Rayleigh–Ritz based expansion method for wakefields in dielectrically lined rectangular waveguides

Highlights

• Eigenmodes of dechirpers with dielectric slabs possess analytical expressions.

• The wake function in a rectangular dechirper is computed by an eigenmode expansion.

• The wake function has an analytical expression.

• The wake function can be effectively computed by a Python-based code.

• The computation algorithm is independent of spatial discretisation.

Abstract

In this work, a semi-analytical method for determining wakefields in dielectrically lined rectangular waveguides is presented. This approach is based on a Rayleigh–Ritz method to analytically identify the eigenmodes of the structure, which is currently studied for the application as a so-called ‘wakefield dechirper’. The electric field is subsequently determined through an eigenmode expansion, and the wakefield is calculated from the electric field. By virtue of using an analytic ansatz throughout the wakefield determination, an expression for the Green's function wakefield is found.

The semi-analytical method is then benchmarked against simulations using purely numerical approaches. Compared to numerical approaches, the advantages of the presented method are the independence from any need of discretisation, the computational efficiency of the method's presented Python-based implementation and finally the opportunity to calculate a true Green's function wakefield. From this Green's function, the wake potentials of different bunch shapes can be obtained via convolution.

Introduction

The efficient reduction of the pulse length of electron beams plays a crucial role in the generation of short pulses in the range of sub-picoseconds at future light sources. At the radiation source ELBE in Dresden–Rossendorf short pulses are required for coherent THz generation and laser-electron beam interaction experiments such as X-ray Thomson scattering.

A first possibility, the generation of very short pulses with magnetic compression, has, however, the severe disadvantage of leading to an increase of the electron beam's energy width due to energy-time-correlation (see, e.g., [1] and [2]). To grant a small energy spectrum, this energy spread has to be counteracted. Conventional methods, like the use of further accelerator cavities, are not suited for application and have several disadvantages. For one, they are very spacious and costly. On the other hand, they cannot reach the necessary energy-time-gradients and thus, a complete compensation of the energy spread is impossible from the very beginning.

The reduction of the energy width can also be carried out passively by letting the beam interact with its own wakefield (cf., e.g., [3], [4] and [5]). Wakefields are generated when a charged particle beam passes through suitable structures, so-called ‘wakefield dechirpers’. The concept behind these structures has been studied for the application at various particle accelerators in the last years, and have also partly been tested at these facilities. The forms of these dechirpers range from corrugated cylindrical pipes ([6] and [7]) over corrugated rectangular waveguides [8] to dielectrically lined cylindrical [9] and rectangular waveguides ([10], [11] and [12]). All proved to induce a reduction of the energy width in the particle beam. The major advantages of these structures are their passive mode of operation and their simplicity (and subsequent cost-efficiency), while they nonetheless remain tunable to a multitude of possible applications due to their highly adaptable geometries.

In this work, a rigorous, self-consistent method is presented with which the wake function (i.e. the Green's function) inside the dechirper can be computed. While also focusing on the expansion of the wakefield into the structure's eigenmodes like in [13] and [14], neither method uses the here presented Fourier-expansion based Rayleigh–Ritz approach to compute the eigenmodes. Both rather consider the solution in the different sub-sections of the waveguide (i.e. dielectric slabs and vacuum).

The idea of using a spatially dependent relative permittivity is similar to the approach presented in [15], which constitutes a rigorous solution to Maxwell's equations in the dechirper structure in presence of a point charge. This then leads to the wake function, in contrast to the expansion-based approach presented here.

Our approach is free of assumptions. The modes computed with the suggested approach are three-dimensional eigenmodes compared to the two-dimensional eigenmodes used in [14]. By using three-dimensional eigenmodes, the influence of the length of the dechirper is still regarded within the analytical description, while a two-dimensional formulation of the problem would automatically assume that the waveguide is infinitely long. As several applications, including the dechirper used at ELBE, include dechirpers of comparatively short lengths, it is not automatically granted that the structure can be assumed to be infinitely long. The open boundary in longitudinal direction is not imposed on the modes, it is rather reached by a superposition of the eigenmodes in a closed structure for PEC and PMC boundary conditions employed in longitudinal direction. In this way, an analytical formulation of the wake function is found which also includes the loss factors (that have not been considered in neither [14] nor [15]) of the structure.

The algorithm presented here can calculate any arbitrary three-dimensional eigenmode in the structure in the same computational time, regardless of its frequency, since the algorithm is free of any spatial or temporal discretisation.

A general layout of a dielectrically lined rectangular waveguide is shown in Fig. 1. The guide itself is composed of a highly conductive material, e.g. copper or aluminium, and is lined symmetrically with two identical dielectric slabs at the top and bottom of the guide. Theoretically, the only requirement on the slabs is that their relative permittivity is higher than one; while practically the chosen materials must be suitable for the use in vacuum as well. The particle beam, here electrons, passes the structure right through its centre.

Compared to cylindrical waveguides, a rectangular structure has the advantage of a higher flexibility: While the cylinder's radius is a quantity that cannot be adjusted once the structure is in production, a rectangular layout still allows for a later tuning. To achieve this, the waveguide is build in a way so that the upper conducting plate with the corresponding dielectric is left unconnected to the remaining guide and is thus movable. This allows for an adjustment of the gap widths between the dielectric plates after production, as has been proposed, for example, in [11] and [12]. Consequently, this also allows for an adjustment of the wakefield.

Next to the technical advantages of using a rectangular waveguide as the structure's basic layout, the complete system of waveguide and dielectric slabs, due to its simplicity, opens up the possibility of an analytic examination of the wakefield. From a mathematical point of view, the insertion of dielectric slabs into the otherwise textbook example of the open, rectangular waveguide, presents a comparatively minor change in geometry. This similarity is used for an analytic determination of the structure's eigenmodes. These eigenmodes can be used as basic functions for an expansion of the electric field generated by a single point charge inside such a waveguide. Calculating the longitudinal wakefield from the electric field, this leads to the possibility of gaining an analytical Green's function in form of the point charge wakefield, which can then be used to determine the wakefield of a variety of differently shaped bunches by means of simple convolutions, instead of performing a complete numerical simulation for every shape.

In total, the structure's geometric simplicity provides the possibility of gaining a fundamental insight in the dechirping qualities of the chosen layout, which should not be left unexploited.