The undulator U50 [1], which is a long-term loan from the ENEA Institute in Frascati, should at ELBE provide far infrared radiation up to roughly 150 mm. Already at 30 mm the transverse extension of the Gaussian mode in an open resonator is larger than allowed by the vacuum chamber located inside the undulator, leading to significant losses of optical power. To minimize these losses we propose a partially waveguided resonator 10 mm high that is guiding the wave along the undulator (240 cm) in the vertical dimension while the chamber is wide enough (40 mm) to allow the propagation of a Gaussian beam in horizontal direction. The currents induced by such a hybrid mode in the waveguide walls are small and the resulting ohmic losses are negligible [2]. Outside the undulator the vacuum chamber is large enough to ensure free propagation in either direction.
Since the horizontal propagation of the optical beam is not affected by the waveguide, the horizontal radius of curvature R_h of the resonator mirrors is determined by the the gaussian optics. We choose a radius corresponding to a Rayleigh range z_R=1 m with a waist located at the center of the undulator.
The radius R_v in vertical direction determines the coupling of the waveguide mode inside the undulator with a free space mode outside of it. A detailed numerical analysis of mode coupling losses has been carried out using the code GLAD [3] and applying the diffraction theory based on the Huygens-Fresnel principle [4].

Fig. 1 Optical beam in a resonator FEL with a waveguide mode inside the undulator and a Gaussian mode outside of it. The resonator mirrors have different radii of curvature R_h and R_v in the horizontal and vertical plane, respectively.

The results of computation (Fig. 2) show that at wavelengths larger than 30 mm the losses can be minimized by a mirror with a fixed radius of curvature R_v which corresponds to the distance D between mirror and waveguide exit. Below 30 mm the Fresnel number becomes so large that the appropriate radius of curvature depends strongly on the wavelength.
The simulated resonator characteristics will be investigated experimentally on a scaled down test resonator structure, excited by the radiation of a CO₂ laser at 10.6 mm. The Q-Factor of the cavity and the associated resonator losses will be then defined by means of a cavity ring down experiment.

Fig. 2 Losses of a single mode conversion in dependence on the wavelength l of the radiation for two different radii of curvature R_v of the mirror. The full line shows the minimum losses which can be obtained with any mirror of constant curvature.

¹ Tel Aviv University, Physical Electronics

References

[1] F. Ciocci et al., Nucl. Instr. Meth. A 250 (1986) 134-137
[2] L. Elias and J. Gallardo, Appl. Phys. B 31 (1983)
[3] G. Lawrence, GLAD, Applied Optics Research, Austin TX, USA
[4] A. E. Siegman, Lasers, University Science Books, Sausalito CA, 1986

 IKH 05/22/01 © R. Wünsch