1. Introduction

High-quality factors (Q-factors) accompanied by near-field enhancement facilitate light-matter interactions in nanocavities. Nanoscale lasers such as distributed feedback plasmonic lasers [1], photonic crystal lasers [2], and microdisk lasers [3] possess a lower threshold owing to their high Q-factors and low mode volumes. Several recently reported low-threshold lasing approaches seek to maximize the Q-factors, such as compound waveguide dielectric gratings [4], eccentric nanoring structures [5], and metallic grating waveguides [6]. However, large Q-factors typically occur at the expense of compactness. Recently, the use of bound states in the continuum (BICs) has paved the way for applications requiring highly confined fields and high Q-factors, such as nanolasers [2,7,8], light absorption [9], refractive index sensing [10,11], and harmonic generation [12]. Theoretically, the Q-factor of the BICs mode can reach infinity if the BICs condition is satisfied. Because it has no radiation channels, its output power is typically low. Thus, energy must be emitted via the finite quasi-BICs, which is switched from BICs by breaking the symmetry of the structures. Based on quasi-BICs, one can obtain high Q-factors that contribute to enhanced electric field and light-matter interactions.

Guided-mode resonance (GMR) supported by dielectric gratings has been used to enhance light-matter interactions [13–17]. In the design of low-threshold lasers, resonant optical pumping (ROP) based on the GMR effect has been proposed [18]. The ROP couples the pump light into the resonant mode using the same pump wavelength as the specific resonance wavelength, which results in an enhanced near-field and further pump absorption of the gain medium. Consequently, the energy conversion efficiency increases and the lasing threshold reduces. To date, most nanolasers are based on a single resonance mode in which the resonance cavity is coupled to the pump or emission wavelength [19]. To design a low-threshold laser, excitation coupling and emission coupling are required simultaneously, that is, two resonance modes matching the pump and emission peaks. Such a design requires a structure to support two resonance modes at different wavelengths [20]. Although GMR gratings can excite multimode resonance, the local electric field is usually limited in waveguides composed of high-refractive index films, making it difficult to fully overlap with the gain medium. In addition, to achieve high Q-factors of the GMR, a routine method is to decrease the amplitude modulation in dielectric gratings [21]. However, it will affect the coupling efficiency of the GMR structure, and weaken the interaction between light and matter. Therefore, it is necessary to introduce a new perspective to avoid this paradigm.

In this study, hybrid resonance modes were realized using slanted guided-mode resonance (SGMR) grating. Symmetry-breaking was introduced to transform BICs into quasi-BICs with a radiation channel and high Q-factor in addition to the GMR mode. We theoretically studied the excitation of hybrid modes, and the proposed nanocavity was used to combine the high Q-factor of the quasi-BICs as the resonant-emitting mode and the GMR as the ROP mode to realize low-threshold lasing. There are several methods that can be also used to reduce the lasing threshold, and the SGMR method proposed in this study is based on the following considerations:

To the best of our knowledge, this is the first study to combine the GMR and BICs modes to facilitate light-matter interactions, and it is believed that our findings could provide guidelines for designing low-threshold nanolasers. This is of importance for the development and application of GMR and BICs effects.

2. Quasi-BICs mode in dielectric SGMR grating

The structure of the SGMR grating is illustrated in Fig. 1(a). Briefly, a slanted grating is patterned on a silica substrate and then covered by a layer composed of silica gel and gain medium Rhodamine 6G (R6G) [22]. The structural symmetry is broken owing to the tilt angle, θ. The grating parameters are denoted as follows: Λ represents grating period, f represents the fill factor, and h represents the groove depth. The refractive indices of the upper cover layer, silica gel, grating, and substrate are denoted as n_c, n_L, n_g, and n_s, respectively. In practice, the preparation of a slanted grating refers to the process of developing the diffraction waveguide structure [23,24]. First, using silica as the substrate, a chrome layer (Cr) and resist layer were coated onto the substrate. The process begins with lithography to generate the grating pattern. The resist layer was then transferred to the Cr metal layer using a chlorine dry-etching process. Thereafter, the resist was removed using the oxygen plasma process, and the Cr layer was used as an etching mask for reactive-ion beam etching. During the etching process, the ionized argon beams were oriented to the surface at the desired oblique incident angle θ. When the curtain etching depth was achieved, the Cr metal layer was removed via chemical wet etching. Finally, a slanted grating with a tilt angle of θ was obtained.

 

Fig. 1. (a) Schematic diagram of the SGMR grating with gain medium. (b) Section view of a grating unit. The dark gray areas indicate the grating made of high-index material, and the pink areas are the gain medium made of silica gel impregnated with R6G. The cover layer is air and the substrate is silica.

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The finite-difference time-domain (FDTD) method was used to calculate the spectral characteristics, electric field, and lasing behavior. Figure 1(b) shows the unit cell and its boundary conditions in the model. The boundary conditions were set to be periodic along the x-axis and y-axis to emulate the infinite planar wave and structural periodicity. A perfectly matched layer (PML) was set at the z-axis boundaries to absorb the outgoing electromagnetic field. The pump light is transverse-electric (TE) polarized with an electric vector oriented along the grating grooves (y-axis). To excite the guided mode, the equivalent refractive index should satisfy the following equation [25]:

To gain insight into the SGMR grating, we present the reflection spectra and electric field in Fig. 2 under normal incidence. Asymmetry parameter Δθ = |90^°– θ| was introduced. We started with a vertical grating (Δθ = 0^°) that was symmetrical to the reflection spectra displayed in the upper panel of Fig. 2(a) (black solid line). A wide GMR peak appears at a wavelength of 510.5 nm with a full width at half maximum (FWHM) of 10.2 nm. Because of the vertical profile, the BICs mode is symmetry-protected without a radiation channel. When a slanted grating (Δθ ≠ 0^°) is introduced, that is, a grating with an asymmetric profile, symmetry-broken transforms BICs into a quasi-BICs. As shown in the lower panel of Fig. 2(a) (red solid line), when θ = 86^° and the other parameters remain the same, a reflection peak appears at λ = 570 nm, except for the inherent GMR. Owing to the same period and unchanged fill factor, the two GMR modes have similar resonance wavelengths. The FWHM of the quasi-BICs is 0.16 nm, which is much narrower than that of the GMR mode. This implies that the quasi-BICs has a radiation channel and a finite high Q-factor value, and most importantly, hybrid resonance modes are obtained in the SGMR grating.

 

Fig. 2. Reflection spectra and electric field for symmetrical and asymmetrical grating. (a) Reflection spectra with θ = 90^° (top) and 86^° (bottom). The solid line indicates the results calculated by using FDTD method and the dashed line represents that calculated by using RCWA. Electric field and Poynting vector of GMR with (b) θ = 90^°, (c) θ = 86^°, and (d) quasi-BICs with θ = 86^°.

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Except for the FDTD method, rigorous coupled-wave analysis (RCWA) method is also commonly used to obtain the resonance points accurately. Therefore, in order to determine the accuracy of the results calculate by FDTD, we also utilize RCWA method to plot the reflection spectra. According to the slab waveguide theory [21], the corresponding eigenfunction of the modulated guide for TE polarization can be expressed as

Meanwhile, the effective waveguide index of refraction is given by Eq. (1). By solving the above equations, the resonance points can be obtained accurately. The result is shown in dashed line in Fig. 2(a), and the resonance wavelengths are almost consistent with that calculated by FDTD. There is a negligible difference in sideband due to distinct principle and accuracy.

For the single GMR mode, the electric fields were symmetrical and distributed in the center and groove in opposite directions as shown in Fig. 2(b). The Poynting vector also showed obvious circumfluence. The sign of the field is identical on both sides of the groove, which means that light can be readily scattered by exciting an out-of-plane radiating wave. Another GMR mode showed similar electric-field distributions, as illustrated in Fig. 2(c). As expected, the quasi-BICs is a radiant mode, and its electric field is distinctly asymmetric with respect to the grooves, as shown in Fig. 2 (d). The energy is concentrated on the grating ridges, and the sign of the field on one side of the groove is opposite to that on the other side. This means that the two energies cannot cancel each other out, so the radiation channel is established. By comparison, the quasi-BICs mode is better able to restrict the localized field, and its maximum field intensity is 51.4, which is about 9 times that of the GMR mode with the same structure.

To investigate the influence of asymmetry on the hybrid modes, spectral analysis was executed via continuous tuning of θ, as shown in Fig. 3(a). Once the groove deviates from 90^°, quasi-BICs resonance appears at λ = 570 nm and red-shifts slightly with an increase in the asymmetry parameter Δθ. The GMR wavelength is also red-shifted, but the rate of change of the wavelength with respect to θ is larger than that of the quasi-BICs mode. It should be noted that the spectra were consistent with the same variations in tilt angles. As shown in Fig. 3(b), the reflection spectra with θ = 100^° and 80^° are identical, and similar spectra can be observed from θ = 95^° and 85^°. Using the Q-factor = λ/Δλ, where λ is the resonance wavelength and Δλ is the FWHM, we investigated the FWHM and Q-factor for quasi-BICs and GMR modes for varying values of the asymmetry parameter Δθ, as shown in Figs. 3(c) and 3(d). When θ = 90^°, BICs is a dark mode with zero FWHM and an infinite Q-factor. As Δθ varies, the linewidth of the quasi-BICs increases gradually. The Q-factor increases up to 1.34 × 10⁴ and the FWHM is 0.04 nm when Δθ = 1^°, while the Q-factor is reduced to 275 and the FWHM is increased to 2.1 nm when Δθ = 15^°, i.e., θ = 75^° or 105^°. The larger the asymmetry degree, the farther away it is from the real BICs. Consequently, the Q-factor is reduced, which can be explained by Q∝β^-2, where β is the degree of asymmetry [14]. This means that the relationship between the asymmetry parameter and the Q-factor of quasi-BICs presents an inverse square dependence; thus, the slanted angle cannot be designed to be too large for a high Q-factor. Overall, a high Q-factor (greater than 10³) and a low FWHM (less than 0.5) can be maintained within Δθ = 7^°. In contrast, the Q-factor and linewidth of the GMR are less affected because they are determined by the coupling strength, which is related to the modulation intensity and fill factor [21]. When θ = 90^°, the GMR mode has the narrowest FWHM of 10.1 nm and increases slightly to 11.1 nm when Δθ = 15^°. The highest Q-factor for the GMR mode is 50.5, which is much less than that of the quasi-BICs mode under weak broken symmetry.

 

Fig. 3. (a) Reflection spectra with respect to angle θ and resonance wavelength. (b) Reflection spectra at θ = 100^°, 95^°, 90^°, 85^°, and 80^°. Dependence of the FWHM (black circles and lines) and Q-factor (red circles and lines) on the tilt angle θ for (c) quasi-BICs mode and (d) GMR mode.

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The spectral response is very sensitive to the incident angle; thus, the inclined incidence is also concerned apart from the above-mentioned normal incidence. Figure 4(a) shows the reflection spectra as a function of incident angle φ with θ = 86^°. The resonance wavelength of GMR mode (left) blue-shifts when φ increases; however, that of quasi-BICs mode (right) is gradually red-shifted with an increase in φ. As shown in Fig. 4(b), we also plotted five reflection spectra with φ = 5^°, 2.5^°, 0^°, -2.5^°, and -5^° in which the negative sign indicates that the light is incident from the other side of the normal. The spectra with symmetrical incident angle are the same. In addition, when φ = 1^°, 1.5^°, -1^°, and -1.5^°, the spectra of quasi-BICs show Fano lineshape and the line-width is calculated as the inset of Fig. 4(c). For quasi-BICs, the Q-factor decreases sharply because the resonance peak becomes wider with an increase in φ. The variation depicted in Fig. 4(d) shows that, contrary to quasi-BICs mode, the resonance peak of GMR mode becomes narrower (and then Q increases) when φ increases. However, the Q value of GMR mode is relatively low, and the range of Q value changing with φ is not large. Additionally, the Q value of quasi-BICs becomes very low when φ is greater than 5^° and the reflection efficiency is sharply decreased when φ deviates from 0^°, which is not favorable for reducing the threshold. Therefore, normal incidence is preferred in terms of lower lasing threshold for the proposed structure.

 

Fig. 4. (a) Reflection spectra with respect to incident angle φ and resonance wavelength. (b) Reflection spectra at φ = 5°, 2.5°, 0°, -2.5°, and -5°. Dependence of the FWHM (black circles and lines) and Q-factor (red circles and lines) on the incident angle φ for (c) quasi-BICs mode and (d) GMR mode.

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Next, we investigated the resonance characteristics of the SGMR grating by varying the fill factor f while keeping the angle θ unchanged, as shown in Fig. 5(a). As f increases, the resonance wavelengths of both modes red-shifted simultaneously. It is obvious that the wavelength spacing between the two modes is affected by the fill factor, and it gradually increases with increasing f. Moreover, the line-width of the quasi-BICs remains almost the same, whereas that of the GMR becomes smaller, which can be explained using coupled-wave equations [26]. The coupling strength depends on sin(πf)/π term, which gives the coupling strength a maximum value at f = 0.50; when f > 0.50 or f < 0.5, the coupling strength decreases. Because the coupling strength is proportional to the line-width, it becomes narrower with an increasing of f. Figure 5(b) depicts the spectra of the structures for different fill factors with f = 0.70, 0.65, 0.60, 0.55, and 0.50. As it is shifted from 0.50 to 0.70 with θ fixed as 86^°, two resonance peaks are red-shifted, and the wavelength spacing is increased from 41.8 nm to 63 nm. Meanwhile, the FWHM of the GMR is reduced from 13.5 nm to 7.1 nm, while the quasi-BICs only creates a slight variation of 0.3 nm. The fill factor has an obvious influence on the spectrum, particularly the wavelength spacing between two modes, which can be used to design structures with specific excitation and emission wavelengths.

 

Fig. 5. (a) Reflection spectrum with respect to fill factor f and resonance wavelength. (b) Reflection spectra at f = 0.70, 0.65, 0.60, 0.55, and 0.50.

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3. Lasing behavior based on the SGMR grating

3.1 Lasing behavior enhanced by hybrid-resonance structure

We chose R6G as the gain medium, which can be doped with silica gel to cover the gratings. In the simulations, the gain medium was described using a four-level energy two-electron model, as shown in Fig. 6 [27]. The four levels involved in lasing action include the ground state level E₀, excited state high-energy level E₃, laser upper level E₂ and laser lower level E₁. Electron transitions were treated as two coupled dipole oscillators. Levels 1 and 2 correspond to dipole P₂₁ and levels 0 and 3 correspond to dipole P₃₀. The external light pumps electrons initially staying at level E₀ to high-energy level E₃. After a fast non-radiative transition from the highest level E₃ to the upper level E₂, amplified emissions occur from the upper level E₂ to the lower level E₁. The pumping and emission transitions of electrons can be driven by the product of the electric field and the population inversion. The dynamics of the population densities at every level can be derived from the rate equation model as follows [27]:

 

Fig. 6. Four-level energy two-electron model.

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The lasing action of the SGMR grating with R6G doped in silica gel is shown in Fig. 7. In the simulations, a 4-ps pump pulse with TE polarization centered at 511 nm was set to excite the dye molecules. By adjusting the structural parameters to θ = 86°, Λ = 320 nm, f = 0.6, and h = 220 nm, the hybrid resonance modes of the GMR and quasi-BICs can spectrally overlap the pump and emission peaks of R6G. This is a crucial step for efficient pumping and low-threshold lasing emissions, and will be explained later. Figure 7(a) shows the normalized emission spectra as a function of pump energy. At low intensities (< 24.6 µJ/cm²), the emission spectra exhibit a larger line-width and weak energy. Above the critical pump intensity (> 24.6 µJ/cm²), a sharp and intense emission emerges close to the position of the quasi-BICs (λ = 570 nm). The line-width was ultimately reduced to 0.19 nm. It can be seen that the line-width of the emission spectra narrowed by almost two orders of magnitude, from about 14.5 nm to 0.19 nm. This phenomenon demonstrates that lasing behavior is coherent. As shown in Fig. 7(b), by extracting the maximum emission intensity and the line-width from Fig. 7(a) and plotting them as a function of the pump energy, the curves further present characteristic lasing threshold behavior with a marked change in slope of around 24.6 µJ/cm². It should be mentioned that the threshold value will not make the R6G molecules undergo the photobleaching effect or quenching phenomenon because the pump energy at the laser threshold greater than that we calculated (24.6 µJ/cm²) [8] and the concentration of R6G greater than that we set (3.8 × 10²⁴) [28] have been reported for efficient laser pumping.

 

Fig. 7. Lasing action of the SGMR grating. (a) Normalized input-output of the lasing emission. (b) The maximum output intensity (black circles and lines) and lasing line-width (red circles and lines) as a function of the pump energy.

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As mentioned above, the wavelengths of the GMR and quasi-BICs are set as 511 and 570 nm, respectively, which match the pump and emission wavelengths, respectively. For comparison, structures that possess only a single-match resonance were also designed. Figure 8(a) shows the reflection spectra of the three cases. Case 1^# represents the structure with λ_GMR = 511 nm and λ_quasi-BICs = 554 nm by changing f = 0.5 and Λ = 328 nm, which only makes the GMR mode match the pump peak of R6G. Similarly, case 2^# only possesses quasi-BICs matching the emission peak of R6G (λ_GMR = 527 nm and λ_quasi-BICs = 570 nm with Λ = 338.5 nm). Case 3# is a hybrid-match structure, as shown in Fig. 7. Figures 8(b) and 8(c) illustrate the lasing action for case 1^# and case 2^#, respectively, which show sudden changes in the emission and line-width at a specific pump energy and are all typical stimulated emissions. We plotted the maximum emission intensity as a function of pump energy for the three cases in Fig. 8(d). The thresholds were 91.1, 56.3, and 24.6 µJ/cm², respectively. The structural parameters and resonance wavelengths for the three cases are listed in Table 1. It is evident that the SGMR grating with hybrid-match resonance, that is, case 3^#, has the lowest threshold. In practice, there is difficulty in precise hybrid-match due to the resonance wavelength shift as a result of fabrication error, but it is likely that the shift can be compensated by coating a thin film or adjusting the incident angle after the grating is prepared. In addition, utilizing advanced technologies such as electron beam etching, nanoimprinting, and ion beam etching, can reduce manufacturing error and wavelength shift as much as possible.

 

Fig. 8. (a) Reflection spectra for three cases. Normalized emission as a function of pump energy and wavelength for (b) case 1^# and (c) case 2^#. (d) The maximum intensity as a function of pump energy for three cases.

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Table 1. Structural parameters, resonance wavelengths, and thresholds for three cases

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The reasons for the aforementioned results are as follows. The GMR mode is accompanied by a change in the photonic density of states and can be suppressed at the resonance wavelength, that is, in the forbidden band. The group velocity at the band edge is close to zero, and the density of states is significantly increased, which increases the interaction between the electric field and gain material. The external excitation of the GMR mode with the same wavelength as the absorption peak of R6G and the pumping wavelength leads to the formation of high-intensity near fields that efficiently excite dye molecules. The above is the ROP technique that is commonly used in photonic crystal nanocavities [18]. Additionally, the quasi-BICs mode has a high Q-factor at resonance, which is beneficial for lasing behavior. By designing a quasi-BICs wavelength that coincides with the emission peak of R6G, the emission efficiency can be further improved. Therefore, the doubly high confinement of light and near-field enhancement in the SGMR grating contributes to an obvious reduction in the lasing threshold. When one of the modes fails to match, the enhancement effect is weakened. It is noteworthy that the threshold for case 2^# is smaller than that of case 1^#, and the former has a higher emission intensity, which is mainly caused by the higher Q-factor and stronger electric field under the quasi-BICs mode.

3.2 Influence of asymmetry on lasing action

Asymmetry is an important factor that affects the hybrid resonance modes, especially the Q-factor and FWHM, which in turn influences the lasing action. We investigated the lasing emission of the SGMR grating at different angles, θ. The emission spectra for θ = 84^°, 82^°, and 80^° are illustrated in Figs. 9(a)–9(c), respectively, and the corresponding input-output curves are shown in Fig. 9(d). Because the resonance wavelengths of both GMR and quasi-BICs modes red-shift with an increase in Δθ, to compare the thresholds under the same hybrid-match conditions (λ_GMR = 511 nm and λ_quasi-BICs = 570 nm), the grating periods change accordingly to compensate for the wavelength shift. It is observed that the threshold is increased and the emission intensity is weakened with an increase in Δθ. The threshold increased from 26.8 to 39.3 µJ/cm² when the slanted angle θ decreased from 84^° to 80^°. Therefore, an increase in the degree of asymmetry results in an increased lasing threshold.

 

Fig. 9. Normalized emission intensity as a function of pump energy for different θ (a) 84°, (b) 82°, and (c) 80°. (d) The intensity as a function of pump energy for three structures.

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The threshold increases with decreasing tilt angle, mainly because of the higher radiation loss and weaker localized electric field. To verify this, the reflection spectra and electric field without the gain medium for three structures with different θ are presented. The line-width for quasi-BICs resonance is 0.35, 0.62, and 0.96 nm, as shown in Fig. 10(a), and the relatively intensity of the electric field decreases gradually from 34.7 to 20.9 as shown in Figs. 10(b)–10(d). These changes lead to an increase in radiation loss; hence, the lasing threshold increases. The closer the quasi-BICs mode is to the true BICs, the easier it is to realize a nanolaser with a lower threshold. Considering the GMR modes, because the variations in asymmetry do not affect the fill factor, the coupling strength is unchanged in the grating layer, and the line-width and electric field are almost stable, as shown in Figs. 10(a) and 10(e)–10(g). For providing successive relationship between the asymmetry parameter and the threshold, we calculated the lasing emission by varying θ from 88^° to 76^° and plotted the threshold as a function of θ as shown in Fig. 11. The results are consistent with the above conclusion. When θ = 88^°, the threshold is reduced to 23.8 µJ/cm². Remarkably, the threshold only increases 3.5 µJ/cm² when asymmetry θ < 83^°. However, the value shifts 84.9 µJ/cm² when θ decreases from 83^° to 76^°, which is in accord with the variation of Q-factor in Fig. 3(c). This characteristic also provides fabrication tolerances for preparation.

 

Fig. 10. (a) Reflection spectrum of SGMR structure with different θ values. Electric field distribution of quasi-BICs with θ = 84^° (b), 82^° (c), and 80^° (d). Electric field distribution of GMR with θ = 84^° (e), 82^° (f), and 80^° (g).

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Fig. 11. Threshold of the proposed structure as a function of the angle θ.

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4. Influence of refractive index of substrate layer on lasing threshold

Considering that the SGMR structure consists of a dielectric grating with a high refractive index, the electric field profile mainly resides within this region, and exponentially decays into its low-refractive-index region that is filled with a gain medium. Because the electric field intensity has an immediate effect on the lasing threshold, we adjusted the refractive index of the substrate to optimize the electric field and to further improve the lasing threshold.

Figure 12(a) shows the reflection spectrum of the SGMR structure with n_s = 1.00, 1.20, and 1.51, where n_s = 1.00 represents a suspended GMR structure, n_s = 1.20 low-refractive-index mesoporous SiO₂, and n_s = 1.51 BK7 glass. The above substrates are commonly used in GMR structures and have been discussed in previous studies [13,29]. When θ = 86^° and ω₃₀ is set as the resonance frequency of the GMR mode of each structure, the structure can meet the requirements of hybrid-match conditions for lasing emission. Figures 12(b)–12(d) show the electric field of the quasi-BICs mode for different values of n_s. The electric field intensity of the quasi-BICs mode decreases with an increase in n_s and the highest intensity is obtained in the suspended SGMR structure, that is, n_s = 1.00. An increase in the electric field intensity indicates a stronger nanocavity, which enhances the interaction between light and matter. In contrast to the quasi-BICs, the electric field intensity and distribution of the GMR mode did not change significantly as shown in Figs. 12(e)–12(g). Additionally, the electric fields of all modes move from the substrate to the grating. This extended penetration is beneficial for the interaction between the localized field and gain medium.

 

Fig. 12. (a) Reflection spectrum of SGMR structure with different n_s. Electric field distribution of quasi-BICs with n_s = 1.00 (b), 1.20 (c), and 1.51 (d). Electric field distribution of GMR with n_s = 1.00 (e), 1.20 (f), and 1.51 (g).

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We also investigated the lasing action corresponding to the SGMR grating above. The lasing behavior can be observed in Figs. 13(a)–13(c) for n_s = 1.00, 1.20, and 1.51, respectively. Compared to the results for n_s = 1.46, as shown in Fig. 7, the maximum intensity as a function of pump energy is illustrated in Fig. 13(d). The threshold decreases from 25.3 to 20.4 µJ/cm² as n_s decreases from 1.51 to 1.00 because the localized energy in the gain medium increases. Therefore, n_s is a critical parameter that affects the electric field intensity and distribution and is helpful to further adjust the lasing threshold of the SGMR laser.

 

Fig. 13. Normalized emission intensity as a function of pump energy and wavelength for different n_s (a) 1.00, (b) 1.20, and (c) 1.51. (d) The maximum intensity as a function of pump energy for the structures with different n_s.

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5. Conclusions

In summary, we provide a novel approach to achieving a low-threshold laser by employing an SGMR grating that supports hybrid resonance modes, that is, GMR and quasi-BICs. When the structure possesses the wavelengths of the GMR and quasi-BICs modes at the pump and emission wavelengths of the gain medium, the threshold can be significantly reduced because of the double enhancement from the resonant optical pumping and high Q-factor nanocavity. In addition, by optimizing the asymmetric parameters and electric fields, the lasing behavior can be controlled and the lower threshold of 20.4 µJ/cm² can be obtained. Our work reveals the potential of a GMR structure with hybrid resonance modes for low-threshold nanolasers and serves as a reference for the design of other light-emitting optoelectronic devices.