1 Introduction

Photonic crystal fibers (PCFs) (Russel 2003) have attracted intense research work all over the world. PCFs have unique characteristics that cannot be achieved using conventional optical fiber such as high birefringence (Hameed et al. 2009a), single mode guidance (Islam et al. 2017), and large effective mode area (Sun et al. 2023). Therefore, PCFs have been used for different applications including polarization rotator (Hameed and Obayya 2010), polarization splitter (Younis et al. 2018), multiplexer–demultiplexer (Hameed et al. 2009b) and optical sensors (Md Leon et al. 2021). Further, PCF sensors (Elhelw et al. 2023; Hameed et al. 2017; Rabee et al. 2019) have high sensitivity, compact size, and flexibility compared to other optical sensors. Nowadays, terahertz (THz) PCFs with hollow core or porous core have been suggested in different applications such as absorbers (Shi et al. 2016), polarization rotator (Hameed et al. 2020), and directional couplers (Bao et al. 2014). Further, terahertz sensors gained great importance because most of the molecular absorption spectra lie in the terahertz range (Saadeldin et al. 2019). Additionally, they are simple and easy to fabricate (Islam et al. 2020) with large dimensions. The relative sensitivity and confinement loss are two main modal characteristics of PCF-based THz sensors. The sensing idea depends on the light interaction with the analyte infiltrated into the air holes in the fiber core and/or cladding region (Sultana et al. 2018a).

The creatinine concentration in human blood is a very important biomarker for kidney function, and its failure detection, filtration rate, determination the effectiveness of hemodialysis treatment, and muscular dystrophy. In addition, high levels of creatinine in the human blood demonstrate many fatal diseases such as pre-eclampsia, diabetic nephropathy, glomerulonephritis, urinary tract obstruction, and renal failure. In contrast, low creatinine levels result in myasthenia and muscular dystrophy (Reddy et al. 2013). In addition, many people around the world, between 11 and 13% suffer from kidney diseases (Luyckx et al. 2018). The Jaffe method is a chemical technique for measuring the creatinine level in blood. But it suffers from low accuracy due to endogenous interferences such as non-creatinine chromogens (Elzen et al. 2018). The second method is an enzyme-based assay. This approach has some drawbacks where the results vary with the environmental conditions (pH, and temperature), and enzyme concentration. Also, it requires a high cost for enzyme production and purification. In addition, problem of endogenous NH+4 interference with blood has been obtained (Iyer et al. 2008). Moreover, Amperometric electrochemical creatinine biosensors depends on high potential for reducing the oxygen potential, and use of three-enzyme systems (Narimani et al. 2021). Furthermore, the colorimetric method is a simple, inexpensive, fast method for measuring the color variations that occur in the sample due to chemical reactions between chemical reagents and creatinine based on the Jaffe method. The fundamental problem in the colorimetric method is the interference of other materials in the chemical reaction and the disturbing color change in the sample (Narimani et al. 2021). Moreover, surface-enhanced Raman scattering (SERS) needs laborious sample preparation steps and expensive instruments (Anselmo et al. 2022). As a result, a susceptible sensor is highly needed for the early detection of creatinine concentrations with a cost-effective, high-speed, portable, and compact size. THz PCF sensors have been widely reported in different sensing applications Moreover, Islam et al. (2021a; b) have reported a THz sensor to detect different human body proteins with a high sensitivity of 99.9%. In addition, Rahaman et al. (2021) have proposed a hollow-core PCF as a chemical sensor with a relative sensitivity of 96.25%, confinement loss of 2.11 × 10−14 cm−1, an effective material loss of 9.16 × 10−4 cm−1, and effective mode area of 1.29 × 106 μm2. Jibon et al. (2021) have reported a poisonous chemical sensor using hollow square core PCF with a sensitivity of 92.7%, EML of 6.83 × 10–3 cm−1, and confinement loss of 2.08 × 10–12 cm−1. Moreover, Daher et al. (2022) have proposed a creatinine level sensor based on 1D ternary Si/ TiN/SiO2/photonic crystal with a sensitivity of 938 nm/RIU.

In this paper, a highly sensitive THz PCF sensor for creatinine level detection in the blood is proposed and analyzed for the first time to the best of our knowledge. In this study, effective index (neff), relative sensitivity, birefringence (B), effective mode area (Aeff), confinement loss (CL), effective material loss (EML), and confined power fractions are studied. The obtained modal properties are used to clarify the effectiveness and performance of the suggested sensor. The reported PCF has a circular sectored cladding and rectangular-shaped core with Gallium Phosphide (GaP) as background material. The modal analysis is carried out by using the full vectorial finite element method (Obayya et al. 2000, 2002; Ibrahim et al. 2020, 2021a, 2021b) via COMSOL Multiphysics software package (http://www.comsol.com). The geometrical parameters are studied to maximize the sensor’s relative sensitivity. The proposed THz-PCF has high relative sensitivity of 93% and 95% for x- and y-polarized modes, respectively. The achieved sensitivity is larger than that reported by (Sen et al. 2021; Rahaman et al. 2022; Gandhi et al. 2022; Asaduzzaman et al. 2022a; Al Mahmud et al. 2023) with a simple design. Further, the reported biosensor can be fabricated using current fabrication technologies (Islam et al. 2020). The sensing technique depends on increasing the light interaction with the infiltrated analyte into the air holes in the fiber core region. This will increase the relative sensitivity coefficient that is proportional to the analyte power fraction. In this regard, most of the light power should be confined through the analyte region. This will increase the relative sensitivity coefficient that is proportional to the analyte power fraction.

2 Design considerations

The two-dimensional cross-sectional view of the suggested sensor with a diameter of 1.7 mm is shown in Fig. 1. The cladding region has two symmetrical air fragments. However, the core region has 9 × 5 identical rectangular holes with a length of L and a width of W. The total length, Lc and width, Wc of the core region are 480 µm, and 400 µm, respectively. These initial parameters are chosen for the fabrication feasibility of the proposed design. The rectangular air holes are used to increase the birefringence between the two polarized beams, reduce the material loss and increase the sensor sensitivity. In this study, the refractive index n, and the absorption coefficient α of the gallium phosphide (GaP) background material at a frequency range from 0.5 to 1.5 THz is given as a function of the frequency by Farooqui et al. (2018) as shown in Fig. 2. The refractive index, nc, and absorption coefficient αc of the blood as a function of creatinine concentrations at 0.4 THz (since the refractive index and absorption coefficient of blood are non-dispersive and nearly constant through the range 0.4–1.5 THz) are shown in Fig. 3. These experimental measurements have been described in detail and reported in (Zhang et al. 2018) where the glucose concentration values are variated from 5.0 to 5.5 µmol/L to keep the blood samples at constant conditions. The GaP (RI ≈ 3.34) is selected as a background material due to its higher refractive index compared to the analyte (RI ≈ 2.76) filled in the air holes. It is aimed to confine the THz electromagnetic waves within the analyte region inside the core region to enhance the sensor sensitivity. The GaP has been widely used in PCF. In this context, high nonlinearity and numerical aperture-based PCFs have been presented with GaP as a background material (Paul et al. 2018; Pandey et al. 2022). Micro-rectangular stripes of GaP have been used in the core region of hexagonal sectored PCF to obtain high birefringence, and low confinement loss (Anas et al. 2018). In addition, THz-based solid core PCF has been suggested by Li et al. (2012). Different fabrications techniques can be employed to fabricate GaP based PCF, e.g., stack and draw technique, sol–gel casting, ultrasonic drilling, extrusion etc., and deep reactive ion etching (Li et al. 2012). The porous rectangular core air holes have an initial length of L = 60 µm, initial width W = 30 µm, and are arranged in a rectangular lattice with a lattice constant Λx = 50 µm in the x-direction, and Λy = 80 µm in the y-direction. It is worth noting that constant values of refractive index and absorption coefficient are used over the studied frequencies range. Also, many researchers have proposed creatinine optical sensor at a single frequency/wavelength due to the constancy of the blood refractive index and absorption coefficient at the studied creatinine concentration range (Aly et al. 2020; Daher et al. 2022; Ahmed et al. 2019; Eid et al. 2020; Islam et al. 2021a, b; Kumar et al. 2021).

Fig. 1
figure 1

Cross-section view of the proposed THz-PCF-based sensor

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Fig. 2
figure 2

The refractive index and absorption coefficient for the Gallium Phosphide as a function of THz frequency (Paul et al. 2018)

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Fig. 3
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The refractive index and absorption coefficient for the creatinine in the blood as a function of concentration at 0.4 THz (Zhang et al. 2018)

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The diameter D of the cladding air fragment is equal to 1460 µm. Further, the structure width, t of the proposed PCF is taken as 30 μm, and the thickness of the fiber’s wall, tw is taken as 240 µm. The initial values of the geometrical parameters are listed in Table 1.

Table 1 The initial values of the geometrical parameters for the proposed sensor
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3 Numerical result and discussion

3.1 Modal properties of the PCF proposed sensor

To study the modal characteristics of the x- and y-polarized beams, the full vectorial finite element method (FVFEM) (Obayya et al. 2000; Rahman and Agrawal 2013) is used. In this investigation, non-uniform meshing is used with a minimum element size of 0.4 µm and a maximum triangular element size of 10 µm, with an element growth rate of 1.25. Additionally, the curvature factor is fixed at 0.1. To achieve high accuracy, a convergence study is made for the meshing parameters where an error of order 10−14 is obtained. Further, a circular scattering boundary condition (SCB) is used to truncate the computational domain and calculate the confinement losses of the supported modes. The normalized electric field distributions of the fundamental mode of the two polarized modes are shown in Fig. 4 at 0.5 THz. In this study, the initial design parameters are used. In Fig. 4, it may be seen that the light is tightly confined inside the core and strongly interacts with the analyte samples.

Fig. 4
figure 4

The electric field distribution of the fundamental mode for a x-polarized mode and b y-polarized mode at f = 0.5 THz and C = 60 µmol/L, respectively

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The effective index of the x-polarized mode and y-polarized mode is shown in Fig. 5, at different creatinine levels. In Fig. 5, as the frequency increases, the mode confinement through the core region increases. Therefore, the effective indices of the two polarized modes increase with increasing frequency as shown in Fig. 5. As the creatinine concentration increases, the analyte refractive index decreases. Therefore, the effective index of the supported modes decreases by increasing the creatinine concentration (Sen et al. 2019).

Fig. 5
figure 5

The frequency-dependent effective index of the fundamental a x-polarized mode, and b y-polarized mode at different creatinine concentrations

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The birefringence is another important parameter for polarization-maintaining fiber that can eliminate the polarization modal dispersion, and is given by (Islam et al. 2018a, b):

$$ B = \left| {n_{x} - n_{y} } \right| $$
(1)

where nx and ny represent the effective index of the x and y polarized modes, respectively. It can be observed that the rectangular holes can cause an effective index difference between the x- and y-polarization modes because of their asymmetrical structure. The birefringence of the proposed sensor as a function of the frequency for different creatinine concentrations is shown in Fig. 6. as the frequency increases, the effective index of the two polarized modes increases. Therefore, mode confinement through the core region will be increased. Hence, the birefringence will decrease as the frequency increases (Islamet al. 2019; Habib et al. 2020). In Fig. 6, it may be seen that the birefringence at a given frequency increases by increasing the analyte concentration. As the creatinine concentration increases, the mode confinement will be decreased. Therefore, the mode filed will be more affected by the inner rectangular shapes which increase the birefringence (Islam et al. 2018a). A maximum birefringence of 0.0328 is obtained at f = 0.5 THz and at a creatinine concentration of 85 µmol/L. The obtained birefringence is higher than that reported by (Hossain et al. 2021c; Bulbul et al. 2021; Gandhi et al. 2022; Juan et al. 2022).

Fig. 6
figure 6

Birefringence of the proposed sensor as a function of the frequency at different creatinine concentrations

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The effective material loss (EML) is a limiting factor of the THz sensor performance. That arises due to the cladding material absorption (Paul et al. 2019). Therefore, it is essential to use a material with low absorption loss. The proposed rectangular hollow core reduces the overall background material, which decreases the EML (Yakasai et al. 2019). The EML of the proposed PCF sensor can be calculated as (Yakasai et al. 2019):

$$ EML = \left( {\frac{{\varepsilon_{0} }}{{\mu_{0} }}} \right)^{\frac{1}{2}} \left( {\frac{{\mathop \smallint \nolimits_{mat}^{{}} n \left| E \right|^{2} \alpha dA}}{{2\mathop \smallint \nolimits_{all}^{{}} S_{z} dA}}} \right) $$
(2)

where εo is the relative permittivity and µo is the relative permeability of the free space. Additionally, n and α are the refractive index and bulk absorption loss of the GaP, respectively. While Sz denotes the z-component of the Poynting vector. Figure 7 shows the frequency dependent EML of the two polarized modes at different creatinine concentrations. As the frequency increases, the absorption coefficient of the Gallium Phosphide is increased (Paul et al. 2018) as shown in Fig. 2. Therefore, the EML of the two polarized modes increases by increasing the operating frequency. Further, due to the neglected absorption of the analyte sample, the creatinine concentration has a slight effect on the EML of the two polarized modes. Also, the EML slightly increases with increasing the creatinine concentration. This may occur because the difference in the refractive index between the core and the cladding regions increases with the increasing analyte’s refractive index. Therefore, a greater amount of light propagates in the core region, and the loss decreases (Sultana et al. 2018a, b; Reza et al. 2020). The suggested design has low EML values of 0.00486 cm−1, and 0.00474 cm−1, for the x- and y-polarized modes, respectively at f = 0.5 THz. The obtained EML values are lower than that reported in (Bulbul et al. 2021; Islam et al. 2021a, b; Hossain et al. 2021c; Rahaman et al. 2022; Gandhi et al. 2022; Al Mahmud et al. 2023). Also, the obtained EML is lower than that for silica-based PCF sensor in THz reported in (Asaduzzaman et al. 2022a).

Fig. 7
figure 7

The frequency-dependent EML of the fundamental, a x-polarized mode, and b y-polarized mode at different creatinine concentrations

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The effective mode area of the fundamental mode within the core region indicates the area of the analyte region that strongly interacts with the light where the mode is confined within the core region (Medjouri et al. 2015) and is given by (Hossain et al. 2020a, b);

$$ A_{eff} = \frac{{\left[ {\smallint I\left( r \right) rdr} \right]^{2} }}{{\smallint I^{2} \left( r \right)rdr}} $$
(3)

where \(I\left( r \right) = \left| {E\left( r \right)} \right|^{2}\) denotes the transverse electric field intensity distribution throughout the cross-section of the fiber. The frequency-dependent effective mode area is shown in Fig. 8 at different creatinine concentrations. It is evident that a large effective area is obtained at a lower operating frequency. As the operating frequency increases, the field will be more confined through the analyte with a small Aeff (Islam et al. 2020; Habib et al. 2020). It may also be noted that the effective mode area is increased at higher concentrations of creatinine where low index contrast between the core and cladding regions occurs and hence low field confinement through the core region (Islam et al. 2018b). The maximum effective mode areas are equal to 0.059 mm2, and 0.0644 mm2 for the x-polarized mode and y-polarized mode, respectively at f = 0.5 THz and the creatinine concentration of 85 µmol/L. The obtained effective mode area is higher than that reported by (Yakasai et al. 2019; Islam et al. 2019; Rahman et al. 2020a, b; Asaduzzaman et al. 2022b; Elhelw et al. 2023), and comparable with that reported by (Islam et al. 2018b; Rahman et al. 2020a, 2020b; Hossain et al. 2021a, b).

Fig. 8
figure 8

The effective mode area of the fundamental, a x-polarized mode, and b y-polarized mode at different creatinine concentrations

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The confinement loss (CL) is an indication of light confinement within the core area (Lee et al. 2015). The CL can be calculated using the imaginary part of the effective index as given by (Tahhan et al. 2020);

$$ CL = 8.686 \times \frac{2\pi f}{C} Im\left( {n_{eff} } \right) \quad {\text{in }}\left( {\text{dB/cm}} \right) $$
(4)

here neff is the effective mode index, f is the operating frequency, and c is the speed of light In Fig. 9, it is revealed that the confinement loss decreases with increasing the operating frequency. This is due to the good light confinement through the core region at higher frequencies (Sultana et al. 2018b; Kumar et al. 2021). At higher creatinine concentrations, low index contrast between the core and cladding regions occurs with low confinement through the ore region (Islam et al. 2018b). Therefore, the confinement loss increases by increasing the creatinine concentrations as shown in Fig. 9. The proposed sensor has very low confinement loss values of 3.035 × 10−11 dB/cm, and 1.408 × 10−11 dB/cm, for x and y-polarized modes, respectively at f = 0.5 THz at a creatinine concentration of 60 µmol/L. The obtained CL value is lower than that reported by (Luo et al. 2021; Hossain et al. 2021a, b; Sen et al. 2021; Hossain et al. 2022; Asaduzzaman et al. 2022a; Rahaman et al. 2022; Gandhi et al. 2022; Al Mahmud et al. 2023).

Fig. 9
figure 9

The frequency-dependent confinement loss for a x- and b y-polarized modes at different creatinine concentrations

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3.2 Sensitivity analysis of the proposed PCF sensor

The relative sensitivity of the proposed PCF is defined by (Kumar et al. 2021),

$$ S = \frac{{n_{r} }}{{n_{eff} }} \times P\% $$
(5)

where the analyte refractive index is nr, the effective mode index is neff, and the power fraction of light confined in the analyte region is P that measures the strength of light–analyte interaction (Paul et al. 2019), and as expressed by (Kumar et al. 2021);

$$ P = \frac{{\mathop \smallint \nolimits_{sample}^{{}} R_{e} \left( {E_{x} H_{y} - E_{y} H_{x} } \right)dxdy}}{{\mathop \smallint \nolimits_{total}^{{}} R_{e} \left( {E_{x} H_{y} - E_{y} H_{x} } \right)dxdy}} \times 100 $$
(6)

The electric field of x- and y- components are Ex and Ey, while the Hx and Hy are the corresponding magnetic field components. Moreover, the field interaction between light and the analyte sample can be measured by the relative sensitivity coefficient. According to the Beer–Lambert law, light is attenuated by the intensity of the evanescent wave absorption (Arif et al. 2017). Figure 10 shows the variation of the power fraction inside the analyte region which affects the sensor sensitivity. It is evident that the power fraction through the analyte region decreases by increasing the creatinine concentrations. This is due to the smaller confinement of the supported modes through the core region at higher analyte concentration and at higher frequency. It is important to clarify that the confinement loss and the effective modal area, which are presented in Figs. 8, and 9, are calculated for the whole core region. The core region contains both the analyte regions and the outer surrounding air holes. Furthermore, the area of the analyte regions is larger than the total area of the surrounding air holes inside the core. For this reason, as shown in Fig. 9 the power fraction inside the analyte region is much larger than that of the surrounding air holes inside the core. Figure 11 shows the relative sensitivity versus the frequency for x- and y- polarized modes. It is evident that the relative sensitivity decreases with increasing operating frequency. This can be attributed to decreasing the analyte power fraction with increasing frequency as shown in Fig. 10. This is because the GaP absorption coefficient increases with increasing the frequency as shown in Fig. 2. This means that part of the light confinement will be absorbed by the GaP material and another part is also leakage into the air holes surrounding the analyte region (Habib et al. 2020; Islam et al. 2021a, 2021b). It is also revealed from Fig. 11 that the relative sensitivity decreases by increasing the creatinine concentration due to the less confinement of the light in the core region at high creatinine concentration (Ahmed et al. 2019). The maximum sensitivity for the proposed sensor is equal to 61%, 63% for x, and y-polarized modes, respectively at f = 0.5 THz and at a creatinine concentration of 85 µmol/L.

Fig. 10
figure 10

Frequency-dependent power fraction of the a analyte x-polarized mode, b analyte y-polarized mode, c core air holes x-polarized mode, and d core air holes y-polarized mode using initial parameters of the proposed PCF

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Fig. 11
figure 11

Frequency-dependent relative sensitivity of the a x-polarized mode and b y-polarized mode using initial parameters of the proposed PCF

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In the optimization process, the effect of different geometrical parameters is studied to clarify the effectiveness of the proposed sensor and to achieve the highest relative sensitivity using the geometrical parameters. Therefore, the effects of the core rectangular air holes length L, width W, rectangular core length Lc, rectangular core width Wc, lattice constant in x-direction \({\Lambda }_{x}\), and lattice constant in y-direction \({\Lambda }_{y}\) are studied to maximize the sensor sensitivity. Figure 12a shows the variation of the relative sensitivity with the core rectangular hole length, L at W = 40 µm. Further, the effect of the rectangular hole width W on the relative sensitivity is shown in Fig. 12b at L = 75 µm at creatinine concentration of 60 µmol/L, and f = 0.5 THz. In this study, the other geometrical parameters are kept at their initial values. As the studied parameter increases, the area of the analyte sample will be increased. Therefore, the light interaction with the samples will be improved which increases the relative sensitivity of the two polarized modes as shown in Fig. 12 (Islam et al. 2018a). High relative sensitivity of 91%, and 90% are obtained for the y-polarized and x-polarized modes at L = 75 µm and W = 45 µm.

Fig. 12
figure 12

Variation of the relative sensitivity with a the core rectangular hole length, L at W = 40 µm, and b the core rectangular hole width, W at L = 75 µm at creatinine concentration equals 60 µmol/L, and f = 0.5THz

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The geometrical parameters are also studied in Fig. 13a–d to improve the sensor sensitivity. Figure 13a, b show the variation of the relative sensitivity with the lattice constants in x-direction, Λx at (Λy = 80 µm), and in y-direction, Λy at (Λx = 49 µm), respectively. In this study, the other geometrical parameters are fixed at L = 75 µm, W = 45 µm, Lc = 450 µm, and Wc = 400 µm. it is revealed that the relative sensitivity increases by decreasing the lattice constants in the x-direction, Λx, and the y-direction, Λy. This is due to the decrease of the GaP background material where the lattice constants decrease at constant core dimensions. Therefore, the Λx = 49 µm and Λy = 79 µm are used with high relative sensitivity of 49.82%.

Fig. 13
figure 13

Variation of the relative sensitivity with a the lattice constant, \({\Lambda }_{x}\) at \({\Lambda }_{y} = 80 \)µm b the lattice constant, \({\Lambda }_{y}\) at \({\Lambda }_{x} = 49\) µm c the length of the rectangular core, Lc, at Wc = 400 µm, and d the width of rectangular core, Wc. at Lc = 450 µm, creatinine concentration of 60 µmol/L, and f = 0.5 THz

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Next, the effect of the rectangular core dimension is investigated as shown in Fig. 13c, d while the other parameters are taken as \({\Lambda }_{{\text{x}}} = 49\) μm, \({\Lambda }_{y} = 79\), \({\text{L}} = 75\) μm, W = 45 µm. It is revealed that the relative sensitivity is nearly constant at different core lengths Lc (at Wc. = 400 µm) and at different rectangular core width Wc. (at Lc = 399 µm), respectively. This is due to the slight effect of the studied dimensions on the field confinement in the analyte region and hence the relative sensitivity. The optimized relative sensitivities are 93%, and 95% for x, and y-polarized modes, respectively. The optimized geometrical parameters are presented in Table 2. It is worth noting that these values match well with that obtained from the modal analysis of the suggested PCF with high birefringence, and low material loss.

Table 2 The geometrical parameters for the optimized sensor
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The effect of the fabrication tolerance of the core rectangular holes length, L, core rectangular holes width W, the rectangular core length, Lc, rectangular core width, Wc, lattice constant in the x-direction, \({\Lambda }_{x}\), and lattice constant in the y-direction, \({\Lambda }_{y}\), are studied with a variation of ± 2% relative to their optimized values. Table 3 shows the modal characteristics and sensitivity values of the proposed PCF sensor at optimum design parameters and with ± 2% variation that may occur due to the fabrication faults for y-polarized mode at 0.5 THz, and C = 60 µmol/L. It is evident from the table that the relative sensitivity decreases with decreasing the width and length of the core rectangular holes. Minimum relative sensitivity of 92.23% and 92.88% will be obtained within a tolerance of -2% in the width and length of the rectangular holes. However, the relative sensitivity increases with increasing the width and length of the core rectangular holes where maximum relative sensitivity of 97.23% and 96.55% will be obtained within a tolerance of +2%. As the dimensions of the core rectangular holes increase, the amount of analyte will be increased. Therefore, the analyte power fraction will be increased, and hence the relative sensitivity will be improved (Sultana et al. 2018a). The variation of relative sensitivity with the rectangular core dimensions is nearly constant, while the maximum relative sensitivity of 97.27% will be obtained within a tolerance of − 2% in the lattice constant in the x-direction. The optimum sensitivity is chosen to be 95% to avoid fabrication faults. The obtained tolerance values ensure the feasibility of the proposed design. Further, the optimized sensor still has a very high birefringence of 0.0438, very low effective material loss of 0.00171 cm−1, and low confinement loss of 6.784 × 10−11 dB/cm. Moreover, the sensitivity cannot be further improved due to the fabrication restrictions, where the maximum separation between adjacent holes is 4 µm (Cordeiro et al. 2020). Also, the high absorption coefficient of the cladding material and analyte limits the sensitivity of the proposed sensor.

Table 3 Fabrication tolerance of different geometrical parameters of the proposed terahertz sensor
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Table 4 shows a comparison between the proposed design and those reported in the most recent literature. It may be seen that our sensor shows better sensitivity than that presented in (Kumar et al. 2021; Islam et al. 2021a, b; Bulbul et al. 2021; Hossain et al. 2021a; Hossain et al. 2021b; Luo et al. 2021; Hossain et al. 2021c; Hossain et al. 2022; Asaduzzaman et al. 2022a; Sen et al. 2021; Rahaman et al. 2022; Gandhi et al. 2022; Al Mahmud et al. 2023). Also, the proposed sensor has the lowest EML compared to the literature reported in Table 4. Moreover, the proposed sensor has lower CL than reported by (Hossain et al. 2021a; Hossain et al. 2021c; Sen et al. 2021; Hossain et al. 2022; Asaduzzaman et al. 2022a; Rahaman et al. 2022; Gandhi et al. 2022; Al Mahmud et al. 2023), and comparable with that reported by (Luo et al. 2021). It may also be seen that the proposed sensor achieves very high sensitivity compared with silica-based PCF in the THz region as reported in (Hossain et al. 2021b; Asaduzzaman et al. 2022a). It is worth noting that the rectangular holes are chosen instead of circular holes to increase the birefringence between the two polarized modes, reduce the effective material loss, reduce the confinement loss, and increase the sensor’s sensitivity as reported in (Islam et al. 2021a, 2021b; Bulbul et al. 2021; Hossain et al. 2021b; Hossain et al. 2022). This is due to asymmetry in the rectangular shape. The advantage of using THz PCF rather than Infra-Red PCF is that the absorption band of most liquids and gases lies in the THz region which improves the detection accuracy of these analytes (Naftaly et al. 2019). Further, many materials including organic substances such as toxic, explosive, pharmaceuticals, and biochemicals, are transparent in the THz region and opaque in the Infrared region (Naftaly et al. 2019), On the other side, the silica-based PCF offers high absorption loss in the THz region (Habib et al. 2022; Asaduzzaman et al. 2022a), and have very low relative sensitivity compared to THz PCF (Hossain et al. 2021b; Asaduzzaman et al. 2022a). Experimentally, the silica-based PCF has a total diameter in the range of 118–270 µm. However, THz PCF has a total size of 3–6 mm which can be easily controlled during the fabrication process (Humbert 2019).

Table 4 Comparison between the proposed PCF and prior design PCFs in the literature with essential guiding properties (NA: Not Available)
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Considering all geometrical parameters, our sensor shows remarkable performance in creatinine sensing with high sensitivity, very low EML, low CL, and very high birefringence. THz-PCFs can be fabricated with the same fabrication techniques of silica-based PCFs (Juan et al. 2022), such as mechanical drilling (Katyba et al. 2018), capillary stack and draw (Rana et al. 2018), sol–gel casting (Paul et al. 2018), and extrusion and 3D printing techniques (Paul et al. 2018). It has been reported that extrusion and 3D printing techniques are suitable for the fabrication of symmetric and asymmetric complex designs (Paul et al. 2018). Additionally, the stack and draw method can fabricate a broad range of PCF with different lattice shapes. Furthermore, the sol–gel casting method creates photonic crystal fibers with any desired structure and the air hole size, shape, and lattice constant to be easily, individually changed. The suggested device length should fulfill the requirement that the total loss for both x-polarized and y-polarized modes will not exceed 0.1 dB (Rahman et al. 2020a, b). Therefore, the device length can be as compact as 5.74 cm and 4.83 cm for the x-polarized mode and y-polarized mode, respectively. The proposed PCF sensor has the nature of repeatability (reuse) after some treatments such as oxygen-plasma ultra-cleaning, preheating, and dry-chemical treatment (Hunt et al. 2011; Tan et al. 2014; Hindal et al. 2018; Hernández et al. 2020; Tong et al. 2021) without bad impact on the device performance which will reduce the diagnostic expenses.

4 Conclusion

In this work, a novel design of THz-PCF sensor is proposed for measuring the creatinine level in the blood. The GaP material is used to construct the suggested PCF sensor due to its index contrast relative to the blood refractive index. The suggested THz-PCF sensor is used in a range of frequencies from 0.5 to 1.5 THz. The relative sensitivity of the y-polarized mode is higher than that of the x-polarized mode with values of 93% and 95%, respectively at f = 0.5 THz. This is due to the confinement in the y-direction being higher than that in the x-direction. This is because of the optimized sensor's physical dimensions (W = 45 µm, L = 75 µm). Additionally, the suggested THz-PCF donates a high birefringence of 0.0438 which is helpful in polarization-maintaining applications with very low confinement loss of 8.1 × 10−12 dB/cm for x-polarized mode and 6.78 × 10−11 dB/cm for y-polarized mode. Furthermore, a low EML of 0.00144 cm−1 is achieved for the x-polarized mode and 0.00171 cm−1 for the y-polarized mode. It is believed that the proposed structure will open the window for high analyte index detection that can help in the earlier detection of kidney diseases with the improved healthcare facility. Further, the fabrication of the proposed sensor is possible by the nowadays technologies.