[Paper Review] Invisibility and perfect reflectivity in waveguides with finite length branches

Lucas Chesnel, С. А. Назаров+1 more|arXiv (Cornell University)|Feb 16, 2017

Photonic and Optical Devices38 references4 citations

TL;DR

This paper investigates invisibility and perfect reflectivity in 2D waveguides with finite-length branches under Neumann boundary conditions. Using asymptotic analysis and scattering matrix theory, it demonstrates how to design geometries where reflection coefficient 𝒫=0 and transmission coefficient 𝒯=1 (perfect invisibility), or 𝒫=0 and 𝒯=1 (non-reflectivity), or |𝒫|=1 and 𝒯=0 (perfect reflectivity), by tuning the branch length and geometry, validated numerically.

ABSTRACT

We consider a time-harmonic wave problem, appearing for example in water-waves and in acoustics, in a setting such that the analysis reduces to the study of a 2D waveguide problem with a Neumann boundary condition. The geometry is symmetric with respect to an axis orthogonal to the direction of propagation of waves. Moreover, the waveguide contains one branch of finite length. We analyse the behaviour of the complex scattering coefficients $\\mathcal{R}$, $\\mathcal{T}$ as the length of the branch increases and we show how to design geometries where non reflectivity ($\\mathcal{R}=0$, $|\\mathcal{T}|=1$), perfect reflectivity ($|\\mathcal{R}|=1$, $\\mathcal{T}=0$) or perfect invisibility ($\\mathcal{R}=0$, $\\mathcal{T}=1$) hold. Numerical experiments illustrate the different results.

Motivation & Objective

• To identify geometric configurations in 2D waveguides with finite-length branches that yield perfect invisibility (𝒯=1, 𝒫=0), non-reflection (𝒫=0), or perfect reflectivity (|𝒫|=1, 𝒯=0).
• To extend existing methods for achieving non-reflectivity to include perfect invisibility, overcoming limitations of smooth perturbation techniques.
• To rigorously analyze the asymptotic behavior of scattering coefficients as branch length increases, using spectral and functional analysis.
• To provide a theoretical framework for designing waveguides with tailored scattering properties using symmetry and branch length control.
• To validate theoretical predictions through numerical experiments demonstrating invisibility and reflectivity phenomena.

Proposed method

• Formulates the waveguide problem as a time-harmonic Helmholtz equation with Neumann boundary conditions in a 2D unbounded domain.
• Employs asymptotic analysis to study scattering coefficients (𝒫, 𝒯) as the branch length L → ∞, deriving decay estimates for the difference between the full solution and the incident wave.
• Uses the scattering matrix 𝔰∞, proven unitary and symmetric, to enforce energy conservation and symmetry constraints on the system.
• Applies functional analysis in H¹ and H¹/² spaces to bound the error between the exact solution and the incident wave, leading to exponential decay estimates.
• Derives the key estimate ‖U−U∞‖H¹(DL) ≤ C e^{√((π/ℓ)²−k²)(d−L)} ‖U∞‖H¹/²(Γd), showing rapid convergence as L increases.
• Leverages symmetry and the structure of eigenmodes (φₙ) to decompose the wave field and control the behavior of boundary terms on the branch.

Experimental results

Research questions

• RQ1 Can finite-length branches in symmetric waveguides be tuned to achieve perfect invisibility (𝒯=1, 𝒫=0) under Neumann boundary conditions?
• RQ2 What geometric and asymptotic conditions allow for perfect reflectivity (|𝒫|=1, 𝒯=0) in such waveguide systems?
• RQ3 How does the scattering behavior (𝒫, 𝒯) evolve as the branch length increases, and can this be used to design non-reflective or invisible configurations?
• RQ4 Why do standard smooth perturbation methods fail to achieve 𝒯=1 (perfect invisibility), and how can singular perturbations overcome this?
• RQ5 What role does symmetry play in enabling perfect invisibility or non-reflection in waveguide systems with finite branches?

Key findings

• Perfect invisibility (𝒯=1, 𝒫=0) is achievable in symmetric waveguides with finite-length branches by tuning the branch length and geometry, even under Neumann conditions.
• The scattering matrix 𝔰∞ is proven to be unitary and symmetric, ensuring energy conservation and physical consistency.
• As the branch length L increases, the solution U converges exponentially fast to the incident wave U∞ in H¹(DL), with the error bounded by C e^{√((π/ℓ)²−k²)(d−L)} ‖U∞‖H¹/²(Γd).
• Non-reflectivity (𝒫=0) can be achieved via symmetric configurations, and when combined with |𝒯|=1, leads to perfect invisibility.
• Numerical experiments confirm the theoretical predictions, demonstrating the emergence of invisibility and perfect reflectivity for specific branch lengths.
• The method overcomes limitations of smooth perturbation techniques by using singular perturbations (e.g., thin rectangles), enabling control over 𝒯 beyond |𝒯|=1.