Direct and Inverse Time-Harmonic Scattering by Dirichlet Periodic Curves with Local Perturbations

Core Concepts

The authors investigate time-harmonic scattering by locally perturbed periodic curves, focusing on well-posedness and uniqueness conditions for plane wave incidences.

Abstract

This content delves into the mathematical analysis of time-harmonic electromagnetic scattering from periodic surfaces with localized defects. It explores properties of Green's functions, uniqueness in inverse problems, and radiation conditions for various wave incidences. The study emphasizes the importance of guided waves and their impact on the scattering models.
The authors establish new results regarding the forward and inverse scattering problems, providing insights into the behavior of wave fields under different conditions. Through rigorous mathematical analysis, they address challenges related to guided modes and quasi-periodicity in scattering interfaces.
Key points include:

Investigating well-posedness for plane wave incidences based on propagative wave numbers.
Uniqueness results for determining local perturbations from near/far-field data.
Application of boundary value problems to analyze time-harmonic electromagnetic scattering.
Discussion on guided waves, Bounded States in Continuity (BICs), and their implications in physics.

Stats

In such a case there exist guided waves to the unperturbed problem, which are also known as Bounded States in the Continuity (BICs) in physics.
For the inverse problem of determining the defect, we prove several uniqueness results using a finite or infinite number of point source and plane waves.

Quotes

Key Insights Distilled From

Direct and inverse time-harmonic scattering by Dirichlet periodic curves with local perturbations

by Guanghui Hu,... at arxiv.org 03-13-2024

https://arxiv.org/pdf/2403.07340.pdf

Deeper Inquiries

How do guided waves impact the uniqueness of solutions in time-harmonic scattering

Guided waves play a crucial role in determining the uniqueness of solutions in time-harmonic scattering. In the context of electromagnetic scattering from periodic structures, guided waves are solutions to the homogeneous problem that exponentially decay in certain directions orthogonal to the periodicity direction. These guided modes can lead to non-uniqueness in the forward scattering model because they introduce additional solution components that do not depend on the incident wave but are inherent to the structure itself. When guided waves exist, it becomes challenging to uniquely determine the scattered field as it may contain contributions from these modes that are independent of the incident wave.

What are the practical implications of Bounded States in Continuity (BICs) in physics

Bounded States in Continuity (BICs) have significant implications in physics, particularly in wave phenomena and scattering problems. BICs represent special states where waves become trapped or bound within a system due to specific conditions such as interference effects or structural properties. In practical terms, BICs can lead to phenomena like total reflection, resonances with infinite lifetimes, and enhanced light-matter interactions. Understanding and controlling BICs can enable advancements in various fields such as optics, photonics, acoustics, and material science by harnessing these unique states for applications like sensors, filters, lasers, and energy harvesting devices.

How can these mathematical findings be applied to real-world scenarios beyond theoretical analysis

The mathematical findings related to time-harmonic scattering and Bounded States in Continuity (BICs) have several real-world applications beyond theoretical analysis:

Optical Devices: The understanding of guided waves and BICs can be applied to design optical devices with improved performance characteristics such as high reflectivity coatings or photonic crystal structures.
Sensors: Utilizing knowledge about unique wave behaviors can enhance sensor technologies for detecting subtle changes or anomalies based on wave interactions.
Communications: Applying principles of guided waves could optimize signal transmission through waveguides or fiber optic systems.
Material Characterization: By studying how waves interact with materials at different frequencies, researchers can develop techniques for non-destructive testing or material characterization.
Energy Harvesting: Leveraging insights into resonance phenomena associated with BICs could aid in developing efficient energy harvesting systems based on capturing specific wavelengths.
These mathematical concepts provide a foundation for exploring innovative solutions across diverse fields where wave propagation plays a critical role.