Abstract
In this paper, we consider the two-dimensional Maxwell's equations with the TM mode in pseudo-chiral media. The system can be reduced to the acoustic equation with a negative index of refraction. We first study the transmission eigenvalue problem (TEP) for this equation. By the continuous finite element method, we discretize the reduced equation and transform the study of TEP to a quadratic eigenvalue problem by deflating all nonphysical zeros. We then estimate half of the eigenvalues are negative with order of \begin{document}$O(1)$\end{document} and the other half of eigenvalues are positive with order of \begin{document}$O(10^2)$\end{document} . In the second part of the paper, we present a practical numerical method to reconstruct the support of the inhomogeneity by the near-field measurements, i.e., Cauchy data. Based on the linear sampling method, we propose the truncated singular value decomposition to solve the ill-posed near-field integral equation, at one wave number which is not a transmission eigenvalue. By carefully chosen an indicator function, this method produce different jumps for the sampling points inside and outside the support. Numerical results show that our method is able to reconstruct the support reliably.
Highlights
The transmission eigenvalue problem (TEP) has attracted a lot of attention recently in the study of direct/inverse scattering problems in inhomogeneous media [3, 4, 5, 6, 9, 11, 12, 20, 25]
The existence of transmission eigenvalues is intimately connected to the “bijectivity” of the far-field operator, which is crucial in some reconstruction methods such as the linear sampling method (LSM) [1, 8] and the factorization method [21]
The model is derived from the Maxwell’s equations with the transverse magnetic (TM) mode in pseudo-chiral media. It turns out the index of refraction of the reduced acoustic equation decreases to negative infinity when we increase the charity parameter
Summary
The transmission eigenvalue problem (TEP) has attracted a lot of attention recently in the study of direct/inverse scattering problems in inhomogeneous media [3, 4, 5, 6, 9, 11, 12, 20, 25]. We will propose a practical numerical method for the reconstruction of the support of the target D based on the LSM and the truncated SVD technique.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
