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

TL;DR

This work investigates backward-propagating surface-defect waves at the surface of a 3D photonic band gap crystal by combining momentum-resolved reflectivity, plane-wave expansion supercell simulations, and a Fresnel-like analytic model. The defect layer, comprised of a 2D periodic pore pattern on an inverse woodpile silicon crystal, supports a narrow mode inside the band gap with a relative linewidth of $Δω/ω = 0.028$ and exhibits negative dispersion along the $k_z$ direction, i.e., backward propagation. Numerical simulations (FDTD and MPB supercells) reproduce the observed dispersion and confirm the backward-propagating nature, while a simple three-medium Fresnel model with a grating explains the mechanism via coupling to grating orders in a negative-$ε'$ medium. The findings demonstrate a route to tunable, directionally dependent photonic emission and provide a framework applicable to other 2D/3D photonic crystals, with potential applications in sensing and quantum-emitter devices. The study highlights momentum-resolved imaging as a powerful tool for rapidly mapping defect-induced dispersion in complex photonic structures.

Abstract

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 ($Δω/ω$ = 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 $ε'$ < 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.

Figures (15)

• Figure 1: Schematic cross-section showing the excitation of waves in a planar surface defect (thickness $d_{\rm{def}}$, pitch $a$, pore diameter $d$) on the surface of a 3D photonic band gap crystal (orange). The surface defect wave (SDW) has a group velocity $\vb{v}_{\rm{SDW}}$ pointing in the opposite direction as the momentum of the incident wave parallel to the surface along z: $k_{\rm{in,z}}$. Part of the incident light is reflected to ${\vb k}{}_{\rm{refl}} \equiv {\vb k}{}$ due to Bragg scattering by reciprocal lattice vector $\vb{G}_{\rm{PBG}}$ where ${\vb k}{} = {\vb k}{}_{\rm{in}}$ + $\vb{G}_{\rm{PBG}}$.
• Figure 2: 3D photonic band gap crystals (A,C) without and (B,D) with surface defect. (A) Model of a periodic crystal which is periodic up to the crystal-to-air interface. (B) Model of the same structure, missing half-pores at the red arrows, creating a surface defect. (C) SEM image of the real photonic crystal after the surface was sliced off, similar to (A). (D) SEM image of the mask of the same structure as (C) before the surface was sliced off, therefore having the surface defect, similar to (B). Scale bars shown are 2 $µm$ wide.
• Figure 3: Optical setup to measure spectrally-resolved reflectivity both in real space and in wave vector space at the detector plane (vertical dashed line, top right). The incident beam has a tunable wavelength between 1000 and 1700 nm. The blue beam path pertains to real-space imaging of the sample. The red beam path pertains to imaging in wave vector space (k-space). Components shown: a flip lens ($\rm{L}_{\it{k}}$), a beam splitter (BS), an objective (OBJ), a sample (S), a mirror (M), and a tube lens ($\rm{L}_{\it{t}}$).
• Figure 4: (Green, dotted) Irreducible Brillouin zone for a 3D photonic band gap crystal with the inverse-woodpile structure. (Gray) First Brillouin zone. (Blue/red) Trajectories in k-space along which we computed bands to compare to the experiment. The trajectories extend until $k_y/|{\vb k}{}|$ = $k_z/|{\vb k}{}|$ = NA = 0.85 for $\tilde{\nu} = 10^4$$cm^{-1}$.
• Figure 5: Momentum-resolved image of the reflectivity of the photonic crystal (left) without and (right) with defect at $\tilde{\nu} = 6850$$\per cm$ ($\lambda = 1460$ nm; $a/\lambda = 0.4657$) for p-polarized light. We observe high reflectivity from the band gap with strong line-like troughs from the defect mode. The NA of the objective determines the maximum relative off-momentum, 0.85.