Laser-induced fluorescence spectroscopy for kinetic temperature measurement of xenon neutrals and ions in the discharge chamber of a radiofrequency ion source

Abstract

Application of single-photon absorption laser-induced fluorescence (LIF) spectroscopy for non-intrusive measurement of neutral xenon and singly charged xenon ion kinetic temperatures in the discharge chamber of a gridded radiofrequency ion source is demonstrated. A LIF spectrum analysis approach including hyperfine structure reconstruction and inverse filtering (Fourier deconvolution) is outlined. Special focus is set on optimization of post-deconvolution filtering as well as retracing of deconvolution result imperfection due to hyperfine structure parameter uncertainty, incorrect natural linewidth, and saturation of the LIF signal. The corresponding contributions to the kinetic temperature estimation error are quantified via simulation of spectral lineshapes. Deconvolution of almost unsaturated LIF spectra recorded in the center of the ion source discharge chamber reveals that the neutral xenon and xenon ion kinetic temperatures range between approximately 500 and 700 K and, respectively, 700 and 1000 K depending on the radiofrequency power supplied to the discharge.

Introduction

Experimental plasma diagnostics are integral means for extending the understanding of the physics of electric propulsion and for improvement of thruster concepts. For instance, the measurement of plasma species density distributions and kinematic properties gives valuable insights into the mechanisms of plasma generation, electric field distributions, interaction between plasma species as well as plasma-wall interaction. This closely relates to aspects of electric thruster efficiency and lifetime, e.g. discharge channel degradation in Hall-effect thrusters or extraction grid erosion in ion engines [1, 2]. To reduce financial efforts in conjunction with time-expensive engineering model tests, estimation of thruster lifetime and performance evolution relies substantially on numerical simulations. Numerical codes for prediction of extraction grid erosion require various plasma quantities for input, e.g. neutral or ion temperatures, ideally measured over a broad range of operating conditions. Plasma diagnostics for determining neutral flow densities, ion velocity distribution functions and ion beam divergence in turn serve for simulation model validation.

Laser-induced fluorescence (LIF) spectroscopy provides a non-intrusive plasma diagnostics framework for spatially resolved measurement of several quantities of neutrals and ions for a wide range of low-pressure inert gas plasmas relevant in electric propulsion. With laser excitation performed along a specific optical transition, the spectral absorption lineshape is recovered from subsequent fluorescence emission measured as function of the laser wavelength, whereas the calibrated fluorescence intensity provides information on the energy level population number density of the investigated plasma species. Excitation of ground state inert gas species, however, requires laser wavelengths in the vacuum-ultraviolet range, which is not accessible for compact continuous-wave (cw) lasers, but can be achieved via multi-photon excitation, e.g. with two-photon absorption laser-induced (TALIF) spectroscopy [3]. Since pulsed dye lasers come into use here, absorption lineshape extraction can be challenging as intrinsic lineshape broadening mechanisms are competing with laser linewidth-related broadening. Alternatively, (single-photon absorption) LIF spectroscopy can probe visible or near-infrared transitions arising from sufficiently populated lower excited states by means of tunable narrow-linewidth diode lasers. This approach generally does not permit ground-state density estimation due to unknown population number statistics under thermal non-equilibrium conditions. Nevertheless, the Doppler-induced broadening component of the transition lineshape maps a velocity-component distribution function representative of the plasma species under investigation. Extracted mean velocities allow for mapping of velocity fields and reconstruction of electric potential gradients, whereas the velocity distribution, if assumed to be Maxwellian, is related to the species’ local temperature.

Aiming at plasma diagnostics, numerous applications of LIF spectroscopy in low-pressure inert-gas plasmas have been reported in the literature. Ion velocity distribution functions of argon [4,5,6,7,8,9], krypton [8], and xenon [6, 8,9,10] were measured in the sheath region of direct-current discharges. Argon [11, 12] and helium neutrals [13] as well as argon ions [13,14,15,16] have been investigated in helicon discharges. LIF spectroscopy within inductively coupled plasma sources was used to estimate argon neutral and ion temperatures [17,18,19,20].

Referring to electric propulsion, application of LIF spectroscopy has been focusing on characterization of xenon-fed Hall-effect thrusters. On the one hand, neutral flow properties within the discharge channel and at the thruster exit plane [21,22,23,24,25,26,27,28,29,30,31] were deduced. On the other hand, strong focus has been set on measuring most probable ion velocities in axial direction for examination of the spatial variation of the electric potential in order to locate the zones of ionization and acceleration [27, 32,33,34,35,36,37,38]. By combining mean values of ion velocities measured with multi-directional LIF spectroscopy approaches, two-dimensional flow-fields were obtained which helped comprehending beam divergence and tracing ion-induced channel-wall erosion [33, 38,39,40,41,42]. Reconstruction of the full ion velocity phase space via tomographic LIF spectroscopy was demonstrated [43]. LIF spectroscopy with time-resolution enhancing approaches allowed for investigation of plasma oscillation phenomena [27, 44,45,46,47,48,49,50]. Moreover, the effects of finite facility background pressures on ionization and acceleration have been addressed [51, 52]. In part, Hall-effect thrusters driven with krypton were characterized from similar viewpoints [27, 32, 53, 54].

Regarding gridded ion engines, in contrast, a considerably smaller number of LIF-based investigations has been reported. In particular, there is still a lack of LIF-based studies with respect to radiofrequency (rf) ion thrusters. LIF spectra of xenon ions and neutrals were captured in the discharge chamber of a Kaufman-type ion thruster to investigate erosion of the integrated hollow cathode [55, 56]. Beam divergence for the same device was studied by performing xenon ion LIF spectroscopy in the thruster plume [57, 58]. Similarly, azimuthal near-field ion velocities were measured to clarify roll torques observed in operation of a microwave ion thruster type [59]. Furthermore, the xenon ion velocity distribution function has been investigated with respect to the interior and plume-region of an ion thruster microwave discharge cathode [60].

Recently, we reported on using LIF spectroscopy for inspection of the radial velocity-component distribution function and extraction of the kinetic temperature of xenon neutrals in the discharge chamber of an rf-driven broad beam ion source [61]. For that, a LIF setup including a cw diode laser system and flexible vacuum-compatible optics has been conceptualized. It is meant to be an extension for the Advanced Electric Propulsion Diagnostics (AEPD) platform, which has been set-up by IOM and partners to provide several diagnostics for in-situ thruster characterization [62, 63]. Relying in first instance on an in-house developed ion source with an unsealed transparent discharge vessel enabled us to qualify our LIF setup without the necessity of expensive modifications to an rf ion thruster for guaranteeing optical access to its discharge chamber. We plan to measure local neutral gas temperatures and ion velocity distribution functions, respectively, in the vicinity of the ion source extraction grids. This is considered to give a solid basis of data required for validation of simulations performed with Dynasim – a numerical code for grid erosion modeling developed at IOM [64, 65].

In this article, further LIF spectroscopy measurements in the rf ion source discharge chamber are presented. Both xenon neutrals and ions have been investigated by taking advantage of two nearby transitions at approximately 834.7 nm. The kinetic temperature of either species has been extracted from the LIF data obtained for different rf power settings. Strong focus is set on the routines for LIF spectrum evaluation. Most probable velocities in parallel to the laser beam propagation still may be read off directly from Doppler-shifted lineshape maxima, whereas proper velocity distribution function (VDF) extraction from LIF spectra is usually not straightforward. This is mainly due to the hyperfine structure inherent to the probed transition and partial saturation of the LIF signal intensity [57, 66,67,68,69]. In the particular case of gridded xenon ion source and ion thruster plasmas (either within the discharge chamber or in the plume) these spectral lineshape features emerge as competing with the Doppler broadening. Thus, the kinetic temperature cannot be inferred from the plain width of structures within a LIF spectrum but requires more sophisticated extraction techniques. Here, hyperfine structure elimination using the inverse filtering (Fourier deconvolution) technique proposed by Smith et al. [57, 68] is demonstrated with simulated LIF spectra. An error analysis is given with respect to the deducible Doppler-induced line broadening width connected with the kinetic temperature. This includes post-deconvolution filtering artifacts, uncertainties concerning the hyperfine structure parameters and the natural linewidth, as well as non-reducible LIF intensity saturation at reasonable laser powers.

Theoretical fundamentals

The theory of single-photon absorption LIF spectroscopy for probing xenon neutrals (Xe I) and single-charge xenon ions (Xe II) has been described in detail elsewhere (see e.g. [57, 66, 70]). In this section, the main aspects, as needed for the LIF spectrum analysis, are summarized.

LIF transition schemes

In the present work, two LIF transition schemes, as given in Fig. 1, apply. These include laser-induced absorption along two nearby transitions at approximately 834.7 nm (in air) or 835.0 nm (in vacuum). Both transitions arise from one of the lower excited states in the $5\textrm{s}^{2}5\textrm{p}^{5}(^{2}\textrm{P}_{1/2}^{\textrm{o}})$ manifold of the Xe I levels and the $5\textrm{s}^{2}5\textrm{p}^{4}(^{3}\textrm{P}_{2})$ manifold of the Xe II levels, respectively. The lower state of the Xe I transition scheme is coupled to the ground state via an electric dipole transition in the VUV range, whereas laser excitation within the Xe II scheme starts at a metastable state. Precise wavelength and frequency data for the laser absorption channels have been summarized in Table 1.

Table 1 Nominal absorption wavelengths (in air) $\lambda _{12}^\mathrm {(air)}$ and corresponding frequencies $\nu _{12}$, estimated natural linewidths $\delta \nu _\mathrm{n}$, as well as fluorescence wavelengths (in air) $\lambda _{23}^\mathrm {(air)}$ for the LIF transition schemes selected for investigation of xenon neutrals (Xe I) and single-charge xenon ions (Xe II); data adopted or calculated on the basis of data from NIST ASD

Fig. 1

Partial Grotrian diagrams for (a) Xe I and (b) Xe II including the single-photon absorption LIF transition schemes applied in this work. Energy level data adopted from NIST ASD

In either case, fluorescence detection is carried out non-resonantly at a transition in the visible range coinciding with a strong radiative decay channel of the laser-excited state. Non-resonant fluorescence detection can be easily achieved by isolating the fluorescence using narrow band-pass filters; this prevents the LIF signal from being distorted by scattered laser radiation.

Spectral absorption lineshape model

When tuning the wavelength $\lambda$ of a narrow-bandwidth laser across the transition of interest, $|1\rangle \rightarrow |2\rangle$, simultaneously measuring the intensity of the (laser-induced) fluorescence at a wavelength corresponding to a transition $|2\rangle \rightarrow |3\rangle$ maps the spectral lineshape of the respective absorption cross-section. The spectral lineshape of the absorption cross-section, $\phi$, is a normalized function of the probing laser frequency $\nu =c/\lambda$. Since laser linewidth broadening is negligible in the present setup, it can be modeled as the convolution of the profiles related to three spectral broadening mechanisms, $\phi = \phi _{\textrm{n}} *\phi _{\textrm{d}} *\phi _{\textrm{hfs}}$ [57]:

• (i) Due to the finite lifetime of excited states, any spectral line is subject to homogeneous natural line broadening following a Lorentzian profile $\phi _{\textrm{n}}$. The natural linewidth $\delta \nu _{\textrm{n}}$, i.e. the FWHM of $\phi _{\textrm{n}}$, corresponds to the sum of the inverse lifetimes of both the upper and the lower state coupled by the transition in question. In terms of the Einstein coefficients $A_{lk}$ for a state $|l\rangle$ decaying into states $|k\rangle$, $E_{l} > E_{k}$, the natural linewidth for the transition $|1\rangle \rightarrow |2\rangle$ is given by [71]

  $$\begin{aligned} 2\pi \, \delta \nu _{\textrm{n}} = \sum \nolimits _{E_1>E_i} A_{1i} + \sum \nolimits _{E_2>E_j} A_{2j}. \end{aligned}$$

  (1)

  Natural linewidth data for the selected transitions are given in Table 1. Enlarged homogeneous line broadening in conjunction with reduced excited state effective lifetimes due to collisional interaction may be resolved depending on the pressure regime.

• (ii) Inhomogeneous line broadening is caused by the Doppler effect. As seen from the laboratory frame, the absorption frequency of an entity of the probed species, which has a velocity component $u_{\textbf{k}}\ll c$ parallel to the laser propagation direction $\textbf{k}/|\textbf{k}|$, is shifted by $\tilde{\nu } \simeq \nu _{12} u_{\textbf{k}} / c$ from the resonance frequency at rest, $\nu _{12}$. Hence, the probed species’ VDF projected onto the laser beam axis, $f_{\textbf{k}}$, translates into a spread of absorption frequency Doppler shifts described by a spectral broadening profile $\phi _{\textrm{d}}$ which is called the Doppler lineshape hereinafter [57]:

  $$f_ {\mathbf{k}}( u_{\mathbf{k }}) \, \mathrm{d}u_{\mathbf{k}} = \phi_{\mathrm{d}}(\tilde{\nu}) \,\mathrm{d}\tilde{\nu} \, , \quad \mathrm{d}u_{\mathbf{k}} = \frac{c}{\nu_{12}} \, \mathrm{d}\tilde {\nu} \quad \Rightarrow \quad \phi_{\mathrm{d}}(\tilde{\nu}) = \frac{c}{\nu_{12}} f_{\mathbf{k}} \! \left (\frac {c }{\nu_{12}} \tilde{\nu} \right)$$

  (2)

  Presuming a VDF of Maxwell-Boltzmann type, $\phi _{\textrm{d}}$ would have the shape of a Gaussian profile potentially shifted by $\triangle \nu _{\textrm{d}}=\nu _{12}\hat{u}_{\textbf{k}}/ c$ according to laser-oriented drift motion of the observed species at the speed $\hat{u}_{\textbf{k}}$. The FWHM of $\phi _{\textrm{d}}$ would be connected with the kinetic temperature T of the probed species (particle mass m):

  $$\begin{aligned} \delta \nu _{\textrm{d}} = \frac{\nu _{12}}{c}\sqrt{8 \ln (2) \frac{k_{\textrm{B}}T}{m}}. \end{aligned}$$

  (3)

• (iii) At typical kinetic temperatures of atoms and ions in low pressure discharges ($\lesssim$ 0.1 eV), Doppler broadening ($<1$ GHz) extends over a similar frequency scale like the intrinsic hyperfine structure features of the absorption lineshape. Since, thus, most hyperfine structure components cannot be spectrally resolved, the sum of their contributions is often treated as another line broadening mechanism. Three phenomena have to be considered: the isotopic composition, the isotope shift, and the nuclear spin splitting (also referred to as hyperfine splitting). By affecting all entities of a given isotope equally, nuclear spin splitting causes homogeneous line broadening, whereas isotope shifts and the abundance distribution do not and, thus, must be considered a combined inhomogeneous line broadening mechanism. The reconstruction of the spectral profile functions $\phi _{\textrm{hfs}}$ corresponding to the hyperfine structures of the two transition lines considered here is extensively described in the Appendix (A.1–A.3) of this article.

Fig. 2

Simulated absorption lineshapes corresponding to the Xe I transition and the Xe II transition each at approximately 835 nm (here and in the following, the spectra are plotted against the laser detuning with respect to the 835.0 nm vacuum wavelength mark). In each case, a projected Maxwell-Boltzmann distribution (kinetic temperature T, no drift along the projection axis) is assumed; the respective (Gaussian) Doppler widths $\delta \nu _{\textrm{d}}$ are given in addition. Note that the hyperfine structures and the naturally broadened spectral profiles, for clarity, are rescaled and displayed with a vertical offset

Figure 2 shows simulated absorption lineshapes for the transitions of Xe I and Xe II at approximately 835 nm for the specific case of Maxwell-Boltzmann distributed velocities and zero net drift in parallel to the direction of laser propagation. The specified kinetic temperatures refer to the nominal (un-shifted linecenter) transition frequencies (see Table 1) and the standard atomic weight of xenon, 131.29 amu [72]. That is, all hyperfine structure components of the respective spectrum have been convolved with the same Doppler lineshape, i.e. a Gaussian profile with uniform FWHM according to Eq. 3, irrespective of the slight isotopic variances in $\nu _{12}$ and m.

Note that, while the plasma in the discharge chamber of an rf ion source approaches partial local thermodynamic equilibrium, which may justify the assumption of Maxwell-Boltzmann distributions for ion and neutral velocities, this model does not necessarily apply to other discharge conditions. In fact, it has been shown invalid for the description of the ionization and acceleration zones in Hall-effect thrusters. Especially, the assignment of (isotropic) neutral and ion temperatures, whilst adhering to the original meaning of the term temperature, turns out to be a misleading concept in that case [23, 40]. Apart from random velocity contributions, broadening in the VDF also maps gradients of directional velocities arising from ionization and acceleration in regions of electric potential variation [21]. The Doppler lineshapes may also reflect VDF narrowing and apparent acceleration of neutrals due to preferential ionization of slowly effusing neutrals [30]. Moreover, time-averaged signal acquisition obscures transient lineshapes of the ion VDF in conjunction with plasma oscillation phenomena and introduces either distinct bimodal structures or just additional blurring [33, 45]. The measurements presented in the following are restricted to the central region of the ion source discharge chamber, though. Hence, Doppler lineshapes composed of a central Doppler shift (according to the bulk velocity component) and a Gaussian-shaped broadening (only accounting for thermalization of the respective plasma species) are assumed within VDF analysis.

In Section 3.1, we describe a method to recover the Doppler lineshape from a measured absorption lineshape directly. This allows for reconstruction of the projected VDF without the necessity of a-priori assumptions on its actual shape.

LIF signal and saturation broadening

Regarding a single channel $|2\rangle \rightarrow |i\rangle$ for decay of the laser-excited state $|2\rangle$ via spontaneous emission of photons with wavelength $\lambda _{2i}$ at the rate $A_{2i}$, the signal S measured in (cw) LIF spectroscopy corresponds to the time-average fluorescence power emitted from the volume cell $\delta {V}$ (probe volume):

$$\begin{aligned} S \propto \frac{h c}{\lambda _{2i}} A_{2i} \left( n_2 + \bar{n}_2 \right) \delta {V}. \end{aligned}$$

(4)

Here, $n_2$ and $\bar{n}_2$ refer to the fractions of the population number density of $|2\rangle$ resulting from laser photon absorption upon the transition $|1\rangle \rightarrow |2\rangle$ and, respectively, collisional excitation of $|2\rangle$ or relaxation into $|2\rangle$. In order to isolate the actual laser-induced fluorescence signal (proportional to $n_2$), the signal from natural plasma emission, which is caused by decay of the species fraction $\bar{n}_2$ within the observed fluorescence channel, has to be suppressed, e.g. by phase-sensitive detection (see Section 4).

A relation between the spectral absorption lineshape and the LIF signal measured when tuning the wavelength of a narrow-linewidth cw laser can be derived on the basis of a steady-state two-level model which comprises the states $|1\rangle$ and $|2\rangle$ coupled by the probed transition [70, 73]. Although strictly valid only for resonant LIF transition schemes involving a metastable lower state or the ground state, this approach gives a reasonable result for non-resonant LIF schemes in qualitative agreement with multi-level models that do not describe $|1\rangle$ and $|2\rangle$ as being isolated from other states [57, 66].

We treat the probed species’ isotopes (nuclear mass numbers A) separately and restrict calculations to a single velocity class $u_{\mathbf{k}} \ldots u_{\mathbf{k}} + \mathrm{d}{u_{\mathbf{k}}}$ (corresponding to a spectral package $\tilde{\nu } \ldots \tilde{\nu } + \mathrm{d}{\tilde{\nu }}$ of Doppler shifts) first [c.f. 70]. With $A_{21}$ and $Q_{21}$, the rate coefficients for spontaneous emission and collisional quenching, as well as $B_{12}$, the Einstein coefficient for stimulated absorption, the laser intensity I, furthermore, $g_1$ and $g_2$ denoting the statistical weights of $|1\rangle$ and $|2\rangle$, Eq. 5 gives the system of rate equations for the (laser-affected) population number density fractions $n_1$ and $n_2$:

$$\begin{aligned} \frac {\mathrm{d}}{\mathrm{d}t} n_ {2,A} = - \frac{\mathrm{d}}{\mathrm{d}t} n_ {1,A} = {} & \frac {B_ {12}}{c} \, I(\nu) \, \psi_ {A}(\nu - \Delta \nu_{A} - \nu_ {12} u_ {\mathbf{k}}/ c) \left(n_ {1,A} - \frac{ g_1 }{ g_2 } n_ {2,A} \right ) \\ {} & - (A_{21} + Q_{21}) \, n_{2,A} \, . \end{aligned}$$

(5)

The lineshape function $\psi _{A}$ contains all homogeneous spectral broadening profiles including natural broadening and the isotope-specific distribution of the nuclear-spin splitting components. In the narrow-bandwidth approximation, it relates to the absorption strength at a given laser frequency $\nu$ for the selected velocity class of a specific isotope whose resonance frequency is shifted by $\triangle {\nu _A}$. Assuming the total population number within the two-level system,

$$n_{1,A}(\nu, u_{\mathbf{k}}) \, \mathrm{d}u_{\mathbf{k}} + n_{2,A}(\nu, u_{\mathbf{k}}) \, \mathrm{d}u_ {\mathbf{k}} = n_{A} f_{\mathbf{k}}(u_{\mathbf{k}}) \, \mathrm{d}u_ {\mathbf{k}} = n_{A} \phi_ {\mathrm{d}}(\tilde {\nu}) \, \mathrm{d}\tilde{\nu} \, ,$$

(6)

to be constant in time, the contribution to the upper state’s stationary population number density establishing with the laser frequency and the intensity, respectively, set to $\nu$ and $I(\nu )$ is given by [c.f. 57, 66, 70]

$$n_{2, A}(\nu, I(\nu), \tilde{\nu}) \, \mathrm{d}\tilde{\nu} = \frac{ g_2 }{g_1 + g_2} \frac{n_A\phi_{\mathrm{d}}(\tilde{\nu}) \, \mathrm{d}\tilde{\nu}}{1 + \left[\dfrac{\pi\delta \nu_{\mathrm{n}}}{2} \dfrac{I(\nu)}{I_{\mathrm {s}}} \psi_ {A}(\nu - \Delta \nu_{A} -\tilde{\nu}) \right ]^{-1}}$$

(7)

where the minimum saturation intensity, normalized to the homogeneous lineshape maximum $2/(\pi \, \delta \nu _{\textrm{n}})$, has been defined as [57, 66]

$$\begin{aligned} I_{\textrm{s}} = \frac{\pi \, \delta \nu _{\textrm{n}}}{2} \frac{g_2}{g_1+g_2} \frac{A_{21}+Q_{21}}{B_{12}/c} = \frac{ 4\pi ^2 h c \, \delta \nu _{\textrm{n}}}{\lambda _{12}^3} \frac{g_1}{g_1+g_2} \left( 1+\frac{Q_{21}}{A_{21}} \right) . \end{aligned}$$

(8)

The homologous laser intensity $I/I_{\textrm{s}}$ (sometimes referred to as saturation parameter) represents the transition-specific strength of optical pumping relative to relaxation of the two-level system.

Integrating Eq. 7 over all Doppeler shifts and summarizing the contributions of all isotopes (abundances $p_A$), yields the LIF signal measured at the laser frequency $\nu$:

$$S(\nu, I(\nu)) \sim n_2(\nu, I(\nu)) = \sum\nolimits_{A} p_{A} \int_{-\infty}^{\infty} n_{2,A}(\nu , I(\nu), \tilde{\nu}) \, \mathrm{d}\tilde{\nu} \, .$$

(9)

Equation 7 reveals the existence of two LIF signal regimes in respect to the strength of the laser excitation:

• (i) With the laser intensity kept well below the minimum saturation intensity, $I \ll I_{\textrm{s}}$, the LIF signal is in proportion to I and the spectral profiles associated with homogeneous and inhomogeneous broadening convolve linearly with each other. Thus, the normalized LIF spectrum converges to the spectral absorption lineshape:

  $$\begin{aligned} S(\nu ,I(\nu )) \sim I(\nu ) \left( \sum \nolimits _{A} \psi _{A} *\phi _{\textrm{d}} \right) \!(\nu ) = I(\nu ) \left( \phi _{\textrm{hfs}} *\phi _{\textrm{n}} *\phi _{\textrm{d}} \right) \!(\nu ). \end{aligned}$$

  (10)

• (ii) For higher laser intensities, $I{\gtrsim } I_{\textrm{s}}$, the rate, at which entities in the lower state absorb laser photons, approaches or exceeds the relaxation rate of the upper state. Depleting the lower state by laser intensity enhancement reduces net absorption; effectively the fluorescence intensity gain is smaller than unity. That is, the LIF signal (at a given laser frequency) scales non-linearly with increasing I and eventually saturates.

Fig. 3

Simulated LIF spectra (Xe I at 835 nm, $\delta {\nu }_{\textrm{d}}=0.55$ GHz) at different homologous laser intensities $I/I_{\textrm{s}}$ normalized to the respective maximum signal in each case. The absolute signal strengths associated with the six distinct peak structures are given in the inset. Note that peak “A” grows approximately linearly by almost three orders of magnitude, whereas the LIF intensity enhancement of peak “D” is relatively small and clearly shows saturation with increasing laser intensity

Figure 3 illustrates the effect of non-linear absorption on LIF spectra taking the Xe I transition at 835 nm as an example. Within the non-linear absorption regime, normalized LIF spectra noticeably depart from the actual absorption lineshape because the saturation threshold intrinsically depends on the homogeneously broadened lineshape components $\psi _{A}$ [66]: Due to preferential absorption, the LIF signal rather saturates in the vicinity of the center of a spectral line component (minimum saturation intensity $I_{\textrm{s}}$). In its wings, conversely, the effective saturation intensity is higher and the LIF signal gain along with increasing laser intensity is larger. Similarly, the individual peak heights in the centers of hyperfine structure component overlap regions do not grow uniformly with increasing laser intensity. In that way, LIF spectra exhibit a laser-intensity dependent, rather complex deformation of the actual absorption lineshape.

Data evaluation and error analysis

LIF spectrum deconvolution

Within the linear absorption regime and in the absence of any spectral broadening mechanisms but hyperfine structure, natural broadening, and Doppler broadening, the spectral absorption lineshape can be written as [68]

$$\begin{aligned} S(\nu ) \propto \phi _{\textrm{w}}(\nu ) = \phi _{\textrm{c}}(\nu ) *\phi _{\textrm{d}}(\tilde{\nu })\ . \end{aligned}$$

(11)

Accounting for finite kinetic temperature (or, more generally, velocity dispersion) represented by the Doppler lineshape $\phi _{\textrm{d}}$, we call $\phi _{\textrm{w}}$ the “warm spectrum” hereinafter, whereas we refer to $\phi _{\textrm{c}} = \phi _{\textrm{hfs}} *\phi _{\textrm{n}}$ as the “cold spectrum” which only comprises the hyperfine structure and natural broadening [57]. Curve fitting to $S(\nu )$ may be applied for extraction of the probed species’ kinematic properties (velocity component mean value and temperature). Apart from modeling the cold spectrum, this approach requires to draw a presumption on the shape of the projected VDF.

An alternative strategy bases on direct deconvolution of the warm spectrum by so-called inverse filtering [57, 68], i.e. elimination of the cold spectrum, which leaves the Doppler lineshape as a remainder and thus allows for extraction of the actual projected VDF: According to the convolution theorem, convolution is equivalent to multiplication in the corresponding Fourier space. Let $\textrm{FT}[{\cdot }]$ and $\textrm{iFT}[{\cdot }]$ denote the forward and the backward Fourier transform, respectively. Then, Eq. 11 turns into

$$\mathrm{FT}[\phi_{\mathrm{w}}](\theta) = \mathrm{FT}[\phi_{\mathrm{c}}](\theta) \times \mathrm{FT}[\phi_ {\mathrm{d}}](\theta) \quad \Rightarrow \quad\phi_{\mathrm{d}}(\tilde{\nu}) = \mathrm{iFT}\!\left[\frac{\mathrm{FT}[\phi_{\mathrm{w}}](\theta)}{\mathrm{FT}[\phi_{\mathrm{c}}](\theta)}\right] \, .$$

(12)

Using Eq. 2, the extracted Doppler lineshape can be transformed into the corresponding velocity component distribution function.

For efficient processing of measured LIF spectra, evaluation routines including hyperfine structure modeling, inverse filtering, and deconvolution result analysis have been created with recourse to several Python libraries [74,75,76]. The deconvolution procedure bases on the SciPy implementation of the discrete Fourier transform taking real inputs. According to Eq. 12, the measured LIF spectrum identified with the warm spectrum and the modeled cold spectrum need to exist on the same laser-frequency domain (which must have regular sample spacing). Since laser-frequency sampling associated with a LIF spectrum scan not necessarily occurs to be uniform, the LIF signal is interpolated to the denser regular frequency domain of the cold spectrum model prior to processing in the Fourier space. The deconvolution result finally is re-evaluated for the originally measured laser-frequency-detuning sequence. This ensures that uncertainties of parameters deduced from curve-fitting to the deconvolution result are not underestimated due to an artificially increased number of the degrees of freedom.

Inverse filtering benchmarking

In order to test and optimize the inverse filtering method, warm spectra for the selected transitions have been simulated in the linear absorption regime assuming Gaussian Doppler lineshapes (Maxwell-Botzmann VDFs) in each case. The simulated spectral lineshapes $\phi _{\textrm{w}}$ have been refined with white noise represented by a sequence $\rho$ of normally distributed random numbers. The noise level attributed to a warm spectrum simulation $\phi_{\textrm{w}}^{\prime}=\phi_{\textrm{w}}+\rho$ is determined by the peak signal-to-noise ratio

$$\begin{aligned} \textrm{PSNR}(\phi _{\textrm{w}}^{\prime }) = \frac{\textrm{Max}{[\phi _{\textrm{w}}(\nu )]}}{\sqrt{\textrm{Var}{[\rho (\nu )]}}}\ . \end{aligned}$$

(13)

Simple inverse filtering

In the presence of noise, inverse filtering does not reveal the true Doppler lineshape $\phi _{\textrm{d}}$ as suggested by Eq. 12 but yields the estimator

$$\begin{aligned} \phi _{\textrm{e}}(\tilde{\nu }) = \textrm{iFT}\left[ {\frac{\textrm{FT}[{\phi _{\textrm{w}}}]\!(\theta ) + \textrm{FT}[{\rho }]\!(\theta )}{\textrm{FT}[{\phi _{\textrm{c}}}]\!(\theta )}}\right] \ . \end{aligned}$$

(14)

A spectral lineshape – even a typical cold spectrum $\phi _{\textrm{c}}$ with relatively sharp structures in the laser-frequency space – has significant amplitudes only at low Fourier frequencies $\theta$, whereas the noise Fourier amplitude typically comes with non-vanishing contributions over the whole range of $\theta$. Therefore, the term $\textrm{FT}[{\rho }]\!(\theta ) / \textrm{FT}[{\phi _{\textrm{c}}}]\!(\theta )$ in Eq. 14 may get dominating at the Nyquist frequency which induces a substantial portion of noise at the frequency-detuning sampling rate within the Doppler lineshape estimator $\phi _{\textrm{w}}^{\prime }$. Figure 4 illustrates this phenomenon with the aid of warm spectrum simulations at different pre-deconvolution noise levels: Proper identification of the Doppler lineshape is only possible if the input noise is practically absent; at more realistic input PSNR, the Doppler lineshape almost vanishes within the deconvolution-induced noise.

Fig. 4

(a) – (c) Warm spectra $\phi_{\textrm{w}}^{\prime }$ ($\delta \nu _{\textrm{d}}=0.55$ GHz) simulated for the Xe I transition at 835 nm with different input peak signal-to-noise ratios (PSNR). (d) – (f) Corresponding Doppler lineshape estimates $\phi _{\textrm{e}}$ extracted via the simple inverse filtering technique according to Eq. 12

To improve the inverse filtering technique, the artificial-noise-inducing high-frequency Fourier components have to be suppressed. In a simple approach, the frequency-detuning sampling rate is reduced prior to computation in the Fourier space. This effectively truncates the Fourier transforms of the cold spectrum model and the measured LIF signal from which three major problems arise: (i) Lowering of the frequency sampling rate reduces the level of detail in the cold spectrum model. (ii) Interpolation of the (noisy) LIF signal from the (non-uniformly spaced) measured laser-frequency samples to a (uniform) frequency domain with too large or even similar sample spacing induces a frequency-jitter-like distortion of the deconvolution result. (iii) Optimization of the frequency-detuning sampling rate requires inefficient re-interpolation of the cold spectrum model and the measured LIF signal at each instance. Figure 5 (b) presents a spectral histogram view of superimposing inverse filtering results repeated for a large number of simulated spectra sharing the same Doppler lineshape. Even though the frequency interpolation spacing has been adjusted for minimizing the output noise level, the spread in estimated Doppler lineshapes is too large to allow for precise extraction of Doppler lineshape-related parameters from a single LIF spectrum. Instead, averaging over several deconvolved LIF spectra would be required.

Fig. 5

(a) Superposition of 500 simulated warm spectra $\phi _{w}^{\prime }$ (Xe I at 835 nm, $\delta \nu _{\textrm{d}}=0.55$ GHz, input PSNR of 100). Spectral histogram of the corresponding deconvolution results $\phi _{\textrm{e}}$ using inverse filtering with near-optimum coarse pre-deconvolution interpolation (b) and with optimized post-deconvolution filtering (c)

Improved inverse filtering

An alternative strategy bases on post-deconvolution filtering with the frequency interpolation spacing kept constant at a sufficiently low level, typically 0.1 to 1 $\textrm{MHz}$ (i.e. one up to two orders of magnitude smaller than the natural linewidth and the minimum spacing between hyperfine structure components). For this approach, the Doppler lineshape estimator is re-defined as

$$\phi_{\mathrm{e}}(\tilde{\nu}) = \mathrm{iFT}\!\left[G(\theta, \delta \theta) \frac{\mathrm{FT}[\phi_ {\mathrm{w}}](\theta) + \mathrm{FT}[\rho](\theta)}{\mathrm{FT}[\phi_{\mathrm{c}}](\theta)} \right]$$

(15)

where $G(\theta , \delta \theta )$ is a suitable low-pass filter window in Fourier space with the effective window edge denoted by $\delta \theta$. Two simple filter windows have been tested here, a rectangular one (sharp edge at $\theta =\delta \theta$) and a Gaussian one (whose HWHM corresponds to $\delta \theta$). Figure 5 (c) shows the result of a simulation performed under the same conditions as specified above for (a), this time however, with a rectangular post-deconvolution filter. Clearly, the variance in Doppler lineshape extraction has been reduced which enables reasonable estimation of VDF parameters at a single instance, i.e. without the necessity of averaging over several deconvolved LIF spectra.

Fig. 6

Estimated Doppler lineshapes obtained from simulated Xe I warm spectra (actual Doppler width of 0.55 GHz, input PSNR of 100) via inverse filtering with post-deconvolution filtering according to Eq. 15 using rectangular windows (a) – (c) and Gaussian windows (d) – (f) with three different window edges $\delta \theta$

The quality of the deconvolution result crucially depends on the choice of the low-pass filter window edge. As shown in Fig. 6, panels (c) and (f), much noise can pass the filter band when using too large filter window edges. On the other hand, too narrow pass-bands induce aliasing effects either (a) in form of side lobes observed with the rectangular filter window or (d) over-smoothing in conjunction with the Gaussian-shaped filter.

Taking inspiration from [57], the following benchmarking procedure has been worked out for determining the properties of both filter window types and for demonstrating the capabilities of the improved inverse filtering technique:

• (i) An ensemble of warm spectra with individual random number noise traces is generated for a given Doppler width (assuming a Gaussian Doppler lineshape) and PSNR. They undergo inverse filtering using rectangular or Gaussian filter windows with various filter window widths.

• (ii) For each parameter set, the post-deconvolution PSNR is calculated analogously to Eq. 13 using the ensemble average maximum value and the frequency-integrated ensemble standard deviation of the Doppler lineshape estimator. The respective noise figure is given by the ratio of the input PSNR and the post-deconvolution PSNR and determines the level of noise amplification.

• (iii) For each sample of a warm spectrum ensemble, the total deviation of the estimated Doppler lineshape $\phi _{\textrm{e}}$ with respect to the true Doppler lineshape $\phi _{\textrm{d}}$ is quantified in terms of the integrated fractional Doppler lineshape estimation error 

  $$\mathrm{IFEE} = \sqrt{\frac{1}{\tilde{\nu}_{\mathrm{max}} - \tilde{\nu}_{\mathrm{min}}}\int_{\tilde {\nu}_{\mathrm{min}}}^{\tilde{\nu}_{\mathrm{max}}} \left\{\frac{\phi_{\mathrm{e}}(\tilde{\nu}) - \phi_{\mathrm{d}}(\tilde{\nu})}{\mathrm{Max}[\phi_{\mathrm{d}}(\tilde{\nu})]} \right\}^{2}\mathrm{d}\tilde{\nu} } \, .$$

  (16)

  The mean IFEE measures the overall distortion of the estimated Doppler lineshape resulting from artifacts of post-deconvolution filtering.

• (iv) A Gaussian profile function is adjusted to each Doppler width estimator sample yielding the deviation between the estimated and the true Doppler width. The mean Doppler width estimation error relates to the systematic uncertainty in kinetic temperatures estimated from filtered deconvolution results.

• (v) When applying inverse filtering to measured LIF spectra, neither minimization of the IFEE nor the Doppler width estimation error can be used for post-deconvolution filter optimization. As an alternative quantity for judging the deconvolution result’s quality the integrated fractional re-convolution discrepancy 

  $$\mathrm{IFRD} = \sqrt{\frac{1}{\nu_{\mathrm{max}} - \nu_{\mathrm{min}}} \int_{\nu_{\mathrm{min}}}^{\nu_ {\mathrm{max}}} \left \{\frac{\phi_{\mathrm{c}}(\nu) \ast \phi_{\mathrm{e}}(\tilde{\nu}) - \phi ^{\prime}_{\mathrm{w}}(\nu)}{\mathrm{Max}[\phi^{\prime}_{\mathrm{w}}(\nu)]} \right\}^{2} \mathrm{d}\nu}$$

  (17)

  is computed by comparing the convolution of the Doppler lineshape estimator $\phi _{\textrm{e}}$ and the cold spectrum model $\phi _{\textrm{c}}$ with the originally measured (here simulated) warm spectrum $\phi _{\textrm{w}}^{\prime }$.

Fig. 7

Noise figure and mean integrated fractional Doppler lineshape estimation error (IFEE) in conjunction with post-deconvolution filtering using (a), (c) rectangular windows and (b), (d) Gaussian windows with variable width (window edge $\delta \theta$). For every tested Doppler width $\delta \nu _{\textrm{d}}$, input PSNR, and low-pass filter edge, 500 Xe I warm spectra have been separately deconvolved

Figure 7 summarizes the noise figures and the IFEE obtained from the outlined inverse filtering benchmarking procedure performed with warm spectrum simulations referring to the Xe I transition at 835 nm. The figures of merit exhibit different absolute values depending on the applied LIF transition scheme. However, simulations for the Xe II transition at 835 nm yield the same general trends and orders of magnitude. Panels (a) and (b) reveal that deconvolution-induced noise amplification does practically not depend on the input PSNR. The minimum noise figure and the respective post-deconvolution filter edge are connected with the given Doppler width. There is essentially no difference in the noise suppression ability between the rectangular filter window (left-hand side) and the Gaussian one (right-hand side). The mean IFEE strongly depends on the filter pass-band size. A pronounced minimum establishes whenever the extracted lineshape is relatively smooth (low noise figure) and gets insignificantly affected by artificial filter broadening at the same time. With well-adjusted filters of either type, the IFEE is found to be roughly 2.5 % even at a relatively poor input PSNR of 20. At higher input PSNR the minimum achievable IFEE can be much lower. However, it cannot drop below a lower limit which probably arises from discretization and finiteness of the laser-frequency domain.

Fig. 8

Optimum rectangular window width (according to the minimum integrated fractional re-convolution discrepancy, IFRD) and corresponding Doppler width estimation error for post-deconvoltion filtering. The shown curves refer to mean values obtained from separate deconvolution of 500 (a), (c) Xe I and (b), (d) Xe II warm spectra simulated for each tested Doppler width $\delta \nu _{\textrm{d}}$ and input PSNR

At least in the case of Gaussian Doppler lineshapes, it turned out that IFEE and IFRD are equivalent. Therefore, the rather general IFRD minimization criterion is used to optimize the filter window edge in regard to a given warm spectrum. Figure 8 shows the optimum rectangular filter edge and the minimum achievable Doppler width estimation error as a function of the given Doppler width for different input PSNR and for both of the selected transitions. As indicated in Panels (c) and (d), the relative Doppler width estimation error is the smaller the larger the actual Doppler width. Slight deviations between the simulation results for either transition scheme are likely to occur because of different hyperfine structures and natural linewidths.

Here, we found the optimum rectangular filter window to perform better as the Gaussian one. For high and moderate input PSNR, the relative Doppler width estimation error obtained for IFRD minimizing Gaussian filter window edge turns out to be higher by approximately one order of magnitude. For that reason, solely the inverse filtering technique using a rectangular filter window has been applied to attain the results reported below.

Hyperfine structure model uncertainty

Parameters required for reconstruction of the hyperfine structure of a transition lineshape are subject to uncertainty. To reproduce error propagation through deconvolution of LIF spectra when using a fixed set of parameters for hyperfine structure modeling, a simple Monte-Carlo simulation has been implemented: For both selected xenon transitions, a large number of potentially true hyperfine structures is computed with respect to the input parameters’ confidence intervals. That is, the isotope shifts as well as the nuclear magnetic dipole and the nuclear magnetic quadrupole interaction constants are randomly sampled assuming normal probability density functions, in each case with the data given in Appendix Tables 2 and 3 interpreted as the most probable values and the respective standard deviations. The isotope abundance distribution as well as the relative intensities computed for each line component arising from nuclear-spin splitting are assumed to be exact. By convolving each hyperfine structure sample with the transition-specific natural lineshape and the same Gaussian Doppler lineshape, we simulate the space of all possible (noiseless) warm spectra at a given Doppler width. All warm spectrum samples in turn undergo inverse filtering each time using the cold spectrum model solely based on the most probable hyperfine structure parameters. This yields the spread in estimated Doppler lineshapes resulting from uncertainty in the true hyperfine structure. The respective distribution of estimated Doppler widths is obtained by adjusting a Gaussian profile to each estimated Doppler lineshape sample.

Fig. 9

(a) Probability density function of the estimated Doppler width $\delta \nu _{\textrm{e}}$ obtained from Monte-Carlo simulation of deconvolution errors due to uncertainties in the parameters determining the hyperfine structure of the selected Xe I and Xe II transition. Each simulation comprises deconvolution of 1 $\times$ 10$^{5}$ warm spectra assuming a Gaussian Doppler lineshape with $\delta \nu _{\textrm{d}}=0.55$ GHz. (b) Standard deviation of the estimated Doppler width $\delta \nu _{\textrm{e}}$ relative to the actual Doppler width $\delta \nu _{\textrm{d}}$ reflecting the hyperfine-structure-associated uncertainty of extracted Doppler widths $\delta \nu _{\textrm{e}}$

Figure 9 (a) shows the simulated probability density functions of the estimated Doppler width for a given (true) Doppler width of 0.55 GHz for both the Xe I and the Xe II transition at 835 nm. There is a noticeably larger spread in estimated Doppler shifts for the Xe II transition than in the Xe I case. This can be attributed to differently sized confidence intervals of the used hyperfine structure parameters, but might be also related to the smaller line component separation in comparison to the Doppler width. Generally, the simulated estimated Doppler width distributions appear to have almost no skew, and thus, the standard deviation may be identified with the absolute Doppler lineshape estimation error. As one can infer from panel (b), the relative Doppler lineshape estimation error decreases with increasing actual Doppler width. In conjunction with the used hyperfine structure parameter sets, the uncertainty can be considered almost negligible in the case of kinetic temperature extraction from Xe I spectra. The actual Doppler width estimation error connected with hyperfine structure uncertainty of the Xe II transition at 835 nm might differ from the simulation results presented here since isotope shifts are not known and must be extrapolated from data available for similar transitions only.

Natural linewidth error

Proper VDF extraction from LIF spectra requires elimination of homogeneous line broadening, i.e. natural line broadening potentially enhanced by collisional quenching and interaction with strong local fields. The latter can be safely neglected here on account of the present gas pressure and plasma density regime [29, 66, 77]. For example, resonance broadening [77], which appears to be the dominating pressure broadening mechanism for the Xe I transition at 835 nm, is estimated to be at least two orders of magnitude smaller than the natural linewidth at $\lesssim$ 1 Pa. However, natural linewidth estimates base on excited state lifetime data that, if any, usually come with significant uncertainty.

The Doppler width estimation error due to natural linewidth uncertainty propagating through the LIF spectrum deconvolution procedure has been simulated in the following way for both transition schemes, each time, assuming Gaussian Doppler lineshapes: A series of cold spectra is computed for a range of potential natural linewidth true values. For each cold spectrum, the corresponding warm spectrum (at a given Doppler width) is calculated and then deconvolved using a fixed cold spectrum model according to the natural linewidth data given in Table 1. Curve fitting to the extracted Doppler lineshape yields an estimate for the actual Doppler lineshape depending on the discrepancy between the true and the assumed natural linewidth.

Fig. 10

Natural linewidth uncertainty-related deconvolution error propagation simulated for warm spectra of the Xe I and the Xe II transition at 835 nm. Each solid line refers to a set of (true) Gaussian Doppler widths $\delta \nu _{\textrm{d}}$ and natural linewidth errors in the respective cold spectrum model yielding equal Doppler width estimates $\delta \nu _{\textrm{e}}$ upon deconvolution

The simulation results have been summarized in the contour plots of Fig. 10. Intuitively, the deconvolution result suffers from overestimation of the Doppler width if the natural linewidth is underestimated (positive natural linewidth discrepancy), and vice versa (negative natural linewidth discrepancy). As an example referring to Panel (a), consider that deconvolution of a Xe I LIF spectrum gives a Doppler linewidth of 0.55 GHz. The corresponding solid line indicates the actual Doppler width could be approximately 0.535 or 0.565 GHz if the true natural linewidth is 50 %, respectively, larger or smaller than the assumed one. For Xe I LIF spectra, the effect of using inaccurate natural linwidth data is not severe; within the $\pm 50$ % range of natural linewidth discrepancy, the Doppler width estimation error is substantially smaller than $\pm 4$ % at all sampled Doppler widths. Dealing with Xe II spectra, the Doppler width estimation error is even much smaller since the true natural linewidth (although not precisely known) is expected to be very small in comparison to the Doppler width.

LIF intensity saturation

As it has been discussed in Sec. 2.3, saturation of the LIF intensity causes a measured LIF spectrum to deviate from the actual spectral absorption lineshape, i.e. the warm spectrum. Deconvolving LIF spectra affected by saturation just by eliminating the intrinsic cold spectrum (thereby, implicitly assuming validity of the linear absorption regime) will result in Doppler lineshape estimates distorted by additional broadening depending on the laser intensity. Although using low laser intensities, in principle, can help to reduce this kind of lineshape distortion, it cannot be suppressed entirely since efficient LIF signal detection is impossible to maintain at arbitrarily low laser intensities.

The effect of LIF intensity saturation on Doppler lineshapes extracted by deconvolution of LIF spectra for the Xe I and the Xe II transition at 835 nm may be simulated as follows: According to Eq. 9, spectral lineshape function are computed for given Gaussian Doppler widths and variable homologous laser intensities $I/I_{\textrm{s}}$ (assumed to be constant throughout the considered spectral range). Straight deconvolution yields a Doppler lineshape estimate whose widths is calculated by means of Gaussian curve-fitting, irrespective of any lineshape distortion. The difference between the Doppler width estimated this way and the actual one represents the saturation-induced Doppler width estimation error.

Fig. 11

LIF intensity saturation-induced deconvolution error simulated for warm spectra of the Xe I and the Xe II transition at 835 nm. Each solid line refers to a set of (true) Gaussian Doppler widths $\delta \nu _{\textrm{d}}$ and homologous laser intensities $I/I_{\textrm{s}}$ yielding equal Doppler width estimates $\delta \nu _{\textrm{e}}$ upon deconvolution

Figure 11 summarizes the simulation results as contour plots. For both xenon transitions, the Doppler lineshape deviation is considerably smaller than 1% at homologous laser intensities below 0.1, i.e. for laser intensities smaller than 1/10 of the transition-specific minimum saturation intensity $I_{\textrm{s}}$. Hence, the saturation effect on the estimated Doppler width is regarded negligible under this condition. However, further increasing the laser intensity is accompanied by a rapidly growing Doppler width estimation error. According to Eq. 8, the minimum saturation intensities for both the Xe I and the Xe II transition at 835 nm amount to, respectively, 0.24 and 0.092 mW mm$^{-2}$ (assuming negligible collisional quenching). With typical net laser intensities expected to range from 0.1 to 5 mW mm$^{-2}$, the corresponding homologous intensities (saturation parameters) are approximately between 0.4 and 20 (Xe I) or between 1 and 50 (Xe II). The calculated relative Doppler width estimation errors obtained by deconvolution of LIF spectra simulated for the specified homologous laser intensity ranges are between 1 to 25 % (Xe I) and 3 to 15 % (Xe II). For both transitions, the relative Doppler width estimation error exhibits only marginal variation with respect to the actual Doppler width. Depending on the laser intensity, LIF intensity saturation, at the same time, can emerge as the dominating source for incorrect Doppler width determination due to systematic errors exceeding the combined uncertainty arising from non-reducible noise amplification as well as the hyperfine structure model and the natural linewidth uncertainty.

Experimental apparatus

As mentioned before, current work at IOM is focused on the investigation and characterization of gridded ion sources and thrusters. Xenon LIF spectroscopy measurements have been carried out in the discharge chamber of an in-house developed radiofrequency (rf) broad beam ion source (ISQ40RF), which relies on inductively coupled plasma excitation at 13.56 MHz [78]. It is equipped with a three-grid extraction system. In order to enable optical access to the discharge chamber, the standard ceramic discharge vessel has been replaced by a fused silica tube. The ion source is placed in a high-vacuum test facility (1.1 m in diameter, 3 m in length). A turbomolecular pump (2000 L s$^{-1}$) backed by a multistage Roots pump (60 m$^{3}$ h$^{-1}$) together with a cryopump (8000 L s$^{-1}$) ensure a base pressure below 1 $\times$ 10$^{-6}$ mbar. With the ion source being operated at 1 sccm xenon supply rate, the facility pressure is approximately 3 $\times$ 10$^{-5}$ mbar. The discharge chamber pressure is larger by one order of magnitude as a result of limited grid flow conductance [78].

Fig. 12

Schematic representation of the LIF setup (adopted from [61])

The LIF spectroscopy setup is partitioned into three sections: the laser system with associated diagnostics, the LIF optics within the vacuum chamber, and the LIF signal detection apparatus. A schematic representation of the setup is given in Fig. 12. Modularity of the setup along with extensive utilization of optical fiber coupling allows for independent component optimization, straightforward extension, and maximum flexibility when used in combination with other diagnostics or when installed in different vacuum facilities.

The cw laser system consists of an extended-cavity diode laser with Littrow-grating stabilization (Toptica DL pro) and a semiconductor optical amplifier with plano-tapered waveguide geometry (Toptica BoosTA). Coarse laser wavelength tuning between 800 and 840 nm is possible by manual rotation of the seed laser grating. Attached piezoelectric actuators allow for automated wavelength scanning without the occurrence of mode hops over a limited range only. For a sufficiently large mode-hop free tuning range, the laser diode injection current must be changed in proportion to the grating piezo voltage. To record the prominent lineshape features of the Xe I transition, a 7 $\textrm{GHz}$ scan range is required; typically, a 4 $\textrm{GHz}$ scan covers the significant hyperfine structure components related to the Xe II transition. In the current configuration, the maximum laser power at the aperture of the optical amplifier is 150 mW at 835 nm. Adjustment of the laser power is carried out by means of externally placed neutral density filters. The laser linewidth nominally amounts to 200 kHz.

Laser wavelength measurement is conducted with a self-calibrating high-precision Fizeau lambdameter (High Finesse WS7-60 UV-I). The wavelength reading is computationally corrected for temperature changes inside the etalon section which yields an absolute accuracy of 60 MHz and a 10 MHz resolution within the relevant laser wavelength range. A digitally controlled photodiode sensor (Thorlabs PM100D and S100C), which terminates a laser beam branch containing approximately 9% of the full optical amplifier output power, is used for monitoring the laser power level. For each wavelength scan, the relative change in the laser intensity connected with variation of the laser diode injection current has to be recorded for subsequent correction of the LIF signal.

By passing a mechanical chopper (Stanford Research Systema SR540), the main laser beam is periodically blocked (50 % duty cycle at ca. 960 Hz) which generates the reference signal required for LIF signal amplification. The modulated beam is transferred into the vacuum chamber via single-mode optical fiber patch cords.

Fig. 13

(a) Schematic representation (not to scale) and (b) photograph of the vacuum-sided LIF apparatus (adopted from [61])

A tiltable fixed-focus collimation unit for fiber decoupling provides a parallel laser beam (1/e diameter $\lesssim 2.5$ mm). Downstream of an anti-reflection coated protection window, the maximum net laser power is approximately 30 mW. For fluorescence detection, the probe volume is imaged on the entrance facet of a multi-mode optical fiber by means of two rigid plano-convex 2$^{\prime \prime }$ lenses. The fiber patch cord mounting can be tilted and shifted with respect to the imaging lens-system axis. The LIF optics are attached to a semi-circular mounting which offers various placement options for both the laser and the fluorescence probe in a common plane defined by their optical axes intersecting at the probe volume (see Fig. 13). The experimental results presented below have been obtained with the LIF optics plane adjusted perpendicular to the direct axis of the ion source and aligned with a gap between the turns of the rf coil. The setup geometry, hence, allows for radial velocity component investigation of xenon neutrals and ions within the ion source discharge chamber.

The fluorescence signal is detected using a fiber-coupled photomultiplier assembly. An optical band-pass filter (center wavelength either at 473 or 543.5 nm, FWHM of 10 nm) preceding the window of the photomultiplier tube (Hamamatsu R7518) isolates the fluorescence channel of interest. To capture the actual LIF signal, the strong background resulting from natural plasma emission needs to be removed from the photomultiplier signal by phase-sensitive detection at the reference frequency determined by the laser beam chopping rate. At each laser wavelength sample, the corresponding time-average LIF signal is obtained by means of a digital lock-in amplifier (Stanford Research Systems SR830). Best signal-to-noise ratios have been found using three internal low-pass filter stages and integration times (tenfold lock-in time constant) of 1 and 3 s per laser wavelength step for recording LIF spectra of the Xe I and the Xe II transition, respectively, at 835 nm.

Results and discussion

A series of Xe I and Xe II LIF spectroscopy measurements has been conducted in the center of the ion source discharge chamber. The probe volume (lateral extent $\lesssim$ 2.5 mm) is located 20 mm upstream the screen grid on the discharge vessel centerline. To minimize LIF intensity saturation-induced broadening, the laser power level was reduced to 6 and 8 % of the maximum achievable laser power for Xe I and Xe II LIF measurements, respectively. At the site of the probe volume, the net laser power therefore is lower than, respectively, 2 and 2.5 mW. Assuming a Gaussian beam with an 1/e waist diameter of 2.5 mm, the corresponding probe volume average laser intensities are 0.4 and 0.5 mW mm$^{-2}$ at maximum. Further reduction of the laser power has not been considered reasonable due to the decreasing signal-to-noise ratio.

LIF spectra for both the Xe I and the Xe II transition at 835 nm were recorded with the ion source started and operated with different rf power settings. For each operation condition, the LIF measurements were performed 90 min after plasma ignition to account for an initial pressure and temperature stabilization period. Figure 14 (a) and (c) show a selected LIF spectrum for each transition scheme. In each case, the LIF signal has been corrected for laser intensity variation during laser wavelength scanning by normalization to the relative laser power measured as a function of the laser wavelength. Thereby, validity of the linear absorption regime (see Eq. 10) has been implicitly presumed.

Fig. 14

LIF spectra for the Xe I transition (a) and for the Xe II transition (c) at 835 nm measured with radial laser injection at the center of the ion source discharge chamber at an absorbed rf power of 19 W. The LIF spectra have been measured 90 min after plasma ignition. The associated reconstructed LIF spectra equal the reconvolution of the Gaussian profiles adjusted to the extracted Doppler lineshapes shown in (b) and (d)

The Doppler lineshape inherent to each measured LIF spectrum was extracted by means of the inverse filtering technique with post-deconvolution filtering as explained in Sec. 3.2. To minimize filter broadening artifacts, a rectangular Fourier-filter window was customized for each LIF spectrum based on the IFRD minimization criterion. Panels (b) and (d) contain the deconvolution results pertaining to the LIF spectra given in (a) and (c), respectively. For both the Xe I and the Xe II spectra recorded at any of the tested ion source operation conditions, the extracted Doppler lineshapes are well approximated by Gaussian profiles implying that at least the radial velocity component is subject to a Maxwellian distribution. To reconstruct the respective LIF spectrum, the best-fit Gaussian profile to the deconvolution result has been reconvolved with the cold spectrum model. In either case, the result is largely consistent with the originally measured LIF spectrum, however, distinct deflections observed in the spectrally resolved discrepancy from the reconstructed spectrum indicate residue LIF intensity saturation and possibly slight inaccuracies of the hyperfine structure model.

With respect to the nominal absorption wavelengths, the recorded Xe I LIF spectra and likewise the extracted Doppler lineshapes are systematically red-shifted by roughly 150 MHz. Conversely, the Xe II LIF spectra consistently show a blue shift within the same order of magnitude. In the discharge chamber center, however, neither neutrals nor ions are assumed to drift in radial direction. Instead, the observed linecenter Doppler shifts most likely result from a systematic error in laser wavelength measurement and also reflect the uncertainty associated with linecenter frequency reference data.

Fig. 15

Kinetic temperature of the xenon neutrals (a) and ions (b) in the discharge chamber center for different rf power settings. The data point symbols refer to converted Doppler widths estimated by adjusting Gaussian profiles to deconvolved LIF spectra recorded 90 min after plasma ignition. Solid error bars refer to the combined uncertainties of curve-fitting as well as hyperfine structure and natural lineshape modeling; dotted error bars indicate the estimated systematic error due to non-reducible LIF intensity saturation

Gaussian curve-fitting to the extracted Doppler lineshapes allowed for estimation of the respective Doppler width. The kinetic temperature of either species at a given ion source operation condition is then obtained by means of Eq. 3, see Fig. 15. Clearly, the kinetic temperatures of both species increase along with the absorbed rf power. The estimated xenon neutral temperatures range from roughly 500 to 700 K, and the ion temperatures are consistently higher by at least 200 K. Due to lower signal-to-noise ratios, the standard errors of the best-fit ion temperatures ($\approx$4.0 %) are higher then in the case of the xenon neutrals ($\approx$1.5 %). Taking the results from Sec. 3, any simulated relative Doppler width estimation error corresponds to half of the relative error in the derived kinetic temperature (according to Eq. 3). As it has been shown above, the inverse filtering-induced bias in the Doppler width or kinetic temperature estimate is small and, thus, it may be considered negligible in conjunction with optimized post-deconvolution filtering. For both transition schemes, the combined error in the estimated kinetic temperature resulting from hyperfine structure and natural broadening uncertainty amounts to 3 % at maximum (when assuming a ±10% error margin for the natural linewidth). For assessment of the LIF intensity saturation-induced error, the simulation results from Sec. 3.5 given with respect to the estimated Doppler width must be interpolated for the experimentally found Doppler widths and inverted to obtain the corrected kinetic temperatures. Referring to the transition specific minimum saturation intensities calculated in Sec. 3.5, the homologous laser intensity for the LIF spectra recorded here is 1.7 for the Xe I transition and 5.4 for the Xe II transition. Compared to the kinetic temperatures derived from deconvolution of the moderately saturated Xe I and Xe II LIF spectra, the actual kinetic temperatures are found at most 8 and 12 % lower, respectively. In Fig. 15, the estimated true-value temperatures are indicated by the arrow tips terminating the dotted error bars.

Taking the correction for LIF intensity saturation into account, the experimentally found kinetic temperatures are in conformity with the common assumption of thermal equilibrium between the neutral gas and the discharge chamber walls including the screen grid. In thruster performance models, the average temperature of thruster body parts close to the plasma is usually supposed to range between 500 and 600 K [79]. Comparative LIF-based kinetic temperature data for xenon low-pressure discharges are available from Hall thruster characterization only. The neutral gas temperature has been measured to range from 1000 to 1600 K in the discharge channel and from 500 to 800 K at the exit plane [31]. Thruster plume investigations, revealed kinetic temperatures at 500 K for the neutrals [21] and between 400 and 800 K for the ions [23, 80]. Besides, a few studies on inductively coupled discharges provide argon neutral gas and ion kinetic temperatures deduced from Doppler broadening in spectral lineshapes. Depending on the gas pressure, the rf power, and the plasma chamber design, the temperature of the neutrals has been observed to vary between 550 and 1000 K [17, 18]. Ion temperatures have been found either in a similar range [17, 19] or to exceed 1000 K significantly [19, 81].

Conclusion

Single-photon absorption LIF spectroscopy with narrow-linewidth cw lasers is a well-suited method for non-intrusive VDF measurement of either neutral or ionized inert gas species in plasmas. This, for example, provides a way to determine local kinetic temperatures of both neutrals and ions in regions that are difficult to access with physical probes. In that way, LIF-based kinetic temperature measurements may support numerical simulation of electric thruster performance and lifetime. In particular, LIF spectroscopy can be conducted in the discharge chamber of rf-driven ion engines provided that the discharge vessel can be prepared to be transparent within the spectral regions relevant for laser excitation and fluorescence detection.

Here, we have demonstrated kinetic temperature extraction from LIF spectra captured in the discharge chamber center of a gridded rf ion source. Unlike most probable values for a specific velocity component, kinetic temperatures usually cannot be inferred from raw LIF spectra directly. In fact, the recorded spectral lineshape must be reduced to the inherent Doppler broadening profile which requires elimination of the respective transition line’s hyperfine structure and natural broadening. In this contribution, we have compiled the necessary information for reconstruction of the hyperfine structures of two nearby transitions at approximately 835 nm that are convenient for investigation of xenon neutrals and ions. We have, moreover, described optimization of the inverse filtering technique for recovering Doppler lineshapes and discussed potential distortion of the deconvolution result due to saturation of the LIF signal. Spectral lineshape simulations and LIF spectroscopy measurements reveal that saturation induces a noticeable discrepancy between the estimated and the actual kinetic temperature even with the laser intensity decreased to a reasonable minimum. Errors arising from uncertainty in the hyperfine structure models or those related to incorrect natural linewidth estimates turn out to be of minor importance.

Further optimization of the experimental apparatus will include improvement of the laser wavelength measurement accuracy and spectral resolution, enlargement of the fluorescence detection efficiency, as well as spatial resolution enhancement. An additional vacuum probe for laser collimation will be integrated to allow for inspection of two velocity components at once without the necessity of ex-situ rearrangement of the LIF optics geometry.

Appendix A Hyperfine structure modeling

In this section, the formalism and the parameter data for hyperfine structure modeling (both of which can be found in detail in the cited literature) is summarized as far as needed to follow the data analysis demonstrated in this article.

A.1 Isotopic composition and isotope shifts

Xenon has six stable and three long-living isotopes whose different neutron numbers imply a variance in the reduced mass of the system consisting of the nucleus and the orbital electrons as well as slightly differing effective nuclear radii. The motion of the nucleus relative to the center of gravity of the atom or ion and the electrostatic interaction potential of a non-point-like nuclear charge distribution are both affecting the electronic energy term distances [82, 83]. In consequence, separate transition line components arise for each isotope with their intensities reflecting the isotope abundance distribution [84]. The abundance of the isotopes $^{124}$Xe and $^{126}$Xe is below 0.1% in either case, which is why their isotope shifts are not entirely known. However, the two corresponding linestructure components are relatively faint and can be omitted without affecting the quality of the hyperfine structure model.

Isotope abundance data and the individual isotope shifts for the selected transitions of Xe I and Xe II, respectively, can be found in Table 2. Due to comparatively compact linestructure component spacing, the isotope shifts of the Xe II transition $(^{3}\textrm{P}_{2})\ 5\textrm{d}[4]_{7/2}\rightarrow (^{3}\textrm{P}_{2})\ 6\textrm{p}[3]_{5/2}^{\textrm{o}}$ at 835 nm still have not been measured precisely [88]. Using the isotope shifts of a similar transition instead, suchlike those of the well-characterized transition $(^{3}\textrm{P}_{2})\ 5\textrm{d}[3]_{7/2}\rightarrow (^{3}\textrm{P}_{2})\ 6\textrm{p}[2]_{5/2}^{\textrm{o}}$ at 605 nm (as proposed in [80]), yields a reasonable estimate for the frequencies of the isotope shift components.

Table 2 Nuclear mass numbers A, nuclear spin quantum numbers I, and natural abundances $p_A$ of all nine (quasi-)stable xenon isotopes; moreover, experimentally found isotope shifts $\triangle \nu _A$ associated with the Xe I transition at 835 nm and data related to the Xe II transition at 605 nm used for reproduction of the (unknown) isotope shifts of the Xe II transition at 835 nm

A.2 Nuclear spin splitting

The two odd-mass number isotopes $^{129}$Xe and $^{131}$Xe have non-vanishing nuclear spin which is connected with a magnetic dipole moment interacting with the magnetic field induced by the orbital electrons. The different (quantized) orientations of the nuclear spin $\textbf{I}$ relative to the total electron shell angular momentum $\textbf{J}$ make an electronic state split into several terms. Each of them is assigned the quantum number F of the mutual angular momentum $\textbf{F}=\textbf{I}+\textbf{J}$ (Russel-Saunders coupling) of the nucleus and the electron shell determined by the quantum numbers I and J in turn: $F \in \{ |I-J|,\, |I-J|+1,\, ...,\, I+J-1,\, I+J \}$; the respective term energy shift corresponds to the magnetic dipole interaction energy [89]

$$\begin{aligned} U_{I(A)}^{|\gamma J\rangle }(F) = \frac{1}{2} \, a_{A}^{|\gamma J\rangle } \, C_{I,J}(F) \end{aligned}$$

(18)

where $C_{I,J}(F) = F\,(F+1) - I\,(I+1) - J\,(J+1)$. The nuclear magnetic dipole interaction constant $a_{A}^{|\gamma J\rangle }$ incorporates the strength of the magnetic field at the site of the nucleus. In addition, nuclei with $I>1$ have a non-vanishing electric quadrupole moment due to their intrinsic charge distribution. The interaction with the electric field of the orbital electrons causes another shift of the term energy [89]:

$$\begin{aligned} W_{I(A)}^{|\gamma J\rangle }(F) = b_{A}^{|\gamma J\rangle } \, D_{I,J}(F) \end{aligned}$$

(19)

where

$$\begin{aligned} D_{I,J}(C) = \frac{\frac{3}{4}\,C\,(C+1) - I\,(I+1)\,J\,(J+1)}{2I\,(2I-1)\,J\,(2J-1)}. \end{aligned}$$

The nuclear electric quadrupole interaction constant $b_{A}^{|\gamma J\rangle }$ again is related to the strength of the inherent electric field gradient. Nuclear magnetic dipole and the electric quadrupole interaction constant data, which have been used for hyperfine structure modeling in the course of this work, are listed in Table 3.

Table 3 Experimental data of the nuclear magnetic dipole interaction constants $a_{A}^{|\cdot \rangle }$ and the nuclear electric quadrupole interaction constants $b_{A}^{|\cdot \rangle }$ used for computation of nuclear-spin splitting

Electric dipole transitions between two nuclear-spin splitting-affected states $|\gamma _{1} J_{1}\rangle$ and $|\gamma _{2} J_{2}\rangle$ (with $\gamma _{i}$ denoting any quantum number but $J_{i}$ and $F_{i}$) follow selection rules in regard to $F_{1}$ and $F_{2}$ [91]:

$$\begin{aligned} F_1 - F_2 \in \left\{ -1, 0, 1 \right\} \quad \text {and} \quad F_1 + F_2 \ge 1\ . \end{aligned}$$

Combining Eqs. 18 and 19, the nuclear-spin splitting-induced shift of the transition frequency  $\nu (|\gamma _{1} J_{1}\rangle \rightarrow |\gamma _{2} J_{2}\rangle )$ is given by

$$\begin{aligned} \triangle \nu ^{\prime }_{I(A)}(J_{1}, F_{1}, J_{2}, F_{2}) = \frac{1}{h} & \left[ U_{I(A)}^{|\gamma _{2} J_{2}\rangle }(F_{2}) + W_{I(A)}^{|\gamma _{2} J_{2}\rangle }(F_{2}) \right. \nonumber \\{} & {} \left. - U_{I(A)}^{|\gamma _{1} J_{1}\rangle }(F_{1}) - W_{I(A)}^{|\gamma _{1} J_{1}\rangle }(F_1)\right] \ . \end{aligned}$$

(20)

The relative intensity of a nuclear-spin splitting component relating to the transition $|\gamma _{1} I J_{1} F_{1}\rangle \rightarrow |\gamma _{2} I J_{2} F_{2}\rangle$ is the product of the respective isotope abundance and the weighted dipole matrix element $p^{\prime }_{I(A)}(J_{1}, F_{1}, J_{2}, F_{2})$ associated with the transition. The latter corresponds to the linestrength s that can be expressed in terms of the appropriate Wigner 6j symbol [82, 91] for which numeric values can be obtained via specialized calculators:

$$\begin{aligned} p^{\prime }_{I(A)}(J_1, F_1, J_2, F_2)&= \frac{s(|\gamma _{1} I J_{1} F_{1}\rangle \rightarrow |\gamma _{2} I J_{2} F_{2}\rangle )}{\sum _{(F_1,F_2)} s(|\gamma _{1} I J_{1} F_{1}\rangle \rightarrow |\gamma _{2} I J_{2} F_{2}\rangle )} \nonumber \\&= \frac{(2F_1+1)\,(2F_2+1)}{2I+1} \, \left\{ \begin{array}{ccc} J_1 &{} F_1 &{} I \\ F_2 &{} J_2 &{} 1 \end{array}\right\} ^{2}\ . \end{aligned}$$

(21)

Alternatively, evaluation of the so-called oscillator strength “sum rules” [91, 92] for multiplets resulting from angular momentum coupling following the Russel-Saunders scheme gives the following relations applying for the linestrength s to be combined with the second expression in Eq. 21: Having defined

$$\begin{aligned} P_{I,J}(F)&= (F + J) \, (F + J +1) - I(I+1) \ , \end{aligned}$$

(22)

$$\begin{aligned} Q_{I,J}(F)&= I(I+1) - (F - J) \, (F - J +1) \ , \end{aligned}$$

(23)

$$\begin{aligned} R_{I,J}(F)&= [ F(F+1) - I(I+1) - J(J+1) ]^{2} \ , \end{aligned}$$

(24)

it follows [57, 66],

• (i) for $J = J_1 = J_2 + 1$ or $J = J_2 = J_1 + 1$,

  $$\begin{aligned}{} & {} s(|\gamma _{1} I J_{1} F_{1}\rangle \rightarrow |\gamma _{2} I J_{2} F_{2}\rangle ) \nonumber \\{} & {} \propto \left\{ \begin{array}{cc} \dfrac{1}{F} \, Q_{I,J}(F) \, Q_{I,J}(F-1) {}&{} \text {if} \quad F = F_1 = F_2 + 1\ ,\\ \dfrac{1}{F} \, P_{I,J}(F) \, P_{I,J}(F-1) {}&{} \text {if} \quad F = F_2 = F_1 + 1\ , \\ \dfrac{2F+1}{F(F+1)} \, P_{I,J}(F) \, Q_{I,J}(F) {}&{} \text {if} \quad F = F_1 = F_2\ , \end{array}\right. \end{aligned}$$

  (25)

• (ii) and for $J = J_1 = J_2$,

  $$\begin{aligned}{} & {} s(|\gamma _{1} I J_{1} F_{1}\rangle \rightarrow |\gamma _{2} I J_{2} F_{2}\rangle ) \nonumber \\{} & {}\propto \left\{ \begin{array}{lc} \dfrac{1}{F} \, P_{I,J}(F) \, Q_{I,J}(F-1) &{} \quad \ \text {if} \quad F\ = F_1 = F_2 + 1 \quad \text{or} \quad F\ = F_2 = F_1 + 1 \ ,\\ \dfrac{2F+1}{F(F+1)} \, R_{I,J}(F)&{} \text {if} \quad F = F_1 = F_2\ . \end{array}\right. \end{aligned}$$

  (26)

A.3 Hyperfine structure spectral profile function

Fig. 16

Broadening-free hyperfine structure spectral profile functions associated with (a) the Xe I transition (b) the Xe II transition at 835 nm. The linestructure components arising from nuclear-spin splitting are assigned the respective (total) angular momentum quantum numbers F of the lower and the upper term. The insets show the central parts of the hyperfine structure profile functions

Given the isotope abundances $p_{A}$ and isotope shifts $\triangle \nu _{A}$ as well as the interaction constants $a_{A}^{|\gamma J\rangle }$ and $b_{A}^{|\gamma J\rangle }$, the hyperfine structure spectral profile function $\phi _{\textrm{hfs}}$ can be reconstructed from a set of weighted delta distributions:

$$\begin{aligned} {\phi }_{\textrm{hfs}}(\nu -\nu _0) = & {} \sum _{A, I=0} p_A \, \delta \!\left( \nu -\nu _0-\triangle {\nu _A} \right) \nonumber \\{} & {} + \sum _{A, I \ne 0} p_{A}\left\{ \sum _{(F_{1},F_{2})} p^{\prime }_{I(A)}(J_{1}, F_{1}, J_{2}, F_{2}) \right. \\{} & {} \left. \vphantom{\sum_{(F_1,F_2)}} \hphantom{+} \cdot \delta \!\left[ \nu -\nu _{0}-\triangle {\nu _{A}}-\triangle {\nu ^{\prime }_{I(A)}}(J_{1}, F_{1}, J_{2}, F_{2}) \right] \right\} .\nonumber \end{aligned}$$

(27)

Here, the linecenter $\nu _0$ is indentified with the strongest linestructure component attributed to the most abundant isotope $^{132}$Xe. In fact, the (weighted) lineshape center practically coincides with $\nu _0$ for both of the modeled transition hyperfine structures. The computed hyperfine structure spectral profile functions are shown in Fig. 16.

The Stark effect and the Zeeman effect trigger further splitting of hyperfine structure components which, however, has been disregarded here as external electric or magnetic fields are considered too weak to cause substantial deviation from the unperturbed linestructure. Details on modeling spectral lineshapes including Zeeman splitting can be found elsewhere [93].