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J. Eur. Opt. Society-Rapid Publ. 21, 4( 2025)
Table 1. Comparison of the focus point FWHM and the error between Mie-theory based simulation [ 16 ] and FRISPmethod for micro-spheres with a diameter of 5 lm, different refractive indices n and wavelength k.
Refractive index n
Focus point with calculation method
1.46 |
Mie-theory [ nm ] |
198 |
256 |
285 |
354 |
|
FRISP [ nm ] |
198 |
254 |
290 |
351 |
|
Deviation [%] |
0.0 |
0.8 |
1.7 |
0.9 |
1.59 |
Mie-theory [ nm ] |
177 |
220 |
275 |
330 |
|
FRISP [ nm ] |
176 |
222 |
271 |
319 |
|
Deviation [%] |
0.6 |
0.9 |
1.5 |
3.4 |
1.93 |
Mie-theory [ nm ] |
126 |
160 |
195 |
216 |
|
FRISP [ nm ] |
128 |
162 |
197 |
223 |
|
Deviation [%] |
1.6 |
1.2 |
1.0 |
3.1 |
2.20 |
Mie-theory [ nm ] |
102 |
120 |
150 |
186 |
|
FRISP [ nm ] |
103 |
125 |
157 |
187 |
|
Deviation [%] |
1.0 |
4.0 |
4.5 |
0.5 |
Figure 4. Comparison of the FRISP method for pyramidal microstructures consisting of two different materials. a) Schematic illustration of the composition of the microstructure. b) Simulation based on FDTD-simulation [ 8 ]( reproduced with permission from author) with L length of the nanojet and c) simulation based on the FRISP approach. The red line highlights the position of the highest intensity of the nanojet.
conditions are applied in all directions to reduce any boundary reflections [ 18 ]. To further reduce reflections, 8 layers of stretched coordinate PML are also used [ 19 ]. In this study, the computation time for the FDTD simulation is almost 5 h using an HP Z400 workstation with 20 GB RAM. To reduce the computation time, symmetric boundary conditions are used instead of PML as suggested in [ 20 ]. However, the calculation time is about 85 min. In contrast, the computational time of the proposed method with a nonoptimized implementation in Matlab is only 38 s. Consequently, the proposed approach achieves a remarkable reduction in computational time with a factor of 135. This allows the use of( e. g. iterative) optimization methods that take into account the dependence of the transmitted field on the actual shape of the micro structure.
Figure 5 provides a comparison of the simulation results between the FRISP method and an FDTD simulation for a microsphere with a diameter of 10 lm anda refractive index of n = 1.5 and a wavelength of 634 nm. It may be seen that good agreement occurs between FRISP results and those obtained by the FDTD. The deviations in the intensity maps a) and b) for regions inside the sphere are due to the backscattering effects that are not captured by the FRISP approach. In order that the intensity distributions are still comparable, the intensities were normalized to their maximum in each case. However, the intensity distribution profiles between the two simulations show a very good agreement and almost identical sizes c) and positions d) of the focal spot. The focal spot is located at a distance of approximately 5.7 lm from the center of the micro sphere for both methods with a deviation of 2.2 % between the FDTD simulation and the FRISP method. The deviation for the Full Width at Half Maximum( FWHM) ofthefocalspotisonly1 % forafocal spot diameter of 399 nm in the shown example.
Table 2 shows the specific values for the position and FWHM of the focus spot for the example shown in Figure 5
Furthermore, the FRISP method does not require a complex preliminary consideration of the boundary conditions and the design of the perfectly matched layer environment, as is necessary for solving Maxwell’ s