Ultracompact high-efficiency polarising beam splitter based on silicon nanobrick arrays

Since the transmission of anisotropic nano-structures is sensitive to the polarisation of an incident beam, a novel polarising beam splitter (PBS) based on silicon nanobrick arrays is proposed. With careful design of such structures, an incident beam with polarisation direction aligned with the long axis of the nanobrick is almost totally reflected (~98.5%), whilst that along the short axis is nearly totally transmitted (~94.3%). More importantly, by simply changing the width of the nanobrick we can shift the peak response wavelength from 1460 nm to 1625 nm, covering S, C and L bands of the fiber telecommunications windows. The silicon nanobrick-based PBS can find applications in many fields which require ultracompactness, high efficiency, and compatibility with semiconductor industry technologies

A sharp version of Kahan's theorem on clustered eigenvalues

AbstractLet the n × n Hermitian matrix A have eigenvalues λ1, λ2, …, λn, let the k × k Hermitian matrix H have eigenvalues μ1 ≤ μ2 … ≤ μk, and let Q be an n × k matrix having full column rank, so 1 ≤ k ≤ n. It is proved that there exist k eigenvalues λi1 ≤ λi2 ≤ … ≤ λik of A such thatmax1≤j≤k|μj − λij| ≤ Cσmin(Q) ‖AQ − QH‖2always holds with c = 1, where σmin (Q) is the smallest singular value of Q, and ‖; · ‖;2 denotes the biggest singular value of a matrix. The inequality was proved for c ≤ √2 in 1967 by Kahan, who also conjectured that it should be true for c = 1