  Optical whispering gallery mode (WGM) resonators are a powerful and versatile
tool used in many branches of science. Fine tuning of the central frequency and
line width of individual resonances is however desirable in a number of
applications including frequency conversion, optical communications and
efficient light-matter coupling. To this end we present a detailed theoretical
analysis of dielectric tuning of WGMs supported in axisymmetric resonators.
Using the Bethe-Schwinger equation and adopting an angular spectrum field
representation we study the resonance shift and mode broadening of high $Q$
WGMs when a planar dielectric substrate is brought close to the resonator.
Particular focus is given to use of a uniaxial substrate with an arbitrarily
aligned optic axis. Competing red and blue resonance shifts ($\sim 30$ MHz),
deriving from generation of a near field material polarisation and back action
from the radiation continuum respectively, are found. Anomalous resonance
shifts can hence be observed depending on the substrate material, whereas mode
broadening on the order of $\sim 50$ MHz can also be simply realised.
Furthermore, polarisation selective coupling with extinction ratios of $> 10^4$
can be achieved when the resonator and substrate are of the same composition
and their optic axes are chosen correctly. Double refraction and properties of
out-coupled beams are also discussed.
