arXiv subjects
Publications and source records attributed to Keyu Tao.
We define and study ordinary differential equations (ODEs) for functions valued in a Banach module $V$ over a finite-dimensional $\Bbbk$-algebra $\mathit{\Lambda}$ by using the tensor of Banach modules. Furthermore, we show that the solution space of a homogeneous linear ODE as above is shown to be a finitely generated $\mathit{\Lambda}$-submodule.
We report collective dark states controlled transmission in metal-dielectric-metal waveguides with a stub coupled to two twin cavities, namely, plasmonic waveguide-stub-dimer systems. In absence of one individual cavity in the dimer, plasmon induced transparency (PIT) is possible when the cavity and the stub have the same resonance frequency. However, it is shown that the hybridized modes in the dimer collectively generate two dark states which make the stub-dimer "invisible" to the straight waveguide, splitting the original PIT peak into two in the transmission spectrum. Simultaneously, the original PIT peak becomes a dip due to dark state interaction, yielding anti-PIT-like modulation of the transmission. With full-wave electromagnetic simulation, we demonstrate that this transition is controlled by the dimer-stub separation and the dimer-stub relative position. All results are analytically described by the temporal coupled mode theory. Our results may be useful in designing densely integrated optical circuits, and in optical sensing and switching applications.