# Exact and variational solutions of 3D eigenmodes in high gain FELs [Exact and variational solutions of 3D eigenmodes in high gain free electron lasers]

## Abstract

Exact solution and variational approximation of eigenmodes in high gain FELs are presented. These eigenmodes specify transverse profiles and exponential growth rates of the laser field before saturation. They are self-consistent solutions of coupled Maxwell–Vlasov equations describing FEL interaction taking into account the effects due to energy spread, emittance and betatron oscillations of the electron beam, as well as diffraction and optical guiding of the laser field. A new formalism of scaling is introduced and based on which solutions in various limiting cases are discussed. Additionally, a fitting formula is obtained from interpolating the variational solution for quick calculation of exponential growth rate of the fundamental mode.

- Authors:

- Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)

- Publication Date:

- Research Org.:
- Lawrence Berkeley National Laboratory, Berkeley, CA (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22)

- OSTI Identifier:
- 751801

- Report Number(s):
- LBNL-44477

Journal ID: ISSN 0168-9002; R&D Project: 455301

- DOE Contract Number:
- AC03-76SF00098

- Resource Type:
- Journal Article

- Journal Name:
- Nuclear Instruments and Methods in Physics Research. Section A, Accelerators, Spectrometers, Detectors and Associated Equipment

- Additional Journal Information:
- Journal Volume: 445; Journal Issue: 1-3; Journal ID: ISSN 0168-9002

- Publisher:
- Elsevier

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 43 PARTICLE ACCELERATORS; 3D; high gain FEL theory; optical guiding; SASE

### Citation Formats

```
Xie, Ming.
```*Exact and variational solutions of 3D eigenmodes in high gain FELs [Exact and variational solutions of 3D eigenmodes in high gain free electron lasers]*. United States: N. p., 2000.
Web. doi:10.1016/S0168-9002(00)00114-5.

```
Xie, Ming.
```*Exact and variational solutions of 3D eigenmodes in high gain FELs [Exact and variational solutions of 3D eigenmodes in high gain free electron lasers]*. United States. doi:10.1016/S0168-9002(00)00114-5.

```
Xie, Ming. Thu .
"Exact and variational solutions of 3D eigenmodes in high gain FELs [Exact and variational solutions of 3D eigenmodes in high gain free electron lasers]". United States. doi:10.1016/S0168-9002(00)00114-5. https://www.osti.gov/servlets/purl/751801.
```

```
@article{osti_751801,
```

title = {Exact and variational solutions of 3D eigenmodes in high gain FELs [Exact and variational solutions of 3D eigenmodes in high gain free electron lasers]},

author = {Xie, Ming},

abstractNote = {Exact solution and variational approximation of eigenmodes in high gain FELs are presented. These eigenmodes specify transverse profiles and exponential growth rates of the laser field before saturation. They are self-consistent solutions of coupled Maxwell–Vlasov equations describing FEL interaction taking into account the effects due to energy spread, emittance and betatron oscillations of the electron beam, as well as diffraction and optical guiding of the laser field. A new formalism of scaling is introduced and based on which solutions in various limiting cases are discussed. Additionally, a fitting formula is obtained from interpolating the variational solution for quick calculation of exponential growth rate of the fundamental mode.},

doi = {10.1016/S0168-9002(00)00114-5},

journal = {Nuclear Instruments and Methods in Physics Research. Section A, Accelerators, Spectrometers, Detectors and Associated Equipment},

issn = {0168-9002},

number = 1-3,

volume = 445,

place = {United States},

year = {2000},

month = {5}

}