A 2018 demonstration by Asano and Noda showed that a convolutional neural network trained on 1000 FDTD-simulated two-dimensional photonic crystal (2D-PC) nanocavities can supply the gradient of the quality factor Q with respect to air-hole displacements so cheaply that gradient ascent over ~10^6 steps in a 50-dimensional design space yields a cavity with theoretical Q = 1.58 x 10^9. The demonstration, however, was reliable only within and near the training distribution: the network underestimates Q in the rare high-Q tail (only 40 of 1000 training samples had Q > 1e8), only hole-position degrees of freedom and the single objective Q were optimized, and all results were theoretical. This problem asks for the completion of that methodology: surrogate models (inexpensive trained stand-ins for the costly electromagnetic simulation) whose high-Q predictions and extrapolation behavior are reliable and controlled during optimization, extension to broader structural degrees of freedom and multi-objective targets, and experimental realization of the theoretically predicted ultra-high Q. Later work has partially resolved the first aspects (iterative dataset regeneration, larger datasets, multi-mode and fabrication-robust designs), but the theory-experiment gap remains open: no fabricated 2D-PC cavity, deep-learning-designed or otherwise, has reached experimental Q near 1e8, roughly two orders of magnitude below the theoretical designs.

Background

A two-dimensional photonic crystal (2D-PC) nanocavity is a wavelength-scale optical resonator formed by a line or point defect in a periodic array of air holes etched into a silicon slab; the periodic lattice forbids in-plane propagation of light in certain frequency bands, so a defect localizes a resonant mode with a very long photon lifetime. That lifetime is quantified by the quality factor Q, the number of optical cycles before the stored energy decays by 1/e; small mode volume V (the effective spatial extent of the field) together with high Q is the figure of merit for applications such as low-threshold Raman lasers, cavity quantum electrodynamics with single emitters, and on-chip sensing. Because out-of-plane radiation leakage from the slab ultimately limits Q, record values have historically been obtained by deliberately shaping the air-hole pattern so that the resonant mode's in-plane wavevector components fall outside the 'leaky' light cone, the region of wavevector space from which light can couple to vertically propagating radiation. Before 2018, manual tuning of a few holes (guided by visualizing the leaky components) had produced theoretical Q up to ~7 x 10^8 and experimental Q up to ~1.1 x 10^7 (Asano et al., Opt. Express 25, 1769 (2017)); the experimental value is limited by nanometer-scale fabrication imperfections in hole radius, shape, and sidewall angle. The stumbling block for going further is that the design space is huge (every hole position, and in principle every radius and shape, is a free parameter) and the gradient of Q with respect to all of them is prohibitively expensive to compute by direct simulation. Asano and Noda (Opt. Express 26, 32704 (2018)) removed this bottleneck with deep learning: they generated 1000 random displacement patterns of the holes of a heterostructure base cavity (a design in which the lattice constant is stepped to confine light by a 'mode gap'), computed each Q by the finite-difference time-domain (FDTD) method, the standard first-principles Maxwell-equation solver, and trained a four-layer network containing a convolutional layer (a neural network whose first layer applies learned spatial filters to the displacement pattern, exploiting locality the way image-processing networks do) to regress log10 Q. Such a trained network is a surrogate model: an inexpensive statistical stand-in for the expensive FDTD computation, from which quantities such as gradients can be extracted essentially for free. Since back-propagation then yields the gradient of the predicted Q with respect to all displacement parameters at almost no cost, gradient ascent over ~10^6 steps became feasible and delivered a design with theoretical Q = 1.58 x 10^9. The same paper, and the adjacent literature it cites (e.g. Minkov, Savona, and Gerace, Appl. Phys. Lett. 111, 131104 (2017), which simultaneously optimized Q/V and vertical radiation coupling, i.e. deliberate out-of-plane emission of the cavity field), make clear what the demonstration did not achieve, and these limitations define the problem stated below.

Problem Statement

How can the deep-learning-based optimization of two-dimensional photonic crystal nanocavities be made reliable and experimentally validated beyond the regime demonstrated by Asano and Noda in 2018? Concretely, the question asks for: (1) a surrogate model and optimization protocol for the cavity quality factor Q whose predictions remain accurate in the ultra-high-Q tail (Q of order 1e9, where training examples are rare because each ground-truth FDTD computation is expensive) and whose reliability outside the training distribution is controlled or quantified, so that gradient-based optimization can safely make large structural modifications rather than only local ones; (2) extension of the optimization from air-hole displacement parameters alone to the other structural degrees of freedom (hole radii, hole shapes, slab thickness) and to multi-objective targets, such as simultaneous optimization of Q and vertical radiation coupling (deliberate out-of-plane emission of the cavity field, needed when the cavity must deliver light to surface-normal emitters or detectors) or of several cavity modes, as required by applications; and (3) fabrication and experimental verification of the predicted ultra-high Q, closing the gap between theoretical designs (Q ~ 1e9-1e11) and measured devices (experimental Q so far at most ~1.1e7 for 2D-PC slab nanocavities), including designs that are robust to realistic fabrication tolerances. These three components serve one objective and are jointly required for its resolution: converting the demonstrated single-scale, single-objective, simulation-only method into a trustworthy, general, experimentally validated design methodology for ultra-high-Q photonic crystal nanocavities; progress on only one or two of the three components does not by itself resolve the task.

A resolution therefore requires each component at the following scope: (1) a surrogate whose accuracy in the ultra-high-Q tail and reliability outside the training distribution are stated quantitatively on held-out ground-truth samples spanning that tail, not only in-distribution; (2) extension of the optimization to all of the named structural degrees of freedom (hole radii, hole shapes, and slab thickness), with the multi-objective targets taken as illustrative examples of design beyond the single objective Q; and (3) a fabricated device whose measured experimental Q reaches the ultra-high-Q regime of order 1e8, beyond the present ~1.1e7 record, without requiring full closure to the theoretical Q ~ 1e9-1e11 designs.

References

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