ROCK: A Flexible Gyrotron Cavity Simulation Toolkit Using an Accuracy-Improved Time-Dependent Self-Consistent Multimode Interaction Model

Before diving into the models, it is important to clarify the term “guiding center.” Generally, the motion of guiding center is solved from the collection of all zero-order terms in the expanded motion equation by Lorentz force. The particular solution of the guiding center equation is a field line of the dominating static magnetic field, which is referred to as the ideal guiding center. Secular effects of the high-order terms, in the presence of an RF field, may cause the actual guiding center,to deviate slightly from the original magnetic field line, where \(\textsf{r}_{\text {e}}\) is the electron position vector and \(\phi \) is the cyclotron phase. This change of field line rarely affects the so-called coupling factor. However, maintaining an accurate description of electron motion, especially tracking the exact cyclotron phase relative to the field, becomes delicate if the guiding center is no longer ideal. It should be noted that Eq. 2 is not intended for direct evaluation; therefore, the actual guiding center is a rather vague concept. Lastly, the term numerical guiding center refers to the program’s state variable.

This paragraph briefly summarizes the classical model for later comparisons. The key parameters of an electron’s motion are its position and velocity. Let \(\varvec{u}:=\gamma \,\varvec{v}\) and \(\mathfrak {i}\) represent the imaginary unit. For convenience, \(\varvec{u}\) is also referred to as velocity in this article. The classical model uses a complex number to represent the two-dimensional (2D) vector \(\varvec{u}_\perp \). This complex representation facilitates the formulation of electron rotation as another amplitude modulation(see Eq. (5) of [7] for example). Determining the vector \(\varvec{u}_\perp \) requires two quantities: its direction and magnitude. In cyclotron coordinates, the direction of \(\varvec{u}_\perp \) consists of the rotor phase \(\omega _{\text {e}}(t-t_0)\), plus the phase of its complex envelope \(u_{\perp ,\text {env}}\) as a corrective angle. The magnitude \(|{}\varvec{u}_{\perp }|\) is given by the modulus \(|{}u_{\perp ,\text {env}}|\), and it is treated as \(|{}\varvec{u}_{\perp }|=|{}u_\phi |\). To determine the position of an electron in cyclotron coordinates, three parameters are required: the cyclotron angle \(\phi \), the Larmor radius \(r_{\text {L}}\), and the guiding center position. They are evaluated as follows: Transversally rotating the angle of velocity by \(90^\circ \) results in the \(\phi \) angle; the Larmor radius is calculated using \(r_{\text {L}}=|{}v_{\perp }/\omega _{\text {e}}|\) where the frequency of the slowly varying \(u_{\perp ,\text {env}}\) can be safely ignored; the numerical guiding center corresponds to the ideal guiding center.

The model of electron motion in ROCK is different from the classical one. To avoid ambiguity between the \(\mathfrak {i}\) associated with the field and the \(\mathfrak {i}\) associated with space, the motion state in the following text is expressed using real vectors instead of complex phasors. Although the final formulae for the Lorentz force are equivalent, this equivalence is not generally guaranteed, as will be discussed in the subsection on field excitation. Similar to the complex formulation, the components of cyclotron velocity are slowly varying, as far as the numerical guiding center position \(\textsf{r}_{\text {g}}\) and the numerical local angular frequency \(\omega _{\text {e}}\) are properly determined. Sorting and collecting terms by their basis vector from the complete equation of Newton’s second law yield more organized expressions:The reader may compare these expressions with Eq. (3.55) of [4]. The conventions used in these equations are explained as follows: Let \(\theta \) be the azimuthal angle, \(z\) the axial coordinate, and \(z_{\text {e}}\) the axial position of an electron. The prime symbol (\('\)) designates the derivative \(\text {d}/\text {d}z_{\text {e}}\). Subscript “\({\phantom{0}}_\phi \)” and “\({\phantom{0}}_{\text {L}}\)” indicate the two perpendicular components of a vector projected onto the cyclotron coordinates, where “\({\phantom{0}}_{\text {L}}\)” stands for the direction of Larmor radius, i.e., \(\varvec{r}_{\text {L}}:=\textsf{r}_{\text {e}}-\textsf{r}_{\text {g}}\). The italic \(r\) designates a radius, while the bold sans-serif \(\textsf{r}\) designates a position vector. The terms \(\alpha _\phi :={}u_\phi /u_z\) and \(\alpha _{\text {L}}:={}u_{\text {L}}/u_z\). The subscripts in \(B_{\text {sg}z}\) indicate the \(z\) component of the static field at the guiding center. \(B_{\text {w}z}\) represents the RF \(B_z\) at \(\textsf{r}_{\text {e}}\), where the subscript “\({}_{\text {w}}\)” stands for wave. The term \(\tilde{\omega }:=\omega _{\text {e}}/ \Omega _{\text {e}}\) serves as the figure-of-merit for the eccentricity, which is integrated byIt is to note that \(\omega _{\text {e}}\) is handled differently than in Eq. (16) of [7]. Ideally, \(\tilde{\omega }=1\). If at any point \(\tilde{\omega }\) deviates significantly from \(1\), it indicates that the numerical guiding center may no longer be appropriate for assuming the validity of slowly varying quantities. However, results using an improper numerical guiding center can still be accurate if the step length is decreased. When this occurs, ROCK calibrates the numerical guiding center using the static \(B\)-field, resetting \(\tilde{\omega }\) to \(1\), \(u_{\text {L}}\) to \(0\), and \(u_\phi \) to \(u_\perp \), without affecting the actual position and velocity from a perspective neutral to the coordinate system. In this article, the local coordinates are still referred to as “cyclotron coordinates,” even though the coordinate basis vectors and the numerical guiding center (the origin of the coordinates) do not necessarily reflect the actual physical values associated with cyclotron motion. Hence, the state vector of an electron contains eight components as function of \(z\):If the static \(B\)-field is axisymmetric, the component \(r_{\text {g}}\) for the guiding center radius can be replaced by the magnetic flux enclosed by a centered transverse circle, taking the aforementioned eccentricity correction into account. In the state vector, the other components, except for \(t\) and \(\phi \), are slowly varying and can be calculated using relatively large steps. Therefore, numerical steps of \(t\) and \(\phi \) include additional terms in the order of \(\delta {}z^2\) to enhance their convergence:where the preceding \(\delta \) denotes differences associated with each step of the electron’s motion. At a higher level, the entire state vector is advanced using the predictor-corrector method to approach global second-order convergence.

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Another highlight of the ROCK model is its incorporation of multiple harmonic terms from the field expansion to enhance the accuracy of electron motion. The transformation from cylindrical coordinates to cyclotron coordinates utilizes Graf’s generalization of Neumann’s addition theorem, as detailed in Chapter 11.3 of [8]. This transformation does not require the numerical guiding center to coincide with the actual one. The addition theorem yields a series with an integer index “\(\texttt {s}\),” which can be physically interpreted as the harmonic number of interaction. For example, the total field applied into Eq. 4 consists of \(E_{\phi }=\sum _{\texttt {k}}E_{\phi ,\texttt {k}}\) and \(E_{\text {L}}=\sum _{\texttt {k}}E_{\text {L},\texttt {k}}\) where \(\texttt {k}\) is an integer indicating a mode from the simulated mode list. The electric field components of the \(\texttt {k}\)’th mode should be evaluated from the seriesIn classical codes, only a single \(\texttt {s}\) is considered for each mode. In contrast, ROCK considers \(\texttt {s}\in \{0,1,2,3\}\) by default. This default range suffices to include both the fundamental (first harmonic) and second-harmonic interactions, while providing additional correction and margin by also incorporating \(\texttt {s}=0\) and \(\texttt {s}=3\). The range of \(\texttt {s}\) is a compile-time variable, offering sufficient flexibility while aiding the compiler in unwrapping and vectorizing the corresponding loops.

A simulation step of the cavity in [9] is performed to demonstrate the effect of incorporating multiple harmonic terms. A fixed RF field with output power of \(200\,\text {kW}\) is manually established, whose frequency and envelope matches the ones of its cold resonance. A probe beam represented by 63 electrons evenly distributed around the same initial guiding center is injected into the cavity. Being compared are the positions of electrons when they are leaving the cavity. For reference, the positions calculated using the Boris [10] or Buneman [11] method in Cartesian coordinates are presented. Figure 1 shows that considering multiple harmonic terms further enhances the accuracy of electron motion. Moreover, stronger interactions tend to exhibit more significant differences, especially when certain electrons bunch and drift extensively, crossing their original guiding centers.

Upon comparing both models, it becomes apparent that the classical model significantly simplifies the situation by assuming \(u_{\perp }=u_\phi \) and anchoring the numerical guiding center to a field line. This was a clever simplification: although the state of a single electron may not be accurately represented, the model performs well statistically in terms of energy exchange for an ensemble of electrons in standard scenarios. In contrast, the ROCK model has the capability to integrate position and velocity accurately while maintaining a similar computational speed by also utilizing slow variables.

In the transformed Helmholtz equation, the source term used to excite a TE-mode is denoted by the symbol “\(S\)” in the following text. Its physical formulation with respect to the slow-varying amplitude-modulated field iswhere \(\varvec{\mathfrak {e}}\) is the TE-mode eigenvector, \(\texttt {k}\) is an index of the considered mode list, \(\mu _0\) is the vacuum permeability, and \(\omega _0\) is the carrier frequency. Numerically, it is evaluated similar to Eq. 10 aswith \( S_{\texttt {k},\texttt {s}}(z_{\text {e}}) :={} S_{\texttt {k},\texttt {s}}^{(0)} + S_{\texttt {k},\texttt {s}}^{(1)} + \ldots \) whose dominating term iswhere \(\phi _0\) is the carrier phase. The next-order term \(S_{\texttt {k},\texttt {s}}^{(1)}\) is implemented in the program but not introduced in this article for the sake of conciseness. Equation 13 is similar to the formulation of SELFT (cf. Eq. (4.45) of [4]). One potential difference compared to some codes is the coefficient of the second term in Eq. 13, whose magnitude should be \( v_z\,\alpha _\phi /r_{\text {L}} = \omega _{\text {e}} = \tilde{\omega }\, \Omega _{\text {e}} \) instead of the approximation with \(\omega _{0,\texttt {k}}\) of the field. Another difference is in the incorporation of the term \((\alpha '_\phi \;\alpha '_{\text {L}})\): both ROCK and SELFT take this term into account, whereas it was ignored in some analyses (e.g., [1, 7]) (see also [3, 4] for further discussions). When examining a single harmonic term, the primary feature of ROCK is the use of vector form rather than a complex phasor. The vector form clearly distinguishes between multiplying by \(\mathfrak {i}\) and rotating the eigenvector by \(90^\circ \). Moreover, the vector form results in slightly different calculations, where \(\alpha _{\text {L}}\) should be multiplied with \(\mathfrak {e}^*_{\text {L}}\). Notably, \(\mathfrak {e}^*_{\text {L}}\) and \(\mathfrak {e}^*_{\phi }\) have different magnitudes.

Similar to the calculation of the Lorentz force, Eq. 12 includes multiple harmonic terms by default. Its effect in a monomode simulation is demonstrated in Fig. 2. In this particular scenario, including multiple terms shows good agreement with the Cartesian reference within the up-taper section and may reveal further details of the after-cavity interaction. Generally, Eq. 12 is capable of handling cases where the harmonic ratio \(\omega _0/ \Omega _{\text {e}}\) is not close to an integer, which may occur with parasite modes. The potential advantages of including multiple harmonic terms must be carefully justified in future investigations.

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The geometry of the simulated coaxial cavity is defined on page 182 of [4]. The quasi-stationary operating points for both mono- and multimode cases from Tab. 7.2 of [4] are referred. To achieve the hard-excitation operating point, the beam voltage \(U_{\text {b}}\) is ramped up from 75 to \(90\,\text {kV}\), while the pitch factor \(\alpha _\phi \) and beam current \(I_{\text {b}}\) are scaled according to Chapter 5.3 of [13]. Further details on the operating point are given on page 184 of [4]. The results are compared in Table 1 and visualized in Fig. 3.

Both operating frequency and power at the designed operating point show very good agreement with the literature values. Furthermore, as discussed on page 188 of [4], the experiment of the gyrotron with this cavity showed a drop of the power in the operating mode TE\(_{28{,}16}\) before losing the mode. This effect can be simulated in ROCK, as shown in Fig. 3. However, the reader should note that conducting an experimental comparison and analyzing uncertainties and error bars would require significant effort. A comprehensive experimental validation will be addressed in a separate article.

To furthermore investigate the improved model, a second-harmonic TE\(_{02}\) example is demonstrated in this section. The code EMPIC is used to provide reference values because it makes minimal assumptions aside from azimuthal symmetry. EMPIC, as part of the in-house beam-optics package ESRAY [14], is a full-wave PIC code incorporating the solution of Maxwell’s equation on a 2D structured, non-orthogonal simulation mesh in cylindrical coordinates using the finite-difference time-domain (FDTD) scheme proposed by Holland [15]. The motion of the particles is computed numerically using the aforementioned Boris method, applied to three velocity components \((v_z, v_r, v_\theta )\) but only two spatial \((z,r)\) components. Consequently, only the interaction with TE\(_{0n}\) modes are considered in the following comparison.

The cavity shape and beam parameters are defined in Table 2, which yields a resonance frequency of approximately \(96.5\,\text {GHz}\) for the second-harmonic main mode TE\(_{02}\). The given beam parameters and the static magnetic field are kept constant for simplicity. Additionally, the space charge effect is disabled to eliminate differences in how the electrostatic effects are handled, ensuring that the comparison focuses solely on the interaction mechanism.

Figure 4 presents the output power during the ramp-up of beam voltage from 60 to \(90\,\text {kV}\) within \(300\,\text {ns}\). ROCK produces results very close to EMPIC, whereas EURIDICE exhibits a slight deviation in the power profile. Various combinations of options have been tested in EURIDICE, but this deviation persists. This comparison indicates that ROCK is more accurate in certain scenarios.