Interference effects on guided Cherenkov emission in silicon from perpendicular, oblique, and parallel boundaries

Waveguide electromagnetic modes excited by swift electrons traversing Si slabs at normal and oblique incidence are analyzed using monochromated electron energy-loss spectroscopy and interpreted using a local dielectric theory that includes relativistic effects. At normal incidence, sharp spectral features in the visible/near-infrared optical domain are directly assigned to $p$-polarized modes. When the specimen is tilted, $s$-polarized modes, which are completely absent at normal incidence, become visible in the loss spectra. In the tilted configuration, the dispersion of $p$-polarized modes is also modified. For tilt angles higher than $\sim 50°$, Cherenkov radiation, the phenomenon responsible for the excitation of waveguide modes, is expected to partially escape the silicon slab and the influence of this effect on experimental measurements is discussed. Finally, we find evidence for an interference effect at parallel $\text{Si}/{\text{SiO}}_2$ interfaces, as well as a delocalized excitation of guided Cherenkov modes.

Figure 1

(a) Simplified schematic representation of Cherenkov emission in a silicon slab. (b) Experimental and modeled spectra for silicon slabs with thicknesses of 315 and 100 nm. Simulated maps displaying the loss probability as a function of the energy (vertical axis) and the scattering vector (horizontal axis) are shown for thicknesses of (c) 315 and (d) 110 nm. On the maps, positive contributions appear dark.

Figure 2

(a) Simplified schematic representation of Cherenkov emission in a silicon slab tilted with respect to the electron trajectory. Experimental (top) and modeled (bottom) spectra for silicon slabs with thicknesses of (b) 115 and (c) 200 nm and for tilt angles of $0°$, $40°$, and $65°$.

Figure 3

Simulated maps displaying the loss probability as a function of the energy (vertical axis) and the scattering vector along the $y$ direction (horizontal axis) for an incidence angle of (a) $0°$, (b) $40°$, and (c) $65°$. The breaking of cylindrical symmetry as a result of tilting results in an inequivalence between $+k_y$ and $-k_y$. The slab thickness is 200 nm.

Figure 4

(a) A simulated map displaying the loss probability as a function of the energy (vertical axis) and the scattering vector along the $z$ direction (horizontal axis) for an incidence angle of $65°$ and for a slab thickness of 200 nm. The surface terms for (b) $s$-polarized and (c) $p$-polarized modes are shown separately.

Figure 5

(A) Experimental and (B) modeled loss spectra for an incident angle of $65°$ and for slab thicknesses of (a) 115 and (b) 200 nm. The integrated $s$ and $p$ polarized surface terms are shown separately (C), together with the integrated loss probabilities inside the light cone (D). For clarity, the D curves have been shifted downward vertically, and their zero intensity corresponds to the position of the energy-loss axes.

Figure 6

(a) Simplified schematic representation of Cherenkov emission in a silicon slab in the presence of a nearby $\text{Si}/{\text{SiO}}_2$ interface. (b) A series of experimental spectra acquired at normal incidence in a 280 nm thick silicon slab at various distances $y_0$ from a parallel $\text{Si}/{\text{SiO}}_2$ interface. The ${\text{SiO}}_2$ layer thickness is around 500 nm.

Figure 7

(a) Loss probability as a function of $Q_{\perp }{y}_0$, where $Q_{\perp }$ is the momentum of the light perpendicular to the electron trajectory, as defined in Ref. 19, and $y_0$ is the impact parameter. Each spectra acquired at distances between 40 and 125 nm from the $\text{Si}/{\text{SiO}}_2$ interface (shown in Fig. 6) is replotted in (a) as a function of $Q_{\perp }{y}_0$ for an energy range from 0.65 to 3.3 eV, and the loss probability was normalized with a spectra acquired far (253 nm) from the $\text{Si}/{\text{SiO}}_2$ interface. The black line is an average of all the spectra. (b) Simulations for the case of two semi-infinite media. For simplicity, the Si and ${\text{SiO}}_2$ media were represented with a dielectric functions with a constant real component, which was taken at 800 nm from optical data (Ref. 29), and no imaginary component. In (b), the loss probability was normalized with the bulk limit.

Figure 8

Experimental spectra acquired in ${\text{SiO}}_2$ at a distance of 5 and 16 nm from the Si slab. A spectrum acquired in Si at 50 nm from the $\text{Si}/{\text{SiO}}_2$ interface is also displayed, and it has been shifted upward vertically for clarity. The slab thickness is 145 nm.