Omnidirectional High Reflectors using 1-D Photonic Crystals

Much interest has been paid to omnidirectional dielectric mirrors based on the 1-D dielectric multilayers because these are better than metal mirrors owing to their low optical loss, high reflectivity and high mechanical robustness. In these applications, the required position and width of the reflection band can be obtained by adjusting the layer thickness and refractive indices of the constituent materials. Here, the omnidirectional reflection characteristics of a 1-D PC formed by TiO2/SiO2 multi-quantum wells (MQW) are investigated in the visible region. It is demonstrated that the omnidirectional high reflector, formed by the proper line-up of four TiO2/SiO2 MQWs, covers the visible band approximately when the number of periodic layers of each quantum well reaches 12. This study can provide guidance in the design of an omnidirectional high reflector in the visible band.


INTRODUCTION
In recent years, omnidirectional reflection i.e. nearly 100% reflectivity at all incident angles for both transverse electric (TE) and transverse magnetic (TM) modes by a periodic 1-D medium has much attention from various research groups of the field.Fink et al. (1998) and Winn et al. (1998) firstly designed the omnidirectional reflection in 1-D photonic crystal and demonstrated it over a wavelength range from 10 to15 μm.A lattice consisting of 19 layers of Na 3 AlF 6 and ZnSe was fabricated by Chigrin et al. (1999a;b) and they observed the omnidirectional band gap in the frequency range 604.3 to 638.4 nm. Lee and Yao (2003) investigated different structures theoretically and experimentally for a realistic fabrication of 1-D photonic bandgap materials, exhibiting omnidirectional band gaps (OBGs) in 2003.
The application of one-dimensional photonic crystals in various optoelectronic devices is found to be cheap and effective because their fabrication is relatively easy.The 1-D PCs can be fabricated by using different deposition techniques such as electron beam evaporation, sol-gel method, holography and r.f.-sputtering etc.The different materials used to fabricate 1-D PCs for OBGs in various frequency ranges are tellurium/polyethylene for infrared (Fink et al., 1998), TiO2/SiO2 (Valligatla et al., 2012) for near infrared (NIR) and TiO 2 /SiO 2 (Lin et al., 2005) for visible light.TiO 2 and SiO 2 are non-absorbing and nondispersive high and low index dielectric oxide materials having good refractive index contrast, and are used for E-mail: jppandeymlk@gmail.com.Tel: +919450517226.
Author(s) agree that this article remain permanently open access under the terms of the Creative Commons Attribution License 4.0 International License visible range.TiO 2 /SiO 2 multilayer is investigated for very high reflectivity band gaps in the visible NIR region and omnidirectional reflectivity in the visible region with nearly 100% reflection (Jena et al., 2016).
The photonic band gap (PBG) of photonic crystals (PCs) can be enlarged by arranging the PCs with an asymmetric topology structure (Giden and Kurt, 2012) when choosing dielectric materials as a constituent (Zhang et al., 2013b) and introducing the metamaterials into PCs (El-Naggar, 2013).In these methods, the first one is a better choice to enlarge the PBGs.Recently the PBGs are enhanced in 1-D plasma photonic crystals (PPCs) with Fibonacci or Thue-Morse quasi-periodic structure (Zhang et al., 2013c;2013d) heterostructure, staggered and cascaded structure (Pandey, 2017).The plasma photonic crystals with fractal structures can also enlarge the bandwidths of PBGs.Segev et al. (2012) reported that the fractal structure has a wide range of applications in optics.A lot of fractal topology structures such as the Cantor, Sierpinski gasket, Menger sponge and Koch fractal structures are found in the nature.The subwavelength or compacted devices can be constructed by introducing the fractal structures (Zhang and Chen, 2017;Zhang and Liu, 2016a).
Hojo and colleagues introduced firstly the plasma into the photonic crystals to realize the plasma photonic band gaps in 2004.The external magnetic field, electrontemperature and plasma density can affect the properties of plasma easily.The PPCs show an enlargement and a tunability of the PBG (Zhang et al., 2009;Zhang and Liu, 2016b) as compared to the conventional dielectric PCs.Various research groups worldwide have studied both theoretically (Prasad et al., 2011;Wu et al., 2010;Zhang et al., 2013a) and experimentally (Fan et al., 2010) during the past ten years about it.Also, the enhancements of the PBGs can be done by using superconductor-dielectric photonic crystal which may be used as broad band reflector and omnidirectional reflector at low temperature below to the critical temperature (Pandey et al., 2016a;b).
Also, 1-D photonic crystal made of Si was theoretically studied for high frequency range of omnidirectional reflection by Singh et al. (Singh et al., 2006;Singh et al., 2007) and Thapa et al. (2006) who investigated 1-D omnidirectional high reflector for infrared wavelengths.They demonstrated that, two Si-based photonic crystals increase the frequency range of OBG by overlapping.Various researches demonstrated a relatively narrow OBG for the visible region and also the lack of suitable materials for omnidirectional high reflectors in visible and near-infrared bands.Two types of reflectors made of metals and multilayers of dielectric are generally found.The reflectivity of metallic reflectors is found to be good over a large frequency range for arbitrary angles of incidence but much power is attenuated due to absorption at visible, optical, or higher frequencies whereas multilayer dielectric reflectors show high reflectivity for p articular frequency ranges and are very low lossy.
The proper choice of high and low refractive index materials (Guan et al., 2011;Kumar et al., 2011;Liu et al., 2015) can broaden the OBG and PBG but there are few materials with high index of refraction in the visible region.So, it is difficult to broaden the OBG and PBG, in the visible spectrum.It is found that the introduction of defects and multi-quantum wells (MQWs) (Liu et al., 2015) are very effective for the broadening of the band gap.The desired range of omnidirectional band gap (OBG) was obtained by Wang et al. (2002) from multiple photonic heterostructures formed by more than onedimensional PCs.The high reflectivity and omnidirectional reflection of multilayer dielectric reflectors can be obtained by using photonic quantum wells (Fink et al., 1998;Guan et al., 2011;Liu et al., 2015).
In this communication, quantum wells are introduced for obtaining the large frequency range of OBG in the visible spectrum.TiO 2 and SiO 2 are taken as two kinds of optical materials having good refractive index contrast and less absorption in the visible region.

MATERIAL AND METHODS
In the Transfer Matrix Method (Born and Wolf 1980;Yeh, 1988), the Maxwell's equations are arranged in the k-space and a mesh is formed.It performs layer by layer calculation of multilayers having finite thickness of photonic crystals.The photonic bandgap structures with defects can be studied by considering a super-cell.The band structures, reflectivity and transmission coefficients can be found by this method easily.Many researchers are using this method because the theoretical data obtained from it is in excellent agreement with experimental data.The limitations of this method are the memory storage and also there is a problem to describe the geometry different from the cubic geometry.
Consider a periodic multilayered structure in which electromagnetic wave is propagating.This multilayered structure consists of alternate layers of materials with different refractive indices.The schematic diagram of a multilayered 1-D PC structure is shown in Figure 1, where ni is the index of refraction of the media of incidence and ns is the index of refraction of substrate.The refractive index profile of the multilayer is given below, (1) where n1 and n2 are the refractive indices of two layers and d1 and d2 are their respective thicknesses.d=d1+ d2 is the period of a structure.
Using the transfer matrix method (Born and Wolf 1980;Yeh, 1988), the photonic spectra of this periodic structure are derived.The equation of the electric field is written with the help of Maxwell's equation.The electric field within layer with index n1 is, (2) and, (3)  and 2  are the ray angles in the two mediums, respectively.
At the interface between layers (x = d1), the solution and its derivative should be continuous.This gives a relation between plane waves amplitude: (5) with ( 6) and, also at x = d, the continuity of the plane waves at the interface is between layers with indices n2 and n1 and its derivative gives (7) where the matrix M21 is the same as (6) interchanging the indices.From the two matrix equations ( 5) and (7), we have; ( 8) where Mi,j = M21M12.The matrix element of the matrix Mi,j is given by, for TE mode, and for TM mode.
The matrix Mi,j is called the transfer matrix of one unit of a periodic lattice.The matrix Mi,j depends on the frequency , and it is unimodular (it is a square matrix with determinant equal to unity).Hence, for each , the matrix Mi,j defines a unique mapping for amplitudes of the plane waves in layer n1 into the amplitude of the next layer with index n2.
The coefficients of right and left hand side propagating waves in both sides of multilayer structures of N periods are calculated by multiplying transfer matrices of each cell as given below, (9) The coefficient of reflection can be found as (10) The reflectance is calculated as, (11) The phenomenon of propagation of electromagnetic waves through multilayered periodic structure is similar to the motion of electrons in solid crystals.So, the electromagnetic waves can be compared by the Bloch waves and the following dispersion relation is satisfied (Yeh, 1988).
is the real number, the electromagnetic wave can transmit through the multilayer.
) , ( , 1 2 is an imaginary number, in this condition, wave cannot transmit through 1-D PC and is totally reflected.This corresponds to the band gap of the structure.When , edges of the band gap of the photonic crystal is found.To understand the condition for the existence of omnidirectional reflection in 1D-PC, the Snell's law is given as, where nH and nL are the high and low refractive indices of media, ni is the index of refraction of medium of incidence.The internal Brewster angle is greater than maximal refracted angle, the incident wave will be totally reflected for all angles of incidence.


is the maximum refractive angle of the light in high index medium.The difference between lower band edge at the normal incidence and the upper band edge at the perpendicular incidence is the OBG for both TE and TM polarizations.

RESULTS AND DISCUSSION
Here, the 1-D photonic crystal is considered as a multilayer periodic structure composed of muti-quantum wells (MQWs).It has TiO 2 as a high refractive index and SiO 2 as a low refractive index material.The structures (AB) m , (CD) n , (EF) l and (GH) s composed of TiO 2 and SiO 2 are considered for the study.
A, C, E and G represent TiO 2 ; B, D, F and H represent SiO 2. m, n, l and s denote the number of periodic layers of structure.The quantum well unit cell constant is represented as d i , and The PBG and OBG of the photonic quantum well of TiO 2 /SiO 2 can be investigated by the transfer matrix method.The thicknesses d i are taken according to the quarter wave stack arrangement as d i = λ 0 /4.n i.The central wavelengths are taken as 800, 680, 560 and 440 nm for the structures (AB) m , (CD) n , (EF)  Figure 2(a) to 6(a) represents the 100% reflection regions of Transverse electric (TE) and Transverse magnetic (TM) modes.If the angle of incidence is 0 0 , the PBGs are completely overlapped in both TE and TM modes.When the angle of incidence is 0° to 30°, the width of the band gap and the reflectivity rate is stable for two polarized modes.But when the angle of incidence is between 30° and 85°, band gap decreases and the gap width of the TM mode becomes narrower.Therefore, the reflectance spectra of the combined structure of the four quantum wells were theoretically investigated.
Firstly, a 1-D PC (AB) m is considered which consists of 12 periodic layers of TiO 2 and SiO 2. Their refractive indices are 2.3 and 1.5, respectively.This 1D-PC structure is on a substrate of glass having refractive index n S = 1.5 and medium of incidence is air n i = 1.0.The thickness of the layers is chosen in accordance with the quarter wave arrangement, where d H and d L are the thicknesses of the TiO 2 and SiO 2 layers.According to this condition, d H = 86.96nm and d L = 133.33nm.For this, photonic crystal structure,

Conclusion
The omnidirectional reflectance spectra of the multilayered dielectric structure composed of TiO 2 /SiO 2 MQW is theoretically investigated by the transfer matrix method in the visible region.It is demonstrated that the omnidirectional high reflector, formed by the proper line-up of four TiO 2 /SiO 2 MQWs, covers the visible band approximately, when the number of periodic layers of each quantum well reaches 12.

Figure 1 .
Figure 1.Schematic diagram of a multilayered dielectric photonic crystal structure.
of omnidirectional reflection is satisfied.The reflectance of TE polarizations at 0°, 45° and 85° angles and TM polarizations at 0°, 45° and 70° angles are represented in Figure 2(a).Figure 2(b) shows the conventional photonic band structure which is obtained by the projection of reflectivity values 1 R  from Figure 2(a).The grey and white regions of

Figure 2
Figure 2(b) represent the forbidden and allowed bands, respectively.The values of frequency regions having nearly 100% reflectivity are shown in the Table 1 and the omnidirectional reflection range (ODR) is represented by the black area between two horizontal lines in Figure 2. In this structure, both the polarizations (TE and TM) possess omnidirectional reflection when angle of incidence is below 70°.ODR is found from H  = 718 to

Figure 2 .
Figure 2. (a) Reflectance of TE and TM polarizations of twelve-pairs of (AB)m 1-D PC at various angles (b) their photonic band structure.

Figure 3 .
Figure 3. (a) Reflectance of TE and TM polarizations of twelve-pairs of (CD)n 1-D PC at various angles (b) their photonic band structure.

Figure 4 .
Figure 4. (a) Reflectance of TE and TM polarizations of twelve-pairs of (EF)l 1-D PC at various angles (b) their photonic band structure.

Figure 5 .
Figure 5. (a) Reflectance of TE and TM polarizations of twelve-pairs of (GH)s 1-D PC at various angles (b) their photonic band structure.

Table 1 .
Region of total reflection and bandgap width for (AB)m 1-D PC having 12 layers.

Table 2 .
Region of total reflection and bandgap width for (CD)n 1-D PC having 12 layers.

Table 3 .
Region of total reflection and bandgap width for (EF)l 1-D PC having 12 layers.