Enhanced the sensitivity of one-dimensional photonic crystals infiltrated with cancer cells

In this work, we use a one-dimensional photonic crystal as a biosensor composed of alternating GaAs and air layers. Within the cavity where they are infiltrated, the Normal, Jurkat, HeLa, PC-12, MDA-MB-231, and MCF-7 cells are bounded by layers of nanocomposite and graphene to increase biosensor sensitivity. The transmission spectrum was calculated using the transfer matrix method. We observed that, when the structural periodicity is broken, defect modes that characterize each cell are created. These defect modes move at a wavelength as the dielectric constant increases. Additionally, the separation between defect modes and bandwidth determines sensitivity, Q factor, and FOM, in which average values of 406.84 nm/RIU, 1765.53, and 535.44 were obtained, respectively, for normal light incidence. Regarding Transverse-Electric (TE) and Transverse-Magnetic (TM) polarization, the defect modes shift toward shorter wavelengths as the angle of incidence increases. For TE polarization, transmittance decreased and the distance between the modes increased. At a 50° angle, sensitivity, Q factor, and FOM increased up to 497.55 nm/RIU, 3182.02, and 1401.25, respectively. Conversely, at a 50° angle in TM polarization, sensitivity remained constant at a value of 407 nm/RIU, along with increased transmittance and decreased performance. Finally, sensitivity and performance were optimized by modifying the cavity thickness value at an incidence angle of 30° for TE polarization, and at an incidence angle of 10° for TM polarization. In both cases, the increased cavity thickness shifted the defect modes toward longer wavelengths while increasing sensitivity up to 495.75 nm/RIU for TE and 451.33 nm/RIU for TM.


Introduction
Photonic Crystals (PC) are nanostructures composed of periodic alternating layers of higher and lower refractive indices. This periodicity creates photonic band gaps (PBG) in which light cannot propagate at a given wavelength range [1][2][3]. Nevertheless, the presence of impurities or defects, such as a layer of different sizes or a new layer that breaks PC periodicity, originates defect modes, thus allowing light to spread throughout certain wavelengths within the PBG [4][5][6][7][8]. PCs can be classified into three categories: one-dimensional (1D), twodimensional (2D), and three-dimensional (3D), each differing in the dimensionality by which the refractive index is changed. 1D-PCs have become popular due to their wide variety of applications, especially as sensors, as well as their affordable costs and easier manufacturing processes, compared to 2D-and 3D-PCs [9,10]. Defect modes are a reference for assessing their shifting based on the sensitivity to small changes as per the refractive index of 1D-PCs when modifying parameters such as pressure, temperature, or refractive index for an analyte infiltrated within the structure [11][12][13]. The fast optical response of 1D-PCs can be used for the early detection of cancer, which may help reduce its mortality rates [14,15]. According to the World Health Organization, cancer could kill up to 19.2 million people by 2040 [16]. Cancer cells contain large volumes of water and protein in their nuclei, thus facilitating their rapid and uncontrollable division. Likewise, they contain thicker cytoplasmic Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. structures and organelles, which translates into a much higher refractive index than healthy cells [17,18]. Cancer cells usually have a refractive index of 1.37 to 1.4, whereas normal cells have an average refractive index of 1.35 [19,20]. This considerable change in the refractive index will allow us to assess biosensor sensitivity and detect analytes based on defect mode shifts within the resonance spectrum. Aly et al proposed using a 1D-PC composed of alternating GaP and SiO 2 layers for detecting bacterial cells such as Pseudomonas aeruginosa and Bacillus anthracia, obtaining a sensitivity value ranging between 333.0882 nm/RIU and 337.3626 nm/RIU [21]. However, Daher et al designed a ternary 1D-PC composed of Si, Bi 4 Ge 3 O 12 , and SiO 2 layers for detecting cancer cells with an oblique incidence of radiation. They obtained a sensitivity of up to 3282.09 nm/RIU and a very low detection limit of 0.0001 RIUs by changing the radiation incidence angle, the cavity thickness, and the number of periods [22]. In this work, the behavior of 1D-PC radiation will be assessed through the Transfer Matrix Method (TMM) [23,24]. Cancer and normal cells are infiltrated into the cavity covered with graphene sheets to increase the sensitivity of the biosensor due to its high-energy transfer efficiency [25][26][27][28]. Biosensor sensitivity is optimized through the modification of parameters such as radiation incidence angle and cavity thickness. Furthermore, the Q factor and the figure of Merit (FOM) will also be determined to assess biosensor performance. This paper is structured as follows: section 2 introduces the theoretical model used and the basic equations governing our study. Section 3 discusses the results from the simulations using the TMM, and, finally, section 4 presents our conclusions. Figure 1 denotes the structure of N-period 1D-PC composed of alternating GaAs (A) and air (B) layers with d 1 and d 2 thicknesses, respectively. The cells infiltrate a cavity (layer D) with a d 5 thickness, filled with nanocomposite (layer C) and graphene (layer G) layers with d 3 and d 4 thicknesses, respectively. The nanocomposite is a mixture of SiO 2 with silver nanoparticles. The transmittance spectrum is calculated using TMM [24], which, at oblique radiation incidence, is characterized by the jth layer matrix (A, B, C, G, and D) with d j (d 1 , d 2 , d 3 , and d 4 ) thicknesses represented by the following:

Theoretical framework
In equation (1) P j is the propagation matrix defined as The j j phase is expressed as ,in which ò j represents the dielectric constant, θ j angle of incidence of light, and d j represents the jth layer thickness. T j is the dynamic matrix, which is given as follows for TE and TM polarization, The total transfer matrix for the structure ( ) ( )( )( )( ) AB CG D GC AB N N can be calculated as follows: where T 0 is the dynamic air matrix used by radiation to enter and exit. Transmittance is calculated using the M 11 element of the M matrix, The dielectric constant of the silver nanoparticles ò m (ω) embedded in silica is determined through the Drude model as follows: where ò 0 = 5 is the relative dielectric constant of Ag, ω p = 13.64 × 10 15 rads is the plasma frequency from free electron gases within an infinite volume, and γ = 3.03 × 10 13 rad/s is the relaxation rate. We can find the effective dielectric constant for the nanocomposite as a homogeneous medium by the Maxwell-Garnett approximation [29][30][31][32], which is represented by the following: where ò c is the dielectric constant of the nanocomposite, ò d is the dielectric constant of silica, ò m (ω) is the dielectric constant of the metal used (in this case, silver), f is the nanoparticle filling factor, and ω is the angular frequency. The dielectric constant for silica ò d and GaAs ò GaAs (λ) is obtained through Sellmeier's empirical equation that relates the refractive index of the material ton a wavelength of λ the incident light on μm [33]: where B 1 = 0.6961663, B 2 = 0.4079426, B 3 = 0.8974794, C 1 = 0.00467914826 μm 2 , C 2 = 0.0135120631 μm 2 and C 3 = 97.9340025 μm 2 are the Sellmeier coefficients determined through experimentation. The dielectric constant of the GaAs semiconductor for incident infrared light at a temperature of T = 300 K is given by [34][35][36], where A = 8.950, B = 2.054 and C 2 = 0.390 μm 2 are the empirical Sellmeier coefficients. The dielectric constant of graphene ò G is described by the following: where ò 0 is the vacuum permittivity, and σ(ω) is the surface conductivity expressed as the sum of intraband (σ intra (ω) ) and interband (σ inter (ω)) conductivity, as given by [37][38][39]: where, e is the electric charge, T is the temperature in Kelvin, k B is the Boltzmann constant, Γ is the phenomenological dispersion rate, and μ C is the chemical potential of graphene.

Results and discussions
The parameters used to calculate the transmission spectrum were the following: The thickness of the GaAs, air, graphene, nanocomposite, and cavity layers are d 1 = 106.1 nm, d 2 = 350 nm, d 3 = 0.34 nm, d 4 = 100 nm, and d 5 = d 1 + d 2 = 456.1 nm, respectively. The layer optical thicknesses were selected from a quarter wavelength according to nd = λ d /4, where n is the index of refraction and λ d (= 1400 nm) is the design wavelength, to increase interference. For graphene, the chemical potential is μ = 0.9 eV, the phenomenological dispersion rate is Γ = 1 THz and the temperature is T = 300 K. For the nanocomposite, the silver nanoparticle filling factor is f = 0.1 and the 1D-PC periodicity is N = 3. Finally, we consider that the radiation enters and exits through the air. Figure 2 shows the transmission spectrum for different cells with a normal incidence of radiation. 1D-PC periodicity generates a 1050 nm-width PBG. However, cell infiltration produces defect modes located at 1330.51 nm, 1346.79 nm, 1347.6 nm, 1348.82 nm, 1350.44 nm, and 1351.2 5 nm within the spectrum for Normal, Jurkat, HeLa, PC12, MDA-MB-231 and MCF-7 cells, respectively. As can be seen, the increase in cell refractive index (see table 1) results in the defect modes shifting toward longer wavelengths. The location of each mode is unique for each type of cell, thus allowing for easy identification. Biosensor sensitivity is based mainly on the separation of cancer cell modes against normal cells: sss the wider their separation, the greater the sensitivity.  To optimize biosensor sensitivity and performance, we changed the radiation incidence angle for TE and TM polarizations. Figure 3 illustrates the transmission spectrum for TE polarized light and incidence angles at θ = 10°( figure 3(a)), 20°( figure 3(b)), 30°(figure 3(c)), 40°(figure 3 (d)), and 50°( figure 3(e)). According to the results, as the incidence angle increases, the mode transmission decreases from 43% to 2.5%. Additionally, the defect modes shift toward much shorter wavelengths. Greater separation between the modes, identifying each cell, can also be noted, resulting in increased sensitivity until reaching 497.65 nm/RIU; sensitivity values are  table 3).
For light with TM polarization, figure 4 shows that the increased incidence angle (10°, 20°, 30°and 50°) causes the defect modes to shift toward shorter wavelengths. However, contrary to the TE polarization, transmittance increases from 47% to 87%, but the separation between the defect modes remains constant at an average value of 407 nm/RIU. Table 4 shows the defect mode position and sensitivity for three angles of incidence. Furthermore, as shown in table 5, performance decreases due to increased defect mode bandwidth, obtaining a Q Factor of 206.91 and a FOM of 67.45 at an angle of 50°. Based on the incidence angle, mode , in which the increase in the angle implies that the light travels to a longer optical path and changes interference conditions.
To achieve greater biosensor optimization, we increased the cavity thickness while holding the incidence angle constant to provide better transmittance, sensitivity, and performance. For TE Polarization, we selected a 30°incidence angle. According to figure 5, the increase in cavity thickness leads the defect modes to shift toward much longer lengths. Additionally, the separation between the modes is greater, thus increasing the sensitivity, which on average reaches 495.75 nm/RIU (see table 6), while the performance is 1757.58 for Q Factor, and 2797.43 for FOM (see table 7). Table 6 shows that an increase of 100 nm in cavity thickness increases sensitivity by approximately 10 nm/RIU. For TM polarization, the incidence angle selected was 10°, observing the same behavior as for TE Polarization, as shown in figure 6. Figure 5 shows that the defect modes move toward longer wavelengths at a greater separation, thus obtaining an increase in sensitivity reaching an average of 451.33 nm/ RIU (see table 8) and performance with a Q Factor of 1757.58 and a FOM of 567.47 (see table 9). Table 9 shows that an increase of 100 nm in cavity thickness increases sensitivity by an average of 9 nm/RIU. Finally, table 10 compares the sensitivity of the sensor proposed with the values reported in other works. The comparative results show that the accuracy of the proposed sensor is better than that of existing sensors.

Conclusions
In this work, the transmission spectrum of a 1D-PC was theoretically calculated using TMM. By infiltrating each cell, we obtained unique defect modes within the PBG. The dielectric constant of cancer cells, whose value is higher than that of normal cells, causes the defect modes to shift toward longer wavelengths, thus increasing the separation of the defect mode that characterizes normal cells. We found that the greater the separation between the defect modes, the greater the sensitivity and performance of the biosensor. Sensitivity optimization was obtained by changing parameter values, such as incidence angle and cavity thickness. When increasing the incidence angle for both TE and TM polarization, the defect modes shift toward shorter wavelengths. However, the sensitivity in TM Polarization remained constant with an increase in transmittance and a decrease in Q Factor and FOM. Meanwhile, for TE Polarization, the sensitivity, Q factor, and FOM increased. Finally, sensitivity was optimized through cavity thickness variations using the incidence angle with the highest sensitivity and performance. The behavior for both TE Polarization and TM Polarization was similar; the defect modes shifted toward longer wavelengths as sensitivity, Q Factor, and FOM increased.