**Karlsruher Forschungsberichte aus dem Institut für Hochleistungsimpuls- und Mikrowellentechnik** **Band 18**

**Tobias Ruess**

**18**

**A First 2 MW 170/204 GHz Multi-Frequency Gyrotron Pre-Prototype**

T. Ruess **A First 2 MW-Class (136)/170/204 GHz Multi-Frequency Gyrotron Pre-Prototype for DEMO: Design, Construction and Key Components Verification**

Tobias Ruess

## **A First 2 MW-Class (136)/170/204 GHz Multi-Frequency Gyrotron Pre-Prototype for DEMO: Design, Construction and Key Components Verification**

## Karlsruher Forschungsberichte aus dem Institut für Hochleistungsimpuls- und Mikrowellentechnik

*Herausgeber: Prof. Dr.-Ing. John Jelonnek*

**Band 18**

# **A First 2 MW-Class (136)/170/204 GHz Multi-Frequency Gyrotron Pre-Prototype for DEMO: Design, Construction and Key Components Verification**

by Tobias Ruess

Karlsruher Institut für Technologie Institut für Hochleistungsimpuls- und Mikrowellentechnik

A First 2 MW-Class (136)/170/204 GHz Multi-Frequency Gyrotron Pre-Prototype for DEMO: Design, Construction and Key Components Verification

Zur Erlangung des akademischen Grades eines Doktors der Ingenieurswissenschaften von der KIT-Fakultät für Elektrotechnik und Informationstechnik des Karlsruher Instituts für Technologie (KIT) genehmigte Dissertation

von Tobias Ruess, M.Sc.

Tag der mündlichen Prüfung: 30. Mai 2023 Referent: Prof. Dr.-Ing. John Jelonnek Korreferent: Assoc. Prof. Dr.-Ing. Konstantinos Avramidis

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ISSN 2192-2764 ISBN 978-3-7315-1309-4 DOI 10.5445/KSP/1000160391

# **Foreword of the Editor**

Nuclear fusion using magnetically conned plasmas is a promising longterm option for production of energy in future. The DEMOnstration power plant (DEMO) will be ITER's successor. It will show the way towards an industry-driven and technology-driven nuclear fusion power plant. A key criterion for DEMO is the production of electricity. It is expected to become operational in the middle of this century.

Electron Cyclotron Resonance Heating (EC) systems are the major plasma heating systems for today's plasma experiments like Wendelstein 7-X and ITER. Even more, future power plants, like DEMO, might rely on EC systems completely. EC systems are used for a variety of tasks, namely auxiliary heating, current drive, plasma stabilization and plasma diagnostics.

Gyrotrons are the only known microwave sources that can deliver an output power in the megawatt range together with a total eciency of around 50 % at operating frequencies that range from below 100 GHz to above 200 GHz. Gyrotrons are used in today's nuclear fusion experiments such as ASDEX-Upgrade, Wendelstein 7-X and will be used in ITER. Those machines use or require gyrotrons with an output power of 1 MW in continuous wave (CW) operation at frequencies of 140 GHz and 170 GHz, respectively. However, gyrotrons with 2 MW output power, operating frequencies up to 240 GHz and eciencies of at least 60 % are required for DEMO. For maximum exibility and multi-purpose operation, they shall be capable of operating at multiple center frequencies. In parallel, also frequency step-tuning in a frequency band of ± 10 GHz around each center frequency shall be possible. The allowable thermal loading in the interaction region and the increased mode competition towards higher frequencies are the main challenges in the gyrotron design to meet those requirements. Automatically, that leads to highly oversized gyrotron cavities with very high-order operating modes. Therefore, KIT is doing research on the coaxial-cavity gyrotron technology, where an insert passes through the interaction cavity. The benet of the coaxial-cavity is a reduced spectral mode density and a lower voltage depression compared to the hollow-cavity technology. This allows the gyrotron to operate with higher order modes and higher nominal operating parameters. Therefore, a more stable operation with 2 MW output power is achieved.

In his work, Dr.-Ing. Tobias Ruess has designed and constructed a rst 2 MW multi-frequency / multi-purpose 170/204 MW coaxial-cavity pre-prototype. The design of this gyrotron contains dierent requirements, including the fact that the multi-frequency gyrotron should be based on the already existing 2 MW 170 GHz single-frequency coaxial-cavity pre-prototype. The main focus was the optimization of dedicated gyrotron key components such as the magnetron injection gun, the coaxial cavity and the quasi-optical output coupler. The assembled pre-prototype gyrotron is the rst to be operated above 200 GHz in Europe in a new 10.5 T super-conducting magnet. In parallel, the ability of frequency step-tuning is investigated and demonstrated for the rst time in simulations using the coaxial-cavity technology. It has been identied that the classical mode selection strategy is not capable, due to violation of the insert loading constraint. A new mode series is identied, which is universally applicable at dierent frequencies. Frequency and mode measurement diagnostics are optimized to enable measurements above 200 GHz, improve measurement quality and reduce measurement time. Special remark is given by the automated mode generator for the verication of quasi-optical output couplers. The new measurement setup reduces the measurement time from weeks/months to days, while providing higher measurement accuracies.

The results of this research and development are major steps towards a rst DEMO-relevant gyrotron. We wish Dr.-Ing. Tobias Ruess much of further success in his future academic and professional career. We are sure that his future activities will benet from his excellent abilities, his knowledge gained and his tools derived within these studies.

# **Kurzfassung**

DEMO soll das erste Fusions-Demonstrationskraftwerk sein, das die Erzeugung von Elektrizität durch den Einsatz des Tokamak-Konzepts demonstriert. Die Reaktionen nden in einem magnetisch eingeschlossenen Plasma zwischen Deuterium und Tritium bei einer erforderlichen Temperatur von etwa 120 Millionen Kelvin statt. Eine mögliche Heizmethode ist die Elektron-Zyklotron-Resonanzheizung (ECRH). Durch den nicht-induktiven Strombetrieb mittels ECRH können potentielle Plasmainstabilitäten kontrolliert werden. Als potentielle Quellen sind bisher nur Hochleistungs-Vakuumröhren (Gyrotrons) bekannt. Sie arbeiten mit Ausgangsleistungen im MW-Bereich und bei Frequenzen im Mikrowellen- bis sub-THz-Bereich. Die für die vorliegende Arbeit relevanten Anforderungen an das europäische (EU) DEMO sind Erweiterungen der bereits für ITER gestellten Anforderungen. Diese beinhalten eine Erhöhung der Ausgangsleistung pro Einheit von 1 MW auf 2 MW, den Betrieb bei mehreren Frequenzen (136/170/204 GHz) statt nur bei 170 GHz und eine stufenweise Durchstimmbarkeit der Frequenz in einer Bandbreite von ±10 GHz in Schritten von 2 − 3 GHz.

In dieser Arbeit soll das erste 2 MW 170/204 GHz Kurzpuls-Gyrotron in koaxialer Geometrie entworfen und auf der Basis eines existierenden 2 MW 170 GHz Kurzpuls-Gyrotrons gebaut werden. Der Elektronenstrahlerzeuger, der koaxiale Resonator und das quasi-optische Auskoppelsystem sollen an das Magnetfeldprol eines neuen 10.5 T supraleitenden Magneten und für den Betrieb bei 170 GHz und 204 GHz adaptiert werden. Die TE40,23- Mode wird als geeignete Betriebsmode für 204 GHz ausgewählt, während die TE34,19-Mode für den Betrieb bei 170 GHz verwendet wird. Die Optimierung des koaxialen Resonators wird mittels eines adaptierten systematischen Verfahrens durchgeführt. Der neu entworfene koaxiale Resonator erreicht die geforderte Ausgangsleistung von ≥ 2 MW bei 170 GHz und 204 GHz. Außerdem wird ein neues quasi-optisches Auskoppelsystem für 170 GHz und 204 GHz vorgestellt, das den Anforderungen an ein Gauß'sches Prol

für den Ausgangsstrahl von besser als 95 % erfüllt. Abschließend zeigen Simulationen der genannten Schlüsselkomponenten einen prinzipiellen Betrieb bei 136 GHz, wobei die DEMO-Anforderungen erfüllt sind.

Die stufenweise Durchstimmbarkeit der Frequenz von koaxialen Resonatoren wird systematisch untersucht. Bei der Auswahl der Moden wird erstmals die Belastung an der Innenwand berücksichtigt. Es wird eine neue Modenserie vorgestellt, die einen Sprung im azimutalen Modenindex von der TE𝑚,𝑝 Betriebsmode zur TE𝑚+2,𝑝−1-Mode enthält. Die vorgeschlagene Modenserie reduziert die Abweichung des relativen Kaustikradius im Vergleich zur bestehenden Modenserien um den Faktor zwei. Simulationen zeigen, dass die Betriebsfrequenzen schrittweise von hohen zu niedrigen Frequenzen und umgekehrt abgestimmt werden können. Dabei variiert die Ausgangsleistung um maximal 5 % bei der Mittenfrequenz von 170 GHz und 18 % bei 204 GHz.

Die bestehenden Messeinrichtungen am KIT werden für den Betrieb von Gyrotrons über 200 GHz und für die Validierung von Komponenten optimiert. Die maximale Betriebsfrequenz der Systeme wird von 175 GHz auf 260 GHz erhöht. Insbesondere der für die Verikation des quasi-optischen Auskoppelsystems notwendige quasi-optische Modenerzeuger wird optimiert. Mechanisch verstellbare Einheiten werden durch hochpräzise computergesteuerte Lineartische und ein Goniometer ersetzt. Eine autonome Messroutine mit einem Algorithmus zur Modenbestimmung wird implementiert. Diese Optimierungen verkürzen die benötige Zeit zur Anregung der gewünschten Gyrotronmode von Monaten auf wenige Tage. Die beiden relevanten Betriebsmoden bei 170 GHz und 204 GHz speisen das neue 170/204 GHz quasi-optische Auskoppelsystem zur Verikation in kalten Messungen. Die Messungen zeigen eine gute Übereinstimmung mit den Simulationen. Die TE40,23-Mode hat den weltweit höchsten Eigenwert einer jemals in kalten Messungen angeregten Mode.

Schließlich werden neue Modenserie untersucht, bei denen eine erhöhte Wandbelastung von 2.5 kW/cm2 durch eine optimierte Kühlung der Resonatorwand akzeptiert wird. Diese Modenserien protieren von einer Reduzierung der Abweichung des relativen Kaustikradius von bisher ca. 5 % auf nun 0.3 %.

# **Abstract**

DEMO is planned to be the rst DEMOnstration fusion power plant that will proof the production of electricity by facilitating the Tokamak concept. The fusion reaction takes place in a magnetically conned plasma that consists of Deuterium and Tritium. A temperature of up to 120 million Kelvin is required for the reaction. Electron Cyclotron Resonance Heating and Current Drive (ECRH&CD) is a possible method to heat and to control the plasma. The sources are high-power vacuum gyrotrons that produce microwave power at MW-levels in the microwave and sub-THz range. The European (EU) DEMO requirements associated with this work are adapted from those of ITER, from 1 MW to 2 MW output power per unit, from 170 GHz single-frequency operation to 136/170/204 GHz multi-frequency / multi-purpose operation and the ability to tune the frequency within a bandwidth of ±10 GHz around the center frequency in steps of 2 − 3 GHz.

In this work, the rst 2 MW 170/204 GHz dual-frequency coaxial-cavity shortpulse pre-prototype has been designed and built on the basis of an existing 2 MW 170 GHz coaxial-cavity short-pulse pre-prototype. The Magnetron Injection Gun, the coaxial cavity and the quasi-optical output coupler are adapted to the magnetic eld prole of a new 10.5 T SC magnet and to the operation at 204 GHz. Existing components, such as the cathode shall be used. The TE40,23 mode is identied as the operating mode at 204 GHz. The mode ts to the parameters that are already determined by the TE34,19 mode at 170 GHz. A new coaxial cavity is designed using an adapted systematic procedure. This design provides an output power of 2.1 MW at 204 GHz, which is an increase of about 20 %. The design allows an output power up to 2.6 MW at 170 GHz. In addition, a new 170/204 GHz dual-frequency quasi-optical output coupler has been developed providing a Gaussian mode content of > 95 %. Finally, simulations at 136 GHz show a principle operation satisfying the DEMO requirements.

Frequency step-tunability around the given center frequencies has been investigated for the rst time considering the insert loading constraint during the mode selection process. A novel mode series is presented that includes a jump in the azimuthal mode index from the nominal TE𝑚,𝑝 mode to the TE𝑚+2,𝑝−1 mode. The proposed mode series reduces the deviation of the relative caustic radii by a factor of two compared to classical approaches. Simulations are performed to tune the gyrotron in steps from high to low frequency and reverse. The deviation of the output power is in maximum 5 % at 170 GHz and 18 % at 204 GHz.

An update of the frequency measurement system for gyrotron operation at KIT is performed to verify gyrotrons and their key components operating above 200 GHz. They are upgraded from the current frequency limitation of 175 GHz to at least 260 GHz. Particularly, the quasi-optical mode generator is optimized. It is used to validate the quasi-optical output coupler. The mechanically controlled linear drivers are replaced by high-precision computer-controlled linear drivers. A goniometer has been added for a full electronically adjustment. A novel automated measurement procedure with mode evaluation algorithms is implemented. The optimizations reduce the time needed to excite the desired gyrotron mode with higher quality from months to a few days. The excited cavity modes operating at 170 GHz and 204 GHz feed the 170/204 GHz dual-frequency quasi-optical output coupler for verication in cold measurement. A comparison shows an excellent agreement with simulations. The TE40,23 mode at 204 GHz is the mode with the highest eigenvalue ever excited in cold measurements.

Finally, a study on alternative mode series is performed considering an extended wall loading constraint of up to 2.5 kW/cm2 . These mode series benet from low deviation of the relative caustic radius. One identied mode series has a beam radius and an insert radius, which is almost identical to the existing mode series. The result is that the most complicated and costly components to manufacture (insert and MIG) can be reused and do not need to be modied. A fast verication of the proposed mode series using several existing components is proposed.

# **Contents**





# **Abbreviations and Symbols**

# **Abbreviations**



# **Constants**


# **Latin variables**




# **Greek variables**


# **1 Introduction**

Nine billion people are expected to live on Earth by 2050 [1]. That are at least 1 billion people more than today. As the population grows, so does the world's demand for energy. In addition, currently common energy sources such as gas, oil and coal are to be replaced by renewable ones. Nuclear ssion is currently being phased out in Germany. However, solar and wind energy are dependent on changing environmental conditions and may not be reliable base energies [2]. An alternative promising approach is to generate energy in a controlled nuclear fusion reaction using a magnetically conned plasma. Recently, the role of nuclear fusion in the industry is increasing providing several start-ups with billion-dollar budgets emerging around the world, like Tokamak Energy Ltd., UK, Commonwealth Fusion Systems, US and TAE Technologies, US. Amongst them are German start-ups as for example Gauss Fusion.

## **1.1 Fundamentals of Fusion and EC RH&CD**

Fusion reactions are running in the sun by fusing atomic nuclei. The most eective fusion reaction under terrestrial conditions takes place with Deuterium ( 2 1 D ) and Tritium ( 3 1 T ) :

$$\,^2\_1\mathrm{D} + \,^3\_1\mathrm{T} \to \,^4\_2\mathrm{He} + n \quad . \tag{1.1}$$

The resulting helium atom is lighter than Deuterium and Tritium together. Thus, according to Einstein's postulation [3], energy 𝐸 is generated by the dierence of the electron rest mass ∆𝑚<sup>𝑒</sup> :

$$E = \Delta m\_{\text{e}} \cdot \text{c}^2 \quad . \tag{1.2}$$

The fusion reaction gains 17.6 MeV, divided into 14.1 MeV of kinetic energy of the neutron 𝑛 and 3.5 MeV of the helium atom. The base material Deuterium can be extracted from water, while Tritium must be gained from specic breeding blankets by a reaction of neutrons with lithium [4].

$$\,\_{1}^{6}\text{Li} + n \rightarrow \mathfrak{a}\_{\text{rad}} + \,\_{1}^{2}\text{T} + \text{4.78 MeV} \tag{1.3}$$

The Lawson criterion species a minimum value for a triple product that the reaction proceeds by itself [5]. This triple product consists of the temperature, the particle density and the energy connement time. A temperature of ≈ 120 − 150 Mio. K and a pressure of about several bars are necessary for the reaction between Deuterium and Tritium on earth. This corresponds to a kinetic energy of ≈ 10 − 13 keV, which is required to overcome the Coulomb barrier. A powerful heating system is needed to achieve such high plasma temperatures. Known heating methods are:


The ECRH describes a mechanism in which the microwave energy is transferred to electrons in the form of thermal kinetic energy. It is based on the cyclotron interaction. ECRH has several advantages compared to the other heating methods. An example is the low-loss microwave transmission via waveguides [10] or via quasi-optical mirror lines[11]. This allows the sources to be located far from the plasma vessel. In addition, the launching mirrors or antennas might be steered by small steps in frequency. Resulting in a power absorption at dierent locations in the plasma. This eect can eliminate plasma instabilities [12]. Additionally, the high operating frequencies lead to smaller port sizes.

Today, MW-class gyrotrons are available for the required high microwave power for ECRH. Gyrotrons are vacuum electron tubes producing microwave output powers of up to 2 MW [13] at frequencies from below 100 GHz up to 300 GHz. In addition to ECRH, gyrotrons can be used for plasma startup, non-inductive Current Drive (CD) [14], Collective Thomson Scattering (CTS) [15] and plasma stability control [16].

Today, the largest ECRH system is installed at the stellarator Wendelstein 7-X (W7-X) in Greifswald, Germany. A total of ten 1 MW CW (1800 s) gyrotrons operate at 140 GHz [17, 18]. The tokamak project ITER in Cadarache, France, is currently under construction. ITER will be equipped with a total of 24 gyrotrons, each providing 1 MW output power at 170 GHz in CW (3600 s) [19]. Other fusion experiments using gyrotrons are WEST, France [20], ASDEX, Germany [21], DTT, Italy [22], TCV, Switzerland [23], DIII-D, US [24], KSTAR, South Korea [25], JT-60SA, Japan [26], J-TEXT, China [27] and T-15 MD, Russia [28].

## **1.2 Requirements for Future Fusion Gyrotrons**

The DEMOnstration power plant (DEMO) shall be the rst fusion power plant the demonstrates the generation of electricity. It is predicted that European (EU) DEMO will generate approximately 2 GW of thermal power [29] and 750 MW of electrical power [30]. A second major objective of EU DEMO is to demonstrate the reliability of a CW fusion power plant and its components for commercial Future Fusion Power plants (FFP). A complete list of all gyrotron requirements for the EU DEMO is described in [31]. A brief summary is given here for completeness. The gyrotrons used in recent fusion experiments such as W7-X and ITER operate at a single frequency such as 140 GHz or 170 GHz. However, rst experimental fusion test facilities have already been equipped with gyrotrons, which are able to operate at multiple frequencies [21]. These multi-frequency gyrotrons can be used for multiple purposes, such as start-up, bulk heating, and noninductive current drive. This advantage is also intended for the EU DEMO. It shall be equipped with multi-frequency/multi-purpose gyrotrons operating at 136 GHz, 170 GHz, 204 GHz, and 238 GHz. In addition, a bandwidth of ±10 GHz shall be covered around each center frequency in steps of 2−3 GHz. Gyrotron frequency tuning can be used for remote steering with a xed launcher or mirror for plasma stability control. Additionally, the microwave output power per gyrotron shall be increased from 1 MW (W7-X, ITER) to 2 MW in CW operation. Increasing the microwave output power per

gyrotron unit results in fewer gyrotrons and associated infrastructure, such as superconducting (SC) magnets. Hence, reducing the number of units reduces the overall cost and the space required. A third major gyrotron requirement is the eciency of the tube. The total gyrotron eciency of > 60 % requires a beam-wave interaction eciency in the cavity of > 35 %.

## **1.3 State of the Art**

State of the art fusion gyrotrons produce 1 MW microwave output power at 140 GHz and 170 GHz in long pulse operation at W7-X and ITER. Those gyrotrons are manufactured by Thales Group, France [32], Gycom Ltd., Russia [33], Canon Electron Tubes & Devices Co., Ltd., Japan [34] and Communication and Power Industries (CPI), US [35]. In [36], short-pulse experiments of a hollow-cavity gyrotron providing 1.5 MW output power at 140 GHz are presented. A CW gyrotron based on the short-pulse design has been fabricated and is currently being validated in experiments. At Karlsruhe Institute of Technology (KIT), the world record in microwave output power of 2.2 MW at 170 GHz in short-pulse operation was achieved with a coaxial-cavity gyrotron [13]. The gyrotron has been upgraded to operate with longer-pulses up to 50 ms [37] and is currently further upgraded for even longer pulses up to 1 s. In parallel, Russian researchers are currently working on injection locking [38, 39] to increase the microwave output power in hollow-cavity gyrotrons to 1.5 − 2 MW.

Besides the intention to increase the output power per tube, the multifrequency operation of gyrotrons is a major part in research and development (R&D). Several fusion experiments are already employing multi-frequency gyrotrons, e.g. ASDEX Upgrade is equipped with eight gyrotrons operating at either 105 GHz or 140 GHz. At JT-60SA the gyrotrons are able to operate at 82 GHz, 110 GHz or 138 GHz and at TCV at 84 GHz or 126 GHz. In addition, theoretical designs and rst experiments up to 2 s have been published for multi-frequency operation at 104 GHz, 136 GHz, 170 GHz, 203 GHz using the hollow-cavity technology [40]. A theoretical design operating at 170 GHz, 204 GHz and 238 GHz using a coaxial cavity is published in [41]. In addition, the frequency step-tunability is part of the research, too. First experiments for stepwise tuning of a hollow-cavity and coaxial-cavity gyrotron at KIT has been successfully performed over a large frequency range in [42, 43] and [44], respectively.

Fusion gyrotrons can also be used for plasma diagnostics using CTS, where an exact frequency is required. Therefore, frequency locking is used to stabilize the frequency as discussed in [45, 46].

A third major topic in gyrotron R&D is the increase of the total eciency. It is dened by the ratio from the output power to the DC power applied to the gyrotron. The total eciency can be separated into the interaction eciency and the collector eciency. The interaction eciency can be increased by a highly optimized cavity design. In [36], an interaction eciency of up to ≈ 40 % has been achieved. The total eciency is typically increased by a Single-Stage Depressed Collector which can result in a total eciency of around 50 %. At KIT, a Multi-Stage Depressed Collector is currently being prepared for initial experiments [47]. It shall allow a total gyrotron eciency of > 60 % in future.

# **1.4 Scope of the Work**

The scope of the work is to build a rst 2 MW multi-frequency / multipurpose coaxial-cavity pre-prototype. The operating frequencies should be 136 GHz, 170 GHz and 204 GHz. Particularly, the focus is on gyrotron operation at 170 GHz and 204 GHz. The design of the multi-frequency preprototype gyrotron bases on the existing 2 MW 170 GHz single-frequency coaxial-cavity pre-prototype, which is designed to operate in the existing SC magnet. A magnetic eld strength of ≈ 8.3 T that is required for operating at 204 GHz cannot be provided by the existing SC magnet. Therefore, a new 10.5 T SC magnet is ordered. The magnetic prole of this new SC magnet is considered in this work. The upgrade of the gyrotron shall considers minimum modications from the existing tube.

The main key components, namely the coaxial cavity, the Magnetron Injection Gun (MIG) and the quasi-optical output coupler, are investigated. An optimized design has to be provided for the pre-prototype that operates at multiple-frequencies. The cathodes of the existing MIGs shall be reused because it is the primary cost driver. All MIGs available at KIT are investigated if the electron beam requirements are satised at 170 GHz and 204 GHz using the new magnetic eld prole. In addition to the ability to operate at multiple frequencies, also frequency step-tunability in a bandwidth of ±10 GHz and in steps of 2 − 3 GHz is investigated at the given center frequencies.

The pre-prototype design presented in this work will be the rst gyrotron at KIT operating above 200 GHz. Three dierent frequency and mode measurement systems are available at KIT. They are used either for the verication of gyrotron components or for recording their behavior during operation. These measurement test stands are the quasi-optical mode generator used for the verication of the quasi-optical output coupler, the gyrotron output window measurement test arrangement and the frequency diagnostic system. All of these have been upgraded from the current limitation of 175 GHz up to a minimum of 260 GHz. A major modication is planned for the upgrade of the quasi-optical mode generator test facility. It was previously adjusted by hand considering usually months of work for adjustment. Electronic linear drivers and evaluation algorithms shall be installed and implemented to reduce the adjustment time drastically.

# **2 Fundamentals and Challenges**

In the rst part of this chapter, the working principle of gyrotron oscillators and their key components are discussed. In the second part, the challenges for a DEMO-relevant design are presented. Three subjects are noted, (i) the wall loading constraints of the cavity and the coaxial insert, (ii) the mode competition, and (iii) the ability to operate the gyrotron at multiple frequencies.

## **2.1 Gyrotron Operation Principle**

Gyrotron oscillators are vacuum tubes that produce MW-level output power at microwave and sub-THz frequencies. A schematic of a coaxial-cavity gyrotron is shown in Fig. 2.1. A hollow electron beam is emitted by the emitter of the MIG and is accelerated by a DC voltage of up to 90 kV. The electrons are guided from the MIG through a beam tunnel to the interaction area (cavity) by an axial magnetic eld. By slowly increasing the magnetic ux in the beam tunnel the guiding center radius of the electrons is compressed and the axial kinetic energy is transferred to transverse kinetic energy. This follows from the Bush theorem considering conservation of energy [48]. The electrons gyrate around the magnetic eld lines on helical trajectories [49]. The interaction bases on a weakly relativistic electron beam and a transverse electric TE𝑚,𝑝 mode. 𝑚 describes the azimuthal and 𝑝 the radial mode index. The maximum magnetic eld strength is related to the operating frequency of that TE𝑚,𝑝 mode. The generated electromagnetic (EM) eld is separated after the interaction from the electron beam using a quasi-optical output coupler. It transforms the high-order rotating TE𝑚,𝑝 mode into a Gaussian beam. This beam can be further collimated and coupled out radially. Finally, the energy of the spent electron beam is dissipated in the collector.

The interaction mechanism in gyrotrons is called electron-cyclotron interaction and is discussed in detail in [51, 52]. The energy exchange takes place

Fig. 2.1: Schematic of the KIT 2 MW 170 GHz short-pulse coaxial-cavity pre-prototype [50].

close to the angular electron cyclotron frequency Ω<sup>c</sup> of the electrons, which can be derived from the Lorentz force equation. The Lorentz factor 𝛾 has to be considered in the equation due to the high electron energy.

$$\frac{\Omega\_{\rm c}}{2\pi} = f\_{\rm c} = \frac{1}{2\pi} \frac{q\_{\rm e} B\_{\rm cav}}{m\_{\rm e} \chi} \approx \frac{28 \cdot B\_{\rm cav}}{\chi} \tag{2.1}$$

The unit of Ω<sup>c</sup> is GHz, if 𝐵cav is given in T. 𝑓<sup>c</sup> describes the electron cyclotron frequency. 𝐵cav is the maximum magnetic eld strength at the center of the midsection of the cavity, 𝑚<sup>e</sup> the electron rest mass and 𝑞<sup>𝑒</sup> the electron charge. The relativistic Lorentz factor is described by,

$$\gamma = \frac{1}{\sqrt{1 - (\upsilon/c\_0)^2}} = 1 + \frac{W\_{\text{kin}}}{m\_\text{e}c\_0^2} \approx 1 + \frac{W\_{\text{kin}}}{511\,\text{[keV]}}\tag{2.2}$$

with the kinetic energy of the electron 𝑊kin, the velocity 𝑣 of the electron and 𝑐<sup>0</sup> the speed of light in vacuum. The distance between the trajectories and the electron guiding center radius is described by the Larmor radius 𝑟L.

$$r\_{\rm L} = \frac{\gamma m\_{\rm e} \upsilon\_{\perp}}{q\_{\rm e} B\_{\rm cav}} \tag{2.3}$$

𝑣<sup>⟂</sup> is the transverse component of the velocity.

The electrons are accelerated or decelerated depending on their relative phase position by the presence of a synchronous EM eld in the interaction region. An example for both cases (accelerating and decelerating phases) is presented in Fig. 2.2. In the accelerating phase (top graph of Fig. 2.2) the kinetic energy increases, which leads to an increase of 𝛾 according to Eq. 2.2. Therefore, the electron cyclotron frequency is decreased according to Eq. 2.1. Those electrons move towards a favorable phase position [52]. In the decelerating phase, the electrons transfer a part of the kinetic energy to the EM eld and become lighter. Thus, the electron cyclotron frequency Ω𝑐 increases. If the electron cyclotron frequency is chosen to be slightly below the angular frequency of the EM wave, the dierence between the frequencies (𝜔 − Ω<sup>c</sup> ) decreases. Therefore, the electrons stay longer in the deceleration phase. As a result, the electrons in the decelerating eld accumulate at the decelerating phase. Finally, net energy is transferred coherently to the EM wave. The accumulation of the electrons is called bunching [53]. The electrons can recover energy from the EM eld region, which is called over-bunching [54], if the interaction region is too long.

Through the movement of the electrons along the EM eld, the frequency is shifted by a Doppler term 𝑘∥𝑣<sup>∥</sup> [54]. The resonance condition is dened by

$$
\omega - k\_{\parallel} \upsilon\_{\parallel} \ge \mathbf{s} \cdot \mathbf{Q}\_{\mathbf{c}} \,. \tag{2.4}
$$

Fig. 2.2: Principle of electron cyclotron interaction mechanism in a gyrotron cavity.

The wavenumber 𝑘 is given by

$$k = \frac{2\pi}{\lambda} = \frac{2\pi f}{c\_0} \,, \tag{2.5}$$

$$k^2 = k\_{\parallel}^2 + k\_{\perp}^2.\tag{2.6}$$

𝑘<sup>∥</sup> and 𝑘<sup>⟂</sup> are the axial and transverse components of the wavenumber. Equation 2.4 is valid for gyrotrons operating at the fundamental (𝑠 = 1) of the electron cyclotron frequency or at a harmonic (𝑠 > 1) . The resonance condition between the transversal electrical TE𝑚,𝑝 mode and the electron beam can be illustrated using the Brillouin diagram [55]. In that diagram, the relation between 𝜔 and 𝑘 is given by,

$$
\omega = c\_0 \sqrt{k\_{\parallel}^2 + k\_{\perp}^2} \,. \tag{2.7}
$$

The beamline follows the condition

$$
\boldsymbol{\omega} - \boldsymbol{k}\_{\parallel} \boldsymbol{\upsilon}\_{\parallel} = \mathbf{s} \cdot \boldsymbol{\Omega}\_{\mathbf{c}} \,. \tag{2.8}
$$

Fig. 2.3: Dispersion diagram of possible operating modes with the electron beamline (red).

A Brillouin diagram is shown in Fig. 2.3. The hyperbolas represent the dispersion curves of two dierent TE modes (blue lines). The intersection points between the dispersion curves and the beamline (red) represent points of possible interaction. In this example, three dierent intersection points are marked with "a", "b" and "c". Point "a" highlights a typical point of a gyrotron interaction at the rst harmonic. The operating frequency is very close to the cut-o frequency. Point "b" represents a backward wave with 𝑘<sup>∥</sup> << 0. Point "c" indicates an interaction at the second harmonic cyclotron frequency.

## **2.2 Gyrotron Key Components**

In the following, the gyrotron key components relevant to this work are discussed. A detailed explanation of all gyrotron sub-components are discussed in [52].

## **2.2.1 Magnetron Injection Gun**

The hollow electron beam is generated by a Magnetron Injection Gun (MIG). There are two major congurations in the MIG design, namely diode-type and triode-type. The basic schematics of the two classic congurations in coaxial-cavity technology are shown in Fig. 2.4. MIGs built in coaxial-cavity technology contain an insert that passes through the cathode. The diode-type MIG has the advantage of a simple and robust design. Only one high-voltage potential is required for the principal operation or two in the depressed collector operation. However, the electron beam parameters can be controlled only by the applied acceleration voltage 𝑉acc = |𝑉<sup>c</sup> − 𝑉b|. MIGs built in triode conguration have an additional modulation anode. It can be used to control the electric eld at the emitter. The triode conguration requires an additional power supply and ramp-up, which slightly increases its complexity and cost. In addition, a third conguration, the Inverse Magnetron Injection Gun (iMIG) exist, which is described in [57, 58, 59].

In the following, the basic operation of the emitter is discussed. The electrons are emitted by thermionic emission, where electrons are released from the heated emitter due to its sucient high temperature 𝑇emt [52]. The MIG

Fig. 2.4: Schematic view of the structure of a coaxial diode-type (left) and triode-type (right) Magnetron Injection Gun [56]. In a triode conguration, the voltages applied are at the cathode 𝑉<sup>c</sup> , the anode 𝑉<sup>b</sup> , the insert 𝑉<sup>i</sup> and at the modulation anode 𝑉mod.

operates in the temperature limited emission area. The total beam current is described by the Richardson-Dushman equation:

$$J\_s = A\_0 T\_{\rm emt}^2 \exp^{-\frac{W\_E}{k\_B T\_{\rm emt}}}.\tag{2.9}$$

𝐽𝑠 is the current density of the emission, 𝐴<sup>0</sup> is the Richardson constant, 𝑊<sup>𝐸</sup> is the work function of the emitter material and 𝑘<sup>B</sup> is the Boltzmann constant. An electric eld 𝐸 at the emitter surface is formed due to the negatively biased emitter of the MIG. The resulting Schottky-Eect lowers the work function of the surface barrier by ∆𝑊. Therefore, the Richardson-Dushman equation must be modied to:

$$J\_s = A\_0 T\_{\rm emt}^2 \exp^{-\frac{W\_E - \Delta W}{k\_B T}},\tag{2.10}$$

where ∆𝑊 is given by

$$
\Delta W = \sqrt{\frac{q\_e^3 \cdot E}{4\pi\varepsilon\_0}}.\tag{2.11}
$$

𝜀0 is the permittivity of vacuum. Today's emitters for fusion gyrotrons are specied with a maximum current density of 4 − 5 A/cm<sup>2</sup> at an emitter surface temperature of around 1000 °C resulting in a lifetime of about 10<sup>4</sup> hours [60]. Thermionic emitters are manufactured from porous tungsten with an impregnated emitter surface to reduce the work function [61]. Commonly, the pores of the emitting tungsten ring are lled with Al2O<sup>3</sup> , BaO and CaO in a certain mixture [56].

The emitted electrons have a transverse velocity 𝑣<sup>⟂</sup> and an axial velocity 𝑣<sup>∥</sup> component. The velocity ratio, or pitch factor 𝛼, is dened by

$$
\omega = \frac{\upsilon\_\perp}{\upsilon\_\parallel} = \frac{\beta\_\perp}{\beta\_\parallel} \,, \tag{2.12}
$$

with the normalized transverse electron velocity 𝛽<sup>⟂</sup> = 𝑣⟂∕𝑐 and parallel electron velocity 𝛽<sup>∥</sup> = 𝑣∥∕𝑐. A high pitch factor is favored leading to a possible higher interaction eciency [62], since the interacting transverse component contains more energy. Typical values for the pitch factor are in the range of 𝛼 = 1.1−1.3 for fusion gyrotrons. It can be varied at the nominal acceleration voltage using a modulation anode in triode-type conguration. Realistic electron beams contain spreads for e.g. the pitch factor ∆𝛼, the kinetic energy ∆𝐸kin and the electron beam radius ∆𝑟beam. The pitch factor and the kinetic energy spread are dened by the standard deviation

$$
\Delta \mathfrak{a} = \frac{1}{\mathfrak{a}\_{\text{mean}}} \cdot \sqrt{\frac{1}{N} \sum\_{l=1}^{N} (\mathfrak{a}\_l - \mathfrak{a}\_{\text{mean}})^2} \tag{2.13}
$$

$$
\Delta E\_{\rm kin, mean} = \frac{1}{E\_{\rm kin}} \cdot \sqrt{\frac{1}{N} \sum\_{l=1}^{N} (E\_{\rm kin, l} - E\_{\rm kin, mean})^2} \tag{2.14}
$$

𝛼mean and 𝐸kin, mean are the mean value of the velocity ratio and ∆𝛼 and ∆𝐸kin are the root-mean-square(rms) values of the spreads. In contrast, the spread of the electron beam radius is uniformly distributed with

$$
\Delta r\_{\text{beam}} = \frac{r\_{\text{emt, max}} - r\_{\text{emt, min}}}{b^{1/2}} \tag{2.15}
$$

The electron motion considers the adiabatic approximation. The adiabatic approximation describes a slow change of the electric and magnetic eld [63] along the gyrotron axis and is valid for a small variation of the electric and magnetic eld [64]. The condition

$$\frac{P\_\perp^2}{B} = \text{const.}\tag{2.16}$$

is an invariant of motion [63]. As presented in Fig. 2.4, the electron beam is compressed from the average emitter radius 𝑟emt to the beam radius at the cavity section 𝑟beam. This eect is due to the adiabatic compression according to the Busch Theorem [48] with respect to the changing magnetic eld ux along the gyrotron axis, between the emitter 𝐵emt and the cavity 𝐵cav. The magnetic compression ratio 𝑏 is dened by

$$b = \frac{B\_{\text{cav}}}{B\_{\text{emt}}} = \left(\frac{r\_{\text{emt}}}{r\_{\text{beam}}}\right)^2. \tag{2.17}$$

## **2.2.2 Cavity**

The cavity is a straight circular waveguide (midsection) with a down-taper at the input and an up-taper at the output with an angle of 5° in maximum. Thus, it forms an open cavity. Figure 2.5 depicts a schematic of the two dierent types of cavities, the hollow-type cavity and coaxial-type cavity. The coaxial cavity has an additional corrugated insert installed. Main energy is transferred from the electron beam to the electromagnetic wave in the midsection. The maximum magnetic eld of the SC magnet is located at the center of the midsection.

#### **Advantages of Coaxial-Cavity Technology**

The main benet of the coaxial conguration is the reduced mode competition, which gets even more relevant for increasing mode eigenvalues at higher operating frequencies [65, 66]. The eigenvalue of a given mode in the coaxial cavity depends on the ratio of the cavity radius to the insert radius. Longitudinal impedance corrugations on the insert lead to an adjustment of the quality factors of the neighboring modes and hence to a reduced mode competition [67]. The second benet is the lower voltage depression in the coaxial-cavity technology. Voltage depression describes the eect of decelerating electrons due to the space charge of the electrons along the beam

Fig. 2.5: Schematic of a hollow-cavity (left) and coaxial cavity (right). Both consist of downtaper, midsection and up-taper. The coaxial-cavity is equipped with a corrugated insert additionally.

path. This creates a negative potential. The voltage depression ∆𝑉 reduces the accelerating voltage 𝑉acc to the beam potential 𝑉beam. It corresponds to the electron energy at the entrance of the cavity. The voltage depression for a coaxial-cavity is described by [52].

$$
\Delta V = |V\_{\rm acc} - V\_{\rm beam}| \left| \tilde{\pi} 60 \frac{I\_{\rm beam}}{\beta\_{\rm \parallel}} \ln \left( \frac{r\_{\rm cav}}{r\_{\rm beam}} \right) \frac{\ln \left( \frac{r\_{\rm beam}}{r\_{\rm i}} \right)}{\ln \left( \frac{r\_{\rm cw}}{r\_{\rm i}} \right)} \right. \tag{2.18}
$$

𝐼beam is the beam current. The cavity and coaxial insert radius is given by 𝑟cav and 𝑟insert, respectively.

#### **Mode Propagation in Circular Waveguides**

The interaction takes place between an electron beam and a mode, which is very close to its cut-o. In a cylindrical waveguide, several dierent types of modes can propagate. Those are the TE𝑚,𝑝 and TM𝑚,𝑝 modes. In case of the coaxial structure the additionally the TEM-mode can propagate. The eld components 𝐸<sup>𝜑</sup> and 𝐸<sup>r</sup> of a TM𝑚,𝑝 mode vanishes operating close to cut-o. Hence, the energy transfer from an electron beam to a TM mode can be neglected [68]. The TEM mode can also be neglected because it provides a very low quality factor. This is due to the fact that the modes do not have a cut-o frequency. Only the TE𝑚,𝑝 modes show a stable operation and an ecient interaction with the electron beam. The azimuthal and radial electrical eld components for a TE𝑚,𝑝 mode can be described by [69].

$$E\_r = -\frac{m}{r\_{\rm cav}} \cdot J\_m \left(\frac{\chi\_{m,p} \cdot r}{r\_{\rm cav}}\right) \cdot e^{-jm\varphi} e^{-jk\_z z} \tag{2.19}$$

$$E\_{\varphi} = j \frac{\mathcal{X}\_{m,p}}{r\_{\text{cav}}} \cdot J\_m' \left(\frac{\mathcal{X}\_{m,p} \cdot r}{r\_{\text{cav}}}\right) \cdot e^{-jm\varphi} e^{-jk\_z z} \tag{2.20}$$

$$E\_z = 0\tag{2.21}$$

$$E = \sqrt{E\_r^2 + E\_\varphi^2} \tag{2.22}$$

The cut-o frequency 𝑓cut of the operating mode can be calculated by

$$f\_{\rm cut} = c\_0 \frac{\mathcal{X}\_{m,p}}{2\pi r\_{\rm cav}} \,. \tag{2.23}$$

The eigenvalue 𝜒𝑚,𝑝 is described by the 𝑝-th root of the derivative of the Bessel function (𝐽 ′ <sup>𝑚</sup>). The beam-wave coupling is given by [67]

$$Ge\_{m,p} = \frac{J\_{m \mp 1} \left(k\_{\perp,m,p} r\_{\text{beam}}\right)}{J\_m \left(\chi\_{m,p}\right) \sqrt{\pi \left(\chi\_{m,p}^2 - m^2\right)}} \,. \tag{2.24}$$

The highest mode coupling is achieved at the beam/guiding center radius 𝑟beam

$$r\_{\text{beam}} = \frac{r\_{\text{cav}} \chi\_{m \mp 1, 1}}{\mathcal{X}\_{m, p}} \,. \tag{2.25}$$

The "-" and "+" in the formula denes the maximum coupling to the co- and counter-rotating mode with respect to the gyrating electron. In general, in this work, a co-rotating mode is dened by a positive azimuthal mode index and a counter-rotating mode by a negative.

#### **Wall Loading Constraints**

The quality factor 𝑄tot of a resonator can be divided into two parts, namely the ohmic 𝑄<sup>Ω</sup> and the diractive quality factor 𝑄dif. The total quality factor can be calculated by

$$\frac{1}{Q\_{\text{tot}}} = \frac{1}{Q\_{\Omega}} + \frac{1}{Q\_{\text{diff}}} \,. \tag{2.26}$$

The Ohmic quality factor is described by [64]

$$Q\_{\Omega} \approx \frac{r\_{\text{cav}}}{\delta} \left( 1 - \frac{m^2}{\chi\_{m,p}^2} \right) \,. \tag{2.27}$$

<sup>𝛿</sup> is the skin depth dened by <sup>𝛿</sup> = 1∕√ 𝜋𝑓𝜇0𝜎. The diractive quality factor is expressed by [64]

$$Q\_{\rm dif} \approx \frac{4\pi K}{n^2} \left(\frac{L\_{\rm eff}}{\lambda\_0}\right)^2. \tag{2.28}$$

17

Where 𝑛 is the longitudinal mode index (typical 𝑛 = 1), 𝐿e the eective length of the cavity and 𝐾 is a constant dened by the reection coecients Γ1,2 at the down- and up-taper [70] described by

$$K/n \approx \frac{1}{1 - |\Gamma\_1| \cdot |\Gamma\_2|} \,. \tag{2.29}$$

A major challenge in the gyrotron design is to achieve low ohmic wall losses, which can be estimated by [64]

$$P\_{\Omega} \approx 2c\_0 \sqrt{\frac{\pi f}{\mu\_0 \sigma}} Q\_{\text{diff}} P\_{\text{out}} \frac{\chi\_{m,p}}{1 - m^2/\chi\_{m,p}^2} \,, \tag{2.30}$$

Assuming the cavity surface 𝑆, the cavity wall loading is given by [71]

$$\varphi\_{\Omega} = \frac{P\_{\Omega}}{S} = \frac{Q\_{\text{diff}} \cdot P\_{\text{diff}}}{Q\_{\Omega} \cdot S} \approx \frac{2\sqrt{\pi}}{c\_{0}^{3}\sqrt{\mu\_{0}}\sigma} \frac{f^{2.5} \cdot Q\_{\text{diff}}}{L\_{\text{eff}}/\lambda\left(\chi^{2}\_{\text{m,p}} - m^{2}\right)} P\_{\text{diff}} \cdot \tag{2.31}$$

The strong dependency of the wall loading and the resonance frequency is given by 𝜌<sup>Ω</sup> ∼ 𝑓 2.5 .

#### **Eiciency**

The eciency 𝜂<sup>⟂</sup> of the power transfer from the kinetic electron beam to the electromagnetic eld is given by

$$
\eta\_{\perp} = \frac{P\_{\text{elec}}}{P\_{\perp}}.\tag{2.32}
$$

𝑃<sup>⟂</sup> is the perpendicular power of the electron beam and 𝑃elec the generated microwave power in the cavity. The theoretical maximum of 𝜂<sup>⟂</sup> is at ≈ 72 % [72]. The total electrical eciency 𝜂elec describing the total energy transfer from the electrons to the TE mode is given by

$$
\eta\_{\rm elec} \approx \frac{\mathfrak{a}^2}{1 + \mathfrak{a}^2} \cdot \eta\_\perp \,. \tag{2.33}
$$

### **2.2.3 Quasi-Optical Output Coupler**

A detailed description of the quasi-optical output coupler can be found in [50]. In this section, the main working principle is discussed. The quasioptical output coupler decouples the electron beam from the EM wave. The EM wave is radially coupled out of the gyrotron, while the electron beam is dissipated in the collector. Such a quasi-optical output coupler consists of a launcher acting like a waveguide antenna, and several mirrors. The rst launchers were proposed in [73, 74]. Several additional designs have been developed, which are summarized and compared in [75]. The quasi-optical output coupler converts the operating gyrotron cavity mode into a linearly polarized Gaussian mode, which has low transmission losses in free-space and in a corrugated waveguide. It is used to guide the electromagnetic wave to the plasma vessel.

In a highly oversized waveguide, the transmission of the EM eld can be approximated using geometric optics [76, 77]. Considering this assumption, a representation of a high-order rotating gyrotron mode is shown in Fig. 2.6. The angle between the unit vector in azimuthal direction (𝑒⃗𝜑) and the ray (direction of Poynting vector) is given by the spreading angle

$$\cos \Theta = \frac{m}{\chi\_{m,p}}.\tag{2.34}$$

The rays are reected by the metallic wall and are all intersecting at the caustic radius 𝑟<sup>c</sup> . The caustic radius is dened by [52]

$$r\_{\rm c} = r\_{\rm cav} \cos \theta = \frac{m}{\mathcal{X}\_{m,p}} r\_{\rm cav} \,. \tag{2.35}$$

Propagation in z-direction, along the gyrotron axis, is represented in the geometrical optics with the Brillouin angle 𝜃<sup>B</sup>

$$\Theta\_{\rm B} = \arcsin\left(\frac{\mathcal{X}\_{m,p}}{k \cdot r\_{\rm cav}}\right) \,. \tag{2.36}$$

Fig. 2.6: Schematic view of quasi-optical propagation in z-direction, along the gyrotron axis, in the left graph and the cross section of the circular waveguide at the right [50]. The red colored arrows describe the rays, which are all intersecting the caustic radius 𝑟<sup>c</sup> (blue).

# **2.3 Challenges Towards a DEMO Relevant Design**

Challenges regarding the design and operation of a hollow-cavity gyrotron for EU DEMO are already discussed and summarized in [31]. In this section, some major challenges with respect to a DEMO-relevant design considering a multi-MW coaxial-cavity gyrotron are discussed.

#### **Cavity and Insert Wall-Loading**

Today, the maximum acceptable cavity wall loading is 2 kW/cm2 and of the insert loading is 0.2 kW/cm2 [78]. The ohmic loading becomes more challenging at higher frequencies with 𝜌<sup>Ω</sup> ∼ 𝑓 2.5 as it is foreseen for an EU DEMO-relevant design. Simulations show that the nominal operation parameters, and thus the resulting performance, degrade if the cavity radius stays constant at higher frequencies. In contrast, increasing the cavity radius leads to operating modes with higher eigenvalues, resulting in an increased mode competition and a reduced mode stability. Hence, a major challenge in future DEMO-relevant gyrotron designs is to nd an appropriate trade-o between performance and mode stability at frequencies above 200 GHz. An alternative approach is to improve the cavity cooling system to relax the wall loading constraints. The maximum limit is given by the utilized cooling technology [79]. However, it is estimated that a maximum wall loading of 2.5 kW/cm2 could be accepted using a mini-channel cooling [56].

#### **Mode Competition**

The eigenvalue of the TE34,19 mode, which is considered as operating mode at 170 GHz, is 𝜒34,19 = 105.2. It is presently the highest operating mode in a gyrotron. This mode, together with a cavity radius of 29.55 mm, meets the gyrotron design requirements providing an output power of 2 MW at 170 GHz with a wall loading below the limit of 2 kW/cm2 . However, the number of propagable modes 𝑁 in a waveguide increases with increasing frequency, which can be estimated with

$$N \approx 2.55 \cdot \left( 2r\_{\text{cav}} / \lambda \right)^2 \,. \tag{2.37}$$

Keeping the existing cavity radius 𝑟cav constant, the eigenvalues of the modes operating at 204 GHz or even higher are far above 100. The coaxial-cavity technology benets from the insert by reduced mode competition. But, it still requires a precise cavity design and fabrication for such high-order gyrotron modes.

#### **Multi-Frequency/Multi-Purpose Operation with Fast Frequency Step-Tunability**

A major part of this work is to design a pre-prototype gyrotron operating at multiple frequencies with the ability to tune the frequencies in steps. The possibility in operating the gyrotron at multiple-frequencies is called in the following multi-frequency operation. Multi-purpose means that the gyrotrons can be used for e.g. start-up, bulk heating, and non-inductive current drive. Several gyrotron key components, such as the MIG, the coaxial cavity with insert and the quasi-optical output coupler must be optimized for operation at dierent modes and frequencies. A key aspect is the mode

selection process, which shall identify modes with similar relative caustic radii for the given center frequencies, e.g. at 136/170/204/238 GHz. In the case of frequency step-tunability the relative caustic radius cannot be similar, but should be in a range of < 4 %. The change of the relative caustic radius makes it more challenging to design a quasi-optical output coupler for those modes. Furthermore, a fast frequency tuning should be possible, which is determined by the change of the maximum magnetic eld [12].

# **3 Design of a First Multi-Frequency Pre-Prototype Gyrotron**

In this chapter, the upgrade of the existing 2 MW 170 GHz TE34,19 mode coaxial-cavity short-pulse pre-prototype gyrotron is considered theoretically towards multi-frequency operation. The relevant gyrotron key components are discussed in detail. The gyrotron has to be updated towards an operation at either 170 GHz or 204 GHz using o-the-shelf components with best possible performance. For the rst time, frequency-step tunability is studied at those center frequencies considering the insert loading in the mode selection process. In addition, rst studies regarding the possibility of cavity operation at 136 GHz are performed, too.

## **3.1 Requirements for Multi-Frequency Operation**

Multi-frequency gyrotrons can be used for dierent purposes in a nuclear fusion experiment, like plasma start-up, bulk heating, non-inductive current drive and plasma stabilization. There are two main design criteria for a multi-frequency operation. First, the corresponding operating frequencies should t to the natural transmission maxima of the gyrotron output window, and second, the launcher should be able to operate properly with all selected gyrotron modes.

## **3.1.1 Determination of the Main Frequencies**

A transparent window is required to couple the microwave beam out of the vacuum tube with a minimum of losses and reection. There are various vacuum window materials that can be used for gyrotrons, such as BeO, Silicon, SiC or chemical-vapor deposition (CVD) diamond [80, 81]. Megawatt-class

gyrotrons operating at long pulses are equipped with a CVD-diamond window. CVD diamond has a high microwave power transmission capability and a low loss tangent of tan 𝛿 ≈ 2 ⋅ 10−5. Its thermal conductivity of about 2000 W/mK is very high and is needed for edge cooling of the window disk [82]. The resonance frequencies 𝑓op are dened by the equation [83]

$$f\_{\rm op} = n \cdot \frac{\lambda\_m}{2} = \frac{n \cdot c\_0}{2 \cdot d\_{\rm window} \cdot \sqrt{\epsilon\_r'}} \quad \forall n = 1, \ldots, \infty \,, \tag{3.1}$$

where 𝜆<sup>𝑚</sup> is the wavelength in the medium (CVD diamond) , 𝑛 an integer number, 𝑑window the thickness of the window and 𝜖 ′ 𝑟 the relative permittivity of the material (𝜖 ′ <sup>𝑟</sup> = 5.67 for CVD-diamond) [82]. Figure 3.1 presents the transmission and reection of a CVD-diamond disk with a thickness of 1.85 mm. This disk is used for the gyrotron and torus windows at ITER [84]. The natural resonances of this disk are at 136.1 GHz, 170.01 GHz, 204.2 GHz and 238.2 GHz in the given frequency range, which corresponds to 4∕2𝜆𝑚, 5∕2𝜆𝑚, 6∕2𝜆𝑚, and 7∕2𝜆𝑚, respectively.

Fig. 3.1: Calculated transmission and reection of a single CVD-diamond disk (𝜖 ′ <sup>𝑟</sup> = 5.67, 𝑑window = 1.85 mm) in the relevant range from 100 − 250 GHz.

Research is currently ongoing to install a diamond disk at the Brewster angle for broadband operation. This means that one polarization direction does not show a resonance behavior and thus the operating frequency does not depend on the CVD-diamond disk [85]. Broadband gyrotron output windows are mandatory for the ability to step-tune the frequency in a range of ±10 GHz in 2 − 3 GHz steps around the center frequencies.

### **3.1.2 Mode Selection Process**

Operating modes have to be identied in the mode selection process, which t to the output window and oer a small deviation in the caustic radii. In [86], a maximum deviation in caustic radii of 2.6 % is proposed to guarantee a proper operation. However, the given launcher is not optimized for multifrequency operation. According to simulations at KIT, a deviation in the relative caustic radii of the selected modes in the mode series of < 4 % is required to guarantee a proper operation of the launcher [41]. Since the upgrade towards a multi-frequency pre-prototype is based in this work on the existing 2 MW coaxial-cavity pre-prototype, the rst mode in the mode series has been dened by the already operating TE34,19 mode at 170 GHz [13]. Therefore, the identication of suitable mode candidates operating at 136 GHz, 204 GHz and 238 GHz are limited. The identied candidates are the TE28,15 mode operating at 136 GHz, the TE40,23 mode at 204 GHz and the TE48,26 mode at 238 GHz. The properties for these modes are summarized in Tab. 3.1. The TE28,15 mode and the TE40,23 mode operating at 136 GHz and 204 GHz operate very closely to the resonance frequency of the diamond disk. It results in a very low reection at the gyrotron window and thus a high transmission. The deviation of the relative caustic radius is in total above 4 %. However, relative to the mode at 170 GHz, the dierence is within the requirements. A study of possible alternative mode series, which are designed from scratch, is discussed in Chapter 6. In the following studies, the main focus is set on a dual-frequency 170/204 GHz operation.

The 𝐸-eld patterns of the chosen TE𝑚,𝑝 modes at 170 GHz and 204 GHz are presented in Fig. 3.2. The mode indices of the TE𝑚,𝑝 mode are given by the azimuthal mode index 𝑚 = 34 and 𝑚 = 40 and the radial mode index by 𝑝 = 19 and 𝑝 = 23 for 170 GHz and 204 GHz, respectively. The graphs in Fig. 3.2 are plotted in absolute values. The radius of the eld maxima is smaller in the case of the TE40,23 mode and the number of maxima in the azimuthal direction is higher.


Table 3.1: Summary of key parameters of suitable modes for the operation at 136 GHz, 170 GHz, 204 GHz and 238 GHz.

Fig. 3.2: 𝐸-eld intensity patterns of the non-rotating TE34,19 mode at 170 GHz (left) and the TE40,23 mode at 204 GHz (right).

## **3.2 Verifying the Existing Pre-Prototype Gyrotron**

The performance of the relevant gyrotron key components of the existing 170 GHz single-frequency pre-prototype design are studied at 204 GHz with the TE40,23 mode. The investigated key components are the coaxial cavity and the quasi-optical output coupler. This study provides an indication about the required modication to achieve a 170/204 GHz dual-frequency gyrotron providing 2 MW output power at both frequencies.

## **3.2.1 Influence of Magnetic Field Profiles**

The existing SC magnet from Oxford Instruments, Abingdon, UK cannot provide the required ≈ 8.3 T to operate the gyrotron at 204 GHz. Simulations presented in this work are based on the dimensions and magnetic eld prole of a future 10.5 T SC magnet. This SC magnet is longer in z-direction due to the higher magnetic eld forces. The distance between the emitter and the center of the cavity is extended from 367 mm for the existing magnet to 400 mm. This implicit that the gyrotron components have to be adapted accordingly. Simulations show that operating the gyrotron without an adaption to the SC magnet leads to a drastic increase of the velocity spreads above 40 % [41]. The electron beam trajectory of the existing SC magnet is compared in Fig. 3.3 to the one of the new 10.5 T SC magnet exemplarily for the iMIG. The distance between the electron beam and the body is increased. The eect of trapped electrons can occur as a consequence. In terms of the longer main coil of the future 10.5 T SC magnet, the magnetic eld prole in the cavity area is more at and the magnetic eld compression is increased. As a result, all available MIGs at KIT have to be validated with the new magnetic eld prole and, if necessary, modied accordingly.

Fig. 3.3: Comparison of the electron beam trajectories of the existing Oxford Instruments (OI) SC magnet and the new 10.5 T SC magnet.

## **3.2.2 Operating the Existing 170 GHz Single-Frequency Pre-Prototype at 204 GHz**

In addition, the coaxial cavity and the quasi-optical output coupler of the 170 GHz single-frequency design are validated for an operation at 204 GHz.

#### **Coaxial Cavity**

The design of the existing 170 GHz single-frequency coaxial cavity has been investigated whether a principle operation at 204 GHz without any modication is possible. For this study, the electron beam parameters of the existing MIG are not needed. Nevertheless, in order to be as realistic as possible, an adiabatic start-up is performed in the interaction simulations carried out with the code package EURIDICE [87]. A general description of the code is given in [88] and in Appendix A.1. A kinetic energy spread of ∆𝐸 = 0.2 % rms and a relatively large velocity spread of ∆𝛼 = 20 % rms is considered. These values correspond to a worst-case approximation according to [31].

Fig. 3.4: Adiabatic start-up scenario at 204 GHz with the existing coaxial cavity, originally designed for 170 GHz single-frequency operation. The circle indicates the operation point of the nominal TE40,23 mode at a beam energy of 79 keV, a beam current of 𝐼beam = 70 A and a magnetic eld strength of 8.15 T .

Initial simulations show that the pitch factor needs to be reduced from the nominal 𝛼 = 1.3 to 𝛼 = 1.2 for operation at 204 GHz to avoid parasitic oscillations. Figure 3.4 presents an adiabatic start-up scenario at 204 GHz. The interaction simulation shows an output power of 1.66 MW and an interaction eciency of 31.2 %. The nominal operation point is at a beam energy of 79 keV and a beam current of 70 A. The operation parameters and thus the performance is limited by the wall loading of 2 kW/cm2 . An insert loading of 0.06 kW/cm2 has been obtained, which is far below the limit. Operation of the existing 170 GHz coaxial cavity is possible at 204 GHz, however, the EU DEMO requirement of 2 MW output power at all possible frequencies is not fullled. Therefore, the coaxial cavity must be modied as proposed in Section 3.3.2.

#### **Quasi-Optical Output Coupler**

The existing quasi-optical output coupler is designed for 170 GHz singlefrequency operation [89] providing a Gaussian mode content of 96.3 %. It has been veried by simulations using TWLDO [90] and the commercial 3D full-wave code SURF3D [91]. The fundamental Gaussian mode content of the single-frequency design operating at 204 GHz has been calculated to be only 91.6 %, which is below the desired 95 %. Figure 3.5 presents the 𝐸-eld intensity distribution of the nominal TE34,19 mode at 170 GHz and the TE40,23 mode at 204 GHz in the window plane. The dierences in the quality of the microwave output beam at the two frequencies are obvious. Therefore, the 170 GHz single-frequency design of the quasi-optical output coupler is not suited for gyrotron operation at 204 GHz. A modication of the quasi-optical output coupler is discussed in Section 3.3.3.

Fig. 3.5: 𝐸-eld intensity distribution of the output beam in the window plane at a) 170 GHz and b) 204 GHz. A gyrotron output window with 49 mm radius is indicated in gray.

# **3.3 Optimization Towards a 170/204 GHz Pre-Prototype Gyrotron**

First simulations show that the MIGs available at KIT have to be modied and adapted to the geometry of the new 10.5 T SC magnet. This step is required to locate the emitter at the correct position in the magnetic eld prole. Other gyrotron key components, such as the coaxial cavity and the quasi-optical output coupler, could in principle be operated at 204 GHz. However, modication of these components is mandatory. In the following, the MIGs are investigated using the magnetic eld prole of the new 10.5 T SC magnet. The resulting electron beam parameters are considered in interaction simulations with the optimized coaxial cavity. A new quasi-optical output coupler is designed for proper operation at 170 GHz and 204 GHz. Finally, a detailed study about frequency step-tunability is performed considering the optimized coaxial cavity and quasi-optical output coupler.

## **3.3.1 MIG Adaption to New Magnetic Field Profile**

Currently, three dierent MIGs are available at KIT, namely the inverse Magnetron Injection Gun (iMIG) [59], the coaxial Magnetron Injection Gun with non-emissive coating of the emitter edges (cMIG) [92] and a coaxial MIG [93]. The iMIG and cMIG are built in a triode conguration, the coaxial MIG is designed in diode conguration. All 3 MIGs are investigated to assess whether the resulting quality of the electron beam is sucient for an operation at 170 GHz and 204 GHz in the new magnetic eld prole. The simulations are performed with the self-consistent, electrostatic trajectory 3D in-house code-package ARIADNE [94]. The objectives of the simulations are to meet the following design criteria:


#### **Optimization of the Coaxial Magnetron Injection Gun**

The performance of the existing coaxial diode MIG is validated at 170 GHz and 204 GHz. The diode MIG benets from its relatively simple construction and operation, since only the acceleration voltage is applied. The coaxial diode MIG that exists at KIT was originally designed and optimized for the magnetic eld prole of the existing SC magnet. Here, only the anode is redesigned to make ecient use of the available components. Therefore, existing components, such as the cathode, are re-used. The modication includes an elongation of the anode shape to adapt the gyrotron to the geometrical conguration of the new 10.5 T SC magnet.

The nominal pitch factors are dened by interaction simulations to be 𝛼170 GHz = 1.3 and 𝛼204 GHz = 1.23 and are thus almost equal [95]. An anode shape optimized for 170 GHz operation reduces the pitch factor to 𝛼204 GHz ≈ 1.0 at 204 GHz, which is too low for an ecient operation. A diode MIG is published in [96, 97] that can operate at two dierent frequencies. The proposed approach is to use co-rotating modes at one frequency and counter-rotating modes at the other. Therefore, a new anode shape is designed that provides the mentioned pitch factors at both frequencies using the counter-rotating TE−34,19 mode at 170 GHz and the co-rotating TE40,23 mode at 204 GHz. The counter-rotating TE−34,19 mode has a beam radius of 𝑟beam = 10.58 mm and is larger compared to the co-rotating TE34,19 mode with 𝑟beam = 10.00 mm. This means that the eect of lowering the pitch factor by increasing the beam radius using a counter-rotating mode at the lower frequency is benecial for dual-frequency operation. However, the polarity of the SC magnet has to be reversed in experiments using a co-rotating mode at the one and a counter-rotating mode at the other frequency. The change of the polarity of the magnetic eld is required to allow a proper operation of the same quasi-optical output coupler. The use of co-rotating and counter-rotating modes is required for the diode MIG only. The nominal parameters like the beam radius 𝑟beam and magnetic eld strength 𝐵cav are summarized in Tab. 3.2.


Table 3.2: Summary of beam radius 𝑟beam and magnetic eld strength 𝐵cav for the TE±34,19 operating at 170 GHz and TE40,23 mode at 204 GHz.

Figure 3.6 presents a schematic of the optimized anode prole. In addition, the magnetic eld lines and the corresponding electron beam trajectories are illustrated for the TE40,23 mode. The halo shield (i. e. the last part of the anode) is optimized according to the magnetic eld lines of the 10.5 T SC magnet. It is used to prevent trapped secondary electrons emitted from the upper rear part of the cathode. The distance between the electron beam and the halo shield is 2.9 mm for the TE40,23 mode, 2.2 mm for the TE−34,19 mode and 1.5 mm for the TE34,19 mode. Therefore, the requirement for the distance between the electron beam and the wall is satised for the TE−34,19 and TE40,23 modes.

Fig. 3.6: Magnetic eld lines (green) of the 10.5 T SC magnet between the MIG (left) and the beam tunnel (right). The electron beam (red) is emitted from the cathode (not modied). A comparison of the shape of the existing (blue dashed) and modied anode (black) is shown.

Two main MIG design criteria are dened in [93]. These are 1) the avoidance of trapped electrons in terms of the emission of secondary electrons from the cathode surface and 2) about potential wells. The avoidance of possible trapped secondary electrons will be discussed rst. The insert is mounted to the bottom of the gyrotron in the coaxial-cavity technology. It adds several challenges to the MIG design. Firstly, it implies that the insert passes through the cathode. Secondly, the gyrotron body and the anode are on the same potential, which creates a high electric eld in the inner region and outer region of the cathode. That can cause signicant diculties in satisfying the rst design criterion. The secondary electrons emitted from the cathode surface are aected by the magnetic compression. They could be guided into the cavity section and can cause issues during operation. This eect can be investigated by using the procedure published in [93] where the whole cathode surface is dened as emitter, as shown in Fig. 3.7. Some of the electrons emitted from the cathode nose are collected by the insert and some of those emitted from the upper rear part are collected by the anode. The eect of possible trapped electrons from the upper part of the cathode can be controlled by the correct choice of the halo shield. The blue dots in Fig. 3.7 indicate the position of emitted electrons that are guided into the cavity and are not collected by a metallic wall (anode or insert). The most critical electrons are those guided into the interaction region. The pitch factor in the cavity is shown as a function of the position of the electron emission (the ones highlighted in blue in Fig. 3.7) in Fig. 3.8 for all three proposed operating modes. Safe operation without electron trapping by the magnetic mirror can be expected, if the maximum pitch factor of all the dierent electrons are smaller than 𝛼max ≤ 3 [93].

The second MIG design criterion is the prevention of potential wells. A potential well is formed, if the electrical equipotential lines cross the magnetic eld lines twice. Then, the electrons are trapped in such an area. The design presented in this work has two potential wells. Neither the cathode nor the anode shape can be modied accordingly. However, this is not critical because of the planned operation with short pulses for the proof-of-principal.

Fig. 3.7: Trajectories of electrons emitted from the cathode surface at 204 GHz. The green lines present the electron trajectories. The blue dots mark the position of emitted electrons reaching the cavity.

Fig. 3.8: Pitch factor 𝛼 at the cavity versus the axial position of the emitting point on the surface of the cathode. The surface area marked in blue in Fig. 3.7 is selected and the main operating modes at 170 GHz and 204 GHz are considered.

The revised coaxial diode MIG is studied for the nominal operating parameters. The pitch factor 𝛼 and the transverse velocity spread 𝛿𝛽<sup>𝑡</sup> of the TE−34,19 mode and the TE40,23 mode are plotted in a wide range of magnetic compression and magnetic eld angles at the emitter in Fig. 3.9 and Fig. 3.10, respectively. The magnetic eld angle and the magnetic compression can be varied independently in experiments. As shown in the graphs, the transverse velocity spread is not signicantly aected by varying the beam radius. However, the pitch factor increases with decreasing beam radius in terms of a larger magnetic compression. A summary of the nominal operation parameters is given in Tab. 3.3.

Table 3.3: Summary of the pitch factor 𝛼 and the transverse velocity spread 𝛿𝛽<sup>𝑡</sup> at 170 GHz and 204 GHz using the coaxial diode MIG for a magnetic eld angle of −0.5°.


Fig. 3.9: a) Pitch factor 𝛼 and b) transverse velocity spread 𝛿𝛽<sup>𝑡</sup> of the TE−34,19 mode at 170 GHz in a wide range of magnetic compression and magnetic eld angles at the emitter 𝜙emt.

Fig. 3.10: a) Pitch factor 𝛼 and b) transverse velocity spread 𝛿𝛽<sup>𝑡</sup> of the TE40,23 mode at 204 GHz versus the magnetic compression and magnetic eld angles at the emitter 𝜙emt.

Figure 3.11 shows a tolerance analysis considering anode displacement in the radial and axial direction. Axial displacement does not signicantly aect the pitch factor and the velocity spread due to the at anode shape around the emitter area. The tolerance for positive axial displacements is limited to 1 mm by secondary electrons. The geometry of the anode itself limits the tolerance to negative displacements to −1.4 mm to keep the distance of the electron beam to the anode above 2 mm. The inuence of the radial displacement is more signicant. A radial shift of up to 0.5 mm is accepted. For larger radial shift, the pitch factor is too high and mode instability can occur. Again, secondary electrons limit the tolerance to positive radial misalignment to 0.3 mm and the negative misalignment is limited by the geometry to −0.5 mm.

The anode with the optimum shape is fabricated. The coaxial diode MIG will be used for the rst proof-of-principal experiments of the 170/204 GHz coaxial-cavity pre-prototype. A measurement with a 3D measurement arm shows manufacturing tolerances of < 60 𝜇m, which is below the specied tolerances dened in the simulations.

Fig. 3.11: Analysis of the tolerances for a shift in a) axial and b) radial direction at 204 GHz.

The diode MIG will be replaced by a triode MIG after the rst proof-ofprinciple experiments. A MIG in triode conguration is intended for a future EU DEMO-relevant design as it provides more exibility in setting the electron beam parameters. The additional modulation anode in a triode conguration can eliminate the issue of maintaining the pitch factor at nominal operating parameters at dierent frequencies. This allows operating modes that rotate in the same direction to be used for multi-frequency operation. Therefore, the polarity of the magnetic eld does not have to be reversed. Two triode MIGs are discussed below.

#### **Magnetron Injection Gun with Coated Non-Emissive Layer (cMIG)**

The design of the triode-type cMIG has been published in [92]. The schematic and the emitted electron beam are presented in Fig. 3.12. It assumes the magnetic eld prole of the new 10.5 T SC magnet. The applied nominal voltages are a cathode voltage of 𝑉<sup>c</sup> = −55 kV, insert and body voltage of 𝑉<sup>b</sup> = 𝑉<sup>i</sup> = 35 kV. The potential of the modulation anode is set to ground. The existing cMIG meets all dened design requirements according to the simulations. But, the geometry still needs to be adapted to t into the new 10.5 T SC magnet.

Fig. 3.12: Schematic of the cMIG.

Figure 3.13 presents the pitch factor during start-up with increasing acceleration voltage. The kinetic beam energy is used at the x-axis for better compatibility. The resulting pitch factor is 𝛼 = 1.3∕1.25 and the transverse velocity spread is 𝛿𝛽<sup>𝑡</sup> = 2.2∕2.5 % at a beam energy of 90.0/87.5 keV, a beam current of 75/68 A and a maximum magnetic eld strength of 6.86/8.27 T for the operation at 170/204 GHz, respectively. The results are summarized in Tab. 3.4. In general, the performance of the cMIG is slightly better than that of the diode MIG.

Table 3.4: Summary of the pitch factor 𝛼 and the transverse velocity spread 𝛿𝛽<sup>𝑡</sup> at 170 GHz and 204 GHz using the cMIG.


Fig. 3.13: Pitch factor for the start-up of the cMIG.

#### **Inverse Magnetron Injection Gun (iMIG)**

The emitter of the iMIG is placed at the outer wall of the MIG. Figure 3.14 presents a schematic of an iMIG [59]. The emitter ring can be much larger without increasing the overall diameter of the MIG design. A larger emitter radius is able to handle higher beam currents, while the maximum current density still does not exceed ≈ 4 A/cm<sup>2</sup> . Therefore, a longer lifetime can be expected at similar beam currents. In addition, a more compact MIG can be constructed because the modulation anode is placed inside resulting in a decreased borehole of the SC magnet. This leads to a better eld homogeneity and reduced magnet cost [98]. Another advantage is the improved cooling, since the heated emitter and the heated surrounding can be cooled by the outer wall. An iMIG was developed in 2015 at KIT for the operation in the existing SC magnet [59]. This iMIG has an average emitter radius of 62 mm and is 3 mm larger than in the other coaxial MIGs at KIT.

The iMIG is a favored conguration for future EU DEMO-relevant fusion gyrotrons. Therefore, the iMIG is adapted to the magnetic eld prole of the new 10.5 T SC magnet. The beamline of the existing and new SC magnet is depicted in Fig. 3.14. The distance of the electron beam trajectories to the

Fig. 3.14: Schematic view of the dierent geometries of the iMIG of the existing design and a modication according to the magnetic eld prole of a 10.5 T SC magnet.

modulation anode is below 2 mm using the new magnetic eld prole. A modication of the modulation anode is performed.

The construction of the modulation anode consists of two parts. One is the modulation anode itself, which is made of molybdenum. The other is a holder made of copper. The new design takes into account that the two parts of the modulation anode can be reworked. The copper rear part has to be shortened to bring the front part closer to the emitter part. At the other part, the rounding at the rear of the modulation anode has to be increased to ensure a small velocity spread. In addition, the radius is decreased to provide a distance of in minimum 2 mm to the electron beam. The resulting pitch factor and the transverse velocity spread at 170 GHz and 204 GHz are presented in Fig. 3.15 and Fig. 3.16. The graphs show the eect of a large range of magnetic compression and the angle of the electron beam at the emitter surface onto the pitch factor and the transversal velocity spread. The results are summarized in Tab. 3.5 for the nominal operation point marked with a red circle. There are combinations with lower velocity spreads providing almost similar pitch factors. However, simulations on secondary electrons show that these combinations can oer trapped secondary electrons.

Table 3.5: Summary of the simulation results of the pitch factor 𝛼 and the transverse velocity spread 𝛿𝛽<sup>𝑡</sup> at the nominal operation point at 170 GHz and 204 GHz using the iMIG.


Fig. 3.15: a) Pitch factor 𝛼 and b) transverse velocity spread 𝛿𝛽<sup>𝑡</sup> of the TE34,19 mode at 170 GHz in a wide range of magnetic compression and magnetic eld angles at the emitter 𝜙emt using the iMIG. The nominal operation points are marked with a red circle.

Fig. 3.16: a) Pitch factor 𝛼 and b) transverse velocity spread 𝛿𝛽<sup>𝑡</sup> of the TE40,23 mode at 204 GHz over a wide range of magnetic compression and magnetic eld angles at the emitter 𝜙emt using the iMIG. The nominal operation points are marked with a red circle.

## **3.3.2 Design of a New Dual-Frequency Coaxial Cavity**

An optimization of the coaxial cavity is required to achieve > 2 MW output power at both frequencies. The optimization considers following requirements:


Figure 3.17 presents a schematic of the coaxial cavity. It consists of a straight midsection 𝐿mid, a down-taper and an up-taper forming an open resonator. The dierent sections are connected by parabolic smoothing to reduce unwanted mode conversion at the transitions [101].

The existing 170 GHz single-frequency design achieves a maximum output power of up to 1.7 MW at 204 GHz. A limiting factor is the wall loading constraint of 2 kW/cm2 that limits the applied acceleration voltage and beam current. This eect results from the dependency of the wall loading and the operating frequency given by 𝜌<sup>Ω</sup> ∼ 𝑓 2.5 . This eect becomes worse at operating frequencies much higher than 200 GHz. A reduction of the wall loading can be achieved by increasing the cavity radius or by choosing a higher volume mode with higher eigenvalues. In both cases, the operating frequencies would not longer t to the resonances of the gyrotron output window. In addition, the mode spectrum becomes more dense. An alternative method is to consider the proportionality between the diractive quality factor and the wall loading, as given in Eq. 2.31. The approach is to tune the diractive quality factor by the geometrical shape of the coaxial cavity to reduce the wall loading. With reduced wall loading, the nominal operating parameters can be increased. Thus, the interaction eciency and the output

Fig. 3.17: Geometry of the coaxial cavity with the normalized transverse electric eld prole.

power can be increased. In addition, a trade-o must be considered in the cavity design for the operation at 170 GHz and 204 GHz.

A systematic hollow-cavity design approach is published in [31]. This approach has been adapted for cavities with a coaxial insert. The proposed design steps are:


**To step 1:** The cavity radius at the midsection is xed to 𝑟cav = 29.55 mm as given in the requirements. The electron beam radii of the two operating modes are calculated with Eq. 2.25 to be 𝑟beam = 10.00 mm for the TE34,19 mode operating at 170 GHz and 𝑟beam = 9.77 mm for the TE40,23 mode operating at 204 GHz.


Table 3.6: Initial geometrical parameters for the coaxial-cavity design.

**To step 2:** The existing coaxial-cavity is chosen as the basis for optimization. A summary of the physical parameters is given in Tab. 3.6.

**To step 3:** A study on the variation of the interaction length is presented in Fig. 3.18 for the TE34,19 mode at 170 GHz and the TE40,23 mode at 204 GHz. The output power, interaction eciency and diractive quality factor are depicted along the variation of the midsection length. The study starts from the initial midsection length of 𝐿mid = 16 mm down to 𝐿mid = 11 mm. This study has been performed for single- and multi-mode operation considering the main competing modes listed in Tab. 6.4. The simulations do not consider parameter spreads. The operating parameters are selected individually at each step to fulll the wall-loading limitation of 2 kW/cm2 and have been optimized for the multi-mode simulations. This explains the better performance compared to single-mode simulations. The performance is increased by reducing the midsection length and thus the diractive quality factor. However, at a midsection length below 13 mm the performance at 170 GHz is decreasing strongly, where the diractive quality factor is close to 𝑄di ≈ 1000. At 204 GHz the mode competition increases strongly with a

Fig. 3.18: Output power, interaction eciency and diractive quality factor for dierent midsection lengths 𝐿mid from 11 mm to 16 mm. The operating parameters are selected individually at each step to fulll the wall-loading limitation of 2 kW/cm2 and to achieve maximum output power.

mid-section length below 13 mm. The nominal midsection length is dened for the further study to be 13.6 mm including some safety margin.

**To step 4:** The corrugation depth of the existing insert is 𝑑 = 0.44 mm. This corresponds to 𝜆0∕4 at 170 GHz. A detailed analysis about longitudinal insert corrugations and insert misalignment is published in [88]. Simulations show that the inuence of the existing corrugations depth is not critical for

an operation at 204 GHz. There is not a signicant improvement using a corrugation depth of 𝑑 = 0.36 mm corresponding to 𝜆0∕4 at 204 GHz. The minor improvement in performance does not justify a new insert for the rst proof-of-principal experiments at 170 GHz and 204 GHz.

**To step 5:** The input angle 𝜃<sup>1</sup> and output angle 𝜃<sup>2</sup> of the down-taper and uptaper are investigated considering a midsection length of 𝐿mid = 13.6 mm. Figure 3.19 presents the eect of the input and output angle onto the output power, eciency and ohmic wall loading. Simulations have been performed using the similar operation parameters. The values in the red box indicates angles with high performance while keeping the ohmic wall loading around 2 kW/cm2 . Further simulations with varied operation parameters have been performed using the angles in the highlighted area. Highest performance is achieved for the angles at 𝜃<sup>1</sup> = 2.5° and 𝜃<sup>2</sup> = 2.5°. About 0.5 % points in the interaction eciency and about 50 kW are gained by changing the down-taper angle from the initial 3° to 2.5°. Considering velocity, guiding center and kinetic energy spreads, the interaction is much more robust using a down-taper angle of 𝜃<sup>1</sup> = 2.5° instead of 𝜃<sup>1</sup> = 3.0°. An input angle in this range can lead to backward waves, which can reduce the performance.

**To step 6 and 7:** The smoothing length of the down-taper 𝐷<sup>1</sup> and up-taper 𝐷<sup>2</sup> has to be optimized and selected to be 𝐷<sup>1</sup> = 𝐷<sup>2</sup> = 2.2 mm to reduce any conversion into competing modes. They are 200 µm longer than in the design of the 170 GHz single-frequency coaxial cavity. The length of the down-taper 𝐿<sup>1</sup> and up-taper 𝐿<sup>2</sup> are xed according to the design requirements. The diractive quality factor 𝑄di of the optimized coaxial-cavity design is given by 1094 and 1647 at 170 GHz and 204 GHz, respectively. An output power of 2.15 MW at an interaction eciency of 35.8 % at 204 GHz is achieved in singlemode simulations. The wall loading is kept to 2 kW/cm2 and the maximum insert loading is 0.05 kW/cm2 . The nominal operation parameters are a beam energy of 𝑈beam = 87.2 keV, beam current of 𝐼beam = 68 A, magnetic eld strength at cavity midsection of 𝐵cav = 8.23 T and a pitch factor of 𝛼 = 1.23.

In addition, a non-linear up-taper is required. It connects the end of the up-taper of the cavity smoothly to the launcher [102]. The wall radius is increased up to 32 mm of the launcher. The allowable mode conversion is

Fig. 3.19: Inuence of the input 𝜃<sup>1</sup> and output angle 𝜃<sup>2</sup> of the down-taper and up-taper for the a) output power, b) eciency and c) resistive wall loading considering the TE40,23 mode at 204 GHz. The area of the red box indicate the best choice regarding the trade-o between high performance and keeping the ohmic wall loading in the limitation.

limited by 0.1 %. The non-linear up-taper of the existing 170 GHz singlefrequency design shows a mode conversion of 0.6 % at 204 GHz from the TE40,23 main mode to the lower order TE40,22 mode. A modication of the geometry reduces the mode conversion to 0.05 % at 204 GHz and keeps it below 0.1 % at 170 GHz. The geometry of the optimized non-linear up-taper is presented in Fig. 3.20.

Fig. 3.20: Geometry of the modication of the non-linear up-taper to reduce mode conversion.

At last, the nal physical parameters of the geometry are presented in Tab. 3.7.

#### **Multi-Mode Time-Dependent Simulations**

Multi-mode simulations are performed with the optimized coaxial cavity and ideal electron beam parameters, i.e. velocity, guiding center and kinetic energy spreads are not included. The start-up scenarios of the given modes are presented in Fig. 3.21a and Fig. 3.21b. In total 63 modes at 170 GHz and 74 modes at 204 GHz are considered. Modes are considered in interaction simulation if their relative coupling is above 0.5 and their operating frequency is in the range between −5 % to +10 % of the center mode frequency. The most dangerous competitors appear in this range. The nominal operation points are indicated with a red circle. The points are dened by the maximum allowed wall loading limit of 2 kW/cm2 . The wall loading are plotted in Fig. 3.21c and Fig. 3.21d, accordingly. The nominal operation points are characterized by a beam energy of 91.7 keV and 89.5 keV, a beam current of 77 A and 70 A and a maximum magnetic eld strength of 6.88 T and 8.27 T for the operation at 170 GHz and 204 GHz, respectively. The optimized coaxial cavity provides an output power of 2.61 MW and an inter-


Table 3.7: Geometrical parameters for the coaxial-cavity design used for the simulations of 170/204 GHz multi-frequency operation towards a rst proof-of-principal experiment.

action eciency of 38.4 % at 170 GHz and 2.22 MW and 37.0 % at 204 GHz. The results demonstrate that the optimized coaxial cavity is able to generate > 2 MW output power at both frequencies using an ideal electron beam.

Fig. 3.21: Oscillation start-up for the optimized coaxial cavity considering an ideal electron beam for the a) TE34,19 mode at 170 GHz and b) TE40,23 mode at 204 GHz. The operation points are indicated with the red circles. They are dened by the wall loading sown in c) and d).

In the following steps, an analysis of the variation of the electron beam radius and the impact of dierent qualities of electron beams are studied. Finally, the electron beam parameters from the previous MIG simulations are included into the interaction simulations.

#### **Analysis of Parameter Tolerances**

Firstly, a variation of the electron beam radius is investigated. The electron beam radius of 10.00 mm for the TE34,19 mode operating at 170 GHz was chosen in previous works [13]. Therefore, this study is focused on the TE40,23 mode. The highest theoretical coupling between the electron beam and the TE40,23 mode is according to Fig. 3.22 at an electron beam radius of 𝑟beam = 9.77 mm. The graph has several maxima corresponding to the radial mode index 𝑝 = 23. However, the maximum interaction eciency can only be achieved at the highest coupling.

The mode spectra of the main competing modes at an electron beam radius of 𝑟beam = 9.70∕9.77∕9.85 and 10.00 mm are investigated. The results are presented in Fig. 3.23. A frequency range of −2% and +3% around the center frequency of 204.15 GHz is selected. The coupling to the next counter-rotating modes, namely the TE−(𝑚−3),𝑛+1 (TE−37,24) and TE−(𝑚−2),𝑛+1 (TE−38,24), decrease with increasing electron beam radius. This behavior is a favorable condition for a proper excitation of the main operating mode. In addition, the coupling to the azimuthally neighboring co-rotating modes with an azimuthal mode index 𝑚 > 40 increases with increasing electron beam

Fig. 3.22: Normalized coupling of the electron beam to the TE40,23 mode and TE−40,23 mode along the radius.

radius. The coupling of the co-rotating TE40,23 mode and counter-rotating TE−40,23 mode is almost identical at an electron beam radius of 10.00 mm, according to Fig. 3.22 and Fig. 3.23d. Therefore, an electron beam radius of 𝑟beam = 10.00 mm sets the upper limit for the beam radii variation. The lower limit is set to 𝑟beam = 9.70 mm. It is dened by the increase of the insert loading in terms of the closer distance of the electron beam to the insert.

Fig. 3.23: Mode spectra with varied electron beam radius 𝑟beam. The main competitors with a relative coupling higher than 0.5 are considered. The beam radius is varied from 9.70 mm to 10.00 mm. The black bars indicate co-rotating modes and the red bars counter-rotating modes.

Fig. 3.24: Variation of the electron beam radius at the cavity midsection and its inuence in respect to the output power and the interaction eciency considering the TE40,23 mode operating at 204 GHz.

Figure 3.24 shows the output power and the interaction eciency of the TE40,23 mode using the given electron beam radii. Maximum performance is achieved at 𝑟beam = 9.77 mm because of the maximum coupling according to Eq. (2.24). A deviation from the maximum leads to a reduced performance. However, a slightly increased electron beam radius improves the mode stability. An electron beam radius of 𝑟beam = 9.85 mm is selected.

A second investigation focuses on the eect of dierent electron beam parameters on the interaction. Dierent values for the guiding center, the pitch factor and the kinetic energy spreads are considered in the interaction simulations. A summary of the simulation results is given in Tab. 3.8. The main operating parameters are xed during the study to 𝑈beam = 91.8 keV, 𝐼beam= 75 A and 𝐵cav = 6.88 T at 170 GHz and 𝑈beam = 89.5 keV,𝐼beam = 68 A and 𝐵cav = 8.27 T at 204 GHz. The velocity, guiding center radius and energy spread are studied separately. During the variation of a certain spread, the other spreads are xed to be ∆𝐸kin = 0.01 %, ∆𝛼 = 6 % and ∆𝑟gc = 1.96 %. In conclusion, the TE34,19 mode and the TE40,23 mode are excited even in the worst-case scenario. However, the velocity spread should be < 20 % to guarantee an output power of 2 MW at 204 GHz.

Table 3.8: Summary of the eect of varying the pitch factor ∆𝛼, guiding center radius ∆𝑟𝑔𝑐 and energy spread ∆𝐸kin on the output power and the interaction eciency. For comparison the beam energy has been xed to 𝑈beam = 91.8 keV at 170 GHz and 89.5 keV at 204 GHz. The reference values are ∆𝛼 = 6 %, ∆𝑟𝑔𝑐 = 1.96 % and ∆𝑟𝑔𝑐 = 0.1 %.


#### **Start-Up Scenario Considering the Coaxial Diode MIG**

Finally, the design of the coaxial cavity is veried including the spreads and the pitch factor from the electron beam of the MIG simulations. The three available MIGs are considered in this study.

Firstly, the coaxial-diode MIG will be installed to the pre-prototype for the rst proof-of-principle experiments. Figure 3.25a presents the start-up of the counter-rotating TE−34,19 mode at 170.01 GHz. The corresponding electron beam parameters from Section 3.3.1 are used in order to be most realistic.

Fig. 3.25: Output power of the start-up scenario using the electron beam parameters from Section 3.3.1 of the coaxial diode MIG for a) TE−34,19 mode at 170 GHz and b) TE40,23 mode at 204 GHz. The black curve indicates the pitch factor versus the beam energy.

The simulation includes 63 modes and considers 2565 electrons. There is no need to distinguish between electrons in cross section or volume, given that a beamlet approach is considered. The spread of the pitch factor is between ∆𝛼 = 5.7 % rms and 6.2 % rms during the start-up. The maximum magnetic eld is 𝐵cav = 6.88 T at the center of the cavity midsection. The nominal mode is the rst excited mode. All other modes cannot get dominant because of the very low pitch factor. A wall loading of 2 kW/cm2 is reached at a beam energy of 𝑈beam = 92.8 keV and a beam current of 𝐼beam = 72 A. The resulting pitch factor is 𝛼 = 1.27 and the insert loading of 0.17 kW/cm<sup>2</sup> is below the insert loading constraint. An output power of 𝑃out = 2.54 MW with an interaction eciency of 𝜂 = 38.8 % is achieved.

The start-up scenario of the TE40,23 mode operating at 204.17 GHz and the corresponding pitch factor is depicted in Fig. 3.25b. The simulation considers 74 modes and 2079 electrons. The spread of the pitch factor is between ∆𝛼 = 5.4 % rms and 6.5 % rms during the start-up. The maximum magnetic eld is 8.27 T. The pitch factor at 204 GHz is higher at lower beam energy compared to the 170 GHz case. The nominal operating parameters are for a beam energy of 𝑈beam = 89.9 keV and a beam current of 𝐼beam = 68 A. The resulting pitch factor is 𝛼 = 1.25 and the insert loading of 0.05 kW/cm<sup>2</sup> is far below

the maximum tolerable constraint. An output power of 𝑃out = 2.08 MW with an interaction eciency of 𝜂 = 34.3 % is achieved. The output power is increased by ≈ 22 % through optimization of the coaxial cavity.

#### **Start-Up Scenario Considering the cMIG**

In the following, the interaction simulations are presented using the electron beam parameters of the cMIG. In contrast to the coaxial diode MIG, the corotating TE34,19 mode is used as operating mode at 170 GHz. The start-up up is given in Fig. 3.26. The nominal operating point is given by a beam energy of 𝑈beam = 90.1 keV, a beam current of 𝐼beam = 77 A and a magnetic eld strength of 𝐵cav = 6.86 T. The resulting pitch factor is 𝛼 = 1.3. The theoretical output power is 𝑃out = 2.63 MW with an interaction eciency of 𝜂 = 37.8 %.

The nominal operation point of the TE40,23 mode is identied at a beam energy of 𝑈beam = 87.4 keV, a beam current of 𝐼beam = 68 A and a magnetic eld strength of 𝐵cav = 8.23 T. The resulting pitch factor is 𝛼 = 1.23. The output power is 𝑃out = 2.05 MW with an interaction eciency of 𝜂 = 34.5 %. In summary, the performance using the cMIG and the diode MIG are quite similar according to similar electron beam parameters.

Fig. 3.26: Start-up scenario considering the electron beam parameters of the coaxial cMIG for a) TE34,19 mode at 170 GHz and b) TE40,23 mode at 204 GHz.

#### **Start-Up Scenario Using MIG Parameters of iMIG**

Finally, the interaction simulation with the beam parameters of the iMIG are presented for 204 GHz. Figure 3.27 shows the start-up scenario for the TE40,23 mode. The nominal operation point is at a kinetic beam energy of 𝑈beam = 86.4 keV, a beam current of 𝐼beam = 68 A and a magnetic eld strength of 𝐵cav = 8.23 T. The resulting output power is 𝑃out = 2.08 MW with an interaction eciency of 𝜂 = 37.8 %.

In conclusion, three MIGs at KIT have been adapted to the magnetic eld prole of the new SC magnet providing sucient performance. An output power of 2 MW is achieved in theory using each MIG. The best results have been obtained using the iMIG. The resulting output power is only slightly above the required 2 MW and therefore has a small reserve. An increase is possible by either increasing the cavity radius, accepting higher mode eigenvalues or tolerating higher wall loading.

Fig. 3.27: Start-up scenario considering the electron beam parameters of the iMIG for the TE40,23 mode at 204 GHz. The black curve shows the corresponding pitch factor.

## **3.3.3 Design of a 170/204 GHz Quasi-Optical Output Coupler**

A quasi-optical output coupler consists of a launcher and several mirrors. It converts the excited high-order rotating gyrotron cavity mode into a linearly polarized Gaussian beam. The existing 170 GHz single-frequency quasioptical output coupler suer from a low Gaussian mode content at 204 GHz. A new quasi-optical output coupler is designed considering the following requirements:


The deviation of the relative caustic radii in a mode series should be < 4 % to ensure a proper operation of the launcher at dierent center frequencies. The two operating modes at 170 GHz and 204 GHz have a deviation of 2 %. The relative caustic radius of the TE34,19 mode is 𝑟c∕𝑟cav = 0.3232 and that of the TE40,23 mode is 𝑟c∕𝑟cav = 0.3167.

A new quasi-optical output coupler is designed, which is optimized for the operation at 170 GHz and 204 GHz [103] using the numerical method described in [90]. First, a dual-frequency launcher has been designed. The launcher input radius is reduced from 32.5 mm to 32.0 mm. In addition, the slope of the inner wall is modied. A graph of the inner contour is plotted in Fig. 3.28. The designed perturbations are in the range of −0.142 < ∆𝑅 < 0.142 mm. The resulting Gaussian mode content at the launcher aperture is 96.5 % for the operation at 170 GHz and at 204 GHz. The maximum ohmic wall loading is 560 W/cm2 at 170 GHz and 540 W/cm2 at 204 GHz at the launcher tip. The microwave beam radiated by the launcher is focused and

Fig. 3.28: Inner wall prole of the dual-frequency launcher [103].

shaped by a set of three mirrors. This system contains a quasi-elliptical and two beam shaping mirrors. The two beam shaping mirrors are designed according to the method described in [104]. The reduced launcher radius leads to a wider microwave beam pattern. The whole microwave beam can be collected either by a larger mirror or a shift of mirror 1. Since a larger mirror does not t into the existing mirror box, the quasi-elliptical mirror with similar width is positioned 20 mm closer to the gyrotron axis. The positions of the beam shaping mirrors are adapted accordingly.

The resulting beam waist at the position of the output window should be around 24.5 mm corresponding to half of the window radius of 49 mm. This ensures an almost homogeneous eld distribution at the center of the window and low truncation losses. The resulting beam waists are 21.6 mm and 24.6 mm for the TE34,19 mode and 19.27 mm and 24.2 mm for the TE40,23 mode in the x- and y-direction. The 𝐸-eld intensity distributions at the window plane are depicted in Fig. 3.29. The calculated stray radiation is given by 1.5 % at 170 GHz and 204 GHz. The components have been manufactured by Reinhold Mühleisen GmbH, Gerlingen, Germany. They are made of high-purity and oxygen-free copper (Cu-OFE) with at least 99.99 % copper. The material does not contain any elements that can evaporate at high temperatures and collapse the vacuum, has a high electrical and thermal conductivity and is therefore suitable for the use in vacuum tubes [105]. Figure 3.30 presents the manufactured components. The manufacturing precision is measured to be < 20 µm for resolving successfully the perturbations.

Fig. 3.29: 𝐸-eld intensity distribution of the microwave output beam in the window plane at a) 170 GHz and b) 204 GHz.

Fig. 3.30: Pictures of the manufactured quasi-optical output coupler. Launcher, quasi-elliptical and two beam shaping mirrors from left to right are depicted.

# **3.4 Frequency Step-Tunability**

Frequency step-tunability in a bandwidth of ± 10 GHz around each center frequency (i.e. at 170 GHz and at 204 GHz) is proposed for EU DEMO. This study is the rst to consider the insert loading in the mode selection process.

### **Dierent Methods for Frequency Tuning**

Theoretically, the operating frequency of a gyrotron can be tuned by a mechanical change of the cavity structure [106]. However, this method is not employed because it requires an opening of the tube. A preferred method is to tune the magnetic eld 𝐵cav and/or the accelerating voltage 𝑉acc [44]. According to Eq. 2.2, tuning the accelerating voltage leads to a strong variation of 𝛾. Additionally, it allows frequency tuning only in the range of several ten MHz. Hence, the most eective method to cover a large frequency bandwidth is to use dierent modes at discrete frequency steps. This can be achieved by tuning the magnetic eld strength. Then, the operating points are tuned individually for each mode for maximum performance. The time required for the frequency tuning depends on the maximum possible speed for changing the magnetic eld strength of the SC magnet. A rst hybrid magnet achieved a tuning speed of 652 mT/s [107]. This corresponds to ≈ 15 GHz/s. However, this magnet is only able to operate in 1 ms pulses and a duty cycle of 0.1 %. Another 7 T SC magnet manufactured by Japan Superconductor Technology Inc. (JASTEC), Hyogo, Japan provided a tuning speed of 40 mT/s (≈ 0.96 GHz/s) in a range of ±200 mT [108].

#### **Broadband gyrotron output windows**

The proposed frequency bandwidth is not covered by classical gyrotron windows. According to Fig. 3.1, high reection occurs outside the resonance of the window. Alternative concepts are double-disk windows or Brewster-windows. A double-disk window with variable spacing in between is used at ASDEX Upgrade [109]. The distance between the disks denes the transparency according to Fabry-Perot resonances [110]. However, this

type of window is only transparent at discrete frequencies. Therefore, a gyrotron window under the Brewster angle 𝜃Brew is under investigation [82, 111, 112], which does not show a resonant transmission behavior for a specic polarization. The Brewster angle of CVD-diamond is 𝜃Brew = 67.2°. The required diameter for a tilted diamond disk has to be consequently larger. The largest diamond disk with a diameter of 180 mm has been manufactured successfully with a thickness of 2 mm, as published in [113].

## **3.4.1 Mode Selection Process**

Equidistant frequency steps can be achieved by a mode series including the azimuthal neighboring modes [114]. This approach is published for hollow-cavity gyrotrons in [31, 42, 44] and for coaxial-cavity gyrotrons in [44]. Azimuthal neighboring modes have a frequency distance of 2.1 GHz in the coaxial-cavity technology, which requires 11 modes to cover the entire bandwidth of ±10 GHz. An initial list of the chosen modes around 170 GHz and 204 GHz is summarized in Tab. 3.9 considering the approach of azimuthal neighboring modes. A detailed list with the important parameters of the modes can be found in Appendix A.2.

Firstly, each mode has been simulated individually to validate its performance and to dene the nominal operating parameters. The insert loading strongly increases towards the lower frequencies in both frequency bands. Five modes exceed the insert loading limitation at the mode series around 170 GHz. These modes are the TE33,19 at 167.9 GHz, TE32,19 at 165.8 GHz, TE31,19 at 163.7 GHz, TE30,19 at 161.6 GHz and TE29,19 at 159.5 GHz. Three modes exceed the insert loading limitation at the mode series around 204 GHz. Here, the modes are the TE37,23 at 197.8 GHz, TE36,23 at 195.7 GHz and TE37,23 at 193.6 GHz. This is due to the decreasing caustic radii and thus the decreasing distance to the insert. Therefore, the covered bandwidth is reduced to in total 10 GHz and 14 GHz around the center frequencies of 170 GHz and 204 GHz. In addition, the caustic radius increases monotonically during the mode series. The deviation of the caustic radius of this mode series exceeds the requirement of 4 %. In conclusion, an azimuthal


Table 3.9: Summary of the initial mode series for frequency step-tunability around 170 GHz and 204 GHz using azimuthal neighboring modes and their frequency distance to the center frequency.

change in the mode index is not sucient to satisfy the requirements for the coaxial-cavity technology.

Hence, an additional change in the radial mode index is required for frequency step-tunability. A new mode series should provide almost equidistant frequency steps. Figure 3.31 presents a mode spectrum with dierent possible modes around the center frequency at 170 GHz. Modes highlighted in red either exceed the insert loading constraint or the deviation limit of the caustic radius, or generally show inecient operation. Modes highlighted in green are suitable candidates for an ecient mode series for frequency-step tunability. The similar study has been performed at 204 GHz.

In conclusion, the best performance can be achieved by a jump of the radial mode index from the TE𝑚,𝑝 mode at the center frequency to the TE𝑚+2,𝑛−1 mode. The modes before and after these two are changed by the azimuthal mode index for equidistant frequency steps of 2.1 GHz. A frequency step of 1.2 GHz is identied at the mode jump. A summary of the proposed mode series is given in Tab. 3.10.

Fig. 3.31: Mode spectrum for the identication of a suitable mode series for ecient frequency step-tunability at the center frequency of 170 GHz. The green bars indicate modes showing in simulations high performance and reliability. The red colored bars indicate modes do not satisfy the requirements.



Figure 3.32 presents a comparison of the caustic radii of the optimized and initial mode series. The optimized mode series has a reduced total deviation in caustic radius compared to the initial mode series. The deviation of the relative caustic radius of the initial mode series is ±9.0 % around 170 GHz and ±7.7 % around 204 GHz. The optimized mode series shows a deviation in relative caustic radius of ±4.1 % around 170 GHz and ±3.7 % around 204 GHz to the mean value.

The performance of the dual-frequency 170/204 GHz quasi-optical output coupler is validated according to the mode series. The number of modes having a Gaussian mode content (GMC) higher than 95 % is summarized in Tab. 3.11. The number of modes satisfying the EU DEMO requirements are doubled using the optimized mode series. In the frame of this research, the used quasi-optical output coupler is optimized for the operation at the center frequencies of 170 GHz and 204 GHz. It is not optimized for frequency-step tunability. However, this quasi-optical output coupler is working with a high performance for a large number of modes. The simulation results of the quasi-optical output coupler are discussed in Appendix A.2.3. It is postulated that a quasi-optical output coupler designed for a relative caustic radius equal to the mean value of Fig. 3.32 will have even better performance.

Fig. 3.32: Relative caustic radii of the mode series using the classical approach in black and the optimized mode series colored in blue, at the center frequency bands around a) 170 GHz and b) 204 GHz.

Table 3.11: Number of modes satisfying the specied Gaussian Mode Content (GMC) of the initial and optimized mode series at 170 GHz and 204 GHz. The numbers in parentheses indicate the number of modes that can be operated if the insert loading is considered.


### **3.4.2 Dynamic Step-wise Frequency Tuning**

Generally, step-wise frequency tuning has to be divided into a) tuning from high to low and b) from low to high frequencies.

#### **Frequency Tuning from High to Low Frequencies**

Continuous magnetic eld tuning is an eective method for frequency steptuning from high to low frequencies as described in [31], [114]. Magnetic eld tuning is performed in the range from 7.32 T to 6.54 T around 170 GHz and from 8.80 T to 7.85 T around 204 GHz to cover the entire frequency bandwidth of ±10 GHz. Figure 3.33 presents a multi-mode simulation of continuous frequency step-tuning from high to low frequencies. It ranges from 180.5 GHz to 159.5 GHz and from 214.7 GHz to 194.6 GHz. The beam current is set to be constant to 75 A and 68 A for the frequency band around 170 GHz and 204 GHz. The beam radii are adjusted to maximize the beam coupling to the corresponding modes. The nominal operation parameters are limited by the maximum wall loading constraint of 2 kW/cm2 . The beam energy is modied for each mode to maintain optimized detuning for maximum output power and eciency. A summary of the nominal parameters and the resulting output powers and eciencies are summarized in Tab. 3.12 and are highlighted in Fig. 3.33. A velocity spread of ∆𝛼 = 6.0 %

and a kinetic energy spread of ∆𝐸 = 0.1 % are considered. The insert loading of the chosen modes does not exceed the specied limit. The maximum deviations of the output power in the mode series are given by 5 % and 18 %.

Table 3.12: Nominal operation parameters (beam energy 𝑈beam, magnetic eld strength 𝐵cav at cavity and electron beam radius 𝑟beam) and simulation results (output power 𝑃out and interaction eciency 𝜂elec) in the frequency band of 170 GHz. and 204 GHz.


Fig. 3.33: Simulation of continuously stepwise frequency tuning using the optimized mode series from high to low frequency around the center frequencies of a) 170.0 GHz and b) 204.1 GHz.

Fig. 3.34: Comparison of the frequency steps of the classical approach and the optimized mode series.

Figure 3.34 depicts the frequency steps along the tuning at 204 GHz. The distances between the operating frequencies are equidistant and correspond to 2.1 GHz. Except, a smaller step of 1.2 GHz, which is due to the jump from the nominal center TE𝑚,𝑝 mode to the TE𝑚+2,𝑛−1 mode. The at tops represent the operation at the dened nominal operating parameters.

#### **Frequency Tuning from Low to High Frequencies**

Continuous magnetic eld tuning is not possible for tuning the frequency from low to high frequencies in terms of hysteresis eects [115, 116]. Increasing the magnetic eld does not lead to a jump to the next higher mode. The dominant mode still operates at low detuning and low output power. This eect has already published theoretically in [31] and experimentally in [116]. Figure 3.35 illustrates this eect for the TE40,23 and TE41,23 modes. The nominal magnetic eld for the TE41,23 mode is 8.40 T. A simulation is performed considering both modes while increasing the magnetic eld strength from 8.21 T to 8.45 T. However, the TE41,23 mode is not excited.

Fig. 3.35: Output power of the TE40,23 and TE41,23 modes for magnetic eld tuning from low to high frequencies. The dashed line indicates the behavior of the TE41,23 mode in single-mode calculation.

The dashed line describes for illustration the behavior of the TE41,23 mode in single-mode calculation.

An alternative approach is to shut down the gyrotron, change the magnetic eld and turn it back on, as published in [31]. Prior to this study, each mode was simulated individually. This coaxial-cavity design is capable to excite each mode with common start-up scenario. Figure 3.36 presents a frequency tuning from low to high frequencies using the TE42,22 mode, the TE40,23 mode and the TE41,23 mode. The graph is limited to 3 modes for better visibility. Each mode is operating at the nominal operation parameter. A shut down can be realized either by ramp down the applied acceleration voltage completely or by reducing the pitch factor [31]. A MIG in triode conguration can vary the pitch factor in a large range. The interaction stops by decreasing the pitch factor to a value of e.g. 0.5. The gyrotron is ramped up in the following by an increase of the pitch factor to its nominal value.

Fig. 3.36: Frequency step-tunability from low to high frequencies. Only the TE42,22 mode, the TE40,23 mode and the TE41,23 mode are plotted in terms of visibility.

## **3.5 Performance Analysis at 136 GHz**

A validation of the performance of the key components at 136 GHz is performed. The key components under investigation are the MIG, the coaxial cavity and the quasi-optical output coupler. A summary of possible operating modes at 136 GHz are summarized in Tab. 3.13. The TE28,15 mode satises the requirements best. The mode operates at a frequency of 135.8 GHz, resulting in a low window reection of 0.03 % and has a deviation in caustic radius of 3.09 %.


Table 3.13: Mode candidates and their key parameters for the operation at 136 GHz.

## **3.5.1 Operating the Magnetron Injection Guns at 136 GHz**

The MIGs at KIT are being investigated for their applicability at 136 GHz. The selected TE28,15 mode has a beam radius of 𝑟beam = 10.35 mm and a maximum magnetic eld strength at the center of the midsection of 𝐵cav = 5.40 T.

#### **Coaxial diode MIG**

The coaxial diode MIG discussed in Section 3.3.1 is not suited to be used at 136 GHz. The velocity spread is well above 20 % and the pitch factor is out of the operating range of 𝛼 = 1.1 − 1.3. There is not a design by changing the anode shape that covers all three frequencies (136, 170 and 204 GHz). The electron beam parameters can be set to nominal values only by modifying the cathode and emitter shapes. This requires re-manufacturing of the emitter ring, which is the most expensive component of a MIG. Therefore, this MIG is skipped as potential candidate at 136 GHz. A triode MIG must be used for this purpose.

#### **iMIG**

The inuence of the modulation anode on the electron beam parameters using the iMIG is presented in Fig. 3.37. In general, the iMIG satises the beam requirements at a voltage of 28 kV applied to the modulation anode. The resulting pitch factor is 𝛼 = 1.2 and the transverse velocity spread is 𝛿𝛽<sup>t</sup> = 3 % for a magnetic eld angle of 𝜙emt = 0° at the emitter. A study on secondary electrons shows that this design prevents trapped electrons. The iMIG is a suitable candidate to be used at 136 GHz.

#### **cMIG**

The triode-type cMIG can be used for operation at 136 GHz. The resulting pitch factor is 𝛼 = 1.28 at a transverse velocity spread of 𝛿𝛽<sup>t</sup> = 2.25 %. Figure 3.38 presents a study about secondary electrons. Some of the electrons emitted from the rear part and the front part of the cathode nose are collected by the anode and the insert, respectively. The other electrons are

guided to the cavity. Figure 3.38b shows the pitch factor at the cavity versus the emitting position of the corresponding electron. Trapped electrons are prevented because the maximum pitch factor is below 𝛼max = 3.

Fig. 3.37: Pitch factor 𝛼 and transverse velocity spread 𝛿𝛽<sup>t</sup> over a wide range of the magnetic eld angle 𝜙emt and applying dierent voltages at the modulation anode 𝑉anode of the iMIG.

Fig. 3.38: a) Trajectories of electrons emitted from the cathode surface. b) Pitch factor 𝛼 at the cavity versus the axial position of the emitting point of the cathode which are highlighted with blue crosses in Fig. 3.38a.

### **3.5.2 Verification of the Interaction Cavity at 136 GHz**

The performance of the coaxial cavity using the TE28,15 mode is investigated. A velocity spread of ∆𝛼 = 6 % and a kinetic energy spread of 0.1 % are considered in the simulations. An adiabatic start-up simulation is presented in Fig. 3.39. The nominal operation point is given by a kinetic energy of the electrons of 𝑈beam = 82.4 keV, a beam current of 𝐼beam = 70 A and a magnetic eld strength of 𝐵cav = 5.40 T. The pitch factor is 𝛼 = 1.3. The resulting output power is 𝑃out = 2.42 MW with an interaction eciency of 𝜂 = 41.9 %. The wall loading is 1.35 kW/cm<sup>2</sup> and the insert loading is > 0.30 kW/cm2 . Thus, the insert loading exceeds the maximum limit. An output power of 1.1 MW with an interaction eciency of 20.7 % is achieved considering an insert loading of 0.2 kW/cm2 . This is the technical limitation of this mode given by the loading constraint published in [78]. However, multi-physics simulations have been performed in [99, 100] for an advanced insert cooling system coming to the result that the insert can theoretically withstand a heat ux of up to 0.39 kW/cm2 . In this case, the TE28,15 mode can be used considering this assumption.

Fig. 3.39: Adiabatic start-up scenario for the TE28,15 mode operating at 135.8 GHz.

### **3.5.3 Operating the Quasi-Optical Output Coupler at 136 GHz**

Both available quasi-optical output couplers, namely the 170 GHz singlefrequency design and the 170/204 GHz multi-frequency design have been evaluated. The input mode is the TE28,15 mode at 135.8 GHz. Both designs provide a Gaussian mode content close to 95 %. Figure 3.40 shows the simulation result of the 𝐸-eld intensity and phase pattern for the 170/204 GHz dual-frequency quasi-optical output coupler at 136 GHz. In conclusion, both designs satisfy the EU DEMO requirements providing a Gaussian mode content of ≥ 95 %. However, the microwave output beam is shifted by 8 mm o center.

Fig. 3.40: a) 𝐸-eld intensity and b) phase pattern of the TE28,15 mode at 135.8 GHz using the quasi-optical mode converter designed for 170/204 GHz dual-frequency operation.

# **4 Upgrade of the Gyrotron Test Systems**

The test systems used in gyrotron development are divided into two categories, i.e. the low-power "cold"-measurements and the high-power "hot" measurements. Gyrotron key components are veried in cold-measurements without the usage of a gyrotron nor the necessity of a SC magnet or a highvoltage power supply. Those systems excite the required input mode to supply the dedicated gyrotron key components for verication. In this way, a pre-validation of e.g. the quasi-optical output coupler or the gyrotron output window is performed [81].

The category of "hot"-measurements includes the monitoring of the gyrotron operation. In this way, an in-direct verication of the design and fabrication quality of its sub-components is performed. Examples are the calorimeters for microwave power measurements and the frequency diagnostics for validating the frequency spectrum purity [117]. The existing frequency diagnostics needs to be adapted to higher operating frequencies due to the increasing demand for gyrotrons operating above 200 GHz.

## **4.1 Measurement of Quasi-Optical Components**

The quasi-optical output coupler separates the electromagnetic wave from the electron beam. In addition, it converts the high-order rotating cavity mode into a linearly polarized Gaussian beam. The Gaussian mode has low transmission losses and is used to guide the microwave beam to the plasma vessel [11, 118]. Proper pre-validation of these quasi-optical output couplers is vital before the installation into the gyrotron. The components, and in particular the launcher, are very sensitive to manufacturing tolerances, which are in the range of a few µm. Therefore, the gyrotron cavity mode needs to

be excited at low power levels for the verication in "cold" measurements. There are dierent methods for the excitation of high-order modes:


The method discussed in (i) is not suitable for exciting gyrotron cavity modes with mode indices 𝑚, 𝑝 ≫ 1. Method (ii) is a purely theoretical design and is not feasible in practice. The excitation of higher order modes by a perforated metal insert of method (iii), is applicable, however, has compared to the quasi-optical method (iv) a lower eciency for the excitation of very high-order modes. This is due to the need of 𝑚 holes on the circumference of the insert. In terms of the small insert radius the holes become smaller with increasing azimuthal mode index 𝑚. Technical feasibility is combined with high mode purity in the quasi-optical mode generator (method (iv)). In the following, its principle and its upgrade including an automated measurement procedure is presented. Then, the TE34,19 mode at 170 GHz and the TE40,23 mode at 204 GHz are excited. Finally, the 170/204 GHz dualfrequency quasi-optical output coupler is veried in cold measurements.

#### **4.1.1 Mode Excitation Principle in a Low Power Test Stand**

Figure 4.1 shows a photo of the quasi-optical mode generator at KIT. The design is based on [125]. A schematic of the measurement setup is given in Fig. 4.2. A step-tapered horn antenna is used to radiate the signal from a vector network analyzer (VNA) as a Gaussian beam (TEM0,0 mode) [124]. Then, a lens system forms an astigmatic Gaussian beam with a plane phase front at the position of a quasi-parabolic (q.-p.) mirror [126]. Spillover losses are reduced by the astigmatism. The quasi-parabolic mirror is adjusted in the vertical and horizontal position and the angle to achieve maximum eciency. The electromagnetic wave is reected at the quasi-parabolic mirror and couples through the perforated cavity wall. The beam is focused close to

drivers

Fig. 4.1: Photo of the mode generator setup used for the TE40,23 mode at 204 GHz.

the caustic radius of the desired cavity mode. A coaxial insert is used and the cavity is designed specically for the low-power tests using a scattering matrix code [127]. The insert improves the separation to competing modes. A non-linear up-taper is assembled to the mode generator setup for the adjustment procedure [128]. This greatly increases the waveguide radius, resolving the measured mode eld intensity patterns with higher quality. The non-linear up-taper is removed and replaced by the quasi-optical output coupler for verication after the desired mode has been successfully excited. The output eld pattern is measured using a rectangular standard waveguide pick-up antenna with a chamfer of 15°. The chamfered antenna edges reduce reections back to the mode generator. The antenna is mounted on a 3D measurement arm and samples the pattern of the generated mode point by point [125]. The received signal is evaluated using the VNA.

In the past, the optimization of the mode quality was a time-consuming task. The sub-components were adjusted by manual optimization. In this work, the mode generator setup has been improved with high-precision linear drivers and a goniometer which are controlled via the computer.

Fig. 4.2: Schematic view of a mode generator setup, VNA, 3D measurement arm and PC. The device under test (DUT) is illustrated in the graph as a launcher. In principle the DUT can be the radiated mode after the non-linear up-taper, the launcher or the complete quasi-optical output coupler.

This upgrade allows the use of new methods for an autonomous mode generator setup.

## **4.1.2 Key Components of a Mode Generator System**

The structure of the quasi-optical mode generator consists of four key components: 1) the horn antenna with lens system, 2) the quasi-parabolic mirror, 3) the coaxial cavity and 4) the non-linear up-taper. The design of the key components for the TE34,19 mode at 170 GHz and TE40,23 mode at 204 GHz are discussed in the following,

#### **Horn Antenna with Lens System**

The mode generator setup requires a Gaussian-like HE<sup>11</sup> hybrid mode as input mode. A corrugated HE<sup>11</sup> horn antenna [129] can be used for this purpose. A cheap alternative is a Potter horn antenna [130] with two straight tapers [131]. Figure 4.3 depicts a schematic of the Potter horn. This type of antenna has a low sidelobe level down to -30 dB. This allows an optimal illumination of the quasi-parabolic mirror and avoids reection artefacts on the mode generator housing. The Potter horn is a dual-mode antenna. The dominant TE<sup>11</sup> and the high-order TM<sup>11</sup> modes are excited in a given ratio. Superposing the TE<sup>11</sup> mode with 85 % of the power, the TM<sup>11</sup> mode with 15 % of the power and a phase shift of 180° between the modes results in a HE<sup>11</sup> mode with 99.19 % purity [132]. The conversion of the TE<sup>11</sup> input mode to the required mode ratio with associated phase shift is not achievable in a wide frequency range. Therefore, a Potter horn is designed at 170 GHz and another at 204 GHz. The geometry is optimized using a scattering-matrix code [133], which is much faster for this purpose in contrast to commercial 3D software tools like CST Microwave Suite [134], since only discretization in axial direction is required. Figure 4.4 and Fig. 4.5 present the simulation results with the parameters given in Tab. 4.1. The ratio at the end of the

Fig. 4.3: Parametric schematic of a Potter horn.

antenna is 85.4 % and 14.0 % at 170 GHz and 85.0 % and 14.2 % at 204 GHz for the TE<sup>11</sup> mode and TM<sup>11</sup> mode, respectively. The phase shift ∆𝜑 between the two modes is ∆𝜑 = 180◦ by selecting the length 𝑙<sup>5</sup> accordingly.


Table 4.1: Parameters of the designed Potter horn antennas operating at 170 GHz and 204 GHz.

Fig. 4.4: a) Relative mode power and b) phase of the Potter horn at 170 GHz according to the parameters of table 4.1. The schematic of the Potter horn is not in scale. The phase dierence corresponds to ∆𝜑 = 180◦ .

Fig. 4.5: a) Relative mode power and b) phase of the Potter horn at 204 GHz according to the parameters of table 4.1.The schematic of the Potter horn is not in scale. The phase dierence corresponds to ∆𝜑 = 180◦ .

The optimum simulation results are validated using CST Microwave Suite. The correct transformation of the input mode into the TE<sup>11</sup> and TM<sup>11</sup> modes in the given ratio is depicted in Fig. 4.6a. The 𝐸-eld distribution at the end of the antenna is perfectly cylindrical, resulting in a transformation into the HE<sup>11</sup> mode. Figure 4.6b depicts the antenna pattern in the 𝐸- and 𝐻-plane at 170 GHz and at 204 GHz. The sidelobe level in the 𝐸-plane is -30 dB and is in agreement with the theoretical value. The half power beam width is 27.0° and 26.6° for the 𝐸- and 𝐻-plane at 204 GHz and 26.8° in both planes at 170 GHz. The directivity is 17.2 dBi for both antennas. Nearly identical results were expected and have been demonstrated since the antennas were scaled with the frequency. Except for the input section which had to be matched to the dierent standardized waveguide anges. The Potter horns are manufactured using a step driller, which has the specic shape of the inner contour of the Potter horn, as discussed in Appendix A.4.

The radiated beam has to be focused to the quasi-parabolic mirror. This can be achieved by using a metallic reector [135, 136] or by a lens system [125, 137]. The mode generator with a lens system benets from a more compact and one-dimensional setup. The quasi-parabolic mirror would be only partially illuminated using a spherical shaped lens. Therefore, and in

Fig. 4.6: a) Intensity prole of the radiated eld at 204 GHz and b) far eld pattern of the antennas at 170 GHz and 204 GHz in 𝐸- and 𝐻-plane using CST Microwave Suite.

order to reduce spillover losses, two cylindrical shaped lenses are used to form an astigmatic beam. One lens is orientated along the horizontal and the second along the vertical direction. A measurement using the Potter horn and the lens system is shown in Fig. 4.13.

#### **Quasi-Parabolic Mirror**

The quasi-parabolic mirror focuses the astigmatic beam in-phase close to the caustic of the desired mode in the coaxial cavity. A schematic is depicted in Fig. 4.7. The shape of the quasi-parabolic mirror can be derived by the requirement of equal optical length between the plane phase front of the incident beam and the phase front at the caustic [138]. The principle of the quasi-parabolic mirror is described in [76]. In this publication, the quasi-parabolic mirror is used in a gyrotron output coupler to convert the high-order gyrotron cavity mode into a free-space Gaussian beam. Here, the mirror operates in reverse direction, i.e. the Gaussian beam is focused close

Fig. 4.7: Schematic of the principle of the quasi-parabolic mirror.

to the caustic. The shape of the quasi-parabolic mirror can be calculated as discussed in [76]:

$$\mathbf{z}(\mathfrak{a}) = 2F \tan\left(\frac{\mathfrak{a}}{2}\right) + r\_c \left(1 + \mathfrak{a} \tan\left(\frac{\mathfrak{a}}{2}\right)\right) \tag{4.1}$$

$$y(\mathfrak{a}) = F\left(\tan^2\left(\frac{\mathfrak{a}}{2}\right) - 1\right) + r\_c\left(\frac{\mathfrak{a}}{2}\tan^2\left(\frac{\mathfrak{a}}{2}\right) + \tan\left(\frac{\mathfrak{a}}{2}\right) - \frac{\mathfrak{a}}{2}\right) \tag{4.2}$$

𝐹 is the desired focal length. 𝛼 is the angle between the incident and reected wave. 𝑟<sup>𝑐</sup> is the caustic radius. The width of the quasi-parabolic mirror must be larger than the midsection of the mode generator cavity to minimize diraction losses. The height is selected based on the limited beam waist of the lens, which is dened by the lens diameter. The caustics of the TE34,19 and TE40,23 modes are similar. Therefore, the identical quasiparabolic mirror can be used at 170 GHz and 204 GHz with 𝐹 = 30 mm and 𝑟<sup>𝑐</sup> = 9.5 mm.

#### **Coaxial Cavity of the Mode Generator**

The cavity for the mode generator consists of a down-taper, a midsection and an up-taper. An increased quality factor is required in comparison to the gyrotron cavity for a more robust mode excitation. There are two dierent methods to increase the quality factor. First, a steeper down- or/and uptaper to increase the reected wave, and second, increasing the midsection length. A wider illumination area of the midsection is selected, in terms of the required geometrical width of the quasi-parabolic mirror. Therefore, the midsection of the cavity of the mode generator is chosen to be 22.0 mm. It is much longer compared to the 13.6 mm of the gyrotron cavity. The schematic is presented in Fig. 4.8. The increased length leads to a quality factor of the mode generator cavity of 3266 compared to 1094 at 170 GHz and 4897 compared to 1647 at 204 GHz of the gyrotron cavity.

The operating frequency depends on the perforated, translucent outer wall. Two dierent cavities are designed to optimize the coupling for each frequency individually. A derivation of the coupling through a translucent wall

Fig. 4.8: Schematic of the coaxial cavity for the mode generator setup.

is given in Appendix A.5. The diameter of the coupling holes has been set to 0.634 mm at 170 GHz and 0.590 mm at 204 GHz. The diameters are in cut-o and are chosen for a coupling of about -20 dB. Due to the coupling holes the eective cavity radius is slightly larger [138]. In the rst publications on quasi-optical mode generators, the coupling holes were placed only in the illuminating region [123, 136]. This results in a cavity without a constant eective cavity radius. The asymmetric cavity leads to an increased counter-rotating amount of the excited mode. Therefore, coupling holes are placed on the complete circumference [137]. The slightly increased eective cavity radius results in a downshift of the frequency. This can be retuned by a reduction of the cavity radius in the design process leading to the cavity radius of 29.415 mm compared to 29.55 mm of the gyrotron cavity.

In the previous work, the outer structure was manufactured from brass, which has good electrical conductivity and therefore low losses compared to other materials. In this work, the cavity is manufactured out of stainless steel, since the cut-o holes in the midsection can be laser cutted. This method has higher precision than the conventionally machining. Therefore, the cut-o holes are almost identical, which reduces also the counter-rotating amount. Later experiments do not show reduced eciency in terms of the used material.

Each mode generator cavity is built in the coaxial structure containing an additional insert for an improved mode selection process [65, 66]. An estimation about the number of modes that can operate in a waveguide with a xed radius is given by Eq. (2.37). About 2800 modes at 170 GHz and 4100 modes at 204 GHz are able to propagate considering a cavity radius of 𝑟cav = 29.415 mm of the proposed cavity. The mode excitation benets from a specic choice of the ratio of insert and cavity radius. The ratio is chosen based on the calculated eigenvalue map with the aim to reduce the inuence of neighboring modes. The eigenvalue curves for the coaxial setup are given by [65, 66].

$$\frac{Y\_m\left(\chi\_{m,p}\frac{r\_{\text{insert}}}{r\_{\text{cur}}}\right)}{Y\_m\left(\chi\_{m,p}\right)} = \frac{J'\_m\left(\chi\_{m,p}\frac{r\_{\text{insert}}}{r\_{\text{cur}}}\right)}{J'\_m\left(\chi\_{m,p}\right)}\quad. \tag{4.3}$$

𝐽<sup>𝑚</sup> and𝑌<sup>𝑚</sup> are the Bessel and Neumann functions of order 𝑚. The eigenvalue 𝜒𝑚,𝑝 is the 𝑝 𝑡ℎ-zero of the Bessel function. Figure 4.9 depicts the eigenvalue spectrum at 170 GHz and 204 GHz. An optimal excitation point for the mode generator is at 𝐶 = 𝑟cav∕𝑟insert = 3.347 for the TE34,19 mode at 170 GHz and 𝐶 = 3.451 for the TE40,23 mode at 204 GHz. This corresponds to an insert radius of 𝑟insert = 8.79 mm and 𝑟insert = 8.52 mm, respectively. It is obvious to see, that the number of modes increase and the distances between adjacent modes decrease with increasing frequency. The inner conductor is still made of brass and mounted with a screw to a Teon cap which is placed at the end of the down-taper. The benet of a Teon cap is a good microwave transparency. Possible unwanted modes are propagating through the Teon cap and are not reected back into the cavity. Therefore, those modes do not disturb the excitation and thus do not reduce the purity of the cavity mode.

Fig. 4.9: Eigenvalue spectrum to identify the optimum ratio 𝐶 = 𝑟 cav 𝑟insert to excite a) TE34,19 mode at 170 GHz (blue) and b) TE40,23 mode at 204 GHz (red).

#### **Non-Linear Up-Taper**

A non-linear up-taper is mounted to the end of the cavity for the verication of the generated cavity mode. The shape of the non-linear up-taper is shown in Fig. 4.10. The non-linear up-taper increases the radius of the waveguide without reducing the mode purity [77]. It helps to achieve a sucient resolution to reliably determine the mode order, because of the limited spatial resolution of the receiving antenna of ≈ 1 mm. The relative mode contents along the non-linear up-taper at 170 GHz and 204 GHz are plotted in Fig. 4.11. The calculations, which have been performed using the scattering matrix code show excellent performance.

Fig. 4.10: Radius prole of the non-linear up-taper at the output of the quasi-optical mode generator.

Fig. 4.11: Relative power of the modes along the z-axis of the non-linear up-taper for the a) TE34,19 mode at 170 GHz and b) TE40,23 mode at 204 GHz.

#### **4.1.3 Novel Automated Measurement Procedure**

The most time-consuming and challenging task is to excite the operating high-order cavity mode with high purity and low counter-rotational amount. In former times, the quasi-parabolic mirror was set manually. These manually adjustable sliding units have low accuracy and high repeatability errors. In addition, as the operating frequency increases, the required adjustment accuracy increases. Therefore, the existing sliding units are replaced by electronically movable high-precision linear-drivers (LTM 60-100-HSM [139]) for the horizontal and vertical positioning each and a goniometer (MOGO 65-40-65-HiSM [140]) for the tilt. The high-precision linear drivers have a position error of < 25 µm/100 mm and a bidirectional repetition failure of < 15 µm. The goniometer has a bidirectional accuracy of < 0.02° in a range of ± 20°. These upgrades were the rst step of an automation process of the mode generator setup [141]. The ow chart of the automatized mode generator adjustment procedure is given in Fig. 4.12. The steps of the procedure are discussed in detail in the following.

Fig. 4.12: Flow-chart of automatized adjustment of the mode generator setup.

#### **Estimation of Starting Parameters for Automation Process**

The feature of the horn antenna, the lens system, the quasi-parabolic mirror and the cavity are implemented in MATLAB [142] as a 2D model. The theoretical distances between 1) the horn antenna and the lens system, 2) the lens system and quasi-parabolic mirror and 3) the quasi-parabolic mirror and cavity are calculated.

To 1): This distance has to be optimized that the complete radiated beam is focused by the lens. This guarantees an illumination of the whole quasiparabolic mirror and thus increase the eciency.

To 2): The focused beam can be represented by Gaussian optics. A plane phase front has to be applied at the position of the quasi-parabolic mirror. The distance between the lens system and the quasi-parabolic mirror is calculated at which this condition is met.

To 3): An initial starting position of the distance between the quasi-parabolic mirror and the cavity is calculated as presented in Fig. 4.7.

#### **Adjustment of the Lens System**

The combination of the Potter horn and the lens system has to be veried. The theoretically calculated distances of 1) and 2) from the previous step are validated experimentally. A measurement of the 𝐸-eld intensity and phase pattern of the Potter horn and the lens system at 204 GHz are depicted in Fig. 4.13. A clear astigmatic eld pattern is visible. Therefore, spillover losses are reduced and thus, the eciency is increased. The relevant illumination part at the quasi-parabolic mirror is 20 mm in horizontal and 30 mm in vertical direction. A phase deviation of 5° in horizontal x-direction and 12° in the vertical y-direction at 204 GHz is measured in the relevant area. Therefore, an almost plane phase front is validated at the position of the quasi-parabolic mirror.

Fig. 4.13: Measured a) 𝐸-eld intensity and b) phase pattern generated by the Potter horn and the lens system at 204 GHz.

#### **Determination of the Operating Frequency**

The operating frequency can only be estimated at the beginning. It is due to the manufacturing tolerances and the eect of the cut-o holes in the cavity wall. The transmission is measured in a given frequency band to identify resonances, which correspond to operating frequencies of specic modes. A measurement is depicted in Fig. 4.14. Therefore, the quasi-parabolic mirror is set according to the calculations. The receiving standard pick-up antenna is positioned at the caustic of the desired mode. Figure 4.14 shows resonances at 203.37 GHz, 204.07 GHz and 204.86 GHz. However, at this stage of the measurement procedure, it is not known which mode corresponds to the measured resonances. The frequency having the highest amplitude peak (here: 204.071 GHz) is selected for the next step.

Fig. 4.14: Amplitude measurement in the frequency range around 204 GHz. The receiving standard waveguide antenna is positioned on the eld maximum of the desired mode. The quasi-parabolic mirror is not yet optimized in this step.

#### **Adjustment of the Quasi-Parabolic Mirror**

The position of the quasi-parabolic mirror is adjusted to excite the cavity mode with a high purity and the correct rotation. The adjustment is highly mode selective. The receiving antenna is positioned at the eld maxima of the desired mode. A eld intensity pattern is recorded for dierent positions of the quasi-parabolic mirror, which is varied in x- and y-direction with a step size of < 100 µm and the angle is varied in 0.2° steps. Figure 4.15 shows as an example a measurement of the eld intensity at two dierent angles. The position of the high-precision linear drivers is selected to provide the highest eld amplitude.

Fig. 4.15: 𝐸-eld intensity pattern at dierent positions of the quasi-parabolic mirror and angle of the goniometer of a) 𝜑 = −1◦ and b) 𝜑 = −2◦ at an operating frequency of 204.07 GHz. The receiving standard waveguide antenna is positioned on the eld maximum of the desired mode.

#### **Acquisition of the Mode Pattern**

First, a low-resolution mode pattern with a pixel size of 0.5 x 0.5 mm is recorded. The selected pixel size is sucient for successful computer-based mode evaluation because the main characteristics can be resolved. This intermediate step is used to reduce the adjustment time. If the correct mode has been excited, a high-resolution mode pattern with a pixel size of 0.2 x 0.2 mm is recorded. However, if the excited mode is not the correct one, an iterative process is started and the next possible frequency is selected according to Fig. 4.14. The success and time of this procedure depends strongly on the starting position of the quasi-parabolic mirror. In general, three cases can be dened in this procedure. Firstly, the selected operating frequency ts to the desired mode. In this case, the mode is successfully excited after the rst iteration. Secondly, the selected operating frequency does not t to the desired mode. In this case, in minimum two iterations are required and in maximum the number of resonances, which are measured in the frequency bandwidth. Thirdly, the starting value of the position of the quasi-parabolic mirror is outside a range in which the mode cannot be excited. In this case, the measurement procedure will fail. This can be avoided by implementing some kind of knowledge after the rst iteration. The measurement in the rst iteration is similar to that described before. It is important to excite a mode with sucient mode quality in the rst iteration so that the mode evaluation process can determine the mode order. The procedure is terminated after the rst iteration if the correct mode is excited. If not, equation (4.4) can be considered to unambiguously calculate the operating frequency of the desired mode 𝑓mode. This formula is approximated by the equation (2.23) taking into account constant mechanical geometrical parameters.

$$\frac{f\_{\text{initial}}}{\mathcal{X}\_{\text{initial}}} \approx \frac{f\_{\text{mode}}}{\mathcal{X}\_{\text{mode}}} \tag{4.4}$$

𝑓initial is the determined frequency and 𝜒initial is the eigenvalue of the validated mode in the rst iteration. 𝜒mode is the eigenvalue of the desired mode and can be calculated since it depends only on the geometrical parameters. A deviation from the theoretical frequency of maximal 1 MHz is obtained using the equation (2.23). This proves the validity of the proposed method. The eect of the translucent cavity wall can be neglected here, since all eigenvalues in the given frequency range are aected equally. This leaves one unknown in the system, which is the position of the quasi-parabolic mirror, which can be solved uniquely. This upgrade makes the algorithm independent of the choice of starting positions. In addition, the upgrade limits the maximum number of iterations to two for successful excitation of the desired mode. The optimized measurement procedure results in a time saving of more than 94 % for the excitation of the TE28,10 mode operating at 140 GHz with high purity compared to the manual adjustment [143].

#### **Mode Evaluation Techniques**

The implemented mode evaluation techniques are discussed based on the example of the TE28,8 mode operating at 140 GHz. This mode of lower order results in a better readability of the shown graphs. A theoretical depiction of the absolute values of the non-rotating eld intensity of the TE28,8 mode is shown in Fig. 4.16a and a simulation of the rotating eld intensity pattern is presented in Fig 4.16b. The simulation includes the eect of the measurement equipment onto the pattern. The receiving antenna is a standardized waveguide guiding the linear polarized fundamental TE1,0 mode. This results in a measurement in cartesian coordinates instead of polar coordinates as described with equations (2.19) and (2.20). The polarization of the receiving antenna aects the mode pattern depending on the orientation, since the receiving antenna measures only one polarization. In the horizontal direction at 𝜑 = 0◦ and 𝜑 = 180◦ there is a dependence of 𝐽 ′ 𝑚( 𝜒𝑚,𝑝⋅𝑟 𝑟cav ), whereas in the vertical plane at 𝜑 = 90◦ and 𝜑 = 270◦ , there is a dependence of 𝐽𝑚( 𝜒𝑚,𝑝⋅𝑟 𝑅 ) [144]. This shows that the graph is not rotational symmetric. The mode pattern in the cross diagonals is blurred, since the eld components 𝐸<sup>𝑟</sup> and 𝐸<sup>𝜑</sup> are measured simultaneously.

(a) Calculated non-rotating eld intensity pattern.

(b) Calculated rotating eld intensity pattern.

Fig. 4.16: a) Calculated non-rotating 𝐸-eld intensity pattern of the TE28,8 mode and b) calculated rotating 𝐸-eld intensity pattern considering the measurement equipment for cold tests of the vertical polarization.

A measurement of the TE28,8 mode at 140.0 GHz is presented in Fig. 4.17. Five dierent evaluation techniques are implemented in the post-processing, to determine


**To (i):** The radial mode index 𝑝 can be determined by counting the number of rings in the radial direction. A cut through the 𝐸-eld intensity pattern is shown in Fig. 4.18. The measured and the simulated patterns are compared. The maximum amplitude levels dier ∆𝐸, which is caused by a 12° rotation of the receiving antenna. The rotation is resolved by optimizing the 3D measurement arm.

Fig. 4.17: Measured 𝐸<sup>𝑦</sup> -eld intensity pattern of the rotating TE28,8 mode operating at 140.0 GHz with a pixel size of 0.25x0.25 mm [144].

Fig. 4.18: Vertical cut of the 𝐸<sup>𝑦</sup> -eld intensity mode pattern of the TE28,8 mode operating at 140.0 GHz [144].

**To (ii)**: The azimuthal mode index 𝑚 is calculated by evaluating the phase jumps in the azimuth. Figure 4.19 depicts a measured phase pattern of the TE28,8 mode. The algorithm can determine whether a co-rotating or counter-rotating mode is excited based on the phase prole. The phase in the center and outside of the mode pattern is random because eld components do not exist there. In post-processing, the algorithm counts the number of phase jumps on a given circle. The circle is indicated in black in Fig. 4.19. Two dierent design criteria are dened for a robust evaluation, 1) counting the number of phase jumps from +180° to -180° and 2) counting the number of zero-crossings divided by two. As published in [128], the counted phase periods are related to the azimuthal mode index 𝑚. The number of phase periods is equal to 𝑚 − 1 for measuring one polarization in Cartesian coordinates. In contrast, the number of phase periods is equal to 𝑚 for a measurement in Polar coordinates. This assumption can be proven starting from the equations (2.19) and (2.20) for 𝐸<sup>r</sup> and 𝐸𝜑. The amplitudes are considered to be 𝑎 = −𝑚∕𝑟 ⋅ 𝐽<sup>𝑚</sup> (𝜒∕𝑟cav ⋅ 𝑟) and 𝑏 = 𝜒∕𝑟cav ⋅ 𝐽 ′ <sup>𝑚</sup> (𝜒∕𝑟cav <sup>⋅</sup> 𝑟). The derivation is done for a xed 𝑧-position at 𝑧 = 0. In the following, the eect

Fig. 4.19: Measured phase pattern of the TE28,8 mode operating at 140.0 GHz. The black circle indicates the circle of evaluation.

is demonstrated for the vertical polarization, considering the coordinate transformation resulting in

$$E\_y = a \cdot \exp(-j \cdot m \cdot \varphi) \cdot \sin(\varphi) + b \cdot \exp(-j \cdot m\varphi) \cdot \cos(\varphi) \quad . \tag{4.5}$$

The trigonometric terms can be expressed by exponential function and under the assumption 𝑎 = 𝑏, which results in

$$\begin{split} E\_y &= a \cdot \exp(-j \cdot m \cdot \varphi) \cdot \frac{\exp(j\varphi) - \exp(-j\varphi)}{2} \\ &+ a \cdot \exp(-j \cdot m\varphi) \cdot \frac{\exp(j\varphi) + \exp(-j\varphi)}{2} \quad , \end{split} \tag{4.6}$$

$$E\_y = a \cdot \exp(-j \cdot \varphi(m-1)) \qquad . \tag{4.7}$$

The argument of this equation is set to zero for calculation of the zerocrossing given by

$$\arg(E\_y) = \arctan\left(\frac{\sin(-\varphi(m-1))}{\cos(-\varphi(m-1))}\right) = 0\tag{4.8}$$

$$
\varphi = \frac{\pi \cdot n}{m - 1} \quad n \in \mathbb{Z} \tag{4.9}
$$

Equation (4.9) conrms that the 𝐸<sup>𝑦</sup> -eld pattern has 2(𝑚 − 1) zero crossings and 𝑚 − 1 phase jumps. In [144], the equation is solved considering 𝑎 ≠ 𝑏.

The dierence between the unwrapped phase around the circle and a linear phase is presented in Fig. 4.20. The mentioned 12◦ rotation of the receiving antenna is observed in the phase dierence. The radial eld component 𝐸𝑟 is measured in the non-highlighted area. So, the phase dierence is almost constant. These areas are around −102◦ (ideally −90◦ ) and +78◦ (ideally +90◦ ). The increase and decrease in phase are due to the fact that the azimuthal eld component 𝐸<sup>𝜑</sup> is additionally measured with a phase dierence of 90◦ .

**To (iii):** The scalar mode content is calculated using the measured 𝐸-eld 𝐸meas in comparison to the theoretical 𝐸theory of the TE𝑚,𝑝 mode using

$$\psi = \frac{\int f\_{\rm S} \left| E\_{\rm meas} \right| \cdot \left| E\_{\rm theory} \right| \, dA}{\sqrt{\int f\_{\rm S} \left| E\_{\rm meas} \right|^2 \, dA \cdot \left| f\_{\rm S} \right| E\_{\rm theory} \right|^2 \, dA}} \tag{4.10}$$

Fig. 4.20: Dierence between the unwrapped phase on the given circle and a linear phase [144].

In case of the TE28,8 mode the scalar mode content amounts to 𝜓 = 94.5 %. The mode diverges between the end of the non-linear up-taper and the receiving antenna. This and any rotation of the receiving antenna will aect the scalar mode content.

**To (iv):** The counter-rotating amount 𝑃−∕𝑃<sup>+</sup> is dened by the power ratio between the power of the unwanted counter-rotating 𝑃<sup>−</sup> and the desired co-rotating mode 𝑃+. It can be calculated by [69]:

$$\frac{P\_{-}}{P\_{+}} = \left(\frac{SWR - 1}{SWR + 1}\right)^{2} \tag{4.11}$$

with

∆ = 20 <sup>⋅</sup> log10(𝑆𝑊𝑅) in dB .

𝑆𝑊𝑅 is the standing wave ratio. ∆ is the dierence between maximum and minimum power in 𝜑-direction. A high percentage of the counter-rotating mode results in the presence of multiple nodes along the circumference. The number of nodes along the circumference corresponds to the azimuthal mode index 𝑚. The counter-rotating amount of the given measurement is 0.33 %. It cannot become zero due to the non-symmetrical mode excitation

**To (v):** The quality factor 𝑄3 dB of the measured coaxial cavity is calculated with the resonance frequency 𝑓res to the 3-dB bandwidth 𝐵<sup>3</sup> dB.

$$Q\_{3\,\mathrm{dB}} = \frac{f\_{\mathrm{res}}}{B\_{3\,\mathrm{dB}}} = 2548\,\tag{4.12}$$

The measured quality factor is in very good agreement with the simulation of 2533 for the TE28,8 mode.

This automatized measurement procedure with the evaluation algorithms is validated successfully. The quasi-optical mode generator is used to excite several modes at dierent frequencies. Those are the TE20,8 mode at 105 GHz [145], the TE28,8 mode [146] and the TE28,10 mode [36], [144] operating both at 140 GHz, the TE36,12 mode at 175 GHz [143], the TE34,19 mode at 170 GHz and the TE40,23 at 204 GHz [147], [148]. In [144], a time saving of up to 94 % has been calculated for the excitation of the TE28,10 mode using the automatized mode generator compared to manual adjustment.

### **4.1.4 Tolerance Analysis of the Mode Generator Setup**

A tolerance study is performed to investigate the inuence of the measurement arm, the temperature changes of the cavity and the inuence of covering the quasi-optical mode generator with absorber material.

#### **Influence of Tolerances coming from the 3D Measurement Arm**

The 3D measurement arm can operate in continuous and in step mode. The continuous mode provides the shortest measurement time. However, this mode suers from high pixel position errors and a blurring of the mode patterns due to the averaging of the measurement points. Therefore, the step mode is preferred. In step mode, the 3D measurement arm vibrates due to the permanent start and stop movement. The vibrations of two measuring arms with dierent lengths of 890 mm and 600 mm are investigated. 600 mm corresponds to the shortest possible length at which the full range of motion can still be used.

Fig. 4.21: Deection 𝛿 of the measurement arm due to the stepwise driving with step size of 0.25 mm.

The transient behavior of the measurement arm is recorded using a distance measurement tool. The arm moves in steps of 250 µm. The vibration is measured for 6 s after the arm stops at the given position. Figure 4.21 compares the vibrations of the two measurement arms. The vibration fades away at around 2 s. The deection 𝛿 is largest after stopping and corresponds to a maximum of ≈ ± 206 µm. It is a trade-o between vibration and measurement time (time per pixel). At a pause of 0.5 s, the deection of the 890 mm long arm is ≈ ±120.5 µm and that of the 600 mm long arm is ≈ ±80 µm. The total deviation of the 600 mm long arm with a pause of 0.5 s is ≈ 160 µm and thus less than the step size of 250 µm. The 600 mm long arm is selected.

#### **Influence of a Temperature Variation During a Measurement**

A measurement of the radiated mode patterns with a resolution of 250 x 250 µm takes up to 48 h for the TE28,8 mode and 120 h for the TE34,19 and TE40,23 modes. During this time the room conditions may change, e.g. due to direct sunlight. This eect needs to be investigated in relation to the high quality factors of the mode generator cavities. A temperature sensor (BME280) is installed on the cavity. The temperature data is stored on a logging system

Fig. 4.22: Temperature prole and the resulting resonance frequency change during ≈3 days measurement time.

(Arduino Mega and SD card). A temperature prole is recorded over 3 days. The maximum temperature change is achieved with direct sunlight during the day and open windows at night. This implicit a possible worst-case scenario. In addition to the temperature, the resonance frequency is recorded for the validation. Figure 4.22 shows the eect of a temperature variation on the operating frequency. The characteristic values are summarized in Tab. 4.2. A maximum temperature drift of ∆𝑇 = 2.1 °C occurs. This results in a change in operating frequency of 4 MHz. The given frequency drift causes an amplitude drop of 0.07 dB, which is negligible. An eect on the measurement can be seen at a frequency drift of ± 14 MHz, corresponding to a temperature drift of ∆𝑇 ≈ ±7.2 °C.

Table 4.2: Characteristic points for the temperature and the related measured quality factor.


#### **Covering the Setup with Absorber Material**

The translucent cavity wall couples only small fraction of the electromagnetic wave into the cavity due to the dimensions of the holes. The reected wave can propagate into the setup and disturb the excitation of the desired cavity mode. The eect is enhanced because the quasi-optical mode generator is mainly made of metal. Covering the quasi-optical mode generator with absorber material has a minor eect on the excited mode in the step of its adjustment. However, the assembly is rotated by 90° after adjustment for the verication of the quasi-optical output coupler. Then, the bottom side is also terminated by the metal of the table. Previously, this side was open to free space where the electromagnetic beam could propagate. This assembly strongly aects the mode and therefore the measurement of the quasi-optical output coupler. Figure 4.23 compares the measurement with a) not covering the beam path and b) covering the beam path with absorber material.

In addition, three identical launchers were veried using a quasi-optical mode generator. The patterns of the 𝐸-eld intensity distribution dier more from launcher to launcher than from dierent measurements using the same launcher. This concludes that the eect of the manufacturing tolerances of the launcher can be measured.

Fig. 4.23: Measured 𝐸-eld intensity of a launcher operating at 105 GHz and a) not covered and b) covered by absorber material.

## **4.1.5 Verification of the Operating Modes and the Quasi-Optical Mode Coupler at 170 GHz and 204 GHz**

Section 4.1.2 discusses a mode generator design for the TE34,19 mode at 170 GHz and the TE40,23 mode at 204 GHz. Then, the automated process and a tolerance study was presented. In the following, the relevant modes are excited and are used to verify the designed 170/204 GHz quasi-optical output coupler.

## **Excitation of the TE**34,19 **mode and the TE**40,23 **mode**

The automated mode excitation procedure is used to excite the TE34,19 and TE40,23 modes. In the rst step, the nominal operating frequencies are determined to be 170.325 GHz for the TE34,19 mode and 204.071 GHz for the TE40,23 mode. A full mode pattern can be taken after the automated adjustment of the quasi-parabolic mirror. Figure 4.24 and Fig, 4.25 show the 𝐸<sup>𝑦</sup> -eld intensity and the phase pattern of the TE34,19 and TE40,23 modes with a pixel size of 250 x 250 µm.

Fig. 4.24: Measured a) 𝐸-eld intensity and b) phase pattern for vertical polarization of the TE34,19 mode operating at 170.325 GHz with a pixel step width of 0.25 mm in both directions.

Fig. 4.25: Measured a) 𝐸-eld intensity pattern and b) phase for vertical polarization of the TE40,23 mode operating at 204.071 GHz with a pixel step width of 0.25 mm in both directions.

The number of rings in the 𝐸<sup>𝑦</sup> -eld intensity patterns correspond to the radial mode index 𝑝, which is 19 at 170 GHz and 23 at 204 GHz. The azimuthal mode index 𝑚 is determined by the number of phase jumps plus one and the azimuthal mode index is determined by 34 and 40, respectively. The scalar mode content 𝜓 and quality factor 𝑄3 dB are determined to be 𝜓 = 92.1 % and 𝑄3 dB = 3603 at 170 GHz and 𝜓 = 91.8 % and 𝑄3 dB = 4835 at 204 GHz. The counter-rotating amount has been calculated to be below < 0.5 % at both frequencies. The generation of the TE34,19 mode has already been described in [149]. A counter-rotating amount of up to 0.7 % was observed there. The quality of the excited modes is, to the best knowledge of the author, highest using the optimized quasi-optical mode generator. The high-order rotating TE40,23 mode is with its eigenvalue of 126.3 the world's highest ever excited rotating mode in cold tests.

#### **Verification of the Dual-Frequency Quasi-Optical Output Coupler**

The cavity modes for 170 GHz and 204 GHz are used as input to verify the quasi-optical output coupler described in Section 3.3.3. The non-linear up-taper is replaced by the launcher, as it is shown in Fig. 4.26. The step size of the 3D measurement arm is set to 1 mm in both directions. The measurements are performed at the selected frequencies of 170.325 GHz for the TE34,19 mode and 204.071 GHz for the TE40,23 mode. Since the contours of the mirrors follow a quadratic function, the most critical component in the manufacturing process of the quasi-optical output coupler is the launcher. Therefore, the rst step is to investigate the performance of the launcher. Figure 4.27a and Fig. 4.28a present the 𝐸-eld intensity of the horizontal polarization (x-direction) of the launcher aperture at 170.325 GHz and 204.071 GHz. The measurements are in very good agreement with the simulations. The vertical disruptions are caused by the counter-rotating part of the excited mode, which cannot be prevented. The measured pattern at the launcher aperture gives a rst indication about the performance. Figure 4.27b and Fig. 4.28b show the measured phase distribution.

Fig. 4.26: 3D schematic of the measurement setup for verication of the launcher.

Fig. 4.27: 𝐸-eld intensity (a) and phase pattern (b) of the x-polarization at 170.33 GHz of the launcher aperture. The pixel step width is selected by 1 mm in both directions.

Fig. 4.28: 𝐸-eld intensity (a) and phase pattern (b) of the x-polarization at 204.07 GHz of the launcher aperture. The pixel step width is selected by 1 mm in both directions.

Figure 4.29 depicts the patterns at the launcher aperture if the counterrotating mode is excited. The mode is not converted properly and the beam is not focused. It is not possible to compare the amplitude levels of the electromagnetic wave emitted from the launcher by the co-rotating and counter-rotating modes.

A conclusive statement about the performance of the designed quasi-optical output coupler is only possible by using the launcher and all three mirrors. Figure 4.30 presents a photo of the 170/204 GHz dual-frequency quasioptical output coupler. The alignment of all components and the virtual gyrotron axis is ensured by mounting them on the mirror plate of the gyro-

Fig. 4.29: 𝐸-eld intensity (a) and phase pattern (b) of the x-polarization at 204.07 GHz of the launcher aperture. The pixel step width is selected by 1 mm in both directions. The launcher is fed with the counter-rotating TE−40,23 mode.

tron. The measurements at each individual mirror position are discussed in Appendix A.3. Most important is the measurement at the gyrotron window plane. The distance from the gyrotron window plane to the gyrotron axis is 300 mm. The position of the window center in the direction of the gyrotron axis corresponds to the center of the third mirror. Figure 4.31 and Fig. 4.32 compare the 𝐸-eld intensity and phase patterns of the simulation and the measurement for the quasi-optical output coupler at 170.325 GHz, respectively. Figure 4.33 and Fig. 4.34 compare the 𝐸-eld intensity and phase patterns of the simulation and a measurement at 204.071 GHz. The measured patterns are in very good agreement with the simulations. The results are summarized in Tab. 4.3. The higher background intensity in the amplitude measurement is due to the fact that a mode with 100 % purity cannot be excited.


Table 4.3: Comparison of measurement results with simulations at 170 GHz and 204 GHz.

Mode generator

Fig. 4.30: Photo of the assembled quasi-optical output system, consisting of the launcher and three mirrors, on the mode generator.

Fig. 4.31: Comparison of simulated (a) and measured (b) 𝐸-eld intensity of the horizontal polarization at 170.325 GHz at the position of the gyrotron window with a pixel step width of 1 mm in both directions.

Fig. 4.32: Comparison of simulated (a) and measured (b) phase pattern of the horizontal polarization at 170.325 GHz at the position of the gyrotron window with a pixel step width of 1 mm in both directions.

Fig. 4.33: Comparison of simulated (a) and measured (b) 𝐸-eld intensity of the horizontal polarization at 204.071GHz at the position of the gyrotron window with a pixel step width of 1 mm in both directions.

Fig. 4.34: Comparison of simulated (a) and measured (b) phase pattern of the horizontal polarization at 204.071 GHz at the position of the gyrotron window with a pixel step width of 1 mm in both directions.

In conclusion, the measurement results conrm that the dual-frequency 170/204 GHz quasi-optical output coupler is expected to operate correctly in gyrotron experiments. The measurements are in excellent agreement with the simulations. The TE40,23 mode is the mode with the world's highest eigenvalue ever excited in cold tests. The upgrade of the quasi-optical output system provides the excitation of co- and counter-rotating modes with a high purity. In addition, the measurement tolerances are below the manufacturing tolerances of the launcher.

## **4.1.6 Possible Future Method for Mode Excitation**

Considering e.g. [150, 151], the trend goes towards operating frequencies of gyrotrons above 200 GHz. The eigenvalue of the modes increases with increasing operation frequency at a given cavity radius. This can be seen in the mode series of the KIT 2 MW coaxial-cavity gyrotron. The corresponding eigenvalues are 𝜒28,15 = 84.0 at 136 GHz, 𝜒34,19 = 105.2 at 170 GHz, 𝜒40.23 = 126.3 at 204 GHz and 𝜒48,26 = 146.8 at 238 GHz. Excitation of a gyrotron cavity mode with high purity becomes even more challenging with increasing operating frequency due to a denser mode spectrum and thus, increased mode competition. A concept for a cavity to improve the output mode purity is discussed in [152]. The cavity provides a step in the cavity section. This transforms the mode according to the step width into a mode with the same azimuthal mode index but a dierent radial mode index. The proposed eect is valid for all axial symmetric radius steps [153].

For example, with the already existing TE40,23 mode generator operating at 204 GHz the TE40,25 mode at also 204 GHz could be excited just with a waveguide radius step. The approach is in terms of its passive structure reciprocal. This means that the same step converts the TE40,25 mode into the TE40,23 mode. The method could be benecial for future EU DEMO-relevant cavity modes operating at frequencies above 200 GHz. However, this needs to be veried experimentally.

## **4.2 Characterization of Gyrotron Windows**

A dielectric output window is used in the gyrotron to separate between vacuum and air condition while coupling out the Gaussian microwave beam with minimal absorption and reection. In the following, a characterization method for dielectric window disks is described.

Other dielectric materials are used in the gyrotron beam tunnel between the indented metal and absorbing ceramic disks for the suppression of parasitic oscillations or in the gyrotron load to dissipate almost the complete produced microwave power in experiments. The manufacturing process and absorption measurements of a thin dielectric layer for a gyrotron load is described in Appendix A.6.

### **4.2.1 Dielectric Measurement Methods**

A dielectric material that is not conductive is dened by its complex relative dielectric permittivity

$$
\varepsilon\_r = \frac{\varepsilon}{\varepsilon\_0} = \varepsilon\_r' - j\varepsilon\_r''.\tag{4.13}
$$

The imaginary part of 𝜖<sup>𝑟</sup> describes the losses in the medium in terms of oscillating dipoles. The loss tangent tan 𝛿 is dened by the ratio between the imaginary and real part of the complex relative permittivity [154]

$$\tan \mathcal{S} = \frac{\mathfrak{e}\_{r}^{\prime\prime}}{\mathfrak{e}\_{r}^{\prime}}.\tag{4.14}$$

One of the most commonly known measurement method is using a coaxial probe measuring liquid and solid dielectrics [155]. This method can be applied up to frequencies in the THz-range. However, it cannot be used for measuring the transmission of the gyrotron window over a large geometric range. Other methods use an open resonator structure [156] or a free space measurement setup [157]. In [158], a measurement setup using an open resonator is presented. The relative permittivity and the loss tangent can be determined very exactly for selected frequency points. However, it is not applicable for a wide frequency range. Therefore, the free space method is used for measuring the reection and transmission behavior. Figure 4.35 shows a measurement schematic on the basis of an existing setup published in [83]. The setup is adapted by a vector network analyzer using frequency extension modules (FCM) to cover the high frequency bands. A TEM0,0 mode is launched by a corrugated circular horn antenna. A set of mirrors collimate the radiated beam to form a Gaussian beam with identical parameters as in the gyrotron. The microwave beam is focused onto the device under test (DUT) using mirror 2. The received signal is picked up by the additional mirror 1 or 3 and the corrugated horn 1 or 2 depending on the reection or transmission measurement, respectively. The evaluation is performed using a vector network analyzer. More details about the mirror design is published in [83].

Fig. 4.35: Schematic of the quasi-optical measurement setup for transmission and reection measurement of a dielectric device under test (DUT).

### **4.2.2 Measurement of the Gyrotron Output Window**

CVD-diamond is used as material for the output window for megawattclass gyrotrons operating at long pulses. A detailed description is given in Section 3.1.1. CVD-diamond has in the millimeter wavelength range a relative permittivity of 𝜖<sup>𝑟</sup> = 5.67 and a loss tangent of tan 𝛿 ≈ 2 ⋅ 10−5 [82].

#### **Measurement of Single Gyrotron Output Windows**

The reection 𝑅 and transmission 𝑇 of a single gyrotron output window is given by [83]

$$R = \frac{R\_0 \left(1 - 2T\_0 \cos\left(2\beta\_{\varepsilon\upsilon} d\_{\text{window}}\right) + T\_0^2\right)}{1 - 2R\_0 T\_0 \cos\left(2\beta\_{\varepsilon\upsilon} d\_{\text{window}}\right) + R\_0^2 T\_0^2} \tag{4.15}$$

$$T = \frac{T\_0 \left(1 - R\_0\right)^2}{1 - R\_0 T\_0 \cos\left(2\beta\_{\varepsilon\nu} d\_{\text{window}}\right) + R\_0^2 T\_0^2} \tag{4.16}$$

$$\text{with}\qquad R\_0 = \left| \rho\_\upsilon \right|^2\tag{4.17}$$

$$T\_0 = \exp\left(-2\alpha\_{\varepsilon\upsilon}d\_{\text{window}}\right) \,. \tag{4.18}$$

The window thickness is given by 𝑑window and the reection factor by 𝜌<sup>𝑣</sup> . The complex wave propagation constant is 𝛾 𝜖𝑣 = 𝛼𝜖𝑣 + 𝑗𝛽𝜖𝑣 with the attenuation coecient 𝛼𝜖𝑣 and the phase constant 𝛽𝜖𝑣. The complete derivation of the formulas is discussed in Appendix A.7. The formulas above are valid for windows mounted almost perpendicular to the gyrotron.

An existing CVD-diamond disk with a thickness of 𝑑window = 1.601 mm is considered. The measurement setup is calibrated using a short (metal plate with high electrical conductivity) for the reection measurement and a through (no disk) for the transmission. Only the amplitude values are calibrated using this calibration method [159]. Time gating is applied additionally to resolve multiple reections that occur in the time domain at 3.54 m. This distance is equal to half of the distance from the transmitting horn to the DUT. In [160], a CVD-diamond window is measured. Figure 4.36 depicts a comparison of the theoretical model and the measurement for the

Fig. 4.36: a) Transmission 𝑇 and b) reection 𝑅 of a perpendicularly assembled CVD-diamond disk in a frequency range of 140 − 200 GHz.

reection and transmission. The measurement and theoretical model are in excellent agreement. The measurement is limited to a frequency range of 140 − 200 GHz, because of the specied frequencies of the isolator and corrugated horn antenna.

## **4.3 Upgrade of the Frequency Diagnostic Systems**

During gyrotron operation dierent characteristic parameters, like the vacuum condition, the accelerating voltage and current, the output power and the operating frequency, are monitored. The measurement setup discussed in this work is related to frequency detection and frequency spectrum monitoring. Besides the interest in the operating frequency, additionally possible parasitic oscillations, mode switching, low-frequency oscillations, modulation of the output frequency and frequency tuning during long pulses can be investigated. A detailed discussion is given in [161]. The existing frequency diagnostic systems at KIT are limited to an operating frequency of 175 GHz. Those are the FMS (Frequency Measurement System) [162]

and PSA (Pulse Spectrum Analyzer) [163]. Concerning the demand towards higher frequencies above 200 GHz, a re-design is required.

## **4.3.1 Gyrotron Frequency Measurement Techniques**

There are dierent methods for frequency measurements. In the past, gyrotrons had an axial output, where the propagating mode mixture can be determined by the use of a k-spectrometer [164]. A measurement takes somehow long, because of a mechanical variation of the position of the diode receivers to the direction of the propagated power and determination of the angle. However, nowadays gyrotrons with radial output coupling system are state of the art. An open-ended waveguide is connected to the relief window guiding a certain amount of the stray radiation to the diagnostic systems for frequency measurements. There are dierent measurement methods for frequency determinations such as using a heterodyne receiver, a Michelson interferometer [165], or dispersive-line systems for analyzing ultra-short pulses [166]. The most appropriate method is to convert the gyrotron frequency with a mixer into a lower frequency band using a heterodyne receiver. Diagnostic equipment operating in these lower frequency ranges around < 20 GHz are more cost-ecient. The requirements for the frequency diagnostic system are:


At KIT, the FMS and PSA system have been developed in the past. They are based on the heterodyne concept. In the heterodyne concept, a received signal with a high frequency (RF) is converted using a mixer. The mixer is pumped with a local oscillator (LO) signal into a lower frequency range (intermediate frequency, IF). The mixer converts the microwave signal at the fundamental or at a harmonic ℎ of the LO signal ℎ ⋅ 𝑓LO. In general, a mixer can be used for down-converting from high to low frequencies and for up-converting from low to high frequencies. In downward mixing, low (LSI) and high side injection (HSI) occurs based on the relative position of the LO and RF frequency [159],

$$\begin{aligned} f\_{\text{IF, LSI}} &= f\_{\text{RF}} - h \cdot f\_{\text{LO}} \quad \text{for } f\_{\text{RF}} > h \cdot f\_{\text{LO}},\\ f\_{\text{IF, HSI}} &= h \cdot f\_{\text{LO}} - f\_{\text{RF}} \quad \text{for } f\_{\text{RF}} < h \cdot f\_{\text{LO}} \end{aligned}$$

In addition, besides the mentioned side bands, all mixing harmonics can convert signals into the desired IF range. Therefore, ambiguities can occur depending on the applied frequencies for the reconstruction of the RF frequency out of the IF and LO signal, because neither the real mixing harmonic nor the side band are known in prior in the gyrotron experiments.

## **4.3.2 Upgrade of the FMS System**

The hardware components are very expensive considering the mentioned requirements for the frequency diagnostic system. Therefore, two independent systems (FMS and PSA) have been developed up to an operating frequency of 175 GHz. Each system has individual objectives to meet nally the overall requirements. In the rst step, the FMS system is discussed. The initial design has been published in [162]. The main concern of the FMS setup was a cost-ecient but real-time capable system. The schematic of the FMS system, which has been modied within this work, is presented in Fig. 4.37. An open-ended waveguide is installed at the relief window of the gyrotron to guide a certain amount of the stray radiation towards the frequency diagnostic system. The oversized waveguide has a length at KIT of up to 15 m having an attenuation of up to 15 dB. In addition, several attenuators are installed

Fig. 4.37: Schematic of the modied FMS system covering a frequency bandwidth from 110-260 GHz [81].

to attenuate the incoming signal for safety reasons of the mixers. The system is modied to cover two waveguide bands from 110 GHz to 260 GHz. Therefore, the input signal is divided into two channels covering each a waveguide band (110 − 170 GHz and 170 − 260 GHz). The existing FMS contains a mixer working on the third harmonic ℎ = 3 having a conversion loss of 20 dB. The new diagnostic setup contains a fullband sub-harmonic mixer (ℎ = 2) for each channel. In terms of the reduced harmonic, the conversion losses are decreased down to 10−15 dB at 110−170 GHz and 170−260 GHz. Therefore, an increase of the dynamic range of up to 7 dB is achieved (taken also the additional 3 dB power splitter into account). The IF bandwidth is in both versions 18 GHz. There is no diagnostic hardware on the market which is real-time capable for a bandwidth of 18 GHz in a cost-ecient way. Therefore, the IF signal is split into two additional channels. The rst channel consists of a lterbank [167] and the second of a FTA (Frequency Time Analyzer) for evaluation of the frequency-time spectra.

### **Filterbank**

The main competing neighboring modes in a coaxial-cavity gyrotron have a frequency separation of ≈ 2 GHz. Therefore, the lterbank divides the input spectrum into 9 adjacent sub-channels with 2 GHz bandwidth each. The insertion losses of the 9 individual sub-channels are plotted in Fig. 4.38b. At each individual sub-channel an RF diode produces a DC voltage, which is proportional to the input RF signal. The DC signals are displayed on an oscilloscope. This setup shows whether a mode is excited in the correct sub-channel, another mode is excited, or even mode switching occurs during a pulse. This channel covers a large bandwidth of 18 GHz in real-time but suers from a frequency resolution of only 2 GHz.

### **Frequency Time Analyzer (FTA)**

The second channel is equipped with a FTA having a frequency resolution of 10 kHz and covers a bandwidth of up to 2.5 GHz. Therefore, the FTA covers one of the 9 sub-channels for with high resolution in real-time. An additional mixer stage and lowpass lter are used to convert the select frequency band below 2.5 GHz. The new FMS system was built up and installed at the FULGOR test stand and has been successfully tested.

(a) Photo of the FMS from 110 − 260 GHz.

(b) Transmission of the lterbank.

Fig. 4.38: a) Photo of the FMS from 110 − 260 GHz and b) the transmission of the lterbank with the 9 adjacent sub-channels from DC to 18 GHz.

#### **4.3.3 Upgrade of the PSA System**

The FMS system gives a sucient indication, if the gyrotron is operating in the nominal regime. However, it suers from the fact that possible mixing eects can lead to unwanted frequency ambiguities. In addition, the FMS system has a low dynamic range of up to 27 dB and there is a trade-o between frequency resolution and covered bandwidth. Therefore, an alternative system calling Pulse Spectrum Analysis (PSA) has been developed during a doctoral thesis [161]. The PSA system meets all requirements except real-time capability because of long post-processing of the data.

A schematic view of the PSA system is presented in Fig. 4.39a. The chosen mixers have an IF bandwidth of ∆𝑓 = 3.5 GHz. The ambiguity issues could be solved by a single mixer and two successive pulses, changing the LO frequency at the dierent pulses. By the direction in which the IF frequency shifts and from the knowledge of the LO frequency, an unambiguous reconstruction of the RF band is possible. However, the behavior of the gyrotron may dier from pulse to pulse due to transient eects. So, an unambiguous reconstruction is required for a single pulse. Therefore, a second channel is built up with a second mixer (see Fig. 4.39a). The LO frequency applied to mixer 2 diers from the frequency applied to mixer 1. This setup guarantees an unambiguous reconstruction with a single pulse. The operating harmonics of mixer 1 and 2 are selected dierently (ℎ1,𝑑 = 10 and ℎ2,𝑑 = 12) to reduce false positives in the reconstruction in the post-processing code [161]. The IF signals are sampled using the oscilloscope RTO1024 from Rohde&Schwarz, Munich, Germany. The sampled signals are transformed using a short-time Fourier transformation into the frequency domain. The resulting spectra of channel 1 𝑆<sup>1</sup> (𝑓, 𝑡<sup>𝑖</sup> ) and channel 2 𝑆<sup>2</sup> (𝑓 ± ∆𝑓<sup>𝑑</sup> , 𝑡𝑖 ) with a dierence in the LO frequency of ∆𝑓<sup>𝑑</sup> are used for the reconstruction in the post-processing. It uses a minimum search on numerically shifted and mirrored spectra. The two side bands can be calculated using [161]:

$$R\_{\rm LSI} = \text{Min}(S\_1(f, t\_l), S\_2(f - \Delta f\_d, t\_l)) \qquad \forall l \tag{4.19}$$

$$R\_{\rm HSI} = \text{Min}(S\_1^{\leftrightarrow}(f - \Delta f\_d, t\_l), S\_2^{\leftrightarrow}(f, t\_l)) \quad \forall l \tag{4.20}$$

The symbol "↔" denotes a mirroring of the frequency axis. The resulting reconstructed spectrum has a blind spot between 0 and ∆𝑓𝐷. The blind spot of the spectrum 𝑅<sup>x</sup> can be eliminated using:

$$R\_{\mathbf{x}} = \text{Min}(S\_1^{\leftrightarrow}(f + \Delta f\_d, t\_l), S\_2(f, t\_l)) \quad \forall l \tag{4.21}$$

A combination of 𝑅LSI, 𝑅HSI and 𝑅<sup>x</sup> results in the complete spectrum in the microwave band. In terms of the deviated LO frequency the eective bandwidth is reduced to 2 ⋅ ∆𝑓 − ∆𝑓<sup>𝑑</sup> . An eective bandwidth of ≈ 6 GHz is available by a channel pair. In practice two independent channel pair are implemented having an eective bandwidth of ≈ 12 GHz. The reconstructed spectrum can be visualized in a spectrogram, which displays the frequency behavior over time. An example is given in Fig. 4.39b presenting a gyrotron pulse. The frequency in the start-up is tuned to the nominal value until the at-top of the power supplies is reached at 260 µs. Mode competition occurs starting from 320 µs ending in a strong modulation and nally a shut down.

The frequency resolution 𝛿𝑓 and the time resolution is inverse proportional 𝛿𝑓 = 1∕𝑇seg. Time resolution in a spectrogram is dened by the time segment 𝑇seg. A common used frequency resolution in practice is given by 𝛿𝑓 = 1 MHz. The time acquisition is limited to 7 ms using both channel pair and maximum sampling rate. The calculated dynamic range is given in [161] by 60 dB up to 2.5 GHz per mixer channel.

Fig. 4.39: (a) Schematic view of one channel pair and (b) example of a spectrogram using the PSA system [163].

An upgrade of the existing PSA system is performed by installing two additional channel pairs with mixers operating in the frequency band of 170 − 260 GHz. The key element of the setup is the selected choice of operating harmonics of the mixers to reduce the number of possible false positives. Therefore, according to the formulation presented in [161], a calculation about all combinations of harmonics and the resulting number of possible false positives is performed in Fig. 4.40. The mixers on the market are operating typically at ℎ = 14, 16, 18 in the given frequency band. Similar harmonics lead to the highest number of false positives. A combination of ℎ1,𝑑 = 14 and ℎ2,𝑑 = 16 has been chosen, because of the lower conversion losses than using ℎ = 18. The additional channels for the PSA system are currently being assembled.

Fig. 4.40: Calculated false positives in the frequency band from 170 − 260 GHz according to the mixer harmonics ℎ of mixer 1 ℎ1,𝑑 and mixer 2 ℎ2,𝑑 [81].

# **5 Preparation of the Pre-Prototype Gyrotron for Experiments**

In this chapter, the preparation of the 2 MW dual-frequency 170/204 GHz coaxial-cavity pre-prototype is presented. In addition, the assembly and preparation of the in-house manufactured iMIG is discussed.

# **5.1 Preparation of the 170/204 GHz Pre-Prototype**

The 2 MW 170/204 GHz dual-frequency pre-prototype gyrotron will be operated in the Fusion Long Pulse Gyrotron Laboratory (FULGOR) [168] at KIT. FULGOR is equipped with a new high-voltage DC power supply providing up to 10 MW. The power supply allows gyrotron output powers up to approx. 4 MW in long pulses and the possibility to operate with a multi-stage depressed collector. In addition, a 10.5 T SC magnet is ordered for FULGOR to overcome the current frequency limitation of 170 GHz. Section 3.3 discusses the modication required to accommodate to the new test stand and the dual-frequency operation. The following components are manufactured for the rst proof-of-principal experiment for 170/204 GHz dual-frequency operation:


Figure 5.1 presents a photo of the assembled 2 MW 170/204 GHz coaxialcavity pre-prototype. The collector is missing in the photo because it is needed in another gyrotron experiment. After installation, a vacuum condition of ≈ 3 ⋅ 10−8 mbar is achieved without baking. First experiments will be performed after the delivery of the 10.5 T SC magnet.

Fig. 5.1: Assembled 2 MW 170/204 GHz dual-frequency coaxial-cavity pre-prototype. It is prepared for proof-of-principal experiments. In this photo the collector is missing.

# **5.2 Initial Operation of the iMIG**

Simulations show promising results for operating the gyrotron with an iMIG. In [59] and [56], the design and fabrication processes of the in-house manufactured iMIG are described. The iMIG has not yet been experimentally validated due to a vacuum leakage.

### **Electron Beam Welding of the Leaking Connection**

The leaky connection between the lament heater of the emitter and the power supply is optimized as part of this work. A new cable feed-through passing through a ange is employed. Electron beam welding is used. A weld between CuCrZn of the iMIG ange and 1.3917 (nickel-iron alloy) of the cable feed-through does not provide a sucient connection. A nickel sleeve is therefore used as intermediate piece. Two welds are required, one weld between the cable feed-through and the nickel sleeve and a second between the nickel sleeve and the CuCrZn ange. The electron beam parameters are determined by welding a dummy. Figure 5.2 presents a photo of the cut dummy. The quality of the welding in the cutting plane under the microscope is given in Appendix A in Fig. A.29. Afterwards, the original component of the iMIG was successfully welded.

Fig. 5.2: Photo of a test weld of a test piece cutted on the cable feed-through.

### **Verification of Temperature Homogeneity on Emitter Surface**

A uniform temperature distribution over the emitter surface is essential. This ensures the highest quality of the electron beam for optimal interaction. The supplier of the emitter performed a temperature homogeneity test of the emitter at the factory acceptance test showing excellent performance. The measured temperature deviation was ±7 °C at the nominal temperature of 1000 °C and a lament current of 24 A [56]. The homogeneity of the emitter temperature is veried at KIT before the installation to a gyrotron to validate shipping and the assembly. A ZnS window is mounted on the top of the iMIG. This allows a view on the emitter ring with an infrared camera to measure the temperature distribution. A ange for the vacuum pump is mounted on the bottom side. This assembly cannot be placed into an oil tank. Therefore, the emitter cannot be operated at nominal temperature due to the missing oil cooling. Figure 5.3 presents the measurements of the emitter temperature. The measurements are performed at 8 dierent positions around the emitter circumference applying three dierent lament currents of 8 A, 9 A and 10 A. The measurements show an excellent homogeneity of less than ±3 °C temperature deviation. Temperatures are slightly lower at position #5. This position corresponds to the feeding point of the heater coil.

Fig. 5.3: a) Photo of the infrared measurement and b) validation of the homogeneity of the heated emitter surface with a lament current of 8 A, 9 A and 10 A.

#### **Conditioning and Initial Operation**

The iMIG is assembled to the 2 MW 170 GHz coaxial-cavity short-pulse gyrotron. A photo is shown in Fig. 5.4. The evacuated gyrotron has an excellent vacuum of 1 ⋅ 10−9 mbar in cold state (without lament heating) and about 5 ⋅ 10−9 mbar with heated emitter.

The high-voltage tests show that arcing does not occur inside the gyrotron structure. This indicates that the internal geometric construction and the isolators have been designed correctly. In the conditioning phase, the lament of the emitter is heated up to the nominal current of 24 A. The emitter must be activated by overheating to achieve sucient electron emission. This is done by increasing the lament current above the nominal value for a few minutes. The rst operation is carried out with a simplied arrangement, i.e. the modulation anode, the insert and the body are grounded. Simulations are performed accordingly to identify an operating point with the given arrangement. An appropriate electron beam quality is achieved at an accelerating voltage of 𝑉<sup>c</sup> = −55 kV and a maximum magnetic eld strength of 𝐵cav = 4 T. The simplied arrangement is proposed for testing a rst emission. A beam current of 3 A is achieved with the rst pulses. From the beginning, an emission of the emitter can be detected. Further conditioning is required to operate the gyrotron at the nominal operation parameters. However, this is not possible due to a drop of the voltage at the lament. The ohmic resistance is reduced from 2 Ω to about 0.2 Ω. This implies a short circuit in the lament coil and further emitter heating is not possible. Infrared measurements show that due to the short-circuit at the beginning of the heating coil no energy can be deposited on the subsequent emitter ring. Therefore, this emitter ring cannot longer be used and a new one must be ordered. An investigation into the cause is currently underway before a replacement can be ordered.

(a) Photo of the 2 MW 170 GHz coaxialcavity gyrotron.

(b) Installation of the gyrotron into the existing SC magnet.

Fig. 5.4: Photos of the installation process of the Inverse Magnetron Injection Gun (iMIG) to the 2 MW 170 GHz single-frequency pre-prototype and into the existing SC magnet.

# **6 Towards an Advanced DEMO Gyrotron**

The mode series discussed in the present work show a high insert loading at 136 GHz and an output power of 2 MW without much reserve at 204 GHz. Alternative mode series are proposed in [31] and [88] for an EU DEMO-relevant gyrotron in hollow- and coaxial-cavity technology, respectively. However, the hollow-cavity design does not provide 2 MW at the relevant operation frequencies. In addition, modes having higher eigenvalues than the present design are selected in the coaxial-cavity design, e.g. 𝜒35,21 = 113.1 compared to 𝜒34,19 = 105.2 at 170 GHz. Here, the focus is on a suitable mode series for multi-frequency operation. Therefore, today's design constraints are discussed with regard to their validity.

# **6.1 Investigation on Alternative Mode Series**

Most important requirements in the mode selection process are already described. The insert loading and its diameter are additional important parameters in the coaxial-cavity technology. A statement about the insert loading can be made after interaction simulations have been performed. The rst step is to select modes according to their operating frequency and their relative caustic radius. Afterwards, the performance of the mode series is validated according to the output power, eciency, wall loading and insert loading.

## **6.1.1 Optimization of the Mode Selection Procedure**

A generic mode selection strategy is proposed in [169]. It is implemented in a program called *Design-o-mat*. The code searches for suitable mode series considering the operating frequency and the deviation of the caustic radii of the modes. Figure 6.1 shows the ow chart of the code. A rating system is implemented to identify suitable mode series. However, the dened rating system has several drawbacks. It is 1) very strict for the deviation of the caustic radii, 2) calculates not directly the operating frequency and 3) is restricted to a relative caustic radii of about 0.32.

The rst scoring criterion is dened by the normalized standard deviation of the caustic radius 𝜎𝑟<sup>𝑐</sup> . It is dened in [169]

$$
\sigma r\_c = \sqrt{\frac{1}{N} \sum\_{n}^{N} \left( \frac{m\_n}{\chi\_{m\_n, p\_n} \overline{r\_c}} - 1 \right)^2} \quad . \tag{6.1}
$$

This rating system prefers modes with a small deviation dened by an upper limit. The limit was originally set at a deviation of 0.5 %. Therefore, the

Fig. 6.1: Flow chart of the *Design-o-mat* to identify suitable modes series for multi-frequency operation [169].

rating system identies mode series with very high eigenvalues, e.g. the mode series published in [88] has an eigenvalue of 𝜒35,21 = 113.1 at 170 GHz. In the future, the eigenvalue shall be decreased compared to the eigenvalue of the existing mode series 𝜒34,19 = 105.2. An adaption of the rating system is made according to the latest requirements in launcher design. This new rating system considers the upper limit at a deviation of 4 % and reaches the highest score at 0.1 %.

The second criterion is the determination of the operating frequency. The existing rating system uses two rating criteria for this purpose. One to calculate the optimum window thickness to the operating frequencies of the modes and the other to check the linear dependence of the eigenvalue series. These two criteria can lead to a mode series that has perfect linearity, but does not t to an appropriate window. An optimization is done by implementing Eq. (4.15) and Eq. (4.16). These formulas accurately calculate the behavior of the gyrotron output window. The rating schemes scores if the operating frequencies is in a given bandwidth ∆𝑓−20 dB with a reection below −20 dB. The highest score is given by a reection of < −40 dB resulting in a reected power below 0.01 %.

The performance of the modied rating system is evaluated with the mode series given in [40]. This mode series has been postulated to be optimal in hollow-cavity technology for an output power level of 1 MW. The identied mode series and its parameters are summarized in Tab. 6.1. This mode series has been validated as a suitable candidate using the modied rating scheme. This demonstrates, that the mode selection procedure is properly working.


Table 6.1: Key parameters of the mode series discussed in [40] for a multi-frequency hollowcavity gyrotron operating at 137, 170 and 203 GHz.

## **6.1.2 Mode Selection Process**

The following study includes two aspects: 1) select a mode series that meets the general requirements of today's gyrotrons, and 2) a mode series that meets the gyrotron requirements with modied loading constraints due to anticipated progress in research. An example is an improved cavity cooling system that allows a wall loading constraint of > 2 kW/cm2 . Additional requirements for the mode selection process are:


The modes are divided into two dierent categories considering a wall loading of ≤ 2 kW/cm<sup>2</sup> and > 2 kW/cm<sup>2</sup> . A mode series is grouped into a wall loading of ≤ 2 kW/cm<sup>2</sup> , if the mode at 170 GHz has an eigenvalue close to 𝜒34,19 = 105.2.

### **Mode Series for Wall Loading** ≤ **2 kW/cm**<sup>2</sup>

Two mode series are identied, which have similar scores. The parameters are listed in Tab. 6.2. The two mode series are referred to mode series 1 and 2 in the following for a better overview. In [99], a thermo-hydraulic simulation is given, which postulates an increase of the insert loading to 0.39 kW/cm2 . Assuming this postulation, the existing mode series of the current coaxial-cavity gyrotron can be considered. The postulation needs to be experimentally veried before the insert constraint is extended. A drawback of the existing mode series is the relatively large deviation of the caustic radii of 5 % in total.

### **Mode Series for Wall Loading up to > 2 kW/cm**<sup>2</sup>

A mini-channel cooling can increase the allowable wall loading constraint to about 2.5 kW/cm2 [56, 79, 170]. A rst investigation about possible mode

Table 6.2: Key parameters of the identied mode series for possible future EU DEMO-relevant gyrotrons for a wall loading ≤ 2 kW/cm<sup>2</sup> .


series that t to this requirement is started. Increased wall loading leads to two approaches that can be followed. Firstly, a higher output power can be achieved with almost identical eigenvalues. Secondly, 2 MW output power can still be achieved with a more compact design that has lower eigenvalues. The future trend is towards more compact gyrotrons. The benet of compact gyrotrons are, e.g. 1.) a SC magnet with a smaller borehole, which drastically reduces the price of the system, and 2.) the mode spectrum is less dense using modes having smaller eigenvalues and thus more robust and reliable operation. The eigenvalue is limited between 90 ≤ 𝜒𝑚,𝑝 ≤ 105 for the modes operating at 170 GHz in the following study. In total, three mode series meet the requirements in this category. The dierent mode series and their key parameters are summarized in Tab. 6.3. For ease of reference, they are referred to mode series 3, 4 and 5, in the following.

Single-mode simulations are performed to investigate the insert loading. The cavity radii are adapted to 28.00 mm, 27.08 mm and 25.60 mm for mode series 3, 4 and 5, respectively, according to Eq. (2.23). The midsection lengths are adjusted to have similar quality factors for comparison. The input 𝜃<sup>1</sup> , output 𝜃<sup>2</sup> and insert angle 𝜃insert, the rounding, the length of



the up-taper, down-taper and the total length are identical to the values given in Table 3.6. A fully optimized cavity is not considered for this initial study. Figure 6.2 presents a comparison of the insert loadings of mode series 3, 4 and 5. The selected beam currents are 75 A, 75 A and 68 A and the selected magnetic eld strengths are 5.45 T, 6.86 T and 8.23 T for the operation around 136 GHz, 170 GHz and 204 GHz, respectively. The simulations do not include parameter tolerances. The insert radius is selected to 𝑟insert = 6.5 mm for a better comparison between the dierent mode series. This insert radius of 6.5 mm sets the lower limit in terms of acceptable heat loading and vibration due to water ow. Any mode that exceeds the insert load, even if the insert radius is minimal, can be withdrawn. The insert loading constraint of 0.20 kW/cm2 , or even 0.39 kW/cm2 as stated in [99], is exceeded by mode series 3 (depicted in blue). The modes in mode series 3 are the most volumetric modes with the lowest relative caustic radius and the lowest beam radius of about 8 mm. All other modes do not exceed the insert loading at the minimum insert radius.

Fig. 6.2: Comparison of the insert loading of mode series 3 (blue), 4 (red) and 5 (green) sweeping the beam energy from 45 keV to 95 keV [171]. The theoretical limit for the insert loading of 0.39 kW/cm2 is highlighted. An insert radius of 6.5 mm, which is dened to be the minimum, is selected for this study.

## **6.1.3 Performance Analysis of Alternative Mode Series**

The performance analysis of the determined mode series is separated into the two categories. They are not comparable with each other with regard to the dierent applied wall loadings.

## **Wall Loading** ≤ **2 kW/cm**<sup>2</sup>

The mode spectrum is analyzed for mode series 1 and 2. Therefore, the main competing modes are considered in simulations. A list of the main competitors to the TE𝑚,𝑝 mode is summarized in Tab. 6.4. The coupling to neighboring co- and counter-rotating modes is lower in mode series 1 compared to mode series 2.

Multi-mode simulations are performed to investigate the performance of each mode. Table 6.5 summarizes the considered velocity, kinetic energy and


Table 6.4: Main competing modes for a TE𝑚,𝑝 mode used in interaction simulation [88].

guiding center spreads. The insert radius is optimized to 𝑟insert = 6.5 mm and 𝑟insert = 9.7 mm for mode series 1 and 2, respectively. Mode series 2 benets from a larger insert radius. It is more robust against vibrations and can withstand higher heat load. The insert radius of mode series 2 is even larger than that of the existing coaxial-cavity gyrotron with 𝑟insert = 7.9 mm.

The results of the interaction simulations for mode series 1 are summarized in Tab. 6.6. The loading constraints of today's requirements are selected. The insert loading constraint of 0.2 kW/cm2 limits the operation of the TE24,17 mode at 136 GHz. The resulting cavity wall loading is 1.43 kW/cm2 . An output power of 𝑃out =2.21 MW with an interaction eciency of 𝜂 =35.7 % is achieved. The insert radius is already set to the minimum value. The performance can be increased by relaxing the cooling constraint. However, the output power at 136 GHz is still above 2 MW. In contrast, the operating points of the TE30,21 mode at 170 GHz and the TE36,25 mode at 204 GHz are

Table 6.5: Summary of the considered velocity, energy and guiding center spreads used in the interaction simulations.



Table 6.6: Summary of the simulation results for mode series 1 (TE24,17 , TE30,21 and TE36,25).

dened by the maximum wall loading constraint. Interaction simulations show a good performance of the TE30,21 at 170 GHz achieving an output power of 2.53 MW with an interaction eciency of 37.8 %. The output power of the TE36,25 mode operating at 204 GHz is calculated to be 1.85 MW, which is below 2 MW. However, it is still higher than with similar midsection length for the TE40,23 mode from the initial mode series (≈ 1.7 MW). Full optimization of the coaxial-cavity, which is not considered, can result in a design above 2 MW. In general, the performance of this mode series is quite similar to the existing one. However, the benet of this mode series is the reduced deviation of the relative caustic radius from in total about 5 % to 0.34 %.

The results of mode series 2 using the TE33,13, TE41,16 and TE49,19 are summarized in Tab. 6.7. Compared to mode series 1, the performance is less considering the wall and insert loading. This eect is obvious regarding the lower eigenvalues. The eigenvalue of mode series 1 at 170 GHz is 𝜒30,21 = 106.5 and of mode series 2 is 𝜒41,16 = 104.1. Nearly identical results are obtained with a wall loading of 2.15 kW/cm2 at 170 GHz and 204 GHz. At 136 GHz, the insert loading can be relaxed due to a 1.8 mm larger insert radius. In general, mode series 2 oers less performance given today's loading constraints. But, the advantages of mode series 2 are a larger insert radius of


Table 6.7: Summary of the simulation results for mode series 2 (TE33,13, TE41,16 and TE49,19). The values in the brackets are to be considered if the constraints for wall and insert loading can be relaxed.

9.7 mm and it provides a small deviation of the relative caustic radius of 0.18 %. Mode series 2 is a suitable candidate if the wall loading constraint is extended to 2.15 kW/cm2 . The simulation results allowing higher wall loadings are listed in brackets in Tab. 6.7.

#### **Wall Loading of up to 2.5 kW/cm**<sup>2</sup>

Even with relaxed constraints, the insert loadings of mode series 3 strongly exceed the limit as presented in Fig. 6.2. Therefore, mode series 3 is not considered as candidate for a future EU DEMO-relevant mode series.

The simulation results of mode series 4 consisting of the TE28,13 mode at 136 GHz, the TE35,16 mode at 170 GHz and the TE42,19 mode at 204 GHz are summarized in Tab. 6.8. The insert radius is selected to be 8 mm, almost identical to the existing one. The beam radii are close to 10.28 mm and are 280 µm larger than in the existing coaxial-cavity gyrotron. This suggests that the existing MIGs at KIT can be used for this mode series without modication. At a wall loading of 2.5 kW/cm2 an almost identical performance to the original mode series is achieved. A relaxed insert loading is considered in the simulations for the TE28,13 mode at 136 GHz. An output power level


Table 6.8: Summary of the simulation results for mode series 4 (TE28,13, TE35,16 and TE42,19).

of > 2 MW at 204 GHz can be achieved with a wall loading constraint of 2.75 kW/cm2 . In addition, improved performance can be achieved considering an optimized coaxial-cavity design. In general, this mode series is a potential candidate for the next step in the gyrotron R&D at KIT. It could be veried in short-pulse operation by modifying the coaxial-cavity and the quasi-optical output system and re-use the insert and the available MIGs. An absolute statement can be made after optimizing the cavity and validating the MIG, which is planned in the next step.

The simulation results of mode series 5 consisting of the TE25,13 mode at 136 GHz, the TE31,16 mode at 170 GHz and the TE37,19 mode at 204 GHz are summarized in Tab. 6.9. This mode series provides the lowest eigenvalues at the corresponding frequencies considered in this work. The insert radius is 6.9 mm and thus 1 mm smaller compared to the initial design. Almost identical performance to the initial mode series is achieved at a wall loading of 2.75 kW/cm2 for the modes operating at 170 GHz and 204 GHz. Output power of > 2 MW at 204 GHz can be achieved if the wall loading constraint is limited to 2.86 kW/cm2 . In addition, an improved performance can be achieved for an optimized coaxial-cavity design.


Table 6.9: Summary of the simulation results for mode series 5 (TE25,13, TE31,16 and TE37,19).

### **6.1.4 Conclusions for Future Coaxial-Cavity Designs**

Here, the results of the simulations are briey summarized. The initial mode series provide the highest performance according to today's wall loading requirements. Mode series 2 is a potential candidate if the wall loading constraint is extended to 2.15 kW/cm2 . The benet of mode series 2 is a signicant reduction in the deviation of the relative caustic radii and a comparable large insert radius. If the wall loading constraint is extended to 2.5 kW/cm2 , mode series 4 shows promising results. The mentioned wall loading can be achieved by the mini-channel cooling [56], which is currently being studied in the EU. The mode series has a small deviation of the relative caustic radii of 0.35 % and has a similar insert and electron beam radius compared to the initial mode series. Therefore, this mode series can be easily veried in short-pulse experiments using the existing 170/204 GHz coaxial-cavity housing.

The mode series, which are discussed within this work are limited to eigenvalues in the range of 90 < 𝜒𝑚,𝑝 < 105 at 170 GHz. Mode series with smaller eigenvalues 𝜒𝑚,𝑝 < 90 at 170 GHz have to be considered, if the wall loading constraint is extended to ≈ 3 kW/cm<sup>2</sup> or even higher.

In addition, the given simulation results can be used to make general statements about multi-frequency gyrotrons built in coaxial-cavity technology. The nominal operation parameters are limited by the insert loading constraint at 136 GHz and by the wall loading constraint at 170 GHz and 204 GHz. The insert loading constraint has to be adapted for a proper design delivering sucient performance at 136 GHz. In addition, initial simulations have been carried out at 238 GHz using the mentioned cavity design. The resulting insert loading of 0.005 kW/cm2 is negligible.

# **6.2 Dual-Beam Launcher**

In [68] a 4 MW, 170 GHz coaxial-cavity is discussed. CVD-diamond disks have a limited power capability of about 2 MW [82, 44] in CW operation. Therefore, two CVD-diamond windows and two microwave beams are used at output power levels above 2 MW [68]. In [68, 172, 173, 174], launchers with two helical cuts forming two microwave beams are described. Figure 6.3 presents the eld distribution at the launcher wall at 170 GHz of a launcher with only one helical cut. The rst design is limited to the TE34,19 mode operating at 170 GHz. The dimensions of the outer geometry are identical to the existing launcher to be compatible. The eld distribution of the radiated pattern at the position of the second mirror is simulated using SURF3D. Figure 6.4a depicts a scheme of the validated quasi-optical output coupler. The eld distribution at the reference plane is shown in Fig. 6.4b. The Gaussian mode content is calculated to be 97.7 % (left beam) and 98.4 % (right beam) [175]. Additional mirrors must be designed to separate the microwave beams.

Fig. 6.3: Field distribution at the launcher wall.

Fig. 6.4: a) Schematic and b) calculated eld intensity pattern of dual beam launcher at 170 GHz using the TE34,19 mode.

# **7 Conclusion and Outlook**

A design of a rst 2 MW 170/204 GHz multi-frequency / multi-purpose and frequency step-tunable coaxial-cavity short-pulse pre-prototype has been proposed and built up. The design bases on the existing 2 MW 170 GHz single-frequency coaxial-cavity short-pulse pre-prototype. The gyrotron has been built using several existing components, such as the cathode of the Magnetron Injection Gun (MIG). This gyrotron will be the rst gyrotron operating at multiple frequencies above 200 GHz in Europe in the new gyrotron test stand FULGOR.

In the design process, there are some restrictions according to the existing pre-prototype, such as the operating mode at 170 GHz, which is xed to the TE34,19 mode. The TE40,23 mode was identied as a suitable candidate for operation at 204 GHz. Its parameters t to the nominal TE34,19 mode at 170 GHz. This mode fullls the criteria of minimum dierence in caustic radius and minimum window reection best. Based on these modes, the gyrotron key components, i. e. the MIG, the coaxial-cavity and the quasioptical output coupler, were investigated and optimized. An output power of > 2 MW at 170 GHz and 204 GHz must be nally provided.

The existing MIGs at KIT have been studied for the dual-frequency operation to assess whether the required electron beam quality is satised. The magnetic eld prole of a new 10.5 T SC magnet is considered. This new magnet provides the required eld strength for an operation at 204 GHz and above. In conclusion, the coaxial triode iMIG is the favored MIG for future fusion gyrotrons because of its large emitter, the ability of tuning the pitch factor by the modulation anode, the cooling performance and the low velocity spread. The anode conguration of the iMIG is modied for a proper operation in the proposed magnetic eld prole. However, the iMIG is still under development. The coaxial triode cMIG can be used too, but is not a suitable MIG for rst proof-of-principal experiments because of its

complex handling. Therefore, the coaxial diode MIG is considered in the rst experiments because of its reliable operation in other experiments and its easier handling. A modication of the anode shape provides a design that operates at 170 GHz and 204 GHz. This design reuses the cathode, which is one of the most expensive component in the short-pulse gyrotron. The electron beam requirements for the coaxial diode MIG are satised using the counter-rotating TE−34,19 mode at 170 GHz and the co-rotating TE40,23 mode at 204 GHz. In the case of counter-rotating modes a reversion of the magnetic eld is required for a proper performance of the quasi-optical output coupler. The designed anode shape has a low eect on the operation due to tolerances. The electron beam parameters of the simulations are considered in the interaction simulations of the coaxial cavity. The design of the existing coaxial cavity provides a maximum output power at 204 GHz of 1.7 MW, which does not satisfy the EU DEMO requirements. Therefore, the coaxial cavity is optimized using a systematic design process, which has been adapted from hollow- to the coaxial-cavity technology. Besides the cavity, a new non-linear up-taper is designed, which provides a mode conversion to neighboring modes of less than 0.1 % at both frequencies. The optimized coaxial cavity achieves in interaction simulations an output power of > 2 MW with an interaction eciency of > 35 % at both frequencies. The output power is increased by 20 % at 204 GHz. Finally, the quasi-optical output coupler is optimized because of the low vectorial Gaussian mode content of the existing one at 204 GHz. A 170/204 GHz dual-frequency quasi-optical output coupler consisting of a launcher and three mirrors has been designed providing a Gaussian mode content of > 96 %.

In addition, a study about frequency-step tunability in a bandwidth of ±10 GHz in 2 − 3 GHz steps at each center frequency has been performed. For the rst time, the insert loading was considered in the mode selection process. A mode series as it is state of the art in hollow cavity gyrotrons uses azimuthal neighboring modes. However, several modes exceed the limitation of the insert loading. A novel mode series with a jump of the operating center TE𝑚,𝑝 mode to the TE𝑚+2,𝑝−1 show a signicantly improved performance in both frequency bands. The deviation of the caustic radius

is reduced in total by a factor of two, which results in an increased performance of the launcher for the given modes.

The gyrotron components designed within this work, such as the anode, the insert, the coaxial cavity and the quasi-optical output coupler, have been manufactured. A validation of the manufacturing tolerances have been performed with a high precision 3D measurement arm and a quasi-optical mode generator, respectively. The quasi-optical mode generator is used to verify the quasi-optical output coupler before the installation into the gyrotron, since its manufacturing tolerances are only a few µm to ensure a proper operation. This system excites the gyrotron cavity mode without the need of an electron beam, a SC magnet or high-voltage power supply ("cold" measurements). The test stand has been equipped with a vector network analyzer and frequency extension modules to cover a frequency range from 90 GHz up to 330 GHz. Manually adjustable linear drivers are replaced by electronic movable linear high-precision drivers and a goniometer. All relevant components of the quasi-optical mode generator are controlled electronically in this way. An automated mode excitation procedure with mode evaluation algorithms are developed. Five dierent evaluation methods are implemented to determine the azimuthal and radial mode index, the quality factor of the cavity, the scalar mode content and the counter-rotating mode amount. The adjustment time of the quasi-optical mode generator is reduced from months to days by the optimizations. The electronic movable linear high-precision drivers and the goniometer make the adjustment reproducible. Nowadays, one quasi-optical mode generator with dierent coaxial cavities is sucient to excite the dierent modes, which reduces the manufacturing eort signicantly. The test stand excited during this work 6 dierent modes at frequencies from 105 GHz to 204 GHz. The relevant TE34,19 mode at 170 GHz and the TE40,23 mode at 204 GHz have been excited with high purity. The TE40,23 mode has the highest eigenvalue, which has ever been excited in cold measurements. The adjusted mode generator is used to verify the 170/204 GHz dual-frequency quasi-optical output coupler. The measurement results are in excellent agreement with the simulations at both frequencies.

After the successful validation, the gyrotron components have been installed to the pre-prototype gyrotron and are waiting for the rst proof-of-principal experiments above 200 GHz after the arrival of the 10.5 T SC at KIT. The expected operation frequency is above the current limitation of 175 GHz of the recent frequency diagnostic systems. Therefore, the existing frequency diagnostic systems (FMS and PSA) were modied in preparation for the rst experiments. The optimized FMS system has already been installed into FULGOR and veried in experiments. The FMS system was upgraded from 110 GHz to 260 GHz for gyrotron operation and covers two waveguide frequency bands. The design of the PSA system has already been completed and all components have been delivered to KIT.

Finally, the selected gyrotron mode series could be extended to an operation at 136 GHz using the TE28,15 mode. Unlike the other modes in the mode series, the operation at 136 GHz is not limited by the wall loading but by the insert loading. The output power is > 2 MW at 136 GHz considering a published insert loading constraint of 0.39 kW/cm2 . There is additional potential in performance, but it is limited by the requirements in using o the-shelf components. Therefore, a study is started to identify alternative mode series for future EU DEMO-relevant gyrotrons. The study is divided into two categories, 1) considering today's requirements with a wall loading of ≤ 2 kW/cm<sup>2</sup> and 2) > 2 kW/cm<sup>2</sup> considering advanced cooling system like mini-channel cooling. The mode series consisting of the TE28,13 mode at 136 GHz, TE35,16 mode at 170 GHz and TE42,19 mode at 204 GHz has been identied as potential alternative mode series considering a wall loading of 2.5 kW/cm2 . This mode series can be veried in experiments because it is expected that the most critical gyrotron components can be reused.

In this work a rst 170/204 GHz coaxial-cavity gyrotron is designed, which will be veried in experiments at KIT. In future R&D, following points have to be considered for an EU DEMO-relevant fusion gyrotron design:

#### **Compact gyrotron design for EU DEMO**

The current design is optimized for the operation at 170 GHz and 204 GHz. However, an EU DEMO gyrotron should probably operate at 136 GHz, 170 GHz, 204 GHz and 238 GHz. A principal operation is shown in simulations at all these frequencies with the proposed cavity. However, a design for an optimized cavity at all frequencies needs to be investigated with the realistic requirements given in this work. In addition, it has been identi ed that the existing mode series does not t best to the requirements of multi-frequency operation. Depending on the wall loading limitation and possible future gyrotron requirements, a dedicated mode series needs to be selected. The aim is to build more compact gyrotrons using lower order modes. In this work, a new mode series is proposed under certain assumptions. However, further simulations about reliability, insert cooling and insert vibration have to be performed.

#### **Harmonic frequency gyrotrons**

A gyrotron can also operate on higher harmonics. One of the main advantages is a reduced magnetic eld strength, which can signicantly reduce costs for the SC magnet. However, in previous experiments, harmonic gyrotrons achieved output power levels only in the kW range [176]. Performance improvement is in progress to become an alternative to fundamental frequency fusion gyrotrons [177], [178].

#### **Preparation of the 2nd prototype of the iMIG**

The iMIG shows excellent results in simulations. In addition, the possibility of an ultra-compact gyrotron using an iMIG is presented in [56]. Regarding the short circuit of the lament heater, a new emitter has to be ordered. A 2nd prototype of the iMIG is being prepared. During the assembly and rst operation of the iMIG, necessary optimizations were identied. These are: i) Replacement of the cable feed-through by an alternative, e.g. a 3D printed version is prepared and passed the leakage tests. ii) A new xture for improved emitter alignment and iii) changing the position of the highvoltage connectors to provide additional space for access.

# **A Appendix**

## **A.1 Code Package EURIDICE**

The interaction simulations are performed with EURIDICE [87], which is a time-dependent, multi-mode, self-consistent code-package. A detailed description of the code is given in [88]. EURIDICE has been veried with the code SELFT from the code package "Cavity" [67]. Both have been benchmarked with the codes TWANG and TWANG-PIC developed at Swiss Plasma Center - École Polytechnique Fédérale de Lausanne (SPC-EPFL) [179, 180]. Additional gyrotron interaction codes are developed at Technical University Hamburg Harburg [38] and MAGY at University of Maryland [181]. These are examples of such codes, a detailed list is presented in [51].

Before the interaction simulation is started, the geometry has to be created in priori with a small discretization of 100 − 200 µm. The "Coldcav" code which is included in the EURIDICE code package can calculate a mode list of possible competing modes based on the selected frequency range and strength of mode coupling. The eigenvalues of each individual mode are calculated with "CHIMP" presented in [67]. The "Coldcav" code is calculating the frequency, mode coupling, quality factors and axial eld-prole without considering the electron beam ("cold" cavity simulations). The interaction simulations are performed with the "Evridiki" code. The equations of electron motion(2nd order Predictor-Corrector scheme) and electromagnetic eld (Crank-Nicolson nite dierence approach) are solved consistently [31]. The interaction mechanism is described in Section 2.1. A number of in maximum 99 modes can be included in the simulations to investigate the eects of possible mode competition, which is an important issue at these high operating frequencies. As the number of included modes and electrons increases, the calculation becomes longer. This is solved by a parallelized computation. In a start-up simulation the behaviour of the gyrotron can be investigated while increasing the beam energy towards the nominal operation value. The beam current and the pitch factor are changing according to the adiabatic approximation and the Schottky formula. However, these parameters can be entered individually, including e.g. simulated MIG results to be as realistic as possible.

## **A.2 Simulation Results of 170/204 GHz Quasi-Optical Output Coupler for Frequency Step-Tunability**

In this section, the simulation results of the 170/204 GHz quasi-optical output coupler are listed. First, a summary of the key parameters of the studied modes given in this work are presented. Then, the eld patterns and the simulation results are shown.

## **A.2.1 Mode Selection Process for Frequency Step-Tunability around 170 GHz**

The key parameters for a mode series with only azimuthally changing mode index at 170 GHz is summarized in Tab. A.1. A deviation of the caustic radius in the range of −9 % to +8 % is investigated. The dierence of the caustic radius is given in relation to the nominal mode. It can be stated, that a quasi-optical output coupler for this mode series with only changing the azimuthal mode index is not suitable. A novel mode series for frequency steptunability for covering a bandwidth of ±10 GHz is summarized in Tab. A.2. This mode series reduces the dierence in the caustic radius in total by a factor of two. The deviation is in the range of 0 % to +8 %. A new type of launcher could be designed for the mean value of the relative caustic radius. Thus, a deviation of ±4 % can be achieved.


Table A.1: Summary of main parameters of the mode series with only change of the azimuthal mode index for frequency step-tunability around 170 GHz.

Table A.2: Summary of main parameters of the mode series with jump of the radial mode index for modes higher than the centre mode for frequency step-tunability around 170 GHz.


## **A.2.2 Mode Selection Process for Frequency Step-Tunability around 204 GHz**

The key parameters for a mode series with azimuthally changing mode index at 204 GHz is summarized in Tab. A.3. A deviation of the caustic radius in the range of −7.7 % to +6.9 % is observed. The dierence of the caustic radius is given in relation to the nominal mode. It can be stated that a mode series with changing the azimuthal mode index is not suitable. A novel mode series for frequency step-tunability for covering a bandwidth of ±10 GHz is summarized in Tab. A.4. This mode series reduces the dierence in the caustic radius in total by a factor of two. The deviation is in the range of 0 % to +6.9 %. A new type of launcher could be designed for the mean value of the relative caustic radius. Thus, a deviation of ±3.5 % can be achieved.


Table A.3: Summary of main parameters of the mode series with only change of the azimuthal mode index for frequency step-tunability around 204 GHz.


Table A.4: Summary of main parameters of the mode series with jump of the radial mode index for modes higher than the centre mode for frequency step-tunability around 204 GHz.

A Appendix

## **A.2.3 Simulation Results of the 170/204 GHz Dual-Frequency Quasi-Optical Output Coupler used for Frequency-Step Tunability**

The tables in the following present the Gaussian mode content (GMC), the shift of the microwave output beam from the centre of the gyrotron output window and the beam waist in x- and y-direction for the initial and novel mode series at 170 GHz and 204 GHz.


Table A.5: Simulation result of 170/204 GHz dual-frequency mode coupler for the initial and optimized mode series at 170 GHz.


Table A.6: Simulation result of 170/204 GHz dual-frequency mode coupler for the initial and optimized mode series at 204 GHz.

## **A.2.4 Field Intensity and Phase Pattern of the Mode Series around 170 GHz for Frequency-Step Tunability**

In the following, the eld intensity and phase patterns of the modes given in Tab. A.5 around 170 GHz are presented.

Fig. A.1: a) Field intensity pattern and b) phase pattern of the TE32,18 mode at 160.4 GHz

Fig. A.2: a) Field intensity pattern and b) phase pattern of the TE33,18 mode at 162.5 GHz

Fig. A.3: a) Field intensity pattern and b) phase pattern of the TE34,18 mode at 164.6 GHz

Fig. A.4: a) Field intensity pattern and b) phase pattern of the TE35,18 mode at 166.7 GHz

Fig. A.5: a) Field intensity pattern and b) phase pattern of the TE36,18 mode at 168.8 GHz

Fig. A.6: a) Field intensity pattern and b) phase pattern of the TE34,19 mode at 170.0 GHz

Fig. A.7: a) Field intensity pattern and b) phase pattern of the TE35,19 mode at 172.1 GHz

Fig. A.8: a) Field intensity pattern and b) phase pattern of the TE36,19 mode at 174.2 GHz

Fig. A.9: a) Field intensity pattern and b) phase pattern of the TE37,19 mode at 176.3 GHz

Fig. A.10: a) Field intensity pattern and b) phase pattern of the TE38,19 mode at 178.4 GHz

Fig. A.11: a) Field intensity pattern and b) phase pattern of the TE39,19 mode at 180.5 GHz

## **A.2.5 Field Intensity and Phase Pattern of the Mode Series around 204 GHz for Frequency-Step Tunability**

In the following, the eld intensity and phase patterns of the modes given in Tab. A.6 for the operation around 204 GHz are presented.

Fig. A.12: a) Field intensity pattern and b) phase pattern of the TE38,22 mode at 194.6 GHz

Fig. A.13: a) Field intensity pattern and b) phase pattern of the TE39,22 mode at 196.7 GHz

Fig. A.14: a) Field intensity pattern and b) phase pattern of the TE40,22 mode at 198.8 GHz

Fig. A.15: a) Field intensity pattern and b) phase pattern of the TE41,22 mode at 200.9 GHz

Fig. A.16: a) Field intensity pattern and b) phase pattern of the TE42,22 mode at 203.1 GHz

Fig. A.17: a) Field intensity pattern and b) phase pattern of the TE40,23 mode at 204.1 GHz

Fig. A.18: a) Field intensity pattern and b) phase pattern of the TE41,23 mode at 206.3 GHz

Fig. A.19: a) Field intensity pattern and b) phase pattern of the TE42,23 mode at 208.4 GHz

Fig. A.20: a) Field intensity pattern and b) phase pattern of the TE43,23 mode at 210.5 GHz

Fig. A.21: a) Field intensity pattern and b) phase pattern of the TE44,23 mode at 212.6 GHz

Fig. A.22: a) Field intensity pattern and b) phase pattern of the TE45,23 mode at 214.8 GHz

# **A.3 Measurements of the Quasi-Optical Output Coupler at dierent Mirror Positions**

In the following, the measurements of the quasi-optical output at the position of mirror 2 and 3 at 170 GHz and 204 GHz are presented.

(a) Field intensity pattern. (b) Phase pattern.

Fig. A.23: a) Field intensity pattern and b) phase pattern of the TE34,19 mode at 170.325 GHz after the launcher and mirror 1.

Fig. A.24: a) Field intensity pattern and b) phase pattern of the TE34,19 mode at 170.325 GHz after the launcher and mirror 2.

(a) Field intensity pattern. (b) Phase pattern.

Fig. A.25: a) Field intensity pattern and b) phase pattern of the TE40,23 mode at 204.071 GHz after the launcher and mirror 1.

Fig. A.26: a) Field intensity pattern and b) phase pattern of the TE40,23 mode at 204.071 GHz after the launcher and mirror 2.

# **A.4 Manufacturing of a Potter Horn**

Fig. A.27: Photo of driller and cut of the manufactured Potter horn.

# **A.5 Coupling through a Translucent Wall**

A derivation of the coupling through the perforated, translucent cavity wall is presented. In terms of mutual interaction between several holes, the diraction theory is taken into account for the coupling [138]. The power balance between the injected beam 𝑃in and the power of the beam, which passed through the coupling holes has to be satised. In a rst approximation, an equation for the coupling through an innite, periodically perforated plane plate is derived. The transmission is given by [138]

$$\left|T\right|^2 = \frac{4\left|\rho\right|^2}{1 + 4\left|\rho\right|^2} \quad . \tag{A.1}$$

𝜌 describes the coupling. The coupling can be calculated, according to Collin [153]. It denes the polarizabilities of a hole. Using the Babinet's principle the polarizability of the periodic system of a plane metallic disk corresponds to the polarizabilities of a dual periodic system of holes in a metallic plate [138]. Therefore the formula

$$\rho = \frac{k}{2\beta \cdot a \cdot b} (1l \cdot P + 1l \cdot aM) \tag{A.2}$$

can be used to calculate the coupling. 𝑃 and 𝑀 characterize the electric and magnetic dipoles for the disk [153]. 𝛼 and 𝛽 are described by the cosine and sine of the incident angle of the input beam, and 𝑎 and 𝑏 are the distances between the holes in both directions. The electric and magnetic dipole moments can be expressed by [153]

$$M = 8/3r\_{\text{hole}}^3\tag{A.3}$$

$$M = -4/3r\_{\text{hole}}^3 \mathfrak{a} \tag{A.4}$$

The interaction between the holes can be included if the electric and magnetic dipole moments are multiplied by (1 − 𝛼𝑒𝐶<sup>𝑒</sup> ) −1 and (1 − 𝛼𝑚𝐶𝑚) −1 , respectively, with [153]

$$\mathcal{C}\_{\varepsilon} = \frac{1.2}{\pi \cdot a} - \frac{8\pi}{b^3} \varkappa\_0 \left(\frac{2\pi \cdot b}{a}\right) \tag{A.5}$$

$$C\_m = -\frac{1.2}{\pi \cdot a} - \frac{1.2}{\pi \cdot b} + \frac{8\pi}{b^3} \varkappa\_0 \left(\frac{2\pi \cdot b}{a}\right) + \frac{8\pi}{a^3} \varkappa\_0 \left(\frac{2\pi \cdot a}{b}\right) \quad . \tag{A.6}$$

𝜅0 is the modied Bessel function of the second kind and 𝛼<sup>𝑒</sup> = 16𝑟 3 hole∕3 and 𝛼<sup>𝑚</sup> = −8𝑟 3 hole∕3.

## **A.6 Measurement Setup of Dielectric Loaded Disks**

In the commissioning phase of a gyrotron, the generated microwave power is transferred to a gyrotron load rather than to the fusion machine. A microwave load should reliably handle powers up to 2 MW in CW operation. The attenuation or even absorption of the microwave beam can be done using a dielectric coating on the inner cylindrical surface of the gyrotron load. The most important characteristics of the dielectric material are according to

[182], high dielectric constant, a reasonable high heat conductivity, high resistance against electromagnetic and thermal loads and a good substrate adhesion. The layer thicknesses are chosen to create almost a resonance structure. This requires around 100 µm thin layers for maximum absorption. Thin coatings are applied by the plasma spraying process. A detailed description of the technology is given in [183]. A common used dielectric material is alumina-titania (AL2O<sup>3</sup> /TiO<sup>2</sup> ) in a given mixture achieving a high absorption of the microwave beam [182, 184].

#### **Measurement Results of Coated Test Samples**

In [184], a mixture of 87/13 wt.% of Al2O<sup>3</sup> /TiO<sup>2</sup> has been used to measured absorption at 140 GHz. In this work, a mixture of 60/40 wt.% is used to be measured at 140 GHz and 170 GHz. Therefore, a set of test samples with dierent thickness from 30 µm to 150 µm are fabricated on a copper disk with a diameter of 124 mm. A photo of a disk is shown in Fig. A.28a. The measurement results and the values published for the 87/13 wt.% mixture are depicted in Fig. A.28b for comparison. The dierent graphs show a clear resonance in its behaviour over the thickness. The resonance occurs at a thickness 𝑡 corresponding to 𝜆𝑚∕4 using:

$$t = \frac{\lambda\_m}{4} = \frac{\lambda\_0}{4 \cdot \sqrt{\varepsilon\_r}} = \frac{c\_0}{4 \cdot f \cdot \sqrt{\varepsilon\_r}}$$

𝜆<sup>𝑚</sup> is the wavelength in the medium. The calculated relative permittivity is 𝜖<sup>𝑟</sup> ≈ 17.5 for the 60/40 wt.% mixture of Al2O<sup>3</sup> /TiO<sup>2</sup> and is in contrast higher than the mixture of 87/13 wt.% having a 𝜖<sup>𝑟</sup> ≈ 14.8. This is due to the larger amount of TiO<sup>2</sup> having a relative permittivity of 𝜖<sup>𝑟</sup> ≈ 169 at 140 GHz [185].

The measurement results are used in the simulation of the gyrotron load. Therefore, a "ballistic" ray-tracing code is used. A large number of single rays are propagated. The powers of the rays are attenuated each time they hit the wall and are reected according to the specied measured values. After multiple reections most of the power is attenuated. Dierent thicknesses over the length are simulated. It was decided to manufacture the gyrotron load with a thin coating having a thickness < 105 µm to avoid any resonance

structures preventing possible arcing. The simulations show the best results when the coating starts with 50 µm, at full power, and is gradually increased to 100 µm. The set of dierent coating thickness results in simulation, that 2.4 % of the input power is reected back to the input of the gyrotron load [183]. It has to be mentioned, that in terms of the internal structure of the gyrotron load the remaining microwave beam does not propagate back towards the gyrotron. After the manufacturing process rst long pulse tests with a pulse duration of 26 min and an injected microwave power of 500 kW have been successfully performed. Visible inspection afterwards did not reveal any defects [183]. The measurements of this study are performed for a plane incident. A study about changing angles are published in [182].

# **A.7 Derivation of Reflection and Transmission of Gyrotron Output Windows**

A plane wave is reected at both boundaries (front and rear) of a perpendicular assembled disk. Both boundaries of the gyrotron output window can

Fig. A.28: a) Photo of the planar test sample with a coating of Al2O<sup>3</sup> /TiO<sup>2</sup> and b) comparison of the absorption measurements with a mixture of 87/13 wt.% (red) [184] and the measured mixture 60/40 wt.% at 140 GHz (blue) and 170 GHz (black) [183].

be described by a port [83], if the incident beam is not converted and the transversal power density is constant. The reectivity 𝑅 and transmission 𝑇 can be described via the S-parameters of a port. The boundaries of the single-disk can be expressed by [83]:

$$\begin{split} \left[ \underline{\underline{S}} \right] &= \frac{1}{1 - \rho\_{\upsilon}^{2} \exp(-2 \underline{\underline{\chi}\_{\varepsilon\upsilon}} d\_{\text{window}})} \cdot \\ & \begin{bmatrix} \rho\_{\upsilon} \left( 1 - \exp(-2 \underline{\underline{\chi}\_{\varepsilon\upsilon}} d\_{\text{window}}) \right) & (1 - \rho\_{\upsilon}^{2}) \exp(-2 \underline{\underline{\chi}\_{\varepsilon\upsilon}} d\_{\text{window}}) \\ (1 - \rho\_{\upsilon}^{2}) \exp(-2 \underline{\underline{\chi}\_{\varepsilon\upsilon}} d\_{\text{window}}) & \rho\_{\upsilon} \left( 1 - \exp(-2 \underline{\underline{\chi}\_{\varepsilon\upsilon}} d\_{\text{window}}) \right) \end{bmatrix} \cdot \,. \end{split} \tag{A.7}$$

𝑑window is the window thickness and 𝜌<sup>𝑣</sup> the reection factor. The microwave output beam of a gyrotron is linearly polarized, whereas the reection factor is calculated with the Fresnel equation to [186]:

$$\rho\_v = \frac{\sqrt{\varepsilon\_1'} \cos(\theta\_{\rm in}) - \sqrt{\varepsilon\_r'} \cos(\theta\_{\rm out})}{\sqrt{\varepsilon\_1'} \cos(\theta\_{\rm in}) + \sqrt{\varepsilon\_r'} \cos(\theta\_{\rm out})} \tag{A.8}$$

𝜃in and 𝜃out are the angles of the incoming and outgoing/transmitted waves. Equation A.8 can be simplied by considering the vacuum condition in the gyrotron with 𝜖 ′ 1 = 1, resulting in

$$\rho\_v = \frac{1 - \sqrt{\epsilon\_r'}}{1 + \sqrt{\epsilon\_r'}} \tag{A.9}$$

The propagation constant 𝛾𝜖𝑣 is described by the attenuation 𝛼𝜖𝑣 and the phase constant 𝛽𝜖𝑣 to

$$
\gamma\_{\varepsilon\upsilon} = \mathfrak{a}\_{\varepsilon\upsilon} + j\beta\_{\varepsilon\upsilon} \quad . \tag{A.10}
$$

The attenuation and the phase constant can be simplied considering a low loss dielectric material tan 𝛿 ≪ 1 to [83]

$$\mathfrak{a}\_{\varepsilon v} \approx \frac{k\_0 \sqrt{\varepsilon\_r'} \tan \delta}{2} \quad , \tag{A.11}$$

$$
\beta\_{\varepsilon\upsilon} \approx k\_0 \sqrt{\tilde{\epsilon'\_r}} \quad , \tag{A.12}
$$

using the free-space wave number 𝑘<sup>0</sup> = 2𝜋𝑓∕𝑐<sup>0</sup> . The reection and transmission can be expressed using the described simplications and Eq. A.7 to [83]

$$R = \frac{R\_0 \left(1 - 2T\_0 \cos\left(2\beta\_{\varepsilon\nu} d\_{\text{window}}\right) + T\_0^2\right)}{1 - 2R\_0 T\_0 \cos\left(2\beta\_{\varepsilon\nu} d\_{\text{window}}\right) + R\_0^2 T\_0^2} \tag{A.13}$$

$$T = \frac{T\_0 \left(1 - R\_0\right)^2}{1 - R\_0 T\_0 \cos\left(2\beta\_{\varepsilon\upsilon} d\_{\text{window}}\right) + R\_0^2 T\_0^2} \tag{A.14}$$

$$\text{with} \qquad R\_0 = \left| \rho\_v \right|^2 \tag{A.15}$$

$$T\_0 = \exp\left(-2a\_{\varepsilon\upsilon}d\_{\text{window}}\right) \tag{A.16}$$

## **A.8 Cutting Plane of Electron Beam Welding**

(a) Cutting plane of cable feed-through and nickel sleeve.

(b) Cutting plane of nickel sleeve and Cu-CrZn ange.

Fig. A.29: Photo of a the cutted planes of the test device under the microscope.

# **Bibliography**


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# **List of Figures**













# **List of Tables**




# **Acknowledgement**

The presented work was conducted during my time as research assistant at the Institute for Pulsed Power and Microwave Technology (IHM) and Institute of Radio Frequency Engineering and Electronics (IHE) at the Karlsruhe Institute of Technology (KIT). Without the support from many people, this work would not have been accomplished.

First and foremost, I would like to express my sincere and deepest gratitude to Prof. Dr.-Ing. John Jelonnek for giving me the opportunity to work in the interesting research eld of gyrotrons. His continuous support and valuable feedback at every stage of my work led to the success of the research. In addition to the technical knowledge, I learned many professional skills from him, which will surely help me in my future career.

Sincere thanks to Prof. Dr.-Ing. Konstantinos Avramidis for being my co-Referee of this thesis. I would also like to thank him for his valuable and continuous support and encouragement.

I would like to express my deep gratitude to Prof. Dr. rer. nat. Dr. h.c. Manfred Thumm for his support and guidance in my work. Throughout the course of this work, discussions with Prof. Thumm initiated many activities and helped me to keep up to date with the latest advancement in the eld of vacuum electron devices.

Additionally, I would like to thank all the members of the IHM and IHE for their support, in particular Dr. Tomasz Rzesnicki, Dr. Gerd Gantenbein, Dr. Stefan Illy, Dr. Jianbo Jin, Dr. Martin Schmid and Dr. Sebastian Stanculovic, for their great support, patience and the friendly working environment at the institute. I would also like to thank the secretariat of the institute, Mrs. Martina Huber, Mrs. Melanie Mai and Mrs. Natascha Zidek for the great support and benecial words during my time at IHM.

I kindly thank all my former and current Ph.D. colleagues at the IHM for providing me homely environment and their continued help, particularly, Dr.- Ing. Alexander Marek, Benjamin Ell, Lukas Feuerstein, Laurent Krier, Julia Hofele, Max Vöhringer, Dr.-Ing. Parth Kalaria and Dr.-Ing. Chuanren Wu.

Many thanks to Dipl.-Ing Jörg Weggen and Dipl.-Ing Thorsten Kobarg who designed the components described in my thesis. This project would not have been possible without their great ideas and perseverance. A big thank goes also to the mechanical and electrical workshop of the IHM, particularly Kai Schäfer, Alexander Sivkovich, Davide Quattrocchi, Klaus Baumann and Kevin Paulus who tirelessly and with the greatest precision manufactured the gyrotron components and were always on hand with help and advice. I would also like to thank the team in building 691, Dipl.-Ing Wolfgang Leonhardt, Daniel Mellein, Rouven Lang and Daniel Papenfuss for your tireless support and the ght against proper vacuum conditions.

I am thankful to Dr. Dietmar Wagner, Max-Planck Institute, Garching, for continuous help and our fruitful discussions about mode generator setups.

Above all, a big thanks goes to my parents for their continuous support in all phases in my life. Many thanks also to my brother for our great childhood we were allowed to spent together and the wonderful and fruitful time at and besides IHM. A special thank to you for putting me in touch with the IHM. The biggest thanks go to my wife for her endless support and her sacrice by giving me the time and space I needed to accomplish this work.

Karlsruhe, July 2023 *Tobias Ruess*

#### KARLSRUHER FORSCHUNGSBERICHTE AUS DEM INSTITUT FÜR HOCHLEISTUNGSIMPULS- UND MIKROWELLENTECHNIK (ISSN 2192-2764)

#### Band 1 MATTHIAS BERINGER

 Design Studies towards a 4 MW 170 GHz Coaxial-Cavity Gyrotron. ISBN 978-3-86644-663-2

#### Band 2 JENS FLAMM

 Diffraction and Scattering in Launchers of Quasi-Optical Mode Converters for Gyrotrons. ISBN 978-3-86644-822-3

#### Band 3 MATTIA DEL GIACCO

 Investigation of Fretting Wear of Cladding Materials in Liquid Lead. ISBN 978-3-86644-960-2

#### Band 4 AMITAVO ROY CHOUDHURY

 Investigations of After Cavity Interaction in Gyrotrons Including the Effect of Non-uniform Magnetic Field. ISBN 978-3-7315-0129-9

#### Band 5 MICHAEL BETZ

 The CERN Resonant WISP Search (CROWS). ISBN 978-3-7315-0199-2

#### Band 6 ANDREAS SCHLAICH

 Time-dependent spectrum analysis of high power gyrotrons. ISBN 978-3-7315-0375-0

#### Band 7 DHIDIK PRASTIYANTO

 Temperature- and Time-Dependent Dielectric Measurements and Modelling on Curing of Polymer Composites. ISBN 978-3-7315-0424-5

#### Band 8 YIMING SUN

 Adaptive and Intelligent Temperature Control of Microwave Heating Systems with Multiple Sources. ISBN 978-3-7315-0467-2

#### Band 9 JIANGHUA ZHANG

Influence of Emitter surface roughness and Emission inhomogeneity on Efficiency and stability of high power Fusion gyrotrons. ISBN 978-3-7315-0578-5

#### Band 10 ANTON MALYGIN

Design and Experimental Investigation of a Second Harmonic 20 kW Class 28 GHz Gyrotron for Evaluation of New Emitter Technologies. ISBN 978-3-7315-0584-6

#### Band 11 JOACHIM FRANCK

Systematic Study of Key Components for a Coaxial-Cavity Gyrotron for DEMO. ISBN 978-3-7315-0652-2

#### Band 12 PARTH CHANDULAL KALARIA

Feasibility and Operational Limits for a 236 GHz Hollow-Cavity Gyrotron for DEMO. ISBN 978-3-7315-0717-8

#### Band 13 CHUANREN WU

Conceptual Studies of Multistage Depressed Collectors for Gyrotrons. ISBN 978-3-7315-0934-9

#### Band 14 MARTIN HOCHBERG

A high-voltage pulsed power modulator for fast-rising arbitrary waveforms. ISBN 978-3-7315-0958-5

#### Band 15 SEBASTIAN RUESS

Pushing the KIT 2 MW Coaxial-Cavity Short-Pulse Gyrotron Towards a DEMO Relevant Design. ISBN 978-3-7315-1024-6

#### Band 16 VASILEIOS RAMOPOULOS

Energy-efficient, scalable and modular industrial microwave applicator for high temperature alkaline hydrolysis of PET. ISBN 978-3-7315-1099-4

#### Band 17 ALEXANDER MAREK

New Type of sub-THz Oscillator and Amplifier Systems Based on Helical-Type Gyro-TWTs. ISBN 978-3-7315-1250-9

#### Band 18 TOBIAS RUESS

A First 2 MW-Class (136)/170/204 GHz Multi-Frequency Gyrotron Pre-Prototype for DEMO: Design, Construction and Key Components Verification ISBN 978-3-7315-1309-4

Die Bände sind unter www.ksp.kit.edu als PDF frei verfügbar oder als Druckausgabe bestellbar.

**Karlsruher Forschungsberichte aus dem Institut für Hochleistungsimpuls- und Mikrowellentechnik**

*Herausgeber: Prof. Dr.-Ing. John Jelonnek*

Nuclear fusion in a magnetically confined plasma is a viable long-term option for providing reliable energy in future. In nuclear fusion devices, gyrotron oscillators are used for auxiliary plasma heating and current drive, plasma stabilization and plasma diagnostics. This work presents the first design and construction of a multi-frequency / multi-purpose coaxial-cavity pre-prototype gyrotron operating at (136)/170/204 GHz with an output power of 2 MW. Key components of a pre-prototype are designed, manufactured and assembled.

Considering the different operating frequencies, all relevant frequency and mode diagnostics are optimized. Of particular note is the automated mode generator. It enables the excitation of the highest order mode ever excited in cold tests and lead to a drastic reduction of the measurement time from weeks/months to days.

Tobias Ruess received the B.Sc. degree in Electrical Engineering from the University Ulm and M.Sc. degree in Electrical Engineering and Information Technology from Karlsruhe Institute of Technology (KIT) in 2015 and 2017, respectively. In 2017, he joined the Institute for Pulsed Power and Microwave Technology (IHM). Tobias Ruess received the Dr.-Ing. degree in Electrical Engineering and Information Technologies in 2023.

ISSN 2192-2764 ISBN 978-3-7315-1309-4 T. Ruess

**18**

**A First 2 MW 170/204 GHz Multi-Frequency Gyrotron Pre-Prototype**