In this subsection, we present our proposed periodic TPnC. Our proposed design adopts the presence of Fano resonance modes inside the PnBGs as a liquid sensor for butanol detection. The sensor structure comprises two identical PnC multilayer designs. Each PnC structure is composed of two-unit cells (N = 2). Then, a defect layer filled with butanol liquid is embedded between the two PnCs. Here, each unit cell is designed from lead and epoxy with thicknesses of d ₁ = 1 µm and d ₂ = 1 μm, respectively. Meanwhile, the suggested design could be configured as, [(Lead/Epoxy)² (Defect layer) (Lead/Epoxy)²]. To sum up, Figure 1 shows a three-dimensions schematic diagram of the proposed sensor in which the lattice constant, or the length of each unit cell, is expressed as a = d ₁ + d ₂. Here, the whole structure is sandwiched between two semi-infinite layers of nylon. Then, transducer one couples the acoustic signal to the structure, whereas transducer two detects the output signal at the opposite end. Table 1 provides an overview of the building materials’ acoustic properties (density and speed). The considerable impedance mismatch between liquids and solids used in these alternating layers makes properly adjusting the thicknesses of layers crucial to optimize the structure’s spectral response and practical feasibility. In other words, the acoustic wave is suppressed by the high thickness of the layers that constitute the proposed structure, resulting in the absence of transmission. Lead and epoxy layers must have a thickness of 1 μm, whereas defect layers must have a thickness of 0.5 µm to attain a pass band in the MHz spectral region. Additionally, the large acoustic mismatch between epoxy and lead facilitates the creation of broad PnBG, which disperses the incident acoustic waves at the interface between layers [8–11].

In addition, we can incorporate a quasi-periodic sequence in the 1D TPnC sensor designs, such as the Fibonacci, Thue-Morse, and two-period sequence, which dramatically changes the acoustic properties of the structures. Quasiperiodic arrangements introduce long-range correlations without translational symmetry [17, 61, 62]. Strong resonances arise spontaneously in a quasiperiodic structure because of the special ordering. In contrast, the emergence of resonant modes through periodic structures exhibits an intentional breach of symmetry. The structure is more resilient to small manufacturing flaws or layer defects according to its intrinsic resonance capability, which also enables increased sensitivity to environmental changes. As a result, the quasiperiodic sequence can thereby increase the TPnC structure’s adaptability and resilience for practical sensing applications [17, 47, 61, 63].

The Fibonacci sequence can be used to display the quasi-periodic PnCs structures in this suggestion. By using the recursive rule [   S n = S n − 1 S n − 2 , for n ⩾ 2 ] to arrange the two fundamental parts, A and B, side by side. It is possible to create a Fibonacci structure [47, 62–64] experimentally. S₀ = B and S₁ = A as a starting point. Consequently, the Fibonacci sequence is produced as S 2 = AB ;   S 3 = ABA ;   S 4 = ABAAB , etc .

On the other hand, mirror symmetry in a periodic TPnC structure can significantly influence its acoustic behaviour, especially regarding the localization of phononic modes and the formation of band gaps [17, 61, 64]. Certain vibrational modes can exhibit symmetric or antisymmetric behavior relative to the mirror plane when a TPnC possesses mirror symmetry [17, 62, 64]. For example, antisymmetric waves may encounter destructive interference, resulting in increased band gaps, whereas symmetric modes may constructively interact throughout the mirror plane, producing areas of high transmission [64], [17, 61, 62]. Additionally, the symmetric arrangement can enhance specific acoustic features, even in the presence of minor imperfections or variations in layer thickness. As a result, mirror symmetry can improve the robustness of band gap properties. This robustness is particularly beneficial for sensor applications, where reliable performance relies on stable band gaps and consistent transmission characteristics.

Herein, the periodic TPnC liquid sensor is designed as shown in Figure 1. Acoustic sound waves could be reflected or transmitted inside the multilayer PnC structures. Meanwhile, multiple methods have been established to investigate the transmission of acoustic waves, such as plane wave expansion [65–68], finite difference time domain [69–71], and transfer matrix method (TMM) as well [72–74]. However, due to its simplicity, and accuracy, TMM could be the best choice that can be applied to evaluate the propagation of incident acoustic waves through multilayer structures [11, 75].

Firstly, the mechanical characteristics of the layers, including sound speed and density varies periodically when an acoustic wave encounters our suggested 1D TPC liquid sensor effectively. Additionally, the incident acoustic wave splits into several waves inside the structure. The equation of motion in each layer for an mechanical wave incident typically from left to right on the 1D PC structure [55, 76, 77], as depicted in Figure 1, can be expressed as follows: ∂ σ ∂ x + f = ρ ∂ 2 u ∂ t 2 (1)

Where σ = σ ( x , t), ρ = ρ ( x , t), f = f ( x , t) and u = u ( x , t) denote the stress, material density, external body force, and displacement, respectively. The elastic material’s stress is determined by: σ = E ∂ u ∂ x (2)

Such that, E = E ( x ) is Young’s modulus of the material. By substituting from Equation 1 into Equation 2, we have: E ∂ 2 u ∂ x 2 = ρ ∂ 2 u ∂ t 2 (3)

In this regard, the response of the incident acoustic waves through the designed 1D PnC structure can be expressed in the vicinity of Equation 3 such as, u x , t = A e i K x − ω t (4) where ω is the angular frequency, K is the wavenumber, and (A) is the wave amplitude. Therefore, the wave equation for a gas or liquid can be simply investigated by substituting from Equation 4 into Equation 3 such as, ∂ 2 u ∂ x 2 = 1 C L 2 ∂ 2 u ∂ t 2 (5)

Examine a 1D PC made up of N unit cells, each of which has two layers. The subscript j = 1, two indicates that the layer j in the unit cell has the following constant material properties: d j , ρ j , E j , a = d 1 + d 2   , λ j , μ j are the thickness, density, Young’s modulus, cell length, Lame constant, and shearing modulus. d 1   and d 2   are the thickness of first and second layer, respectively. Each layer’s acoustic wave velocity is C L = E / ρ . The following is an expression for the solution to Equation 5: u x , t = A + j e i K j X + A − j e − i K j X e − i ω t (6)

Where the wavenumber is K j = 2 π f C L , the incident wave frequency is f, and the transmitted and reflected wave amplitudes are denoted by the coefficients A + and A − , respectively.

Equation 7 expresses the formula for the incident wave equation through the gas layer. We are expected to introduce a defect layer within the perfect phononic structure in order to address the interaction between gases and PC. Acoustic waves are widely recognized as one type of pressure fluctuation that can occur in any compressible gas. The equation that describes the propagation of acoustic waves in gases is Equation 7, and it states that sound waves can be transmitted through the gas medium by neutral atoms and molecules [55, 78]. ∂ 2 p ∂ x 2 − ρ β ∂ 2 p ∂ t 2 = 0 ∂ 2 p ∂ x 2 − 1 C 0 2 ∂ 2 p ∂ t 2 = 0 (7)

Where p is the gas’s instantaneous pressure, and C 0 = β ρ is the speed of sound in the gas layer, ρ is its mass density; β is its bulk modulus. Equation 8 provides the solution to Equation 7: p x , t = P + 0 e i K j X + P − 0 e − i K j X e i ω t (8)

Both local pressure and density changes are produced when a sound wave passes through a gas medium. Equation 9 establishes a relationship between the acoustic impedance of gas and the speed of sound within it as follow [55]. Z = ρ × C 0 (9) where C 0 is the speed of sound in gas and ρ is the density of gas. Consequently, when the medium density and sound speed increase, so does the acoustic impedance that is working against the wave’s propagation. At the interfaces between layers, two conditions must be met: the continuation of stress and the continuity of displacement. By changing Equation 6 into Equation 2, we may provide the stress equation as follows: σ x = E i K j A + 0 e i K j X − A − j e − i K j X σ x = i Z j A + 0 e i K j X − A − j e − i K j X where Z j = E j K j refers to the acoustic impedance. Consequently, the two-state vectors representing the entire propagation of acoustic waves at the left and right sides of layer j in the kth unit cell can be written as follows: V j L k = u X R j , σ X R j V j R k = u X L j , σ X L j   (10)

Where the left and right sides of the layer are labeled by the subscripts R and L, respectively. As a result, in the kth unit cell, the left and right state vectors of layer j have the following relationship: V jR k = D j ′ V jL k

Where D j describes the transfer matrix for layer j . After that, we explain the theoretical framework for applying the transfer matrix method (TMM) to investigate stress and displacement in PnC structures begins with defining Young’s modulus for each material used in the PnC. Stress is then related through an acoustic impedance formula. A wave matrix connects the displacement and stress at the boundaries of each layer. We also present the propagation matrix, which describes how acoustic waves travel through a single layer. In the supplementary data, we detail how the transfer matrix for each layer is derived and how it can be utilized to relate stress and displacement at the interfaces of multiple layers in a multilayer PnC sensor. The transfer matrices for each layer are multiplied sequentially to obtain the overall transfer matrix for the entire structure. This approach allows us to calculate displacement and stress throughout the PnC structure based on the acoustic impedances and material properties. Thus, we can derive comprehensive insights into the system’s behavior by calculating the transfer matrix ( D j ) for the layer j as presented in the following equation: D j = cos   K j d j 1 / Z j ⁡ sin   K j d j − Z j ⁡ sin   K j d j cos   K j d j (11)

Each material’s acoustic impedances Z j and E j were used to build the PnC structure and ascertain the behaviour of the D j matrix. Equations 12–14 were utilized to represent the entire transfer matrix components D i j = D i , j given in Equation 11. D j 1 , 1 = D j 2 , 2 = cos   K j d j (12) D j 1 , 2 = 1 / Z j ⁡ sin K j d j (13) D j 2 , 1 = − Z j ⁡ sin   K j d j (14)

Given that d j denotes the layer thickness and Z j = E j K j denotes the acoustic impedance. The following formula represents the relationship between two consecutive state vectors in the kth and (k-1)th unit cells: V 2 R k = D k V 2 R k − 1 (15)

Consequently, D k can be expressed as a transfer matrix that links the next two-unit cells, such that: D k = D 2 D 1 ′

Then, the following formula can be used to determine the transmission coefficient of our PnC structure [77, 79]: U e U 0 = 2 E 0 D 11 D 22 − D 12 D 21 E 0 D 11 − E e D 21 − D 12 − E e D 22

Here, the two semi-infinite solid Young’s moduli to the left and right of the PnC structure are denoted by E 0 and E e , respectively, then the incident and transmitted wave amplitudes are denoted by U 0 and U e , respectively.

Using Bloch-Floquet theorem, the band theory, and the energy spectrum of wave propagation can be calculated and plotted through the infinite 1D multilayered PnC structures [17, 61, 62]. In this case, the state vectors in Equation 10 at the right boundary of each unit cell satisfy the following equation: V 2 R i = V 2 R i − 1 ⁡ exp i K a i = 2 , . . . . . , n + 1 (16)

Given that in every layer and unit cell, K is the wave vector and its equal to kj. Equation 15 and Equation 16 can be compared to obtain the following eigenvalue problem: T i − e i K a I = 0

It can also be written in another format as follows: T i V 2 R i − 1 = λ V 2 R i − 1 (17)

Where, V 2 R i − 1 describes a complex eigenvector and λ = e i K a expresses the complex eigenvalue. Then, Equation 17 can be used to compute and plot the band structure and dispersion relation of the wave propagation through the infinite 1D periodic structures in the irreducible first Brillouin zone. The wave vector K is given as a complex wave number, comprising real and imaginary parts as follows:

K = K real − i   K imaginary The following form for Equation 16 will be used if K = K real and K real > 0: V 2 R i = V 2 R i − 1   e i K real a

The phase difference between the state vectors at the boundaries of each unit cell and every two successive unit cells is equal to e i K real a , and the resulting bands are referred to as pass bands. Equation 16 will be expressed as follows if K = K imaginary and K imaginary < 0. V 2 R i = V 2 R i − 1   e − K imaginary a

The state vectors at the boundaries of each unit cell and every two successive unit cells exhibit no phase difference; rather, their spatial phase experiences an exponential attenuation with a magnitude of K imaginary . Therefore, the resulting bands are referred to as stop bands.

In Figure 2A, the band structure is plotted for one unit cell (n = 1) of the two building blocks of the PnC (Pb and epoxy) with equal thickness ( d 1 = d 2 = 1 μm) as specified in Table 1. Any crystal structure can be identified by its band diagram, which only needs one unit cell to display the properties of wave propagation through the multilayer stack. The band diagram of the proposed design of TPnC structure is plotted between the non-dimensional frequency q (3.142 ≤ q ≤ 4.157) and the non-dimensional wavenumber ζ = K ×a [25–27]. Here, q = q L B = ωa/ c L B , where c L B is an arbitrary acoustic wave speed, taken as the longitudinal wave velocity in epoxy. In Figure 2B, the band diagram is shown for different thicknesses ( d 1 = 1.05 μm and d 2 = 0.95 μm). Both figures illustrate the same acoustic band gap with nearly identical bandwidths, confirming that the inclusion of lead and epoxy layers in the proposed phononic crystal (PnC) design serves as a solid foundation for the suggested tapered phononic crystal (TPnC) structure. Even with variations in material thickness, the PnC structure maintains edge mode stability, promotes strong wave propagation, and shows resilience to local disturbances. The proposed TPnC structure indicates that certain waveguide properties remain unaffected by minor physical disturbances or distortions. The real part of the wavenumbers is represented by the weighted colors of the passbands, which are shown as solid lines in the band diagram. The band gaps, corresponding to the imaginary part of the wavenumber, are illustrated in gray with dotted lines. Both band diagrams reveal that the phononic band gap (PnBG) has a similar bandwidth (∆ω), as seen in Figure 2. The interaction between wide and narrow bands within the same crystal leads to Fano modes, which are enhanced by this wide bandgap. When alcohol is added, its low acoustic properties allow the wide band to interact with the confined evanescent modes in butanol, thereby improving the localization and propagation of Fano modes.

The acoustic properties of butanol liquid, particularly its response to acoustic waves, play an important role in different applications, such as ultrasonic sensing, medical imaging, and industrial processes. Density, viscosity, and compressibility are some of the variables that can be effective on the acoustic properties of butanol, a form of alcohol with four carbon atoms. We first considered the experimental data to highlight the relationship between the temperature and acoustic properties (density and sound velocity) of butanol liquid [80]. Subsequently, we numerically fitted these data to introduce such response in our study. Meanwhile, the experimental data for the mass density of butanol liquid were fitted using the following fitting equation: ρ kg m 3 = α 1 + α 2 × T ° C + α 3 × T ° C 2 where ρ refers to the density, T refers to the temperature; and  α 1 , α 2   and   α 3 are the coefficients of the fitting relationship. The values of the fitting coefficients are α 1 = 335.0969 , α 2 = −3.50009 and   α 3 = 0.00948. Figure 3 illustrates that density increases in parallel with temperature. In this regard, Equation 26 investigates the polynomial fitting of this relationship in the form of a second-order polynomial. The compressibility and density of butanol dictate the speed of sound [80]. In general, for a liquid, its sound speed decreases as its density and compressibility increase. Therefore, the acoustic waves can also be attenuated as they travel through a liquid due to the viscosity of butanol. High viscosity tends to dampen the propagating acoustic waves, which in turns might be effective on their ability to travel over long distances [80]. Furthermore, acoustic impedance, which is the result of multiplying density by sound speed, offers some information about how well acoustic waves may enter or exit the butanol medium [80]. Comprehending these properties is essential to design and improve ultrasonic sensors that use butanol as a coupling or sensing medium. Next, the experimental results for the sound speed of the butanol liquid are fitted with the following equation: v m s = β 1 + β 2 × T ° C where v is the speed of sound, T refers to the temperature, and α  and  β are the coefficients of the fitting relationship. The values of the fitting coefficients are β 1 = 299.87912 and β 2 = −0.4978. Figure 3 shows that when temperature rises, sound velocity decreases. This behaviour has a linear fit based on the previous equation. Usually, tests and experiments are conducted to describe the acoustic characteristics of certain liquids, such as butanol. In these experiments, the frequency-dependent behaviour of acoustic waves in the liquid and optical approach is analysed by using some methods, such as ultrasonic spectroscopy [80]. The data obtained could contribute to the development of some technologies, such as PnC sensors, that rely on butanol’s acoustic properties by allowing the construction of models and comprehending the interactions of acoustic waves with the material.

We adopt a novel strategy to demonstrate the topological state and stability of our proposed PnC structure: a methodical investigation of the Fano resonance peak in response to changes in the butanol layer’s thickness. Given that the thickness of the defect layer is equivalent to 0.5 μm, the Fano resonance peak appears at a normalizing frequency of 0.0497, as shown in Figure 4A. We controlled the changes in the butanol layer’s thickness and tracked the associated modifications in the Fano resonance peak to assess the robustness of the proposed structure, as illustrated in Figures 4A, C. When the thickness decreases to 0.4 μm, the peak moves slightly to a normalized frequency of 0.0509. The noteworthy aspect despite this small advancement is the stability exhibited by the PnC structure under this perturbation. Further varying butanol layer thickness to 0.45, 0.55, and 0.6 µm gives rise to the shifting in Fano resonance peak downwards lower normalized frequencies of 0.0503, 0.0492, and 0.0486, respectively, as shown in 4a. Despite this change in normalized frequency, we note that the proposed PnC structure retains its stability and exhibits a consistent response at the Fano resonance peak given the negligible shift in normalization frequency. We thus confirm the proposed PnC structure’s validity and topological integrity by showing that it is resistant to changes in the thickness of the butanol layer, as demonstrated by the steady and minute movements of the Fano resonance peak under various conditions.

Here, we examined how the thicknesses of the defect layers affect our TPnC structure’s resonance frequency as shown in Figure 4A. Figure 4C illustrates that as the defect layer thickness changes from 0.4 to 0.6 µm, the position of the resonant peak remains almost at 50 MHz, providing strong evidence for the proposed topological structure. Therefore, we believe that this structure displays the topological edge’s condition and is robust against external distortions. Unique electrical or vibrational states that form at the limits or edges of a topologically nontrivial material or structure are referred to as topological edge states (TES). Regarding periodic structures, such as PnC, these edge states exhibit robustness and stability against certain disturbances or changes in the material. These states are topological because they are resistant to local perturbations and can persist even when defects are present.

Additionally, we have introduced the effect of lead and epoxy layers’ thicknesses on the transmission spectrum to demonstrate the topological state of our proposed PnC structure. When the lead–epoxy layer is 1 μm thick, a Fano resonance peak appears at a normalized frequency of 0.0497. The Fano resonance peak then shifts following carefully controlled variations in lead and epoxy layers’ thicknesses, as seen in Figure 4B. The peak of the lead and epoxy layers’ together shifts slightly to a normalized frequency of 0.05 when the thickness decreases to 0.9 µm. Even with this slight displacement, the stability of the PnC structure under this disturbance is remarkable. Figure 4B shows that further varying on the thicknesses of the lead and epoxy layers to 0.95, 1.05, and 1.1 µm results in the movements of the Fano resonance peaks to new normalized frequencies of 0.0499, 0.0496, and 0.0494, respectively. Despite the slight shift in normalizing frequency, the proposed PnC structure remains stable and exhibits a consistent response at the Fano resonance peak. Therefore, we established the validity of our proposed TPnC structure and its topological integrity by demonstrating its flexibility to variations in lead–epoxy layers’ thicknesses, as investigated by the consistent and small movements of the Fano resonance peak across different conditions. Accordingly, we examined how the thicknesses of the lead and epoxy layers affect the resonance frequency of the TPnC structure, as shown in Figure 4D. The proposed topological structure is strongly supported by the fact that the resonance frequency stays constant at 50 MHz when the defect layer thickness increases from 0.9 to 1.1 μm. This structure provides a TES that is robust against external deformations.

Finally, we thoroughly analysed the effect of symmetric and asymmetric structures on the transmission spectrum, as illustrated in Figure 5, to demonstrate the topological state of the proposed PnC structure. Firstly, the arrangement and regularity of the layers or unit cells that make up a crystal are classified as symmetric and asymmetric PnC structures. A symmetric PnC structure features materials and layers that are mirrored across a central point or plane, creating a configuration on one side that is essentially a reflection of the other. In such structures, the layers follow a consistent, repeating pattern, and physical properties such as density, mass, and stiffness are evenly distributed. For instance, a symmetric structure with a defect layer (D) between two-layer B units could be represented as {ABDBA}, when layers A and B are used. In contrast, an asymmetric PnC structure does not exhibit this mirroring. There is no central point or plane where the layers and materials are symmetrical. Instead, the layers in an asymmetric structure are arranged in an uneven or non-repetitive manner. As a result, the physical properties may vary, leading to differences in wave propagation characteristics. An example of an asymmetric structure, lacking a point of symmetry, could be organized as {ABDAB} using the same layers A, B, and the defect layer (D).

Here, we investigated that the transmission spectra of symmetric and asymmetric structures provide Fano resonance peak located at the same normalized frequency of 0.0985 as shown in Figures 5A, B, respectively. This finding demonstrate that the structure is flexible and supports the topological integrity of our PnC sensor.

In our proposed TPnC structure, both narrow discrete resonant mode and broader, continuous background mode interfere to produce Fano resonance. We incorporate localized resonators, like narrow cavities or defects, into a periodic or quasiperiodic array while designing the PnC [11, 81, 82]. At specific frequencies, these resonators produce a limited number of resonant modes that interact with the wider transmission spectrum of the surrounding PnC layers. The distinctive asymmetric line shape of Fano resonance arises from the interference between the discrete resonant mode (originating from the narrow resonator) and the background mode (from the broader PnC structure) that a wave encounters [83–85]. Because of the constructive and destructive interference patterns that are produced in the transmission spectrum by the interaction between the discrete and continuous modes, this line shape appears. Sharp peaks and dips are thus seen in the response, which might increase sensitivity to slight changes in frequency [81, 82, 85, 86].

As described in this section, we delved into the effect of the elevated butanol temperature on the properties of Fano resonance peaks. Figure 6 provides a comprehensive overview of the correlation between different temperatures (170°C–270 °C) and corresponding variations in the sound speed and density of butanol liquid. The subsequent analysis, depicted in Figure 6, highlights the dynamic shift in the positions of Fano resonance peaks at various temperatures. The frequency of the Fano resonance peak exhibits a discernible reduction that ranges from 48.706 MHz to 36.946 MHz as butanol liquid temperatures increase. This behaviour is due to the strong dependence of the density and sound speed of butanol on temperature, as depicted in Figure 3. Concurrently, the position of the Fano resonance peaks varies. Meanwhile, a gradual increase in temperature from 190 °C to 270 °C corresponds to an observed enhancement in sensitivity from 102.9 KHz to 117.6 KHz. This insight underscores the intricate interplay between the changes in temperature and acoustic properties of butanol liquid, offering valuable implications for optimizing periodic PnC structures in temperature-sensitive applications.

This section introduces the performance of our sensing tool by several related parameters, such as the FOM, QF, and S. These parameters are believed to have a potential importance for characterizing sensor’s performance. Specifically, defect mode characteristics serve as the primary basis for their values. The change in the position of the Fano resonance mode as a function of temperature variation is known as sensor sensitivity [17, 53, 87]. Its values can therefore be established by using the following relationship: S = ∆ f ∆ T where ∆ T indicates the change in butanol temperature, and ∆ f indicates the resonance peak or transmission frequency at each temperature. Then, we computed some additional performance metrics, including QF. This metric is highly affected by the location of the Fano resonance peak [17, 87, 88] such that: Q = f r f H B W where f H B W refers to the peak’s half bandwidth frequency, and f r represents the Fano resonance peak frequency. In this case, sharp Fano resonance peaks are indicated by a high QF, increasing frequency resolution [17, 87]. Additionally, we derive a parameter called FOM, which forecasts the capability of the sensor to measure any variation in resonant frequency [17, 87, 88]. The following formula can be used to determine FOM: FOM = S f H B W

When the half bandwidth frequency decreases, FOM increases. Lastly, we consider the damping rate, another relevant performance parameter. The sharpness of transmitted Fano resonance peaks is determined by damping, which describes how the acoustic waves in the designed PnC decay after a disturbance across the structure [53, 87, 88]. The following formula can be used to determine the damping rate: ζ = 1 2 * Q (18)

The effect of varying temperatures on the Fano resonance peaks of the topological butanol liquid sensor, which eventually influences sensitivity, is depicted in Figure 7. The Fano resonance modes shift linearly towards high frequencies when the butanol temperature drops. This response also increases the sensor’s sensitivity. Here, we can confirm that within the temperature range of 170°C–270°C, our PnC sensor offers excellent sensitivity and performance. Figure 7 shows that as the butanol temperature increases from 170 °C to 190 °C, our proposed design offers a relative high sensitivity of 102,900 Hz/°C. As butanol temperature rises, sensitivity increases till reaches 119,233.33 Hz/°C at 230 °C. Meanwhile, the investigated sensitivity of our periodic TPnC design is promising compared with those of other designs in some previous work [22, 89–93]. In particular, previously reported sensors based on 1D PnC structures provided a maximum value of 4,400 Hz/°C, whereas our proposed structure provides a maximum value of 119,233.33 Hz/°C.

Figure 8 presents the effect of the QF of our proposed sensor and the damping rate of acoustic waves. It shows that the minimum value of the damping rate is observed at the maximum value of QF. Specifically, Equation 18 establishes an inverse relationship between QF and ζ. Interestingly, high QF values are obtained when FWHM values are low [11, 17, 91, 94]. Consequently, low damping rates are obtained because FWHM values are low. Low FWHM values indicate the limited ability of the proposed structure to absorb incident acoustic waves [11, 17, 91, 94]. The Fano resonance peak sharpens as the damping rate decreases. As shown in Figure 8, the resonance peak has its highest sharpness at 250 °C, leading to a QF of 386.85. Subsequently, at this temperature, we obtain the lowest damping rate of 0.00129. However, the peak has the maximum damping rate of 0.0021 and the lowest sharpness (Q) of 240.04 when the temperature is equivalent to 190 °C. Nonetheless, the QF, which ranges from 240.04 to 386.85, is highly acceptable at all butanol temperatures. This result suggests that all Fano resonance peaks are sharp, thus increasing the frequency resolution of the proposed sensor. The numerical results provided in Figure 8 depict the unique characteristics of our sensor when compared with other sensors and their equivalents with 1D PnC designs [11, 17, 91, 94]. Comparison with those of earlier 1D PnC sensors shows that the QF of our sensor may increase to 386.85.

Figure 9 shows the response of the butanol liquid sensor’s detection limit and FOM to temperature variation. Figure 9 illustrates that FOM values decrease when butanol temperature increases. We discovered that as the temperature rises from 190 °C to 250 °C, FOM changes from 0.5295/°C to 1.173/°C. Furthermore, FOM exhibits a reaction that is comparable to that shown by sensitivity, particularly for this parameter. This reaction is also known as the reduced sensitivity. Additionally, the link between various temperatures and the detection limit of the butanol liquid sensor is depicted in Figure 9. The lowest temperature or analyte concentration in a sample that can be identified with a certain probability is known as the limit of detection [17, 22, 87, 91]. The values of the detection limit decrease from 0.0944 to 0.043 when the temperature rises from 190 °C to 250 °C. These results are acceptable, distinguishing our sensor from earlier liquid sensors of similar design [17, 22, 87, 91].