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

## 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 superscript (mn) subscript (pq) for any incoming mode u(mn) into the outgoing u(pq) are obtained as functions of the deflection angle phi, the reflected beam spot size W(o), and the mirror size. The coefficient R superscript (00) superscript (00) for the lowest-to-lowest mode reflection is determined analytically. The spot size W(0) can then be selected, depending on the mirror size, to maximize R superscript (00) subscript (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 firstmore »

- Authors:

- Publication Date:

- Research Org.:
- National Bureau of Standards, Washington, DC (USA)

- OSTI Identifier:
- 6158919

- Report Number(s):
- AD-A-201778/8/XAB

- Resource Type:
- Technical Report

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 42 ENGINEERING; FREE ELECTRON LASERS; REFLECTION; OSCILLATORS; CORRECTIONS; FUNCTIONS; MATRICES; MIRRORS; OPTICAL PROPERTIES; RESONATORS; SURFACES; ELECTRONIC EQUIPMENT; EQUIPMENT; LASERS; PHYSICAL PROPERTIES; 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*. United States: N. p., 1987.
Web.

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

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

```
@article{osti_6158919,
```

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

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 superscript (mn) subscript (pq) for any incoming mode u(mn) into the outgoing u(pq) are obtained as functions of the deflection angle phi, the reflected beam spot size W(o), and the mirror size. The coefficient R superscript (00) superscript (00) for the lowest-to-lowest mode reflection is determined analytically. The spot size W(0) can then be selected, depending on the mirror size, to maximize R superscript (00) subscript (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 diffraction angle theta(d) 1.},

doi = {},

journal = {},

number = ,

volume = ,

place = {United States},

year = {1987},

month = {9}

}