Surface scattering and band gaps in rough waveguides and nanowires

The boundaries of waveguides and nanowires have drastic influence on their coherent scattering properties. Designing the boundary profile is thus a promising approach for transmission and band-gap engineering with many applications. By performing an experimental study of microwave transmission through rough waveguides we demonstrate that a recently proposed surface scattering theory can be employed to predict the measured transmission properties from the boundary profiles and vice versa. A new key ingredient of this theory is a scattering mechanism which depends on the squared gradient of the surface profiles. We demonstrate the nontrivial effects of this scattering mechanism by detailed mode-resolved microwave measurements and numerical simulations.

Figure 1

Experimental setup with exchangeable rough boundaries. The photo shows the considered waveguide with the surface disordered region in the center and the lifted top plate. Step motors are attached to the top plate to shift the input and output antenna in $z$ direction. Absorbers at both ends suppress reflection (indicated in the sketch below).

Figure 2

The inverse backscattering lengths $1/L_2^{\left(b\right),\left(\mathrm{AS}\right)}$ (gray curve) and $1/L_2^{\left(b\right),\left(\mathrm{SGS}\right)}$ (black curve) for the second mode, $n=2$ are shown; see Eqs. (5) and (6). The corresponding power spectra are calculated directly from the rough boundaries shown in Fig. 1. Regions of dominating SGS and AS mechanisms are shaded.

Figure 3

Measured transmittance $T_n$ (gray curve), for (a) the first ($n=1$) and (b) the second mode ($n=2$). The corresponding theoretical prediction is shown as solid black curve [see Eq. (3)]. For the dotted line the SGS is neglected explicitly. The shaded areas indicate the gaps caused by the dominating scattering mechanism. (c) Numerically calculated transmittance for the second mode $T_2$ (dashed gray line), compared to measured transmittance (solid gray). The markers indicate the positions of the numerically calculated wave functions shown in Fig. 4. (d) Numerically calculated transmittance (dashed gray curve) compared to the theoretical prediction (black solid curve) for reduced disorder strength $\sigma$ (50$\%$ of experimental value).

Figure 4

Numerically calculated wave functions for transmission in the second mode. The corresponding $k_2$ values of (a) $23.46$ m${}^{-1}$, (b) $64.02$ m${}^{-1}$, and (c) $84.39$ m${}^{-1}$ are indicated in Fig. 3c.