Towards Promising Platform by Using Annular Photonic Crystals to Simulate and Design Useful Mask

: Human masks are considered the mainstay in air ﬁltration and puriﬁcation technologies and against the spreading of bacterial and viral infections. This paper introduces a novel design of a human mask to increase the ultraviolet germicidal irradiation effect on pathogens. The proposed design consists of a tube with an annular photonic crystal (APC) attached to the mask’s oriﬁce, and a UV source is located in the tube’s center. The main role of this study is the enhancement of UV doses based on the reﬂectivity of the proposed APC. Therefore, increasing pathogens’ inactivation level in the incoming air to the mask’s oriﬁce could be investigated. The numerical investigations demonstrated that the proposed APC could provide a complete photonic bandgap with a high reﬂectivity in the wavelength regime from 207 to 230 nm. In addition, we have considered the roles of the thickness of layers, inner core radius, and the azimuthal number. Meanwhile, the results showed the ability to use a wide range of core radius values without almost any variations in the optical properties of the proposed design. Such results could grant the advantage of using this design by the manufacturing of human masks with different sizes besides the inclusions in other ultraviolet germicidal irradiation applications.


Introduction
The spread of airborne-transmitted infections, such as the SARS-CoV-2 (COVID- 19) outbreak, urgently highlights the need for personal protective equipment (PPE) to eliminate such biological hazards. One of the essential PPE is the human mask, as it can effectively prevent transferring aerosol pathogens via the respiratory route [1]. Masks contain spreading contaminated respiratory droplets generated through coughing and sneezing even through speaking [2]. In addition, they represent a physical barrier between aerosolized pathogens and the human respiratory system with various levels of protection [3]. Ultraviolet germicidal irradiation (UVGI) can be used to disinfect medical equipment and surgical masks. UVGI can effectively inactivate pathogens, as it destroys their genetic materials [4][5][6][7]. The level of inactivation depends on the type of the pathogen and ultraviolet (UV) doses as well. According to the FDA guidance, three log levels or more are required to accomplish an optimal inactivation dose [8]. The level of single-stranded RNA viruses reduces with the increase of UV dose [9]. UVGI devices, such as medical disinfectors, usually use 254 nm UV light, which has health hazards on the skin and eyes. An alternative approach to building UVGI could be obtained by using UV light in the range from 207 up to 222 nm, which efficiently inactivates pathogens, without damaging human tissues or cells [10,11]. An optimal dose is required to achieve a sufficient inactivation level, which can be accomplished by a longer exposure time or a stronger UV source. Another way to increase UV dose with the same UV source and exposure time is to use an effective reflector. In this regard, photonic crystals (PCs) receive considerable attention due to its high reflectivity [12,13]. PCs are periodic arrangement structures with periodic refractive index functions that represent a building block in many optical, physical, and biomedical applications [14]. A range of electromagnetic frequencies is prohibited from propagating through PCs; this range is called the photonic bandgap (PBG) [15,16]. Bragg scattering at the interfaces of these periodic arrangements explains the physical interpretation of the PBG's appearance. PCs can be classified into one-dimensional, two-dimensional, and three-dimensional based on the structure periodicity, which leads to the formation of a PBG in a certain direction [17,18]. Although three-dimensional PCs express a complete PBG capable of reflecting electromagnetic waves in any direction, one-dimensional PCs are researched extensively with numerous designs due to the ease of fabrication [19,20].
In this regard, researchers have used numerous methods to fabricate PCs, such as atomic layer deposition (ALD), chemical vapor deposition (CVD), and different lithographic techniques [12,21]. Recently, new methods for building PCs have been explored, for example, the self-assembly of nanoparticles to form colloidal PCs [22,23]. In particular, the ability of photon localization has been explored as a significant property of PCs. Such a property is promising in the design and experimental verifications of many technological applications in many different aspects. In this context, photon localization is formed by introducing a defect in this periodic arrangement, which causes a high transmittance peak inside the PBG [24,25]. Such a defect peak is sensitive to the change in the refractive index of the defect layer [26,27]. By using this feature, numerous sensors are designed to measure various conditions based on the changes in the refractive index under these conditions [28][29][30][31][32][33].
An alternative method to build two-dimensions PCs with a one-dimension periodicity is verified by using concentric cylindrical quarter-wavelength periodic structures. Such structures are known as annular photonic crystals (APCs), circular photonic crystals, or cylindrical Bragg reflectors [34,35]. APCs have attracted significant interest, since Kaliteevski et al., adopted the transfer matrix method (TMM) in cylindrical coordinates [36]. APCs offer low-loss (high Q-factor) resonators, making them suitable for chemical and biological sensing applications [37]. One of APCs' prime applications is the annular Bragg lasers, which can be integrated into optical fibers or other optical components for optoelectronics and optical communications [38]. APCs' desired properties, such as small modal volume, offer strong atom-photon interaction with applications including cavity quantum electrodynamics (QED) [39].
In this paper, we introduce a novel design based on APCs with high reflectivity or almost near-zero transmissivity in the wavelengths ranging from 207 to 222 nm. The ability to investigate a PBG with high reflectivity in this range of wavelengths could efficiently inactivate pathogens without damaging human tissues or cells. Moreover, we choose polycarbonates as the material of the mask's tube, which is one of the most common materials for face mask production. Combining our structure with an active UV source for the disinfection of the proposed mask eliminates the need for disposable (single-use) face masks. Thus, the proposed design is eco-friendly, as it reduces plastic pollution coming from disposable masks. In this regard, such a design could be of interest to improve the performance of human masks against viral and bacterial infections. Our theoretical and numerical investigations are mainly based on the fundamentals of the TMM in cylindrical coordinates. Moreover, the effects of the periodicity number, the core radius, the thickness of constituent materials, and the azimuthal number are considered in detail.

Theoretical Analysis
In this section, we present an overview of the proposed mask and the basis of our theoretical modeling. The proposed mask consists of two parts: conventional parts like the mask's body, the rim that seals to the patient face, and the orifice. Non-conventional part: a tube attached to the mask's orifice coated by co-centric cylindrical multi-layers materials that form APC and UV source in its center as illustrated in Figure 1a. The incoming air enters the mask through the open end of a tube. As the air flows in the tube, it exposes to a suitable UV dose by the UV source and its reflection by the APC. The main purpose of APCs is to achieve a PBG in the wavelengths of interest with high reflectivity. Thus, an effective UV dose can be achieved without the need for a more powerful UV source that requires large-energy sources. Each unit cell in the proposed APCs consists of two cylindrical layers with refractive indices n 1 and n 2 , respectively, and thicknesses d 1 and d 2 , respectively, as shown in Figure 1b. The distance from the center of APC and j-th layer is defined as: where ρ 0 is the radius of the air core layer with refractive index n 0 = 1. Here, each unit cell is repeated for a period number N = n 2 , and the outer medium has a refractive index n f . Our modeling is essentially based on the TMM in the cylindrical coordinates [36]. The two possible polarization modes for cylindrical Bragg waves are TE and TM polarizations. The non-zero fields at TE polarization modes are E z , H φ , and H ρ . The electric field E z and the magnetic field H φ can be expressed as: where J m and Y m are Bessel function and Numann function, respectively, and J m , and Y m are their respective derivatives; the intrinsic admittance of the material is given as p = µ/ε , while the wave number inside the material is described as k = ω √ µε; A and B are constants, and m is the azimuthal number. Then, the connection between U(ρ) and V(ρ) functions at the interface between two adjacent layers, such that jth and (j − 1)th layers can be described by the following expression: where the matrix M j is given as: where the elements of the matrix M j can be described as: In the above expression, we have q j−1 = k j ρ j−1 and q j = k j ρ j . The whole APC structure can be described by matrix M, which relates the fields at the inner core interface with radius ρ 0 to the outer medium ρ f . Such matrix M could be described as: where matrix M is the product of the individual layers matrix, which can be written as: Therefore, the reflectance of the proposed structure can be calculated in the vicinity of Equation (11) as following: where M 11 , M 11 , M 12 , M 21 , and M 22 are the elements of the APC's matrix M, and p = √ ε h µ h , and h = 0 and f are for the inner core and the final media, respectively; C (1,2) ml is obtained by the following equation: where H (1) m and H (2) m are the first and the second kinds of Hankel function, respectively. The equations for TM polarization mode can be acquired by replacing every ε ↔ µ and i ↔ −i in all equations of TE polarization mode.
where matrix is the product of the individual layers matrix, which can be written as: 21 22 ] .
Therefore, the reflectance of the proposed structure can be calculated in the vicinity of Equation (11) as following: where 11 , 11 , 12 , 21 , and 22 are the elements of the APC's matrix , and = √ ℎ ℎ , and ℎ = 0 and are for the inner core and the final media, respectively; (1,2) is obtained by the following equation: where (1) and (2) are the first and the second kinds of Hankel function, respectively. The equations for polarization mode can be acquired by replacing every ↔ and ↔ − in all equations of polarization mode.

Results and Discussions
In this section, we present the results of our modeling of the proposed APC with a UV source in the center of the mask. The proposed APC structure is embedded into a tube, with a real refractive index taken from reference [40] and an inner air core with refractive index 0 = 1, as shown in Figure 1a. Polycarbonate is selected, due to its wide use in face masks, light weight, and reasonable cost [41]. The APC consists of a bi-layer unit cell repeated for periodicity number , as illustrated in Figure  1b. The first layer in each unit cell is an 2 3 layer with thickness 1 and refractive index 1 which is taken form reference [42]. The second layer is composed of a

Results and Discussions
In this section, we present the results of our modeling of the proposed APC with a UV source in the center of the mask. The proposed APC structure is embedded into a polycarbonates tube, with a real refractive index n f taken from reference [40] and an inner air core with refractive index n 0 = 1, as shown in Figure 1a. Polycarbonate is selected, due to its wide use in face masks, light weight, and reasonable cost [41]. The APC consists of a bi-layer unit cell repeated for periodicity number N, as illustrated in Figure 1b. The first layer in each unit cell is an Al 2 O 3 layer with thickness d 1 and refractive index n 1 which is taken form reference [42]. The second layer is composed of a SiO 2 layer with thickness d 2 and refractive index n 2 , taken from reference [43]. The initial parameters for the proposed APC are d 1 = 29 nm, d 2 = 33.5 nm, N = 20, m = 0, and ρ 0 = 5 cm. Frist, we studied the reflectance of our structure for TE polarization, as illustrated in Figure 2. The reflectance spectrum shows the appearance of the PBG between 207.7 and 229.6 nm with an average reflectance intensity around 94% and a central wavelength of 218 nm. Moreover, the reflectivity at the center of the PBG reaches 99%. The high reflectance promises a significant increase in UV dose without the need of more powerful UV sources or longer exposure time. Here, the location of the PBG covers the entire far-UVC light range of 207-222 nm, which efficiently inactivates pathogens without harmful effects on human tissues or cells [44]. Moreover, the harmful ozone production by UV light is deficient, as the absorption of UV light by oxygen in the chosen UV range is very low [45]. In this regard, the incident UV radiation could interact with the proposed design with its full intensity and almost no dissipation due to the production of the harmful Ozone. In particular, the mainstay of our study depends on the presence of an optimal dose to achieve a sufficient inactivation level.
Photonics 2021, 8, x FOR PEER REVIEW 5 of 12 0 = 5 cm. Frist, we studied the reflectance of our structure for TE polarization, as illustrated in Figure 2. The reflectance spectrum shows the appearance of the PBG between 207.7 and 229.6 nm with an average reflectance intensity around 94% and a central wavelength of 218 nm. Moreover, the reflectivity at the center of the PBG reaches 99%. The high reflectance promises a significant increase in UV dose without the need of more powerful UV sources or longer exposure time. Here, the location of the PBG covers the entire far-UVC light range of 207-222 nm, which efficiently inactivates pathogens without harmful effects on human tissues or cells [44]. Moreover, the harmful ozone production by UV light is deficient, as the absorption of UV light by oxygen in the chosen UV range is very low [45]. In this regard, the incident UV radiation could interact with the proposed design with its full intensity and almost no dissipation due to the production of the harmful Ozone. In particular, the mainstay of our study depends on the presence of an optimal dose to achieve a sufficient inactivation level. We considered the effects of altering the azimuthal number on the reflectance properties and the PBG of the proposed structure for both TE and TM polarizations. Figure 3 shows the reflectance of the APC structure, as the azimuthal number changes from 1 up to 20. The reflectance was calculated at an inner core radius 0 = 5 cm for both TE polarization (left) and TM polarization (right) with unit cell layer thicknesses 1 = 29 nm and 2 = 33.5 nm and periodicity number = 20. The reflectivity of our structure and the properties of its PBG are almost unaffected by the changes in the values of the azimuthal number for both TE and TM polarizations, as shown in Figure 3a-c. Therefore, the investigated result could grant our design another advantage over traditional multilayer structures. In particular, the optical properties of the traditional one-dimensional PCs are significantly affected with the changes in the values the angle of incidence, especially in reducing the PBG width for TM polarization. Here, we could avoid such a problem. In particular, the changes in the values of the azimuthal number have no effects on the optical properties of our design, especially at large values of the core inner radius. Therefore, the choice of APCs in the designs and fabrications of the human masks could be of increased concern. We considered the effects of altering the azimuthal number on the reflectance properties and the PBG of the proposed structure for both TE and TM polarizations. Figure 3 shows the reflectance of the APC structure, as the azimuthal number changes from 1 up to 20. The reflectance was calculated at an inner core radius ρ 0 = 5 cm for both TE polarization (left) and TM polarization (right) with unit cell layer thicknesses d 1 = 29 nm and d 2 = 33.5 nm and periodicity number N = 20. The reflectivity of our structure and the properties of its PBG are almost unaffected by the changes in the values of the azimuthal number for both TE and TM polarizations, as shown in Figure 3a-c. Therefore, the investigated result could grant our design another advantage over traditional multilayer structures. In particular, the optical properties of the traditional one-dimensional PCs are significantly affected with the changes in the values the angle of incidence, especially in reducing the PBG width for TM polarization. Here, we could avoid such a problem. In particular, the changes in the values of the azimuthal number have no effects on the optical properties of our design, especially at large values of the core inner radius. Therefore, the choice of APCs in the designs and fabrications of the human masks could be of increased concern.
Photonics 2021, 8, x FOR PEER REVIEW 6 of 12  Next, we discussed the effect of changing the thicknesses of the constituent materials on the reflectivity of our APCs, as shown in Figure 4. Here, the values of other related parameters are as following: the inner core radius 0 = 5 cm, azimuthal number = 0, and periodicity number = 20. Figure 4a illustrates the reflectance of the suggested APC structure as the thickness of 2 3 layer 1 changes from 29 up to 35 nm and the thickness of the 2 layer is fixed at 33.5 nm. The increase of the 2 3 layer thickness leads to a significant shift in the position of the PBG towards the longer wavelengths with a little increase in its width. However, the average intensity declines slightly with increasing the value of 1 . The left and the right edges of the PBG also shift towards longer wavelengths as well as the PBG's central wavelength. Then, we considered the effect of changing the thickness of the 2 layer 2 on the reflectance of the proposed structure, as illustrated in Figure 4b. Thickness 2 varies from 31 to 37 nm, while the thickness of 2 3 layer 1 is fixed at 29 nm. Similar to the effect of 1 , the PBG is shifted towards the longer wavelengths with the increase in the value of d2. Thus, Next, we discussed the effect of changing the thicknesses of the constituent materials on the reflectivity of our APCs, as shown in Figure 4. Here, the values of other related parameters are as following: the inner core radius ρ 0 = 5 cm, azimuthal number m = 0, and periodicity number N = 20. Figure 4a illustrates the reflectance of the suggested APC structure as the thickness of Al 2 O 3 layer d 1 changes from 29 up to 35 nm and the thickness of the SiO 2 layer is fixed at 33.5 nm. The increase of the Al 2 O 3 layer thickness leads to a significant shift in the position of the PBG towards the longer wavelengths with a little increase in its width. However, the average intensity declines slightly with increasing the value of d 1 . The left and the right edges of the PBG also shift towards longer wavelengths as well as the PBG's central wavelength. Then, we considered the effect of changing the thickness of the SiO 2 layer d 2 on the reflectance of the proposed structure, as illustrated in Figure 4b. Thickness d 2 varies from 31 to 37 nm, while the thickness of Al 2 O 3 layer d 1 is fixed at 29 nm. Similar to the effect of d 1 , the PBG is shifted towards the longer wavelengths with the increase in the value of d 2 . Thus, the effect of thickness could be of interest in providing the different regions of UV sources with high reflectivity. longer wavelengths with increasing both 1 and 2 or either of them; however, the shift in the PBG becomes larger with rising both 1 and 2 concurrently. Such effects, including the width of the PBG and its central wavelength, are as a result of the change in the optical path length of the incident radiation. In other words, the changes in the thicknesses of the constituent materials require a similar change in the position, width, and central wavelength of the PBG to verify the condition of the constant phase [45]. In addition, we investigated the effects of the number of periods on the reflectivity of our design for both TE and TM polarizations, as demonstrated in Figure 5. At low values of the number of periods especially smaller than 5, the PBG is not formed due to the limited chances of Bragg's scattering at the interfaces, as shown in Figure 5a. As the number of periods increases to 7, the PBG with little sharp edges begins to appear between 205 and 234 nm, as shown in Figure 5a,b. For further increase in the number of Then, we analyzed the effects of varying d 1 and d 2 on the PBG's width. The simulation data, which describe the relationship between the width and thicknesses d 1 and d 2 , can be fitted as following: where W PBG is the width of the PBG in nanometers at Al 2 O 3 layer thickness d 1 in nanometers and SiO 2 layer thickness d 2 in nanometers. Equation (14) shows that the PBG's width grows with increasing either the values of d 1 or d 2 and the increase in the PBG's width becomes more prominent with increasing d 1 and d 2 simultaneously. Additionally, the PBG's central wavelength versus the thicknesses of the unit cell are fitted to the following equation: where λ 0 is the PBG's central wavelength, and d 1 and d 2 are the unit cell layer's thicknesses. Equation (15) shows that the central wavelength of the PBG moves towards longer wavelengths with increasing both d 1 and d 2 or either of them; however, the shift in the PBG becomes larger with rising both d 1 and d 2 concurrently. Such effects, including the width of the PBG and its central wavelength, are as a result of the change in the optical path length of the incident radiation. In other words, the changes in the thicknesses of the constituent materials require a similar change in the position, width, and central wavelength of the PBG to verify the condition of the constant phase [45].
In addition, we investigated the effects of the number of periods on the reflectivity of our design for both TE and TM polarizations, as demonstrated in Figure 5. At low values of the number of periods especially smaller than 5, the PBG is not formed due to the limited chances of Bragg's scattering at the interfaces, as shown in Figure 5a. As the number of periods increases to 7, the PBG with little sharp edges begins to appear between 205 and 234 nm, as shown in Figure 5a,b. For further increase in the number of periods, the edges of the PBG become sharper, and its width slightly decreases. At periodicity number N = 8 for TE polarization, the PBG appears between 204.5 and 233.8 nm with a central wavelength of 219.17 nm. Thus, the PBG's width equals 29.3 nm, and its average intensity is 71.1%. The PBG's width shrinks to 26 nm with an average intensity of 82%, as the periodicity number reaches N = 11. The PBG's intensity is approximately 98% with a width of 19 nm, as the periodicity number increases to 40; meanwhile, the PBG central wavelength is almost stable at 218 nm. Besides, the PBG parameters for TM polarization are almost identical to their values TE polarization, as shown in Figure 5b,c. Moreover, the peaks around the PBG become more intense and less wide, with a shift in their position towards the PBG as the periodicity number increases. Additionally, there is no significant difference between the reflectance in TE polarization and that in TM polarization with increasing the periodicity number. Then, the PBG's parameters were evaluated as written in Table 1. The PBG's left edge λ le f t is shifted towards the longer wavelengths while the right edge λ right moves to shorter wavelengths, as the periodicity number increases. The PBG's right and left edges were calculated when the reflectance's intensity reaches 50% of the maximum intensity in the PBG, while the PBG average intensity I PEG was calculated by the normalized sum of all simulation points between the right and left edges of the PBG. Therefore, the periodicity number can be adjusted to optimize the intensity of reflectance and the ease of fabrication.
233.8 nm with a central wavelength of 219.17 nm . Thus, the PBG's width equals 29.3 nm, and its average intensity is 71.1%. The PBG's width shrinks to 26 nm with an average intensity of 82%, as the periodicity number reaches = 11. The PBG's intensity is approximately 98% with a width of 19 nm, as the periodicity number increases to 40; meanwhile, the PBG central wavelength is almost stable at 218 nm. Besides, the PBG parameters for polarization are almost identical to their values polarization, as shown in Figure 5b,c. Moreover, the peaks around the PBG become more intense and less wide, with a shift in their position towards the PBG as the periodicity number increases. Additionally, there is no significant difference between the reflectance in TE polarization and that in TM polarization with increasing the periodicity number. Then, the PBG's parameters were evaluated as written in Table 1. The PBG's left edge is shifted towards the longer wavelengths while the right edge ℎ moves to shorter wavelengths, as the periodicity number increases. The PBG's right and left edges were calculated when the reflectance's intensity reaches 50% of the maximum intensity in the PBG, while the PBG average intensity was calculated by the normalized sum of all simulation points between the right and left edges of the PBG. Therefore, the periodicity number can be adjusted to optimize the intensity of reflectance and the ease of fabrication.  Finally, we have investigated the effect of inner core radius on the characteristics of the PBG and the reflectivity of our design for both TE and TM polarizations. Here, we have considered that, N = 20, d 1 = 29 nm d 2 = 33.5 nm and azimuthal number m = 0. Figure 6 describes the effect of the inner core radius on the properties of the PBG and its reflectivity. Figure 6a illustrates the reflectance of the proposed APC at TE and TM polarizations with varying the inner core radius from 1 cm up to 30 cm. The reflectance has no apparent changes, which adds more flexibility and simplicity from the fabrication point of view of human faces with different sizes. As the inner core radius changes from 1 cm up to 30 cm, the structure reflectivity, width, and position of the PBG are almost unaffected, as shown in Figure 6b,c. Meanwhile, the PBG's central wavelength is located at 218.6 nm for both TE and TM polarizations and is almost unaffected by the increase in the values of the core radius. In particular, the increase in the core radius over a few micrometers is not almost effecting on the reflectivity of APC because the structure behaves like planer PCs and the intensity no longer depends on radius [35,[46][47][48]. Thus, the reflectivity and the characteristic of the PBG of our structure provide a significant stability with the variations in the values of the inner core radius. Such result could be of potential interest from the experimental point of view in the manufacturing of human masks with different sizes to be compatible with the different faces. In addition, this result could open the way towards the inclusions of such design in many different applications that concern ultraviolet germicidal irradiation applications.

Conclusions
To sum up, we have introduced, in this research paper, a novel design of APCs

Conclusions
To sum up, we have introduced, in this research paper, a novel design of APCs with a high reflectivity in the wavelengths ranging from 207 to 230 nm and an average reflectance intensity of around 94%. The proposed structure could be useful in producing an effective UV dose without the need for a more powerful UV source that requires highenergy sources. Therefore, the inclusion of such a design with human masks could leads to a significant improvement against viral and bacterial infections. In this context, the numerical results showed the appearance of a complete PBG with reflectivity is more than 99% at the center of the PBG. Moreover, the reflectivity and the PBG of the proposed design are almost unaffected by the azimuthal number change from 1 up to 20. Additionally, the effects of the core radius on the reflectivity and the PBG are negligible, as the inner core radius changes from 1 to 30 cm. Therefore, the flexibility of using this structure through the different sizes of human masks is applicable. Moreover, the PBG could be easily shifted based on the variations of the thicknesses of the constituent materials, which allows the production of the different types of UV sources.