Trapped-mode-induced Fano resonance and acoustical transparency in a one-dimensional solid-fluid phononic crystal

We investigate theoretically and numerically the possibility of existence of Fano and acoustic-induced transparency (AIT) resonances in a simple though realistic one-dimensional acoustic structure made of solid-fluid layers inserted between two fluids. These resonances are obtained by combining appropriately the zeros of transmission (antiresonance) induced by the solid layers and the local resonances induced by the solid or combined solid-fluid layers with surface free boundary conditions. In particular, we show the possibility of trapped modes, also called bound states in continuum, which have recently found a high renewal interest. These modes appear as resonances with zero width in the transmission spectra as well as in the density of states (DOS). We consider three different structures: (i) a single solid layer inserted between two fluids. This simple structure shows the possibility of existence of trapped modes, which are discrete modes of the solid layer that lie in the continuum modes of the surrounding fluids. We give explicit analytical expressions of the dispersion relation of these eigenmodes of the solid layer which are found independent of the nature of the surrounding fluids. By slightly detuning the angle of incidence from that associated to the trapped mode, we get a well-defined Fano resonance characterized by an asymmetric Fano profile in the transmission spectra. (ii) The second structure consists of a solid-fluid-solid triple layer embedded between two fluids. This structure is found more appropriate to show both Fano and acoustic-induced transparency resonances. We provide detailed analytical expressions for the transmission and reflection coefficients that enable us to deduce a closed-form expression of the dispersion relation giving the trapped modes. Two situations can be distinguished in the triple-layer system: in the case of a symmetric structure (i.e., the same solid layers) we show, by detuning the incidence angle $\theta$, the possibility of existence of Fano resonances that can be fitted following a Fano-type expression. The variation of the Fano parameter that describes the asymmetry of such resonances as well as their width versus $\theta$ is studied in detail. In the case of an asymmetric structure (i.e., different solid layers), we show the existence of an incidence angle that enables to squeeze a resonance between two transmission zeros induced by the two solid layers. This resonance behaves like an AIT resonance, its position and width depend on the nature of the fluid and solid layers as well as on the difference between the thicknesses of the solid layers. (iii) In the case of a periodic structure (phononic crystal), we show that trapped modes and Fano resonances give rise, respectively, to dispersionless flat bands with zero group velocity and nearly flat bands with negative or positive group velocities. The analytical results presented here are obtained by means of the Green's function method which enables to deduce in closed form: dispersion curves, transmission and reflection coefficients, DOS, as well as the displacement fields. The proposed solid-fluid layered structures should have important applications for designing acoustic mirrors and acoustic filters as well as supersonic and subsonic materials.

Figure 1

Schematic representation of a solid layer of thickness $d_s$ sandwiched between two semi-infinite fluids $F_0$. $\theta$ is the angle of incidence.

Figure 2

Transmission coefficient (in color scale) versus the dimensionless frequency $\mathrm{\Omega }$ and the angle of incidence $\theta$. White dotted curves represent the frequencies of the transmission zeros (total reflection), whereas black and cyan curves are associated to symmetric and antisymmetric modes of the free surface solid layer, respectively.

Figure 3

Variation of the transmission coefficient (a), phase (d), and DOS (in arb. units) (g) versus the dimensionless frequency $\mathrm{\Omega }=\frac{\omega d_s}{v_0}$ for the structure depicted in Fig. 1 for $\theta =33.2<{\theta }_I=33.{49}^{\circ }$. The phase is represented in the restricted frequency domain around ${\mathrm{\Omega }}_I=6.77$. (b), (e), (h) Represent the same curves as (a), (d), (g) but for $\theta ={\theta }_I=33.{49}^{\circ }$. (c), (f), (k) Represent the same curves as (a), (d), (g) but for $\theta =33.8>{\theta }_I=33.{49}^{\circ }$.

Figure 4

(a)–(c) Variation of the transmission coefficient versus the dimensionless frequency $\mathrm{\Omega }=\frac{\omega d_s}{v_0}$ for the structure depicted in Fig. 1 for $\theta ={24}^{\circ }<{\theta }_{II}=26.{34}^{\circ },\theta ={\theta }_{II}$, and $\theta =27.5^{\circ }>{\theta }_{II}=26.{34}^{\circ }$, respectively. (d)–(f) Characterize the DOS (in arb. units) versus $\mathrm{\Omega }$ for the same values of $\theta$ as in Figs. 3–3, respectively.

Figure 5

Schematic representation of a triple-layer structure made of a fluid layer $F$ of thickness $d_f$ sandwiched between two solids $S_1$ and $S_2$ of thicknesses $d_{s1}$ and $d_{s2}$, respectively. The whole system is inserted between two fluids $F_0$. $\theta$ is the angle of incidence and $L=d_f+d_{s1}+d_{s2}$ is the total length of the triple layer.

Figure 6

(a) Variation of the transmission coefficient versus the dimensionless frequency $\mathrm{\Omega }=\frac{\omega d_f}{v_f}$ for the structure depicted in Fig. 5. The width of the layers is taken such as $d_s=d_f$. The modes $I$ at $\mathrm{\Omega }={\mathrm{\Omega }}_0=3.52$ correspond to the trapped mode where two transmission zeros induced by the two solid layers coincide with the hidden Fano resonance. (b) The same as (a) but for the variation of the phase. (c), (e) The same as (a) but for $\theta =30.8{}^{\circ }$ and $34.4{}^{\circ }$ with their phases (d) and (f), respectively. Modes 1 and 3 correspond to Fano resonances, whereas modes 2 and 4 correspond to the transmission zeros.

Figure 7

Transmission coefficient versus the dimensionless frequency $\mathrm{\Omega }$ and the angle of incidence $\theta$. Color areas indicate the magnitude of the transmission rate. White dotted curves represent the frequencies of the transmission zeros (total reflection).

Figure 8

(a) Square modulus of the displacement field versus the space position for the mode labeled 2 (transmission zero) in Figs. 6 and 7. (b) The same as in (a) but for the mode labeled 1 (Fano resonance) in Figs. 6 and 7. (c) The same as in (a) but for the mode labeled I (BIC mode) in Figs. 6 and 7.

Figure 9

Comparison between the transmission curves associated to exact results (solid line) and fitted results (open circles) obtained from Eq. (31) versus the dimensionless frequency $\mathrm{\Omega }$ for $d_s=d_f$ and $\theta =32.6^{\circ }$.

Figure 10

(a) Variation of the dimensionless frequency $\mathrm{\Omega }$ of the Fano resonance and the transmission zero versus $\theta$ as in Fig. 7. (b), (c) Variation of the Fano parameter $q$ and the quality factor $Q$ versus $\theta$.

Figure 11

(a) Comparison between exact results (solid line) and fitted results (open circles) obtained from Eq. (38) of the transmission coefficient versus $\mathrm{\Omega }$ for a triple layer characterized by the thicknesses $d_{s_1}=1.1d_f$ and $d_{s2}=0.9d_f$. The incidence angle is fixed to $\theta =32.6^{\circ }$. The insets in (a) give the transmission for $\theta ={32}^{\circ }$ and ${33}^{\circ }$. (b), (c) The same as in (a) but for the phase and the delay time, respectively.

Figure 12

(a), (b) Square modulus of the displacement field versus the space position for the modes labeled 1, 2 (transmission zeros) in Fig. 11. (c) The same as in (a) and (b), but for the mode labeled I (AIT resonance) in Fig. 11.

Figure 13

(a) Variation of the transmission coefficient as a function of $\mathrm{\Omega }$ and the angle of incidence $\theta$. Color areas denote different transmission rates. The white dotted curves show the positions of the transmission zeros (total reflection) induced by the first and second solid layers with thicknesses $d_{s_1}=1.1d_f$ and $d_{s_2}=0.9d_f$, respectively. The solid and fluid media are composed of Plexiglas and water, respectively.

Figure 14

Transmission coefficient versus $\mathrm{\Omega }$ for $\theta =0{}^{\circ }$ (a) and $\theta =4{}^{\circ }$ (b). The structure is the same as in Fig. 13.

Figure 15

Variation of the transmission coefficient versus the dimensionless frequency $\mathrm{\Omega }$ for $d_{s2}=0.9d_f$ fixed whereas $d_{s1}$ is variable: $d_{s1}=1.1d_f$ (full curve), $d_{s1}=1.2d_f$ (dashed curve), and $d_{s1}=1.3d_f$ (dashed-dotted curve). The incidence angle is fixed at $\theta =32.6{}^{\circ }$. (b) The same as in (a) but the AIT resonances are brought back to unity by choosing appropriately the angles of incidence as indicated in the inset. The open circle on the abscissa indicates the position of the transmission zero induced by the solid layer of thickness $d_{s2}=0.9d_f$.

Figure 16

(a) Variation of the transmission coefficient versus the dimensionless frequency $\mathrm{\Omega }$ for the same structure as in Fig. 11 (full curve) and when the water layer is replaced by an alcohol layer (dashed curve). (b) The same as in (a) but the Fano resonance in (a) is brought back to a unitary AIT resonance by choosing appropriately the incidence angle $\theta =35.6^{\circ }$.

Figure 17

Same as in Fig. 16, but one Plexiglas layer is replaced by a PMMA layer.

Figure 18

Dispersion curves $\mathrm{\Omega }$ versus the incidence angle $\theta$ for an infinite superlattice (SL) made of Plexiglas and water layers (hatched areas). The thin solid curves show the dispersion curves obtained from the maxima of the transmission (zero reflection), whereas the open circles give the position of the total reflection (transmission zero).

Figure 19

Variation of the transmission coefficients as a function of $\mathrm{\Omega }$ for a finite SL composed of $N=2$ [(a), (d), and (g)] and $N=4$ [(b), (e), and (h)] periods. The left, middle, and right panels correspond to incident angles: $\theta =32.6^{\circ },{34}^{\circ }$, and ${30}^{\circ }$, respectively. (c), (f), (k) Give the dispersion curves (i.e., $\mathrm{\Omega }$ versus the dimensionless Bloch wave vector $KD)$ inside the first Brillouin zone (i.e., $0<KD<\pi )$.

Figure 20

Variation of the transmission coefficients as a function of $\mathrm{\Omega }=\frac{\omega d_s}{v_0}$ for the structure depicted in Fig. 1 for $\theta =32.5^{\circ }$ (blue curve) and $\theta =34.5^{\circ }$ (red curve). The curves are plotted for a Plexiglas layer without absorption (a), small absorption (b), and high absorption (c).

Figure 21

Variation of the transmission coefficients as a function of $\mathrm{\Omega }=\frac{\omega d_f}{v_f}$ for a solid-fluid-solid triple layer (Fig. 5) where the thicknesses of the layers are chosen such that $d_{s_1}=d_{s2}=d_f$. The incidence angle is taken as $\theta ={25}^{\circ }$ (blue curve) and $\theta ={35}^{\circ }$ (red curve). The curves are plotted for a Plexiglas layer without absorption (a), small absorption (b), and high absorption (c).

Figure 22

Variation of the transmission coefficients as a function of $\mathrm{\Omega }=\frac{\omega d_f}{v_f}$ for a solid-fluid-solid triple layer (Fig. 5) where the thicknesses of the layers are chosen such that $d_{s_1}=1.2d_f$ and $d_{s2}=0.8d_f$. The incidence angle is taken as $\theta =32.2^{\circ }$. The curves are plotted for a Plexiglas layer without absorption (red curve), small absorption (blue curve), and high absorption (black curve).