In the section titled “Simulation results” of our manuscript,1 the material system employed in the simulations was different from that described in the text. This was an unintentional mistake. The material system was supposed to be silicon-on-silica with air surroundings, as shown in Fig. 3(b), and thus is the same material system that is used for the fabricated samples on which the experimental results in Sec. V are based. Instead, the simulations were performed employing a silicon slab completely surrounded by air.
By conducting additional simulations, we found that the deviation of the substrate parameters has an influence on the simulated energy decay in the integrating cells. However, these changes do not influence the impact and conclusions of our publication. In particular, the experimental results are not affected by the changes of the simulated material system, and the experimental and simulation results are even in better agreement.
Hence, the following are the revisions made to the article:
On p. 096102-6, the sentence “By comparative simulations, it has been found that the oxide layer below the silicon slab has almost no influence on the reflectivity of the PhC boundaries, so it was kept in place” is not valid.
Figure 5.4(c) shows the simulation results for a silicon waveguide layer with air surroundings. The simulated energy decay in an integrating cell with uniform boundaries, employing a silicon slab waveguide on top of a silica substrate and with air surroundings on top, is shown in Fig. 1(b). In addition, the maximum and minimum slopes of the energy decay are indicated by the red and blue lines, respectively. They are used to estimate an upper and a lower limit for the reflectivity of the PhC boundaries in the integrating cell.
As a consequence of the different simulated energy decays, the calculated reflectivities need to be adjusted in the text. The minimum and maximum expected reflectivities are calculated from the smallest and largest slopes of the energy decay, respectively, as indicated by the red and blue lines. On p. 096102-7, the text “Nonetheless, we use the steepest slope of the energy decay which is indicated by a red line in the logarithmic plot in order to get a lower estimate of the average reflectivity of the PhC boundaries. For a cell with uniform hole radii, we determine a maximal energy decay of −1.32 dB/ps. Using Eq. (10), we obtain an average energy loss per reflection of −0.08 dB, corresponding to a reflectivity of ρ = 98.2% for the cell boundaries.” should be corrected to “The steepest and the lowest slopes of the energy decay are evaluated to obtain a minimum and a maximum estimate for the reflectivity of the PhC boundaries. For a cell with uniform hole radii, a maximal energy decay of −3.07 dB/ps is determined after 1 ps as indicated by a red line in Fig. 5.4(c). Using Eq. (5.12), an average energy loss per reflection of −0.18 dB is obtained, corresponding to a reflectivity of ρ = 95.8% for the cell boundaries. In analogy, the minimal energy decay of −1.20 dB/ps, as indicated by a blue line in Fig. 5.4, gives rise to an average energy loss per reflection of −0.07 dB, which corresponds to a reflectivity of ρ = 98.4% for the cell boundaries.”
Based on the new simulations with a silica substrate, the calculated reflectivities in Table 5.1 should be modified to the values reported in Table I.
On p. 096102-7, the maximal achievable path length was calculated for a reflectivity of ρ = 99.7%. However, according to the new simulations results, the maximum reflectivity predicted for taper 2 on a silica substrate is ρ = 99.2%. This would result in an optical path length of 16 cm in an integrating cell with 1 mm radius.
In Sec. V, the experimental results are compared to the simulation results. Due to the correction of the substrate material in the simulations, experimental and simulation results are in better agreement. The old simulation results gave rise to a much higher improvement of the PhC reflectivity of up to 99.7% for taper 2, which could not be confirmed by the experimental data. By adjusting the substrate in the simulations, the experimental data match with the simulation results. The text on p. 096102-9 “Comparing the experimental and simulation results, we observe that the reflectivities of the cells with uniform hole radii are in good agreement. For taper 1, the experimental results give rise to a vertical scattering loss that is higher than in the simulations by a factor of 2, and for taper 2, the scattering loss seen in our experiments is even higher, by a factor of 3, compared to the scattering loss expected from the simulations. This can be explained by the limitations of the fabrication process. The holes in the uniform cells are homogeneously etched with a smooth interface between air and silicon. However, as hole radii become smaller, the roughness of the interface, the variation in the hole shape, and etching depth increase, thereby resulting in a higher vertical scattering loss.” is not valid anymore. Instead, a better text would be as follows: “By comparing the experimental and simulation results, we observe that the reflectivities obtained from the experimental results lie within the range expected from the simulations.”
In Sec. VI, the numbers for the reflectivities obtained from the simulations should be corrected:
(a) Top view of a hexagonal integrating cell with a side length of 3.78 µm and holes of uniform radius. An electromagnetic dipole is used to excite a TE mode inside the integrating cell. The position and orientation of the dipole only has a negligible influence on the energy decay. One exemplary position of the electromagnetic dipole is indicated by the red triangle. (b) Cross-sectional view. A silicon-on-insulator (SOI) platform with air as top cladding is used for the simulations. (c) Field energy depending on the simulation time plotted on a logarithmic scale for the cells with uniform hole radii shown in (a). The steepest slope of the energy decay is indicated by a red dashed line and is related to a reflectivity of ρ = 95.8%, while the smallest slope of the energy decay is shown by a blue dashed line and corresponds to a reflectivity of ρ = 98.4%.
(a) Top view of a hexagonal integrating cell with a side length of 3.78 µm and holes of uniform radius. An electromagnetic dipole is used to excite a TE mode inside the integrating cell. The position and orientation of the dipole only has a negligible influence on the energy decay. One exemplary position of the electromagnetic dipole is indicated by the red triangle. (b) Cross-sectional view. A silicon-on-insulator (SOI) platform with air as top cladding is used for the simulations. (c) Field energy depending on the simulation time plotted on a logarithmic scale for the cells with uniform hole radii shown in (a). The steepest slope of the energy decay is indicated by a red dashed line and is related to a reflectivity of ρ = 95.8%, while the smallest slope of the energy decay is shown by a blue dashed line and corresponds to a reflectivity of ρ = 98.4%.
Simulation results for integrating cells with uniform holes and two different adiabatic tapers.
| Uniform | Taper 1 | Taper 2 | |
|---|---|---|---|
| Smallest hole radius rmin | 126 nm | 90 nm | 65 nm |
| Taper length | … | 5 holes | 7 holes |
| Resulting reflectivity ρ | 95.8%–98.4% | 96.6%–98.8% | 97.3%–99.2% |
| Uniform | Taper 1 | Taper 2 | |
|---|---|---|---|
| Smallest hole radius rmin | 126 nm | 90 nm | 65 nm |
| Taper length | … | 5 holes | 7 holes |
| Resulting reflectivity ρ | 95.8%–98.4% | 96.6%–98.8% | 97.3%–99.2% |
“Using FIT simulations, it was shown that a reflectivity of up to 99.2% is possible for integrating cells with a linear taper of the radii in the PhC boundary holes and a minimal hole radius of 65 nm.”
The authors would like to thank Hendrik Preuß from the Institute of Optical and Electronic Materials at the Hamburg University of Technology for his support in calculating the corrected reflectivities and for the fruitful discussion of the results.
