Observation of miniaturized bound states in the continuum with ultra-high quality factors

Light trapping is a constant pursuit in photonics because of its importance in science and technology. Many mechanisms have been explored, including the use of mirrors made of materials or structures that forbid outgoing waves, and bound states in the continuum that are mirror-less but based on topology. Here we report a compound method, combing mirrors and bound states in the continuum in an optimized way, to achieve a class of on-chip optical cavities that have high quality factors and small modal volumes. Specifically, light is trapped in the transverse direction by the photonic band gap of the lateral hetero-structure and confined in the vertical direction by the constellation of multiple bound states in the continuum. As a result, unlike most bound states in the continuum found in photonic crystal slabs that are de-localized Bloch modes, we achieve light-trapping in all three dimensions and experimentally demonstrate quality factors as high as $Q = 1.09 \times 10^6$ and modal volumes as low as $V = 3.56~ \mu m^3$ in the telecommunication regime. We further prove the robustness of our method through the statistical study of multiple fabricated devices. Our work provides a new method of light trapping, which can find potential applications in photonic integration, nonlinear optics and quantum computing.

While most light-trapping methods rely on the use of mirrors to forbid radiation, it is recently realized that optical bound states in the continuum (BICs) provide an alternative approach. BICs are localized wave functions with energies embedded in the radiation continuum, but, counterintuitively, do not couple to the radiation field. So far, BICs have been demonstrated in multiple wave systems, including photonic, phononic, acoustic, and water waves, and found important applications in surface acoustic wave filters and lasers 9,10,20,27,[31][32][33][34][35] . In many cases, BICs can be understood as topological defects [36][37][38][39][40] : for example, they are fundamentally vortices in the far field polarization in photonic crystal (PhC) slabs, each carrying an integer topological charge 41 .
Manipulations of these topological charges have led to interesting consequences, including resonances that become more robust to scattering losses 13 and unidirectional guided resonances that only radiate towards a single side without the use of mirrors on the other 42 .
So far, most BICs studied in PhC slabs are only localized in the vertical (thickness) direction, but remain de-localized in the transverse direction across the slab, rendering them less ideal in enhancing light-matter interaction with localized emitters or quantum applications. While it is known that perfect BICs localized in all three dimensions cannot exist 43 , it is of great interest to explore the limit of BIC miniaturization. A simple truncation of the PhC slab can reduce modal volume V , but also drastically degrades the quality factor Q, as it introduces leakage in both 2 lateral and vertical directions. A common relationship between Q and V for BICs with out-going boundary conditions in plane has been derived 44 , showing good agreements with experiments 45 .
Here we theoretically propose and experimentally demonstrate a method to achieve miniaturized BICs (mini-BICs) in PhC slabs, through the proper arrangement of multiple topological charges in the momentum space. Specifically, we start by enclosing the mini-BIC with a photonic band-gap mirror, using a lateral hetero-structure, to forbid transverse leakage. Similar to electronic quantum dots, the continuous photonic bands of an infinite PhC turn into discrete energy levels, due to the momentum quantization according to the size of the mini-BIC. For the same reason, the out-of-plane leakage of the mini-BIC is also dominated by a few directions that satisfy the momentum-quantization condition. As the PhC unit cell design is varied, multiple BICs 9, 10, 46, 47each carrying a topological charge and together composing a topological constellation -evolve in the momentum space, and eventually match with the major leakage channels. At this point, the out-of-plane radiation of the mini-BIC is topologically eliminated, giving rise to an ultra-long lifetime and a small modal volume.
Design and topological interpretation -As a specific example, we consider a PhC slab ( Fig. 1a), where circular air holes (radius r = 175 nm) are defined in a silicon layer of h = 600 nm thick. The mini-BIC design consists of a square cavity region A surrounded by a boundary region B with a gap size of g = 541 nm in between. The cavity A has a length of L in each side. Region A and B have different periodicities, a = 529 nm and b = 552 nm, to form a heterostructure in-plane.
We focus on the lowest-frequency TE band in region A above the light line (black line in Fig. 1b), whose energy is embedded in the bandgap of region B. By tuning parameters b and g, the interface between region A and B is almost perfectly reflective, minimizing the lateral leakage of the mini-BIC. In a finite sized structure, the continuous band of a PhC (left panel of Fig. 1b) splits into a series of discrete modes (right-panel) as the continuous momentum space is quantized into isolated points with a spacing of δk = π/L in between (Fig. 1c). This is analogous to to what happens in an electronic quantum dot. Each mode can thus be labelled by a pair of integers (p, q), indicating that its momentum is mostly localized near (pπ/L, qπ/L) in the first quadrant. Next, we show that the radiation loss of each mode can be strongly suppressed through the topological constellation of BICs. Fundamentally, BICs are topological defects in the far-field polarization, which carry integer topological charges: (1) Here φ(k) is the angle between the polarization major axis of radiation from the mode at k and the x−axis. C is a simple closed path that goes around the BIC in the counter-clockwise (CCW) direction. As shown in Fig. 2a, for an infinite PhC with a = 526.8 nm (case W ), there are 9 BICs: one is at the center of the Brilluion zone (BZ), and the other 8 form an octagonal-shaped topological constellation, which is denoted by their distance to the BZ center (k BIC ). The position of the topological constellation can be controlled by varying the periodicity a: for example, as a increases from 526.8 nm (case W in Fig. 2a) to 534 nm (case Z), the topological constellation shrinks and merges together before it turns into a single topological charge. The evolution of the Qs of infinite PhCs is shown in the lower panel of Fig. 2a.
Whenever the topological constellation matches with the main momenta of a mode M pq , i.e.
k BIC L/π = p 2 + q 2 , its radiation loss is strongly suppressed, as its major underlying Bloch mode components are now BICs with infinitely high Qs. This is confirmed by our simulation results in The scanning electron microscope images of the samples are shown in Fig. 3. The underlying SiO 2 layer is removed before measurements to restore the up-down mirror symmetry, required by the off-normal BICs. The footprint of the each sample is about 19.8 µm × 19.8 µm, including a cavity region that is 11.9 µm × 11.9 µm in size. The periodicity of the cavity region a is varied from 518 to 534 nm to satisfy the matching-condition and maximize Qs of M 11 , M 12 and M 22 . The periodicity of the boundary region and the gap distance are fixed at b = 552 nm and g = 541 nm.
The experimental setup is schematically shown in Fig. 3d, which is similar to our previously reported results 13,42 . A tunable laser in the telecommunication band is first sent through a polarizer in the y-direction (POL Y) before it is focused by a lens (L1) onto the rear focal plane (RFP) of an infinity-corrected objective lens. The incident angle and beam diameter of the laser are fine tuned by L1 to maximize the coupling efficiency. The reflected beam is collected by the same objective, and further expanded by 2.67 times through a 4f system to best fit the camera. A X-polarizer (Pol X) is used to block direct reflection from the sample, while allowing the resonance's radiation to pass. See Methods for more details on the experimental setup.
Experimental results -Whenever the excitation laser wavelength becomes on-resonance with a mode, the scattered light from the sample, captured on the camera, is maximized, which allows us to measure the resonance frequencies and Qs of different modes. Furthermore, under on-resonance condition, the far-field radiation pattern of each mode can also be recorded by the camera with polarizers (Fig. 4a). In particular, the far-field radiation of mode M 11 , M 12 and M 22 is measured to be donut-, dipole-and quadrupole-shaped, respectively, showing good agreements with numerical simulation. Furthermore, by placing a pin hole (not show in Fig. 3d) at the image plane of the RFP of the objective to reject stray light, scattered light intensity is recorded using a 6 photo-diode as the wavelength of the tunable laser is scanned. Distinct and sharp resonance peaks are found (mid-panel in Fig. 4b), corresponding to the 4 modes, M 11 through M 22 . We note that the fabrication imperfection slightly breaks the C 4 symmetry and causes a minimal energy difference Furthermore, to demonstrate the suppression of radiation loss originates from topological constellation, we vary periodicity a between 518 and 534 nm and track how Q changes. The measured wavelength of all modes agree well with simulation results (Fig. 5a). We see that, indeed, their measured Qs are always maximized when the topological-constellation-matching condition is met, which happen when a = 529.1, 525.9, and 522.8 nm for M 11 , M 12 and M 22 , respectively ( Fig. 5b). This finding shows good agreement with our simulation results in Fig. 2b. Finally, we prove the robustness of our method by measuring 87 different samples fabricated under the same design and through the same process. The histogram of their measured Qs of mode M 11 is shown 7 in Fig. 5c, featuring an averaged Q of 6.65 × 10 5 with a modest standard deviation of 1.22 × 10 5 .
See Supplementary Information Section VII to IX for more details.
To summarize, we present a type of ultra-high-Q and ultra-compact mini-BICs by combining in-plane mirrors and out-of-plane BICs in an optimized way. We experimentally demonstrate a record-high quality factor for BICs of Q = 1.09 × 10 6 and a small modal volume of 3.56 µm 3 . Our finding can potentially lead to on-chip lasers with ultra-low thresholds 18,22,49 , chemical or biological sensors [50][51][52][53] , nonlinear nanophotonic devices [23][24][25] , and elements for quantum computing 28-30 .
Furthermore, our method of achieving ultra-high-Q and ultra-low-V are proven to be robust, owing to their topological nature, which paves the way to further improving the performance of optoelectronic devices.

Methods
Numerical simulation All simulations are performed using the COMSOL Multiphysics in the frequency domain. Three-dimensional models are created with photonic crystal slabs placed between two perfect-matching layers. In other words, we have periodic boundary condition in-plane and out-going boundary condition in the vertical direction. The spatial meshing resolution is adjusted to adequately capture resonances with Qs of up to 10 9 . The eigenvalue solver is used to compute the frequencies and the quality factors of the resonances. The far-field emission patterns are computed by first retrieving the complex electric fields E 0,j (j = x, y) just above the PhC 8 surface and then calculating the emission fields as: x,y E 0,j (x, y)e −ik 0 (tan θx+tan φy) dxdy.

(M1)
Sample fabrication. We fabricate the sample on a silicon-on-insulator (SOI) wafer with e-beam lithography (EBL) followed by induced coupled plasma (ICP) etching. For EBL, we first spin-coat the cleaved SOI chips with a 500nm-thick layer of ZEP520A photo-resist before it is exposed with EBL (JBX-9500FS) at beam current of 400 pA and field size of 500 µm. Then we etch the sample with ICP (Oxford Plasmapro Estrelas 100) using a mixture of SF 6 and CHF 3 . After etching, we remove the resist with N-Methyl-2-pyrrolidone (NMP) and the buried oxide layer using 49% HF.