Investigation of an annular photonic crystal sensor for real-time monitoring of calcium carbonate scales in water distribution systems

Abstract

This research exhibits a new configuration of photonic crystals known as annular photonic crystals (APCs) for real-time detection of calcium carbonate (CaCO₃) scale in the water pipeline. The proposed sensor features a circular arrangement of porous silicon materials with varying levels of porosity. A central defect layer is incorporated into the design to capture the target analyte, allowing it to detect changes in the refractive index caused by scale formation. To analyze the reflectance spectrum of this structure, a modified transfer matrix method is utilized. An extensive optimization process is conducted based on the characteristics of the defect mode, focusing on various geometric parameters, including layer thickness, porosity levels, core circle radius, and structural periodicity, to achieve optimal sensor performance. The simulation outcomes revealed that at the optimized structure parameters, the sensor offers a remarkable QF, sensitivity, and FoM of 1215, 176.85 nm/RIU, and 350.5 1/RIU, respectively. Moreover, the proposed structure is simple, cost-effective, and compact, which makes it an ideal candidate for the detection of calcium carbonate scales formed in pipes and devices in water supply networks.

1. Introduction

The limited solubility of numerous inorganic salts like calcium carbonate, calcium sulfate, and calcium fluoride is the main cause of the formation of scale in water pipes, which leads to several problems in using the geothermal water for bath, kitchen, and laundry [1,2]. In addition to this, variations in physical parameters like temperature and pressure also significantly affect the steadiness, which causes scales to form on the walls of equipment like flow pipes, wells, turbines, and valves [3,4]. Over time, the flow rate of water in pipelines decreases due to scale formation. This issue leads to significant expenses for maintenance, personnel, and cleaning the accumulated scale in the machinery. To tackle this problem, researchers have proposed various methods, including the use of organic phosphorus acid, Ethylenediaminetetraacetic acid, pH adjustments, and gamma ray or ultrasonic irradiation [[5], [6], [7]]. However, it is crucial to evaluate the effectiveness of these methods for preventing scale development, especially given the limited research data available. Direct visual inspections of the internal pipe surfaces may provide insights into scale formation, but this approach is time-consuming and labor-intensive, making it unsuitable for real-time monitoring. To address this challenge, we propose utilizing photonics technology for real-time scale monitoring within water pipelines. Over the past two decades, several photonic structures have been successfully developed for various sensing applications, with photonic crystal structures standing out due to their unique properties [[8], [9], [10], [11]].

Photonic structures are essential for developing sensing applications. Various types, including photonic crystal structures (PhCs), optical fibers, photonic crystal fibers (PCFs), and plasmonic structures, have been extensively researched for sensing both liquid and gaseous analytes [12,13]. Photonic crystals (PhCs) represent a unique category of optical structures characterized by a periodic variation in the refractive index of their constituent materials. Owing to this periodicity, a specific range of wavelengths are forbidden from transmitting in such structures, which is termed as photonic band gap (PBG) [14,15]. The PBG property leads to realization of high degree of reflectivity with minimal loss, which makes PhC an ideal candidate to perceive numerous sensing and filtering applications. The periodicity can be created along one, two and three dimensions, which gives rise to 1D, 2D, and 3D PhCs [16,17]. Over the past two decades, 1D PhC has garnered significant interest from researchers due to its exceptional capacity to regulate the passage of electromagnetic waves within the structure. Notably, if a defect layer is included in the PhC design, its periodicity is interrupted, which leads to the creation of strongly localized allowed states, known as defect modes, within the PBG [18]. The defect mode primarily corresponds to a specific wavelength that can propagate within the structure. Both the wavelength and intensity of this defect mode can be controlled by varying the thickness and refractive index of the defect layer. Additionally, the characteristics of the defect mode are highly sensitive to even slight changes in the refractive index of the defect layer. By appropriately tuning these characteristics, defect-based photonic crystals (PhCs) can be employed in various critical applications across biomedical, agricultural, food safety, and defense sectors [19,20].

With the remarkable advancement in nanotechnology, it is now possible to practically realize many geometrical configurations of PhCs by modifying the nature of arrangement of materials inside the structure. In the recent decade, researchers can draw a novel configuration of 1D PhCs known as annular photonic crystals (APCs), where the successive layers of different permittivity are arranged in a cylindrical ring fashion [21]. A team led by M.A. Kaliteevkki studied the flow of electromagnetic waves in an APC for the first time in 1999, where the authors reported the fabrication procedures of APC, and investigated its transmittance and reflectance characteristics by employing a modified TMM [22]. M.M. Abadlla et al. designed an 1D APC comprising of alternate materials of Si and SiO₂ for temperature sensing application. Using modified TMM, the authors analyzed the shift in the spectra location within the band gap by changing the temperature and obtained a sensitivity of 0.033 nm/^°C [23]. A.A. Ameen et al. proposed an APC radiating a high inactivation UV dose of 99 %, which can help in air purification and restrict the spread of airborne diseases [24]. Srivastava et al. demonstrated a ternary APC comprising of semiconductor, superconductor, and air, where the authors studied its reflectance spectrum for temperature sensing applications [25]. Later in 2016, the same team came forward with a double negative metamaterial-based APC, where the authors investigated the reflectance spectrum of the structure by varying the azimuthal mode number and starting radius of the APC [26]. A.A. Ameen and his team reported an APC designed with alternating materials of Al₂O₃ and SiO₂ for application as a useful mask to protect from numerous deadly diseases. They optimized various geometrical parameters of the structure to attain a high reflectivity in the UV band in the range from 207 nm to 230 nm [27]. A.A. Ameen et al. proposed a 1D defected APC-based pressure sensor, where the defect layer is realized with polymer materials like polystyrene (PS) and polymethyl methacrylate (PMMA). They optimized the structure parameter by studying the reflectance characteristic and obtained a maximum pressure sensitivity of 51.29 nm/GPa [28]. Recently, J. Sui et al. reported a non-reciprocity sensor designed with multilayers of dielectric materials to achieve multi-scale sensing of bio-analytes like cancer cells and blood glucose concentrations [29]. JT Zhang et al. demonstrated a magnetized Plasma-based cylindrical PhC for the detection of multiphysics quantity [30,31]. Applying the transfer matrix method, the authors studied the transmittance characteristics, and analyzed the shift in the defect mode with respect to different multiphysics quantity. W. Li et al. reported optical grating and metamaterial based tunable absorber devices for refractive index-based sensing applications [32,33].

This paper presents several innovative concepts related to the sensor's configuration, computational approach, and target analyte. To the best of the authors' knowledge, we are introducing a novel APC configuration utilizing porous silicon materials specifically for detecting CaCO₃ scale in water pipelines for the first time. The modified transfer matrix method is employed to investigate the reflectance characteristics of the sensor. All geometric parameters of the designed sensor are carefully optimized to ensure high performance. Notably, the sensor is easy to fabricate, compact, highly stable, non-intrusive, and label-free, making it suitable for installation within water pipelines to gather real-time data on CaCO₃ scale concentration.

2. Proposed structure and theoretical analysis

The schematic of a 2D cross-sectional view of the projected annular photonic crystal (APC) is demonstrated in Fig. 1. The structural configuration of the APC consists of arrangement of materials of different refractive indices in a circular ring fashion. A TE polarized light incident normally onto the proposed APC structure. The proposed design starts with a central circular ring having a radius of ${\rho }_0$. Further, the design of APC is comprised of periodic arrangement of bi-layer unit cell of period N, followed by a defect layer, and then the same periodic arrangement of bi-layer unit cell of period N. So, the entire structure configuration is ${\left[laye{r}_{a}laye{r}_b\right]}^{N}Laye{r}_{\left(defect\right)}{\left[laye{r}_{a}laye{r}_b\right]}^N$. The value of N is considered as 10. The layers (a and b) are designed with porous silicon (PSi) material, but with different porosity. The first layer (a) is realized with PSi having a porosity of 0.7, whereas the second layer (b) is designed with PSi having a porosity of 0.4. The thickness of layer a, layer b, and the defect layer is denoted as $d_1$, $d_2$, and $d_D$ respectively, whose values are taken as 55 nm, 70 nm, and 120 nm, respectively. The main purpose of the defect layer is to realize a defect mode in the transmittance spectrum. The radius of layer a, layer b, and the defect layer is denoted as ${\rho }_1$, ${\rho }_2$, and ${\rho }_D$ respectively. The whole structure is immersed in air of refractive index n₀. The defect layer is filled with water samples containing different concentrations of CaCO₃.

Fig. 1.

Schematic of the proposed annular photonic crystal.

With the advancement in nanofabrication technology, the proposed APC structure can be definitely fabricated. The simple fabrication techniques like atomic layer deposition and sacrificial etching method can be employed to fabricate the designed APC structure [34]. Additionally, these techniques achieve atomic level precision, low cost, and shows high stability. Further, the different layers of the proposed APC are designed using porous Si materials, which is cost-effective material. Hence, the cost of fabricating the entire structure is less. Its also worth noting that, from the fabrication point of view, APCs are capable of easy measuring of the concentrations of different liquids compared with the planar ones. In particular, the fabrication process of the 1D APCs is more realistic than the case of the 1D planar PCs [24,25].

The optical properties of APC are investigated through the analysis of electric field and magnetic field distribution in the structure by employing transfer matrix method (TMM) in the cylindrical coordinate system. It is considered that the cylindrical waves appeared from the core and continued to move away towards the upper layers in the APC. Here, the Helmholtz wave equations can be employed to numerically describe the electric field mode (Ez, Hϕ, H $\gamma$), which is stated as below [35,36],

$$\frac{1}{\sigma }\frac{\partial }{\partial \sigma }\left(\rho \frac{\partial E_z}{\partial \rho }\right)+\frac{1}{{\sigma }^2}\frac{{\partial }^2{E}_z}{\partial {\varnothing }^2}=-{\epsilon }_0{\mu }_0ϵ\mu w^2{E}_z$$ (1)

By expressing $E_z$ as $E_z=F\left(r\right)\theta (\varnothing )$, and modifying the above equation, equation (1) can be written as,

$$E_z\left(\sigma ,\varnothing ,z\right)=\left[X{W}_v\left(\beta \sigma \right)+Y{Z}_v\left(\beta \sigma \right)\right]e^{iv\varnothing }$$ (2)

Where, $W_v$ and $Z_v$ express the Bessel's function of the 1st and 2nd order respectively. $\beta$ denotes the propagation constant of the EM wave, which is stated as ${\beta }^2={\epsilon }_0{\mu }_0ϵ\mu w^2={k_0}^2{n}^2$. The tangential component is expressed as follows [37],

$$H_{\varnothing }=\frac{i}{\omega \mu }\frac{\partial E_z}{\partial \sigma }=\frac{i}{\omega \mu }\left[X{W}_v^{\prime }\left(\beta \sigma \right)+Y{Z}_v^{\prime }\left(\beta \sigma \right)\right]e^{iv\varnothing }$$ (3)

Here, we study only the fundamental mode to simplify the field equations. The simplified equations are written as,

$$E_z=\left[X{W}_0\left(\beta \sigma \right)+Y{Z}_0\left(\beta \sigma \right)\right]$$ (4)

$$H_{\varnothing }=-\frac{iq}{\omega \mu }\left[X{W}_1\left(\beta \sigma \right)+Y{Z}_1\left(\beta \sigma \right)\right]=-ip=\left[X{W}_1\left(\beta \sigma \right)+Y{Z}_1\left(\beta \sigma \right)\right]$$ (5)

We can express the field equations in the form of a matrix, which can be expressed as [38]:

$$\left(\begin{array}{c}{E}_z\\ H_{\varnothing }\end{array}\right)=\left(\begin{array}{cc}{W}_0\left(\beta \sigma \right)& Z_0\left(\beta \sigma \right)\\ -ip{W}_1\left(\beta \sigma \right)& -ip{Z}_1\left(\beta \sigma \right)\end{array}\right)\left(\begin{array}{c}X\\ Y\end{array}\right)$$ (6)

Where $p=\raisebox{1ex}{$q$}\!\left/ \!\raisebox{-1ex}{$\omega \mu $}\right.\text{.}$ The terms X and Y are related to the amplitude factor, which can be represented as,

$$\left(\begin{array}{c}X\\ Y\end{array}\right)={\left(\begin{array}{cc}{W}_0\left(\beta \sigma \right)& Z_0\left(\beta \sigma \right)\\ -ip{W}_1\left(\beta \sigma \right)& -ip{Z}_1\left(\beta \sigma \right)\end{array}\right)}^{-1}\left(\begin{array}{c}{E}_z\\ H_{\varnothing }\end{array}\right)$$ (7)

By substituting an equivalent field at $\sigma ={\sigma }_0$, we can write,

$${\left(\begin{array}{c}{E}_0\\ H_0\end{array}\right)}_{\sigma ={\sigma }_0}={\left(\begin{array}{c}{E}_1\\ H_1\end{array}\right)}_{\sigma ={\sigma }_0}=\left(\begin{array}{cc}{W}_0\left({\beta }_1{\sigma }_0\right)& Z_0\left({\beta }_1{\sigma }_0\right)\\ -i{p}_1{W}_1\left({\beta }_1{\sigma }_0\right)& -i{p_{1}Z}_1\left({\beta }_1{\sigma }_0\right)\end{array}\right)\left(\begin{array}{c}{X}_1\\ Y_1\end{array}\right)$$ (8)

Within a specific layer, the amplitude value is constant, so we can write,

$$\left(\begin{array}{c}{X}_1\\ Y_1\end{array}\right)={\left(\begin{array}{cc}{W}_0\left({\beta }_1{\sigma }_1\right)& Z_0\left({\beta }_1{\sigma }_1\right)\\ -i{p}_1{W}_1\left({\beta }_1{\sigma }_1\right)& -i{p_{1}Z}_1\left({\beta }_1{\sigma }_1\right)\end{array}\right)}^{-1}{\left(\begin{array}{c}{E}_2\\ H_2\end{array}\right)}_{\sigma ={\sigma }_1}$$ (9)

By substituting eq. (9) in eq. (8), we can write,

$${\left(\begin{array}{c}{E}_0\\ H_0\end{array}\right)}_{\sigma ={\sigma }_0}=\left(\begin{array}{cc}{W}_0\left({\beta }_1{\sigma }_0\right)& Z_0\left({\beta }_1{\sigma }_0\right)\\ -i{p}_1{W}_1\left({\beta }_1{\sigma }_0\right)& -i{p_{1}Z}_1\left({\beta }_1{\sigma }_0\right)\end{array}\right){\left(\begin{array}{cc}{W}_0\left({\beta }_1{\sigma }_1\right)& Z_0\left({\beta }_1{\sigma }_1\right)\\ -i{p}_1{W}_1\left({\beta }_1{\sigma }_1\right)& -i{p_{1}Z}_1\left({\beta }_1{\sigma }_1\right)\end{array}\right)}^{-1}{\left(\begin{array}{c}{E}_2\\ H_2\end{array}\right)}_{\sigma ={\sigma }_1}=M_1{\left(\begin{array}{c}{E}_2\\ H_2\end{array}\right)}_{\sigma ={\sigma }_1}$$ (10)

Where, $M_1$ is the transfer matrix. Upon simplifying $M_1$, we can write the subsequent equations,

$$M_{11}=-\frac{\pi q_1}{2}\left[W_0\left({\beta }_1{\sigma }_0\right)Z_1\left({\beta }_1{\sigma }_1\right)-W_1\left({\beta }_1{\sigma }_1\right)Z_0\left({\beta }_1{\sigma }_0\right)\right]$$ (11a)

$$M_{12}=\frac{i\pi q_1}{2p_1}\left[W_0\left({\beta }_1{\sigma }_0\right)Z_0\left({\beta }_1{\sigma }_1\right)-W_0\left({\beta }_1{\sigma }_1\right)Z_0\left({\beta }_1{\sigma }_0\right)\right]$$ (11b)

$$M_{21}=\frac{i\pi p_1{q}_1}{2}\left[W_1\left({\beta }_1{\sigma }_0\right)Z_1\left({\beta }_1{\sigma }_1\right)-W_1\left({\beta }_1{\sigma }_1\right)Z_1\left({\beta }_1{\sigma }_0\right)\right]$$ (11c)

$$M_{22}=\frac{\pi q_1}{2}\left[W_1\left({\beta }_1{\sigma }_0\right)Z_0\left({\beta }_1{\sigma }_1\right)-W_0\left({\beta }_1{\sigma }_1\right)Z_1\left({\beta }_1{\sigma }_0\right)\right]$$ (11d)

By carrying out similar analysis for the rest of the layers, the field at the 1st layer and the last layer (S^th layer) can be linked as,

$${\left(\begin{array}{c}{E}_0\\ H_0\end{array}\right)}_{\sigma ={\sigma }_0}=Z_1{Z}_2{Z}_3....Q_S{\left(\begin{array}{c}{E}_f\\ H_f\end{array}\right)}_{\sigma ={\sigma }_S}=N{\left(\begin{array}{c}{E}_f\\ H_f\end{array}\right)}_{\sigma ={\sigma }_S}$$ (12)

Where, the term N indicates the transfer matrix of the entire structure, which can be acquired by multiplying the transfer matrix equations of each layer.

2.1. Refractive index of constituent materials

We consider Psi material of different levels of porosity to design the proposed APC. Porous materials are preferred in the design of photonic sensors because these materials contain numerous pores, which makes it easy and convenient to hold the liquid analytes inside the pores. This leads to a higher interaction of analyte with the light travelling inside the material, hence it gives improved sensing performance. Porosity level (% of porosity) must be chosen wisely because a high porosity level may decrease the strength of the material. The refractive index of PSi material ($n_{PSi}$) can be evaluated by using Bruggeman's effective medium approximation, which is given as [37],

$$n_{PSi}=0.5\sqrt{\psi +\sqrt{{\psi }^2+8{\left(n_{Si}\right)}^2{\left(n_{analyte}\right)}^2}},\psi$$ (13)

$$=3P\left(n_{analyte}^2-n_{Si}^2\right)+\left({2n}_{Si}^2-n_{analyte}^2\right)$$

Where, $n_{Si}$ represents the refractive index of silicon and $n_{analyte}$ denotes the refractive index of the analyte i.e. various concentrations of CaCO₃ in water. The numerical expression of $n_{Si}$ is given as [38],

$$n_{Si}\left(\lambda \right)=\sqrt{11.45248+\frac{1.2372}{0.90584{\lambda }^2}}$$ (14)

Such that, λ defines the wavelength of the incident electromagnetic waves in nm scale from 210 nm to 830 nm [38]. In fact, it's also worth noting that, the dependence of PSi of different porosity in PhCs could provide good matching through the fabrication process besides its high surface area and diversity of pores [37] unlike the dependence on some different materials which in some times may give rise to some mismatching. Moreover, PSi materials are mechanically stable for a porosity level smaller than 80 %, which, in turn, can be effective in controlling the effective refractive index based on the porosity level [37]. In this research, we attempt to measure the concentrations of calcium carbonate (CaCO₃) pigment in water, which is the measure cause of formation of scale in the pipeline. As refractive index is considered as the most important biophysical parameter, we used the refractive index data of different concentrations of CaCO₃ for accurate measurement. The experimentally obtained refractive index of different concentrations of CaCO₃ is taken from Ref. [39] and shown graphically in Fig. 2.

Fig. 2.

Refractive index of different concentrations of CaCO₃.

2.2. Performance parameters

It is essential to assess various sensor performance metrics to determine the efficiency of the sensor. Sensitivity ($S_{\lambda }$) is considered as the most vital parameter to evaluate the sensor's performance, which can be computed as the ratio of the change in wavelength (${\Delta \lambda }_{res}$) to the change in the refractive index ($\Delta n$) of analyte. Quality factor (QF) and figure of merit (FoM) indicate the ability of the sensor to accurately detect the resonant mode wavelength. Sensitivity, QF and FoM can be numerically stated mathematically as follows [40,41],

$$S_{\lambda }=\frac{{\Delta \lambda }_{res}}{\Delta n}$$ (15)

$$QF=\frac{{\Delta \lambda }_{res}}{FWHM}$$ (16)

$$FoM=\frac{S_{\lambda }}{FWHM}$$ (17)

3. Results and discussion

The proposed structure is realized with alternatively organized circular layers of porous silicon (PSi) of different porosity level. A defect layer is inserted at the center, which serves as the channel to fill the target analyte in the structure. The layers have a repeatability of factor ‘N’ both before and after the defect layer. The configuration of the whole structure is ${\left[laye{r}_{a}laye{r}_b\right]}^{N}Laye{r}_{\left(defect\right)}{\left[laye{r}_{a}laye{r}_b\right]}^N$. The initial geometric parameters are set with a porosity of 0.4 for layer A and 0.7 for layer B, with N defined as 10. The thicknesses are initially considered as follows: layer A at $d_1$ = 55 nm, layer B at $d_2$ = 70 nm, and the defect layer at $d_D$ = 120 nm. The thickness of the starting medium (${\rho }_0$) is established at 500 nm, which is significantly greater than that of the other layers, indicating that the proposed APC functions similarly to a planar photonic crystal. To simplify the computational process, light is assumed to be incident normally onto the structure, with the azimuthal number (m) set to zero. The modified transfer matrix method (TMM) is employed to optimize various geometric parameters by analyzing the reflectance spectrum of the structure. A key focus of this research is to investigate the shift in defect mode wavelength in relation to changes in various geometric parameters. All structural parameters are optimized based on the sensor's performance metrics, including sensitivity, quality factor (QF), and figure of merit (FoM), evaluated across CaCO₃ concentrations ranging from 0 g/mL to 0.5 g/mL.

3.1. The effect of changing the defect layer thickness (${\mathit{d}}_{\mathit{D}}$)

Initially, we study the effect of variation in the thickness of the defect layer ($d_D$) at a constant $d_1=55nm\text{,}$ $d_2=70nm\text{,}$ and N = 10. Fig. 3(a) demonstrates the reflectance of the proposed APC structure by varying $d_D$ in the range from 100 nm to 150 nm. It is noticed that increasing in the value of $d_D$ leads to a significant shift in the characteristics of the PBG and the defect mode wavelength. Although, there is a marginally shift in the wavelength of PBG towards the higher wavelengths, but the defect mode wavelength experiences a substantial red shift (i.e. moves to higher wavelength value) with increase in the value of $d_D$. From Fig. 3(b), it can be clearly seen that, with an increase in the value of $d_D$ from 100 nm to 150 nm, the defect peak wavelength is shifted from 600 nm to 630 nm. The nature of shift in the wavelength is almost linear, which deems that the proposed sensor has good stability. Fig. 3(c), (d), 3(e), and 3(f) represent the analysis of sensing performance parameters like FWHM, QF, sensitivity, and FoM respectively. The FWHM of the defect mode remains almost constant at 0.51 for variation in $d_D$ from 100 nm to 120 nm, and FWHM gradually increases for higher values of $d_D$. For a variation in the value of $d_D$ from 100 nm to 120 nm, the QF remains almost same at 1200, but for $d_D>120nm$, QF decreases rapidly. Similarly, FoM maintains at constant value of approximately 345 1/RIU up to $d_D=120nm$, but for $d_D>120nm$, FoM decreases swiftly. We analyzed the sensitivity with respect to different thicknesses of the defect layer. It is found that the sensitivity increases with an increase in the value of $d_D$. Notably, the sensitivity is 174 nm/RIU at $d_D=120nm$, after which sensitivity comparatively slowly for higher values of $d_D$. So, considering the above-mentioned analysis, and to reduce the cost of fabrication and to make the sensor more compact, we chose the optimized value of $d_D$ as 120 nm.

Fig. 3.

(a) Reflectance spectrum, (b) Analysis of the peak wavelength of the defect mode, (c) Analysis of the FWHM, (d) Analysis of the QF, (e) Sensitivity and (f) FoM for different values of $d_D$.

3.2. The effect of changing the thickness of the first layer (${\mathit{d}}_{\mathbf{1}}$)

Afterwards, we move on to optimize the thickness of the first layer ($d_1$) by fixing the other parameters at $d_2=70nm\text{,}$ $d_D=120nm$, and N = 10. We studied the reflectance of the proposed APC sensor by varying the thickness of layer a ($d_1$) in the range from 52 nm to 57 nm, which is shown in Fig. 4(a). The appearance of the defect mode can be clearly seen within the PBG in the reflectance spectrum. It is observed that the PBG is shifted to larger wavelength range for a rise in the value of $d_1$. Also, the defect mode wavelength undergoes a significant red shift for higher values of $d_1$. Fig. 4(b) depicts that as we increase the value of $d_1$ from 52 nm to 57 nm, the wavelength of the defect mode increases linearly from 598 nm to 622 nm. Fig. 4(c)–4(f) illustrate the analysis of FWHM, QF, sensitivity and FoM respectively, with respect to the variation in $d_1$. It is perceived that FWHM of the sensor shows a very marginal change with respect to increase in the value of $d_1$. A minute observation reveals that for $d_1>55$ nm, the FWHM increases to 0.51. Nevertheless, the QF shows a very little variation with respect to change in $d_1$, but a deep insight into the figure shows that QF attains a maximum value of 1216 at $d_1=55$ nm. Sensitivity increases from 172.6 nm/RIU to 179.5 nm/RIU as the value of $d_1$ increases from 52 nm to 57 nm. FoM increases up to 351 1/RIU at $d_1=55$ nm, after which FoM starts falling for higher values of $d_1$. By considering a trade-off between QF, FoM, and sensitivity, we select the optimized value of $d_1$ as 55 nm.

Fig. 4.

(a) Reflectance spectrum, (b) Analysis of the peak wavelength of the defect mode, (c) Analysis of the FWHM, (d) Analysis of the QF, (e) Sensitivity and (f) FoM for different values of $d_1$.

3.3. The effect of changing the thickness of the second layer (${\mathit{d}}_{\mathbf{2}}$)

Afterwards, we studied the effect of the variation in the thickness of the layer b ($d_2$) on the sensor's performance. The shift in PBG characteristics and defect mode position is analyzed by changing $d_2$ in the range from 67 nm to 73 nm at a constant $d_1=55nm\text{,}$ $d_D=120nm$, and N = 10, which is represented in Fig. 5(a). Like the previous simulation results, in this case also it is noticed that both the lower edge and upper edge of the PBG shift towards higher wavelength region. Additionally, the defect mode is red shifted from 600 nm to 624 nm with respect to increase in the value of $d_2$ from 67 nm to 73 nm, which is shown in Fig. 5(b). Fig. 5(c)–5(f) depict the analysis of FWHM, QF, sensitivity and FoM respectively, with respect to the variation in $d_2$. Here, it is indicated that with a variation in $d_2\text{,}$ there is little change in the FWHM, sensitivity, QF, and FoM. Here, we consider the optimized value of $d_2$ as 70 nm, where we get FWHM, sensitivity, QF, and FoM as 0.504 nm, 176.8 nm/RIU, 1210, and 351 1/RIU respectively.

Fig. 5.

(a) Reflectance spectrum, (b) Analysis of the peak wavelength of the defect mode, (c) Analysis of the FWHM, (d) Analysis of the QF, (e) Sensitivity and (f) FoM for different values of $d_2$.

3.4. The effect of changing the porosity of the first layer (${\mathit{P}}_{\mathbf{1}}$)

Further, we investigated the effect of level of porosity of the 1st layer on the sensor's performance. We studied the reflectance of the structure by varying the porosity of layer a ($P_1$) from 0.68 to 0.72 at an interval of 0.1, which is depicted in Fig. 6. Interestingly, it is seen that the lower edge of the PBG shows a little variation, whereas the upper PBG edge shows a significant variation (red shift) with an increase in $P_1$. The defect mode wavelength shows a relatively lower variation with a rise in $P_1$. As perceived in Fig. 6(a), the defect mode wavelength increases from 607.63 nm to 616.42 nm, when $P_1$ increases from 0.68 to 0.72. The nature of variation in defect wavelength with change in $P_1$ is linear, which indicates the linearity and stability of the proposed sensor. Fig. 6(c)–6(f) represent the analysis of FWHM, QF, sensitivity and FoM respectively, with respect to the variation in the value of $P_1$. The FWHM shows a downtrend with respect to increase in $P_1$. In particular, the FWHM decreases from 0.67 to 0.38 with rise in $P_1$ from 0.68 to 0.72. Regarding the quality factor (QF), sensitivity, and figure of merit (FoM), it is observed that all these parameters increase as the value of $P_1$ rises. While higher porosity levels generally enhance performance, it is important to note that excessive porosity can weaken the material and reduce the strength of the particles. As a compromise, we select a porosity level of 0.7 for layer A, which yields a QF of 1210, sensitivity of 176.58 nm/RIU, and FoM of 350 1/RIU.

Fig. 6.

(a) Reflectance spectrum, (b) Analysis of the peak wavelength of the defect mode, (c) Analysis of the FWHM, (d) Analysis of the QF, (e) Sensitivity and (f) FoM for different values of $P_1$.

3.5. The effect of changing the porosity of the second layer (${\mathit{P}}_{\mathbf{2}})$

After that, we move on to optimize the porosity level of the layer b ($P_2$), where we varied $P_2$ in the range from 0.38 to 0.42, while keeping other geometrical parameters constant. We examined the reflectance of the sensor, which is illustrated in Fig. 7(a), where it is observed that the lower edge of the PBG shows a significant shift but the upper edge of the PBG shows a little shift with respect to increase in $P_2$. As shown in Fig. 7(b), the defect mode appears within the PBG, which is red shifted from 607 nm to 618 nm with respect to increase in $P_2$ from 0.38 to 0.42. Interestingly, it is seen that unlike the previously stated optimization of $P_1$, in case of optimization of $P_2$ the sensing parameters like FWHM, QF, sensitivity, and FoM decreases with an increase in the value of $P_2$, which are illustrated in Fig. 7(c), (d), 7(e), and 7(f) respectively. Even if good results are obtained at very low porosity, we avoid the low porosity level because the porous diameter will be too small to contain the analyte. So, as a tradeoff, we consider the optimized porosity level of layer b as 0.4, where we attain a FWHM, QF, sensitivity, and FoM of 0.5 nm, 1200, 177 nm/RIU and 348 1/RIU respectively.

Fig. 7.

(a) Reflectance spectrum, (b) Analysis of the peak wavelength of the defect mode, (c) Analysis of the FWHM, (d) Analysis of the QF, (e) Sensitivity and (f) FoM for different values of $P_2$.

3.6. The effect of changing the inner core radius (${\mathit{\rho }}_{\mathbf{0}}$)

Another important geometrical parameter is the radius of the inner core, which can have a significant impact on the sensor's reflectivity and performance. The reflectance of the suggested APC is shown in Fig. 8(a), where the inner core radius (${\rho }_0$) is varied from 200 to 700 nm. It is perceived that with an increase in ${\rho }_0\text{,}$ there are no discernible fluctuations in the reflectivity, width, and position of the PBG, which makes the sensor fabrication versatile and simple. This concludes that the proposed APC behaves like a planar PhC and the intensity no longer depends on the radius of the inner circle. In addition to this, other significant parameters such as shift in defect mode's wavelength, FWHM, QF, sensitivity, and FoM remain stable for change in the value of ${\rho }_0$, which are shown in Fig. 8(b), (c), 8(d), 8(e), and 8(f), respectively. In this regard, we select the optimized value of ${\rho }_0$ as 500 nm, where the value of FWHM, QF, sensitivity, and FoM are 0.505 nm, 1215, 176.85 nm/RIU, and 350.5 1/RIU respectively.

Fig. 8.

(a) Reflectance spectrum, (b) Analysis of the peak wavelength of the defect mode, (c) Analysis of the FWHM, (d) Analysis of the QF, (e) Sensitivity and (f) FoM for different values of $\mathrm{N}$.

3.7. The effect of changing the period (N)

Finally, we examined the impact of the period (N) of the APC on the reflectance, which is depicted in Fig. 9(a). For a very low value of N, the PBG is not realized in the structure owing to the less chance of Bragg's scattering at the interface of the consecutive layers. For N > 8, the intensity of the defect mode appears to be more prominent. Also, it is noticed that the width of the PBG shrinks as the value of N increases. On the other hand, intensity of the reflectance increases with a rise in N. As represented in Fig. 9(b), the defect mode wavelength increases from 612.038 nm to 612.12 nm as the value of N increases from 8 to 15. We consider the optimized value of N as 10, because for higher values of N, the structure becomes bulky in nature. As shown in Fig. 9(c)–9(f), at N = 10, the value of FWHM, QF, sensitivity, and FoM are 0.5 nm, 1215, 176.85 nm/RIU, and 350.5 1/RIU respectively.

Fig. 9.

(a) Reflectance spectrum, (b) Analysis of the peak wavelength of the defect mode, (c) Analysis of the FWHM, (d) Analysis of the QF, (e) Sensitivity and (f) FoM for different values of N.

Finally, we compared the sensor's performance with some recently published similar works, which is shown in Table 1. It can be clearly seen that the proposed APC sensor outperforms in the sensitivity values compared with other works as listed in the table, which proves the effectiveness of the proposed sensor over these designs. Moreover, due to the annular property, the proposed sensor is compact in size, and can be easily fitted in a small area.

Table 1.

Comparison of between the current work with previous studies in sensitivity.

Sl. No.	Sensor configuration	Sensing analyte	Sensitivity (nm/RIU)	Ref.
1	1D PhC	Biodiesel	136.8	[42]
2	1D PhC	Cancer cells	43	[43]
3	1D binary PhC	Blood hemoglobin	167	[44]
4	1D binary PhC	Fat concentration in milk	106.7	[45]
5	1D Annular PhC	Calcium carbonate concentration in water	176.85	This work

4. Conclusion

In conclusion, an annular photonic crystal is designed and analyzed for the detection of the concentration of CaCO₃ in water flowing in the pipeline. As the high concentration of CaCO₃ leads to the formation of scale in the water pipes, this research carries a lot of importance in the water distribution sector. The proposed APC is realized with a configuration of ${\left[laye{r}_{a}laye{r}_b\right]}^{N}Laye{r}_{\left(defect\right)}{\left[laye{r}_{a}laye{r}_b\right]}^N$. The focus of this research is to investigate the characteristics of the defect mode that emerges within the bandgap of the reflectance spectrum. A thorough optimization process is conducted for all design parameters to achieve optimal sensor performance. The porosities of layers A and B are optimized at 0.7 and 0.4, respectively, while the thicknesses of layers A, B, and the defect layer are optimized to 55 nm, 70 nm, and 120 nm, respectively. The period of the APC is set to 10, and the inner core radius (${\rho }_0$) is chosen as 500 nm to ensure peak performance. With these optimized geometric parameters, we achieved impressive sensing metrics, including a quality factor (QF) of 1215, a sensitivity of 176.85 nm/RIU, and a figure of merit (FoM) of 350.5 1/RIU. Additionally, the designed sensor features a simple configuration, compact size, and ease of fabrication, making it an ideal candidate for real-time detection of CaCO₃ scale in pipes and devices within water supply networks.

CRediT authorship contribution statement

Ayman A. Ameen: Software. Abinash Panda: Writing – original draft. Ahmed Mehaney: Writing – review & editing, Project administration. Mehdi Tlija: Supervision, Funding acquisition. Stefano Bellucci: Supervision. Mostafa R. Abukhadra: Supervision. Hussein A. Elsayed: Writing – review & editing, Supervision.

Consent for publication

Not applicable.

Ethics approval and consent to participate

Not applicable.

Availability of data and material

Requests should be addressed to corresponding author on a reasonable request.

Funding

The authors extend their appreciation to King Saud University for funding this work through Researchers Supporting Project number (RSPD2024R685), King Saud University, Riyadh, Saudi Arabia.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.