# Reflection matrix for Gaussian light beams in FEL oscillators

## Abstract

The transformations of Gaussian radiation beams caused by reflection off mirrors is an important issue for FEL's operating as oscillators. The reflected radiation from a single incident Gaussian mode will contain other modes due to the finite 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/sup mn//sub pq/ for an incoming mode ..mu../sub mn/ into the outgoing ..mu../sub pq/ are obtained as functions of the deflection angle phi, the reflected beam waist w/sub o/, 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 mirrormore »

- Authors:

- Publication Date:

- Research Org.:
- Science Applications International Corp., McLean, VA (US); Plasma Theory Branch, Plasma Physic Div., Naval Research Lab., Washington, DC (US)

- OSTI Identifier:
- 6044975

- Resource Type:
- Journal Article

- Journal Name:
- IEEE J. Quant. Electron.; (United States)

- Additional Journal Information:
- Journal Volume: 25:4

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 42 ENGINEERING; LASER MIRRORS; BEAM OPTICS; ANALYTICAL SOLUTION; COMPUTER CALCULATIONS; CORRECTIONS; FREE ELECTRON LASERS; GAUSSIAN PROCESSES; LASER RADIATION; R MATRIX; REFLECTIVITY; ELECTROMAGNETIC RADIATION; LASERS; MATRICES; MIRRORS; OPTICAL PROPERTIES; PHYSICAL PROPERTIES; RADIATIONS; SURFACE PROPERTIES; 420300* - Engineering- Lasers- (-1989)

### Citation Formats

```
Riyopoulos, S, Tang, C M, and Sprangle, P.
```*Reflection matrix for Gaussian light beams in FEL oscillators*. United States: N. p., 1989.
Web. doi:10.1109/3.17345.

```
Riyopoulos, S, Tang, C M, & Sprangle, P.
```*Reflection matrix for Gaussian light beams in FEL oscillators*. United States. doi:10.1109/3.17345.

```
Riyopoulos, S, Tang, C M, and Sprangle, P. Sat .
"Reflection matrix for Gaussian light beams in FEL oscillators". United States. doi:10.1109/3.17345.
```

```
@article{osti_6044975,
```

title = {Reflection matrix for Gaussian light beams in FEL oscillators},

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

abstractNote = {The transformations of Gaussian radiation beams caused by reflection off mirrors is an important issue for FEL's operating as oscillators. The reflected radiation from a single incident Gaussian mode will contain other modes due to the finite 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/sup mn//sub pq/ for an incoming mode ..mu../sub mn/ into the outgoing ..mu../sub pq/ are obtained as functions of the deflection angle phi, the reflected beam waist w/sub o/, 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 intheta/sub d/.},

doi = {10.1109/3.17345},

journal = {IEEE J. Quant. Electron.; (United States)},

number = ,

volume = 25:4,

place = {United States},

year = {1989},

month = {4}

}