1. Introduction

Since the photonic bound states in the continuum (BICs) was first realized in two arrays of parallel dielectric gratings and cylinders in 2008 [1], there has been growing interest in the such field due to the fruitful physics, and numerous advanced applications including ultra-sensitive sensing [2–4], ultralow-threshold lasering [5–10], and enhanced nonlinear phenomena [11–20]. Conventionally, there are two types of BICs, i.e. symmetry-protected [21–23] and Friedrich-Wintgen BICs [24–26]. The symmetry-protected BICs appear at the gamma (Γ) point of symmetric nanostructures under normal incidence. The modes near the Γ point are not allowed to form the free space propagating waves due to the symmetry incompatibility, and hence are completely localized [21]. The Friedrich-Wintgen BICs are also called “accidental” BICs, which appear at an off-Γ point, and are the consequence of destructive interference of the resonances in the nanostructures [24–26]. For a true BIC mode, it has an infinite life-time, zero line-width, and infinite Q-factor, and thus is unobservable as a dark mode. Only quasi-BICs with finite spectral width and Q-factor are observed experimentally.

Among these applications of BICs, the enhanced nonlinear phenomena draw more attention due to the important practical applications in generation of new frequency via efficient harmonic generation [16–20] or four wave mixing [27], ultra-low threshold all-optical switching [28], and optical bistability [14,29,30]. In the previous studies on enhanced nonlinear phenomena via BICs, the quasi-BICs in symmetry-protected BICs which can be accessed by breaking the symmetry structure or using the oblique illumination, are conventionally employed [31–37]. The huge enhancement of localized field in the nanostructures at quasi-BICs was straightly used for the enhancement of nonlinear response. Friedrich-Wintgen BICs can be obtained by continuously tuning the parameters of a system, such as the geometry of nanostructures or the angle of incidence. At a set of particular parameters, the destructive interferences of resonance modes may be satisfied to present an avoided crossing with a Rabi splitting due to the strong coupling. The local field inside the nanostructures, and thus the nonlinear response, should be also dynamically varied during the change of parameters. Until now, the nonlinear response during the formation of Friedrich-Wintgen BICs is few investigated.

In this paper, we demonstrate the evolution of optical harmonic generation near BICs in hybrid plasmonic-photonic structures. The plasmonic resonance mode from a 1D periodic Au grating and photonics modes in a dielectric slab waveguide were formed and could be coupled with each other at some angles of incidence in the hybrid system. The symmetry-protected BICs and Friedrich-Wintgen BICs were both observed at the normal illumination and at a particular angle of incidence, respectively. The evolution of second- and third-harmonic generation (SHG and THG) in the system as the variation of the angle of incidence was investigated near the BICs.

2. Numerical structure and method

A schematic of the proposed structure is depicted in Fig. 1(a), the same as the reported structure in Ref. [37]. The structure is composed of an Au relief grating with period $\mathrm{\Lambda}=580\,{\rm nm}$ fabricated on a silica glass substrate covered with a silicon nitride (SiN) slab with thickness d = 500 nm. The height and width of ridges of the Au grating are 30 nm and 100 nm, respectively, while the thickness of the back Au film is 100 nm to fully prevent the light transmission. The light of transverse magnetic (TM) polarization and an angle of incidence θ shines on the nanostructure in the x-o-z plane. Except as otherwise noted, the intensity of light 100 MW/cm², and thus, the amplitude E₀ around 2.75×10⁷ V/m, was used for all the simulations.

 

Fig. 1. (a) A hybrid plasmonic-photonic structure with an Au grating covered with a SiN layer under the illumination of a TM-polarized light. (b) Dependence of reflectance spectrum on the angle of incidence of the coupled system. BICs are marked by dotted circles. Two symmetry-protected BICs due to plasmonic (blue dotted circle) and photonic (red dotted circle) modes occur at the normal incidence, and a Friedrich-Wintgen BIC appears due to the interference of plasmonic and photonic resonances (white dotted circle). (c) Reflectance spectra at angles of incidence 0° and 2°. Blue and red arrows represent the plasmonic and the photonic collapsed symmetry-protected BICs, respectively. (d) Avoided crossing of Rabi splitting about 58 meV due to the strong coupling between plasmonic and photonic modes at different angles of incidence. Bold red and blue lines denote photonic and plasmonic modes, respectively.

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The governing equations of fundamental, SHG and THG waves in the frequency domain were given in details in Ref. [17]. The parameters of SiN including the linear refractive index, the tensor components of second-order susceptibility χ⁽²⁾ and third-order susceptibility χ⁽³⁾ are the same as given in Ref. [17], which are all from the experimental data [38,39]. Briefly, under the coordinate system of Fig. 1(a), the non-zero tensor components of the second order susceptibility of SiN are $\chi _{xx\textrm{z}}^{(2)} = \chi _{xzx}^{(2)} = \chi _{yyz}^{(2)} = \chi _{yzy}^{(2)}$=0.49 pm/V, $\chi _{zxx}^{(2)} = \chi _{zyy}^{(2)}$=0.47 pm/V, $\chi _{zzz}^{(2)}$=2.47 pm/V [38] and the tensor components of the third order susceptibility are $\chi _{iiii}^{(3)}$=3×10⁻²⁰ m²/V², and $\chi _{iijj}^{(3)} = \chi _{ijij}^{(3)} = \chi _{ijji}^{(3)} = {1 / 3}\chi _{iiii}^{(3)}$ with i, j = x, y, z [39]. The dielectric constant of Au was taken from the Ref. [40]. The numerical simulation was conducted using finite element method via Comsol Multiphysics in the frequency domain. The detailed settings can be referred [17].

3. Results and discussion

The dependence of calculated reflection spectrum on the angle of incidence is shown in Fig. 1(b). The symmetry-protected BICs appears at the normal incidence, as labeled by red and blue dotted circles, respectively. The reflectance spectrum at angles 0° and 2° are particularly plotted, as shown in Fig. 1(c). The sudden appearance of Fano resonances (marked by red and blue arrows) was induced by breaking the symmetry-protected BICs at the incline illumination 2°. Notably, the Friedrich-Wintgen BIC occurs at the angle of incidence of around 18° (marked by white dotted circle in Fig. 1(b)) due to the interference and strong coupling between the plasmonic and photonic modes. The reflectance spectrum at selected incident angles around Friedrich-Wintgen BIC are shown in Fig. 1(d). The plasmonic and photonic modes are denoted using bold blue and red lines, respectively. The modes are exchanged due to the strong coupling, and the resonance completely vanishes on the right branch at the incident angle around 18°. The avoided crossing with a Rabi splitting of about 58 meV can be observed. The interaction between the two modes near the BIC regime will strongly affect the local field distribution (See below figures), and thus the nonlinear optical response.

We will first focus on SHG and THG behaviors near the Friedrich-Wintgen BIC regime to demonstrate how the local field distribution influenced by the interaction of photonic and plasmonic modes to further affect the nonlinear response in the hybrid system. The calculated SHG and THG conversion efficiencies as a function of the angle of incidence and fundamental wavelength are shown in Fig. 2. On the right branch of resonance modes, SHG and THG response both almost vanish with the destructive interference of two modes at the angle of incidence 18°. At such the condition, the local field collapses with almost less enhancement, so the nonlinear response is quite weak, as shown in Fig. 3. SHG and THG both increase when the angles of incidence are far away from the critical angle. The phenomenon is ascribed to the gradually enhanced local field inside the SiN layer at the angles far away the destructive interference of the two modes (See the local field distribution at 12° and 24° in Fig. 3). It is noted that the plasmonic modes give rise to stronger nonlinear response than the photonic modes performed, because the enhancement of local electric field induced by plasmonic modes around Au grating and in SiN layer is greater than that at the photonic modes (See the local field distribution at 12° and 24° in Fig. 3). The SHG and THG fields at angle of incidence 18°, 12° and 24° are shown in Fig. 3, respectively. For the plasmonic modes on the left branch of resonance modes, the trend of SHG and THG response is totally converse with those on the right branch of resonance modes, i.e. SHG and THG both increase with the angles of incidence close to the critical angle 18°. The behavior also can be explained by the local field distributions. However, there is no monotonical change for SHG and THG response on the left branch of resonance modes when the angles of incidence are less than 18°. Deep insight on the local field distribution near 18° in the nanostructures will find that the field dose not definitely belong to plasmonic mode or waveguide mode, but a mixing mode due to the interaction, as shown in Fig. 4 for the field distribution at typical angles 13° and 15°. There exist the optimized angles for SHG and THG response. The maximum SHG and THG conversion efficiency occurs at the angle of incidence 13° and 15°, respectively. The different angle of incidence for maximum SHG and THG response is ascribed to two aspects, one is that the nonlinear response is not only determined by local field distribution, but the tensor components of nonlinear susceptibility should be also considered simultaneously; the other is that the loss and radiation efficiency of the structure are different at SHG and THG wavelengths under different angles. Thus, the maximum SHG and THG response is possible to happen at the different angle of incidence. The distribution and magnitude of SHG and THG field at these two angles of incidence are shown in Fig. 4. So, the interaction of the resonance modes near the critical angle of incidence of BIC introduces the different evolution behavior of SHG and THG response, and the optimized nonlinear response exists at some certain angles.

 

Fig. 2. (a) SHG and (b) THG conversion efficiency near the Friedrich-Wintgen BIC regime.

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Fig. 3. Electric fields |E_ω/E₀| of fundamental wave ((a), (d), (g)), SHG (|E_2ω|) ((b), (e), (h)) and THG (|E_3ω|) ((c), (f), (i)) at the corresponding resonant wavelengths at incident angles of 18°, 12° and 24°, respectively. The units of SHG and THG fields are V/m.

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Fig. 4. Electric fields |E_ω/E₀| of fundamental wave ((a), (d)), SHG (|E_2ω|) ((b), (e)) and THG (|E_3ω|) ((c), (f)) at the corresponding resonant wavelengths at incident angles of 13° and 15°, respectively. The units of SHG and THG fields are V/m.

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We finally investigate SHG and THG responses near the symmetry-protected BICs at the normal incidence, as marked by blue and red dotted circles representing plasmonic and photonic BICs, respectively, as shown in Fig. 1(b). SHG and THG conversion efficiency in the range of incident angle 0°−10° are shown in Fig. 5. At the normal incidence, there are only two resonance modes, photonic mode at around 920 nm and plasmonic mode at around 1128 nm, to enhance the SHG and THG response. It is interesting to notice that the SHG response at the plasmonic mode is larger than that at the photonic mode, but the THG has a converse response behavior. From the perspective of enhancement of local field, the local field under photonic mode has a larger enhancement than that at the plasmonic mode (not shown here), so traditionally the nonlinear response at photonic mode should be stronger than that at plasmonic mode. However, as aforementioned, the nonlinear response is not only determined by local field but also the tensor components of nonlinear susceptibility and extraction efficiency affected by the loss of Au and the coupling out efficiency by the structure at different wavelengths. Here we consider the loss of Au is the main reason for such phenomena. The imaginary part of Au at the SHG wavelength of plasmonic mode (around 564 nm) is much smaller than that at the SHG wavelength of photonic mode (around 460 nm), so the loss induced by ohmic loss is much weaker to produce the stronger SHG at the plasmonic mode. At the THG wavelength of the plasmonic mode and photonic mode, the imaginary part of Au has no much difference and the local field enhancement is dominate for THG response. Especially, the largest THG conversion efficiency arrives up to 1.5×10⁻⁷.

 

Fig. 5. (a) SHG and (b) THG response near the symmetry-protected BICs at the normal incidence.

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When the angle of incidence is nonzero, the quasi-BICs of finite linewidths occur, as shown in Fig. 1(c), to enhance the nonlinear response. The electric field distributions at the fundamental, SHG and THG wavelength at a typical angle of incidence 3° are shown in Fig. 6. Generally, SHG and THG conversion efficiency at the quasi-BICs increase with the increase of angle of incidence. The largest SHG conversion efficiency at 10° of the quasi-plasmonic mode reaches up to 2.5×10⁻⁷. However, we notice that the SHG response at the quasi-photonic mode is not monotonically increasing with the increase of incident angles. It is also ascribed to the local field distribution and tensor components of nonlinear susceptibility of SiN, especially the mixing of photonic mode and plasmonic mode becomes more obvious with the increase of incident angles.

 

Fig. 6. Electric fields |E_ω/E₀| of fundamental wave ((a)-(d)), SHG (|E_2ω|) ((e)-(h)) and THG (|E_3ω|) ((i)-(l)) at the corresponding resonant wavelengths at the incident angle of 3°, respectively. The units of SHG and THG fields are V/m.

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4. Conclusions

We investigate the nonlinear optical harmonic generation behaviors near the BICs regime. The evolution of SHG and THG are highly dependent on the local field distribution, tensor components of nonlinear susceptibly of nonlinear materials, and extraction efficiency of nonlinear signals from the nanostructure. Especially, the hybrid modes give rise to the optimized SHG and THG efficiency. The results are important to understand the nonlinear response behaviors and to design the nonlinear photonic devices.