Optimization of Q factor in optical nanocavities based on free-standing membranes

We express the quality factor of a mode in terms of the Fourier transforms of its field components, and prove that the reduction in radiation loss can be achieved by suppressing the mode's wave-vector components within the light cone. Although this is intuitively clear, our analytical proof gives us insight into how to achieve the Q factor optimization, without the mode delocalization. We focus on the dipole defect mode in free standing membrane and achieve Q > 10^4, while preserving the mode volume of the order of one half of cubic wavelength in material. The derived expressions and conclusions can be used in optimization of Q factor for any type of defect in planar photonic crystals.