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Abstract
We reveal the potential of bound states in the continuum (BIC) to enhance the nonlinear response in specialty optical resonators in the presence of gain and loss. We demonstrate this phenomenon in a square core–shell AlGaAs nanowire having a proper engineered spatial variation of gain and loss to sustain quasi–BICs. The presence of these high-quality modes at both fundamental and second-harmonic wavelengths leads to an extremely high enhancement in second harmonic generation, thus preluding a framework to fabricate composite media with high effective nonlinearity.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The field of engineering photonic systems towards enhanced and efficient nonlinear response has attracted tremendous attention since the dawn of nonlinear optics. Metal–less platforms [1,2], owing to their advantages in terms of near-zero dissipation and strong magnetic multipole resonances [1], play a critical role in linear and nonlinear optics for several applications, such as, for example, ultrashort pulse generation [3], processing of quantum information [4], directional dependent scattering [5] and vibrational spectroscopy [6]. Several works have been devoted to the understanding of nonlinear effects in a variety of dielectric media with different operation principles [2,7–11]. Contrary to the situation which takes place in bulk media [12–14], using nanoantennas, it is possible to enhance the nonlinear response by increasing the parameter Q/V, i.e. the ratio of the quality factor (Q-factor) over the volume V of a given mode. Super-resonances, aiming to enhance Q/V, have been identified even in simple isolated or coupled cavities [15–19].
Avoided resonance crossing (ARC) can play a critical role to identify modes with a high Q-factor such as quasi–bound state in continuum (BIC) [20–24]. This phenomenon has been studied in detail in various structures including hexagonal shaped microcavites [22] and one-dimensional Fabri-Perot type optical resonators [21] with adequate gain and loss, suggesting its robustness and generality [19,25,26]. A BIC is an ideal eigenmode without any leakage radiation outside the cavity and hence has an infinite quality factor [27,28]. Although a true BIC is a mathematical abstraction, it has been shown that photonic cavities can support resonant modes in strong coupling regime whose radiated field almost completely suppress each other [17–19]. This condition results in one of the two modes to evolve in a high-Q mode. Since, this mechanism is similar to the BIC physics described in [29], such high-Q modes are called a quasi–BIC. Physical realization of a BIC is still a field of extensive research and analysis [19,26,30,31]. Recently, the existence of a BIC was demonstrated using a pair of quantum dots coupling to their respective reservoirs [30].
In this paper, we show that it is possible to engineer gain and loss in a simple resonator and obtain record high nonlinear response driven by quasi–BIC at both the pump and emission wavelengths. In the presented work we look out to engineer the structure where there is no interference between Mie and Fabry Perot modes, like reported in [31]. The novelty of our approach is that is the presence of gain and loss, that under specific topologically parameters optimization, create two resonances that undergo the mechanism of avoided resonance crossing and thus the formation of a quasi-BIC mode. Avoided resonance crossing are first observed to guide the design of the system; with the proper tuning of gain and loss within the resonator so as to sustain the quasi–BIC, we show that the enhancement of conversion efficiency of second-harmonic generation (SHG) is interlinked with the presence of the quasi–BIC at both fundamental and harmonic wavelength.
The paper is divided into 3 sections. In Section 2, we exploit the concept of ARC to identify quasi–BIC and present an example of a core-shell nanowire with gain and loss engineered to sustain quasi-BIC at multiple wavelengths. In Section 3, we demonstrate the enhancement in SHG efficiency that can be obtained for an overall lossy structure. Finally, in Section 4, conclusion and some future prospectives are given.
2. Using ARC to identify quasi–BIC modes sustained by gain and loss
Provisionally, the nature of coupling between any two proximity resonances can be analysed by considering a second-order Hamiltonian system. This system is characterized by two passive energy eigenstates, $E_1$ and $E_2$, and is subjected to an external perturbation. For the sake of simplicity, we take only the off-diagonal terms of perturbation such that:
For the physical implementation of ARC in an optical system, we employ a spatial variation of judiciously optimized gain and loss in the structure in a constant dielectric background. Accordingly, we take the background refractive index of the structure as $n_{B}$, wherein there exist a spatial distribution of gain/loss along its length in case of a one-dimensional (1-D) system, along its area in case of a two-dimensional (2-D) system or along its volume in case of a three-dimensional (3-D) system. We limit our discussion to the 2-D system that is represented in Fig. 1(a). In this core-shelled nanowire with a square cross-section, the gain and loss complex refractive indices are defined as $n_G=n_B-i\gamma$ and $n_L=n_B+i\gamma \tau$, respectively, where $\gamma$ is the gain coefficient and $\tau$ is defined as the ratio of loss to the gain coefficient. The system in consideration respects causality which guarantees the independent ability to tune and optimize the $\Im (n)$ for a particular operating frequency as per the Kramer-Kronig relation [32]. The choice of considering a square shaped resonator was taken for sake of simplicity, however the conclusion that are being reported is extendable to any other geometry. The design of the nanoscale resonator was taken such that the core comprised of the region with gain while the latter consists of region with loss. In the $x,y-$ plane for $0 \leq x,y \leq L_1$ defines the region governed by gain and the region $L_1 \leq x,y \leq L_2$ defined the lossy region and was then surrounded with air as can be seen in Fig. 1(a). We select an optimized lengths $L_1,L_2$ to be $487.8$ $nm$ and $835.9$ $nm$ respectively. This optimization was carried out to achieve highest Quality factor at a given operating fundamental frequency. With regards to the real part of the background refractive index, we used the values of $Al_{0.18}Ga_{0.82}As$ at a wavelength of $1594 \, nm$ ($n=3.3$) and $797 \, nm$ ($n=3.5$) that correspond to the fundamental and second harmonic (SH) wavelengths, respectively.
Fig. 1. (a) Schematic of square core–shell AlGaAs nanowire configured with gain in the region $0 \leq |x,y| \leq L_1$ and with loss in the region $L_1 \leq |x,y| \leq L_2$. The nanowire is free standing in air. Two plane waves ($\psi _p^+$ and $\psi _p^-$) are impinging on the structure. The structure is invariant in the $z$-axis. (b) The scattering efficiency versus operating wavelength: the black line shows the behaviour for the optimized structure and the peak at 1594 $nm$ is a signature of a quasi–BIC; the green line represents the behavior of the structure without gain and loss. The variation of quality-factors [(c), (d)] and real parts of the poles N$_{real}$ in the complex frequency plan [(e), (f)] versus operating gain for the two interacting modes at the fundamental and second harmonic frequency respectively. The blue lines represent the modes with the higher Q-factor instead the red ones the modes with lower Q.
In the system represented by Eq. (1), we vary the parameters $(\gamma ,\tau )$ to determine the appropriate coupling strength necessary for the enhancement of lifetime of one of the resonances near an ARC, with the subsequent decay of the latter. To completely understand the explicit interdependence between gain-loss with the quality factor of the proximity resonances, we look at the conventional scattering matrix (S-matrix) of the system which can be written of the form [21,33]:
where $A$, $D$ are the complex valued amplitude of the incident wavefunctions and $B$, $C$ are the complex amplitude of the scattered wave outputs. Utilizing this, we can extract the poles of the S-matrix. Inside the nanocavity, the proximity resonances contemplate dynamic confinement of light scattering source particle and hence is considered to be equivalent to a pole of the scattering matrix in the complex frequency plane (${k}$-plane). However, to maintain the conservation of the causality of the system, the poles in the lower half of the ${k}$-plane are considered. In the presence of properly optimised gain and loss in the system, the real part of the poles, $\Re [k]$, would be depicting the frequencies at which linear scattering is enhanced. The imaginary part, $\Im [k]$, represents the output coupling loss [21,33]. We thus consider the quality of the proximity resonance to be $1/2\Im [k]$, where $k$ is the complex pole of the S-matrix. Theoritically, a BIC is said to exhibit no leakage radiation, or in other terms the resultant coupling loss would be converging to 0. Thereby, for the quasi–BIC we could understand that the quality factor diverges to achive ultra-high values when the $\Im [k]$ converges to almost null value.To achieve this, we configure the gain and loss profile with carefully selected parameters values $\gamma =0.0875$ and $\tau = 1.09$ at the fundamental wavelength and $\gamma = 0.114$ and $\tau = 1.09$ at the SH wavelength. The selection of gain was based on the characteristic of the structure that would attain a quasi–BIC with the maximum of Q-factor for a fixed $\tau$. An important point to note is that our selection of $\tau$ is greater than 1, implying that our the system is overall lossy. Therefore, any enhancement in Q-factor is solely due to the quasi-BIC condition present at that particular parametric preset. This enhancement in Q-factor of the resonance at fundamental frequency is recognized to have the characteristic of a quasi–BIC-driven resonance which is optimized in terms of $(\gamma ,\tau )$, such that it has a more rapid increase compared to any other pair of $(\gamma ,\tau )$. Similarly, for a different set of parameters, i.e $\gamma =0.114$ and $\tau =1.09$ as prescribed for SH freqency, we also establish the same behaviour at the SH wavelength.
Both the quasi–BICs in both fundamental and SH wavelengths with such high Q-factor are expected to result in a significant increase in the field enhancement inside the structure. However this Q-factor depends on a variety of factors such as the coupling strength, background refractive index, the frequency of operation and the gain and loss character of the system. Furthermore, the ARC analysis that we performed considers closed systems whereas typical photonic dielectric or semiconductor resonators are constituted by open systems. Therefore, to confirm the results of ARC analysis, we performed full-vectorial numerical simulations implemented in COMSOL. At the fundamental wavelength we consider an incident field constituted by two plane waves with the electric field linearly polarized along the $x-$ axis and with opposite direction of propagation as background field $E_{b}=E_0 (e^{jk_0y}+e^{-jk_0y})$. The value of $E_0=8.6802 \times 10^7 ,\, V/m$ was taken corresponding to the measure of intensity of one plane wave to be $1 GW/cm^2$. We clarify that the excitation of the BIC is possible with a single input source. However, in the case of one plane wave exciting beam the gain that guarantee the BIC mode would be higher, roughly 0.57. For this reason, we have opted to use two counter propagating waves. The scattering efficiency $\sigma$ of the structure, exhibits a peak at a wavelength of 1594 $nm$ which was identified as a signature of a quasi–BIC at this operating wavelength, see Fig. 1(b). By definition, the Q-factor of an ideal BIC is infinite. Thus, in the scattering spectrum, BIC is associated with a dip in the scattering efficiency. Hence, although true BICs are invisible in the scattering spectrum, we can claim that a quasi-BIC can be tracked roughly by controlling peak shapes in the scattering power because (due to the finite Q factor) it is possible to couple light with an external field [18]. The fundamental study at the prescribed wavelength is performed in the structure with the appropriate amount of gain/loss as well as in a structure without gain/loss.
For the optimized parameters, at the fundamental harmonic regime, we show a crossing in the real part of the poles of the S matrix. The cross of the real part of the poles (N$_{real}$) is observed for a gain approximately equal to 0.0612, which is identified as the point of avoided resonance crossing. At the ARC there is an exchange of energies between the modes so that one achieves an enhancement in Quality factor while the other undergoes a decay in the quality, see Figs. 1(c) and (e). This can be easily understood to be the destructive interference between the two resonances followed by the subsequent redistribution of mode energy after the ARC point. In a similar fashion, we achieve the ARC at the second harmonic wavelength for a gain of 0.1, as depicted in Figs. 1(d) and (f). In entirety, similar behavior has been depicted in [21] where it has already been demonstrated that the ARC leads BIC with extremely high quality. We define the field enhancement due to the quasi–BIC as $max(E_{out1})/max(E_{out2})$ where $E_{out1}$ is the electric field distribution in the structure assisted with the gain and loss while $E_{out2}$ is the electric field distribution in an AlGaAs nanowire with the same dimensions (non-quasi-BIC assisted). The field enhancement due to quasi-BIC at a wavelength of 1594 nm was found to be approximately 3.49 as can be deduced from Figs. 2(a) and 2(b), which was higher than any other pair of $(\gamma ,\tau )$ parameter values. This field enhancement can be used for enhancing nonlinear optical effects such as SHG [31]. Moreover, from the comparison between the scattering of the structure with optimized parameters and the one without gain and loss, we have verified that the Q factor of the resonance at the fundamental frequency moves from approximately 70 to 220.
Fig. 2. (a) and (b) Distribution of absolute value electric field in the core-shell nanostructure, (a), assisted by gain and loss at a wavelength of 1594 nm (quasi–BIC) and, (b), without gain and loss. (c) and (d) Electric field map of the two proximity resonances undergoing the ARC at a wavelength of $1594 \, nm$. Resonant mode shown in (c) is the one that experiences the strong Q-factor enhancement.
Performing an eigenfreqeuncy analysis of our structure with COMSOL, we further locate and identify the modes that undergo ARC that lead to the formation of the quasi–BIC. The distribution of the magnitude of electric field can be seen in Figs. 2(c) and 2(d). We point out that the maps are in arbitrary units because are obtained from an eigenfrequency analysis. These modes were taken to be the ones undergoing ARC as they satisfied the condition of weak coupling of ARC near the operating fundamental wavelength. As a result, the mode that we identify as BIC is found to have a quality factor of $Q_{BIC}=1130.6$ while the decaying resonance was found to has a quality of $Q_{decay}=41.79$. Upon observation, the inclusion of gain and loss in the structure has led to the enhancement in the magnitude of electric field distribution as compared to the structure without the gain and loss as can be seen in Figs. 2(a) and 2(b). However, the distribution of the electric field remains unhampered but we demonstrate that the quasi–BIC induces an field enhacement via gain and loss as can be seen by comparing the Figs. 1(a) and 1(b).
This process was repeated for study of BIC at the SH wavelength where it was found that $Q_{BIC}=195.35$ and the decaying mode had $Q=42.23$.
3. Enhancing second harmonic generation of a quasi BIC-driven resonance
In this paragraph we study the SHG in the AlGaAs core-shell nanowire. We consider the optimized parameters of Section 2 which guarantee the quasi–BIC at both fundamental and second harmonic wavelengths. The zincblende crystal structure of AlGaAs determines that the induced nonlinear currents can be expressed by $J_i=i\omega_{SH}\varepsilon_0\chi^{(2)}E_{FF,j}E_{FF,k}$ where $i \neq j \neq k$, $\varepsilon _0$ is the permitivity of vacuum, $\omega _{SH}$ is the SH frequency, and the electric fields $E_{FF,j}$ and $E_{FF,k}$ are the electric field in the $j^{th}$ and $k^{th}$ direction at the fundamental wavelength, respectively. The value of $\chi ^{(2)}$ is taken to be $1 \times 10^{-12} C/N$[2]. The SHG efficiency is defined as: $\eta =P_{SH}/I_0 L_{eff}$ where $L_{eff}$ is the effective length upon which the incident intensity is illuminated and in this case is $2L_2$ since, at the fundamental wavelength, we set two counterpropagating plane waves of equal intensity $I_0$, $P_{SH}$ is the output SH power per unit length that is scattered from the structure at the SH wavelength. The SHG efficiency for the quasi–BIC is found to be $4.82\times 10^{-2}$ as can be seen in Fig. 3(b). The spectral plot of the SHG efficiency, $\eta$, exhibits a peak at the wavelength of the quasi–BIC. A small shoulder in the SHG efficiency appears due to coupling to a different mode at the fundamental frequency that is not in phase coherence with the other mode. The efficiency of the system without any gain and loss is found to be $2.88\times 10^{-5}$ (see Fig. 3(c)) which is comparable to the SHG efficiency obtained in GaAs nanowires in similar conditions [34]. We verify that the peak in the SHG efficiency in the structure without any gain and loss is due to the presence of a resonant mode of the nanowire at the pump wavelength of 1596 nm that exhibit a quality factor of 36.18. The increase of about 3 orders of magnitude in SHG conversion efficiency of double quasi–BIC illustrates the ability to boost optical nonlinear phenomena with a injection of such low-valued pair of gain and loss in an overall lossy dominated structure. We point out that, although the considered process is a non-linearity of the second order, the presented approach can be used and extended to boost the efficiency of other nonlinear effects by performing the same optimization procedure as presented in Section 2 for the specific fundamental and harmonic wavelengths.
Fig. 3. (a) Magnitude of electric field distribution at the second harmonic when injected with the non-linear current. (b) SHG efficiency as a function of pump wavelength in optimized structure designated by $\eta _{BIC}$. (c) SHG efficiency designated as $\eta _{non-BIC}$ as a function of pump wavelength without gain and loss.
4. Conclusion
In summary, we use ARC approach to design quadratic AlGaAs nanowires with a core with gain and a shell with loss. By varying the amount of loss and gain we engineered the nanowires to support quasi–BICs at multiple wavelengths. We use this technique to design a structure supporting quasi–BIC at both fundamental and SH wavelength and demonstrate an enhancement of 3 orders of magnitude in the conversion efficiency of SHG compared to an unstructured AlGaAs nanowire. This engineering ability of structures to further tune the modes undergoing the nonlinear phenomenon to give high nonlinear response opens up a huge potential in the field of nonlinear photonics to device and design optical systems.
Funding
Education, Audiovisual and Culture Executive Agency (5659/002-001); Università degli Studi di Padova (STAR-StG project PULSAR); Ministero dell’Istruzione, dell’Università e della Ricerca (Law 232/2016).
Disclosures
The authors declare no conflicts of interest.
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