Description of electromagnetic fields in inhomogeneous accelerating sections. IV couplers

A new approach to incorporating coupling elements into a generalized coupled mode theory is presented. The simplest model of coupling of a structured waveguide with an external RF power source and load through loops and transmission lines was used. Even such a simple model significantly complicated the system of coupled equations. It turned into a coupled integro-differential system of the Barbashin type with degenerate kernels. Since the integral kernels are degenerate, this system is reduced to three independent systems of differential equations. Instead of solving a system of coupled integro-differential equations, we need to find solutions to three systems of ordinary differential equations. Two systems describe the distribution of the field excited by one loop and the specified value of the excitation current in it. In the first system the loop is located at the section’s input, and in the second, at the section’s output. The third system does not depend on the loop parameters at all. It describes the distribution of the field excited by an electron beam in a section without loops. Based on the developed analytical model, the computer code was developed for matching the loop couplers for the uniform accelerating sections of X-band. The calculation results were used to simulate the non-uniform section. Without additional matching, we obtained an input reflection coefficient of 8E-3.

💡 Research Summary

The paper presents a novel extension of generalized coupled‑mode theory (CMT) that explicitly incorporates loop couplers and transmission‑line feeding networks into the electromagnetic description of non‑periodic, structured accelerating waveguides. Starting from a modal expansion of the fields in terms of generalized eigenvectors (Eq. 1), the authors derive a set of coupled integro‑differential equations (Eq. 2) that contain the loop current and the voltage on the feeding line as additional unknowns. Because the integral kernels are degenerate (Barbashin type), the system can be reduced analytically to three independent ordinary differential‑equation (ODE) subsystems:

1. The field excited by a single loop placed at the entrance of the section with a prescribed excitation current.
2. The analogous problem for a loop located at the exit.
3. The field generated by the electron beam alone, i.e., a section without any loops.

These subsystems are governed by Eqs. (25)–(26) together with boundary conditions (Eq. 22) that enforce perfect electric walls at the “cut‑off” waveguide ends. The loop is modeled as a thin conductor with a geometric factor (G_r) (Eq. 15) and is fed by a transmission line of characteristic impedance (Z_0) (Eq. 14). The magnetic flux through the loop is expressed as a sum of one‑dimensional integrals (Eq. 16), which are evaluated using Simpson’s rule.

For numerical solution the authors employ a fourth‑order Runge‑Kutta scheme combined with adaptive, non‑uniform meshing. Because the disk‑loaded waveguide (DLW) cells vary in radius, iris aperture, and diaphragm thickness, a constant‑step mesh would be inefficient; instead each cell is divided into a fixed number of sub‑segments (typically (D_N=60)), yielding a variable step size that follows the geometry. The resulting sparse linear system (Eq. 32) is solved with the LSLZG routine from the IMSL Math Library, providing the constants (\Gamma_{1,2}^{\pm}) and the modal amplitudes (C_s^{\pm}(z)).

The methodology is applied to an X‑band (11.994 GHz) DLW accelerating structure with a phase advance of (2\pi/3) per cell. The structure is deliberately non‑uniform: cell diameters, iris radii, and diaphragm thicknesses all change gradually along the length. Using the developed analytical model and a dedicated computer code, the loop dimensions and transmission‑line parameters are optimized to achieve impedance matching at both ends. The calculated input reflection coefficient is (|S_{11}| = 8\times10^{-3}), a very low value that demonstrates the effectiveness of the matching without requiring additional tuning elements.

Key contributions of the work include:

• Demonstration that degenerate integral kernels allow the transformation of a complex integro‑differential CMT problem into a set of tractable ODEs, dramatically simplifying both analysis and computation.
• Introduction of loop current and transmission‑line voltage as explicit variables within the CMT framework, enabling direct treatment of external RF feeding.
• Development of a robust numerical scheme that handles strong geometric non‑uniformity via adaptive meshing and efficient sparse‑matrix solvers.
• Validation of the approach on a realistic X‑band DLW structure, achieving an input reflection of less than 1 % (8 × 10⁻³), which is competitive with, or superior to, conventional matching techniques.

The presented theory is not limited to X‑band DLW; it can be extended to other frequency bands, different waveguide geometries, and multi‑mode situations. Future work may incorporate beam loading non‑linearities, higher‑order mode coupling, and tolerance analysis for manufacturing imperfections, thereby broadening the applicability of the method to high‑gradient linear accelerators, free‑electron lasers, and advanced accelerator concepts.