Nanometer-scale photon connement inside dielectrics


 Optical nanocavities confine and store light, which is essential to increase the interaction between photons and electrons in semiconductor devices, enabling, e.g., lasers and emerging quantum technologies. While temporal confinement has improved by orders of magnitude over the past decades, spatial confinement inside dielectrics was until recently believed to be bounded at the diffraction limit. The conception of dielectric bowtie cavities (DBCs) shows a path to photon confinement inside semiconductors with mode volumes bound only by the constraints of materials and nanofabrication, but theory was so far misguided by inconsistent definitions of the mode volume and experimental progress has been impeded by steep nanofabrication requirements. Here we demonstrate nanometer-scale photon confinement inside 8 nm silicon DBCs with an aspect ratio of 30, inversely designed by fabrication-constrained topology optimization. Our cavities are defined within a compact device footprint of 4 lambda^2 and exhibit mode volumes down to V = 3E-4 lambda^3 with wavelengths in the lambda = 1550 nm telecom band. This corresponds to field localization deep below the diffraction limit in a single hotspot inside the dielectric. A crucial insight underpinning our work is the identification of the critical role of lightning-rod effects at the surface. They invalidate the common definition of the mode volume, which is prone to gauge meretricious surface effects or numerical artefacts rather than robust confinement inside the dielectric. We use near-field optical measurements to corroborate the photon confinement to a single nanometer-scale hotspot. Our work enables new CMOS-compatible device concepts ranging from few- and single-photon nonlinearities over electronics-photonics integration to biosensing.

emitters 5 or material nonlinearities 12 . Besides the fundamentally 46 different confinement mechanism, DBCs differ from previous 47 cavity paradigms in several ways. First, the small mode volume 48 of nanometer-scale DBCs implies strong light-matter interaction 49 without resorting to extremely high quality factors, , thus en-50 abling applications requiring wide bandwidths such as nanoscale 51 light-emitting diodes, few-photon nonlinearities 12 , quantum op-52 tics with broadband emitters 31 , and optical interconnects 20 . Sec-53 ond, the modes of DBCs are strongly confined and therefore very 54 sensitive to the size of the dielectric bridge 11-15 at the center of 55 the bowtie. Smaller bridges reduce , immediately implying that 56 a new frontier of nanocavity research is concerned with reducing 57 the smallest feature size allowed by the nanofabrication process. 58 This is in contrast to previous work that aimed to increase , 59 which required reducing structural disorder rather than the critical 60 dimension 32,33 . Finally, the commonly used definition of the 61 mode volume is not generally applicable to DBCs because it 62 can pick up unintended surface effects rather than the effect of 63 confining light inside the material, as discussed in further detail 64 below and in Supplementary Section 1. 65 Inverse design and nanofabrication 66 We use carefully measured fabrication constraints as input to size-67 and tolerance-constrained topology optimization 13,16 aiming to 68 maximize the projected local density of optical states 5 (LDOS) at 69 the geometric center of the domain. Before running the optimiza-70 tion algorithm, we establish the smallest possible feature size of 71 our nanofabrication process, see Supplementary Section 2. We 72 fabricate 240 nm crystalline (100) silicon membranes ( = 3.48) 73 suspended in air using electron-beam lithography, dry etching, 74 and selective vapour-phase hydrofluoric acid etching. We op-75 timize a cyclic dry-etching process 34 to minimize the critical 76 dimension while tolerating periodic sidewall roughness in the 77 form of scallops, see Methods. We note that surface roughness 78 and the size of the scallops could be reduced by hard etching 79 masks. The fabrication constraints are quantified as a set of 80 critical dimensions, which we define through minimum attainable 81 radii. For our process, we find the radius of curvature of any 82 solid feature, ≥ 10 nm, and any void feature, ≥ 22 nm. The 83 critical radii are limited by proximity effects during electron-beam 84 lithography but it is possible to go below these limits with manual 85 shape modifications of the exposure mask, see Supplementary 86 Section 3. From systematic tests we find that it is possible to 87 obtain a mean bowtie bridge width of 8 nm in a localized area, 88 which we include as a third critical radius of curvature, ≥ 4 nm, 89 at the center of the design domain. The topology optimization 90 targets a maximum LDOS around = 1550 nm by tailoring the 91 material layout in a small square domain with 2 side length. 92 We obtain the highly optimized DBC shown in Fig. 1a, which 93 strongly enhances the vacuum field in the center of the cavity 94 as shown in Fig. 1b, and calculate ∼ 0.08 ( /(2 )) 3 and 95 ∼ 1100, around = 1551 nm. We fabricate DBCs based . a, Rendering of the DBC design generated by tolerance-constrained topology optimization. The normalized |E|-field is projected on the faces defining the three symmetry planes of the design. b, Zoom-in of the solid silicon bowtie exhibiting a strong field confinement due to the bowtie bridge dimension of 8 nm. c, 40°tilted scanning electron microscopy (SEM) image of a fabricated cavity. d, Global geometry-tuning, . Each air (black) pixel (1 nm 2 ) inside a -outline is exposed uniformly with electron-beam lithography; hence, air features defining the device are uniformly tuned. e-g, 40°tilted SEM images of bowtie region for = {−2, −4, −6} nm. We measure the mean width of the fabricated bowties to be (8 ± 5) nm, (10 ± 5) nm, and (17 ± 5) nm for figures e, f, and g, respectively, noting the variation in width along the -direction caused by the scallops and ∼ 1°negative sidewall angle represented by the uncertainty as discussed in the main text. agreement with the designed geometry as displayed in Fig. 1c. We 98 stress that the bowtie, along with all other details, are emergent 99 features arising entirely from the inverse design process 9,10,13 .

100
Similarly, the fact that the mode volume falls deep below the 101 diffraction limit of = ( /(2 )) 3 is a result of our algorithm 102 aiming to optimize the LDOS. 103 We calculate the quasi-normal mode of the structure shown 104 in Fig. 1a (including the tethers used to suspend the cavity, cf. with E(r) and (r) the electric field and dielectric constant at the bridge width of a few nanometers is close to the practical 119 resolution limit of conventional microscopy methods, such as 120 scanning electron microscopy (SEM). We therefore fabricate 121 three sets of DBCs, each of which subject to a global geometry-122 tuning, , of the entire mask, thereby shrinking the exposed 123 areas (air) in incremental steps of 2 nm as shown in Fig. 1d. 124 In order to further validate the yield and reproducibility, we 125 fabricate and characterize six nominally identical copies of each 126 geometry-tuned device. Representative SEM images of each of 127 the three geometry-tuned devices are shown in Figs. 1e-g and the 128 2 nm systematic variations are clearly observed in the change of 129 the fabricated bowtie dimensions. We measure a mean bowtie 130 bridge widths of 8 nm, 10 nm, and 17 nm, for the three geometry-131 tuned devices, respectively. See Methods and Supplementary 132 Section 4 for further details on the SEM characterization, and 133 Supplementary Section 7 for an overview of devices characterized 134 in this work.

135
Far-and near-field measurements 136 We characterize the devices using confocal cross-polarized mi-137 croscopy (see Methods) and a representative reflection spectrum 138 is shown in Fig. 2a. This spectrum shows the cavity mode as 139 a feature around 1520 nm. The DBC mode interferes with the 140 low-vertical cavity mode formed by the (∼ 3 µm) air gap be-141 tween the silicon device layer and the silicon substrate. This 142 results in a Fano resonance, which is well known from confocal 143 characterization of nanocavities 36 . The Fano line shape takes the 144 form where is the frequency, 0 is the DBC resonant frequency, 146 Γ is the linewidth, 0 ( ) is a linear function representing the 147

187
When continuously exciting the DBC with a laser tuned to the 188 cavity resonance while scanning the position, we can map out the 189 spatial structure of the cavity mode. The result, which is shown in 190 Fig. 3c shows that the mode is strongly localized at a single hotspot. 191 The near-field measurements in Fig. 3c show enhanced fields at 192 the edges of the void features on the sides of the silicon bridge. 193 These scattering fields arise because the tip goes down into the 194 holes and therefore scatters not only surface fields but a complex 195 combination of the surface field and the field in the voids, see 196 Supplementary Section 5. We disregard the data obtained when 197 the tip falls into the voids in the following analysis to facilitate a 198 direct comparison between the measured field above the device 199 to theoretical predictions. Figure 3d shows a high-resolution map 200 of the measured normalized scattered field amplitude with the 201 regions above the voids blacked out. The measured field cannot 202 be compared directly to the calculated quasi-normal mode shown 203 in Figs. 1a-b because the measured amplitude probes the intensity 204 of the cavity mode and, in addition, because of the influence of the 205 tip. We model the tip instrument function, ( ), as a Gaussian of 206 standard deviation and maximize the overlap between measured 207 and calculated field through the Bhattacharyya coefficient, = 208 | | · (| | 2 * ( )), where * denotes convolution and 209 ( ) is the measured (calculated) field 15 nm above the surface. 210 The tip has a nominal radius of curvature of 10 nm and probes 211 the field when the edge is 5 nm above the surface. Both fields are 212 normalized, | | = | | 2 * ( ) = 1, with the sum being 213 over all pixels. This analysis yields = (37±5) nm and = 0.984 214 indicating an excellent agreement between theory and experiment. 215 The instrument function is broader than the DBC mode size, so the 216 measurement gives an upper bound to the mode volume. Notably, 217 we find that the mode is confined to dimensions much smaller than 218 37 nm in agreement with our theoretical predictions. Additional s-219 SNOM measurements (see Supplementary Section 5) on a device 220 of different global geometry-tuning, = −6 nm (corresponding 221 to a mean bowtie width of 17 nm), also yields the largest overlap, 222 = 0.991, for the same instrument function = 37 nm. The 223 overlap between the two measurements is = 0.996, which further 224 confirms that the DBC mode is localized below the instrument 225   Scanning electron microscope characterization 493 We measure the dimensions of the fabricated structures by com-494 paring a combination of top-view and tilted SEM images analyzed 495 with detailed image analysis, presented and discussed in Supple-496 mentary Section 4. We measure the width of the bowties as 13 nm, 497 15 nm, and 21 nm from the top-view SEM images of the 3 sets of 498 devices presented in Fig. 1e-g, respectively. Furthermore, we use 499 multiple tilted views to estimate the width at the bottom of the 500 bowties, which we find is ∼ 10 nm narrower than at the top. This 501 implies a negative sidewall angle ∼ 1°of all devices and a mean 502 width of the bowtie bridges of 8 nm, 10 nm, and 17 nm, for the 503 three geometry-tuned devices, respectively, consistent with the 504 critical radius of curvature imposed on the topology optimization. 505 Supplementary Section 1 presents careful numerical simulations 506 of the fabricated dimensions, which both includes the sidewall 507 angle as well as variations of the dimensions of the calculated 508 structure. This confirms that the mode volume in the center of our 509 tolerance-constrained DBC-design remains robust to variations 510 and is deep below the diffraction limit.

511
Confocal cross-polarized microscopy setup 512 A supercontinuum laser (NKT Photonics SuperK Compact) is 513 focused on the cavity through a NA = 0.4 microscope objective. 514 The scattered light is collected through the same objective and 515 measured with an optical spectrum analyzer (AQ6370D Yoko-516 gawa), wavelength range = [1200, 1700]nm. The excitation 517 polarization is controlled with a /2-plate and light is collected 518 through a linear polarizer rotated 90°to reduce specular reflec-519 tions. Both excitation and collection is rotated 45°to the main 520 optical axis of the cavity (along x in Fig. 1a).

521
Near-field optical measurements 522 We use an s-SNOM (Neaspec, neaSNOM), equipped with a 523 pseudo-heterodyne module, in reflection mode to map the DBC 524 modes in the near-field. The incident light from a tunable 525 continuous-wave laser (Santec, TSL-710) is focused on a sil-526 icon AFM probe (NanoWorld, Arrow-NC) with a nominal tip 527 radius of 10 nm. The probe is used in intermittent contact mode 528 at a frequency 0 = 280 kHz oscillating with an amplitude of 529 60 nm. The amplitude of the scattered signal depends nonlinearly 530 on the height above the sample due to the near-field contribution, 531 therefore, demodulating at 4 0 with a lock-in amplifier yields the 532 near-field signal at the smallest height (∼ 5 nm) above the surface 533 while strongly suppressing contributions from the far-field back-534 ground. The laser is s-polarized, which is aligned along the -axis 535 of the DBC (see Fig. 1a), to minimize the perturbation from the 536 tip and to excite the cavity most efficiently. A polarizer is placed 537 in front of a photoreceiver (New Focus, 2053-FS) to select the 538 s-polarization of the scattered field. We determine the resonant 539 wavelength in the near-field from a Lorentzian fit to the near-field 540 spectrum obtained from a fixed position in several spatial maps 541 obtained around the bowtie for a number of wavelengths in a 542 20 nm band, see Supplementary Section 5 for further details.