Optical fibres have successfully been used to couple high-resolution spectrographs to telescopes for many years. As they allow the instrument to be placed in a stable and isolated location, they decouple the spectrograph from environmental influences. Fibres also provide a substantial increase in stability of the input illumination of the spectrograph, which makes them a key optical element of the two high-resolution spectrographs of CARMENES. The optical properties of appropriate fibres are investigated, especially their scrambling and focal ratio degradation (FRD) behaviour. In the laboratory the output illumination of various fibres is characterized and different methods to increase the scrambling of the fibre link are tested and compared. In particular, a combination of fibres with different core shapes shows a very good scrambling performance. The near-field (NF) shows an extremely low sensitivity to the exact coupling conditions of the fibre. However, small changes in the far-field (FF) can still be seen. Related optical simulations of the stability performance of the two spectrographs are presented. The simulations focus on the influence of the non-perfect illumination stabilization in the far-field of the fibre on the radial velocity stability of the spectrographs. We use ZEMAX models of the spectrographs to simulate how the barycentres of the spots move depending on the FF illumination pattern and therefore how the radial velocity is affected by a variation of the spectrograph illumination. This method allows to establish a quantitative link between the results of the measurements of the optical properties of fibres on the one hand and the radial velocity precision on the other. The results provide a strong indication that 1ms-1 precision can be reached using a circular-octagonal fibre link even without the use of an optical double scrambler, which has successfully been used in other high-resolution spectrographs. Given the typical throughput of an optical double scrambler of about 75% to 85 %, our solution allows for a substantially higher throughput of the system.