## Abstract

In this work, we report the design of topological filter and all-optical logic gates based on two-dimensional photonic crystals with robust edge states. All major logic gates, including OR, AND, NOT, NOR, XOR, XNOR, and NAND, are suitably designed by using the linear interference approach. Moreover, numerical simulations show that our designed all-optical logic devices can always work well even if significant disorders exist. It is expected that such robust and compact logic devices have potential applications in future photonic integrated circuits.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

In recent years, there has been a great deal of interest in studying all-optical logic gates due to their important applications in all-optical networks, optical computing and optical signal processing. Since these important applications, many schemes have been proposed to realize such devices [1–8]. Traditionally, all-optical logic gates are constructed by using nonlinear optical fibers [9,10] and semiconductor optical amplifiers [2,11,12]. While, the large size of these conventional apparatus makes the proposed all-optical logic gates cannot be integrated. Recently, engineering all-optical logic gates based on photonic crystals (PhC) has received increasing attentions. With the utilization of nonlinear [13–19] and linear [20–31] PhCs, the wavelength-scale optical logic gates can be achieved. Additionally, these optical devices present the feature of lower-power consumption, only about microwatts, and faster time response, less than a few picoseconds [32–34]. Hence, all-optical logic gates based on PhCs offer the possibility of creating miniaturized optical circuits with lower losses and higher speed. However, the non-negligible backscattering always occurs when the PhC waveguide possesses sharp corners [35,36]. In this case, logical output errors are inevitable. This limits the application of PhC based all-optical logic gates.

On the other hand, the study of topological phases in photonic artificial systems has received an increasing amount of attentions [37–60]. By using the robust boundary states, many novel phenomena have been revealed, such as topological power splitters [61], selective filtering with edge solitons [62] and topological optical isolators [63]. In particular, the spatial-symmetry protected topological PhCs, which do not need of external magnetic fields and are easily to be fabricated [52,60], have wider applications. It is therefore relevant to ask whether the robust edge state in PhCs can be used to construct the all-optical logic gates.

In this work, we report the design of topological filter and all-optical logic gates based on two-dimensional PhCs with robust edge states. All major logic gates, including OR, AND, NOT, NOR, XOR, XNOR, and NAND, are suitably designed by using the linear interference approach. The properties of fault tolerance and immunity from disturbance for these devices are also demonstrated. It is expected that such robust and compact logic devices have potential applications in future photonic integrated circuits.

## 2. Modes analysis of topological photonic crystal cavity

Here, all of the optical logic devices are designed by using two-dimensional Si PhCs with triangular lattices, as shown in the inset of Fig. 1(a). The lattice constant is *a *= 1µm. The unit cell is composed of six Si dielectric cylinders with the corresponding diameter and relative permittivity being *d = *0.24*a* and *ε*=12, respectively. In addition, the absorption of Si (expressed by complex permittivity *ε _{1}*-i

*ε*) [64] at near-infrared frequency is able to be ignored (the value of

_{2}*ε*is less than 6×10

_{2}^{−3}). The distance between the center of the unit cell and the center of dielectric cylinders is

*R*. By suitably tuning the ratio between the intra- and inter-cell coupling strength (

*a*/

*R*), the topological phase transition can be realized [52,60]. When

*a*/

*R*> 3, the trivial photonic band gap appears. And, the topological phase transition occurs at the point with

*a*/

*R =*3 where the bulk band gap gets closed. Beyond the transition point (

*a*/

*R*< 3), the non-trivial bulk gap gets opened. In this work, the ratio

*a*/

*R*= 2.8 (

*a*/

*R*= 3.14) is used for the topological PhC1 (trivial PhC2).

In Fig. 1(a), we plot the dispersion relation of the ribbon-shaped supercell composed of PhC1 and PhC2. It is clearly shown that the edge state appears in the band gap. Moreover, the regular hexagon ring cavity [65,66] (length of side is 6*a*) can also be formed between the boundaries of the PhC1 and PhC2, as shown in Fig. 1(b). In Fig. 1(c), we calculate the eigenvalues of the ring cavity mode and mark them from the 1*st* to 9*th*. It is important to note that the cavity modes should be classified into two categories: traveling modes (marked by the red dots) and standing modes (marked by the green dots) [67]. This can be clearly shown by the distribution of the Poynting vectors of these cavity modes. As shown in the left chart of Fig. 1(d), the distribution of Poynting vectors for the 8*th* cavity mode (traveling mode at frequency *f _{t}* = 0.466488

*c*/

*a*corresponding to the 8

*th*mode marked in Fig. 1(c)) is plotted. We find that Poynting vectors exhibit counterclockwise feature along the cavity. In contrast, the distribution of Poynting vectors for the 3

*rd*cavity mode (standing mode at eigenfrequency

*f*= 0.448310

_{s}*c*/

*a*corresponding to the 3

*rd*mode marked in Fig. 1(c)) are perpendicular to the boundary of the cavity, as shown in the right chart of Fig. 1(d). In this case, the electromagnetic fields can only oscillate at a fixed position and cannot flow along the cavity. These different properties of cavity modes can play a key role in coupling with robust edge states.

In fact, the cavity modes we studied are whispering gallery modes (WGMs). According to the previous work [68], most modes of WGMs are traveling modes in nature. Because of the rotational symmetry, the cavity can support a pair of traveling modes with counter-propagation directions, which are degenerate in frequency. These different modes can be named as the clockwise mode (CWM) and the counterclockwise mode (CCWM), and these modes cannot couple to each other in general. When some scatterers exist, such as the 120° corner in the cavity, the degeneracy of CWM and CCWM will be lifted, and two new modes called standing modes can be formed.

We would like to point out the detail of the simulation. All the numerical simulations in this paper are supported by finite element methods (COMSOL Multiphysics) [69]. The size of meshes is set as less than 1/10 of the effective wavelength to make sure the correctness of simulations. The boundary conditions of supercell are periodic in *x* direction and finite in *y* direction, as shown in the inset of Fig. 1(a). The finite system in Fig. 1(b) is analyzed by using finite element numerical solver [69] and the wave vector is set as *k *= 0.

## 3. Coupling between topological photonic crystal cavity and edge states

To study the coupling properties between the topological cavity and edge states, we numerically study the transmission (and reflection) characters of the mixture system as shown in Fig. 2(a). Here, two topological edge states (called receiver and bus) are symmetrically positioned at two sides of the topological cavity. Two ports of the receiver (bus) are named as *T* and *R* (*DL* and *DR*), respectively. We can suitably excite the left-transferred edge states by using a modal matched light source nearby the port *R*. The intrinsic orbital angular moment of the input source (*E _{z}=E_{0}(x + iy)e^{iωt}*) ensures the effective excitation of the one-way edge state. When the cavity mode is resonantly excited, the propagated edge state within the bus can effectively couple into the receiver. However, completely different behaviors appear when the excited cavity mode is either traveling or standing mode.

As shown in Fig. 2(b), we plot output intensity (normalized) from four ports as functions of the source frequency around *f _{t}* = 0.466488

*c*/

*a*(the eigenfrequency of traveling mode). The black, red, blue and green lines represent the results of

*DR*,

*T*,

*R*and

*DL*ports, respectively. When

*f*=

*f*, the edge state can completely couple toward the

_{t}*DR*port. The corresponding steady-state distribution of electric field is shown in Fig. 2 (d). These phenomena are consistent with the functionality of a topological optical filter, where the selective filtering of specific frequency can be realized. Additionally, when

*f*=

*f*, the output energy toward

_{half}*DR*and

*T*ports possesses equal magnitude, as shown in Fig. 2(e). In Fig. 2(g), we plot the phase difference of the electric fields after (toward to

*DR*port) and before (toward to

*T*port) coupling to the receiver via travelling cavity mode. When

*f*<

*f*(

_{t}*f*>

*f*), the phase difference is +

_{t}*π*/2 (-

*π*/2). By using this phase difference, we can suitably construct all major optical logic gates based on liner interference effects. The detailed descriptions are provided in the following parts.

The coupling phenomena are completely different when the edge states are coupled with standing cavity modes. Figure 2(c) plots output intensity from four ports as a function of the source frequency near the standing mode (*f _{s}* = 0.448310

*c*/

*a*). We find that the output fields from four ports are almost equivalent when the standing cavity mode is excited. The corresponding steady-state field distribution is shown in Fig. 2 (f). In fact, similar phenomena for two waveguide modes coupling to the cavity modes (traveling and standing modes) have been explained in [67] using the coupling mode theory. According to such a theory, the equation for the evolution of the resonator mode in time domain is given by:

*a*is the amplitude of the resonator mode,

_{0}*S*is the input wave function from the

_{R-in}*R*port, it has the time dependence of

*e*. In Eq. (1), we suppose that the rate of decay is equal to the coupling coefficient in every port, and the loss is out of consideration. By solving Eq. (1), we can get:

^{ift}*S*,

_{R-out}*S*,

_{T-out}*S*, and

_{DL-out}*S*are the output wave function from the

_{DR-out}*R*,

*T*,

*DL*, and

*DR*ports respectively,

*β*is propagation constant in the waveguide, and

*d*is coupling distance. By substituting Eq. (2) into Eqs. (3)–(6), the output intensities of four ports can be expressed as:

*O*,

_{R}*O*,

_{T}*O*, and

_{DR}*O*represent the transmittivities (or reflectivity) on

_{DL}*R*,

*T*,

*DR*and

*DL*ports, respectively. Here,

*к*,

_{T}*к*,

_{R}*к*, and

_{DL}*к*are the coupling coefficients between the cavity and four ports.

_{DR}*f*is the frequency of the source and

*f*is the resonant frequency of the cavity mode.

_{0}By fitting the numerical results (Figs. 2(b) and 2(c)) with Eqs. (7)–(10), the values of *к _{T}*,

*к*,

_{R}*к*, and

_{DL}*к*can be determined. For the traveling cavity mode, we have

_{DR}*к*=

_{R}*к*= 0 and

_{DL}*к*=

_{DR}*к*= 9.3×10

_{T}^{−6}

*c*/

*a*. In Fig. 2(h), we plot the analytical results, which are consistent with the numerical simulations. Additionally, we have

*к*=

_{T}*к*=

_{R}*к*=

_{DL}*к*= 8.985×10

_{DR }^{−7}

*c*/

*a*for the standing cavity mode. The coupling coefficients are determined by the overlapping region size of the evanescent waves of the cavity and the waveguides. So, the values of the coupling coefficients increase (decrease) when the gapped region becomes smaller (bigger). The corresponding analytical results are given in Fig. 2(i), which are also consistent with the numerical simulations.

In addition, zig-zag boundary and scattering boundary conditions are used in order to reduce the reflection influence at the end of waveguides.

## 4. Realization of topological all-optical logic gates

Based on the effective coupling between the traveling cavity mode and topological edge states, in the following, we design seven topological all-optical logic gates, including OR, AND, NOT, NOR, XOR, XNOR, and NAND gates. The working frequency is around *f _{half}*, in this case, the phase variation with the field coupled to the other edge channel via the traveling cavity mode is +

*π*/2, as shown in Fig. 2(g).

#### 4.1 XOR, OR, and NOT gates

Firstly, we focus on the design of XOR and OR gates. The schematic diagram is shown in Fig. 3(a). The light source L1 ($E{e^{i{\varphi _1}}}$) and L2 ($E{e^{i{\varphi _2}}}$) are placed at the left-upper and right-lower ports, respectively. The phase difference between L1 and L2 is $\varDelta \varphi = {\varphi _1} - {\varphi _2} = {\pi \mathord{\left/ {\vphantom {\pi 2}} \right.} 2}$ to realize the ideal interference effect. Here, we set ${\varphi _1} = {\pi \mathord{\left/ {\vphantom {\pi 2}} \right.} 2}$ and ${\varphi _2} = 0$. The input amplitude of *E* (0) corresponds the logical input state being 1 (0). The intensity of output fields from upper-right and lower-left ports marks the output logical estimation of OR and XOR gates, respectively. By suitably engineering the interference between the two sources, the ideal XOR and OR gates can be realized.

When L1 and L2 are both turned-off (input 00), it is easy to show that the amplitudes of output fields at XOR and OR ports are both zero (output 0), as shown in Fig. 3(e). Based on the wave interference analyzes, as shown in Figs. 3(b) and 3(c), when one of the input sources is turned-off and the other one is turned-on (input 01 or 10), the output amplitudes of XOR and OR ports are both ${{\sqrt 2 E} \mathord{\left/ {\vphantom {{\sqrt 2 E} 2}} \right.} 2}$ (output 1). This can be further demonstrated with simulations, as shown in Figs. 3(f) and 3(g). Moreover, when L1 and L2 are both turned-on (input 11), the destructive and constructive interference happen on the XOR and OR ports, respectively, as shown in Figs. 3(d) and 3(h). In this case, the output fields on the XOR and OR ports become 0 (output 0) and $\sqrt 2 E{e^{{{i\pi } \mathord{\left/ {\vphantom {{i\pi } 2}} \right.} 2}}}$ (output 1), respectively. The above four kinds of input and output operations are summed up in Table 1, which is identical with the actual truth table for the XOR and OR gates.

The designed logical devices can work well in a wide frequency range near *f _{half}*. The transmission spectrums of OR and XOR ports are shown in Figs. 3(i) and 3(j), respectively. We can measure the efficiency of the logical gate by calculating the extinction ratio (ER), which is defined as 10log(

*I*/

_{1}*I*). Here,

_{0}*I*(

_{1}*I*) is the output energy corresponding to the 1 (0) logical operation. As shown in Fig. 3(k), a large ER of the XOR gate (>20dB) exists in a wide frequency range. The footprint of this device is 50µm×45µm.

_{0}We would like to point out that the relationship between the output power (*P _{out}*) and the amplitude of the output wave function (

*E*) is expressed as:${P_{out}} = E_{}^2$, where

*P*is shown as the intensity in Figs. 3(i) and 3(j);

_{out}*E*is shown in Table 1.

As for the NOT gate, it can be achieved with the same system in Fig. 3(a). We suppose L1 as a bias light (BL) and always turned-on. L2 corresponds to the input signal. The intensity of output field from lower-left port marks the output logical estimation of the NOT gate. The numerical results in Figs. 3(g) and 3(h) correspond to the case when the input signals are 0 and 1, respectively. We list these two different input and output operations in Table 2, which are identical with the actual truth table for the NOT gate.

#### 4.2 XNOR gate

The XNOR gate can be formed by integrating two basic devices, which are connected by a middle waveguide with phase accumulation being *ψ*, which is determined by the length of the waveguide, as shown in Fig. 4(a). Three light sources L1 ($E{e^{i{\varphi _1}}}$), L2 ($E{e^{i{\varphi _2}}}$) and BL (${{\sqrt 2 E{e^{i(\psi - \pi )}}} \mathord{\left/ {\vphantom {{\sqrt 2 E{e^{i(\psi - \pi )}}} 2}} \right.} 2}$) should be used to realize the ideal interference effect. It is important to note that the accumulated phase (*ψ*) during wave propagation in the middle waveguide possesses a key role in the formation of the ideal destructive interference between two different basic devices. In this case, if the length of the middle waveguide is changed, the initial phase of the BL source should also be altered to realize the required destructive interference effect. Actually, when the length of the middle waveguide is chosen to be 7µm, the accumulated phase is *π*. For simply, we set *φ _{1}*=

*π*/2,

*φ*0, and

_{2}=*ψ=π*.

When L1 and L2 are both turned-off (input 00), the output amplitude toward the XNOR port is *E*/2 (output 1) resulting from the BL source coupled via the cavity. The corresponding simulation result is presented in Fig. 4(b). When one of the input sources is turned-off and the other one is turned-on (input 01 or 10), the completely destructive interference happens on the XNOR port (output 0). This can be clearly seen in Figs. 4(c) and 4(d). Additionally, when L1 and L2 are both turned-on (input 11), the output field from the left part of the device is $\sqrt 2 E{e^{{{i\pi } \mathord{\left/ {\vphantom {{i\pi } 2}} \right.} 2}}}$. It can interfere with the source BL and the output amplitude from the XNOR port is *E*/2 (output 1). The simulation result is plotted in Fig. 4(e). The above four kinds of input and output operations are summed up in Table 3, which is identical with the actual truth table for the XNOR gate.

The designed XNOR gate can also possess a wide operation regime. In Fig. 4(f), we show the transmission spectrum of the XNOR port with four different input signals. The ER of the XNOR gate is calculated in Fig. 4(g). It is clearly shown that a large ERs of the XOR gate (>20 dB) exists near the operation frequency *f _{half}*. The footprint of this device is 100µm×45µm.

#### 4.3 NAND gate

The scheme for the realization of the NAND gate is shown in Fig. 5(a). The designed parameters are the same to the XNOR gate, except for the amplitude of BL is changing to be $\sqrt 2 E$. In this case, the ideal destructive interference appears only when both sources are turned-on (input 11). The output amplitude is nonzero for other cases (inputs 00, 01 and 10). The analytical processes and numerical simulations are shown in Figs. 5(b)–5(e). Four different input and output operations are summed up in Table 4, which is identical with the truth table for the NAND gate. Similarly, the designed the NAND gate can also operate in a wide regime, as shown in Figs. 5(f) and 5(g). The footprint of this device is 100µm×45µm.

#### 4.4 NOR gate

The schematic diagram for the realization of NOR gate is shown in Fig. 6(a). The NOR gate is composed of two basic parts (OR and NOT gates), which are proposed in previous sections. The only difference for the used parameter is that the BL amplitude is set as ${{3\sqrt 2 E} \mathord{\left/ {\vphantom {{3\sqrt 2 E} 4}} \right.} 4}$ to achieve the ideal logical operation.

The interference analyzes and numerical simulations with four different logical inputs (00, 01, 10, 11) are shown in Figs. 6(b)–6(e). When L1 and L2 are both turned-off (input 00), the output amplitude toward the NOR port is 3*E*/4 (output 1) resulting from BL source coupled via the cavity. The corresponding simulation result is presented in Fig. 6(b). When one of the input sources is turned-off and the other one is turned-on (input 01 or 10), the destructive interference happens at the NOR port. This can be clearly seen in Figs. 6(c) and 6(d). The output amplitude is *E*/4, which is almost an order of magnitude smaller than that of output 1. Hence, it is feasible to set *E*/4 as logic 0. Additionally, when L1 and L2 are both turned-on (input 11), the output amplitude from the NOR port is also *E*/4 (logic 0). The simulation result is plotted in Fig. 6(e). The above four kinds of input and output operations are summed up in Table 5, which is identical with the actual truth table for the NOR gate. In Fig. 6(f), It is shown that the transmission spectrum of the NOR port with four different input signals. The ERs of the NOR gate is calculated in Fig. 6(g). The footprint of this device is 100µm×45µm.

#### 4.5 AND gate

In Fig. 7(a), we propose the scheme to construct the AND gate, which is composed of three basic parts. The left two parts correspond to the NAND gate and the right part is the NOT gate. It is noted for the used parameter is that the BL_{1} (BL_{2}) amplitude is set as $\sqrt 2 E$ (3*E*/4) to achieve the ideal interference effect.

The interference analyzes and numerical simulations with four different logical inputs (00, 01, 10, 11) are shown in Figs 7(b)–7(e). Similar to the NOR gate, the output amplitude of logic 1 (0) is set as ${{3\sqrt 2 E} \mathord{\left/ {\vphantom {{3\sqrt 2 E} 8}} \right.} 8}$ (${{\sqrt 2 E} \mathord{\left/ {\vphantom {{\sqrt 2 E} 8}} \right.} 8}$), so the intensity ratio between output logic 1 and 0 is 9:1. The four different input and output operations are summed up in Table 6 and is identical with the actual truth table for the AND gate. Fig. 7(f) is the transmission spectrum of the AND port with four different input signals. The ERs of the AND gate is shown in Fig. 7(g). The footprint of AND gate is 150µm×45µm.

## 5. Demonstration of defect-immune of topological all-optical logic gates

Our designed topological all-optical logic gates possess ultrafast operation speed and low power consumption, which are the same as the original trivial all-optical logic gates proposed in [13–30]. Moreover, our proposed all-optical logic gates are robust with some disorders due to the topologically-protected edge state. In this part, we prove these phenomena by adding some defects in the waveguide and cavity.

#### 5.1 Defect in waveguide

We take an example of the NAND gate to demonstrate the robust property in all-optical logic gate. As shown in Fig. 8, a defect is added in the topological waveguide, but the correct logic outputs still appear. It's important to note that the phase of BL should be adjusted to fulfill the ideal interference effect. This is due to the fact that the phase difference between BL and the light from the middle waveguide must be *π*/2.

#### 5.2 Defect in cavity

We use XOR and OR gates to demonstrate robust logical operation when a defect exists in the topological cavity. As shown in Fig. 9(a), a unit cell is missed in the cavity. The numerical results for the input state being 10, 01 and 11 are plotted in Figs. 9(b)–9(d). We find that although the cavity is disturbed, the XOR and OR gates still work well.

The above results are only for Si dielectric cylinder structure, which supports TM mode. We propose a set of realizable schemes of topological all-optical logic gates. They can be fabricated experimentally [70,71], although it is a little difficult [72–74]. We can also use a traditional air-hole type structure [53] (TE mode) in order to reduce the difficulty in experimental verification (see Appendix A for details).

We plot a table in Appendix B to compare previously reported researches (non-topological logic gate) and ours [14,17,20,28,30,75–77] as shown in Table 7. It shows that our gates realize all logic functions and possess the robust property.

## 6. Summary

Based on the coupling mode theory and accurate numerical simulations, we have investigated the coupling properties between the topological PhC cavity and edge states. The different coupling characteristics between various cavity modes and topological edge states have been disclosed. Topological filter has been realized by using the traveling cavity mode. Furthermore, we have designed seven all-optical logic gates, including OR, AND, NOT, NOR, XOR, XNOR and NAND. The corresponding logic functions have been demonstrated by accurate numerical simulations. Importantly, our numerical simulations have shown that these all-optical logic devices can always work well even if significant disorders exist, which exhibit strongly topologically protected properties. We believe that these robust and compact logic devices have potential applications in future photonic computing.

## Appendix A

Here, we give a scheme of all-optical topological logic gates by using two-dimensional Si PhCs (triangular air-holes drilled in Silicon) [53]. As shown in Fig. 10(a), we use a lattice constant of *a* = 1µm, an edge length of the equilateral triangle of *s* =${{\sqrt {3} a} \mathord{\left/ {\vphantom {{\sqrt {3} a} 6}} \right.} 6}$, the distance *R* from the center of a cell to the centroid of a triangle. In this structure, it is topological when *R *> *a*/3 and trivial when *R *< *a*/3. We concentrically shift the triangular holes by increasing *R* to 1.06*a*/3 to get topological PhC1, and by decreasing *R* to 0.88*a*/3 to get trivial PhC2. These parameters ensure opening roughly the same band gap of PhCs structure. It is noted that TE mode is supported in this model. For demonstrating the robust property of the device, a defect is added at OR port.

The topological cavity is built and one of its traveling modes is at *f _{0}=*116.15THz, we choose the excitation frequency at

*f*116.31Thz. It is same as Section 4.1, we propose the schematic diagrams of OR and XOR gates in Fig. 10(b). The numerical simulation results are given in Figs. 10(c)–10(e) corresponding to input states 01,10, and 11.

_{half}=The gate can operate in a wide regime (116.28THz–116.36THz) as shown in Fig. 10(f) and the ER is given in Fig. 10(g). By integrating two or three basic devices, AND, NAND, XNOR, and NOR gates are suitably designed via the methods mentioned in Section 4.

The above discussions only focus on the topological all-optical logic gates based on the 2D air-hole structure. In fact, the finite thickness sample with the air-hole structure is important to the optical experimental design. The corresponding results in 3D PhC slab with the finite thickness may be also obtained in the same method as described in Ref.[53], and the experimental confirmation can be expected in the future.

## Appendix B

The comparison table with the previously main reported researches [14,17,20,28,30,75–77].

## Funding

National key R&D Program of China (2017YFA0303800); National Natural Science Foundation of China (91850205, 61421001); Graduate Technological Innovation Project of Beijing Institute of Technology (2019CX20046).

## References

**1. **T. Tanabe, M. Notomi, S. Mitsugi, A. Shinya, and E. Kuramochi, “Fast bistable all-optical switch and memory on a silicon photonic crystal on-chip,” Opt. Lett. **30**(19), 2575–2577 (2005). [CrossRef]

**2. **S. Ma, Z. Chen, H. Sun, and N. K. Dutta, “High speed all optical logic gates based on quantum dot semiconductor optical amplifiers,” Opt. Express **18**(7), 6417–6422 (2010). [CrossRef]

**3. **S. D. Smith, “Optical bistability, photonic logic, and optical computation,” Appl. Opt. **25**(10), 1550–1564 (1986). [CrossRef]

**4. **D. Hillerkuss, R. Schmogrow, T. Schellinger, M. Jordan, M. Winter, G. Huber, T. Vallaitis, R. Bonk, P. Kleinow, F. Frey, M. Roeger, S. Koenig, A. Ludwig, A. Marculescu, J. Li, M. Hoh, M. Dreschmann, J. Meyer, S. Ben Ezra, N. Narkiss, B. Nebendahl, F. Parmigiani, P. Petropoulos, B. Resan, A. Oehler, K. Weingarten, T. Ellermeyer, J. Lutz, M. Moeller, M. Hübner, J. Becker, C. Koos, W. Freude, and J. Leuthold, “26 Tbit s^{−1} line-rate super-channel transmission utilizing all-optical fast Fourier transform processing,” Nat. Photonics **5**(6), 364–371 (2011). [CrossRef]

**5. **L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E. J. Geluk, T. Vries, P. Regreny, D. V. Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photonics **4**(3), 182–187 (2010). [CrossRef]

**6. **K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, “Sub-femtojoule all-optical switching using a photonic-crystal nanocavity,” Nat. Photonics **4**(7), 477–483 (2010). [CrossRef]

**7. **A. E. Willner, S. Khaleghi, M. R. Chitgarha, and O. F. Yilmaz, “All-optical signal processing,” J. Lightwave Technol. **32**(4), 660–680 (2014). [CrossRef]

**8. **Q. Wang, G. Zhu, H. Chen, J. Jaques, J. Leuthold, A. B. Piccirilli, and N. K. Dutta, “Study of all-optical XOR using Mach-Zehnder interferometer and differential scheme,” IEEE J. Quantum Electron. **40**(6), 703–710 (2004). [CrossRef]

**9. **J. W. M. Menezes, W. B. De Fraga, A. C. Ferreira, K. D. A. Saboia, A. F. G. F. Filho, G. F. Guimarães, J. R. R. Sousa, H. H. B. Rocha, and A. S. B. Sombra, “Logic gates based in two-and three-modes nonlinear optical fiber couplers,” Opt. Quantum Electron. **39**(14), 1191–1206 (2007). [CrossRef]

**10. **J. R. R. Sousa, A. C. Ferreira, G. S. Batista, C. S. Sobrinho, A. M. Bastos, M. L. Lyra, and A. S. B. Sombra, “Generation of logic gates based on a photonic crystal fiber Michelson interferometer,” Opt. Commun. **322**, 143–149 (2014). [CrossRef]

**11. **K. E. Stubkjaer, “Semiconductor optical amplifier-based all-optical gates for high-speed optical processing,” IEEE J. Sel. Top. Quantum Electron. **6**(6), 1428–1435 (2000). [CrossRef]

**12. **J. H. Kim, Y. M. Jhon, Y. T. Byun, S. Lee, D. H. Woo, and S. H. Kim, “All-optical XOR gate using semiconductor optical amplifiers without additional input beam,” IEEE Photonics Technol. Lett. **14**(10), 1436–1438 (2002). [CrossRef]

**13. **F. Mehdizadeh and M. Soroosh, “Designing of all optical NOR gate based on photonic crystal,” Indian J. Pure Appl. Phys. **54**(01), 35–39 (2016).

**14. **Y. Liu, F. Qin, Z. M. Meng, F. Zhou, Q. H. Mao, and Z. Y. Li, “All-optical logic gates based on two-dimensional low-refractive-index nonlinear photonic crystal slabs,” Opt. Express **19**(3), 1945–1953 (2011). [CrossRef]

**15. **Q. Liu, Z. Ouyang, C. J. Wu, C. P. Liu, and J. C. Wang, “All-optical half adder based on cross structures in two-dimensional photonic crystals,” Opt. Express **16**(23), 18992–19000 (2008). [CrossRef]

**16. **B. M. Isfahani, T. A. Tameh, N. Granpayeh, and A. R. M. Javan, “All-optical NOR gate based on nonlinear photonic crystal microring resonators,” J. Opt. Soc. Am. B **26**(5), 1097–1102 (2009). [CrossRef]

**17. **P. Andalib and N. Granpayeh, “All-optical ultracompact photonic crystal AND gate based on nonlinear ring resonators,” J. Opt. Soc. Am. B **26**(1), 10–16 (2009). [CrossRef]

**18. **M. Ghadrdan and M. A. Mansouri-Birjandi, “Concurrent implementation of all-optical half-adder and AND and XOR logic gates based on nonlinear photonic crystal,” Opt. Quantum Electron. **45**(10), 1027–1036 (2013). [CrossRef]

**19. **H. Alipour-Banaei, S. Serajmohammadi, and F. Mehdizadeh, “All optical NOR and NAND gate based on nonlinear photonic crystal ring resonators,” Optik **125**(19), 5701–5704 (2014). [CrossRef]

**20. **Y. Zhang, Y. Zhang, and B. Li, “Optical switches and logic gates based on self-collimated beams in two-dimensional photonic crystals,” Opt. Express **15**(15), 9287–9292 (2007). [CrossRef]

**21. **T. T. Kim, S. G. Lee, H. Y. Park, J. E. Kim, and C. S. Kee, “Asymmetric Mach-Zehnder filter based on self-collimation phenomenon in two-dimensional photonic crystals,” Opt. Express **18**(6), 5384–5389 (2010). [CrossRef]

**22. **R. Fan, X. Yang, X. Meng, and X. Sun, “2D photonic crystal logic gates based on self-collimated effect,” J. Phys. D: Appl. Phys. **49**(32), 325104 (2016). [CrossRef]

**23. **D. Zhao, J. Zhang, P. Yao, X. Jiang, and X. Chen, “Photonic crystal Mach-Zehnder interferometer based on self-collimation,” Appl. Phys. Lett. **90**(23), 231114 (2007). [CrossRef]

**24. **W. Zhang and X. Zhang, “Backscattering-Immune computing of spatial differentiation by nonreciprocal plasmonics,” Phys. Rev. Appl. **11**(5), 054033 (2019). [CrossRef]

**25. **F. Parandin, M. R. Malmir, M. Naseri, and A. Zahedi, “Reconfigurable all-optical NOT, XOR, and NOR logic gates based on two dimensional photonic crystals,” Superlattices Microstruct. **113**, 737–744 (2018). [CrossRef]

**26. **P. Rani, Y. Kalra, and R. K. Sinha, “Design of all optical logic gates in photonic crystal waveguides,” Optik **126**(9–10), 950–955 (2015). [CrossRef]

**27. **C. Tang, X. Dou, Y. Lin, H. Yin, B. Wu, and Q. Zhao, “Design of all-optical logic gates avoiding external phase shifters in a two-dimensional photonic crystal based on multi-mode interference for BPSK signals,” Opt. Commun. **316**, 49–55 (2014). [CrossRef]

**28. **P. Rani, Y. Kalra, and R. K. Sinha, “Realization of AND gate in Y shaped photonic crystal waveguide,” Opt. Commun. **298–299**, 227–231 (2013). [CrossRef]

**29. **Y. Ishizaka, Y. Kawaguchi, K. Saitoh, and M. Koshiba, “Design of ultra compact all-optical XOR and AND logic gates with low power consumption,” Opt. Commun. **284**(14), 3528–3533 (2011). [CrossRef]

**30. **W. Liu, D. Yang, G. Shen, H. Tian, and Y. Ji, “Design of ultra compact all-optical XOR, XNOR, NAND and OR gates using photonic crystal multi-mode interference waveguides,” Opt. Laser Technol. **50**, 55–64 (2013). [CrossRef]

**31. **V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, *Handbook of nonlinear optical crystals* (Springer).

**32. **A. Salmanpour, S. Mohammadnejad, and A. Bahrami, “Photonic crystal logic gates: an overview,” Opt. Quantum Electron. **47**(7), 2249–2275 (2015). [CrossRef]

**33. **Y. Fu, X. Hu, and Q. Gong, “Silicon photonic crystal all-optical logic gates,” Phys. Lett. A **377**(3–4), 329–333 (2013). [CrossRef]

**34. **Q. Xu and M. Lipson, “All-optical logic based on silicon micro-ring resonators,” Opt. Express **15**(3), 924–929 (2007). [CrossRef]

**35. **A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. **77**(18), 3787–3790 (1996). [CrossRef]

**36. **B. Wang, S. Mazoyer, J. P. Hugonin, and P. Lalanne, “Backscattering in monomode periodic waveguides,” Phys. Rev. B **78**(24), 245108 (2008). [CrossRef]

**37. **T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. **91**(1), 015006 (2019). [CrossRef]

**38. **A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. **12**(3), 233–239 (2013). [CrossRef]

**39. **M. I. Shalaev, W. Walasik, A. Xu, Y. Tsukernik, and N. M. Litchinitser, “Robust topologically protected transport in photonic crystals at telecommunication wavelengths,” Nat. Nanotechnol. **14**(1), 31–34 (2019). [CrossRef]

**40. **S. Stützer, Y. Plotnik, Y. Lumer, P. Titum, N. H. Lindner, M. Segev, M. C. Rechtsman, and A. Szameit, “Photonic topological Anderson insulators,” Nature **560**(7719), 461–465 (2018). [CrossRef]

**41. **Y. Yang, Z. Gao, H. Xue, L. Zhang, M. He, Z. Yang, R. Singh, Y. Chong, B. Zhang, and H. Chen, “Realization of a three-dimensional photonic topological insulator,” Nature **565**(7741), 622–626 (2019). [CrossRef]

**42. **C. He, X. C. Sun, X. P. Liu, M. H. Lu, Y. Chen, L. Feng, and Y. F. Chen, “Photonic topological insulator with broken time-reversal symmetry,” Proc. Natl. Acad. Sci. **113**(18), 4924–4928 (2016). [CrossRef]

**43. **F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. **100**(1), 013904 (2008). [CrossRef]

**44. **S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A **78**(3), 033834 (2008). [CrossRef]

**45. **Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljačić, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. **100**(1), 013905 (2008). [CrossRef]

**46. **K. Fang, Z. Yu, and S. Fan, “Realizing effective magnetic field for photons by controlling the phase of dynamic modulation,” Nat. Photonics **6**(11), 782–787 (2012). [CrossRef]

**47. **L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics **8**(11), 821–829 (2014). [CrossRef]

**48. **L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological states in photonic systems,” Nat. Phys. **12**(7), 626–629 (2016). [CrossRef]

**49. **S. A. Skirlo, L. Lu, Y. Igarashi, Q. Yan, J. Joannopoulos, and M. Soljačić, “Experimental observation of large Chern numbers in photonic crystals,” Phys. Rev. Lett. **115**(25), 253901 (2015). [CrossRef]

**50. **Y. Yang, H. Jiang, and Z. H. Hang, “Topological valley transport in two-dimensional honeycomb photonic crystals,” Sci. Rep. **8**(1), 1588 (2018). [CrossRef]

**51. **M. I. Shalaev, S. Desnavi, W. Walasik, and N. M. Litchinitser, “Reconfigurable topological photonic crystal,” New J. Phys. **20**(2), 023040 (2018). [CrossRef]

**52. **L. H. Wu and X. Hu, “Scheme for achieving a topological photonic crystal by using dielectric material,” Phys. Rev. Lett. **114**(22), 223901 (2015). [CrossRef]

**53. **S. Barik, H. Miyake, W. DeGottardi, E. Waks, and M. Hafezi, “Two-dimensionally confined topological edge states in photonic crystals,” New J. Phys. **18**(11), 113013 (2016). [CrossRef]

**54. **T. Ma and G. Shvets, “All-Si valley-Hall photonic topological insulator,” New J. Phys. **18**(2), 025012 (2016). [CrossRef]

**55. **L. Xu, H. X. Wang, Y. D. Xu, H. Y. Chen, and J. H. Jiang, “Accidental degeneracy in photonic bands and topological phase transitions in two-dimensional core-shell dielectric photonic crystals,” Opt. Express **24**(16), 18059–18071 (2016). [CrossRef]

**56. **F. Liu, H. Y. Deng, and K. Wakabayashi, “Topological photonic crystals with zero Berry curvature,” Phys. Rev. B **97**(3), 035442 (2018). [CrossRef]

**57. **Z. Chai, X. Hu, F. Wang, C. Li, Y. Ao, Y. Wu, Y. Wu, K. Shi, H. Yang, and Q. Gong, “Ultrafast on-Chip Remotely-Triggered All-Optical Switching Based on Epsilon-Near-Zero Nanocomposites,” Laser Photonics Rev. **11**(5), 1700042 (2017). [CrossRef]

**58. **L. Zhang and S. Xiao, “Design of terahertz reconfigurable devices by locally controlling topological phases of square gyro-electric rod arrays,” Opt. Mater. Express **9**(2), 544–554 (2019). [CrossRef]

**59. **X. T. He, E. T. Liang, J. J. Yuan, H. Y. Qiu, X. D. Chen, F. L. Zhao, and J. W. Dong, “A silicon-on-insulator slab for topological valley transport,” Nat. Commun. **10**(1), 872 (2019). [CrossRef]

**60. **Y. Yang, Y. F. Xu, T. Xu, H. X. Wang, J. H. Jiang, X. Hu, and Z. H. Hang, “Visualization of a unidirectional electromagnetic waveguide using topological photonic crystals made of dielectric materials,” Phys. Rev. Lett. **120**(21), 217401 (2018). [CrossRef]

**61. **X. D. Chen, Z. L. Deng, W. J. Chen, J. R. Wang, and J. W. Dong, “Manipulating pseudospin-polarized state of light in dispersion-immune photonic topological metacrystals,” Phys. Rev. B **92**(1), 014210 (2015). [CrossRef]

**62. **D. Leykam and Y. D. Chong, “Edge solitons in nonlinear-photonic topological insulators,” Phys. Rev. Lett. **117**(14), 143901 (2016). [CrossRef]

**63. **R. El-Ganainy and M. Levy, “Optical isolation in topological-edge-state photonic arrays,” Opt. Lett. **40**(22), 5275–5278 (2015). [CrossRef]

**64. **M. A. Green and M. J. Keevers, “Optical properties of intrinsic silicon at 300 K,” Prog. Photovoltaics **3**(3), 189–192 (1995). [CrossRef]

**65. **Y. Yang and Z. H. Hang, “Topological whispering gallery modes in two-dimensional photonic crystal cavities,” Opt. Express **26**(16), 21235–21241 (2018). [CrossRef]

**66. **G. Siroki, P. A. Huidobro, and V. Giannini, “Topological photonics: From crystals to particles,” Phys. Rev. B **96**(4), 041408 (2017). [CrossRef]

**67. **C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. **35**(9), 1322–1331 (1999). [CrossRef]

**68. **Y. Zhi, X. C. Yu, Q. Gong, L. Yang, and Y. F. Xiao, “Single nanoparticle detection using optical microcavities,” Adv. Mater. **29**(12), 1604920 (2017). [CrossRef]

**70. **M. A. Gorlach, X. Ni, D. A. Smirnova, D. Korobkin, D. Zhirihin, A. P. Slobozhanyuk, P. A. Belov, A. Alù, and A. B. Khanikaev, “Far-field probing of leaky topological states in all-dielectric metasurfaces,” Nat. Commun. **9**(1), 909 (2018). [CrossRef]

**71. **S. Peng, N. J. Schilder, X. Ni, J. van de Groep, M. L. Brongersma, A. Alù, A. B. Khanikaev, H. A. Atwater, and A. Polman, “Probing the Band Structure of Topological Silicon Photonic Lattices in the Visible Spectrum,” Phys. Rev. Lett. **122**(11), 117401 (2019). [CrossRef]

**72. **A. Adibi, Y. Xu, R. Lee, A. Yariv, and A. Scherer, “Properties of the slab modes in photonic crystal optical waveguides,” J. Lightwave Technol. **18**(11), 1554–1564 (2000). [CrossRef]

**73. **C. Monat, B. Corcoran, D. Pudo, M. Ebnali-Heidari, C. Grillet, M. Pelusi, D. Moss, B. Eggleton, T. White, and T. Krauss, “Slow light enhanced nonlinear optics in silicon photonic crystal waveguides,” IEEE J. Sel. Top. Quantum Electron. **16**(1), 344–356 (2010). [CrossRef]

**74. **V. Jandieri, R. Khomeriki, and D. Erni, “Realization of True All-Optical AND Logic Gate based on the Nonlinear Coupled Air-hole Type Photonic Crystal Waveguide,” Opt. Express **26**(16), 19845–19853 (2018). [CrossRef]

**75. **M. H. Rezaei, A. Zarifkar, M. Miri, and O. Materials, “Ultra-compact electro-optical graphene-based plasmonic multi-logic gate with high extinction ratio,” Opt. Mater. **84**, 572–578 (2018). [CrossRef]

**76. **M. H. Rezaei and A. Zarifkar, “Dielectric-loaded graphene-based plasmonic multilogic gate using a multimode interference splitter,” Appl. Opt. **57**(35), 10109–10116 (2018). [CrossRef]

**77. **X. Wu, J. Tian, and R. Yang, “A type of all-optical logic gate based on graphene surface plasmon polaritons,” Opt. Commun. **403**, 185–192 (2017). [CrossRef]