  We present an optical study of various device designs for electrically
contactable circular Bragg grating cavities in labyrinth geometries. To create
an electrical connection between the central disk and the surrounding membrane,
which are separated through air gaps, we introduce connections between the
adjacent rings. We propose to rotate these connections, creating a labyrinth
like structure, to disable waveguiding and keep the mode confinement. To
investigate how different arrangement and size of the connections affect the
optical properties and to find the optimal design, six different layouts with
either 3-fold or 4-fold symmetry and one with 2-fold symmetry are investigated
experimentally and by numerical simulations. Reflectivity measurements and
simulations show that rotating the connections improves the mode confinement,
far-field pattern and Purcell factor compared to layouts with connections
arranged in straight lines. We compare results between different layouts for
different connection widths and perform polarization resolved measurements to
investigate whether the connections create asymmetries in the photonic
confinement that would impede the performance of the device.
