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Abstract
In this article we present modelling results of the amplification of High Order Harmonics (HOH) carrying orbital angular momentum (OAM) in plasma amplifiers created from krypton gas and silver solid targets. The resulting amplified beam is characterized in terms of intensity, phase and decomposition in helical and Laguerre-Gauss modes. Results show that the amplification process conserves OAM, although some degradation is apparent. Several structures appear in the intensity and phase profiles. These structures have been characterized with our model and related to refraction and interference with the plasma self-emission. Thus, these results not only demonstrate the capability of plasma amplifiers to deliver HOH amplified beams carrying OAM but also pave the way towards using HOH carrying OAM as a probe beam to diagnose the dynamics of hot, dense plasmas.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Since the demonstration of saturated amplification of High Order Harmonics (HOH) in plasmas [1], this source of extreme ultraviolet (XUV) and soft X-rays has demonstrated several properties of interest for applications: sub-femtosecond pulses [2,3], spatial filtering [4], correction of aberrations (coma and astigmatism) [5] and delivering circularly polarized beams [6], paving the way towards applications in such fields as imaging [7], non-linear laser-matter interaction [8], and plasma diagnosis [9].
The fact that plasma amplifiers conserve the polarization (i.e. the spin angular momentum, SAM) of the HOH seed [6] is a remarkable fact that could improve current imaging techniques based on circularly polarized probes [10]. The use of not only SAM in diagnosis techniques, but also of orbital angular momentum (OAM) [11] would boost the use of plasma-based seeded soft X-ray lasers in different applications [12], such as soft X-ray microscopy [13], spectroscopy [14], coherent imaging [15], atomic physics [16] and techniques that take advantage of helicoidal dichroism [17].
Here we will present the 3D modelling of the amplification of high order harmonics carrying orbital angular momentum (topological charge $l = \pm 25$) in Ni-like krypton (Kr$^{8+}$) and Ni-like silver (Ag$^{19+}$) plasmas. We use a suite of different codes to fully model this soft X-ray source [18], taking into account the different multiscale physical phenomena involved: creation and hydrodynamic evolution of the plasma, propagation of intense infrared (IR) pulses throughout it, collisional and atomic processes and propagation and amplification of the HOH seed.
The results show that the amplification process conserves the OAM carried by the HOH seed, in plasmas from gas and solid targets. However, both intensity and phase footprints are distorted in some regions due to the presence of strong amplified spontaneous emission (ASE) and steep electron density gradients. Since these structures depend on the topological charge $l$ (both magnitude and sign) of the HOH, the reconstruction of its wavefront will also provide valuable data about several properties of the plasma amplifier (electron density gradients, ionization,…), enhancing current diagnostics methods for hot, dense plasmas [9].
The article is structured as follows. First, we will briefly describe the codes used to carry out the modelling of the amplification and diagnostics of the resulting beams. Afterwards, we will show the results of our study in plasmas from symmetric gas targets (krypton) and from solid targets (silver) that present steep electron density gradients and break the symmetry of the seed. Then, we will discuss the origin of the structures that appear in the intensity and phase footprints, relating them to electron density gradients and plasma self-emission. Finally, we will conclude with a short section in which we will outline the potential future applications that HOH carrying OAM promise as a diagnostic tool for hot, dense plasmas.
2. Methodology
In this section we will briefly explain the computational multiscale framework used to model the amplification of HOH carrying OAM in plasma amplifiers from gas (Kr) and solid (Ag) targets. This framework encompasses the multiscale/multiphysics codes used to model the full process (from plasma creation and hydrodynamic evolution to amplification of a seed) [18] and the specific tools developed to postprocess the results, asessing the conservation of OAM throughout all the amplification process.
2.1 Modelling of the plasma amplifier evolution and amplification
The experimental setup for amplification of HOH in an Optical Field Ionization (OFI) Ni-like krypton plasma modelled in this article is based in the seminal experiment of Zeitoun et al. [1]. An intense infrared pulse is focused into a krypton gas cell. The strong electromagnetic field of the laser ionizes the medium, creating the lasing ion Kr$^{8+}$, and heating the free electrons so they create a population inversion in the $3d^9~4d_{J=0}\rightarrow 3d^9~4p_{J=1}$ transition. The Ni-like silver experimental setup modelled is similar to the one described in [19]. A Transient Collisional Excitation (TCE) Ni-like silver amplifier using Grazing-Incidence Pumping (GRIP) geometry was created by using several IR pulses.
In both cases, the amplification of the HOH beam is modelled with our Maxwell-Bloch code Dagon [20]. Dagon solves a wave equation for the electric field in the paraxial and slowly varying envelope approximations:
The constitutive relation for the polarization is obtained from the non-diagonal density matrix element of the two-level system. Its dynamics is given by Bloch equations. The resulting differential equation, also in the slowly varying envelope approximation, is
Finally, the temporal dynamics of the diagonal terms of the density matrix, as given by Bloch equations, allows us to write the rate equations for the two levels of the transition. These equations have been enhanced with populating terms from another levels.
Since in the case of OFI krypton amplifiers plasma hydrodynamics can be decoupled from the creation and pumping of the lasing ion [18], collisional rates and level populations for krypton plasmas were computed using a collisional-radiative code: OfiKinRad [21]. This is not the case for TCE amplifiers. Thus, we conducted 1.5D hydrodynamic and collisional-radiative simulations of silver plasmas using eHybrid [22]. These codes compute the temporal evolution of atomic populations in paralell with the relaxation of the electron density function (EDF), which initially is strongly non-Maxwellian, by solving a Fokker-Planck equation and rate equations.
2.2 Diagnosing the OAM carried by the amplified seed
The OAM carried by the HOH is characterized by computing its Azimuthal Fourier Transform (AFT) and by a decomposition in Laguerre-Gauss modes. The Azimuthal Fourier Transform computes the coefficients arising from the expansion of the field into helical modes. The probability of a photon to be found in a l-order OAM state can be computed from these coefficients [23]. In order to obtain the AFT, the cartesian complex field $U(x,y,z)$ obtained in the simulation is interpolated into a polar grid, with a resolution in the azimuthal dimension $k\pi$ times proportional to the number of l-modes desired to evaluate. For these simulations, the selected range was $l \in \{-50, 50\}$. Since we are interested in comparing the topological charge of the OAM before and after amplification, the dependence of the axial direction can be removed, obtaining $U(r, \phi )$. The resulting complex field expressed in polar coordinates can be expanded into a sum of fundamental helical modes with helicity $l$ (Eq. (4)):
where the $c_l(r)$ coefficients are obtained from the Azimuthal Fourier Transform (Eq. (5))The $c_l$ coefficients retain the information of the topological charge across the radial dimension. The integral of the squared modulus of $c_l$ along the radial coordinate is known as the OAM power spectrum. Since the OAM seed used in the simulations is injected as a pure Laguerre-Gauss beam, one step further in the topological analysis of the OAM is to decompose the polar complex field $U(r, \phi )$ into Laguerre-Gauss LG$_{p,l}$ modes (Eq. (6))
where $b_{p.l}$ coefficients, in contrast with the previous $c_l(r)$ decomposition, do not have a radial dependence since the LG$_{p,l}$ modes have a well defined dependence. The coefficients $b_{p,l}$ can be computed as a 2D Fourier Transform (Eq. (7)).In our modelling, only the LG$_{0,l}$ modes were considered.
2.3 Simple interference and refraction model
In order to explicitly assess the influence of amplified spontaneous emission (ASE) and electron density gradients in the OAM conservation (i.e. in the phase profile of the amplified seed), we have developed a simplified model taking into account only these two effects in order to compare its results with those of our full 3D Maxwell-Bloch model.
Refraction is taken into account by solving a simplified eikonal equation, that governs the trajectory of rays (Eq. (8))
where $n(x)$ is the complex refractive index, that can be related to the electron density by considering $n(x)$ as the square root of the dielectric function. The dielectric function can be expressed in terms of the electron density, obtaining Eq. (9):Within these approximations, the gradient of the refractive index is proportional to the gradient of the electron density. The angle of curvature varies with the propagation length and is proportional to the electron density gradient. Since the strongest electron density gradients appear in the transverse plane ($x-y$ plane), the resolution of Eq. (10) yields a parabolic trajectory in z, whose leading coefficient is proportional to the electron density gradient (Eq. (11)).
where $\alpha _0$ is the initial angle and $x_0$ the initial $x$-position at $z$=0. From this relation, the angle of propagation can be obtained numerically. This angle, when applied to the wave number in the phase part of both complex fields, allows to retrieve the patterns that are apparent in the amplification simulations.Interference is modelled in a simple way. The HOH seed is considered as a pure Laguerre-Gauss beam LG$_{0,25}$ and ASE has been characterized as a gaussian beam TEM$_{0,0}$. The resulting field is obtained as the sum of the two complex fields $E_{f}=E_{LG}+E_{G}$ after taking into account refraction effects with the aforementioned model. While approximating the ASE beam by a TEM$_{0,0}$ mode is a crude approximation, it will prove enough to explain the interference patterns observed in the results of Maxwell-Bloch simulations and stochastic ASE.
3. Amplification in plasmas from gas targets: Ni-like krypton amplifier
In this section, we will show the modelling results of the amplification of a $\lambda = 32.8$ nm, E = 1 pJ, $\Delta \tau$ = 100 fs FWHM HOH seed carrying OAM with a topological charge $l=25$ (a pure LG$_{0,25}$ mode) through a $L=5$ mm Ni-like Kr plasma amplifier. The seed spatial width is chosen to overlap the gain region. The computational domain is a 100 $\mu$m $\times$ 100 $\mu$m $\times$ 5 mm box. We have varied the density of Kr atoms, from $n_{Kr} = 9.88\times 10^{17}$ cm$^{-3}$ [1] to $n_{Kr} = 1.5\times 10^{19}$ cm$^{-3}$ [2] to study the robustness of OAM conservation when the amplifier presents strong amplification and density gradients.
Figure 1 shows a 2D slice, at $y= 50$ $\mu$m, of the amplified HOH and ASE at increasing densities. The seed has propagated throughout 4.74 mm of plasma and it is barely noticeable (the position of the seed is marked by a black arrow in Fig. 1(a)). Due to the mismatch between the seed duration and the plasma response, the seed is not amplified but develops an intense wake. The peak and duration of the wake strongly depends in the plasma electron density [24,25]. From the images shown, it is apparent that the amplified spontaneous emission (ASE) is well developed in the central region of the plasma, while the HOH is amplified in the periphery. This is a logical consequence of the Laguerre-Gauss profile of the HOH: when the seeded HOH is Gaussian, it inhibits the central ASE [9].
Fig. 1. Intensity slice (y = 50 $\mu$m, the $x-z$ plane is shown, being z the propagation direction) of the HOH seed after $\approx$ 4.74 mm propagation (the position of the seed, barely visible, is marked by an arrow in panel a) for different densities: a) $n_{Kr} = 9.88\times 10^{17}$ cm$^{-3}$, b) $n_{Kr} = 3.5\times 10^{18}$ cm$^{-3}$, c) $n_{Kr} = 7.0\times 10^{18}$ cm$^{-3}$, d) $n_{Kr} = 1.0\times 10^{19}$ cm$^{-3}$, e) $n_{Kr} = 1.5\times 10^{19}$ cm$^{-3}$. The intensity scale is different for each case in order to appreciate the varied structures and duration of the resulting beams. The ASE develops in the central part of the amplifier (25 $\mu$m < r < 75 $\mu$m) while the seed dominates the outer regions. When both signals overlap, a characteristic interference pattern appears in the beam footprint ($x-y$ plane).
It is clearly shown that the density strongly affects the longitudinal spatial distribution - or equivalently, the temporal structure - of the propagated beam. The pulse duration shortens as the density increases, in conformity with experimental results [2,3]. Furthermore, Rabi oscillations are clearly defined in low density (i.e. $n_{Kr} = 9.88 \times 10^{17}$ cm$^{-3}$ and $n_{Kr} = 3.5 \times 10^{18}$ cm$^{-3}$) amplifiers.
The influence of density in the beam footprint and OAM carried can be assessed by studying the transverse intensity and phase patterns (Fig. 2). These footprints have been obtained at the z coordinate where the amplified HOH beam has its maximum intensity. Regarding the intensity, the original clean-cut Laguerre-Gauss shape of the pulse at low density tends to blur at higher densities, while the noisy central pattern, that is clearly visible at higher densities, is the signature of strong ASE. The intensity structures observed in the inner ring of the HOH beam, defined more sharply at low densities, are a result of the interference between the ASE and the HOH as it will be shown in section 5.
Fig. 2. Intensity (left) and phase (right) slices ($x-y$ plane) after propagating > 4.5 mm throughout increasing density plasma amplifiers: a) initial seed, b) $n_{Kr} = 9.88\times 10^{17}$ cm$^{-3}$, c) $n_{Kr} = 3.5\times 10^{18}$ cm$^{-3}$, d) $n_{Kr} = 7.0\times 10^{18}$ cm$^{-3}$, e) $n_{Kr} = 1.0\times 10^{19}$ cm$^{-3}$, f) $n_{Kr} = 1.5\times 10^{19}$ cm$^{-3}$ The z coordinate of each slice varies since they are taken at the maximum intensity of the beam: a) z = 0 mm, b) z = 4.5 mm, c) z = 4.71 mm, d) z = 4.67 mm, e) z = 4.69 mm, f) z = 4.71 mm
As for the phase (Fig. 2), the observed distributions show the conservation of the OAM at all density levels. Nevertheless, whereas for $n_e = 9.88 \times 10^{17}$ cm$^{-3}$, the phase pattern is very similar to that of the original seed, when the density increases it starts to acquire a certain level of curvature. In fact, in Fig. 2, it can be observed how this curvature increases at greater radii. This is consistent with the fact that electron density gradients are greater at higher densities and radii, for the gaussian density profile used. In addition to this, it can be shown that the sign of the curvature and the gradient are closely related, so that in a plausible situation such as a waveguide [2], the curvature would change its orientation. The same effect (inversion of the phase curvature) is observed when the sign of the azimuthal mode ($l$) of the Laguerre-Gauss seed is changed.
In order to quantitatively characterize the conservation of OAM, the AFT of the amplified field has been computed in order to obtain the spatial mode decomposition, and thus, characterize the decomposition in helical modes [23]. Figure 3 shows the radial dependence of the azimuthal coefficients, as given by Eq. (5). Within the depicted range of $l$, the $l = 25$ mode is predominant. It is worth mentioning that the $l = -25$ is an artifact of the decomposition algorithm, without physical meaning. The $l=0$ mode is mainly present in the central region ($r < 20$ $\mu$m) of the amplifier. This is an expected consequence of the spatial distribution of the seed, such that OAM is only observed in the Laguerre-Gauss beam shape. The observation of the $l = 0$ mode at small radii is compatible with the presence of ASE, and it has been confirmed by performing simulations without seed and without ASE.
Fig. 3. $c_l(r)$ coefficients resulting from the Azimuthal Fourier Transform (logarithmic scale) of the HOH beams amplified in plasmas of increasing density: a) initial seed, b) $n_{Kr} = 9.88\times 10^{17}$ cm$^{-3}$, c) $n_{Kr} = 3.5\times 10^{18}$ cm$^{-3}$, d) $n_{Kr} = 7.0\times 10^{18}$ cm$^{-3}$, e) $n_{Kr} = 1.0\times 10^{19}$ cm$^{-3}$, f) $n_{Kr} = 1.5\times 10^{19}$ cm$^{-3}$
OAM conservation through the propagation and amplification processess is confirmed computing the coefficients $b_{p,l}$ of the Laguerre-Gauss decomposition (Eq. (7)). This decomposition shows the predominance of the LG$_{0,25}$ mode in the amplified beam for all densities tested. A small degradation, increasing with density, is observed, attributable to the presence of strong ASE and to the curvature distortion in the phase profile, due to strong electron density gradients.
Finally, in order to test the robustness of the amplification of HOH carrying OAM, we have used experimentally measured intensity and phase profiles of an $l=25$ HOH (that we will name "real seed"), instead of the analytic formula for the corresponding LG$_{0,25}$ Laguerre-Gauss mode. These profiles, along its decomposition in helical and LG modes, are shown in Fig. 4.
Fig. 4. Experimentally measured intensity (a) and phase (b) of a $l=25$ HOH ("real seed") and its corresponding $c_l(r)$ (logarithmic scale) coefficients (c) and Laguerre-Gauss decomposition (d).
It is clearly seen that the intensity profile, while presenting an annular shape, differs from an ideal LG$_{0,25}$ profile. The phase is also distorted, although presenting the 25 phase jumps characteristic of the OAM carried. The decomposition in helical and Laguerre-Gauss modes are shown in Fig. 4(c) and Fig. 4(d) respectively. The HOH clearly carries OAM, but it has a broad range of helical modes, centered at $l = 25$. The dominant mode is still the LG$_{0,25}$ although with non-negligible contributions from other modes. It is thus necessary to assess the effect of amplification in this complex decomposition in helical modes.
This HOH seed is injected in a 5 mm Ni-like Kr plasma amplifier with a density of $n_{Kr} = 9.88\times 10^{17}$ cm$^{-3}$, with the same computational parameters as in the previous cases. The resulting amplified beam characterization is depicted in Fig. 5. The intensity and phase profile are shown in the upper panels, a) and b). The ring structure of the seed is preserved, although intensity contrasts have increased. The phase conserves its initial structure, a clear signal of the OAM conservation through amplifcation.
Fig. 5. Intensity (a) and phase (b) of an $l=25$ "real seed" after amplification throughout a L = 5 mm plasma and corresponding $c_l(r)$ (logarithmic scale) coefficients (c) and Laguerre-Gauss decomposition (d).
The decomposition in helical and Laguerre-Gauss modes is shown in Figs. 5(c) and 5(d). The decomposition in helical modes (Fig. 5(c)) is still centered around the $l=25$ mode and it is narrower than the initial one. A strong $l=0$ component, signature of ASE, is clearly observed at small radii. The Laguerre-Gauss decomposition (Fig. 5(d)) still conserves its initial structure, with a peak in the LG$_{0,25}$ component. However, some modes, already present in the seed, have been amplified throughout the process. It is worth mentioning that the $b_{0,0}$ component is not noticeable in Fig. 5(d). This is caused by the central region of the amplified beam. As it is seen in Fig. 5(b), the phase presents some structure in the central region that results in several higher order Laguerre-Gauss modes instead of the expected $b_{0,0}$ coefficient of the ASE.
To summarize this section, we can assert that the modelling carried out shows that amplification in plasmas conserve the OAM of the seed beam. Even when the seed beam has a complex OAM structure and ASE is strong enough to interfere with the HOH, the amplified beam still carries OAM and its decomposition in helical modes is similar to the initial one. However, one caveat could still be raised: the Ni-like krypton amplifier modelled has cylindrical symmetry, as the Laguerre-Gauss modes injected. Will the OAM be conserved when the amplifier has no symmetries and, in addition to this, presents strong density gradients? This is the case of plasma amplifiers created from solid targets, the subject of next section.
4. Amplification in plasmas from solid targets: Ni-like silver amplifier
This section will focus on the amplification of XUV HOH carrying OAM in plasmas created from a silver slab. The OAM seed ($\lambda = 13.9$ nm) carries a topological charge l = 25 (a pure LG$_{0,25}$ mode). We have not changed neither the topological charge nor the Laguerre-Gauss mode to ease the comparison with krypton results. It has a spatial FWHM of 5.76 $\mu m$ and it is centered at $x=13~\mu$m $y=15~\mu$m in order to fit in the plasma’s population inversion zone. The simulation window was also resized to 26 $\mu$m $\times$ 26 $\mu$m ($x-y$ plane, transverse to the propagation) to fully seize grid resolution. To asses the effect and relevance of the ASE, simulations with and without ASE were conducted.
The main results are summarised in Figs. 6 and 7. Figure 6 shows the intensity (a) and phase (b) frootprints, along the $c_l(r)$ coefficients (c) and the LG decomposition (d) for the Laguerre-Gauss seed before amplification. We can see the annular intensity shape and the 25 phase jumps, characteristic of the LG$_{0,25}$ mode. The decomposition in helical modes shows a strong $l=25$ component and its harmonics and the LG decomposition shows the predominance of this mode.
Fig. 6. Intensity (a), phase (b), $c_l(r)$ (logarithmic scale) coefficients (c) and Laguerre Gauss decomposition (d) of the HOH seed before amplification.
Fig. 7. Intensity (a), phase (b), $c_l(r)$ (logarithmic scale) coefficients (c) and Laguerre Gauss decomposition (d) of the HOH seed after amplification. The electron density (x-y plane) is shown in (e) and a 1D cut in (f).
Figure 7 depicts the HOH seed after amplification. The amplified beam intensity profile shows a non-symmetrical beam amplification on the right side of the seed. The less intense ASE beam appears near the central region of the OAM beam. The intensity profile shows periodic structures, similar to the ones observed in Kr simulations (see Fig. 2), related to the interference between OAM and ASE, since they are only apparent when ASE is taken into account. Regarding the phase pattern, it retains the topological charge of the initial seed after amplification. The phase in the region close to the target (0 $\mu$m $< x <$ 10 $\mu$m) shows a strong curvature, related to refraction effects induced by the strong electron density gradient on this region of the plasma (shown in Fig. 7(e) and Fig. 7(f)). This refraction-induced curvature is also apparent in Fig. 2. Further analysis on the topological charge conservation is shown in Figs. 7(c) and 7(d), in which the helical and LG decomposition of the amplified beam is provided.
The impact of the amplification in the LG decomposition is quite noticeable. The LG decomposition of the seed unveils a precise $l=25$ topological charge. After amplification, the LG decomposition shows a strong component around $l=25$, although several other components are noticeable. This is better observed in the decomposition in helical modes, where the $l=0$ mode dominates at small radii while the $l=25$ mode starts to be noticeable for $r > 5~\mu$m. The degradation of the LG profile and appearance of new modes may be a consequence of both the asymmetric amplification and strong refraction near the target, that distorts the intensity and phase profile, in contrast with the symmetric amplification of the aforementioned krypton amplifier.
The robustness of the amplification of OAM XUV beams in inhomogeneous and strongly refractive plasmas can be asessed by seeding an experimentally measured ("real seed") HOH carrying OAM instead of the ideal Laguerre-Gauss mode. The intensity and phase profile of the seed are the same as in the case of krypton amplifier, shown in Fig. 4. The OAM seed is centered at $x = 13~\mu$m $y = 15~\mu$m, in order to replicate previous simulation conditions.
The intensity and phase profile after amplification are depicted in Fig. 8. Similar periodic structures, related to the interference between seed and ASE, appear in the intensity footprint. Refraction also distorts strongly the phase profile near the target, where electron density gradients are steeper, in a similar way as when the ideal Laguerre-Gauss beam was amplified.
Fig. 8. Intensity (a), phase (b), $c_l(r)$ (logarithmic scale) coefficients (c) and LG decomposition (d) of the "real seed" after amplification.
The decomposition in helical modes and the Laguerre-Gauss decomposition, depicted in Fig. 8, show a similar behaviour as in the previous cases. A strong $l=0$ component at small radii appears, denoting the presence of ASE in this region. The Laguerre-Gauss decomposition shows a small number of components, as before amplification, but displaced towards lower topological charges. Now, the maximum appears around $l=20$ while the initial seed decomposition peaks at $l=25$. The asymmetries in amplification and the strong refraction are at the origin of this effect.
In conclusion, asymmetries in the amplification, interference with the plasma self-emission and strong electron density gradients distort both intensity and phase footprints of the amplified seed. However, the seed still carries its initial OAM, allowing us to assert its conservation throughout the amplification process.
5. Analysis of the phase distortion and structures: the role of refraction and interference
To properly explain the structures in intensity and the curvature in the phase patterns, two main phenomena should be analyzed in further detail; the interference between the HOH and ASE beams, and the electron density of the plasma. The structures in the intensity profile, together with the circular distortion in the center of the phase pattern are explained by the former, while the curvature distortion of the phase is due to the latter. In silver simulations, the plasma is created from a solid state target. Hence, the electron density profile is completely different from the symmetric profile that the krypton simulations had, with even sharper electron density gradients. To characterize the relevance of these two phenomena, simulation results are contrasted with numerical predictions based on the interference of the OAM electromagnetic field $E_{LG}$ with the ASE electromagnetic field $E_G$. The OAM is considered as a pure Laguerre-Gauss beam LG$_{0,25}$ and ASE has been characterized as a gaussian beam TEM$_{0,0}$. The resulting field is obtained as the sum of the complex fields $E_{f}=E_{LG}+E_{G}$ and the comparative with the amplified OAM is portrayed in Figs. 9 and 10.
Fig. 9. Intensity (a) and phase (b) of a HOH seed amplified in a $n_{Kr} = 9.88\times 10^{17}$ cm$^{-3}$ Kr amplifier, as given by our Maxwell-Bloch model. Intensity (c) and phase (d) of a HOH seed as given by our refraction and interference model.
Fig. 10. Intensity (a) and phase (b) of a HOH seed amplified in a silver amplifier, as given by our Maxwell-Bloch model. Intensity (c) and phase (d) of a HOH seed as given by our refraction and interference model.
If none of the beams is significantly more intense than the other, interference patterns between both beams can be appreciated, as shown in Figs. 9 and 10. The interference with the ASE beam produces intensity structures around the LG$_{0,25}$ ring. The number of these structures is directly related to the topological charge of the LG beam, appearing 25 nodes in both the simulation and numerical results, although asymmetries in amplification and refraction effects can blur some of them, as shown in Fig. 10.
Refraction effects are more apparent in silver amplifiers, as shown in Fig. 10. The ASE has a linear phase pattern that results in a propagation direction not parallel to the $z$ axis, but with a slight angle of propagation, due to refraction. On the other hand, the HOH seed shows a non-symmetrical curvature in the phase pattern, mainly related to the electron density gradient. Sharp density gradients translate in an increased phase deformation and thus the left side of the simulations presents bigger refraction effects, since near the target electron density gradients are steeper.
In conclusion, our simplified interference and refraction model explains the structures that appear in the amplified seed intensity and the distortions in the phase curvature. These effects are thus related to interference with the plasma self-emission and with refraction due to steep electron density gradients.
6. Potential applications in dense plasma diagnosis
The combined measurement of intensity and phase of an XUV amplified seed has recently allowed to unveil the ionization dynamics of a dense plasma waveguide [9] while spectrally resolved transmission measurements of HOH have been used to diagnose warm dense matter [26]. OAM, like polarization (SAM), is an additional degree of freedom that can be used to extract additional information about the plasma.
The results presented in this article show the potential of HOH carrying OAM in the field of dense plasma diagnosis. Since the distortion of the phase profile is related to both the electron density gradient and the sign of the topological charge $l$, it is possible to estimate the electron density gradients of the plasma. Moreover, some regions of the beam have an undisturbed phase profile (as small radii regions in krypton amplifiers, or regions further away from the solid target in silver amplifiers) that can be used to asses not only the electron density gradients but also the electron density itself. The structures induced by the interference of the probe beam and the plasma self emission (ASE) have the potential to unveil the spatial and temporal dynamics of ASE, by varying the delay and position of the injected probe beam. Thus, the use of HOH carrying OAM has the potential to be an invaluable diagnostic tool for hot, dense plasmas.
7. Conclusions
In this article we present the modelling of the amplification of HOH carrying OAM in plasma amplifiers created from krypton and silver targets. We show that the OAM is conserved throughout the amplification process (similarly to the SAM [6]). However, distortions in the phase and structures in the intensity appear, broadening the decomposition in helical modes and increasing the components in the Laguerre-Gauss decomposition. These effects are related to refraction and interference with the plasma self emission and reproduced by our models. Once understood, these effects provide an invaluable tool to use the propagation of XUV HOH carrying OAM through dense plasmas as a diagnostic method of electron density gradients and self emission of the plasma, enhancing the potential of this source and promising to unveil the inner dynamics of hot dense plasmas.
Funding
Ministerio de Ciencia e Innovación (Ramón y Cajal RYC2018-026238-I, PID2021-124129OB-I00); Comunidad de Madrid (Apoyo a jóvenes investigadores, CROM); Euratom Research and Training Programme (CfP-FSD-AWP21-ENR-01-CEA-02); FP7 People (627191, project DAGON); Fundación BBVA (Beca Leonardo 2021).
Acknowledgements
The authors acknowledge support from COST (European Cooperation in Science and Technology - www.cost.eu) for funding the Action TUMIEE (CA17126) supporting this work; the People’s Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme under REA Grant Agreement 627191, project DAGON; EUROFUSION Enable Research project Advancing shock ignition for direct-drive inertial Fusion CfP-FSD-AWP21-ENR-01-CEA-02; Plan Estatal de Investigación Científica, Técnica y de Innovación, grant PID2021-124129OB-I00 funded by MCIN/AEI/ 10.13039/501100011033/ERDF, EU; the Universidad Politécnica de Madrid and the Comunidad Autónoma de Madrid, línea de actuación estímulo a la investigación de jóvenes doctores, project CROM and the Spanish Ministerio de Ciencia e Innovación through a Ramón y Cajal RYC2018-026238-I fellowship and a Beca Leonardo a Investigadores y Creadores Culturales 2021 de la Fundación BBVA.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper are publicly available upon request to the corresponding author (EO).
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