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Abstract
High-Q factor can enhance the interaction between light and matter, which is an important parameter to decrease the threshold of nanolasers. Here, we theoretically propose an eccentric nanoring structure with a high and controllable Q factor to realize a low-threshold and controllable nanolaser by amplifying the quasi-bound states in the continuum (quasi-BIC). The designed nanostructure supports a quasi-BIC because of the symmetry protection-breaking of the nanostructure. The quasi-BIC has a very high Q factor of about 9.6×104 and can also be adjusted by changing structural parameters. We use the energy level diagram of the four-level two-electron system to study the lasing action of the eccentric nanoring structure. The results show that the nanolaser has a relatively low threshold of about 6.46 μJ/cm2. Furthermore, the lasing behavior can be tuned by controlling the structural parameters of the eccentric circular ring structure.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
With the development of science and technology, miniaturization and integration of photonic devices (including lasers) are indispensable. As a coherent light source of optoelectronic-integrated chips, nanolasers have been widely used in optical communication, biomedical and other fields [1,2]. The types of nanolasers include band-edge surface-emitting lasers [3], VCSELs (Vertical-Cavity Surface-Emitting Lasers) [4–6], microdisk lasers [7,8], photonic crystal lasers [9–11], etc. They have relatively lower threshold because of the high quality factors (Q factors) of the structure.
High-Q factor is an important parameter to decrease threshold of nanolasers. Higher-Q factor indicates that the micro-cavity has a strong ability to restrict light, which yields a higher density of states, in turn, decreases the threshold of lasing action [3]. Some applications of reducing nanolaser threshold by increasing Q factor have been proposed and demonstrated in the past few years [12,13]. Z. Zhang et al. demonstrated Q factor could be flexibly tuned by changing the filling factors of compound waveguide dielectric grating, so as to tune the lasing threshold [3]. Y. Xiao et al. reported a low-threshold microlaser in a high-Q asymmetrical microcavity [13]. Some research used the dark mode with higher Q factor to reduce the lasing threshold of nanolaser [14–17]. However, the Q factor of these nanostructures only can be tuned in a limited range.
Recently, the hot research on quasi bound states in the continuum (quasi-BIC) of all-dielectric materials demonstrated it has very high Q factor and very narrow line-width, which can enhance the interaction between light and matter [18–21]. It is widely used in biosensors, high harmonic generation and so on [22–27]. Quasi-BIC can also be used in nanolaser. A. Kodigala et al. reported lasing action from an optically pumped BIC cavity at room temperature [28]. S. T. Ha et al. demonstrated directional lasing with a lower threshold and high-Q factor in active dielectric nano-antenna arrays [29]. Huang et al. introduced a vortex microlaser used topological symmetry breaking of perovskite structure [30]. However, these quasi-BIC of the micro/nano-laser nanostructure mentioned above are influenced by the electromagnetic coupling (periodic effect), and they have higher requirements on the period of periodic structure. In 2020, V. Mylnikov et al. experimentally demonstrated a nanolaser based on quasi-BICs of a single semiconductor nanocylinder that took advantage of the destructive interference between Fabry-Perot and Mie modes, which has lower requirements on the period of periodic structure [26]. However, the Q factor of this quasi-BIC mode can only be adjusted in a limited range.
In this paper, we theoretically demonstrate a nanolaser based on quasi-BIC mainly influenced by the individual dielectric resonator. The Q factor of quasi-BIC in our nanostructure can reach about 9.6×104 and can be flexibly tuned in a wide range. Combined with a gain medium, we study the lasing action of the eccentric nanoring structure. The results show that not only does the nanolaser have a relatively low threshold about 6.46 μJ/cm2, but also the lasing behavior can be tuned by controlling the structural parameters of the eccentric circular ring structure.
2. Numerical structure
Figure 1(a) shows the schematic diagram of the periodic eccentric nanoring structures, which are formed by a solid disk with a decentered penetrating hole. The center of the hole shifts an offset d from the center of the disk along the x-axis. The radii of every circular disk and hole are R1 and R2. The height of the nanoring is h = 200 nm. The nanoring structures are deposited on a silica substrate with a refractive index n = 1.5. In order to satisfy the resonant conditions, a high refractive index contrast between the nanoring and matrix must be ensured. We choose the refractive index of the nanoring as n = 3.3, such as indium phosphide (InP), silicon, germanium and other semiconductor materials [27,31–33]. The eccentric nanorings are arranged periodically with a period of Px = Py = 600 nm. The incident light propagating along the z-axis is polarized and parallel to the x-axis. In the following calculations, we use the finite-difference time-domain (FDTD) to accurately calculate the spectral characteristics and laser behavior of the periodic eccentric ring nanostructures.
Fig. 1. (a) Structure diagram of the periodic eccentric ring on a silica substrate. $\vec{k}$, $\vec{E}$ and $\vec{H}$ are the wavevector, electric and magnetic field of the incident light, respectively. (b) Reflected spectrum of the eccentric ring nanostructure with the offset of d = 0 nm (red dashed line) and d = 5 nm (black solid line), the other structural parameters are R1 = 220 nm and R2 = 90 nm. (c-h) Represent cross-section patterns of electric field and current distribution (red arrow) at their resonance wavelength in the nanostructure with offset d = 0 (c-e) and 5 nm (f-h) in x-y plane (c, f), z-x plane (d, g) and y-z plane (e, h).
3. Results and discussion
Figure 1(b) shows the reflection spectrum of the eccentric nanoring structure with different offset d = 0 nm and d = 5 nm. When d = 0 nm, i. e. the nanoring is symmetrical in every plane, there is only a broad spectrum appears, which is a bright mode of the nanoring. The electric field distribution and the current distribution of this bright mode are shown in Fig. 1(c)-(e). It can be seen that the current distributions of this resonance mode are two symmetrical ring currents in y-z plane and a single symmetrical ring current in x-z plane. However, when the center of the hole shifts a small distance along x-axis, a narrow transmission peak appears on the broad spectrum. This is a quasi-BIC mode of the eccentric nanoring caused by the symmetry protection-breaking. The electric field distribution and the current distribution of the quasi-BIC mode of the nanoring with offset d = 5 nm are shown in Fig. 1(f)-(h). It's obvious that the currents in x-z plane have become two asymmetrical ring currents, which caused by symmetry breaking of the nanoring in x-axis. When there is an offset in the y-axis, the narrow transmission peak doesn't appear, i.e. there is no quasi-BIC appears. Because the current distribution in y-z plane can still be two ring currents, which will not be destroyed by the destruction of structural symmetry when the incident light is polarized in x-axis.
We study the effect of the offset d, the radius of disk R1 and the radius of hole R2 on the quasi-BIC mode in Fig. 2. In Fig. 2(a), the radii R1 and R2 are set as 220 and 90 nm, respectively. Once the hole center deviates from the center of the disk in the x-axis, the quasi-BIC immediately appears in the reflection spectrum. The smaller d, the closer to the real-BIC, the smaller the radiation loss [27]. As the offset d increasing, the line-width of quasi-BICs increasing gradually. Because once the symmetry is broken, BIC becomes a radiant bright mode (i.e. quasi-BIC), which has a larger radiation loss. Remarkably, the resonance wavelength of the quasi-BIC keeps almost the same when offset d < 5 nm. When d > 5 nm, resonance wavelength blue shifts only 3 nm when d increases from 5 nm to 30 nm, as shown by the red dotted line in Fig. 2(a), which are caused by the almost same effective volume of the cavity.
Fig. 2. (a-c) Reflection spectrum of the eccentric nanoring as a function of wavelength for different offset d (a), different hole radius R2 (b) and different disk radius R1 (c). (d) Q factor as a function of the offset d and radius of hole R2. Inset: the diagram of line-width calculation method.
Figure 2(b) shows the reflection spectrum of the eccentric nanoring changed as the hole radius R2. The offsets d and R1 of the eccentric nanoring structure are set as 20 and 220 nm. It can be seen that with the increasing of the hole radius, the resonance wavelength of the quasi-BIC blue-shifts from 973.3 to 920 nm, as shown by the red dotted line in Fig. 2(b). For the resonance wavelength of nanocavity λ ∝ V, V is the effective volume of the cavity [34]. When the hole radius increases, the effective volume of the cavity decreases, which supports the resonance mode with a shorter resonance wavelength. Furthermore, the line-width of the quasi-BIC also gradually increases as the hole radius R2 increasing. When the hole radius is increased to 130 nm, the line-width will increase to 4 nm. Because when the offset d keeps constant and the radius R2 increases, the asymmetry degree of the structure increases. By the way, there appears another narrow resonance peak in the reflection spectrum when R2 ≤ 70 nm, as marked by the blue arrow. They originate from the other different quasi-BIC mode of the eccentric nanoring structures.
The Q factor of the quasi-BIC in Fig. 2(a) and (b) are calculated in Fig. 2(d), where Q = λres / △λ, λres and △λ is the resonance wavelength and the line-width (which is calculated as the inset of Fig. 2(d) shown) of the quasi-BIC. The Q factor can be tuned from 367 to 9.6×104 when the offset d increases from 1 to 30 nm. When R2 increases from 50 to 130 nm, the Q factor only can be tuned from 230 to 1947. It can be seen that the closer the quasi-BIC to the true BIC, the higher Q factor. Thus, the eccentric nanoring is very suitable to realize a nanolaser with a lower threshold.
Figure 2(c) shows reflection spectrum of the eccentric nanoring changed as the disk radius R1. The offsets d and hole radii R2 are set as 10 and 90 nm. When R1 increases from 210 to 260 nm, the wavelength of the quasi-BIC red shifts from 928 to 1068 nm, as shown by the red dotted line in Fig. 2(c). It proves that the eccentric nanoring structure has good tuning performance.
The electric field distribution of the eccentric nanoring structure at the respective resonant wavelength is plotted in Fig. 3. The electric field distribution in Fig. 3(a)-(c) corresponds to the quasi-BIC mode in Fig. 2 (a) with the offsets d of 5, 10, and 30 nm. The electric field intensities and the electric field localized in the cavity of disk decrease as the offset d increasing. As a result, the radiation loss increases with the increase of d. Figure 3(d)-(f) shows the electric field distribution of the quasi-BIC in Fig. 2(b) with R2 of 50, 90, and 110 nm. The electric field intensities and the electric field localized in the cavity of disk also decrease as R2 increasing. And the radiation loss also increases with the increase of R2. According to the analysis above, the asymmetry increases with the increase of the offset d and the hole radius R2, which will cause the higher radiation loss and the weaker electric field intensity localized in the cavity. However, the radius R2 variation has relatively small adjustion on radiation loss. Thus, the closer the quasi-BIC mode is to true BIC, the easier it is to realize a nanolaser with lower threshold.
Fig. 3. Electric field distribution in x-y plane of the eccentric nanoring structure with different offsets d (a-c) and hole radii R2 (d-f), respectively. The other parameters: (a-c) R1 = 220 nm, R2 = 90 nm. (d-f) d = 20 nm, R1 = 220 nm.
We choose III-V semiconductors Indium Phosphide (InP) [35] as the gain materials because there is very little absorption in the near-infrared region. We use the energy level diagram of the four-level two-electron system to verify the lasing action of the nanoring structure. Here, the four-level two-electronic system to describe the time evolution of the total density of the gain material, the rate equations are as follows [36]:
Where N1, N2, N3 and N4 are the electron population density corresponding to 0, 1, 2 and 3 energy levels, respectively. τij is the decay time between levels i and j. ωa and ωb represent frequencies of the transitions between levels 1 and 2, 0 and 3, respectively. $\overline {{P_a}} $ and $\overline {{P_b}} $ correspond to the net macroscopic polarization resulting from transition from level 1 to level 2 and level 0 to level 3. And finally, $\bar{E}$ represents the total electric field. In this paper, the material parameters are obtained according to the Ref. [36]. The decay time is of τ21 = τ30 = 300 ps, τ32 = τ10 = 100 fs and total molecular density is of N0 = 5×1023 m-3, ωa = 1.97×1015 Hz and ωb = 2.248×1015 Hz.
Figure 4 calculates the normalized emission spectrum of the eccentric nanoring structure with d = 5 nm, R1 = 220 nm and R2 = 90 nm as a function of the incident pump fluence and wavelength. The resonance wavelengths still appear at about 960 nm. At the beginning, the resonance peak intensity increases slowly with the increase of the pump fluence. And the line-width keeps nearly unchanged, it is still a broad spectrum. When the pump fluence 6.46 μJ/cm2, the spectral line suddenly becomes very sharp and the intensity suddenly increases. This is typical stimulated emission behavior in a laser system. This pump fluence is defined as the threshold of the nanolaser, which is clearly shown in Fig. 4(b), where we plot the normalized maximum emitted intensity as a function of the input pump fluence. This lasing threshold is lower than that of the other nanolasers. Table 1 summarizes the lasing threshold of several different nanolasers. It can be seen that the lasing threshold of our proposed nanolaser is two or three orders of magnitude lower than that of nanolaser based on the surface plasmon [37–39], several tens of times lower than that of the nanolaser based on the quasi-BIC of a single semiconductor nanocylinder [26], and even lower than that of the DFB nanolaser [40]. In order to further indicate the behavior of the lasing, we also plot the line-width as a function of pump fluence in Fig. 4(b). It can be seen that the line-width of the emission spectra narrowed by almost two orders of magnitude, from about 10 nm to 0.2 nm. This phenomenon also clearly demonstrates that is a coherent lasing behavior. Figure 4(c) and (d) show the electric field distribution of the nanoring at the pump fluence is less than and greater than the threshold fluence, respectively. When the pump energy is less than the threshold, the quasi-BIC cannot be excited at all. However, when the pump energy is greater than the threshold, the electric field of the quasi-BIC is excited and can be amplified about 1.24×105 times compared to the electric field of the nanostructure with no gain.
Fig. 4. (a) Lasing actions of the eccentric nanoring structure with R1 = 220 nm, R2 = 90 nm, d = 5 nm. Evolution of the normalized emission intensity as a function of pump fluence and wavelength. (b) Emission spectra line-width and the maximum intensity as a function of pump fluence. (c-d) Electric field distributions when pump fluence is of 6.1 μJ/cm2 and 7.7 μJ/cm2.
Table 1. A comparison of the threshold of several nanolasers
The Q-factor of the quasi-BIC decreases as the increasing of offset d and the hole radius R2 as shown in Fig. 2, which will influence the lasing behavior of nanoring structures. We investigate the lasing behavior of nanoring structures with different offsets d and R2 in Fig. 5, which plots the maximum emission intensity as a function of pump fluence. The resonant wavelength of the quasi-BIC shifts only a few nanometers as the changing of the offset d. The parameters setting of the gain material is the same as the Fig. 4. However, the resonant wavelength of the quasi-BIC shifts a lot as the changing of R2. When investigated the lasing behavior of nanoring structures as the changing of R2, ωa is set at the quasi-BIC resonance frequencies of every eccentric nanoring structure, the other parameters are also set the same as the Fig. 4. It can be seen that the curves have the same characteristic. They are all typical of stimulated emission behavior. However, the threshold value increases and the maximum emission intensity decreases as the offset d and the hole radius R2 increasing, which are caused by the larger the radiation loss and the weaker electric field of the eccentric nanoring structure with the larger d and R2. It's just that the threshold value and the maximum emission intensity controlled by R2 is relatively small because of the small modulation on Q factors of R2.
Fig. 5. Maximum emission intensity of the eccentric nanoring structure with different offsets d, R1 = 220 nm, R2 = 90 nm (a) and with different hole radii R2, R1 = 220 nm, d = 20 nm (b) as a function of the input pump fluence.
4. Discussion
The reflection spectra of the quasi-BIC resonances above are calculated by ignoring the imaginary part of the dielectric constant. Considering the low absorption of semiconductor materials in the near-infrared region, the imaginary part of the dielectric constant is considered in the example of InP in Fig. 6 [41]. The geometric parameters are d = 5 nm, R1 = 220 nm and R2 = 90 nm. When we set the imaginary part of InP refractive index Imag (n) = 3.6×10−5, the resonance wavelength of the quasi-BIC basically keeps constant, but the line-width increases from 0.115 to 0.15 nm, meaning the absorption loss increases, as shown in Fig. 6(a). The maximum emission intensity as a function of pumping fluence is also compared in Fig. 6(b). The eccentric nanoring still has the typical of stimulated emission behavior in a laser system when we consider the imaginary part of the material. However, the threshold value of the eccentric nanoring nanolaser increase from 6.46 to 6.63 μJ/cm2, and the maximum emission intensity also decreases slightly.
Fig. 6. Comparison of reflection spectrum (a) and the lasing actions (b) of InP eccentric nanoring structure with imaginary part of refractive index Imag (n) ≠ 0 and Imag (n) = 0.
5. Conclusions
In summary, we demonstrated a lasing action based on the quasi-BIC mode of the eccentric nanoring structure in theory. The eccentric nanoring structure supports a quasi-BIC caused by symmetry-breaking of the nanoring. The quasi-BIC has very high Q factor of about 9.6×104, which can be adjusted by changing the offsets d and the hole radius R2. Combined with a gain medium, we use the energy level diagram of the four-level two-electron system to study the lasing action of the eccentric nanoring structure. The results show that the nanolaser not only has a relatively low threshold, but also the lasing behavior can be tuned by controlling the structural parameters of the eccentric nanoring structure. Besides our nanostructure does not suffer from large intrinsic losses due to all-dielectric configurations.
Funding
National Natural Science Foundation of China (11504209, 11404195, 91950106, 11674199); Natural Science Foundation of Shandong Province (ZR2019MA024).
Disclosures
The authors declare no conflicts of interest.
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