J. Eur. Opt. Society-Rapid Publ. 22, 1( 2026) 9
Figure 12. 0th( black) and �1st( red) orders diffraction efficiencies measurements( points) and theoretical curves( lines), the grating period is 390 nm, the depth 69 nm, the duty cycle 0.57 and the waveguide thickness 268 nm.
The simulated and experimental curves are quite similar for low incidence angles( h s < 38 °); resonances appear at the same angles for the 0th reflected order( h s = 2.8 ° for point A and h s = 33 ° for point B), but for h s > 38 °, an increased shift of the experimental values is observed towards smaller incident angles. The first intersecting point( C) and the second( E) appear experimentally at h s = 44 ° and h s = 61 ° instead of 46.8 ° and 69.5 ° theoretically. The angle corresponding to the 0th and �1st orders extrema efficiencies is also experimentally smaller than the theoretical Littrow angle( h s = 51 ° instead of 56 °). The greater is the incident angle, the angular shift increases. This shift is explained by the opto-geometrical configuration of the experimental set-up since the reflected 0th and �1st orders are not centered on the half sphere center. As shown in Figure 13, the output angles h e and h e�1 of the reflected 0th and �1st orders are determined by the substrate refractive index n s = 1.5 and its thickness t = 1 mm( BK7) and to the BK7 half sphere radius R = 12.7 mm by the equation( 6) obtained by geometric optics calculations:
h e � h s ¼ h 1 � h 2 ¼ 0 ð6Þ h �1d � h s ¼ �1
with n s sin h 1 ¼ sin h 2, R sinh 1 ¼ 2t sin h s, R sinh �1d ¼ t cos h s ðtan h ds � tan h s Þ and sin h ds ¼ sin h s � k
Kn s
. This
leads to an angular shift for the first and second crossing points of D 0 = D �1 = 3 ° and D 0 = D �1 = 4 ° respectively. Similarly, the Littrow angle is calculated to be shifted by D 0 = D �1 = 4 ° from its actual value. Experimentally, the angular shifts at these different points are in close agreement with the calculated ones. The shifted theoretical curves are plotted in thick lines( black and red for the 0th and �1st orders) in Figure 12 and correspond to the experimental data although the half sphere and the indexmatching fluid introduce aberrations that become more significant as the incidence angle increases.
Figure 13. 0th and �1st reflected orders paths when taking into account the substrate thickness t and the half-sphere.
In addition, compared to the theoretical curve, the relative amplitudes of both orders are experimentally lower than the prediction; in other words, the efficiencies are less important than expected. Furthermore, the grating profile may not be perfectly square and its roughness was not considered in the simulation, which may lead also to reduce diffracted orders efficiencies due to optical losses. Nevertheless, the experimental proof of the energy transfer is clearly demonstrated, showing the two expected intersection angles at C and E positions between 0th and �1st reflected orders efficiencies and energy transfer in between them. These two particular angles could then be used for optical sensing in future applications. Since their corresponding efficiencies can be unbalance due to a change in the medium to be probed, its refractive index variation can be measured at these two angles. This configuration and implementation will be exploited in further developments as a highly sensitive refractometry-based sensor.
5 Conclusion
A comprehensive study has demonstrated the feasibility of optical energy transfer between the 0th and �1st orders diffracted by a grating in all-dielectric resonant structure. Using optimized a-Si: H layers, which combine a high refractive index with low infrared absorption, the structure was fabricated via e-beam lithography and plasma etching, and its angular diffraction behavior was experimentally characterized. For the first time, such a structure was shown to allow energy transfer between the 0th and �1st diffraction orders, while conventional high-index materials( e. g., TiO 2, HfO 2, Si 3 N 4) do not allow this effect because higher-order diffracted modes arise within the guiding layer