Elsevier

Optik

Volume 123, Issue 2, January 2012, Pages 125-131
Optik

Design of an omnidirectional mirror using one dimensional photonic crystal with graded geometric layers thicknesses

https://doi.org/10.1016/j.ijleo.2011.03.010Get rights and content

Abstract

We propose a flexible design for one-dimensional photonic crystals (1D-PCs) with a controllable omnidirectional band gap covering the optical telecommunication wavelengths which are 0.85 μm, 1.3 μm and 1.55 μm. We used for this design the chirped grating. Chirping is applied to geometric thicknesses of layers. It takes two forms, one is linear and the other is exponential. We exploit this technique to have the suitable omnidirectional band gap covering the maximum of optical telecommunication wavelengths. With a quarter wave structure, we can have an omnidirectional band gap generating only one of these wavelengths. With graded structure, we obtain, as is reported in this paper, an omnidirectional band gap which covers the wavelengths 1.3 μm and 1.55 μm at the same time with a large bandwidth. We also achieve an omnidirectional band gap containing the wavelength 0.85 μm and which is obviously larger than that of the quarter wave stack.

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, (ΔdH)j and (ΔdL)j take the forms shown in (17), (18).(ΔdH)j=(j1)*δdH(ΔdL)j=(j1)*δdLFor j = 1, the layer of high refractive index and that of low refractive index have thicknesses shown in respectively (19), (20).dH=λ04*nHdL=λ04*nl

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),y=β*exp(α*x)

Each layer of high refractive index can take the form (22)(dH)j=λ04*nH+β*exp(α*j)

In the same way for the layers of low index, their thicknesses can vary according to their order as shown in (23)(dL)j=λ04*nL+β*exp(α*j)

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

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