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Holographic optical elements (HOEs) are based on the principle of holography, which can implement arbitrary functions such as convex lenses and concave mirrors. The performance of HOEs is expected to be enhanced by a cooperative operation of multiple HOEs. However, the design of multiple HOEs is difficult to achieve with conventional design methods such as ray tracing software. We will introduce the HOE design of the cooperative operation by machine learning. In this work, we implemented a diffractive deep neural network (D2NN) to realize the cooperative operation by multiple HOEs at the visible wavelengths. D2NN is a kind of optical neural network that is represented by light propagation, and it is implemented by multiple DOEs that can represent arbitrary optical functions. However, multiple-layer HOEs cause noise to be overlapped on the output wavefront since the HOE generates unnecessary lights such as the direct light and high-order lights. Therefore, we implemented the D2NN consisting of two layers of HOEs by an off-axis D2NN, which avoids this obstacle. The two-layer HOEs were trained to perform a classification task of handwritten digits as a task. The trained D2NN model with HOEs was evaluated in a numerical simulation, achieving 87.1% accuracy in the simulation. The method enables the design of cooperative operation of multiple HOEs, it enables HOEs to achieve more complex and higher performance functions.
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