arXiv subjects

Tim Gillespie

Publications and source records attributed to Tim Gillespie.

4 recordsLinked to original sources

arXiv2024

Detectorless 3D terahertz imaging: achieving subwavelength resolution with reflectance confocal interferometric microscopy

Terahertz imaging holds great potential for non-destructive material inspection, but practical implementation has been limited by resolution constraints. In this study, we present a single-pixel THz imaging system based on a confocal microscope architecture, utilising a quantum cascade laser as both transmitter and phase-sensitive receiver. Our approach integrates laser feedback interferometry detection to achieve a two-fold improvement in lateral resolution and a two-order-of-magnitude enhancement in axial resolution over conventional imaging through precise interferometric phase measurements. This translates to a lateral resolution near $\lambda/2$ and a depth of focus better than $\lambda/5$, significantly outperforming traditional confocal systems. The system can produce a 0.5 Mpixel image in under two minutes, surpassing both raster-scanning single-pixel and multipixel focal-plane array-based imagers. Coherent operation enables simultaneous amplitude and phase image acquisition, and a custom visualisation method links amplitude to image saturation and phase to hue, enhancing material characterisation. A 3D tomographic analysis of a silicon chip reveals subwavelength features, demonstrating the system's potential for high-resolution THz imaging and material analysis. This work sets a new benchmark for THz imaging, overcoming key challenges and opening up transformative possibilities for non-destructive material inspection and characterisation.

physics.optics↗

arXiv2022

Double square moments and bounds for resonance sums of cusp forms

Let $f$ and $g$ be holomorphic cusp forms for the modular group $SL_2(\mathbb Z)$ of weight $k_1$ and $k_2$ with Fourier coefficients $\lambda_f(n)$ and $\lambda_g(n)$, respectively. For real $\alpha\neq0$ and $0<\beta\leq1$, consider a smooth resonance sum $S_X(f,g;\alpha,\beta)$ of $\lambda_f(n)\lambda_g(n)$ against $e(\alpha n^\beta)$ over $X\leq n\leq2X$. Double square moments of $S_X(f,g;\alpha,\beta)$ over both $f$ and $g$ are nontrivially bounded when their weights $k_1$ and $k_2$ tend to infinity together. By allowing both $f$ and $g$ to move, these double moments are indeed square moments associated with automorphic forms for $GL(4)$. By taking out a small exceptional set of $f$ and $g$, bounds for individual $S_X(f,g;\alpha,\beta)$ will then be proved. These individual bounds break the resonance barrier of $X^\frac58$ for $\frac16<\beta<1$ and achieve a square-root cancellation for $\frac13<\beta<1$ for almost all $f$ and $g$ as an evidence for Hypothesis S for cusp forms over integers. The methods used in this study include Petersson's formula, Poisson's summation formula, and stationary phase integrals.

math.NT↗

arXiv2009

Solvable Base Change and Rankin-Selberg Convolutions

Given unitary automorphic cuspidal representations $π$ and $π'$ defined on $GL_n(\mathbb{A}_E)$ and $GL_m(\mathbb{A}_F)$, respectively, with $E$ and $F$ solvable algebraic number fields we deduce a prime number theorem for the Rankin-Selberg L-function $L(s,AI_{E/\mathbb{Q}}(π)\times AI_{F/\mathbb{Q}}(π'))$ under a self-contragredient assumption and a suitable Galois invariance condition on the representations, where $AI_{K/\mathbb{Q}}$ denotes the automorphic induction functor for any number field $K/\mathbb{Q}$.

math.NT↗

arXiv2009

A Prime Number Theorem for Rankin-Selberg L-functions over Number fields

We prove a prime number theorem first for the classical Rankin-Selberg L-function $L(s,π\timesπ')$ over any Galois extension with $π$ and $π'$ unitary automorphic cuspidal representations of $GL_n$ and $GL_m$ respectively with at least one of the representations subject to a self-contragredient assumption. We then extend these results to two representations $π$ defined on $GL_n/E$ and $π'$ defined on $GL_m/F$ with $E$ and $F$ cylic algebraic number fields of coprime degree where $π$ and $π'$ admit a base change lift from $\mathbb{Q}$ again given a self-contragredient assumption on at least one of the representations which lift to $π$ or $π'$.

math.NT↗