All-optical image denoising using a diffractive visual processor.

Image denoising, one of the essential inverse problems, targets to remove noise/artifacts from input images. In general, digital image denoising algorithms, executed on computers, present latency due to several iterations implemented in, e.g., graphics processing units (GPUs). While deep learning-enabled methods can operate non-iteratively, they also introduce latency and impose a significant computational burden, leading to increased power consumption. Here, we introduce an analog diffractive image denoiser to all-optically and non-iteratively clean various forms of noise and artifacts from input images - implemented at the speed of light propagation within a thin diffractive visual processor that axially spans <250 × λ, where λ is the wavelength of light. This all-optical image denoiser comprises passive transmissive layers optimized using deep learning to physically scatter the optical modes that represent various noise features, causing them to miss the output image Field-of-View (FoV) while retaining the object features of interest. Our results show that these diffractive denoisers can efficiently remove salt and pepper noise and image rendering-related spatial artifacts from input phase or intensity images while achieving an output power efficiency of ~30-40%. We experimentally demonstrated the effectiveness of this analog denoiser architecture using a 3D-printed diffractive visual processor operating at the terahertz spectrum. Owing to their speed, power-efficiency, and minimal computational overhead, all-optical diffractive denoisers can be transformative for various image display and projection systems, including, e.g., holographic displays.

Diffractive optical computing in free space.

Structured optical materials create new computing paradigms using photons, with transformative impact on various fields, including machine learning, computer vision, imaging, telecommunications, and sensing. This Perspective sheds light on the potential of free-space optical systems based on engineered surfaces for advancing optical computing. Manipulating light in unprecedented ways, emerging structured surfaces enable all-optical implementation of various mathematical functions and machine learning tasks. Diffractive networks, in particular, bring deep-learning principles into the design and operation of free-space optical systems to create new functionalities. Metasurfaces consisting of deeply subwavelength units are achieving exotic optical responses that provide independent control over different properties of light and can bring major advances in computational throughput and data-transfer bandwidth of free-space optical processors. Unlike integrated photonics-based optoelectronic systems that demand preprocessed inputs, free-space optical processors have direct access to all the optical degrees of freedom that carry information about an input scene/object without needing digital recovery or preprocessing of information. To realize the full potential of free-space optical computing architectures, diffractive surfaces and metasurfaces need to advance symbiotically and co-evolve in their designs, 3D fabrication/integration, cascadability, and computing accuracy to serve the needs of next-generation machine vision, computational imaging, mathematical computing, and telecommunication technologies.

Data-Class-Specific All-Optical Transformations and Encryption.

Diffractive optical networks provide rich opportunities for visual computing tasks. Here, data-class-specific transformations that are all-optically performed between the input and output fields-of-view (FOVs) of a diffractive network are presented. The visual information of the objects is encoded into the amplitude (A), phase (P), or intensity (I) of the optical field at the input, which is all-optically processed by a data-class-specific diffractive network. At the output, an image sensor-array directly measures the transformed patterns, all-optically encrypted using the transformation matrices preassigned to different data classes, i.e., a separate matrix for each data class. The original input images can be recovered by applying the correct decryption key (the inverse transformation) corresponding to the matching data class, while applying any other key will lead to loss of information. All-optical class-specific transformations covering A → A, I → I, and P → I transformations using various image datasets are numerically demonstrated. The feasibility of this framework is also experimentally validated by fabricating class-specific I → I transformation diffractive networks and is successfully tested at different parts of the electromagnetic spectrum, i.e., 1550 nm and 0.75 mm wavelengths. Data-class-specific all-optical transformations provide a fast and energy-efficient method for image and data encryption, enhancing data security and privacy.

Snapshot multispectral imaging using a diffractive optical network.

Multispectral imaging has been used for numerous applications in e.g., environmental monitoring, aerospace, defense, and biomedicine. Here, we present a diffractive optical network-based multispectral imaging system trained using deep learning to create a virtual spectral filter array at the output image field-of-view. This diffractive multispectral imager performs spatially-coherent imaging over a large spectrum, and at the same time, routes a pre-determined set of spectral channels onto an array of pixels at the output plane, converting a monochrome focal-plane array or image sensor into a multispectral imaging device without any spectral filters or image recovery algorithms. Furthermore, the spectral responsivity of this diffractive multispectral imager is not sensitive to input polarization states. Through numerical simulations, we present different diffractive network designs that achieve snapshot multispectral imaging with 4, 9 and 16 unique spectral bands within the visible spectrum, based on passive spatially-structured diffractive surfaces, with a compact design that axially spans ~72λm, where λm is the mean wavelength of the spectral band of interest. Moreover, we experimentally demonstrate a diffractive multispectral imager based on a 3D-printed diffractive network that creates at its output image plane a spatially repeating virtual spectral filter array with 2 × 2 = 4 unique bands at terahertz spectrum. Due to their compact form factor and computation-free, power-efficient and polarization-insensitive forward operation, diffractive multispectral imagers can be transformative for various imaging and sensing applications and be used at different parts of the electromagnetic spectrum where high-density and wide-area multispectral pixel arrays are not widely available.

Super-resolution image display using diffractive decoders.

High-resolution image projection over a large field of view (FOV) is hindered by the restricted space-bandwidth product (SBP) of wavefront modulators. We report a deep learning-enabled diffractive display based on a jointly trained pair of an electronic encoder and a diffractive decoder to synthesize/project super-resolved images using low-resolution wavefront modulators. The digital encoder rapidly preprocesses the high-resolution images so that their spatial information is encoded into low-resolution patterns, projected via a low SBP wavefront modulator. The diffractive decoder processes these low-resolution patterns using transmissive layers structured using deep learning to all-optically synthesize/project super-resolved images at its output FOV. This diffractive image display can achieve a super-resolution factor of ~4, increasing the SBP by ~16-fold. We experimentally validate its success using 3D-printed diffractive decoders that operate at the terahertz spectrum. This diffractive image decoder can be scaled to operate at visible wavelengths and used to design large SBP displays that are compact, low power, and computationally efficient.

Cascadable all-optical NAND gates using diffractive networks.

Owing to its potential advantages such as scalability, low latency and power efficiency, optical computing has seen rapid advances over the last decades. Here, we present the design and analysis of cascadable all-optical NAND gates using diffractive neural networks. We encoded the logical values at the input and output planes of a diffractive NAND gate using the relative optical power of two spatially-separated apertures. Based on this architecture, we numerically optimized the design of a diffractive neural network composed of 4 passive layers to all-optically perform NAND operation using diffraction of light, and cascaded these diffractive NAND gates to perform complex logical functions by successively feeding the output of one diffractive NAND gate into another. We numerically demonstrated the cascadability of our diffractive NAND gates by using identical diffractive designs to all-optically perform AND and OR operations, which can be formulated as [Formula: see text] and [Formula: see text], respectively. We also designed an all-optical half-adder that takes two logical values as input and returns their sum and the carry using cascaded diffractive NAND gates. Cascadable all-optical NAND gates composed of spatially-engineered passive diffractive layers can serve optical computing platforms.

Polarization multiplexed diffractive computing: all-optical implementation of a group of linear transformations through a polarization-encoded diffractive network.

Research on optical computing has recently attracted significant attention due to the transformative advances in machine learning. Among different approaches, diffractive optical networks composed of spatially-engineered transmissive surfaces have been demonstrated for all-optical statistical inference and performing arbitrary linear transformations using passive, free-space optical layers. Here, we introduce a polarization-multiplexed diffractive processor to all-optically perform multiple, arbitrarily-selected linear transformations through a single diffractive network trained using deep learning. In this framework, an array of pre-selected linear polarizers is positioned between trainable transmissive diffractive materials that are isotropic, and different target linear transformations (complex-valued) are uniquely assigned to different combinations of input/output polarization states. The transmission layers of this polarization-multiplexed diffractive network are trained and optimized via deep learning and error-backpropagation by using thousands of examples of the input/output fields corresponding to each one of the complex-valued linear transformations assigned to different input/output polarization combinations. Our results and analysis reveal that a single diffractive network can successfully approximate and all-optically implement a group of arbitrarily-selected target transformations with a negligible error when the number of trainable diffractive features/neurons (N) approaches [Formula: see text], where Ni and No represent the number of pixels at the input and output fields-of-view, respectively, and Np refers to the number of unique linear transformations assigned to different input/output polarization combinations. This polarization-multiplexed all-optical diffractive processor can find various applications in optical computing and polarization-based machine vision tasks.

Classification and reconstruction of spatially overlapping phase images using diffractive optical networks.

Diffractive optical networks unify wave optics and deep learning to all-optically compute a given machine learning or computational imaging task as the light propagates from the input to the output plane. Here, we report the design of diffractive optical networks for the classification and reconstruction of spatially overlapping, phase-encoded objects. When two different phase-only objects spatially overlap, the individual object functions are perturbed since their phase patterns are summed up. The retrieval of the underlying phase images from solely the overlapping phase distribution presents a challenging problem, the solution of which is generally not unique. We show that through a task-specific training process, passive diffractive optical networks composed of successive transmissive layers can all-optically and simultaneously classify two different randomly-selected, spatially overlapping phase images at the input. After trained with ~ 550 million unique combinations of phase-encoded handwritten digits from the MNIST dataset, our blind testing results reveal that the diffractive optical network achieves an accuracy of > 85.8% for all-optical classification of two overlapping phase images of new handwritten digits. In addition to all-optical classification of overlapping phase objects, we also demonstrate the reconstruction of these phase images based on a shallow electronic neural network that uses the highly compressed output of the diffractive optical network as its input (with e.g., ~ 20-65 times less number of pixels) to rapidly reconstruct both of the phase images, despite their spatial overlap and related phase ambiguity. The presented phase image classification and reconstruction framework might find applications in e.g., computational imaging, microscopy and quantitative phase imaging fields.

All-optical synthesis of an arbitrary linear transformation using diffractive surfaces.

Spatially-engineered diffractive surfaces have emerged as a powerful framework to control light-matter interactions for statistical inference and the design of task-specific optical components. Here, we report the design of diffractive surfaces to all-optically perform arbitrary complex-valued linear transformations between an input (Ni) and output (No), where Ni and No represent the number of pixels at the input and output fields-of-view (FOVs), respectively. First, we consider a single diffractive surface and use a matrix pseudoinverse-based method to determine the complex-valued transmission coefficients of the diffractive features/neurons to all-optically perform a desired/target linear transformation. In addition to this data-free design approach, we also consider a deep learning-based design method to optimize the transmission coefficients of diffractive surfaces by using examples of input/output fields corresponding to the target transformation. We compared the all-optical transformation errors and diffraction efficiencies achieved using data-free designs as well as data-driven (deep learning-based) diffractive designs to all-optically perform (i) arbitrarily-chosen complex-valued transformations including unitary, nonunitary, and noninvertible transforms, (ii) 2D discrete Fourier transformation, (iii) arbitrary 2D permutation operations, and (iv) high-pass filtered coherent imaging. Our analyses reveal that if the total number (N) of spatially-engineered diffractive features/neurons is ≥Ni × No, both design methods succeed in all-optical implementation of the target transformation, achieving negligible error. However, compared to data-free designs, deep learning-based diffractive designs are found to achieve significantly larger diffraction efficiencies for a given N and their all-optical transformations are more accurate for N < Ni × No. These conclusions are generally applicable to various optical processors that employ spatially-engineered diffractive surfaces.

Spectrally encoded single-pixel machine vision using diffractive networks.

We demonstrate optical networks composed of diffractive layers trained using deep learning to encode the spatial information of objects into the power spectrum of the diffracted light, which are used to classify objects with a single-pixel spectroscopic detector. Using a plasmonic nanoantenna-based detector, we experimentally validated this single-pixel machine vision framework at terahertz spectrum to optically classify the images of handwritten digits by detecting the spectral power of the diffracted light at ten distinct wavelengths, each representing one class/digit. We also coupled this diffractive network-based spectral encoding with a shallow electronic neural network, which was trained to rapidly reconstruct the images of handwritten digits based on solely the spectral power detected at these ten distinct wavelengths, demonstrating task-specific image decompression. This single-pixel machine vision framework can also be extended to other spectral-domain measurement systems to enable new 3D imaging and sensing modalities integrated with diffractive network-based spectral encoding of information.

Ensemble learning of diffractive optical networks.

A plethora of research advances have emerged in the fields of optics and photonics that benefit from harnessing the power of machine learning. Specifically, there has been a revival of interest in optical computing hardware due to its potential advantages for machine learning tasks in terms of parallelization, power efficiency and computation speed. Diffractive deep neural networks (D2NNs) form such an optical computing framework that benefits from deep learning-based design of successive diffractive layers to all-optically process information as the input light diffracts through these passive layers. D2NNs have demonstrated success in various tasks, including object classification, the spectral encoding of information, optical pulse shaping and imaging. Here, we substantially improve the inference performance of diffractive optical networks using feature engineering and ensemble learning. After independently training 1252 D2NNs that were diversely engineered with a variety of passive input filters, we applied a pruning algorithm to select an optimized ensemble of D2NNs that collectively improved the image classification accuracy. Through this pruning, we numerically demonstrated that ensembles of N = 14 and N = 30 D2NNs achieve blind testing accuracies of 61.14 ± 0.23% and 62.13 ± 0.05%, respectively, on the classification of CIFAR-10 test images, providing an inference improvement of >16% compared to the average performance of the individual D2NNs within each ensemble. These results constitute the highest inference accuracies achieved to date by any diffractive optical neural network design on the same dataset and might provide a significant leap to extend the application space of diffractive optical image classification and machine vision systems.

Terahertz pulse shaping using diffractive surfaces.

Recent advances in deep learning have been providing non-intuitive solutions to various inverse problems in optics. At the intersection of machine learning and optics, diffractive networks merge wave-optics with deep learning to design task-specific elements to all-optically perform various tasks such as object classification and machine vision. Here, we present a diffractive network, which is used to shape an arbitrary broadband pulse into a desired optical waveform, forming a compact and passive pulse engineering system. We demonstrate the synthesis of various different pulses by designing diffractive layers that collectively engineer the temporal waveform of an input terahertz pulse. Our results demonstrate direct pulse shaping in terahertz spectrum, where the amplitude and phase of the input wavelengths are independently controlled through a passive diffractive device, without the need for an external pump. Furthermore, a physical transfer learning approach is presented to illustrate pulse-width tunability by replacing part of an existing network with newly trained diffractive layers, demonstrating its modularity. This learning-based diffractive pulse engineering framework can find broad applications in e.g., communications, ultra-fast imaging and spectroscopy.

All-optical information-processing capacity of diffractive surfaces.

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.

Analysis of Diffractive Optical Neural Networks and Their Integration with Electronic Neural Networks.

Optical machine learning offers advantages in terms of power efficiency, scalability and computation speed. Recently, an optical machine learning method based on Diffractive Deep Neural Networks (D2NNs) has been introduced to execute a function as the input light diffracts through passive layers, designed by deep learning using a computer. Here we introduce improvements to D2NNs by changing the training loss function and reducing the impact of vanishing gradients in the error back-propagation step. Using five phase-only diffractive layers, we numerically achieved a classification accuracy of 97.18% and 89.13% for optical recognition of handwritten digits and fashion products, respectively; using both phase and amplitude modulation (complex-valued) at each layer, our inference performance improved to 97.81% and 89.32%, respectively. Furthermore, we report the integration of D2NNs with electronic neural networks to create hybrid-classifiers that significantly reduce the number of input pixels into an electronic network using an ultra-compact front-end D2NN with a layer-to-layer distance of a few wavelengths, also reducing the complexity of the successive electronic network. Using a 5-layer phase-only D2NN jointly-optimized with a single fully-connected electronic layer, we achieved a classification accuracy of 98.71% and 90.04% for the recognition of handwritten digits and fashion products, respectively. Moreover, the input to the electronic network was compressed by >7.8 times down to 10×10 pixels. Beyond creating low-power and high-frame rate machine learning platforms, D2NN-based hybrid neural networks will find applications in smart optical imager and sensor design.

Design of task-specific optical systems using broadband diffractive neural networks.

Deep learning has been transformative in many fields, motivating the emergence of various optical computing architectures. Diffractive optical network is a recently introduced optical computing framework that merges wave optics with deep-learning methods to design optical neural networks. Diffraction-based all-optical object recognition systems, designed through this framework and fabricated by 3D printing, have been reported to recognize hand-written digits and fashion products, demonstrating all-optical inference and generalization to sub-classes of data. These previous diffractive approaches employed monochromatic coherent light as the illumination source. Here, we report a broadband diffractive optical neural network design that simultaneously processes a continuum of wavelengths generated by a temporally incoherent broadband source to all-optically perform a specific task learned using deep learning. We experimentally validated the success of this broadband diffractive neural network architecture by designing, fabricating and testing seven different multi-layer, diffractive optical systems that transform the optical wavefront generated by a broadband THz pulse to realize (1) a series of tuneable, single-passband and dual-passband spectral filters and (2) spatially controlled wavelength de-multiplexing. Merging the native or engineered dispersion of various material systems with a deep-learning-based design strategy, broadband diffractive neural networks help us engineer the light-matter interaction in 3D, diverging from intuitive and analytical design methods to create task-specific optical components that can all-optically perform deterministic tasks or statistical inference for optical machine learning.

Non-iterative phase hologram computation for low speckle holographic image projection.

Phase-only spatial light modulators (SLMs) are widely used in holographic display applications, including holographic image projection (HIP). Most phase computer generated hologram (CGH) calculation algorithms have an iterative structure with a high computational load, and also are prone to speckle noise, as a result of the random phase terms applied on the desired images to mitigate the encoding noise. In this paper, we present a non-iterative algorithm, where simple Discrete Fourier Transform (DFT) relations are exploited to compute phase CGHs that exactly control half of the desired image samples (those on even - or odd - indexed rows - or columns) via a single Fast Fourier Transform (FFT) and trivial arithmetic operations. The encoding noise appearing on the uncontrolled half of the image samples is reduced by the application of structured, non-random initial phase terms so that speckle noise is also kept low. High quality reconstructions are obtained under temporal averaging of several SLM frames. Interlaced video within half of the addressable image area is readily deliverable without frame rate division. Our algorithm provides about 6X and 20X reduction in computational cost compared to IFTA and FIDOC algorithms, respectively. Simulations and experiments verify that the algorithm constitutes a promising option for real-time computation of phase CGHs.