J. Eur. Opt. Society-Rapid Publ. 21, 4( 2025) 33
Figure 3. Comparison of the FRISP methode for the simulations based on Rayleigh-Sommerfeld diffraction followed by step-wise propagation and FDTD-simulations based on [ 12 ]( Ó 2018 The Japan Society of Applied Physics). a) to d) results of FDTDsimulations, e) to h) results of the FRISP method. Refractive index n = 1.3 for( a) and( e), n = 1.5 for( b) and( f), n = 1.65 for( c) and( g) and n = 1.9 for( d) and( f) of a microsphere with a diameter of 10 lm.
lations published in the literature is shown in the following. To validate the FRISP method, we started our analysis by using a microsphere characterized with a fixed diameter of 10 lm, while systematically varying its refractive index. This scenario, previously explored by Zhou et al. in their seminal paper [ 12 ], was investigated using FDTD simulation within the CST MWS software package.
Figure 3 visually compares results derived from the FDTD-based simulations in the paper from Zhou et al( a – d) and our approach( e – h) based on the Rayleigh- Sommerfeld diffraction integral and a layer-wise propagation. The visual representation in the figure shows a striking consistency for the focal width and position of the focal spot between the two methods. This compelling agreement underscores the effectiveness and accuracy of the FRISP approach in capturing the intricate optical nuances associated with microsphere behavior. However, it is important to note one difference between the simulations. The FRISP approach lacks the capability of describing backward propagation. This limitation manifests itself as a loss of energy at the boundaries of the microsphere, which prevents the method from calculating the interference pattern of the backscattered light within the sphere. The influence on the energy caused by the disregard of reflections can be estimated by observations of refractive index transition. For typical refractive index differences, the losses due to reflections are in the lower single-digit percentage range per interface. Despite this difference, the overall congruence in the essential focal characteristics highlights the practicability of our method in realistically representing the optical behavior of the microstructures.
In order to prove the validity of the FRISP method for different refractive indices, the results of the work by Lee et al. [ 16 ] based on the Mie theory were compared with the FRISP method. The comparison results for the FWHM are shown in Table 1. The deviations between the methods are less than 5 % over a wide range of wavelengths and refractive indices, which demonstrates the validity of the FRISP method.
Figure 4 illustrates the potential of the FRISP approach by showing simulation results of a pyramidal structure composed of two different materials. a) Shows a schematic sketch of the simulated microstructure with 2 different materials, b) shows the result of an FTDT simulation by Ge et al. [ 8 ] while c) shows the result of the FRISP method. A comparative analysis with the simulation performed by Ge et al. [ 8 ] shows noticeable deviations in the intensity distribution within the pyramid structure, especially below the boundary layer separating the two materials. The significant refractive index contrast induces back reflection in the silicon nitrate. It’ s noteworthy that the FRISP approach, which lacks the capability for backpropagation, results in energy loss at the boundaries within the pyramid, specifically at the interface between silicon nitrate( left) and silicon dioxide( right). In addition, the absence of interference effects within the pyramid structure is a distinctive feature observed in our simulation, which distinguishes it from the simulation by Ge et al. These differences show the limitations of the FRISP method with respect to backscattering and backreflections. However, FRISP can be used to determine the position and shape of the nanojet generated by the microstructure.
3.2
Performance and computational time
In this study, we evaluate the computational effort associated with the FRISP method and compare it to finitedifference time-domain( FDTD) simulations [ 17 ]. The computational complexity of the FRISP approach depends primarily on factors such as the number of discrete layers, the pixel density per layer and the number of materials with different indices. The simulation domain, tailored to the typical scenario of studying photonic nanojets, spans dimensions of 25 lm 25 lm 25 lm, with two different refractive indices delineating the inside and outside regions of the microstructure. Using a mesh resolution of 50 nm 50 nm 50 nm, the computational time of the FRISP method implementation in MATLAB is less than 1 min. However, the FDTD based on the Ansys / Lumerical software package is used with a computational domain of 25 lm 25 lm 38 lm andafixed mesh size of 50 nm. In addition, perfect matched layer( PML) boundary