Fig. 1. Description of the encoding method (a) Binary image representation of a nanostructure. (b) Illustration of a level set function $\phi (x,y)$. The topology shown in (a) (encircled by red line) is represented by the zero-level set. (c) Encoded sparse representation of the nanostructure shown in (a). (d) The outline of the encoding method. For the encoding process, a level set function is first constructed from the given binary image. The spare representation of the image is derived from the Fourier transform of the level set function. Decoding the binary image from the sparse representation is the inverse of the encoding process.

Fig. 2. Properties of the encoding method (a) – (d) Dimensionality reduction using the proposed encoding method. Initial binary image (a) is encoded to the sparse representation (b). The low-dimensional representation (latent vector) can be achieved by deleting the high-frequency components as shown in (c). The latent vector can be recovered to the initial structure without substantial loss of information. (e) – (i) Continuously varying two topologies by linearly interpolating the latent vectors. (j) – (n) Generated samples with various geometric symmetries. The shown images are tiled unit cells of the generated patterns. (o) – (p) Adding fine features to initial pattern (o) by gradually expanding the dimensions of latent vectors from $7\times 7$ to $15\times 15$.

Fig. 3. Configuration of the DOE and the optimization method (a) The cross section of the DOE. The grating pattern and the substrate share the same material with a refractive index of $1.566$. The period of the DOE is $p=2.83$ $\mu$m and the thickness of the grating pattern is $t=840$ nm. Light with a wavelength $\lambda _{0} = 940$ nm is incident from the substrate side. Our aim is to optimize the grating pattern such that the central $3\times 3$ order diffractions present various intensity distributions. The angle between 0 and +1 order diffraction is $21^{\circ }$. (b) Architecture of neural network simulator for predicting the diffraction intensities of DOEs. The input is the encoded vectors of the DOEs, and the output is the vector containing normalized diffraction intensities and maximum intensity of all diffraction orders. The network is an eight-layer fully connected networks, and each hidden layer has 128 neurons. (c) Schematic of the evolution strategy. Randomly generated latent vectors are evaluated by the network simulator. Elites whose performance is closed to the design objectives are selected for subsequent reproduction and mutation. The algorithm iterates until some encoded vectors satisfy the design objectives or the maximum iteration is reached.

Fig. 4. Training of the network and statistic of the optimized results (a) The variation of training (blue) and validation (orange) loss versus the training epoch. The validation loss reaches 0.03 after 100 epochs of training. (b) Uniformity errors of 150 designed DOE structures with the objective of all diffraction intensities being equal. The blue bars represent the distribution of $U_{err}$ calculated with the network simulator during the optimization, and the oranges bars are the evaluated $U_{err}$ using RCWA for validation. The final design is selected from the validated DOE with the minimum $U_{err}$.

Fig. 5. Examples of designed DOEs with various diffraction intensity distributions. In each panel, the leftmost figure represents a tiled unit cell of the designed DOE. The middle image represents the simulated efficiencies of all the diffraction orders. The rightmost plot compares the objective intensities (blue) versus the RCWA simulated intensities (orange) of the design. All the designed DOEs are able to diffract light with intensity distributions essentially replicating the design objectives. The uniformity errors of the displayed designs are (a) 0.035, (b) 0.045, (c) 0.073, (d) 0.068, (e) 0.194, (f) 0.036, (g) 0.352, (h) 0.079, respectively. By the definition of Eq. (10), when the design objectives include diffraction orders with small intensities, a tiny disagreement of actual diffraction and objectives induces large uniformity errors. This leads to a large $U_{err}$ for the design shown in (e), (g), and (h).