society for Optics and Photonics. Vol. 0813. SPIE; 1987. pp. 329-330 - 12.
Takeda M. Spatial-carrier fringe-pattern analysis and its applications to precision interferometry and profilometry: An overview. Industrial Metrology. 1990; 1 (2):77-99 - 13.
Takeda M. Optical metrology: Methodological analogy and duality revisited. In: Mahajan VN, Kim D, editors. Tribute to James C. Wyant: The Extraordinaire in Optical Metrology and Optics Education, International Society for Optics and Photonics. Vol. 11813. SPIE; 2021. pp. 134-141 - 14.
Nakayama S, Toba H, Fujiwara N, Gemma T, Takeda M. Enhanced Fourier-transform method for high-density fringe analysis by iterative spectrum narrowing. AIP Advances. 2020; 59 (29):9159-9164 - 15.
Hipp M, WoisetschlÃger J, Reiterer P, Neger T. Digital evaluation of interferograms. Measurement. 2004; 36 (1):53-66 - 16.
Grava J, Purvis MA, Filevich J, Marconi MC, Dunn J, Moon SJ, et al. Soft X-ray laser interferometry of a dense plasma jet. IEEE Transactions on Plasma Science. 2008; 36 (4):1286-1287 - 17.
Doyle LA, Martin GW, Al-Khateeb A, Weaver I, Riley D, Lamb MJ, et al. Electron number density measurements in magnesium laser produced plumes. Applied Surface Science. 1998; 127–129 (716):720 - 18.
Patra AS, Phukan TD, Khare A. Measurement of two-dimensional electron density profile in a low current spark using interferometry. IEEE Transactions on Plasma Science. 2005; 33 (5):1725-1728 - 19.
Han R, Li C, Yuan W, Ouyang J, Wu J, Wang Y, et al. Experiments on plasma dynamics of electrical wire explosion in air. High Voltage. 2022; 7 (1):117-136 - 20.
Sotnikov VI, Hamilton A, Malkov MA. Collision of expanding plasma clouds: Mixing, flow morphology, and instabilities. Physics of Plasmas. 2020; 27 (12):122113 - 21.
Sarkisov G. Shearing interferometer with an air wedge for the electron density diagnostics in a dense plasma. Instruments and Experimental Techniques. 1996; 09 (39):727-731 - 22.
Sarkisov GS. Anomalous transparency at 1064 nm of a freely expanding gas cylinder in vacuum during fast electric explosion of thin metal wires. Journal of Applied Physics. 2022; 131 (10):105904 - 23.
Zaraś-Szydłowska A, Pisarczyk T, Chodukowski T, Rusiniak Z, Dudzak R, Dostal J, et al. Implementation of amplitude–phase analysis of complex interferograms for measurement of spontaneous magnetic fields in laser generated plasma. AIP Advances. 2020; 10 (11):1-20 - 24.
Fitzpatrick R. Plasma Physics an Introduction. Boca Raton, London and New York: CRC Press; 2015 - 25.
Hickstein DD, Gibson ST, Yurchak R, Das DD, Ryazanov M. A direct comparison of high-speed methods for the numerical Abel transform. Review of Scientific Instruments. 2019; 90 (6):065115 - 26.
Smith RF, Moon S, Dunn J, Nilsen J, Shlyaptsev VN, Hunter JR, et al. Interferometric diagnosis of two-dimensional plasma expansion. AIP Conference Proceedings. 2002; 641 (1):538 - 27.
Fan J, Lu X, Xu X, Zhong L. Simultaneous multi-wavelength phase-shifting interferometry based on principal component analysis with a color CMOS. Journal of Optics. 2016; 18 (5):055703 - 28.
Vargas J, Sorzano COS, Estrada JC, Carazo JM. Generalization of the principal component analysis algorithm for interferometry. Optics Communications. 2013; 286 :130-134 - 29.
Dubois A. A simplified algorithm for digital fringe analysis in two-wave interferometry with sinusoidal phase modulation. Optics Communications. 2017; 391 :128-134 - 30.
Zhong J, Weng J. Spatial carrier-fringe pattern analysis by means of wavelet transform: Wavelet transform profilometry. Applied Optics. 2004; 43 (26):4993-4998 - 31.
Maaboud N, El-Bahrawi M, Abdel-Aziz F. Digital holography in flatness and crack investigation. Metrology and Measurement Systems. 2010; XVII (4):583-588 - 32.
Sur F, Blaysat B, GrÃ©diac M. Determining displacement and strain maps immune from aliasing effect with the grid method. Optics and Lasers in Engineering. 2016; 86 :317-328 - 33.
Hu E, Zhu Y, Shao Y. Defect information detection of a spare part by using a dual-frequency line-scan method. Optik. 2014; 125 (3):1255-1258 - 34.
Rivera-Ortega U, Pico-Gonzalez B. Computational tool for phase-shift calculation in an interference pattern by fringe displacements based on a skeletonized image. European Journal of Physics. 2016; 37 (015303):1-9 - 35.
Takeda M. Fourier fringe analysis and its application to metrology of extreme physical phenomena: A review invited. Applied Optics. Jan 2013; 52 (1):20-29 - 36.
Popiolek-Masajada A, Borwinska M, Przerwa-Tetmajer T, Kurzynowski P. Application of the Fourier analysis methods to the three beam interferometry. Optics Laser Technology. 2013; 06 (48):503-508 - 37.
Zhang H, Lu J, Ni X. Optical interferometric analysis of colliding laser produced air plasmas. Journal of Applied Physics. 2009; 106 (6):063308-1-063308-5 - 38.
Singh G, Mehta DS. Measurement of change in refractive index in polymeric flexible substrates using wide field interferometry and digital fringe analysis. Applied Optics. 2012; 51 (35):8413-8422 - 39.
Ye S, Takahashi S, Michihata M, Takamasu K, Hansen H, Calaon M. Quantitative depth evaluation of microgrooves on polymer material beyond the diffraction limit. Precision Engineering. 2019; 59 :56-65 - 40.
Micali JD, Greivenkamp JE. Dual interferometer for dynamic measurement of corneal topography. Journal of Biomedical Optics. 2016; 21 (8):85007 - 41.
van Werkhoven TIM, Antonello J, Truong HH, Verhaegen M, Gerritsen HC, Keller CU. Snapshot coherence-gated direct wavefront sensing for multi-photon microscopy. Optics Express. 2014; 22 (8):9715-9733 - 42.
Huntley JM. Automated fringe pattern analysis in experimental mechanics: A review. The Journal of Strain Analysis for Engineering Design. 1998; 33 (2):105-125 - 43.
Jesus C, Arasa J, Royo S, Ares M. Design of adaptive digital filters for phase extraction in complex fringe patterns obtained using the Ronchi test. Journal of Modern Optics. 2012; 04 (59):721-728 - 44.
Zhao B, Asundi AK. Criteria for phase reconstruction using Fourier transformation method. In: Fiddy MA, Millane RP, editors. Image Reconstruction from Incomplete Data, International Society for Optics and Photonics. Vol. 4123. SPIE; 2000. pp. 269-278 - 45.
Kujawińska M. The architecture of a multipurpose fringe pattern analysis system. Optics and Lasers in Engineering. 1993; 19 (4):261-268 - 46.
Saville MA. Unpublished Notes on Fourier Transform-Based Phase Reconstruction with Smoothed and Leveled Imagery; 2022 - 47.
Feister S, Nees JA, Morrison JT, Frische KD, Orban C, Chowdhury EA, et al. A novel femtosecond-gated, high-resolution, frequency-shifted shearing interferometry technique for probing pre-plasma expansion in ultra-intense laser experiments. Review of Scientific Instruments. 2014; 85 (11):11D602 - 48.
Nilsen J, Johnson WR. Plasma interferometry and how the bound-electron contribution can bend fringes in unexpected ways. Applied Optics. Dec 2005; 44 (34):7295-7301