Diffraction-based Optical Architectures for Processing One-dimensional Data

Abstract

The demands of proliferating big data and massive deep learning models, against a backdrop of a mounting climate emergency and the abating of Moore’s law, have pushed technologists to develop high-speed, high-throughput, low-energy, cloud on-demand and scalable computer hardware, with photonic methods at the centre of emerging solutions. Using light as a fundamental resource, free-space optical computing and integrated photonic computing devices provide, respectively, powers of natural parallelism and miniaturisability. Recent work harnessing diffraction within slab waveguides has been proposed, on its face combining the best elements of free-space and integrated photonic technology. Yet even on its own terms, the descent to the diffractive plane implemented by slab waveguides comes at the expense of one of the diffractive degrees of freedom available in free space, with corresponding reductions in the dimensionality of individual transformations that can feasibly be implemented at speed and at scale. Questions remain as to the extent to which diffractive photonic integrated chip (DPIC) architectures can be made to function well as intended at high operating frequencies, and actually mark an improvement over classical optics (CO) apparatus. This thesis seeks to answer several of these questions, through a variety of analytic, simulation-based and experimental methodologies. It is shown analytically, first in a scalar and then in a vectorial framework, that rigorous time-dependent solutions reveal a temporal source of error in the expected performance of DPIC devices, with the error emerging from the gap between solutions to the wave equation over two spatial dimensions and presumed time-separable harmonic solutions to the 2D Helmholtz equation. The size of this error, for typical physical scales of DPIC architecture observed in the literature, is examined, with concrete bounds on device operating frequency justified. The temporal error is then shown to effectively disappear when temporal bandlimiting constraints are placed on photonic signal inputs, opening the door to appropriate time-separability assumptions and to valid time-independent methods. Filling in gaps in understanding of time-independent modelling strategies, the key results of free-space Fourier optics are proved in the planar context, with Green’s functions given in terms of Hankel functions, including the Rayleigh-Sommerfeld diffraction formulae, Fresnel and Fraunhofer approximations, and angular spectrum formulations, with the Fourier relationships between various formulations elucidated. Results are expanded further to extend the range of applicability of the planar Rayleigh-Sommerfeld diffraction formulae to arbitrary boundary surfaces, rather than the usual straight boundaries incorporated in the theory. This enables the rigorous analysis of Star Coupler DPIC architectures self-consistently within the Fourier optical framework. The tools developed to extend the domain of validity of Rayleigh- Sommerfeld diffraction formulae, based on anticonformal maps, are then extended further to introduce new methods for designing aberration-free gradient index lenses. Parameter sweeps are demonstrated to find lenses within the framework, with minimal dynamic range of refractive index distributions found from two families of aberration-free collimation lenses. As such, the theoretical tools developed for the thesis are shown to not only support further understanding of DPIC devices, but also provide constructive frameworks for the design of future devices. Towards the understanding of CO apparatus, methods are demonstrated for extracting 1D Fourier operations from standard free-space Fourier lens systems (which more naturally produces two-dimensional relationships). The validity of a family of irregular sampling schemes, in input or output planes, for producing 1D relationships is demonstrated analytically, and in simulation, via a Python Jupyter notebook attached to the thesis in an appendix. Finally an encoding pipeline that enables the implementation of convolutions of 1D signals in a standard free-space JTC setup is revealed, then tested end-to-end in simulation with point-spread functions (PSFs) incorporated. Multiplications of hundred-bit integers, encoded as binary vectors, are demonstrated to high precision using simulated apparatus featuring a binary SLM, diffraction-limited lenses, and a limited bit-depth camera. Out of 1000 pairs of randomly generated integers, up to 94% accuracy is achieved, with further measures to improve success rate proposed. Connecting theory to practice, the design and experimental construction of a proof-of- concept joint transform correlator (JTC) architecture, doubling as a Fourier optical holo- graphic projector, is presented. The architecture, which undergoes two iterations of design and physical implementation, is made up of a laser source, collimating optics, a high-speed ferroelectric liquid crystal binary spatial light modulator (SLM), a Fourier-transform lens, a camera and other ancillary optical elements. Near diffraction-limited Fourier lenses are demonstrated, designed using a maximum of three optical elements, all of them taken from stock catalogues. With the aid of computer-generated holography algorithms, long exposure techniques and then high-speed camera synchronisation, holographic projections are demonstrated to qualitatively validate the performance of the refined optical systems. These high quality outputs illuminate a path towards the full-scale JTC-based implementation of high-throughput multiplication systems in future work.