shows the comparison of OPBG as a function of maximum refractive index (nmax), for the structures with sinusoidal, Gaussian, and Bragg refractive index profiles for different OT and PT.
Figure 1 OPBG as a function of nmax for the sinusoidal, Gaussian, and Bragg profiles. Panels (a), (b), and (c) correspond to the OT of 24, 25, and 26 μm, respectively. Panels (d), (e), and (f) correspond to the PT = 7.76 μm for three different (more ...)
The nmax was varied from 2.2 to 2.9, while the miminum refractive index (nmin) was adjusted to keep the OT constant as (a) 24, (b) 25, and (c) 26 μm. The computed range of nmax was limited by the experimental capability to obtain high refractive indices (keeping PS as a possible reference material) and the adjusted values of nmin to keep the same OT of all the structures. Figure
a,b,c demonstrates that for each OT, one can find a particular value of nmax at which the profile corresponding to the higher value of OPBG changes. For example, in Figure
b, the largest OPBG for nmax range of 2.25 to 2.45, the Bragg-type profile has to be the preferred choice. For 2.45 <nmax < 2.57, the sinusoidal profile has the largest OPBG, but the Gaussian profile prevails for nmax > 2.57. A similar behavior is observed for higher OTs (Figure
c). For the OT of 24 μm, the Bragg-type profile fails to demonstrate any OPBG (Figure
a). Although the Gaussian structure shows the largest OPBG, the corresponding value of nmax is also very high.
d,e,f shows the comparison of the OPBG for the structures with the same PT, i.e., 7.76 μm. The nmax was varied from 2.3 to 2.9, while the nmin was kept constant as (a) 1.1, (b) 1.35, and (c) 1.5. Figure
a,b,c demonstrates that the Gaussian refractive index profile always requires higher refractive index values to obtain the same OPBG as compared to the sinusoidal refractive index profile. Equivalently, the OPBG obtained for the sinusoidal profile is always higher as compared to that for the Gaussian profile for a given nmax. In spite of the failure of the Bragg-type profile to demonstrate any OPBG for nmin = 1.1 (see Figure
d), the tunability to increase/decrease the OPBG for nmin = 1.35 as compared to the sinusoidal and Gaussian profiles is shown in Figure
e. One can identify three particular intervals for the Bragg profile (2.35 <nmax < 2.51, 2.51 <nmax < 2.72, and 2.72 <nmax < 2.9) at which the OPBG is higher/lower as compared to the sinusoidal and Gaussian profiles (Figure
e). For a higher nmin, Figure
f shows a significant enhancement for the Bragg-type structure, revealing a larger OPBG as compared to the other profiles. Hence, one can obtain the tunability of the OPBG in a certain refractive index range, depending on the available refractive indices and the profile of the photonic structure.
The result shows that no particular profile can be designated as the best profile for the complete range of maximum refractive indices discussed in this work. Apart from that, one can obtain the tunability of the OPBG in a certain refractive index range, depending on the available refractive indices and the profile of the photonic structure. The vertical dashed line in Figure
b corresponds to nmax = 2.5 and the particular OT incorporated in the forthcoming experimental and simulated results.
shows the experimental (fabricated with PS multilayers) and simulated reflectivity spectra for the three types of photonic structures at 8° and 68° of incidence angle. As mentioned earlier, the results are obtained for nmax = 2.5 and 25 μm of OT (dashed vertical line in Figure
b). OPBG is shown as a vertical gray band. Good agreement between the calculated (dashed line) and the experimental spectra (solid line) is observed. The experimental OPBG was taken with more than 90% of the reflectivity for each multilayered structure. The sinusoidal profile (Figure
a,d) shows a 95-nm photonic bandgap, while the Gaussian (Figure
b,e) and Bragg (Figure
c,f) profiles show 45 and 63 nm of OPBGs, respectively. Hence, for the given value of OT (25 μm) and nmax (2.5), the sinusoidal profile was shown to have almost twice the OPBG than the other two profiles under discussion.
Figure 2 Theoretical (dashed line) and experimental (solid line) reflectivity spectra for structures with the same OT. The PBG generated using the (a) sinusoidal, (b) Gaussian, and (c) Bragg profiles for 8°. The corresponding reflectivity spectra at 68° (more ...)
On the other hand, Figure
shows the experimental and theoretical results for the photonic structures with the same PT. A good agreement is observed between theoretical and experimental results. The overlapping of PBG for different angles was measured as 177 nm for the sinusoidal profile (Figure
a,d), while the Gaussian (Figure
b,e) and Bragg (Figure
c,f) profiles show an OPBG of 130 and 80 nm, respectively. To verify the mechanical stability of such structures, the surface images of the PS multilayered structure corresponding to each profile are shown as insets. The surface fractures observed on the Bragg-type structure (see inset in Figure
c) are attributed to the high-porosity contrast between two consecutive layers
]. For the sinusoidal and Gaussian refractive index profiles, the inset images (see inset in Figure
a,b) show a flat-uncracked surface due to the gradual variation of the porosity between consecutive layers, which helps in reducing the stress and enhances the mechanical stability
]. Therefore, a significant reduction in the intensity of the reflectivity spectra observed for the Bragg-type photonic structure (Figure
c,f), as compared to the theoretical simulations, is attributed to the cracked structure which provokes a higher dispersion of the incident light.
Figure 3 Theoretical (dashed line) and experimental (solid line) reflectivity spectra. (a) Sinusoidal, (b) Gaussian, and (c) Bragg refractive index profiles at 8°. The corresponding reflectivity spectra at 68° are shown in panels (d), (e), and (more ...)
shows the theoretical contour plots for the reflectivity spectra as a function of the wavelength and the incident angle for the sinusoidal (Figure
a,d), Gaussian (Figure
b,e), and Bragg (Figure
c,f) mirrors. Figure
a,b,c corresponds to the photonic structures with the same OT, while Figure
d,e,f corresponds to the photonic sutructures with the same PT. As the angle of incidence is increased, the PBG (red region) decreases for all the photonic structures. In spite of the largest PBG at 0° (over the other profiles) for the Bragg mirror, the ability for keeping a semi-constant stop band, independent of the incident angle, is better demonstrated for the sinusoidal and Gaussian structures, showing a more pronounced fall of the PBG (after 45°) for the Bragg structure, as compared to the other mirrors. Hence, depending on the application, the refractive index profile can be selected to have a larger PBG within a certain angular range (e.g., from 0° to 45°, Bragg mirrors are a better choice) or a small PBG but for any possible incidence angle.
Contour plot of the reflectivity spectra as a function of the angle and wavelength. (a, d) Sinusoidal, (b, e) Gaussian, and (c, f) Bragg refractive index profiles. The color scale indicates the reflectivity percentage from 0% (blue) to 100% (red).