[Paper Review] Reconstruction of dielectric constants of multi-layered optical fibers using propagation constants measurements

Evgenii M. Karchevskii, Larisa Beilina+2 more|arXiv (Cornell University)|Dec 18, 2015

Advanced Fiber Laser Technologies6 citations

TL;DR

This paper presents a novel Tikhonov regularization-based method for reconstructing the variable refractive index profile of multi-layered circular optical fibers using measured propagation constants of eigenwaves. By leveraging a theoretically derived initial approximation and minimizing a Tikhonov functional, the method achieves stable and accurate reconstruction even with up to 20% noise in propagation constant measurements, demonstrating robustness and practical feasibility for inverse waveguide problems.

ABSTRACT

We present new method for the numerical reconstruction of the variable refractive index of multi-layered circular weakly guiding dielectric waveguides using the measurements of the propagation constants of their eigenwaves. Our numerical examples show stable reconstruction of the dielectric permittivity function $\\varepsilon$ for random noise level using these measurements.

Motivation & Objective

• To develop a stable numerical method for reconstructing the refractive index profile of multi-layered circular dielectric waveguides from measured propagation constants of eigenwaves.
• To address the inverse problem of determining variable dielectric permittivity in waveguides where traditional methods are limited by waveguide-specific assumptions.
• To ensure numerical stability and accuracy in the presence of measurement noise, particularly for weakly guiding, multi-layered structures.
• To provide a theoretically justified and computationally effective approach applicable to real-world optical fiber design and characterization.

Proposed method

• Formulates a Tikhonov functional to regularize the inverse problem of reconstructing the dielectric permittivity function from propagation constant measurements.
• Uses a nonlinear, non-self-adjoint eigenvalue problem based on weakly singular integral equations to model the forward problem and guide the inverse reconstruction.
• Employs a good initial approximation for the refractive index derived from theoretical analysis of the forward problem, enhancing convergence and stability.
• Applies numerical optimization to minimize the Tikhonov functional, matching computed propagation constants to measured ones.
• Validates the method using synthetic data with controlled noise levels (5% and 20%) to test robustness.
• Uses a system of boundary integral equations to represent the waveguide's eigenmode characteristics, linking refractive index and propagation constants.

Experimental results

Research questions

• RQ1 Can the refractive index profile of a multi-layered circular optical fiber be stably reconstructed from measured propagation constants of its eigenwaves?
• RQ2 How does the choice of initial approximation affect the convergence and accuracy of the reconstruction algorithm?
• RQ3 To what extent is the proposed method robust to random noise in propagation constant measurements?
• RQ4 Can the Tikhonov regularization approach be effectively applied to non-self-adjoint, nonlinear eigenvalue problems arising in waveguide inverse problems?
• RQ5 Is the theoretical justification of the inverse problem's well-posedness sufficient to support stable numerical reconstruction in practice?

Key findings

• The method achieves stable reconstruction of the dielectric permittivity function ε even with 20% random noise in the measured propagation constants, as confirmed by numerical experiments.
• Relative reconstruction errors remained within [0, 0.20] for 20% noise, with approximated values of ε differing from exact values by no more than 20%.
• For 5% noise, the relative error was consistently below 0.05, indicating high accuracy under low-noise conditions.
• The use of a theoretically grounded initial approximation significantly improved convergence and stability of the optimization process.
• The Tikhonov functional minimization successfully reconstructed ε across multiple test cases, including complex multi-layered configurations.
• The algorithm demonstrated robustness and reliability across various initial guesses, confirming the method's practical feasibility for inverse waveguide problems.