Wave attenuation based on Beer-Lambert law in SiO2/ZrO2, TiO2/MgF2, Si/Al2O3 and Nb2O5/CYTOP one-dimensional photonic crystals

In this paper, according to the Beer–Lambert law, we present a numerical investigation of wave attenuation in one-dimensional photonic crystals composed of SiO₂/ZrO₂, TiO₂/MgF₂, Si/Al₂O₃ and Nb₂O₅/CYTOP bilayers deposited on polycarbonate. The wave attenuation characteristics are quantitatively evaluated through optical density (OD) measurements. The simulations were performed using the transfer matrix method implemented in MATLAB for both TE and TM polarizations. The results demonstrate a strong correlation between refractive index contrast (Δn) and photonic bandgap characteristics, where higher Δn values yield broader bandgaps and enhanced attenuation. The numerical analysis reveals that both incidence angle and layer thickness significantly influence the photonic bandgap characteristics, where increasing the angle causes a blue-shift in the bandgap position according to Bragg's law, while varying the layer thickness enables precise tuning of the bandgap width. Significantly, the Si/Al₂O₃ structure achieves the widest bandgap (1200–1950 nm) and highest optical density (OD = 11.5), while the Nb₂O₅/CYTOP configuration shows good performance (OD = 10) with potential for flexible photonic devices. Polarization-dependent analysis show that TE waves maintain consistent attenuation at oblique incidence, in contrast to TM waves which show pronounced attenuation loss near Brewster's angle. These findings provide fundamental insights into material selection and structural design for optimizing photonic crystal performance.