Microring-based programmable coherent optical neural networks

We design, simulate, and train a coherent optical neural network fully based on microring resonators including the linear multiplication and the reconfigurable nonlinear activation components, which shows advantages in terms of device footprint and energy efficiency.

Optics, due to its speed and energy efficiency, has attracted attention as a potential platform for the next generation computing hardware [1,2].Programmable photonic circuits have been demonstrated to perform linear operations for a variety of applications such as an optical neural network for deep learning [2,3] and quantum information processing [4].Many programmable photonic circuits are based on Mach-Zehnder interferomenters (MZIs) [2].However, in order to achieve large phase tuning ranges, the driving voltage of an MZI is relatively high and the device length is on the order of 100 μm.In large-scale on-chip integrated circuits for complex applications, the device footprint and power consumption then become major considerations.A natural idea is to employ resonant structures which can increase the light-matter interactions and thus reduce the device footprint, driving voltages, and power consumption.
In this work, we propose a coherent optical neural network constructed with microring resonators (Fig. 1), which shows advantages in device footprint and energy efficiency when compared with existing optical neural networks constructed using MZI-based architectures.We describe the input and output relationship of our architecture using the transfer function method and directly train the tunable parameters with automatic differentiation., where f is a nonlinear function.The black ring of the nonlinear activation unit is used as a directional coupler to route the α portion of the optical energy for electrical signal processing.The diode is a photo-detector.The blue ring provides modulation to the signal.M denotes an electronic circuit, which takes the electronic output from the photo-detector to generate a modulation signal for the right ring.cascades multiple layers to increase the approximation capability, where information moves in one direction from input to output data without cycles.For the l th layer, the input and output relationship can be represented as:

Ring-based coherent optical neural network (RONN) architecture
In Eq. ( 1), x l and x l+1 are, respectively, the input and output vectors comprising signal amplitudes.W l is a matrix that performs a linear transformation on the input vector.f l is an element-wise nonlinear activation function.In our design, the linear matrix multiplication layer to perform W l is constructed by cascading multiple linear units, each consisting of a serially coupled double ring resonator for signal mixing of different ports (Fig. 1(b)) and a single ring resonator for phase tuning (Fig. 1(a)).For the nonlinear unit, which performs element-wise activation f l at each port, we employ the optical-modulator-based design concept proposed in [5] but use ring resonators as directional couplers and optical modulators, as shown in Fig. 1(c).In this design, a small portion of the optical signal is tapped off using a directional coupler from the bus waveguide and converted to the electric signal.One then performs a nonlinear transformation of the electric signal to generate a voltage that is applied on a modulator in order to influence the transmission of the optical signal in the bus waveguide.Here, we assume that all of the ring resonators have the same diameter, whereas the separation distances between rings and waveguides can vary depending on the functionality.Additionally, we assume all components are working under continuous wave conditions at a single operating frequency, ω 0 , such that we can control the phase and amplitude of transmitted signals by tuning the refractive index of each component.
Neural network training based on transfer matrix method -After choosing appropriate coupling distances, the RONN can be trained to perform different tasks by adjusting the tunable elements in the linear devices.The training process involves two stages: 1. Component analysis: During the training process, we will need to get the derivative of the response function to the tunable parameters.Thus, for each tunable device, we need to perform a component analysis to characterize its response to tunable parameters such as temperatures, voltages, and forces.Such a characterization equation can be done through fitting.2. Neural network design: We can then design the network structures and the optimization algorithms for the system with the characterization equation of each component.The gradient with respect to each tunable parameter is obtained using automatic differentiation, and then gradient methods can be used to search for the appropriate tunable parameters for different applications.After training, we apply the trained tunable parameters to the corresponding physical devices.The system now behaves as a programmed neural network for specific tasks.
As a representative example, we demonstrate the operation of the network for information processing tasks such as operating as an Exclusive OR (XOR) gate or performing handwritten digit recognition on the MNIST dataset.The performance of RONN is comparable with the performance of MZI architecture, but our proposed architecture has a nearly 10-fold reduction in the overall device footprint area and the dynamic energy consumption for the nonlinear part reduces at least by a factor of 10.
In conclusion, we introduce an on-chip ring-based optical neural network architecture.We study the properties and transfer functions of different components including the phase tuning components and signal mixing components for matrix multiplications and modulated ring resonators for nonlinear activation.With the parameterized transfer functions, we train the RONN for different tasks such as a XOR gate and MNIST handwritten classification.This method can be extended to different physical computing platforms with arbitrary tunable transfer functions such as electric, thermal, or mechanical control systems.

Fig. 1 .
Fig. 1.Layout of a single layer, three-port, ring-based coherent optical neural network.Coherent light sources are injected from the left ports and the transmitted signals at the right ports will be detected or become the input of the next layer.(a) All-pass single ring resonator acting as a phase tuning component.(b) Serially coupled double ring resonators as a signal mixing component between ports.(c) Nonlinear activation unit to convert input signal x n to f (x n ), where f is a nonlinear function.The black ring of the nonlinear activation unit is used as a directional coupler to route the α portion of the optical energy for electrical signal processing.The diode is a photo-detector.The blue ring provides modulation to the signal.M denotes an electronic circuit, which takes the electronic output from the photo-detector to generate a modulation signal for the right ring.
-A standard feed-forward neural network Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.