Sensitivity enhancement of an optical sensor based on a binary photonic crystal for the detection of Escherichia coli by controlling the central wavelength and the angle of incidence

A one-dimensional binary photonic crystal with the structure air/(GaAs,SiO2)N/D/(GaAs,SiO2)N/glass is proposed as an optical sensor to detect E. coli bacteria, where D is the defect layer. Water and E. coli bacteria are treated as the defect layer. The sensing mechanism of the proposed detector is based on the refractive index difference between pure water and waterborne bacteria samples. The transmission spectra of the photonic crystal are investigated and the sensitivity to E. coli bacteria is calculated. The effects of the central wavelength and the angle of incidence on the sensitivity and sensor performance parameters are studied. It is found that the central wavelength increase can enhance the sensor sensitivity and most of the performance parameters. Increasing the incidence angle can improve the sensitivity and all the performance parameters such as full width at half maximum, quality factor, detection limit, sensor resolution, signal-to-noise ratio, dynamic range, detection accuracy and figure of merit. The sensitivity, quality factor and detection limit that have been obtained are 213.259 nm/RIU, 10,010 and 0.0318, respectively.


Introduction
Varying types of bacteria have different water body contamination levels. The presence detection of bacteria in water, therefore, plays a significant role in ensuring drinking water in a safe manner. Escherichia coli (E. coli) is a prokaryotic bacterium which in humans and other animals is known to cause diarrhea. This is a key bacterial indicator for environmental monitoring of water quality and food security (Hu et al. 2017). At present, the The main objective of the current work is to design a PC-based sensor for monitoring the E. coli bacteria. The defect mode analysis is employed for the refractive index monitoring. It is also intended to investigate the effect of variation of central wavelength and angle of incidence on the performance of the PC detector. When Bragg quarter-wave condition is applied to determine the layer thicknesses of the PC, then changing the central wavelength is expected to have a significant effect on the performance of the PC-based detector. Moreover, the angle of incidence is one of the most significant parameters affecting the transmission spectrum of a PC. Therefore, it is expected to significantly affect the defect mode and hence the PC-based sensor.
2 Theoretical model Figure 1 shows a schematic diagram of a one-dimensional binary PC with a defect layer of the form air/(AB) N D(AB) N /glass, where A and B are GaAs and SiO 2 respectively, and layer D is the defect layer. The transmission spectra are investigated for water and E. coli bacteria as defect layers. The transfer matrix method is employed to study the transmission properties through the PC. The characteristic matrix of one layer is given by  where l is the phase in layer l and is given by l = 2 n l d l cos l , where n l and d l are the refractive index and thickness of layer l. cos l = √ 1 − n 2 0 sin 2 0 n 2 l is the angle in layer l. n 0 and 0 are the refractive index of ambient (air) and initial incident angle. For TE mode l = n l cos l . The total transfer matrix M system of the proposed PC is given by (Taya et al. 2020) where M s is the transfer matrix of the substrate layer. The transmission coefficient can be written in terms of the total matrix elements as  (1) with, 0 = n 0 cos 0 and s = n s cos s . The transmittance through the PC can be obtained as To determine the refractive indices of both water and E. coli, the Sellmeier equation can be used.
For water, the index of refraction is given by (Hoang et al. 2019) and the refractive index of E. coli is given as (Hoang et al. 2019) where the wavelength of the incident radiation is used in µm to find the indices in Eqs. (5) and (6). The sensitivity (S) is the main parameter to estimate the proposed structure performance. The sensitivity is defined as the ratio of the change in the wavelength position of the transmission peak, Δ R = bacteria − water , to the change in the refractive index, Δ n = n bacteria − n water . Then, it is given by Several parameters that determine the optical detector performance can be investigated. These parameters determine the quality and accuracy of the optical sensor. Among these parameters are the full width at half maximum (FWHM), quality factor (QF), detection limit (DL), sensor resolution (SR), signal-to-noise ratio (SNR), dynamic range (DR), detection accuracy (DA) and figure of merit (FoM). The following mathematical equations are used to find these parameters where Δ R 1 2 is the difference between defect mode right and left edges at half maximum.

Numerical results
Based on the transfer matrix method and using MAPLE 2017 software, the transmission spectra of the proposed PC are simulated. The spectral range of 300 to 800 nm is adopted. At the interfaces separating the neighboring layers of the PC, incident radiation at an angle of incidence (θ 0 ) encounters sequential transmissions and reflections. The incident beam's electric field vector is considered to be perpendicular to the plane of incidence (s-polarization or TE mode). The materials used in the proposed structure are nonmagnetic (relative permeability equals unity). The refractive indices of the defect layers (water and E. coli) are calculated using the Sellmeier approximation formula at room temperature (Eqs. 5 and 6, respectively). The refractive indices of GaAs, SiO 2, and glass are 3.36, 1.46, and 1.5 respectively. The thicknesses of GaAs and SiO 2 layers are selected based on Bragg quarterwave conditions such as d i = c 4n i , where c is the central wavelength. The thickness of the defect layer is taken as d D = d GaAs + d SiO2 . In Fig. 2, the variations of the refractive indices of water and E. coli bacteria with the incident radiation wavelength are shown. The sensing mechanism of the proposed PC is based on the refractive index difference between the two samples. This index variation leads to a resonant peak shift in the transmission spectra. Figure 3 illustrates the transmission spectrum of the proposed PC without a defect layer. The PC is considered to have the structure air/(GaAs,SiO 2 ) N (GaAs,SiO 2 ) N /glass. N and λ c are taken as 4 and 450 nm, respectively. Normal incidence is assumed in which θ 0 = 0°. Figure 4 illustrates the transmission spectra through the structures air/(GaAs,SiO 2 ) N /water/ (GaAs,SiO 2 ) N /glass and air/(GaAs,SiO 2 ) N /E. coli/(GaAs,SiO 2 ) N /glass, when water and E. coli bacteria are treated as defect layers at λ c = 450 nm and θ 0 = 0°. The resonant peaks are found at wavelength positions of 488.3 and 494.5 nm for water and E. coli bacteria, respectively. The sensitivity of the proposed PC to the E. coli bacteria is calculated as 128.923 nm/RIU, where RIU is the refractive index unit. The quality factor (QF) of the proposed sensor is 1052.17 which is calculated using Eq. (9). It is found that the defect mode gets shifted towards longer wavelengths when E. coli bacteria instead of water is used as a defect layer. This shift is due to the refractive index-dependent resonant wavelength of the defect layer. Bragg quarter-wave condition is used to determine the layer thicknesses of the PC. Therefore, any variation of the central wavelength has a significant effect on the layer's thicknesses and hence on the detector performance. The transmission spectra at different central wavelengths (λ c ) and normal incidence angle (θ 0 = 0°) are investigated in Fig. 5. The central wavelength is varied from 350 to 600 nm by a step of 50 nm. Some enlarged views of the defect mode of these results (at λ c = 350, 400, 450, and 500 nm) are shown in  Table 1 which presents the full width at half maximum (FWHM), the defect mode position and the PC sensitivity at different central wavelengths. The defect mode gets shifted towards a longer wavelength region when E. coli bacteria replace pure water samples in the defect layer. Moreover, the increase of λ c at constant  θ 0 leads to a redshift of the wavelength positions of the defect modes. As can be seen from the table, the FWHM of both structures increases as the central wavelength increases, and the defect mode position of both analytes gets shifted towards a higher wavelength region as λ c increases. The sensitivity enhances from 100.183 nm/RIU to 180.422 nm/RIU as the central wavelength changes from 350 to 600 nm. The variation of the defect mode FWHM of both analytes with the central wavelength variation is shown in Fig. 6. As the central wavelength increases for the same incidence angle, the FWHM enhances linearly for both structures. The central wavelength dependence of the transmission peak positions is linear for both analytes as shown in Fig. 7. The structure with E. coli bacteria as a defect layer has higher transmission peak positions when compared to those of water. Figure 8 shows that the sensitivity variation of the proposed detector with the central wavelength variation. It is linearly dependent on the central wavelength. It can be enhanced from 100.183 nm/RIU to 180.422 nm/RIU by increasing the central wavelength from 350 to 600 nm. The effect of the central wavelength variation on the E. coli sensor performance parameters is studied in Table 2. The quality factor does not show a considerable dependence on the central wavelength. With the variation of λ c , the quality factor shows some fluctuations in the rage 1031 to 1052.17 which can be considered as small oscillations. The sensor is more efficient as the figure of merit gets higher. For λ c = 350 nm, the figure of merit is 270.8 and it then shows a slight improvement to 286.4 as λ c = 600 nm. The detection limit, which is the smallest change in the refractive index that can be precisely detected, decreases as the central wavelength enhances. It starts at 1.318 when λ c = 350 nm and becomes 1.211 when λ c = 600 nm, which is a good sign. Smaller detection limit values correspond to better performance of the sensor. Detection accuracy is defined as the reciprocal of the full width at half maximum. The sensor has a good performance when the full width at half maximum is narrow and the detection accuracy is as high as possible. With the increase of the central wavelength, the detection accuracy changes from 2.7 at λ c = 350 nm to 1.59 at λ c = 600 nm. The signal-to-noise ratio is a measure of the resonant peak signal strength relative to the background noise. The higher signal-to-noise ratio means a narrower resonant peak and a minute shift in wavelength is measurable. As can be seen from the table, signalto-noise ratio is enhanced from 12.16 at λ c = 350 nm to 13.65 at λ c = 600 nm. The sensor resolution describes the smallest possible spectral shift that can be estimated precisely. The smaller the value of the sensor resolution the better is its performance. As the central wavelength increases from 350 to 600 nm, the sensor resolution increases from 0.132 to 0.219, respectively. Dynamic range is the ratio between the maximum value of amplitude and root mean square of the noise floor. It starts at 633.1 when the central wavelength = 350 nm and attains a value of 829.88 at λ c = 600 nm.
In conclusion to Table 2, increasing the central wavelength can improve the sensitivity, the quality factor, the detection limit, the signal-to-noise ratio, the dynamic range, and the figure of merit.
The transmission spectra of the proposed PC are investigated at different incident angles at a constant central wavelength of 450 nm. The angle of incidence is varied from 0° to 70° by steps of 5°. Enlarged views of the defect mode of some of these results (at θ 0 = 10°, 20°, 30° and 40°) are plotted in Fig. 9. It is found that the defect mode gets shifted towards longer wavelengths when E. coli bacteria sample replaces the pure water. Increasing the incidence angle at constant λ c leads to a blue shift of the resonant modes. Table 3 shows the variation of FWHM, defect mode position and PC sensitivity with the angle of incidence variation. It is shown that the FWHM for both analytes decreases as the angle of incidence increases as shown in Fig. 10 with the FWHM of the E. coli sample is greater than that of water. The variation of the defect mode position as a function of the angle of incidence for both analytes is plotted in Fig. 11. In contrary to its variation with the central wavelength, the defect mode position does not vary linearly with the incident angle. As the incident angle increases, the defect mode positions shift towards a lower wavelength region. The sensitivity variation as a function of the incident angle is illustrated in Fig. 12. As the incident angle increases from 0° to 70°, the sensitivity is considerably enhanced from 128.923 nm/RIU to 213.259 nm/RIU.  Table 2 Variation of the quality factor, figure of merit, detection limit, detection accuracy, signal-to-noise ratio, standard deviation, sensor resolution and dynamic range with the central wavelength Central wavelength   Table 4 presents the effect of the angle of incidence variation on the E. coli detector performance parameters. The quality factor, figure of merit, detection accuracy, signal-tonoise ratio and dynamic range show considerable enhancements with the increase of the angle of incidence. With the variation of incidence angle from 0° to 70°, the quality factor exhibits an improvement from 1052.1 to 10,010, the figure of merit gets higher from 274.3 to 5331, the detection accuracy also enhances from 2.13 to 25, the signal-to-noise ratio shows an improvement from 13.19 to 240 and dynamic ranges also enhances from 721.3 to 2002. All of these enhancements are good signs for the detector efficiency. As the angle of incidence increases, the detection limit and sensor resolution get smaller and smaller. When θ 0 = 0, the detection limit and sensor resolution are 1.28 and 0.1644 and they continue decreasing to 0.0318 and 0.0068 as the angle increases to 70°. As the detection limit and sensor resolution get smaller, the sensor gets more efficient.
In conclusion of Table 4, increasing the angle of incidence can improve the sensitivity, the FWHM, the quality factor, the figure of merit, the detection limit, the detection accuracy, the signal-to-noise ratio, and the dynamic range.
It is worth mentioning that the proposed E. coli sensor can be used for any biosensing application. It can be employed to detect various concentrations of human creatinine in the blood, hemoglobin concentration and glucose concentrations in the blood. It can also be used as ethanol and methanol detector. This is because the sensor operating   principle is dependent on the refractive index variations between a reference sample and the sample under investigation.
A comparison of the current work sensitivity with that of the most recent published biosensors is presented in Table 5 which shows the proposed structure and the sensitivity obtained in each work. As can be seen, the current sensor achieves the highest sensitivity.

Conclusion
A binary PC having the structure air/(AB) N D(AB) N /glass has been investigated as an optical sensor for the E coli bacteria, where A and B are GaAs and SiO 2 , respectively. D is the defect layer which can be pure water or waterborne bacteria samples. The transfer matrix method has been employed to study the transmission spectra of the structure. There is a minute difference between the refractive index of pure water and that of E. coli bacteria which is the principle of operation of the proposed detector. This index variation leads to a shift in the resonant peak position. All the sensor parameters such as sensitivity, full width at half maximum, quality factor, detection limit, sensor resolution, signal-to-noise ratio, dynamic range, detection accuracy and figure of merit have been investigated with the central wavelength and the angle of incidence. Many interesting findings have been found. Any increase of the central wavelength can enhance the sensor sensitivity and most of the performance parameters. The increase of the incidence angle is found to be more influential and significant than that of the central wavelength. The increase of the incidence angle can improve the sensitivity and all the performance parameters. The sensitivity, quality factor and detection limit that have been obtained are 213.259 nm/RIU, 10,010 and 0.0318, respectively.
The experimental design of the proposed sensor is very simple since the number of periods is only four. Moreover, the defect layer is a gap between the two periods which is filled by pure water sample. Then the pure water sample is replaced by the contaminated water sample. In each of the two cases, the defect mode position is determined and the sensitivity is calculated.  Table 4 Variation of the quality factor, figure of merit, detection limit, detection accuracy, signal-to-noise ratio, standard deviation, sensor resolution and dynamic range with the angle of incidence Incident angle   Table 5 Comparing the sensitivity of the current work with those of the most recent published biosensors