A protocol for mapping transport current density of REBCO-coated conductor by magneto-optical imaging

REBa₂Cu₃O_7−x (REBCO, where RE refers to rare-earth elements)-coated conductors (CCs) have a multilayered-film structure and serve as one of the key superconducting wires for applications of high-field superconducting magnets [1–3], superconducting transmission cables [4], and superconducting energy storage [5], etc. The critical current of CCs, especially in high magnetic fields, is a significant parameter in their engineering design and applications [6–8]. However, in addition to temperature and applied magnetic field, the critical current is influenced by strains [9–11], grain angles [12], pinning centers [13, 14], defects [15] and damages [16] in superconducting layer. Improving the properties of the superconducting layer so as to achieve a high critical current of CCs requires sufficient investigation of these factors on the CC's current density. Unlike common conductors, the current density distribution for a CC is not uniform but varies along its width [17, 18]. Since there are no direct methods to measure the current density, mapping of the current density distribution is usually reconstructed from the inversion of the magnetic field at the CC surface according to the Biot–Savart law [19–22].

Among the approaches to obtain the magnetic field distribution with high spatial resolution are scanning techniques, such as SQUID [23] or micro Hall-probe array [24], as well as magneto-optical imaging (MOI) based on the Faraday effect [25, 26]. Due to the sub-nanosecond response time [27], MOI is the preferred application for real-time visualization of the magnetic field, including magnetic flux avalanches [28], quench propagation [29], and crack-induced flux penetration [16, 30], etc. For this reason, MOI has mostly been used for quantitativecurrent density analysis of superconducting films, especially for the time-dependent case [31].

Since MOI directly provides the intensity data as a two-dimensional picture, mapping of the current density must start from the inversion of the magnetic field. There have been sufficient studies on the magnetic-induction current density of superconducting films, in which the magnetic field only in the same direction. The magnetic field with different directions of superconducting samples has been observed through light and dark regions or color changes [32–38], but most of these studies only provided only an intensity analysis. Nevertheless, the mapping of the transport current density by MOI has rarely been studied, and only a few related works on the current-carrying micro superconducting bridge have been reported [34, 35, 37, 39]. Investigation of the transport current density distribution of CC by MOI on the macroscopic scale still needs to be well researched, as well as its calibration. It should be noted that there are still several problems to be solved in the reconstruction of the transport current density by MOI: (1) the magnetic field induced by the transport current contains both variables of magnitude and sign. In this case, the common crossed setting of polarizers no longer applies. The adoption of a white light source and uncrossed polarizers in MOI to obtain greenish or yellow pictures [36] was just able to provide qualitative analysis, i.e. on the direction rather than the magnitude of the magnetic field. The other triplet-rotation-angle approach [40] was effective to obtain a high resolution of the magnetic field with magnitude and sign simultaneously, with the aid of an external device to precisely control the rotation plane of polarizing light, which to some extent complicated the MOI system. By practically calibrating a polynomial fit to the relationship between the intensity and magnetic field, we can get rid of the main hindrance of non-uniform illumination [41, 42]. The parameters for calibration strongly depend on the conditions of the light source and camera settings, making it hard to be comparable between different utilizers. (2) The routine MOI experiment starts by recording MOI pictures of samples in the superconducting state below T_c , followed by a calibration process at elevated temperatures above T_c . Though the working temperature range is wide for widely used MOI indicators made from Bi-doped iron garnet, the temperature influence on the properties of the MOI indicator is often neglected during tests. Thus, there is a need to develop a calibration procedure that considers the temperature effect. (3) Physically speaking, the magnetic field generated by a CC with the transport current diffuses through the infinite space; however, information on the detected magnetic field is limited due to the size limit of the MOI indicators, and thus a correlation between the measured space size of the magnetic field and the accuracy of the current density needs to be revealed.

In this work, we address these problems. First, a systematic non-linear calibration process was developed to get rid of non-uniform illumination. In this method, only two parameters of the MOI indicator were determined, both independent of the light and camera setting conditions. In addition, the temperature effects on these two parameters were studied, and an improved procedure for MOI measurement was provided. Second, the relationship between the measured window space size and the accuracy of the current density was investigated, while a practical method for magnetic field extension was proposed for MOI testing in the case where the sample size was comparable to the size of the MOI indicator. Finally, current density reconstructions of the CC were realized with different applied transport currents.

2.1. Sample parameters

The CC sample used in this experiment is made by Suzhou Advanced Material Research Institute Co., Ltd. Its specifications are listed in table 1.

Table 1. Sample specifications.

Width	Fabrication technique	Structure	Critical current at 77 K
2 mm	Ion beam assisted deposition and metal organic chemical vapor deposition	Ag (2 $\mu {\text{m}}$) YBCO (1 $\mu {\text{m}}$) Buffer (0.2 μm) Hastelloy (50 $\mu {\text{m}}$)	88 A

2.2. Experimental setup

The experimental setup contains two subsystems: one is a common MOI system, the other a specially-designed cryostat that can apply strain, transport current, and external magnetic field. Details of the cryostat can be found in our previous work [43, 44]. The MOI system consists of an assembled 530 nm LED light source, a pair of polarizers, a beam-splitter prism, and an 8-bit digital camera with macro lens for large-area imaging. The MOI indicator used has a Bi-doped magneto-optical sensor layer of 5 μm, a gadolinium gallium garnet substrate of 500 μm, and a mirror layer of 0.2 µm. The details of the system are shown in figure 1.

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2.3. Calibration

The sensor layer of the MOI indicator has an in-plane spontaneous magnetization M_s without an external applied magnetic field B. If there is an applied B at an angle α to the film plane, the M_s tilts out of the plane with the angle φ, as shown in figure 2.

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The interaction energy contains magnetocrystalline anisotropy energy and magnetostatic energy [21], as expressed in equation (1):

$$\begin{equation}{E_{\operatorname{int} }} = {E_A}\left(1 - \cos \phi \right) + B{M_s}\left[ {1 - \cos \left( {\alpha - \phi } \right)} \right]\end{equation} \tag{ 1 }$$

where ${E_A}$ is the anisotropic energy and α is the angle between the applied field and the film plane. At a given α, φ varies with the magnetic field, and the whole process satisfies a minimum energy change, i.e.

$$\begin{equation}\delta {E_{\operatorname{int} }} = \frac{{\partial {E_{\operatorname{int} }}}}{{\partial \phi }}\delta \phi = 0.\end{equation} \tag{ 2 }$$

Equation (2) requires $\frac{{\partial {E_{\operatorname{int} }}}}{{\partial \phi }} = 0$, from which we get

$$\begin{equation}\phi = \arctan \left( {\frac{{{B_z}}}{{{B_{xy}} + {B_A}}}} \right)\end{equation} \tag{ 3 }$$

where ${B_A} = {E_A}/{M_S}$ is the anisotropic magnetic field, ${B_{xy}} = B\cos \alpha$ is the in-plane magnetic field component, and ${B_{\text{z}}} = B\sin \alpha$ is the perpendicular magnetic field component. It should be noted that when there is an applied field strictly perpendicular to the CC, the in-plane magnetic field component will be generated by the in-plane current density flowing in the CC. The Faraday rotation angle α_F , proportional to the z-component of magnetization, is denoted as

$$\begin{equation}{\alpha _F} = c{M_S}\sin \phi \end{equation} \tag{ 4 }$$

where c is a constant that relates to the thickness of the MOI indicator. Using a common MOI system, the light intensity received by the camera is expressed according to Malus' law:

$$\begin{equation}I\left(x,y\right) = {I_0}\left( {x,y} \right) + {I_{\max }}\left( {x,y} \right){\cos ^2}\left[ {{\alpha _F}\left( {x,y} \right) + \theta } \right]\end{equation} \tag{ 5 }$$

where I_max is the incident light intensity, I₀ is the background light intensity, and θ is the initial cross-angle between the polarizer and the analyzer. In an MOI photo, I_max and I₀ are usually non-uniform because of the inhomogeneous illumination of the light source and the background intensity of the camera. Here, we develop a new way to get rid of this effect. First, at a set angle θ, the I_max is calculated as I_{θ, max} without applied magnetic field by

$$\begin{equation}{I_{\max }} = {I_{\theta ,\max }}\left(x,y\right) = \frac{{{I_{\theta ,B = 0}}\left(x,y\right) - {I_{{{90}^ \circ },B = 0}}\left(x,y\right)}}{{{{\cos }^2}\left( \theta \right)}}\end{equation} \tag{ 6 }$$

where I_{θ,B =0} is the intensity with θ of polarizers and I_90°,B=0 is the intensity recorded with the crossed setting of polarizers. Then, by combining equations (2)–(5), the normalized light intensity I_N (x,y) as a function of the applied field B is obtained as

$$\begin{align}{I_N}\left(x,y\right) & = \frac{{I\left(x,y\right) - {I_{{{90}^ \circ },B = 0}}\left(x,y\right)}}{{{I_{\theta ,\max }}\left(x,y\right)}}\nonumber\\ & = {\cos ^2}\left[ {\frac{{c{M_S}{B_z}\left(x,y\right)}}{{\sqrt {{{\left( {{B_A} + {B_{xy}}\left(x,y\right)} \right)}^2} + B_z^2\left( {x,y} \right)} }} + \theta } \right].\end{align} \tag{ 7 }$$

Accordingly, the z-component of the magnetic field is calculated as

$$\begin{align}{B_z}\left( {x,y} \right) & = \left[ {{B_A} + {B_{xy}}\left(x,y\right)} \right]\tan\nonumber\\ &\quad\times \left\{ {\arcsin \left[ {\frac{1}{{c{M_s}}}\left( {\arccos \sqrt {{I_N}\left( {x,y} \right)} - \theta } \right)} \right]} \right\}.\end{align} \tag{ 8 }$$

Equations (7) and (8) are used to calibrate the intensity versus the applied magnetic field and extract the magnetic field from the intensity, respectively.

MOI measures the z-component of the magnetic field perpendicular to the plane of the MOI indicator, and can be used to reconstruct the two-dimensional current density. Calculation in Fourier space according to the Biot–Savart law is an efficient method. The average current densities J_x and J_y along the thickness of the superconducting film are [22]

$$\begin{align}{J_x}\left( {x,y} \right) = {\mathcal{F}^{ - 1}}\left[ {\frac{{i{k_y}}}{{{\mu _0}}}{e^{kh}}\operatorname{csch} \left( {\frac{{\sqrt {k_x^2 + k_y^2} }}{2}d} \right){{\tilde B}_z}\left( {{k_x},{k_y},h,d} \right)} \right]\end{align} \tag{ 9 }$$

$$\begin{align}{J_y}\left( {x,y} \right) = {\mathcal{F}^{ - 1}}\left[ { - \frac{{i{k_x}}}{{{\mu _0}}}{e^{kh}}\operatorname{csch} \left( {\frac{{\sqrt {k_x^2 + k_y^2} }}{2}d} \right){{\tilde B}_z}\left( {{k_x},{k_y},h,d} \right)} \right]\end{align} \tag{ 10 }$$

where k_x and k_y are coordinate indices in the Fourier space, h is the interval between the magneto-optical layer and the superconducting layer, and d is the thickness of the superconducting layer. In addition, the expressions for the in-plane components of the magnetic field are

$$\begin{align}{B_x}\left( {x,y} \right) = {\mathcal{F}^{ - 1}}\left[ {\frac{{{\mu _{\text{0}}}{e^{ - kh}}}}{k}{{\tilde J}_y}\left( {{k_x},{k_y}} \right)\sinh \left( {\frac{{kd}}{2}} \right)} \right]\end{align} \tag{ 11 }$$

$$\begin{align}{B_y}\left( {x,y} \right) = {\mathcal{F}^{ - 1}}\left[ { - \frac{{{\mu _{\text{0}}}{e^{ - kh}}}}{k}{{\tilde J}_x}\left( {{k_x},{k_y}} \right)\sinh \left( {\frac{{kd}}{2}} \right)} \right].\end{align} \tag{ 12 }$$

Generally speaking, the magnetic field is distributed over the infinite space with a source current. It should be noted that the acquired magnetic field information for MOI is limited because of the limited size of the MOI indicator. Equations (9) and (10) have proven to work well for the reconstruction of the induced superconducting current density with the limited measured magnetic field by MOI. Nevertheless, for the case of transport current density, i.e. only self-field information is provided, there is a need for careful scrutinization when choosing different window sizes of the measured magnetic field. Here, the analytical results from Brandt and Indenbom [17] were simulated as a numerical experiment. Considering a current I that transports along the y direction in an infinite long superconducting strip as in figure 3(a), the current density along the width (x direction) is

$$\begin{align}J\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {\left( {\frac{{2{J_c}}}{\pi }} \right)\arctan {{\left[ {\frac{{{a^2} - {b^2}}}{{{b^2} - {x^2}}}} \right]}^{1/2}},\left| x \right| < b} \\[4pt] {{J_c},b \unicode{x2A7D} \left| x \right| < a} \end{array}} \right.\end{align} \tag{ 13 }$$

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where $b = a{\left( {1 - {{{I^2}} \mathord{\left/ \right. } {{I_c}^2}}} \right)^{{1 \mathord{\left/ \right. } 2}}}$. The z component of the magnetic field is

$$\begin{equation}B\left( x \right) = \left\{ {\begin{array}{*{20}{l}} \begin{gathered} 0,\qquad\qquad\qquad\qquad\quad {\text{ }}\left| x \right| < b{\text{ }} \hfill \\[4pt] \frac{{{B_c}x}}{{\left| x \right|}}{\text{arctgh}}{\left[ {\frac{{{x^2} - {b^2}}}{{{a^2} - {b^2}}}} \right]^{1/2}},b < \left| x \right| < a \hfill \\[4pt] \end{gathered} \\ {\frac{{{B_c}x}}{{\left| x \right|}}{\text{arctgh}}{{\left[ {\frac{{{a^2} - {b^2}}}{{{x^2} - {b^2}}}} \right]}^{1/2}},\left| x \right| > a} \end{array}} \right.\end{equation} \tag{ 14 }$$

where ${B_c} = {{{\mu _0}{J_c}} \mathord{\left/ \right. } \pi }$. It should be noted that the current density calculated by equation (13) is continuous, while the magnetic field by equation (14) has two singularities at both strip edges. Thus, a window with a size 500 times as large as the width of the strip was selected here, with which a continuous z-component of the magnetic field was reconstructed with equation (10). This agrees well with the theoretical value except the singularities, as shown in figures 3(b) and (c). Next, magnetic fields with windows of different sizes were selected to reconstruct the current density using the mentioned Fourier calculation, simulating the loss of magnetic-field information during the MOI measurement. The results are plotted in figure 3(d). It was found that the value of the constructed current density did not agree well with the theoretical one unless the window size of the magnetic field was as large as possible. For a small window size, e.g. where the size was just two times the width of the strip, the current density was less than the theoretical one within the strip and had a false value outside the strip.

Here, a practical approach to solve this problem was proposed by supplementing the magnetic field information outside for the case of a small window. Considering that the magnetic field far away from the strip decays scale to x⁻¹ (x denotes the distance from the origin) according to Ampere's law, the decaying behavior can be well approximated using a power function as follows:

$$\begin{equation}{B_{{\text{extended}}}} = \frac{{{P_1}}}{{x - {P_2}}}\end{equation} \tag{ 15 }$$

where A and B are fitted parameters. Thus, the magnetic field window is expanded to a sufficiently large extent and the current density is reconstructed using the extended magnetic field. For example, as shown in figure 4(a), a magnetic field window size of two times the strip width was first selected and assumed to be measured magnetic field, i.e. within the window, the true magnetic field values were used, while outside the window, the magnetic field values (assumed to be not measured by MOI) were calculated from the data within the outmost magnetic field window of width of 0.4a by the fitted function of equation (15), in such a way, the magnetic field window was then extended from 2 to 500 times of sample width, as in figures 4(b) and (c), followed by the current density reconstruction. Table 2 lists the integrated current results and relative errors with different extended magnetic window sizes. In comparison, the current density with an extended magnetic field window size of 500 times the sample width agrees well with the theoretical one as plotted in figure 4(d), with a relative error of 1.01%.

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Table 2. Results of integrated current and their relative errors with different extended magnetic window sizes.

Extended magnetic field window size ($\frac{x}{a}$)	2	10	200	500
${I_{\operatorname{int} }}(A)$	73.2024	110.5498	118.5426	118.7835
$\frac{{{I_0} - {I_{\operatorname{int} }}}}{{{I_0}}}$	39.00%	7.88%	1.21%	1.01%

4.1. Magnetic field extraction from MOI picture

Considering that the maximum Faraday rotation angle of the polarizing plane in the MOI indicator under an applied magnetic field is in the range of ±10°, the common orthogonal polarizer configuration, i.e. θ equals 90° in figure 5(a), is only able to measure the magnitude of the magnetic field without information of its direction, according to equation (5). Thus, initial angles between 115° and 150° were set so that a monotonic change of intensity versus the applied magnetic field from negative to positive was acquired. As shown in figure 5(b), at the maximum applied magnetic field, the intensity value was set close to the saturation (value of 255 in our 8-bit camera) of the image. It was clear that the largest variation range of intensity existed at 115° and the smallest at 150°; in contrast, the sensitivity at the negative maximum applied magnetic field was smallest at 115°, hence there was a need to find a balance between the intensity contrast and sensitivity during the MOI test.

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Taking an initial angle of 125° as an example, a MOI image of 87.5 mT at room temperature is shown in the inset of figure 6(a), in which 16 sub-areas were chosen to show their dynamic responses under the applied magnetic field.

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Different dynamic ranges were caused by the non-uniform illumination. By utilizing the proposed calibration method, all the curves displayed nearly the same behavior, as shown in figure 6(b).

From the calibrated curve that gets rid of non-uniform illumination, the data points are fitted by equation (7) with the least-squared method, resulting in only two parameters, i.e. B_A and cM_s , which are nearly independent optical measure conditions but only related to the properties of the MOI indicator itself, as plotted in figure 7. However, the results of calibration at different temperatures manifest that the values of B_A and cM_s depend on different temperatures. When comparing the values of B_A and cM_s at different temperature, the value of B_A is smaller and cM_s larger at cryogenic temperatures than they are at room temperature, which as shown in figure 8.

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The origin MOI pictures of a CC sample with a width of 2 mm and different transport currents under 10 K self-field are displayed in figure 9(a). Using the proposed calibration approach, the extraction of the magnetic field distributions are drawn in figure 9(b).

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4.2. Current density reconstruction of CC with transport current

After the extraction of the magnetic field data in the measured window of MOI, the current density is reconstructed through the Fourier calculation as mentioned in section 3. Here, the magnetic field data along the red dotted line in figure 9(a) is focused. If the current density is directly inverted by the measured magnetic field as in figure 10(a), the current density result plotted as in figure 10(b) is very similar to the simulated case of 2x = 4a in figure 3(d). Therefore, the magnetic field outside the MOI indicator was extended with the approach presented in section 3, shown with blue complemented data points as in figure 10(c). Then the current density was obtained as in figure 10(d), eliminating a large part of the false current density outside the sample.

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However, there are still some false current densities near the edge of the CC. This is thought to be caused by the in-plane field effect, i.e. the influence of the current-induced magnetic field components parallel to the plane of the MOI indicator. The problem here was solved by the approach proposed by Laviano et al [22], in which the z-component of the magnetic field data inside the sample was iteratively corrected as in figure 11(a). The corresponding corrected current density is plotted in figure 11(b).

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In the calibration process, equation (8) is used to calcalate the magnetic field on the basis of eliminating the non-uniform effect of optical illumination. In comparison with other calibration methods, such as the polynomial-fitting , the proposed approach can realize the conversion from intensity to magnetic field only by providing parameters B_A and cM_s about the physical properties of MOI indicator. The merits of the MOI indicator parameters remain unchanged with changes in optical measure conditions, making the results to be comparable between different users.

In addition, for common MOI tests on superconducting samples, MOI pictures are usually taken at temperatures below T_c , and calibration is done at temperatures above T_c , hence temperature-induced changes on the MOI indicator as well as the induced corresponding errors were always neglected. In section 2.3, we develop a calibration method, which just needs to be conducted without a superconducting sample at cryogenic temperature, then the calibrated parameters of B_A and cM_s can directly be reused for the superconducting case at the corresponding temperature, minimizing the temperature effect on the measured results.

The size of the MOI indicator determines the window size of the measured magnetic field data, in the case of just magnetic field-induced current, and the influence of the measuring window size on current density construction was seldom considered and often neglected. It is physically explained that the magnetic field outside the superconducting sample decays quickly, as shown in figure 12(a), because the directions of current within the two halves of the strip are opposite to each other, resulting in the magnetic fields outside the strip being compensated by the opposite currents. The further away from the strip, the more complete the compensation, thus the reconstruction of the induced current density is less sensitive to the measuring window size, as depicted in figure 12(c). For the case of transport current (where the currents in the strip are the same), the measuring window size effect needs to be considered so that proper construction of the transport current density is attained. Based on the investigations above, a flow chart for mapping the transport current density of CCs is drawn in figure 13.

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In the case of a CC sample with increased applied current, the magnetic field along the width of the CC was plotted as in figure 14(a), showing the elevated magnitude of the magnetic field near the edge of the sample. Using a common method to perform current density inversion with a limited magnetic field window, the false current density outside the sample becomes remarkable when increasing the applied current, as in figure 14(b). By using the protocol presented, a reasonable current density distribution was obtained, as shown in figure 14(c).

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In this paper, we present a protocol for mapping the current density of CCs with a transport current. A calibration method was developed based on the non-linear physical governing function for the MOI indicator, in which only two parameters, B_A and cM_s , are obtained. Both parameters are independent of the measuring conditions, but dependent on the temperature. The MOI measurement with the superconducting sample should use parameter values of B_A and cM_s as close as possible to the corresponding temperature.

Current density inversion for the case of transport current should be implemented with a large window size of the magnetic field. Choosing the long strip as a simple geometry, it was found that little error in inverted current density was obtained with a magnetic field window size 500 times large as the width of the strip. For practical MOI measurements, the magnetic field outside the MOI window was extended by fitting with a power function according to Ampere's law, through which the false current density is automatically suppressed to a large extent. It is believed that this work will advance MOI for a more precise mapping of the transport current density.

All the authors thank Suzhou Advanced Material Research Institute Co., Ltd for the free samples.

Raw experimental data shall be provided for proper requirement.

All data that support the findings of this study are included within the article (and any supplementary files).

The authors declare no conflicts of interest.

This work was supported by the Fund of Natural Science Foundation of China (Nos. 12272155, U2241267, and 12232005) and Gansu Association for Science and Technology of China (No. GXH202220530-18).