[Paper Review] Scattering, trapped modes and guided waves in waveguides and diffraction gratings

V. É. Grikurov|ArXiv.org|Jun 2, 2004

Optical Coatings and Gratings8 references3 citations

TL;DR

This paper presents a numerical method for detecting trapped modes and guided waves in waveguides and diffraction gratings by analyzing scattering matrix components. It introduces a criterion based on the determinant of a submatrix of the scattering coefficients to identify localized solutions, enabling the discovery of new trapped modes in complex geometries with high accuracy and generality.

ABSTRACT

We suggest the numerical approach to detect eigenfrequencies of trapped modes in waveguides or guided waves in diffraction gratings. At the same time, the approach works perfectly for computation of systems with finitely many scattering channels. The most attractive example concerns the possibility of control on electron transport in nano-dimensions system consisting of a resonator and finitely many adjoined channels due to external variable electric field.

Motivation & Objective

• To develop a general numerical method for identifying trapped modes and guided waves in waveguides and diffraction gratings with finitely many scattering channels.
• To provide a computationally robust criterion for the existence of localized solutions based on scattering matrix components.
• To overcome numerical instability in asymptotic mode decomposition by avoiding direct computation of exponentially growing/decaying coefficients.
• To extend the applicability of scattering matrix methods to systems with complex or perturbed geometries where traditional methods fail.
• To demonstrate the method on new examples, including bent waveguides and domains with central holes, where no prior numerical estimates existed.

Proposed method

• Truncate infinite waveguide or grating channels beyond a radius R > R₀ to create a bounded computational domain.
• Compute the scattering matrix 𝕊 = (ℂ⁻)⁻¹ℂ⁺ from the asymptotic behavior of solutions in the truncated channels.
• Partition the scattering coefficient matrix ℂ⁺ into blocks, particularly isolating the C(22) block corresponding to non-propagating (evanescent) modes.
• Apply the criterion det C(22) = 0 as a necessary and sufficient condition for the existence of a solution decaying at infinity, indicating a trapped mode.
• Use the full ℂ⁺ and ℂ⁻ matrices to simultaneously extract scattering and trapping information without separate solvers.
• Leverage the unitarity of the scattering matrix and the analytic structure of mode amplitudes to ensure numerical stability and accuracy.

Experimental results

Research questions

• RQ1 Under what conditions does a solution to the Helmholtz or Schrödinger equation decay at infinity, indicating a trapped mode?
• RQ2 How can the scattering matrix be used to detect localized solutions without directly computing exponentially growing/decaying modes?
• RQ3 Can the determinant of a submatrix of the scattering coefficient matrix reliably predict the existence of trapped modes?
• RQ4 How does the proposed method perform on systems with complex or non-perturbative geometries where standard asymptotic or variational methods fail?
• RQ5 What new trapped mode configurations can be discovered using this general and robust numerical framework?

Key findings

• The condition det C(22) = 0 serves as a reliable and numerically stable criterion for the existence of trapped modes in waveguides and diffraction gratings.
• Numerical results for a waveguide with a one-side indentation show good agreement with asymptotic predictions from [8], validating the method’s accuracy even in the narrow applicability range of asymptotic formulas.
• A trapped mode in a bent waveguide with a radius of curvature matching the structure in Fig. 3 was numerically found at kd ≈ 3.091, slightly below the sub-threshold conductance frequency.
• For a symmetric three-channel waveguide with a central disk hole, a Dirichlet trapped mode was found at kd ≈ 2.594 when the hole radius a = 0, and the eigenfrequency approaches the threshold as the hole radius increases.
• The method successfully identifies new trapped modes in geometries previously unexplored numerically, including systems with Neumann conditions on internal boundaries and complex symmetries.
• The approach enables the detection of resonances in the complex plane, suggesting future extension to resonance analysis in scattering systems.