# All-Optical Synthesis of an Arbitrary Linear Transformation Using Diffractive Surfaces

arXiv: https://arxiv.org/abs/2108.09833 · HTML: https://mlsanity.com/paper/2108.09833/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 50 references that are in the map.
- First posted on arXiv 22 Aug 2021.

## Builds on (most cited first, 6 of 6)

- [All-Optical Machine Learning Using Diffractive Deep Neural Networks](https://mlsanity.com/paper/1804.08711/) (2018)
- [Misalignment Resilient Diffractive Optical Networks](https://mlsanity.com/paper/2005.11464/) (2020)
- [Design of Task-Specific Optical Systems Using Broadband Diffractive Neural Networks](https://mlsanity.com/paper/1909.06553/) (2019)
- [Massively Parallel Amplitude-Only Fourier Neural Network](https://mlsanity.com/paper/2008.05853/) (2020)
- [Ensemble learning of diffractive optical networks](https://mlsanity.com/paper/2009.06869/) (2020)
- [Scale-, shift- and rotation-invariant diffractive optical networks](https://mlsanity.com/paper/2010.12747/) (2020)

## Built on by (most cited first, 24 of 24)

- [Diffractive all-optical computing for quantitative phase imaging](https://mlsanity.com/paper/2201.08964/) (2022)
- [To image, or not to image: Class-specific diffractive cameras with all-optical erasure of undesired objects](https://mlsanity.com/paper/2205.13122/) (2022)
- [Nonlinear Processing with Linear Optics](https://mlsanity.com/paper/2307.08533/) (2023)
- [Snapshot Multispectral Imaging Using a Diffractive Optical Network](https://mlsanity.com/paper/2212.05217/) (2022)
- [Super-resolution image display using diffractive decoders](https://mlsanity.com/paper/2206.07281/) (2022)
- [Unidirectional Imaging using Deep Learning-Designed Materials](https://mlsanity.com/paper/2212.02025/) (2022)
- [Polarization Multiplexed Diffractive Computing: All-Optical Implementation of a Group of Linear Transformations Through a Polarization-Encoded Diffractive Network](https://mlsanity.com/paper/2203.13482/) (2022)
- [Multispectral Quantitative Phase Imaging Using a Diffractive Optical Network](https://mlsanity.com/paper/2308.02952/) (2023)
- [All-Optical Phase Conjugation Using Diffractive Wavefront Processing](https://mlsanity.com/paper/2311.04473/) (2023)
- [Diffractive Interconnects: All-Optical Permutation Operation Using Diffractive Networks](https://mlsanity.com/paper/2206.10152/) (2022)
- [All-optical image denoising using a diffractive visual processor](https://mlsanity.com/paper/2309.09215/) (2023)
- [All-optical image classification through unknown random diffusers using a single-pixel diffractive network](https://mlsanity.com/paper/2208.03968/) (2022)
- [Time-lapse image classification using a diffractive neural network](https://mlsanity.com/paper/2208.10802/) (2022)
- [Pyramid diffractive optical networks for unidirectional image magnification and demagnification](https://mlsanity.com/paper/2308.15019/) (2023)
- [EEG Opto-processor: epileptic seizure detection using diffractive photonic computing units](https://mlsanity.com/paper/2301.10167/) (2022)
- [All-optical complex field imaging using diffractive processors](https://mlsanity.com/paper/2401.16779/) (2024)
- [Multiplane Quantitative Phase Imaging Using a Wavelength-Multiplexed Diffractive Optical Processor](https://mlsanity.com/paper/2403.11035/) (2024)
- [Integration of Programmable Diffraction with Digital Neural Networks](https://mlsanity.com/paper/2406.10688/) (2024)
- [Unidirectional imaging with partially coherent light](https://mlsanity.com/paper/2408.05449/) (2024)
- [Snapshot multi-spectral imaging through defocusing and a Fourier imager network](https://mlsanity.com/paper/2501.14287/) (2025)
- [Spatially Parallel All-optical Neural Networks](https://mlsanity.com/paper/2509.23611/) (2025)
- [Snapshot 3D image projection using a diffractive decoder](https://mlsanity.com/paper/2512.20464/) (2025)
- [Wavelength-multiplexed massively parallel diffractive optical information storage and image projection](https://mlsanity.com/paper/2604.02624/) (2026)
- [HAMON: Passive Optical Sequence Mixing for Long-Horizon Forecasting](https://mlsanity.com/paper/2606.17028/) (2026)

## Abstract (arXiv)

We report the design of diffractive surfaces to all-optically perform arbitrary complex-valued linear transformations between an input (N_i) and output (N_o), where N_i and N_o 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 N_i x N_o or larger, 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 < N_i x N_o. These conclusions are generally applicable to various optical processors that employ spatially-engineered diffractive surfaces.