Enabling low threshold laser through an asymmetric tetramer metasurface harnessing polarization-independent quasi-BICs

Fig. 2a shows the mapping of the reflection spectra while changing the radius $r$ from 0 to 150 nm (correspondingly, $0\leq\alpha\leq 0.58$) while keeping R constant at 260 nm. When $\alpha=0$, the reflection spectrum in the range of 1400 to 1600 nm is close to zero, therefore presenting a high transmittance (none of the adopted materials is lossy). This result can be attributed to the presence of SP-BICs, which display resonances with infinitely narrow linewidth Zhang2018 ; Doeleman2018 . This is further confirmed by the observation that, as $\alpha$ increases, the symmetry of the structure is disturbed and two resonances gradually appear in the reflection spectrum. This particular phenomenon can be attributed to the coupling of bound states to radiative channels, resulting in the excitation of two quasi-BICs. These modes have been marked by QBIC₁ and QBIC₂, the latter with a longer emission wavelength.

Interestingly, we notice a blue-shift of both BIC₁ and BIC₂ upon increase of $\alpha$, possibly due to the decrease of the effective refractive index of the structure Zhou2018 ; Lustig2023 . At the same time, their line widths become larger, indicating the degradation of the Q-factors of these two quasi-BIC modes. Finally, when the size of the air hole is increased even further, these two modes merge into one another, resulting in a phenomenon known as “degeneracy”. This implies that the characteristics of the two individual modes become indistinguishable or interchangeable. This merging of the modes can occur due to the increasing similarity in their respective wavelengths, allowing them to overlap and share common features Wang2020 . Fig. 2b shows the representative reflection spectra of the metasurface of the tetramer when $\alpha=$0 (black line), 0.15 (pink line), 0.32 (blue line) and 0.58 (green line), which confirms the analysis of Fig. 2a.

To confirm the resonance and calculate the Q-factor of BIC modes $Q=\omega_{0}/2\gamma$, we employ the conventional Fano formula Saadatmand2023 :

Fig. 2e and f show the dependence of the Q factor of the two quasi-BIC modes on the asymmetry parameter $\alpha$ (in log-log scale). The Q factors of these two resonant modes become very similar when $\alpha>0.4$, suggesting a mode degradation possibly leading to a loss of confinement. This is confirmed by the green line in Fig. 2b ($\alpha=0.58$), clearly showing two very broad linewidths associated to the two resonance modes. For this reason, here we shall focus on the region where $\alpha<0.4$. Furthermore, the two figures further stress the enhanced radiation loss occurring for high alpha values, by showing a steady Q factor decrease upon $\alpha$ increase. When $\alpha=0$, both modes are BIC resonances and exhibit infinite Q factors. When a degree of asymmetry $\alpha=0.1$ is introduced, the Q factors of both QBIC₁ and QBIC₂ remain high with values above $10^{4}$. However, for a larger degree of asymmetry $\alpha=0.2$, the Q factor substantially decreases to $10^{3}$, suggesting an augmented radiation loss of the tetramer metasurface. Furthermore, our study reveals that the Q factors of both modes exhibit a clear dependence on $\alpha$ following an inverse quadratic law when $\alpha>0.2$, which is in line with the findings of previous studies Koshelev2018 ; Liu2019 , hence suggesting to be in the presence of SP-BICs.

Fig. 3 shows the simulated intensity distributions of the electromagnetic field of QBIC₁ and QBIC₂ in the ($x$, $y$)-plane, providing a clear visual representation of the spatial distribution of both modes under asymmetric conditions ($\alpha=0.15$). This visualization provides a better understanding of the resonances QBIC₁ and QBIC₂. As shown in Fig. 3a, for the QBIC₁ mode we observe high electromagnetic intensity especially within the InGaAsP disks, with a distribution clearly resulting from a x-polarized impinging radiation. Conversely, for the QBIC₂ mode, the electromagnetic radiation is primarily concentrated within the air holes (highlighted by dashed lines) and in the air region surrounding the InGaAsP disks. As already depicted in Fig. 2a, this results in the QBIC₁ resonance being blue-shifted compared to the QBIC₂ resonance (InGaAsP refractive index $>1$) Vaity2021 ; Algorri2024 .

To better understand the properties of QBIC modes, the cartesian multipole decomposition method is employed to compute the scattering power for diverse multipole excitations. By integrating the displacement current density $j(r)$ across the unit cell, a comprehensive portrayal of the far-field responses of electromagnetic sources is achieved. Subsequently, for a specified frequency, the total far-field scattered power is determined for all multipoles. Among these, the eight multipoles including the electric dipole (ED), the magnetic dipole (MD), the magnetic toroidal dipole (MTD), the electric toroidal dipole (ETD), the electric quadrupole (EQ), the magnetic quadrupole (MQ), the electric toroidal quadrupole (ETQ), and the magnetic toroidal quadrupole (MTQ), are defined by the following equations respectively Saadatmand2023 ; Tang2017 :

The multipole decompositions of the structure for alpha=0.15 are depicted in Fig. 4. Specifically, Fig. 4a illustrates that for QBIC₁ mode, the MQ is the dominant multipole, accompanied by partial contributions from the TD and MD. Notably, the contributions from other higher-order multipoles are significantly suppressed in this instance. Analogously, QBIC₂ mode presents a similar property, with the MTQ remaining the dominant multipole and the TD contributing partially. In this case as well, the contributions from other higher-order multipoles are negligible, as shown in Fig. 4b.

The multipole analysis provides insights into the underlying mechanisms responsible for the dark mode behaviour associated with QBIC₁ and QBIC₂ at $\alpha$ = 0 and for small $\alpha$ values (see Fig. 2a). Specifically, from Fig. Fig. 4c, the evolution of the electric dipole (ED) multipole to QBIC₁ and QBIC₂ with changing $\alpha$ is evident. The analysis reveals that the ED multipole increases with $\alpha$, which explains why a linearly polarized incident radiation cannot effectively excite the structure at $\alpha$ = 0 or for small $\alpha$ values, as observed in Fig. 2a. This is because linearly polarized radiation corresponds to an ED-like mode. Considering that, in addition to energy and momentum parallel to the structure surface, the symmetry of the incident radiation must also be conserved during mode transformation (such as when an incident radiation excites a mode), it becomes apparent that a linearly polarized radiation can best excite ED modes Algorri2022 . In turn, this means that a linearly polarized radiation can properly couple to the structure shown in Fig. 1 only at high $\alpha$ values. This finding is confirmed by Fig. 4d, where the evolution of the MQ multipole is shown. The results say that at $\alpha$ = 0 the main multipole is MQ (see Fig. 4a and Fig. 4b), a contribution that decreases by increasing $\alpha$, exactly the opposite found for the ED multipole.

In order to gain an even deeper understanding of the impact of the asymmetry, we further conduct an analysis on the electromagnetic field distributions of the QBIC₁ and QBIC₂ modes in ($x$, $y$)-plane for the structures with air hole size of $\alpha=$ 0.08, 0.23, and 0.39. As shown in Fig. 5a-c, we find that the maximum intensity of QBIC₁ mode decreases with the size of the air holes (similar behaviour occurs also for QBIC₂, as shown in Fig. 5d-f). The decreasing trend can be attributed to the reduction in the factor $Q$ of the QBIC₁ mode as already observed in Fig. 2c and d. Indeed, a lower $Q$-factor implies a weaker confinement for the electromagnetic field within the structure, thus resulting in the observed weakening of the field intensity. Furthermore, when comparing Fig. 5c and f, it is observed that the two resonances tend to converge towards a similar pattern, namely QBIC₁ and QBIC₂ somehow merge in a unique kind of mode, a behaviour matching with the results of Fig. 2a.

To verify the polarization independent behaviour of the quasi-BIC resonances in the metasurface, it was calculated their reflection spectra with different excitation angles and for $\alpha=$ 0.15. The direction of polarization of the incident linearly polarized beam is defined by the angle ($\theta$) between the incident electric field and the $x$-axis (see Fig. 6a). As shown in Fig. 6b, when the incident beam is polarized along the $x$ axis (i.e., $\theta=0^{\circ}$), the resonant wavelengths of QBIC₁ and QBIC₂ are 1512 nm and 1557 nm, respectively. As the polarization angle $\theta$ varies from $0^{\circ}$ to $90^{\circ}$ (see dashed arrow in the figure), the resonant wavelengths of both quasi-BIC modes remain nearly unchanged. This observation clearly indicates that the proposed metasurface shows polarization insensitivity when exciting the quasi-BIC resonances. This unique characteristic can be exploited to reduce, if not eliminate, the need for polarization calibration, thus offering greater flexibility for practical applications such as wireless communication systems, radar technology, and optical sensing Cui2024 ; Zhang2023 .

Lasing action based on BICs or quasi-BICs in two-dimensional metasurface structures has recently garnered significant attention in the field of photonics Kodigala2017 ; Zhang2022 ; Heilmann2022 . These unique structures exhibit remarkable properties that enable the confinement and manipulation of light at the nanoscale, leading to potential advancements in lasers and optical devices Pavlov2018 . In the present context, the selected structure for realizing lasing from both the quasi-BIC modes is defined by $\alpha$ = 0.15. Furthermore, since InGaAsP is a natural gain material, the imaginary part of its refractive index is considered through all the numerical simulations.

The gain medium can be well described as a four-energy-level system, and a semi-quantum framework is adopted to simulate the interaction between the electromagnetic fields and gain medium. Details about the population dynamics of each energy level within the gain medium, including the ground state and the three excited states, can be found in the Ref. Di2023 ; Zhang2022 . Similarly to the calculation of the quasi-BIC modes, the FDTD method is employed also for simulating the lasing effect, where a 4 ps plane wave pump pulse is deployed on the metasurface along $z$-direction. The detailed parameters used in the simulation are described in Di2023 .

Fig. 7 shows the gain spectrum of the designed metasurface (black curve), the QBIC modes reflection spectra (blue curve) and their lasing emission (red curve), the latter being red-shifted against the reflection spectra due to the quantum efficiency of the gain medium. It is evident that the gain spectrum exhibits a wide linewidth, and its central region overlaps the peaks associated to the QBIC modes. So, this characteristic can guarantee that cavity losses are effectively compensated, enabling optimal laser operation Wang2017 ; Sun2021 .

Fig. 8 shows the normalized (i.e., divided by the maximum output value) input-output function curves of the QBIC₁ (a) and QBIC₂ (b) modes, providing a comprehensive understanding of their characteristic lasing behaviors. From these results, it is clear that the QBIC₁ mode has a significantly lower lasing threshold compared to the QBIC₂ mode, indicating that the QBIC₁ mode requires substantially less input power to initiate the lasing action. In addition, both modes exhibit a consistent and proportionate growth with input power within the specified power range. This implies that the output of the system experiences a steady and uniform increase as the input power surpasses a certain threshold. The linear increment in output power enhances the overall performance of both modes and reaffirms their reliability and effectiveness in practical applications.

Fig. 9 presents the variation of lasing emission peaks with respect to pump power. These spectra provide valuable insights into the dynamic evolution of the lasing behavior of QBIC₁ and QBIC₂ modes. Specifically, when the laser operates below threshold at a pump power of $4.5\times 10^{5}$ V/m, no discernible mode is observed. As the pump power increases to $8.0\times 10^{5}$ V/m, the peak describing the lasing effect associated to the QBIC₁ mode can be clearly observed, whereas no lasing effect associated to the QBIC₂ mode is noticed due to the insufficient input power. The result is consistent with input-output function curves in Fig. 8. As the pump power is further elevated to $1.5\times 10^{6}$ V/m, the QBIC₂ mode starts lasing as well, yet the QBIC₁ mode remains the dominant mode, with its intensity continuously increasing and linewidth steadily decreasing (as indicated by the blue curve, representing the spectrum of the device at a pump power of $1.5\times 10^{6}$ V/m). Finally, when the pump power reaches $2.5\times 10^{6}$ V/m, the intensity of the QBIC₂ mode surpasses that of the QBIC₁ mode, signifying a mode-switching event.