Design of a Thickness Sensor Based on a One-Dimensional Phononic Crystal

Design of a Thickness Sensor Based on a One-Dimensional Phononic Crystal

Ahmed Kahlouche* Mounir Bouras Abdessalem Hocini

Department of Electronics Faculty of Technology, ‎Signals and Systems Analysis Laboratory, University of Mohamed Boudiaf-M’sila, PO Box.166, Road of Ichebilia, M’sila 28000, Algeria

Corresponding Author Email:
13 April 2022
9 June 2022
30 June 2022
| Citation



Nowadays, sensor technology has attracted great interest in various domains. In this ‎work, we have analyzed a one dimensional phononic crystal made by the stack of N bilayers of ‎‎(LiNbO3/SiO2). The sensor design consists of a one dimensional phononic crystal structure with a ‎defect layer inserted in the middle. Using the Transfer Matrix Method (TMM), the transmission ‎spectrums of acoustic waves are calculated and plotted. In this work, we are interested in the ‎resonance peak that is transmitted inside the phononic band gap. The results obtained show clearly that the resonant frequency of the measured ‎transmission peak is very sensitive to the layer defect properties. This proves that such structure offers a new platform for sensing ‎applications.


1D phononic crystal, band gaps, transfer matrix method (TMM), sensors materials, acoustic waves

1. Introduction

Phononic crystals (PnCs) are a novel type of artificial materials that exhibit periodic ‎distributions in their densities and mechanical properties in one, two or three dimensions of space ‎‎[1, 2]. Such crystals help to guide, control, and modify the transmission of acoustic and elastic ‎waves in fluids and solids, respectively. An interesting feature of these structures is the formation ‎of phononic band gaps, which prohibit the propagation of elastic/acoustic waves in certain ‎directions and frequency ranges. These phononic band gaps have the same behavior as photonic ‎band gaps with respect to the propagation of electromagnetic waves in photonic crystal ‎structures [3, 4]. Phononic bands are created principally by spatial modulation of the acoustic ‎impedance of the materials that compose the structure. Therefore, when designing phononic ‎crystal structures, geometrical and physical parameters such as fill factor, mass density, sound ‎speed, and angle of incident must be considered [5-9]. ‎

In recent years, phononic crystal structures have attracted considerable interest due to their ‎potential applications in various technological fields, such as acoustic filtering, cavities, ‎waveguides, barriers sonic and sensors [10-14]. On the other hand, great attention concerns the ‎multilayer phononic structures, which have been widely studied and proposed as new ‎configurations for measuring the acoustic properties of fluids [15-18]. The characterization of these ‎sensors is commonly based on the resonance frequency, transmission ratio, or width of the ‎transmission peak [15, 19-21]. In this work, the resonance frequency of the defect mode is used to ‎detect changes in the defect layer properties, especially the thickness. Firstly, we have studied ‎the propagation of the longitudinal acoustic waves through a regular one-dimensional phononic ‎crystal in order to determine the phononic band gaps. Secondly, a special interest was devoted to ‎the phenomenon of local resonance inside the phononic band gap in order to use such a structure ‎for sensing the variations in thickness of the layer defect. The transmission coefficients are ‎calculated and plotted by using the Transfer Matrix Method (TMM), which is usually adopted ‎for multilayer structures [15, 22-24].

2. One-Dimensional Phononic Crystal Structure

The proposed 1D-PnC structure is shown in Figure 1. The structure consists of five unit ‎cells. Each unit cell is composed of two different materials; LiNbO3 and SiO2 with thicknesses a1=aLiNbO3= 0.5 mm ‎and a2=aSiO2= 0.5 mm respectively. The lattice constant of every unit cell is a=a1+a. In ‎this work, we used the transfer matrix method to study the acoustic/elastic wave propagation in ‎one-dimensional phononic crystal. TMM is based on the calculation of the transmission or reflection coefficients.

Figure 1. A schematic diagram of a perfect 1D-phononic crystal structure

For a longitudinal acoustic wave with normal incidence on the phononic crystal, the ‎pressure of the acoustic wave in the medium is governed by the following equation:‎

$\nabla^{2} \emptyset=\frac{1}{C_{l}^{2}} \frac{\partial^{2} \emptyset}{\partial t^{2}}$               (1)

where, Ø is the displacement and Cl is the longitudinal speed of sound in each ‎layer [22-24]. The elastic constants of the matrix and of the inclusions constituting the 1D crystal ‎are illustrated in Table 1 [25].‎

Table 1. The elastic properties of materials


Density $\rho$(Kg/m3)

Transverse Celerity Ct (m/s)

Longitudinal Celerity Cl (m/s)













3. Results and Discussions

3.1 1D Phononic band gap

First, we consider the one dimensional perfected phononic structure illustrated in Figure ‎‎1. We also consider the propagation of longitudinal elastic waves across the structure. Figure 2(a) ‎shows the transmission spectrum as a function of the frequency obtained with the TMM for 2, 4 ‎and 5 unit cells. It’s clear that the number of oscillations on the left and right sides of the gap ‎depends on the number of unit cells. On the other hand, the transmittance in the gap decreases ‎sharply to 0% for a high number of cells. Figure 2(b) shows the variations of the band gap width ‎and location as a function of the filling ratio.

Figure 2. (a) Transmission spectrum for a 1D-PnC for 2, 4 and 5 unit ‎cells; (b) Variations of phononic bandgap width as a function of the filling fraction ratio

The value of the gap width of the first band increases with the increase of the value of the ‎LiNbO3 material thickness aLiNbO3 (a1) until it reaches a critical value and then decreases. The ‎largest phononic band gap appears at a filling factor of ff = 0.50. For this value, the frequency of ‎the band gap goes from 2.251 MHz to 3.935 MHz. Then, as the filling ratio of any layer increases ‎to the critical value, the cutoff frequencies of the higher modes of the incident wave also ‎decrease until the band is completely closed. Therefore, the value of the filling fraction of any ‎materials is necessary to obtain a wide bandgap, which prevents the propagation of waves.‎

In this section, we will study the influence of the acoustic impedance defined by the ‎product of the density and speed of sound on the size and location of the phononic band gaps.

Figure 3. Width and position of 1D phononic band gap as a function of density contrast and ‎speed of sound for 5 unit cells

Figure 3 reports the variations in the phononic band gap ∆f (MHz) and band position f0 ‎‎(MHz) as a function of mass density contrast (red and blue curves, respectively). While the green ‎and black curves show the influence of the longitudinal velocity contrast on the properties of the ‎phononic band gap. The results clearly demonstrate that the density and the longitudinal velocity ‎of sound have a great influence on the location (f0) and the width (∆f) of the phononic band gap.‎

The obtained results can be explained as follows, more the ratio between the acoustic ‎impedance of the two materials constituting the structure, the speed of the waves at the ‎interfaces is modified, which leads to the formation of a forbidden band. Therefore, a good ratio ‎between the acoustic impedance and a good filling factor can lead to the appearance of a band ‎gap in which the propagation of acoustic or elastic waves is not allowed.‎

3.2 Sensing the thickness

In order to use our structure as a thickness sensor. In the present section, we are studying ‎the influence of the thickness of the defect layer on the phononic band properties. Our ‎design [(LiNbO3/SiO2)2/(defect)/(LiNbO3/SiO2)2], is a defective one dimensional phononic ‎structure composed of two different layers of (LiNbO3/SiO2) repeated in N=4 unit cells with ‎equal thickness. The defect material is a layer of LiNbO3 with a thickness of D =1 mm. It is ‎inserted in the middle of the periodic structure to form a mirror around it, as shown in Figure 4.

Figure 4. A schematic diagram of a defect 1D-PnC structure‎

In contrast to the periodic structure, the insertion of a defect layer in a periodic structure ‎induces the breaking of the periodicity of the crystal, which causes a different interaction with ‎the acoustic wave. Therefore, a phenomenon of localization of the acoustic wave inside the band ‎gap appeared. The transmission spectrum of the defected 1D-PnC structure obtained with TMM ‎is shown in Figure 5.

Figure 5. Transmission spectrum of a defected 1D-PnC with geometrical defect

According to the results, the phononic band has expanded and a defect mode with ‎intensity nearly equal to 100% has appeared inside the phononic band gap at the reference frequency f=3.2 MHz. ‎The transmitted peak within the phononic band gap is caused by wave localization within the ‎defect layer. The basic idea in the process of detecting changes in the defect layer parameters is ‎usually the resonant frequency of the defect mode.‎

Based on the resonant frequency of the defect mode of our defected phononic crystal ‎structure, we will study now the influence of the defect thickness on the resonant mode (RM) ‎appeared inside the phononic band gap. ‎

Figure 6(a) shows the effects of the thickness changes on the resonant mode. We can ‎observe that the thickness of the defect layer has a pronounced influence on the resonance ‎frequency of the transmitted peak. Thus, by increasing the thickness D from 0.85 mm to 1.15 ‎mm, the position of resonant mode (f0) can be adjusted inside the band gap. Figure 6(b) shows the ‎relationship between the thickness of the defect layer and the position of the resonant mode ‎‎(RM) inside the band gap. With the increase in thickness, the resonant mode frequency (f0) shifts ‎towards lower frequencies.

The simulated results indicate that the change in the thickness of the defect layer can be ‎used to adjust the position of the defect mode while maintaining a high effective transmission of ‎approximately 100% of all results.

Figure 6. (a) The effect of the defect layer thickness on the transmission spectrum; (b) ‎Relationship between the thickness of the defect layer and the position of resonant mode

4. Conclusions

In this work, an analysis of a perfect 1D-PnCs formed by the stack of N bi-layers of ‎LiNbO3/SiO2 has been studied, using the Transfer Matrix Method. The various calculations show ‎clearly that the physical and geometrical properties; in particular mass density contrast, ‎longitudinal speed in different materials, the number of unit cells and filling fraction ratio have a ‎great influence on the position and on the size of forbidden phononic band gap. This proves the ‎ability to use the defective 1D-PnCs with defect layer in sensors applications. Here, all results are ‎achieved with TMM. This simulation method was widely used to simulate the resonant structure ‎and the one-dimensional phononic crystal sensors.‎

Secondly, one layer of LiNbO3 with thickness D is inserted in the middle of the perfect ‎‎1D-PnC structure in order to form a mirror. The results revealed the generation of a sharp ‎resonant peak inside the phononic band gap for a given thickness of the defect layer. Moreover, ‎the position of the transmitted peak is significantly affected by the changes in defect layer ‎thickness. It can be moved toward lower frequencies with an increase in thickness. Thus, the ‎results prove the capability of such a structure to provide a promising candidate for sensing the ‎thickness of the defect layer. In summary, our design can be used in the domain of ultrasound ‎sensors to measure the mechanical properties of materials. It is especially suitable for piezo-‎electric materials, whose geometric parameters change depending on environmental parameters.‎


This work was supported by the Algerian Ministry of Higher Education and Scientifc Research ‎and The General Direction of Scientific Research and Technological Development (GDSRTD) ‎via funding through the PRFU Project No. A10N01UN280120190001. Acknowledgements ‎belong here.‎


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