ψ -shaped phononic demultiplexer with three channels based on fano and acoustically induced transparency resonances for wave routing application

Abstract

In this research, we give an analytical demonstration of the possibility to realize a simple phononic demultiplexer based on Fano and AIT resonances for application of wave routing. Our proposed demultiplexer contains a $Ψ$ shaped waveguides with an input line and three output lines, each output line contains two resonators grafted at two positions far from the input line. We derive the analytical expressions for a selective transfer of a single propagation mode through one output line by keeping the other output lines unaffected. The AIT resonance; induced by two resonators inserted in the same position or by the segment that separates the two resonators, is characterized by a maximum transmission peak wedged between two transmission zeros. On the other hand, the existence of a maximum transmission peak beside a transmission zero enables to select a given frequency in the first output line, by canceling the transmission in the second and third output lines and the reflection in the input line, this resonance type is called Fano resonance in which the resonators lengths d_i2 and d_i4 must have the same values. These resonances are obtained by detuning the lengths of the two resonators. The position and the quality factor of these two resonances (Fano and AIT) are extremely sensitive to the geometrical parameters of the segments and resonators which should be chosen appropriately. Our results are found out using the Green function method. We demonstrate that our proposed three-channel demultiplexer can be used to design a multifunctional device that functions as a wave routing splitter for different frequencies coming from the input line in order to separate each frequency at each output line. This system is capable to realize various wave guiding characteristics with perfect transmissions and good quality factors of resonant modes.

Keywords

Phononic demultiplexer fano resonance AIT resonance channel wave routing

Introduction

Wave routing application has always been a great challenge for many physical applications. Numerous works focus on the design of multi-channel devices capable to guide waves from a specific input line to different output lines. Such devices include power divider or combiner (which divides or combines one or several inputs into one or several channels) or multiplexer systems which select one of several inputs and transfers it into a specific channel. For example, in microwave engineering, guiding devices such as power dividers, circulators, filters, couplers and multiplexer are commonly used.¹ Multi-channel devices can be realized in different ways employing resonant structures,^1–3 photonic crystals,^4–6 metamaterials^7,8 as well as waveguides with multimode interferences.^9,10 In electromagnetics, selected filters and multiplexers have also been implemented by using structured materials as nonuniform waveguide structure^11,12 and complex photonic circuits are studied and designed for optical signal processing or computing in integrated optics.¹³

Fano and EIT (Electromagnetically Induced Transparency) type resonances^14–16 have an atomic origin, but they have been the subject of several studies in classical systems such as coupled micro-resonators,^17–19 photonic waveguides,^20–24 acoustic slender tube waveguides and solid-liquid multilayers^25–34 as well as plasmonic nonostructures.^35–39 Fano resonance can be explained as the product of two processes of constructive and destructive wave interferences. This phenomenon gives rise to a resonance followed by an antiresonance over a narrow frequency range and can be manifested by an asymmetrical profile shape. In the transmission spectra, the Fano profile appears as a maximum near to a transmission zero.^40,41 When the Fano resonance falls between two antiresonances (two transmission zeros) it becomes an EIT resonance. In optics, this phenomenon has shown potential applications to realize slow light and data storage of optical information.^42–45 Fano and EIT resonances are originally the product of a coupling between one or more discrete states and a continuum.⁴⁰ In general, to create this type of resonances in classical systems, one connects two or more resonators with a waveguide. Among the simple structures giving a clear theoretical and experimental demonstration of such resonances, one can cite a guide connected with two lateral resonators at the same position (called a cross structure) or at two different positions (called U-shaped structure) to demonstrate the possibility to realize Fano and AIT (Acoustically induced transparency analogue of EIT) resonances.³⁶ A few years later, the same structure was studied by Santillan et al.³⁷ to demonstrate AIT resonance and delayed sound. More recently, cross and U-shaped structures have been the subject of interest to experimentally show AIT and Fano resonances as well as the possibility to realize perfect absorption with such structures.³⁸ Similar structures with multiple stubs have been proposed to realize multiband and broadband absorbers for low-frequency sound.⁴⁶ In addition, A. Mouadili et al. have studied acoustic demultiplexer based on Fano and induced transparency resonances in slender tubes.⁴⁷ The concept of electromagnetically induced transparency (EIT), originally developed in quantum optics to describe interference-based transparency in multi-level atomic systems, has inspired a broad range of classical analogues. This includes acoustic induced transparency (AIT), where similar interference mechanisms arise from the coupling between resonant elements in subwavelength acoustic structures. The transition from electromagnetic EIT to acoustic AIT is particularly relevant because acoustic systems allow more direct control of geometric parameters, enabling clearer observation and tuning of destructive-interference phenomena. In this context, slender tube waveguides are especially suitable for demonstrating AIT and Fano-type resonances: their simple geometry, low-loss propagation, and strong confinement of acoustic modes make them more favorable than solid or photonic structures for isolating and analyzing coupling effects between resonators. These features provide a clean platform for studying interference-driven transparency and facilitate the realization of high-contrast, tunable resonance behaviors.

Recently, three-port acoustic system has been used to study subwavelength control of absorption using resonators on each channel.⁴⁸ Also, the multi-channel systems have shown the possibility to generate outgoing waves only in certain channels by controlling the incoming waves. Then, I. El kadmiri et al. have investigated Y-shaped branch structure using asymmetric resonators for phononic demultiplexing.⁴⁹ A. Rostami et al. have studied two ports acoustic demultiplexer based on fluid-fluid phononic crystal ring resonators. They used the effects of scatterer rods of the ring resonator, then compared inner rods at different radii, pressures, and temperatures. It is shown that by using appropriate inner rods with suitable radii, the quality factor can be enhanced. Moreover, the operating frequencies can be tuned appropriately by varying the cavities operating temperatures and pressures.⁵⁰ O. Richoux et al. have demonstrated that asymmetric three port devices can be used to design a multi-functional set-up operating as a symmetric combiner and splitter at the same frequency. They employed resonant acoustic three ports device operating in a sub-wavelength regime, showing splitting/combining abilities with a nearly perfect transmission.⁵¹ J. Han et al. have designed a tunable four-channel wavelength demultiplexer with approximately 60 nm channel spacing by using objective-first inverse design method. They have achieved approximately 70% transmission and less than −13 dB channel crosstalk. The results realized through using a finite difference time domain (FDTD) showed that the wavelength of each channel is capable of being tunable by changing the refractive index of the structure. They discussed the design process, the optical transmission characteristics, as well as the tunability for splitting different wavelengths.⁵² P. Moradi et al. have presented the design procedure for the elastic GHz ranged demultiplexer based on a solid-solid phononic crystal platform. The designed demultiplexer can separate three frequencies in the range of GHz.⁵³ B. R. Dogolsara et al. have presented the design program of switchable acoustic demultiplexer based on fluid-fluid phononic crystal (PnC). It consists of a T-shaped PnC waveguide, which is coupled to two output waveguide ports through two different point defect cavities. The PnC platform consists of a periodic array of infinitely long water (inclusion) rods embedded in a mercury background. The waveguide is made by removing a row or column of inclusions from PnC, and the fluid in one of the two different cavities is methyl nonafluorobutyl ether (MNE), and the other cavity is ethyl nonafluorobutyl Base ether (ENE). The difference in sound velocity between MNE and ENE provides sufficient difference in resonance modes of different cavities, which is required for the demultiplexing function of the design structure.⁵⁴ They also proposed a tunable four-channel (four output ports) acoustic demultiplexer, based on a fork-shaped phononic crystal (PnC).⁵⁵

In this present research, we propose Ψ-shaped acoustic demultiplexer based on the U-shaped structure (Figure 1(b)). The demultiplexer in this device is based on AIT and Fano resonances.^56,57 Our objective consists of demonstrating the possibility of finding analytically the appropriate lengths of the different waveguide resonators in order to reach a total transmission in one output line by keeping the other lines unaffected. The demultiplexer proposed in this study have several advantages over those based on phononic crystals,^58,59 such as: the simplicity of the device manufacturing where two resonators are inserted in each output line, the simplicity of the structure enables a full analytical calculation, which allows to deduce the exact expressions of the different lengths of the waveguides to achieve a perfect demultiplexing, and the possibility of increasing the quality factor of the filtered resonances. This property is a feature of Fano and induced transparency resonances that does not exist in standard phononic crystals with defects, in which filtering is performed using finite width Breit-Wigner resonances.^58–61 The validity of our results is related to the requirement that the cross section of the slender tubes should be negligible compared to their length and to the propagation wavelength. The main of the paper is organized as follows: in Section 2, we give the analytical expressions for a demultiplexer based on U-structure. These analytical results are obtained from an analysis of the transmission and reflection coefficients in order to obtain Fano and Acoustically Induced Transparency resonances. In section 3, we give numerical results and discussions of our proposed demultiplexer in standard acoustic slender tubes. The last section contains the conclusion of our work.

Figure 1.

(a) Schematic illustration of the acoustic waveguide of length d₁₃ with dangling resonators on both sides, the lengths of the dangling resonators are noted d₁₁ and d₁₃. (b) Schematic representation of a Ψ-shaped demultiplexer with one input line and three output lines, two resonators are grafted at two different positions along each output line.

Theoretical analysis

Transmission and reflection coefficients through a simple structure composed of two grafted resonators at two sites

Our theoretical analysis is performed with the help of the interface response theory^62,63 of continuous media, which allows to calculate the Green function of any composite materials, as well as the transmission and reflection rates. The simple structure presented in this work (Figure 1(a)) is composed of two grafted resonators at the sites 0 and 1. Each resonator induces zero of transmission. Between the two zeros of transmission, the global system induces a complete transmission following an AIT type resonance. We obtain a Fano type resonance if the two grafted resonators have the same lengths. To visualize these resonances in our structure, we calculate the transmission and reflection coefficients. In this case, it is helpful to know the Green function of its elementary constituents (finite segment of length d₁₂, the resonators of lengths d₁₁ and d₁₃ and semi-infinite guides).

The inverse Green function of semi-infinite waveguides constituting the input and the output lines is given by:

g^{- 1} (0, 0) = g^{- 1} (1, 1) = - F_{s}

(1)

The Green function of a finite segment of length d₁₂ is a matrix $(2 \times 2)$ in the interface space (0,1)⁴⁹:

\overset{⃡}{g_{13}^{- 1}} (0; 1) = (\begin{array}{c} \frac{- F_{12} C_{12}}{S_{12}} & \frac{F_{12}}{S_{12}} \\ \frac{F_{12}}{S_{12}} & \frac{- F_{12} C_{12}}{S_{12}} \end{array})

(2)

With:

C_{12} = \cosh (α_{12} d_{12})

;

S_{12} = \sinh (α_{12} d_{12})

;

F_{12} = α_{12} = - j k_{12} = - j \frac{ω}{Z_{12}}, j = \sqrt{- 1} a n d Z_{12} = ρ \frac{v}{s_{12}} (v = 342 m / s)

While the inverse Green function of the closed resonators $g_{11}^{- 1} (0, 0)$ and $g_{13}^{- 1} (1, 1)$ are given by:

g_{12}^{- 1} (0, 0) = \frac{- S_{11} F_{11}}{C_{11}}

(3a)

g_{14}^{- 1} (1, 1) = \frac{- S_{13} F_{13}}{C_{13}}

(3b)

With: $C_{1 i} = \cosh (α_{1 i} d_{1 i}); S_{1 i} = \sinh (α_{1 i} d_{1 i}) (i = 1, 3); {F_{1 i} = α}_{1 i} = j \frac{ω}{Z_{1 i}}$

We suppose all that the guides and resonators have the same characteristics (i.e., F_s = F₁₁ = F₁₂ = F₁₃ = F).

The Green’s function of the whole system in the interface space M = {0,1} can be obtained from a linear superposition of the above inverse Green’s functions of the constituent:

g^{- 1} (M, M) = - F (\begin{array}{c} 1 + \frac{S_{11}}{C_{11}} + \frac{C_{12}}{S_{12}} & - \frac{1}{S_{12}} \\ - \frac{1}{S_{12}} & 1 + \frac{C_{12}}{S_{12}} + \frac{S_{13}}{C_{13}}) \end{array})

(4)

The transmission coefficient through the structure is given by the following expression:

t = - 2 F g (0, 1) w i t h g (0, 1) = \frac{a}{\det (g^{- 1} (M, M))}

(5)

From the expression of t (equation (5)), one can deduce the transmission rate as follows:

T = {| t |}^{2}

(6)

The reflection coefficient is given by : r = - 1 - 2 F g (0, 0) w i t h g (0, 0) = \frac{b}{\det (g^{- 1} (M, M))}

(7)

with a = - \frac{F}{S_{12}}; b = - F - F \frac{C_{12}}{S_{12}} - F \frac{S_{13}}{C_{13}}; \det (g^{- 1} (M, M)) = (- F - F \frac{C_{12}}{S_{12}} - F \frac{S_{11}}{C_{11}}) (- F - F \frac{C_{12}}{S_{12}} - F \frac{S_{13}}{C_{13}}) - \frac{F^{2}}{S_{12}^{2}}

From the expression of r (equation (7)), one can deduce the reflection rate:

R = {| r |}^{2}

(8)

One can show easily the conservation energy, namely:

T + R = 1

(9)

Transmission and reflection coefficients through a phononic demultiplexer contains three channels

In this part, we consider the structure shown in Figure 1(b), this structure is composed of an input line and three output lines, all fixed at point 1. The first output contains a length d₁₁ and two resonators of lengths d₁₂ and d₁₄ separated by a distance d₁₃. Likewise, the second output line contains a length d₂₁ and two resonators of lengths d₂₂ and d₂₄ separated by a distance d₂₃. Similarly, the third output contains a length d₃₁ and two resonators of lengths d₃₂ and d₃₄ separated by a distance d₃₃. As mentioned above, the U-shaped resonator along a waveguide was the subject of several studies in acoustics by different authors.^25,27 In particular, this structure can present two types of resonances: AIT type resonance when the two resonators have different lengths and Fano type resonance when the two resonators have the same lengths. For simplicity, all waveguides are assumed being characterized by the same characteristic impedance $= \frac{v}{s}$ , where $ρ$ = 1.2 Kg/m³ and $v$ = 342 m/s are respectively the density and velocity of the fluid inside the slender resonators (air) and $s$ is the section of the guide. However, the resonators are supposed to be simple stubs with narrow neck in order to get analytical expressions for a perfect demultiplexing, otherwise the calculation becomes simple and only numerical simulation can be performed. In order to calculate the transmission and reflection rates, we should form the expression through the Green function’s inverse of the whole system (Figure 1(b)) in the interface spaces M’ = {1,2,3,4,5,6,7}. This process claims the knowing of the Green function’s inverse of elementary constituents.

The inversed Green function of the three semi-infinite waveguides:

{g_{s}}^{- 1} (1, 1) = {g_{s}}^{- 1} (3, 3) = {g_{s}}^{- 1} (5, 5) = {g_{s}}^{- 1} (7, 7) = A_{0} = - F

(10)

The inversed Green function of finite segments (d₁₁ and d₁₃) in the interface spaces {1,2} and {2,3} along the first output line⁴⁹:

g^{- 1} {1, 2} = (\begin{array}{c} \frac{- F C_{11}}{S_{11}} & \frac{F}{S_{11}} \\ \frac{F}{S_{11}} & \frac{- F C_{11}}{S_{11}} \end{array}) a n d g^{- 1} {2, 3} = (\begin{array}{c} \frac{- F C_{13}}{S_{13}} & \frac{F}{S_{13}} \\ \frac{F}{S_{13}} & \frac{- F C_{13}}{S_{13}} \end{array})

(11)

The inversed Green function of the closed resonators $g_{12}^{- 1} (2, 2)$ and $g_{14}^{- 1} (3, 3)$ along the first output line are given by:

g_{12}^{- 1} (2, 2) = \frac{- F S_{12}}{C_{12}}

(12a)

g_{14}^{- 1} (3, 3) = \frac{- F S_{14}}{C_{14}}

(12b)

The inversed Green function of finite segments (d₂₁ and d₂₃) in the interface spaces {1,4} and {4,5} along the second output line⁴⁹:

g^{- 1} {1, 4} = (\begin{array}{c} \frac{- F C_{21}}{S_{21}} & \frac{F}{S_{21}} \\ \frac{F}{S_{21}} & \frac{- F C_{21}}{S_{21}} \end{array}) a n d g^{- 1} {4, 5} = (\begin{array}{c} \frac{- F C_{23}}{S_{23}} & \frac{F}{S_{23}} \\ \frac{F}{S_{23}} & \frac{- F C_{23}}{S_{23}} \end{array})

(13)

The inversed Green function of the closed resonators $g_{22}^{- 1} (4, 4)$ and $g_{24}^{- 1} (5, 5)$ along the second output line are given by:

g_{22}^{- 1} (4, 4) = \frac{- F S_{22}}{C_{22}}

(14a)

g_{24}^{- 1} (5, 5) = \frac{- F S_{24}}{C_{24}}

(14b)

The inversed Green function of finite segments (d₃₁ and d₃₃) in the interface spaces {1,6} and {6,7} along the third output line⁴⁹:

g^{- 1} {1, 6} = (\begin{array}{c} \frac{- F C_{31}}{S_{31}} & \frac{F}{S_{31}} \\ \frac{F}{S_{31}} & \frac{- F C_{31}}{S_{31}} \end{array}) a n d g^{- 1} {6, 7} = (\begin{array}{c} \frac{- F C_{33}}{S_{33}} & \frac{F}{S_{33}} \\ \frac{F}{S_{33}} & \frac{- F C_{33}}{S_{33}} \end{array})

(15)

The inverse Green function of the closed resonators $g_{32}^{- 1} (6, 6)$ and $g_{34}^{- 1} (7, 7)$ along the third output line are given by:

g_{32}^{- 1} (6, 6) = \frac{- F S_{32}}{C_{32}}

(16a)

g_{34}^{- 1} (7, 7) = \frac{- F S_{34}}{C_{34}}

(16b)

The linear superposition of the inversed Green functions of the previous constituents gives us the inversed Green function of the composite structure in the interface spaces M’ = {1,2,3,4,5,6,7}:

g^{- 1} (M^{'}, M^{'}) = (\begin{array}{c} (- F + A_{11} + A_{21} + A_{31}) & B_{11} & 0 & B_{21} & 0 & B_{31} & 0 \\ B_{11} & (A_{11} + A_{12} + A_{13}) & B_{13} & 0 & 0 & 0 & 0 \\ 0 & B_{13} & (- F + A_{13} + A_{14}) & 0 & 0 & 0 & 0 \\ B_{21} & 0 & 0 & (A_{21} + A_{22} + A_{23}) & B_{23} & 0 & 0 \\ 0 & 0 & 0 & B_{23} & (- F + A_{23} + A_{24}) & 0 & 0 \\ \begin{array}{l} B_{31} \\ 0 \end{array} & \begin{array}{l} 0 \\ 0 \end{array} & \begin{array}{l} 0 \\ 0 \end{array} & \begin{array}{l} 0 \\ 0 \end{array} & \begin{array}{l} 0 \\ 0 \end{array} & \begin{array}{c} \begin{array}{c} (A_{31} + A_{32} + A_{33}) \\ B_{33} \end{array} & \begin{array}{l} B_{33} \\ - F + A_{33} + A_{34}) \end{array} \end{array} \end{array})

(17)

with : A_{0} = - F; A_{11} = \frac{- F C_{11}}{S_{11}}; A_{13} = \frac{- F C_{13}}{S_{13}}; A_{21} = \frac{- F C_{21}}{S_{21}}; A_{23} = \frac{- F C_{23}}{S_{23}}; A_{31} = \frac{- F C_{31}}{S_{31}}; A_{33} = \frac{- F C_{33}}{S_{33}}

B_{11} = \frac{F}{S_{11}}; B_{13} = \frac{F}{S_{11}}; B_{21} = \frac{F}{S_{21}}; B_{23} = \frac{F}{S_{23}}; B_{31} = \frac{F}{S_{31}}; B_{33} = \frac{F}{S_{33}}

A_{12} = \frac{- F S_{12}}{C_{12}}; A_{14} = \frac{- F S_{14}}{C_{14}}; A_{22} = \frac{- F S_{22}}{C_{22}}; A_{24} = \frac{- F S_{24}}{C_{24}}; A_{32} = \frac{- F S_{32}}{C_{32}}; A_{34} = \frac{- F S_{34}}{C_{34}}

The reflection coefficient in the input line of the phononic demultiplexer (Figure 1(b)) is given by:

r = - 1 - 2 F g (1, 1) with g (1, 1) = \frac{A}{\det (g^{- 1} (M^{'}, M^{'}))}

(18)

The expression of the reflection rate R in the input line is given by:

R = {| r |}^{2}

(19)

The transmission coefficients in the first, second and third output lines are given respectively by:

t_{1} = - 2 F g (1, 3) with g (1, 3) = \frac{B}{\det (g^{- 1} (M^{'}, M^{'}))}

(20a)

t_{2} = - 2 F g (1, 5) with g (1, 5) = \frac{C}{\det (g^{- 1} (M^{'}, M^{'}))}

(20b)

t_{3} = - 2 F g (1, 7) with g (1, 7) = \frac{D}{\det (g^{- 1} (M^{'}, M^{'}))}

(20c)

with:

A = (A_{11} + A_{12} + A_{13}) (((A_{21} + A_{22} + A_{23}) (A_{0} + A_{23} + A_{24}) - B_{23} {}^{2}) (A_{31} + A_{32} + A_{33}) (A_{0} + A_{33} + A_{34}) - ((A_{21} + A_{22} + A_{23}) (A_{0} + A_{23} + A_{24}) - B_{23} {}^{2}) B_{33} {}^{2})

B = B_{11} B_{13} (((A_{21} + A_{22} + A_{23}) (A_{0} + A_{23} + A_{24}) - B_{23} {}^{2}) (A_{31} + A_{32} + A_{33}) (A_{0} + A_{33} + A_{34}) - ((A_{21} + A_{22} + A_{23}) (A_{0} + A_{23} + A_{24}) - B_{23} {}^{2}) B_{33} {}^{2})

C = (A_{11} + A_{12} + A_{13}) B_{21} B_{23} ((A_{31} + A_{32} + A_{33}) (A_{0} + A_{33} + A_{34}) - B_{33} {}^{2})

D = (A_{11} + A_{12} + A_{13}) ((A_{21} + A_{22} + A_{23}) (A_{0} + A_{23} + A_{24}) - B_{23} {}^{2}) B_{31} B_{33}

\det (g^{- 1} (M^{'}, M^{'})) = (((((A_{11} + A_{12} + A_{13}) (A_{0} + A_{11} + A_{21}) - {B_{11}}^{2}) (A_{21} + A_{22} + A_{23}) - (A_{11} + A_{12} + A_{13}) {B_{21}}^{2}) (A_{0} + A_{23} + A_{24}) - ((A_{11} + A_{12} + A_{13}) (A_{0} + A_{11} + A_{21}) - {B_{11}}^{2}) {B_{23}}^{2}) (A_{31} + A_{32} + A_{33}) - (A_{11} + A_{12} + A_{13}) ((A_{21} + A_{22} + A_{23}) (A_{0} + A_{23} + A_{24}) - {B_{23}}^{2}) {B_{31}}^{2}) (A_{0} + A_{33} + A_{34}) - ((((A_{11} + A_{12} + A_{13}) (A_{0} + A_{11} + A_{21}) - {B_{11}}^{2}) (A_{21} + A_{22} + A_{23}) - (A_{11} + A_{12} + A_{13}) {B_{21}}^{2}) (A_{0} + A_{23} + A_{24}) - ((A_{11} + A_{12} + A_{13}) (A_{0} + A_{11} + A_{21}) - {B_{11}}^{2}) {B_{23}}^{2} {B_{33}}^{2}

The transmission rates in the three output lines are given respectively by:

T_{1} = {| t_{1} |}^{2}; T_{2} = {| t_{2} |}^{2}; T_{3} = {| t_{3} |}^{2}

(21)

The transmission and reflection rates that satisfy the energy conservation:

T_{1} + T_{2} + T_{3} + R = 1

(22)

Results and discussions

AIT and fano resonances in the U structure

Before discussing the problem of the whole structure described above (Figure 1(b)), let us first recall briefly the results of a particular case (two resonators grafted at two sites different) which is necessary to understand the acoustic wave propagation in the structure shown in Figure 1(a) in order to obtain the Fano and AIT resonances in slender tubes. Figure 2 gives the variation of the transmission T and reflection R rates as a function of the reduced frequency Ω (Ω = $\frac{ω D}{c}$ ) for two values of d_1j [j = 1,3] by keeping the segment length d₁₂ = 1D. From Figure 2(a) (in the case where d₁₁ = d₁₃ = 0.46D) and Figure 2(b) (in the case where d₁₁ = d₁₃ = 0.54D), we observe that both indicated resonances are of Fano type resonance. In addition, these resonances present different asymmetric line shapes (i.e., opposite Fano parameters) in order to achieve a full transmission at a particular frequency. For example, in the first case (a), we get a maximum transmission for Ω₃ = 3.27 near to a transmission zero for Ω₂ = 3.37. In the second case (b), we notice that there is a maximum transmission peak for Ω₃ = 3.02 near to a transmission zero for Ω₁ = 2.91. In the third case (c) which is related to AIT resonance (d₁₁ = 0.46D and d₁₃ = 0.54D), we notice that the resonance at Ω₃ = 3.13 (T = 1) is squeezed between two transmission zeros (Ω₁ = 2.91 and Ω₂ = 3.35) induced by the dangling resonators, as illustrated in the variation curve of the phase (Figure 2(d)). The width of this resonance increases as d₁₁ and d₁₃ deviate from 0.5D. In this case, the AIT resonance as well as the two transmission zeros induced by the resonators fall at two different position, so the resonance collapses, the transmission coefficient vanishes and the phase drops by π. The transmission zeros induced by the different lengths of resonators fall at two frequencies; therefore, the phase (Figure 2(d)) shows a drop of π at these frequencies. The phase drops give rise to negative peaks in the transmission phase time as illustrated in Figure 2(e). A recent similar work has been performed by Rotter and Sadreev⁶⁴ on two single quantum dots coupled by a wire of finite length and neglecting the absorption. They found that at the transmission zeros, the phases jump by π mostly. These results are similar to those obtained by E H. El Boudouti et al. when they studied a simple acoustic device consisting of two dangling side resonators grafted at two sites on a slender tube⁶⁵ (Figure 3).

Figure 2.

Variation of the transmission and reflection rates versus the reduced frequency $Ω$ for the structure indicated in Figure 1(a). (a) d₁₁ = d₁₃ = 0.46D and d₁₂ = 1D. (b) d₁₁ = d₁₃ = 0.54D and d₁₂ = 1D. (c) Variation of the transmission and reflection rates versus the reduced frequency $Ω$ for the structure presented in Figure 1(a) with d₁₁ = 0.46, d₁₃ = 0.54D and d₁₂ = 1D. (d) and (e) represent the phase spectrum and phase time in the same situation of the curve (c).

Figure 3.

Color map of the transmission rate as a function of the resonator lengths d₁₁ and d₁₃ with d₁₂ = 1D and 2.9 $< Ω <$ 3.4. The red color indicates the high value of the transmission rate, the green indicates the average value and the dark blue color indicates the low value.

To obtain the adequate geometrical parameters leading to a desirable performance of the studied system, with a high transmission rate of the AIT resonance, we plot the color map of the transmission rate belonging to the reduced frequency Ω interval [2.9-3.4], as a function of resonator lengths d₁₁ and d₁₃ with d₁₂ = 1D. The map below shows that for each point of d₁₁ and d₁₃, we obtain a transmission maximum. Firstly, the red color represents a high transmission rate for different values of resonator lengths. The transmission maxima are symmetrical with respect to the center of resonator lengths (d₁₁ = d₁₃ = 0.5D). In this case, we obtain a Fano resonance when d₁₁ = d₁₃ = 0.46D and d₁₁ = d₁₃ = 0.54D (same lengths of resonators), as well as the AIT resonance is obtained when d₁₂ = 0.46D and d₁₄ = 0.54D (different lengths of resonators). Secondly, the green color indicates a medium transmission rate but the dark blue color indicates the transmission minima in case of the medium values of d₁₁ and d₁₃ (0.4D<d₁₁ < 0.6D and 0.4D<d₁₃ < 0.6D). To summarize, the ideal situation to operate on is the first one (red color) because the transmission rate has a maximum value. As for the last one (dark blue color), the performed system could be used as a total reflection filter.

Demultiplexer based on $ψ$ structure

In this part, we consider the $ψ$ -shaped demultiplexer composed of an input line and three output lines, all connected at the same point 1 (Figure 1(b)). The calculation of the transmission and reflection rates for the demultiplexer based on U-shaped structure is shown in section 2.2 using the Green function method. However, the corresponding expressions are more complicated; therefore, we give only the conditions that should be satisfied by all the lengths of guides and resonators in order to realize a perfect phononic demultiplexer. Also, we distinguish two different cases depending on whether the resonators are slightly different along each output line which gives rise to AIT type resonance, or identical length of resonators which gives rise to Fano type resonance.

Demultiplexer based on fano resonances

As mentioned in reference 25, in order to achieve a Fano resonance, we should take two identical lengths of resonators, but slightly different from d₁₃/2 (i.e., d₁₂ = d₁₄≠d₁₃/2) along the first output line. Similarly, we should take d₂₂ = d₂₄≠d₂₃/2 along the second output line and we should take d₃₂ = d₃₄≠d₃₃/2 along the third output line. The explicit expressions of the twelve different lengths d₁₁, d₁₂, d₁₃, d₁₄, d₂₁, d₂₂, d₂₃, d₂₄, and d₃₁, d₃₂, d₃₃, d₃₄ should satisfy the following equations in order to obtain a total transmission rate along one output line by keeping the other lines unaffected:

d_{11} = d_{12} = \frac{d_{0}}{2} + ϵ; d_{13} = 1 D; d_{14} = d_{11}

(23)

d_{21} = d_{22} = d_{11} - \frac{ϵ}{2}; d_{23} = d_{13} + 3 ϵ; d_{24} = d_{22}

(24)

d_{31} = d_{32} = d_{11} - \frac{ϵ}{4}; d_{33} = d_{23} + 3 ϵ; d_{34} = d_{31}

(25)

Where

ϵ

represents the detuning between the lengths of the different resonators constituting the phononic demultiplexer.

In this part, we study on one hand the evolution of the reduced frequency Ω of Fano resonances versus the parameter ϵ with d₀ = 1.3D (Figure 4(a)). The Fano resonances get closer to each other along the output lines when ϵ takes low values (0D $<$ ϵ $<$ 0.02D), as well as these resonant modes are separated with the increase of the parameter ϵ. Let us mention that the Fano resonances are deplacing to low frequencies through the variation of the parameter ϵ. In order to achieve a Fano resonance with high performance, the lengths of the resonators must be identical and the parameter ϵ must take low values. Figure 4b–(d) gives the variation of transmission rates T₁, T₂, T₃ and reflection rate R as a function of the reduced frequency for three values of the parameter ϵ. From Figure 4, one can see that three resonances are of Fano type resonance. In addition, these resonances present different asymmetric line shapes (i.e., opposite Fano parameters¹⁴) in order to achieve a full transmission in one output line. We can notice that for ϵ = 0.05D (Figure 4(b)), the transmission along the first output line (black color) reaches unity (T₁ = 1) at Ω = 2.62, the transmission along the second and third output lines T₂, T₃ and the reflection R vanish (i.e. T₂ = T₃ = R = 0). Similarly, when the transmission along the second output line reaches unity (T₂ = 1) at Ω = 2.51, the transmission along the first and third output lines T₁, T₃ and the reflection R vanish (i.e., T₁ = T₃ = R = 0). Consequently, when the transmission along the third output line reaches unity (T₃ = 1) at Ω = 2.39, the transmission along the first and second output lines T₁, T₂ and the reflection R vanish (i.e., T₁ = T₂ = R = 0). We observe that when we increase the parameter ϵ (Figure 4c–(d)), the three Fano-type resonances move towards low frequencies and when the parameter ϵ takes the low values, the three Fano resonances move closer to each other and move towards the high frequencies. We can see that the frequencies of Fano resonances depend on the parameter ϵ. Also, the quality factors of the Fano resonances are almost identical for the three cases presented in Figure 4.

Figure 4.

(a) Evolution of the reduced frequency as a function of the parameter ϵ with d₀ = 1.3D. Evolution of the transmission rates T₁, T₂ and T₃ as a function of the reduced frequency for ϵ = 0.05D (b), ϵ = 0.091D (c) and ϵ = 0.121D (d).

On the other hand, we present in Figure 5 a the variation of the reduced frequency Ω as a function of the parameter d₀. We remark that the Fano type resonances decrease to low frequencies with the variation of d₀. Figure 5b–(d) indicate the variation of the transmission rates T₁, T₂ and T₃ as a function of the reduced frequency for three values of d₀ with ϵ = 0.0925D. We can see that for d₀ = 1.3D, the transmission along the first output line (black curve) reaches unity (T₁ = 1) at Ω = 2.54, the transmission along the second and third output lines T₂ (red curve) and T₃ (Blue curve), and the reflection R are cancelled (i.e., T₂ = T₃ = R = 0). Similarly, when the transmission along the second line (red curve) reaches unity (T₂ = 1) at Ω = 2.35, the transmission along the first and third lines T₁ (black curve) and T₃ (Blue curve) and the reflection R vanish (i.e., T₁ = T₃ = R = 0). Similarly, when the transmission along the third output line (Blue curve) reaches unity (T₃ = 1) at Ω = 2.16, the transmission along the first and second lines T₁ (black curve) and T₂ (red curve) and the reflection R are cancelled (i.e., T₁ = T₂ = R = 0). We have seen that this Fano resonance is very sensitive to the parameter d₀ and specially the resonator length. The full width at half maximum of the Fano resonances depend strongly on the difference between the resonator lengths. For example, the quality factor of the Fano resonance along the first output line is very high when the parameter d₀ = 0.9D. Similarly, the quality factors of the Fano resonances along the second and third output lines are very high when the parameter d₀ = 1.3D. The results presented in this section can be easily studied and similar to those found by Mouadili et al when they studied acoustic demultiplexer based on Fano and AIT resonances in slender tubes.⁴⁷

Figure 5.

(a) Evolution of the reduced frequency as a function of the parameter d₀ with ϵ = 0.0925D. Evolution of the transmission rates T₁, T₂ and T₃ as a function of the reduced frequency for d₀ = 0.9D (b), d₀ = 1D (c) and d₀ = 1.3D (d).

To realize a perfect phononic demultiplexer with a high performance (i.e., total transmission along each output line with a high quality factor), it is necessary that equations (15)–(17) must be satisfied. However, if one of the above conditions is not satisfied, the transmission rate cannot reach unity and the demultiplexer becomes imperfect. Figure 6 gives an example making it possible to explain this deviation from perfect demultiplexing when we choose the same parameters as those of Figure 5(d), except varying d₁₄. We show in Figure 6 a the variation of the reduced frequency Ω as a function of the resonator length d₁₄, we take d₀ = 1.3D and ϵ = 0.0925D. The first, second, and third output lines are indicated by black, red, and blue curves, respectively. We note that the transmission maximums along the second and third output lines remain constant for two reduced frequencies (Ω = 2.16 for the blue line and Ω = 2.35 for the red line) with the variation of the resonator length d₁₄, this is due to the constant parameter of d₀ and ϵ, which means that the lengths of guides and resonators along the second and third output lines are constant and finally the fano type resonances fall at the same frequencies in the second and third channels. Therefore, if we change the resonator length d₁₄, the Fano type resonance moves towards low frequencies. Then, we present in Figure 6(b)–(d) the variation of the transmission rates T₁, T₂ and T₃ as a function of the reduced frequency for different values of the resonator length d₁₄ with d₀ = 1.3D and ϵ = 0.0925D. One can notice that the transmission in the first output line (black curve) does not reach unity at Ω = 2.86 (for d₁₄ = 0.4D) and Ω = 2.79 (for d₁₄ = 0.5D), but we observe that the transmission T₁ reaches unity when the resonator length d₁₄ takes the value 0.7D. Indeed, the two fano type resonances keep the same reduced frequency along the second and third output lines with high transmission rates (T₂ = T₃ = 1) and good quality factors for the ideal parameters of d₀ and ϵ.

Figure 6.

(a) Evolution of the reduced frequency as a function of the parameter d₁₄ with ϵ = 0.0925D and d₀ = 1.3D. Evolution of the transmission rates T₁, T₂ and T₃ as a function of the reduced frequency for d₁₄ = 0.4D (b), d₁₄ = 0.5D (c) and d₁₄ = 0.7D (d).

In summary, we present an explanatory table to analyze the sensitivity of the parameter ε and its impact on the efficiency of demultiplexing between channels.

Perfect ψ-shaped phononic demultiplexer	Effect of ε (Figure 4)	- When ε increases, the three resonances shift toward lower frequencies, with a well-defined resonance frequency for each output channel. -The quality factor of the first and third output channels decreases as ε increases, whereas the transmission of the second output channel increases with increasing ε.
Perfect ψ-shaped phononic demultiplexer	Effect of d0 (Figure 5)	-When d₀ increases, the three resonances shift toward lower frequencies, with a well-defined resonance frequency for each output channel. -The quality factor of the three output channels increases as d₀ increases.
Imperfect ψ-shaped phononic demultiplexer	Effect of d14 (Figure 6)	-The transmission through the second and third output lines remains fixed at the well-defined frequencies Ω₂ and Ω₃. In contrast, the transmission through the first output line shifts toward lower frequencies as d₁₄ increases. -The quality factor of the second and third output lines remains constant, whereas the quality factor of the transmission in the first output line increases with increasing d₁₄.

Demultiplexer based on AIT resonances

As already demonstrated in previous works,^25,27 in order to achieve an AIT resonance with such structures, we should take two different lengths of resonators along each output line (i.e., $ϵ$ = d_i4-d_i2 with i indicates the output line). In addition, we fix the total length of the two resonators along the first output line such as d₀ = d₁₂ + d₁₄ which fixes the position of the AIT resonance along this output line. Now, in order to achieve a total transmission rate for two neighboring frequencies along the three output lines, the lengths of the different guides and resonators should be taken as follows:

d_{11} = \frac{d_{0}}{2} - \frac{ϵ}{2}; d_{12} = \frac{d_{0}}{2} + \frac{ϵ}{2}; d_{13} = d_{0}; d_{14} = d_{11}

(26)

d_{21} = \frac{d_{0}}{2}; d_{22} = d_{21}; d_{23} = d_{0} - ϵ; d_{24} = \frac{d_{0}}{2} - ϵ

(27)

d_{31} = \frac{d_{0}}{2} - \frac{ϵ}{2}; d_{32} = d_{31}; d_{33} = d_{23} - ϵ; d_{34} = \frac{d_{0}}{2} - 2 ϵ

(28)

Figure 7(a)–(f) present the variation of the transmission rates T₁ (black curve), T₂ (red curve) and T₃ (blue curve) as a function of the reduced frequency Ω for different values of ϵ with d₀ = 0.875D. We can see that for each value of ϵ, when the transmission along the first output line reaches unity (T₁ = 1), the transmission along the second and third output lines cancel out each other (i.e., T₂ = T₃ = 0). Similarly, when the transmission along the second output line reaches unity (T₂ = 1), the transmission along the first and third output lines vanish (i.e., T₁ = T₃ = 0). Similarly, when the transmission along the third output line reaches unity (T₃ = 1), the transmission along the first and second output lines vanish (i.e., T₁ = T₂ = 0). We observe that the AIT resonance in the first output line falls at the same frequency for all values of the parameter ϵ, its width decreases as ϵ decreases and disappears for ϵ = 0, giving rise to detuned of resonators. As ε→0, the detuning vanishes and the destructive interference of the coupled modes becomes perfect, leading to a vanishing linewidth (Q→∞). This behavior reflects the theoretical limit of AIT resonances, where perfectly matched resonators produce extremely sharp, high-Q spectral features. In addition, the shape and width of the AIT resonance change slightly when ϵ becomes negative (i.e., for a permutation of both resonators lengths d_i2 and _di4). The position and width of the AIT resonances along the second and third output lines strongly depend on the parameter ϵ. When ε≠0, the detuning ε breaks the symmetry of the system, resulting in asymmetric splitting of the resonances. This explains why the second and third channels shift differently. Indeed, the first AIT resonance (black curve) has two transmission zeros around a maximum transmission peak at Ω = 3.59, the positions of the second and third AIT resonances (red and blue curves) fall above Ω = 3.6 for ϵ > 0 and disappear for ϵ = 0. Similarly, the second and third AIT resonances fall below Ω = 3.59 for ϵ < 0 and disappear for ϵ = 0. We point out that the AIT resonance along the first output line falls at the same frequency whatever the value of ϵ, whereas the AIT resonances along the second and third output lines are asymmetrical with respect to the center of the parameter ϵ (opposition of the AIT resonance when ϵ becomes negative). So, the AIT resonance along the second and third output lines move towards the high frequencies from ϵ > 0, and move towards the low frequencies from ϵ < 0 and disappear from ϵ = 0. Also, the variation of the quality factors Q of the three AIT resonances in the three outputs lines increases when the absolute value of ϵ decreases.

Figure 7.

Evolution of the transmission rates T₁, T₂ and T₃ as a function of the reduced frequency for ϵ = 0.01D (a), ϵ = −0.01D (b), ϵ = 0.05D (c), ϵ = −0.05D (d), ϵ = 0.08D (c) and ϵ = −0.08D (f). (g) Evolution of the reduced frequency as a function of the parameter ϵ with d₀ = 0.875D.

These results are summarized in Figure 7(g) where we plot the frequencies of three AIT resonances as a function of the parameter ϵ. The frequencies of three AIT resonances depend on the parameter ϵ. Also, the quality factors of the three AIT resonances are high and tend to infinity when ϵ tends to zero. These results are original and identical to those obtained by Mouadili et al.⁴⁷

On the other hand, we present in Figure 8a–(d) the variation of the transmission rates T₁, T₂ and T₃ as a function of the reduced frequency Ω for different values of d₀ with ϵ = 0.01D. We remark that the AIT resonances decrease to low frequencies with the increase d₀. We can see that for each value of d₀, the transmission along the first output line (black curve) reaches unity (T₁ = 1), the transmission along the second and third output lines T₂ and T₃ (red and Blue curves) and the reflection R vanish (i.e., T₂ = T₃ = R = 0). The conservation energy (T₁₊T₂+T₃+R = 1) must be satisfied for each output line. We can see that the frequencies of AIT resonances depend on the parameter d₀. Also, the quality factors of the three resonances are almost identical.

Figure 8.

Evolution of the transmission rates T₁, T₂ and T₃ as a function of the reduced frequency for d₀ = 0.5D (a), d₀ = 0.65D (b), d₀ = 0.8D (c) and d₀ = 0.95D (d) with ϵ = 0.01D.

To achieve an AIT type resonance with high performance (i.e., total transmission along each output line with a high quality factor), it is necessary that equations (18)–(20) must be satisfied. However, if one of the above conditions is not satisfied, the transmission rate cannot reach unity and the phononic demultiplexer becomes imperfect. Figure 9 gives an example making it possible to explain this deviation from a perfect demultiplexer when we choose the same parameters as those of Figure 7(c), except varying d₂₂. We show in Figure 9 a the variation of the reduced frequency Ω as a function of the resonator length d₂₂, we take d₀ = 0.875D, ϵ = 0.05D and d₁₄ = 0.42D. The transmission maximum along the first, second and third output lines are indicated by the black, red and blue curves, respectively. We note that the transmission maximums along the first and third output lines remain constant for two reduced frequencies (Ω = 3.57 for the black line and Ω = 4.11 for the blue line) with the variation of the resonator length d₂₂, this is due to the constant parameter of d₀ and the parameter ϵ which means that the lengths of guides and resonators along the first and third output lines are constant, and finally the AIT type resonances fall at the same frequencies. Therefore, if we change the resonator length d₂₂, the AIT type resonance moves towards low frequencies. Then, we present in Figure 9b–(c) the variation of the transmission rates T₁, T₂ and T₃ as a function of the reduced frequency Ω for two values of the resonator length d₂₂ with d₀ = 0.875D and ϵ = 0.05D. We notice that the transmission in the third output line (black curve) does not reach unity at Ω = 4.11 for d₂₂ = 0.13D and d₂₂ = 0.43D, but the transmission T₂ reaches unity when the resonator length d₂₂ = 0.43D. The AIT resonance along the first output line reaches unity (T₁ = 1) and remains constant for Ω = 3.57 with a high transmission rate. It is noted that the filtered resonance has a shape of AIT type resonance. The width at the half maximum of the AIT resonances depends strongly on the parameters d₀ and ϵ. Our finding agrees with that obtained by A. Mouadili et al.^3,66

Figure 9.

(a) Evolution of the reduced frequency as a function of the resonator length d₂₂ with d₀ = 0.875D, ϵ = 0.05D and d₁₄ = 0.42D. Evolution of the transmission rates T₁, T₂ and T₃ as a function of the reduced frequency for d₂₂ = 0.13D (b) and d₂₂ = 0.43D (c).

In summary, low crosstalk and high quality factor^67,68 ensure clear channel separation and minimize signal degradation, which is essential for demultiplexer efficiency in phonon routing applications.

Conclusion

In this research, we have studied a $ψ$ -shaped acoustic demultiplexer based on a U-shaped structure for applicaion of wave routing. Our proposed demultiplexer contains a $ψ$ -shaped waveguides with an input line and three output lines (three channels). Each output line consists of two resonators grafted at two positions far from the input line. We have given the analytical expressions of the lengths of the different guides and resonators to achieve a total transmission in one output line by canceling at the same time the reflection and transmission in the other lines. The maximum transmission peak has a shape of AIT (characterized by a maximum transmission peak wedged between two transmission zeros) type resonance, which is induced by the segment that separates the two resonators. On the other hand, the existence of a maximum transmission peak beside a transmission zero enables to select a given frequency in one output line, by canceling the transmission in the other output lines and the reflection in the input line. This resonance type is called Fano resonance in which the resonators lengths d_i2 and d_i4 must have the same values. These resonances are obtained by detuning the lengths d_i2 and d_i4 (i.e. $ϵ$ = d_i4-d_i2 with i indicating the number of channels) of two resonators. Therefore, the two resonators induce two transmission zeros which fall at two neighboring frequencies f_i1 and f_i2. The frequency f_i3 is stuck between f_i1 and f_i2 in which the transmission reaches unity. The position and the quality factor of these two resonances (Fano and AIT) are extremely sensitive to the geometrical parameters of the segments and resonators which should be chosen appropriately. We have shown two possibilities of phononic demultiplexer by using either AIT resonance or Fano resonance. Indeed, we have shown that the frequencies of the filtered resonances as well as their widths (i.e., the quality factors) can be tuned by appropriately choosing the lengths of the different waveguides constituting this system. The obtained numerical results can be easily validated through simple experiments in the audible frequency range.^26,27,46

Footnotes

Appendix

Finally, we present in the Table 1 the comparaison summary of different demultiplexer systems that are cited in the part of references [69,70,71,72,73,74]. Table 1.

Comparative summary of recent demultiplexer systems.

Reference	Structure type	Number of channels	Resonance mechanism	Methodology	Validation domain	Key features
⁶⁹	Y-shaped waveguide with detuned stubs	2	Fano and electromagnetically induced transparency (EIT) resonances	Analytical (Green’s function) + experimental	Radiofrequency (RF)	High channel selectivity, dual resonance control, experimental validation
⁷⁰	A demultiplexer based on a waveguide system containing segments and loops with two geometric defects.	2	Electromagnetically induced transparency (EIT)	Fully analytical (Green’s function)	Photonic	Two-channel demultiplexing, tunable via geometry
⁷¹	Y-shaped demultiplexer based on asymmetric loop photonic waveguides	2	Fano	Fully analytical (Green’s function)	Photonic	Two-channel demultiplexing, tunable via geometry
⁷²	T-shaped plasmonic demultiplexer	2	Fano and plasmonic induced transparency (PIT)	Analytical (Green’s function) and numerical simulations using a 2D finite element method	Plasmonic	Two-channel demultiplexing, tunable via geometry
⁷³	ψ-shaped waveguide with double resonators	3	Fano and electromagnetically induced transparency (EIT) resonances	Fully analytical (Green’s function)	Photonic	Compact and scalable design, three-channel demultiplexing, tunable via geometry
⁷⁴	Phononic crystal with ring resonators.	Multiplexer: 2 × 1 et 4 × 1. Demultiplexer: 1 × 2 et 1 × 4.	Localized resonant modes in the ring resonators within the phononic crystal; coupling and interference (acoustic resonance) enable the multiplexing/demultiplexing behavior.	Numerical simulations, likely solving acoustic wave propagation in the phononic crystal, using ring resonator coupling, mode analysis, etc. (The paper is based on modeling rather than experimental work).	Simulation/theoretical	- Compact architecture with ring resonators in a phononic crystal. - Capability to perform both multiplexing and demultiplexing acoustically. - Multiple channels (2 and 4) for input/output. - Potential for integrated acoustic signal processing.
This work	ψ-shaped waveguide with inserted two resonators in two sites in each channel	3	Fano and acoustically induced transparency (AIT) resonances	Formalism analytical: Interface response theory based on the green function method	Phononic	Compact and scalable design, three-channel demultiplexing, tunable via geometry

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Younes Errouas

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ψ -shaped phononic demultiplexer with three channels based on fano and acoustically induced transparency resonances for wave routing application

Abstract

Keywords

Introduction

Theoretical analysis

Transmission and reflection coefficients through a simple structure composed of two grafted resonators at two sites

Transmission and reflection coefficients through a phononic demultiplexer contains three channels

Results and discussions

AIT and fano resonances in the U structure

Demultiplexer based on ψ structure

Demultiplexer based on fano resonances

Demultiplexer based on AIT resonances

Conclusion

Footnotes

Appendix

Declaration of conflicting interests

Funding

ORCID iD

References

Demultiplexer based on $ψ$ structure