Nonlinear Theory: Optical Mode Analysis

Abstract

The previous chapter dealt with the nonlinear theory in the steady-state regime based on the slowly varying envelope approximation (SVEA). Most of the time-dependent free-electron laser simulation codes that are in use at the present time deal either with an extension of the SVEA in order to solve the wave equation or a particle-in-cell simulation where Maxwell’s equations are solved using a finite-difference time-domain (FDTD) algorithm. The time-dependent formulation presented in this chapter is an extension of the SVEA, in which the SVEA is extended by allowing the slowly varying amplitude to vary in both axial position and time. A time-dependent formulation is necessary to simulate short-wavelength free-electron lasers employing radio-frequency linear accelerators (RF linacs) or storage rings. RF linacs produce high-energy beams with picosecond pulse times and bunch charges of at most several nano-Coulombs. In X-ray free-electron lasers, the actual bunch charge used is about 250 pC or less. Since the growth rate depends upon the peak current, it is desirable to produce bunches with peak currents of several hundred to several thousand amperes, and this requires compression of the bunch to sub-picosecond pulse times. As a result, the slippage of the optical field relative to the electrons can be significant. In addition to describing the slippage of the optical pulse, time dependence is also needed to study the spectral properties of the optical field such as the temporal coherence, linewidth, sideband production, etc. Furthermore, in contrast to the guided-mode analysis used for the steady-state formulation presented in the preceding chapter, the three-dimensional formulations presented in this chapter make use of superpositions of Gaussian optical modes to represent the radiation fields.