  Integrated $\chi^{(2)}$ devices are a widespread tool for the generation and
manipulation of light fields, since they exhibit high efficiency, small
footprint and the ability to interface them with fibre networks. Surprisingly
many things are not fully understood until now, in particular the fabrication
of structures in potassium titanyl phosphate (KTP). A thorough understanding of
the fabrication process and analysis of spatial properties is crucial for the
realization and the engineering of high efficiency devices for quantum
applications. In this paper we present our studies on rubidium-exchanged
waveguides fabricated in KTP. Employing energy dispersive X-ray spectroscopy
(EDX), we analysed a set of waveguides fabricated with different production
parameters in terms of time and temperature. We find that the waveguide depth
is dependent on their widths by reconstructing the waveguide depth profiles.
Narrower waveguides are deeper, contrary to the theoretical model usually
employed. Moreover, we found that the variation of the penetration depth with
the waveguide width is stronger at higher temperatures and times. We attribute
this behaviour to stress-induced variation in the diffusion process.
