Edge scanning reflectometry for density profile measurement on the SPARC tokamak

The SPARC tokamak is designed as a high-field (B₀ = 12.2 T), compact (R₀ = 1.85 m, a = 0.57 m), superconducting D-T tokamak.¹ A full set of diagnostics is required to support its first experimental campaign.² Edge scanning reflectometry (“ESRL” in the SPARC term) is one of the diagnostics to fulfill the Tier 1 requirements for the SPARC operation. In particular, ESRL aims to measure the edge density profiles, which are necessary for determining the plasma outer gap and calculating the pedestal pressure.

Reflectometry has been widely implemented to measure electron density profile in fusion devices.³ Frequency modulation continuous wave (FMCW), a standard and mature radar technique, is commonly chosen for profile reflectometry. In FMCW, the group delay of the reflected signal is converted to the beat frequency F_b. After measuring F_b for a range of millimeter-wave frequencies, the density profile can be obtained by inverting the theoretically known dependence of the cutoff frequencies vs the electron density and the magnetic field in the wave path.

SPARC is in a unique regime for profile reflectometry because of its high density ⟨n_e⟩ ∼ 3 × 10²⁰ m⁻³ and high B field. Although the right-hand X-mode cutoff (RHC) is preferable for profile reflectometry in most tokamaks, it would be approaching 300 GHz even at the edge of SPARC and thus prohibitive in terms of technology and cost. As a result, we have chosen the other two modes: O-mode cutoff and the left-hand X-mode cutoff (LHC). To cover the large density range from the scrape-off layer to the top of the pedestal, ESRL needs to have multiple millimeter wave bands. In the latest design, K, Ka, U, and E bands will be used.

The ESRL design is in development and the current status is reported. This paper concentrates on the physics requirement and design choices (Sec. II), back-end electronics (Sec. III), and optimization of the horn antenna geometry via COMSOL Multiphysics^® modeling (Sec. IV).⁴ Detailed engineering analysis on horn antennas, windows, and waveguides is discussed in another paper of this conference.⁵ 

Figure 1 shows the cutoff frequencies for a typical SPARC plasma. The left-hand cutoff f_LHC ranges from 0 to 100 GHz, the O-mode cutoff f_OC from 0 to 220 GHz, and the right hand-cutoff f_RHC from 280 GHz and above.

Most profile reflectometry systems use f_RHC and f_OC. One of the advantages of f_RHC is that at n_e = 0, f_RHC is non-zero, and thus, it can determine the cutoff location independently and give profile inversion with a clear starting point. Neither f_OC nor f_LHC has this advantage, and the profile below the lowest frequency of the system has to be obtained from other diagnostics or from some assumptions. However, the magnetic field is above 10 T on SPARC even near the plasma edge. The high field renders f_RHC to be >280 GHz, which would be too costly for an active mm-wave system. If using O-mode alone, f_OC would also be above 140 GHz at the pedestal top due to its slow density dependence.

Because of the SPARC n_e/B value near the edge, f_LHC is at the appropriate frequency range for practical usage. It should be noted that f_LHC is much lower than the upper-hybrid frequency but above the lower-hybrid frequency so there is no absorption layer preventing the probing wave from reaching the cutoff. Although f_LHC was not a designated cutoff frequency, reflectometers on ASDEX-Upgrade and NSTX-U have reported to have detected signals reflected from this cutoff under certain plasma conditions.^6,7 Moreover, density profiles were reconstructed using the signal. Therefore, using f_LHC as the designated cutoff for profile reflectometry will provide SPARC with a practical reflectometry diagnostic.

To measure plasma density from the SOL to the top of the pedestal, we have chosen the frequency range of 18–90 GHz, covering mm-wave frequency bands, K (18–26 GHz), Ka (26–40 GHz), U (40–60 GHz), and E (60–90 GHz). If necessary, F-band could be added later. With this choice of bands, the entire density coverage is shown in Fig. 2. For each band, both O-mode and X-mode polarization will be used. The K-band O-mode is necessary in order to have a more accurate group delay to the lowest density limit. O-mode cutoffs can cover densities from 0.03 × 10²⁰ to 1 × 10²⁰ m⁻³. LHC X-mode can cover from ∼0.5 × 10²⁰ to ∼4.0 × 10²⁰ m⁻³. The density profiles obtained from the O-mode measurement can be used as the starting points for the X-mode measurement. The O-mode cutoff density for the E-band partially overlaps the LHC X-mode cutoff density for the K-band. For 8 T operation, the density coverage is 0.03 × 10²⁰ m⁻³ to 3.0 × 10²⁰ m³, with more overlapping O-mode and X-mode. This overlapping is not only beneficial for profile inversion but also for cross-checking for potential system errors and/or calibration errors.

The 4-band O–X mode system is adequate to provide density information to help determine the outer gap. It is also able to reach the top of the pedestal region for typical H-mode plasmas. If necessary, it can provide some useful information for densities below the ICRF fast-wave evanescent density (∼10¹⁹ m⁻³). In addition, ESRL can provide real-time density information to be implemented to assist plasma operation.

Although the O-mode and X-mode cutoff locations are always separated in the plasma for the same wave frequency, the difference in their beat frequencies is often too small to be separable in the electronics or in the data should they share the same transmitting and receiving wave paths. As a result, O-mode and X-mode from the same band need to be separated either physically in space or temporally in time. The port space available is not enough for eight separate pairs of horns. We have chosen to have four pairs of horn antennas by combining two adjacent bands for the same modes, i.e., K/Ka O-mode, K/Ka X-mode, U/E O-mode, and U/E X-mode.

As in most FWCW reflectometry systems, the probing millimeter wave signals are generated from a common frequency sweep source (e.g., a voltage-controlled oscillator) plus frequency multiplication chains. Heterodyne IF signals are added onto the carriers so that the phase information can be detected at the IF frequencies. The challenges for ESRL are the following: cost, availability, intermodulation signals, cross-talking between bands, and the method of bands combination.

We have followed the design philosophy of the 100–140 GHz profile reflectometer previously installed on Alcator C-Mod^8,9 and expanded it to a multiple-band system. A simplified diagram of the latest back-end electronics is shown in Fig. 3, including the common sources for all the bands and the receiving circuit for the E-band O-mode. The circuits for other bands are similar in design. Because we plan to share multiple bands in the same path, it is critical to have separate IF frequencies (and not harmonics to each other). In the current scheme, three phase-locked DROs (dielectric resonance oscillators) are used to mix with the VCO signal to generate these IF frequencies.

A VCO in the frequency range of 4.3–7.6 GHz provides the frequency sweep for all the bands. A DRO at 3.5 GHz up-converts the signal to 7.8–11.1 GHz, followed by a frequency doubler to 15.6–22.2 GHz. Two phase-locked DROs are mixed with the primary signal to create two pairs of signals 8.95–15.55 GHz and 8.75–15.5 GHz. Heterodyne frequencies are synthesized from different combinations of DRO-1, DRO-2, DRO-3, and frequency multipliers. The IF frequencies for each band are 700 MHz (K), 450 MHz (Ka), 1400 MHz (U), and E-band = 900 MHz (E). Intermodulation frequencies from the mixing process and frequency multiplications have been carefully analyzed. With custom band-pass filters, clean signals can be generated from each band.

In the receiving arm, the reference signal not affected by the plasma is produced by mixing the VCO signal x2 and the sampled signal from a directional coupler before frequency multiplication. In the example shown in Fig. 3, the reference IF signal at 150 MHz for E-band is then multiplied by 6 to 900 MHz, matching the frequency of the heterodyne signal returned from the plasma. Multiplying the frequency in the IF stage would have lower cost than sampling after the frequency multiplier.

Another reference signal is generated from the VCO passing delay lines and then frequency-multiplied. For E-band, it is multiplied to 30–45 GHz and sent to a sub-harmonic mixer. The 30–45 GHz signal is then mixed with the millimeter wave signal back from the plasma. If sub-harmonic mixers do not meet the performance requirement, a fundamental mixer will be used. For all other bands, we plan to use fundamental mixers. The resulting 900 MHz signal has the group delay embedded and carries the information of plasma density.

The harmonic suppression, LO leakage, conversion losses of mixers, and other important parameters will be considered in the next step of electronic design.

The phase difference of the two 900 MHz IF signals is then measured by using a quadrature phase detector (I/Q) and sent to fast digitizers.

Equation (6) shows the relation between the resolution of the beat-frequency Δf_b, range resolution ΔL, full-band width f_max − f_min, and full-band sweep time τ_sw. In plasma reflectometry, the resolution is dictated by the density gradient (the WKB limit), the turbulence, and the microwave source quality. Reducing the sweep time does not increase resolution, but it does help with turbulence. In the current design, technologically feasible full-band sweep time τ_sw = 5 µs is chosen. To minimize the plasma fluctuations, we plan to average over N_sw = 20 sweeps to reach a temporal resolution of Δt = 0.1 ms per profile.

The bandwidth of the beat-frequency is proportional to the variation of the range that the system is designed to measure. Here, f_max − f_min ≈ 30 GHz for E band, if the delay line can be adjusted to be within 0.25 m in comparison with the plasma path, the baseline variation of Δf_b would be >10 MHz. Therefore, digitizer sampling rate >20 MHz is necessary, and higher sampling rate is preferred to cover different plasmas.

The method to combine and split multiple bands is still being explored. For the U and E bands, most likely we will choose the quasi-optical approach similar to the one on JET.^10,11 For K and Ka bands, we may choose wideband power combiner/splitter or quasi-optical boxes.

The distance from the port to ESRL in the diagnostic lab is ∼20 m. This is similar to the case for the Alcator C-Mod reflectometry,^8,9 where WR-90 tall guides were successfully used for 100–150 GHz. WR-90 tall-guide would be good enough for power budget. Other waveguide options are being explored. More detailed analysis is presented in Ref. 5.

A COMSOL Multiphysics^®,⁴ RF model in 2D has been setup to calculate the electric field with the plasma and to optimize the arrangement of the horn antennas. The plasma and horn antennas are modeled as a two-port network. The COMSOL RF model calculates the electromagnetic field at a given discrete frequency and the S-parameters of the network, in particular, the magnitude and phase of S₂₁ for coupling and phase change of the received signal, respectively. Horns are modeled as copper material, and the boundary of the computation box uses the perfect-matched-layer to minimize the artifact reflections for outgoing waves. Non-uniform triangular meshes are automatically created using the default COMSOL setup.

Simulations in the poloidal cross section and toroidal cross section have been carried out separately. The poloidal simulations are used to optimize the horn sizes, the tilting angles, the distance between the plasma, and the horn opening. The toroidal simulations are to optimize the distance to the side-plate and the minimum separation for the O-mode and X-mode horns.

The plasma is treated as a special material with particular dielectric properties. The index of refraction is calculated from the density profiles and magnetic fields using the O-mode or X-mode dispersion equations. In poloidal cross section simulation, plasma is modeled as an ellipse in 2D using the typical SPARC shape, and in toroidal cross section simulation, plasma is shaped as a circle. The horn parameters are optimized via frequency scan and plasma scan. The optimizing criterion is mainly from |S₂₁| for coupling analysis.

In Fig. 4, the electric field contour (in log scale) of a poloidal cross section simulation is shown for an optimized case of 36 GHz. The horn on the left is the transmitting horn and on the right is the receiving horn. The distance from the horn openings to the plasma edge is (vertical axis) 18.5 cm; the horn opening width 6.4 cm, height 5.1 cm, and length 11 cm; and the tilting angle of the horn is 7°. The computation domain width is 50 times the wavelength of 26 GHz. The maximum mesh size is 10⁻³ m and minimum size is 3 × 10⁻⁵ m. The PML thickness is two times the wavelength of 26 GHz. The line of y = 0 is the lower border of the computation domain with PML extended to Y < 0. The curved line is the plasma/vacuum boundary, and waves penetrate the plasma edge and reflected at the cutoff density. After optimization of multiple parameter scans, we are able to achieve |S₂₁| ∼ −5 dB (in the range of −8 to −4 dB for the entire K/Ka band range 18–40 GHz). In Fig. 5, the case for 70 GHz is shown. The domain width is 80 times the wavelength of 40 GHz, and the PML thickness is two times. In this setup, the distance from the horn openings to the plasma edge is 18.5 cm; the horn opening width 4.3 cm, height 3.5 cm, and length 12 cm; and the tilting angle of the horn is 6°, |S₂₁| ∼ −5 dB (in the range of −7 to −3 dB for the entire U/E band range 40–90 GHz).

In Fig. 6, the electric field contour (in log scale) of a toroidal cross section simulation is shown for a case of 40 GHz (using the smaller horns of the U + E bands). In this setup, the minimum distance from the horn opening to the port side-wall appears to be 3 cm, and the horns should not be farther behind the sidewall cutout by more than 2 cm. The cross coupling from O-mode horns to X-mode horns are also assessed, and the minimal distance between the O-mode horn pairs and X-mode horn pairs are shown to be no less than 10 cm.

The 2D setup is insufficient to accurately calculate the true coupling efficiency. This is being helped by Ansys Electronics Desktop simulation (without plasma).⁵ The actual antenna coupling and group delay sensitivity will also be measured in a bench test.

RF analysis is one of the required analyses. Detailed engineering analyses have been carried out for the horn antennas and other in-vessel components to ensure survivability under expected mechanical, magnetic, and thermal loads and a tritium environment.⁵ 

Combining the O-mode cutoff and left-hand X-mode cutoff, FMCW reflectometry on SPARC (18–90 GHz) can measure density profiles from SOL to the top of the H-mode pedestal. Preliminary design review of this system has been completed.

This work was supported by Commonwealth Fusion Systems at MIT via RPP031.

The authors have no conflicts to disclose.

Y. Lin: Conceptualization (equal); Formal analysis (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). V. Nikolaeva: Conceptualization (equal); Formal analysis (equal); Project administration (equal). D. Hachmeister: Formal analysis (equal); Writing – review & editing (equal). E. Kowalski: Formal analysis (equal); Writing – review & editing (equal). M. L. Reinke: Project administration (equal).

Data available on request from the authors.