Extension of the Transfer Matrix Method for Photonic Band Gap Engineering in Optimized TiO₂ Nanomaterial Bragg Reflectors

INTRODUCTION 
At present, Nano-materials stand as the most advanced type of material in scientific knowledge and commercial applications. Inorganic nanoparticles (NPs), including silver, copper, titanium, and zinc, are among the most interesting NPs because of their applications and positive impact against pathogenic microorganisms. Recently, TiO2 NPs have gained special attention because of their unique optical, electrical, and chemical properties. TiO2 is an excellent photo catalyst that has been widely used for antibacterial activity due to its high photosensitivity, high efficiency, nontoxic nature, strong oxidizing power, relative cheapness, and chemical stability In order to enhance the photocatalytic activity, intensive interdisciplinary researches have been made on TiO2 It is known that the photocatalytic activity of NPs depends on their crystalline structure, doping, surface area, and hydroxyl group. One of the most significant developments in laser physics was the ultrafast laser technology. The duration of the pulse of the pulse laser is ultra-short, measured in picoseconds (ps) or femtoseconds (fs) [10,11]. An ultrashort pulse optical system reflector needs to have reflection extremely high across the bandwidth of the pulse. Two methods were used to realize an omnidirectional reflector for both normal and oblique light incidents: one was to use thin sheets of gold or silver as mirrors, and the other was to use entirely dielectric materials. Metallic mirrors do not allow light to pass through a dispersive material and maintain the shape of the pulse after reflection; however, their reflection is too low for applications that involve ultrashort pulse use [12].High-intensity, femtosecond pulses can be reflected by metallic reflectors; however, absorption in the metallic reflector leads to damage and eventual destruction of the assembly. Because all-dielectric mirrors have very low losses (less than 0.01%) at optical and infrared wavelengths, they were used over regular metallic mirrors [13]. Most probably Penselin and Steudel used the dielectric high reflection mirror for the first time [14]. They used thicknesses with harmonic progression to make an all-dielectric high-reflecting mirror with reflectivity near 95% and bandwidth of 430–450 nm for Ti: sapphire laser systems [15]. Baumeister and Stone made a basic computer-based method [16], maximize the reflectivity of the multilayer reflector to about 94% over the 1.5–1.6 µm wavelength range. Pelletier et al. [17] reported computer design and improvement of reflectors in 1971. The results showed that ripple appeared in the high reflection zone within 95.5% to 98.8% over a wide spectrum range of 1.0-2.0 µm. Numerical approaches were split into two categories: refinement methods and synthesis methods. The former was said to have better potential in building filter coatings of the specificities complex spectrum, which cannot be designed in the standard ways. The synthesis method used in this paper to arrive at the best design was ZEMAX optimization operand — Hammer optimization. It is, therefore, the objective of this work to present a design methodology for 1D-photonic wideband omnidirectional reflectors for application in the near-infrared region.

MATERIALS AND METHODS
Design of DBR
It is generally known that a stack of quarter-wave dielectric layers with basic building blocks (HL) or LH that alternately has high and low indices can provide a high reflectivity. This is because the beam reflected from every phase when it reached the front surface, where it combined constructively [18]. The following equation for maximum reflectance and width was presented. The characterize matrix of the basis stack (HL) at an incidence angle of θ in an incident medium of refractive index n0 is provided by [19] if the refractive index at the thickness of two materials is represented as (nH, dH) and (nL, dL) respectively. 

At quarter wave length:

Then, the reflectance for TE and TM polarization is:

here Ʌ represents a polarization of TE and TM, ZEMAX, which is based on the elegant transfer matrix approach (TMM) for optical performance analysis, was used in this investigation [20]. As shown in Fig. 1 and omnidirectional reflection were specified by two factors (Center wavelength λc, Omnidirectional bandwidth ∆λOmni).
For optimal reflection (Bragg condition) from a pair of low and high refractive index layers at a specific angle and polarization, the optical thickness at the stop band center must be half wavelength. [21]:

λc/2=nL dL + nH dH 

For TM-polarization light, λc is also a very excellent approximation for the reflectance band center [22] in the following:

λc=[(1-sin2θ0/nL2)1/2(1-sin2θ0/nH2)1/2

Accordingly, the normalized omnidirectional band width ∆λOmni with respect to the omnidirectional band’s center wavelength λc was determined to be [22].

Optimization method
Nearly all numerical techniques for designing optical multilayer coatings rely on the merit function MF, which must reflect the discrepancy between the needed Ti and the calculated spectral characteristic Vi. Merit functions are one-value representations of the system’s current performance that guide the course of upcoming computations [23-24]. The square root of the weighted sum of the squares of the difference between each operand’s actual and target value for coating determines the merit function. Since this is how the merit function is defined, zero is the ideal value. Because the optimization method will try to minimize the value of this function, the merit function should reflect the desired outcome of the system. The default merit function is not required. Dobrowolski’s [25] merit function form is the one that is most commonly used: 

where the operand number (or spreadsheet row number) is indicated by the subscript “i” and “W”, “V”, and “T” stand for the operand’s weight, current value, and target value, respectively [26]. The aforementioned formula makes it very evident that the closer the system performs to the aim, the smaller the MF value. Additionally, all of the layer systems’ construction factors affect MF [17], i.e. MF= MF[ns, ks, n0,(ni, ki, di), i:1,2,q].By using ZEMAX, the optimization method computes but disregards the operand when the weight is zero. This is especially helpful if the value is used as a check or monitored parameter, or if the result is calculated without a defined objective but may be used elsewhere in the merit function [27]. In this work Hammer Optimization was used. It was a customized algorithm made to look for different designs that are based on the first design. The main purposes of the Hammer method are to avoid local minimum or to confirm that there isn’t a better design in the vicinity of the solution space.

RESULTS AND DISCUSSION
XRD Result for TiO2 Nano-layer 
Fig. 2 shows the XRD pattern for TiO2. The diffraction peaks at 25,50o, 37,89o 48,40o, 53,98o, 55,29o, 62,65o, 68,47o, 70,57o. were indexed to (101), (004), (200), (105), (211), (204), (220), (220) and (215) crystalline planes respectively. This corresponded to the anatase phase of TiO2 according to JCPDS Card No. (21-1272). The anatase phase of TiO2 is preferred over rutile and brookite for photocatalytic degradation of organic compounds [28]. Table 1 is shown the average of the grain size of TiO2-NPs.

Effective index η
Fresnel equations are the main cause of coating polarization dependency at oblique incidence. It is possible to think of each polarization behavior mode as a normal incidence performance of two designs. The effective index η was the cause of this. The behavior of η_TE and η_TM as a function of incidence angle and material refractive indices was shown in Fig. 3. Effective indices for TM and TE polarization are consistently higher or lower (in the comparable design) than in the actual coating, as seen in Fig. 3. The behavior were shown for three optical commonly used nano- materials (TiO2, SiO2) with indice 2.45,1.44 deposited on glass of index 1.50 at 1.55 µm.
Design Bragg reflector for dielectric material
The two alternating layers with refractive indices nH(TiO2) and nL(SiO2) that make up our multilayer, one-dimensional photonic crystal were chosen because they are non-absorbing, isotropic, and homogenous dielectric materials. The reflectance of TM-polarization light at θ=89ois greater than 97%, according to the study of the initial design Air ǀ [LH]10 ǀ glass period. The reflectance was plotted against wavelength in Fig. 4. The reflectance 84.8% reflection drop that appears in the near-infrared areas can be interpreted as either “noise” or as a narrow photonic bandgap function acting as an omnidirectional narrow band filter
The Fig. 5 is shown between the reflectance and angle of incident (0o – 90o ) for the design Air ǀ [LH]10 ǀ glass at wave length design 1.55 µm in the following.

Infrared region
Fig. 6 represent the conventional photonic band structure for TE an TM polarizations of both design structures: 

using Teraplot program, the wavelength design λ0=1.55 µm, which can be obtained by the projection of R≈1 the spectra are plotted in terms of wavelength and for incident angle θ0. The red region represents the forbidden band and the other colored regions the allowed. The area between the two horizontal lines gives the TODR band.

Optimum design Bragg reflector
Numerical approaches were separated into two categories: refinement methods and synthesis methods. The former was described as having a high potential to construct filter coatings of the specificities complex spectrum that cannot be designed in the standard ways [29].To get to the best design, the ZEMAX optimization operand (Hammer optimization) is used as a synthesis approach in this work. The RTM of the photonic band gap of the original design was extended and improved using ZEMAX optimization operands of coating at (89⸰) angle of incidence for Air|(LH) 10|Glass. The optimal choice of target, numeric merit function, and operand coating were necessary to achieve such goals like TM-polarization mode with normal incidence curves. Only “thickness multipliers” are discussed as a variable in this section. With design construction, we were able to meet our goal for NIR:
Design #
Air|0.239322L 0.222700H 0.477447L 0.197263H 0.044163L 0.220264H 0.153981L 0.555462H 0.126097L 0.550452H 0.255334L 0.067249H 0.161282L 0.169167H 0.537156L 0.213032H 0.081301L 0.149266H 0.234298L 0.534108H |glass
Shown in Fig. 7 when the reflectance in infrared region behavior was plotted as a function of wavelength and angle of incidence 
It was found that TODR were arrived for NIR regions with photonic band gap (0.625 µm). The total optical thickness for before optimization is equal (5µm), then after optimization the total optical thickness for infrared region of design # is (5.189344µm). The Hammer optiumization was made cancellation some stack at optimization work for get to optimum design and thus reduced for number of layers, this process needed a high weight to reach the required target.The reflectance versus angle of incidence, for the infared region where R is approximation ~ 0.94% at angle of incidence 2θ = 89⸰. Therefore, a remarkable improvement in the performance of omnidirectional reflector for both grazing angle and TM mode of polarization, where the reflected rays from the stack nearly in phase thus, a constructive interference yield within the wavelength infrared range (1-2 µm).

CONCLUSION
An omnidirectional 1D-phtonic bandgap is very well provided by quarter-wave stacks. When the matching layer thickness was used as the initial design in the optimization process, the omnidirectional reflection band was broadening. Furthermore, the results demonstrate that the suggested optimal approach is not dependent on the specific example being studied and, in certain situations, results in designs that are difficult to solve using other established methods. This technique predicts the ideal thickness that can deviate from the typical quarter-wave design to provide a high performance 1D-photonic band gap, and it is readily adaptable to other materials and wavelengths.

CONFLICT OF INTEREST
The authors declare that there is no conflict of interests regarding the publication of this manuscript.