Designing low-loss cavities across the band-gap of photonic crystal slabs

Abstract

Photonic crystal cavities (PCCs) are defects in host photonic crystals (PCs) which create bound states in the PC band gap. These bound states are resonant states of the electromagnetic field with a complex resonance frequency and can have very small mode volumes. PCCs are attractive for a variety of applications, from cavity quantum electrodynamics to biosensing. A PC slab geometry is advantageous given its superior manufacturability compared to three-dimensional crystals, and the accessibility of the surface allows sensing and coupling. However, the emission into the half spaces above and below the slab limits the bound state lifetime. Controlling this emission is thus crucial for applications, most of which benefiting from a long lifetime. A range of methods to find defect geometries suppressing the emission to increase the lifetime have been demonstrated in the past. However, they do not cater for a designed resonant frequency covering a wide addressable range, as needed for multiplexed devices. Here, we demonstrate a design method controlling both resonance frequency and emission, by minimising a cost function including both losses and target frequency. We show applications on PCCs in GaAs PC slabs immersed in water, relevant for biosensing. The reduced refractive index contrast in these structures compared to previously studied PCCs embedded in vacuum renders the emission suppression more challenging. We optimize the quality factor of a standard L3 cavity from 1000 to 10^4-10^5, with an addressable resonance frequency range covering 12% relative bandwidth, spanning more than half of the band gap. We furthermore report optimised structures of H1 cavities, and provide the optimisation code for widespread use.

Figures (9)

• Figure 1: Flowchart of the iterative optimisation algorithm.
• Figure 2: First minimisation iteration from a nominal L3 cavity, showing $F$ as function of step number $m$. The holes modified during the optimisation are sketched. For $m>0$ the holes defined by the two mirror symmetries of the structure indicated by dashed lines are omitted for clarity. The shifts of the holes have been tripled for clarity. The red circle indicates the position of the minimum cost function determined by a quadratic fit (dashed line) to the lowest three values of the cost function obtained during the minimisation.
• Figure 3: Optimisation trajectory for an L3 cavity starting at $\nu_\mathrm{c}=0.26212$ with unchanged remaining holes of the PC, to a target frequency of $\nu_\mathrm{t}=0.27102$. The reduced linewidth $\Gamma_\mathrm{c}$ is shown versus the reduced frequency $\nu_\mathrm{c}$ for each simulation made. The clustered points represent the gradient calculations, which are numbered by the gradient step $n$ up to $n=5$ for clarity. The weighting $\alpha$ changes from 10 (squares), to 20 (circles), 40 (stars), 80 (triangles) and 160 (diamonds) for the final stage. The symbol colour hue changes for increasing simulation from red to green to blue to purple for clarity. The yellow larger symbols identify the end of the optimisation for the corresponding value of $\alpha$.
• Figure 4: As Fig. \ref{['fig:L3Opt_250THz']}, but for a target frequency $\nu_\mathrm{t}=0.26018$. The weighting $\alpha$ changes from 100 (squares) for $n=0..26$ to 200 (circles) for $n=27..30$, to 400 (stars) for $n=31..33$.
• Figure 5: Comparison of the nominal L3 cavity (left) and the optimised cavity with $\nu_\mathrm{t}=0.26018$ (right). The geometries of the parametrised holes surrounding the cavity are shown in a) and b) with grey circles indicating the nominal geometry. The magnitude of the electric field of the RS at the central plane of the slab is shown in c) and d) in real space. The corresponding $x$ and $y$ polarised field components in $\mathbf{k}_{||}$ space are shown in e) and f), with the edge of the radiative cone $|\mathbf{k}_{||}|=n_\mathrm{m}\omega_\mathrm{c}/c$ shown as red circles. The colour scale is given, and data are shown normalised to their maximum. Relative values of maxima are given.