Optical Bound States in the Continuum with Nanowire Geometric Superlattices

Perfect trapping of light in a subwavelength cavity is a key goal in nanophotonics. Perfect trapping has been realized with optical bound states in the continuum (BIC) in waveguide arrays and photonic crystals; yet the formal requirement of infinite periodicity has limited the experimental realization to structures with macroscopic planar dimensions. We characterize BICs in a silicon nanowire (NW) geometric superlattice (GSL) that exhibits one-dimensional periodicity in a compact cylindrical geometry with a subwavelength diameter. We analyze the scattering behavior of NW GSLs by formulating temporal coupled mode theory to include Lorenz-Mie scattering, and we show that GSL-based BICs can trap electromagnetic energy for an infinite lifetime and exist over a broad range of geometric parameters. Using synthesized NW GSLs tens of microns in length and with variable pitch, we demonstrate the progressive spectral shift and disappearance of Fano resonances in experimental single-NW extinction spectra as a manifestation of BIC GSL modes.

Figure 1

Optical BICs in a NW GSL. (a) Geometry of a NW GSL under TE-polarized plane wave illumination, where the length of each segment is $p/2$. (b) $Q$ factors of two GSL eigenmodes with varying $p$ in a NW GSL with $d=200\mathrm{nm}$, $e=170\mathrm{nm}$. Modes are labeled with angular numbers $m=0$ or $m=1$. (c) $H_y$ pattern of $m=0$ GSL eigenmode. (d) $H_y$ (upper) and $E_y$ (lower) patterns of $m=1$ GSL eigenmode. (e) $Q_{\text{sca}}$ spectrum of a NW GSL with $d=200\mathrm{nm}$, $e=170\mathrm{nm}$, and $p=400\mathrm{nm}$ (solid black curve), and of a NW with $d=185\mathrm{nm}$ (gray dashed curve). (f),(g) Heat maps of (f) $Q_{\text{sca}}$ and (g) log($U/U_0$) for a NW GSL with varying $p$ for fixed $d=200\mathrm{nm}$ and $e=170\mathrm{nm}$. Single spectra for a uniform NW with $d=185\mathrm{nm}$ are presented on top of each heat map.

Figure 2

$Q_{\text{sca}}$ from TCMT and from full wave simulations. (a)–(c) Total $Q_{\text{sca}}$ calculated by modified TCMT (blue circles) overlaid with numerical calculations (red curve) for a NW GSL with (a) $d=200\mathrm{nm}$, $e=170\mathrm{nm}$, and $p=220\mathrm{nm}$; (b) $p=260\mathrm{nm}$; or (c) $p=320\mathrm{nm}$. Insets show magnified views near Fano resonances. Peaks marked with asterisks result from higher-order GSL modes not included in the TCMT. (d) Total $Q_{\text{sca}}$ (dashed curve) from (c) decomposed into each angular channel (circles).

Figure 3

Geometric dependence of $Q$ factor. (a) Plot of $Q$ factor as a function of $kp/2\pi$. (Inset) Band structure of the $m=1$ mode. (b) $Q$ factors of an $m=1$ mode as a function of $p$ with $d=200\mathrm{nm}$ and various $e$.

Figure 4

Experimental extinction measurement of NW GSLs. (a) SEM image (upper panel) of a NW containing five GSL sections; scale bar, $10\mu \mathrm{m}$. Magnified views (lower panels) of each GSL segment corresponding to the boxed regions in the upper panel; scale bars, 200 nm. Geometric parameters are $d=189\pm 2\mathrm{nm}$, $e=141\pm 2\mathrm{nm}$, $p=201\pm 4\mathrm{nm}$ (GSL 1); $d=185\pm 1\mathrm{nm}$, $e=135\pm 2\mathrm{nm}$, $p=250\pm 4\mathrm{nm}$ (GSL 2); $d=186\pm 1\mathrm{nm}$, $e=147\pm 2\mathrm{nm}$, $p=300\pm 3\mathrm{nm}$ (GSL 3); $d=185\pm 1\mathrm{nm}$, $e=153\pm 2\mathrm{nm}$, $p=348\pm 4\mathrm{nm}$ (GSL 4); and $d=183\pm 1\mathrm{nm}$, $e=148\pm 2\mathrm{nm}$, $p=400\pm 6\mathrm{nm}$ (GSL 5). (b) Simulated $Q_{\text{sca}}$ (spectra offset by 1) and (c) measured extinction (spectra offset by 5%) of GSLs. Red traces in both graphs represent spectra of a GSL at the BIC condition. Arrows and asterisks indicate the $m=0$ and 1 Fano resonances, respectively.