Photonic crystals (PhCs) are synthetic materials that can uniquely manipulate light propagation across different length scales [1], and are therefore useful for a wide range of applications such as optical filters, lasers, waveguides, solar energy applications, optical switches, and light-emitting diodes (LEDs) [2] [3] [4] [5] [6]. Because of their low cost and ease of fabrication, lD PhCs composed of alternating layers of high and low dielectric constant material are the most researched and applied PhCs, and have been gaining in popularity as Bragg mirrors, thermal sensors, energy-saving spectrally-selective coatings, anti-glare side view car mirrors, transparent heat reflectors, thermal collectors, optical filters, and structural colors because of their intriguing optical properties [7] - [12]. For 1D PhCs, the position of the transmission spectra and the intensity of the reflectance peak are influenced by the thickness, number, and refractive index of the layers. The greater the dielectric constant difference between the structural layers, the fewer the layers required to achieve a given reflectance value, which allows for optical filtering applications to be targeted [13]. From a lattice dynamics standpoint, the Bragg scattering bandgap and local resonant bandgap principles are confirmed for 1D PhCs [14].

Optical filters allow us to extract required signals without the use of electrical circuits, and common optical filters include Fabry–Perot interferometers, waveguide Bragg gratings, and Mach–Zehnder interferometers [15]. However, these are not suitable for optical integrated circuits due to their large area. On the other hand, PhC filters perform significantly better than previous systems, with benefits such as ease of fabrication, quick modulation, low-cost material composition, and sharp and smooth optical transitions [16] [17]. For example, SP Singh et al. [18] used the transfer matrix method (TMM) to create 1D optical filters from GaP and GaSb. The effect of disorder on the transmission spectrum of electromagnetic waves has been investigated using the transfer matrix method in one-dimensional multi-layered structures containing ferroelectric materials, such as LiTaO₃ [19]. This and other studies have shown that there is a high level of agreement between TMM simulations and physical experiments [20]. This suggests that TMM simulations could be used to elucidate the promise of other materials as 1D PhC optical filters.

Theoretical investigations have recently enabled the development of novel types of PhC devices, such as all-optical switches, two-state and many-state memory, all-optical limiters, all-optical modulators, and all-optical transistors. Significant research has focused on designing innovative PhC architectures into each functional layer of the solar cell to boost device performance. These optical structures feature one-of-a-kind qualities that bring up new possibilities for a wide range of applications. However, there are holes in the literature regarding comparative studies of 1D PhCs structures in regards to the influence of layer thickness, number, and refractive index on the ability of the PhCs to control light transmission. Metal oxides such as TiO₂, SiO₂, and SnO₂ have emerged as critical materials for high-performance optoelectronics [21] [22] [23], and we, therefore, explored their optical filtering performance as 1D PhCs using TMM simulations. The main objective of this theoretical analysis was to define and compare the influence of layer thickness, number, and refractive index on the ability of the PhCs to control light transmission for optical filtering and photovoltaic applications. We found stark differences between layered 1D PhCs composed of different combinations of the three metal oxides that helped identify materials trends potentially promising for crating optical filters with different controllable properties. This work paves the way for improved optical filters and confirms the value of TMM simulations for screening the properties of various theoretical 1D PhCs.

We must develop methods for examining multilayers in order to understand their fundamental properties [24] [25], and plane-wave propagation in an isotropic homogeneous medium is the most basic case. An important approach for studying the interaction between incident electromagnetic waves and 1D PhC layers is TMM, as it is the most widely used method for mathematically studying wave transmission in 1D materials [26].

TMM can be used to measure the transmission and reflection of incident electromagnetic waves across a multilayer periodic system, such as 1D PhCs made up of two layers of two different dielectric materials that are repeated in N unit cells in a periodic pattern. We utilize different materials and 1D PhCs structures (Figure 1) and theoretically compute the transmission of light from 300 nm up to 2500 nm using TMM. Specifically, our structures are made up of layers that are repeatedN times and are only periodic in the x-direction, hence they are 1D PhCs. We specify the thickness (d₁, d₂) and refractive index (n₁, n₂) of two PhC dielectric materials (chosen from TiO₂, SiO₂, and SnO₂) with different refractive indexes (A and B layers), and the medium is homogeneous in the z-direction such that Snell’s law is followed at each interface (i.e., n 1 sin θ 1 = n 2 sin θ 2 ).

The following are the dynamical matrices for the transverse electric, TE, mode used in our TMM simulations:

D m T E = [ 1 1 n m − n m cos θ m ] , m = 1 , 2 (1)

where m = 0, 1, 2 denotes air, and the first and second layer, respectively, θ_m is the angle of the incidence for each layer, ωis the angular frequency, c is the electromagnetic (EM) wave speed in a vacuum, and n_m is the refractive index.

Each layer’s propagation matrix is given as:

P m = [ e − i k m b m 0 0 e − i k m b m ] , m = 1 , 2 (2)

where k m = ω n m cos θ m / c is the is the wave vector value.

Each periodic layer’s transfer matrix is written as:

M p = D 1 P 1 D 1 − 1 D 2 P 2 D 2 − 1 (3)

After multiplying all of the individual transfer matrices for the overall periods (N) of the structures, we get:

M = ( M 11 M 12 M 21 M 22 ) = D 0 − 1 M p N D 0 (4)

where D₀ is the air dynamical matrix.

The characteristic matrix M[d] of one period is given by:

M [ d ] = ∏ ​ i = 1 l [ cos φ i − i sin φ i p i − i p i sin φ i cos φ i ] = [ M 11 M 12 M 21 M 22 ] (5)

where φ = k|d|, and l represents the layers of refractive index.

The characteristic matrix of the medium is given by:

M N = [ M 11 U N − 1 ( q ) − U N − 2 ( q ) M 12 U N − 1 ( q ) M 21 U N − 1 ( q ) M 22 U N − 1 ( q ) − U N − 2 ( q ) ] ≡ [ m 11 m 12 m 21 m 22 ] (6)

The Chebyshev polynomials of the second kind are:

U N ( q ) = sin [ ( N + 1 ) cos − 1 q ] [ 1 − q 2 ] 1 / 2 (7)

where q = 1 / 2 [ M 11 + M 22 ] .

The following equation gives the transmission value:

T = ( 1 M 11 ) 2 (8)

In terms of the transmission coefficient, t, and the transmissivity of this structure can be stated as:

T = p s p 0 | t | 2 (9)

The transmission coefficient, t, of the multilayer is given by:

t = 2 p 0 ( m 11 + m 12 p 0 ) p 0 + ( m 21 + m 22 p 0 ) (10)

where p₀ = n₀cosθ₀ = cosθ₀ is the propagation vector. The absorptance, A, is defined as the fraction of energy released and is calculated by A = 1 − R − T, whereR is the reflectance. The formulas for r and t may be used to show that R + T = 1 for dielectric systems with real n₁ and n₂ according to energy conservation.

To simplify the calculations, we focus on the electromagnetic transmission of the PhC’s. This study concentrates on transmission, as it is the most relevant for various optical applications including the design of optical filters. MATLAB was used and code was written to calculate various factors that affect the performance of the PhCs, such as the ambient medium (n₀), the refractive index of the substrate (n_s), the incidence angle (θ), the wavelength range (λ), the number of bilayers (N), and the film thickness (d).

PhCs have refractive indexes that change in the same order as the wavelength of light, where the electromagnetic radiation cannot propagate for a specified range of energies and wave vectors [27]. PhCs rely on the phenomenon of slow group velocity photons, also known as “slow” light that occurs when the group velocity of light is reduced near the photonic bandgap, which can increase the degree of light absorption is useful for optical filtering [1] [28]. While PhCs can be prepared from different materials, the most common wide-bandgap oxides used in electronics and optical devices are TiO₂, SiO₂, and SnO₂ because of their high sensitivity to a wide range of optical wavelengths, simple manufacturing methods, low cost, and excellent compatibility with other parts and processes. We therefore explore the theoretical modeling of PhCs composed of alternating layers of these metal oxides thin films. All materials used herein are non-magnetic, homogeneous, and isotropic, and the different parameters included in this study are summarized in Table 1.

1) 1D TiO₂/SiO₂ PhCs

Photonic multilayer films such as TiO₂/SiO₂ are commonly manufactured for third-generation photovoltaic cells by alternately evaporating these high and low refractive index materials under high vacuum conditions [29] [30] [31] [32] [33]. TiO₂/SiO₂ has a high refractive index contrast, good passivity, and the ability to provide a conductive pathway, and has therefore been widely used experimentally [34]. Moreover, various SiO₂/TiO₂ stacks have been simulated using TMM for near-ultraviolet reflective and near-infrared anti-reflective filters [35]. Generally, these earlier studies only covered wavelengths up to 1500 nm, which suggested there was a need to examine a longer range of wavelengths. We therefore

use TMM to simulate the optical properties of the TiO₂/SiO₂ layers from 300 - 2500 nm (Figure 1). TMM is regarded as one of the most suitable methods for investigating the interaction between incident electromagnetic waves and 1D PhC structures composed of different layers (N). In most studies, we can see only one photonic bandgap in the visible region, but by controlling the number of layers and the thickness, we were able to observe different bands in our study.

We investigated the effect of increasing the number of TiO₂/SiO₂ layers from 1 to 8 and the corresponding transmission curves. We chose a simple structure a TiO₂/SiO₂ thickness of 100 nm (50 nm/50 nm) for each layer in order to investigate the effect of multiple layers on the position and width of photonic bandgaps. There were visible differences in transmission spectra as the number of layers increased from 1 to 8 (Figure 2). At 8 layers, we observed the lowest transmission curve through the photonic bandgap area, while the rest of the curve still had a high transmission value. While 1 and 2 layers did not show photonic bandgaps, there was no clear interference through the rest of the transmission curves. The position of the photonic bandgaps in these two cases begins at longer wavelengths and then shifts to shorter wavelengths as the number of layers increases. We can also see that the width of the photonic bandgap was wider at higher layer numbers (i.e., N = 8). After 8 layers, there was no discernible differences in bandgap width, and therefore we utilized N = 8 in the following sections.

2) 1D TiO₂/SnO₂ PhCs

Metal oxides such as SnO₂ have promise as primary materials in advanced applications in the optical, electronic, optoelectronic, and biological domains [36]. SnO₂ has a bulk bandgap of 3.6 eV (at room temperature), and has long been used as an opacifier and white colorant in ceramic glazes. Moreover, according to Diego Lopez-Torres et al. [37], SnO₂ is highly sensitive to humidity variations

and can be used to increase the sensitivity of sensors based on PhC fibers. For example, SnO₂ does not require heating to function on an optical fiber, while the majority of metallic oxides require temperatures greater than 150˚C to function [38].

We used TMM to explore the changes in transmission curves when increasing the number of TiO₂/SnO₂ layers from 1 to 8. Like for TiO₂/SiO₂, we used a thickness of 100 nm (50 nm/50 nm) for each layer in order to investigate the effect of multiple layers on the position and width of photonic bandgaps (Figure 3). Increasing the number of layers from 1 to 8 caused the bandgap to widen. At N = 8, the transmission curve through the area of the photonic bandgap is the lowest, while the transmission of the curve was still high. At N = 1 and 2, the transmission curve began to appear at 450 nm with a modest peak. However, between N = 6 and 8, this peak flips into a broad photonic bandgap. Overall, the peaks are shifted to lower wavelengths at higher layer numbers. Interestingly, the band-edge is not as sharp as it is in the SiO₂ system, as the transition is smooth. Moreover, N = 8 has the lowestT value, which is close to zero and may reflect nearly 100% of the light.

3) Comparative study between TiO₂/SiO₂ and TiO₂/SnO₂ 1D PhCs

We see different behaviors when we compare the different 1D PhCs studied above with 1, 2, and 4 layers (Figure 4). We saw the bandgap appear first for TiO₂/SiO₂ at N = 2, and a clear shift to a lower wavelength is also observed. As the number of layers increase to 6 and 8, the width of the photonic bandgap becomes wider for TiO₂/SiO₂ when compared to TiO₂/SnO₂ (Figure 5). Moreover, at N = 8 the edges become even sharper and wider for TiO₂/SiO₂ when compared to TiO₂/SnO₂, and the photonic bandgaps were zero T and ~10%T, respectively. In general, this result agrees with the experimental and theoretical work of F. Javier Ramos et al. [39].

While we previously restricted our analysis to systems with a fixed layer thickness, different thicknesses should produce different properties. For example, we see a clear difference in the position and width of the bandgaps and transmission peaks when the thickness is changed from 50 nm/50 nm to 150 nm/150 nm or 300 nm/150 nm (Figure 6). Specifically, TiO₂/SiO₂ has a wider photonic bandgap from 350 - 500 nm, whereas TiO₂/SnO₂ has a bandgap from

450 - 550 nm without sharp edges that shifted slightly to longer wavelengths under the same conditions. Moreover, we could better observe two photonic bandgaps for TiO₂/SiO₂ when the thickness was increased to 150 nm/150 nm, where higher thicknesses had an even more pronounced effect on the transmission behavior. The TiO₂/SiO₂ films exhibited high transmittance and a wide bandgap for systems with thicknesses of 500 nm/150nm and 800 nm/150nm (Figure 6), which supports a wide range of optical filter applications. However, in the case of TiO₂/SnO₂, these thicker structures could support a single optical detector. The dependence on layer number and thickness motivated us to investigate SiO₂/SnO₂.

4) 1D SiO₂/SnO₂ PhCs

SiO₂/SnO₂ materials have previously been studied in a variety of applications such as sensors, thin films, and transparent ceramic electrodes due to their transparency in the visible and near-infrared parts of the electromagnetic spectrum. Several studies have been carried out in order to realize and demonstrate the photorefractive effect of SiO₂/SnO₂ glass-ceramics for photonic applications, as well as the role of SnO₂ nanocrystals as rare-earth luminescence sensitizers [40]. Specifically, UV irradiation causes a change in the refractive index, which allows for the direct writing of channel waveguides and Bragg gratings. We continued our comparison study of SiO₂/SnO₂ using TMM and the same structural parameters used above. By increasing the number of layers, we did not see a clear photonic bandgap at N = 1 or 2. The lowest T was observed at N = 8, while the highest T occurred at N = 4. Specifically, increasing the layer number enhances the homogeneity of the films due to the high interference probability (Figure 7). It is interesting to note that there is some consistency in the optical

waves around λ = 700 nm.

5) Comparative study of the three different 1D PhCs

The SiO₂/SnO₂ PhC almost had a higher T than the TiO₂-based PhCs, particularly in the visible region. Specifically, the SiO₂/SnO₂ peak was at lower wavelengths, i.e., the UV region, compared to the TiO₂ systems. Interestingly, the bandgap for TiO₂/SiO₂ appears at N = 2, but for the other two systems does not appear until N = 4. The SiO₂/SnO₂ transmission curve was more concentrated in the UV region when compared to the other curves. The TiO₂/SnO₂ curve had a higher wavelength shift, whereas the TiO₂/SiO₂ curve had the lowestT value (Figure 8). The smooth/single wavelength transition was present in all three curves for N = 4, confirming they are compatible with single wavelength devices and applications. When the number of layers is increased to N = 6, there was no change in the bandgap position, but there was a widening of the bandgap that was magnified further at N = 8. The photonic bandgap in SiO₂/SnO₂ was centered at 350 nm and extends through the UV region, while the TiO₂/SnO₂ bandgap was centered at 480 nm and the TiO₂/SiO₂ photonic bandgap at 430 nm, both in the visible region.

The effect of layer thickness on the transmittance of the different 1D PhCs was studied, however, the thickness of the bottom layer produced no discernible differences. Therefore, we decided to test varying the top thicknesses at N = 8 (Figure 9). At a layer thickness of 150 nm/150 nm, we can see that the bandgap shifted to the NIR region with two narrow bandgaps and one wide bandgap. When compared to the other two systems, SiO₂/SnO₂ had a higher transmittance. Generally, the photonic bandgap shifted to a longer wavelength at a layer thickness of 300 nm/150 nm. Surprisingly, the SiO₂/SnO₂ structure had only one photonic bandgap, while the other structures had multiple bandgaps. As a result, 1D SiO₂/SnO₂ PhCs could potentially be applied as one-photonic bandgap photonic devices. TiO₂/SiO₂ had a multi-channel photonic bandgap at all times and the T values were always greater than those of the other structures, making it particularly promising for multi-channel optical devices. Also, differences in angular frequency and transmission dependency for three structures were also reported in (Figure 10).

PhCs are arranged in periodic high-low dielectric patterns to control the penetration of light through structures, which opens the door to a wide range of nanotechnology and photovoltaic applications. Despite the fact that 3D PhCs are widely used in various applications, 1D PhCs are gaining popularity due to their fascinating optical properties and ease of fabrication. We conducted a theoretical analysis of 1D PhCs composed of TiO₂/SiO₂, TiO₂/SnO₂, and SiO₂/SnO₂. Specifically, we found that the bandgap appears first when using TiO₂/SiO₂, and there is a clear shift to lower wavelengths. By increasing the number of layers toN = 6 or 8, we see that the edges grow wider and sharper and the width of the photonic bandgaps becomes wider for TiO₂/SiO₂ when compared to TiO₂/SnO₂. Moreover, in TiO₂/SiO₂, the photonic bandgap reaches zeroT, whereas it does not reach this for TiO₂/SnO₂. Thick TiO₂/SiO₂ films exhibit high transmittance for wide bandgaps, which supports their use in a wide range of optical filter applications. On the other hand, TiO₂/SnO₂ can support a single wavelength optical detector. Interestingly, we did not see a clear photonic bandgap at N = 1 or 2 when combining SiO₂ and SnO₂, and at higher layer numbers only had one bandgap centered at 350 nm through the UV region, whereas the other systems have many. Specifically, TiO₂/SnO₂ shows a bandgap centered at 480 nm, while TiO₂/SiO₂ is centered at 430 nm. In terms of the effect of thickness, for large thicknesses, the TiO₂/SiO₂ film showed high transmittance for wide bandgaps, and the SiO₂/SnO₂ structure only had one photonic bandgap, whereas the others have many. TiO₂/SiO₂ always showed a multi-channel photonic bandgap.

Authors are thankful to Alfaisal University for providing MATLAB^®.

The authors declare no conflicts of interest regarding the publication of this paper.

Alamrani, F. and Alsharaeh, E. (2022) Controlling the Bandgaps of One-Dimensional TiO₂/SiO₂, TiO₂/SnO₂, and SiO₂/SnO₂ Photonic Crystals Using the Transfer Matrix Method. Optics and Photonics Journal, 12, 171-189. https://doi.org/10.4236/opj.2022.127013

θ_m: Angle of incidence for each layer.

ω: Angular frequency.

λ: Wavelength.

A: Absorptance.

c: The speed of light in a vacuum.

d: Thickness.

D₀: Air dynamical matrix.

D_m: Dynamical matrices.

K: Bloch wave number constant.

l: Layers of refractive index.

P_m: Propagation matrix.

PBG: Photonic band gap.

n₀: Refractive index at the interface.

n_m: Refractive index of the “m” layer.

n_s: Refractive index at the substrate.

N: Number of periods.

r: Reflection amplitude.

R: Reflectance.

SiO₂/SnO₂: Silicon dioxide/Tin dioxide.

t: Transmission coefficient.

T: Transmissivity.

TiO₂/SiO₂: Titanium dioxide/Silicon dioxide.

TiO₂/SnO₂: Titanium dioxide/Tin dioxide.

TMM: Transfer matrix method.

TE: Transverse Electric.