Observation of miniaturized bound states in the continuum with ultra-high quality factors
Zihao Chen, Xuefan Yin, Jicheng Jin, Zhao Zheng, Zixuan Zhang, Feifan Wang, Li He, Bo Zhen, Chao Peng
aa r X i v : . [ phy s i c s . op ti c s ] F e b Observation of miniaturized bound states in the contin-uum with ultra-high quality factors
Zihao Chen , Xuefan Yin , Jicheng Jin , Zhao Zheng , Zixuan Zhang , Feifan Wang , Li He , BoZhen , & Chao Peng , ∗ State Key Laboratory of Advanced Optical Communication Systems and Networks, Departmentof Electronics & Frontiers Science Center for Nano-optoelectronics, Peking University, Beijing100871,China DepartmentofPhysicsandAstronomy,UniversityofPennsylvania,Philadelphia,PA19104,USA
Light trapping is a constant pursuit in photonics because of its importance in science andtechnology. Many mechanisms have been explored, including the use of mirrors made of ma-terials or structures that forbid outgoing waves , and bound states in the continuum thatare mirror-less but based on topology . Here we report a compound method, combingmirrors and bound states in the continuum in an optimized way, to achieve a class of on-chipoptical cavities that have high quality factors and small modal volumes. Specifically, light istrapped in the transverse direction by the photonic band gap of the lateral hetero-structureand confined in the vertical direction by the constellation of multiple bound states in thecontinuum. As a result, unlike most bound states in the continuum found in photonic crystalslabs that are de-localized Bloch modes, we achieve light-trapping in all three dimensions andexperimentally demonstrate quality factors as high as Q = 1 . × and modal volumesas low as V = 3 . µm in the telecommunication regime. We further prove the robust- ess of our method through the statistical study of multiple fabricated devices. Our workprovides a new method of light trapping, which can find potential applications in photonicintegration , nonlinear optics and quantum computing . While most light-trapping methods rely on the use of mirrors to forbid radiation, it is recentlyrealized that optical bound states in the continuum (BICs) provide an alternative approach. BICsare localized wave functions with energies embedded in the radiation continuum, but, counter-intuitively, do not couple to the radiation field. So far, BICs have been demonstrated in multiplewave systems, including photonic, phononic, acoustic, and water waves, and found important ap-plications in surface acoustic wave filters and lasers
9, 10, 20, 27, 31–35 . In many cases, BICs can beunderstood as topological defects : for example, they are fundamentally vortices in the farfield polarization in photonic crystal (PhC) slabs, each carrying an integer topological charge .Manipulations of these topological charges have led to interesting consequences, including reso-nances that become more robust to scattering losses and unidirectional guided resonances thatonly radiate towards a single side without the use of mirrors on the other .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 inenhancing light-matter interaction with localized emitters or quantum applications. While it isknown that perfect BICs localized in all three dimensions cannot exist , it is of great interest toexplore the limit of BIC miniaturization. A simple truncation of the PhC slab can reduce modalvolume V , but also drastically degrades the quality factor Q , as it introduces leakage in both2ateral and vertical directions. A common relationship between Q and V for BICs with out-goingboundary conditions in plane has been derived , showing good agreements with experiments .Here we theoretically propose and experimentally demonstrate a method to achieve minia-turized BICs (mini-BICs) in PhC slabs, through the proper arrangement of multiple topologicalcharges in the momentum space. Specifically, we start by enclosing the mini-BIC with a photonicband-gap mirror, using a lateral hetero-structure, to forbid transverse leakage. Similar to electronicquantum 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 themomentum-quantization condition. As the PhC unit cell design is varied, multiple BICs
9, 10, 46, 47 —each carrying a topological charge and together composing a topological constellation — evolvein 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-longlifetime 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 nmthick. 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),3hose energy is embedded in the bandgap of region B . By tuning parameters b and g , the interfacebetween 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 aseries of discrete modes (right-panel) as the continuous momentum space is quantized into isolatedpoints with a spacing of δk = π/L in between (Fig. 1c). This is analogous to to what happens inan electronic quantum dot. Each mode can thus be labelled by a pair of integers ( p, q ) , indicatingthat its momentum is mostly localized near ( pπ/L, qπ/L ) in the first quadrant. Four modes, M through M , are shown as examples, where M and M are degenerate in frequency due to the90-degree rotation symmetry of the structure ( C ). These modes exhibit distinctly different near-and far-field patterns (Fig. 1d), which are determined by their quantized momenta accordingly.The modal volumes of M through M are calculated as . µ m , . µ m and . µ m ,respectively. More details on the theory and design are presented in Section I through V of theSupplementary Information.Next, we show that the radiation loss of each mode can be strongly suppressed through thetopological constellation of BICs. Fundamentally, BICs are topological defects in the far-fieldpolarization, which carry integer topological charges: q = I C d k · ∇ k φ ( k ) . (1)Here φ ( k ) is the angle between the polarization major axis of radiation from the mode at k andthe 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 . nm (case W ), there are 9BICs: one is at the center of the Brilluion zone (BZ), and the other 8 form an octagonal-shaped4opological constellation, which is denoted by their distance to the BZ center ( k BIC ). The positionof the topological constellation can be controlled by varying the periodicity a : for example, as a increases from . nm (case W in Fig. 2a) to nm (case Z ), the topological constellationshrinks and merges together before it turns into a single topological charge. The evolution of the Q s 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 p + q , its radiation loss is strongly suppressed, as its major underlying Bloch modecomponents are now BICs with infinitely high Q s. This is confirmed by our simulation results inFig. 2b (see Methods for more details): the Q of M (red line) is maximized in case X when thematching condition is met. The maximum Q exceeds × . Similarly, the Q s of M (blue) andM (black) are also maximized when the matching conditions are satisfied, labelled by blue andblack dashed lines, respectively. Here we note that all localized modes penetrate, partly, into theboundary region, so the effective cavity length L eff is calculated as . a , which is slightly largerthan the physical length of the cavity L = 17 a . More details on the simulation are presented in theSection VI of the Supplementary Information. Sample fabrication and experimental setup —
To verify our theoretical findings, wefabricate PhC samples using e-beam lithography and induced coupled plasma etching processeson a 600 nm thick silicon-on-insulator wafer (see Methods for more details on the fabrication).The scanning electron microscope images of the samples are shown in Fig. 3. The underlyingSiO layer is removed before measurements to restore the up-down mirror symmetry, required by5he off-normal BICs. The footprint of the each sample is about . µ m × . µ m, including acavity region that is . µ m × . µ m in size. The periodicity of the cavity region a is variedfrom to nm to satisfy the matching-condition and maximize Q s of M , M and M . Theperiodicity 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 previouslyreported results
13, 42 . A tunable laser in the telecommunication band is first sent through a polarizerin the y -direction (POL Y) before it is focused by a lens (L1) onto the rear focal plane (RFP) of aninfinity-corrected objective lens. The incident angle and beam diameter of the laser are fine tunedby L1 to maximize the coupling efficiency. The reflected beam is collected by the same objective,and further expanded by 2.67 times through a f system to best fit the camera. A X-polarizer (PolX) is used to block direct reflection from the sample, while allowing the resonance’s radiation topass. See Methods for more details on the experimental setup. Experimental results —
Whenever the excitation laser wavelength becomes on-resonancewith a mode, the scattered light from the sample, captured on the camera, is maximized, whichallows us to measure the resonance frequencies and Q s of different modes. Furthermore, underon-resonance condition, the far-field radiation pattern of each mode can also be recorded by thecamera with polarizers (Fig. 4a). In particular, the far-field radiation of mode M , M and M is measured to be donut-, dipole- and quadrupole-shaped, respectively, showing good agreementswith numerical simulation. Furthermore, by placing a pin hole (not show in Fig. 3d) at the imageplane of the RFP of the objective to reject stray light, scattered light intensity is recorded using a6hoto-diode as the wavelength of the tunable laser is scanned. Distinct and sharp resonance peaksare found (mid-panel in Fig. 4b), corresponding to the 4 modes, M through M . We note that thefabrication imperfection slightly breaks the C symmetry and causes a minimal energy difference( ≈ and M Higher resolution measurements near each mode yield results shown in the left and rightpanels of Fig. 4b. The design is optimized for mode M ( a = 529 nm). The Q of each mode isextracted by numerically fitting the scattering spectrum to a Lorentzian function. As shown, themeasured Q s of modes M and M are . × and . × , respectively. Meanwhile, thehighest Q for mode M reaches . × , corresponding to a full width half maximum of . pm. To the best of knowledge, this is a record-high quality factor in small modal volume of BICs,which is about 60-fold enhancement of Q and 4-fold shrinking of V comparing with previouslyreported results .Furthermore, to demonstrate the suppression of radiation loss originates from topologicalconstellation, we vary periodicity a between and nm and track how Q changes. Themeasured wavelength of all modes agree well with simulation results (Fig. 5a). We see that, indeed,their measured Q s are always maximized when the topological-constellation-matching conditionis met, which happen when a = 529 . , . , and . nm for M , M and M , respectively(Fig. 5b). This finding shows good agreement with our simulation results in Fig. 2b. Finally, weprove the robustness of our method by measuring 87 different samples fabricated under the samedesign and through the same process. The histogram of their measured Q s of mode M is shown7n Fig. 5c, featuring an averaged Q of . × with a modest standard deviation of . × .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 combin-ing in-plane mirrors and out-of-plane BICs in an optimized way. We experimentally demonstrate arecord-high quality factor for BICs of Q = 1 . × and a small modal volume of . µ m . Ourfinding can potentially lead to on-chip lasers with ultra-low thresholds
18, 22, 49 , chemical or biolog-ical sensors , nonlinear nanophotonic devices , and elements for quantum computing .Furthermore, our method of achieving ultra-high- Q and ultra-low- V are proven to be robust, owingto their topological nature, which paves the way to further improving the performance of optoelec-tronic devices. MethodsNumerical simulation
All simulations are performed using the COMSOL Multiphysics in thefrequency domain. Three-dimensional models are created with photonic crystal slabs placed be-tween two perfect-matching layers. In other words, we have periodic boundary condition in-planeand out-going boundary condition in the vertical direction. The spatial meshing resolution is ad-justed to adequately capture resonances with Q s of up to . The eigenvalue solver is used tocompute the frequencies and the quality factors of the resonances. The far-field emission patternsare computed by first retrieving the complex electric fields E ,j ( j = x, y ) just above the PhC8urface and then calculating the emission fields as: F j ( θ, φ ) ∝ (cos θ + cos φ − Z Z x,y E ,j ( x, y ) e − ik (tan θx +tan φy ) dxdy. (M1) Sample fabrication.
We fabricate the sample on a silicon-on-insulator (SOI) wafer with e-beamlithography (EBL) followed by induced coupled plasma (ICP) etching. For EBL, we first spin-coatthe cleaved SOI chips with a 500nm-thick layer of ZEP520A photo-resist before it is exposed withEBL (JBX-9500FS) at beam current of pA and field size of 500 µ m. Then we etch the samplewith ICP (Oxford Plasmapro Estrelas 100) using a mixture of SF and CHF . After etching, weremove the resist with N-Methyl-2-pyrrolidone (NMP) and the buried oxide layer using HF.
Measurement setup.
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Correspondence
Correspondence and requests for materials should be addressed to Chao Peng. (email:[email protected]). yyyy θ M11M12M21M22Near field Far field -30° 30°30°-30° φ φφφ a.u. -18-88-88-88-8 8-8 ( μ m ) a.u. M22M11M12/21 k (2π/a) k ( π / a ) -0.08 0 0.08-0.08 Reciprocal space y (1,1) (2,2)(2,1)(1,2) θ φ xyz A B L RadiationSideleakage ... ~~ ~~~~ PBG
M11M12/21M22-0.08 0 0.08 k (2π/a) X ← Γ → M ω (2πc/a) ω TE A a bc d
Figure 1: | Mini-BIC modes. a,
Schematic of a mini-BIC (region A) surrounded a photonicbandgap (PBG, region B). b, A continuous band (TE-A) of an infinitely large PhC with periodicboundary condition (left) turns into a set of discrete modes under the PBG boundary condition(right). c, The momentum distribution of each mode is highly localized to points that form asquare lattice in the momentum space with a spacing of π/L . Modes are labeled as M pq , accordingto their momentum peak positions in the first quadrant at ( pπ/L, qπ/L ) . d, The near-field modeprofiles of four modes M through M (left) and their far-field emission patterns (right).17 L ~22a eff W X Y Z Q M11M12M22
520 524 528 532 536 a (nm) k ·L / π BIC eff ∞ Q -0.060.060 k (2π/a) k ( π / a ) W (a=526.8 nm) X (a=529.1 nm) Y (a=531.4 nm) Z (a=534.0 nm) k BIC x k (2π/a) x k (2π/a) x k (2π/a) x y k ( π / a ) y ab Figure 2: | Maximizing the Q s of mini-BICs by properly arranging topological charges inthe momentum space. a, Multiple BICs appear on bulk band TE-A in momentum space, inwhich 8 off- Γ ones with q = ± topological charges compose an octagonal-shaped topologicalconstellation, denoted by the radius k BIC . When unit cell periodicity a varies from . nm(W) to . nm (Z), the topological constellation shrinks, merges, and annihilates to a singletopological charge (upper panel). The quality factor Q for each unit cell design is shown in thelower panel. b, The quality factor Q of modes, M through M , as functions of periodicity a (upper axis) and topological constellation k BIC (lower axis). Q for M (red line) maximizes whenits quantized momentum √ π/L matches the topological constellation k BIC , corresponding to caseX ( a = 529 . nm) in a . Similar maxima are also observed for M (blue) and M (black) underother designs, when k BIC matches √ π/L and √ π/L , respectively.18 Y X Y L2L34 f Camera/PDRFP LaserPol YL1Sample BSObjective Pol X
529 nm 350 nm500 nm Si n m
10 μm
L ~11.9 μm eff
A Ba bcd
Figure 3: | Fabricated sample and experimental setup. a,b,c
Scanning electron microscopeimages of the fabricated samples from top and side views. The underlying SiO layer is removedbefore measurements. The chosen structural parameters correspond to case X in Fig. 2a to maxi-mize Q for mode M . d, Schematic of the experimental setup. L, lens; RFP, real focal plane; PD,photodiode; POL, polarizer; BS, beam-splitter; Lens L2 and L3 are confocal.19
M22 Q = 161,121 Wavelength (nm) Q = 336,126 M12
Experiment Fit I n t en s i t y ( a . u . ) Wavelength (nm)1.44 pm Q = 1,093,824 Experiment Fit
Wavelength (nm)
M11 M12M21 M22M11
ExEyExperiment Simulation
M115°
Experiment Simulation Experiment Simulation
M12 M22
ExEy ExEy
5° 5°θ θ φ θ ab Figure 4: | Observation of mini-BIC modes. a,
The far-field emission patterns ( x -, y -polarizedand overall) of modes M through M , measured with a camera (gray color map), show goodagreements with simulation results (hot color map). b, Middle panel: measured scattered lightintensity as the laser wavelength scans from nm to nm. Four clear peaks are observedand identified as M through M . The Q of M reaches . × (left panel). In the samesample, the Q s of M and M are measured as . × and . × , respectively (rightpanel). 20
18 522 526 530 534 a (nm) W a v e l e n g t h ( n m ) Q
520 524 528 532 a (nm) ×10 k ·L /π BIC / eff Q S a m p l e c o u n t s ×10 M11M12M22
ExperimentPolynomial fitExperimentSimulation
M11M12M22 Q distribution of M11 Gaussian fit a b c
Figure 5: | Demonstration of mini-BIC robustness against fabrication errors. a,
Measuredresonance wavelengths (circles) in samples with different periodicities a show good agreementswith simulation results (dashed lines). b, Measured Q s (circles) in samples with different peri-odicities a (upper axis) and, therefore, different BIC constellation ( k BIC , lower axis). Polynomialfittings are shown in solid lines. Each curve is maximized when the matching condition is satisfied,indicated as dashed vertical lines. c, Histogram statistics of measured Q s of M11