# Reflection matrix for optical resonators in FEL (free electron lasers) oscillators. Interim report

## Abstract

The transformations 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 found to be a paraboloid. Integral expressions for the reflection coefficients R/sub pq//sup mn/ for any incoming mode u/sub mn/ into the outgoing u/sub pq/ are obtained as function of the reflection angle phi, the reflected beam spot size W/sub 0/, and the mirror size. The coefficient R/sub 00//sup 00/ for the lowest-to-lowest mode reflection is determined analytically. The spot size W/sub 0/ can then be selected, depending on the mirror size, to maximize R/sub 00//sup 00/. The ratio of the mirror size to the spot size is the dominant factor determining the reflection coefficient. The effects of deflecting the light beam enter as small corrections, of first order in the diffractionmore »

- Authors:

- Publication Date:

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

- OSTI Identifier:
- 6523540

- Report Number(s):
- AD-A-199775/8/XAB; NRL-MR-6285

- Resource Type:
- Technical Report

- Country of Publication:
- United States

- Language:
- English

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

### Citation Formats

```
Riyopoulos, S, Sprangle, P, Tang, C M, and Ting, A.
```*Reflection matrix for optical resonators in FEL (free electron lasers) oscillators. Interim report*. United States: N. p., 1988.
Web.

```
Riyopoulos, S, Sprangle, P, Tang, C M, & Ting, A.
```*Reflection matrix for optical resonators in FEL (free electron lasers) oscillators. Interim report*. United States.

```
Riyopoulos, S, Sprangle, P, Tang, C M, and Ting, A. Thu .
"Reflection matrix for optical resonators in FEL (free electron lasers) oscillators. Interim report". United States.
```

```
@article{osti_6523540,
```

title = {Reflection matrix for optical resonators in FEL (free electron lasers) oscillators. Interim report},

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

abstractNote = {The transformations 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 found to be a paraboloid. Integral expressions for the reflection coefficients R/sub pq//sup mn/ for any incoming mode u/sub mn/ into the outgoing u/sub pq/ are obtained as function of the reflection angle phi, the reflected beam spot size W/sub 0/, and the mirror size. The coefficient R/sub 00//sup 00/ for the lowest-to-lowest mode reflection is determined analytically. The spot size W/sub 0/ can then be selected, depending on the mirror size, to maximize R/sub 00//sup 00/. The ratio of the mirror size to the spot size is the dominant factor determining the reflection coefficient. The effects of deflecting the light beam enter as small corrections, of first order in the diffraction angle theta/sub d/ < <1.},

doi = {},

journal = {},

number = ,

volume = ,

place = {United States},

year = {1988},

month = {9}

}