Mapping electron dynamics in highly transient EUV photon-induced plasmas: a novel diagnostic approach using multi-mode microwave cavity resonance spectroscopy

From a historical perspective, MCRS is based on a series of publications from the 1950s [24–26]. The last of these, in particular, proposes the use of interaction between electromagnetic fields in the microwave frequency regime and plasmas to probe the density of free electrons. Ever since, MCRS has been further developed and used to investigate the properties of many types of plasma. For instance, to measure the density of electrons in low-pressure radiofrequency-driven gas discharges [27]. In concert with laser-induced photodetachment, MCRS has even appeared to be an advantageous diagnostic to measure the density of negative ions in, for instance, etching plasmas [28–30] and powder-forming plasmas [31–34]. Since 2015, MCRS has been used to study the highly transient phenomena in plasmas generated by the irradiation of gas by ionizing (EUV) photons [13–15, 17–19, 35]. Full theoretical considerations regarding the MCRS technique have been described extensively in literature [27]. In this section, the working principle of this diagnostic is explained by mentioning only its key aspects. Readers are referred to the aforementioned literature for more details and mathematical background.

The essence of MCRS is that the plasma under investigation is produced inside a cylindrical pillbox cavity. The geometry of this cavity is designed such that resonant modes can be excited at resonance frequencies in the microwave (MW) range (a few GHz). The exact frequency at which a certain resonant mode exists is defined mainly by the geometrical configuration of the cavity (which is fixed during experiments) and the permittivity $\newcommand{\e}{{\rm e}} \epsilon ={{\epsilon }_{0}}{{\epsilon }_{r}}$ of the medium inside the cavity. For a fixed cavity, multiple resonant modes may exist over a certain frequency range. Two types of resonant modes are usually recognized: transverse electric, TE_mnp, and transverse magnetic, TM_mnp, with m  =  0, 1, 2, ... (for TE: m  =  1, 2, 3, ...), n  =  1, 2, 3, ... and p  =  0, 1, 2, ...).

The quantitative diagnostic approach of MCRS is based on the fact that the presence of free charge carriers (in this case plasma) directly gives rise to a finite increase in _r as [30]

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{\epsilon }_{r}}=1-\frac{\omega _{pe}^{2}}{{{\omega }^{2}}+{{\upsilon }^{2}}}+i\frac{1}{{{Q}_{0}}}\approx 1-\frac{\omega _{pe}^{2}}{{{\omega }^{2}}}+i\frac{1}{{{Q}_{0}}}.\nonumber \end{align} \tag{ 1 }$$

In this equation, the term $i\frac{1}{{{Q}_{0}}}$ (with i the imaginary unit) accounts for losses due to non-idealities of (or in) the cavity while the collision frequency $\upsilon$ between electrons and other particles—which is usually considered as a loss term as well—is negligible in our case due to the low gas and electron densities. As will be described later in this article, the quality factor 'Q' is a measure of the rate at which a cavity can be loaded or reloaded with microwave energy and therefore determines the response time of the diagnostic under discussion. As can be seen in equation (1), _r depends on the angular frequency $\omega$ of the microwave field used and the electron plasma frequency ${{\omega }_{pe}}$ given by:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{\omega }_{pe}}=\sqrt{\frac{{{n}_{e}}{{e}^{2}}}{{{m}_{e}}{{\epsilon }_{0}}}},\nonumber \end{align} \tag{ 2 }$$

where ${{m}_{e}}$ and $e$ are the mass of an electron and the elementary charge respectively, and $\newcommand{\e}{{\rm e}} {{\epsilon }_{0}}$ the permittivity of vacuum. The diagnostic force becomes immediately clear when we realize that the electron plasma frequency—and therefore _r as well—is directly affected by the electron density ${{n}_{e}}$.

Since the diagnostic is operated in the microwave regime, the inertia of ions and heavier charge carriers (e.g. nanometre and micrometre-sized particles) is too large to allow these particles to follow the electric field oscillations. Unlike ions, electrons are sufficiently mobile to follow the microwave fields applied and therefore to determine the permittivity of the plasma.

Overall, compared with the situation in vacuum, the presence of free electrons inside the cavity affects , while in turn affects the resonance frequency ${{f}_{0}}={}^{{{\omega }_{0\!\!}}}\diagup{}_{\!2\pi }\;$ of the resonant mode used. In the diagnostic presented here, a lossless and isotropic cavity medium with permittivity $\newcommand{\e}{{\rm e}} \epsilon ({\bf x})$ and permeability $\mu ({\bf x})$ is assumed. This means that the electric field E and electric displacement field D of a resonant mode are related by $\newcommand{\e}{{\rm e}} {\bf D}=\epsilon {\bf E}$. For the magnetic fields, a similar relationship holds: ${\bf B}=\mu {\bf H}$. The current application focuses on unmagnetized low-pressure and non-thermal plasmas, for which it is common to assume $\mu \left({\bf x} \right)={{\mu }_{0}}$ and that the ions are immobile and cold. It can be derived from [36] that the electric-field-averaged electron density ${{\bar{n}}_{e}}$ inside a cavity relates to the resonance frequencies f₀ and f of a specific resonant mode in an empty cavity and in a cavity partly or completely filled with free electrons, i.e. plasma, respectively as:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{\bar{n}}_{e}}=\frac{8{{\pi }^{2}}{{m}_{e}}{{\epsilon }_{0}}}{{{e}^{2}}}\frac{{{f}^{2}}\Delta f}{{{f}_{0}}},\nonumber \end{align} \tag{ 3 }$$

with $\Delta f\underline{\underline{{\rm def}}}\,f-{{f}_{0}}>0$ the resonant frequency shift due to the presence of plasma. Note that ${{\bar{n}}_{e}}$ is an electric-field-averaged electron density taking into account both the spatial distribution of the free electrons over the cavity volume ${{n}_{e}}({\bf x})$ and the local value of the electric-field component ${\bf E}\left({\bf x} \right)$ of the MW resonant mode.

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{\bar{n}}_{e}} {\underline{\underline{{\rm def}}}}\frac{\int{\int{\int_{{\rm cav}}{{{n}_{e}}({\bf x}){{\left| {\bf E}({\bf x}) \right|}^{2}}{{{\rm d}}^{3}}x}}}}{\int{\int{\int_{{\rm cav}}{{{\left| {\bf E}({\bf x}) \right|}^{2}}{{{\rm d}}^{3}}x}}}}.\nonumber \end{align} \tag{ 4 }$$

This means that, depending on the resonant mode used, electrons at different positions in the cavity are probed to a different extent. Hence, MCRS yields electron densities which are inherently cavity-averaged and electric-field weighted.

Until now, this has always been experienced as a disadvantage of the MCRS technique. However, the analysis and comparison of multiple modes can serve to resolve the electron density distribution spatially. Successful application of this method is demonstrated in the current article.

As already mentioned in the previous section, the extent to which free electrons are probed by the MCRS diagnostic scales directly with the local value of ${\bf E}({\bf x})$ of the specific resonant mode. Hence, the resonance frequency shift $\Delta f$ induced by the local presence of electrons is different for each resonant mode. Six examples of computed ${\bf E}({\bf x})$ components of different resonant modes in an ideal cylindrical cavity are given in figure 1.

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Taking into account equation (4), it can be reasoned that electrons present on the axis of this cavity are probed with a high weighting factor (large $\Delta f$) by the resonant TM₀₁₀ and TM₀₂₀ modes because ${\bf E}\left({\bf x} \right)$ is at its maximum on the axis. At the same time, the same electrons are hardly detected by the other modes shown in figure 1 because these show ${\bf E}\left({\bf x} \right)=0~$ on the axis. It is precisely this difference in the spatial sensitivity of the different resonant modes to the presence of free electrons which is used to make multi-mode MCRS spatially resolved. The procedure to achieve this is elaborated in the following section.

2.2.1. Reconstruction of electron density distribution profile.

To obtain spatially resolved electron density profiles, MCRS is applied in N resonant modes and for each time step and resonant mode the shift in resonance frequency $\Delta {{f}_{1}}$; $\Delta {{f}_{2}}$; ... ; $\Delta {{f}_{N}}$ caused by the presence of free electrons is measured. From these values, the cavity-averaged and electric-field-weighted electron densities ${{\bar{n}}_{e,1}}$; ${{\bar{n}}_{e,2}}$; ... ; ${{\bar{n}}_{e,N}}$ can be calculated using equation (3). For each $(i,j)$ combination of resonant modes $(\,j>i)$, the ratio between the respective ${{\bar{n}}_{e,i}}$ and ${{\bar{n}}_{e,j}}$ values is determined as $r_{ij}^{{\rm meas}}={{\bar{n}}_{e,i}}/{{\bar{n}}_{e,j}}$. In parallel, from a theoretical perspective, the similar ratios $r_{ij}^{{\rm theory}}$ are determined by assuming a certain spatial n_e profile shape. Corresponding with measurements of the intensity profile of the initial irradiation profile (see figure 12(b)), a Gaussian profile has been considered here. However, it should be noted that every other possible profile shape (e.g. Bessel-like) might and can be used if the actual plasma physical situation demands so.

The reconstructed profile corresponds with the smallest root-mean-square error ${{\xi }_{k}}$ between $r_{ij}^{{\rm meas}}$ and $r_{ij}^{{\rm theory}}$,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{\xi }_{k}}=\sum\limits_{j>i}^{N}{{{\left(\frac{r_{ij}^{{\rm theory}}-r_{ij}^{{\rm meas}}}{r_{ij}^{{\rm theory}}} \right)}^{2}}},\nonumber \end{align} \tag{ 5 }$$

which is brute-force calculated over a range of centre positions $({{x}_{0}},{{y}_{0}})$ and widths $\sigma$ of the 2D Gaussian electron density profiles,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{n}_{e}}\left(x,y \right)=n_{e}^{\max }\exp \left(-\frac{{{\left(x-{{x}_{0}} \right)}^{2}}+{{\left(y-{{y}_{0}} \right)}^{2}}}{2{{\sigma }^{2}}} \right).\nonumber \end{align} \tag{ 6 }$$

Note that, since the modes used in this work all have field components which are axially independent (despite small deviations close to the entrance and exit holes of the cavity), and also the xy intensity profiles of the beam used does not change along the z-direction inside the cavity, ${{n}_{e}}\left(x,y \right)$ is independent of z.

This reconstruction procedure can be run for each time step for which experimental data for $\Delta {{f}_{1}}$; $\Delta {{f}_{2}}$; ... ; $\Delta {{f}_{N}}$ is available. Therefore, the output of the reconstruction method we have developed in-house is the centre position (${{x}_{0}}(t)$, ${{y}_{0}}(t)$) of the maximum of the electron density profile and its width σ(t) with—in the case of reproducible plasma events—a temporal resolution limited by the Q of the cavity and the resonant mode used. The spatial resolution (of the reconstruction algorithm) of the centre position is 100 µm and that for the width is 10 µm. Volume integrals are calculated numerically, with electric fields and electron densities averaged along the axis of the cavity (z-axis) since we are only interested in and calculate 2D ${{n}_{e}}(x,y)$ profiles (equation (6)),

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle &\int{\int{\int_{{\rm cav}}{{{n}_{e}}({\bf x}){{\left| {\bf E}({\bf x}) \right|}^{2}}{{{\rm d}}^{3}}{\bf x}\to }}}\int{\int_{{\rm cav}}{{{n}_{e}}\left(x,y \right){{\left| {\bf E}\left(x,y \right) \right|}^{2}}}}\nonumber \\ &{{{\rm d}x{\rm d}y\approx {{\delta }^{2}}}}\sum\limits_{i=1}^{{{N}_{xy}}}{{{n}_{e}}({{x}_{i}},{{y}_{i}}){{\left| {\bf E}({{x}_{i}},{{y}_{i}}) \right|}^{2}}},\nonumber \end{align} \tag{ 7 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \int{\int{\int_{{\rm cav}}{{{\left| {\bf E}({\bf x}) \right|}^{2}}{{{\rm d}}^{3}}{\bf x}}}}\to \int{\int_{{\rm cav}}{{{\left| {\bf E}(x,y) \right|}^{2}}~{\rm d}x~{\rm d}y\approx {{\delta }^{2}}}}\sum\limits_{i=1}^{{{N}_{xy}}}{{{\left| {\bf E}({{x}_{i}},{{y}_{i}}) \right|}^{2}}},\nonumber \end{align} \tag{ 8 }$$

where ${{N}_{xy}}$ is the number of grid points in the xy-plane and $\delta =100\,\mu {\rm m}$ the constant spacing of the grid points in both the x- and y-direction; and

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {\bf E}\left(x,y \right)\underline{\underline{{\rm def}}}\frac{1}{{{N}_{z}}}\sum\limits_{i=1}^{{{N}_{z}}}{{\bf E}\left(x,y,{{z}_{i}} \right)},\nonumber \end{align} \tag{ 9 }$$

where ${{N}_{z}}$ is the number of grid points in the z-direction.

For ${\bf E}\left(x,y,z \right)$, fields simulated with CST Microwave Studio (see appendix A for detailed information and the computed E-field profiles) have been used as input. Furthermore, it is assumed that the electron density is zero outside the probing volume (see figure 3).

2.2.2. Restoring absolute values.

Since the reconstruction algorithm optimizes ${{x}_{0}},$ ${{y}_{0}}$ and σ for the smallest value of ${{\xi }_{k}}$, the maximum value of the profile $n_{e}^{\max }$ is lost and has to be determined during post-processing. To this end, equation (4) is solved for $n_{e}^{\max }$ but now using the reconstructed 2D n_e profile (with numerical approximations similar to those above),

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle n_{e,i}^{\max }={{\bar{n}}_{e,i}}\frac{\int{\int_{{\rm cav}}{{{\left| {{{\bf E}}_{i}}(x,y) \right|}^{2}}~{\rm d}x{\rm d}y}}}{\int{\int_{{\rm cav}}{\exp \left(-\frac{{{\left(x-{{x}_{0}} \right)}^{2}}+{{\left(y-{{y}_{0}} \right)}^{2}}}{2{{\sigma }^{2}}} \right){{\left| {{{\bf E}}_{i}}(x,y) \right|}^{2}}~{\rm d}x{\rm d}y}}},\nonumber \end{align} \tag{ 10 }$$

where $i$ signifies a certain resonant mode. Irrespective of the combination of modes used for the reconstruction, $n_{e}^{\max }$ is calculated for all N modes that show good agreement between experimentally determined E-field profiles using the bead-scanning method (see appendix B) and the ones simulated using CST Microwave Studio (see appendix A). Subsequently, the mean value $\left\langle n_{e}^{\max } \right\rangle$ and standard error $s$ associated with it is calculated as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \left\langle n_{e}^{\max } \right\rangle =\frac{1}{N}\sum\limits_{i=1}^{N}{n_{e,i}^{\max }},\nonumber \end{align} \tag{ 11 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle s=\frac{{{\sigma }_{{\rm sample}}}}{\sqrt{N}},\nonumber \end{align} \tag{ 12 }$$

where ${{\sigma }_{{\rm sample}}}$ is the uncertainty on each value. The 95% confidence interval of $\left\langle n_{e}^{\max } \right\rangle$ is given by roughly $\pm 2\,{\rm s}$.

The experimental configuration used is depicted schematically in figure 2. Mechanically, broadly speaking the setup consists of three chambers: the source chamber, the collector chamber and the measurement chamber. The radiation source—housed in the source chamber—is a discharge-produced pinch plasma in xenon gas, producing pulsed radiation including contributions with wavelengths in the EUV range. All the features of this source have been described extensively in [14, 37]. In the configuration used, the pulsed radiation produced by the EUV source has a repetition rate of 500 Hz, a pulse duration of roughly 100 ns and a pulse energy of 125 µJ in the 10–20 nm wavelength range.

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The collector chamber (1.5 m in length and 1 m in diameter) houses the collector, which is a set of grazing-incidence multilayer mirrors in 'Wolter' configuration (aligned in an 'onion-like' structure) as described in, for instance, [38]. This collector focuses the light emitted by the EUV source in the intermediate focus IF in the measurement chamber. In the IF, the beam waist is approximately 4 mm and the beam has a divergence of 10° [37]. Although this setup has the option to install a spectral purity filter between the collector and the measurement chamber, none was used during the experiments presented here. The measurement chamber houses the microwave resonant cavity through which EUV photons can be directed without releasing any electrons from the walls by photoelectric effects. The spectrally integrated EUV power is monitored by a thermal EUV power sensor with an experimental error of 6% [37].

A schematic diagram and a photograph of the cylindrical cavity developed for this work are provided in figure 3. The base of the cavity is an aluminium body with a screw-on brass lid. Fully mounted, the inner diameter of the cavity is 29 mm and its inner height is 15 mm. Two concentric holes in the brass lid and in the base of the cavity, aligned at the cavity axis, allow irradiation of the gas inside with EUV photons without creating photoelectrons from the cavity walls. These photon entrance and exit holes both have a diameter of 10 mm. Based on the work of Lassise et al [39, 40] on making small and power-efficient cavities, the cavity is partly filled with a ZiTiO₄ ceramic between a radius of 5 mm and 12.5 mm. This ceramic from T-CERAM ('E-37') was chosen for its low loss term $\tan \delta =2\times {{10}^{-4}}$ and its high $\newcommand{\e}{{\rm e}} {{\epsilon }_{r}}=37$ (at 10 GHz) [40]. Whilst using a material with a low loss term is largely relevant for these applications, a high $\newcommand{\e}{{\rm e}} {{\epsilon }_{r}}$ could be advantageous for two reasons. First, the high permittivity makes the cavity larger in a virtual sense, meaning that the resonant modes used can be excited at lower resonant frequencies. This in turn means that the holes in the lid and base can be drilled larger without causing significant 'leakage' of microwave radiation from the cavity. Second, the mode shows field concentration around the axis of the cavity while the ratio between the volume with which plasma is created by the EUV irradiation and the total cavity volume is also much larger. This combination results in a significant improvement overall in the signal-to-noise ratio of the diagnostic. It is impossible to neatly drill a small hole in the ceramic, so it was decided instead to position the (straight) antenna used to excite the MW modes outside the ceramic material, very close to the inner cavity wall. To keep the ceramic in place, a Teflon centring ring fills the gap between the cavity wall and ceramic (between a radius of 12.5 mm and 14.5 mm). The antenna used, which slides into the slot in the Teflon ring, is a straight and fixed piece of copper wire (diameter 1 mm, length approximately 20 mm) connected to the MW source by a fixed SMA feedthrough. To be able to heat the cavity to a set temperature (and keep it stable there), the aluminium housing includes four slots housing cartridge heaters and one slot that can house a Pt1000 temperature sensor. It also has four M4 holes in the outer side wall (at half height), to be able to fix it in the setup.

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A schematic diagram of the acquisition system is provided in figure 4. Unlike past MCRS measurements by the EPG research group, which used two loop antennas as a sender–receiver pair for transmission measurements, ours are performed in reflection mode using a single straight antenna. A microwave generator (Stanford Research Systems SG386) produces a sinusoidal microwave signal at 16 dBm power and at a frequency set by the PC. This output is connected to the output port of a directional coupler (Mini-Circuits ZHDC-10-63-S+) and in principle passes unhindered to its input port, which is connected to the antenna inside the cavity. In this way, resonances can be excited in the cavity by applying the correct output frequency from the microwave generator. If the cavity is off resonance, power coupling to it is very inefficient and most of the power reaching the antenna reflects back to the microwave generator, where it is dissipated. Part of this signal (10%), however, is rerouted by the directional coupler to the measurement leg of the detection system, where it is first converted to DC by a logarithmic power detector (Hittite 602LP4E, 10 ns rise and fall time) and subsequently fed to a high-frequency (up to 250 MHz, but for the current measurements set to 50 MHz) transient recorder (Spectrum M3i.4121-exp) inside the same PC. The transient recorder continuously samples its input port and stores the data in its internal memory. Only upon an external TTL trigger, for which the trigger signal produced by the EUV source related to the discharge between the electrodes is used, is data made available to the user for subsequent analysis. This arrangement allows us to analyse part of the signal before the actual trigger occurs. If the cavity is at resonance, hardly any signal reflects back (most of the energy is dissipated in the electromagnetic field inside the cavity) and that probed by the transient recorder is low. Data analysis is similar to that discussed in previous work [33, 37, 41, 42], differing only in the 'direction' of the resonance peaks (maxima versus minima, see figure 5).

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During the measurements presented in this paper, the microwave source was set to a certain frequency and kept at this frequency during the entire duration of the EUV pulse and the measurement time after this pulse. Just before the gas in the cavity is exposed to the next EUV pulse, the microwave source is set to the next value. To obtain sufficient accuracy, a resonant peak is probed by measurements at 25 different frequencies located in the frequency domain closely around the resonant frequency.

In order to translate the measurements taken into quantitative results, (i) each resonant mode has to be identified and (ii) the electric-field profile of each resonant mode used must be spatially resolved in a known manner. To this end, the electric-field profiles are obtained in two ways: numerically, using the commercially available Microwave Studio simulation package (see appendix A), and experimentally using the 'bead-scanning' method (see appendix B).

Figure 5(a) shows the cavity-response spectrum in the frequency range 1.5 to 4.0 GHz. In this spectrum, 10 resonant modes can be identified. But only those (modes 1, 2, 5, 6, 8, 9 and 10) for which the electric-field profiles obtained by the bead-scanning method and from CST Microwave Studio show reasonable agreement are taken into account for further multi-mode MCRS analysis. As an example, figure 6 shows the experimentally and numerically determined field profiles for the first two modes; results for the others used can be found and compared in figure C1 in appendix C.

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Figure 5(b) shows those (simulated) electric-field profiles in the cavity probing volume to be used for further analysis.

In MCRS in general, the Q-factor (or 'Q') is a crucial parameter when it comes to limitations in terms of resolution in time and accuracy in ${{\bar{n}}_{e}}$ determination. This factor is defined as the ratio between the energy stored in the cavity and the energy dissipated per cycle due to non-idealities. It can be deduced from the measured FWHM Γ and the frequency f at which this resonance curve shows its maximum

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle Q\underline{\underline{{\rm def}}}\frac{f}{\Gamma }.\nonumber \end{align} \tag{ 13 }$$

The higher the 'level of perfectness' or Q of a cavity, the more energy can be stored in it and the lower the energy-dissipation rate due to imperfections. This means that cavities with a high Q have a narrow resonance curve which results in accurate determination of ${{\bar{n}}_{e}}$. At the same time, a high Q results into a slow response by the cavity to changes in permittivity and hence in ${{\bar{n}}_{e}}$. The 1/e response time τ of the cavity relates to Q and resonance frequency of a specific resonant mode as:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \tau =\frac{Q}{\pi f}.\nonumber \end{align} \tag{ 14 }$$

In practice, when designing a cavity one should tweak Q in order to find a suitable optimum between accuracy in ${{\bar{n}}_{e}}$ determination and the time resolution of the diagnostics. The resonant modes of our cavity have Q's listed in table 1 together with the corresponding cavity response times, which are in the order of 100 ns.

The current diagnostic has a huge dynamic range but still is limited. For instance, the lower detection limit is noise-determined. The upper detection limit of MCRS is determined by the moment at which the perturbation theory of Maier and Slater [43] is no longer valid ($\newcommand{\e}{{\rm e}} {{\epsilon }_{r}}\ll 1$). The lower detection limit has—up until now [14, 15, 18, 19]—been demonstrated to be in the order of 10¹³ m⁻³, while in our work the sensitivity is improved by one order of magnitude. The upper detection limit is 10¹⁸ m⁻³. Expected values of ${{\bar{n}}_{e}}$ in the EUV-induced plasma under the current conditions are between the lower detection limit and 10¹⁶ m⁻³. Hence, MCRS is a suitable diagnostic for studying this kind of plasma system.

The cavity-averaged and E-field-weighted density of free electrons ${{\bar{n}}_{e}}$ obtained from traditional MCRS using different resonant modes is depicted on the left-hand side of figure 7 as a function of time before and after irradiation of the gas with a pulse of EUV photons. On the right-hand side are shown simulated electric-field profiles through the whole cavity volume (including the ceramic and Teflon) for each of these resonant modes. Note that modes 8 and 9 are basically identical, with the only dissimilarity being that their axes of symmetry are tilted slightly differently in the xy-plane. Note that in this case the cylindrical symmetry of the excited modes in the cavity is broken by the presence of the antenna. From these cavity-averaged results, two interesting features can already be identified. First, whereas van der Horst et al [17] found MCRS suitable for detecting ${{\bar{n}}_{e}}$ down to 10¹³ m⁻³, in our work better cavity design and fitting procedures have resulted in a lower detection limit improved by one order of magnitude (down to ${{\bar{n}}_{e}}\approx$10¹² m⁻³). This opens the door for the study of plasma dynamics at much lower gas pressures, as demonstrated in this article. Second, absolute values of ${{\bar{n}}_{e}}$ and its development as a function of time appear to depend largely on the resonant mode used. This indicates that spatial information about the distribution of ${{\bar{n}}_{e}}$ over the cavity volume is indeed hidden within the data. As already explained in section 2, time-resolved MCRS measurements of ${{\bar{n}}_{e}}$ at multiple resonant modes form the basic 'raw' data for further analysis.

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In the procedure we have developed, this 'raw' data is used to derive the absolutely calibrated maximum value of the electron density $\left\langle n_{e}^{\max } \right\rangle$, in a temporally resolved fashion. Note that—although only maximum values are presented in this section—our procedure is able to reconstruct absolutely calibrated and fully spatially resolved electron density profiles (as will be demonstrated in the next section). For reconstruction and absolute calibration, the field profiles of respectively five, six and seven different microwave resonant modes have been combined. For each set, figure 8 plots the results together with the upper and lower limit of the corresponding 95% confidence interval and the temporal evolution of the radial width σ of the calculated electron density distribution.

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From this figure, it can be concluded that—especially during the first 220 ns of the profiles shown—the signals contain too much noise to deliver valuable information. This noise is caused by EMC from the EUV source just before it emits the main in-band EUV radiation and is more dominant in our measurements than in those in other literature [14] because no spectral purity filter (SPF) has been used here (SPF normally blocks part of the EMC form the source). Between 220 ns and 380 ns, the general trend for the different sets of resonant modes used appears to be the same but the absolute values are not consistent. The reason for this is that timescales for plasma dynamics in this range are of the same order as the response time τ of the cavity used (roughly 100 ns, see table 1). Since τ is different for different resonant modes, it is not surprising that the $\left\langle n_{e}^{\max } \right\rangle$ and σ curves diverge from one another when different sets of resonant modes are used to construct them. In this range, it is important to note that, although absolute values of the electron density and its spatial distribution width might not be very accurate, trends in temporal evolution can be made visible at sub-100 ns timescales. The reason for this is that MCRS reacts to trends even when the specific resonant modes are not yet fully loaded. Beyond 380 ns, the same results are produced when taking into account different sets of resonant modes. This indicates the reliability—in terms not only of temporal evolution, but also of absolute values—of the multi-mode MCRS approach we have developed.

A zoomed-in picture of the curves reconstructed using modes 1, 2, 5, 6 and 8 is presented in figure 9 with—for clarity—independently measured [37] (but synchronized in time with the other data) out-of-band and in-band EUV-radiation intensity delivered by the same EUV source at the same position in the cavity.

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In order to discuss the plasma dynamics involved, the time evolution of the system has been divided in four different phases (denoted I–IV in figure 9). As already explained in the previous section, dynamic processes in phases III and IV are sufficiently slow, compared with the cavity response time τ, to deliver accurate and absolute results. Processes and time behaviour in phases I and II are of the same order as or faster than τ, meaning that although dynamics can be probed in relative sense using this method, absolute values may be less accurate.

As can be observed from figure 9, from approximately 220 ns onwards the measured signal-to-noise ratio is sufficient to obtain $\left\langle n_{e}^{\max } \right\rangle$ and the radial width σ of the electron density profile as a function of time.

In phase I, the increase in ${{n}_{e}}(t)$ is directly related to the upcoming in-band EUV irradiation creating free electrons in the cavity by photoionization of the background gas. Assuming that the majority (~80%) of the gas present is N₂, the majority of these electrons primarily have an energy (76.4 eV) equal to the difference between the photon energy (92 eV at 13.5 nm) and the ionization potential (15.6 eV for N₂). Note that the electron-energy distribution function (EEDF) is highly non-Maxwellian shortly after the electrons have been produced. In this phase, σ is observed to decrease until the in-band EUV intensity has reached its maximum. At very short timescales (a few ns), the first electrons reach and charge the inner cavity wall while the created ions remain are still fixed in position due to their high inertia. This creates an elevated plasma potential and radially directed electric fields confining additional electrons and preventing them to be lost from the plasma. The observed contraction of the spatial electron distribution is explained by two processes. First, continued production of electrons on the axis of the cavity while its inner surface is already in quasi-equilibrium with the elevated plasma potential results in a spatial electron density distribution which is higher at the axis of the cavity. Second, electrons which have not been involved in charging the inner walls of the cavity start to oscillate through it without reaching the walls. Due to collisions with background molecules, these electrons lose part of their energy. This eventually confines the electron density distribution by the radial electric fields present. For initial electrons with energies in the order of ε  = 76 eV, it can be estimated that the collision time of those with the mainly N₂ background gas molecules is indeed in the order of 80-100 ns, as calculated with [45],

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{\tau }_{e{-}{{{\rm N}}_{2}}}}\left(\varepsilon \right)=\frac{1}{v\left(\varepsilon \right){{\sigma }_{e{-}{{{\rm N}}_{2}}}}\left(\varepsilon \right){{n}_{{{{\rm N}}_{2}}}}},\nonumber \end{align} \tag{ 15 }$$

where v(ε) and ${{\sigma }_{e{-}{{{\rm N}}_{2}}}}\left(\varepsilon \right)$ are the electron velocity and the electron-N₂ collision cross-section respectively, and ${{n}_{{{{\rm N}}_{2}}}}$ is the N₂ density. At the same time, computing the inverse ion plasma frequency by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \omega _{pi}^{-1}=\sqrt{\frac{{{\epsilon }_{0}}{{m}_{{\rm N}_{2}^{+}}}}{{{e}^{2}}{{n}_{{\rm N}_{2}^{+}}}}},\nonumber \end{align} \tag{ 16 }$$

yields timescales of the same order (~80 ns). This indicates that, on the same timescales, ions start to move together with the electrons towards the walls, where they eventually recombine.

In phase II, the in-band EUV irradiation intensity has passed its maximum value and decreases as a function of time. Consequently, the production of free electrons also decreases while the loss rate due to plasma recombination at the wall continues. Overall, $\left\langle n_{e}^{\max } \right\rangle$ tends to decline over time while the width of this distribution increases due to the expansion of the plasma.

In phase III, we observe a special behaviour which has not been seen before. Even though the main in-band EUV irradiation has now faded out, electrons continue to be produced. Since the width of the electron distribution decreases in this phase, these electrons must be produced or gathered along the cavity axis. Although the exact process is not yet fully understood, two processes may be responsible for this behaviour.

First, it is very likely that out-of-band radiation containing photons with energies higher than the ionization threshold of the background gas is responsible for significant photoionization even after the in-band EUV radiation has faded out. As can be seen from the orange curve in figure 9, the shape and timescales of the out-of-band emissions match the observed time evolution of the $\left\langle n_{e}^{\max } \right\rangle$ profile. The fact that a considerable number of electrons is produced even though the out-of-band emission intensity is just a fraction of the in-band one is easily explained by the photoionization cross-section of N₂, which is more than an order of magnitude higher just above the ionization energy threshold (2.5  ×  10⁻²¹ m² at 15.6 eV [46]) when compared with its value in the in-band regime (2.5  ×  10⁻²² m² at 92 eV [46]). Second, a significant number of the electrons produced by the in-band radiation (in phases I and II) remain confined in the positive plasma potential during this phase. Whereas the energy loss of 76 eV electrons is initially very fast (<100 ns [45]), electron energy-loss times in 1 Pa N₂ just above ionization thresholds are computed by van de Ven [45] to be ~2  ×  10² µs. Scaling linearly with gas density, this translates to roughly 2 µs for the pressures involved in the experiments reported here.

As during the second part of phase I, the creation of additional electrons on the axis of the cavity combined with cooling of the electron distribution narrows the width of that distribution.

In phase IV, production of electrons decreases while the plasma is transported towards the walls, where it decays due to wall recombination.

Output parameters of the multi-mode MCRS measurements are the position (${{x}_{0}}(t)$, ${{y}_{0}}(t)$) of the maximum $\left\langle n_{e}^{\max } \right\rangle$ of the electron density profile and its width σ(t). Once these have been derived from the measurements, absolutely calibrated electron density profiles can be retrieved for every time step. This means that, from an application point of view, the multi-mode MCRS diagnostic presented here can be used as a non-intrusive and in-line beam monitor for ionizing radiation. Parameters that can be derived from this method and used in, say, feedback loops in the source control are: beam power or energy per pulse, cross-sectional beam-intensity profile and pointing stability. This is all due to the fact that the initial electron density profile translates to the cross-sectional shape of the EUV beam and that the number of electrons created translates directly to the EUV pulse energy, i.e. the number of photons (with known photoionization cross-sections, gas purity and gas pressure). As shown by our measurements, the background gas pressure can be kept low (10⁻⁴ mbar, or even lower as long as the MCRS signal remains above the lower detection limit) while obtaining sufficient signal for the diagnostics to work.