Linearized spectrum correlation analysis for thermal helium beam diagnostics

A thermal helium beam (THB) diagnostic is capable of measuring the electron density n_e and the electron temperature T_e in the scrape-off layer (SOL) and the confined region near the last closed flux surface (LCFS) in hot magnetized plasmas.^1–4 In this diagnostic technique, a neutral helium gas is injected into the plasma, and the active emissions of multiple He I lines are observed. Generally, the ratio of a triplet to a singlet He I line is more sensitive to T_e than to n_e, while the ratio of the same spin state lines has a stronger dependence on n_e than on T_e. Thus, local measurements of n_e and T_e can be made by evaluating properly chosen line ratios.⁵ Unfortunately, some of these lines often have weak emission intensities, which necessitate time averaging of data in order to robustly evaluate n_e and T_e, resulting in reduced time resolution. In principle, the maximum achievable time resolution in the THB is set by the atomic relaxation time,⁶ provided that the optical system has an arbitrary high throughput and photon-detection efficiency. However, practically, observable light intensities limit the time resolution under numerous circumstances, e.g., when the region near the LCFS is measured.

A diagnostic technique that is similar to the THB but more often used for fast measurements is gas puff imaging (GPI),^7,8 which observes the two-dimensional profile of a single line intensity. Since brightness is one of the main considerations in selecting a line for GPI, a sufficient amount of light can be collected relatively easily within a short time interval. While GPI is able to resolve fast time scale fluctuations up to several hundred kHz, interpreting data is not straightforward because the active line emission intensity depends on multiple parameters.⁹ Extending the time resolution of the THB and directly investigating plasma parameter fluctuations up to high frequencies relevant to turbulent transport will help understand important topics in the edge of fusion plasmas, such as power exhaust and pedestal physics.^10–12 

Linearized spectral correlation analysis (LSCA)^13,14 overcomes a similar issue in ion Doppler spectroscopy, whose time resolution is also typically limited by observable light intensities. The principle of LSCA is based on the ensemble average and cross correlation. In short, this procedure is given by

where

Here, d_1,f and d_2,f are frequency components of two measurable time series, which have the same signal component s_f but different noise X_1,f and X_2,f. The triangle bracket 〈⋯〉 stands for an ensemble average. When X_1,f and X_2,f are independent of each other, all terms with noise contribution converge to zero. Note that an ensemble-averaged auto-power of d_1,f is

Thus, measuring 〈|s_f|²〉 is difficult when 〈|X_1,f|²〉 is not negligible. The most well-known application of Eqs. (1) and (2) is probably the correlation electron cyclotron emission diagnostic, which provides T_e fluctuation power spectra even when the signals are smaller than wave noise.^15,16 LSCA is designed to extract physically meaningful frequency power spectra from signals that depend on multiple parameters in complicated fashions. The first step of LSCA is to manipulate the dependence of each parameter by performing arithmetic operations on available time series. Then, the cross-spectra, which become less sensitive to noise as the size of an ensemble increases, are calculated.

The formalism of LSCA discussed in Ref. 13 is developed specifically for ion Doppler spectroscopy. In this paper, we reformulate LSCA for the line-ratio analysis in the THB and introduce a technique to calculate the fluctuation power spectra of n_e and T_e. This paper is organized as follows: In Sec. II, we apply arithmetic operations on time series of emission intensity data and manipulate the n_e and T_e dependence. We also take into account the noise reduction in this process. The validation of the new LSCA for the THB is given in Sec. III by using synthetic data. We evaluate cross-spectra of the quantities introduced in Sec. II and compare the results with the input spectra. The application of LSCA to real experimental data is discussed in Sec. IV. Finally, we summarize the analysis procedure, validation, and results in Sec. V.

In this section, we manipulate the n_e and T_e dependencies of time series and the signal-to-noise (S/N) ratios. We consider a case where the plasma is quasi-stationary, and n_e and T_e can be decomposed into equilibrium values and small fluctuations, i.e., $ne=ne,0+ñe$ and $Te=Te,0+T̃e$⁠. When the relaxation time between the spin states can be neglected, the intensity of the He I line is given by

where n_He is the helium atom density. The subscript λ represents the wavelength of the He I line. PEC_λ(n_e, T_e) is called the photon emissivity coefficient, which differs for each He I line. In this paper, we use PEC_λ provided by Ref. 6. We assume that the measurements of λ = 587, 667, 706, and 728 nm lines are available. These four He I lines are currently measured by the THB system at ASDEX Upgrade (AUG).² However, the methodology presented herein is also applicable to different combinations of He I lines, such as in Ref. 4.

First, we divide the observable He I lines in two groups A and B and construct the following quantity:

where $cxA$ and $cxB$ are arbitrary coefficients we can freely choose. The subscript …₀ indicates that the function is evaluated at n_e = n_e,0 and T_e = T_e,0, e.g., I_λ,0 ≡ I_λ(n_e,0, T_e,0). Note that R_I depends only on n_e and T_e, and the contribution from n_He, which may hamper the n_e and T_e fluctuation measurements, is removed.

Next, we Taylor-expand R_I around n_e,0 and T_e,0 as follows:

where

By adjusting the elements of A and B, $cxA$ and $cxB$⁠, the n_e and T_e dependence of R_I can be controlled. To illustrate this, we consider a case where n_e,0 = 7 · 10¹⁸ m⁻³ and T_e,0 = 50 eV and evaluate

and

Table I shows $αne$ and $βTe$ for each R_I, while Fig. 1 shows R_Is as functions of n_e and T_e. When making A and B and choosing $cxA$ and $cxB$ for $RI(1)$⁠, we maximized $|αne|$ while keeping $|βTe|$ sufficiently small with respect to $|αne|$⁠. As a result, the contour plot of $RI(1)$ shown in Fig. 1 has a small dependence on T_e near n_e,0(=7 · 10¹⁸ m⁻³) and T_e,0(=50 eV). On the other hand, in the case of $RI(2)$⁠, we increased the sensitivity to T_e under the condition that $|αne|$ is smaller than $|βTe|$ by many orders of magnitude. Figures 1(a) and 1(b) show that $RI(1)$ and $RI(2)$ can be treated as functions of only n_e and T_e, respectively, near n_e,0 and T_e,0. As for $RI(3)$⁠, we selected A, B, $cxA$⁠, and $cxB$ that lead to a high $|αne+βTe|$⁠.

There are multiple combinations of A, B, $cxA$⁠, and $cxB$ that cancel the n_e or T_e dependence. In order to determine the combination that is most useful for measurements, we estimate the S/N ratio of R_I. Fast spectroscopic measurements typically employ photosensors with an internal signal amplification mechanism, such as avalanche photodiodes,^17,18 photomultiplier tubes,^2,19 and silicon photomultipliers.⁴ Photon noise is usually dominant over other noise sources for those sensors when properly designed electronics are used. For this reason, we evaluate the photon noise. We introduce a coefficient G that converts the light intensity into photon counts. Since the Poisson process describes the photon counting statistics, the photon noise for the signal GI_λ is $GIλ$⁠. While each spectral channel can have different G in reality, we assume that G is the same for all channels for simplicity. The following discussion can be easily modified for the case where G is different for each channel. Considering the error propagation, the photon noise for $∑k∈AckAIk/∑j∈AcjAIj,0$ is

We can similarly define the photon noise for $∑l∈BclBIl/∑i∈BciBIi,0$⁠. Again, considering the error propagation, the photon noise for R_I is given by

Since $1/GnHene$ is independent of A, B, $cxA$⁠, and $cxB$⁠, we use

at n_e = n_e,0 and T_e = T_e,0 for noise estimation.

When determining $RI(1)$⁠, we tried out all possible combinations of elements in A and B and adjusted $cxA$ and $cxB$ in such a way that the T_e dependence of R_I is removed near n_e,0 and T_e,0. Then, we employed A, B, $cxA$⁠, and $cxB$ that lead to the highest $|αne/γ|$⁠. Similarly, the choice of A, B, $cxA$⁠, and $cxB$ for $RI(2)$ has the highest $|βTe/γ|$ among the ones that cancel the sensitivity to T_e. Likewise, $RI(3)$ is constructed such that it provides the highest $|(αne+βTe)/γ|$⁠. Typically, the phase difference between n_e and T_e fluctuations is small in magnetized plasmas. Thus, the sensitivity to the plasma fluctuations is amplified when $αne$ and $βne$ have the same sign. As will be discussed in Sec. III, a high S/N ratio can be achieved by using $RI(3)$⁠.

Here, we introduce and validate a technique to calculate power spectral densities and a relative phase between n_e and T_e from noisy data. In real measurements, the He I line intensities are subject to both plasma parameter fluctuations and noise. In order to compare the analysis results and true values, we generate synthetic THB data by using quantities shown in Table II. The equilibrium electron density and temperature are set to n_e,0 = 7 · 10¹⁸ m⁻³ and T_e,0 = 50 eV. These values are close to experimentally measured data that will be discussed in Sec. IV. The fluctuations of n_e and T_e are totally coherent. We add a source of fluctuations at 50 kHz, where the relative phase between n_e and T_e is −π/6. The power spectral densities of n_e and T_e given by the other source of fluctuations $Sa1/f$ are inversely proportional to the frequency. $σne/ne,0$ and $σTe/Te,0$ are typical values for near the LCFS.²⁰ The n_He fluctuations, which we try to cancel in this analysis, are also modeled by a 1/f power spectrum. This fluctuation is incoherent with those of n_e and T_e.

We generate synthetic data for four He I lines λ = 587, 667, 706, and 728 nm. We choose the sampling frequency of 1 MHz, which is close to that of the THB diagnostic at AUG (900 kHz). The signal amplitude at each time point is the number of photons integrated over 1 µs. The line-intensity-to-photon-count conversion factor G is adjusted such that the sum of all He I line signals leads to the photon flux of around 500 counts/μs for n_e = n_e,0, T_e = T_e,0, and n_He = 1. An example of the time series of GI_λ obtained through this procedure is shown in the red dotted lines in Fig. 2. Then, we add photon noise by resampling all the time points from Poisson distributions with the means equal to the original photon counts and generate synthetic He I line intensity data d_λ. Time traces of d_λ are also shown in the blue solid lines in Fig. 2.

Next, we redefine Eq. (5) using d_λ,

We Taylor-expand Eq. (15) around n_e,0 and T_e,0 in analogy with Eq. (5). Then, we consider the frequency component of R_d as follows:

where X_f,1 is a noise term. $αne$ and $βTe$ are given by Eqs. (7) and (8). In order to remove the noise contribution, we prepare another time series as follows, which measures the same spatial point as the first one by using a different photon detector:

where X_f,2 is again a noise term. We assume that X_f,1 is not correlated with X_f,2, which is the case for photon noise. When we choose the same A, B, $cxA$⁠, and $cxB$ as $RI(1)$⁠, the cross-spectrum between R_d,f and R_d,f′ becomes

where the superscript …⁽¹⁾ indicates that it has the same A, B, $cxA$⁠, and $cxB$ as $RI(1)$⁠. We use the same notation for $RI(2)$ and $RI(3)$⁠. Note that any term with X_1,f and/or X_2,f converges to zero when an ensemble average is taken. Since $αne$ is given by Eq. (7), in general, or by Table I for this specific case, the power spectral density of n_e can be calculated from Eq. (18). Similarly, the T_e power spectral density is given by

The power spectral densities of n_e and T_e obtained by Eqs. (18) and (19) are shown in Figs. 3(a) and 3(c). The input spectra and the ones calculated by auto-power of R_d,f and R_d,f′ are also shown. The noise floors of auto-spectra are much higher than the input signal except 50 kHz for both n_e and T_e power spectra. However, cross-spectra remove the noise contributions and provide the power spectral densities that agree reasonably well with the input spectra. The slight overestimation of the n_e power spectrum is due to the linear approximation since this feature remains when the photon counts are increased. The total n_e fluctuation level calculated by integrating over all frequencies is 22.3%, which is 2.1% larger than the input value shown in Table II. The T_e power spectrum is also slightly underestimated due to the linearization. Note that the range of fluctuation amplitudes within which a linear approximation holds depends on the equilibrium values of n_e and T_e.

In addition to the n_e and T_e power spectra, we calculate the cross power between the n_e and T_e fluctuations and their relative phase by using

The results are shown in Fig. 4. The cross-spectrum reproduces the input spectrum. Furthermore, the phase lag between n_e and T_e at 50 kHz is successfully resolved in Fig. 4(b). The uncertainties of cross phase depend on coherence shown in Fig. 4(c).²¹ Above 100 kHz, the cross-phase measurements suffer large errors since the coherence level drops to near the statistical significance level.

In the end of the synthetic data analysis, we evaluate $<Rd,f(3)*Rd,f(3)′>$⁠. Neither n_e nor T_e dependence is canceled for $Rd,f(3)$ and $Rd,f(3)′$⁠. This allows for a higher flexibility in choosing A, B, $cxA$⁠, and $cxB$⁠, and a larger reduction in photon noise becomes possible. Figure 5(a) shows the cross-spectrum between $Rd,f(3)$ and $Rd,f(3)′$⁠, the corresponding input power spectrum, and auto-power spectra. The auto-spectra are closer to the input compared with Fig. 3(a) or Fig. 3(c), indicating a higher S/N ratio. Reduction in the noise contribution is also shown in Fig. 5(b). The coherence stays well above the statistical significance level up to the Nyquist frequency (=500 kHz). While $Rd,f(3)$ and $Rd,f(3)′$ are sensitive to both n_e and T_e, their contributions are specified by $αne$ and $βTe$⁠, and quantitative comparisons can still be made.

In this section, we apply LSCA to the THB data at AUG. Equation (4) is applicable only when the helium atoms travel with the equilibrated spin states. When observation points are in low density regions, care must be taken. Distances that He atoms need to travel to reach the spin state equilibrium are given in Table IV in the  Appendix. We choose a radial location of ρ_pol = 1.02 in an enhanced D_α emission (EDA) H-mode plasma,¹⁰ where n_e ∼ 7 · 10¹⁸ m⁻³ and T_e ∼ 50 eV. As shown in Table IV, the spin states almost equilibrate within 1 mm of the traveled distance for these parameters. Therefore, Eq. (4) is expected to be a good approximation. In regard to this point, Ref. 6 reports that n_e and T_e calculated by assuming equilibrated spin states converge to the ones calculated by taking into account the time dependence of He atoms around n_e = 2 · 10¹⁸ m⁻³. There are two spatial channels, ch. 1 and ch. 2, with the separation of ∼1 mm at the selected radial position. We correlate those spatial channels and calculate the n_e and T_e power spectral densities.

In the THB, 667, 706, and 728 nm lines are most commonly used to determine n_e and T_e.^3,22,23 In addition to these three lines, a 587 nm line is also measured by the AUG THB system, as already mentioned. We calculate the equilibrium n_e and T_e from the line ratios of the standard pairs, 667, 706, and 728 nm lines, and also from all four lines, 587, 667, 706, and 728 nm. When we utilize four lines, we conduct a least-square optimization by assuming the same relative uncertainties in the line intensity measurements. Note that due to a long integration time (∼5 ms), the statistical uncertainties are negligible when determining equilibrium values. The results are shown in Table III. The differences of n_e,three and T_e,three between ch. 1 and ch. 2 are within the expected range of statistical or systematic errors. However, when all four lines are utilized, n_e is underestimated, and T_e is overestimated compared to the three line cases. The optical opacity of the helium gas is not responsible for this discrepancy since its escape factor was already evaluated,^24–26 and the helium gas flow was controlled so that the photon absorption was negligible for the lines of interest. Atomic modeling for plasma diagnostics is still an active field of research, and in general, modeling is not necessarily in agreement with experimental observations while significant progress has been made.^5,27 The uncertainties of all the atomic rate coefficients and their influence onto the observed line radiation have been investigated in a new paper, which has been submitted.²⁶ 

Even though validating an atomic model is not within the scope of this paper, it is important to characterize how the uncertainty in the atomic model affects the n_e and T_e power spectrum measurements. To this end, we compare the power spectral densities calculated by using all four lines with the ones obtained by using only 667, 706, and 728 nm lines. Note that the formalism discussed in Secs. II and III is still applicable even when only three lines are available. For each spatial channel and for each use of He I emission lines, we determine the groups of lines A and B and coefficients $cxA$ and $cxB$ by following the linearization procedure at n_e,three and T_e,three or n_e,four and T_e,four. Due to the differences in the equilibrium n_e and T_e values, $αne$ and $βTe$ slightly differ between ch. 1 and ch. 2. When the four lines are utilized, we rescale the line intensities so that their relative intensities become the same as the atomic model calculations for the equilibrium n_e and T_e values. The corrections are less than 13%. Figure 6 shows the n_e and T_e power spectral densities. While R_d,f and $Rd,f′$ are expanded and approximated by linear functions around different points, similar power spectrum densities are obtained. The peaks around 30 kHz are a quasi-coherent mode, which is usually observed near the LCFS in an EDA H-mode. When all four lines are included in the analysis, the coherence levels shown in Figs. 6(b) and 6(d) are higher, and associated statistical uncertainties are reduced. In the four line case, the coherences are above the statistical significance level up to around 90 kHz, indicating that the correlated signals originating from the n_e or T_e fluctuations are resolved below this frequency. The total n_e and T_e fluctuation levels integrated up to 90 kHz are 18.6% and 9.62%, respectively, for the three line case and 18.0% and 10.5% for the four line case. Given these fluctuation amplitudes and the equilibrium parameters in Table III, which are close to the input values in Table II, the linear approximation is not expected to introduce significant errors based on the discussion in Sec. III. Depending on the discharge scenario and the radial position, the n_e fluctuation amplitude can be larger than 20% in the SOL.²⁰ When applying LSCA, we need to keep the limit of the linear approximation in mind as well as the spin state equilibration. The applicability of LSCA can be checked by performing synthetic data analysis, such as in Sec. III.

There are several other candidates of He I lines for the THB diagnostic. If those lines are measured, a further increase in coherence is expected, which helps resolve particularly high frequency ranges where the coherence level tends to be low. In addition, simultaneous measurements of more He I lines will facilitate the validation of the helium atomic model and improve the reliability of the THB.

We have introduced linearized spectral spectrum correlation analysis (LSCA) for the thermal helium beam diagnostic. By performing arithmetic operations on He I line intensities, the contributions from neutral helium atom density fluctuations are canceled, and either the dependence on the electron density or the electron temperature is maximized or minimized. The time series with the desired parameter dependence can be expressed as a linear function of the electron density and the electron temperature around its mean values. By cross-correlating two time series constructed through this procedure and by taking ensemble averages, we can remove the noise contribution and calculate the power spectral densities of meaningful parameters.

The proposed analysis method is validated by using synthetic data. We have shown that the input power spectral densities of the electron temperature and the electron density can be measured even under low-photon-count conditions. When we allow the mixture of the electron density and temperature fluctuations, the signal-to-noise ratio can be improved compared with the cases where a single parameter is measured. In addition, we have applied LSCA to experimental data at ASDEX Upgrade and demonstrated that the power spectral densities of the electron density of the temperature in the scrape-off layer can be resolved up to 90 kHz. LSCA allows for quantitative comparisons between experiments and plasma edge simulations,^28,29 which will provide insights into the edge physics in hot magnetized plasmas.

The authors would like to thank Dr. J. Boedo and Dr. E. M. Hollmann for valuable discussions and Dr. J. M. Muñoz Burgos for providing the photon emissivity coefficients for He I lines. This work has been carried out within the framework of the EUROfusion Consortium and has received funding from the Euratom research and training programme 2014–2 018 and 2019–2 020 under Grant Agreement No. 633053. The views and opinions expressed herein do not necessarily reflect those of the European Commission.

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

This appendix includes Table IV providing the distances required for the spin states to reach 95% of the equilibrium ratio when all atoms are initially in the ground state.