Mode evolution and nanofocusing of grating-coupled surface plasmon polaritons on metallic tip

We present a detailed analysis on mode evolution of grating-coupled surface plasmonic polaritons (SPPs) on a conical metal tip based on the guided-wave theory. The eigenvalue equations for SPPs modes are discussed, revealing that cylindrical metal waveguides only support TM₀₁ and HE_m1 surface modes. During propagation on the metal tip, the grating-coupled SPPs are converted to HE₃₁, HE₂₁, HE₁₁ and TM₀₁ successively, and these modes are sequentially cut off except TM₀₁. The TM₀₁ mode further propagates with drastically increasing effective mode index and is converted to localized surface plasmons (LSPs) at the tip apex, which is responsible for plasmonic nanofocusing. The gap-mode plasmons can be excited with the focusing TM₀₁ mode by approaching a metal substrate to the tip apex, resulting in further enhanced electric field and reduced size of the plasmonic focus.