In recent years, the human demand for information has increased exponentially, necessitating faster data transmission capacity in the transmission network, thereby also necessitating faster processors and optic fiber systems. The existing semiconductor devices are about to reach a bottleneck because of the influence of electronic speed, and optical devices have the characteristics of high-speed processing and of not being easily affected by electromagnetic noise, so the future use of optical devices is foreseeable^[1-2]. Compared to traditional electronic devices, photonic crystal logic devices have lower energy consumption. Through research on the design and optimization of photonic crystal logic devices, their energy consumption can be further reduced and more energy-efficient information processing systems can be achieved. Photonic crystal logic devices have a wide range of application potential, such as photon computing, photon communication, quantum computing, etc.

All-optical logic devices are very important in optical signal processing^[3]. In recent years, optical logic devices^[4-5] have been implemented by some teams for spatial coding of optical fields^[6-7], semiconductor optical amplifiers^[8-9], highly nonlinear optical fibers^[10], micro and nanoscale waveguides^[11], one-dimensional magnetized InSb photonic crystals layered topology^[12], Janus metastructures^[13] and photonic crystal structures^[14-15]. At present, all-optical logic devices designed with photonic crystal structure are more popular among designers with smaller occupied area and manufacturing cost. Photonic crystal is a synthetic material in which several materials with different dielectric constants are arranged according to certain rules in space, and has two characteristics: photonic band gap and photonic local area, and a structure divisible into one, two and three dimensions^[16].The introduction of defects into two-dimensional photonic crystals can guide the transmission of light waves. If the frequency of incident light is within the photonic band gap, the light can be trapped in the photonic crystal structure, and the resulting defective photonic crystals are applied to many optical devices^[17], such as filters^[18], wavelength division multiplexers^[19-20], optical fibers^[21], beam splitters^[22] and optical logic devices^[23-24]. In addition to some photonic crystal applications in need of solving, such as sensors^[25], solar cells^[26], photodetectors^[27] and so on. Photonic crystals can be used to realize optical logic devices by a variety of methods, such as multimode interference^[28], autocollimation transmission^[29], nonlinear effect and linear interference, among which the optical logic devices designed using linear interference effect and nonlinear effect are simple in structure and suitable for large-scale integration. In order to remove the interference frequency signal and enhance the coupling efficiency of the output end, a filter structure can be added to the output end to select the frequency of the output light wave. Commonly used structures include microcavity structure and ring cavity structure, and the filtered frequency can be adjusted by setting the radius of the medium column. The designers of the device’s basic logic gate structure make use of linear interference effect so that the designed device has a simple structure and fast response speed. Designing devices solely based on linear interference effects generally cannot achieve high efficiency and high contrast output, so the use of nonlinear materials is necessary. In 2020^[29], Bouaouina M S et al. designed a common logic gate using photonic crystal structure on the basis of defect interference and resonance coupling. The structure of the gate is complex and the contrast between non, same or gate logic is too low. In 2022^[30], Tanay Chattopadhyay et al. proposed a series of logic gates based on complementary photonic crystal logic devices. The overall area of the devices is large and the logical contrast is not high. In 2023^[31], Vadivu N S et al. proposed a triangular lattice two-dimensional photonic crystal structure based on the linear interference effect. The designed logic gate structure is simple and works well in both the transverse magnetic mode (TMM) and the transverse electric mode (TEM) polarization modes, but the phase of the input signal needs to be controlled while operating, and the logical contrast of the device is low.

In this paper, an all-optical logic gate with simple structure and high logical contrast is designed by combining nonlinear effect and linear interference effect. XOR, NOT and two-input AND logic gates are designed in TEM polarization mode. For more complex logic expressions, it converts into a cascaded combination of existing logic gates through the inversion theorem. Thus, a more complex or non-sum four-input AND gate logic device is realized. The proposed device can also be designed as a logic gate with more input by expanding the input.

A nonlinear ring cavity is used in the device structure designed in this paper. When the light intensity of the output waveguide changes, the refractive index of the light-sensitive material will change and the coupled light wave will change in conjunction therewith, so that the wavelength that can be coupled by the nonlinear ring cavity will change^[32-33]. The refractive index of many nonlinear materials depends on the light intensity, and its refractive index n is expressed as follows:

$ n = n_0 + n_2I\quad, $ (1)

where n₀ is the weak-field refractive index, and the product factor n₂ is the nonlinear refractive index^[34]. The above formula shows that the refractive index of the nonlinear material increases as a result of the increase of the light intensity I. Assuming that the field is linearly polarized, the total polarization is caused by second- and third-order nonlinearities:

$ P = {\varepsilon _0}({\chi ^{\left( 1 \right)}}E + {\chi ^{\left( 2 \right)}}EE + {\chi ^{\left( 3 \right)}}EEE) \equiv {P^{(1)}} + {P^{(2)}} + {P^{(3)}}, $ (2)

where χ⁽²⁾ and χ⁽³⁾ are second- and third-order nonlinear polarizability. Assuming that the nonlinear material has crystal symmetry, the second-order magnetic susceptibility can be ignored, and the nonlinear refractive index n₂ can be obtained:

$ n_2 = \frac{{3{\chi ^{\left( 3 \right)}}}}{{4n_0^2}}Z_0 \quad,$ (3)

where Z₀=376.7 Ω, is the free space impedance. The photonic crystal has two kinds of structure: air column and medium column. The air column is easy to manufacture, and the medium column has the advantage of strong coupling. All the designs in this paper adopt the photonic crystal with a cubic lattice dielectric column structure. The lattice constant a=0.54 μm and the filling factor r/a=0.18 are set. The material of dielectric column is Si, and the refractive index n is 3.46. The characteristics of the nonlinear ring cavity can be understood through the study of the output characteristics of the structure in Figure 1(a) (color online), where R is a nonlinear ring cavity and the material is doped glass, whose weak-field refractive index n₀=1.4 and nonlinear Kerr coefficient χ⁽³⁾=10⁻¹⁴ m²/V², the radius of the dielectric column is 0.18a.^[35-36] The structure in Figure 1 was simulated and analyzed using the FDTD method. The Gaussian light source was incident on the IN port on the right side of the structure, and the normalized power curve of the output port of the light source in the band 1.42−1.52 μm was shown in Figure 1(b) (color online). It can be seen that the light wave with a wavelength of 1.47 μm will not be output from the O₁ and O₂ ports after coupling through the ring cavity. Figure 2 (color online) shows the influence of the light wave power on the output of Figure 1(a) structure. Set the wavelength of light source to 1.47 μm. When the power of the incident light source is low, the refractive index of the nonlinear ring cavity does not change much. It can be seen from the normalized power curve in Figure 2(c) that only small power light waves are output from O₁ and O₂ ports. When the power of the incident light source increases, the coupling characteristics of the nonlinear ring cavity change because of the large increment of the refractive index of the nonlinear ring cavity. As can be seen from Figure 2(d), light waves are output from O₁ and O₂ ports. The waveband coupled by the structure of Figure 1(a) and output from the IN port can be controlled by adjusting parameters. When the lattice constant or medium column radius is increased, the overall image will be redshifted (increasing in wavelength); likewise, when it is decreased, the image will be blueshifted (decreasing in wavelength).

When there is a phase difference between two beams of light in the same path, there will be constructive interference or destructive interference. When the phase difference is 2mπ (m = 0, 1, 2, 3, ···), the output signal is enhanced because of constructive interference between light waves; when the phase difference is (2m+1)π, the output signal is weakened because of destructive interference^[34]. In the design of all-optical logic devices, the phase difference can be generated between the input signals by introducing optical path difference, so as to achieve the purpose of destructive interference. Figure 3 shows the XOR gate structure designed in this paper, with I₁ and I₂ as input terminals and OUT as output terminal. The wavelength of the signal light source is set as 1.47 μm and the power as P_in. The function of the XOR gate is that when the input is logic '01' or '10', the output is logic '1', and when the input is '00' or '11', the output is logic '0'. In the XOR gate structure, the distance between signal I₁ and signal I₂ to the output port OUT waveguide is 2a, and the phase difference is generated when the two signals propagate to the output waveguide. The FDTD method was used to simulate the structure in Figure 3, and the results were shown in Figure 4 (color online). Figure 4(a) and Figure 4(d) show the steady-state electric field diagrams and the normalized power curves at the output end of OUT when the input logic is '01'. When the light source is incident from I₂, part of the light wave will be reflected back, part of the light wave will be output from the OUT port with a power of 0.442 P_in, and the other part will be output from the I₁ port. The same is true when the input is '10', but because of the asymmetric structure, the output power is 0.503 P_in. Figure 4(c) shows the steady-state electric field diagram when both signal light sources are logic '1'. The phase difference between the two light waves at the output port is (2m+1)π, and the two input signals generate destructive interference. The output power at the OUT port is only 0.012 P_in, and the output is '0' when the input is logic '11'.

Logic Contrast (CR) is an important measure of device performance by calculating the ratio of the minimum output power P₁ of the logic '1' to the maximum output power P₀ of the logic '0'. The calculation formula is:

$ CR = 10{\mathrm{log}}\left( {{P_1}/{P_0}} \right) \quad.$ (4)

The minimum output power of XOR gate logic '1' is 0.442 P_in, the maximum output power of logic '0' is 0.012 P_in, and the calculated CR is 15.7 dB. The response time of the logic gate is 0.25 ps and the data transfer rate is 4 Tbits/s. The XOR gate logic structure can also be changed into a non−logical gate, for example, the input I₁ of the XOR gate in Figure 3 is fixed as a logic '1', $ A\oplus 1=\bar{A} $, so that the XOR gate becomes a NOT gate with the input I₂, and the calculated CR is 16.2 dB.

Out of the basic logic gates, AND, OR, and NOT, implementation of the NOT gate has thus far been discussed in this paper. This section uses nonlinear effects to design a two-input AND gate, as shown in Figure 5. The OR gate only needs the input signal to be structurally symmetrical, which means that the optical path difference is the same. For example, in the structures of Figure 5 and Figure 6 (color online), the two input signals I₁ and I₂ at the position of the box exhibit constructive interference with each other, which can be considered as the implementation of OR logic. On the left side of the two-input AND gate structure, R is a nonlinear annular cavity, and the O₁ port is the output end of the AND gate. To reduce reflection, a silicon dielectric column is added at the right-angle waveguide. If the structure of Figure 5 is simulated using the FDTD method, with a light source signal of 1.47 μm continuous wave, and power of 1.06 W/μm², as shown in Figure 6. When there is no signal or only one signal input, then the refractive index of the nonlinear material does not change much, which is not enough to change the characteristics of the nonlinear ring cavity. Therefore, the output power at the O₁ port is very small at only 0.0168 P_in. Because of the symmetrical structure of the device, the output power of the O₂ and O₃ ports is the same, so only the O₂ curve is plotted in the normalized power curve. When the input is logic '11', the optical path difference between the two input signals is the same, and there is a constructive interference in the O₁ waveguide. Because of the increase in power, the coupling characteristics of the nonlinear ring cavity change. Most of the light waves can be output from the O₁ port, with an output power of 1.0440 P_in. The CR of the device is 17.9 dB, the response time of the logic gate is 0.67 ps, and the data transfer rate is 1.5 Tbits/s. The power of the light source signal has a significant impact on the logical function of the device. For example, in logic devices, when the light source signal power is small, even if the input is all logic '1', it cannot change the characteristics of the nonlinear ring cavity. The simulation results are shown in Table 1. The power of the signal light source that can enable the normal operation of the logic device has an on threshold and an off threshold, which will be discussed in Section 3.4.

Any complex logical expression can be expressed through the combination of AND, OR, and NOT, and this idea can also be applied in the design of optical logic devices. In the above work, XOR, NOT, OR and AND logic gates were designed. For more complex logical expressions, we can convert them into existing simple logic through inversion theorem. When designing a NOR gate from the logical expression $ \overline{A+B} $, the inversion theorem can be used to obtain $ \overline{A+B}=\bar{A}\bar{B} $. Figure 7 shows the OR gate structure designed in this paper, with Ref as the logical '1' reference light source and the light source signal set to 1.47 μm continuous wave with a power of 0.6 W/μm². R is a nonlinear ring cavity, and the output end of the device is O₁. In order to improve output efficiency, a microcavity coupling structure has been added to the overall structure. From the box in the figure, it can be seen that the distance difference between the input signals I₁ and I₂ and the reference light source Ref to the waveguide interference point is 2a. As shown in section 2.2, the input signals I₁ and I₂ can work together with the reference light source to achieve nonlogical functions. The nonlinear annular cavity in the middle of the structure is partially functional, realizing the idea of cascading complex logic expressions with simple logic gates. The simulation results are shown in Figure 8. When the input logic is '00', the two reference light sources undergo phase interference because of the same optical path difference in the O₁ waveguide. At high power, the nonlinear ring cavity’s characteristics change, and the light wave can be coupled to the O₁ port for output, with an output power of 0.483 P_in. When the input logic is '10', the phase difference between the signal I₁ and the reference light source in the output waveguide is (2m+1)π, and only the light wave of the right reference light source can enter the output waveguide, so the output power is low, with only 1.5×10⁻³P_in on the O₁ port. When the input logic is '11', because of the cancellation interference of light waves on both sides, the output power is only 1.1×10⁻³P_in. The simulation results are shown in Table 2. The proposed NOR gate occupies an area of 231 μm² and has logical contrast CR of 25.1 dB. The response time of the logic gate is 1.67 ps and the data transfer rate is 0.6 Tbits/s.

Based on the existing two-input AND logic gates, a four-input AND gate structure is designed by expanding one input to two, as shown in Figure 9 (color online). Only when all four inputs are logic '1', can the optical wave power be sufficient to change the coupling characteristics of the nonlinear ring cavity, and the optical wave can be output from the O₁ port. The four inputs of the four-input AND gate are structurally symmetrical, and the light waves propagating to the output waveguide can be considered the same. Therefore, the number of inputs through logic '1' is classified into four categories. Set the light source signal to 1.47 μm continuous wave with power of 1.47 W/μm², the steady-state electric field diagram and normalized power output curve obtained through simulation using the FDTD method are shown in Figure 10 (color online). When the input is logic '1000', '1100' and '1110', the output power of O₁ port is lower than 0.039 P_in. When the input is logic '1111', the output power can change the nonlinear ring cavity’s characteristics, resulting in O₁ port output of 1.47 μm light wave, with an output power of 1.221 P_in. The simulation results are shown in Table 3. The designed four-input AND logic gate CR is 15 dB, the response time of the logic gate is 1 ps and the data transfer rate is 1 Tbits/s.

The structure proposed in this paper is compared with similar structures proposed in recent years in Table 4.

Taking the four-input AND logic gate as an example to analyze the impact of light source power on device performance, it can be seen from Section 2.1 that when the input power reaches a certain value, the characteristics of the nonlinear ring cavity will change. Therefore, as long as the light source power is high enough, the original output logic '0' can become logic '1'. In a four-input AND gate, the effect of light source power on device logic function is explored by comparing the normalized output power changes of logic inputs '1110' and '1111' as the light source power increases. The simulation results are shown in Figure 11. Based on the logical contrast of the device, the output of the designed four-input AND gate structure can be divided into three stages: when the signal power is less than the lower threshold power of 1.1 W/μm², even if the input is a logic '1111', the output power is not enough to change the characteristics of the nonlinear ring cavity, and the output power is lower, resulting in less obvious logical function of the device; when the signal input power reaches the lower threshold power of 1.1 W/μm² and is less than 3.4 W/μm², the input power of logic input '1110' cannot change the characteristics of the nonlinear ring cavity, and its output power is lower; however, the input power of logic input '1111' can change the characteristics of the nonlinear ring cavity and output higher power. At this point, the device has good output performance, with a logic contrast of over 10 dB; when the signal power is greater than the upper threshold power of 3.4 W/μm², the output power of logic inputs '1111' and '1110' can change the characteristics of the nonlinear ring cavity. Because of the high output power of logic input '1110' at this time, the device logic function will be lost.

This utilization of nonlinear effects and linear interference effects to design all simple logic gates using photonic crystal structures is described in this paper. After proposing the design of AND, OR, and NOT logic gates, NOR logic gates were implemented through cascading, and four-input AND logic gates were implemented by expanding the input on the basis of two-input AND gates, and it was shown how NOR logic gates can also be expanded. The structure proposed in this paper can achieve cascading of multi-level logic gates, but the optical loss will also increase accordingly. After the inversion theorem transformation, the devices involved in logic use nonlinear ring cavities, which are characterized by the presence of opening and closing threshold power. The device’s higher power consumption is due to the low nonlinear coefficient of the nonlinear materials used. Replacing the low nonlinear materials with high nonlinear materials can significantly reduce the power consumption. The designed structure can achieve destructive interference of light waves by introducing optical path differences between different input signals, avoiding the need to control the input signal phase. The device occupies an area of less than 248 μm², and it has an advantage in response speed because of its simple structure. Compared to all-optical logic gates designed solely using linear interference, the devices designed in this paper have certain advantages in cascading, area, and logical contrast, which helps to design and implement devices with practical application value in fields such as all-optical integration, all-optical information processing, and optical computing.