# Minimization of the Ohmic Loss of Grooved Polarizer Mirrors in High-Power ECRH Systems

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## Abstract

A set of two corrugated polarizer mirrors is typically used in high-power electron cyclotron resonance heating (ECRH) systems to provide the required polarization of the ECRH output beam. The ohmic losses of these mirrors can significantly exceed the losses of plane mirrors depending on the polarization of the incident beam with respect to the orientation of the grooves. Since polarizer mirrors incorporated into miter bends of a corrugated waveguide line are limited in size, active water cooling can become critical in high-power cw systems like the one for ITER. The ohmic loss of polarizer mirrors has been investigated experimentally at high power. A strategy to minimize the losses for given mirror geometries has been found.

## Keywords

Electron cyclotron resonance heating Polarizer Ohmic loss## 1 Introduction

In modern high-power electron cyclotron resonance heating (ECRH) systems, the power transmission capability may be limited by high losses in some components of the transmission line. In corrugated HE11 waveguide lines, one of the critical elements are polarizer mirrors inserted into miter bends. Typically, a set of two corrugated polarizer mirrors is applied to provide the required polarization (usual elliptical) of the ECRH beam injected into the plasma. For single frequency systems, it usually consists of an elliptical and a linear polarizer with effective groove depths corresponding to ≈λ_{0}/8 and ≈λ_{0}/4 (universal polarizer) with λ_{0} being the free space wavelength [1, 2, 3]. Such a universal polarizer can provide any arbitrary polarization in the output beam of the transmission line. In quasi-optical transmission lines, these polarizers can be made large [4, 5], but polarizers in miter bends of HE11 transmission lines have a limited diameter and thus a very high incident power density. With increasing power and pulse length, the ohmic loss of the polarizer mirrors cannot be neglected anymore. In particular, for systems like in ITER with high-power beams of 1 MW and up to 3600 s pulse length, effective cooling of these mirrors can become a serious issue.

The ohmic loss of corrugated polarizer mirrors has been calculated using a 2D FDTD algorithm [6] and space harmonic calculations [2]. Cold tests with grooved polarizer mirrors made out of copper have been performed using a quasi-optical three-mirror resonator [7]. In [2], it was shown that the loss of such mirrors can vary considerably depending on their setting.

We present here high-power tests on similar polarizer mirrors made out of stainless steel which were inserted into an HE11 miter bend. The measurements were performed using a gyrotron of the ASDEX Upgrade (AUG) ECRH system [8] at 140 GHz. The measurements with the corrugated polarizer mirrors were cross-calibrated with similar measurements using a plane mirror made of the same material. A simple formalism could be revealed which allows the calculation of the ohmic loss of each polarizer mirror for any given incident polarization and rotation angle. In Chapter 4, we show that there are not more than four possible combinations of the angular settings of the two mirrors of a universal polarizer for any wanted output polarization [9]. Using this formalism, the combination of polarizer angles resulting in the lowest possible ohmic attenuation can be found for each required polarization of the output beam.

## 2 Experimental Arrangement

## 3 Test Results

### 3.1 A. Plane Mirror

*φ*= 0° to

*φ*= 180° using the polarizer mirrors in the MOU box. The angle

*φ*is defined as the angle of the incident electric field vector with respect to the incidence plane on the miter bend (Fig. 3).

*φ*= 0° and

*φ*= 90°, we have theoretical results [7] and

*R*

_{S}is the surface resistance, and

*Z*

_{0}is the wave impedance.

*a*

_{⊥}to correspond to

*a*

_{⊥mV }= 573 mV (respectively, 3.438 K). In the following graphs, we normalize the measured ∆T values to

*a*

_{⊥mV }as

*∆E*

_{0}per unit area in the center of the mirror of thickness

*d*as

*p*

_{0}on the center of the mirror surface can be calculated from

*P*= 375 kW is the total injected power, 2

*R*= 87 mm is the diameter of the waveguide, \( \sqrt{2}/2 \) is due to the 45° incidence on the mirror, and

*∆E*

_{0}per unit area absorbed in the center of the mirror during a pulse of

*τ*= 50 ms is then

Together with Eq. (4), and using the data for stainless steel, heat capacity *c* = 0.5 J/(g⋅K), specific mass *ρ* = 7.9 g/cm^{3}, and the value of *∆T* = 3.438 K corresponding to *a* _{⊥}, we can estimate the absorption coefficient as *a* _{⊥} = 0.82 %, which gives for *E*-plane polarization (*φ* = 0°) *a* _{E} = 1.16 % and for *H*-plane polarization (*φ* = 90°) *a* _{H} = 0.58 %. These estimated values agree reasonably well with theory which gives 0.993 and 0.496 % for *E*-plane and *H*-plane incidence, respectively [7, 10]. Note that some increase of the ohmic loss due to surface roughness with respect to theory is expected.

### 3.2 B. Polarizer Mirrors

*z*-axis aligned along the incident

*k*-vector of the mirror under test and the

*x*-axis lying in the incidence plane. The polarization of an incident linearly polarized wave is given by the angle

*φ*of the electric field vector with respect to the

*x*-axis. The other coordinate system, Fig. 5b, relates to the mirror under test with the

*x*2-

*z*2-plane being the mirror surface, the

*x*2-axis oriented perpendicular to the grooves, and the

*z*2-axis along the grooves. The orientation of the grooved mirror is described by the angle

*α*between the grooves and the incidence plane.

*φ*of a linear polarized incident wave, while the orientation

*α*of the grooves of the mirror is constant at either

*α*= 0 or 90°. In Fig. 7, the graphs show a scan of the groove angle

*α*, while the angle

*φ*of the incident linear polarization is constant at

*φ*= 0° or 90°.

There is no absorption coefficient *a* _{ norm } < 1. A prominent feature in all these curves is that they fit to a dependence of cos(2*φ*), respectively, cos(2*α*), suggesting a dependence on the square of the incident field components, which scale as sin*φ* or sin*α*.

### 3.3 C. Special Cases

*ε*, the orientation of the ellipse

*β*with respect to the confining magnetic field, and the sense of rotation of the electric field

*r*. To realize this in a waveguide transmission line, a universal polarizer with two polarizer mirrors of ≈λ/8 and ≈λ/4 groove depth, mounted in 90° miter bends, i.e., with an incidence angle of 45°, is required. For a single frequency application (in this paper 140 GHz), there are up to four possible combinations of their groove angles

*α*for each output polarization. Examples are shown in Figs. 12 and 13. Similar results were obtained for broadband polarizers with a steeper incident angle (17.2°) at 140 GHz with groove depths that corresponded to λ/8 and λ/4 at their design frequency of 122.5 GHz [9]. From the above measurements, we have to expect that these combinations have different losses. In the following, we compare for a few examples for the losses for cases with four possible combinations. To calculate the necessary combinations of mirror angles, we assume the arrangement shown in Fig. 8 where the λ/8-mirror is followed by the λ/4-mirror. All calculations are performed for incident angles of 45°.

The apparently complicated arrangement in Fig. 8 takes into account that in our present setup, we have only one place where we can insert a rotatable polarizer mirror into a miter bend. Therefore, we proceed in the following way: we calculate the settings required for a two-mirror arrangement as in Fig. 8 with a linear polarized incident field. Then, we insert the λ/8 mirror, set it to the calculated orientation *α* _{λ/8}, and determine its normalized loss. Next, we insert the λ/4 mirror, set it to the calculated value of *α* _{λ/4}, feed it with an elliptic input polarization corresponding to the calculated output of the λ/8 mirror, and determine its loss. This elliptic input polarization can be set by the polarizer mirrors in the MOU box. The total loss is the sum of the individual losses.

*φ*= 0° direction (

*E*-plane incidence), and the right figure with an incident field in

*φ*= 90° direction (

*H*-plane incidence). The four possible settings in the first case give absorption coefficients in the range 4.2 to 5.4. In the second case, the absorption coefficients vary between 2.2 and 5. In order to verify that we get the proper output polarization, we also recorded the detector signal obtained at the directional coupler at the end of the transmission line (Fig. 1) which must be constant in all eight cases.

*ε*= 0.418,

*β*= 76.4°, and

*r*= −1, again with a linear input polarization either horizontal or vertical. Here too, we see a variation of the absorption coefficients in the range of 2.9 to 4.7 for the different possible settings.

We see a proper choice among the four possible mirror settings can greatly reduce the heat load on the polarizer mirrors and thus increase the power capability of the transmission line.

## 4 Model Calculation

*φ*) and cos(2

*α*) dependencies of the losses suggest that they can be described as depending on the square of the incident field components. We therefore make an empirical Ansatz:

considering only the tangential fields as defined for the mirror coordinate system shown in Fig. 5. We also make the assumption that a calculation based on infinite plane waves is a good approximation for the center part of the beam.

*E*

_{0}and

*H*

_{0}are the field amplitudes in the waveguide, and the factor \( \sqrt{2}/2 \) is due to the 45° incidence on the mirror.

This relation simplifies for *α* = 0° or 90° and for *φ* = 0° or 90°. Inserting this into Eq. (8) and setting *E* _{0} = *H* _{0} = 1, we get:

*α*= 0° and

*φ*= 0°:

*α*= 0° and

*φ*= 90°:

*α*= 90° and

*φ*= 0°:

*α*= 90° and

*φ*= 90°:

*a*

_{ norm }at these

*α*and

*φ*combinations, we can derive the loss coefficients A, B, C, and D. We have done this for the data of Fig. 6 with the result given in Table 1.

Loss coefficients for the plane and corrugated mirrors under test

Plane mirror | λ/8-mirror | λ/4-mirror | |
---|---|---|---|

A | 0 | 0.60 | 2.83 |

B | 0 | 0.01 | 0.01 |

C | 1.41 | 2.01 | 2.45 |

D | 1.41 | 2.25 | 16 |

*α*and

*φ*dependencies of the losses of the respective mirrors. The results are shown in Figs. 4, 6, and 7 by the lines named model calculation. We see an astonishingly good agreement with this simple model. In Fig. 11, we show once more the calculated results of Figs. 4, 6, and 7 together with the detailed contributions of

*E*

_{ x }

^{2},

*H*

_{ x }

^{2}, and

*H*

_{ z }

^{2}. The contribution of

*E*

_{ z }

^{2}is not shown because it is in all cases negligibly small. This makes sense because

*E*

_{ z }must vanish along the ridges of the grooves and cannot excite a wave inside the grooves.

For the plane mirror, the result is clear: The surface current is proportional to the squared tangential magnetic field, which at *φ* = 0° incidence has the full amplitude H_{0}; while at *φ* = 90°, the tangential x-component has only a magnitude of \( 1/2\cdot \sqrt{2}\cdot \) *H* _{0}. And there is no *E* _{ x } contribution as there is no tangential electric field. The corrugated mirror which behaves somewhat similar to the plane mirror is the λ/8-mirror with *α* = 90°, but the *H* _{ x } and *H* _{ z } contributions are higher than in the plane mirror case and there is a small *E* _{ x } contribution. In the other cases, the *E* _{ x } contribution is considerably higher. All these components can excite a wave propagating in the grooves, thus increasing the losses with respect to the plane mirror.

Our model can also be applied to the special cases of Chapter 3.3 with combinations of λ/8 and λ/4 mirrors. The results are also shown in Figs. 9 and 10 and reproduce the experimental data pretty well. The simple model given in Eq. (8) is thus well suited to calculate the losses of a pair of corrugated mirrors for any setting and for any geometrical arrangement, once the coefficients A, B, C, and D for the individual mirrors are determined. It can be integrated in a polarization matrix code as described in [9].

*α*of the λ/8 and λ/4 mirrors. In Fig. 12, we show, as an example, calculations of the output polarization for an input with horizontal linear polarization and the configuration given in Fig. 8. Shown are 2D plots of ellipticity

*ε*, orientation

*β*, rotation

*r*, and losses

*a*

_{ norm }. These figures are similar to the ones given in [9], but now including losses. The data points in these graphs show the possible settings for a scan of the orientation

*β*from 0 to 180° of the output polarization ellipse with

*ε*= 0.2 and rotation

*r*= +1 (ctr. clockwise) with a horizontal linear input polarization. The dots represent solutions for the required output polarization with a cross-polarization content <0.1 % of the total power.

*ε*, the dots again representing a

*β*-scan. We see that at low ellipticities of

*ε*= 0 and 0.2, there are four solutions for the mirror settings at any value of

*β*, whereas at the ellipticities

*ε*= 0.4 and 0.6, we find for some

*β*-values only two solutions. As an example, the black dots show the solutions for

*β*= 60°.

*β*from 0° to 180°, but for the ellipticities

*ε*= 0 (linear) and

*ε*= 0.2, and both for a linear input polarization either horizontal or vertical.

We see that the four possible settings, as shown in Fig. 14, to realize a wanted output polarization have quite different total loss, both for the linear and the elliptical output case. Lower loss can be obtained for a vertical input polarization (*H*-plane incidence). This result is, however, true only for the polarizer arrangement, Fig. 8, as discussed here.

## 5 Discussion

Our model is not a theoretical one, rather an empirical one. The coefficients A, B, C, and D for each mirror were obtained experimentally. Normalized to perpendicular incidence on a plane mirror they are valid for any material, not just stainless steel as used here. But they will depend on the corrugation parameters like profile shape, depth, and periodicity [7]. Nevertheless, they give a guideline how to choose a setting of the polarizer mirrors which leads to lower losses. A numerical code for polarizers like the one described in [6] should be able to calculate the coefficients A, B, C, and D where the calculation of only four special cases according to Eq. (8) is necessary. Experimentally, the coefficients can also be obtained at low power in a three-mirror-resonator setup as in [7]. In such a setup, only cases with linear polarization equal for input and output, and thus also only cases with parallel or perpendicular orientation of the grooves w.r.t., the E-field vector can be studied. These are just the cases required to determine the four parameters A, B, C, and D.

Applied to a universal polarizer consisting of two corrugated mirrors as done above the result for the total losses will also depend on the specific arrangement: We used here the arrangement in Fig. 8, but other arrangements where the two incidence planes are parallel or perpendicular to each other can also be treated with this model in the same way. One needs only to know the parameters A, B, C, and D for the applied mirrors. In general, for the proper choice of the mirror settings, we need also to consider the changes of the polarization along the downstream transmission line including the launcher mirror. There is no doubt that in the case of a complex but fixed transmission line geometry, there is also an optimum position for the polarizer mirrors which can be found using this model.

## Notes

### Acknowledgments

We would like to thank our colleagues F. Monaco and H. Schütz for their valuable help in setting up and performing the high-power experiments and for proposing to use a PT-100 detector instead of a thermocouple.

Open access funding provided by Max Planck Society.

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