Abstract
It is shown that Jacobi elliptic function solutions of the nonlinear Duffing equation model the radiation field in a high-gain free electron laser through early saturation. After initial start-up, the field can be expressed equivalently with a hyperbolic secant. The model is derived for arbitrary detuning from resonance, which enables study of the spectral properties in the early nonlinear regime.
Highlights
Linear free electron laser (FEL) theory has been extremely successful at describing the basic properties of the single pass, high-gain FEL in the start-up and exponential gain regimes
Solutions to the timedependent FEL equations in the limit of strong slippage have been obtained in the form of hyperbolic secant functions that evolve self- [11,12]
We show that the field can be modeled by (1) and find relatively simple Jacobi elliptic function solutions that closely match numerical solutions for arbitrary detuning
Summary
Linear free electron laser (FEL) theory has been extremely successful at describing the basic properties of the single pass, high-gain FEL in the start-up and exponential gain regimes. Werkhoven and Schep in [10] obtained somewhat involved but closed-form expressions for the resonant FEL field amplitude in terms of Jacobi elliptic functions that match well with numerical solutions deep into saturation. We show that the radiation field in a timeindependent high-gain FEL can be described from startup (either from electron beam bunching or from an external radiation field) through early saturation by the Jacobi elliptic function solutions to the unforced, undamped Duffing equation,. We show that the field can be modeled by (1) and find relatively simple Jacobi elliptic function solutions that closely match numerical solutions for arbitrary detuning This provides an analytic description of the spectral evolution of the field as it saturates. We show that the resonant solutions for the complex field extend to the electron beam bunching factor and energy spread
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