# Three-dimensional free electron laser dispersion relation including betatron oscillations

## Abstract

We have developed a 3-D FEL theory based upon the Maxwell-Vlasov equations including the effects of the energy spread and emittance of the electron beam, and of betatron oscillations. The radiation field is expressed in terms of the Green's function of the inhomogeneous wave equation and the distribution function of the electron beam. The distribution function is expanded in terms of a set of orthogonal functions determined by the unperturbed particle distribution. The coupled Maxwell-Vlasov equations are then reduced to a matrix equation, from which a dispersion relation for the eigenvalues is derived. In the limit of small betatron oscillation frequency, the present dispersion relation reduces to the well-known cubic equation of the one-dimensional theory in the limit of large beam size, and it gives the correct gain in the limit of small beam size. Comparisons of our numerical results with other approaches show good agreement. We present a handy empirical formula for the FEL gain of a 3-D Gaussian beam, as a function of the scaled parameters, that can be used for a quick estimate of the grain. 5 refs., 2 figs.

- Authors:

- Publication Date:

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

- Sponsoring Org.:
- USDOE; USDOE, Washington, DC (United States)

- OSTI Identifier:
- 5039720

- Report Number(s):
- LBL-30673; CONF-9108118-19

ON: DE92002307; TRN: 91-031886

- DOE Contract Number:
- AC03-76SF00098

- Resource Type:
- Conference

- Resource Relation:
- Conference: 13. international free-electron laser (FEL) conference, Santa Fe, NM (United States), 25-30 Aug 1991

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 43 PARTICLE ACCELERATORS; 42 ENGINEERING; FREE ELECTRON LASERS; DISPERSION RELATIONS; BEAM EMITTANCE; BETATRON OSCILLATIONS; BOLTZMANN-VLASOV EQUATION; DISTRIBUTION FUNCTIONS; ELECTRON BEAMS; GREEN FUNCTION; THREE-DIMENSIONAL CALCULATIONS; WAVE EQUATIONS; BEAM DYNAMICS; BEAMS; DIFFERENTIAL EQUATIONS; EQUATIONS; FUNCTIONS; LASERS; LEPTON BEAMS; OSCILLATIONS; PARTIAL DIFFERENTIAL EQUATIONS; PARTICLE BEAMS; 430200* - Particle Accelerators- Beam Dynamics, Field Calculations, & Ion Optics; 426002 - Engineering- Lasers & Masers- (1990-)

### Citation Formats

```
Chin, Y H, Kim, K J, and Xie, M.
```*Three-dimensional free electron laser dispersion relation including betatron oscillations*. United States: N. p., 1991.
Web.

```
Chin, Y H, Kim, K J, & Xie, M.
```*Three-dimensional free electron laser dispersion relation including betatron oscillations*. United States.

```
Chin, Y H, Kim, K J, and Xie, M. 1991.
"Three-dimensional free electron laser dispersion relation including betatron oscillations". United States. https://www.osti.gov/servlets/purl/5039720.
```

```
@article{osti_5039720,
```

title = {Three-dimensional free electron laser dispersion relation including betatron oscillations},

author = {Chin, Y H and Kim, K J and Xie, M},

abstractNote = {We have developed a 3-D FEL theory based upon the Maxwell-Vlasov equations including the effects of the energy spread and emittance of the electron beam, and of betatron oscillations. The radiation field is expressed in terms of the Green's function of the inhomogeneous wave equation and the distribution function of the electron beam. The distribution function is expanded in terms of a set of orthogonal functions determined by the unperturbed particle distribution. The coupled Maxwell-Vlasov equations are then reduced to a matrix equation, from which a dispersion relation for the eigenvalues is derived. In the limit of small betatron oscillation frequency, the present dispersion relation reduces to the well-known cubic equation of the one-dimensional theory in the limit of large beam size, and it gives the correct gain in the limit of small beam size. Comparisons of our numerical results with other approaches show good agreement. We present a handy empirical formula for the FEL gain of a 3-D Gaussian beam, as a function of the scaled parameters, that can be used for a quick estimate of the grain. 5 refs., 2 figs.},

doi = {},

url = {https://www.osti.gov/biblio/5039720},
journal = {},

number = ,

volume = ,

place = {United States},

year = {1991},

month = {8}

}