  For fiber-fed spectrographs with a stable external wavelength source,
scrambling properties of optical fibers and, homogeneity and stability of the
instrument illumination are important for the accuracy of radial-velocimetry.
Optical cylindric fibers are known to have good azimuthal scrambling. In
contrast, the radial one is not perfect. In order to improve the scrambling
ability of the fiber and to stabilize the illumination, optical double
scrambler are usually coupled to the fibers. Despite that, our experience on
SOPHIE and HARPS has lead to identified remaining radial-velocity limitations
due to the non-uniform illumination of the spectrograph. We conducted tests on
SOPHIE with telescope vignetting, seeing variation and centering errors on the
fiber entrance. We simulated the light path through the instrument in order to
explain the radial velocity variation obtained with our tests. We then
identified the illumination stability and uniformity has a critical point for
the extremely high-precision radial velocity instruments (ESPRESSO@VLT,
CODEX@E-ELT). Tests on square and octagonal section fibers are now under
development and SOPHIE will be used as a bench test to validate these new feed
optics.
