# Recirculated Wave Undulators for Compact FELs

## Abstract

**:**

## 1. Introduction

## 2. The Optical Cavity

## 3. Solution of the Liouville Equation for Longitudinal Beam Dynamics in Magnetic Chicanes

## 4. Dynamics of Relativistic Electrons Inside an Electromagnetic Undulator

## 5. 1D FEL Equations

## 6. Discussions on the Recirculated Wave Undulator FEL Scheme

## 7. Coherent Synchrotron Radiation

## 8. FEL Radiation: Numerical Simulations

## 9. FEL Radiation: Comparison with the Analytic Model

## 10. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Appendix A. Derivation of the 1D FEL Equations for the Wave Undulator

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**Figure 2.**Evolution of the longitudinal phase-space of the electron beam passing from one interaction point to the other. The evolution through the magnetic chicane is calculated via the analytic solution of the Liouville equation. The longitudinal phase spaces shown above will be commented in Section 8.

**Figure 4.**Simulation results with FEL1D code in Python, concerning the first interaction point of Figure 2.

**Figure 5.**Simulation results with FEL1D code in Python, at the entrance of the second interaction point.

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Curcio, A.
Recirculated Wave Undulators for Compact FELs. *Appl. Sci.* **2021**, *11*, 5936.
https://doi.org/10.3390/app11135936

**AMA Style**

Curcio A.
Recirculated Wave Undulators for Compact FELs. *Applied Sciences*. 2021; 11(13):5936.
https://doi.org/10.3390/app11135936

**Chicago/Turabian Style**

Curcio, Alessandro.
2021. "Recirculated Wave Undulators for Compact FELs" *Applied Sciences* 11, no. 13: 5936.
https://doi.org/10.3390/app11135936