# Transformations of Gaussian light beams caused by reflection in FEL (free electron lasers) resonators. Interim report

## Abstract

The transformation of Gaussian radiation beams caused by reflection off mirrors is an important issue for FELs operating as oscillators. The reflected radiation from a single incident Gaussian mode will contain other modes due to the finite mirror size, the deflection of the beam and mismatches in the curvature. A method for analytic computation of the reflection matrix is developed by taking the convolution of the source function at the surface of the mirror with the paraxial propagator. The mirror surface that reflects spherical incoming wavefronts into spherical outgoing is determined analytically. Integral expressions for the reflection coefficients R(mn) sub pq for any incoming mode u sub mn into the outgoing u sub pq are obtained as functions of the deflection angle phi, the reflected beam waist sub 0 and the mirror size. Analytic expansions of the reflection matrix in powers of the diffraction angle theta sub d <<1 are then obtained in both Hermite and Laguerre representations. The cross coupling among vector components of the radiation field, caused by the curvature of the mirror surface, is included. It is shown that the lowest-order contribution to the off-diagonal matrix elements is caused by the finite mirror size. The effects ofmore »

- Authors:

- Publication Date:

- Research Org.:
- Naval Research Lab., Washington, DC (USA)

- OSTI Identifier:
- 6332741

- Report Number(s):
- AD-A-202472/7/XAB; NRL-MR-6347

- Resource Type:
- Technical Report

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 42 ENGINEERING; FREE ELECTRON LASERS; LASER MIRRORS; CORRECTIONS; OSCILLATORS; DIFFRACTION; OPTICAL PROPERTIES; PROGRESS REPORT; REFLECTION; COHERENT SCATTERING; DOCUMENT TYPES; ELECTRONIC EQUIPMENT; EQUIPMENT; LASERS; MIRRORS; PHYSICAL PROPERTIES; SCATTERING; 420300* - Engineering- Lasers- (-1989); 420800 - Engineering- Electronic Circuits & Devices- (-1989)

### Citation Formats

```
Riyopoulos, S, Tang, C M, and Sprangle, P.
```*Transformations of Gaussian light beams caused by reflection in FEL (free electron lasers) resonators. Interim report*. United States: N. p., 1988.
Web.

```
Riyopoulos, S, Tang, C M, & Sprangle, P.
```*Transformations of Gaussian light beams caused by reflection in FEL (free electron lasers) resonators. Interim report*. United States.

```
Riyopoulos, S, Tang, C M, and Sprangle, P. Thu .
"Transformations of Gaussian light beams caused by reflection in FEL (free electron lasers) resonators. Interim report". United States.
```

```
@article{osti_6332741,
```

title = {Transformations of Gaussian light beams caused by reflection in FEL (free electron lasers) resonators. Interim report},

author = {Riyopoulos, S and Tang, C M and Sprangle, P},

abstractNote = {The transformation of Gaussian radiation beams caused by reflection off mirrors is an important issue for FELs operating as oscillators. The reflected radiation from a single incident Gaussian mode will contain other modes due to the finite mirror size, the deflection of the beam and mismatches in the curvature. A method for analytic computation of the reflection matrix is developed by taking the convolution of the source function at the surface of the mirror with the paraxial propagator. The mirror surface that reflects spherical incoming wavefronts into spherical outgoing is determined analytically. Integral expressions for the reflection coefficients R(mn) sub pq for any incoming mode u sub mn into the outgoing u sub pq are obtained as functions of the deflection angle phi, the reflected beam waist sub 0 and the mirror size. Analytic expansions of the reflection matrix in powers of the diffraction angle theta sub d <<1 are then obtained in both Hermite and Laguerre representations. The cross coupling among vector components of the radiation field, caused by the curvature of the mirror surface, is included. It is shown that the lowest-order contribution to the off-diagonal matrix elements is caused by the finite mirror size. The effects of the mirror curvature and the deflection of the light beam enter the reflection matrix as first-order corrections in theta sub d.},

doi = {},

journal = {},

number = ,

volume = ,

place = {United States},

year = {1988},

month = {10}

}