1. Introduction

The manipulation of light flow is the essential technique to realize the photonic functionalities including the optical communication, optical computing, and optical signal processing. With the development of techniques, photonic devices which can be integrated on a chip to realize the miniaturization and multi-function are required. Besides the traditional on-chip waveguides to guide the light, photonic crystals have draw increasing attentions to manipulate the flow of light [1]. However, the backscattering and loss in the traditional photonic crystals are always unavoidable due to the defects introduced during nanofabrications or the sharp corners of waveguides [2,3], which limits the applications of photonic crystals based optical devices.

Driven by the concept of topology in condensed matter physics, topological states in optical systems have also been discovered and developed rapidly, attracting widespread attentions in the photonics community. Topological photonics have topological protection features, which provide a novel and powerful platform for designing photonic devices with robustness [4–7]. There are several ways to implement topological photonics, such as using bi-anisotropic metamaterials [8], gyromagnetic materials [9], and all-dielectric materials [10,11]. Particularly, the spatial-symmetry protected topological states in all-dielectric photonic crystals which do not require the application of external magnetic or electric fields, and can be easily fabricated for integration, draw more attentions. Many topological all-dielectric photonic devices have been designed, including routers [12,13], filters [14,15], logic gates [16], and rainbow devices [17–19].

However, most of the topological structures in topological photonic devices with superior performances commonly have fixed optical properties. This means that their functionalities cannot be dynamically tuned after they are fabricated. In recent years, with the development of topological photonics, tunable topological photonic devices have been designed and implemented, such as using the thermal phase-changing materials [20,21], liquid crystals [22], and free-carrier excitation [23]. Dynamic tunability not only allows the device to achieve more complex functions but also helps to investigate other features of the topological phase. However, the response of phase-changing materials and liquid crystals is relatively slow, which limits the advanced applications. Otherwise, topological all-optical logical gates based on photonic crystals were numerically designed and experimentally realized recently [24,25]. The linear interference approach is performed in principle. Dynamically tuned topological photonic devices based on the intrinsic nonlinearity of materials have not been studied yet.

In the present work, we explore the all-optical self-manipulation of light flow in on-chip topological photonic crystal waveguides using the intrinsic Kerr nonlinearity of silicon. Topological edge states are constructed at the wavelength around 1550 nm using the concept of synthetic dimension [26]. The path of the light flow in topological waveguides is controlled by varying the intensity of incident light, and thus the output of light from $O_1$ or $O_2$ is controlled as switching. The properties of fault tolerance and immunity from disturbance for these devices are also demonstrated. This compact, robust all-optical and self-manipulating device has potential applications in future photonic integrated circuits.

2. Construction and regulation of topological boundary states

Topological edge states are constructed using the concept of the synthetic dimension method [27]. Synthetic dimension provides additional degrees of freedom for topological photonics, provides a platform for the construction of on-chip topological nanophotonic devices, and has been validated both theoretically and experimentally [17,19]. The synthetic dimension here is constructed by introducing a translational degree of freedom in a two-dimensional photonic crystal [17]. The introduction of an additional degree of freedom in the photonic system allows the system to break the original geometric dimensional limitations to study higher dimensional problems and simplify the photonic device structure. This facilitates our subsequent manipulation of the light flow.

The schematic diagram of the extension of translational deformation from a one-dimensional photonic crystal to a two-dimensional photonic crystal in the $x$-direction is depicted in Fig. 1(a). The displacement parameter $\eta$ in the $x$-direction and the Bloch vector $(k_x, k_y)$, form a three-dimensional parameter space $(k_x, k_y, \eta )$. When $k_y$ is fixed, for each $\eta$, the phase of the $n$th band $\theta ^{(Zak)}_n (k_y,\eta )$is defined as [28]

Because $k_y$ is fixed, the two-dimensional photonic crystal can be effectively reduced to a one-dimensional system. Similar to the one-dimensional case, the Zak phase of a two-dimensional lattice with the fixed $k_y$ can be viewed as the center of the "quasi-one-dimensional" Wannier function,

 

Fig. 1. (a) Structural diagram for constructing topological edge states by introducing displacement parameters in photonic crystals. $a$ is the lattice constant, $d$ is the cylinder diameter, and $\eta$ is the displacement parameter. (b) Dispersion relations of a unit cell of the photonic crystal. The dark area is silicon ($n_0$=3.48) and the light area is air ($n_0$=1). (c) Band diagram of supercells with different displacement parameters (the red points present edge modes, the black points are bulk modes).

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When the displacement parameter is introduced to move the photonic crystal along the lattice direction, the center of the quasi-one-dimensional Wannier function also changes, causing the Zak phase to vary with the lattice translational deformation. As $\eta$ changes from 0 to $a$, the Zak phase increases with translation by $2\pi$ [17,27]. Here the topology is characterized by the Chern number, and as $\eta$ varies from $-a/2$ to $a/2$, the winding of the Zak phase is divided by $2\pi$ to obtain the Chern number of the band [29]

The introduction of the displacement parameter causes the Chern number of the original structure to change, so that topological edge states are generated between the undeformed structure and the structure after translation. The Chern number for each isolated band is calculated to be equal to 1, and the total Chern number associated with the gap is equal to the cumulative sum of the Chern numbers of the bands located below that gap. In our work, since we have selected the third gap, its Chern number is equal to 3. According to the bulk-edge correspondence in translational deformation, during the translational of $\eta$, $n$ modes always raise their energies out of the $n$th bulk energy band and go across the band gap between the $n$th and $(n+1)$th bands [27]. So for one cycle of change in $\eta$, three topological edge states appear in the gap.

The bulk dispersion of the silicon photonic crystal of lattice constant $a = 0.697$ $\mathrm{\mu}\textrm{m}$ and the diameter of the silicon cylinders $d = 0.6a$ at the transverse magnetic (TM) mode is shown in Fig. 1(b). The silicon cylinders have a linear refractive index of $n_0$ =3.48. The band gap of frequency $0.406c/a - 0.510c/a$ occurs between the third and the fourth band. When $\eta$ varies, topological edge states appear according to the bulk-edge correspondence, as shown in Fig. 1(c). The frequency range of topological edge states highly depends on the displacement parameter $\eta$, and thus the topological photonic devices of the desired frequency can be designed by adjusting $\eta$.

3. Realization of all-optical self-manipulation of optical flow in topological waveguides

All simulations are conducted using the finite element method in COMSOL Multiphysics software. We first verify the all-optical self-manipulation of light flow in a topological photonic crystal waveguide of simple one input and one output configuration, as shown in Fig. 2(a). It is composed of a topological photonic crystal waveguide with different displacement parameters on the two sides of the dashed line. The displacement parameter is set as $\eta =0.54a$ and $\eta =0.433a$ on the left and right side of the dashed line, respectively. The corresponding topological edge states are shown in Fig. 1(c). The linear transmission spectrum of the structure is shown in Fig. 2(b), which allows the light of wavelength around 1550 nm to pass through. In the process of device design, in order to better carry out optical injection, part of the medium barrel is removed at the beginning of the interface [30].

 

Fig. 2. (a) Structure diagram of the topological photonic crystal waveguide of one input and one output configuration. (b) Linear transmission spectrum of topological optical switches (The red dashed line is the position of 1550 nm). (c) Nonlinear transmission spectrum of the topological optical switch when the incident light wavelength is 1550 nm. (d) The distribution of |E| in the topological optical switch, corresponding to the three markers in (c).

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Next, we introduce the Kerr-type nonlinearity of silicon in the structure. It implies that the refractive index of silicon can be modified by the intensity of light as $n(x,y)=n_0+n_2 I (x,y)$, where $n_2 =6.8\times 10^{-18}$ ${\rm m}^2/W$ is the nonlinear coefficient of the silicon and $I (x,y)$ is the local light intensity inside the silicon cylinders [31]. Increasing the electric field of incident light $E_0$ causes the refractive index of silicon to become larger, which eventually shifts the topological edge state and the corresponding wavelength of transmission. The dependence of the normalized transmission of the incident light of wavelength 1550 nm on the electric field $E_0$ is shown in Fig. 2(c). It is clear that the transmission becomes smaller with the increase of $E_0$, so the all-optical switching ON/OFF states can be realized by modifying the intensity of incident light. Figure 2(d) shows the distribution of |E| at the different $E_0$, and the process of the optical switching behavior from ON to OFF state. Such simple model is helpful for understanding the realization of light flow self-manipulation in topological waveguides under the combined effect of synthetic dimensionality and nonlinear effects.

Based on the above work, we next design a new structure of one input and two outputs to realize a more complex self-manipulated light flow, which can control the output of light flow from different ports by controlling the $E_0$ of incident light. The structure is shown in Fig. 3(a), which consists of three topological waveguides. It starts with a bus waveguide with a displacement parameter of 0.467$a$, which allows the passage of light with wavelengths from 1396 nm to 1639 nm. Then there are two secondary topological waveguides $O_1$ and $O_2$ with a displacement parameter of 0.542a and 0.429$a$, respectively. Figure 3(b) shows the linear transmission spectrum from the outputs $O_1$ and $O_2$. In the linear case, the light of wavelengths 1549-1639 nm will output from the $O_1$ port, and light of wavelengths 1396-1540 nm will output from the $O_2$ port. This means that in the linear case, it is a wavelength router, which can spatially separate the light of different wavelengths and output them from different ports [32]. Interestingly, when we introduce Kerr nonlinearity of silicon, its functionality undergoes a significant transformation, resulting in the achievement of numerous novel and intricate functions.

 

Fig. 3. (a) Topological waveguides structure for all-optical self-manipulation of optical flow. (b) Linear transmission spectrum of topological waveguides. (c) Nonlinear transmission spectrum of the topological waveguides. The incident light wavelength is 1550 nm. (d) The distribution of |E| in the topological waveguides with different $E_0$.

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From the transmission spectrum, it can be seen that the incident light at 1550 nm will output from the $O_1$ port in the linear case. When we keep increasing the $E_0$ of incident light and consider the Kerr nonlinearity of silicon, the transmission spectrum of the structure will be shifted, as demonstrated in the first configuration. The shift of the transmission spectrum will make the transmission of the incident light at 1550 nm decrease at the $O_1$ port while increase at the $O_2$ port. The relationship between the normalized transmission and $E_0$ at the $O_1$ and $O_2$ ports is shown in Fig. 3(c). It can be seen that when the $E_0$ of the incident light at 1550 nm reaches $1.42\times 10^8$ V/m, the output port, as well as the transmission path, will be changed from the $O_1$ to $O_2$. Figure 3(d) shows the distribution of |E| in the topological photonic crystal waveguide at the different $E_0$ of incident light of wavelength 1550 nm. It is clear that the light flow transmission path changes as the $E_0$ varies, and finally the self-manipulation of the light flow in the topological photonic crystal waveguide is realized.

All-optical self-manipulation of optical flow in topological photonic crystal waveguides has many application scenarios. In optical communication systems, it can be used as a more flexible and reliable optical switch that can easily route and switch optical signals. It can also be used in optical sensors, where the detection of light intensity can be realized by detecting the output signals of two ports. In addition, through further optimization and design, it is also possible to realize the function of optical logic gate, which can control the output signal by adjusting the incident light intensity. In conclusion, because of its light intensity sensitivity and robustness, it has a wide range of applications in the field of optical communication and optical computing.

4. Demonstration of defect-immune of device

We finally test the robustness of topological photonic crystal waveguides by introducing some defects into the aforementioned structures. As shown in Fig. 4(a), we randomly introduce perturbations in the waveguides, including missing silicon cylinders, position shifts, and shape changes. The position offset is introduced randomly in the $x$- or $y$-direction, and the offset ranges from $0.03a$ to $0.05a$. The transmission spectrum of the incident light at 1550 nm is calculated, as shown in Fig. 4(b). We can see that the self-manipulation of the light flow by changing the intensity of incident light is still kept, and the perturbations only slightly change the switch threshold of electric field of incidence light. This is because the introduction of more perturbations inevitably has some effects on the linear transmission spectrum of the device itself, which will affects the variation of its nonlinear transmission spectrum. We further analyze the electric field distribution at different $E_0$, as shown in Fig. 4(c). It can be seen that the introduction of the perturbations does not cause serious backscattering and the device still works well.

 

Fig. 4. (a) Topological waveguide structure for all-optical self-manipulated light flow after the introduction of defects. On the right side are the enlarged pictures of the defects. (b) Nonlinear transmission spectra of topological waveguides after the introduction of defects. (c) The distribution of |E| in the topological waveguides with defects.

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5. Conclusion

We demonstrate the self-manipulation of light flow in topological photonic crystal waveguides based on the intrinsic Kerr nonlinearity of materials and the topological edge state designed by the principle of synthetic dimension. The path of the light flow in the waveguide can be controlled so that it can be output from different ports by adjusting the $E_0$ of the incident light at 1550 nm. In addition, our numerical simulations show that even if there is a significant disorder in the structure, it still does not affect the self-manipulation of the light flow, and exhibit good topological protected properties. Besides the silicon, other materials of a larger Kerr nonlinearity work for the all-optical self-manipulation of light flow with even a lower intensity of incident light. The results indicate that such all-optical self-manipulation of light flow in topological photonic crystal waveguides has potential applications in future integrated photonic networks.