Analytical Solutions for Transient and Steady State Beam Loading in Arbitrary Traveling Wave Accelerating Structures

Analytical solutions are derived for both transient and steady state gradient distributions in the travelling wave (TW) accelerating structures with arbitrary variation of parameters over the structure length. The results of the unloaded and beam loaded cases are presented. Finally the exact analytical shape of the RF pulse waveform was found in order to apply the transient beam loading compensation scheme during the structure filling time. The obtained theoretical formulas were crosschecked by direct numerical simulations on the CLIC main linac accelerating structure and demonstrated a good agreement. The proposed methods provide a fast and reliable tool for the initial stage of the TW structure analysis.


Introduction
The steady state theory of beam loading in electron linear accelerators was developed in the '50s by a number of authors both for constant impedance [1,2,3] and constant gradient [4] accelerating structures. They considered the equation for energy conservation in a volume between any two cross sections; the power gained by the beam or lost in the walls due to the Joule effect results in a reduction of the power flow. Later on, transient behavior was studied following a similar approach, but in this case, in addition to the power dissipated in the walls and gained by the beam, the transient change in the energy stored in the volume contributes to the power flow variation along the structure. Again, only constant impedance [5,6,7] or constant gradient [8,9] accelerating structures were considered.
However, traveling wave accelerating structures with arbitrary (neither constant impedance nor constant gradient) geometrical variations over the length are widely used today in order to optimize the acceleration structure and linac performance [10,11]. The relationships between structure length, input and average accelerating gradients are obtained by solving the energy conservation equation numerically. For the first time an analytical solution of the gradient profile in a loaded arbitrary TW structure was recently proposed in [12] but for the steady-state regime only. The comprehensive numerical analysis of an arbitrary TW structure including the effects of a signal dispersion was recently published in [13] using the circuit model and mode matching technique.
In this paper, generalized analytical solutions of the gradient distribution in the TW accelerating structure with an arbitrary variation of parameters over the structure length are presented for both steady state and transient regimes. It is based on the method suggested earlier by one of the coauthors [14] and is similar to the classical approach [1−9]. Finally a simple analytical relation is derived that allows the input power ramp needed to create, at the end of the filling time, the field distribution inside the TW structure that coincides to the loaded field distribution in the presence of the beam to be determined. The compact analytical formulas so obtained give us a better understanding of the physics of TW structures and provide a tool for a fast preliminary structure optimization.
The following definitions are used throughout the paper: The following assumptions are used: a) the structure is perfectly matched at both ends and has no internal reflections, b) all dispersion effects that limit field rise time: where c is the speed of light, are neglected, c) time separation between two neighboring bunches and time of flight of the beam through the structure are much less than the filling time of the structure.

Steady State Regime
The basic traveling wave structure relations are: Energy conservation including wall losses and the interaction with the beam gives: Using Eq. (2.2) in the derivation of the power flow Eq. (2.1) yields: Substituting Eq. (2.6) into Eq. (2.5) and using Eq. (2.7) yields: Integrating Eq. (2.8) gives: ) and z' is a local integration variable.
Therefore the general solution of Eq. (2.5) is: The solution for the homogeneous Eq. (2.7) is: is a gradient at the beginning of accelerating structure and can be found from initial conditions : The integral of function ) (z  can be simplified using analytical solutions: Finally we can rewrite Eq. (2.10) as: 14) The first term on the right hand side of Eq. (2.14) is the solution of the homogeneous equation for the unloaded gradient obtained above in Eq. (2.13). The second term is the so-called beam induced gradient which is the difference between the loaded and unloaded gradient distributions. Parameters of the CLIC main linac accelerating structure are summarized in the Table 1 [11]. They have been used to compare an accurate solution for an arbitrary variation of the TW structure parameters given by Eq. (2.14) to an approximate solution given in [4] where it has been assumed that the shunt impedance and Q-factor are constant in the range over which the group velocity changes and that they are both equal to their respective averages over the structure.  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27  The unloaded gradient has been calculated for a 3D model of the structure using Ansoft HFSS [15], a frequency-domain finite-element code which takes into account internal reflections [11]. First of all the parameters of individual cells were calculated for the given phase advance, shunt impedance, group velocity and maximum EM-field strength on the surface. The result of individual cell optimization is shown in Fig. 1. Next, the input and output RF couplers were designed in order to match the TW structure with (c) (d) feeding waveguide and RF loads. The detailed procedure of RF coupler design using Ansoft HFSS code is described in [16]. After that we made the simulation of full CLIC main linac accelerating structure and verified RF phase advance per cell and internal reflections using the well-known "Kroll's" method [17] (see Fig. 2). Finally we derived the secondary values (stored energy and RF power flow per cell) necessary for the unloaded gradient calculation. simulation. In contrast, the approximate solution is quite different from the accurate solution due mainly to a significant (~30%) variation of the shunt impedance along the structure, see Table 1.

Transient Regime
The transient regime can also be derived analytically. The instantaneous energy conservation is given by:

Substituting Eqs. (2.1), (2.2) and (2.4) into Eq. (3.1) yields:
We assume the following initial conditions: Using the Laplace transformation of a function ) (t G : and taking into account Eqs. (3.3) we can write Eq. (3.2) as follows: First, we consider the unloaded case ( 0  I ). In this case Eq. (3.4) becomes a homogeneous differential equation: . The solution of Eq. (3.5), obtained in a similar manner to the solution of Eq. (2.7), is: where ) (z g is defined in Eq.
is the Heaviside step function and is the signal time delay. Thus, the distribution of the unloaded gradient in time-domain along the structure is: or taking into account Eqs. (2.11) and (2.13) it can be expressed as a function of the input RF power: The Then Eq. (3.4) becomes: Substituting Eqs. (3.5) and (3.11) into Eq. (3.12) yields: and furthermore using Eqs. (3.7) and (3.8) 14) The solution of Eq. (3.14) can be obtained by integration in the form: . Finally, the general solution of Eq. (3.4) is derived using Eqs. (3.11 and 3.15): Thus the time-dependent solution of Eq. (3.1) is obtained by applying the inverse Laplace transform to Eq. (3.16). Here again the time shifting property has been used: where, ) (z  is a function of the coordinate z and given by Eq. (3.8).
The first term on the right hand side of Eq. For the CLIC main linac accelerating structure with the parameters from the Table 1, the time-dependent solution given by Eq. (3.17) during the transient related to structure filling and to beam injection is illustrated in Fig. 4 (a) and (b) respectively. In Fig. 5, the corresponding input power and beam-current time dependences are shown together with the unloaded, loaded and beam voltages defined as:

Compensation of the transient beam loading
The idea of transient beam loading compensation was proposed in 1993 at SLAC (USA) [18], where a linear ramp of the input RF amplitude has been applied to compensate the bunch-to-bunch energy variation to first order. Later a sophisticated is the steady-state value of the input gradient after injection. The input gradient in Eq. (4.2) indirectly depends on time. Introducing the function ) (t z as a solution of the following integral equation: Eq. (4.2) becomes an explicit function of time: An expression for the input RF power is derived using Eq. (2.11): is the steady-state value of the input RF power after injection. The solution of Eq. (4.5) is shown in Fig. 6 (blue) together with the beam current (green) injected exactly at the end of the ramp and the corresponding unloaded (black), loaded (red) and beam (light blue) voltages. The gradient distribution at different moments of time is presented for the compensated case in Fig. 7 (a) and (b) for the structure filling transient and the beam injection transient, respectively.

Summary
Analytical expressions for unloaded and loaded gradient distributions in travelling wave structures with arbitrary variation of parameters were derived in steady state and in transient. They were applied to the case of the CLIC main linac accelerating structure. The analytical solution agrees very well with the numerical solution obtained using finiteelement code. On the other hand, it differs from the approximate solution obtained using expressions derived earlier in [4]. Finally the exact analytical solution was found for the wavefront of input RF pulse which theoretically provides exact compensation of the beam loading effect. The derived analytical formulas are very useful during the preliminary stages of structure design and later for structure efficiency optimization.