Introduction

The development and manufacturing of photonic crystal fiber (PCF)-based sensor have received much attentiveness of numerous researchers across the globe in the current years because of their versatile applications. It has sparked a great deal of interest because of its unique light guiding features and the prospect of light interaction involving gases or liquids. PCF sensor based on PbS has emerged nowadays because of the small size, low price, robustness, the capacity to execute in adverse conditions such as high electromagnetic and chemically exposed fields [1,2,3]. Sensing of vapor including gases, biochemical sensing, and medicinal analysis, tracking of factory output, ocean and atmospheric research, bio-process control, including automotive control are some of the key characteristics of PCF-based sensors. A. Ramola et al. have reported maximum confinement loss of 40.56 dB/cm at wavelength of 1100 nm for cervical cancer cells using PCF-SPR sensor [4].

Hollow core as biosensor is preferred over the other core types like solid core, as well as porous core since it permits a larger analyte capacity into its core region as measured to other core types. Additionally, since the guided mode in hollow core fibers gets tightly restricted inside the core region, the influence of a background material on the fiber's wave guiding characteristics being effectively decreased [5, 6].In sensing application, the air inside the air hole core gets replaced by one liquid analyte.

Breast cancer results because of the abnormal growth of breast cells. This illness is highly heterogeneous. Whenever a patient is diagnosed with breast cancer, the breast tissues take on the shape of a lump, which appears as an aggregation and is usually detected via X-rays [7, 8]. The severity of breast cancer has been reflected in the data on cancer reported in the paper [9]. According to the statistics, Breast cancer is the commonest form of cancer among people between the ages 30 to 39 [10].

Despite breast, cancer has been almost exclusively a female disease, it could also affect men. Breast tumors of any kind are rarely cancerous. The certain growth or swelling in the breast should be examined to see that the cells are cancerous. There are few other methods for testing cancerous tissue, within each variety of attributes [11]. Miller et al. suggested a technique for identifying breast cancer cells using photonic crystal fibers [9, 12]. In recent times, researchers have shown considerable interest in utilizing PCF for biochemical sensing and identification purposes. PCF is a remarkable category of optical fiber that has definite unique features, like optical energy propagation through the PCF having minimum confinement loss (CL) and dispersion having improved nonlinearities [13,14,15].

Sensitivity and CL are the two vital leading parameters in liquid and chemical analysis. Birefringence would be a primary remarkable feature of PCF, and an increase in its value is foreseen. Asymmetry within fiber [16, 17] could be used to achieve this. PCF with a greater birefringence has been extensively used during telecommunications, chemical detection, and other areas [18,19,20]. Silica, Teflon, and Zeonex are now just a few forms of fiber materials which could be accepted as bulk materials [18, 19]. Chemical properties of Zeonex have made things exceptionally well enough and distinctive. Various experiments have already shown that Zeonex does have a higher chemical resistance; hence, it is the best fiber material [22, 23].

PCF has been employed to create a wide diversity of chemical and biosensors so far. Arif et al. described another PCF sensor based on PbS [24].The cladding area with this PbS has hexagonal pattern-oriented air holes; whereas, the core was elliptical. They have reported birefringence as 0.005, the confinement loss as 2.17 × 1012 cm−1and the sensitivity as 41%.The effective material loss (EML) for the studied sensor has still not been measured [24].Leon et al. [25] has reported similar as 41%. Its cladding air holes are placed identically to those in the sensor proposed by Arif et al. [24].Its core is organized into four small circular holes. The sensor has sensitivity approximately 49% and a slightly greater birefringence of around 0.008. Leon et al. have found larger confinement loss, but they have not examined the EML characteristic of the proposed sensor. Hasan et al. in 2019 reported a heptagonal shaped PCF and achieved a sensitivity of 63.24, 61.05, and 60.03% respectively for benzene, ethanol, and water by using TOPAS as a background material. They have suggested alternative PbS having heptagonal organized air holes inside the cladding area [26]. Sensitivity of the proposed PbS remains slightly greater, about 63.24%. However, the confinement loss and effective area (EA) of the reported sensor were larger.

Hasan et al. [26] have studied a PCF-based chemical sensor in which the cladding holes are heptagonal arranged. At 1.0 THz frequency, the relative sensitivity achieved for the sensor is only 63.24% having low confinement loss. Various optical characteristics of the sensor along with effective material loss (EML) and numerical aperture (NA) have not been evaluated. Yakasai et al. attained 64% relative sensitivity for liquid chemicals (sarin and tabun liquids) in which quad-elliptical core is used [27]. Number of PbS have been reported by other workers [28], but with poor sensitivity, which encourages to explore the possibility to construct a PbS with improved sensitivity, a manufacturing-viable structure, less effective area, birefringence, confinement loss, nonlinearity, and dispersion. Various type of sensors like physical sensors [29, 30], curvature/bend sensors [31], displacement/strain sensors [32], electric and magnetic field sensors [33], pressure sensors [34], temperature sensors [35], torsion/twist sensors [36], transversal loading sensors [37], refractive index sensors [38], vibration sensors [39], multiparameter sensing [40], gas sensors [41], molecular sensors [42], humidity and pH sensors [43] have already been reported in the literature.

The utilization of a rectangular PCF coupled with Zeonex material in the terahertz regime introduces a range of innovative sensing capabilities. Terahertz waves, being non-ionizing and able to penetrate diverse materials, prove valuable for non-destructive testing, imaging, and sensing in fields such as material characterization, security screening, and medical diagnostics. The distinct sensor design can harness the exceptional features of the rectangular PCF to amplify sensitivity, resolution, or directional sensitivity, specifically tailored for certain applications[44].

The objective of the current work is to implement and evaluate a PbS for identifying cancerous cells. Various parameters have been examined, including relative sensitivity on both the x and y-axis, effective area (EA), confinement loss (CL) and birefringence, effective refractive index, dispersion, nonlinearity, and detection performance. The fabrication possibility of the designed sensor has also been explored. The approach for numerically evaluating the performance of the studied sensor has also been described in detail in Fig. 1. Eventually, the outcomes are provided, with a convincing commentary.

Fig. 1
figure 1

Diagram illustrates the systematic process of designing and analyzing the sensor model

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Design of the PCF sensor model

A Photonic crystal fiber (PCF) sensor with a rectangular core and rectangular air holes as the cladding area has been constructed. The sensor is created employing software based on finite element technique (FEM). Consecutive steps are considered in the implementation and improvement of models utilizing FEM for the creation of proposed PCF model. Subsequently, both in the core and cladding sections, materials have been allocated. Mesh creation is completed after expressing its initial conditions. Ultimately, performance evaluations get retrieved employing partial differential equation (PDE) algorithms. Parameters such as CL, EA, birefringence, relative sensitivity, dispersion are examined, and the process continue until the desired outcomes are achieved. Figure 2a and b depicts a partial aspect and mesh resolution of the studied sensor. Inside the core area, normal and cancerous cells are used as analytes.

Fig. 2
figure 2

a Partial representation of the presumed PCF sensor. b An identical Mesh resolution of suggested PCF sensor

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The entire fiber has a radius of 1650 µm. The perfectly matched layer (PML) has a width of 132 µm given the boundary conditions. An electromagnetic signal traveling more toward the surface is intercepted by PML in this case. Twelve rectangular holes surround the cladding; in which air is injected to analysis. The rectangular air holes have the same width (W) as 500 µm, but not the same length. The rectangle core consists of same height and width as 500 µm. R1, R2 and R3 have lengths 500, 2500 and 1500 µm respectively. The separation connecting two holes in the cladding section is invariable. The pitch (distance joining two adjacent rectangles) of the PCF sensor has been taken 15 µm. The measure joining the centers of two adjacent rectangles has been taken 515 µm. Zeonex has been implemented as fiber substance. In the terahertz band, Zeonex has relatively low absorption loss than few alternative materials like Topas, Silicon, and others. Zeonex is being used to draw metamaterial fibers for the THz region because of its high material loss. Zeonex is worth using as a cladding material but pairing Zeonex with various kinds of different materials allows acquiring the appropriate RI variation to use it as core material. In comparison with Topas, Zeonex has greater chemical resistance and biocompatibility. Zeonex has bulk absorption loss (BAL) 0.2 cm−1 in 0.1–10 THz range, with a RI of 1.53 [45]. All over the terahertz spectrum, the above RI maintains a seemingly linear value [13]. A physics-controlled mesh size is employed in the simulation to achieve the most precise results. Complete mesh consists of 68,646 domain elements and 4146 boundary elements of the fiber. The resolution of mesh analysis can be seen in Fig. 2b.

Various articles explain the experimental configuration of sensor applications using PCF-based systems [18, 46,47,48].The trial and error method has been used to produce the structure shown in Fig. 2. This model is selected because of its prevailing outcomes after modeling several designs and comparing its optical properties. This model is referred to as the optimum (OPT) model.

The OPT model has been modified by 2% while keeping the strut value fixed to show a comparability of the optical properties and verify the choice of the OPT model as the recommended sensor in the present paper. Table 1 shows the design parameters for two models that came from the optimization, as well as the OPT model; it contains the width (W) and height (H) of each rectangle.

Table 1 Optical characteristics for the studied PCF-based breast cancer cell sensor
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Cancer cells and normal cells possess light propagation via their analytes inside the x and y directions, as depicted in Fig. 3. Because only a little intensity of light escaped the core region, there was a little loss. The maximum confinement of light is shown in red. As become closer to the core barrier, the intensity starts to fade. The red arrow has indicated the light propagation direction.

Fig. 3
figure 3

a and c Mode field distribution along x-polarized light, b and d Mode field distribution along y-polarized light, for normal and cancer cell respectively

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Manufacture viabilities

Stacking, sol–gel, 3D printing, and extraction are few of the most familiar techniques of PCF production today. Drilling and slurry casting support sub-micron scale fabrication [49,50,51,52]. Circular air holes could be made using the stacking and sol–gel techniques. Extraction and 3D printing techniques, on the other hand, are essential for making asymmetrical PCF sensors, like rectangles. The even and uneven structures contribute to achieving the desired efficiency and used for a variety of sensor applications. Manufacturing of rectangular air holes have already been reported in the literature [50, 53, 54]. Extrusion using fabrication technique could produce executed design of the PCF of the present study.

Fabricating rectangular PCFs can be challenging but achievable through various methods, such as the stack-and-draw technique or the drilling technique. These methods involve creating a preform with the desired geometry and then drawing it into a fiber while maintaining the rectangular arrangement of air holes. Specialized equipment and precision control are necessary to ensure the accuracy of the final PCF structure. Once the rectangular PCF is fabricated, it can be integrated into a sensing setup to explore its sensing capabilities. The sensing experiments can be carried out in various ways depending on the specific application. Here are some examples: Chemical Sensing, Temperature Sensing, Strain sensing, Bio sensing, Gas Sensing.

Analytical techniques

The experiment is performed in three separate conditions: OPT −2%, OPT, and OPT + 2% for the purpose of an experiment. To achieve these parameters, the width and length of the core are both changed by 2%. The core's height and width have been adjusted to 500 and 300 µm respectively, for the best performance. The width has been altered for OPT −2% and OPT + 2% as height remains same as in the optimum case. The width has been set at 294 µm for OPT −2% and 306 µm for OPT + 2%.

Initially an analyte liquid is placed in the PCF sensor's core. The refractive index (RI) of material controls the speed of light as it passes through it. It depends entirely on the wavelength of the light inside the liquid analyte as well as the fiber material. Some sensing characteristics, such as birefringence, confinement loss (CL), effective area (EA), effective refractive index, power ratio, relative sensitivity, nonlinearity, and dispersion is used to assess the efficiency of the studied sensor.

Birefringence

Birefringence relates to the inequality in the configuration of the core and the cladding region. This is a fiber material's optical property. Polarization and light propagation have an impact on this optical feature. Birefringence is represented by the RI difference in the two regions, and it is evaluated using following Eq. (1) [55, 56].

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

In this context, B represents the birefringence of the polarized mode, whereas \({n}_{\mathrm{x}}\) and \({n}_{\mathrm{y}}\) correspond to the actual values of the effective refractive index for x and y-polarization, respectively.

Confinement loss

Confinement loss (CL) arises when the optical confinement of the PCF is lower because of the core-cladding configuration. The quantity of light absorption by the air holes in the cladding around the core is used to evaluate the loss. If the CL were lower, the sensing quality would be improved. CL is denoted by LConf, and it could be calculated using Eq. (2) [55, 56].

$$L_{conf}=8\cdot 686\times {{K}}_{0}\times \mathrm{Img}\left[{{n}}_{\mathrm{effect}}\right]\times {10}^{6}$$
(2)

Where \({K}_{0}\) =\(\frac{2\uppi }{\uplambda }\), λ represents the wavelength of light and Img \(\left[{n}_{\mathrm{effect}}\right]\) indicates the imaginary value of modal RI.

Effective area

Dependent upon the electric field, light could sometimes travel outside the central region from the analytes. The effective area (EA) is measured as the region where analyte sensing is most efficient. A low EA combined with a substantial nonlinear effect might lead to high power density in the core region.

Equation (3) allows for the assessment of the EA represented as Aeffect [55].

$${{A}}_{\mathrm{effect}} =\frac{{\left(\int \int {\left|{E}\right|}^{2}d{x}d{y}\right)}^{2}}{\int \int {\left|{E}\right|}^{4}d{x}d{y}}$$
(3)

The proposed fiber sensor's electric field distribution is denoted by |E|.

Relative sensitivity

The relative sensitivity of a sensor is a crucial characteristic as it signifies the amount of analyte needed to interact with the sensing components. In typical cases, sensing is achieved by adjusting the refractive index (RI). To assess the relative sensitivity (RS), one needs to evaluate the light intensity directly interacting with the sensing analyte [25]. The Eq. (4) is utilized to compute the relative sensitivity (Rs) of the analytes [57].

$${{R}}_{\mathrm{s}}=\frac{{{n}}_{\mathrm{A}}}{{{n}}_{\mathrm{B}}}{{P}}_{\mathrm{R}}$$
(4)

The refractive index (RI) of the analyte is represented by \({{n}}_{\mathrm{A}}\); whereas, the effective modal refractive index is denoted by \({{n}}_{\mathrm{B}}\). Additionally, \({{P}}_{\mathrm{R}}\) indicates the power ratio.

Power ratio

The power ratio (\({{P}}_{\mathrm{R}}\)) serves as a parameter to quantify the amount of light emitted by the core's sensing materials. Equation (5) can be applied to calculate the power ratio [57].

$${{P}}_{\mathrm{R}}= \frac{{\int }_{\mathrm{analyte }}\mathrm{Re}\left({{E}}_{\mathrm{x}}{{H}}_{\mathrm{y}}-{{H}}_{\mathrm{x}}{{E}}_{\mathrm{y}}\right)d{x}d{y}}{{\int }_{\mathrm{total }}\mathrm{Re}\left({{E}}_{\mathrm{x}}{{H}}_{\mathrm{y}}-{{H}}_{\mathrm{x}}{{E}}_{\mathrm{y}}\right)d{x}d{y}}$$
(5)

The numerator indicates the amount of light transmitted within the core region of the fiber; while, the denominator represents the total cross sectional area. The electric and magnetic field elements are denoted as Ex, Ey, Hx, and Hy, respectively, with subscripts x and y representing the polarizing mode directions.

Effective refractive index

The effective refractive index is the ratio of the speed of light in a vacuum to the speed of light in the medium for a given polarization in the guided design direction (neff). The Eq. (6) allows one to determine the effective refractive index of any material [58].

$$n_{{{\text{effpm}}}} = \left( {\frac{c}{{vzpm}} = \frac{{\lambda _{0} v}}{{v\lambda _{{zpm}} }} = \frac{{\lambda _{0} }}{{\lambda _{{zpm}} }} = \frac{{2\Pi }}{{k_{0} }} = \frac{{\frac{{2\Pi }}{{k_{0} }}}}{{\frac{{2\Pi }}{{K_{{zpm}} }}}} = \frac{{K_{{Zpm}} }}{{k_{0} }} = \frac{{\beta _{{pm}} }}{{k_{0} }}} \right)$$
(6)

where \({{K}}_{\mathrm{zpm}}=\) \({\upbeta }_{\mathrm{pm}}\), p denotes the polarization (TM or TE) and m denotes the mode of the polarization, \({\upbeta }_{\mathrm{pm}}\) denotes the propagation constant and \({k}_{0}=\frac{\omega }{C}\)=\(\frac{2\Pi v}{C}\)=\(\frac{2\Pi }{{\lambda }_{0}}\) denotes the number of waves in vacuum, the symbol \({\lambda }_{0}\) represents the wavelength, and \(v\) represents the frequency in a vacuum. By examining Eq. (6), it can deduce that the effective refractive index is dependent on both the frequency and propagation constant.

Nonlinearity

The proportion of light intensity to confine is largely determined by the nonlinear coefficient of the PCF, which is denoted as ‘ɣ’. Nonlinearity is closely related to effective area of the fiber. The nonlinear coefficient can be evaluated by using the Eq. (7) [24].

$$\gamma =\frac{2\Pi }{\lambda }\left(\frac{{n}_{2}}{{{A}}_{\mathrm{effect}}}\right)$$
(7)

The nonlinearity of the fiber material is denoted by n2, and \({{A}}_{\mathrm{effect}}\) represents the effective area. The higher level of nonlinearity is accountable for the phenomenon of supercontinuum generation (SCG).

Dispersion

The adoption of Zeonex like a fiber material may result in low material dispersion [52]. Consequently, the dispersion may be avoided, and the only dispersion that is the waveguide dispersion must be considered. PCF has the ability to inherit multichannel transmissions, which could be examined more easily with flat dispersion. As a result, the performance of signal propagation from the transmitter to the receiver can be enhanced. The dispersion (D) of suggested sensor is calculated using the Eq. (8) [56, 58].

$$D\; = \;\left[ {\frac{2}{C}\frac{{d{\text{neff}}}}{{d\omega }} + \frac{\omega }{c}\frac{{d^{2} {\text{neff}}}}{{d\omega ^{2} }}} \right],{\text{ }}\left( {{\text{ps}}/{\text{THz}}/{\text{cm}}} \right)$$
(8)

where \(\omega\), c and neff denote the angular frequency, speed of light and the effective refractive index, respectively.

Results and discussion

A series of information of the proposed biosensor are acquired from finite element method (FEM)-based software for detecting CL characteristic, EA, birefringence and relative sensitivity characteristic along the x and y axes. The operating frequency is 1.8 THz. In the present paper, two sorts of cells: cancerous and normal cells have been identified and examined.

Figure 4a represents the birefringence of cancerous cells while 4(b) shows normal cells’ birefringence. The calculated birefringence is 0.00205, 0.02048, 0.000194 for cancerous cells, and 0.00205, 0.00233, and 0.00302 in case of normal cells for OPT + 2%, OPT and OPT -2% case, respectively.

Fig. 4
figure 4

a Birefringence plot of Breast cancer cells and b Normal cells

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The birefringence property of the designed biosensor has been calculated and presented in Fig. 4a and b by using Eq. (1). The projected PCF's birefringence decreases as the working frequency increases, as shown in Fig. 4a and b. With increasing frequency, the index contrast between the polarization modes increases, which results to decrease in birefringence. Zeonex is a transparent thermoplastic material commonly used in various optical applications due to its excellent optical properties and low birefringence. Unlike most anisotropic materials, Zeonex has a negligible intrinsic birefringence, which means that its refractive indices for the two orthogonal polarization components are very close to each other across a wide range of frequencies. The reason behind Zeonex's low birefringence lies in its molecular structure. Zeonex is designed to be optically isotropic, meaning that its molecules are arranged randomly without any preferred direction or alignment. As a result, the refractive indices of Zeonex do not significantly vary with the direction of light polarization, leading to minimal birefringence.

Similarly, Fig. 5a, b and c, d represents the cancerous cells’ confinement loss plot and normal cells’ confinement loss plot in x and y-direction, respectively. The cancerous cells’ confinement loss values are 17.3328 × 10–9, 66.844 × 10–9, and 14.5444 × 10–10 and normal cells’ confinement loss values are 8.3649 × 10–8, 24.26592 × 10–10, and 29.5411 × 10–10 for OPT + 2%, OPT % and OPT- 2% case, respectively.

Fig. 5
figure 5

a and b Confinement loss in breast cancer cell in x and y-axis and c and d normal cell in x and y-axis

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The optically confined result, as well as a part of the designed PCF structure, can be seen in Fig. 5a, b and c, d. The smaller the confinement loss (CL), the higher the optical confinement into the PCF's core region. Light confinement within the core region greatly enhances if air holes occur in the cladding region of PCF. If an infinite number of air holes are deployed in PCF, the CL must be zero, as per the mathematical framework. Numerical air holes were required in real-world applications, and therefore appear to be simple to generate inside a PCF.

From Fig. 5, it is observed that a rise in frequency reduces confinement loss because wavelengths are lowered as frequency increases, and all these relatively small wavelengths are more concisely confined within the core. It can be concluded that y-polarization suffers less loss than x-polarization, which could be attributed to the fact that y-polarization strongly confining more light in the core area than x-polarization [24].

Figure 6a represents the cancerous cells’ effective area and (b) depicts normal cells’ effective area. The cancerous cells’ effective areas are 3.04 × 10−8μm2, 3.41 × 10−8μm2, and 2.81 × 10−8μm2 and normal cells’ effective areas are 2.20 × 10–8 μm2, 2.56 × 10−8μm2, and 2.93 × 10−8μm2 for OPT + 2%, OPT and OPT-2% case respectively.

Fig. 6
figure 6

a Effective area in breast cancer cell and b normal cell

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Effective area decreases with increasing frequency for cancerous cell at different OPT + 2%, OPT, OPT −2%. Similarly, the effective area of normal cell decreases with increasing frequency for cancerous cell at different OPT + 2%, OPT, OPT −2%.

The variation beta frequency and effective area of the studied PCF can be seen in Fig. 6a and b. Effective area of the studied PCF with analyte reduces almost linearly with a rise in operating frequency, as shown in Fig. 6a and b. At 1.8 THz, the photonic crystal fiber experiences a transition from one band gap to another (a change in the mode profile due to a shift in the photonic band structure). This leads to a sudden reduction in the effective area as the guided mode becomes more confined.

Fundamentally, the effective area of a PCF in the transverse dimension is the proportion of cross sectional area it covers. The electric fields in the fiber have determined effective area, which decreases as frequency increases. At lower frequencies (longer wavelengths), the photonic crystal lattice structure can effectively confine light to a smaller region, resulting in a smaller effective area. This occurs because longer wavelengths allow light to interact more strongly with the periodic structure, leading to enhanced mode confinement.

Conversely, at higher frequencies (shorter wavelengths), the photonic crystal lattice may become less effective at confining light, leading to a larger effective area. In other words, as the wavelength decreases, light tends to spread out over a larger area, reducing mode confinement and thus increasing the effective area.

Figure 7a, b and c, d describes the effective RI of cancerous cells and normal cells, respectively. The effective refractive indices (neff) of cancerous cells are 0.89995, 0.88728, and 0.89950. Normal cells’ effective refractive indices are 0.89205, 0.89087, and 0.89356 for OPT  + 2%, OPT and OPT −2% case correlatively.

Fig. 7
figure 7

a and b Effective refractive index in breast cancer cell along x and y-direction. c and d Effective refractive index in normal cell along x-direction and y-direction

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Figure 7a, b, c, and d describes the alteration between the frequency and effective refractive index. The air holes of core in the proposed PCF are asymmetric in design, resulting into a considerable variation in refractive index between the polarization modes and therefore attaining high birefringence. With the progressive increase in frequency, it has been observed that the variation in effective RI is finite, and this property of the studied fiber gives very low dispersion.

For light signals with a range of frequencies, the effective refractive index is a weighted average of the refractive indices corresponding to those frequencies. Since higher-frequency light tends to experience a higher refractive index, it can contribute more to the effective refractive index, resulting in an increase in the effective refractive index with increasing frequency.

Figure 7a and b depicts that the effective refractive index (neff) rises with rising frequency for cancerous cell. From Fig. 7c and d, it is evident that the effective RI increases as the frequency increases, which could be because a higher frequency (lower wavelength) electromagnetic signal chooses to pass through a higher refractive index region. At 1.8 THz, the photonic crystal fiber experiences a significant change in its photonic band structure, leading to a sharp change in the effective refractive index. This is due to the alignment of certain periodicities in the PCF structure with the wavelength of light at that specific frequency.

Figure 8a, b represents the cancerous cells’ and normal cells’ power ratio. The cancerous cells’ power ratio values are 48.26562, 47.01512 and 16.1884 and normal cells’ power ratio values are 44.99257, 28.68942 and 18.20241 for OPT + 2%, OPT and OPT-2% case respectively. Power increases with increasing frequency and get constant at 1.8 THz frequency for breast cancerous cell. Similarly, for normal cell, Power increases with increasing frequency and get constant at 1.8 THz frequency.

Fig. 8
figure 8

a Power fraction in breast cancer cell and b normal cell

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The variation between power ratio of proposed PCF and the operating frequency is shown in Fig. 8a and b. It has been noticed that PCF’s power fraction increases as frequency rises. At 1.8 THz, the photonic crystal fiber experiences a change in its photonic band structure, leading to an abrupt change in the power ratio. The abrupt decrease in the power ratio is attributed to change in light confinement, resulting in a stronger interaction with the material being sensed. For the propagation mode, the optical power passes through the proposed core, and as frequency rises, light leaks from the core volume to the cladding, causing the progressive attitude [59].

Figure 9a, b and c, d represents the cancerous cells’ relative sensitivity and normal cells’ relative sensitivity, respectively. The cancerous cells’ relative sensitivity values are 65.5347%, 24.5076% and 18.2448% and normal cells’ relative sensitive values are 60.7770%, 48.6416% and 36.5164% for OPT  + 2%, OPT and OPT −2% case, respectively.

Fig. 9
figure 9

a Relative sensitivity for breast cancer cell along x-direction and b y-direction and c For normal cell in x-direction d y-direction

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With increasing frequency, relative sensitivity increases and becomes constant at 1.8 THz frequency for different cases optimum  + 2%, optimum, optimum-2% for breast cancer cell. Similarly, for normal cell with increasing frequency relative sensitivity increases and becomes constant value at 1.8 THz frequency for different cases optimum  + 2%, optimum, optimum-2%.

As the input frequency deviates from the system's resonance frequency, the system's sensitivity diminishes. However, as the input frequency approaches the system's resonant frequency, the sensitivity becomes more pronounced, leading to an increase in relative sensitivity. The abrupt change in sensitivity at 1.8 THz is related to the photonic band structure and the existence of resonances within the photonic crystal fiber. At this specific frequency, the sensor experiences a change in the photonic band structure, leading to significant variations in the mode profiles and the interaction of light with the surrounding medium.

The relative sensitivity of the x-polarized mode is always greater than that of the y-polarized mode. It is possibly because the x-polarized mode has a lower portion of energy distributed outside of the analyte-filling area, such as in strut and the air region in the cladding neighboring the core. Especially, light interacts more strongly with x-polarized mode. So, the x-polarized mode has been selected as the optimum mode for proposed PCF and the characteristics of the x-polarized and y-polarized mode have been exhibited [60].

Figure 10a represents the nonlinearity of the breast cancerous cells and (b) depicts normal cells respectively. The cancerous cells’ nonlinear coefficient values are 6.3769 × 1016, 4.32567 × 1016, 4.8837 × 1016 and the normal cells’ nonlinear coefficient values are 6.1493 × 1016, 5.5801 × 1016 and 4.9343 × 1016 for OPT + 2%, OPT and OPT −2% case, respectively. Similarly, both figures demonstrate the variation between nonlinear coefficient and operating frequency of the designed PCF sensor. It is seen that nonlinear coefficient of the PCF increases nonlinearly with increase in frequency.

Fig. 10
figure 10

a Nonlinearity for cancerous cell and b normal cell

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From Eq. (7), it is evident that nonlinearity has an inverse relation with effective area, which could be seen by comparing Fig. 6a and b. At 1.8 THz, the photonic crystal fiber encounters specific bandgap effects that affect the propagation of light through the fiber. These bandgap effects and resonances leads to changes in the effective refractive index, dispersion properties, and mode interactions. As a result, the nonlinearity exhibits an abrupt change at this frequency, causing it to decrease.

The nonlinear frequency dependence in PCFs is a result of the interaction between light and the material's response at different frequencies. It arises due to complex interactions between the light and the electronic and atomic properties of the medium. At higher frequencies (shorter wavelengths), light carries more energy and interacts more strongly with the material's electrons and atoms. This increased interaction leads to enhanced nonlinear effects, resulting in a higher nonlinear coefficient at higher frequencies.

Figure 11a shows the breast cancerous cells’ dispersion plot and (b) normal cells’ dispersion plot. The cancerous cells’ dispersion values are 0.1588 ± 0.0121, 0.1632 ± 0.0113, 0.1660 ± 0.0241 and the normal cells’ dispersion values are 0.1852 ± 0.014, 5.5801 × 1016 and 4.9343 × 1016 for OPT + 2%, OPT and OPT-2% case respectively.

Fig. 11
figure 11

a Dispersion for the breast cancer cell and b normal cell

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From the Eq. (8), it is evident that D relies on the alteration of the effective refractive index. The dispersion variation relies only on the effective refractive index of the waveguide. The obtained dispersion is very small and practically flat, allowing for much easier correlation signal transmission with equivalent pulse expanding. As the effective refractive index varies with frequency, different frequency components of light experience different velocities, leading to group velocity dispersion and affecting the temporal characteristics of light pulses. Understanding and managing dispersion are critical in various optical communication and laser applications to optimize performance and signal integrity.

Table 1 presents the optical characteristics of the proposed PCF based on PbS along with its two variations. This table illustrates that a minor change in structural model throughout fabrication would not result in any issue. Thereby ensure uniform range across all optical parameters but apart from a large degree of variation in the optical characteristics the projected PCF based on PbS is compared to existing PCF based on PbS in Table 2. From Table 2, it is evident that execution of the assumed PbS is finer as analyzed to recently proclaimed PCF-based sensors hence the proposed PCF based on PbS performs exceptionally well in detecting cancerous cells.

Table 2 Performance comparison among different PCF based on PbS
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Conclusion

For optical characteristics accompanying great yielding, the advised photonic crystal fiber sensor for breast cancerous cell identification displays guided treasure. The excellent sensing execution of the studied sensor helps in identifying cancerous cells, which indicates the efficiency and efficacy of the proposed sensor. At frequency range of about 2.0 THz, the electromagnetic field resonate with the specific vibrational modes of the cancer cells or biomolecules, leading to enhance the phenomena of absorption or scattering. Furthermore, the resonant frequency corresponds to the bandgap or photonic crystal mode of the fiber, which affects the transmission or reflection of light. By which the PCF sensor is more applicable in this range to detect the cancer cells. The operating conditions for this PCF sensor model are as follows: the related optical properties at the frequency of 1.8 THz are the best breast cancer sensor models possessing 17.33 × 10 −9 dB/cm CL, 3.04 × 10 −8 µm2 EA and 0.0020 birefringence, effective refractive index 1.376, 65.53% and 53.63% relative sensitivity in the y and x -axes, nonlinearity 6.38 × 10 16, and dispersion 0.1588 ± 0.0121. Furthermore, the suggested sensor model is fabricable adequately by putting existing fabrication strategies to the test. Hence the reported sensor a satisfactory candidate for the detection of cancerous cell.