Figure 3

Transparency appearing and disappearing. (a) Acoustic impedance ratio in the forbidden band. The blue dashed and green dash-dot lines denote the reflection impedance ratios at the interfaces of WI and WII, respectively. The red solid line represents their summation and the zero point at ψ = 0.5π indicates the extraordinary transparency appearing. (b) Transparency frequency shift from the Bragg frequency vs the phase difference between WI and WII. The frequency detuning δ_N is normalized by the half width of the forbidden band. Thus, δ_N = ±1 indicates that the transmitted peak falls at the edge of the forbidden band and the extraordinary transparency disappears when the phase matching Δφ = 0. Only when the phases mismatch, the frequency falls in the forbidden band and we can observe the extraordinary transparency. (c) Transmission for the waveguides with different junction structures, which are formed by changing the phase of WII φ_ΙΙ and fixing that of WI φ_Ι at π/2. The circles connected by the dashed lines indicate the peak frequency of the transparency shifting to the high frequency when increasing φ_ΙΙ and a frequency jump arising when the phases match at φ_ΙΙ = φ_Ι = π/2.