The Cyclotron Auto-Resonance Maser: Analytical and numerical study

Nuclear Inst. and Methods in Physics Research, A 955 (2020) 163310

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The Cyclotron Auto-Resonance Maser: Analytical and numerical study G. Dattoli, E. Di Palma ∗ Enea - Frascati Research Center, Fusion and Technology for Nuclear Safety and Security Department Via Enrico Fermi 45, 00044, Frascati, Rome, Italy

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Keywords: Self amplified spontaneous emission Gyrotron CARM Free electron laser

ABSTRACT We study the small signal regime of Cyclotron Auto-Resonance Maser (CARM), using different computational tools. The analysis is motivated by the necessity of an accurate benchmarking of the design parameter for the CARM device under construction at ENEA Frascati center. The main goal is that of comparing the results from three different procedures, the analytical formulation and two different numerical items: the home-made code GRAAL (Gyrotron Radiation Auto-Resonance Amplification Laser) and the, commercially available, software package CST. We prove the substantial consistence between the different methods and comment on the reasons leading to the discrepancies. The extension to the non-linear regime is finally touched on.

1. Introduction The CARM [1] is gaining attention because it may be exploited as a source of additional heating in Fusion Plasma experiments [2]. Within such a context a program has been started at the ENEA Frascati center for the construction of a CARM source operating at 250 GHz [3]. The motivation for the choice of a CARM, instead of the much more technologically mature Gyrotrons [4], is suggested by the fact that the latter exhibits a significant efficiency loss above 200 GHz. The operating wavelength of the ENEA experiment, chosen above this threshold, meets the requirement for an appropriate wave plasma coupling in high field Tokamaks [2,3,5]. The line of research opened with the ENEA CARM project is aimed at exploring the feasibility of such a device, which has been designed to be driven by a highly quality, moderately relativistic electron beam, injected into a wave guide enclosing an intense longitudinal magnetic field. The geometry of the interaction is reported in Fig. 1(a). The electron beam, injected at a pitch angle with respect to the magnetic field, executes a helical path allowing the coupling with the TE component of the field co-propagating with the beam itself. The coupling induces an energy modulation, followed by a density modulation (bunching), which is the crucial condition for the coherent emission. Since the process does not occur in vacuum, the frequency selection mechanism is fixed by the intersection of the waveguide and e-beam dispersion curves (see Fig. 1(b)). According to the quadratic nature of the cavity dispersion relations, two frequencies are selected namely the lower 𝜔− and up-shifted 𝜔+ frequencies which can be written as [3,6] (for a refractive index close to unity) 𝜔+ ≈

𝛺 1 − 𝛽𝑧

𝜔− ≈

𝛺 1 + 𝛽𝑧

(1)

with 𝛽𝑧 being the reduced longitudinal velocity of the electrons; the subscripts +, − denotes respectively up-shifted CARM and gyrotron frequencies. In the previous equation 𝛺 = 𝛺0 ∕𝛾 is the relativistic cyclotron frequency, with 𝛺0 = 𝑒𝐵∕𝑚𝑒 being 𝑒 the electron charge and 𝑚𝑒 the electron mass. Regarding the up-shifted case, considering the relativistic limit, denoting by 𝛽⟂ the transverse reduced longitudinal velocity, we end up with the following approximation: 𝜔+ ≅

𝛺 = 1 − 𝛽𝑧

𝛺 √ 1−

1−

1 𝛾𝑧2

≅ 2 𝛾𝑧2 𝛺, (2)

𝛾

𝛾𝑧 = √ , 𝛼̄ = 𝛾𝛽⊥ 1 + 𝛼̄ 2 The resonance frequency 𝜔+ including the correct phase velocity writes 𝜔 𝑅 = 𝛺 + 𝑘 𝑧 𝑣𝑧

(3)

The above equation exhibits the 𝛾 2 Doppler up-shift term and a complete analogy with the Undulator (U-) based FEL can be drawn, provided that 𝛺 be understood as the angular frequency associated with the undulator (for further comments see Ref. [7] where a thorough comparison has been accomplished). In order to support the analogy we do not make explicit the dependence of 𝛺 on the relativistic factor 𝛾. If it is included the up-shift amounts to a factor 𝛾 only, above the cyclotron frequency. The analogy has allowed the development of a theoretical point of view, which has provided a fairly straightforward comparison with (U-FEL) [6]. This paper is a further contribution in this direction.

∗ Corresponding author. E-mail addresses: [email protected] (G. Dattoli), [email protected] (E. Di Palma).

https://doi.org/10.1016/j.nima.2019.163310 Received 31 July 2019; Received in revised form 1 December 2019; Accepted 16 December 2019 Available online 23 December 2019 0168-9002/© 2019 Elsevier B.V. All rights reserved.

G. Dattoli and E. Di Palma

Nuclear Inst. and Methods in Physics Research, A 955 (2020) 163310

Fig. 1. (a) Geometry of the CARM interaction; (b) CARM Brillouin diagram and position of the CARM upshifted 𝜔+ and Gyrotron 𝜔− frequencies.

We explore the CARM small signal behavior by exploiting three computational means:

It characterizes how the axial momentum varies with beam energy. The larger 𝑏 is, the more rapidly the axial momentum decreases with decreasing 𝛾. The change of the axial momentum due to a slow axial variation of the wave amplitude is accounted for by the term 𝑄(𝜁) in Eqs. (4). This term includes also the effect of the longitudinal magnetic field of the interacting TE wave [11,12]. This quantity changes when in the cavity interaction the magnetic transverse (TM) mode will be considered [8]. The next two parameters are the normalized detuning 𝛥 and the small-signal gain coefficient 𝐼𝑔 given by ( ) 𝛽 2 1 − 𝛽𝑧0 ( 𝑝 𝜔 ) 𝛥 = 1− 𝑅 (7) 2 (1 − 𝛽 −2 ) 𝜔 𝛽⟂0 𝑝 )3 ( 𝛽 1 − 𝛽𝑧0 𝑝 2𝜇0 |𝑒| 𝛽𝑝 [𝐶𝐽 ]2 𝐼0 𝐼𝑔 = 𝑚𝑐𝛾0 𝛽 4 (1 − 𝛽𝑝−2 )

(1) the FEL-CARM high gain equation; (2) the home-made code GRAAL [7] based on the numerical integration of the CARM equation derived in Refs. [8]; (3) the three dimensional particle-in cell code distributed by CST Microwave Studio [9]. Apart from the academic interest, the study aims, among the other things, at benchmarking the design criteria developed for the ENEA FRASCATI CARM project, in which the precise determination of the small signal gain and of its dependence on the beam and magnetic field inhomogeneities is a quantity to be fixed with a high degree of confidence. The paper consists of three further sections: in the forthcoming section we discuss the comparison between semi-analytical scaling formulae and the numerical 1-D GRAAL code, which are also confronted with results reported in literature, the reliability of the numerical and analytical protocols is then (sec. 3) further benchmarked with a full 3-D code; the last section is devoted to final comments and plane for future investigations.

⟂0

where 𝐼0 is the beam current and 𝐶𝐽 is a kind of a filling factor, accounting for the overlapping of the beam with the transverse field distribution of the waveguide mode; 𝛾0 is the relativistic factor of the beam at the entrance of the cavity interaction. In the last equation of the system (4), the averaging over the different initial conditions (distributions of initial phases and energies) of an ensemble of electrons is denoted by ⟨⋯⟩ and is used in order to calculate the growth of the field amplitude 𝐹 induced by the beamwave interaction. It should be noted that the normalization constant contains the initial velocities, when we consider non monoenergetic beams they should be replaced by the corresponding average value. The GRAAL code has been developed as a tool going beyond the analytical procedure [13] and allows to analyze more adequately the influence of the design parameters on the operational performance of the CARM. It is a one dimensional macro-particle code, based on the integration of the Eqs. (4) using a fourth order Runge–Kutta method. The growth of the field amplitude 𝐹 induced by the beam-wave interaction is reconstructed by averaging, the rhs of the first of Eqs. (4) over the uniform phase distributions and normal transverse velocity distribution centered at 𝛽⟂0 or 𝛾0 for a fixed pitch factor 𝛽⟂ ∕𝛽𝑧 (for a more accurate description of the CARM equation see Ref. [8,11] and for a deeper discussion about GRAAL see [14]). The key parameters accounting for the beam-wave interaction are specified by the dimensionless quantities (𝑏, 𝐼𝑔 , 𝛥), providing the inputs for the GRAAL simulation. The code outputs consists of two files containing the signal power and the system efficiency along the longitudinal direction converted in physical unit (𝑘𝑊 , 𝑚) by the use of the following relations

2. The CARM equation, the small signal gain regime and the GRAAL code The equations describing the CARM interaction of an electron beam with a transverse electrical (TE) mode in a circular cavity have been established in Refs. [8,10] 1

[1 − 𝑢] 2 𝑑𝑢 = 𝑅𝑒(𝐹 𝑒−𝑖𝜃 ) 𝑑𝜁 1 − 𝑏𝑢 [ ] 1 𝑑𝜃 1 1 = 𝛥 − 𝑢 − 𝑏𝑄(𝜁) + [1 − 𝑢]− 2 𝑅𝑒(𝑖𝐹 𝑒−𝑖𝜃 ) 𝑑𝜁 1 − 𝑏𝑢 2

(4)

1 2

𝑑𝑄(𝜁) [1 − 𝑢] 𝑑𝐹 −𝑖𝜃 = − 𝑅𝑒(𝑖 𝑒 ) 𝑑𝜁 1 − 𝑏𝑢 𝑑𝜁 ⟨ ⟩ 1 [1 − 𝑢] 2 𝑖𝜃 𝑑𝐹 = 𝐼𝑔 𝑒 𝑑𝜁 1 − 𝑏𝑢 where all variables are suitable normalized and have the following meanings: 𝑢 is the energy variable of an electron with a phase 𝜃, 𝐹 is the dimensionless amplitude of the electromagnetic wave and 𝜁 is linked to the axial coordinate through the identity 𝜁=

2 (1 − 𝛽𝑝−2 ) 𝜔𝑧 𝛽⟂0 ( ) 2𝛽𝑧0 𝑐 𝛽 1 − 𝛽𝑧0

(5)

2

⎤ ⎡ 3 (1 − 𝛽 −2 )1∕2 𝛾0 𝛽⟂0 ⎥ ⎢ 𝑝 1 𝑃 (𝑧𝑛 ) = ⎢𝐹 ((𝑛 − 1)𝛿) ( ) ⎥ 𝛽𝑧0 2 ⎥ 2[𝐶𝐽 ]2 𝛽𝑝 ⎢ −3 2.41 ⋅ 10 1 − 𝛽𝑝 ⎦ ⎣ ( ) 𝛽 1 − 𝛽𝑧0 𝑝 2𝛽𝑧0 𝑐 𝑧𝑛 = (𝑛 − 1)𝛿 2 −2 𝜔 (1 − 𝛽 𝛽⟂0 𝑝 )

𝑝

𝑐 the speed of the light and 𝛽𝑝 , 𝛽⟂0 , 𝛽𝑧0 are the reduced velocities associated to the wave phase and electrons (longitudinal and transverse, at the entrance of the interaction region) respectively. The parameter 𝑏, some time called recoil term, is defined as 𝑏=

2 𝛽⟂0

2𝛽𝑧0 𝛽𝑝 (1 − 𝛽𝑧0 ∕𝛽𝑝 )

(6)

being 𝑛 and 𝛿 the index and discretization step, respectively. 2

(8)

G. Dattoli and E. Di Palma

Nuclear Inst. and Methods in Physics Research, A 955 (2020) 163310

The GRAAL code has been validated by comparing its results with various analytical solutions (e.g. based on a linearized system of Eqs. (4) and benchmarking it against other codes. The used linearization procedure is based on the lowest order expansion in the field amplitude 𝐹 . In order to underscore the analogy with the undulator based devices, we consider a moderately CARM regime thereby considering moderate 𝑢 variation (𝛿𝑢∕𝑢 less than 30% and 𝑏 < 1), we expand all functions containing the normalized energy 𝑢 to the lowest order and neglect the terms like 𝑢𝐹 . In this way we can reduce the previous set of Eqs. (4) to the pendulum like form ( ) ⟨ ⟩ ⟨ ⟩ 1 𝑑𝐹 ≈ 𝐼𝑔 𝑒𝑖𝜃 𝜃 + 𝐼 𝑔 𝑢𝑒𝑖𝜃 𝜃 , 𝐼 𝑔 = 𝑏 − 𝐼 0 0 𝑑𝜁 2 𝑔 𝑑𝑢 ≈ 𝑅𝑒[𝐹 𝑒−𝑖𝜃 ] (9) 𝑑𝜁 { } 𝑑2𝜃 𝑑 1 𝑑 ≈ (𝑏𝛥 − 1)|𝐹 |𝑐𝑜𝑠(𝜃 + 𝜙) − 𝑏 𝑄(𝜁) − 𝑅𝑒(𝑖𝐹 𝑒−𝑖𝜃 ) 𝑑𝜁 2 𝑑𝜁 𝑑𝜁 2

Fig. 2. The solution of Eq. (13) without initial bunching (𝐹 ′ (0) = 0) dashed line and with an initial bunching setting 𝐹 ′ (0) = 10−4 continuously line.

The linearization of Eqs. (4) yields the unsaturated behavior of the CARM dynamics and in this regime the evolution of the field amplitude 𝐹 is ruled by the integro-differential equation 𝜁

Table 1 The CARM amplifier parameters with the GRAAL simulation settings.

𝜁

𝐼𝑔 ′ ′ 𝑖 𝑑𝐹 = (𝑏𝛥 − 1)𝐼𝑔 (𝜁 − 𝜁 ′ )𝐹 (𝜁 ′ )𝑒𝑖𝛥(𝜁−𝜁 ) 𝑑𝜁 ′ + 𝑖 𝐹 (𝜁 ′ )𝑒𝑖𝛥(𝜁−𝜁 ) 𝑑𝜁 ′ ∫0 𝑑𝜁 2 2 ∫0 (10) The use of the formalism of negative derivative operators can be useful to cast Eq. (10) in the form of a third order ordinary differential equation. By noting indeed that [15] 𝑥

𝐷̂ −𝑛 [𝑔(𝑥)] =

1 (𝑥 − 𝜉)𝑛−1 𝑔(𝜉)𝑑𝜉 (𝑛 − 1)! ∫0

(11)

Cavity radius Beam current Beam Voltage Transverse/axial velocity Axial Magnetic field Operating mode

3.5 mm 100 A 2 MeV 0.3 2.9 T 𝑇 𝐸11

𝛽𝑝ℎ Recoil parameter b Normalized current 𝐼𝑔 Normalized detuning 𝛥

1.00523 6.25 ⋅ 10−1 5.65 ⋅ 10−3 0.16

and interpreting therefore the integrals in Eq. (10) as second and first order negative derivatives, we can write Eq. (10) as 𝑒−𝑖𝛥𝜁

𝑑 𝑖 𝑖 𝐹 (𝜁) = (𝑏 𝛥 − 1) 𝐼𝑔 𝐷̂ −2 [𝑓 (𝜁)𝑒−𝑖 𝛥𝜁 ] + 𝐼̄𝑔 𝐷̂ −1 [𝑓 (𝜁)𝑒−𝑖 𝛥𝜁 ] 𝑑𝜁 2 2

relevant validity to a regime in which the relative energy, lost by the electrons, is limited to few percents (less than 5%). The second approximation is that yielding the ‘‘unbounded’’ intensity growth implicit in Eq. (13). The small signal regime yields the linear growth only. Our interest for this regime is to get an accurate benchmarking of the small signal gain, to be compared with GRAAL and CST codes. An idea of the evolution of the CARM field square amplitude vs. 𝜁, with and without bunching, is given in Fig. 2 where we have reported |𝐹 |2 for seed and bunching triggered growth. The distinctive feature of the two behaviors is the start-up. In the case of a pre-bunched beam the field grows up from the vacuum with a kind of blow up and then follows the characteristic exponential. In the second case the intensity enters the exponential regime after a lethargic section. The prebunching can be effectively impressed to the e-beam at the entrance of the cavity using e.g. multistage device, which foresees the use of a beam cavity prebuncher and a drift section, before the injection in the CARM cavity. In the following we will make reference to initial bunching without any reference to a specific prebuncher tool. The prebunching mechanism will not be included as a design optional but only to benchmark the consistency between different computational tools. The analytical solution of the CARM linearized equations has been compared with the numerical results achieved with the GRAAL code applied to an amplifier operating with the parameters of Table 1, relevant to a 250 GHz CARM experiment driven by an induction LINAC at Lawrence Livermore National Laboratory [17]. In Fig. 3 we have reported the intensity growth of the 𝑇 𝐸11 mode. The figure, drawn for different values of the initial bunching, contains a comparison between linearized analytic solution and GRAAL code. The prebunching has been fixed by different choices of the initial phase distribution (in GRAAL), which has then been exploited to fix the initial conditions for the high gain CARM equation (13). The agreement

(12)

By deriving twice both sides of Eq. (12) with respect to 𝜁 we find (the apexes denote the relevant order of derivative) ( ) ] [ ( )] [ 𝑖 1 𝑖 𝛥 1 𝐹 ′′′ −2𝑖𝛥𝐹 ′′ − 𝛥2 − 𝑏− 𝐼𝑔 𝐹 ′ − (𝑏 𝛥 − 1) − 𝑏− 𝐼𝑔 𝐹 = 0 2 2 2 2 2 (13) The term 𝑄(𝜁) does not give a significant contribution in the small signal regime. As also reported in [11] its role for interaction far form the cutoff is not particularly significant. It is however worth noting that the GRAAL code does not employ any of these approximations and deals with the integration of the Eqs. (4). The integration of Eq. (13) is almost straightforward and can be obtained by the use of standard means, thus getting for 𝐹 (for further comments see [4]) 𝐹 (𝜁) =

3 ∑

𝑓𝑗 𝑒𝛿𝑗 𝜁

(14)

𝑗=1

where 𝛿𝑗 are the roots of the associated cubic equation (equivalent to the CARM dispersion relation), 𝑓𝑗 depends on the initial conditions which are fixed by the input seed 𝐹 (0), by the first and second derivatives 𝐹 ′ (0), 𝐹 ′′ (0) associated with the initial bunching at the fundamental harmonics (for further comments see [15,16]). Before proceeding further a few clarifications are necessary. As noted in Ref. [4] (chapter VI) the use of the linearized equation has only limited usefulness. It is indeed decoupled from the e-beam dynamics and the field intensity goes indeed linearly without undergoing any saturation. The linearization process we have followed is based on two steps. The first in which we reduce the original equations, derived in [4,8, 10], in a form closer to the FEL pendulum equation. This hampers the 3

G. Dattoli and E. Di Palma

Nuclear Inst. and Methods in Physics Research, A 955 (2020) 163310

Fig. 3. Comparison between numerical (dot line) and analytical (solid line) computation of the dimensionless field intensity growth. The GRAAL simulation has been executed using 15 000 macroparticles, with gaussian energy distribution and the phases arranged to provide different bunching.

Fig. 5. The gain versus 𝛥 evaluated using the definition given by Eq. (16) for 𝜁1 = 29 and 𝜁2 = 40 with 𝐹 the solution of Eq. (13) (continuously line) and the solution of the GRAAL code using the parameters reported in Table 1 (dot line).

consistency between the different computations. In the present study we use an analogous definition, except that we include the bunching and calculate the fractional intensity variation between two different points 𝜁2 and 𝜁1 on the growth curve taken in the region after the blow up zone and before the onset of the saturation (see Fig. 3 solid line), namely

between numerical and analytical solutions is fairly good in the linear regime (GRAAL includes the non linear effects eventually leading to saturation). As already stressed, the gain is a quantity of paramount importance and should be properly defined; according to the ordinary convention, in the case of seeded devices, it is provided by the fractional intensity variation |𝐹 (𝜁2 )|2 − |𝐹 (0)|2 | 𝐺≡ | |𝐹 (0)|2

|𝐹 (𝜁 )| − |𝐹 (𝜁 )| 2 | 1 | | 𝐺(𝛿𝜁 ) ≡ | |𝐹 (𝜁1 )|2 | | 2

(15)

2

with 𝛿𝜁 = 𝜁2 − 𝜁1 > 0

(16)

The inclusion of pre-bunching makes the things slightly more complicated as it is shown in Fig. 5. The occurrence of peaks in the gain function does not hide any particular physical meaning. Its appearance can however be interpreted as a computational fake, if not properly checked. The appearing of the same peaks with different procedure is an indication ensuring that the problem is treated accurately, at least from the computational point of view. The comparison in Fig. 5 (the red dots denote data from the GRAAL code) yields a substantial agreement.

with |𝐹 (0)|2 being associated with the input seed power. The transition from the low to high gain condition occurs when 𝜁 increases, as illustrated in Fig. 4 (a), (b) in which the gain curve looses the antisymmetric oscillator shape to take an almost symmetric form dominated by exponential terms. The Fig. 4 (c), (d) includes also the bunching contributions, that are responsible for a peculiar behavior, which, albeit devoid of any physical meaning, can be exploited to check the

Fig. 4. The gain versus 𝛥, defined in Eq. (15), for different values of the initial bunching (𝐹 ′ (0)) and 𝜁2 : (a) 𝜁 = 10, F’(0) = 0, (b) 𝜁 = 20, F’(0) = 0, (c) 𝜁 = 5, 𝐹 ′ (0) = 10−4 , (d) 𝜁 = 15, 𝐹 ′ (0) = 10−4 .

4

G. Dattoli and E. Di Palma

Nuclear Inst. and Methods in Physics Research, A 955 (2020) 163310

Table 2 The Enea CARM Frascati parameters used for GRAAL and CST simulations. Cavity radius Beam current Beam Voltage Transverse/axial velocity Axial Magnetic field Operating mode

7.5 mm 20–35 A 0.6 MeV 0.53 5.3 T 𝑇 𝐸53

𝛽𝑝ℎ Recoil parameter b Normalized current 𝐼𝑔

1.06452 3.936 ⋅ 10−1 (4.56–7.99) ⋅ 10−4

CST has furthermore confirmed that along with modes, whose intensity growth is shown in Fig. 6, other modes become excited (see Fig. 7 ). Therefore, it is very likely that the lower saturated power exhibited by the mode 𝑇 𝐸53 in Fig. 6a) is due to power lost by the beam into other modes inside the waveguide. In Fig. 7 we have reported the growth of the parasitics modes 𝑇 𝐸52 and 𝑇 𝑀52 , different (but close) to the desired operating mode 𝑇 𝐸53 . The run with CST shows that they reach a not insignificant amount of power which spoils the saturation performances of 𝑇 𝐸53 . Regarding Fig. 6(b) we note that the linear growth looks faster this is due to a more efficient beam wave coupling allowing larger gain. The concluding comment is that the computational tools we have developed in support of the CARM development activity are sufficiently reliable. Furthermore the indications by CST goes in the direction that a suitable strategy to suppress unwanted modes in oscillator configuration is necessary. In that case parasitics modes grow inside the cavity and if they have sufficient gain may reach enough power level to spoil the efficiency toward the operating mode.

In this section we have shown that the CARM gain is sufficiently well benchmarked at least regarding two computational methods. There are obvious pitfalls in the analysis we have developed, which should be further refined to become an accurate design tool and an appropriate device for the analysis of the experimental results. This aspect of the discussion will be analyzed in the forthcoming section, where we will discuss the use of the CST code.

4. Conclusions

3. 1-D vs 3-D code simulations This section is devoted to a comparative description including the use of a full 3-D (dimensional) code, commercially distributed by CST Microwave Studio [9]. We believe the use of this code of pivotal importance to model the experiment because it contains all the elements of an actual experimental configuration. The analysis we have developed in the previous sections is 1-D and does not account for one of the most delicate points, associated with the fact that the waveguide, in which the CARM field grows, is overmoded. In real life devices the beam may lock more than one transverse modes, thus determining a loss of efficiency of the device itself. The presence of an intra waveguide transverse mode-locking may cause early saturation and a significant power reduction. In the following we describe the code operation and performances. The particle in cell module realized inside the CST Microwave Studio implements the Maxwell equations in time domain: the entire volume is subdivided with an hexahedral mesh were the charge particles move under the effect of the relativistic Lorentz force equation. The associated current and charge distribution become the source terms driving the field evolution through the integration of the Maxwell equations. The current flow in the loop algorithm is controlled by checking the charge conservation during the iteration cycles. The simulation is executed in such a way that the electric and magnetic fields are derived from the Maxwell equation at each integration step. The method employs a finite integration technique or a finite difference time algorithm. The fields are calculated by means of an interpolation technique, in correspondence of the particle positions. They are exploited at the next step to evaluate the current, induced by the particle motion, which is interpolated on the mesh. Then the procedure is iterated. In Fig. 6 we have shown the comparison of the intensity growth computed with the three different tools. The simulations have been run for the set of parameters reported in Table 2 and corresponding to the ENEA Frascati CARM design. In this case the operating mode is no more 𝑇 𝐸11 but the high order counterpart 𝑇 𝐸53 . The dot red line corresponds to the intensity evaluated with CST, the comparison is satisfactory as to the linear regime but a discrepancy is evident regarding the saturation length and the output power level. The GRAAL code yields indeed a significantly larger amount of power. The reasons of the discrepancy are very likely due to the fact that

This motivations of this paper are manyfold however, we would like to emphasize that our primary interest was to develop a numerical tool which together with the semi-analytical scaling formulae might provide a reliable guide to get a first understanding of the dynamical behavior of a CARM and to specify a very first design criterion. The comparison we have developed confirms the reliability of GRAAL code and analytical tools for routinely cheek of the CARM gain and unsaturated behavior. The possibility of using them, without continuously resorting to use the CST code (limited by the large amount of computational resources), represents a significant simplification to fix the main design parameters. The final parameters assembly is eventually checked by a start-to-end CST run. We must finally underscore that the analysis of the transverse field dynamics accomplished by the use of CST is an issue of not secondary importance. The code does not employ any orthogonal set. The cavity mode content should be carefully derived using a projection of the CST output field on the orthogonal set chosen to characterize the cavity. A massive simulation campaign is being developed, in such a respect. The relevant conclusions will be published elsewhere. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. CRediT authorship contribution statement G. Dattoli: Supervision, Conceptualization. E. Di Palma: Formal analysis, Software. Acknowledgments The Authors express their sincere appreciation to dr. Svilen Sebachevsky for clarifying discussions on the theory of CARM and GYROTRON devices. It is also a pleasure to recognize the critical remarks by Drs. N. Ginzburg, N. Peskov and A. Savilov. The kind assistance of Dr. I. Spassovsky during any stage of the paper is gratefully recognized.

(a) GRAAL (and the analytical model) are single mode and 1D; (b) CST takes the cavity as it is, namely with its geometry and all the possible transverse modes which are likely to be excited and amplified by the interaction with the beam (see Fig. 7). 5

G. Dattoli and E. Di Palma

Nuclear Inst. and Methods in Physics Research, A 955 (2020) 163310

Fig. 6. The intra-cavity CARM growth signal evaluated with different code using the parameters reported in Table 2 with a beam current of 20 𝐴 in (a) and 35 𝐴 in (b). The system has been optimized with 𝛥 = 0.025 in (a) and 𝛥 = 0.10 in (b) which matches a frequency of 260.5 GHz and 263.5 GHz respectively.

Fig. 7. The output signal from CST particle-in-cell simulation using the parameters of Table 2 with a beam current of 20 𝐴 for the mode 𝑇 𝐸52 in (a) and 𝑇 𝑀52 in (b).

References

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