Tunable Bloch Wave Resonances and Bloch Gaps in Uniform Materials with Reconfigurable Boundary Profiles

We study wave propagation in uniform materials with periodic boundary profiles and introduce for the first time Bloch resonances and Bloch gaps. Bloch resonances are due to transverse phase matching, i.e., the coupling of two transverse standing waves corresponding to different harmonics. These are distinct from well-known Bragg resonances that result from longitudinal phase matching. We show that Bloch gaps can be engineered over the entire first Brillouin zone up to an infinite wavelength, i.e., $k_x=0$, where $k_x$ is the wave number in the direction of propagation. This is in contrast to Bragg gaps that open at a fixed wavelength, twice the period of the structure. Bloch resonances and gaps can be tuned by reconfiguring the boundary profile and we derive analytical expressions that predict these phenomena when the amplitude of the profile is small. The theory is fundamental as it broadly applies to wave phenomena that span the quantum to continuum scale with applications that range from condensed matter to acoustics. We validate the theory experimentally for the electromagnetic field at GHz frequencies. We also discuss potential photonic and electronic applications of the theory such as a white-light distributed feedback laser and a two-dimensional electron gas transistor.

Figure 1

Experimental setup showing the $x\text{−}y$ cross section of the waveguide. Polarization of the ${\mathrm{TE}}_{01}$ wave radiated by the horn is along the $z$ axis. Vector network analyzer (VNA) and $\mathrm{\Delta }x$ (lateral shift between the plates) are shown.

Figure 2

(a) $E$ field amplitude distribution along the $y$ axis for degenerate states $k_{y,0,2}$ and $k_{y,\pm 1,1}$ with the same frequency at Bloch resonance ($\mathrm{\Delta }\mathrm{x}=0$); (b) dispersion $\omega (\mathit{k})$ for symmetric waveguide ($\mathrm{\Delta }\mathrm{x}=a/2$); and (c) for the asymmetric case at $\mathrm{\Delta }\mathrm{x}=0$, and for $d=a\sqrt{3}/2=2.72\mathrm{cm}$.

Figure 3

Transformation of Fig. 2 dispersion due to the Bloch geometric resonance at $d=a\sqrt{3}/2=2.72\mathrm{cm}$ and $\mathrm{\Delta }x=0$. The degenerate cutoff frequency 11 GHz in Fig. 2 splits into two:$f_{0,2}^{+}=12,6 \mathrm{GHz}$ and $f_{1,1}^{+}=9.3\mathrm{GHz}$_, opening a 3.35 GHz Bloch gap (shaded) at $k_x=0$.

Figure 4

Measured $S_{21}$ vs frequency for two cases: (a) $\mathrm{\Delta }x=0$ (black line), where a 3.2 GHz Bloch gap opens at 11.2 GHz and $k_x=0$; and (b) $\mathrm{\Delta }x=a/2$ (blue line), where a 2 GHz Bragg gap opens at 7.6 GHz and $k_x=q/2$ but the Bloch gap closes. The insets show the geometry of the waveguide for the two cases.