# Regeneration of 200 Gbit/s PAM4 Signal Produced by Silicon Microring Modulator (SiMRM) Using Mach–Zehnder Interferometer (MZI)-Based Optical Neural Network (ONN)

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## Abstract

**:**

^{−2}) can be achieved at 200 Gbit/s transmission, and the proposed ONN has nearly the same performance as an artificial neural network (ANN) implemented using traditional computer simulation.

## 1. Introduction

^{−2}) can be achieved at 200 Gbit/s transmission, and the proposed ONN has nearly the same performance with the artificial neural network (ANN) implemented using computer software.

## 2. Theory of the MZI-Based ONN

**Y**represents the output optical field matrix,

**X**is the input optical field matrix, and

**H**denotes the Hilbert space matrix. This operation is like the fully connected layer shown in Figure 3.

_{i}to y

_{j}can be written as ${x}_{i}\text{}{w}_{i,j}+{b}_{i,j}$, where ${w}_{i,j}$ and ${b}_{i,j}$ are the weight and bias value at connect line, respectively. The relationship between x

_{i}and y

_{j}is illustrated in Equation (7). Using a matrix to express this relationship, we can obtain Equation (8), where

**Y**is output matrix,

**X**is input matrix,

**W**is weight matrix, and

**b**is the bias matrix. Comparing Equation (8) with Equation (6), it can be observed that they are very similar.

**H**matrix value in the lower loss function value as shown in Equation (9),

**[S**of each MZI is equal to its conjugate transpose as Equation (10)

_{MZI}]^{−1}**H**is equivalent to the reverse arrangement of MZIs. This leads to successive products culminating in the eventual formation of the identity matrix as shown in Equation (11). Through the sequential multiplication of

**H**by

**[D**in a defined order, the off-diagonal elements in both the upper and lower triangles of the matrix would eventually become 0. Subsequently, Gaussian elimination can be applied to determine the phase shift values $\phi $ and $\theta $ at each phase shifter.

_{n}]^{−1}_{b}to input to the MZI phase shift. The operation of electro-optic nonlinear activation function is illustrated in Equation (12), with the two internal components defined in Equations (13) and (14).

_{π}is the voltage of the MZI phase shift π, G is the gain of the electric amplifier, and R is the responsivity. Hence, by controlling the V

_{b}, we can conveniently modify Equation (13) to a different nonlinear activation function. By connecting the electro-optic nonlinear activation function in series after the MZI network mesh, a neural network with an activation function can be realized.

## 3. Experimental Setup

## 4. Result and Discussion

^{17}bauds. We use 20% data for training and 80% for testing. In the proof-of-concept demonstration illustrated in Figure 6, the input data are experimentally generated by a bandwidth-limited SiMRM chip. This experimental ISI-distorted optical PAM4 signal will be detected by a separated PD, and a RTO will store the electrical PAM4 signal as shown in Figure 6. Hence, this stored electrical PAM4 signal can be used for the ONN simulation. In the future ONN chip implementation, the ISI distorted optical PAM4 signal can be directly launched into the ONN chip “RX signal” port as shown in Figure 2; hence, no additional OE conversion by the PD is needed. In this case, four on-chip PDs on the ONN chip are used as shown in Figure 2. The optical amplification can be realized by the pumping light as discussed before; hence, VOA and EDFA may not be necessary. Figure 7 shows the accuracy and loss curves for the proposed ONN. It is evident from the results that the ONN exhibits convergence at approximately 100 epochs.

^{−2}) can be up to 200 Gbit/s.

_{π}of the MZI phase shift is 5 V, the V

_{b}is set to be −5 V, G is set to be 20, and the responsivity R is set to be 1. Therefore, ${\phi}_{b}$ is set to be -π, and ${g}_{\phi}$ is set to be 0.4π. Figure 11 shows the transmission coefficient (i.e., $\frac{{\left|f\left(z\right)\right|}^{2}}{{\left|z\right|}^{2}}$) of the electro-optic nonlinear activation function with normalized input field Z. We can observe that the electro-optic nonlinear activation function defined exhibits similarities to the sigmoid function but shifted towards the positive x-axis. In the simulation work here, the α = 0.1 is used for reducing the loss for electro-optic nonlinear activation function. The electro-optic nonlinear activation function will have different characteristics under different ${\phi}_{b}$ and ${g}_{\phi}$. Here, we found that the nonlinear activation function as illustrated in Figure 11 has a better performance in our model. Therefore, ${\phi}_{b}$ is set to be −π, and ${g}_{\phi}$ is set to be 0.4π.

^{−2}) can be up to 200 Gbit/s. This reveals that when the input signal power is high enough, no additional bit error will be introduced for the ONN without the electro-optic nonlinear activation function. However, the introduction of activation function increases the robustness of the proposed ONN. We analyze the impact of the phase shift error on MZI ONN performance. To simulate the phase error of phase shift, we introduce a random normal distribution $N\left(0,{\sigma}^{2}\right)$ and add it to the final training results of the phase shift value for each phase shifter in the MZIs. Here, σ is the standard deviation of the phase error. Therefore, the θ and φ in Equation (1) are now written as $\widehat{\theta}$ and $\widehat{\phi}$as shown in Equations (15) and (16).

## 5. Conclusions

^{−2}) can be achieved at 200 Gbit/s transmission, and the proposed ONN has nearly the same performance with ANN implemented using traditional computer simulation. Moreover, we also discussed the effect of electro-optic nonlinear activation function on the ONN model. By comparing the ONN model with and without electro-optic nonlinear activation function in different input signal amplitudes, it can be observed that the accuracy and loss can be significantly improved at low input signal amplitudes. Even at the normalized input signal amplitude of 0.1, the accuracy can still achieve 99.7%. Furthermore, we analyzed the impact of the phase shift error of MZI to the ONN model. Both ONN model with and without electro-optic nonlinear activation function can still achieve SD-FEC threshold under a 1° phase shift error.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**A typical 2 × 2 MZI used in the ONN. It consists of two 3-dB couplers, a phase shifter θ=, and a phase shifter φ.

**Figure 5.**The structure of electro-optic nonlinear activation functions. MZI: Mach–Zehnder Interferometer; DC: directional coupler; PD: photodetector.

**Figure 6.**The experimental setup to obtain the PAM4 optical signal. AWG: arbitrary waveform generator; DFB: distributed feedback laser diodes; PC: polarization controller; EDFA: erbium-doped fiber amplifier; VOA: variable optical attenuator; PD: photodetector; RTO: real-time oscilloscope. Inset: photo of the SiMRM.

**Figure 8.**BER performances of ONN and ANN used for classifying the distorted PAM4 signal without the activation function.

**Figure 9.**Accuracy and loss performance of different normalized input signal amplitudes without activation function.

**Figure 10.**Modified ONN model with electro-optic nonlinear activation functions. MZI: Mach–Zehnder Interferometer; EO: electro-optic nonlinear activation function; PD: photodetector.

**Figure 12.**Accuracy and loss performance of different normalized input signal amplitudes with an activation function.

**Figure 13.**BER performance under various standard deviation phase errors at a data rate of 160 Gbit/s.

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**MDPI and ACS Style**

Hung, T.-Y.; Chan, D.W.U.; Peng, C.-W.; Chow, C.-W.; Tsang, H.K.
Regeneration of 200 Gbit/s PAM4 Signal Produced by Silicon Microring Modulator (SiMRM) Using Mach–Zehnder Interferometer (MZI)-Based Optical Neural Network (ONN). *Photonics* **2024**, *11*, 349.
https://doi.org/10.3390/photonics11040349

**AMA Style**

Hung T-Y, Chan DWU, Peng C-W, Chow C-W, Tsang HK.
Regeneration of 200 Gbit/s PAM4 Signal Produced by Silicon Microring Modulator (SiMRM) Using Mach–Zehnder Interferometer (MZI)-Based Optical Neural Network (ONN). *Photonics*. 2024; 11(4):349.
https://doi.org/10.3390/photonics11040349

**Chicago/Turabian Style**

Hung, Tun-Yao, David W. U Chan, Ching-Wei Peng, Chi-Wai Chow, and Hon Ki Tsang.
2024. "Regeneration of 200 Gbit/s PAM4 Signal Produced by Silicon Microring Modulator (SiMRM) Using Mach–Zehnder Interferometer (MZI)-Based Optical Neural Network (ONN)" *Photonics* 11, no. 4: 349.
https://doi.org/10.3390/photonics11040349