As the plasma beta (*β*) increases in high-performance tokamaks, electromagnetic turbulence becomes more significant, potentially constraining their operational range. To investigate this turbulence, a cross-polarization scattering (CPS) diagnostic system is being developed on the HL-3 tokamak for simultaneous measurements of density and magnetic fluctuations. In this work, a quasi-optical system has been designed and analyzed for the Q-band CPS diagnostic. The system includes a lens group for beam waist size optimization, a rotatable wire-grid polarizer for polarization adjustment, and a reflector group for measurement range regulation and system response enhancement. Laboratory tests demonstrated a beam radius of order 4 cm at the target measurement location (near the plasma pedestal), cross-polarization isolation exceeding 30 dB, and poloidal and toroidal angle adjustment ranges of ±40° and ±15°, respectively. These results verify the system’s feasibility through laboratory evaluations. The quasi-optical system has been installed on the HL-3 tokamak during the 2023 experimental campaign to support the development of CPS diagnostics.

## I. INTRODUCTION

The quasi-optical design for the cross-polarization scattering (CPS) diagnostic^{1} to measure electromagnetic turbulence on the HL-3 tokamak (formerly known as HL-2M) is presented in this paper. Theoretical and experimental results indicate that as the plasma *β* = 2*μ*_{0}*p*/*B*^{2} (*p* is the plasma pressure and *B* is the magnetic field) increases, the intensity of electromagnetic turbulence in the plasma significantly increases, potentially constraining the operational range of high-performance tokamaks.^{1,2} Therefore, to directly measure local electromagnetic fluctuations in plasma, CPS has been developed in lots of magnetic fusion devices, such as Tore-Supra,^{1,3} Tokamak Fusion Test Reactor,^{4,5} GAMMA-10,^{6,7} FT-1,^{8} Mega Ampere Spherical Tokamak(MAST),^{9} MAST Upgrade,^{10} DIII-D,^{11–14} C-2W,^{15} Experimental Advanced Superconducting Tokamak,^{16} and HL-2A.^{17,18}

^{1,14,19}

*ɛ*

_{0},

*ω*

_{pe},

*E*

_{i},

*ω*

_{i},

*ne*,

*σ*, and

*B*are the vacuum permittivity, the local plasma frequency, the electric field of the incident wave, the incident wave frequency, the local plasma density, the unperturbed conductivity tensor, and the local magnetic field, respectively. The first and second terms on the right side of Eq. (1) correspond to the Doppler backscattering (DBS, parallel to

*E*

_{i}, scattered by $n\u0303$) and CPS (orthogonal to

*E*

_{i}, scattered by $B\u0303$) processes, respectively. Notably, there is a clerical error in Refs. 1 and 18. The numerator of the second term should be

*ω*

_{ce}instead of

*ω*

_{i}.

However, there are still some challenges in the application of CPS. First, the intensity of magnetic fluctuations $B\u0303$ is significantly smaller than that of density fluctuations $n\u0303$ ($B\u0303/B2/n\u0303/n2$ is about −20 to −50 dB), which introduces inherent difficulties in signal extraction and interpretation.^{1,5,9,12} Another challenge is the polarization match, which is crucial to ensure DBS signals do not contaminate the CPS signals.^{9,14} In addition, the ray-tracing analysis confirms that strict accordance with the toroidal and poloidal launch angles of the probe microwave is indeed necessary to improve the received signal level.^{18}

Therefore, the quasi-optical system plays a key role in the CPS diagnostic. First, it allows for beam width control, which is essential for improving spatial and wavenumber resolution. Second, it assists in launch angle control, enabling flexible regulation of the measurement wavenumber and position range while also improving the received scattering power and thereby the signal-to-noise ratio. Finally, it enables precise control of polarization, a critical factor for differentiating magnetic from density fluctuations.

Following the introduction section, quasi-optical design, including the Keplerian beam expander lens groups, the ellipsoidal and plane mirror group, and the rotatable polarizer, is described in Sec. II. The results of laboratory tests are presented in Sec. III. Finally, a summary is given in Sec. IV.

## II. QUASI-OPTICAL DESIGN

The general geometry and layout of the quasi-optical components of the system are shown in Fig. 1.

The overall design goal is to form a Gaussian beam at the edge of HL-3 plasmas for Q-band frequency operation, with the ability to control beam width, launch angle, and polarization. The quasi-optical system consists of the orthomode transducer (OMT) based dual polarized antennas, the transmitting lens group, the rotatable polarizer, and the reflector group.

The OMT-based antennas have both vertical and horizontal ports, allowing them to receive or transmit waveforms in varying polarizations. In Fig. 1, antennas 1 and 2 serve as the respective receive and transmit antennas.

### A. Transmitting lens group

As illustrated in Fig. 2, the transmitting lens group is designed for beam width optimization. In terms of wavenumber resolution, $\Delta k\u22a5=22w1+w2k0\rho eff2$, an optimal beam spot radius *w*_{04} exists.^{20} This radius is typically moderate, ranging around 2.5–3.5 cm. Straying from this optimal range could result in less accurate wavenumber measurements. Finally, the beam spot radius *w*_{04} also affects the spectral resolution: $R=\sigma fD=\Delta u\u22a5u\u22a5+\Delta k\u22a5k\u22a5$. It influences Δ*k*_{⊥}, which in turn affects the broadening of the Doppler shift peaks.^{21} Hence, controlling the beam spot radius is crucial for accurate spectral measurements.

^{22}The relationship between the focal length (

*f*), the radius of curvature (

*r*

_{1},

*r*

_{2}), the refractive index (

*n*), and the thickness (

*d*) of the thick lens is given by the Lensmaker’s equation,

^{23}

The refractive index of HDPE, ∼1.52,^{24–26} was estimated by the free space measurement method, which comprises a vector network analyzer (VNA) and two antennae facing each other.^{27–29} The phase of the transmitted signal-beams was recorded under two conditions for a 10 mm thick HDPE sample: (a) sample in place and (b) sample removed. The refractive index *n* was then computed from the observed phase shift using the formula *n* = Δ*ϕλ*_{0}/*d* + 1, where *λ*_{0} is the free-space wavelength, *d* is the thickness of the sample, and Δ*ϕ* is the measured phase shift in radians. Figure 3 presents the results of refractive index estimation within the frequency range of 30–40 GHz. The variations observed in the refractive index are possibly attributed to the multiple-reflection Fabry–Perot phenomenon.^{27}

^{30}

*d*

_{in}and

*d*

_{out}represent the distances from the lens to the input beam waist

*w*

_{in}and the output beam waist

*w*

_{out}, respectively. The focal length of the thin lens is represented by

*f*. In addition, $zR=\pi \omega 02\lambda $ denotes the Rayleigh distance or Rayleigh range, and

*w*

_{0}signifies the beam waist.

*d*

_{1}=

*f*

_{1}). Then get

Therefore, the Keplerian beam expander system enables frequency-independent Gaussian beam transformation. The beam waist at the output, represented by *w*_{03}, is determined by the ratio of the focal lengths of lens 2 to lens 1 and is proportional to the initial beam waist *w*_{01}.

### B. Reflector group

As shown in Fig. 2, the reflector group includes an ellipsoidal mirror and a plane mirror. The reflective nature of the ellipsoidal mirror enables beam direction manipulation, providing convenience for subsequent angle adjustments. In this design, the tilt angle of the ellipsoidal mirror is specifically chosen to ensure a 90° angle between the outgoing and incoming beams.

It is important to note that the beam waist of the microwave emitted by the horn antenna varies with the frequency. Therefore, additional adjustments are necessary when utilizing a Keplerian beam expander system. An effective approach is to incorporate an ellipsoidal mirror, which serves as the equivalent of a converging lens, providing a similar focusing effect as expressed in Eq. (5).

*w*

_{0}=

*k*

_{ant}

*λ*). The half-power beam width (HPBW) of the Gaussian beam’s intensity profile, denoted as HPBW(

*z*) or

*θ*

_{HPBW}, can be expressed as

*k*

_{ant}can be calculated as

*d*

_{in}=

*f*), similar to the concept in Eq. (6), the frequency-dependent characteristics introduced by the antenna can be effectively mitigated,

*d*

_{out}=

*d*

_{in}=

*f*. The beam waist size

*w*

_{04}is then set to the desired value, determined by the required wavenumber resolution. The specific distance

*d*

_{out}is determined by the relative position of the measured location and the ellipsoidal mirror. With these parameters, the radius of curvature

*R*

_{out}can be calculated by Eq. (14). Using Eq. (15), the radius of curvature

*R*

_{in}can then be computed

However, *R*_{out} and *R*_{in} are related to wavelength *λ*. For the center wavelength *λ*_{m} within the frequency range, we can calculate the corresponding values of *R*_{out,m} and *R*_{in,m} to determine the parameters of the ellipsoidal mirror.

Based on the aforementioned constraints and calculations, the parameters of the ellipsoidal mirror are determined to be a major axis (a) of 1272.1 mm, a minor axis (b) of 625.2 mm, and a tilt angle of 37.55°.

The ray-tracing code BORAY^{31} is utilized to estimate the scattering location and wavenumbers of the density and magnetic fluctuations at various launch angles. Results show that the toroidal and poloidal launch angles should cover the range of 5°–25° and 11°–40°, respectively.^{18} In addition, it is important to note that in order to improve the received scattering power and, hence, the signal-to-noise ratio, the toroidal launch angles should align with the poloidal launch angles.^{18,32,33}

Therefore, the plane mirror is utilized to select the wavenumber by adjusting the poloidal launch angle and to reduce mismatch attenuation by modifying the toroidal launch angle. It receives the reflected microwave beam from the ellipsoidal mirror and directs it toward the plasma. It is specifically designed to rotate in two orthogonal directions, enabling simultaneous adjustment of both the toroidal (±15°) and poloidal (±40°) launch angles in order to facilitate the desired adjustments.

The distance between the plane mirror and the ellipsoidal mirror is designed to be adjustable based on the size of the window and the optical aperture, allowing for flexible installation.

### C. Rotatable polarizer

Polarization control is crucial for distinguishing the scattered signals from density fluctuations $(n\u0303)$ and magnetic fluctuations $(B\u0303)$. The relative fluctuation level $B\u0303/B2/n\u0303/n2$ is reported to be ∼−20 to −50 dB.^{1,5,9,12} Hence, a minimum polarization isolation exceeding 20 dB is necessary. The beam is focused between the two lenses of the Keplerian beam expander system, which facilitates enhanced polarization control. Considering the diminished diameter of the required polarizer, the integration of a higher speed rotatable polarizer at the beam’s focal point can significantly improve the temporal precision of polarization control. Therefore, a 150 mm hollow-core motorized rotary stage along with a wire grid polarizer is utilized to achieve the desired polarization control. The wire grid polarizer, composed of tungsten wires with a diameter of 10 *μ*m and a wire spacing of 25 *μ*m, provides excellent polarization isolation.^{17} In addition, the motorized rotary stage allows for precise adjustment of the polarizer angle, as shown in Fig. 1.

As shown in Fig. 4, antenna 1 is oriented with its *H*-plane (*H*_{1}) aligned along the *x*-direction and its *E*-plane (*E*_{1}) aligned along the *y*-direction. Antenna 2 is symmetric to antenna 1 with respect to the polarizer plane, but its polarization deviates from antenna 1 by an angle of *γ* due to the installation error.

The wire grid polarizer is installed at an angle of *β* = 45° between the normal and the optical axes. By rotating the polarizer, the orientation of the beam’s polarization can be adjusted. In Fig. 4, the blue arrow represents the wire direction, and its projection on the *x–y* plane is denoted by the red vector *H*_{in}. An electric field vector parallel and perpendicular to the *H*_{in} is reflected and transmitted, respectively. *E*_{in} represents the transmitted electric field vector, which is perpendicular to *H*_{in}. It is important to note that the polarizer angle, denoted as *θ*_{grid}, should not be confused with the polarization angle, denoted as *θ*. The relationship between the two angles is described by the equation tan(90 − *θ*) = cos(*β*)tan(90 − *θ*_{grid}).

*B*

_{0}represents the direction of the magnetic field lines at the edge of the plasma, with the direction perpendicular to

*B*

_{0}being the pure X-mode electric field direction. The angle

*α*represents the orientation of the magnetic field

*B*

_{0}with respect to the

*x*-axis, and

*δ*represents the mismatch angle between the X-mode and

*E*

_{in}. It is important to note that

*α*=

*θ*+

*δ*,

The microwave beam emitted from antenna 1 undergoes polarization selection as it passes through the polarizer. Before entering the plasma, the incident microwave beam can be decomposed into two orthogonal modes (O mode and X mode) based on the relationship between its electric field and the background magnetic field. Inside the plasma, both the O-mode and X-mode waves interact with turbulents, leading to DBS and CPS scattering simultaneously. The scattering efficiencies for these four processes are denoted as polarization scattering matrix *S* = [*S*_{oo}, *S*_{ox}; *S*_{xx}, *S*_{xo}].^{34} The scattered microwaves propagate through the plasma and return to the polarizer. A portion of the scattered waves transmits through the polarizer and re-enters antenna 1, while another portion is reflected by the polarizer and enters antenna 2.

*θ*=

*α*, the rotation matrix

*R*(

*α*−

*θ*) becomes the identity matrix

*I*, effectively decoupling the scattering matrix. Considering the reflection and transmission efficiency (

*η*) of the polarizer, for high polarization isolation, we have

*η*

_{t,‖}→ 1,

*η*

_{t,⊥}→ 0 and

*η*

_{r,⊥}→ 1,

*η*

_{r,‖}→ 0. In addition, by rotating antenna 1 to

*α*= 0, we have

*R*(

*α*) =

*I*. Similarly, rotating antenna 2 to

*γ*= 0, results in

*R*(

*γ*) =

*I*. With these considerations,

*S*

_{xo}and

*S*

_{oo}could be distinguished,

## III. LABORATORY TESTS

A full-scale quasi-optical system has been implemented for laboratory testing. A series of tests were conducted to verify the design performance before the final installation of HL-3.

### A. Test platform

As illustrated in Fig. 5(a), a three-dimensional displacement platform was utilized to facilitate precise control over the movement of the receiving antenna in the *x, y,* and *z* directions. The typical parameters of the receiving antenna include a 20 dBi gain and a 3 dB beamwidth of 10°. In the measurement setup, the *x*-axis represents the horizontal direction, the *y*-axis represents the vertical direction, and the *z*-axis corresponds to the optical axis direction.

The received microwave signal (with power *P*_{in}) from the movable antenna was transmitted into the radio frequency (RF) port of the mixer through a 4 m coaxial line, then mixed with the local oscillator (LO) signal and down converted to an intermediate frequency (IF) signal. The LO chain is configured as an active ×4 frequency multiplier with a frequency tunable synthesizer to provide sufficient drive power and cover the entire Q band. A logarithmic detector with a dynamic range exceeding 50 dB is employed to measure the power of the IF signal, providing a low frequency (DC) output (*V*_{out}) that is logarithmically (“linear in dB”) related to *P*_{in} level.^{35}

Then calibration was performed, and the results are depicted in Fig. 5(b). The power entering the receive antenna (*P*_{in}) was systematically increased from −60 to 0 dBm in 1 dBm increments for three different frequencies: 34 GHz (black), 40 GHz (blue), and 48 GHz (red). In addition, the voltage output of the logarithmic detector (*V*_{out}) is plotted as a function of the power entering the receive antenna (*P*_{in}) in Fig. 5(b). As one can see, the calibration results indicate that the power response of the test platform can be categorized into three regions: linear response region (−35–0 dBm), nonlinear response region (−50 to −35 dBm), and saturation region (<−50 dBm).

### B. Beam radius

Gaussian beam radius measurements were conducted for three representative frequencies in the band of interest (34, 40, and 48 GHz). The receiving antenna was scanned in the *x–y*, *y–z*, and *x–z* planes to measure the power at various distances from the quasi-optical system.

The output voltage matrix *V*_{out}(*x*, *y*, *z*) obtained from the measurements was interpolated using the calibration results shown in Fig. 5(b) to derive the power distribution [*P*_{in}(*x*, *y*, *z*)] in units of dBm. Then, it was converted to power distribution in units of mW using the formula $PmW=10(PdBm/10)$. For each *z* value, representing the distance from the quasi-optical system, the horizontal and vertical power distributions [*P*_{in}(*x*) and *P*_{in}(*y*)] were fitted using a Gaussian function to determine the half-power beam width [HPBW_{H}(*z*) and HPBW_{V}(*z*)]. The beam radius [*w(z)*] was then calculated using Eq. (8). Note that the input power for the transmitting antenna was set to 10 dBm to ensure that the testing was conducted within the linear response region of the test platform.

The representative spatial power distributions in the vertical *y–z* plane are depicted in Figs. 6(a), 6(c), and 6(e), providing insight into the beam patterns at 34, 40, and 48 GHz. Furthermore, the beam profiles at three different *z* positions (*z* = 30, 35, and 40 cm) are displayed in Figs. 6(b), 6(d), and 6(f), demonstrating a close resemblance to a Gaussian distribution. These profiles offer a detailed view of the beam intensity distribution at different distances from the plane mirror. For instance, at a distance of 30 cm (black lines) from the plane mirror, the peak power for the 34 GHz beam is ∼0.05 mW (−13 dBm), with a gradual decrease as the distance increases. The spatial power distributions in the horizontal *x–z* plane are similar to those in the *y–z* plane. Note that the power at the Gaussian beam width is reduced by 8.68 dB relative to the peak power, as 86.5% of the total energy is contained within the Gaussian beam width.^{36}

The calculated Gaussian beam width *w(z)* is shown in Fig. 7. The beam widths for the 34 GHz (black square) and 40 GHz (blue triangle) emission frequencies exhibit similarity in the *z* = 20–40 cm range, but the 40 GHz beam width is narrower for *z* > 40 cm. Moreover, the beam width for 48 GHz (red circle) is lower than the other two frequencies across the entire measurement range. Within the target measurement area in the plasma (*z* = 30–40 cm, corresponding to *R* = 2.35–2.45 m, covering from the plasma edge to the pedestal top), the radial beam widths are of order 4 cm. In addition, the fluctuation in beam width with a change in distance might be attributed to multipath effects, specifically the constructive or destructive interference between signals from direct and reflected paths.^{37} This interference might affect power distribution and potentially influence the calculated beam width.

In summary, the quasi-optical microwave system achieves a beam width of 3–4 cm. However, there is still frequency dependence in the beam width, which suggests further optimization in future studies.

### C. Polarization isolation

The polarization isolation of the rotating polarizer was tested using the above test platform, with the receiving antenna fixed at a distance of *z* = 20 cm from the wire grid polarizer. During the test, the wire grid polarizer was rotated at a constant speed of 200° per second, and the corresponding changes in received power were measured.

The solid lines in Fig. 8 illustrate the test results of polarization isolation. Figures 8(a) and 8(b) show the normalized outputs from the receiving antenna for transmission and reflection, respectively. These plots illustrate how the outputs change with the angle of the wire grid polarizer, *θ*_{grid}, at a launching frequency of 34 GHz.

The electromagnetic wave launched from the transmitting antenna is set to be vertically polarized. In Fig. 8, the black and red lines correspond to the vertically and horizontally polarized outputs of the receiving antenna, respectively. The maximum transmission of the polarizer occurs when there is alignment between the polarization of the receiving antenna and the rotatable polarizer, as indicated by the maximum vertically polarized outputs at *θ*_{grid} = 0*°* or 180*°* in Fig. 8(a).

Any misalignment between the polarization of the receiving antenna and the polarizer results in polarization loss. At *θ*_{grid} = 90*°*, the polarization of the receiving antenna and the rotatable polarizer are perpendicular, resulting in the minimum transmission [Fig. 8(a)] and maximum reflection [Fig. 8(b)]. The minimum transmission is 39 dB lower than the maximum, while the minimum reflection is 35 dB lower than the maximum. As a result, the polarization isolations for the transmission and reflection of the wire grid polarizer are 39 and 35 dB, respectively. Furthermore, the difference in the vertically and horizontally polarized outputs at *θ*_{grid} = 180*°* for transmission and at *θ*_{grid} = 90*°* for reflection indicates a polarization isolation exceeding 40 dB for the receiving antenna.

In particular, the shape differences of the black curves in Fig. 8(a) (narrow) and (b) (wide) can be attributed to the differences between the polarizer angle *θ*_{grid} and the polarization angle *θ*.

*P*∝

*E*

^{2}) and the polarization angle is described by Eqs. (20) and (21), as illustrated by the dash lines in Fig. 8. The measured results align well with the calculated values across a wide range of angles,

*γ*

_{1}and

*γ*

_{2}represent the installation error angles between the transmitting and receiving antennas for transmission and reflection measurements, respectively.

The polarization isolations of two additional frequencies, 40 and 48 GHz, were also tested. The results indicate that the polarization isolations for the transmission and reflection of the wire grid polarizer are 35 and 30 dB at 40 GHz and 34 and 30.25 dB at 48 GHz, respectively.

In summary, the rotating wire grid polarizer achieves polarization isolation exceeding 30 dB throughout the Q-band frequency range, meeting our requirements.

## IV. SUMMARY

In this work, the quasi-optical system of the CPS diagnosis, including the Keplerian beam expander lens group, the ellipsoidal and plane mirror groups, and the rotatable polarizer, has been designed for the HL-3 tokamak. The Keplerian beam expander lens group enables frequency-independent Gaussian beam transformation, while the ellipsoidal mirror compensates for the variation of the beam waist in the antenna with respect to frequency. The quasi-optical microwave system achieves a beam width of order 4 cm within the target measurement area in the plasma. Based on the previous simulation, the toroidal and poloidal launch angles are designed to have an adjustment range of ±15° and ±40°, respectively. The rotating wire grid polarizer achieves polarization isolation exceeding 30 dB throughout the Q-band frequency range, while the polarization isolation of the OMT antenna is higher than 40 dB. In summary, the designed quasi-optical system meets the requirements for simultaneous measurements of density and magnetic fluctuations. At present, the CPS diagnostic with this quasi-optical system has been installed on the HL-3 tokamak during the 2023 experimental campaign. Further tests and experimental results of CPS will be shown in the near future.

## ACKNOWLEDGMENTS

This work was supported by the National Natural Science Foundation of China (Grant Nos. 12105087, 12175113, and 12275096), the Joint Funds of the National Natural Science Foundation of China (Grant No. U21A20440), and the Sichuan Key R&D Project (Grant No. 2023YFG0139).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Y. Zhou**: Conceptualization (equal); Data curation (equal); Investigation (equal); Software (equal); Visualization (equal); Writing – original draft (equal). **R. H. Tong**: Writing – review & editing (lead). **W. L. Zhong**: Funding acquisition (equal); Project administration (equal); Supervision (equal); Writing – review & editing (equal). **Y. Tan**: Funding acquisition (equal); Supervision (equal). **M. Jiang**: Supervision (equal). **Z. B. Shi**: Writing – review & editing (equal). **Z. C. Yang**: Methodology (equal); Software (equal); Validation (equal). **Y. Q. Shen**: Methodology (equal); Software (equal); Validation (equal). **J. Wen**: Methodology (equal); Validation (equal). **A. S. Liang**: Methodology (equal); Validation (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## REFERENCES

*Fusion Plasma Diagnostics with mm-Waves: An Introduction*

*Dynamics in Microwave Chemistry*

*Quasioptical Systems: Gaussian Beam Quasioptical Propogation and Applications*