  Recently, gradient index cavities, or so-called transformation cavities,
designed by conformal transformation optics have been studied to support
resonant modes with both high Q-factors and emission directionality. We propose
a new design method for transformation cavities to realize a narrower width of
the refractive index profile, a great advantage in experimental
implementations, without losing the benefits of conformal mapping. We study the
resonant modes with both high Q-factor and directional emission in the newly
designed transformation cavities, where the refractive index profile is 50
percent narrower than in previously proposed transformation cavities. By
varying a system parameter with a fixed maximal value of the refractive index
profile inside the cavity, the width of the refractive index profile narrows,
the Q-factors become higher, and the near and far field patterns maintain their
properties, namely conformal whispering gallery modes and bidirectional
emission, respectively.
