1 Introduction

The body naturally produces cholesterol, which is one of the biomolecules, a steroid found in lipids, and a form of alcohol known as a sterol. Cholesterol is a key component of membranes and a type of fat. Blood also contains cholesterol, which comes from two main sources: diet (25%) and liver (75%) respectively. Foods with high cholesterol can refer to meat, liver, brain, eggs, and dairy products; in contrast, plant foods have no cholesterol [1,2,3].

Following a meal, cholesterol is absorbed by the intestine and then enclosed in a protein coating together with triglycerides. This mixture of proteins and fats is known as a chylomicron [4,5,6]. The liver can both create and eliminate cholesterol from the blood [7,8,9]. The growth of arterial plaques and atherosclerosis development can be influenced by blood cholesterol levels. These plaques build up on the blood vessel walls, especially the coronary vessels, causing them to narrow. This narrowing ultimately reduces blood flow, plugs the vessels, and results in heart attacks. A 4-ring molecule containing 27 carbon atoms makes up cholesterol. This lipid compound has a steroid structure created by four hydrocarbon rings. A hydrocarbon tail is attached to one end of the cholesterol molecule, and a steroid and a hydroxyl group are attached to the other. The hydroxyl group can form a hydrogen bond with the carbonyl oxygen in phospholipids and sphingolipids. Due to its composition of steroids and alcohol, cholesterol falls within the category of sterols [10,11,12].

High-density lipoprotein (HDL): The HDL carries cholesterol from the cell to the liver or carries it outside the body as waste. Because of this, HDL is referred to as "good cholesterol," and the more HDL present, the better.

Low-density lipoprotein (LDL): The cells that require cholesterol are transported there by it. If there is more cholesterol available than what is needed by the cells, excess cholesterol is deposited in the walls of the arteries, which leads to artery-related disorders. LDL is known as harmful cholesterol for this reason. The LDL-blocked arteries are depicted in Fig. 1a [13,14,15].

Fig. 1
figure 1

a phases of vascular obstruction and the development of cholesterol plaques in the veins (b) Plaque in the arteries is kept from forming due to HDL

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According to the Table 1, a normal 70 kg person has 140 g of cholesterol in their body.

Table 1 Cholesterol content of human tissue
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The envisioned structure comprises dielectric rods within an air medium, featuring a resonator nanoring positioned centrally amidst two input and output waveguides. This nanoring serves as the focal point for sensing, with specific rods designed as cylinders or tubes to enable biomolecule attachment. Subsequent iterations involved increasing the diameter of these measuring cylinders, thereby influencing analyte-light interaction and subsequently augmenting the sensor's quality factor and sensitivity. Concurrently, densification of the nanoring's outer wall was achieved by augmenting rod numbers, thereby enhancing light confinement and interaction with target biomolecules. The entrance of the nanoring is adorned with three rows of dielectric rods, where a higher count of these coupled rods correlates with an increased sensor quality factor albeit at the expense of reduced incoming light intensity. To mitigate this intensity reduction, the radius of the third coupled rod was decreased.

1.1 Photonic crystals

By periodically altering the insulating material's refractive index or by forming a regular row of holes or cylinders, a photonic crystal can be made [16,17,18,19,20]. The photonic crystals are alternative micro-nano-structures of dielectric/metal-dielectric, which influence the pathway of the electromagnetic wave. This reflection happens over a wide spectrum of wavelengths. In other words, the photonic band gap (PBG), the foundation of operation in photonic crystals, is the wavelength or frequency of light that cannot pass through the material [21,22,23].

Photonic crystals can confine light better than other optical structures, and this is one of the suitable properties for making photonic crystal sensors. It is one of the photonic crystal structures in the design of biosensors such as blood component detection sensors that can detect blood parameters [24,25,26,27,28,29,30,31,32,33]. Another biosensor created with a photonic crystal structure is the detection of different types of cancer cells [34,35,36] and potential malaria detection [37,38,39,40,41]. Another application of photonic crystals, Salinity, and temperature detection for seawater [42,43,44]. Photonic crystals can also be used to detect chemical gases and dangerous toxins [45,46,47].

2 Materials and methods

Maxwell equations should be solved assumptions of an isotropic, non-magnetic, and charge-free medium to study the propagation of EM waves in a 2D periodic medium [48,49,50,51,52,53].

$$\mathbf{D}(x,z,t)={\varepsilon }_{0}{\varepsilon }_{r}\left(x,z\right){\varvec{E}}(x,z,t)$$
(1)
$$\mathbf{B}(x,z,t)={\mu }_{0}{\varvec{H}}(x,z,t)$$
(2)

where \({\varepsilon }_{r}\) is the relative permittivity of the material and \({\varepsilon }_{0}\) and \({\mu }_{0}\) are the permeability and permittivity of the free space, respectively.

$$\nabla \times H=\frac{\partial D}{\partial t}+{j}_{e}$$
(3)
$$\nabla \times E=-\frac{\partial B}{\partial t}-{j}_{m}$$
(4)

In these equations, E and H are respectively the electrical and magnetic field intensity, additionally, D and B are respectively the electrical and magnetic flux density. \({j}_{e}\) electric current source and \({j}_{m}\) electromagnetic current source.

The following are the two curl formulas for harmonic waves moving through periodic media at frequency \(\omega\):

$$\nabla \times {\varvec{E}}\left(x,z\right)={i\omega \mu }_{0}{\varvec{H}}(x,z)$$
(5)
$$\nabla \times {\varvec{H}}\left(x,z\right)={-i\omega \varepsilon }_{0}{\varepsilon }_{r}(x,z){\varvec{E}}(x,z)$$
(6)

The finite-difference time-domain (FDTD) approach can be used to determine the electromagnetic fields inside Photonic Crystals [54,55,56]. This method uses the FDTD updating equation to calculate the electromagnetic field within the photonic crystal at every instant of time using the field's past and current values [57,58,59]. The FDTD updating equations' source-free Maxwell curl equation in the time domain is:

$$\nabla \times {\varvec{E}}\left(x,z,t\right)={\partial }_{t}{\varvec{B}}(z,z,t)$$
(7)
$$\nabla \times {\varvec{H}}\left(x,z,t\right)={\partial }_{t}{\varvec{D}}(x,z,t)$$
(8)
$${H}_{x}^{p}=\left(m,n\right)={H}_{x}\left(m\Delta x,n\Delta y,p\Delta t\right)$$
(9)
$${H}_{z}^{p}=\left(m,n\right)={H}_{z}\left(m\Delta x,n\Delta y,p\Delta t\right)$$
(10)

is calculated by applying the central difference to the Eqs. (1)–(2). In this case, the integer indexes \(m\), \(n\), and \(p\) are connected to the \(x\), \(y\), and \(t\), respectively. The following are the pertinent FDTD updating equations:

$${E}_{y}^{p+1}=\left(m,n\right)={E}_{y}^{p}\left(m,n\right)+\frac{\Delta t}{{\varepsilon }_{y}(m,n)\Delta \text{z}}\left[{H}_{x}^{p+\frac{1}{2}}\left(m,n\right)-{H}_{x}^{p+\frac{1}{2}}(m,n-1)\right]-\frac{\Delta t}{{\varepsilon }_{y}(m,n)\Delta \text{x}}\left[{H}_{z}^{p+\frac{1}{2}}\left(m,n\right)-{H}_{z}^{p+\frac{1}{2}}(m-1,n)\right]$$
(11)
$${H}_{x}^{p+\frac{1}{2}}={H}_{x}^{p-\frac{1}{2}}\left(m,n\right)+\frac{\Delta t}{{\mu }_{x}(m,n)\Delta \text{z}}\left[{E}_{y}^{p}\left(m,p+1\right)-{E}_{y}^{p}(m,n)\right]$$
(12)
$${H}_{z}^{p+\frac{1}{2}}={H}_{z}^{p-\frac{1}{2}}\left(m,n\right)+\frac{\Delta t}{{\mu }_{z}(m,n)\Delta \text{x}}\left[{E}_{y}^{p}\left(m+1,p\right)-{E}_{y}^{p}(m,n)\right]$$
(13)

FullWave offers a versatile simulation package designed to analyze the propagation of light waves in diverse geometries. It relies on the finite-difference time-domain (FDTD) technique, a well-established method for solving Maxwell's equations without approximations or theoretical limitations. Widely utilized in integrated optics, FDTD excels in scenarios where alternative methods like the Beam Propagation Method (BPM) struggle with complex geometries or fail to provide satisfactory solutions. By directly solving Maxwell’s curl equations, FDTD captures a broader range of effects compared to approximate methods. Imagine a region of space which contains no flowing currents or isolated charges. Maxwell's curl equations can be written in Cartesian coordinates as six simple scalar equations [71,72,73,74]:

$$\frac{{\partial H}_{x}}{{\partial }_{t}}=\frac{1}{\mu }\left(\frac{{\partial E}_{y}}{{\partial }_{z}}-\frac{{\partial E}_{z}}{{\partial }_{y}}\right)$$
(14)
$$\frac{{\partial E}_{y}}{{\partial }_{t}}=\frac{1}{\varepsilon }\left(\frac{{\partial H}_{x}}{{\partial }_{z}}-\frac{{\partial H}_{z}}{{\partial }_{x}}\right)$$
(15)

The remaining four are analogous counterparts derived by cyclically interchanging the subscripts and derivatives of x, y, and z. Maxwell’s equations depict a scenario where the evolution of the electric field (E) over time is linked to the spatial fluctuations of the magnetic field (H), and conversely.

The boundary conditions at the spatial boundaries of the computational domain require meticulous attention. Numerous simulations utilize absorbing boundary conditions to remove outward-propagating energy reaching the domain boundaries. Among the most efficient methods is the perfectly matched layer (PML), where electric and magnetic conductivities are introduced to maintain a constant wave impedance, effectively absorbing energy without causing reflections. The use of the perfectly matched layer (PML) technique is being considered for boundary settings along the structure's edges. Employing PML enables enhanced absorption of electrical and magnetic energy, minimizing reflections [76,77,78].

In this structure PML width is 0.5 µm and boundary-max, boundary-min are 7.2 and −7.2 Respectively.

There are two methods used by photonic crystal sensors' measurement mechanisms:

  • Resonance wavelength shift measurement

  • Determining the transmission spectrum's intensity at its resonant wavelength

The resonance wavelength in the structure will change in the first method when a physical sensor parameter is changed or when a biomolecule is connected to the biosensor. Such sensors are improved by reducing the full width at half maximum (FWHM).

In the second method, the resonant intensity of wavelength changes as the analyte is added to the sensing surfaces, and a physical sensor parameter is changed. Compared to the first procedure, these take fewer tools. This method's limitation to a narrow range of wavelengths is one of its disadvantages. The following has been done to analyze certain photonic crystal sensor parameters [68,69,70,71,72]:

2.1 Quality factor

One of the main parameters in a photonic crystal biosensor is the quality factor. This parameter can be obtained according to the following equation:

$$\text{Q}=\frac{{\lambda }_{0}}{{\Delta \lambda }_{FWHM}}$$
(16)

In the above relation \({\lambda }_{o}\) is the central resonance wavelength of the structure and \({\Delta \lambda }_{FWHM}\) is the full width at half maximum for the central spectrum.

2.2 Sensitivity

Changes in the connection of the analyte to a sensor will lead to changes in the signal transduction of the photonic crystal biosensor. The sensitivity of a biosensor can be obtained according to the following two equations.

$$s=\frac{{\Delta \lambda }_{0}}{m}$$
(17)

In the above relation, \({\Delta \lambda }_{0}\) is the displacement of the transmission spectrum and m is the biomolecule weight.

$$s=\frac{{\Delta \lambda }_{0}}{\Delta n}$$
(18)

In the above relation, Δn is the refractive index change.

2.3 Full-width at half-maximum (FWHM)

Due to this parameter, the wavelength distance between half of the output power can be obtained.

2.4 Free spectral range (FSR)

This range indicates the distance between the two resonance peaks at the output. This parameter plays an important role in the selectivity of a photonic crystal biosensor.

The next important parameter is DL, which is obtained according to Equation 14-16.

$$\text{DL}=\frac{\lambda }{10SQ}$$
(19)

In the above relation, λ is equal to the resonance wavelength, S is equal to the sensitivity and Q is equal to the quality factor.

The next parameter is the competency factor or FOM, which is obtained according to Eqs. (14–16).

$$\text{FOM}=\frac{SQ}{\lambda }$$
(20)

\(S\) is the sensitivity, \(Q\) is the quality factor, and \(\lambda\) is the same as the resonance wavelength in the equation above.

The design and simulation of PCRR (Photonic Crystal Ring Resonators) are facilitated by RSoft software. The simulations and calculations are conducted employing the finite-difference time-domain (FDTD) method. To utilize this method effectively, precise meshing and time calculations are imperative. The meshing size for the structure is set to \({\Delta }_{x}\)=\({\Delta }_{z}\)=25 nm, corresponding to the fixed size of the network, which is 600 nm. The time step equation is also taken into consideration [75].

$${\Delta }_{t}=\frac{1}{c}\sqrt{{(\frac{1}{c})}^{2}+{(\frac{1}{{\Delta }_{z}})}^{2}}$$
(21)

Here, \({\Delta }_{t}\), \({\Delta }_{x}\), and \({\Delta }_{z}\) represents the time step, mesh size in the x-direction (\({B}_{x}\)), and mesh size in the z-direction, respectively. C represents the speed of light. With the number of time steps set to 5000 for each simulation, the time step required for calculations is determined as \({B}_{x}\)=0.016.

2.5 Photonic crystal biosensor

Electronics, optical communication, and biosensors are just a few fields in which photonic crystals are used. In photonic crystal-based biosensors, the refractive index of the photonic crystal structure is initially changed by the identification of the measured parameter. Accordingly, including the desired substance in the photonic crystal structure will change that region's refractive index, which will change the transmission spectrum [60,61,62,63, 79].

Based on the changes in the refractive index in the FDTD method [64, 80, 81], a photonic crystal biosensor is created to identify cholesterol, HIV, and sickle cell anemia. A blood sample is placed within the sensor's hole cavity, and a Gaussian light source with a 1.5–1.6 mm wavelength is injected into the waveguide.

A biosensor, based on dielectric rods in the air is introduced in Ref [65]. Between the waveguide input and output is an embedded oval-shaped nanoring in this structure. A dielectric rod is positioned between the input waveguide and the nano ring to raise the quality factor.

A one-dimensional photonic crystal defect-based biosensor is proposed in Ref [66]. The TMM approach has been used in this biosensor to investigate its performance. The concentration of blood keratin, defect layer thickness, and other information can be obtained using the parameters that were obtained.

A biosensor based on air holes in the silicon plate is introduced in Ref [67]. Defects in the shape of a capsule between two waveguides make up this biosensor. The center of the structure is where these breaches, which resemble capsules, are found. This sensor aims to determine the amount of concentration glucose in the human body.

The proposed structure consists of silicon dielectric rods in the air with a refractive index of 1. This structure is placed on a silicon dioxide plate with a refractive index of 1.46. In order to alleviate challenges related to absorption and material dispersion, it is preferable to use a silicon dielectric in an air configuration. In the RSoft software, this configuration is referred to as "pure silicon.

The intended structure comprises two waveguides, including an input and an output waveguide, with a nano-ring resonator sandwiched between them, which is a detecting region inside the nanoring. The lattice constant of the structure is 600 nm, while the cylinders' radius is 120 nm. We conducted over 300 manual simulations to fine-tune the structural parameters, carefully scrutinized the resulting values, conducted 100 sweeps to identify the optimal lattice constant and non-defect rods radius value based on bandgap analysis, ultimately determining them to be 600 nm and 120 nm, respectively.

This structure, along the X and Z axes, has \(23\times 17\) cylinders. The photonic crystal’s band structure with a direct bandgap along the \(\Gamma -\text{\rm M}-K-\Gamma\) of the first Brillouin zone is depicted in Fig. 2, and the desired sensor structure is shown in Fig. 3.

Fig. 2
figure 2

The structure photonic band gap

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Fig. 3
figure 3

The proposed structure

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The band gap of this photonic crystal structure is in the range \(0.39<\text{a}/\uplambda <0.47\), which is within the range of the third telecommunication window. The gap has a wavelength distance of 1276 nm to 1538 nm. At the top of the band gap, the frequency levels in the \(\Gamma -\text{\rm M}\) path have a steeper slope and have less fluctuation in the \(\text{\rm M}-K\) path, which shows that the \(\text{\rm M}-K\) region has contributed more to the photonic gap. Also, at the edges of M and K, the frequency levels are perpendicular, indicating the formation of optical standing waves.

Two cylinders are placed in the input waveguide just before the light enters the resonator nano ring to increase the quality factor. The size of R0 in Fig. 3 is equal to R0 = radius × 0.5. First, a biomolecule of blood cholesterol with a refractive index of 1.5250 is connected to the sensing region, and the biosensor output is inspected. This output has been evaluated using 0.1 step changes in the blood cholesterol refractive index. In the first case, Nano cavity cylinders' radii are the same size as the structure's other cylinders. Table 2 displays the biosensor output after attaching the cholesterol molecule to the sensing region.

Table 2 The first case of cholesterol binding to the sensing region
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As shown in Table 2, we have a good resonance wavelength shift, high-quality factor, reasonably good sensitivity, high figure of merit, and low detection limit if the detecting cylinders around the sensing region are the same size as the other rods of the biosensor structure. The sensitivity, figure of merit, and resonance wavelength shift all rise as the cholesterol refractive index steps rise, but the detection limit and quality factor decrease. When cholesterol is attached to the detecting region, Fig. 4 shows each desired biosensor output separately.

Fig. 4
figure 4

The Output of proposed structure in different RI (a) air (b) 1.025 (c) 1.125 (d) 1.225 (e) 1.325 (f) 1.425 (g) 1.525

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As mentioned earlier, the refractive index of blood cholesterol is 1.5250 [64, 82], and in this structure, by connecting the biological molecule to the sensing region or changing the refractive index of the sensing cylinders, there are changes in the resonance wavelength shift and quality factor. Refractive index changes occur in steps of 0.1 from 1.025 to 1.5250. The biosensor's output is measured and its changes are noted for each change. Figure 5 depicts the outcomes of step changes to the structure.

Fig. 5
figure 5

The Output of the proposed structure

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As seen in the image above, a high resonant wavelength shift occurs when the biomolecule is connected to the biosensor sensing region, which improves and facilitates the identification of the target biomolecule.

The second stage's lattice constant is still 600 nm, but the sensing region's cylinders' radius has increased. The size of these cylinders is equal to R1 = R2 = R3 = R4 = R5 = R6 = radius × 1.5, which is equal to 180 nm. The desired structure is shown in Fig. 6.

Fig. 6
figure 6

Increasing the radius of sensing region cylinders

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All simulation processes were repeated after increasing the sensing region's cylinders, and the results are shown in Table 3 along with the output parameters.

Table 3 Binding of cholesterol to the sensing region in the second stage
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Table 3 shows how increasing the radius of the cylinders surrounding the sensing region improves the structure's output results. The resonance wavelength shift and the detection limit have increased with an increase in the refractive index of blood cholesterol, whereas the figure of merit, sensitivity, and quality factor has decreased. When cholesterol is connected to the sensing region, Fig. 7 shows each output of the biosensor separately (stage 2).

Fig. 7
figure 7

The Output of the proposed structure in different RI (a) air (b) 1.525 (c) 1.425 (d) 1.325 (e) 1.225 (f) 1.125 (g) 1.025

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As seen in the above figure, in addition to the resonance wavelength shift, the output power has also increased, which improves the ability of the detector section to identify the output wavelengths. Additionally, for a clearer comparison, Fig. 8 shows the resonance wavelength shift of the structure close to one another (stage 2).

Fig. 8
figure 8

The Output of the proposed structure

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Figure 8 shows that the resonance wavelength shift improves and increases when the refractive index approaches 1.5250. However, as the refractive index changes are gradually lessened, the output power increases.

Table 4 compares the proposed structure to prior works and evaluates the simulation results and other aspects.

Table 4 Reviewing and comparing with previous works
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Table 4 shows that the proposed structure has a higher quality factor and a higher figure of merit than other structures. Additionally, the detection limit has a low value. These parameters result from the change in cholesterol refractive index from 1.025 to 1.5250.

The possible implementation structure is depicted in Fig. 9. A laser is used as a broadband light source in this experimental setup. An objective lens couples the light into the polarization controller. The biosensor is fixed in a silicon dioxide plate. The transmitted light is then collected by a polarizer lens, and it will be obtained by a laptop after being detected by a detector.

Fig. 9
figure 9

Schematic diagram of the experimental setup of the proposed sensor for practical sensing application

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3 Conclusion

A photonic crystal biosensor based on a nanoring resonator between the input and output waveguide is presented. In this biosensor, the dispersion and optical characteristics of light under changing refractive indices are calculated using the plane wave expansion (PWE) and finite difference in time domain (FDTD) methods. The refractive index in this biosensor gradually rose till it reached 1.525, and changes in the refractive index have been studied. Also, by increasing the radius of the cylinders around the sensing region, the biosensor output parameters have been improved. The obtained quality factor has a high value of 4988, and the figure of merit has also increased and has a value of 57.14 RIU−1. The detection limit in this structure is low and has a value of 0.001. The sensitivity obtained for this biosensor is equal to 32 (nm/RIU).

This paper has not reported experiments on live vertebrates or higher invertebrates.