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All-Optical Information-Processing Capacity of Diffractive Surfaces Onur Kulce1,2,3,Denizmengu1,2,3,Yairrivenson1,2,3 and Aydogan Ozcan 1,2,3

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All-Optical Information-Processing Capacity of Diffractive Surfaces Onur Kulce1,2,3,Denizmengu1,2,3,Yairrivenson1,2,3 and Aydogan Ozcan 1,2,3 Kulce et al. Light: Science & Applications (2021) 10:25 Official journal of the CIOMP 2047-7538 https://doi.org/10.1038/s41377-020-00439-9 www.nature.com/lsa ARTICLE Open Access All-optical information-processing capacity of diffractive surfaces Onur Kulce1,2,3,DenizMengu1,2,3,YairRivenson1,2,3 and Aydogan Ozcan 1,2,3 Abstract The precise engineering of materials and surfaces has been at the heart of some of the recent advances in optics and photonics. These advances related to the engineering of materials with new functionalities have also opened up exciting avenues for designing trainable surfaces that can perform computation and machine-learning tasks through light–matter interactions and diffraction. Here, we analyze the information-processing capacity of coherent optical networks formed by diffractive surfaces that are trained to perform an all-optical computational task between a given input and output field-of-view. We show that the dimensionality of the all-optical solution space covering the complex-valued transformations between the input and output fields-of-view is linearly proportional to the number of diffractive surfaces within the optical network, up to a limit that is dictated by the extent of the input and output fields-of-view. Deeper diffractive networks that are composed of larger numbers of trainable surfaces can cover a higher-dimensional subspace of the complex-valued linear transformations between a larger input field-of-view and a larger output field-of-view and exhibit depth advantages in terms of their statistical inference, learning, and generalization capabilities for different image classification tasks when compared with a single trainable diffractive surface. These analyses and conclusions are broadly applicable to various forms of diffractive surfaces, including, e.g., plasmonic and/or dielectric-based metasurfaces and flat optics, which can be used to form all-optical processors. 1234567890():,; 1234567890():,; 1234567890():,; 1234567890():,; Introduction harnessed engineered light–matter interactions to perform The ever-growing area of engineered materials has computational tasks using wave optics and the propaga- – empowered the design of novel components and devices tion of light through specially devised materials25 38. that can interact with and harness electromagnetic waves These approaches and many others highlight the emerging in unprecedented and unique ways, offering various new uses of trained materials and surfaces as the workhorse of – functionalities1 14. Owing to the precise control of mate- optical computation. rial structure and properties, as well as the associated Here, we investigate the information-processing capa- light–matter interaction at different scales, these engi- city of trainable diffractive surfaces to shed light on their neered material systems, including, e.g., plasmonics, computational power and limits. An all-optical diffractive metamaterials/metasurfaces, and flat optics, have led to network is physically formed by a number of diffractive fundamentally new capabilities in the imaging and sensing layers/surfaces and the free-space propagation between – fields, among others15 24. Optical computing and infor- them (see Fig. 1a). Individual transmission and/or reflec- mation processing constitute yet another area that has tion coefficients (i.e., neurons) of diffractive surfaces are adjusted or trained to perform a desired input–output transformation task as the light diffracts through these Correspondence: Aydogan Ozcan ([email protected]) 1Electrical and Computer Engineering Department, University of California, Los layers. Trained with deep-learning-based error back- Angeles, CA 90095, USA propagation methods, these diffractive networks have 2 Bioengineering Department, University of California, Los Angeles, CA 90095, been shown to perform machine-learning tasks such as USA Full list of author information is available at the end of the article image classification and deterministic optical tasks, These authors contributed equally: Onur Kulce, Deniz Mengu © The Author(s) 2021 Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a linktotheCreativeCommons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/. Kulce et al. Light: Science & Applications (2021) 10:25 Page 2 of 17 operator that multiplies an input field vector to create an a Output plane output field vector at the detector plane/aperture. This (N ) o operator is designed/trained using, e.g., deep learning to transform a set of complex fields (forming, e.g., the input data classes) at the input aperture of the optical network into another set of corresponding fields at the output Diffractive optical network d (K layers, N neurons/layer) K+1 aperture (forming, e.g., the data classification signals) and d K is physically created through the interaction of the input light with the designed diffractive surfaces as well as free- d3 space propagation within the network (Fig. 1a). d2 In this paper, we investigate the dimensionality of the all-optical solution space that is covered by a diffractive d1 network design as a function of the number of diffractive Input plane (Ni) surfaces, the number of neurons per surface, and the size of the input and output fields-of-view (FOVs). With our b H theoretical and numerical analysis, we show that the dimensionality of the transformation solution space that can be accessed through the task-specific design of a Td (M) diffractive network is linearly proportional to the number of diffractive surfaces, up to a limit that is governed by the extent of the input and output FOVs. Stated differently, d ≥ adding new diffractive surfaces into a given network design increases the dimensionality of the solution space that can be all-optically processed by the diffractive net- work, until it reaches the linear transformation capacity dictated by the input and output apertures (Fig. 1a). T (N) Beyond this limit, the addition of new trainable diffractive features/layer neurons/layer N M surfaces into the optical network can cover a higher- * H dimensional solution space over larger input and output FOVs, extending the space-bandwidth product of the all- M features/layer, K layers Number of modulation parameters : K × M optical processor. Dimensionality of solution space: K × N – (K–1) Limit of dimensionality: N × N Our theoretical analysis further reveals that, in addition i o to increasing the number of diffractive surfaces within a Fig. 1 Schematic of a multisurface diffractive network. a network, another strategy to increase the all-optical pro- fi Schematic of a diffractive optical network that connects an input eld- cessing capacity of a diffractive network is to increase the of-view (aperture) composed of N points to a desired region-of- i number of trainable neurons per diffractive surface. interest at the output plane/aperture covering No points, through K-diffractive surfaces with N neurons per surface, sampled at a period However, our numerical analysis involving different image of λ/2n, where λ and n represent the illumination wavelength and the classification tasks demonstrates that this strategy of refractive index of the medium between the surfaces, respectively. creating a higher-numerical-aperture (NA) optical net- = Without loss of generality, n 1 was assumed in this paper. b The work for all-optical processing of the input information is communication between two successive diffractive surfaces occurs through propagating waves when the axial separation (d) between not as effective as increasing the number of diffractive these layers is larger than λ. Even if the diffractive surface has deeply surfaces in terms of the blind inference and generalization subwavelength structures, as in the case of, e.g., metasurfaces, with a performance of the network. Overall, our theoretical and much smaller sampling period compared to λ/2 and many more numerical analyses support each other, revealing that degrees of freedom (M) compared to N, the information-processing deeper diffractive networks with larger numbers of capability of a diffractive surface within a network is limited to propagating modes since d ≥ λ; this limits the effective number of trainable diffractive surfaces exhibit depth advantages in neurons per layer to N, even for a surface with M >> N. H and H* refer terms of their statistical inference and learning capabilities to the forward- and backward-wave propagation, respectively compared with a single trainable diffractive surface. The presented analyses and conclusions are generally applicable to the design and investigation of various including, e.g., wavelength demultiplexing, pulse shaping, coherent all-optical processors formed by diffractive – and imaging38 44. surfaces, such as, e.g., metamaterials, plasmonic or The forward model of a diffractive optical network can
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