Absorption and eigenmode calculation for one-dimensional periodic metallic structures using the hydrodynamic approximation

We develop a modal method that solves Maxwell's equations in the presence of the linearized hydrodynamic correction. Using this approach, it is now possible to calculate the full diffraction for structures with a period of the order of the plasma wavelength, including not only the transverse but also the longitudinal modes appearing above the plasma frequency. As an example for using this method we solve the diffraction of a plane wave near the plasma frequency from a bimetallic layer, modeled as a continuous variation of the plasma frequency. We observe absorption oscillations around the plasma frequency. The lower frequency absorption peaks and dips correspond to the lowest longitudinal modes concentrated in the lower plasma frequency region. As the frequency is increased, higher order longitudinal modes are excited and extend to the region of higher plasma frequency. Moreover, an examination of the propagation constants of these modes reveals that the absorption peaks and dips are directly related to the direction of phase propagation of the longitudinal modes. Furthermore, we formulate a variant of the plane wave expansion method, and use it to calculate the dispersion diagram of such longitudinal modes in a periodically modulated plasma frequency layer.

Figure 1

Schematic showing two unit cells of the diffraction problem geometry, with light incident from left to right. The air/metal interface is located at $z=0$, and the direction of periodicity is along the vertical (i.e., $x$) axis. The region with reduced free-carrier density is in the center of the unit cell (bright), whereas the boundaries of the unit cell are the region with higher free-carrier density (dark).

Figure 2

(a) Absorption spectrum of a semi-infinite metallic layer with ${\lambda }_p=\frac{2\pi c}{{\omega }_{p,0}}=5L$. Blue line: ${\omega }_{p,1}^2=0.1{\omega }_{p,0}^2.$ Green line: ${\omega }_{p,1}^2=0.05{\omega }_{p,0}^2$. Red line: ${\omega }_{p,1}^2=0$. (b) Real part of the propagation constant with an absolute value closest to zero, calculated for the case where ${\omega }_{p,1}^2=0.1{\omega }_{p,0}^2.$

Figure 3

(a), (b) Real and imaginary parts of $\nabla ·J$, calculated at the absorption peaks, located at (a) $\frac{\omega }{{\omega }_{p,0}}=0.9605$ and (b) $\frac{\omega }{{\omega }_{p,0}}=0.9772$. (c), (d) Real and imaginary parts of $\nabla ·J$, calculated at the absorption dips, located at (c) $\frac{\omega }{{\omega }_{p,0}}=0.9674$ and (d) $\frac{\omega }{{\omega }_{p,0}}=0.9841$. The air/metal interface is at $z=0$.

Figure 4

(a) Absorption spectrum of an Ag/Au grating with $L=35$ nm as a function of $\omega$. (b) Absorption spectrum of an Ag/Au grating with $L=10$ nm as a function of $\omega$. (c) The distribution of ${\omega }_p$ in the unit cell assumed for the calculation. (d) The real part of $\nabla ·J$, calculated for $\omega =8.592$ eV and $L=35$ nm.

Figure 5

Dispersion of the longitudinal modes. (a) Periodic case with ${\omega }_{p,1}^2=0.02{\omega }_{p,0}^2$. (b) Uniform case $({\omega }_{p,1}^2=0)$.

Figure 6

The profile of $|J_x{|}^2$ for the first to third bands of the dispersion diagram plotted in Fig. 5a. (a) $\omega =0.9936{\omega }_{p,0}$, (b) $\omega =1.0005{\omega }_{p,0}$, (c) $\omega =1.0064{\omega }_{p,0}$. All modes are calculated for $k_{x,0}=0.$

Figure 7

Schematic showing a nonlocal layer embedded in a local environment.