Implementation of amplitude–phase analysis of complex interferograms for measurement of spontaneous magnetic fields in laser generated plasma

Spontaneous magnetic field (SMF) research plays an important role in understanding various phenomena and processes associated with the interaction between laser radiation and matter, producing a new trend in high-density energy physics. One interesting area of research is the use of SMFs for generating magnetized plasma streams. This is useful in both research related to the implementation of inertial confinement fusion (ICF) and that related to laboratory astrophysics.

Previous studies show that SMFs above 1 MG can significantly change plasma transport coefficients, thus likely affecting electron concentration, plasma temperature distribution, absorption of laser radiation, and constraining plasma ablation pressure.^1,2

Therefore, with regard to important elements in ICF, of which the most important are shock ignition (SI)^3,4 and fast ignition (FI),^5,6 a deep understanding of SMF generation mechanisms and their role in fusion ignition realization is very important.

In astrophysical research, one of the main difficulties associated with modeling objects and phenomena in the universe is the generation of magnetized hot plasma streams. The use of SMFs mechanisms for this purpose is an interesting alternative to traditional magnetic field generators. A new idea in this field is using optical generators based on capacitor-coil targets to magnetize laser plasma streams.⁷ 

It should be emphasized that a detailed knowledge of SMFs space–time distributions is crucial in ICF concept implementation. Therefore, improvements in SMF measurement methods and quantitative data analysis are particularly important in order to increase the reliability of results.

SMFs measurements in laser plasmas are a difficult experimental task mainly because such fields are generated in the initial phase of ablative plasma expansion during the interaction between the laser pulse and the plasma generated by this pulse. Under these conditions, to obtain information on the amplitude and structure of SMFs, a sophisticated diagnostic tool, with sufficiently high spatial and temporal resolution, is required. The most common methods of SMFs measurements are (i) optical techniques based on the Faraday effect^8–12 and the Stark–Zeeman effect,^13,14 (ii) magnetic^15,16 and current¹⁷ probes, and, more recently, (iii) proton deflectometry.^18,19 The polaro-interferometric method based on the Faraday magneto-optical effect obtains a sufficient amount of information on a magnetic field, with high space–time resolution, over the entire area of investigated plasma.

However, the method based on the Faraday effect is difficult to implement, and it requires simultaneous measurement of the distribution of the angle of probe beam polarization plane rotation and electron-density distribution. The former may be obtained with polarimetric measurement, and the latter may be determined from the phase-shift distribution of a probe beam acquired from interferometric measurements.

SMFs measurements using the method based on the Faraday magneto-optical effect were carried out for the first time in 1975 by Stamper.¹¹ The magnetic fields were generated on an illuminated plane of thick targets with neodymium laser radiation at an intensity of about 10¹⁵ W/cm². SMFs with a maximum amplitude of several MG were observed, independent of the target material. In later studies,^20,21 the methodology of SMFs measurements was improved. In Ref. 20, to effectively eliminate plasma self-illumination (which would hinder measurement) generated by the second harmonic of the neodymium laser, the diagnostic beam was tuned by means of a Raman shift to a wavelength of λ = 633 nm. In Ref. 21, interferometry was used for the first time together with measurements of the angle of polarization plane rotation, permitting not only an assessment of the SMF amplitude but also the determination of the SMF profile.

SMFs studies were also conducted for the spherical heating of microspheres.^8,22,23 The results presented in Refs. 22 and 23 were obtained in the Rutherford laboratory in Chilton in the United Kingdom. These studies demonstrated that, provided the symmetrical irradiation of microspheres, SMFs attain amplitudes of several hundred kG.²² Two mechanisms are responsible for SMFs generation: (i) thermoelectric instabilities and (ii) thermal instabilities. In addition, the SMFs have a subtle structure²² confirmed by theoretical predictions^24,25 and polaro-interferometric measurements carried out on the DELFIN laser system at the Lebedev Physical Institute in Moscow.⁸ 

One of the few papers concerning the methodology of polarimetric measurement is Ref. 9. This paper presents the fundamentals of magnetic field measurement based on the magneto-optical Faraday effect and analyzes in detail the impact of factors, such as probe wave depolarization and refraction, inhomogeneities in the intensity of a diagnostic beam, and the effect of self-illumination of the investigated plasma on the measurement error of an angle of polarization plane rotation.

After 1990, new ways of conducting measurements in laser plasmas became available using the imaging diagnostics of CCD cameras. This greatly extended the capabilities of the automated measurement, recording, and analysis of data. The attempt to automate polaro-interferometric measurement is presented in Ref. 26. The subject of automation was a three-channel polaro-interferometer, equipped with CCD cameras, with a matrix of resolution of 512 × 512 pixels, allowing a simultaneous recording of a polarogram, an interferogram, and a shadowgram.

SMFs measurements at the Prague Asterix Laser System (PALS) laser system were successfully carried out using a setup with a two-channel polaro-interferometer.²⁷ This system made it possible to simultaneously obtain information on the distribution of the angle of polarization plane rotation (in the polarimetric channel) and on the phase-shift distribution (in the interferometric channel). The polaro-interferometer was irradiated by a Ti:Sa laser pulse, with a wavelength of 808 nm and a duration of about 40 fs. This enabled retrieval of information on the angle of polarization plane rotation and the phase in the very early stages of ablation plasma expansion in the time interval covering the PALS iodine laser pulse length. The subject of the investigation was the SMFs generated from the interaction between the fundamental-frequency PALS iodine laser pulse, with an energy of about 250 J and a duration of 300 ps, and flat massive targets of Cu or plastic. The obtained results in Ref. 27 have differences in the structures of the SMFs, depending on the atomic number of the target material, what has not been shown in previous SMF-related research.²⁸ Based on the results from Ref. 27, current density distributions in the ablative plasma generated from flat thick Cu targets have been calculated and published in Ref. 10. The structure of the SMFs and the current-density distribution in the ablative plasma showed that (i) fast electrons produced as a result of resonance absorption are responsible for SMFs generation and (ii) a significant part of the current flowing in the ablative plasma, the so-called direct current, is carried by streams of fast electrons (with energies in the range of several dozens to hundreds of keV) ejected from the target surface.

However, a key problem in continuing SMFs studies at PALS using the magneto-optical Faraday effect is the reliability of measurements of the angle of polarization plane rotation and the phase shift of probe radiation. Accordingly, the most important task is to choose the appropriate measurement methodology, which includes choosing the appropriate construction and parameters of a measurement system, as well as the appropriate method of quantitative data analysis.

When compared to classical polaro-interferometry, complex interferometry seems to be a more reliable method of measuring SMFs because information about the SMFs distribution in the plasma is obtained using the amplitude–phase analysis of one image (a complex interferogram) instead of two images (a polarogram and an interferogram), registered in separate channels. A complex interferogram is obtained by merging a polarimetric and an interferometric channel into one channel. In this channel, a distribution of the angle of polarization plane rotation is represented by an interference-fringe-intensity distribution, while interference-fringe shifts on the interferogram represent a phase distribution. Despite the mathematical foundations of complex interferometry being known,²⁹ it has not been realized to a significant degree in SMF-related research.

In order to apply complex-interferometric measurements, a two-channel femtosecond polaro-interferometer setup, used in investigations at the PALS system,^10,27 was modified. In addition, based on the modified polaro-interferometer, three-frame SMFs measurements have been performed (described in Sec. III A).

To obtain information on the phase shift and the angle of polarization plane rotation of a probe beam from complex interferograms, a procedure of quantitative analysis of the phase–amplitude structure of interference fringes has been developed and is described in this paper. The main part of the paper constitutes the SMFs measurement results acquired from the PALS experiment.

The Faraday effect is caused by a magnetic field and results in a polarization plane rotation of an electromagnetic wave that propagates through a medium along magnetic field lines (see Fig. 1).

In a medium of laser plasma, the Faraday effect is attributed to birefringence that occurs there in the presence of the magnetic field. The plasma can, therefore, be characterized by two refractive indices n₁ and n₂, where subscript 1 is related to an extraordinary component of the probing wave and subscript 2 is related to an ordinary component of the probing wave.

In general, n₁ and n₂ depend on the magnitude of the magnetic field, the electron density of the plasma, and also the probing wave frequency and propagation with respect to the magnetic field lines. Assuming that the probe wave frequency ω is much higher than the ion cyclotron frequency Ω_B (ω ≫ Ω_B), ignoring the electron–ion collision frequency, and neglecting absorption, the refractive indices n₁ and n₂ can be described by the following equation:^30,31

and the polarization coefficient of each wave is described by the following equation:

In Eqs. (1) and (2), α is an angle between the direction of probe-wave propagation and the magnetic field lines, while parameters u and v are represented by the following proportions:

where $ωe=4πnee2/me$ is the electron plasma frequency, $ωB=eB/mec$ is the electron cyclotron frequency, e is an electron charge, m_e is the electron mass, n_e is the electron density in a plasma, and c is the speed of light.

Equations (1) and (2) lead the general case (when α ≠ 0), where both the wave components (extraordinary and ordinary) and the resultant wave exhibit elliptical polarization, as illustrated in Fig. 2.

Extraordinary and ordinary waves have left- and right-hand polarization, respectively. The degree of polarization of these waves and the rotation angle of the resultant wave result from the direction of propagation of the probe wave with respect to the magnetic-field force lines, as well as the frequency of the probe wave (ω) with respect to the characteristic plasma frequencies (ω_e and ω_B), which are functions of parameters u and v.

We can therefore distinguish two extreme cases:

1. The probing wave propagates parallel to the direction of the magnetic-field force lines in the plasma (α = 0°).

2. The direction of wave propagation is perpendicular to the direction of the magnetic-field force lines (α = 90°).

The first case, shown in Fig. 1, concerns the Faraday effect. Equation (2) can be simplified, and as a consequence, the value of the polarization coefficient for the extraordinary component is K₁ = i, which means left-circular polarization; the value of the polarization coefficient for the ordinary component is K₂ = −i, which means right-circular polarization—the resultant wave has linear polarization. After passing through the plasma, the wave is rotated by an angle equal to a half of the phase shift Δδ between the component waves,

where L is the path of the probe wave in the plasma and λ is the wavelength of the probe wave; Eqs. (1) and (2) are simplified to give³¹ 

Based on Eqs. (4) and (5), we can describe the angle of polarization plane rotation as a function of the magnetic-field induction B_∥ and electron density n_e in the plasma,

The instance when α = 90° is related to the Cotton–Mouton effect. Both the wave components (ordinary and an extraordinary) are linearly polarized, and in contrast to the Faraday effect, the resultant wave, after passing through the plasma, can take any polarization state: linear, circular, or elliptical, depending on the phase difference between constituent waves according to following formula:³⁰ 

Figure 3 shows the Faraday and the Cotton–Mouton effects in axially symmetrical laser plasma when the linearly polarized wave propagates at different angles to the magnetic-field force lines.

Figure 3 shows that to determine the angle of polarization plane rotation (φ), using azimuthal geometry for the magnetic field, both the Faraday and Cotton–Mouton effects should be taken into account. Then, Eqs. (1) and (2) should be used, depending on the angle between the direction of wave propagation and the direction of the magnetic-field force lines.

However, under certain conditions, it is possible to use the simplified expressions of Eqs. (6) and (7) representing the Faraday and Cotton–Mouton effects, respectively. In the case of the Faraday effect, the conditions for quasi-longitudinal wave propagation are given by³⁰ 

In order to fulfill the conditions in Eqs. (8) and (9), the parameters u and v must be much smaller than unity (u, v ≪ 1): Using the proportions in Eq. (3), this is achieved when the probe wave frequency ω is much higher than the electron cyclotron frequency ω_B and the electron plasma frequency ω_e(ω ≫ ω_B, ω_e). Then, the Faraday effect predominates the whole area of investigated plasma, the Cotton–Mouton effect is negligible, and a simplified form of Eq. (1) may be used,

The final expression for the angle of polarization plane rotation φ of the probe beam is given as follows:

As the angle φ is proportional to the square of the wavelength λ, when choosing the optimum frequency of the probing beam, one has to remember that a higher frequency results in a smaller angle, requiring higher sensitivity in the measuring setup. In the case of a laser plasma with an electron density in the range of 10¹⁸–10²⁰ and an estimated SMF induction in the order of a few MG, the optimum frequency of the probe beam is within the visible band of the electromagnetic spectrum.

According to Eq. (11), the magnetic field induction B is related not only to the angle φ of polarization plane rotation but also to the electron density n_e; the latter being determined using the phase distribution δ obtained from interferometric measurement of laser plasma.

In the case of axially symmetrical laser plasma (illustrated in Fig. 4), the distributions of angle φ and phase δ are described by the following Abel integral equations:⁹ 

where $Bφr$ is the azimuthal magnetic field distribution, n_e(r) is the radial electron-density distribution, λ is the wavelength of the probe wave, and R is the plasma radius.

Solving the set of Eqs. (12) and (13), the SMF distribution can be calculated according to the following formula:⁹ 

where the distribution function f_B(r) is calculated by solving the Abel equation of the distribution of the normalized angle of polarization plane rotation S_B(y) = φ(y)/y and the distribution function f_n(r) is obtained by solving the Abel equation of the distribution of the phase S_n(y) = $δy$⁠.

The normalized angle distribution is here defined as

where φ(y, z) is the angle of polarization plane rotation at (y, z), r(y) is the radius—the distance of a point from the plasma symmetry axis, z is the distance of a point from the target’s surface, and R_max is the maximum radius of the plasma.

The functions $fBr$ and $fnr$ are determined using numerical methods, since the experimental functions φ(y) and $δy$ are given in the form of measured values at many points, the number of which depends on the CCD resolution.

There are two basic types of approach for the numerical Abel equation solution: interpolation and quadratic mean approximation methods.^9,32–35

Among the interpolation methods, the most basic, yet reliable and useful, are the ones that use the Lagrange interpolation polynomial of degree n = 0 and 1:

1. n = 0, the Mach–Schardin method (linear interpolation),^9,32,35 and

2. n = 1, the Van–Vorish method (interpolation by trapeze).^9,32,34

In the quadratic mean approximation, even polynomials of different degrees^34,35 are used to approximate the sought-after function f(r).

However, in high-resolution interferograms recorded using CCD cameras, both approaches deliver unsatisfactory results due to vast amounts of data that adversely affect the quality of reproduction of the functions. In interpolation methods, the error determining the distribution function increases with the number of points and is the largest on the plasma axis. In the average square approximation using polynomials, too high ratio of the resolution to the polynomial degree causes smoothing and loss of detail in the sought-after distribution function.

A more appropriate approach for solving the Abel equation involves fast Fourier transform (FFT),^29,35,36 since, contrary to other methods, its accuracy increases with the number of measured points of an experimental function. The FFT method is similar to the method based on quadratic mean approximations, but the sought-after functions (f_B and f_n) are approximated using the Fourier series.

A common method of measuring the angle of polarization plane rotation involves the use of two crossed polarizers.⁹ A schematic diagram of the diagnostic setup is presented in Fig. 5.

Behind the first polarizer, the probe beam is polarized horizontally. Then, while passing through the plasma, the polarization of the beam is rotated by an angle φ due to the presence of a magnetic field. The second polarizer—an analyzer—transmits the vertical component of polarization. As a result, the intensity (I) detected behind the analyzer depends on the angle φ according to the Malus’s law,

where I₀ is the intensity of the probe beam in front of the analyzer.

Under real measurement conditions, there are three main factors that negatively influence the accuracy of measurement:

1. Depolarization of the probe beam.

2. Inhomogeneity of the energy distribution in a cross section of the probe beam.

3. Influence of parasitic plasma self-luminosity.

Depolarization of the probe beam may occur when the conditions of quasi-longitudinal propagation [Eqs. (8) and (9)] are not achieved, but it can also be caused by scattering and refraction due to the plasma density and temperature gradients, imperfections in optical elements, and a setup misalignment. Depolarization cannot be completely eliminated; however, it may be reduced using a probe beam with a short wavelength, high-quality optical elements, and scrupulous setup alignment.

Errors related to inhomogeneities in cross-sectional intensity of the probe beam are difficult to eliminate. Such errors can be minimized by multi-channel registration, e.g., by registering the additional image (shadowgram) as in Ref. 37. Another solution, implemented by the authors of this paper, is to record a reference polarogram with plasma absent. This does not neutralize random shot-to-shot intensity fluctuations but eliminates inhomogeneities that repeat from shot to shot, e.g., caused by imperfections in optics and by setup misalignment.

The influence of parasitic plasma self-luminosity may be reduced by using narrow-band color or interference filters, spatial filters (diaphragms), detectors with optimal spectral sensitivity, and high-speed shutters (e.g., electro-optical filters³⁸).

An additional difficulty affecting the reliability of polaro-interferometric measurement is the refraction and absorption of the probe beam by the plasma. The associated errors are difficult to eliminate and can only be minimized by multi-channel registration,³⁷ significantly complicating measurement.

Because the complete elimination of parasitic plasma illumination and probe wave depolarization is practically impossible, polarimetric measurements are usually carried out with an initial rotation of the analyzer by some angle φ₀, at which the ratio of the useful signal to the parasitic signal (resulting from depolarization and self-luminosity of plasma) is highest.

The balance of intensity registered by the detector⁹ is expressed by the following equation:

where K is a polarization coefficient of the probe beam [Eq. (2)], k is the contrast of the polarizers, and I_pl is the intensity of plasma self-luminosity.

The parasitic background signal is described as

The optimum angle of the initial rotation of the analyzer at which the ratio of useful signal to background signal (I/I_B) is largest (for small polarization plane rotation angles in plasma) is described as⁹ 

where ε = I₀/I_pl is the ratio of the probe beam intensity to the intensity of plasma self-luminosity.

The dependence of the initial rotation angle of the analyzer φ₀ on ε, when the probe wave does not experience depolarization (K = 0) and the contrast of polarizers in the measuring system is k = 5 × 10⁻⁶, is presented in Fig. 6.

Figure 6 shows that increasing parasitic plasma self-luminosity causes the need to register a polarogram with increasing initial rotation angle of the analyzer. In the absence of self-illumination of the plasma, the best registration conditions occur when the polarizers are completely crossed; this is provided that the probe wave does not depolarize (K = 0), and the polarizer contrast in the polarimeter is high enough.

In experiments carried out at PALS (presented later in the paper), φ₀ = 2° was selected as the optimum angle of initial rotation of the analyzer. The selection was made by taking into account the highest possible suppression of self-illumination of the plasma at ε = 10³, as well as the polar–interferometer parameters.

A two-channel femtosecond polaro-interferometer (see Fig. 7), which was hitherto used in investigations at the PALS laser system, ^10,26 provided for SMFs measurements in the complex interferometry regime.

The system shown in Fig. 7 can operate in two regimes, depending on the configuration of the polarimetric channel.²⁷ In the first regime—classical polaro-interferometry, a polarogram and an interferogram are recorded separately. In the second regime—complex interferometry, a complex interferogram is recorded in the polarimetric channel by placing the wedge, as shown in Fig. 7.

The polaro-interferometer is irradiated by a femtosecond Ti:Sa laser beam with a wavelength of 808 nm and a pulse duration of 40 fs. To obtain optimal registration conditions both in the polarimetric and in the complex interferometry channel, the initial rotation angle of the polarizer (φ₀) was determined using Eq. (19). Taking into account the parameters ε ≈ 10³, K = 0, and k = 5 × 10⁶, measurements were carried out for φ₀ = 2°. Examples of a polarogram, an interferogram, and a complex interferogram demonstrating the operation of a two-channel polaro-interferometer in both regimes are shown in Fig. 8.

The complex interferogram in Fig. 9(b) shows Faraday illumination in the lower half of the plasma stream. This asymmetry of the Faraday effect is caused by the initial rotation angle of the analyzer φ₀ = 2° in relation to the polarizer, and it confirms the azimuthal geometry of the magnetic-field force lines. Moreover, interferometric fringe shift in the complex interferogram contains information on the electron-density distribution, and the intensity modulation contains information on the angle of the polarization plane rotation; therefore, one complex interferogram gives the same amount of information that is obtained from a polarogram and an interferogram. Using complex interferometry simplifies the measurement system, since instead of two channels, only one, operating in complex interferometry mode, is needed. Consequently, a measurement system for complex interferogram registration, shown in Fig. 9, was constructed, which is a modified two-channel polaro-interferometer (Fig. 7).

The Wollaston prism polarizer is replaced by a Glan–Thompson one, mounted crosswise with respect to the first polarizer and additionally rotated by an angle of φ₀. The wedge located after the telescope induces the interferometric fringes, which are modulated in both amplitude and phase. The diameter of the probe beam should be at least twice the size of the investigated plasma because only then can the undisturbed part of the probe beam be used as a reference beam.

The measurements were carried out using a lens of focal length f = 500 mm and a wedge with a breaking angle γ = 3°, the position of which was tuned to obtain width of an interference fringe of approximately 40 µm and optimal separation of beams.

Using three single-channel modules for registration of the complex interferograms, multi-frame measurements of the SMFs at the PALS laser system were performed. The optical scheme of the three-frame complex interferometer is presented in Fig. 10.

To record three temporally separated interferograms, the polaro-interferometer is irradiated by a Ti:Sa femtosecond laser, with a full width at half maximum of 40 fs and an energy of 10 mJ, through an optical delay line, as illustrated in Fig. 10. Each of the three beams pass through a plasma plume perpendicularly to the normal line to a target, and the separation angle between the beams is 5°. Assuming that the laser plasma is axially symmetrical, the angular separation does not affect the probing conditions of the three beams. Taking into account the dimensions of the experimental chamber and window locations, this system enables the registration of three consecutive complex interferograms, with a time interval between frames of about 400 ps. The moment in time for the first frame of the sequence is selected in relation to the moment of maximum intensity of the main laser pulse. The heating and diagnostic laser pulses are synchronized with an accuracy of ±100 ps by the electronic synchronization method, described in Ref. 39.

In each channel, interferograms are registered by means of a high-resolution CCD camera with a matrix of 2048 × 2048 pixels, a pixel size of 6 µm, and 12-bit dynamics. The cameras support the GigEV protocol; they are connected to a computer through a gigabit ethernet interface and controlled by custom-built software, PALS Vision GigEV.

Basic functions of the software are as follows:

1. Camera operation control (e.g., change in resolution and time parameters for internal pulse generators).

2. Real-time image preview.

3. Image acquisition in preview and triggered modes.

4. Export of recorded images to popular graphic formats, e.g., RAW and TIF.

5. Processing of recorded images (modification of Fourier-spectrum images, histogram corrections, and scaling).

Figure 11 shows three-frame sequences of raw, complex interferograms obtained for the interaction between a laser beam with an energy of about 500 J and a Cu thick planar target. The full set of interferograms obtained in the experiment and used for detailed analysis covers the range t = −200 ps to about t = 1000 ps. All complex interferograms exhibit the anti-symmetry related to the combination of the Faraday effect with the initial rotation of the analyzing polarizer at φ₀ = −2° (counter-clockwise). For this initial rotation direction of the analyzer, the brightening connected with the Faraday effect is noticeable in the upper half of the expanding plasma in Fig. 11. Here, the rotation direction of the probe beam polarization is the same as the direction of initial rotation of the analyzer. In the lower part of the expanding plasma, darkening occurs due to rotation of the diagnostic beam polarization in the opposite direction. This observation is consistent with the expected physical azimuthal structure of the SMF distribution with respect to the target normal.

To obtain information on the space–time distributions of the electron density and the SMF, amplitude–phase analysis of the complex interferograms was conducted.

The theoretical basis of complex interferometry has been thoroughly described elsewhere, e.g., in Ref. 30, but is described here briefly. The intensity of a recorded complex interferogram may be described by the following formula:

where ω₀ and ν₀ are the spatial frequencies in the horizontal and vertical directions and functions $by,z$ and $vy,z$ are background and visibility functions, respectively,

The background function $by,z$ contains information about the amplitudes of the reference beam A_r and the probe beam A_p(y, z), and the visibility function $vy,z$ contains information about the phase shift of the probe beam δ(y, z).

In order to extract the background and the visibility functions from Eq. (20), a Fourier transformation should be performed. Then, the intensity distribution of a complex interferogram may be described in the frequency domain $ω,υ$⁠,

The terms $Vω−ω0,υ−ν0$ and $V*ω+ω0,υ+ν0$ represent the visibility function and they correspond to the side lobes of the graph in Fig. 14; the term $Bω,υ$ represents the background function and corresponds to the central lobe of the graph in Fig. 14.

Information on the visibility functions is obtained as follows:

1. The selection of a V or V* function in order to eliminate the influence of other bands.

2. The shift of the selected function to zero frequency.

3. Performing an inverse Fourier transform to obtain the space (y, z) domain.

The procedure may be mathematically described as

The procedure to obtain the background function is similar because, as the B function is already located around zero frequency, it does not have to be shifted:

1. Select a B function.

2. Perform an inverse Fourier transform to obtain the space (y, z) domain,

Both of the above procedures are illustrated in Fig. 14.

When the functions v(y, z) and $by,z$ are determined, one may calculate the phase shift δ(y, z) and the ratio of amplitudes of the probe and the reference beam γ = A_p/A_r, according the following equations:

where ρ is the ratio of energies of the probe and the reference beams, b_p(y, z) is the background function extracted from the complex interferogram recorded in the presence of plasma, and b_r(y, z) is the background function extracted from the reference interferogram, recorded in the absence of plasma. The analysis of a complex interferogram, with a corresponding reference interferogram, allows the determination of the phase and amplitude distributions with high accuracy, even in the case of poor beam quality.

The polarization plane rotation angle can be calculated as follows:

where φ₀ is the initial angle of rotation of the analyzing polarizer.

A signal-free reference interferogram is used to improve amplitude reconstruction and is also used for phase reconstruction.

Software for complex interferogram analysis was written in the Python Spyder environment and tested on computer-generated synthetic interferograms; these were created according to Eq. (20) using phase and fringe-intensity distributions typical for laser plasma. The software with its preliminary tests has been presented in Ref. 40 and described in detail in Ref. 36.

Figure 13(a) shows a synthetic interferogram, whose phase distribution δ(y, z) and amplitude ratio γ(y, z) are described by

For each complex interferogram, a reference interferogram with the same frequency of fringes but without distortion was generated. Figure 13 shows a Fourier spectrum obtained from a Fourier transform of the synthetic interferograms in Figs. 12(a) and 12(b).

Side lobes represent the visibility functions, and the center lobe represents the background function. The algorithm described in Sec. III B was implemented in order to determine the phase and amplitude distributions of the synthetic interferograms. The process of visibility and background function determination based on the Fourier spectra of the main and reference synthetic interferograms is illustrated in Fig. 14.

In order to carry out visibility determination [see Fig. 14(b)], a side lobe should be selected from the main interferogram spectrum. Simultaneously, the same area of spectrum is selected as a reference interferogram. Then, for both interferograms, all values beyond the selected areas are reduced to zero, and the areas are shifted to zero frequency. Next, an inverse Fourier transform is performed. Using both the obtained visibility functions, the phase-shift distributions for the main [δ(r, z)] and for the reference [δ_ref(r, z)] interferograms are calculated according to Eq. (26), and then, the reference phase is subtracted from the phase obtained from the main interferogram.

A similar procedure is used for the determination of the background function, which is illustrated in Fig. 14(c). The central lobe is selected on the Fourier spectrum of the main interferogram.

Exactly the same area is selected on the Fourier spectrum of the reference interferogram. All values beyond the selected area are then reduced to zero, and an inverse Fourier transform is performed. The two obtained background functions b_p(y, z) and b_r(y, z) are used in Eq. (27) in order to calculate the amplitude γ(y, z). Then, the angle of polarization plane rotation is calculated according to Eq. (28).

Phase determination is sensitive to the inappropriate selection of the side-lobe contour. When the selected area is too wide and it overlaps the central lobe area, the amplitude modulation affects phase reconstruction, meaning that interferometric fringes are visible in the reconstructed phase. Conversely, when the selected area is too narrow, some information on the phase is lost, and as a result, some non-linear artifacts occur in the reconstructed phase close to the target surface. When selecting the side-lobe contour, the widest area that does not overlap the central lobe should be chosen. Examples of inappropriate lobe-contour selection and its consequences in phase reconstruction are presented in Fig. 15. Additionally, as part of resultant phase verification, the reconstructed phase distribution must always be compared with the interferogram, as the phase distribution shape should correspond to the interferometric fringe shape.

In order to assess the usefulness of the developed software for phase–amplitude analysis of complex interferograms, the reproduced functions for phase distribution δ(y, z) and amplitude distribution γ(y, z) were compared with their corresponding analytical functions. Figure 16 shows a comparison of the phase and amplitude distributions used to create synthetic data with those calculated from synthetic complex interferograms using the described methodology.

Both the shape and the values of the phase and amplitude distributions are satisfactorily reproduced with the developed software. The distribution of absolute error of the calculated phase and amplitude is shown in Fig. 17. The relative error of reproduction of the maximum value of phase is Δδ/δ ≈ 7%, while the relative error of reproduction of the maximum value of amplitude is Δγ/γ ≈ 1.4. It should be noted that when comparing the reproduced amplitude distribution in Fig. 16(b) with the absolute amplitude error in Fig. 17(b), the absolute error of maximal amplitude reconstruction is lower than 0.04.

The presented software was tested for various assumed phase distributions, typical for laser plasma, and for various amplitude distributions that corresponded to polarization plane rotation angles expected from experiment.

A comparison of the assumed and calculated phase distributions is presented in Figs. 18 and 19.

Figure 18 shows test results for different phase distributions. In Fig. 18(a), the assumed phase is described by

and in Fig. 18(b), the assumed phase is described by

The amplitude distribution in both figures is described by the same equation,

where C = 1.026, and the maximum intensity corresponds to the value of angle of polarization plane rotation φ = 1.5°.

The software was also tested for different amplitudes, but the same phase, the results of which are presented in Fig. 19. The assumed phase distribution is created using Eq. (32), and the amplitude distributions are described by Eq. (33). The parameter C is adjusted so that the maximum values of intensity represent the maximum values of polarization plane rotation angle φ: C is equal to 0.685 for φ = 1° and 2.736 for φ = 4°. Based on the assumed and calculated amplitude distributions, the angle of polarization plane rotation was determined according to Eq. (28). Figure 19 shows a comparison of the assumed and calculated angles of polarization plane rotation distributions for different maximum values of the angle. The assumed initial angle of rotation φ₀ is 2°. When the angle maximum is 4°, the accuracy of reproduction decreases due to saturation of the Faraday signal; however, the relative error of reproduction is still in the range of 10%.

It should be mentioned that the different assumed amplitudes, presented in Fig. 19, do not affect phase reconstruction because Faraday brightening does not overexpose the interferometric fringes, meaning that they have enough definition to be analyzed. When Faraday brightening attenuates the interferometric fringes so they are not visible, the information regarding phase is lost. With the experimental conditions at the PALS facility and the diagnostic beam parameters given in Sec. III A, the expected value of rotation of the polarization plane is in the range of 4°, and the Faraday brightening related to this value does not overexpose interferometric fringes. If an angle of rotation greater than 4° occurs, one may change the initial angle of rotation between the polarizer and analyzer in order to change the sensitivity of the setup; this allows the setup to be adapted for larger angles of polarization plane rotation detection.

Figure 17(b) shows that maximal absolute errors appear for minimal values of amplitude, which correspond to the part of the interferogram obscured by the Faraday effect. Polarimetric measurement is more sensitive to an increase in illumination rather than to a decrease. This means that angle reproduction is more accurate in the area of Faraday brightening. Because of the axial symmetry of the laser plasma, the angle distribution calculated using the half of the interferogram where Faraday brightening occurs is symmetrized with respect to the z-axis for further analysis. Figure 20 shows the angle distribution after symmetrization, which corresponds to the rotation angle distribution in Fig. 19(a).

A developed methodology for the phase–amplitude analysis of complex interferograms³⁷ was used to quantitatively interpret SMF measurements performed during the PALS experiment, where a multi-frame system was applied. The purpose of this research was to implement this new diagnostic tool because it is more reliable than classic polaro-interferometry. In order to demonstrate the quantitative analysis of complex interferograms using the software described in Sec. III C, a sequence of complex interferograms is selected, shown in Fig. 11(c).

An important feature of the software is that it performs data preparation: appropriate cropping (with the selected resolution) and symmetrization and equalization of the intensity of both main and reference interferograms.

Figure 21 shows complex interferograms and their corresponding reference interferograms and Fourier spectra; Fig. 22 shows the distributions of phase and the angle of polarization plane rotation calculated using the software.

Figure 22 confirms that the angle of polarization plane rotation is largest for t = 19 ps, when the laser beam intensity reaches its maximum value. Later, after the end of the pulse, the rotation angle decreases several times due to the radial expansion of plasma and remains at about 1°.

According to the methodology presented in Sec. II, to calculate the magnetic-field distribution using Eq. (14), both the field distribution function $fBr$ and the electron-density distribution function $fnr$ should be determined. The function $fnr$ is determined by solving an Abel equation [Eq. (13)] for the phase distribution δ(y, z); this is performed using the software for Abel equation solution described in Ref. 41.

In order to determine the normalized angle distribution φ(y, z)/y using Eq. (15), the maximum radius R_max is assumed to constitute the border of the area limited by the plasma envelope at a level of 5 × 10¹⁷ cm⁻³. All values beyond this area are reduced to zero. The field distribution function $fBr$ is obtained by solving an Abel equation for the normalized angle distribution φ(y, z)/y. The results of the calculation are presented in Fig. 23.

Finally, having determined the distribution function of the normalized angle and the distribution function of the electron density, the distributions of SMFs are calculated using Eq. (14). The results of the calculations are shown in Fig. 24.

Time changes in the electron-density distributions, shown in Fig. 24, clearly demonstrate the spherical character of ablative plasma expansion, with a minimal value on the axis, which deepens with time. The SMF distributions, shown in Fig. 24(b), are characterized by maximal magnetic-field values at the plasma front in the area near the symmetry axis. The amplitude reaches a value of 9 MG for t = 19 ps, when the laser intensity is highest, and then decreases to about 2 MG (for t = 449 ps) due to the radial expansion of the plasma and weakening of SMFs generation mechanisms.

An important issue is the accuracy of the obtained data. A deviation in axial symmetry, which is inevitable in experimental data, leads to an error of about 5%–10% for both the extracted phase and rotation angle distributions. The assumption of axial symmetry works in this procedure as an uncertainty factor, mainly in the central region, thus increasing the error in the magnetic field near the axis.^10,27 To minimize the error, a procedure of creating synthetic interferograms based on the distributions of rotation angle and phase shifts was introduced in Sec. III B.

Based on the space–time distributions of the SMFs, the current density distributions, related to electrons moving both from the target (the so-called direct current) and toward the target (the so-called return current), are calculated. To calculate the current density distributions, Ampere’s law is used,¹⁰ 

where $Bφr,z≡B(r,z)$⁠, if we assume the magnetic field to be exactly azimuthal $B⃗0,Bφ,0$ and disregard the displacement currents.

The integrated (total) current along the z-axis at different moments in time is estimated using the following equation:

where j_z(r, z) corresponds to either the direct or return current.

The results of the calculation of current density and total current are shown in Fig. 25.

Similar to what is discussed in Ref. 10, two currents are observed: (i) a direct current for j_z < 0, corresponding to electrons moving away from the target, and (ii) a return current for j_z > 0 corresponding to electrons moving toward the target. In both cases, the absolute value of the current density j = |j| is shown. The distributions shown in Fig. 25(a) clearly demonstrate that a significant part of the direct current is concentrated close to the symmetry axis and can reach values larger than 1014 A/m². Such values correspond to electrons with energies of several tens or even hundreds of keV, recorded by spectroscopic measurements.⁴² As for the total current, as expected, it reaches a maximum value of about 800 kA when the laser intensity is highest and then gradually decreases, reaching a value about twice as low at t = 834 ps, after the intensity maximum.

In order to correctly determine the SMF distribution using complex interferometry with the developed software, the complex interferograms obtained in the experiment were compared with reconstructed synthetic interferograms. To create the latter, a calculated SMF distribution was converted back to a polarization plane rotation angle using Eq. (12), and then, together with the determined experimental phase, it was substituted into Eq. (20), describing the intensity distribution of the complex interferogram. This permitted us to control non-signal values that appeared in the synthetic interferograms and—in the case of significant error—to discard them, especially in the close-axis regions. A comparison of complex interferograms obtained in the experiment with their synthetic reconstructions is shown in Fig. 26. The figure is evidence of the differences between the real Faraday effect and the reconstructed one.

In this paper, the results of SMF measurements conducted using complex interferometry are presented. Thanks to the implementation of this unique diagnostic tool at the PALS facility, we conducted measurements in a multi-frame regime. This allowed us to obtain information on the space–time distribution of SMFs and the electron density in ablative plasma generated by Cu thick planar targets.

It should be emphasized that multi-frame complex interferometry creates greater measurement possibilities compared with classical polaro-interferometry in laser plasma studies in ICF and astrophysics. Complex interferometry is more reliable because, unlike classical polaro-interferometry, it obtains information on the distribution of the magnetic field and electron density on the basis of one (the same) interferogram—a complex interferogram.

In shock-ignition, information on the magnetic-field distribution is crucial for understanding the processes associated with the generation of hot electrons. These electrons are responsible for the transport of laser energy to the shock wave, which ignites the thermonuclear fuel. In laboratory astrophysics, a particularly interesting area of research involves the use of knowledge of SMFs for the production of magnetized plasma streams and the study of the processes that occur during magnetic-field reconnection.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

This research was supported by the access to the PALS RI under the EU LASERLAB IV project (Grant Agreement No. 654148), the Ministry of Science and Higher Education, the Republic of Poland (Decision No. 3880/H2020/2018/2), the Ministry of Education Commission, the Ministry of Education, Youth, and Sports of the Czech Republic [Project Nos. LM2015083 (PALS RI) and EF16_013/0001552], and the Czech Science Foundation (Grant Nos. 19-24619S and P205/11/P712).

This research was partially supported by the Competitiveness Program of NRNU MEPhI and by a project (Grant No. FSWU-2020-0035; Ministry of Science and Higher Education of the Russian Federation).