## Abstract

We theoretically realize the Fano resonance with a high quality factor of 10^{6} using a structure, which is constructed from three one-dimensional photonic crystals and a defect layer. The emerged Fano resonance can be attributed to the weak coupling between a Fabry-Perot cavity mode and a topological edge state mode provided by the topological photonic crystal heterostructure. Moreover, we experimentally reproduce this Fano resonance in the optical communication range with a high quality of 10^{4}. This may be useful reference for the study of applications of photonic topological states in integrated photonic devices and information processing chips.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Photonic topological insulators play an increasingly important role in integrated optics due to their unique properties of topologically-protected edge states, i.e. topological protection and immune to structural disorders [1–3]. Various schemes have been proposed to construct topological photonic modes, including using plasmonic nanoparticles [4, 5], one-dimensional (1D) optical waveguides [6, 7], 1D photonic crystals (PhC) [8], two-dimensional (2D) PhC made of rods [9, 10] and three-dimensional (3D) PhC [11, 12]. However, due to the complicated design and fabrication, the potential applications of topological photonic modes in 2D and 3D topological PhC have been greatly limited, especially in fields of integrated circuits and integrated photonic devices. Recently, Fano resonances have attracted much attention to be applied in many photonic devices [13–16], such as Fano-filters for polarized electrons [17, 18], Fano-transistors [19], demultiplexer [20] and optical switching [21] because of the large transmission change experiencing within a narrow wavelength range. As known, the Fano resonance results from the interference between the discrete bound state and the continuum of states [22]. Illuminated by this mechanism, we build a structure consisting of two parts to generate the Fano resonance response. One is the 1D topological PhC which provides a topological edge mode, and the other is the Fabry-Perot cavity. Using this structure, we theoretically realize a Fano resonance with a high quality factor of 10^{6}. Moreover, our experiment reproduces such Fano resonance with a high quality factor reaching 10^{4} in the optical communication range. Our study combines topology with Fano resonance for the first time through the interference coupling of a Fabry-Perot cavity mode and a topological edge mode. So the application of topological photonics could be promoted. This certainly has important implications for the applications of photonic topological states in integrated photonic devices and information processing chips.

## 2. Calculation

The whole structure consists of two PhC 1 (alternative layers of silicon and silicon dioxide), one PhC 2 (alternative layers of silicon dioxide and silicon), and the defect layer made of silicon dioxide, as shown in Fig. 1(a). The PhC 1 in the left-hand side and PhC 2 construct a heterostructure providing the topological state situated at the heterostructure interface. While the PhC 2, the defect layer and the PhC 1 in the right-hand side construct a Fabry-Perot cavity. The PhC 1 and 2 (shown in Fig. 1(b) and Fig. 1(c), respectively) are composed of four period units and both can support wide forbidden gaps [23, 24].

The transmission spectrum of the PhC heterostructure composed of the PhC 1 and 2 is calculated with a commercial finite element solver package – COMSOL Multiphysics. In our calculation, the effect refractive index *n _{eff}* is set to be 1.46 for silicon dioxide, 2.82 for silicon [25]. In PhC 1, the silicon layer is 680-nm-thick and the silicon dioxide layer is 815-nm-thick. While in PhC 2, the thicknesses are 1290 nm for the silicon dioxide layer, 685 nm for silicon layer and 1045 nm for silicon dioxide defect layer, respectively. It is pointed out that the topological edge modes can exist in the 1D PhC heterostructure interface, when two PhC whose bandgaps are in the same wavelength range have different topological properties [26]. We show the calculated transmission spectra of the PhC 1 and 2 in Fig. 1(d). The forbidden band of PhC 1 situates in the same wavelength range around 1540 nm as that of PhC 2. In addition, we respectively calculate the band diagrams [Fig. 1(e)] of PhC 1 and 2 by the conjugate gradient minimization of a Maxwell operator based on a plane-wave basis [26]. Four photonic bandgaps can be observed for both PhC 1 and PhC 2 within the frequency range from 20 to 220 THz. Here, the red band indicates a bandgap with a positive topological phase, while the blue implies a bandgap with a negative topological phase. The position of the fourth gap of PhC 1 coincides with that of PhC 2, which is located at the operating frequency of 195 THz. Besides, the fourth gaps of PhC 1 and PhC 2 are both broad bandgaps and have different signs of the topological property. This provides a possibility for the existence of a topological edge mode at the heterostructure interface around 195 THz by calculating the sum of the Zak phases below fourth bandgap [27]. In other words, there may emerge a transmission peak in the interface of the PhC heterostructure in the vicinity of the forth bandgaps [28].

As shown in Fig. 2(a), we illuminate the PhC heterostructure from left to right. The calculated electric-field distribution and transmission spectrum of the PhC heterostructure are shown in Fig. 2(b) and Fig. 2(c), respectively. Obviously, the electric fields are confined around the heterostructure interface, indicating the existence of topological edge mode. Besides, a sharp peak with 99.9% transmittance at the wavelength of 1538 nm (corresponding to the frequency of 195 THz) is revealed on the transmission spectrum. Actually, the center frequency of the sharp transmission peak is exactly located at the overlap parts of the fourth gaps in PhC 1 and PhC 2. A zoom-in of the sharp peak together with its fit is shown in Fig. 2(d). The fitting curve has a typical symmetric Lorentzian line shapes. This demonstrates that a Lorentzian resonance has arisen at the heterostructure interface [28], with a center wavelength *λ*_{1} of 1538.1 nm and a full width at half maxima *∆λ*_{1} (FWHM) of 2.3 nm.

Now we build a Fabry-Perot cavity. As illustrated in Fig. 3(a), the cavity consists of the PhC 2, the defect layer, and the PhC 1. Likewise, after the light illumination we theoretically obtain the electric-field distribution and the transmission spectrum of the Fabry-Perot cavity. In Fig. 3(b), the calculated electric-field distribution of the Fabry-Perot cavity are obviously confined around the defect layer. And the fitting of the transmission spectrum of Fabry-Perot cavity exhibits a typical symmetric Lorentzian line shapes, as shown in Fig. 3(c). This transmission spectrum has is a center wavelength *λ*_{2} of 1534.6 nm and an FWHM *∆λ*_{2} of 1.4 nm. According to the definition of quality factor, i.e. *λ*/*∆λ*, we obtain the quality factors, which are 669 for the photonic topological edge mode centered at 1538.1 nm and 1096 for the Fabry-Perot cavity mode centered at 1534.6 nm, respectively. More results about the quality factors can be found in Appendix I.

We combine the PhC heterostructure and Fabry-Perot cavity as a whole structure to study the interaction between the two parts. In Fig. 4(a), we show the calculated electric-field distribution of the whole structure. One can observe the electric field are mainly distributed around the PhC heterostructure interface and defect layer, which can be regarded as a simple superposition of the electric fields in the PhC heterostructure and the Fabry-Perot cavity. The transmission spectrum of the whole structure is calculated, as seen in Fig. 4(b). For comparison, we also display the corresponding spectra of the PhC heterostructure and the Fabry-Perot cavity (depicted by red and blue curves, respectively). Surprisingly, an extremely narrow peak, highlighted by a gray region with the blue edge, can be observed on the transmission curve of the whole structure. For the better visualization, we zoom in this peak in Fig. 4(c) and find it has an asymmetry profile resembling that of Fano resonance [29], i.e.

Here, *q* is the asymmetry parameter of the Fano resonance and *γ* is the line width of the Fano resonance. Then we fit the sharp peak with Eq. (1). When *q* = 5.3 and *γ* = 2.3 × 10^{7}, the fitting curve matches the transmission peak best. This fit demonstrates that the sharp peak appeared in the transmission spectrum happens to be a sharp Fano resonance. The Fano resonance occurs at 1532.345 nm with a high spectral contrast, and its profile evolves from a dip of 0% to a peak of 95.5% transmission within a very narrow wavelength range. The quality factor of the Fano resonance can reach 10^{6}. In the limit of *|q|*→*∞*, the transmission of the continuum states is very weak, and the line shape is only determined by the transmission of the discrete state [30]. Then a standard Lorentzian profile will emerge. While in our calculation, the asymmetry parameter *q* is 5.3. This means the transmission of the discrete states are comparable to that of the continuum states. These two kinds of states jointly result in the asymmetric Fano profile. The existence of the Fano lineshape is conformed in Appendix II.

As mentioned above, a topological edge mode arises at the heterostructure interface while the Fabry-Perot cavity generate another mode with the Lorentz profile. An approximate perturbation theory was applied to clarify the appearance of asymmetric resonances and this resonance has been considered as a special state generated by coupling between discrete bound states and continuum states [22]. In Fig. 4(b), the line width of the mode generated by the Fabry-Perot cavity is narrower than that in the PhC heterostructure. Thus we can assume that the Fabry-Perot cavity provides the discrete bound state, while the PhC heterostructure provides the continuum states. If we combine the PhC heterostructure and the Fabry-Perot cavity together as shown in Fig. 4(a), their corresponding modes will couple with each other and the symmetry of the transmission spectrum will be broken. It is the interference of above two states that generate the Fano resonance [22]. Therefore, the transmission spectrum of Fano resonance experiences a huge change within a very narrow wavelength range due to the destructive interference between the two states.

## 3. Experiments

To verify our simulation, we carried out the experiment as follows. As shown in Fig. 5(a), we fabricated the sample to obtain the Fano resonance experimentally. Firstly, a 100 nm-thick Au film was deposited by the E-beam evaporation. And then the SiO_{2} and Si layers with the thickness of 300 nm and 220 nm were deposited using the chemical vapor deposition (CVD) on the Au film successively. To acquire the grating structure as mentioned in Fig. 1(a), we used the focused ion beam (FIB) technology to etch the Si layer. The SEM image of the etched grating structure can be seen in Fig. 5(b). Two 150 μm-long grooves etched beside the grating structure were used to confine the light in the Si waveguide with the width of 12 μm. At both ends of this waveguide, the coupling gratings (duty-cycle of 0.58 and a grating constant of 560 nm) were etched to couple the light in and out the waveguide. Figure 5(c) showed a SEM image of the whole structure with a 220 nm etching depth. Finally, another 300 nm-thick SiO_{2} layer was covered on the whole sample with the CVD, and after that, a 100 nm-thick Au layer was deposited again only on the grating structure using the E-beam evaporation. We have fabricated gold layer on both bottom and top to form the gold mirrors. These metal layers contributed to the confinement of the light within the structure as well as constraining the electric field to be normal to the parallel plates [31]. It can efficiently reduce the radiation loss of the light transmission in the grating, and meanwhile better reproduce the propagation properties of the 1D PhC in simulation [25].

The light source was a tunable fiber laser (Santec-TSL-510), whose output power was 20mW. One single-mode-fiber was coupled to the left coupling grating while another was coupled out at the right coupling grating. The coupling efficiency of the fiber was more than 90%. The fiber spectrometer (Avantes-AvaSpec-NIR256-1.7) was used to measure the signal intensity. The measuring accuracy of the fiber spectrometer was 0.01 nm and we chose the working wavelength from 1520 nm to 1550 nm to measure it.

As shown in Fig. 5(d), a dramatically narrow peak with high quality factor of 10^{4} could be obtained and the experimental transmission was 69.4% at 1530.11 nm. The difference between the transmission of experiment and calculation was caused by inevitable experiment deviation such as surface roughness, scattering and radiation loses in other dimensions. A further calculation of a similar structure in two-dimensional can be seen in Appendix III. Experiment can verify the theoretical calculation well, which makes integrated photonic devices possible.

## 4. Conclusion

In this paper, Fano resonance was realized in a 1D topological PhC heterostructure. High quality factors of 10^{6} and 10^{4} were reached theoretically and experimentally for the Fano resonance, respectively. The transmittance experiences a huge change within a very narrow wavelength range around the optical communication band. This characteristic of the Fano resonance has implication for the low-power optical switching. Our study may be useful reference for the study of applications of photonic topological states in integrated photonic devices and information processing chips.

## Appendix 1 Quality factor

To find out the change regulation of the quality factor Q in the 1D PC heterostructure and the Fabry-Perot cavity, we carry out the following calculation. According to the definition of quality factor, i.e. λ/∆λ, we obtain the quality factors, which are 669 for the photonic topological edge mode centered at 1538.1 nm and 1096 for the 1D defect PC centered at 1534.6 nm, respectively. This difference could also be confirmed from the electric-field distribution shown in Fig. 2(b) and Fig. 3(b). As for the second comment, we calculate the transmission spectrum of the 1D defect PC with different distances d_{0} between the PhC 2 and PhC 1, as shown in the following Fig. 6, together with its quality factor. When d_{0}=900nm, 960nm, 1020nm, 1080nm, 1140nm and 1200nm, correspondingly Q=423, 715,897, 1033, 749, 408. According to these results, we may conclude that 1D defect PC has higher Q when it locates in the center of the forbidden band, while it has lower Q when locating in the band edges of the forbidden band. Therefore, it is not a universally applicable conclusion that the topological PCs always have higher Q.

## Appendix 2 Confirmation of the existence of Fano linetype

To make sure that the Fano linetype will persist to exist under different conditions, we have calculated the transmission spectrum of the whole structure with different distances d_{0} of the defect layer. As shown in Fig. 7, although the calculated transmittance and central wavelength are changing with respect to d_{0}, the Fano linetype still exists. This demonstrates that Fano profile can persist to exist under various conditions.

## Appendix 3 2D Calculation with FDTD

In order to better explain the experiment, we try to change our model from pure one-dimensional simulation to two-dimensional which closely resembles the experiment geometry. Here we propose that the dielectric gratings could replace 1D PhC in a similar way. The 1D PhC band structure derives from Bragg scattering of photons in periodic layers, which is the same phenomena occurring for the case of light propagating in gratings. Our 2D geometry [shown in Fig. 8(a)] is just like the cross-section of our experiment sample in Fig. 5(a). First consider a 220-nm dielectric gratings clad with 300-nm silicon dioxide layers top and bottom, with two 100 nm-thick metallic cladding layers over the silicon dioxide layers. The grating structure consists of silicon and silicon dioxide.

Then we carry out the finite-difference time domain simulation (FDTD) with the geometry. As shown in Fig. 8(a), the electric field of Ez component of the 2D geometry is not only distributed around the PhC heterostructure interface and defect layer, but also distributed at the center of the 2D geometry. This distribution may be caused by the difference between the 1D PhC and the 2D dielectric gratings. Besides, we calculate the transmission spectrum of the 2D geometry with the finite-difference time domain simulation (FDTD), as shown in Fig. 8(b). We find this spectrum is different from the transmission spectrum of the 1D PhC heterostructure [shown in Fig. 4(c)], which is calculated by commercial finite element solver package-COMSOL Multiphysics. Both the transmittance and the quality factor (Q≈4×10^{4}) calculated by FDTD reduce because of the radiation losses in the 2nd dimension.

## Funding

973 Program of China Grants 2013CB328704 and 2014CB921003; National Natural Science Foundation of China (NSFC) Grants 61475003, 61775003, 11734001, 11527901; National Postdoctoral Program for Innovative Talents Grant BX201700011.

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