Dispersion of backward-propagating waves in a surface defect on a three-dimensional photonic band-gap crystal

We experimentally study the dispersion relation of waves in a thin quasi-two-dimensional (2D) defect layer with periodic nanopores that sits on a 3D photonic band-gap crystal made from silicon by CMOS-compatible methods. The nanostructures are probed by momentum-resolved broadband near-infrared imaging of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi>p</a:mi></a:math>-polarized reflected light as a function of off-axis wave vectors. We identify surface defect modes at frequencies inside the 3D photonic band gap with a narrow relative linewidth (<b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:mi mathvariant="normal">Δ</b:mi><b:mi>ω</b:mi><b:mo>/</b:mo><b:mi>ω</b:mi><b:mo>=</b:mo><b:mn>0.028</b:mn></b:mrow></b:math>), which are absent in defect-free 3D photonic band-gap crystals. We calculate the dispersion of the states with relevant mode symmetries using a plane-wave-expansion supercell method, with structural parameters directly extracted from scanning electron microscope images. The calculated bands match very well with the measured data. The slope of the dispersion curves of the surface defect states is negative in one off-axis direction, corresponding to backward-propagating waves in that direction where the phase velocity and the group velocity point in opposite directions, as confirmed by finite-difference time-domain simulations. We also present a didactic and analytic model of a 2D grating sandwiched between vacuum and a negative real effective <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mrow><d:msup><d:mi>ɛ</d:mi><d:mo>′</d:mo></d:msup><d:mo><</d:mo><d:mn>0</d:mn></d:mrow></d:math> that mimics the 3D photonic band gap. The model's dispersion agrees with the experiments and with the full 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 embedded quantum emitters in response to their emission frequency.