As discussed previously, interfacial roughness in one-dimensional photonic crystals (1DPCs) can have a significant effect on their normal reflectivity at the quarter-wave tuned wavelength. We report additional finite-difference time-domain (FDTD) simulations that reveal the effect of interfacial roughness on the normal-incidence reflectivity at several other wavelengths within the photonic bandgaps of various 1DPC quarter-wave stacks. The results predict that both a narrowing and red-shifting of the bandgaps will occur due to the roughness features. These FDTD results are compared to results obtained when the homogenization approximation is applied to the same structures. The homogenization approximation reproduces the FDTD results, revealing that this approximation is applicable to roughened 1DPCs within the parameter range tested (rms roughnesses < 20% and rms wavelengths < 50% of the photonic crystal periodicity) across the entire normal incidence bandgap.
© 2005 Optical Society of America
Previously, we reported the results of several finite-difference time-domain (FDTD) simulations investigating the effect of interfacial roughness on the normal reflectivity of one-dimensional photonic crystals (1DPCs) . These simulations were all done at the quarter-wave tuned wavelength of the 1DPCs. However, many devices that use 1DPCs, for example band-pass filters or broad-band reflectors, utilize their reflectivity behavior across the entire bandgap. Indeed, roughness is an important issue in some of the structures that are currently being investigated in research settings. For example, tunable 1DPC devices are currently being studied through the incorporation of liquid crystals into their multilayer configuration [2,3]. These structures have extremely high interfacial roughness values (so high that the layers are discontinuous). Furthermore, many groups are utilizing macroporous silicon to produce 1DPC devices [4,5]. Again, these structures have some interfacial roughness due to the porosity of the material. Therefore, this paper will present the results from several additional FDTD simulations that were done to investigate the effect of roughness on the reflectivity of the entire normal incidence band gap. In addition, it was found that the roughened structures presented in the previous study could be represented accurately with the homogenization approximation [6–9]. Thus, this approximation will also be applied in this study and compared to the FDTD results in order to test its validity across the entire normal incidence bandgap.
As in the previous study, the photonic crystal configurations modeled in this study were free-standing quarter-wave stacks, with the high index layer on top (nearest the light source). Again, the simulated structures were characterized by their rms roughness, defined as
where y0 is the average interface height, y is the actual interface height, and n is the number of discrete units (cells) on each interface. All distances in the structure were normalized by the characteristic periodicity of the photonic crystal, a, in order to make the results independent of absolute length scale. The rms roughness is shown schematically in Fig. 1, along with a second characteristic parameter, the rms wavelength. Because the previous study showed that the reflectivity from the roughened structures was independent of the rms wavelength over the simulated parameter space, that parameter was not systematically varied in this study. All simulated structures had rms wavelengths between 30% and 50% of the photonic crystal periodicity.
2. FDTD reflectivity results
The generation of the structures, and the simulation and analysis method pertaining to this study were detailed in a previous publication . Briefly, the structures were generated by first creating quarter-wave stacks with flat interfaces and subsequently displacing each interface node by a random amplitude. A new random structure was generated for each RMS roughness value. The optical response of the structures was then simulated with a two-dimensional FDTD code utilizing the Yee algorithm [10–12]. The cell size in all simulations was 0.01a, constraining the minimum roughness feature size to also be 0.01a. Both periodic and absorbing boundary conditions were used in order to simulate 1DPCs that were infinite in length with a finite number of bilayers [11–13]. Independence of the simulation with respect to the distance between periodic boundaries was checked by calculating results for longer periodic cells and verifying that the results were statistically equivalent. The number of time steps used in the simulation varied between 7,000 and 20,000, depending on the simulation domain size (which was determined by the number of bilayers in the structure and the incident wavelength). The reflectivity, r, of the structures was found from the time-averaged squared electric field after initial transient behavior had disappeared. The relative (or percent ) change in reflectivity from the perfect structure was then calculated:
where r rough is the reflectivity of the roughened structure and r smooth is the reflectivity of the equivalent perfect (smooth interfaces) structure. In order to map out the reflectivity in the bandgap, several wavelengths within the bandgap of each structure’s perfect analogue were chosen. Approximately twenty distinct wavelengths spanning the bandgap were simulated at four different roughness values. On each of the Figs., these wavelengths are reported in normalized units (λ/a).
Approximately 400 total FDTD simulations were performed on several different 1DPC configurations. The specific refractive indices used in this study were chosen to roughly correspond to typical values for materials in the visible and near infrared regimes. The reflectivities from a TE-polarized normal incidence plane wave impinging on the roughened quarter-wave stacks were obtained. For comparison, these reflectivities were then plotted with the normal incidence reflectance spectrum for the corresponding perfect structure, which was computed with a one-dimensional transfer matrix calculation . The FDTD reflectance data were also fit with a polynomial expression in order to better observe any resulting trends.
The simulated reflectance spectra corresponding to several 4-bilayer systems are shown in Fig. 2. Each point corresponds to the calculated reflectivity results obtained from a single simulation. The curves are the polynomial fits to the reflectivity data described above, which visually reveal the resulting band structure at each roughness value. Consistent with the previously reported results, Fig. 2 shows that the reflectivity decreases with increasing roughness across the entire normal incidence band gap for all the modeled structures. However, the magnitude of this decrease is not the same for all wavelengths. The blue end of the band gap (smaller wavelengths) shows a much larger change in reflectivity than the red end (larger wavelengths). This results in a narrowing and red-shifting of the normal incidence band gap with increasing rms roughness. By using the term “red-shifting” here, we mean that the reflectivity becomes weighted towards the red end of the spectrum due to the unbalanced reflectivity change across the band gap. Although this effect is seen in all the structures presented in Fig. 2, the lowest index contrast structure (n1/n2=1.25, n1=2.25, n2=2.0) appears to be more sensitive to this red-shifting. This is illustrated by the fact that the red-shifting is already evident in the lowest index contrast structure at the smallest R value tested (3.71%), whereas the same roughness value in the other structures produces no such effect.
Figure 3, which illustrates Δr across the bandgap, demonstrates this red-shift more clearly. Due to the steep slope of the reflectivity at the edges of the band gap, the Δr value in this region is more sensitive. As a guide to ranges of practical use, the region where the perfect band gap is within 10% of its maximum value (i.e. where the band gap reflectance is reasonably flat) is shaded in the plots. Again, Fig. 3 shows that all simulated systems experience a decrease in the percent change in reflectivity within this shaded region as the wavelength increases. However, the lowest index contrast structure shows evidence of this red-shifting at even the smallest R value. This same roughness produces a nearly uniform change in the reflectivity across the shaded region in the higher index contrast structures. Additionally, the magnitude of the reflectivity change near the center of the bandgap is larger in the lowest index contrast structure, and decreases as the index contrast increases. This is consistent with our previously reported results, which showed that the higher index contrast structures were more tolerant to interfacial roughness.
Additional simulations were done to investigate the effect of increasing the number of bilayers in the structure. Figure 4 shows the reflectance spectra for two n1/n2=1.5 systems with differing bilayer numbers. Again, a red-shifting is observed in all the systems. However, unlike the previous study, there does not appear to be one system that is more sensitive to this red-shifting than another. This is further illustrated in Fig. 5, which shows the relative change in reflectivity (Δr) for the n1/n2=1.5 systems. The lowest R value shows no red-shift in any of the systems. However, the trend of the reflectivity change is again consistent with the previously reported results - the system with more bilayers experiences a lower relative change in the reflectivity near the center of the bandgap.
3. Homogenization approximation reflectivity results
The red-shift in the normal incidence bandgap could be readily explained using classical scattering theory, which states that the magnitude of scattering increases as the incident wavelength decreases . Thus, if the decrease in reflectivity from these structures was largely due to diffuse scattering losses, then the blue end of the bandgap would show a larger reflectivity change. However, as previously reported, the fact that the homogenization approximation can accurately predict the FDTD reflectivity at the quarter-wave tuned wavelength implies that the amount of incoherent reflected power from diffuse scattering in these structures is extremely small . In order to determine if this physical hypothesis remains valid at wavelengths other than the quarter-wave tuned wavelength, the homogenization approximation was applied to this study as well.
The homogenization approximation is a method used to estimate the amount of coherent reflection from a rough surface. The rough surface is approximated by a smooth surface with a diffuse refractive index profile. This approximation, which is similar to effective medium theory for bulk materials , is typically valid when the distances between roughness features are small compared to the incident wavelength, causing the incident wave to sample an averaged refractive index at the surface. The exact procedure used to apply homogenization theory to the roughened quarter-wave stacks is detailed in a previous publication . It involves spatially averaging the dielectric constant at the interfaces of the rough structure to produce a corresponding 1D structure with smoothly varying refractive index profiles at all the interfaces. A transfer matrix calculation is then performed on the representative 1D structure to estimate the coherent reflectivity of the rough structure. The results of the homogenization approximation applied to the 4-bilayer structures presented above are shown in Figs. 6 and 7. As before, the points on each plot correspond to the calculated reflectivity from the approximated structures, while the curves are polynomial fits of this data that represent the band structure at each roughness value.
A comparison between these Figs. and Figs. 2 and 3 reveals that the homogenization approximation results match the FDTD calculations across the entire normal incidence band gap. Specifically, it reproduces the red-shifting of the FDTD data, and moreover, it accurately predicts the increased red-shifting sensitivity of the lowest index contrast structure. The validity of the homogenization approximation in this regime is further supported with the comparison of Figs. 8 and 9 with Figs. 4 and 5. Again, the red-shifting behavior is clearly predicted and the magnitude of the reflectivity change matches the FDTD calculations. There is some fluctuation present in the FDTD data that is not captured by the homogenization approximation. These fluctuations arise from variations in the amount of incoherent reflection from the rough 1DPC structures. The homogenization approximation does not capture these fluctuations because it can only provide information on the amount of coherent reflection from rough surfaces [6–9]. Nonetheless, it appears that the decrease in the coherent reflectivity, due to the roughness features collectively smearing-out the refractive index profile in the vicinity of the interfaces, vastly dominates the total amount of reflectivity loss in these structures. Thus, the homogenization approximation can be used to estimate the reflectivity of roughened structures within the tested parameter space (rms roughnesses < 20% and rms wavelengths < 50% of the 1DPC periodicity, index contrasts < 1.75) for all wavelengths spanning the normal incidence band gap. This presents a significant improvement in computing time over the FDTD simulations. For example, the time required to obtain the reflectivity spectrum of a roughened 1DPC is over three orders of magnitude longer using the FDTD method. Thus, these results provide a more efficient scheme for calculating roughness effects in 1DPCs. Furthermore, these results imply that reflectivity loss in these structures is dominated by the influence of the roughness features on the effective index profile of the structure, rather than by diffuse scattering from the roughened interfaces.
It is interesting that the homogenization approximation can also capture the red-shifting behavior obtained by the FDTD simulations. One explanation for this could be that the effective widths of the smoothed interfaces in the approximated structure are larger for bluer light than for redder light. Therefore, the interfaces are effectively sharper (and, thus, closer to the ideal 1DPC structure) for longer wavelengths. Thus, the red end of the band gap shows a smaller decrease in the reflectivity than the blue end, resulting in a red-shift. Furthermore, the increased sensitivity of low index contrast structures to roughness can be explained by the fact that the smearing of the refractive index profile at the interfaces causes low index contrast structures to deviate more from their perfect analogue than high index contrast structures. This can be seen by considering the slope of the index profile at the interfaces of the approximated structures. Perfect structures have infinite slopes at each interface for any index contrast. However, the index profiles of the approximated structures have finite slopes at the interfaces. For a given RMS roughness, the index profile at an interface will be graded over the same distance, independent of the index contrast of the structure. Thus, the slope of the index profile for a high index contrast structure will be larger (and therefore, closer to the perfect analogue) than that for a low index contrast structure at the same RMS roughness value.
Several 2D FDTD simulations were done in order to determine the effect of interfacial roughness on the normal incidence band gap. Many 1D photonic crystal configurations were tested, with systematic variations in the index contrast and number of bilayers. In all systems tested, a narrowing and red-shifting of the normal incidence band gap was observed. Furthermore, the lowest index contrast system exhibited a higher sensitivity to the red-shifting. This suggests that lower index contrast structures are less tolerant to structural changes than higher index contrast systems.
Although the red-shifting observed in the reflectance spectra could be explained using classical scattering theory, the previous success of the homogenization approximation in reproducing the FDTD results  suggests that the amount of diffuse scattering in these systems is extremely small. Thus, the homogenization approximation was applied to this study in order to determine if it could also reproduce the red-shifting. The results of these calculations revealed that the homogenization approximation could be used to accurately predict the reflectance behavior of the structures tested here (rms roughnesses < 20% and rms wavelengths < 50% of the 1DPC periodicity, index contrasts < 1.75) for all wavelengths spanning the normal incidence band gap. These results imply that the reflectivity loss in these structures is dominated by the effective smearing-out of the refractive index profile near the interfaces in the structure by the roughness features, which can be used to explain both the red-shifting and the increased sensitivity of the low index contrast structures to interfacial roughness.
This research was supported by the Los Alamos National Laboratory Directed Research and Development Program and the U.S. Army through the Institute for Soldier Nanotechnologies under contract DAAD-19-02-D-0002 with the U.S. Army Research Office. The content does not necessarily reflect the position of U.S. government, and no official endorsement should be inferred.
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14 . Previously, we have referred to the quantity reported in Eq. (2) as the “percent” change in reflectivity. Here, we have changed the wording to “relative” as it is a more accurate description of the quantity in Eq. (2). Please note this change when comparing this manuscript to our previous ones.
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