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J. Eur. Opt. Society-Rapid Publ. 22, 20( 2026)
Figure
4. For the optimized result, the normalized frequencies obtained from the Bloch wave solutions are plotted as a function of the wave vector. For the best-performing geometry, the mode profile of the third band corresponding to the normalized frequency xa ⁄ 2pc = 0.55 is illustrated.
Figure 5. The steady-state electromagnetic wave distribution in the periodic square lattice pattern of the optimized unit cell.
bands( 1400 – 1600 nm). This result demonstrates that the final performance criterion of achieving a low – refractiveindex electromagnetic medium over the optical communication bandwidth has been successfully satisfied.
An electromagnetic wave is incident at an angle of 45 ° onto a structure formed by arranging the geometry shown in Figure 5 in a square lattice( number of layers: 10) at a normalized frequency of xa / 2pc = 0.55. According to Snell’ s law, unlike the case of a positive-refractive-index medium, the refracted wave is expected to transition from Region III to Region I when passing from one medium to another; however, it is clearly observed to propagate within Region II.
The reason for this behavior is explained by the relation:
j sgnðn 1 Þ sinðh 2 Þ¼sinðsgnðn 1 Þh 2 Þ¼ n 1jsin ðh 1 Þ
; ð3Þ n 2
which indicates that, depending on the sign of the refractive index of Medium 1, sgn( n 1), the sign of the angle formed between the wave in Medium 2 and the surface normal is
reversed. Consequently, the attainment of a negative refractive index is confirmed by the condition sgn( n 1)< 0.
In periodic structures such as all dielectric photonic crystals, the scattering response of each individual metaatom typically corresponds to a positive effective refractive index. However, the collective behavior emerging from Bloch-wave solutions [ 53 ] in the frequency domain, as well as from time-domain simulations of electromagnetic wave propagation( combination of destructive / constructive interferences), can exhibit characteristics consistent with an effective negative refractive index. In other words, the phase velocity and group velocity become oppositely directed. Such behavior is generally expected to arise in higherorder bands, where the dispersion relation significantly deviates from a simple plane-wave character( e. g., third band and above). In the present study, this phenomenon is observed in the third band.
It has been demonstrated that a negative effective refractive index can be achieved through photonic medium design. As a general methodology, models were developed by learning the input parameters using different