2.1. Model geometry

Figure 1.

The 2-D model circular geometry, which adopts NSTX parameters, consists of a circular core plasma with $n_{e,core}$  = $3\times 10^{19} \,\mathrm{m}^{-3}$ and the outer annular SOL region with $n_{e,SOL}$  = $3\times 10^{18}\,\mathrm{m}^{-3}$ . The radius of the core plasma is 0.9 m, while the annular SOL has a width of 0.1 m, representing a nominal distance from the LCFS and antenna on NSTX. $B_0$  = 0.4 T. In this configuration, the antenna is located at r = 1.0 m on a finite azimuthal arc angle of $\alpha$  = 72 $^\circ$ for $|\theta | \leqslant 36^{\circ }$ on the right-hand-side (or the `low-field’ side) of the model geometry, indicated by the $J_{ant}$ vectors. The wave $B_z = B_\parallel$ profile is shown. In the annular SOL, a small region is introduced at the opposite side of the antenna ( $175^{\circ } \leqslant \theta \leqslant 185^{\circ }$ ) with a high level of conduction loss ( $\sigma$  = 1) as a sink region to prevent wave interferences between the two propagating SOL modes.

Figure 1 presents a 2-D circular model constructed based on NSTX parameters. The plasma is assumed to consist of electrons and a single ion species (deuterium). The geometry is infinite in the axial (z-) direction, and the plane wave variation ( $\exp (-ik_zz)$ ) is assumed in that axial direction. The axial wave wavenumber ( $k_z$ ) is an input parameter. Only the uniform axial background magnetic field, $B_{z}$  = 0.4 T, is considered in this section ( $k_\parallel = k_z$ ), which will be adjusted in §§ 3 and 4. The core plasma has a constant density of $n_{e,core}$  = $3\times 10^{19}\, \mathrm{m}^{-3}$ , while the outer annular SOL density is reduced to $n_{e,SOL}$  = $3\times 10^{18}\, \mathrm{m}^{-3}$ . The SOL density chosen is sufficiently high to allow a propagating wave solution above the R-cutoff density for a wide range of $k_{\parallel }$ in order to isolate the geometric constraint. (The HHFW dispersion is shown in figure 5.) The density magnitude and its step-like profile are a simple adoption of the NSTX 130608 density profile. The annular SOL has a width of 0.1 m, representing a nominal distance from the last closed flux surface (LCFS) and antenna on NSTX. The core plasma radius is 0.9 m. In this geometry, the antenna is located at r = 1.0 m on a finite azimuthal arc from $\theta = -36^{\circ }$ to $\theta = +36^{\circ }$ at the right-hand side (‘low-field side’) of the model, which produces a tangent current of $\boldsymbol{J}_{\!{ant}} = -\sin \theta \, \hat {x}+\cos \theta \, \hat {y}$ . This antenna excites a spectrum of azimuthal modes (m = $k_{\theta } r$ ) at $f_0$  = 30 MHz, whose amplitude dependence follows a sinc-type function, as shown in figure 2. The full width of the main lobe is 4 $\pi /\alpha$ , where $\alpha$ is the antenna arc angle in radians. As pointed out in the annulus resonance study (Perkins et al. Reference Perkins, Hosea, Bertelli, Taylor and Wilson2016), high-m modes can be selectively coupled and amplified by the SOL resonance condition, and can be a major contributor to the wavefield in the SOL. Since the SOL annular width remains the same outside the antenna azimuthal extent, these resonant poloidal modes are also responsible for the waveguide mode with slightly modified boundary conditions because the outer-wall boundary is changed from the antenna boundary to a perfect electric conductor (PEC), as will be discussed below.

Figure 2.

Azimuthal spectrum of a circular-arc-shaped antenna with an angular extent of $\alpha$  = 72 $^\circ$ : $\tilde {J}_\theta (m) = \sin (m\alpha /2)/(m\pi )$ , where m is the azimuthal mode number. In the $m \rightarrow$ 0 limit, $\tilde {J}_\theta = \alpha /(2\pi )$ .

In addition to the HHFW wavefield in the plasma core, figure 1 shows the wavefield structure localised in the annular SOL, propagating azimuthally within that region. In the upper part of the antenna region, the poloidal phase front propagates counterclockwise, while in the lower part it propagates clockwise. The input axial wavenumber is $k_z = k_{\parallel } = 3\,\mathrm{m}^{-1}$ . With the antenna exciting a tangential current, the wavefield excited is dominantly the transverse electric (TE) mode, with a large azimuthal electric field $E_\theta$ and axial magnetic field ( $B_z$ ). In the low density, particularly just above the R-cutoff density, the dominant power-flow direction can be axial (parallel to B), in line with the fast-wave dispersion (Hosea et al. Reference Hosea2008). In the core, the standard fast-wave propagation is visible. The uniform conductive loss is intentionally introduced in the plasma core ( $\sigma$  = 0.02) to minimise the wave interference and cavity effects. This effectively simulates a high single-pass absorption scenario. In the annular SOL, a small region is introduced at the opposite side of the antenna ( $175^{\circ } \leqslant \theta \leqslant 185^{\circ }$ ) with a high level of conduction loss ( $\sigma$  = 1) as a sink region to prevent wave interferences between the two propagating SOL modes.

Figure 3.

Poloidal spectrum of the wave $E_\theta$ along the circumference at r = 0.95 m in the middle of the annular SOL at different input $k_\parallel$ .

2.2. Impact of $k_\parallel$ on SOL waveguide mode

The input $k_z=k_\parallel$ plays an important role in generating the SOL waveguide mode by governing the wave perpendicular wavelength. To identify the corresponding azimuthal mode number of the propagating SOL mode at each $k_\parallel$ , figure 3 shows the poloidal mode spectrum of the theta component of the wave electric field ( ${E}_{\theta }$ ) measured along the circumference at r = 0.95 m, located in the middle of the annular SOL between the core/SOL interface (r = 0.9 m) and the outer-wall boundary (r = 1 m). The input $k_\parallel$ is varied from 2 to 7 $\mathrm{m}^{-1}$ . Both positive and negative azimuthal mode numbers $m = k_\theta r$ are found, consistent with two oppositely propagating structures in figure 1. As $k_\parallel$ increases, however, the dominant azimuthal mode number (|m|) decreases, and no clear mode is observed for $k_{\parallel } \simeq 7\,\text{m}^{-1}$ . The corresponding mode amplitude decreases to zero, indicating no waveguide mode is supported at a high $k_{\parallel }$ .

Figure 4.

Comparison between the peak poloidal mode numbers obtained from numerical modelling ( $m_{peak}$ ) and the analytic estimates ( $m_{estimated}$ ) as a function of $k_{\parallel } = k_z$ .

Figure 4 shows that the numerically observed azimuthal mode numbers agree with the analytical estimates based on the radial half-wavelength condition. The perpendicular wavenumber consists of the radial and azimuthal terms: $k_\perp ^2 = k_\theta ^2 + k_r^2$ . In the analysis, $k_r = \pi /\varDelta$ is imposed as a radial eigenmode constraint. At the same time, the poloidal mode number respects the HHFW dispersion relation, and $m_{estimated}$ is found

(2.1) \begin{equation} m_{est} = r \sqrt {k_\perp ^2-(\pi /\varDelta )^2}, \end{equation}

where $k_\perp = (\omega /c)n_\perp$ can be found from the HHFW dispersion relation

(2.2) \begin{equation} n_\perp ^2 = \frac {(n_{\parallel }^2-R)(n_{\parallel }^2-L)}{S-n_{\parallel }^2}, \end{equation}

where $L = 1 - {\sum}_j ({\omega _{pj}^2}/{(\omega (\omega - \varOmega _{cj}))})$ and $S = 1 - {\sum}_{j} ({\omega _{pj}^2}/{(\omega ^2 - \varOmega _{cj}^2)})$ .

Figure 5.

The HHFW dispersion that shows the wave $k_\perp$ as a function of density for a given $k_\parallel$ at $B$  = 0.4 T and $f_0$  = 30 MHz. Two horizontal lines represent the two perpendicular wavenumbers that provide half-wavelength ( $\lambda _\perp /2$  = 0.1 m) and quarter-wavelength ( $\lambda _\perp /2$  = 0.2 m) fits, respectively, for the SOL gap of 0.1 m at $n_e = 3\times 10^{18}\,\mathrm{m}^{-1}$ . For the case considered here, $k_\parallel = 7$ and 13 $\mathrm{m}^{-1}$ meet the respective conditions.

Figure 5 shows the corresponding $k_\perp$ values for various input wave $k_{\parallel }$ . Note that, in the HHFW regime, the dispersion relation can be approximated to be $\omega \approx k_\perp v_A \sqrt (1+c^2k_{\parallel }^2/\omega _{pi}^2)$ , where $v_A = B/\sqrt {\mu _0n_im_i}$ is the Alfvén speed. Increasing the wave $k_{\parallel }$ decreases $k_\perp$ , and therefore limits the available azimuthal mode numbers since the radial wavenumber is constrained to be $k_r = \pi /\varDelta$ , regardless of $k_{\parallel }$ . As shown in figure 4, the predictions from the dispersion relation agree well with the mode numbers found in the numerical modelling. Above $k_{\parallel } \approx 7\, \mathrm{m}^{-1}$ , no propagating waveguide mode can be supported because the HHFW perpendicular wavelength becomes too long to satisfy the half-wavelength condition. This cutoff mechanism is to be distinguished from the homogeneous fast-wave density R-cutoff condition, for which $n_\perp ^2 \lt 0$ . For the given SOL plasma, the R-cutoff occurs at $k_{\parallel } \gt 17\, \mathrm{m}^{-1}$ , well above the geometric cutoff.

2.3. Analysis of azimuthal mode-number structure

In this subsection, the azimuthal mode-number analysis is continued for $k_\parallel$  = 3 and $7\,\mathrm{m}^{-1}$ . In addition to the Fourier transform analysis, another practical approach to identify the azimuthal mode number responsible for the radially standing-wave structure is to conduct a numerical mode-number analysis by applying the azimuthal phase factor, $\exp (-i m_0 \theta )$ , to the antenna tangential current density

(2.3) \begin{equation} \boldsymbol{J}_{\!{ant}} (m_0) =[-\sin \theta \hat {x}+\cos \theta \hat {y}]\exp (-im_0\theta ), \end{equation}

which will shift the centre of the antenna spectrum, and dominantly excite the $m_0\hbox{}$ th harmonic Fourier mode. Figure 6 shows an example of the resulting wave $E_\theta$ for $\boldsymbol{J}_{\!{ant}} (m_0 = \pm 34)$ at $k_\parallel = 3\,\mathrm{m}^{-1}$ . The antenna power coupled is dominantly trapped in the SOL in both cases. A radially standing wavefield is formed in front of the antenna and propagates azimuthally, clockwise for $m_0=-34$ and counterclockwise for $m_0=+34$ .

Figure 6.

The response of the wave tangential electric field ( $E_\theta$ ) to the antenna current source: $\boldsymbol{j}_\theta =\hat {\theta } \mathrm{exp}(im_0\theta )$ , where $\hat {\theta }=-\mathrm{sin}\theta \,\hat {x}+\mathrm{cos}\theta \,\hat {y}$ , for (a) $m_0$  = –34 and (b) $m_0$  = +34. The antenna azimuthal extent is indicated by a set of antenna current vectors. Here, $k_z$  = $3\,\mathrm{m}^{-1}$ .

At these resonant $m_0$ , the power coupled to the SOL is very high, indicating high antenna loading, while not necessarily meaning good core coupling (Hosea Reference Hosea2017). Here, ‘resonant’ refers to modes whose radial structure satisfies the half-wavelength condition in the SOL, leading to enhanced field amplitude and antenna loading. Figure 7 shows the area-integrated axial Poynting flux $S_z = (1/2) \int (E_x H_y^* - E_y H_x^*){\rm d}A$ in both the core and annular SOL regions as a function of $m_0$ . Here, $S_z$ is used as a measure of power coupling for the given antenna current magnitude, which represents the degree of bare loading at each $m_0$ . The power flowing to the annular SOL (red curve) is maximum at the resonant values $m_0 = \pm 34$ , where it exceeds the nominal $m_0 = 0$ case by an order of magnitude.

Please note that, in the absence of explicit wave power dissipation physics in the present model, the area-integrated axial Poynting flux strictly represents wave power flow and antenna loading, rather than being a direct measure of absorbed power in the plasma, which is a limitation of the cold-plasma treatment employed in this study. For example, for a realistic antenna spectrum including symmetric $\pm k_{\parallel }$ components, this approach would cause the net axial power flow to cancel, even though finite power is coupled to and absorbed by the plasma. In the limit of large conductive loss in the model, Poynting flux would vanish as well. Another possible approach is to consider power absorption from the conductive loss implemented in our model, but the chosen parameter is phenomenological (constant conductive loss in the core and non-uniform conductive loss in the annular SOL (please see § 2.1)). Therefore, throughout the manuscript, the term ‘power coupling’ is used in a loose sense, since it is expected that the power-flow quantity is proportional to the absorbed power under appropriate absorption models. For example, those wave powers present in the SOL can serve as a proxy for SOL parasitic power absorptions in the experiment, as they are expected to drive a range of parasitic wave–edge interactions, such as collisions, sheath rectification (Perkins et al. Reference Perkins2012, Reference Perkins, Hosea, Jaworski, Bell, Bertelli, Kramer, Roquemore, Taylor and Wilson2017b ), filament-assisted mode conversion to slow waves (Tierens et al. Reference Tierens, Bilato, Bertelli, Shiraiwa, Myra and Colas2022) and parametric decay instabilities (Wilson et al. Reference Wilson, Bernabei, Biewer, Diem, Hosea, LeBlanc, Phillips, Ryan and Swain2005).

Figure 7.

Area-integrated axial Poynting flux $S_z = (1/2) \int (E_x H_y^* - E_y H_x^*){\rm d}A$ in both the core and annular SOL regions as a function of $m_0$ in the antenna current source. Here, $k_z$  = $3\,\mathrm{m}^{-1}$ and $B$  = 0.4 T.

Figure 8.

Area integrated axial Poynting flux between the core and annular SOL regions as a function $k_{\parallel }$ , showing the dependence of antenna loading on $k_\parallel$ . The secondary x-axis shows the corresponding $(\lambda _{\perp }/2)/\varDelta$ .

As shown in figure 8, when the half-wavelength condition is satisfied at low $k_\parallel$ , a substantial fraction of the coupled power is transported by the SOL waveguide mode. However, above $k_\parallel \approx 7\,\mathrm{m}^{-1}$ , the power coupled to the plasma core starts to exceed the power coupled to the annular SOL. At $k_\parallel \approx 13\,\mathrm{m}^{-1}$ , close to the quarter-wavelength condition as shown in the HHFW dispersion in figure 5, the total amount of power coupled into the plasma increases by an order of magnitude, reaching its maximum. This resonant dependence of the loading on $k_\parallel$ has implications for future experiments and will be explored in a separate work. Finally, when the fast-wave R-cutoff condition is not met above $k_\parallel = 16\,\mathrm{m}^{-1}$ , the loading is reduced.

Figure 9 shows the corresponding wave $E_\theta$ at different $k_\parallel =$ 3, 7, 13 and 18 $\mathrm{m}^{-1}$ . In the figures, the colour bar range for the wave $E_\theta$ is kept the same for visualisation purposes. At $k_{\parallel }$  = 3 $\mathrm{m}^{-1}$ , the impact of resonant high-m components, $m\approx \pm 34$ , is apparent. Because the magnetic geometry is aligned with the model axial geometry, the azimuthal mode number supported by the SOL plasma $m$ matches the input antenna phase factor $m_0$ . At $k_z$  = 7 $\mathrm{m}^{-1}$ , the propagating SOL mode is no longer observed outside the antenna as the SOL cannot naturally support the radial eigenmode half-wavelength structure. At $k_z$  = 13 $\mathrm{m}^{-1}$ , the azimuthal distribution of the wavefield within the antenna extent is nearly flat ( $m\approx 0$ ). Finally, a high-m surface-wave-like mode is observed at $k_{z} = 18\, \mathrm{m}^{-1}$ , and the overall power coupling to plasma is strongly reduced.

Figure 9.

The $E_\theta$ profile at different $k_z$ : (a) 3 $\mathrm{m}^{-1}$ , (b) 7 $\mathrm{m}^{-1}$ , (c) 13 $\mathrm{m}^{-1}$ and (d) 18 $\mathrm{m}^{-1}$ .

Figure 10.

(a) Fractional SOL power to the total power (SOL + core) and (b) fractional annulus power outside the antenna extent to the total SOL power as a function of $k_\parallel$ .

As shown in figure 10(a), the fractional power in the annular SOL ( $S_{z,SOL}$ ) relative to the total power ( $S_{z,SOL}+S_{z,core}$ ) is more than $50\,\%$ of the total power at a low $k_z \lt 5$ $\mathrm{m}^{-1}$ . Such a large fraction coupled to the SOL is consistent with the experimental trend that power loss is aggravated at low $k_{\phi }$ . Furthermore, the power carried by the waveguide mode outside the antenna region can be non-negligible, as shown in figure 10(b). More than 50 $\%$ of the SOL power is accounted for by the propagating mode. This is due mainly to a longer extent of the annular SOL outside the antenna, as the antenna azimuthal extent spans only 20 $\%$ of the total circumference ( $|\theta |\lt 36^{\circ }$ ) in the model. The SOL mode is continuously supported in the annular SOL until it reaches the artificial sink region at the opposite side of the antenna. In real-world experiments, the magnetic and physical geometries should be taken into account, as discussed in later sections, and the details of these power fractions are expected to vary. Past modelling (Bertelli et al. Reference Bertelli2016) using the NSTX geometry has reported parasitic power losses of up to 40 $\%$ in the SOL, so these power fractions are not considered unreasonable.

As was shown in figure 9(b), the propagating SOL mode is no longer observed outside the antenna at $k_z$  = 7 $\mathrm{m}^{-1}$ . However, a poloidal structure is still observed in front of the antenna, which exhibits a higher mode-number structure than $m \approx 2$ from the analytic mode estimates in figure 4. Compared with the PEC boundary ( $E_\theta =0$ ) at the outer wall for the SOL waveguide mode, the boundary condition is expected to be modified in the presence of the antenna current boundary, and can be generalised to

(2.4) \begin{align} E_\theta + Z\frac {{\rm d}E_\theta }{{\rm d}r}=0. \end{align}

This is as a general form of the boundary condition based on Ampere’s law that gives $H_z \propto {\rm d}E_\theta /{\rm d}r$ , as the tangential antenna current drives the wave $B_z$ from the surface boundary condition: $\boldsymbol{n} \times (\boldsymbol{H}_{plasma}-\boldsymbol{H}_{wall})=J_\theta \hat {\theta }$ . In the PEC boundary, Z = 0, and the radial wavenumber condition in the annular SOL recovers the standard half-wavelength condition. With the antenna boundary, however, the antenna current may impose a specific phase shift to the wavefield, effectively reducing $k_r$ less than $\pi /\varDelta$ , thus availing a finite level of $k_\theta$ to induce a poloidal structure. This driven mode is not naturally supported outside the antenna azimuthal extent, and is absent in this case. If a functional form of $E_\theta (\xi ) \,\sim$ sin( $k_r\xi$ ) is assumed in the annular SOL with $E_\theta (\xi =0) = 0$ and $E_\theta (\xi =\varDelta ) + Z({{\rm d}E_\theta }/{{\rm d}r})(\xi =\varDelta )=0$ , this boundary condition leads to

(2.5) \begin{align} k_r \varDelta +\tan^{-1}(Zk_r)=n\pi , \end{align}

where n is a multiple of integers. Based on the observed half-wavelength fit structure in the annular SOL, $E_{\theta } (\xi =0) = 0$ is imposed at the core–SOL interface. It is analogous to a PEC boundary at the outer wall where the tangential electric field vanishes. Here, $\xi$ denotes the radial coordinate across the SOL region, where $\xi$  = 0 corresponds to the core–SOL interface and $\xi$  = $\varDelta$ corresponds to the outer edge of the SOL. The half-wavelength case, n = 1, becomes

(2.6) \begin{equation} k_r =\frac {\pi - \phi }{\varDelta }, \end{equation}

where $\phi = \tan^{-1}(Zk_r)$ is an antenna-introduced phase-shift factor with Z being an effective impedance ( $\propto E_\theta / H_z$ ) at a surface impedance boundary. For $k_{z}=7\,\mathrm{m}^{-1}$ , the poloidal mode number seen is $m\approx 17$ , which corresponds to $k_r \approx 26.7\, \mathrm{m}^{-1}$ , which is less than the wave $k_\perp = 31.6\,\mathrm{m}^{-1}$ . This gives $Z = \tan(\pi -k_r \varDelta )/k_r = 0.02$ .

Figure 11.

Area-integrated axial Poynting flux $S_z = (1/2) \int (E_x H_y^* - E_y H_x^*){\rm d}A$ in both the core and annular SOL regions as a function of $m_0$ in the antenna current source: $\boldsymbol{j}_\theta =\hat {\theta } \mathrm{exp}(im_0\theta )$ , where $\hat {\theta }=-\mathrm{sin}\theta \hat {x}+\mathrm{cos}\theta \hat {y}$ . Here, $k_z$  = $7\,\mathrm{m}^{-1}$ .

As shown in figure 11, the mode-number analysis is repeated for $k_z$  = 7 $\mathrm{m}^{-1}$ by biasing the antenna spectrum with the azimuthal antenna phase factor $m_0$ . The resonant power coupling at $m_0 \approx 17$ matches the azimuthal structure seen in the wave electric field structure above. Also, the coupling is asymmetric. The resonant poloidal mode number reverses when the background axial B-field direction is reversed, indicating a role of the plasma response to the applied antenna field when the SOL mode is forced to be excited. This behaviour arises from the magnetic-field-sign-dependent off-diagonal term in the plasma dispersion, $D=1-\varSigma _j(\varOmega _{cj}/\omega )/(\omega _{pj}^2/(\omega ^2-\varOmega _{cj}^2)$ . This term selectively couples one sign of the m-component, and its effect increases with a plasma inhomogeneity and an m-number, such that $\propto \pm m{\rm d}D/{\rm d}r$ (Swain et al. Reference Swain, Carter, Wilson, Ryan, Wilgen, Hosea and Rosenberg2003; Tierens et al. Reference Tierens2026); this term is responsible for the up–down (poloidal) asymmetry of the wavefield structure, which can also influence the parallel wavenumber spectrum in the presence of the poloidal magnetic field. This asymmetric resonance coupling is also confirmed by the complex Fourier transform analysis by taking the $E_\theta$ signal azimuthally in the middle of the SOL, as shown in figure 12. The asymmetric azimuthal mode-number structure at m = 17 appears in both the polarisation components of the TE field.

Figure 12.

(a) Azimuthal amplitude spectrum of $E_\pm$ along the circular arc at r = 0.95 m for $k_z = 7$ $\mathrm{m}^{-1}$ within the antenna azimuthal extent.

2.4. Control of SOL waveguide mode

Based on the analysis so far, the propagating SOL mode outside the antenna extent can be controlled in several approaches. In the previous full-wave simulation, large SOL fields were observed once the fast-wave cutoff opened in front of the antenna. The present results show that this condition is necessary but not sufficient, as the SOL waveguide mode occurs only when the annular SOL can support a waveguide eigenmode satisfying the half-wavelength condition. Therefore, approaches to control the SOL mode include maintaining the SOL density below the R-cutoff density, increasing wave $k_\parallel$ , as discussed so far, and increasing B or reducing the SOL width. The latter approach is shown in figure 13(a). When the annular width of the SOL plasma outside the antenna is reduced from 10 to 5 cm at $k_z = 3\,\mathrm{m}^{-1}$ , with all other parameters kept constant, a waveguide mode outside the antenna is no longer supported, while annulus resonance in front of the antenna is still formed locally. Conversely, when the antenna width is reduced, annulus resonance is suppressed while the waveguide mode remains supported, as shown in figure 13(b). Another approach is to increase the background B-field, as shown in figure 14. Here, the B-field is increased from 0.4 to 0.8 T, thereby increasing the HHFW perpendicular wavelength. A high-m mode structure is no longer observed in the annular SOL in front of the antenna, nor outside the antenna azimuthal extent. The wave coupling and propagation characteristics resemble a typical low-harmonic ICRF coupling scenario.

Figure 13.

Control of the propagating SOL waveguide mode (a) by a reduction of the SOL width outside of the antenna extent and (b) by a reduction of the SOL width within the antenna extent from 10 to 5 cm. Here, $k_z$  = 3 $\mathrm{m}^{-1}$ . Other parameters are identical to those in figure 1.

Figure 13. Long description

Two contour plots illustrate the wave Bz in Tesla, showing the impact of reducing the Scrape-Off Layer width on wave propagation. Panel A: The contour plot shows wave Bz with the Scrape-Off Layer width reduced outside the antenna extent. The x-axis ranges from -1 to 1 meters, and the y-axis ranges from -1.2 to 1.2 meters. The color scale ranges from -6.67e-6 to 7.17e-6 Tesla. Panel B: The contour plot shows wave Bz with the Scrape-Off Layer width reduced within the antenna extent. The x-axis ranges from -1 to 1 meters, and the y-axis ranges from -1.2 to 1.2 meters. The color scale ranges from -3.96e-6 to 3.9e-6 Tesla. Both plots indicate changes in wave propagation patterns due to the reduction in Scrape-Off Layer width.