Spectral Form Function with Applications in Beam Physics

TL;DR

The paper introduces the spectral form function (SFF) as the 6D Fourier transform of the beam phase‑space density $F(K)=∫ ψ(X) e^{-i K·X} dX$, extending the 1D bunching factor to capture full 6D structures. It develops a coupled Liouville–Fokker–Planck framework for SFF in general linear lattices with damping and diffusion, yielding explicit transport rules for F and ψ via the deterministic transfer Ř and diffusion, including a fundamental Gaussian kernel with covariance σ. The framework is applied to electron storage rings and laser‑driven microbunching, deriving single‑stage HGHG and double‑stage EEHG spectral forms and showing how diffusion damps high‑frequency structure while lattice transport shapes the spectrum. This provides a unified, quantitative tool for predicting coherent radiation properties and guiding microbunching‑based light source design.

Abstract

To describe longitudinal fine structure like microbunching within a particle beam, a classical approach is to define a bunching factor which is the Fourier transform of the particle longitudinal density distribution. Such a 1D definition of bunching factor can be generalized to a 6D spectral form function (SFF) to describe more complicated structure in phase space. The complex SFF is another complete description of beam in spectral domain and can offer complementary and valuable insight in beam dynamics study which usually invokes the real particle density distribution. The basic property and Fokker-Planck equation of the SFF is presented, along with its solution in a general coupled linear lattice. The example applications of SFF in electron storage ring physics and laser-induced microbunching are presented.