Theoretical Study on Polycarbonate-Based One-Dimensional Ternary Photonic Structures from Far-Ultraviolet to Near-Infrared Regions of Electromagnetic Spectrum

Abstract

In the present research work, we have theoretically analyzed the photonic band-gap properties of one-dimensional photonic structures composed of polycarbonate and non-glass materials. These photonic structures, PC₁, PC₂, PC₃ and PC₄, are composed of alternating layers of polycarbonate/Al₂O₃, polycarbonate/MgF₂, polycarbonate/BaF₂ and polycarbonate/TiO₂ materials, respectively. The period of each photonic structure is made up of a thin non-glass material layer sandwiched between two identical polycarbonate layers. The transfer matrix method has been used to investigate the transmission properties of PC₁ to PC₄. The comparison between the transmission spectra of PC₁ to PC₄ shows that the polycarbonate and TiO₂-based photonic structure (PC₄) possess three PBGs of zero transmission located at far-ultraviolet, visible and near-infrared regions of the electromagnetic spectrum at normal and oblique incidence (θ₀ = 55°), both corresponding to TE wave only. The index of refraction of all five materials used in this study was obtained by applying the Sellmeier-type dispersion relationship to ensure accuracy in the results. The purpose of selecting polycarbonate along with Al₂O₃, TiO₂, MgF₂ or BaF₂ as constituent materials of these photonic structures is due to the heat resistance properties of polycarbonate and the unique optical properties of oxide and fluoride materials with wide transparency from the ultraviolet to the near-infrared regions of the electromagnetic spectrum. The proposed work can be used to design some influential wavelength-selective reflectors composed of 1D PCs behind the active region of the solar cells for improving the photovoltaic performance of solar panels. This study can further be utilized for the fabrication of advanced solar cell designs consisting of 1D photonic mirror-based luminescence and reflection concentrators. The low temperature problem which arises in satellites may also be overcome with the help of smart windows based on the proposed multilayer structures.

1. Introduction

Photonic crystals (PCs) are periodic structures with a tremendous ability to control the propagation of electromagnetic waves. These PCs possess spectral regions which rein the propagation of electromagnetic waves and are commonly known as photonic band gap (PBG) materials due to multiple Bragg scattering [1,2,3,4,5]. Such structures have emerged as a novel optical material due to their potential applications in the field of optical engineering and technology since the initial predications of Yablonovitch and John [6,7]. Nowadays, lots of attention has been given to photonic devices due to their tremendous ability to control and manipulate the light passing through them [8]. The interaction of electromagnetic waves with the various periodic structures consisting of dielectrics, meta-materials, superconductors, plasma, etc., have been investigated by various research groups for the designing of several photonic devices such as sensors, super capacitors, switches, etc. [9,10,11,12]. Moreover, PCs have potential applications in the field of optics and optoelectronics due to the development of tunable filters, zero-phase shift filters, omnidirectional filters and tunable resonators based on PCs which could work in the millimeter and sub-millimeter ranges [13,14,15,16,17,18,19]. Recently, photonic devices, due to nanotechnology-based easier fabrication techniques, have emerged as a key technology for addressing global energy requirements through the development of solar cells composed of selective back reflectors based on photonic crystals for concentrating solar radiation falling on them to improve their efficiency [20,21,22,23]. Moreover, such structures can be used to fabricate photonic crystal mirror-based luminescence and reflection concentrators for improving the efficiency of solar cells. These structures open a new gateway for the development of photonic crystal-based smart windows which could address the decreasing temperature problem of satellites [24]. For example, the experimental work of Gondek et al. showed that the 1D PC-based selective back reflector can be used as a sunlight concentrator to design improved photovoltaic systems [20]. Saravanan et al. showed how the performance of amorphous silicon-based solar cells can be enhanced by using back reflectors composed of 1D PC [21]. Apart from the piece of excellent research work discussed above, the photonic devices which could work in an ultraviolet to visible region are still difficult to realize because of the low impact strength, moderate transparency, poor dimensional stability and high temperature resistance of the constituent materials [25,26,27].

On the other hand, polycarbonate (PYC), an amorphous thermoplastic polymer material with low moisture absorption and temperature-independent dielectric constant properties, is a very suitable material for fabricating PCs due to additional degrees of freedom to overcome the above-mentioned limitations. The polycarbonate provides the mechanical stability to the constituent material layers during their deposition on the substrate, which is cleanly thermally removed. Although they are made commercially available in a variety of colors, perhaps translucent, perhaps not, the raw material allows for the internal transmission of light nearly in the same capacity as glass. Polycarbonate polymers are used to produce a variety of materials and are particularly useful when impact resistance and/or transparency are a product requirement (e.g., in bulletproof glass). Polycarbonate is commonly used for plastic lenses in eyewear, in medical devices, automotive components, protective gear, greenhouses, digital disks (CDs, DVDs and Blu-ray) and exterior lighting fixtures. Polycarbonate also has very good heat resistance and can be combined with flame retardants without significant material degradation. Polycarbonates are engineered plastics which are typically used due to their heat resistance capabilities and robust nature, in applications such as “glass-like” surfaces [28,29,30].

Polycarbonate has been considered as a major constituent material of our proposed design due to the above remarkable properties. Apart from polycarbonate, we have also selected some oxide and fluoride materials as other constituent materials because of their unique characteristics and wide transparency from the ultraviolet to near-infrared region of the electromagnetic spectrum [31,32,33,34]. Moreover, the optical absorption loss along with the thermal expansion coefficient of these materials is also very low. They have good resistance to temperature, corrosion and radiation as well. In addition, the reason behind the selection of oxide and fluoride materials along with polycarbonate for constructing our structures is to ensure the small refractive index difference between the constituent material layers. The fulfillment of this requirement makes the fabrication process of PC easier by maintaining the high Q factor. The high Q factor is the mandatory requirement in a real time-controlled deposition process which allows a precise deposition rate and the maintenance of uniformity in the thickness of the film to be deposited [35]. There are several deposition techniques for the fabrication of 1D photonic structures with a variety of materials, such as the sol-gel method, electron beam evaporation and glance angle deposition in the ultraviolet to near-infrared region [36]. Motivated by the above-mentioned qualities of polycarbonate, oxide and fluoride materials, we proposed 1D photonic structures composed of these materials to work from the far-ultraviolet to near-infrared region of the electromagnetic spectrum. To the best of our knowledge, such structures have been rarely reported.

The paper is organized as follows: The theoretical formulation of the proposed problem and corresponding analytical formulae such as wavelength-dependent refractive index of constituent materials along with the transmission coefficient are introduced in Section 2. The device fabrication requirements and its spectrum analysis are discussed in Section 3. Numerical results pertaining to the dispersion properties of constituent materials and transmission characteristics of PC₁ to PC₄ are discussed in Section 4. Finally, conclusions are presented in Section 5.

2. Structural Design and Theoretical Formulation

In this section, we introduce the schematic view of a 1D ternary photonic structure (TPC) [air/(PXP)^N/air] as shown in Figure 1. The letters P and X are being used to represent two different layers of materials, polycarbonate and non-glass, respectively. The period of the proposed structure is composed of sandwiching a thin layer of non-glass material of refractive index n_X between two identical polycarbonate layers of refractive index n_P. The period of the structure is given by d = d_P + d_X + d_P; here, d_P and d_X are representing the thickness of polycarbonate and non-glass material layers, respectively. The period number of the 1D ternary photonic structure is given by N. Suppose a plane wave is injected into 1D TPC at an incident angle θ₀ with respect to the stratification direction, which is along the positive direction of the z-axis. The layers are situated in an xy plane as shown in Figure 1. In general, electric and magnetic field vectors of incident light at two interfaces of any layer j can be correlated by a transfer matrix method (TMM) [24,37] as:

$$M_j=\left(\begin{array}{cc}\cos {\delta }_j& -\frac{i}{p_j}\sin {\delta }_j\\ -i{p}_j\sin {\delta }_j& \cos {\delta }_j\end{array}\right)$$

(1)

where ${\delta }_j=\frac{2\pi }{{\lambda }_0}{n}_j{d}_j\cos {\theta }_j,{\lambda }_0$ is the wavelength of the incident electromagnetic wave, d_j is the thickness of the jth layer, θ_j is the ray angle inside jth layer of refractive index $n_j$ and $\cos {\theta }_j=\sqrt{1-{\left(\frac{n_0\sin {\theta }_0}{n_j}\right)}^2}$. Here, n₀ is the refractive index of the environment wherein the incident wave tends to enter into the structure at an angle θ₀. For the TE and TM waves, $p_j=n_j\cos {\theta }_j$ and $p_j=\frac{\cos {\theta }_j}{n_j}$, respectively. For a finite-layer 1D periodic structure as shown in Figure 1, we used TMM, which is one of the most effective techniques to analyze the transmission properties of various 1D photonic structures. The total transfer matrix of the whole structure, as depicted in Figure 1, which connects the electric and magnetic fields of the first and last layers, can be derived as:

$$\left(\begin{array}{c}{E}_1\\ H_1\end{array}\right)=\left(M_P{M}_X{M}_P\right)---{\left(M_P{M}_X{M}_P\right)}^{}\left(\begin{array}{c}{E}_{n+1}\\ H_{n+1}\end{array}\right)=M^N\left(\begin{array}{c}{E}_{n+1}\\ H_{n+1}\end{array}\right)$$

(2)

Here, M_P and M_X are the transfer matrix of polycarbonate and various non-glass material layers, respectively. Here, M^N is the total transfer matrix connecting electric and magnetic fields at the incident end (E₁, H₁) and exit end (E_n+1, H_n+1).

The transmission coefficient of the whole structure is given by:

$$t=\frac{2p_0}{\left(m_{11}+m_{12}{p}_{n+1}\right)p_0+\left(m_{21}+m_{22}{p}_{n+1}\right)}$$

(3)

Here, p₀ and p_n+1 are representing the first (z < 0) and last (z > L) media of the structure. The alphabet L is used to represent the total length of the structure. For a TE wave, $p_0=p_{n+1}=n_0\cos {\theta }_0$, whereas for a TM wave, $p_0=p_{n+1}=\frac{\cos {\theta }_0}{n_0}$.

In order to study the transmission characteristics of various 1D photonic structures composed of polycarbonate and non-glass materials such as Al₂O₃, MgF₂, BaF₂ and TiO_2, we used the Sellmeier approximation formulae to calculate the wavelength-dependent refractive index of the materials polycarbonate, Al₂O₃, MgF₂ and BaF₂ at room temperature to obtain more realistic results, as in [29,30,31]:

$$n_X=n_X^{\prime }-i{n}_X^{″}=\sqrt{A+\frac{B_1{\lambda }_0^2}{{\lambda }_0^2-C1}+\frac{B_2{\lambda }_0^2}{{\lambda }_0^2-C_2}+\frac{B_3{\lambda }_0^2}{{\lambda }_0^2-C_3}}$$

(4)

The dispersion properties of the refractive index of the TiO₂ material layer are also taken into account by considering another type of Sellmeier approximation relationship, as in [31]:

$$n_{Ti{O}_2}=n_{Ti{O}_2}^{\prime }-i{n}_{Ti{O}_2}^{″}=\sqrt{5.913+\frac{0.2441}{{\lambda }_0^2-0.0803}}$$

(5)

In Equations (4) and (5), $n_X^{\prime }$ and $n_{Ti{O}_2}^{\prime }$ are used to represent a real part of the complex refractive index of the materials polycarbonate, Al₂O₃, MgF₂, BaF₂ and TiO_2, respectively, whereas the imaginary parts $n_X^{″}$ and $n_{Ti{O}_2}^{″}$ are known as extinction coefficients, which measure the absorption in the medium.

Figure 1. The schematic design of 1D ternary photonic structures considered in this work. Here, the letters P and X are being used to represent polycarbonate and other material layers (Al₂O₃, TiO₂, MgF₂ and BaF₂) of thickness d_p and d_x, respectively.

Figure 1. The schematic design of 1D ternary photonic structures considered in this work. Here, the letters P and X are being used to represent polycarbonate and other material layers (Al₂O₃, TiO₂, MgF₂ and BaF₂) of thickness d_p and d_x, respectively.

3. Experimental Section Pertaining to the Device Fabrication and Spectrum Analysis

The proposed structures can be realized by using a spin-coating fabrication technique in addition to a sol-gel method, because the spin-coating process can allow the fabrication of a periodic layered structure composed of many materials, including nano-particles and polymer solutions. To initiate the spin-coating process, a precursor solution is dropped over the flat substrate, then the solvent is evaporated from the substrate with the help of the spin process. Before spinning the next layer, we use an annealing procedure to solidify the earlier deposited layer on the substrate. The aforementioned process is repeated until we obtain a stack of the required number of layers. The thickness of each film of the multilayer periodic stack can be controlled with the variation of either rotational speed or cast solution concentration. Since the sol-gel technique does not require any complex installations associated with technology, this minimizes the fabrication cost of the photonic structure. The experimental setup required to analyze the performance of the coating is shown in Figure 2 below. The broad-band light source is used to launch the light into the proposed photonic structures through an input port with the help of a single-mode fiber (SMF) in addition to existing splicing techniques available for SMFs. The transmission spectra of the proposed photonic structures are coupled into the computer for measurements through an optical spectrum analyzer (OSA) as shown in Figure 2 below.

4. Numerical Results and Discussion

In this section, numerical results showing the transmission properties of photonic structures PC₁ to PC₄ at normal and oblique incidence, both in the wavelength range 300–900 nm corresponding to TE wave only, are presented. PC₁, PC₂, PC₃ and PC₄ are composed of polycarbonate and other non-glass materials, Al₂O₃, MgF₂, BaF₂ and TiO₂, respectively. The thickness of polycarbonate and other non-glass material layers are taken as 94.4 nm and 50 nm, respectively. The thicknesses of the material layers are identical for all structures under consideration. The period of the structures PC₁ to PC₄ is fixed to 20. Before we discuss the transmission behavior of PC₁ to PC₄ at incident angles; ${\theta }_0=0^{°} \mathrm{and} {\theta }_0={55}^{°}$, it is necessary to examine the wavelength-dependent behavior of polycarbonate, Al₂O_3, MgF₂, BaF₂ and TiO₂ materials. For this purpose, we used the Sellmeier approximation formulae as defined in Equations (4) and (5) to obtain the wavelength-dependent refractive index of polycarbonate, Al₂O₃, MgF₂, BaF₂ and TiO₂ materials at room temperature, respectively.

Table 1. Sellmeier coefficients of polycarbonate, Al₂O₃, MgF₂ and BaF₂ materials used in the proposed 1D ternary photonic designs, PC₁ to PC₃.

Materials	A	B1	B2	B3	References
		C1	C2	C3
Polycarbonate	1	1.4182	-	-	[38]
		0.0213	-	-
Al2O3	1	1.43135	0.65055	5.3414	[39,40]
		0.07266	0.11932	18.0283
MgF2	1	0.48755	0.39875	2.31204	[32]
		0.04338	0.09461	23.7936
BaF2	1.3397	0.8107	0.19652	4.52469	[34]
		0.10065	29.87	53.82

summarizes the Sellmeier coefficients associated with polycarbonate, Al₂O₃, MgF₂ and BaF₂ materials as given below.

First, we discuss the wavelength-dependent dispersive properties of polycarbonate, Al₂O₃, MgF₂, BaF₂ and TiO₂ materials used in the proposed photonic designs. Figure 3 shows variation in the refractive index and extinction coefficient of polycarbonate, Al2O₃, MgF₂, BaF₂ and TiO₂ materials with respect to wavelength, which varies from the far-ultraviolet to near-infrared region of the electromagnetic spectrum. Figure 3a shows the dependence of the refractive index and extinction coefficient of polycarbonate material on the wavelength of incident light. As evidenced in Figure 3a, the variation in the refractive index of polycarbonate material exhibits asymptotic behavior near the far-ultraviolet (FUV) region and varies from 1.645 to 1.575 in the wavelength range 370–900 nm. The extinction coefficient of polycarbonate in this range is almost zero, i.e., polycarbonate is working as a lossless medium in the region of investigation that varies from 300 nm to 900 nm. This is one reason for selecting polycarbonate as a major contributor to our structures. Almost similar observations have been noticed for Al₂O₃, MgF₂ and BaF₂ materials as evidenced in Figure 3b–d, respectively. In addition, the dependence of the refractive index and extinction coefficient of TiO₂ material on wavelength is also plotted in Figure 3e. It shows that the refractive index of material TiO₂ varies from 2.814 to 2.26 in the wavelength range 300 nm to 900 nm. On the other hand, its extinction coefficient also varies from 0.025 to 0.000 between the wavelengths 300 and 900 nm. The losses in TiO₂ material are relatively prominent between 300 nm and 500 nm. After 500 nm, the TiO₂ serves as a lossless medium [41,42,43,44]. The above discussion justifies our choice of selecting polycarbonate, Al₂O₃, MgF₂, BaF₂ and TiO₂ as constituent materials to design PC₁ to PC₄ structures.

Next, we will examine the transmission properties of the proposed ternary photonic structures PC₁, PC₂, PC₃ and PC₄ at θ₀ = 0^°. These structures were made by sandwiching the Al₂O₃, MgF₂, BaF₂ and TiO₂ material layers between two identical polycarbonate layers of thickness of 94.4 nm, one by one, to make PC₁, PC₂, PC₃ and PC₄ structures, respectively. The thickness of the sandwiched layer was fixed to 50 nm in all four structures. In order to compare the optical properties of the proposed designs in a true sense, all the structural parameters of PC₁ to PC₄ were made identical, except the index of the refractive layers of materials Al₂O₃, MgF₂, BaF₂ and TiO₂ to be sandwiched between two identical polycarbonate layers to form PC₁ to PC₄, respectively.

The wavelength-dependent transmission properties of PC₁ to PC₄ at normal incidence are presented in Figure 3 with the help of TMM. Figure 4a–d show the transmission properties of PC₁, PC₂, PC₃ and PC_4, respectively. Figure 3 shows the number of PBGs of different widths, depths and central wavelengths depending upon the refractive index contrast between the constituent materials used in the respective photonic designs. It is evident from Figure 4a–c that PC₁, PC₂ and PC₃ each possess two PBGs of different widths and depths. Out of these two PBGs, one exists between 300 nm and 410 nm and the other one exists between 700 nm and 820 nm. The PBGs located at the lower and higher side of the electromagnetic spectrum are narrower and wider, respectively. The comparison in Figure 4a–c also shows that only PC₂ possesses a narrow PBG of negligible transmission located between 300 nm and 410 nm. This is due the higher refractive contrast between the polycarbonate and MgF₂ material layers, as evidenced in Figure 3a,c. Thus, the higher value of the refractive contrast between the constituent material layers of the structure determines the width of PBGs of negligible transmission. On the other hand, the transmission spectra of PC₄ show three PBGs of negligible transmission. The width of these PBGs varies from smaller to larger as we move from lower to higher wavelength sides, respectively, as shown in Figure 4d. The existence of these three PBGs can be explained on the basis of the refractive index variation in TiO₂ from 2.826 to 2.084 between the wavelengths 300 nm and 900 nm, as shown in Figure 3e. The reason behind the existence of these PBGs is the multiple reflections that occur at internal interfaces of the structure. The wavelengths of incident light, which are satisfied by Equation (6), will be reflected back and are not allowed to pass through the structure [45].

$$2\left(n_1{d}_1\cos {\theta }_1+n_2{d}_2\cos {\theta }_2+n_3{d}_3\cos {\theta }_3\right)=m{\lambda }_0$$

(6)

Here, λ₀ is the free-space wavelength of incident light and m = 1, 2, 3, … represents the order of the reflection. This is the well-known Bragg’s law and is responsible for the formation of PBG or the Bragg gap. The wavelengths that lie inside the PBGs of these structures can be determined with the help of the structural parameters by using Equation (6). The major parameters that decide the central wavelengths of the Bragg gap are refractive indices (n_j), thickness (d_j) and ray angles (θ_j) inside the jth layer of the period of the structure. In order to find the central wavelength of the Bragg gap we can modify Equation (6) as [45]:

$${\lambda }_0^c=\frac{2\left(n_1{d}_1\cos {\theta }_1+n_2{d}_2\cos {\theta }_2+n_3{d}_3\cos {\theta }_3\right)}{m}$$

(7)

Since the period of the proposed photonic structures PC₁ to PC₄ is composed of three layers of two materials, we can write $n_1{d}_1\cos {\theta }_1=n_3{d}_3\cos {\theta }_3$. Thus, the central wavelength of each PBG can be obtained with the help of modified Equation (8) as:

$${\lambda }_0^c=\frac{n_2{d}_2\cos {\theta }_2}{1+\frac{2d_1}{d_2}}\left(1+\frac{2n_1{d}_1\cos {\theta }_1}{n_2{d}_2\cos {\theta }_2}\right)\frac{1}{m}$$

(8)

Next, we examine the behavior of the transmission properties of PC₁ to PC₄ at θ₀ = 55° corresponding to the TE wave only. Figure 5a–d shows the transmittance of PC₁, PC₂, PC₃ and PC₄ at θ₀ = 55° corresponding to the TE wave only. As evidenced in Figure 5, the PBGs of structures PC₁ to PC₄ shifted to the lower wavelength side as the angle of incidence changed from 0° to 55° corresponding to the TE wave. The width and height of their PBGs of finite transmission are also relatively reduced. On the other hand, the transmission properties of PC₄ show that both the edges of all three PBGs shifted to the lower wavelength side. Additionally, their width increases in contrast to the properties, as depicted in Figure 4d. Thus, the variation in the angle of incidence can be used for tuning the PBG position as per our desire.

As discussed above, we studied the transmission properties of four different kinds of PCs made up of polycarbonate and other oxide and fluoride materials: TiO₂, Al₂O₃, MgF₂ and BaF₂. The above study was carried out to explore the utility of polycarbonate for the designing of photonic structures suitable to work in far-ultraviolet to near-infrared regions of the electromagnetic spectrum. The polycarbonate-based photonic structures can be used for designing selective back reflectors, luminescence, and reflection concentrators to improve the performance of solar cells.

5. Conclusions

In conclusion, we studied the polycarbonate-based one-dimensional ternary photonic structures, which can be used from the far-ultraviolet to near-infrared region of the electromagnetic spectrum. These structures were designed by sandwiching Al₂O₃, MgF₂, BaF₂ and TiO₂ material layers between two identical layers of polycarbonate to form PC₁, PC₂, PC₃ and PC₄ structures, respectively. We studied the transmission properties of PC₁ to PC₄ at normal and oblique incidence, both with the help of a transfer matrix method. These findings may be helpful for designing durable and heat-resistant thin-film coatings with mechanical stability based on such 1D polycarbonate photonic structures with the ability to work in far-ultraviolet to near-infrared regions of the electromagnetic spectrum. For improving the photovoltaic performance of solar cells, we could design some significant selective back mirrors made up of the proposed 1D PC designs. This study may also open new gateways for the designing of luminescence as well as reflection concentrators based on 1D PCs, which can notably improve the efficiency of solar cells. Moreover, the proposed work may also provide some insight for designing 1D PC-based smart windows, which could efficiently handle the low temperature satellite problem, affecting the working efficacy.