Design of an omnidirectional mirror using one dimensional photonic crystal with graded geometric layers thicknesses
Introduction
A photonic crystal is a periodic dielectric structure that owns a range of wavelengths for which light propagation is forbidden. Conventional photonic crystals are fabricated with structure periods that are comparable to the wavelength of the incident electromagnetic waves. To have the required forbidden band gap, there are great efforts to obtain tenability of band gaps. So, we deal with changing the conventional structure using defects [1], quasi-periodic systems [2], heterostructures [3], negative refractive index [4], disordered structure [5], [6], etc.
The use of a chirped grating structure (chirped mirror) instead of a uniform period one offers also the benefit of a larger bandwidth [7]. Several researches on chirped mirrors have been reported. Tehranchi and Kashyap have proposed a chirped and an apodized gratings for broad band frequency doublers based on quasi-phase matched second harmonic generation in lithium niobate waveguides [7]. Bi and Wang have showed that with the chirped structure, the photonic band gap was effectively extended, they analysed numerically the photonic crystals with two-dimensional chirped structure and with apodized structure [8]. Chirped mirrors are used in applications to reflect a wider range of light wavelengths than ordinary dielectric mirrors, or to compensate for the dispersion of wavelengths that can be created by some optical elements because they have the property to retard light depending on its wavelength [9].
One of the most important applications of chirped structure in photonic crystals is the chirped fibre grating. Several researches report that chirped fibre grating exhibits features of wide reflected band [7], [8], [10]. The core of the fibre Bragg grating has a periodic dielectric structure. That of the chirped fibre Bragg grating has a chirped grating structure. Chtcherbakov and Swart have proposed a sub-carrier phase detection scheme for interrogation of a chirped Bragg grating strain sensor. They demonstrate the concept experimentally with a linearly chirped grating [9].
Besides, scientists seek for attaining high reflection at any incident angle for both polarizations, the transversal electric (TE) and transversal magnetic (TM) polarization. Therefore, it is interesting to design omnidirectional mirrors in any optical range according to the users’ requirements. It is known, in the past, that omnidirectional mirrors just exist in three-dimensional photonic crystals (PCs), which are difficult to fabricate. Since omni-directional band gap of one dimensional photonic crystals was proposed in 1998 [10], they have received much attention and applications about this omnidirectional band gap have been developed [11], [12], [13], [14].
Within this background, the present work proposes some designs of an omnidirectional mirror using a chirped structure. We deal with selecting the appropriate functions of chirping and the appropriate coefficients that are suitable for our requirement which is in this work broadening the omnidirectional PBG which covers the telecommunications wavelengths 0.85 μm, 1.3 μm and 1.55 μm. The idea consists on increasing gradually the geometric thicknesses of layers according to a linear gradation at the first and later an exponential gradation. The results show that by choosing the convenient chirping functions, we can extend the complete PBG according to our requirements. The 1D PCs considered here consist of a periodic arrays of two alternating layers, Si and SiO2 with high and low refraction indices (nH and nL) and different thickness values (dH and dL) with one or two of them are gradually changed. For our study, refractive indices of these materials are assumed to be constant in the wavelength region of interest. The numerical method employed to obtain the transmission response of the structure is the transfer matrix method.
Section snippets
Model and formalism
For the calculation of system reflection and transmission, we employed the transfer matrix method (TMM). This technique is a finite difference method particularly well suited to the study of PBG materials and it can solve the standard problem of the photonic band structures and the scattering (transmission, reflection, and absorption) spectrum [15].
It is based on Abeles method in terms of forward and backward propagating electric field, that is, E+ and E− which were introduced to calculate the
Linear gradation
Linear gradation means that the difference between a layer and the next one of the same material is constant. So, and take the forms shown in (17), (18).For j = 1, the layer of high refractive index and that of low refractive index have thicknesses shown in respectively (19), (20).
We first consider the reference wavelength 0.8 μm. So, the thicknesses values are dH = 0.054 μm and dL = 0.1379 μm. We report in Fig. 1a the transmission spectra
Exponential gradation
We apply an exponential gradation of layers thicknesses, so the increasing of thicknesses has the following form (21),
Each layer of high refractive index can take the form (22)
In the same way for the layers of low index, their thicknesses can vary according to their order as shown in (23)
The first two layers of high and low indices will have as thicknesses those showed respectively in (19), (20). The choice of the coefficients α and β
Conclusion
In this work, an approach is used to enlarge the omnidirectional PBG of one dimensional multilayer structure. This approach is based on increasing gradually the geometric layers thicknesses according firstly to a linear function, and secondly to an exponential function. We show by numerical simulations, the efficiency of this approach to control the omnidirectional PBG by selecting the appropriate gradation degrees. The structure can take place in the practice since its fabrication seems
References (16)
- et al.
Optical transmission spectra in symmetrical Fibonacci photonic multilayers
Phys. Lett. A
(2009) - et al.
Negative refraction in 1D photonic crystals
Solid State Commun.
(2008) - et al.
Band-gap extension of disordered 1D binary photonic crystals
Physica B
(2000) - et al.
Experimental evidence of photonic band gap extension for disordered 1D photonic crystals based on Si
Opt. Commun.
(2006) - et al.
Eye-protection glasses against YAG laser injury based on the bandgap reflection of one-dimensional photonic crystal
Opt. Laser Technol.
(2007) Principles of the plane-wave transfer-matrix method for photonic crystals
Sci. Technol. Adv. Mater.
(2005)- et al.
Emulation of two-dimensional photonic crystal defect modes in a photonic crystal with a three-dimensional photonic band gap
Phys. Rev. B
(2001) - et al.
Optimized design and experiment of one-dimensional omnidirectional reflector using P-wave angle domain compensated overlapping method
Appl. Phys. Lett.
(2002)
Cited by (28)
Tunable filter properties in 1D linear graded magnetized cold plasma photonic crystals based on Octonacci quasi-periodic structure
2020, Photonics and Nanostructures - Fundamentals and ApplicationsComparative study of two thermal tunable filters with and without symmetrical linear gradation of thickness
2019, Physica B: Condensed MatterAnalysis of transmittance properties in 1D hybrid dielectric photonic crystal containing superconducting thin films
2018, Physica B: Condensed MatterRealization of compatible stealth material for infrared, laser and radar based on one-dimensional doping-structure photonic crystals
2017, Infrared Physics and Technology