Poloidal magnetic field reconstruction by laser-driven ion-beam trace probe in spherical tokamak

1.   Introduction

The poloidal magnetic field (${B}_{\mathrm{p}}$) is one of the most basic physical quantities in magnetic confinement fusion plasma. As the safety factor q is determined by poloidal magnetic field, the equilibrium profile of ${B}_{\mathrm{p}}$ determines the fundamental configuration of plasma and is crucial to both equilibrium and stability of plasma. Given its significance, diagnosing of ${B}_{\mathrm{p}}$ accurately is essential for the study of various instabilities and the steady operation of magnetic confinement fusion devices. Considering the importance and difficulty of ${B}_{\mathrm{p}}$ diagnosis, multiple diagnostic methods should be developed to complement each other. The traditional diagnostic methods of ${B}_{\mathrm{p}}$ are the motional Stark effect [1–3] and polarimetry using the Faraday rotation effect [4–6], but they require complex systems and high standards for calibration and denoising.

Based on the rapidly developing technology of laser-driven ion acceleration, the laser-driven ion-beam trace probe (LITP) has been proposed to diagnose the ${B}_{\mathrm{p}}$ and ${E}_{r}$ profiles of tokamaks by detecting the laser-driven ion beam (LIB) in recent years (see figure 1) [7, 8]. LITP utilizes a laser to hit the target and produce ion beam, which then travels through the plasma and is finally collected by the detector. Since ${B}_{\mathrm{p}}$ and ${E}_{r}$ in the plasma affect the traces of ion beam, it is theoretically feasible to reconstruct ${B}_{\mathrm{p}}$ and ${E}_{r}$ in the plasma from the detection of ion beam. LIB has the properties of large energy spread, short pulse lengths, and multiple charge states, which make it possible for LITP to achieve a 2D and high temporal resolution diagnostic. Proof-of-principle experiments have been conducted on the linear device PKU Plasma Test (PPT) with the LIB produced by Compact LAser Plasma Accelerator (CLAPA) [9], and the feasibility of ion detection system of LITP has been validated on the tokamak HL-2A [10]. Notably, LITP has also been adapted to diagnose the ${B}_{\mathrm{p}}$ profile of field-reversed configurations (FRCs), with a corresponding reconstruction method proposed [11]. However, while theoretically feasible [12], the application of LITP to diagnose the electromagnetic field of spherical tokamaks (STs) remains incomplete.

Figure  1.  (a) Schematic diagram illustrating the LITP. (b) Detected strike-lines of ion beams in the presence and absence of ${B}_{\mathrm{p}}$.

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Possessing properties of high plasma beta and natural large elongation, STs are promising to realize a relatively compact fusion reactor. Unlike traditional tokamaks, STs have a strong poloidal magnetic field that matches the toroidal magnetic field, which invalidates the linear reconstruction relationship. To tackle this challenge, a new method based on nonlinear principles has been proposed for STs. This method uses an iterative approach, employing Newton’s method to solve the nonlinear equation effectively.

We design the experimental setup of LITP for EXL-50, which is a middle-sized ST [13]. The simulation results show that the reconstructed ${B}_{\mathrm{p}}$ relative error is below 10%. Even with measurement errors of 5 mm for beam traces or 1 cm for flux surface shape errors, the average relative error of reconstruction remains below 15%. These results prove the feasibility of LITP diagnosing ${B}_{\mathrm{p}}$ profiles in STs.

In this article, the reconstruction method of LITP diagnosing ${B}_{\mathrm{p}}$ in STs is presented in section 2. In section 3, the experimental setup of LITP on EXL-50 is proposed, and corresponding reconstruction procedure is simulated. Section 4 assesses the impact of detector measurement errors and flux surface shape errors on the reconstruction results, which prove the feasibility of LITP diagnosing the poloidal magnetic field in STs. The discussion and conclusion are included in section 5.

2.   The principle of LITP

The ion beam generated by laser devices is injected into the plasma and deflected by the electromagnetic field. The traces of LIB are not affected by the plasma because additional ionization or collision of LIB in the plasma hardly happens. This is chiefly due to the LIB’s high energy, which results in a flight time (less than 1 ms) significantly shorter than the collision time (seconds). Consequently, the fraction of LIB colliding with the plasma is negligible.

With the advantage of short pulse length, ions can be detected by a detector with high spatial and temporal resolution. Ions with different energy bend in the poloidal cross-section due to the Lorentz force of toroidal field (${B}_{\mathrm{t}}$) and their traces cover a certain area. Meanwhile, ions are deflected in the toroidal direction due to the Lorentz force caused by ${B}_{\mathrm{p}}$ and the toroidal displacement is related to the ${B}_{\mathrm{p}}$ profile (figure 1(b)). Consequently, by measuring this toroidal displacement, the ${B}_{\mathrm{p}}$ profile can be reconstructed.

2.1   Linear theory of LITP method

In conventional tokamaks, the magnetic configuration satisfies ${B}_{\mathrm{t}}\gg {B}_{\mathrm{p}}\gg E/v$ (E is electric field and $v$ is the speed of injected ion beam) and the corresponding linear theory of LITP is obtained [7]. The poloidal motion is totally decided by ${B}_{\mathrm{t}}$ and the toroidal displacement, $\mathrm{\Delta }\varphi$, is caused by ${B}_{\mathrm{p}}$. $\mathrm{\Delta }\varphi$ can be obtained by performing twice integration over time on the ion motion equation in the toroidal direction, $\mathrm{d}{\boldsymbol{v}}/\mathrm{d}t=\dfrac{q}{m}\left({\boldsymbol{E}}+{\boldsymbol{v}}\times{\boldsymbol{B}}\right)$. After ignoring the toroidal effect and assuming no initial toroidal velocity, we transform the time integral into an integral along the traces, $\mathrm{\Delta }\varphi$ is linearly dependent on ${B}_{\mathrm{p}}$, discretized as:

Here i represents different reconstruction pixel along the ion trace, and j denotes the ion traces of varying energy. Detailed derivation of equation (1) is shown in reference [3]. The coefficients ${P}_{ij}$ can be calculated by numerical method. Consequently, ${B}_{\mathrm{p}}$ profiles can be reconstructed by solving the linear equations in conventional tokamaks [7].

Though linear reconstruction method has proven effective for conventional tokamaks, the unique configuration of ST invalidates the linear assumption of this method. In STs, the ${B}_{\mathrm{p}}$ is comparable to or even larger than the ${B}_{\mathrm{t}}$ in the outboard region [14]. Specifically, the parameters of EXL-50 are $B_{\mathrm{t}}=0.48\mathrm{\ T}$ and $I_{\mathrm{p}}=0.2\mathrm{\ M}\mathrm{A}$, which corresponds to $B_{\mathrm{p}\mathrm{M}\mathrm{a}\mathrm{x}}\ \sim\ 0.05\mathrm{\ T}=0.1B_{\mathrm{t}}$. Considering that ${B}_{\mathrm{t}}$ decreases faster in the outboard in ST than traditional tokamak and most of the traces are in the outboard, the relation should be $\dfrac{{B}_{\mathrm{p}\mathrm{M}\mathrm{a}\mathrm{x}}}{{B}_{\mathrm{t}}}\geqslant 0.1$. Previous numerical simulation concludes that the nonlinear effect is obvious in toroidal device with ${B}_{\mathrm{p}\mathrm{M}\mathrm{a}\mathrm{x}}$ > 0.1B_t [12]. As a result, the linear assumption underlying traditional reconstruction methods is no longer applicable in STs. Therefore, a nonlinear reconstruction method is required for LITP diagnosing the ${B}_{\mathrm{p}}$ of STs.

2.2   Nonlinear reconstruction method

As ${B}_{\mathrm{p}}$ in STs is no longer much smaller than ${B}_{\mathrm{t}}$, the linear relationship between the toroidal displacement $\mathrm{\Delta }\varphi$ and ${B}_{\mathrm{p}}$ is no longer effective. To handle the nonlinear effect, a new nonlinear reconstruction method for diagnosing ${B}_{\mathrm{p}}$ in STs is developed. ${E}_{r}$ is about 1 kV/m for $\ \sim10^2\mathrm{\ k}\mathrm{A}$ plasma current in STs [15], which means that ${B}_{\mathrm{p}}$ is two orders of magnitude larger than ${E}_{r}/v$ for LIB with energies around the MeV level. Therefore, the effect of electric field can still be ignored.

Following the common principle of reconstruction, the objective is to find the solution that minimizes the residual error of direct measured signals $\mathrm{\Delta }\varphi$. Since the derivative of the nonlinear relationship between $\Delta \varphi$ and ${B}_{\mathrm{p}}$ can be calculated numerically, the improved Newton’s method is used to get the solution, i.e. the minimum point of the sum of squared residuals:

here, ${\varphi }_{j}\left({B}_{i}\right)$ is the displacement of ion beams calculated from the iterative ${B}_{\mathrm{p}}$, while ${\varphi }_{\mathrm{e}\mathrm{x}\mathrm{p},j}$ is the detected displacement of ion beams in experiments.

This function,$f\left({B}_{i}\right)$, can be expressed as

where ${F}_{j}\left({B}_{i}\right)={\varphi }_{j}\left({B}_{i}\right)-{\varphi }_{\mathrm{e}\mathrm{x}\mathrm{p},j}$. Therefore, Newton’s method is utilized to obtain the minimum value of $f\left({B}_{i}\right)$ by finding the zeros of multi-dimensional function ${F}_{j}\left({B}_{i}\right)$. The iterative formula is as follows:

where ${J}_{ij}^{n}=\dfrac{{\partial F}_{j}\left({B}^{n}\right)}{\partial {B}_{i}}$ is the Jacobian matrix of the function ${F}_{j}\left({B}^{n}\right)$, and ${\text{[}{J}_{ij}^{n}]}^{+}$ is its Moore-Penrose pseudo-inverse matrix. Superscript n indicates the iteration number.

Due to the error of calculation, the multidimensional function ${F}_{j}\left({B}_{i}\right)$ is usually non-zero, necessitating improvements to Newton’s method. It can be proved that in the iterative formula of Newton’s method the direction of the change of ${B}_{i}$, $\delta {B}^{n}={B}_{i}^{n+1}-{B}_{i}^{n}$, must be the decreasing direction of $f\left({B}_{i}\right)$. The product of the direction and the gradient of $f\left({B}_{i}\right)$ can be written as

By the Moore-Penrose pseudo-inverse property, the $J\cdot {J}^{+}$ is a positive-definite matrix, ensuring that equation (5) remains negative. Therefore, the direction of the iteration obtained by Newton’s method must be the decreasing direction of $f\left({B}_{i}\right)$.

To avoid divergence and improve the convergence rate, a one-dimensional search algorithm is utilized at each iteration to find one-dimensional minimum point following the determination of the decreasing direction. Additionally, a numerical dissipation term ${(\nabla }^{2}{B)}^{2}$ is added to the residual error in equation (2). This term serves to filter the components with high wave number and smooth the reconstructed profile, which makes it closer to the actual physical profile [16].

Based on the above methods, a nonlinear reconstruction method is established, as depicted in the flow chart shown in figure 2. At each iteration, the toroidal displacement ${\varphi }_{j}$ is obtained by calculating traces with the iterative ${B}_{i}^{n}$. If the residual error with ${\varphi }_{\mathrm{e}\mathrm{x}\mathrm{p},j}$ is significant, the decreasing direction at ${B}_{i}^{n}$ should be calculated using the Newton’s method. The Jacobian matrix ${J}_{ij}^{n}$ can be calculated from $\dfrac{{F}_{j}\left({B}^{n}+\delta {b}_{i}\right)-{F}_{j}\left({B}^{n}\right)}{{\delta b}_{i}}$, where $\delta {b}_{i}$ is a much smaller quantity than ${B}_{\mathrm{p}}$. Subsequently, the minimum point in this direction is found with one-dimensional linear search algorithm. Finally, the minimum point in this direction is set as ${B}_{i}^{n+1}$ for next iteration. When the convergence condition ||${\varphi }^{n}-{\varphi }_{\mathrm{e}\mathrm{x}\mathrm{p}}$|| < $\varepsilon$ is satisfied, iteration ends, and the output ${B}_{i}^{n}$ is the minimum point of all directions.

Figure  2.  ${B}_{\mathrm{p}}$ reconstruction flow chart.

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3.   Reconstruction of B_p in the ST

3.1   Experimental setup design

The experimental setup of LITP on EXL-50 is designed. EXL-50 is a middle-sized spherical tokamak of which major and minor radii are approximately 0.58 m and 0.41 m, ${B}_{\mathrm{t}}$ is approximately 0.5 T at $r\sim \mathrm{ }0.48\;\mathrm{m}$ [13].

The calculation of ion beam traces is crucial to ensure that ions can be incident from the port, pass through the center of the plasma, and finally still exit from the port under the limitations of LIB energy and the port space. Given these considerations, two distinct setups with different ion traces have been proposed, as illustrated in figure 3.

Figure  3.  The geometry of the proposed experimental setups at EXL-50. The traces of ion beams are represented by the colored lines. The red dashed line is the plasma edge. The green line is the scintillator position. The blue lines are vacuum vessel and the ports. The pseudo-color plot in the background shows the distribution of ${B}_{\mathrm{t}}$, while the arrow plot depicts the distribution of ${B}_{\mathrm{p}}$. Poloidal view (a) and top view (b) of scheme I. Poloidal view (c) and top view (d) of scheme II.

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In both schemes, the preset poloidal magnetic field corresponds to a plasma current of 200 kA. The laser-driven ions are injected from the mid-plane port of EXL-50 and detected either in the lower port (scheme I) or the same mid-plane port (scheme II). The parameters of ion injection geometry are detailed in table 1. Specifically, ${\alpha }_{\mathrm{p}\mathrm{o}\mathrm{l}}$ is the poloidal injection angle and ${\alpha }_{\mathrm{t}\mathrm{o}\mathrm{r}}$ is the toroidal injection angle. ${\alpha }_{\mathrm{p}\mathrm{o}\mathrm{l}}=0 {\text{°}}$ means injection along the mid-plane and the positive value means upwards injection. ${\alpha }_{\mathrm{t}\mathrm{o}\mathrm{r}}=0 {\text{°}}$ means perpendicular injection and the positive direction is towards ${B}_{\mathrm{t}}$ (clockwise from a top view). These two injection angles are carefully selected to ensure that the ions can reach the core of plasma and return to the detection port. The position of launching point is limited by the geometry of port and the chamber of laser target.

Table  1.  The parameters of ion injection geometry.

	Ion species	Energy range (MeV)	${\alpha }_{\mathrm{p}\mathrm{o}\mathrm{l}}$	${\alpha }_{\mathrm{t}\mathrm{o}\mathrm{r}}$	$Z\ \left(\mathrm{m}\right)$	$R\ \left(\mathrm{m}\right)$
Scheme I	Proton	2.4–3.0	25 °	−7.8 °	0	2.08
Scheme II	Proton	1.06–1.3	60 °	−11.0 °	−0.2	1.93

 | Show Table

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From the top view, the ions with different energy spread along the toroidal direction, facilitating the reconstruction of the poloidal magnetic field profile through toroidal displacement. In practical applications, ions with lower energy are favored due to a lower power need for the laser. Additionally, the spatial compatibility of all the components like the target chamber, the scintillator, the coils, and the necessary vacuum devices also needs to be considered. Having taken all the above factors into account, both proposed schemes hold promise for future experimentation and need to be simulated in advance.

3.2   Reconstruction model

Following the nonlinear reconstruction method, a numerical reconstruction model is built. Fourth-order Runge-Kutta difference in time and bilinear interpolation of rectangular coordinate for field have been taken. The model consists of three main parts: calculating the ion traces for the iterative ${B}_{\mathrm{p}}$ profiles in STs, calculating the coefficient matrix ${J}_{ij}^{n}$ and reconstructing the ${B}_{\mathrm{p}}$ profile using the nonlinear reconstruction method.

The input physical parameters ${B}_{\mathrm{t}}$ and flux surface shape for the model can be acquired from other basic diagnostics. The ${B}_{\mathrm{t}}$ distribution within the poloidal cross-section and the ${B}_{\mathrm{p}}$ distribution outside the plasma can be derived from the currents in coils and external magnetic measurements. Notably, the ${B}_{\mathrm{p}}$ profile is not poloidally symmetric because the flux surfaces of STs are D-shaped, which poses a challenge for the reconstruction. However, tokamak discharges can use poloidal field coils to shape the flux surfaces, adjusting parameters such as the triangularity and the elongation. Approximate flux surface shapes are provided by equilibrium calculation programs, enabling the derivation of two-dimensional ${B}_{\mathrm{p}}$ profile from one-dimensional ${B}_{\mathrm{p}}$ profile. Subsequently, LIB traces can be calculated to get the corresponding toroidal displacement $\varphi \left(B\right)$.

The ion displacement ${\varphi }_{\mathrm{e}\mathrm{x}\mathrm{p}}$ can be obtained from the scintillator. An arbitrary initial poloidal magnetic field profile, denoted as ${B}_{i}^{0}$, is input into the iterative reconstruction procedure, which is typically set as a uniform profile. When the iterative reconstruction procedure finished, the output ${B}_{\mathrm{p}}^{n}$ is the reconstructed ${B}_{\mathrm{p}}$ profile.

3.3   Simulation results

In the simulated reconstruction process, a preset ${B}_{\mathrm{p}}$ profile is calculated, with its magnetic axis situated at $R=0.5\mathrm{\ }\mathrm{m}$, using a specified toroidal current distribution. The resulting ion displacements corresponding to this preset ${B}_{\mathrm{p}}$ profile are calculated and serve as input to the model, representing ${\varphi }_{\mathrm{e}\mathrm{x}\mathrm{p}}$ (during experiments, ${\varphi }_{\mathrm{e}\mathrm{x}\mathrm{p}}$ can be obtained from the scintillator). The number of ions with different energy is 300. The radial resolution is 4 cm within the diagnostic area.

Two schemes with different traces are both simulated. The residual error ${\chi }^{2}$ (figures 4(a) and (e)) and the relative error of ${B}_{\mathrm{p}}$ profile (figures 4(b) and (f)) decrease rapidly as the iterations proceed. The reconstructed region is divided into two parts: one tracing the path incident from the target to the plasma core, and the other outgoing from the core to the detector, considering that ${B}_{\mathrm{p}}$ profiles of these two parts are markedly distinct while in the same part ${B}_{\mathrm{p}}$ profiles of different traces are similar. Reconstructed ${B}_{\mathrm{p}}$ profiles of these two parts along the trace of the highest energy are shown separately (figures 4(c), (g) and (d), (h)). The overall error is within 5% except near the magnetic axis, where the relative error is large due to the small ${B}_{\mathrm{p}}$. These results demonstrate that LITP is feasible for the ${B}_{\mathrm{p}}$ diagnostic in STs.

Figure  4.  The evolutions of residual ${\chi }^{2}$ (a) and the relative error of ${B}_{\mathrm{p}}$ profile (b) during the iterations. Reconstruction results of ${B}_{\mathrm{p}}$ profile along the incident trace (c) and along the outgoing trace (d) for scheme I. (e)–(h) Reconstruction results of the scheme II. The blue lines are the preset ${B}_{\mathrm{p}}$ profile along the trace. The black dots are reconstructed ${B}_{\mathrm{p}}$. The red marks are the relative errors of reconstructed ${B}_{\mathrm{p}}$.

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4.   The effect of measurement error on the reconstruction

4.1   Detector error

In actual experiments, the strike points of ion beams are detected by the scintillator detectors. The measurement errors caused by the finite beam size and spatial solution of scintillator and imaging system, bring interference to the reconstruction of ${B}_{\mathrm{p}}$ profiles. To assess the impact of measurement errors on reconstruction, various degrees of measurement errors are introduced to the displacement of ions.

During the reconstruction simulation, a random number uniformly distributed within the interval [−σ, +σ] is added to the calculated toroidal displacement of ions, where σ represents errors in measured toroidal displacements ${\varphi }_{\mathrm{e}\mathrm{x}\mathrm{p}}$. Simulated reconstructions are conducted with σ ranging from 0.5 to 5 mm. The relationship of the reconstruction error and the measurement error is roughly linear.

Figure 5 shows the reconstruction results with measurement error of σ = 5 mm. Comparing these results with those obtained without measurement errors, it is evident that the reconstruction error predominantly increases in the area within 20 cm near the magnetic axis, while remaining relatively small in other regions (mostly below 20%). The average relative error of the whole reconstructed area is 12%. Given that the spatial resolution of detector system is typically under 1 mm, LITP method proves robust enough for the detector measurement error.

Figure  5.  Reconstruction results of ${B}_{\mathrm{p}}$ profile with 5 mm error of detection (a) along the incident part trace and (b) along the outgoing part trace.

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4.2   Flux surface error

As mentioned earlier, the LITP reconstruction method for STs requires the shape of flux surface as a known quantity for reconstruction. In general, the shape of each flux surface, including its triangularity, elongation and other parameters, is obtained from the equilibrium calculation program. However, the error of flux surface shape depends on the assumed model of equilibrium calculation program, and it can be significant. Since the shape of flux surface remains constant during the LITP iterative reconstruction procedure and can directly affect the reconstruction results, it is crucial to evaluate the impact of the errors caused by flux surface shapes on the reconstruction of ${B}_{\mathrm{p}}$ profile.

To assess the impact of flux surface shape, shape errors of flux surfaces are introduced by changing triangularity and elongation, which is equivalent to shaping the flux surfaces to circles. The triangularity and elongation of errored flux surfaces are expressed as

where $\mathrm{\Delta }$ represents the change ratio of triangularity and elongation, and $\delta$ and $\kappa$ represent the triangularity and elongation of accurate flux surfaces. Figure 6 shows the comparison between errored and accurate flux surfaces. Figure 7(a) shows that the reconstruction error of ${B}_{\mathrm{p}}$ profiles gradually increases with the error of the flux surface shape. Notably, when the flux surface shape exhibits a 10% error (black lines in figure 6), the average error of the reconstructed ${B}_{\mathrm{p}}$ reaches 15%. The corresponding reconstruction results are shown in figures 7(b) and (c). The overall trend of the reconstructed ${B}_{\mathrm{p}}$ profile is still consistent with the preset magnetic field, but the reconstruction error increases significantly in the area near the magnetic axis.

Figure  6.  The red lines are the flux surfaces without errors and the black lines are the flux surfaces with 10% error. The magenta circle represents the diagnostic area.

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Figure  7.  (a) Dependence of the average error of reconstruction on the shape error of flux surface. The horizontal axis represents the ratio of the triangularity and elongation change (100% means that the input flux surface is circular), and the vertical axis represents the average relative error of the reconstructed ${B}_{\mathrm{p}}$ profile. Reconstruction results of ${B}_{\mathrm{p}}$ profile with 10% shape error of flux surface (b) along the incident part trace and (c) along the outgoing part trace.

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Experiments on tokamaks and STs have shown that the uncertainty in the position of the last closed flux surface (LCFS) or separatrix, obtained from equilibrium reconstruction procedure, is typically around 1 cm [17–20]. Referring to figure 6, when the flux surface shape error is 10%, the large radii R of both the outboard and inboard sides of the LCFS differ from those of the exact flux surfaces by approximately 1.4 cm. This discrepancy exceeds the error typically associated with equilibrium reconstruction procedures. It illustrates that LITP is capable to tolerate the error caused by flux surface shape, offering resilience in the face of such uncertainties.

5.   Discussion and summary

A nonlinear reconstruction method for diagnosing the ${B}_{\mathrm{p}}$ profiles with LITP has been developed for a spherical tokamak. Simulated reconstructions of the ${B}_{\mathrm{p}}$ profiles for EXL-50 have been conducted and the errors of reconstruction results are mostly within 5%. Moreover, even with a measurement error of 5 mm or a flux surface shape error of the LCFS within 1.4 cm in the R direction, the average reconstruction errors remain within 15%. These findings illustrate the experimental feasibility of using LITP to diagnose the ${B}_{\mathrm{p}}$ profiles in spherical tokamaks.

In future studies, the combination with the plasma equilibrium reconstruction will be considered to develop a reconstruction program independent of input flux surface. The reconstructed ${B}_{\mathrm{p}}$ profiles from LITP can be used as a constraint for equilibrium reconstruction like the way motional Stark effect [21, 22] and Faraday rotation effect [23–25] do to obtain reliable flux surface shapes and two-dimensional ${B}_{\mathrm{p}}$ profiles. Additionally, LITP reconstruction and equilibrium reconstruction can iterate mutually to converge to a more accurate ${B}_{\mathrm{p}}$ profile. Furthermore, based on the multiple charge states of LIB, diagnostics of electric fields will also be developed.