The optical resonators of the ELBE FELs
The optical resonator consists of two focusing mirrors located on the undulator axis facing each other. The distance L_{R} = 11.53 m between the two mirrors (resonator length) is determined by the repetition rate (13 MHz) of the electron pulses. This length ensures that the reflected optical pulse, which was produced by the preceding electron pulse, just hits the following electron pulse. A small deviation (resonator detuning) in the order of the radiation wavelength λ may increase the final power of the laser pulses.
At lower wavelengths the cross section of an open resonator mode is small enough to ensure both high laser gain and small diffraction losses. So we use a symmetric, nearly concentric, open resonator with two spherical mirrors for the U27 undulator. Its Rayleigh range z_{R}=1 m, which corresponds to a radius of curvature R=5.94 m, is a tradeoff between minimum mode area and resonator stability.
Outside the undulator the vacuum chamber is large enough to let pass the optical beam without loss. Inside the undulator the chamber is smaller (10 mm high and 35 mm wide) and may affect the optical beam. Moreover the outcoupling hole in the upstream mirror affects the beam and increases the diffraction loss. The larger the hole is the stronger it influences the beam propagation. We have estimated the losses by means of the code GLAD. The result is shown in fig.1.
The optical losses in the resonator have experimentally been determined by measuring its ringdown time. They fit the theoretical prediction quite well (Fig.1a)
In the U100 undulator working at wavelengths up to 280 μm the cross section of an open resonator mode is much larger and the laser gain is considerably smaller than for a narrower optical mode at shorter wavelength. To reduce the vertical beam size we shall install a partial waveguide inside and downstream the undulator (see fig.2). It is 10 mm high and wide enough to allow a free propagation. It compresses the optical field vertically, increases the laser gain and allows a small undulator gap.
The resonator is symmetric and the undulator is located in its center. The wave is guided only by the horizontal walls while it freely propagates in horizontal direction (parallelplate waveguide). Such a waveguide favors hybrid modes characterized by a Gaussian dependence in the horizontal dimension and a sinuidal dependence in the vertical dimension. If the wave is properly polarized (electric field horizonatlly; E^{π}mode) such a hybrid mode does practically not induce currents in the waveguide walls and the ohmic losses are negligible over a propagation hundreds of meters (see ref.[1]).
Fig.2: Scheme of a partial waveguide resonator with a bifocal mirror (M1) and a cylindrical mirror (M2). On the left side the beam propagates freely while it is vertically confined in the waveguide on the right side of the resonator.
In horizontal direction the waveguide walls do not influence the beam and its profile is a Gaussian one. Waist and Raygleigh range are determined by the horizontal radii of curvature R_{h} of the resonator mirrors. We choose radii coresponding to a waist in the undulator center with a Rayleigh range somewhat smaller than half the undulator length. Fig.3 shows the calculated beam diameter (4*w) in the resonator in horizontal and vertical direction.
Fig.3: Horizontal (upper side) and vertical (lower side) of the beam diameter as a function of the position in the U100 resonator calculated for the indicated wavelengths.
The vertical radius of curvature R_{v} of the left mirror has to be chosen such that the coupling losses beetween the waveguide mode inside the undulator and the Gaussian modes outside of it are minimal.
Using the code GLAD we have studied the propagation from the waveguide to the upstream mirror with an outcoupling hole with diameter D_{h} and back to the waveguide. Fig. 4 shows the losses of intensity on this way, where we have added 1.2% absorption on each resonator mirror. It has turned out that the hole does not only couple out a fraction of the beam intensity but also distorts the reflected beam leading to additional beam losses approximately in the same order of magnitude.
Fig.4: Calculated roundtrip losses for the indicated hole diameters D_{h} as a function of the wavelength λ. The black line shows the losses without any outcoupling. The losses including outcoupling are well approximated by the sum of losses without outcoupling and twice the outcoupled fraction (broken lines).
We found that the losses are minimal when the radius R_{v} equals to the distance between waveguide exit and mirror (farfield approximation). In this case (R_{v}=361 cm) the roundtrip losses are around 5%. At short wavelength below 30 mm the losses increase noticeably.
U27  U100  

Resonator length  L_{R}  1153 cm  
Rayleigh range  z_{R}  1 m  1.8 m (hor.)  
Mirror radius of curvature  M1  R_{h} R_{v} 
594 cm  633 cm 361 cm 
M2  R_{h} R_{v} 
594 cm  633 cm ∞ 

Mirror diameter  M1  horizontal vertical 
7 cm  13 cm 17 cm 
M2  horizontal vertical 
7 cm  13 cm 1 cm 

Outcoupling hole diameter  1.5 mm 2.0 mm 3.0 mm 4.0 mm 
2 mm 4.5 mm 7 mm 
[1] L.R. Elias and J.C. Gallardo, Appl.Phys. B31 (1983) 229233