Fast dispersion tailoring of multimode photonic crystal resonators

We introduce a numerical procedure which permits us to drastically accelerate the design of multimode photonic crystal resonators. Specifically, we demonstrate that the optical response of an important class of such nanoscale structures is reproduced accurately by a simple, one-dimensional model within the entire spectral range of interest. This model can describe a variety of tapered photonic crystal structures. Orders of magnitude faster to solve, our approach can be used to optimize certain properties of the nanoscale cavity. Here we consider the case of a nanobeam cavity, for which the confinement results from the modulation of its width. The profile of the width is optimized in order to flatten the resonator dispersion profile (so that all modes are equally spaced in frequency). This result is particularly relevant for miniaturizing parametric generators of nonclassical light, optical nanocombs, and mode-locked laser sources. Our method can be easily extended to complex geometries, described by multiple parameters.

Figure 1

Common design of high-$Q$ resonators based on gentle confinement: (a) tapered distributed feedback grating [18], (b) 1D nanobeam with parabolic width [25], (c) 1D nanobeam bichromatic resonator [26], and (d) two-dimensional bichromatic resonator [11] and the corresponding tapering parameter $\mathrm{\Delta }$.

Figure 2

(a) Dispersion diagram of a periodic structure (see inset) with width $w=450\mathrm{nm}$ and period $a=465\mathrm{nm}$, centered on the $K$ point of the reduced Brillouin zone ($k_0=\pi /a$). The reduced model considers coupled forward and backward waves (dashed lines), generating the valence and conduction bands (solid lines). The solid circles represent the valence band calculated by periodic 3D MEs. (b) The corresponding residuals $\sigma /2\pi$ of the fit.

Figure 3

Calculated structural parameters for the RM vs the control parameter $w$. (a) Averaged residual of the fit and (c)–(f) extracted parameters (black squares), polynomial fit (blue dashed line), and polynomial fit on the updated RM (solid red line). In (e) the group velocity is reported in units of $c_0$, i.e., the speed of light in vacuum. (b) Residual of the fit of ${\omega }_0$ (dashed blue line) and after updating the RM (red solid line).

Figure 4

Tapered nanobeam optical cavity. (a) Simplified layout and (b) spatial dependence of the valence band edge at $k=k_0$ (dashed black line), the normalized distribution of the squared electric field corresponding to the calculated Bloch modes (3D MEs) of the cavity along the axis $y=0$, $z\left(d\right)=0$ (solid gray line), and envelopes calculated from the reduced model of the cavity (blue dashed line) and after the model update (red solid line). The vertical offset corresponds to the frequency at resonance. (c) Integrated dispersion $D_{\mathrm{int}}$ calculated by solving the 3D MEs (squares), with the reduced cavity model (blue dashed line) and after the update (red solid). (d) Spatial dependence of $w$. (e) Frequency deviation for each mode ${\sigma }_{\mathrm{RM}}/2\pi$ of the RM (blue dashed line) and the updated RM (red solid line) relative to the 3D ME calculations.

Figure 5

Design of a nanobeam multimode resonator with flat dispersion. (a) Bloch modes (from the 3D ME) and envelopes from the RM after optimization (as in Fig. 4). (b) Integral dispersion $D_{\mathrm{int}}$ for the resonator with parabolic tapering $w_0\left(x\right)$ (gray) and after the first $w_1\left(x\right)$ (red) and second $w_2\left(x\right)$ (blue) optimizations. Symbols represent the solution of the 3D ME, and lines correspond to the RM solutions. Note the logarithmic vertical scale. (c) Corresponding tapering profiles $w_i\left(x\right)$. (d) Residuals (difference between RM and 3D MEs) computed after updating the RM with $w_0\left(x\right)$ (red dashed line), after the second update and $w_1\left(x\right)$ (red solid line), and with $w_2\left(x\right)$ (blue line).

Figure 6

Calculated $Q$ factors for the reference cavity with parabolic tapering $w_0\left(x\right)$ (orange) and the dispersion-flattened cavity with an optimized profile (blue). Circles correspond to the radiation-limited $Q$; error bars are an estimate of $Q$ accounting for fabrication disorder. On the $x$ axis the frequency of the resonances is reported.

Figure 7

Numerical solution of 3D ME with the nanobeam cavity: convergence of FEM (red) and FDTD (blue) methods against the number of elements and size of the grid, respectively; the error bar stands for the stochastic simulation on an ensemble of 20 realizations. (a) Fundamental mode frequency, (b) the integral dispersion up to the seventh-order mode, and (c) the CPU time in seconds.

Figure 8

Flowchart of the design algorithm: the band diagram calculation of a perfectly periodic structure (I) gives a first estimate of the structure parameters. A reference $w\left(x\right)$ PhC profile is taken into account (II) and numerically solved (III). After the calibration of the RM (IV), we proceed with the actual optimization (V). A loop across steps IV and V might be necessary if the accuracy is not sufficient upon optimization (red arrows).