Optimized quasi-optical cross-polarization scattering system for the measurement of magnetic turbulence on the DIII-D tokamak

This paper reports on a new design quasi-optical system for cross-polarization scattering (CPS)^1–12 measurements of magnetic turbulence on the DIII-D tokamak. Previous studies^1–12 describe in detail the CPS theory and experimental method, experimental measurements, validity tests, etc., while this paper focusses on a new optimized system. The system presented here employs equipment and techniques not heretofore used in these types of measurements (e.g., internal and independent poloidal and toroidal steering). These improvements provide a new ability to: selectively detect CPS signals from all parts of the probe beam, access an expanded range of detected wavenumbers, and access higher wavenumbers. The design is based upon plasma and quasi-optical simulations along with laboratory tests. The goal of this new design is to demonstrate the full spatial and wavenumber capabilities of the CPS diagnostic.

The CPS technique is based upon the scattering of an incident microwave beam into an orthogonal polarization by magnetic fluctuations $B̃$⁠. The scattered cross-polarized signal provides a measure of the internal magnetic fluctuations that induce the scattering. In the CPS process, the incident electric field E_i accelerates plasma electrons that interact with the magnetic perturbations $B̃$⁠. The resulting induced current $J̃⊥$ is perpendicular to E_i, $J̃⊥∼Ei×B̃$⁠.³ The induced current radiates an electric field E_s with a polarization orthogonal to both E_i and $B̃$⁠. It is this scattered field E_s, proportional to $B̃$⁠, that is detected. The CPS method was first used on Tore-Supra^3,6 and was followed by experiments on GAMMA-10,^7,8 FT-1,⁹ MAST,¹¹ and DIII-D.^10,12 The new design presented here is a significant improvement over that in Refs. 10 and 12 having larger radial and wavenumber ranges as well as significant aiming control. Multiple tests demonstrating the validity of the CPS data as well as the first data from the original CPS system can be found in Refs. 10 and 12.

In the initial design of the CPS technique on DIII-D,¹⁰ a Doppler backscattering (DBS) probe beam was chosen for its ability to improve the spatial localization and to provide simultaneous density turbulence and flow data. First introduced on TUMAN-3M¹³ and W7-AS,¹⁴ DBS is now a widely used technique that measures intermediate wavenumber (typically trapped electron modes to lower-k electron temperature gradient modes), density turbulence ñ, and flows.^13–21 The DBS method consists of injecting a probe beam of a given frequency and polarization into a plasma which contains a cutoff for that frequency and polarization. The radiation is injected at an angle with respect to the cutoff layer (see Fig. 1) and scatters from internal density fluctuations. The beam propagates toward the cutoff and as the cutoff is approached refraction bends the beam upwards. Near the cutoff layer, a long-wavelength electric field pattern is formed roughly parallel to the flux surface (e.g., see Ref. 14). It is this field pattern that interacts strongly with the density fluctuations propagating in the poloidal direction, i.e., the ñ fluctuations just in front of the cutoff layer.

CPS and DBS are scattering techniques and as such have well known physical properties and constraints.²² The most familiar constraint arises from conservation of momentum and energy. These conservation laws lead to a Bragg relation for the scattering process,² viz.,

where k_i and k_s are the incident and scattered wavevectors, respectively, and $kB̃$ is the wavevector of the magnetic fluctuation (similarly for the frequencies ω_i, ω_s, and $ωB̃$⁠). It is worth noting that, in the CPS process, k_i and k_s are of orthogonal polarizations (e.g., X-mode and O-mode) and, due to the different indices of refraction, can have very different wavelengths and propagation properties (e.g., see the vector diagram inset in Fig. 1). Since the Bragg relation is a vector relation, scattering only occurs if the $kB̃$ wavevector is both present (i.e., there needs to be turbulence with that wavevector) and aligned such that it scatters the incident wavevector into the desired scattered wavevector (Fig. 1 illustrates this desired matching condition). Note that the desired scattered wavevector k_s should be in the physical direction and within the antenna pattern of the receiver otherwise little or no signal will be detected. The scattering process and the effect of this wavevector alignment have been examined by multiple authors^11,22 and can be mathematically expressed as

where E_i and E_s are the incident and scattered electric fields, a_θ, a_r are the incident beam electric field half-widths (the beams are assumed to be Gaussian shapes in the directions perpendicular to the axis of propagation), and the directions [θ, r] are perpendicular to the local magnetic field but within the flux surface and perpendicular to both the local magnetic field and the local flux surface, respectively. In this example, the wavenumbers parallel to the magnetic field k_B,||, k_i,||, and k_s,|| are assumed to be negligibly small. From Eq. (3), it can be seen that if the differences in wavenumbers are not small (i.e., if k_i,r + k_B,r − k_s,r and k_i,θ + k_B,θ − k_s,θ are not separately near zero), the resulting scattered field E_s will be substantially reduced, possibly to effectively zero.

In Fig. 1, the DBS probe beam is X-mode and the cross-polarized scattered signal is O-mode. In this illustration, the CPS O-mode cutoff location is assumed to be far to the left in the figure, i.e., well away from the illustrated X-mode cutoff location. The X-mode probe beam is refracted upwards by the plasma, whereas the O-mode scattered beam has little refraction due to the assumed differences in indices of refraction. The wavenumber matching/scattering diagram illustrates the incident X-mode wavevector k_i, the scattered O-mode wavevector k_s, and the magnetic fluctuation scattering wavevector $kB̃$⁠. In this illustration, the O-mode wavevector is much longer than the X-mode consistent with the larger O-mode index of refraction at this location, a common occurrence in lower density DIII-D plasmas.

In the following, Sec. II presents the scientific and engineering requirements, followed by 3-D raytracing calculations simulating cross-polarization scattering from an X-mode probe beam (DBS) to an O-mode scattered beam (CPS) in Sec. III. These simulations were run for a variety of plasmas to optimize the launch–receiver geometries. This is followed by Gaussian beam calculations using realistic optical elements in Sec. IV. These calculations were evaluated for a variety of focal lengths, distances, etc. Section V details how these elements come together into an integrated quasi-optical design that meets the scientific and engineering requirements.

The scientific requirements for the system were (1) poloidal and toroidal angular access of order ±20° and ±10°, respectively; this provides for DBS access to higher density turbulence wavevectors k_θ (i.e., higher poloidal angle) as well as to match the toroidal pitch angle of these wavevectors which is known to be important at higher k;^11,15 (2) independent polarization control of launch and receive signals, an important requirement as both CPS and DBS are polarization dependent; (3) independently aimed DBS and CPS antennas; this allows the CPS system to look at arbitrary locations along the probe beam, thus adjusting both the wavenumber probed as well as the spatial location; and (4) a controlled and optimized beam size within the plasma; the beam size at the reflection position is known to affect the wavenumber and spatial resolutions.²³ Engineering constraints included (1) component setback from the plasma to avoid plasma heating and/or plasma contamination, and possible interaction with fast ions; (2) a reliable method of in vacuo mirror control; and (3) protection from ≥6 MW of 110 and 117 GHz electron cyclotron heating.

The design was arrived at by first examining the angular (poloidal and toroidal) requirements for both CPS and DBS. The radial, vertical, and antenna separations were varied to determine the optimal locations. The 3-D raytracing code GENRAY²⁴ was utilized. The inputs to this code were measured density profiles (from Thomson scattering and profile reflectometry), and magnetic equilibria (from the EFIT code²⁵) with toroidal symmetry are assumed. A variety of plasmas were examined for this optimization including L-mode, H-mode, and QH-modes, and the launch and receive locations were varied for each case. An L-mode discharge from this set of plasmas is displayed for illustration in Figs. 2 and 3. The magnetic equilibrium and cutoffs and resonances and the example of the 3-D raytracing results are shown in Figs. 2 and 3, respectively. The eight DBS cutoff locations are shown in Fig. 2 from which the 70 GHz frequency is selected to display in Fig. 3. In Fig. 3, the red curves are the launched X-mode DBS probe beam (70 GHz) and the blue curves are the scattered O-mode CPS rays. The solid curves are within the plasma, while the dashed lines are outside the last closed flux surface, where GENRAY uses an index of refraction N = 1. The antennas are indicated and the 70 GHz right-hand cutoff location within the plasma is shown. The DBS probe ray is launched upwards [Fig. 3(a)] and is refracted by the plasma, while the scattered O-mode ray is relatively unaffected by the plasma due to its larger index of refraction (i.e., the O-mode cutoff is much interior to that of the X-mode as can be seen in Fig. 2). Figure 3 illustrates two important aspects of this study. The first is the range of poloidal angles needed by the CPS antenna in order to sample the DBS probe beam. Table I shows how the detected magnetic wavenumber and scattering position vary with the receiver angle. Using these simulations, it was determined that the vertically displaced CPS/DBS antennas produced a superior wavenumber range as well as greater spatial control. Note that the toroidal propagation of the probe beam is shown in Fig. 3(b). This is partially due to launching the probe beam with a toroidal angle so as to maximize the scattering efficiency [i.e., maximize relation (3)] but also partly due to the natural tendency of a ray to travel toroidally (with the direction depending upon the ray polarization, O- or X-mode).

The requirements to be met by the quasi-optical design were that it should (1) form a Gaussian profile beam of the desired width within the plasma over the range of frequencies used (i.e., 55–75 GHz), (2) be simple with as few moving parts as possible, (3) propagate arbitrary polarizations without significant polarization scrambling, and (4) fit into the available space. Based upon the previous experience at DIII-D as well as the published literature,¹⁷ a beam radius of ∼3 cm (1/e radius of magnitude electric field) was a reasonable starting point. While multiple focusing elements are possible, single element systems are less prone to alignment or other failure and generally have lower insertion losses. A metal lens²⁶ was selected for its frequency dependent index of refraction which focusses higher frequencies further from the lens and thus deeper into the plasma [other options include dielectric lenses (e.g., quartz), focusing mirrors, either parabolic or ellipsoidal]. This better matches the focus to the reflection point being deeper in the plasma for higher frequencies. Other important features of the metal lens are its resistance to thermal damage, low differential expansion with respect to its metal mounting frame, and being unaffected by thin coatings of the various metals and compounds that internal components often suffer. Figure 4 shows a design drawing of a large metal lens. Although the shape is concave, the lens is a converging lens due to the index of refraction in the small circular metallic waveguides (i.e., the holes). The index of refraction for the metallic lens varies as $n=[1−(λ/λcutoff)2]1/2$⁠, where λ is the vacuum wavelength of the selected frequency, λ_cutoff = 1.706D, and D is the diameter of the holes in the metal lens. For this design, the hole diameter is chosen to be 4.41 mm, resulting in a cut-off frequency of 39.946 GHz. The radius of curvature of the lens is 17.89 cm. This results in focal lengths of 57.25 cm and 116.5 cm for 55 and 75 GHz frequencies, respectively.

A Gaussian beam radius code (based upon a standard formulation²⁷) was used to calculate the beam widths at various probe frequencies and distances from the lens. This was used to optimize the beam sizes for the frequency range 55-75 GHz and the distances of interest within the plasma. Note that this code does not take into account refraction due to the plasma. This will generally result in modest increases in the beam size with distance.²⁸ This code was used to optimize the beam size in the plasma region of interest by varying the focal length and lens to antenna distance. The iteration of this process for selected frequencies covering the frequency range of interest allowed a selection of the best overall lens focal length and lens to antenna distance. Figure 5 shows the result of these optimizations for three representative frequencies in the band of interest (50, 60, and 70 GHz). The metallic lens is located at the circular waveguide aperture (see Sec. V for a presentation of the resulting design). The frequency dependence of the index of refraction results in the different focal lengths for different frequencies, as shown in Fig. 5. The beam radii minima are at ∼20, 27, and 35 cm from the lens surface (for 50, 60, and 70 GHz, respectively). The last closed flux surface (the approximate edge of the plasma) and normalized radial locations ρ = 0.5 and ρ = 0 are indicated. As can be seen, the radial beam widths are of order 3 cm for the radial range extending from the edge to ρ = 0.5. The higher frequency widths (≥70 GHz) remain near 3 cm interior to ρ = 0.5. Also shown in Fig. 5 are measurements of the beam profile widths for the frequencies shown. These measurements were made using a HDPE dielectric lens with a fixed focal length of 57 cm and serve to illustrate beam radius variation with frequency.

The above elements were integrated into the design shown in Fig. 6, which illustrates the major elements. The DBS probe radiation enters the lower waveguide from the left via a quartz vacuum window (tilted 5° to avoid direct back reflections) and propagates down the waveguide to a 45° miter bend and then to a metal lens (forming the correct beam shape and size within the plasma, Sec. IV). The radiation is reflected from the adjustable flat mirror into the desired poloidal and toroidal angles. The corrugated circular waveguide (8.9 cm inner diameter) supports arbitrary polarization making it ideal for this application. The received CPS radiation enters the upper waveguide [Fig. 6(a)], passes through a metal lens and into the circular waveguide, and is then detected. The DBS retro-reflected signal retraces its outward propagation path back into the waveguide where it is also detected. The polarization, both launched and received, is set by large, external wire grid polarizers (not shown) similar to those in Refs. 10 and 12. The components closest to the plasma are recessed approximately 13 cm from the tile surface, thus avoiding interactions with either fast neutral beam particles (calculated to have approximately 3° deviation or maximum 1.8 cm penetration into the port box) or the plasma itself. The mirror movement is effected by the mirror control rod assembly shown in Fig. 6(b). The mirror is mounted on a “U” shaped yoke by a rotating hinge as shown by the arrow in Fig. 6(b). This movement is controlled by a sliding arm connected to the mirror providing one axis adjustment. The second movement axis is provided by a rotation about the long axis of the mirror assembly [arrow in Fig. 6(b)]. The sliding arm rotates with this long axis rotation allowing independent control in both directions. The mirror control design itself was strongly based upon the successful ECH aiming mechanism.²⁹ 

This work was supported by the US Department of Energy under Nos. DE-FG02-08ER54984 and DE-FC02-04ER54698. Discussions with Dr. John Lohr and Dr. Mirela Cengher are gratefully acknowledged.

This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. DIII-D data shown in this paper can be obtained at https://fusion.gat.com/global/D3D_DMP.