Introduction

Presently there is no medicine which can eradicate cancer completely. The single option to fight cancer is to diagnose it at slightly earlier stages accurately. Cancer diagnosis can be done by employing the optical and biochemical performance of cancerous cells [1]. Because various cells’ responses to light are different, this property could be investigated to identify different cells proficiently. Furthermore, early cancer detection could play an important role in lowering cancer-related mortality rate. As a result, recently research related to an efficient and effective method for early cancer diagnosis has drawn much attention across the globe. In fast paced life, cancer––the unlimited development of cancer cells in the human body has taken place unhealthy, fatal illness specified by high daily death rate [2, 3]. This kind of uncontrolled tumor growth is a collaborative result of excessive interaction to toxic materials, emitting sources, and chronic materials, strong liquor consumption, greasy health, an unbalanced diet, low physical activity, nutritional insufficiency, and so on [4]. If the damaged cells are not spotted at a primitive phase, the cancer could develop all over the body, eventually killing the patient. Various blood analysis (e.g., blood protein testing, tumor marker, complete blood count, tumor marker testing etc.) and urine analysis have been generally performed to detect cancer, but the consequences could not explicitly state the negatively damaged organ [4]. Biopsy has been the standard method for recognizing cancerous cells, and it requires taking samples of suspected cells for testing [5]. Positron emission tomography (PET), ultrasound scan, magnetic resonance imaging (MRI) and computed tomography (CT) might be other prevalent tests for detecting cancer [6, 7]. Moreover, these tests have been time-consuming (varying from 2 to 30 mins to 2–3 days), that could have negative consequences for cancer patients who require prompt diagnosis [8]. Consequently, optoelectronics researchers have been attempting to develop fast cancer cell identifiers. Cancerous and normal cells have distinct refractive indices (RIs) and show different propagation properties when light given from a similar source. Therefore, to detect the characteristics of cell could be detected by investigating the received signal [9].

The terahertz frequency spectrum has been known as the IR-microwave transmission bridge. THz waves are a subcategory of the electromagnetic spectrum from 0.1 to 10 THz frequency range. The terahertz frequency’s outstanding proficiency makes it relevant in a wide area of usage. The most essential of these applications have been explosive threat detection, medical, sensing and real-time imaging, environmental [10, 11].

Photonic crystal fibers (PCF) are a kind of optical fiber in which air holes have been designed randomly or systematically across the cross section [12]. PCF not only reduces the risk of uncontrollable losses through free space [13], but these are also small, inexpensive robust, and relevantly, extremely delicate as well as relevant [14]. There have been two kinds of PCF based on their guiding techniques: photonic bandgap fibers that use the photonic bandgap mechanism [15], and index guiding fibers, which use modified total internal reflection [16]. Due to the extremely low energy of each photon, THz-based sensors have been strongly efficient of penetrating porous samples except damaging the test layer [12, 17].

The versatility in developing the cross section represents the most beneficial characteristic of PCFs which characterizes it from conventional fiber optics [18]. This enables researchers to achieve the desired waveguide characteristics, such as reduced confinement loss [19] and tremendously high sensitivity [20], high birefringence [21] and numerical aperture [22], large modal area [23], and ultra-fat near-zero dispersion [24, 25]. So, plenty of complicated PCF cross sections have been developed over the years, such as hexagon shaped [26], octagon shaped [27], spiral [28], elliptical [29], honeycomb [30], and hybrid [31] claddings.

The representative material in PCF has also been a key factor of investigations. However, metallic optical fibers were previously utilized. Current research is concentrating on optical polymers such as SILICA [32], PMMA [33], ZEONEX [34], TEFLON [35], TOPAS [36] and TELLURITE [37]. Optical polymers have been well-known for their low cost, impact resistance, and ease of integration with optical and mechanical systems [38]. THz sensors based on PCF have been broadly employed in air quality monitoring of our surroundings [39], identification of toxic substances and explosive materials such as cyanides [40],blood component evaluation[41, 42], alcohol identification in beverages [43] and leaking nature of highly flammable and dangerous gases such as hydrogen sulfide and methane in industry sectors because of their enormous popularity.

Sharan et al. [44] proposed the photonic bandgap technique to identify cancer cells in 2014. Moreover, the photonic band gap technique has complexity in the manufacturing procedure. The thermal profile has an influence on the result. Sharma et al. [45] reported a 2D photonic crystal fiber-based sensor for the analysis of cancer cells in cervical, breast and basal cases. The working wavelength region has been extremely narrow. Furthermore, the manufacturing tolerance was not presented. Ramanujan et al. reported a geometrical covered nanocomposite material-based photonic crystal fiber for identifying cancer cells in 2018 [46].They were using surface plasmon resonance (SPR)-based sensor and obtained extremely small sensitivity. Shimohammadli et al. [47] developed a microfluidic apparatus for analyzing breast cancer in the middle of 2018. Ayynar et al. [48] reported a dual-core PCF model that could detect breast, basal and cervical cancers during the same year. Despite the enhanced sensitivity response, the identification limit has been extremely small. Jabin et al. [49] proposed a cancer model based on surface plasmon resonance for the rapid identification of several kinds of cancer cells with enhanced sensitivity performance and identification limit in 2019. Furthermore, a novel amoeba-shaped design has been constructed, and major optical parameters have been evaluated, but the relative sensitivity performance did not come as satisfactory compared to the other parameters [50]. To overcome the ongoing difficulties and to rise the relative sensitivity the present model has been designed and analyzed. As per the knowledge of authors, eminently sensitive mono-rectangular photonic crystal fiber (MRC-PCF)-based sensor for cancer cell detection in THz region is not reported so far.

In the present work, a close-packed cladding PCF has been proposed. The proposed model is simple in structure with high relative sensitivity and reduced confinement loss which delivers simple design with enhanced outcome for the identification of cancer cells and normal cells. Several characterizations have been investigated in the proposed work like relative sensitivity, confinement loss, effective area, effective refractive index, birefringence, and propagation constant. The process for mathematically calculating the performance of the studied model is also presented in Fig.1. Finally, the results are shown, along with a conclusive analysis.

Fig.1
figure 1

Schematic sketch of proposed sensor design and analysis

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Design methodology

In this design, the finite element method (FEM) has been used. Firstly, the geometry of the model is designed. Furthermore, material has been applied as per their respective area. Zeonex is employed in the background material as well as perfectly matched layer (PML). Air has been inserted in the cladding region and analyte in the core region. The normal cells and cancer cells as analytes have been filled into core region. The PML aims to prevent light from reflection and to prevent the PCF from the consequences of the surrounding environment, but it makes no contribution to practical analysis. Boundary conditions have been initialized and mesh setup has been done. Thereafter, the partial differential equation (PDE) algorithm has been done. From the PDE solutions, optical performance parameters have been analyzed. This consecutive process has been shown in Fig.1.

The pictorial depiction of the constituents symbolic for our predictable MRC-PCF model has been represented in Fig.2. The primary aim of this model is to make a unique manufacturer-friendly PCF design. A further essential aim is to obtain standard values for the optical characteristics. The PCF using only rectangles has been designed which confirms that such manufacturing is viable now. The optimized MRC-PCF shown in Fig. 2 has been observed after various investigations of altering the design (shifting the rectangle numbers, adjusting the width and length, modifying the rectangle locations, etc.). The presented MRC-PCF has a complete radius of 1750 µm and it is associated with a perfectly matched layer (PML) on the internal surface having thickness 8% of the radius. PML acts as an anti-reflection coating and avoid signals departing the fiber [51]. The core area has been shown in the diagram by a single rectangle represented by Rc. The core’s height and width have been taken as 500 and 920 µm, respectively. Fourteen rectangular air holes have been organized irregularly in the cladding region. Same symbols have been used to represent the similar rectangles. For example, the core’s nearest right and left rectangles have been similar and therefore indicated by a single symbol R1. Going to follow this strategy, designed six different rectangle structures are indicated by R1, R2, R3, R4, R5, and R6 , the height is (H) and width is (W) for every rectangle represented in Fig.2. The studied design has a pitch of 310 µm and a strut of 10 µm between two sequential rectangles. Zeonex has been chosen as our fiber component due to its unique properties. In the THz region, Zeonex has key beneficial property as its immobile RI of 1.53. Furthermore, Zeonex has been acquired with high chemical resistance, glass transmigration warmth, and humidity insensitivity [52].

Fig.2
figure 2

Configuration of the proposed PCF model a representative and b fine mesh analysis

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Various fabrication processes of PCF have been used so far. For example, 3D printing, extraction, Sol–gel etc. [53]. To fabricate irregular design, both extraction and 3D printing have been examined favorably. PCF designs as rectangles have already been manufactured in Planck Institute. The manufacturing viabilities of the studied optimized PCF are described in the literature [54].

Model analysis

In this study, physics-controlled mesh size is employed for better outcome. Complete mesh consists of 55584 domain elements and 3390 boundary elements of the fiber. Popular FEM has been employed for the studied PCF. Finite element method changes the microfabricated model into the ordinary differential equation. Finite element method is precise and gives designing of complicated structures before manufacturing. It also gives the complete analysis through which the design can be enhanced. Figure 2 shows the mode field distribution at 2.1 THz frequency for normal and cancer cell as analyte and gives best result. It can be noticed from Fig.3 that field intensity is more at the midpoint and then reducing continuously within the radius of the fiber which shows a Gaussian property. It can also be noticed that light has been airtight limited within the core area, which establishes a strong interaction between material and field to be sensed. As a result, the relative sensitivity of the proposed sensor fiber would be high. For model analysis, evaluation of the waveguide characteristics against important parameter, the operating frequency of the THz range has been chosen. For analysis, x polarization has been chosen and y polarization has been ignored because light confinement has been improved significantly in the x polarization mode, with improved relative sensitivity behavior and reduced confinement loss.

Fig.3
figure 3

Mode field distribution of proposed PCF model a x-polarized and b y-polarized

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Using FEM in 1.0–2.5 THz frequency range, the optimum MRC-PCF has been proposed. To analyze the performance of the model, implementing the material to be examined into the core, various performance parameters have been calculated. The relative sensitivity (RS), effective area (Aeff), confinement loss (CL), birefringence (B) and effective refractive index (neff), propagation constant (β) have been examined.

It shows that intensity of light has strong interaction with the analytes, while exhibiting less interaction in the cladding area. The results obtained for studied PCF show enhanced sensitivity. Sensitivity is directly proportional to the analytes’ refractive index.

Confinement loss

The confinement loss (CL) fundamentally arises because of optical power leakage from core to cladding because of introduction of a finite air holes. The CL can be evaluated using effective refractive index imaginary part. It can be lowered by appropriately ordering the core holes and cladding. The numerically CL is calculated using Eqn. (1) [55, 56].

$$CL = \frac{4\pi f}{c} \times {\text{Img}}\left( {n_{{{\text{eff}}}} } \right),{\text{ cm}}^{ - 1}$$
(1)

where neff and f represent the effective refractive index and frequency in THz region, respectively.

Figure 4 indicates that the confinement loss is very low. It indicates that confinement loss reduces as the frequency rises. From Fig. 4, it can be observed that an inverse relation between confinement loss and frequency holds. The confinement loss recorded at 2.1 THz frequency are 5.828 × 10−25 cm−1, 3.072 × 10−27 cm−1 for cancer and normal cells, respectively.

Fig.4
figure 4

Confinement loss vs frequency between the cancer cell and normal cell

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Confinement loss occurs by optical mode leakage from the core of the fiber. The air holes inside the cladding region have been organized so that the refractive index difference has been sufficient to preserve the optical field within core region. Light travels away from the core and into the cladding region due to insufficient refractive index contrast, expanding confinement loss. As a result, creating effective cladding with adequate air pitch and air-hole diameter values will assist in lowering confinement loss, which may be lowered to minimal levels by appropriate cladding structure design [57,58,59].

Birefringence

Birefringence is the most essential parameter of optical characteristics which has been employed widely in coherent optical communications and in the manufacturing of fiber optic sensors. The electric field of a propagating electromagnetic wave has always been associated in a specific direction, which is known as polarization. When a linearly polarized light penetrates a fiber core with an anisotropic refractive index distribution, it divides into two mutually orthogonal polarized modes. These two orthogonal modes propagate at varying speeds based on the refractive index of the fiber along the direction of polarization. The birefringence is the difference in refractive indices along two mutually perpendicular directions [57]. This parameter represents the polarization-controlling ability of photonic crystal fiber. Birefringence can be evaluated using the refractive index (RI) difference between the x (nxplane) and y (nyplane) plane as shown in Eqn. (2) [55, 56].

$$B = \left| { \, n_{{{\text{xplane}}}} - \, n_{{{\text{yplane}}}} } \right|$$
(2)

where nxplane and nyplane indicate the effective refractive index of x and y polarizing modes, respectively.

From Fig.5, it is obvious that the change of birefringence can be represented with the variation of frequency range. Because of the tight confinement at higher frequencies, the refractive index mode varies in a decreasing direction after a specific frequency. The modeling of an asymmetric PCF design produces a noteworthy refractive index difference. For the studied PCF, birefringence has been obtained as 4.709 × 10−4 and 4.885 × 10−4 for cancer and normal cells, respectively.

Fig.5
figure 5

Birefringence vs frequency plot for the cancer cell and normal cell

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Because the birefringence in PCFs is caused by a non-axisymmetric allocation of the effective refraction around the core, they are extremely temperature insensitive. Because of such immunity, PCF with high birefringence is very appealing for telecommunication and sensing applications. Such PCF can compensate the polarization mode dispersion in optical fibers.

Effective area

Effective area measures the analyte identification area of any PCF-based sensor. For reducing the deterioration increased by the nonlinear effect, a wide effective area is essential. A low effective area allows for the high-power density required for nonlinear impacts to be considerable. Extension of the electromagnetic wave is based on the strut size and holes size of the fiber. The mathematical parameters of the designed model can be arranged in the similar way. The effective area has been calculated as [55].

$$A_{{{\text{effect}}}} = \frac{{\left( {\smallint \smallint \left| E \right|^{2} dxdy} \right)^{2} }}{{\smallint \smallint \left| E \right|^{4} dxdy}}$$
(3)

Here, the distribution of electric field over the cross section of fiber can be represented by |E|.

From Fig.6, it is noticed that the effective area decreases as the frequency increases. With increasing frequency, it is observed that light confinement takes place through the core region, consequently lowering the effective area of the fiber. At 2.1 THz frequency, the effective area can be attained as 4.445 × 10−8 µm2 and 2.549 × 10−8 µm2 for normal and cancer cells, respectively.

Fig.6
figure 6

Effective area vs frequency for the cancer cells and normal cells

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Relative sensitivity

In MRC-PCF major light travels through the core area. A very small portion of the beam specified as the evanescent wave transfers through the cladding region and the light corresponds with the material to be sensed. The sensitivity evaluating factor measures the light interaction with the selected materials. It can be expressed using Beer–Lambert law as [55].

$$I\left( f \right) = I_{0} \left( f \right)\exp [ - r\alpha_{m} l_{c} ]$$
(4)

where I0(f) and I(f) represent light intensity before and after putting the analyte at the assigned place, respectively. lc denotes the length of the channel, \(\alpha\) m denotes the absorption coefficient, where lc inversely proportional to the \(\alpha\) m, while f denotes the operating frequency and the relative sensitivity ‘r’.

The absorbance of the liquid concentration or evanescent wave can be evaluated by Eqn. (6) [53].

$$A = \log \left[ {\frac{I\left( f \right)}{{I0\left( f \right)}}} \right] = - r\alpha_{m} l_{c}$$
(5)

The relative sensitivity of presented PCF can be calculated by [55].

$$r = \left[ {\frac{{n_{r} }}{{n_{{{\text{eff}}}} }}} \right] \times P$$
(6)

where nr represents the refractive index of assigned materials, neff represents the effective refractive index of guided mode and P shows the ratio of interaction between the light intensity and directed analytes given by Eqn. (7) [55].

$$p = \frac{{\mathop \smallint \nolimits_{{\text{analyte }}} {\text{Re}} \left( {E_{x} H_{y} - H_{x} E_{y} } \right)dxdy}}{{\mathop \smallint \nolimits_{{\text{total }}} {\text{Re}} \left( {E_{x} H_{y} - H_{x} E_{y} } \right)dxdy}}$$
(7)

where Ex and Ey, Hx and Hy shows the electric field and magnetic field elements along x and y modes, respectively.

Figure 7 shows that the studied PCF gives the maximum sensitivity for cancer cell and minimum for normal cell since these holds the higher and lower refractive index, respectively. Relative sensitivity relates to the operating frequency too. With increase in frequency, compression of beam of light occurs and light is slightly constrained within the core with a high intensity which improves the light matter interaction, so sensitivity is improved. With rise in frequency the sensitivity rises till a particular frequency. Because of higher frequency, the beam of light has been more concisely constrained in the core area and strong interaction with the related analyte occurs which improves the power ratio of the core. Furthermore, with rise in frequency the refractive index also increases resulting an increase in sensitivity. The sensitivity is achieved as 81.38 and 65.83% for cancer cell and normal cells, respectively, for the designed PCF.

Fig.7
figure 7

Relative sensitivity vs frequency between the cancer cell and normal cell

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Effective refractive index

The effective refractive index is defined by Eq. (8) [53].

$$n_{{{\text{eff}}}} = \frac{{\lambda_{0} }}{\lambda }$$
(8)

Here, \(\lambda\) represents wavelength of the em wave in the analyte and \({\lambda }_{0}\) shows the wavelength in the vacuum.

The refractive index of the core of the fiber is larger than the cladding so that the wave of the light can travel by the total internal reflection and the maximal quantity of power of the wave can be confined into the core.

Figure 8 shows the variation of the effective refractive index with frequency. Electromagnetic spectrum having larger frequency possess smaller wavelength, resulting in a larger neff, and light propagating in an analyte with a larger neff lessens the wavelength and increases the neff because of the related Eqn. (8). The neff can be achieved as 1.475 × 10−18 and 7.773 × 10−21 for cancer cell and normal cells, respectively, for the designed PCF.

Fig.8
figure 8

Effective refractive index vs frequency for the cancer cells and normal cells

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A high-refractive index core area has been bounded by a reduced refractive index material that serves as the cladding. The refractive index of the cladding has been reduced by constituting a microstructure of air holes in a predictable pattern which follows the length of the fiber. Because the interface between the reflecting core and the cladding is not made clear, the guiding principle has been the modified total internal reflection from the core and effective cladding [57].

Propagation constant

To calculate the number of losses that occurs due to leakage can be predicted by propagation constant (β). Using the imaginary part of the neff, propagation constant can be evaluated with the help of Eqn. (9) [53].

$$\beta = \frac{2\pi f}{c}{\text{Img}}\left( {n_{{{\text{eff}}}} } \right)$$
(9)

where f denotes the frequency, Img (neff) imaginary value of neff and c speed of light.

The propagation constant is evaluated using the consistent values that arises due to the dispersion of incident light from the core to the cladding. Figure 9 shows the variation of β for the cancer cells and normal cells in the 1.0–2.5 THz frequency range. With increase in frequency, propagation constant slightly decreases then becomes almost zero. At 2.1 THz, propagation constant values are − 1.01961 × 10−24 and 9.44071 × 10−27 for cancer cells and normal cells, respectively.

Fig.9
figure 9

Propagation constant vs frequency for the cancer cell and normal cell

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Nonlinearity

The density of high optical power of fiber has been given by a low effective area for which the nonlinear effects are prominent. Nonlinearity Ɣ, known as nonlinear coefficient, calculates the capability of constraining the high-intensity light. It is defined as the inverse of the effective area of the fiber. It can be evaluated using Eq. (10) [30].

$$V = \frac{2\pi f}{c}\frac{{n_{2} }}{{A_{{{\text{effect}}}} }}$$
(10)

where \({n}_{2}\) shows the effective refractive index of the fiber material, \({A}_{\mathrm{effect}}\) denotes the effective area, c represents the speed of light and f represents the frequency in the THz range.

Optical nonlinearity can be employed to construct feasible optical fiber devices. By making large air holes or lowering the core dimension, PCFs could be constructed with a high-refractive index contrast between both the core and the cladding. This design promotes strong confinement of the waveguides, that increases the nonlinear coefficient [57].

Figure 10 depicts that with the increase in frequency nonlinear coefficient increases. As it follows the inverse relation to the effective area (see Fig.6). The nonlinear coefficients are 1.59833 and 1.63352 W−1Km−1 for normal and cancer cells, respectively.

Fig.10
figure 10

Nonlinearity vs frequency for the cancer cell and normal cell

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Conclusion

In the present work, a square core PCF biosensor has been proposed for sensing the cancerous cells as analyte. The studied PCF model sensor shows extremely remarkable high relative sensitivity of 81.38% for cancer cells and 65.83% for normal cells. Furthermore, other noteworthy propagation characteristics––reduced confinement loss of and large effective area has been attained for the analyte at design parameters of 2.1 THz. This model has been preferably satisfactory for sensing of cancer cells and normal cells. The studied PCF gives noteworthy results and could be employed in biosensing applications.