Topological photonic crystal nanocavity laser

Topological edge states exist at the interfaces between two topologically distinct materials. The presence and number of such modes are deterministically predicted from the bulk band topologies, known as the bulk-edge correspondence. This principle is highly useful for predictably controlling optical modes in resonators made of photonic crystals (PhCs), leading to the recent demonstrations of microscale topological lasers. Meanwhile, zero-dimensional topological trapped states in the nanoscale remained unexplored, despite its importance for enhancing light–matter interactions and for wide applications including single-mode nanolasers. Here, we report a topological PhC nanocavity with a near-diffraction-limited mode volume and its application to single-mode lasing. The topological origin of the nanocavity, formed at the interface between two topologically distinct PhCs, guarantees the existence of only one mode within its photonic bandgap. The observed lasing accompanies a high spontaneous emission coupling factor stemming from the nanoscale confinement. These results encompass a way to greatly downscale topological photonics. For most lasing and photonic applications, it is essential to control the number of lasing modes that are present. In this work, an interface between two topologically distinct photonic crystals is used to ensure single-mode lasing with enhanced light-matter interactions due to a near-diffraction-limited mode volume.

T he localization of waves in low dimensions constitutes a basis for diverse applications in various physical systems, including electron, sound and light. Given bulk band topologies, the bulk-edge correspondence, a physical principle that is originally developed for condensed matter physics 1 , provides a deterministic route to localizing the waves: the difference in topological invariants between the two materials in contact is associated with the number of existing localized interface modes. Accordingly, zero-dimensional (0D) topological interface states 2 , which function as cavities for the waves, can be defined a priori with knowing the bulk band topologies 3-12 . Regarding photonics, in which the control of the number of confined modes is vital for its applications, such deterministic design based on topology [13][14][15][16] is highly desired. Indeed, the mode-number control in a micro/nanoscale cavity lacks a rigid strategy and still often relies on empirical approaches. This fact makes a stark contrast with the impressive progress in designing ultra-high Q factor and/or ultra-small mode volume (V) nanocavities based on photonic crystals (PhCs) [17][18][19] and plasmonics 20 . Recently, topological microcavity lasers designed based on the bulk-edge correspondence have been demonstrated [4][5][6]21,22 . While the topological microcavities exhibited single-mode lasing, the cavity designs do not totally deny the existence of other confined modes resonating near the lasing frequencies. Moreover, the spatial confinements of the microscale cavities are relatively weak, hampering the realization of strong light-matter interactions.
In this work, we report a topology-based, deterministic design of a single-mode PhC nanocavity and its application to lasing. We show that a 0D interface state is predictably formed between two PhC nanobeams with distinct Zak phases 23 , topological invariants in the one-dimensional (1D) systems. The 0D edge state functions as a high Q factor nanocavity with a small V close to the diffraction limit 24 . With semiconductor quantum dot gain, the nanocavity exhibits single-mode lasing with a high spontaneous emission coupling factor (β) of~0.03, which is orders of magnitude larger than those for conventional semiconductor lasers (β~10 −6 ) and can be regarded as a hallmark consequence of the tight optical confinement [25][26][27][28] .

Results
Design of a topological nanocavity. Figure 1a shows a design schematic of the topological PhC nanobeam cavity investigated in this study. Two airbridge PhC nanobeams 19 Fig. 1 Topological nanocavity design concept. a Schematic of the nanocavity formed by interfacing two photonic crystals (PhCs) with a common bandgap and distinct Zak phases. The Zak phases denoted in the bracket are respectively for the lowest and the second lowest band. b Detailed description of the design. The two PhCs are in essence the same but differ in the way that the unit cells are defined. Blue unit cell arranges the small airhole with a width of d 1 at the center, while red unit cell has the large airhole with d 2 at the center. The centers of the two holes respectively correspond to the inversion centers of the double periodic PhC. The sum of d 1 and d 2 equals to 0.5a. c Band structures and wave functions at the band edges calculated for the PhC with the blue and red unit cell. The color of the band curve expresses its Zak phase (blue = π, red = 0). A band inversion occurs when swapping the unit cells. For the case of d 1 = 0.25a ( = d 2 ), a Dirac point appears at the band edge colored in red and blue are interfaced at the cavity center. They share a common photonic band structure but are distinct in terms of the band topology characterized by the Zak phase, which is defined as an integral of the Berry connection over the first Brillouin zone 29 . Further details of the design are found in Fig. 1b. The nanobeam is composed of GaAs-based unit cells (refractive index, n = 3.4) patterned with a period of a and is formed with a width of 1.6a and a thickness of 0.64a, which are so small that the nanobeam supports a single transverse mode. The unit cells respectively contain two square-shape airholes with a width of 0.5a, which are separated by half a period. The two airholes differ in their lengths, d 1 and d 2 , which relate each other via an equation d 1 + d 2 = 0.5a. The blue unit cell places the d 1 airhole at the center, while the red unit cell has the d 2 airhole at the center instead. In the following discussion, we consider situations with d 1 ≤ d 2 , such that d 1 ≤ 0.25a. The two airholes are arranged within the unit cells so as to keep inversion symmetry, leading to the quantization of Zak phase to either 0 or π 23 . This allows for predicting the existence of an interface state solely based on the Zak phase 29 , which is associated with the symmetry of the relevant bulk band wavefunction 30 . Figure 1c shows calculated 1D band structures using the threedimensional (3D) plane wave expansion method for the blue and red unit cell when d 1 = 0.19a (left and right panel, respectively) and d 1 = 0.25a (center). At d 1 = 0.25a, the two airholes become identical and the nanobeam forms a gapless band structure with a Dirac point at the zone edge. Meanwhile, for d 1 = 0.19a, a photonic bandgap appears around a normalized frequency of 0.25a/λ, where λ is the wavelength of light. The blue and red nanobeam share the same band structure, but the associated wave functions at the band edge have different parities: the blue (red) nanobeam supports a p(s)-wave-like magnetic mode with a function node at the unit cell center for the lowest band, while a s (p)-wave-like magnetic mode for the second lowest band. Since the optical mode at the zero frequency is always s-wave like, the lowest band for the blue unit cell should have a "twist" in its band nature somewhere in the momentum space. This makes the band topologically nontrivial in the sense of its finite Zak phase of π 30 , whereas the lowest band for the red unit cell can be regarded as topologically trivial. Now, we consider a situation where the two 1D PhC nanobeams are put in contact as schematically shown in Fig. 1a, b. When considering the lowest-energy bandgap of the 1D system, there predictably exists a single 0D edge state at the interface, when and only when the interface is composed of two 1D PhCs with distinct Zak phases for the lowest bands 29 . Suppose that only d 1 and d 2 vary in the system, the single modeness is robustly preserved as long as the two PhCs are topologically distinct, i.e., d 1 ≠ d 2 . The abrupt interface formed with the inversion-symmetric points eliminates the existence of other localized modes (Tamm modes) other than that of the topological origin (Shockley mode 30 ). In contrast, the conventional defect PhC nanocavities relying on index modulation sensitively vary the number of confined modes depending on the modulation pattern and in general cannot guarantee the single modeness 19,28 . We simulate the predicted interface modes of the designed nanobeams for different d 1 values by the 3D finite difference time domain method. Figure 2a shows the calculated electric field profiles of modes appear within the bandgaps. All the field distributions peak at the interfaces, with the field amplitude decreasing towards the exterior, demonstrating the existence of the localized cavity modes at the interfaces. We numerically Q=59,700 f=0.24492 (a/ ) c Computed Q factors, showing a peak value of~60,000 at d 1 = 0.19a. The quick decay of Q factor for d 1 > 0.19a originates from the photon leakage from the finite-size simulation region, while the reduction for d 1 < 0.19a is due to the light leakage into free space due to the tighter optical confinement confirmed the existence of a single cavity mode within the bandgap of each structure through its excitation spectrum (see Supplementary Figure 2). The spatial extent of the cavity mode depends largely on d 1 value, which essentially determines the depth and size of the corresponding bandgap and hence the optical penetration length into the PhC. In our design, the PhC with d 1 = 0a possesses the largest bandgap and thus realizes the tightest optical confinement, resulting in a small mode volume of 0.23(λ/n) 3 , which is close to the conventional diffraction limit 24 of~(λ/2n) 3 . This mode volume is at least an order of magnitude smaller than those reported in the previous topological microcavity designs [4][5][6] . Figure 2b summarizes the evolution of Vs when increasing d 1 , showing an exponential-like increase. Meanwhile, interestingly, the calculated resonant frequencies, shown as the red balls in the same plot, do not exhibit a significant dependence on d 1 31 .
In addition, we calculated the Q factors of the edge modes for different d 1 values and plotted them in Fig. 2c. The Q factor peaks at d 1 = 0.19a and decreases for both higher and lower d 1 values. For d 1~0 a, the optical confinement is too strong to suppress leaky components violating the condition for the total internal reflection, resulting in low Q factors of only about 1000. When increasing d 1 from 0a, the Q factor improves at the expense of enlarged mode volume. For d 1 > 0.2a, Q factor rapidly degrades since the mode size becomes so large compared to that of the allocated simulation domain (70 periods in the x direction), which leads to the light leakage through the domain boundaries. Interestingly, the existence of the 0D edge mode is robust: it resonates even when the patterned PhC area is harshly reduced to containing only a few unit cells 31 . In the current design and simulation size, at d 1 = 0.19a, the topological nanocavity support the maximum Q factor of 59,700 with a mode volume of 0.67(λ/ n) 3 , which are comparable to those for existing PhC nanobeam cavities employed for laser application 28 . The topology-based design enables the wide control of Q factor and mode volume while strictly keeping the single modeness of the nanocavity. This property is highly beneficial to develop single-mode lasers using broadband gain media, including semiconductor quantum dots, since such inhomogeneously broadened gain media often hamper the selection of a single lasing mode by controlling gain properties.
Sample fabrication and optical characterization. We fabricated the designed topological nanocavities with a = 270 nm into a 180 nm-thick GaAs slab containing InAs quantum dots (see Methods) using standard semiconductor nanofabrication processes. The total length of the nanobeam was set to 20 μm (including 68 unit cells). A scanning electron microscope image of a nanocavity designed with d 1 = 0.19a is shown in Fig. 3a. Unlike the theoretical model, the airholes are not perfect squares and possess rounded edges. In the topology-based design, however, the hole shape does not critically influence the existence of the cavity mode as far as the topological bandgap remains opened.
To characterize the fabricated nanocavity, we performed micro-photoluminescence (μPL) measurements at a low temperature of 10 K (see Methods). We used a modulated diode laser oscillating at 808 nm (repetition rate 5 MHz, pulse width 20 ns) and an objective lens for the optical pump of the samples. PL signals were analyzed with a grating spectrometer and an InGaAs camera. Figure 3b-d shows a series of PL spectra measured with an average pump power of 2.3 μW for the nanocavities designed using different d 1 s of 0a, 0.19a and 0.23a. Each spectrum exhibits a single strong peak resonating around a common wavelength of 1040 nm, as indicated by a colored arrow in the figure. The peak arises from the designed edge mode localized at the interface, confirmed through the experimental evidences provided below. In the figure, we also plotted PL spectra measured for homogeneous PhC nanobeams without the topological interfaces using gray solid lines. The spectra show the frequencies of the band edges of the respective designs (indicated by gray arrows). As expected from the design principle, the bandgaps that span between the pairs of the band edge peaks actually enclose the emission peaks originated from the interfaces. The measured peak frequencies of the emission peaks are summarized in Fig. 3e. All the peak positions agree well with those simulated using the plane wave expansion method (solid lines). A highlight of the plot is the nearly fixed resonant frequencies of the in-gap mode, which are in line with the predicted behavior of the topological edge states. Then, we investigate the spatial extents of the edge modes by measuring the pump position dependences of the PL spectra. The pump spot size in this experiment was about 2 μm. Figure 3f shows the normalized peak intensities of the interface modes taken along the x direction. The three curves are of the nanocavities with different d 1 s and clearly exhibit the mode sizes being dependent on d 1 . The case of d 1 = 0a and d 1 = 0.23a respectively show the smallest and largest mode distribution, as expected from the simulations. We note that the measured intensity profiles compare well with those simulated, after taking into account the spatial resolution of our measurement setup (see Supplementary Figure 3 and Supplementary Note 2). For detailed characterization of the field profiles, it will be important to directly evaluate them using, e.g., scanning near-field optical microscopes 32 . We also characterized the Q factors of the interface modes of different d 1 s and the corresponding PL spectra are depicted in the insets in Fig. 3b-d. The Q factors for d 1 = 0a and d 1 = 0.23a are measured to be only 700 and 600, respectively. In contrast, we observed a high Q factor of at least over 2000 for d 1 = 0.19a. The measured Q factor for the nanocavity of d 1 = 0.19a is limited by the spectrometer resolution (~400 μeV) used in this particular measurement as well as by the absorption originated from the embedded quantum dots. Therefore, its precise characterization requires further measurements, which we will discuss in the following. Now, we study the pump power dependence of the topological edge state formed in the d 1 = 0.19a nanocavity, in order to verify its lasing oscillation using the quantum dot gain. For the measurements, we reduced the repetition rate of the pump pulses to 0.5 MHz. Figure 4a displays two spectra taken with respective peak pump powers of 5 μW and 150 μW. For the low pump power PL spectrum, we observe broad background emission stemming from the quantum dot spontaneous emission, together with a sharp peak of the localized cavity mode. It is apparent that, with increasing the pump power, the cavity mode emission grows dramatically and dominates the spectrum. In Fig. 4b, c, we summarize the peak integrated intensities of the cavity mode measured under various pump powers. The light-in-light-out curve indeed exhibits an intensity jump giving a s-like shape, which is typical for lasers and is well fitted using a semiconductor laser model 26,33 (see Methods). Through the fit, we deduce a peak threshold pump power of 46 μW and a spontaneous emission coupling factor (β) of 0.03 for this laser: this high β value, which is orders of magnitude larger than those for conventional semiconductor lasers, can be read as a hallmark of nanolasers with tight optical confinement [25][26][27][28] . Nevertheless, the estimated β is relatively smaller than those reported for similar PhC nanolasers 28 . We consider that this arises from the use of the excited state of the quantum dots (QDs) for the laser gain, where fast non-radiative carrier recombination processes may reduce β in appearance 28,33 (see also Methods). In Fig. 4c, concomitant to the sharp intensity increase, we observed a clamp of the background emission (see green points), as another expected  Figure 4d shows the evolution of the measured cavity linewidths as a function of pump power, evaluated by using a higher-resolution spectrometer. A significant linewidth narrowing by nearly an order of magnitude is observed, further confirming the lasing oscillation in the investigated topological nanocavity. At a transparency pump power of 11 μW (estimated from the laser model used for the fitting), we estimated a cold cavity Q factor of~9600, experimentally demonstrating the formation of a high Q factor mode by the topology-based design. The current limiting factor of the achievable Q factor would be fabrication imperfection, since the current topological design requires smaller airhole sizes than those of conventional PhC defect-cavity designs resonating at the same wavelength 28 . It is noteworthy that the observed lasing can be regarded as quasicontinuous wave since all the dynamics within the laser are much faster than the pump pulse duration.

Discussion
In the current work, we have demonstrated single-mode lasing using a topological nanocavity with QD gain. At the design level, the single modeness of the nanocavity within the corresponding bandgap is guaranteed as far as the bandgap remains opened, thanks to the topological protection associated with the bulk-edge correspondence. Experimentally, even under harsh pumping to the system, the topological cavity mode will be sustained unless the pumping becomes stronger than that closing the bulk bandgap. Meanwhile, there are some possibilities of emerging confined modes due to focused optical pumping. Such carrier injection locally modifies the refractive index of the host material, which may lead to the formation of cavity modes within the mode gaps 34 . However, we did not observe the rise of such modes in the current experimental conditions. This would be due to very-low refractive index change in the current carrier pumping conditions. The low refractive index change is indirectly confirmed via the marginal change in the laser wavelengths during measuring pump power dependence (see Supplementary Figure 4 and Supplementary Note 3). Additionally, we note that we did not observe lasing when using the same 1D nanobeams without the topological interfaces. As seen in Fig. 3b-d, they support band edge modes, the Q factors of which, however, are too low to achieve lasing with the gain of our QDs. We note that the suppression of lasing from such band edge mode is easy, since they spread the entire PhC and thus their Q factors rapidly degrades The scattered data points above the threshold originate from the noisy PL spectra for the QD emission, due to strong cavity emission and the limited dynamic range of our detector. d Measured evolution of cavity linewidths as a function of peak pump power. The linewidths were extracted by fitting to the high resolution spectra with Lorentzian peak functions convolved with a Gaussian function representing the spectrometer response (~40 μeV). Linewidths shows a significant narrowing of nearly an order of magnitude. The increase of the linewidth above the lasing threshold is likely due to heating in the nanocavity. Error bars express standard errors deduced by fitting by reducing the number of periods of the PhC. Overall, all these properties facilitate the robustness of single-mode lasing in our topological nanocavity. In addition, the robust single-mode lasing observed in the current work infers that gain/loss and optical nonlinearities accompanied with optical pumping have only a minor influence on the properties of the topological nanocavity. This is largely due to the weak pumping powers in the current experiments, which is confirmed via the marginal changes of cavity resonant wavelengths. Meanwhile, for future studies, it will be of great interest to investigate the nature of topological edge states under the presence of strong gain and loss [35][36][37] . The influence of nonlinearities on the characteristics of topological PhC lasers [38][39][40] will also be of importance from a perspective of practical applications.
In summary, we realized a topological PhC nanocavity laser. We employed the bulk-edge correspondence to deterministically define single-mode PhC nanocavities, which support high Q factors and small mode volumes, both of which were confirmed theoretically and experimentally. The tight optical confinement effect manifested itself as the observed high-β lasing. Our results showcase an example on how topological physics penetrates into nanophotonics and on how to downscale topological photonics into the nanoscale. We believe that topological photonics 15,16 will renew and advance the understanding and the engineering of nanoscale resonators and lasers.

Methods
QD wafer used for sample fabrication. We fabricated the designed topological nanocavities into a 180 nm-thick GaAs slab. The slab is grown by molecular beam epitaxy and contains 5 layers of InAs QDs with an areal density of 3.6 × 10 10 cm −2 per layer. At 10 K, the ground state emission of the QD ensemble is peaked at 1100 nm with a spectral full width half maximum of 45 meV. Under strong optical pumping, the QD ensemble exhibits spectrally broad PL emission with additional two peaks associated with the excited state emission at~1030 nm (the first excited state) and at~970 nm (the second excited state).
Laser model used for fitting of the measured light output curve. We fitted the measured light output curve shown in Fig. 4b, c using a conventional semiconductor laser model 33 described below.
where n and N are the cavity photon and carrier number, respectively. N t is the transparent carrier number and estimated to be 3500. κ is the transparent cavity leakage rate of 125 μeV; γ is the total spontaneous emission rate of 0.66 μeV; P is the pump power; and β describes the spontaneous emission coupling factor and is treated as a main fitting parameter. Since the pump duration (20 ns) is much longer than all the laser dynamics, the laser is assumed to be operated under continuous wave pumping. As such, we can set dt = 0. The κn corresponds to light output intensity, which is to be fitted with the experimental output curve. Using a nonlinear least squared method, we found the best fitting when β = 0.03. Here, we did not take into account non-radiative carrier recombination processes, which are known to reduce β value in appearance: i.e., the non-radiative processes result in a larger intensity jump in the output curve across the threshold, which can be interpreted as a reduction of β when not considering the non-radiative processes. Accordingly, the estimated β of 0.03 can be regarded as the lowest bound for the current laser device.
Optical measurement setup. Figure 5 shows a schematic of the μPL setup used for the experiments. The fabricated samples are located within the cryostat cooled with liquid helium. The cryostat is placed on a stepping motor for course positioning and on piezo stages for fine adjustment as well. We access each sample with an objective lens with a magnification of 50. By inserting a beam splitter, we introduce excitation laser beam focused on the sample. The diameter of the focusing spot was estimated to be~2 μm, relating a peak pump power of 1 μW to a pump density of~30 W cm −2 . The PL signal is collected by the same lens and sent to a spectrometer equipped with a cooled InGaAs camera. The spectrometer resolution can be switched by changing diffraction gratings. In the detection optical path, we inserted a long-wavelength-path filter in order to reject the pump laser beam oscillating at a wavelength of 808 nm. The entrance slit of the spectrometer is narrowed down to~50 μm, allowing for spatial filtering with a width of~2 μm along with the nanobeam. The optical path is aligned so that PL signal at the pump position is guided into the spectrometer. For performing the position-dependent PL measurements, we solely shifted the sample position with respect to the x-axis using the piezo stage. The relationship between the pump and detection position is fixed during the measurements. The spatial resolution of this measurement is roughly assumed to be 2 μm.

Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.