Detailed analysis and simulation verification for reconstruction of plasma optical boundary with spectrometric technique on HL-2M tokamak

Zhihao SU; Jinming GAO; Zhe GAO

doi:10.1088/2058-6272/acb97d

Plasma Science and Technology > 2023 > 25(7): 075103. > DOI: 10.1088/2058-6272/acb97d

Zhihao SU, Jinming GAO, Zhe GAO. Detailed analysis and simulation verification for reconstruction of plasma optical boundary with spectrometric technique on HL-2M tokamak[J]. Plasma Science and Technology, 2023, 25(7): 075103. DOI: 10.1088/2058-6272/acb97d

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Detailed analysis and simulation verification for reconstruction of plasma optical boundary with spectrometric technique on HL-2M tokamak

1.
Department of Engineering Physics, Tsinghua University, Beijing 100084, People's Republic of China
2.
Southwestern Institute of Physics, Chengdu 610041, People's Republic of China

More Information

Corresponding author:
Zhihao SU, E-mail: su-zh19@mails.tsinghua.edu.cn
Received Date: October 03, 2022
Revised Date: February 05, 2023
Accepted Date: February 05, 2023
Available Online: December 05, 2023
Published Date: March 28, 2023

Graphical Abstract

Abstract

Abstract

The plasma optical boundary reconstruction technique based on Hommen's theory is promising for future tokamaks with high parameters. In this work, we conduct detailed analysis and simulation verification to estimate the 'logic loophole' of this technique. The finite-width effect and unpredictable errors reduce the technique's reliability, which leads to this loophole. Based on imaging theory, the photos of a virtual camera are simulated by integrating the assumed luminous intensity of plasma. Based on Hommen's theory, the plasma optical boundary is reconstructed from the photos. Comparing the reconstructed boundary with the one assumed, the logic loophole and its two effects are quantitatively estimated. The finite-width effect is related to the equivalent thickness of the luminous layer, which is generally about 2‒4 cm but sometimes larger. The level of unpredictable errors is around 0.65 cm. The technique based on Hommen's theory is generally reliable, but finite-width effect and unpredictable errors have to be taken into consideration in some scenarios. The parameters of HL-2M are applied in this work.

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1. Introduction

When a toroidal fusion system reaches a relatively stable state, there will be a thin but stable luminous layer existing at the edge of plasma, which is known as a plasma optical boundary. The mechanism of luminescence in this region is believed to be related to electronic transitions of edge ions or atoms. An idea on reconstructing a plasma optical boundary from charge-coupled device (CCD) camera images was introduced [1, 7, 8, 12] which has been widely recognized as a promising diagnostic method for future reactors.

Further, G Hommen et al have developed a curve-mapping theory [2] which maps the characteristic curves on a CCD camera image onto the corresponding plasma optical boundary curves in 3D space. Usually, as long as the camera maintains enough distance from the plasma volume, there will be characteristic curves (CCs) on the image brightly and clearly. Any straight line passing through a point of a CC and the focal point of the camera is tangent to the plasma optical boundary surface, so the shape and position of the plasma optical boundary can be identified by solving the mapping equations with CCs.

There are at least four advantages of using this technique. Firstly, this method greatly reduces the computational load compared with full-image inversion transformation. Secondly, it mitigates and nearly eliminates the effects and interference caused by the reflection from device wall, which is also one of the greatest challenges in the full-image inversion transformation. Thirdly, this technique relies on simple, reliable, and quick-response diagnostic tools situated outside the vacuum chamber. It is almost the ideal diagnostic technique for reactors with high parameters. Fourthly, the high efficiency in reconstruction enables this technique to operate online, and thus can be used as a real-time feedback-control signal source [3, 11].

This technique has been successfully applied to TCV [3] and EAST [4]. In these works, self-consistency, efficiency, and reliability were analyzed. However, discussions on reliability mainly focused on comparing the reconstructed plasma boundary with the last closed flux surface (LCFS) from EFIT, and are still not sufficient to guarantee reliability. Further, this technique suffers from a 'logic loophole' consisting of finite-width effect and unpredictable errors. These two effects will reduce the reliability of the reconstruction results. At present, there is little discussion on these two effects. In fact, it is almost impossible to analyze the two effects through experiments on tokamak devices, even with the EFIT tool. There are two main reasons. Firstly, there is no strict correspondence between the magnetic structure and the luminous structure, and the maximum luminous intensity layer (plasma optical boundary) also deviates from the LCFS. Secondly, the LCFS calculated by the EFIT tool [9] also deviates from the actual one, especially in the region near X points. However, simulation makes it possible to quantitatively assess these two effects. In this paper, these two effects will be illustrated by analytical and simulation calculation in sections 2 and 3 separately.

In order to assess the two effects and eliminate the logic loophole, a new computer program named Plasma Boundary Identification System (PBIS) is adopted. PBIS was originally developed for the HL-2A/HL-2M tokamak by the authors of this article, and can be easily applied to other tokamak devices due to its modular design and flexible interface. PBIS is able to reconstruct plasma optical boundary with CCD camera images from the HL-2A/HL-2M tokamak, and can conduct simulation verification work. This paper is based on the simulation functions of PBIS.

The general idea of simulation verification goes as follows. First, assume a set of original plasma boundary curves; then assume a profile of luminous intensity on a toroidal cross-section; next, simulate and create a camera image by integration as happens in real physics scenes; finally, reconstruct the plasma boundary from this image and compare the results with the originally assumed boundary curves. In principle, the plasma optical boundary assumed can be arbitrary, and we adopt a LCFS calculated by EFIT on the HL-2M tokamak [10].

This paper is organized as follows. In section 2, the imaging theory, the characteristic curve theory, and Hommen's theory are reevaluated, including the two effects of the logic loophole. In section 3, the processes and results of simulation verification are introduced. The reliability of the reconstruction technique and the quantitative estimations of the two effects are discussed as well. In section 4, conclusions and topics for further investigation are given.

2. Basic theory

2.1 Imaging theory

For a transparent and isotropic spatial luminous body, the imaging process is essentially a line integral. In an imaging scene, as shown in figure 1, the spherical coordinate frame $(ρ, θ, ϕ)$ is defined as follows. The origin is fixed on the lens focus, and the z-axis is fixed on the optical axis. Considering a volume element, its luminous intensity is $f (ρ, θ, ϕ) d V = f (ρ, θ, ϕ) d ρ ρ d θ ρ \sin θ d ϕ .$

Figure 1. Schematic diagram of the imaging scene of a spatial luminous body.

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The ratio of the part reaching the camera to the total intensity is under the approximations $r_{\text {len }} \ll \rho$ and $f_{\text {len }} \ll \rho$ . here, and refer to the area, radius, and focal length of the lens, respectively. Therefore, the light intensity contribution of this volume element to the lens is

$δ I = \frac{S_{lens}}{4 π} \sin θ \cos θ f (ρ, θ, ϕ) d ρ d θ d ϕ$

(1)

The corresponding area element on image is

$\begin{matrix} δ S = w_{c} \tan θ d ϕ d (w_{c} \tan θ) \\ = w_{c}^{2} \frac{\tan θ}{\cos^{2} θ} d θ d ϕ \end{matrix}$

(2)

where $w_{c}$ refers to the distance from the lens focus to the equivalent imaging plane.

Considering the following correspondence between volume element and area element,

$\int_{\rho_{\min }}^{\rho_{\max }} \delta I=F \delta S$

(3)

the light intensity distribution on the imaging plane is

$F=\int_{\rho_{\min }}^{\rho_{\max }} \frac{S_{\text {lens }}}{4 \pi w_{\mathrm{c}}^2} \cos ^4 \theta f(\rho, \theta, \phi) \mathrm{d} \rho$

(4)

where $ρ_{\min}$ and $ρ_{\max}$ refer to the lower and upper bounds of the integral, respectively.

In Hommen's theory and its practical applications, the Cartesian coordinate system $(u, v, w)$ is defined as follows. The v-axis coincides with the symmetry axis of the toroidal device. The w-axis is parallel to the camera optical axis. In this coordinate system, a new integral variable $α,$ the normalized $ρ$ in equation (4), is introduced, which is defined as follows:

$α \equiv \frac{ρ}{\sqrt{{(u - u_{c})}^{2} + {(v - v_{c})}^{2} + w_{c}^{2}}}$

(5)

Within equation (5), $(u_{c}, v_{c}, w_{c})$ refers to the position of camera optical focus. $(u, v)$ refers to the position of a pixel on the equivalent imaging plane, the u-v plane. Then, the coordinates of a space volume element can be expressed as

$(u_{ele}, v_{ele}, w_{ele}) = (u_{c}, v_{c}, w_{c}) + α [(u, v, 0) - (u_{c}, v_{c}, w_{c})]$

(6)

The $(R, Z)$ coordinates of this volume element on toroidal cross-section are

$\{\begin{matrix} R = \sqrt{u_{ele}^{2} + w_{ele}^{2}} = \sqrt{{[α u + (1 - α) u_{c}]}^{2} + {[(1 - α) w_{c}]}^{2}} \\ Z = v_{ele} = α v + (1 - α) v_{c} \end{matrix}$

(7)

Supposing that the luminous intensity distribution on the toroidal cross-section is $f = f_{CS} (R, Z),$ equation (4) is rewritten as follows:

$\begin{aligned} & F(u, v)=\frac{S_{\text {lens }} w_{\mathrm{c}}^2}{4 \pi\left[\left(u-u_{\mathrm{c}}\right)^2+\left(v-v_{\mathrm{c}}\right)^2+w_{\mathrm{c}}^2\right]^{\frac{3}{2}}} \\ & \times \int_{\alpha_{\mathrm{m}}}^{\alpha_{\mathrm{M}}} f_{\mathrm{CS}}\left[\sqrt{\left[\alpha u+(1-\alpha) u_{\mathrm{c}}\right]^2+\left[(1-\alpha) w_{\mathrm{c}}\right]^2}\right. \\ & \left.\alpha v+(1-\alpha) v_{\mathrm{c}}\right] \mathrm{d} \alpha \end{aligned}$

(8)

Within equation (8), $a_{m}$ and $α_{M}$ are the lower and upper bounds, respectively, of $α$ , and the physical process of imaging in the tokamak can be simulated by equation (8).

2.2 Characteristic curve theory and finite-width effect

A simplified model of the plasma optical structure is considered, which is shown in figure 2(a). In this model, light sources are uniformly distributed in a thin layer of equal thickness at the edge of a cylinder. The corresponding luminous distribution function is written as

$f_{\text {simp }}(x)= \begin{cases}1 & \boldsymbol{x} \in \text { thin layer } \\ 0 & \boldsymbol{x} \notin \text { thin layer }\end{cases}$

(9)

Figure 2. Subfigure (a) is the schematic diagram of a simplified model of the plasma luminous structure and integral paths. The light yellow ring-shaped area represents the plasma luminous layer. The orange lines represent the integral paths of the imaging.

$r_{i}$ and

$r_{o}$ represent the inner and outer radii, respectively, of the luminous layer.

$x$ is the chord-center distance. Subfigure (b) shows the relationship between the equivalent light intensity (effective length of integral path) and the chord-center distance of the integral path. In this case,

$r_{o}$ = 2 m and

$r_{i}$ = 1.95 m. The dashed line I, line C, and line O refer to the positions of the inner surface, center layer, and outer surface of the luminous layer, respectively.

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For simplicity, in equation (4), the angle effect $\cos^{4} θ$ of imaging is ignored, so the integral of $f_{simp}$ can be approximately reduced to the effective length $L$ of the integral path through the luminous layer, which corresponds to the bold parts of the orange lines in figure 2(a):

$\begin{aligned} F & \approx \int_{\rho_{\min }}^{\rho_{\max }} \frac{S_{\text {lens }}}{4 \pi w_{\mathrm{c}}^2} f_{\text {simp }}(\rho, \theta, \phi) \mathrm{d} \rho \\ & \sim \int_{\rho_{\min }}^{\rho_{\max }} f_{\text {simp }}(\rho, \theta, \phi) \mathrm{d} \rho=L \end{aligned}$

(10)

According to the geometric relationship shown in figure 2(a), the effective integral path length $L$ is equivalent to the difference in chord length between the inner surface and the outer surface:

$L = \{\begin{matrix} 0 & x > r_{o} \\ 2 \sqrt{r_{o}^{2} - x^{2}} & r_{i} < x \leq r_{o} \\ 2 \sqrt{r_{o}^{2} - x^{2}} - 2 \sqrt{r_{i}^{2} - x^{2}} & 0 < x \leq r_{i} \end{matrix}$

(11)

where $x$ is the chord-center distance, and $r_{o}$ and $r_{i}$ are the inner and outer radii of the luminous layer, respectively. The relationship between $L$ and $x$ is shown in figure 2(b).

There is a significant peak of $L$ value when $x = r_{i} .$ This indicates that when the integral path is tangent to the inner surface of the equivalent luminous layer, the maximum light intensity on the image will appear. When the integral path rotates along the tangential direction of the luminous layer, this maximum point will draw bright curves on image. Such curves are called characteristic curves.

Therefore, strictly speaking, the characteristic curves correspond to the inner surface of the equivalent luminous layer, rather than the maximum layer of the luminous layer. Further, the optical boundary reconstructed by the ideal CCs is the inner surface, not the maximum layer. The deviation is of the same order of the thickness of the equivalent luminous layer, which is called finite-width effect. If the structure of the luminous layer is not too complicated and the luminous intensity is evenly distributed, the deviation is exactly 1/2 of the luminous layer thickness.

Assuming $r_{\mathrm{o}}-r_{\mathrm{i}} \ll r_{\mathrm{o}}$ , this deviation can be ignored, and the inner surface can be regarded the same as the maximum layer. Previous studies have tacitly accepted that such an assumption is reasonable, and this effect is not important. However, sometimes it is necessary to take this effect into consideration, especially when the thickness of the layer increases.

2.3 Hommen's theory

In Hommen's theory [2], finite-width effect is neglected and the geometric relationship in characteristic curve theory is used to reconstruct the plasma boundary. A CC on the u-v plane is represented as $f_{CC} (u_{e}, v_{e}) = 0,$ and a plasma boundary curve on the cross-section is represented as $f_{PB} (R_{e}, Z_{e}) = 0 .$ The mapping relationship between the plasma boundary and CCs is

$\begin{matrix} u_{c} \cos φ_{e} + w_{c} \sin φ_{e} = R_{e} - (Z_{e} - v_{c}) \frac{d R_{e}}{d Z_{e}} \\ u_{e} - u_{c} = \frac{- w_{c} (R_{e} \cos φ_{e} - u_{c})}{R_{e} \sin φ_{e} - w_{c}} \\ v_{e} - v_{c} = \frac{- w_{c} (Z_{e} - v_{c})}{R_{e} \sin φ_{e} - w_{c}} \end{matrix}$

(12)

and

$\begin{matrix} \tan φ_{e} = \frac{(u_{e} - u_{c}) - (v_{e} - v_{c}) \frac{d u_{e}}{d v_{e}}}{w_{c}} \\ R_{e} = \frac{w_{c} u_{e}}{[(u_{e} - u_{c}) \sin φ_{e} + w_{c} \cos φ_{e}]} \\ Z_{e} = \frac{w_{c} v_{e} \cos φ_{e} + (v_{c} u_{e} - u_{c} v_{e}) \sin φ_{e}}{[(u_{e} - u_{c}) \sin φ_{e} + w_{c} \cos φ_{e}]} \end{matrix}$

(13)

Within equations (12) and (13), $(u_{e}, v_{e})$ refers to the position of a CC point on the image. $(R_{e}, Z_{e})$ refers to the position of a point of a plasma boundary curve. $d u_{e} / d v_{e}$ refers to the tangent slope of the CC at $(u_{e}, v_{e}) .$ $d R_{e} / d Z_{e}$ refers to the tangent slope of the plasma boundary curve at $(R_{e}, Z_{e}) .$ $φ_{e}$ is the toroidal coordinates of the tangent point between the integral path and plasma boundary surface.

Based on equation (12), only points that satisfy the following conditions on plasma optical boundary curves have corresponding points on CCs, which is called validity criteria:

$|R_{e} - (Z_{e} - v_{c}) \frac{d R_{e}}{d Z_{e}}| \leq \sqrt{u_{c}^{2} + w_{c}^{2}}$

(14)

With equations (12) and (13), the transformation between CCs and plasma boundary curves can be realized.

Hommen's theory is general and has no strict restrictions on the coordinates of the camera other than equation (14). Therefore, in the following discussions, when the camera is mounted at the top region, bottom region, or equatorial plane, Hommen's transformation equations can be used directly.

2.4 Failure of numerical differentiation and unpredictable errors

In equation (13), the calculation of derivative $d u_{e} / d v_{e}$ shall be transformed into numerical derivative $Δ u_{e} / Δ v_{e}$ in practice. However, the CCs identified from the image are sequences of pixels arranged in 4-connected or 8-connected form [5], as shown in figure 3. For the 4-connected pixel series (Von Neumann neighborhood), pixels are connected horizontally or vertically. For the 8-connected pixel series (Moore neighborhood), pixels are connected horizontally, vertically, or diagonally. Further, it indicates that the values of $Δ u_{e} / Δ v_{e}$ will be a bunch of zeros and infinities for 4-connected form, or a bunch of zeros, ones, and infinities for 8-connected form. This is clearly unacceptable, and is called failure of numerical differentiation.

Figure 3. Schematic diagram of two typical types of the curve fill algorithm; the left one is in 4-connected form, and the right one is in 8-connected form.

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Therefore, fitting CCs is necessary before calculation for numerical derivative. Usually, quadratic and hyperbolic fitting are found to be simple and stable, and are adapted in the cases in this paper.

However, fitting only alleviates the numerical differential failure; it does not eliminate the error and even introduces new uncertainties. The deviations between the true values and the calculated values of $R_{e}$ and $Z_{e}$ consist of three components: the deviations of $u_{e},$ $v_{e},$ and $d u_{e} / d v_{e} :$

$\left\{\begin{array}{l} \Delta R_{\mathrm{e}}=\frac{\partial R_{\mathrm{e}}}{\partial u_{\mathrm{e}}} \Delta u_{\mathrm{e}}+\frac{\partial R_{\mathrm{e}}}{\partial v_{\mathrm{e}}} \Delta v_{\mathrm{e}}+\frac{\partial R_{\mathrm{e}}}{\partial\left(\frac{\mathrm{d} u_{\mathrm{e}}}{\mathrm{d} v_{\mathrm{e}}}\right)} \Delta \frac{\mathrm{d} u_{\mathrm{e}}}{\mathrm{d} v_{\mathrm{e}}} \\ \Delta Z_{\mathrm{e}}=\frac{\partial Z_{\mathrm{e}}}{\partial u_{\mathrm{e}}} \Delta u_{\mathrm{e}}+\frac{\partial Z_{\mathrm{e}}}{\partial v_{\mathrm{e}}} \Delta v_{\mathrm{e}}+\frac{\partial Z_{\mathrm{e}}}{\partial\left(\frac{\mathrm{d} u_{\mathrm{e}}}{\mathrm{d} v_{\mathrm{e}}}\right)} \Delta \frac{\mathrm{d} u_{\mathrm{e}}}{\mathrm{d} v_{\mathrm{e}}} \end{array}\right.$

(15)

Within equation (15), $Δ R_{e}, Δ Z_{e}, Δ v_{e}, Δ u_{e}, Δ d u_{e} / d v_{e}$ are deviations between numerical values and true values of the corresponding physical quantities. Deviations $Δ v_{e}, Δ u_{e}, Δ d u_{e} / d v_{e}$ consist of two parts sampling error and fitting error:

$\begin{matrix} Δ u_{e} = Δ_{fitting} u_{e} + Δ_{sampling} u_{e} \\ Δ v_{e} = Δ_{fitting} v_{e} + Δ_{sampling} v_{e} \\ Δ \frac{d u_{e}}{d v_{e}} = Δ_{fitting} \frac{d u_{e}}{d v_{e}} + \frac{d Δ_{sampling} u_{e}}{d v_{e}} + \frac{d u_{e}}{d v_{e}} \frac{d Δ_{sampling} v_{e}}{d v_{e}} \end{matrix}$

(16)

The sampling error is local and mainly depends on the pixel spacing. The fitting error is global and difficult to estimate analytically. Further, the error of the reconstructed plasma boundary is more difficult to estimate, which is thus called unpredictable error, and simulation is the only method to define the error level and the reliability of the reconstruction technique.

3. Simulation verification work

3.1 Program flowchart of simulation verification

There are six processes in PBIS designed for simulation verification, as shown in figure 4. Construct Light Intensity Profile is designed for creating a profile of luminous intensity on a cross-section with assumed plasma optical boundary and related artificial parameters. Calculate Curvilinear Integral of Light Intensity is designed for simulating an image on a virtual camera with equation (8). This process is more computation-intensive compared to other alternatives. The paths corresponding to each pixel should be finely segmented, and the luminous intensity at each discrete point should be calculated successively and then integrated. Delineate Regions of Interest is designed for delineating regions of interest on a simulated image, making it possible to apply a more properly optimized algorithm to each sub-region. Extract Characteristic Curves is designed for extracting CCs from each sub-region. It allows users to choose among the watershed method, slicing and searching method [5], etc. according to practical needs. Reconstruct Plasma Boundary is designed for calculating plasma optical boundary curves with equation (13). The computational efficiency of this process is quite high, as it only depends on analytic transformation expressions. Compare is designed for comparing the reconstructed plasma boundary curves with the originally assumed one, which will estimate the magnitude of finite-width effect and unpredictable errors, and will define the reliability of the plasma boundary reconstruction technique as well.

Figure 4. Program flowchart of the whole process of simulation verification.

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3.2 Classic demo

In this section, a classic demo is used to explain the details of the simulation verification work. As explained in the introduction, a LCFS calculated by EFIT on the HL-2M tokamak is chosen as the assumed plasma optical boundary, which is shown in figure 5(a). Then a corresponding hypothetical distribution of luminous intensity on toroidal cross-section is created. In this demo, the distribution is simulated by SOLPS-ITER [6], as shown in figure 5(a).

Figure 5. Subfigure (a) is the assumed distribution of the luminous intensity on the toroidal cross-section, and the red dashed curve represents the assumed optical boundary. Subfigure (b) is the photo taken by the virtual camera, that is, the light intensity distribution obtained by integration.

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Setting the position of the virtual camera to (1.685 m, 0 m, 1.98 m), an image is created by simulation as shown in figure 5(b). The length and width of this image are 2.5 m, and the center coordinate is (1.685 m, 0 m, 0 m). It has 1000 × 1000 pixels.

Then, two rectangular sub-regions are delineated in this case. The coordinates of two vertices of the sub-region on the high-field side are (1.0858 m, −0.3841 m) and (1.5187 m, 0.3342 m). The ones on the low-field side are (1.9516 m, −0.8296 m) and (2.7649 m, 0.7495 m). CCs are extracted from each sub-region and fitted separately, as shown in figure 6(a).

Figure 6. Case 1, in which the camera is on the equatorial plane. Subfigure (a) is the camera photo obtained by simulation, and the red scatters represent the identified characteristic curves. In subfigure (b), the blue scatters represent the reconstructed plasma boundary and the red curve represents the originally assumed plasma boundary.

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Then, plasma boundary curves are reconstructed by Reconstruct Plasma Boundary, as shown in figure 6(b), and marked with blue scatters. For comparison, the originally assumed plasma boundary curves are superimposed on figure 6(b) and marked with red curves.

4. Results and discussion

4.1 Reconstruction results of a SOLPS-ITER-simulated distribution model

Using a luminous intensity distribution generated by simulation of SOLPS-ITER, as in section 3.2, is close to the real scenario. In order to verify the reconstruction reliability for different camera positions, the virtual camera was placed in three different locations: equatorial plane region (1.685 m, 0 m, 1.98 m), top region (1.649 m, 0.8254 m, 1.98 m), and bottom region (1.46 m, −1.1 m, 1.98 m), as shown in figure 7. The operations and other parameters are the same as the ones adopted in section 3.2, except for a few minor adjustments for the regions of interest. The relevant results are shown in figures 6, 8, and 9.

Figure 7. Spatial diagram of camera installation position. The cubes represent the camera positions, and the corresponding squares on the u-v plane represent image positions. From top to bottom are the top case, equatorial plane case, and bottom case.

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Figure 8. Case 2, in which the camera is at the top. Subfigure (a) is the simulated image, and the red scatters are CCs. In subfigure (b), the blue scatters represent the reconstructed boundary, and the red curve is the originally assumed boundary.

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Figure 9. Case 3, in which the camera is at the bottom. Subfigure (a) is the simulated image, and the red scatters are CCs. In subfigure (b), the blue scatters represent the reconstructed boundary, and the red curve is the originally assumed boundary.

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4.1.1 Identification of CCs

As has been shown in the panels (a) of figures 6, 8, and 9, the extraction of CCs is stable and robust. It is worth noting that the parabolic fit is used in both the equatorial plane case and the bottom case, while the hyperbolic fit $a x + b / (x + c)$ is used in the top case.

4.1.2 Reconstruction quality

Comparing panel (a) with panel (b) in figures 6, 8, and 9, the reconstructed plasma boundary is generally consistent with the assumed one. However, there is a slight deviation from the assumed boundary towards the central column. This deviation is several centimeters (2‒4 cm), as shown in figure 10. This is not a fault of this technique, but a result of the finite-width effect and unpredictable errors. As shown in figure 10(a), the reconstructed boundary coincides with the equivalent inner edge of the luminous layer. It indicates that the interference of the unpredictable errors is less than the finite-width effect. However, quantitative discussion needs to be based on a more simplified model; see section 4.2 for details.

Figure 10. Subfigure (a) is the assumed luminous intensity distribution, and the red curves represent the plasma boundary reconstructed by three different cameras from Case 1 to Case 3. Six subfigures (b)‒(g) show the deviations of reconstructed boundary curves of different regions from the originally assumed one. (b)‒(d) correspond to the boundary of the high-field side. (e)‒(g) correspond to the boundary of the low-field side. (b) and (e) are for Case 2. (c) and (f) are for Case 1. (d) and (g) are for Case 3.

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4.1.3 Feasibility of multi-camera joint reconstruction

A single camera mounted on the equatorial plane is less able to reconstruct the plasma boundary close to the top or bottom. In this work, by fixing the camera at various heights away from the equatorial plane, it is proved that a reconstruction of high quality can be achieved for the top or bottom plasma boundary. Moreover, the reconstructed results of these different cameras are compatible with each other. This proves the feasibility of joint reconstruction of multiple cameras.

4.2 Reconstruction results of a simplified distribution model

To evaluate the effect of unpredictable errors, the assumed luminous intensity distribution was adjusted to a uniform and equal thickness model, as shown in figure 11(a). In this simplified model, the luminous layer is 10 cm thick. It is worth noting that in order to highlight the finite-width effect, this thickness is larger than the average experimental level. With operations and parameters similar to those in section 4.1, the reconstruction results for three cameras are obtained, as shown in figures 11(b)‒(d).

Figure 11. Subfigure (a) is the assumed simplified luminous intensity distribution, and the red dashed curve represents the assumed optical boundary. Subfigures (b)‒(d), corresponding to Cases 4 to 6, are the photos taken by three virtual cameras in different positions, and the dashed red curves represent the CCs. The installation locations of the cameras are consistent with the three cases in section 4.1.

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4.2.1 Finite-width effect

As shown in figure 12, due to the larger thickness value, the total deviations of the reconstructed boundary are larger than the results of the cases in section 4.1, and the reconstructed boundary coincides well with the inner surface of the luminous layer. This is fully consistent with the conclusions in section 2.2 that the finite-width effect is exactly half of the luminous layer thickness, in this case, 10 cm/2 = 5 cm.

Figure 12. Subfigure (a) is the assumed simplified luminous intensity distribution, the red curves represent the reconstructed plasma boundary on three cameras from Cases 4 to 6, and the black dashed curve is the assumed boundary. Six subfigures (b)‒(g) show the deviations of reconstructed boundary curves of different regions from the originally assumed one, the black curves represent the total deviations, the blue curves represent the deviations of finite-width effect, and the red curves represent the deviations of unpredictable errors. (b)‒(d) correspond to the boundary of the high-field side. (e)‒(g) correspond to the boundary of the low-field side. (b) and (e) are for Case 4. (c) and (f) are for Case 5. (d) and (g) are for Case 6.

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4.2.2 Assessment of unpredictable errors

Based on the certain finite-width effect in this model, the magnitude of unpredictable errors can be assessed as follows:

$\Delta_{\text {unpredictable errors }}=\Delta_{\text {total }}-\Delta_{\text {finite-width effect }}$

(17)

As shown in figure 12, the average magnitude of unpredictable errors is about 0.16 cm, and the maximum magnitude is about 0.65 cm. For different regions, the magnitudes of unpredictable errors are close, but are slightly larger in the equatorial plane case. This is mainly because in this case, the range of CCs is large and the accuracy of fitting is slightly decreased. In contrast, the level of unpredictable errors at the bottom is relatively low, about 0.1 cm on average.

5. Conclusions

The optical boundary reconstruction technique based on Hommen's theory is of great significance to the operation and control of future high-parameter tokamak devices. However, this technique has a 'logic loophole' that requires detailed analysis and verification. In this paper, the loophole is discussed in section 2 and assessed by simulation in section 3. In the assessment, a new system, PBIS, is developed with imaging simulation and boundary reconstruction functions.

The logic loophole is mainly composed of finite-width effect and unpredictable errors. The finite-width effect can be estimated using half of the equivalent thickness of the luminous layer. The corresponding deviations are usually around 2‒4 cm towards the center column on HL-2M. Unpredictable errors can only be estimated by simulation. The deviations of this effect are about ±0.65 cm on HL-2M, which is acceptable in most cases.

In practical application, before using the optical boundary reconstruction technique, the first step is to evaluate the accuracy tolerance of the experimental target. If the accuracy required is lower, such as on the order of 5 cm, the technique can be used directly. If the required accuracy is higher, such as on the order of 1 cm, it is necessary to specifically design post-processing programs to eliminate finite-width effects, which will need further exploration. If the required accuracy is relatively high, such as on the order of 0.1 cm, the technique might fail unless new treatments for unpredictable errors are developed.

In general, reconstruction technique based on Hommen's theory is reliable. The reconstruction results for cameras in different positions are similarly reliable and compatible with each other. Therefore, the multi-camera joint reconstruction technique is feasible. This can greatly expand the application range of the optical boundary reconstruction technique.

In this work, the estimation of the finite-width effect depends on a simplified luminous structure. For a more complex structure, as in the cases in sections 3.2 and 4.1 of this paper, it may be not difficult to estimate the total effect of the logic loophole but quite difficult to estimate the finite-width effect, and therefore the unpredictable error. Therefore, the equivalent width estimation for complex luminous structures is an important topic for the future. Furthermore, the treatments for finite-width effect and the optimized fitting methods deserve further study to reduce the logic loophole of the optical boundary reconstruction technique.

Acknowledgments

This work was supported by the Tsinghua University 2021 Doctoral Summer Project. This work was also supported by the National Key R & D Program of China (No. 2018YFE0301102) and National Natural Science Foundation of China (Nos. 11875020 and 11875023).

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Article Contents

Abstract

1. Introduction

2. Basic theory

3. Simulation verification work

4. Results and discussion

5. Conclusions

Acknowledgments

References

Detailed analysis and simulation verification for reconstruction of plasma optical boundary with spectrometric technique on HL-2M tokamak