# Coupled resonator acoustic waveguides-based acoustic interferometers
designed within two-dimensional phononic crystals: experiment and theory

###### Abstract

The acoustic response of defect-based acoustic interferometer-like designs,
known as Coupled Resonator Acoustic Waveguides (CRAWs), in two-dimensional
phononic crystals (PnCs) is reported. The PnC is composed of steel cylinders
arranged in a square lattice within a water matrix with defects induced by
selectively removing cylinders to create Mach-Zehnder-like (MZ) defect-based
interferometers. Two defect-based acoustic interferometers of MZ-type are
fabricated, one with arms oriented horizontally and another one with arms
oriented diagonally, and their transmission features are experimentally
characterized using ultrasonic spectroscopy. The experimental data are
compared with finite element method (FEM) simulations and with tight-binding
(TB) calculations in which each defect is treated as a resonator coupled to
its neighboring ones. Significantly, the results exhibit excellent agreement
indicating the reliability of the proposed approach. This comprehensive match
is of paramount importance for accurately predicting and optimizing resonant
modes supported by defect arrays, thus enabling the tailoring of phononic
structures and defect-based waveguides to meet specific requirements. This
successful implementation of FEM and TB calculations in investigating CRAWs
systems within phononic crystals paves the way for designing advanced acoustic
devices with desired functionalities for various practical applications,
demonstrating the application of solid-state electronics principles to
underwater acoustic devices description.

###### keywords:

phononic crystals, defects, acoustic interferometer, tight-binding model

David Martínez-Esquivel Rafael Alberto Méndez-Sánchez Hyeonu Heo Angel Marbel
Martínez-Argüello* Miguel Mayorga-Rojas Arup Neogi* Delfino Reyes-Contreras*

D. Martínez.
Programa de Doctorado en Ciencias, Facultad de Ciencias, Universidad Autónoma
del Estado de México, Campus “El Cerrillo”, Toluca, Estado de México, México

R. A. Méndez-Sánchez
Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México,
Cuernavaca, Morelos,
México

H. Heo
Graduate Program in Acoustics, The Pennsylvania State University, University
Park, Pennsylvania 16802, USA

A. M. Martínez-Argüello
Instituto de Física, Benemérita Universidad Autónoma de Puebla, C. P. 72570
Puebla, Pue., México
*E-mail<EMAIL_ADDRESS>
M. Mayorga
Facultad de Ciencias, Universidad Autónoma del Estado de México, Campus “El
Cerrillo”, Toluca, Estado de México, México

A. Neogi
Institute of Fundamental and Frontier Sciences, University of Electronic
Science and Technology of China, Chengdu, 610054, China
*E-mail<EMAIL_ADDRESS>
D. Reyes
Laboratorio de Acústica y Nanomateriales (LAN), Facultad de Ciencias,
Universidad Autónoma del Estado de México, Campus ”El Cerrillo”, Toluca,
Estado de México, México
*E-mail<EMAIL_ADDRESS>
##  1 Introduction

Defects engineering in phononic crystals (PnCs) has convincingly demonstrated
the feasibility of guiding waves within their acoustic bandgap when defects
are optimally induced [1, 2, 3]. PnCs, also known as acoustic bandgap
materials, consist of sound scatterers periodically arranged in a matrix in
which both components exhibit different physical properties such as mass
density, speed of sound, and Young’s modulus. This spatially periodic mass
distribution enables the observation of acoustic or elastic bandgaps. Similar
to stopbands in electronic or photonic crystals, phononic bandgaps are
frequency intervals where wave propagation is forbidden across the crystal. By
introducing defects that break the crystal symmetry, typically induced by
removing scatterers, defect modes become allowed within the bandgap [3, 4, 5,
6]. While a single defect acts as a resonator, promoting a unique in-gap
eigenmode, a group of defects gives rise to multiple coupled eigenmodes
leading to miniband transmission within the stopband interval [7].

The coupling of resonant modes in defect-based PnCs has been successfully
described approximating these as Coupled Resonator Acoustic Waveguides (CRAWs)
within the framework of tight-binding models (TBM) [8, 9, 10, 11, 12, 13].
TBM, widely employed in solid-state physics to study the electronic properties
of materials, can be extended to investigate other excitations, such as
phonons which represent the quantized vibrations of crystal lattices [7, 13].
In this approach, a PnC-based CRAW is discretized into separate sites and the
dynamics of phonons are described by considering the interactions between
neighboring sites. The propagation of waves in CRAWs through the TBM
approximation can be analyzed by solving the equations of motion for phonons
in the crystal lattice and considering the interaction strengths between
adjacent lattice sites [13, 14, 15]. The dynamical matrix of the phononic
crystal, which incorporates the interactions between neighboring sites,
provides information about the allowed phononic modes [15, 16].

CRAWs engineered within phononic crystals consist of multiple waveguide
segments coupled through resonators or defects forming a structure of
interconnected unit cells. Each unit cell contains the waveguide segment and
resonator or defect composing the CRAW [4, 17]. These structures enable
precise manipulation and control of acoustic waves, facilitating the creation
of waveguides with desirable transmission properties and the formation of
localized modes within band gaps. The theoretical framework based on TB theory
allows for constructing the Hamiltonian that describes the dynamics of
acoustic waves propagating along these coupled resonant waveguides. The
Hamiltonian incorporates kinetic terms representing the wave´s energy in the
waveguide segment and potential terms accounting for the coupling between
adjacent segments through the resonators or defects [13, 15]. This approach
facilitates determining the coupling strength between the waveguide segments
which directly impacts the transmission properties and formation of localized
modes [7]. As a result, it becomes feasible to predict the resonant
frequencies of the coupled waveguide structure. These resonant frequencies are
associated with the formation of localized modes representing waveguide modes
confined within the resonator or defect regions [16, 18].

The application of the TB model in the design and optimization of CRAWs
systems in PnCs with specific functionalities such as enhanced transmission
within certain frequency ranges, strong localization of waveguide modes, and
efficient filtering of acoustic waves, is experiencing significant growth in
the field of acoustic and mechanical phononic crystal structures [18, 19, 20,
21, 22, 23, 24]. Besides that, 1D and 2D Phononic meta-materials can be also
used to manipulate acoustic wave propagation, neither modeling 1D phononic
structures with hyperbolic dispersion [25, 26], nor 2D phononic structures
with tailorable equifrequency contour for focusing [c], or beam steering [27,
28].

This study presents the adaptation of the TBM to describe the acoustic
response of defect-based acoustic interferometer-like designs, referred to as
CRAWs systems in two-dimensional PnCs. The phononic crystal is composed of
steel cylinders, arranged squarely in a water matrix, with defects induced by
removing specific cylinders to create Mach-Zehnder-like defects-based
interferometers. Two defect-based acoustic interferometers of MZ-type are
constructed, one with arms oriented horizontally and another one with arms
oriented diagonally, and their transmission features are experimentally
measured using ultrasonic spectroscopy. The tight-binding approximation is
applied to both models considering each defect as a resonator coupled to its
neighboring ones. Finite element method (FEM) simulations and bandstructure
calculations based on the linearized Navièr-Stokes equation using the COMSOL
platform are also considered. The obtained results are compared with the
experimental data, demonstrating the application of solid-state electronics
principles to underwater acoustic-devices analysis. Remarkably, our results
show excellent agreement with the experimental and simulation data. This
comprehensive match is crucial for predicting and optimizing resonant modes
supported by defect arrays, enabling the customization of phononic structures
and defect-based waveguides to specific requirements.

##  2 Results

###  2.1 Ultrasonic spectroscopy results

A phononic crystal (PnC) composed of a $23\times 21$ arrangement of stainless-
steel cylinders of radius 0.8 mm is immersed in a water matrix with a lattice
parameter of 2 mm, as depicted in Figure 1 a). Two defect-based
interferometers, inspired by Mach-Zehnder-type designs, as shown in Figure 1
b) are constructed. Those are labeled as the diagonal model (DM) and the
horizontal model (HM), placed on the left and center side of Figure 1 b),
respectively. The DM-interferometer consists of a defect input and two arms
built with defects diagonally arranged, which coincide with another defect, as
illustrated on the left side of Figure 1 b). The HM-interferometer has one
input defect and two linear arms, as observed on the center side of Figure 1
b). The right side of Figure 1 b) are schematic representations of the DM and
HM interferometers; red and orange arrows describe the input and output
points, respectively, and the yellow ones are the defect-paths within the
phononic structure. To maintain their positions without perturbation, all
cylinders are stacked on perforated ABS plastic bases. Experimental results
have demonstrated that groups of defects continuously induced in a phononic
crystal can function as waveguides for waves whose frequencies fall within
their phononic bandgap [3, 5].

Figure 1: (Color online) Actual interferometers and experimental setup for
ultrasonic spectroscopy analysis. a) Lateral-actual view of the perfect PnC.
b) Top-actual view of the DM (left) and HM (center) interferometers,
(right)-schematic of both designs showing the defect paths. c) Ultrasonic
spectroscopy setup.

The experimental transmission spectra for both designed defect-based acoustic
interferometers, DM and HM, are depicted in Figure 2 a) in red and blue lines,
respectively, covering the frequency range from 400 to 460 kHz. Acoustic
response was experimentally recorded in dBm units, which have its logarithmic
equivalence in watt (+20 dBm (0.1 W)), and represent power quantity. It was
selected to use this unit, which is related to the wave intensity when
considering the transducer area, by simplicity as the raw information from the
spectrum analyzer. It is evident that the transmitted acoustic power of the
models differs significantly depending on the spatial disposition of the
defects forming each interferometer, which yields different transmitted power
at the same frequencies. It has been reported the perfect crystal exhibits a
full bandgap from approximately 380 to 480 kHz, as will be shown below, and
previously reported by our group [3, 5, 27].

Figure 2: (Color online) Experimental and theoretical (FEM) transmission
characterization for the designed interferometers. a) Experimental
transmission spectra for both DM (blue line) and HM (red line)
interferometers: inset corresponds to the FEM simulated results. b) FEM
spectra from the COMSOL platform for the two models.

For the DM array, a narrower transmission band is induced within the acoustic
gap compared to the response recorded for the HM. This behavior indicates the
selective transmission of waves through the first model. The observed
differences in the acoustic response between both models are due to the
spatial configuration of the defects, as it has been previously demonstrated
that nine defects configured diagonally allow for a single peak approximately
16 times narrower ($\sim 426$ \- 429 kHz) than when defects are arranged
horizontally (relative to the propagation axis), which results in multiple
peaks ($\sim 405$ \- 460 kHz) [5]. The DM interferometer exhibits a miniband
with at least two well-defined in-gap resonances at 427.4 and 431.2 kHz along
with a smaller one at 434.8 kHz, see the red line in Figure 2 a). In contrast,
multiple resonant peaks were recorded for the HM design, resulting in a new
well-defined, and broader in-gap transmission band. These multiple resonant
peaks arise from the collectively coupled resonant modes supported by each
individual defect [3, 5].

Figure 3: (Color online) Pressure maps for the DM and HM interferometers.
Obtained from COMSOL for the a) DM and b) HM interferometers for the frequency
of 425.0 kHz. As observed, the acoustic energy is being transported across the
defect paths. c) Experimental pressure map for the HM model, where the
acoustic energy distribution has a similar distribution as for the FEM
results.

###  2.2 FEM Calculations

The experimentally analyzed PnCs structures are designed using the COMSOL
Multiphysics platform (version 5.3), and their acoustic properties are
calculated employing the Finite Element Method (FEM) which serves as the
foundation of the cited platform. The 2D phononic structures consist of
stainless-steel scatterers of radius 0.8 mm arranged in a $23\times 21$ square
lattice within a water matrix with a lattice parameter of 2 mm. The density
and the longitudinal sound velocity values for stainless steel and water are
obtained from the library included in the COMSOL platform:
$\rho_{{}_{\mathrm{S}}}=7800$ kg/$\mathrm{m}^{3}$,
$\rho_{{}_{\mathrm{W}}}=1000$ kg/$\mathrm{m}^{3}$, $c_{{}_{\mathrm{S}}}=5790$
m/$\mathrm{s}$, and $c_{{}_{\mathrm{W}}}=1480$ m/$\mathrm{s}$, all of them at
20${}^{\circ}\mathrm{C}$. For water and steel, we used the density and
longitudinal speed of sound. Additionally, Young’s modulus and Poisson ratio
were incorporated for the solid steel, 205 GPa and 0.28, respectively, which
are demandant for the simulations and were taken from the library. The
simulations are obtained for the same frequency range as that of the
experimental ones. The DM and HM interferometers are depicted in Figure 5,
panels a) and b) respectively. The acoustic-solid interaction model is
utilized to simulate the acoustic response of each model with a frequency
sweep using a plane wave radiation source. Figure 2 b) corresponds to the FEM
spectra, which nearly agree with the experimental results recorded through
ultrasonic spectroscopy (Figure 2 a)). For the DM interferometer three
numerical peaks were found at 425, 426.8, and 429 kHz, and are associated with
the experimentally recorded peaks. It could be convenient to clarify that in
COMSOL, the input signal is given in Pa, which is a unit of pressure that is
equivalent to 93.9794 dB Sound Pressure Level (SPL), and that is the reason
because the transmitted signal is in dB, representing the SPL.

Additionally, pressure maps were generated to gain insight into the acoustic
energy distribution inside and outside the phononic structures. Figure 3 a)
shows the pressure map for the DM interferometer at a frequency of 425.0 kHz
which can be associated with the experimental in-gap resonant peak observed at
428.4 kHz. In the simulations, the source is placed at a distance of two
millimeters from the crystal which is consistent with the experimental setup
and has a size of 25 mm, corresponding to the actual transducer size. Figure 3
b) contains the pressure map for the HM model at the same frequency, 425.0
kHz. These pressure maps are represented as SPL in dB, which has been
clarified above. Pressure naps provide valuable insights into the behavior of
the acoustic energy for this frequency in each model, revealing how the waves
follow the designed defect paths to guide them. This can be visually observed
as the sound pressure at the center of both models has the lowest levels (blue
color), which implies that no wave propagation at these frequencies occurs due
to the bandgap. The largest sound pressure levels (red color) are observed at
the input and output points and along the defect paths due to the defect array
being a CRAW by itself working as a waveguide to allow transmission at the in-
gap frequencies.

To obtain an insight into the acoustic pressure distribution in the actual
models, an experimental pressure map was obtained for the HM model based on
the temporal evolution and equipment requirements (see experimental section).
The experimental pressure map for this model is displayed in Figure 3 c),
where the acoustic pressure in dBm units has similar characteristics as that
observed in Figure 3 b). The sound pressure was measured 2mm away from the
transducer facet. It was -17.86 dBm for a 20 Vpp from the frequency generator,
implying a power loss of over ten dBm. Here, 20 Vpp means the sinusoidal wave
amplitude, which induces an input power (2mm in front of the transducer used
as emitter) measured in the Spectrum analyzer of -17.86 dBm, which reduces to
-29 dBm at the output point of the HM model, indicating a reduction in the
experimentally measured power over 10 dB. This reduction in power is higher in
the rest of the selected area, due to the green color meaning -40 dBm. The
local acoustic energy measured using a needle hydrophone as a receiver drops
in intensity compared to the global pressure recorded using a transducer
(Figure 2 a)). However, it is still detectable by the used equipment, which
means it is viable for practical measurements in actual devices or possible
applications.

Figure 4: (Color online) Band structure calculations for the selected unit
cells. a) Unit cells where the k-vectors are sketched, b) Band structure for
the perfect phononic crystal, where the corresponding unit cells are included.
Zoomed bandstructure diagrams for the c) DM-UC-1, d) DM-UC-2, e) HM-UC-1, and
f) HM-UC-2 unit cells for better visualization.

###  2.3 Band structure calculations

Figure 4 contains the bandstructure calculations performed to understand and
correlate the observed results. For both DM and HM interferometers, two
separate unit cells (UC) containing a group of defects were calculated and
assigned as UC-1 and UC-2, as shown in Figure 4 a). The band structure of each
interferometer was calculated utilizing Pressure Acoustics in conjunction with
Solid Mechanics and applying the Floquet periodic boundary condition. The band
structure was calculated employing a wavevector normalized by the period of
the crystal. Specifically, we defined the X and M points within the Brillouin
Zone (BZ) as illustrated in 4 a). Notably, we established the center of the
superlattice as $\Gamma$, and the definitions of the X and M points are in
accordance with the C4 symmetry axis. In Figure 4 b), the black squares
represent the calculated band structure for the perfect phononic crystal
between 1 - 800 kHz. The other points in Figure 4 b) correspond to each
model’s unit cells, which are zoomed in Figures 4 c)-f).

For the two-unit cells of each interferometer, DM, and HM array, in-gap
solutions were obtained in the 400-460 kHz frequency interval. For the PnC, it
has been reported that one defect induces an alone resonant defect mode at 427
kHz [5], while several aligned defects induce defect modes whose number is
exactly the number of defects; the diagonal disposition of 5 defects induced a
narrow band 4 kHz-width centered at around 427 kHz. It is hard to capture the
entire coupling of all the induced resonant modes due to the presence of the
18 or 34 defects composing both, DM and HM models, respectively. In the band
structure diagram of the four UC, an in-gap solution in the frequency range of
424-429 kHz was recorded, which can be related to the fact that at each of
these, there is diagonal coupling between at least two defects. As observed in
Figures 4 c), d), and e), two or three allowed modes appear between 424-238
kHz, indicating the diagonal coupling is dominating. This is more visible in
Figures 4 d), where only modes in this frequency interval were obtained, which
agrees with the unit cell DM-UC-2 (Figure 4 a)), where only diagonal coupling
can be assigned; the inter-period separation between the first and last defect
is three, which reduce the coupling strength.

However, for the HM, in both unit cells the allowed modes are separated,
covering the entire range of 400-460 kHz, as seen in Figure 4 f). The modes
near 410 and 460 kHz in Figures 4 c), and e) and f) could be the result of the
coupling between the two horizontal continuous defects, which are present in
these three unit cells. Even when the band structure diagrams do not
faithfully reproduce results in Figure 2, the in-gap induced modes for the
selected unit cells agree with the fact that the induced transmission band due
to the HM array extends along a larger frequency interval compared with the DM
model. Larger unit cells must be necessary to fully reproduce the entire
behavior.

Figure 5: (Color online) Schematic top-view of the models under study. The
unoccupied sites represent defect cavities in the array and define a) the DM
and b) the HM interferometer. These models can effectively be described by a
tight-binding model defined by the red and blue dots for c) the DM and d) the
HM models.

###  2.4 Tight Binding Model

As previously mentioned, the tight-binding approximation is employed to
theoretically describe the experimental and FEM simulation results for the
designed CRAWs, which constitute the defects-based Mach-Zehnder-type acoustic
spectrometer. The schematic representation of the models, consisting of steel
cylinder arrays (top view) is illustrated in Figure 5. The unoccupied sites
within that two-dimensional locally periodic structure or finite PnC
correspond to defect cavities or impurities that define the diagonal and
horizontal models shown in panels a) and b), respectively. These defect
cavities are intentionally designed such that their corresponding resonant
frequencies lie within the bandgap of the finite PnC, thereby allowing the
localization of the resonant modes associated with the respective impurities.

The weak coupling between neighboring resonators, via evanescent Bloch waves,
yields the nearest-neighbor tight-binding approximation usually found in solid
state physics [29]. Note that this description is the same found in the
coupled-resonator optical [30, 31], acoustical [3, 5, 11, 12, 32, 33], and
elastic [7, 13, 15, 16, 34, 35] waveguides. Panels c) and d) in Figure 5, show
a schematic of the tight-binding model of the corresponding system defined by
the red and blue dots for the diagonal and horizontal model respectively.
These models can be described by the following nearest-neighbor tight-binding
Hamiltonian

$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{n,m}f_{n,m}|n,m\rangle\langle n,m|$ $\displaystyle+$
$\displaystyle\sum_{n,m}\big{[}\nu_{(n,m),(n+1,m)}|n,m\rangle\langle
n+1,m|\big{.}$ $\displaystyle+\,\nu_{(n,m),(n-1,m)}|n,m\rangle\langle n-1,m|$
$\displaystyle+\,\nu_{(n,m),(n,m+1)}|n,m\rangle\langle n,m+1|$
$\displaystyle\big{.}+\nu_{(n,m),(n,m-1)}|n,m\rangle\langle n,m-1|\big{]},$

where $f_{n,m}$ is the resonance frequency of the defect cavitie at positions
$|n,m\rangle$ and $\nu_{i,j}$ is the hopping frequency to nearest neighbours.

####  2.4.1 Scattering matrix approach

The isolated diagonal and horizontal samples of Figure 5 c) and d) can be
opened by attaching to them two semi-infinite single-mode perfect leads with
coupling strength $\gamma^{\mathrm{L,R}}$ to the left (L) and right (R) sites,
respectively. The $2\times 2$ scattering matrix, $S$-matrix, can be written as
[36, 37]

$S(f)=\left(\begin{array}[]{ccc}r&t^{\prime}\\\
t&r^{\prime}\end{array}\right)=\textbf{1}_{2}-2\mathrm{i}\,\sin(k)W^{T}(f-H_{\mathrm{eff}})^{-1}W,$
(2)

where $r$($r^{\prime}$) and $t$($t^{\prime}$) are the reflection and
transmission amplitudes when the incidence is from the left (right),
$\textbf{1}_{2}$ is the unit matrix of dimension $2$, $k=\arccos(f/2)$ is the
wave vector at frequency $f$ supported in the leads, and $H_{\mathrm{eff}}$ is
the effective non-Hermitian Hamiltonian given by

$H_{\mathrm{eff}}=H-\frac{\mathrm{e}^{\mathrm{i}k}}{2}WW^{T},$ (3)

where $H$ is the $N\times N$ Hamiltonian matrix that describes the sample with
$N$ resonant states, see Eq. (2.4). In equations (2) and (3), the matrix $W$,
where the superscript $T$ indicates the matrix transposition operation, is an
$N\times 2$ matrix that couples the $N$ resonant states of the closed sample
to the $2$ propagating modes in the leads. Its elements are defined by

$W_{ij}=2\pi\sum_{c=\mathrm{L,R}}A_{i}^{c}(f)A_{j}^{c}(f)$ (4)

with the coupling amplitudes given by

$A_{i}^{\mathrm{L,R}}(f)=\sqrt{\frac{\gamma^{\mathrm{L,R}}}{\pi}}\left(1-\frac{f^{2}}{4}\right)^{1/4}\big{(}\delta_{i,\mathrm{L}}+\delta_{i,\mathrm{R}}\big{)}$
(5)

with $\delta_{i,j}$ the usual Kronecker delta. Furthermore, the frequency
dependence in $H_{\mathrm{eff}}$ can be neglected since $\arccos(f/2)$ changes
slightly at the center of the band. Then, from the two-channel $S(f)$-matrix
of equation (2), the transmission is obtained from the transmission amplitude
as $|t|^{2}$.

Figure 6: (Color online) Transmission as a function of frequency obtained from the tight-binding model for both the diagonal and horizontal model in red and blue solid lines, respectively. The inset shows the transmission in the frequency range of interest. Interferometer |  $f_{0}$ (kHz) |  $\nu$ (kHz) |  $\nu^{\prime}$ (kHz) |  $\gamma$ (kHz)
---|---|---|---|---
Horizontal | 432.0 | 12.0 | 12.0 | 10.0
Diagonal | 430.7 | 57.0 | 4.2 | 0.005
Table 1:  Parameters for constructing Hamiltonian (2.4) and $S$-matrix (2)
along with equations (3)-(5) for both the horizontal and diagonal
interferometer.

The tight-binding model and the scattering approach, as described above, are
applied to elucidate the behavior of both interferometers. For this purpose,
further simplifications of Hamiltonian of Eq. (2.4) can be made by noticing
first that for both models (DM and HM) the defect cavities are roughly
equivalent and then have the same resonance frequency, i.e.,
$f_{nm}=f_{mn}=f_{0}$. Second, the hopping frequency depends on the distance
between defects since the solutions in the PnC are evanescent, then for the
diagonal and horizontal interferometers those hoppings are expected to be
different [see Figs. 5 c) and d)]. In addition, the hopping frequencies
between nearest neighbors in the bulk defects can be considered the same,
i.e., $\nu_{i,j}=\nu$, while the ones at the left and right of the
interferometers can be considered as $\nu^{\prime}$. Thus, the TB model of Eq.
(2.4) only requires the knowledge of the parameters $f_{0}$, $\nu$, and
$\nu^{\prime}$ for each interferometer. Now since $f_{0}$ determines the
center of the band, from the experimental spectrum we obtain that
$f_{0}\approx 432.0\,(430.7)$ kHz for the H (D) interferometer (see Fig. 2
a)). Meanwhile, the nearest-neighbor frequency hopping determines the width of
the band associated with the defects: the bandwidth is equal to four times the
hopping frequency [38].

On the one hand, from the experimental frequency spectrum for the horizontal
model, the bandwidth is approximately 48.08 kHz and then $\nu\approx 12.0$ kHz
with $\nu=\nu^{\prime}$ as the distance between the defects is the same. For
the diagonal model, on the other hand, $\nu\neq\nu^{\prime}$ since the
distance between the defects for this model is different. Furthermore, from
the bandwidth of the miniband observed in the frequency spectrum ($\sim 16.0$
kHz), $\nu^{\prime}\approx 4.2$ Hz and $\nu$ can be estimated as 57.0 kHz. The
latter could be obtained from the full experimental spectrum of the diagonal
model that unfortunately we do not have, as it is necessary to experimentally
measure the power at each cavity.

Figure 7: (Color online) Power as a function of frequency for the horizontal
and diagonal models. Red and blue continuous lines are the experimental
results for the DM and HM, respectively. The red and blue dashed lines are the
results for the respective tight-binding model of equation (2.4) together with
equations (2) - (5).

In order to obtain the $S$-matrix for each H (D) model, the coupling strength
to the exterior, $\gamma^{\mathrm{L(R)}}$, at the left (L) and right (R) sides
has to be determined. For simplicity, we consider that for each model this
coupling strength is the same, that is $\gamma^{\mathrm{L(R)}}=\gamma$. Here,
there are two main leaking mechanisms of each interferometer to the exterior.
One is coming from the input and output defects and the other one coming from
the interferometer arms. Notice that the leaking from the diagonal
interferometer is much smaller than that from the horizontal interferometer
since the defects forming the arms of the H-based interferometer are closer to
the borders of the PnC. Then, the coupling strength $\gamma$ is fixed by
adjusting the width of the resonances within the band associated with each
interferometer and corresponds to $\gamma=10.0\,(0.005)$ kHz for the
horizontal (diagonal) interferometer. Table 1 summarizes the TB and $S$-matrix
parameters for each interferometer.

Figure 6 displays the normalized transmission as a function of frequency for
the diagonal and horizontal models in red and blue lines, respectively,
obtained through the TB model of Eq. (2.4) together with the $S$ matrix of
(2). The parameters for each model are given in Table 1. The bandwidth is
equal to four times the hopping frequency, then, from the experimental
spectrum for the HM, this bandwidth is approximately 48.08 kHz. For the DM we
only have an experimental spectrum from 400 to 460 kHz which covers only the
of interest miniband of approximately 16.8 kHz shown in Fig. 2 a). This allows
us to obtain the hopping frequency of 4.2 kHz. To obtain the hopping frequency
nu experimentally, the full bandwidth is needed. However, by using TB
calculations the value of $\nu=57.0$ kHz was obtained, which could be slightly
different for an experimental spectra of 300 - 500 kHz, for example. The inset
in Figure 6 provides the normalized transmission on a logarithmic scale. For
the horizontal model (blue line), the resonant frequencies around 430 kHz
align with the in-gap resonances obtained from the experiment for this model.
Conversely, for the diagonal model (red line), it is noteworthy that due to
the weak coupling to the outside, the resonances observed are narrower than
those obtained for the horizontal model, which is in complete agreement with
the experimental spectra and bandstructure results described above.

Finally, Figure 7 summarizes the behavior of the transmitted intensity (power)
as a function of frequency for both DM and HM interferometers experimentally
obtained using ultrasonic spectroscopy (continuous lines), and through the
tight-binding model of equation (2.4) together with equations (2) - (5)
(dashed lines). The plots have been overlapped in the same frequency range for
a better comparison, and the intensities were adjusted to demonstrate the
agreement and tendencies of the acoustic response of the designed
interferometers for each experimental and theoretical analysis. It is
convenient to appoint out the obtained TB parameters and the coupling strength
$\gamma$ are used in Eqs. (3) - (5) and then into the S-matrix of Eq. (2)
which now is only a function of frequency $f$. The transmitted power as a
function of $f$ is obtained from the S-matrix element $|t|$ as
$10Log_{10}(|t|)$ varying the frequency range from 400 to 460 kHz, as shown in
Fig. 7 for each interferometer. This gives the TB acoustic power similar to
the experimental ones, as previously described

In the experimental results, slight fluctuations due to errors in locating the
cylinders on the respective arrays can be observed. Additionally, a certain
degree of power loss is expected, as seen in Figure 2. These effects can be
accounted for in the tight-binding model by introducing a certain degree of
disorder, such as in the frequency resonance of the defect cavities, and
incorporating an imaginary part in the frequency resonance of the impurity to
consider the effect of absorption. However, even without the inclusion of
these effects, the tight-binding model presented here notably captures the
essential features of the horizontal and diagonal models, as observed in both
figures 6 and 7, showing high concordance with the other methods of analysis
discussed here. When comparing the results with FEM calculations using COMSOL,
a left-shifting of 3.5 kHz can be observed. This shifting is attributed to the
finite size of the simulated crystal in the FEM, which differs from the
idealized infinite crystal assumption in the tight-binding model, and has been
previously observed by our group [5]. Figure 7, which shows consistent
results, leads to propose the possibility of predicting resonant peaks in
CRAWs forming defects-based Mach-Zehnder-type acoustic spectrometers with
different and optimized configurations for the selective transmission of
ultrasonic waves.

##  3 Conclusion

In conclusion, this study successfully adapted the tight-binding model (TBM)
to effectively describe the acoustic response of defect-based acoustic
interferometer-like designs, specifically Coupled Resonator Acoustic
Waveguides (CRAWs) in two-dimensional phononic crystals (PnCs). The phononic
crystal was fabricated with steel cylinders arranged in a square lattice
immersed within a water matrix, and defects were induced by removing cylinders
to create Mach-Zehnder-like (MZ) defects-based interferometers. Two MZ-type
defect-based acoustic interferometers were constructed, with arms oriented
horizontally or diagonally, and their transmission features were thoroughly
characterized through experimental ultrasonic spectroscopy. By employing the
tight-binding approximation, each defect was treated as a resonator coupled to
its neighboring ones, which allowed for accurate predictions of the acoustic
response within the CRAWs system. The obtained results were compared with
experimental ultrasonic spectroscopy data, FEM simulations, and bandstructure
calculations. Remarkably, our results exhibited outstanding agreement with the
experimental and simulation data, providing robust evidence of the reliability
and validity of the proposed TBM description, an approach of solid-state
electronics principles successfully proposed for underwater acoustic devices
performance. The achieved comprehensive match between the theoretical
predictions and the experimental observations holds crucial significance, as
it offers an essential tool for precisely predicting and optimizing resonant
modes supported by defect arrays within the phononic crystal. Finally, the TBM
formulation applied here for describing the CRAWs can straightforwardly be
extended to study wave propagation in three-dimensional configurations [39]
which will be a subject of research for future works.

##  4 Experimental Section

To observe the interferometric behavior of the guided waves occurring in the
designed defects-based acoustic interferometers, the experimental setup
illustrated in Figure 1 c) is used. The excitation of waves is provided by a
Frequency Generator equipment (Teledyne Lecroy, Wave Station-2012) using an
unfocused immersion transducer (Olympus V301, 0.5 MHz) as the emitter element.
Another transducer, connected to a Spectrum Analyzer (Tektronix-MDO 3024b),
serves as the receiver which records the acoustic response. For the analysis,
both transducers are positioned face to face at a distance of 50 mm, with the
PnC placed in between. The three elements are then immersed in a tank filled
with 40 liters of deionized water for measurements at room temperature (22
${}^{\circ}\mathrm{C}$). The utilized PnC has been thoroughly characterized
and possesses a bandgap approximately spanning from 380 to 480 kHz [27]. The
acoustic response for both models DM and HM, along with the perfect crystal,
is recorded in the frequency range from 390 to 475 kHz, corresponding to the
bandgap of the perfect crystal.

The experimental pressure maps were obtained using the same set as in Figure 1
c), however, the transducer used as the receiver was changed by a needle
hydrophone (Muller) of 0.5 mm diameter connected to an amplifier. The
hydrophone was moved using a translation stage to record the pressure map
covering an area of 40x60 mm in front of the HM interferometer (see Figure 2
c)), recording the transmitted signal each 2 mm in both directions, starting 2
mm away from the phononic structure. A group of 600 data points were used to
generate the experimental pressure map for the frequency of interest.

Acknowledgements

DME acknowledges CONAHCyT for the grant to develop Ph.D. studies at UAEMex.
A.M.M-A. acknowledges financial support from CONAHCyT under the program
“Estancias Posdoctorales por México 2022”. DR Thanks to UAEMex for the partial
financial support through project 6753/2022CIB. AN acknowledges the support
from the Distinguished Professorship start-up funds from UESTC and the
Ministry of Science and Technology of China (MOST) International Collaboration
Grant No. 2022YFE0129000 entitled “Cavity Acoustotodynamics for nonreciprocal
wave propagation”. RAMS was supported by DGAPA-UNAM under project PAPIIT-
IN111021.

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