Thermo-optical properties of binary one dimensional annular photonic crystal including temperature dependent constituents
Introduction
Photonic crystals (PCs) are novel composite structures in 1D, 2D or 3D that have the capability to control and attenuate electromagnetic waves in certain frequencies. The range of frequencies, within which the incident electromagnetic waves are effectively prohibited is called the photonic band gap (PBG), which is considered the novel characteristic of PCs [[1], [2], [3], [4]]. The structure of PCs always comprises two or more materials with lower and higher refractive indices in a periodic alteration. Due to their remarkable physical properties, many attempts were devoted to using these novel structures in new engineering, physical and biomedical applications over the past two decades. PCs are capable of realizing new types of optical fibers, optical waveguides, Metamaterials, filters and biosensors [[4], [5], [6], [7]].
Nowadays, it is possible to accomplish PCs in different geometrical structures due to progress in the experimental fabrication techniques and theoretical models [3,6,8]. The studies of the PBGs and PCs properties in periodic 1D multilayer structures are mainly dependent on the well-known Transfer Matrix Method (TMM). Based on this technique, the study of the PBGs in dielectric, metallic and superconducting multilayer PC structures have been introduced [[9], [10], [11], [12]]. In addition to the multilayer planar 1D PCs, the propagation of electromagnetic waves in a new geometry of PCs called cylindrical multilayer structures (CMS) has attracted many researchers due to their interesting results in recent years [[13], [14], [15], [16]]. Cylindrical photonic crystals (CPCs) are constructed from multilayers stacked in a cylindrical periodic alteration. This is also known as cylindrical Bragg reflectors, annular photonic crystals (APCs) or circular PCs. The APC structure is produced with alternate cylindrical rings in an innermost region (starting medium) [17].
Similar to the planer multilayer PCs, the transmission and reflection spectra of APCs can be analytically deduced based on a modified version of TMM. Basically, the propagation of electromagnetic waves in APCs was initially studied by Kaliteevski et al. [18]. The investigations of TMM for cylindrical multilayer are based on Abeles theory in Cartesian coordinates [19]. The characteristics of the PBGs in superconducting, metallic and dielectric APCs have been studied as well [13]. Moreover, annular Bragg lasers are produced in the APCs by creating a ring defect into a periodic APCs structure by Jacob Scheuer et al. [20].
Recently; many papers have been dedicated on temperature sensing by PCs structures. For example, Yang-Hua Chang et al. studied theoretically a thermally modulated tunable filter based on the defect modes in a 1D defective PCs in the visible region [21]. Qiang Liu et al. introduced a highly sensitive temperature sensor based on PCs fiber using defect mode and liquid core mode [22]. Also, Dena M. El-Amassi et al. presented a 1D ternary PCs temperature sensor with a polymer layer sandwiched between Si and SiO2 layers [23]. In a different manner, R. Boufenar et al. studied the temperature effects and presented the regular planar PCs as a temperature sensor based on the variation in the PBG width [24]. Although temperature effects have been studied extensively in many planar PCs and PCs fibers. There is still a need to study temperature effects on APCs and to introduce the APCs structure as a temperature sensor due to their remarkable role in the relevant PCs applications. Therefore, the object of the present study is to give a detailed theoretical explanation of the electromagnetic waves propagation in the APCs structure. Also, the proposed APCs can work as a highly sensitive temperature sensor comparable with the previously introduced planar PCs.
In dealing with APCs, both electromagnetic waves polarization modes, E-polarization and the H-polarization can be possibly used [18]. As reported in many previous studies [18,25,26], thermal properties and sensitivity values will not be much different and are almost applicable to the other H-type component. Without loss of generality, we here implement cylindrical APCs with E-polarization only. Our paper is organized as follows: In section 2 we give a detailed theoretical argument of the electromagnetic waves propagation through APC structures. Section 3 is devoted to numerical simulations and results about the transmission spectra of the APCs at different temperatures. The transmittance of the APCs versus temperatures at specific wavelengths are also studied. Moreover, the transmittance of the electromagnetic waves in APCs structures are plotted at different materials, different number of periods and different core radii. Finally, the conclusion remarks and output findings are given in section 4.
Section snippets
Theoretical framework
The first two periods of the structure of consideration is shown in Fig. 1. A fiber of core and refractive index is coated with alternative layers of thicknesses d1 and d2 and refractive indices and ; respectively. The radius of the jth layer can be given as follow:
Such that: is the period of the coatings. All media in our structure are nonmagnetic so that: . As temperature changes, both the radii of the layers and refractive indices
Numerical results and discussion
The previous investigations of the theoretical analysis could be helpful here to present the transmittance characteristics of the one dimensional APCs. Otherwise mentioned, the proposed structure is designed from Si and SiO2 dielectric layers. The main idea of our work is essentially based on the demonstration of the thermal properties of the transmittance spectrum. Thus, the thermal characteristics of the constituent materials are described in the form of the thermo-optic and thermal expansion
Conclusion
In summary, we investigated theoretically the effects of temperature on the transmittance properties of one dimensional APCs. The designed structure is composed of Si and SiO2 for N = 10 periods. The numerical investigations are demonstrated using the modified transfer matrix method. The numerical results emerged the significant effects of temperature on the transmittance properties of the APC structures compared with the planar PCs. We could obtain a perceptible increase in the width of the
Declaration of competing interest
The authors declare that they have no conflict of interest.
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2021, Optical MaterialsCitation Excerpt :Furthermore, the reflection spectra with periodic sequences were analyzed by Refs. [25–27]. In Ref. [28], lower edge increasing and decreasing of band gaps were observed in transmittance spectra through a change of temperature. Other types of materials, as the superconductor YBCO interspersed periodically by MgO were targeted in Ref. [29], and spectra behavior of reflectance using frequency-dependent permittivity or permeability in context of Metamaterials with single-negative materials was aimed by El-Naggar [30].