## 1 Controlling Thermal Radiation in Photonic Quasicrystals Containing Epsilon-Negative Metamaterials  
**[PERSON] 1,2,*, [PERSON] 3, [PERSON] 3 and [PERSON] 1,* 2**  
Footnote 1: [[https://www.mdpi.com/journal/applsci](https://www.mdpi.com/journal/applsci)]([https://www.mdpi.com/journal/applsci](https://www.mdpi.com/journal/applsci))  
Footnote 2: [[https://www.mdpi.com/journals/applsci](https://www.mdpi.com/journals/applsci)]([https://www.mdpi.com/journals/applsci](https://www.mdpi.com/journals/applsci))  
Footnote 3: [[https://www.mdpi.com/journals/applsci](https://www.mdpi.com/journals/applsci)]([https://www.mdpi.com/journals/applsci](https://www.mdpi.com/journals/applsci))  
Footnote 4: [[https://www.mdpi.com/journals/applsci](https://www.mdpi.com/journals/applsci)]([https://www.mdpi.com/journals/applsci](https://www.mdpi.com/journals/applsci))  
Footnote 5: [[https://www.mdpi.com/journals/applsci](https://www.mdpi.com/journals/applsci)]([https://www.mdpi.com/journals/applsci](https://www.mdpi.com/journals/applsci))  
Footnote 6: [[https://www.mdpi.com/journals/ap](https://www.mdpi.com/journals/ap)]([https://www.mdpi.com/journals/ap](https://www.mdpi.com/journals/ap))  
==WARNING: Truncated because of repetitions==
its associated electric and magnetic fields. The transfer matrix of the \(j^{th}\) layer, which relates the electric and magnetic components of the incident beam at the left and right interfaces of the \(j^{th}\) layer, is given as follows [33]:  
\[M_{j}= \begin{pmatrix}\cos\left(k_{j2}d_{j}\right)&-ip_{j2}^{-1}\sin\left(k_{j2 }d_{j}\right)\\ -ip_{j2}\sin\left(k_{j2}d_{j}\right)&\cos\left(k_{j2}d_{j}\right)\end{pmatrix} .\] (4)  
where \(d_{j}\left(j=A,B,C\text{ or }D\right)\) is the thickness of \(j^{th}\) layer, \(k_{j2}=(\omega/c)\sqrt{\varepsilon_{j}\mu_{j}}\sqrt{1-\left(\sin^{2}\theta/ \varepsilon_{j}\mu_{j}\right)}\) is the \(z\) component of local wave vector in \(j^{th}\) layer, \(\omega\) indicates the angular frequency of the incident wave, and Equations (5) and (6) represent the parameters associated with the transfer matrix for TE and TM modes, respectively:  
\[P_{j2}=\sqrt{\frac{\varepsilon_{j}}{\mu_{j}}}\sqrt{1-\left(\sin^{2}\theta/ \varepsilon_{j}\mu_{j}\right)}.\] (5)  
\[P_{j2}=\sqrt{\frac{\mu_{j}}{\varepsilon_{j}}}\sqrt{1-\left(\sin^{2}\theta/ \varepsilon_{j}\mu_{j}\right)}.\] (6)
+++  
The total transfer matrix of the multilayered structures is obtained by multiplying the corresponding matrix of each individual layer defined as  
\[M_{Total}=\prod_{j=1}^{N}M_{j}= \begin{bmatrix}M_{11}&M_{12}\\ M_{21}&M_{22}\end{bmatrix} . \tag{7}\]  
Accordingly, the transmission and reflection coefficients can be derived as Equations (8) and (9):  
\[t(\omega)=\frac{2p_{i}}{(M_{11}+p_{F}M_{12})p_{i}+(M_{21}+M_{22}p_{F})}. \tag{8}\]  
\[r(\omega)=\frac{(M_{11}+M_{12}p_{F})p_{i}-(M_{21}+M_{22}p_{F})}{(M_{11}+M_{12} p_{F})p_{i}+(M_{21}+M_{22}p_{F})}\;. \tag{9}\]  
where \(p_{i}\) and \(p_{F}\) are relevant to the initial and final environment corresponding to air \(\varepsilon_{i}=\varepsilon_{F}=\varepsilon_{0}\). Therefore, \(\mu_{i}=\mu_{F}=\mu_{0}\) and \(p_{i}=p_{F}=\sqrt{\varepsilon_{0}/\mu_{0}}\) for TE and TM polarization, respectively. Subsequently, the total transmission and reflection can be reached as follows:  
\[T(\omega)=\frac{p_{F}}{p_{i}}\abs{t(\omega)}^{2},R(\omega)=\abs{r(\omega)}^{ 2}. \tag{10}\]  
Regarding the dielectric layers Teflon and ZrO\({}_{2}\), both materials have unique optical properties, such as high transparency in the visible, near-infrared (NIR), and THz regions, low refractive index, and low absorption coefficient [42; 43]. It can be used as an optical or photonic material for lenses, filters, waveguides, coatings, and substrates. Teflon is a synthetic polymer that has many benefits for various purposes. It has low friction, high strength, and excellent electrical insulation. It can resist high heat, strong chemicals, and corrosion [42]. ZrO\({}_{2}\) or Zirconia is a type of ceramic oxide that has remarkable properties.