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This project addresses key challenges arising from the application of randomized CUR compression methods to compress impedance matrices resulting from the application of the Method of Moments to integral equations typical of electromagnetic scattering prob- lems. Our primary focus lies in the pivot selection process, which regulates the resulting dimensions of the compressed impedance matrix. To address this concern, pivot allocation strategies are proposed. Once the overall best strategy is identified, the number of pivots required to achieve acceptable compression errors for a given interaction is determined. For simple geometries and uniform meshing, a basic weighted equation based on the 2D definition of the electric field degree of freedom is formulated to easily derive the opti- mum number of pivots. This simplified approach leads to compression times up to four times faster than those of Adaptive Cross Approximation (ACA), the competing simple compression method. However, this strategy proves inadequate for complex geometries with irregular meshing, and a priori knowledge is required to establish an upper bound on the resulting compression error. To handle complex geometries and automatically set this upper bound without prior information, a random forest model is introduced. The random forest approach aims to establish connections between the most critical geomet- ric parameters of the interaction under evaluation and the expected optimal number of pivots to achieve the predefined error. The utilization of random forest regression results in compression times faster than ACA for the same compression error.

In this work, we propose a GPU parallel implementation of the randomized CUR (or Pseudo Skeleton) Approximation to compress the -matrices of linear systems that arise in the discretization of integral equations modeling electromagnetic scattering problems. This compression method is highly parallelizable, in contrast with other similar methods such as the Adaptive Cross Approximation. It involves dense linear algebra computations that can be efficiently implemented on a GPU device. Besides, a stochastic convergence criterion is introduced to minimize the communication between the host and the device. Testing the code with standard cases shows the efficiency and accuracy of the method., This work was partly funded by the Ministerio de Ciencia e Innovacion (MICINN) under projects PID2019- 107885GB-C31 / AEI / 10.13039/501100011033, PID2020-113832RB-C21 / AEI / 10.13039/501100011033 and PID-2020-118410RB- C21 / AEI / 10.13039/501100011033, and Catalan Research Group 2017 SGR 219 and grant 2021 FI B2 00096., Peer Reviewed, Postprint (author's final draft)

We propose a randomized Pseudo Skeleton (or CUR) approximation method to compress the H-matrices of linear systems that arise in the discretization of integral equations in electromagnetic scattering. This method is essentially the Matrix Decomposition Algorithm (MDA) of Michielssen and Boag (1994) with equivalent basis and testing functions equal to a random subset of the original ones. It is highly parellelizable and well suited for fine-granularity architectures as GPUs. As the Adaptive Cross Approximation (ACA), CUR is purely algebraic. The method is tested with standard cases to assess its quality and efficiency., This work was partly funded by the Ministerio de Ciencia e Innovacion (MICINN) under projects PID2019-107885GBC31 / AEI / 10.13039/501100011033, PID2022-136869NBC31 / AEI / 10.13039/501100011033, PID2020-113832RBC21 / AEI / 10.13039/501100011033 and PID-2020-118410RB-C21 / AEI / 10.13039/501100011033, PDC2022-133091-I00 / AEI / 10.13039/501100011033, and Catalan Research Group 2017 SGR 219 and grant 2021 FI B2 00096., Peer Reviewed, Postprint (published version)

We propose a randomized Pseudo Skeleton (or CUR) Approximation method to compress the H-matrices of linear systems that arise in the discretization of integral equations in electromagnetic scattering. This method is highly parellelizable and well suited for fine-granularity architectures as GPUs. As the Adaptive Approximation Method (ACA), the method is purely algebraic. The method is tested with standard cases to assess its quality and efficiency, This work was partly funded by the Ministerio de Ciencia e Innovacion (MICINN) under projects PID2019-107885GBC31 / AEI / 10.13039/501100011033, PID2020-113832RBC21 / AEI / 10.13039/501100011033 and PID-2020-118410RB-C21 / AEI / 10.13039/501100011033, PDC2022-133091-I00 / AEI / 10.13039/501100011033, and Catalan Research Group 2017 SGR 219 and grant 2021 FI B2 00096., Peer Reviewed, Postprint (published version)

Numerical results on the implementation of a computationally efficient algorithm for finding the instantaneous photocurrent across the gap volume in Photoconductive antennas (PCAs) through solution of one-dimensional (1D) drift-diffusion equations along the bias field lines are presented. Excellent agreement with a reference two-dimensional (2D) numerical algorithm is found and the relative error is seen to decrease with the number of lines used in the computation. © 2022 IEEE., Peer Reviewed, Postprint (published version)

A numerical approach for the modelling of dipole Photoconductive Antennas (PCAs) is presented, based on the classical Drude model with extension to include the bias field screening due to both the space charge and the THz radiation fields. By employing a fixed-point iterative algorithm, good computational efficiency and convergence are obtained for typical PCA configuration parameters. Results unveiling the role played by each of the different phenomena and the radiated THz spectra for different dipole lengths are shown and discussed. © 2022 IEEE., Peer Reviewed, Postprint (published version)

In this work, we propose Monte Carlo sampling as a technique to avoid communication overhead in the numerical solution of complex electromagnetic scattering problems. The method is based on statistically sampling a formal representation of the solution of the scattering problem, based on a modified Born series. The sampling process can be independently carried out in different processors with no need for communication between them. This lack of communication makes the algorithm suitable for massively parallel computational environments., This work was partly funded by the Ministerio de Ciencia e Innovacion (MICINN) under projects PID2019-107885GB-C31 / AEI / 10.13039/501100011033, PID2020-113832RB- C21 / AEI / 10.13039/501100011033 and TEC2017- 84817-C2-2-R / AEI / 10.13039/501100011033 and the Unidad de Excelencia Maria de Maeztu MDM-2016- 0600/AEI/10.13039/501100011033, which is financed by the Agencia Estatal de Investigaci ´on, Spain, and Catalan Research Group 2017 SGR 219, and Ag`encia de Gesti ´o d’Ajuts Uni- versitaris i de Recerca with grant 2021 FI B2 00096., Peer Reviewed, Postprint (published version)

Photoconductive antennas are devices capable of generating THz waves. In this thesis, we propose a model of THz wave generation in photoconductive antennas based on the Dude model. The derived equations explain the THz radiation when an optical pulse illuminates a photoconductor with an applied bias field. Furthermore, we derive an additional term appearing in the Drude equation that accounts for the generation of carriers with zero velocity. The model includes space-charge screening through a geometrical factor and radiation screening through an effective antenna impedance. The system is solved numerically using the iterative fixed-point method and Anderson acceleration. The fixed-point method convergence is limited, but Anderson acceleration can converge for a broader range of parameters. Analyzing the photocurrent at different optical powers, we find that it saturates at high optical powers due to the slowdown of the dynamics by the screening fields.

In this work, we develop a novel Monte Carlo method for solving the electromagnetic scattering problem for clusters of photonic particles. The method is based on a formal solution of the scattering problem as a modified Born series whose coefficients are found by a conformal transformation. The terms of the Born series are approximated by sampling random elements of its matrix representation, computed by the Method of Moments. Unlike other techniques like the Fast Multiple Method, this Monte Carlo method does not require communication between processors, which makes it suitable for large parallel executions., This work was partly funded by the Ministerio de Ciencia e Innovacion (MICINN) under projects PID2019- 107885GB-C31 / AEI / 10.13039/501100011033, PID2020-113832RB-C21 / AEI / 10.13039/501100011033 and TEC2017-84817-C2-2-R / AEI / 10.13039/501100011033 and the Unidad de Excelencia Maria de Maeztu MDM- 2016-0600/AEI/10.13039/501100011033, which is financed by the Agencia Estatal de Investigación, Spain, and Catalan Research Group 2017 SGR 219 and grant 2021 FI B2 00096., Peer Reviewed, Postprint (author's final draft)

The solution of the electromagnetic scattering problem by the Method of Moments often leads to huge systems of equations that require large computational resources. In parallel computing environments, communication between computing units constitute the main bottleneck that limits the scalability of the problem. In this work, we use randomized techniques to relax the need for communication, thus achieving better scalability. The method is based on a formal solution of the scattering problem as a Neumann series modified by a set of coefficients who ensure its convergence. In the case of the Volume Integral Equation for dielectric objects, these coefficients are obtained by a conformal transformation that depends on the spectrum of the scattering operator, which is known to be confined in a certain region of the complex plane thanks to the spectral localization theorem. Then, the terms of the Neumann series are approximated by randomly sampling elements of its matrix representation computed by the method of moments. We show that this estimator is unbiased and, unlike other Monte Carlo techniques based on the Neumann series, always converges to the solution. The main advantages of this method are its ease of formulation and its potential parallelism. Since the coefficients of the modified Neumann series are known beforehand, the Monte Carlo samples can be computed independently without the need of communication between different processors or computing units, apart from a final reduction operation. Thus, the method is scalable and well suited for highly parallel architectures. Besides, the granularity of the parallel method can be adapted in a versatile way to exploit the capabilities of a particular computational environment.

The conventional scattering analysis of perfectly conducting sheets ignores the scattering contribution of the rim. The so-called thin-surface scheme arises from the discretization of the Electric-field Integral Equation by assuming the sheet thickness to be null. This gives rise to many less unknowns than the full approach, arising from modelling the whole plate. Although good accuracy is observed in many practical applications, the thin-plate scattering analysis of thick enough sheets, especially under oblique incidences, with low grazing angles, exhibits great inaccuracies. A recent approach, so-called thick-surface, has proved to show similar accuracy as the full scheme, also in those cases where the thin-surface scheme fails, with moderate computational effort. In this paper, we show RCS results computed with the thick-surface approach for several sheets with nonzero thickness where the thin-surface scheme fails. Several types of meshes are adopted, triangular, quadrangular or hybrid. The required symmetries for the generation of the meshes are pointed out in each case.

In this work, we propose a method to enhance the convergence of the Born series. The Born series is widely used in scattering theory, but its convergence is only guaranteed under certain restrictive conditions which limit the cases where this formulation can be applied. The proposed method, based on modifying the singularities of the resolvent operator by a change of variables, accelerates the convergence of the series or even achieves convergence in otherwise divergent cases. Due to its low computational cost and ease of implementation, the method preserves the advantages of the Born series while it extends its range of application. It can also be applied to a variety of methods relying on the same mathematical principles as the Born series. Numerical examples are provided to show that a significant improvement can be achieved with simple techniques and with a limited knowledge about the operator spectrum., This work was partly funded by the Ministerio de Ciencia e Innovacion (MICINN) under projects TEC2016-78028-C3-1-P, MDM2016-0600, TEC2017-83343-C4-2-R, TEC2017-84817-C2-2-R, PID 2019-107885GB-C31 and Catalan Research Group 2017 SGR 219; and by Secretary of Universities and Research of the Ministry of Business and Knowledge of the Generalitat of Catalonia and European Social Fund under scholarship 2019 FI B00788., Peer Reviewed, Postprint (author's final draft)

In this work, we develop a novel Monte Carlo method for solving the electromagnetic scattering problem. The method is based on a formal representation of the solution as a modified Born series whose coefficients are found by a conformal transformation. The terms of the Born series are approximated by sampling random elements of its matrix representation, computed by the Method of Moments. This method requires almost no communications between cores, which makes it suitable for large scale parallelization., This work was partly funded by Unidad de Excelencia Maria de Maeztu MDM-2016- 0600/AEI/10.13039/501100011033, ID2019-107885GBC31/ AEI/10.13039/501100011033, TEC2017-83343-C4- 2-R/AEI/10.13039/501100011033, TEC2017-84817-C2-2- R/AEI/10.13039/501100011033, Catalan Research Group 2017 SGR 219 and Secretary of Universities and Research of the Ministry of Business and Knowledge of the Generalitat of Catalonia and European Social Fund under scholarship 2019 FI B00788., Peer Reviewed, Postprint (published version)

En aquest treball, tenim com a objectiu estudiar els processos d’emissió i detecció de THz que es produeixen als substrats fotoconductors. Aquests estan alimentats òpticament per polsos làsers ultracurts o per senyals làser contínues formades per la superposició de dues frequències, per espectròmetres temporals/freqüencials, respectivament. Analitzarem els processos fonamentals relatius a la dinàmica de portadors fotogenerats. En particular, ens interessen models simplificats que descriguin la generació de portadors i equacions de transport que tinguin en compte l’estructura i les propietats del material. Les simulacions resultants, ens permetran clarificar els processos darrere la generació de fotocorrent i ens proporcionaran intuició per entendre els comportaments observats, per finalment optimitzar el disseny de les antenes per tal de millorar l’eficiència., En este trabajo, tenemos como objetivo estudiar los procesos de emisión y detección de THz que se producen en los sustratos fotoconductores. Estos están alimentados ópticamente por pulsos láser ultracortos o por señales láser continuas formadas por la superposición de dos frecuencias, para espectrómetros temporales/frecuenciales respectivamente. Analizaremos los procesos fundamentales relativos a la dinámica de portadores fotogenerados. En particular nos interesan modelos simplificados que describan la generación de portadores y ecuaciones de transporte que tengan en cuenta la estructura y las propiedades del material. Las simulaciones resultantes, nos permitirán clarificar los procesos subyacentes a la generación de fotocorriente y nos proporcionarán intuición para entender los comportamientos observados, para finalmente optimizar el diseño de las antenas para mejorar su eficiencia., In this work we aim at studying THz emission and detection processes in photoconductive substrates, optically fed by either ultrashort laser pulses, or by two frequency continuous laser signals, respectively for time/frequency domain spectrometers. We will review the fundamental photogenerated carrier dynamic processes. In particular, we are interested in simplified models for the carrier generation and transport equations that account for the structure geometry and the material properties. The resulting simulation methods will allow to shed some light into the underlying effects of photocurrent generation and to provide intuitive insights on the observed behaviors, and ultimately to optimize antenna designs in order to improve the optoelectronic THz conversion efficiency.

The laser-induced current carrier generation and transport under large biasing voltages in photoconductive materials located in electrode gap region of photoconductive antennas is a complex process involving many different phenomena that interact with each other in intricate ways. Given the large biasing voltages employed, a simple model that assumes that the carriers travel along the biasing field lines allows to retain the basic features of the photocarrier generation process. The model is presented in detail and results for a typical interdigitated electrode are shown to well agree with measures found in the literature., This work was supported in part by the Spanish Inter-Ministerial Commission on Science and Technology (CICYT) under projects TEC201678028-C3-1 and, by FEDER and the Unidad de Excelencia Maria de Maeztu MDM-2016-0600, which is financed by the Agencia Estatal de Investigacion, Spain, and by Catalan Research Group 2017 SGR219., Peer Reviewed, Postprint (published version)

Geometric Algebra provides a mathematical frame-work that allows to express many equations of physics in a more compact way [1]. It unifies the electric and the magnetic field in a single mathematical object: a multivector. Then, the four Maxwell’s equations can be written as a single one. Here we study how the integral formulation of the electromagnetic field in Geometric Algebra naturally embeds the Calderon Projector when the field is evaluated in a surface., This work was supported by FEDER and the "Spanish Plan Estatal de Investigación Científica y Técnica y de Innovación", under projects TEC2016-78028-C3-1-P and TEC2017-84817-C2-2-R and the Unidad de Excelencia Maria de Maeztu MDM-2016-0600, which is financed by the Agencia Estatal de Investigación, Spain, and Catalan Research Group 2017 SGR219., Peer Reviewed, Postprint (published version)

We propose an extension of the Ulam-Neumann algorithm for solving system of equations arising from photonic problems. This method has good parallel properties and allows to implement acceleration techniques.

En este proyecto se pretende comprobar si la miniaturización del tamaño de las antenas utilizando geometría fractal es superior con antenas plasmónicas que con antenas convencionales debido a que la alta localización de los modos plasmónicos podría reducir el efecto "atajo" que se observa en antenas convencionales y que limita la posibilidad de miniturización. Se estudiará la teoría y se realizarán simulaciones de nano-antenas plasmónicas de geometría fractal. http://en.wikipedia.org/wiki/Surface_plasmon http://en.wikipedia.org/wiki/Surface_plasmon_polariton Para las, We study the effects of using fractal shapes on the design of nanoantennas at optical geometries., Estudiamos los efectos de emplear geometrías fractales en el diseño de nanoantenas en frecuencias ópticas, Estudiem els efectes d'emprar geometries fractals a nanoantenes a frequències òptiques.