Fig. 1: Theoretical analysis of the merging bound state in the continuum (BIC) laser.<\/b>![\"Fig.](\"https:\/\/www.europesays.com\/ie\/wp-content\/uploads\/2026\/05\/41377_2026_2355_Fig1_HTML.png\")
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a<\/b> Schematic representation of the photonic crystal slab and the emission from the BIC mode. b<\/b> Simulated band structure of the photonic crystal devices. The BIC band is indicated by a red line, and the inset shows the top-view electric field distribution (at the center plane of the quantum well) of the BIC mode at \u0393 point in a unit cell. The first Brillouin zone of the square lattice is also shown in the inset. c<\/b> Simulated far-field Q-factor distribution of the BIC modes, illustrating the evolution from before merging, to at the merging point, and after merging cases. As the hole diameters of the photonic crystal slab increase, the eight isolated off-\u0393 BICs move toward the at-\u0393 BIC and merge into a single BIC. Simulation parameters: 940\u2009nm period, and air hole diameters are 484, 488, and 510\u2009nm for before merging, merging, and after merging BIC scenarios, respectively. d<\/b> Q-factor scaling along the \u0393X direction of the BIC mode before merging (green), at the merging point (red), and after merging (blue). The merging BIC case (red line and dots), characterized by a high Q-factor near the \u0393 point, follows a scaling behavior of \\(Q\\propto 1\/{k}^{6}\\), whereas the before and after merging cases follow \\(Q\\propto 1\/{k}^{2}\\)<\/p>\n
Figure 1c<\/a> shows the Q-factor distribution of the band in k-space. Generally, there exists a symmetry-protected BIC at the \u0393 point along with eight accidental BICs. These nine BIC modes exhibit topological charges of \u00b11, as illustrated in Fig. S2<\/a>. As the size of the air holes of the photonic crystal slab increases, the eight off-\u0393 accidental BICs gradually move towards the \u0393 point in momentum space and finally merge with the at-\u0393 symmetry-protected BIC. Upon further increasing the diameter of the air holes past the merging BIC condition, a single BIC at the \u0393 point remains. This merging of BICs significantly impacts the overall loss at the BIC resonance27<\/a>. In the case of an isolated BIC mode, the Q-factor exhibits a quadratic decay in k-space; while in the merging scenario, this scaling transitions to 1\/k6, as depicted in Fig. 1d<\/a>. This behavior creates a region of higher Q-factor around the \u0393 point for the merging BIC, with the Q-factor decreasing more gradually away from the \u0393 point compared to that of an isolated BIC mode. Such characteristics suggest an increased robustness against symmetry-breaking perturbations and improved tolerance to nanofabrication variations.<\/p>\n BIC laser performance of the 20\u2009\u00d7\u200920-period devices<\/p>\n To experimentally verify this concept, we fabricated suspended photonic crystal slabs with a 20\u2009\u00d7\u200920 array of unit cells. Fabrication details are provided in Supplementary Text and Fig. S1<\/a>. A scanning electron microscope (SEM) image of the sample is shown in Fig. 2a<\/a>, where the period is 940\u2009nm, the quantum well layer thickness is 320\u2009nm, and the air hole diameters range from 400 to 600\u2009nm. (Due to the dry-etching process in the fabrication, the etched hole side wall in the quantum well is never perfectly perpendicular. So we refer to the hole diameters as effective diameter d, which is the average value of the hole diameters on the top and bottom surface of the quantum well). A 1064-nm pulsed fiber laser (5\u2009ns, 50\u2009kHz) is used to optically excite the photonic crystal cavity. Using two-dimensional k-space imaging, we measured the photoluminescence (PL) band structures of samples with varying air hole sizes, as shown in Fig. 2b<\/a>. The air hole diameters range from 410 to 514\u2009nm in Fig. 2b-I\u2013IV<\/a>, representing the progression from before merging, to pre-merging, merging, and after-merging BIC regimes, respectively. The white dashed ellipses highlight the designed photonic band, which matches the simulated band structures in Fig. 2c<\/a> and Fig. 1b<\/a> very well. Due to the finite size of the photonic crystal, the measured band structure exhibits discrete energy levels corresponding to different mode distributions. Similar finite-size-induced discretization of BIC bands has also been analyzed numerically in previous works25<\/a>,38<\/a>,41<\/a>, where a continuous BIC band is quantized into a ladder of cavity-like s- and p-type modes in real space. Here, the two-dimensional PL band mapping in Fig. 2b<\/a> directly visualizes the discretized BIC dispersion as a set of well-separated peaks in momentum space, providing experimental evidence of finite-size quantization of BIC bands.<\/p>\n Fig. 2: Scanning electron microscope (SEM) image and band structure analysis of the fabricated photonic crystal slabs.<\/b> a<\/b> SEM image of a photonic crystal slab with a 20\u2009\u00d7\u200920 array of unit cells. The right panel shows a close-up view with a lattice constant a\u2009=\u2009940\u2009nm and hole diameter d\u2009=\u2009467\u2009nm. b<\/b> Experimental normalized angle-resolved photoluminescence (PL) spectra for slabs with hole diameters d\u2009=\u2009410, 467, 490, and 514\u2009nm, labeled I\u2013IV, representing conditions from before to after merging. The BIC mode is highlighted by white dashed circles. c<\/b> Simulated normalized transmission spectra corresponding to the experimental cases in b<\/b> calculated using rigorous coupled-wave analysis (RCWA) open-source electromagnetic solver S450<\/a> with the same parameters in b, which produced the same band structure as COMSOL (Fig. S3<\/a>). Due to the high quality factor of the BIC modes, these BIC modes have a near-zero linewidth. The symmetry-protected and accidental BICs can be better identified from the zoomed-in transmission spectra shown in Fig. S4<\/a><\/p>\n Power-dependent micro-PL spectroscopy was performed to evaluate the impact of the merging BIC condition on the laser threshold. Figure 3a<\/a> presents the power-dependent PL spectra for the sample shown in Fig. 2b-II<\/a>, with an air hole diameter d\u2009=\u2009467\u2009nm, which is near the merging BIC condition. The corresponding log\u2013log intensity plot and the observed linewidth narrowing indicate the clear laser behavior. The linewidth of the BIC mode around the laser threshold is ~0.2\u2009nm, corresponding to an estimated laser Q of ~7600. By measuring samples with different air hole diameters, we mapped the laser thresholds and BIC mode distributions as a function of the effective hole diameter d (Fig. 3b<\/a>), showing a distinct reduction in the laser threshold near the merging BIC condition. This observation is consistent with simulation predictions (Fig. 1<\/a>) and confirms the merging BIC\u2019s role in lowering the lasing threshold.<\/p>\n Fig. 3: Experimental characterization of the BIC laser.<\/b> a<\/b> Log-log plot of PL intensity and the emission linewidth as a function of pumping power from a photonic crystal slab with hole diameter d\u2009=\u2009467\u2009nm, representing a slightly pre-merging condition. The linewidth of the BIC mode around the laser threshold is ~0.2\u2009nm, corresponding to an estimated laser Q of ~7600. b<\/b> Laser threshold and BIC mode wavelength distributions as a function of hole diameter of the photonic crystal slab, showing a reduction in laser threshold near the merging BIC condition. c-I<\/b>\u2013IV<\/b> Normalized laser emission spectra as a function of pumping power for hole diameters d\u2009=\u2009410, 467, 490, and 514\u2009nm, respectively. In the pre-merging BIC condition (II), the laser achieves optimal single-mode performance, sustained up to at least 80 Pth. Another set of devices also shows a similar lasing characteristics (Fig. S7<\/a>)<\/p>\n Upon increasing the pump power above the lasing threshold, the single-mode characteristics of different samples became evident. The BIC mode at the high-Q \u0393 point was consistently the first to lase, as designed. Under the before-merging condition (Fig. 3c-I<\/a>), other non-BIC modes also began lasing at approximately three times the threshold power (Pth), suggesting that the BIC mode did not offer clear advantages over other modes at this power. However, as the air hole diameter approached the merging BIC condition, the single-mode characteristics of the merging BIC were significantly enhanced. Surprisingly, the best single-mode performance was observed under the pre-merging BIC condition. As shown in Fig. 3c-II<\/a>, lasing remained strictly single-mode even at 80\u00d7 Pth. In contrast, under the merging BIC condition in Fig. 3c-III<\/a>, a higher-order BIC mode started to lase at approximately 50\u00d7 Pth. A similar trend was observed in a separate set of devices (Fig. S7<\/a>).<\/p>\n This anomalous single-mode behavior is a direct consequence of the finite-sized photonic crystal slab46<\/a>. In the 20\u2009\u00d7\u200920-period devices, the BIC band exhibits discrete energy levels, as shown in the measured band diagram in Fig. 2<\/a>. These energy levels correspond to distinct field distributions, with the ground state s-state and p-state depicted in Fig. 4a, b<\/a>, respectively. Owing to comparable Q-factors stemming from fabrication imperfections, mode competition occurs under high excitation, which in turn influences the laser\u2019s single-mode characteristics. To quantitatively characterize this behavior, we performed finite-difference time-domain (FDTD) simulations of the finite 20\u2009\u00d7\u200920 photonic crystal array and extracted the resonance wavelength and Q-factors of the lowest-order s-like BIC mode and the first excited p-like mode under both the pre-merging and merging conditions. The underlying simulated spectra used to identify these resonances are provided in Fig. S8<\/a>.<\/p>\n Fig. 4: Analysis of the BIC mode in finite-size photonic crystal slabs.<\/b> a<\/b>, b<\/b> Simulated normalized electric field distributions of the merging BIC mode in a photonic crystal slab with 20\u2009\u00d7\u200920 array of unit cells. Due to restrictive boundaries, the ideal BIC band exhibits s- and p-state electric field distributions, corresponding to the discrete energy levels measured in Fig. 2b<\/a>. These closely spaced energy levels result in mode competition, which directly impacts single-mode lasing performance. c<\/b> Simulated threshold gain of the s- and p-state modes under the pre-merging and merging BIC conditions. These two modes correspond to the BIC lasing mode and the competing mode observed experimentally, i.e., the discrete peaks marked by the white dashed circles in Fig. 2b<\/a>, whose energy splitting agrees well with the numerically calculated response spectra in Supplementary Fig. S8<\/a>. For the pre-merging BIC, the s- and p-like modes exhibit threshold gains of 7.89\u2009cm\u22121 and 25.5\u2009cm\u22121, respectively, whereas for the merging BIC the corresponding values are 7.96\u2009cm\u22121 and 10.66\u2009cm\u22121. These results indicate that under high excitation powers, the competing mode is more likely to lase in the merging BIC scenario due to the small threshold gain difference. On the other hand, the larger threshold gain difference in the pre-merging case suggests a larger pump power range of single-mode lasing of the BIC mode, consistent with our experimental observations<\/p>\n The lasing threshold gain \\({g}_{{\\rm{th}}}\\) is defined as \\({g}_{{\\rm{th}}}=n\/\\lambda \\varGamma Q\\)47<\/a>, where n is the refractive index of the gain material, \u03bb is the mode wavelength, \u0393 is the mode\u2013gain overlap factor (~0.93 for both the BIC and competing modes), and Q is the mode\u2019s quality factor. The calculated threshold gains are summarized in Fig. 4c<\/a>. Under the pre-merging condition, the calculated threshold gains for the s-like BIC and the p-like competing modes are 7.89\u2009cm\u22121 and 25.5\u2009cm\u22121, respectively \u2013 more than a threefold difference. In contrast, under the merging condition, the threshold gains are 7.96\u2009cm\u22121 and 10.66\u2009cm\u22121, yielding only a\u2009~\u20091.3\u00d7 difference. Although the calculated threshold-gain values under ideal conditions differ from experimentally measured threshold values due to unavoidable fabrication imperfections, the relative trends remain the same. In the pre-merging regime, the fundamental BIC mode requires substantially lower modal gain than the nearby competing modes, whereas at the merging condition the discrete s- and p-like modes exhibit much more similar threshold gain values. This occurs because merging broadens the high-Q region around the \u0393 point, enhancing not only the fundamental s-like mode but also nearby quantized modes such as the p-like state, thereby reducing modal discrimination in the finite lattice. In semiconductor lasers, the carrier density is strongly clamped near the threshold of the mode with the lowest \\({g}_{{\\rm{th}}}\\); therefore, increasing the external pump power first primarily increases the photon number in that mode rather than the material gain available to competing modes. As a consequence, in the pre-merging regime, the large threshold-gain contrast prevents the p-like mode from ever reaching its higher \\({g}_{{\\rm{th}}}\\), and strictly single-mode lasing of the s-like BIC mode is maintained even at pump powers up to 80\u00d7 Pth. In contrast, at the merging condition the reduced threshold-gain contrast allows the p-like mode to reach lasing at a significantly lower pump level (experimentally around 50\u00d7 Pth), leading to the onset of multimode emission.<\/p>\n These results reveal a new physical approach for robust single-mode lasing in finite-size photonic crystal BIC lasers: the pre-merging condition enhances single-mode robustness by maximizing the Q-factor contrast between the fundamental mode and nearby competitors. This effect arises from the spectral discretization and field confinement inherent in finite lattices, and is expected to occur broadly in systems exhibiting BIC merging behavior. While super-\/merging-BIC concepts have largely been pursued to suppress radiative loss and reduce lasing thresholds by enhancing the radiative Q factor39<\/a>, the primary limitation for high-power single-mode operation in finite BIC lattices is instead set by mode competition among discretized states. Crucially, we find that the optimal condition for robust single-mode operation far above threshold is not the merging regime, but rather the pre-merging regime, where the threshold gain of the nearest competing mode is maximally separated from that of the fundamental BIC mode. This counter-intuitive condition enables a substantially expanded single-mode dynamic range, reaching up to 80\u00d7 Pth.<\/p>\n A hallmark of BIC is its topological nature, manifested as a polarization vortex in k-space15<\/a>,16<\/a>,17<\/a>. To experimentally confirm this feature, we conducted far-field self-interference measurements using a Michelson interferometer (Fig. 5<\/a> and Fig. S5<\/a>). In this configuration, the far-field emission pattern is interfered with a slightly tilted, inverted replica of itself, and phase singularities in the underlying field appear as two inverted fork-like dislocations in the interference fringes. The \u201cdouble-fork\u201d patterns highlighted by red arrows in Fig. 5c<\/a> are characteristic of such phase singularities and are consistent with the unit topological charge of the polarization vortex predicted for the BIC mode in Fig. 1c<\/a> and Fig. S2<\/a>. Importantly, the donut-shaped far-field emission in Fig. 5<\/a> indicates that the radiation channel at \u0393-point is strongly suppressed, consistent with the polarization singularity. In a practical finite-size photonic crystal slab, unavoidable symmetry breaking (finite aperture, boundary scattering, and fabrication imperfections) turns an ideal BIC into a quasi-BIC with a large but finite radiative Q21<\/a>,48<\/a>, so the out-coupled emission is dominated by small non-zero k components, naturally producing a donut-like distribution. Meanwhile, the high-contrast fringes obtained in the self-interference experiment demonstrate the high spatial coherence of the BIC laser emission, confirming its excellent lasing performance. Due to the finite size of the photonic crystal, additional interference arising from the device boundaries is visible away from the central singularity. Polarization intensity measurements are also provided in Fig. S6<\/a>, which are consistent with the polarization vortex nature.<\/p>\n Fig. 5: Far-field characterization of the BIC laser.<\/b> a<\/b>–I<\/b>\u2013III<\/b> Simulated far-field images of the 20\u2009\u00d7\u200920 array photonic crystal slabs with hole diameters d\u2009=\u2009410, 467, and 490\u2009nm, respectively. b<\/b>–I<\/b>\u2013III<\/b>, Experimental far-field images corresponding to the simulations in (a<\/b>). c<\/b>–I<\/b>\u2013III<\/b>, Experimental self-interference patterns of the far-field images in (b<\/b>). The double-fork patterns, highlighted by red arrows, indicate the presence of a vortex in the BIC laser emission. The white dashed circles in a<\/b>\u2013c<\/b> represent the numerical aperture (NA) of 0.32<\/p>\n BIC laser performance of the 5\u2009\u00d7\u20095-period devices<\/p>\n Despite the exceptional performance of the merging BIC laser with a 20\u2009\u00d7\u200920 array, achieving BIC lasing in even smaller devices is critical for advancing miniaturized photonic technologies. Leveraging the concept of merging BIC, we also experimentally demonstrate an ultra-compact photonic crystal slab laser comprising only 5 periods, as shown in Fig. 6a<\/a>. However, due to the extremely small photonic crystal size, the BIC modes experience significant leakage at the sample edges, losing their advantage over competing modes. Additionally, the highly discretized band structure in such a small device prevents continuous tuning of the merging BIC condition. To address these limitations, we employed edge-engineering by slightly reducing the air-hole diameter at the device boundaries to ~75% of that in the center, as illustrated in Fig. 6a<\/a>. Simulations confirm that this edge-engineering structural optimization significantly enhances the Q-factor of the BIC mode in the 5\u2009\u00d7\u20095 array compared to the unmodified BIC design, as shown in Fig. S9<\/a>. Nevertheless, the increased boundary loss inherent to such small cavities inevitably leads to a higher lasing threshold. To compensate for this, we utilized the 300-nm thick quantum well gain medium and designed the BIC mode near 1400\u2009nm, where the gain spectrum peaks under high excitation power. The resulting device has a period of 760\u2009nm.<\/p>\n Fig. 6: Characterization of the BIC laser from a 5\u00d75 array photonic crystal slab.<\/b> a<\/b>–I<\/b>\u2013II<\/b>, Top and side views of SEM images of the photonic crystal slab with a lattice constant a\u2009=\u2009760\u2009nm. The air-hole diameters are smaller at the edges to suppress mode leakage and enhance the Q factor. b<\/b> Log-log plot of PL intensity and the emission linewidth as a function of pumping power from a sample with d\u2009=\u2009366\u2009nm. c<\/b> The emission spectrum at 10\u00d7 Pth of the sample in b<\/b> which maintains single-mode performance. d<\/b> Far-field images and self-interference patterns of the sample in (b<\/b>). Double-fork patterns, highlighted by red arrows, indicate the vortex structure of the BIC emission. The white dashed circles denote the NA of 0.42<\/p>\n Power-dependent PL intensity and linewidth measurements confirm lasing behavior in Fig. 6b<\/a>, with the laser maintaining single-mode operation even at excitation powers exceeding 10 times the threshold (Fig. 6c<\/a>). Although the mere 5-period size prevents direct band structure measurements, far-field interference patterns reveal distinct vortex structures in momentum space, verifying that lasing originates from the BIC mode. Moreover, the small physical footprint leads to a broader distribution in momentum space, further supporting its unique BIC-based origin of the lasing mechanism. It is worth noting that previous mini-BIC lasers38<\/a>,41<\/a> achieve footprint reduction by defining a mini-BIC cavity core within a much larger photonic-crystal heterostructure, where the surrounding photonic bandgap provides in-plane confinement. By contrast, in our approach the entire patterned photonic-crystal slab is limited to only 5\u2009\u00d7\u20095 periods, and the Q enhancement is realized by direct edge engineering without increasing footprint, offering a route toward BIC single-mode lasing with the smallest possible total patterned active footprint.<\/p>\n","protected":false},"excerpt":{"rendered":"Theoretical analysis of the merging BIC laser Figure 1a shows the schematic of our merging BIC laser. 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