The functionality of the proposed 1-bit AOCMP was analyzed and simulated with three optical pulses at the input ports. Because the proposed AOCMP was made of linear materials, the power of these light waves is very low (0.001 W/µ\({\text{m}}^{2}\)) at a central wavelength of 1550 nm. The FDTD method has been used for calculating the optical wave propagation throughout the proposed AOCMP, in which the incident electric field (\({E}_{y}\)) was considered parallel to the y-axis and the propagation plane was the x-z plane. In this procedure, Maxwell’s equations (\(\nabla \times E\left(r,t\right)=-\mu \frac{\partial }{\partial t}H(r,t)\),\(\nabla \times H\left(r,t\right)=\epsilon \left(r\right)\frac{\partial }{\partial t}E\left(r,t\right)\)) can be discretized in terms of position and time using Yee’s Algorithm and written as follows [33, 34]:

$${E}_{y}\left|\genfrac{}{}{0pt}{}{n+\frac{1}{2}}{i-\frac{1}{2},k+\frac{1}{2}}\right.=\left(\frac{2\varDelta t}{2\epsilon +\sigma \varDelta t}\right)\left(\frac{{H}_{x}\left|\genfrac{}{}{0pt}{}{n}{i-\frac{1}{2},k+1}-{H}_{x}\left|\genfrac{}{}{0pt}{}{n}{i-\frac{1}{2},k}\right.\right.}{\varDelta z}-\frac{{H}_{z}\left|\genfrac{}{}{0pt}{}{n}{i,k+\frac{1}{2}}-{H}_{z}\left|\genfrac{}{}{0pt}{}{n}{i-\frac{1}{2},k+\frac{1}{2}}\right.\right.}{\varDelta x}\right)+\left(\frac{2\epsilon -\sigma \varDelta t}{2\epsilon +\sigma \varDelta t}\right){E}_{y}\left|\genfrac{}{}{0pt}{}{n-\frac{1}{2}}{i-\frac{1}{2},k+\frac{1}{2}}\right.$$

1

$${H}_{x}\left|\genfrac{}{}{0pt}{}{n+1}{i-\frac{1}{2},k+1}\right.=\left(\frac{2\varDelta t}{2\mu +\rho \varDelta t}\right)\left(\frac{{E}_{y}\left|\genfrac{}{}{0pt}{}{n+\frac{1}{2}}{i-\frac{1}{2},k+\frac{3}{2}}-{E}_{y}\left|\genfrac{}{}{0pt}{}{n+\frac{1}{2}}{i-\frac{1}{2},k+\frac{1}{2}}\right.\right.}{\varDelta z}\right)+\left(\frac{2\mu -\rho \varDelta t}{2\mu +\rho \varDelta t}\right){H}_{x}\left|\genfrac{}{}{0pt}{}{n}{i-\frac{1}{2},k+1}\right.$$

2

$${H}_{z}\left|\genfrac{}{}{0pt}{}{n+1}{i,k+\frac{1}{2}}\right.=-\left(\frac{2\varDelta t}{2\mu +\rho \varDelta t}\right)\left(\frac{{E}_{y}\left|\genfrac{}{}{0pt}{}{n+\frac{1}{2}}{i+\frac{1}{2},k+\frac{1}{2}}-{E}_{y}\left|\genfrac{}{}{0pt}{}{n+\frac{1}{2}}{i-\frac{1}{2},k+\frac{1}{2}}\right.\right.}{\varDelta x}\right)+\left(\frac{2\mu -\rho \varDelta t}{2\mu +\rho \varDelta t}\right){H}_{z}\left|\genfrac{}{}{0pt}{}{n}{i,k+\frac{1}{2}}\right.$$

3

where \(i\) and \(k\) present the discretized grid point in x–z planes, respectively, index \(n\) ,H ,\(\epsilon\), \(\mu\), \(\sigma\) and \(\rho\) denote the discrete time step, magnetic field, permittivity, permeability, electric conductivity and equivalent magnetic conductivity, respectively. \(\varDelta x\), \(\varDelta z\) and \(\varDelta t\) are the spatial steps in the x and z directions and time interval, respectively, which are related by the following equation:

$$\varDelta t\le \frac{1}{c}\sqrt{{\left(\frac{1}{\varDelta x}\right)}^{2}+{\left(\frac{1}{\varDelta z}\right)}^{2}}$$

4

The simulation process has the following four stages because, as mentioned, the proposed scheme has two input ports, during which the initial phase of the light waves of port REF is 180 degrees.

**CASE 1**

(X = Y): When the initial phase (IP) of the optical signals launched into the optical structure of the proposed 1-bit AOCMP through X and Y ports is 180 degrees (i.e., X = Y = 0 and REF = 0). It means that the phase difference between X and Y ports is 0 radians. Therefore, the optical beams inside OWG2 and OWG3 waveguides have destructive interference at their junction and destroy each other and a small part of them go to port O2. In the other part of the structure, a large portion of the optical waves coming from REF port will be travel toward O1 by the OWG1 and OWG7 waveguides and the point defects located at their junction. Figure 3a depicts the Field distribution in waveguide paths for this case of optical comparator. Also, Fig. 3b shows the normalized power at O1 and O2 which are 73% and 9%, respectively. As a result, the logic levels of ports O1 and O2 are 1 and 0, respectively (i.e., O2 = 0 and O1 = 1), which means that X is equal to Y.

**CASE 2**

(X < Y): When the IP of the optical signals launched into the structure of the proposed 1-bit AOCMP through X and Y are 180 and 0 degrees, respectively. (i.e., X = 0 and Y = 1 and REF = 0). It means that the phase difference between X and Y ports is π radians. Therefore, the optical beams inside OWG2 and OWG3 waveguides have a constructive interference at their junction and amplify each other and the resulting light waves move inside the OWG4 waveguide. Then, these waves are divided into two parts, some of them go to port O2 and the other part is directed to the OWG5 waveguide.

The REF port light waves inside the OWG1 waveguide have a destructive interference with the light waves inside the OWG5 waveguide at their junction and destroy each other. Therefore, a small part of these waves is transmitted to port O1. Figure 4a depicts the Field distribution in waveguide paths for this case of optical comparator. Also, Fig. 4b shows the normalized power at O1 and O2 which are 15% and 246%, respectively. As a result, the logic levels of ports O1 and O2 are 0 and 1, respectively (i.e., O2 = 1 and O1 = 0), which means that X is smaller than Y.

**CASE 3**

(X > Y): When the IP of the optical signals launched into the structure of the proposed 1-bit AOCMP through X and Y are 0 and 180 degrees, respectively. (i.e., X = 1 and Y = 0 and REF = 0). As in case 2, the phase difference between X and Y ports is π radians. Therefore, the optical beams inside OWG2 and OWG3 waveguides have a constructive interference at their junction and amplify each other and the resulting light waves move inside the OWG4 waveguide. Then, these waves are divided into two parts, some of them go to port O2 and the other part is directed to the OWG5 waveguide. In this case, the REF port light waves inside the OWG1 waveguide have a constructive interference with the light waves inside the OWG5 waveguide at their junction and amplify each other and the resulting light waves move to the O1 port. Figure 5a depicts the Field distribution in waveguide paths for this case of optical comparator. Also, Fig. 5b shows the normalized power at O1 and O2 which are 160% and 129%, respectively. As a result, the logic levels of ports O1 and O2 are 1 and 1, respectively (i.e., O2 = 1 and O1 = 1), which means that X is larger than Y.

**CASE 4**

(X = Y): When the IP of the optical signals launched into the optical structure of the proposed 1-bit AOCMP through X and Y ports is 0 degrees (i.e., X = Y = 1 and REF = 0). As in case 1, the phase difference between X and Y ports is 0 radians. Therefore, the optical beams inside OWG2 and OWG3 waveguides have destructive interference at their junction and destroy each other and a small part of them go to port O2. In the other part of the structure, a large portion of the optical waves coming from REF port will be travel toward O1 by the OWG1 and OWG7 waveguides and the point defects located at their junction. Figure 6a depicts the Field distribution in waveguide paths for this case of optical comparator. Also, Fig. 6b shows the normalized power at O1 and O2 which are 79% and 3%, respectively. As a result, the logic levels of ports O1 and O2 are 1 and 0, respectively (i.e., O2 = 0 and O1 = 1), which means that X is equal to Y. The obtained results of the proposed 1-bit AOCMP including initial phase (IP) and logic state (LS) of input ports, and the normalized intensity (NI) and logic state of output ports have been summarized in Table 3. This table indicates that the minimum and maximum values of the NI at ON and OFF states ( and) for output ports are 73% and 15%, respectively. Also, according to Figs. 3 to 6, the maximum,, and BR are about 0.15ps, 0.05ps, 0.45ps and 2.22 Tb/S, respectively.

Table 3

Obtained results of the proposed 1-bit AOCMP.

X | Y | O2 | O1 |

IP1 | LS2 | IP | LS | NI3 | LS | NI | LS |

π | 0 | π | 0 | 0.09 | 0 | 0.73 | 1 |

π | 0 | 0 | 1 | 2.46 | 1 | 0.15 | 0 |

0 | 1 | π | 0 | 1.29 | 1 | 1.6 | 1 |

0 | 1 | 0 | 1 | 0.03 | 0 | 0.79 | 1 |

IP1: Initial Phase |

LS2: Logic State

NI3: Normalized Intensity

As described, these values (\({P}_{1}\),\({P}_{0 }\),\({T}_{r}\), \({T}_{f}\), \({T}_{d}\)and BR) are optimal and are obtained by repeated simulation processes with optimal refractive index and radius of the rods. To obtain the optimal refractive index of the rods of the proposed 1-bit AOCMP, we performed four stages of the simulation process for different refractive indices and in each step we calculated the ON-OFF contrast ratio (CR) based on the following formula [35–41].

$$CR=10\text{*}\text{l}\text{o}\text{g}({P}_{1}/{P}_{0})$$

5

After evaluating these values, the optimal refractive index value was 3.46, as shown in Fig. 7. In this Figure, the horizontal axis is a different size for the refractive index of rods and the vertical axis is the ON-OFF contrast ratio of the proposed 1-bit AOCMP.

Also, to obtain the desired radius of the rods, we performed four steps of the simulation process for different radii and calculated the CR in each step. Figure 8 shows the CR of the proposed 1-bit AOCMP, proportional to the changes in the radius of rods from 112 to 124nm. In this Figure, the horizontal axis is a different size for the radius of rods and the vertical axis is the CR. As shown in the Figure, the optimal value of the radius is 118 nm.

Table 4 compares the designed 1-bit AOCMP with previous schemes. The proposed 1-bit AOCMP was smaller than previous comparators with three output ports and offered a higher switching speed. Also, it has less \({T}_{d}\) than them [26–30]. As shown in the Table 4, the proposed 1-bit AOCMP has a higher CR than similar structures with two output ports [31, 32]. Also, this comparator based on LPhCs has a simpler structure than other designs, which has less point defect in its structure and made of completely linear materials [26–32].

Table 4

Comparison of the results of the suggested 1-bit AOCMP with the previous schemes

Ref. | No. of output port | Point Defects | \({T}_{d}\) (ps) | TFP (\({ {\mu }\text{m}}^{2}\)) | CR (dB) |

[26] | 3 | 128 | 6 | 1705 | 10.26 |

[27] | 3 | 60 | 6 | 2399 | 12.04 |

[28] | 3 | 0 | 0.8 | 2056 | 6.53 |

[29] | 3 | 275 | 4 | 1136 | 9.71 |

[30] | 3 | 45 | 1.5 | 624 | 10.73 |

[31] | 2 | 11 | 0.31 | 55 | 5.26 |

[32] | 2 | 5 | 0.07 | 60 | 6.43 |

This work | 2 | 3 | 0.45 | 351 | 6.87 |