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Publication# Computer generated optical volume elements by additive manufacturing

Abstract

Computer generated optical volume elements have been investigated for information storage, spectral filtering, and imaging applications. Advancements in additive manufacturing (3D printing) allow the fabrication of multilayered diffractive volume elements in the micro- scale. For a micro-scale multilayer design, an optimization scheme is needed to calculate the layers. The conventional way is to optimize a stack of 2D phase distributions and implement them by translating the phase into thickness variation. Optimizing directly in 3D can improve field reconstruction accuracy. Here we propose an optimization method by inverting the intended use of Learning Tomography, which is a method to reconstruct 3D phase objects from experimental recordings of 2D projections of the 3D object. The forward model in the optimization is the beam propagation method (BPM). The iterative error reduction scheme and the multilayer structure of the BPM are similar to neural networks. Therefore, this method is referred to as Learning Tomography. Here, instead of imaging an object, we reconstruct the 3D structure that performs the desired task as defined by its input-output functionality. We present the optimization methodology, the comparison by simulation work and the experimental verification of the approach. We demonstrate an optical volume element that performs angular multiplexing of two plane waves to yield two linearly polarized fiber modes in a total volume of 128 mu m by 128 mu m by 170 mu m.

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Neural network

A neural network can refer to a neural circuit of biological neurons (sometimes also called a biological neural network), a network of artificial neurons or nodes in the case of an artificial neur

Plane wave

In physics, a plane wave is a special case of wave or field: a physical quantity whose value, at any moment, is constant through any plane that is perpendicular to a fixed direction in space.
For

Simulation

A simulation is the imitation of the operation of a real-world process or system over time. Simulations require the use of models; the model represents the key characteristics or behaviors of the se

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The design and minaturization of microwave circuits involves the analysis and optimization of a large varity of structures, which, in general, are based on arbritrary objects embedded in also arbritary media. The most common situation is found when optimizing terminal antennas, whose main constraint is the antenna's volume, which must be fitted into a specified space, which in general is the phone or laptop case. This kind of antennas can also include stratified media, oblique metallizations, dielectric objects, waveguide components, in their definition making their analysis more difficult from the numeric point of view. The complete treatment of this kind of problems has been performed traditionally using Finite Differences or Finite Elements software, while it is more rare to find Method of Moments implementations for the solution of this kind of general problems. The Method of Moments has a certain number of advantages as the inclusion of radiation conditions or stratified media in the Green's function of the problem. Due to this it is possible to mesh only the surfaces or boundary conditions that are not included in the Green's function definition, obtaining, for instance, a more natural radiation boundary condition compared to other fullwave methods. The other big concern is the optimization of this kind of structures. In general the structures we are dealing with in this work can have very irregular shapes, in order to fit within the required volume constraints. The traditional optimizations techniques based on the gradient or conjugated gradient are not efficient when optimizing functions containning multiple minima and a large search space, which is the case in the optimization of terminal antennas. When miniaturizing terminal antennas or just a simple microstrip filter, the number of unknowns can be very large, and the different optimization variables can vary within wide limits in the search space. In this situation the Genetic Algorithms are ideal for seeking the global optimum solution of our problem. This work covers the analysis and optimization of microwave circuits, including antennas and filters in microstrip and waveguide technologies, paying more attention to the implementation of the Method of Moments (MoM) in conjugation with the Mixed Potential Integral Equation (MPIE) and Electric/Magnetic Field Integral Equation (EFIE/MFIE). The work has been split into 3 main goals: The analysis of 3 dimensional structures in stratified media, including arbritary shaped dielectric bodies using the Method of Moments, (MPIE,EFIE,MFIE). The analysis of shielded environments and more precisely the analysis of retangular waveguide cavities with the Method of Moments. The implementation of several optimization techniques, emcompassed in the frame of genetic algorithms, for the design and miniaturization of terminal antennas and microstrip filters. Some relevant examples linked to the efficiency and accuracy of the different methods and techiques explained in this work are included in each chapter, yielding to practical results suitable to be used in real life design problems. The main contributions of this work can be summarized as follows: In Chapter 2 we have developed the interpolation of 3D Green's function to the general solution of dielectric objects embedded in multilayered media. The interpolation method itself is preceeded by a spectral extraction of the quasi-static terms. This technique, very often applied in complex images techniques, has been applied to our specific Green's function problem yielding very good results in terms of accuracy and Green's function computation time. Another contribution, also emcompassed in Chapter 2, is the integration of the static part of the field dyadic G̿EM, performed by splitting the field dyadic into TM and TE components, allowing then an analytical integration of the dyadic terms in stratified media. In the frame of shielded environments in Chapter 3, the most important contribution is the extension of Ewald's acceleration technique to full electric and magnetic problems, allowing the acceleration of the field dyadic G̿EM and thus permit the extension of the MoM to the simulation of arbritary metallic shapes coupled to apertures through Ewald's approach. In Chapter 4 the most important contribution is the implementation of a Bayesian optimiser based on the estimation of probability made by dependencies trees. The dependency tree based method found in [1] and later in [2] has been adapted to the terminal antenna miniaturization yielding to excellent results in optimisation time and quality of the solution provided by the optimiser. Our implementation of the dependency tree method was found to be better than the traditional method used in this kind of problems. Chapter 4 contains also an implementation of the growing cells method presented in [3] for the specific problem of optimising pseudo periodic structures, like microwave filters, yielding to a powerful interpolating method which accelerates the optimisation process of microwaves devices whose fitness response is very time consuming.

Optical tomography has been widely investigated for biomedical imaging applications. In recent years, it has been combined with digital holography and has been employed to produce high quality images of phase objects such as cells. In this Thesis, we look into some of the newest optical Diffraction Tomography (DT) based techniques to solve Three-Dimensional (3D) reconstruction problems and discuss and compare some of the leading ideas and papers. Then we propose a neural-network-based algorithm to solve this problem and apply it on both synthetic and biological samples. Conventional phase tomography with coherent light and off axis recording is performed. The Beam Propagation Method (BPM) is used to model scattering and each x-y plane is modeled by a layer of neurons in the BPM. The network's output (simulated data) is compared to the experimental measurements and the error is used for correcting the weights of the neurons (the refractive indices of the nodes) using standard error back-propagation techniques. The proposed algorithm is detailed and investigated. Then, we look into resolution-conserving regularization and discuss a method for selecting regularizing parameters. In addition, the local minima and phase unwrapping problems are discussed and ways of avoiding them are investigated. It is shown that the proposed learning tomography (LT) achieves better performance than other techniques such as, DT especially when insufficient number or incomplete set of measurements is available. We also explore the role of regularization in obtaining higher fidelity images without losing resolution. It is experimentally shown that due to overcoming multiple scattering, the LT reconstruction greatly outperforms the DT when the sample contains two or more layers of cells or beads. Then, reconstruction using intensity measurements is investigated. 3D reconstruction of a live cell during apoptosis is presented in a time-lapse format. At the end, we present a final comparison with leading papers and commercially available systems. It is shown that -compared to other existing algorithms- the results of the proposed method have better quality. In particular, parasitic granular structures and the missing cone artifact are improved. Overall, the perspectives of our approach are pretty rich for high-resolution tomographic imaging in a range of practical applications.

In this thesis, we advocate that Computer-Aided Engineering could benefit from a Geometric Deep Learning revolution, similarly to the way that Deep Learning revolutionized Computer Vision. To do so, we consider a variety of Computer-Aided Engineering problems, including physics simulation, design optimization, shape parameterization and shape reconstruction. For each of these problems, we develop novel algorithms that use Geometric Deep Learning to improve the capabilities of existing systems. First, we demonstrate how Geometric Deep Learning architectures can be used to learn to emulate physics simulations. Specifically, we design a neural architecture which, given as input a 3D surface mesh, directly regresses physical quantities of interest defined over the mesh surface. The key to making our approach practical is re-meshing the original shape using a polycube map, which makes it possible to perform computations on Graphic Process Units efficiently. This results in a speed up of 2 orders of magnitude with respect to physics simulators with little loss in accuracy: our main motivation is to provide lightweight performance feedback to improve interactivity in early design stages. Furthermore, being a neural network, our physics emulator is naturally differentiable with respect to input geometry parameters, allowing us to solve shape design problems through gradient-descent. The resulting algorithm outperforms state of-the-art methods by 5 to 20% for 2D optimization tasks and, in contrast to existing methods, our approach can be further used to optimize raw 3D geometry. This could empower designers and engineers to improve the performance of a given design automatically, i.e. without requiring any specific knowledge about the physics of the problem they are trying to solve. To perform shape optimization robustly, we develop novel parametric representations for 3D surface meshes that can be used as strong priors during the optimization process. To this end, we introduce a differentiable way to produce explicit surface mesh representations from Neural Signed Distance Functions. Our key insight is that by reasoning on how implicit field perturbations impact local surface geometry, one can ultimately differentiate the 3D location of surface samples with respect to the underlying neural implicit field. This results in surface mesh parameterizations that can handle topology changes, something that is not feasible with currently available techniques. Finally, we propose a pipeline for reconstructing and editing 3D shapes from line drawings that leverages our end-to-end differentiable surface mesh representation. When integrated into a user interface that provides camera parameters for the sketches, we can exploit our latent parametrization to refine a 3D mesh so that its projections match the external contours outlined in the sketch. We show that this is crucial to make our approach robust with respect to domain gap. Furthermore, it can be used for shape refinement given only single pen strokes. This system could allow engineers and designers to translate legacy 2D sketches to real-world 3D models that can readily be used for downstream tasks such as physics simulations or fabrication, or to interact and modify 3D geometry in the most natural way possible, i.e. with a pen stroke.