J. Eur. Opt. Society-Rapid Publ. 22, 20( 2026) 193
Figure 10.( a) Variation of the group index distributions due to under / over-etching in Band 3.( b) Sensitivity of the response to relative permittivity variations( 11:5 e r 12:5). The results demonstrate the robustness of the optimized 4 4 symmetric geometry against systematic fabrication-induced deviations across the considered wavelength range.
Si as the target material. Dielectric constants were always kept fixed for all optimizations, if the target material is not dispersionless then this condition should be considered in simulations or should be analyzed as post-optimization process. Moreover, if the material has non-negligible extinction coefficient, then total loss due to absorption of a finitesize device should be analyzed in time-domain simulations via transmission efficiency calculations.
Another concern on a realistic design covers manufacturing errors. In topology-optimization-based fabrication processes, one of the most common error sources is underetching and over-etching [ 63 ]. These deviations typically manifest as systematic, unidirectional fabrication errors that depend on the specific etching process, leading to lateral feature growth or shrinkage on the order of ± 10 nm.
Additional factors such as material impurities or temperature variations may induce changes in the dielectric constant. However, both GaP and Si is known to be relatively robust against temperature-induced variations. In particular, at 1550 nm the thermo-optic coefficient is quite
small, approximately 1:2 10 �4 n
C �1 and 1:8
T
10 �4 n
C �1 for GaP and Si [ 64, 65 ], respectively. These
T
results directly indicate weak temperature sensitivity in the telecom band.
Considering all these factors, it is reasonable to expect that in realistic, fabrication-inclusive implementations, performance degradation will not primarily originate from material dispersion or absorption. Instead, it is more likely to stem from impurity-induced variations in the dielectric constant and from geometric deviations introduced during fabrication.
Keeping in mind all the factors above, unlike previous optimization studies, a 44 lattice exhibiting repeating symmetry was considered in this work. The geometry is formed by uniformly sized square blocks. A total of 2500 randomly generated until cells were simulated. Dielectric constant of the material was selected as e r ¼ 12:1toensure compatibility with Si in the telecommunications frequency range. Furthermore, the design process focused on identifying a geometry that mitigates the impact of systematic fabrication-induced deviations, thereby enhancing robustness against manufacturing tolerances. Repeating pattern of the meta-atoms is given as subsets in Figure 10( Black: Si, White: Vacuum, 1a = 500nm).
To evaluate the practical feasibility and robustness of the design, we first performed simulations covering systematic fabrication error. The variation in group refractive index induced by under- and over-etching is shown in Figure 10a. Then material parameter sweep over the range 11:5 e r 12:5 was done, and the results are presented in Figure 10b. Figures show that lateral expansion, namely overgrowth( s > 0) causes a similar orientation on index deviation as that of increasing permittivity and vice versa. This can be explained by filling ratio of background and effective medium theories. Final geometry is tolerant to both error types such that even the worst cases have operating bandwidths greater than 60 nm within the target operation band( 1500 – 1600 nm).
In a negative refractive index( NRI) medium, the wave vector k, the electric field E, and the magnetic field H follow a left-handed orientation. Physically, this implies that the direction of phase propagation is opposite to the direction of energy flow. In practice, this behavior can be verified through two main conditions:( i) the dot product between the group velocity and the wave vector must be negative( v g k < 0), indicating that energy flows opposite to the phase velocity; and( ii) the Poynting vector( which represents the direction of energy transport) must be in opposite direction with respect to the wave vector. When both conditions are satisfied, the medium does not merely exhibit a backward Bloch mode, but genuinely demonstrates left-handed behavior with an effective negative refractive index.
( i) v g k < 0 is satisfied automatically since v g ¼ @ x
@ k < 0 and k > 0 within operation bandwidth.
( ii) Poynting vector field within a single unitcell should be analyzed as well. P k < 0 should also be satisfied for any( w, k) within operation bandwidth. See Sx Poynting distribution and Poynting Vector Field distributions for varying wave vector and diagram bands. P k < 0 condition is satisfied under band-3 within k 125 10 �3 such that k > 0 while P < 0.
Extensive studies on broadband negative refractive index( NRI) metamaterials have been reported in the literature