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Figure 8. The dispersion distribution for an incident wave with an angle of 0 ° [ arctan( 0.0)].
The group velocity model exhibits strong performance when enhanced by deep learning. The predicted versus actual relationship closely follows a linear trend, and no systematic bias is observed. In the residual – prediction plot, the data points are largely distributed around zero. Although the residuals are concentrated within a narrow range, a spread toward extreme values is also observed, which may increase the error margin in predicting group velocities closer to �1.
The residual histogram forms an almost symmetric bellshaped distribution. The ratio comparison of the residuals follows the reference line over a broad range, with deviations appearing only in the extreme regions. Overall, deep learning demonstrates a balanced and statistically consistent performance in group velocity prediction by accurately scaling a low-variance variable.
The deep learning model developed for bandwidth prediction successfully captures the overall trend, similar to the case of group velocity. In the predicted versus actual plot, the data points lie close to the ideal 45 ° line, indicating that the model accurately scales the output. In the residual – prediction plot, the residuals are distributed around zero without forming a pronounced pattern, suggesting the absence of systematic error. The residual histogram exhibits an approximately normal distribution with a mean close to zero, providing further evidence of the model’ s effectiveness. Aside from minor deviations observed in the ratio comparison, the majority of the residuals closely follow the reference line, indicating statistically consistent performance. Overall, this model produces significantly more balanced, scale-independent, and highly accurate predictions compared to the previous GPR model.
Figures 8 and 9 present the dispersion diagrams of the first four bands for waves incident at two different angles( a tanð0:0Þ ¼0 and a tanð0:5Þ 26:6). While generating these plots, the deep learning model was constrained to yield the following target values in the third band: v g ¼�0:5, n eff ¼�2:0. The magnitude of the wave vector( k mag) and the third-band frequency are first considered in order to identify the data point closest to the target frequency. By selecting the immediate neighboring points before and after this point, the changes in angular frequency( dx) and wave vector( dk) are computed, yielding an approximate derivative given by: dx / dk. As a result, the
Figure 9. The dispersion distribution for an incident wave with an angle of 26.6 ° [ arctan( 0.5)].
approximate values of k and x at the target point are reported. Using equation( 2), the resulting bandwidth Df for normal incidence( 0 °) corresponds to the wavelength range of 1429 – 1578 nm, which encompasses the S, C, and L bands. This outcome further confirms that the final performance criterion – achieving a low-refractive-index electromagnetic medium over the optical communication bandwidth – has been successfully satisfied.
3.1 Fine tuning: robustness and feasibility
In this subsection we give remarks on target material properties and present robustness and feasibility of previous geometries. In this manner, an alternative design with an improved methodology is proposed.
Let us begin with the target relative permittivity( e r = 9) of the previous analyses in which zinc-blende Gallium Phosphide( GaP) emerges as one of the most fabrication-compatible materials. GaP exhibits negligible absorption in the 1400 – 1600 nm wavelength range at high material purity, while its refractive index remains nearly constant at n 3:05 [ 59, 60 ]. Nanophotonic devices based on GaP have also been successfully fabricated recently using topology-optimization design methodologies [ 61 ], further supporting its technological viability. Nevertheless it is challenging to obtain vertically thick( that corresponds to out of plane direction in 2D simulations) blocks and fabricateddevicemaycauseoutofplanelossesandmodaldeviations due to limited thickness and sidewall tapering. Besides telecommunication band, same relative permittivity coincides with that of Alumina( Al 2 O 3) whichshowslow loss tangent and no remarkable dispersion within 1 – 20 GHz [ 62 ]. In this case, none of the abovementioned issues will be encountered and negative index refraction is expected to show similar behavior between computations and experimental results.
On the other hand Silicon( Si) corresponds to target e r ¼ 12:1 relative permittivity. 1550 nm optical communication wavelength is already the most suitable region where Si almost does not present dispersion such that refractive index varies from 3.46 to 3.44 between 1400 and 1600 nm. It also has extinction coefficient less then 1:6 10 �12 in the same region. Accordingly, this subsection presents the results obtained for an alternative geometry based on