Dispersion of backward-propagating waves in a surface defect on a 3D photonic band gap crystal

We experimentally study the dispersion relation of waves in a two-dimensional (2D) defect layer with periodic nanopores that sits on a three-dimensional (3D) photonic band gap crystal made from silicon by CMOS-compatible methods. The nanostructures are probed by momentum-resolved broadband near-infrared imaging of p-polarized reflected light that is collected inside the light cone as a function of off-axis wave vectors. We identify surface defect modes at frequencies inside the band gap with a narrow relative linewidth ($\Delta\omega/\omega$ = 0.028), which are absent in defect-free 3D crystals. We calculate the dispersion of modes with relevant mode symmetries using a plane-wave-expansion supercell method with no free parameters. The calculated dispersion matches very well with the measured data. The dispersion is negative in one of the off-axis directions, corresponding to backward-propagating waves where the phase velocity and the group velocity point in opposite directions, as confirmed by finite-difference time-domain simulations. We also present an analytic model of a 2D grating sandwiched between vacuum and a negative real $\epsilon'$ < 0 that mimics the 3D photonic band gap. The model's dispersion agrees with the experiments and with the fuller theory and shows that the backward propagation is caused by the surface grating. We discuss possible applications, including a device that senses the output direction of photons emitted by quantum emitters in response to their frequency.