  We describe instrumentation designed to perform millimeter-wave conductivity
measurements in pulsed high magnetic fields at low temperatures. The main
component of this system is an entirely non-metallic microwave resonator. The
resonator utilizes periodic dielectric arrays (photonic bandgap structures) to
confine the radiation, such that the resonant modes have a high Q-factor, and
the system possesses sufficient sensitivity to measure small samples within the
duration of a magnet pulse. As well as measuring the sample conductivity to
probe orbital physics in metallic systems, this technique can detect the sample
permittivity and permeability allowing measurement of spin physics in
insulating systems. We demonstrate the system performance in pulsed magnetic
fields with both electron paramagnetic resonance experiments and conductivity
measurements of correlated electron systems.
