Abstract
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 bulkedge 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, zerodimensional topological trapped states in the nanoscale remained unexplored, despite its importance for enhancing light–matter interactions and for wide applications including singlemode nanolasers. Here, we report a topological PhC nanocavity with a neardiffractionlimited mode volume and its application to singlemode 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.
Introduction
The 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 bulkedge 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, zerodimensional (0D) topological interface states^{2}, which function as cavities for the waves, can be defined a priori with knowing the bulk band topologies^{3,4,5,6,7,8,9,10,11,12}.
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 modenumber 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 ultrahigh Q factor and/or ultrasmall mode volume (V) nanocavities based on photonic crystals (PhCs)^{17,18,19} and plasmonics^{20}. Recently, topological microcavity lasers designed based on the bulkedge correspondence have been demonstrated^{4,5,6,21,22}. While the topological microcavities exhibited singlemode 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 topologybased, deterministic design of a singlemode 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 onedimensional (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 singlemode 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^{19,28} respectively 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 GaAsbased 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 squareshape 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)wavelike magnetic mode with a function node at the unit cell center for the lowest band, while a s(p)wavelike magnetic mode for the second lowest band. Since the optical mode at the zero frequency is always swave 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 lowestenergy 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 inversionsymmetric 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}. Further discussions on the existence of topological edge modes at a variety of interfaces composed of various 1D PhC nanobeams can be found in the supplementary material (see Supplementary Figures 1 and 2 and Supplementary Note 1).
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 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 exponentiallike 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} ~0a, 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 topologybased 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 singlemode 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 nmthick 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 topologybased 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 microphotoluminescence (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 ingap 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 nearfield 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 lightinlightout curve indeed exhibits an intensity jump giving a slike 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 nonradiative 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 phenomenon for lasing. Figure 4d shows the evolution of the measured cavity linewidths as a function of pump power, evaluated by using a higherresolution 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 topologybased 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 defectcavity 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 singlemode 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 bulkedge 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 verylow 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 by reducing the number of periods of the PhC. Overall, all these properties facilitate the robustness of singlemode lasing in our topological nanocavity.
In addition, the robust singlemode 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 bulkedge correspondence to deterministically define singlemode 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 highβ 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 nmthick 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 nonradiative carrier recombination processes, which are known to reduce β value in appearance: i.e., the nonradiative 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 nonradiative 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 longwavelengthpath 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 positiondependent PL measurements, we solely shifted the sample position with respect to the xaxis 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.
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Acknowledgements
The authors thank Y. Hatsugai, C.F. Fong and I. Kim for fruitful discussions. This work was supported by JSPS KAKENHI GrantinAid for Specially promoted Research (15H05700), KAKENHI(17H06138, 15H05868) and JSTCREST and was based on results obtained from a project commissioned by the New Energy and Industrial Technology Development Organization (NEDO).
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Y.O. and S.I. designed the topological nanocavity with the assistance of R.K. Y.O. performed the optical characterization of the nanocavities fabricated by R.K. Y.O. analyzed the data and wrote the manuscript with input from all the authors. K.W. grew the QD wafer. S.I. and Y.A. supervised the project.
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Ota, Y., Katsumi, R., Watanabe, K. et al. Topological photonic crystal nanocavity laser. Commun Phys 1, 86 (2018). https://doi.org/10.1038/s4200501800837
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DOI: https://doi.org/10.1038/s4200501800837
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