J. Eur. Opt. Society-Rapid Publ. 21, 4( 2025) 31
optimization procedures that employ a significant amount of parameter variations.
In this paper, we present a novel approach for the fast simulation of the influence of a refractive free-form micro structure on a wave field based on scalar diffraction theory. Our approach is based on the Rayleigh-Sommerfeld diffraction integral which is a powerful tool to exactly calculate the propagation of wave fields even in the near field regime. We also incorporate a thin element approximation to the diffraction integral, to propagate the wave field through successive slices of the refractive element. The method allows us to predict the wave field in the near-field region of the refractive elements. Additionally, by incorporating the thin element approximation, we can simplify the mathematical complexity of the problem, making the method more accessible for practical applications. Our approach therefore provides a reduced computational time compared to conventional methods such as Mie theory and FDTD-simulations.
In order to demonstrate the suitability of the FRISP approach, comparisons with simulations of photonic nanojets shown in the literature based on FDTD-simulations [ 8, 12 ] and measurements on real micro-spheres are shown. Due to the reduced computation time, the effect of deviations between the desired and the real geometry of microstructures can be determined using suitable parameter studies.
2 Scalar diffraction theory applied to micro structures
2.1 Modeling
The FRISP approach is based on scalar diffraction theory, in which the propagation can be described by the Rayleigh- Sommerfeld diffraction integral [ 13 ]. The validity of the scalar diffraction theory for determining the focusing properties of microstructures is based on the assumptions that the microstructures have a homogeneous predefined refractive index distribution and that the focusing properties are independent from polarization( birefringence). The Rayleigh-Sommerfeld diffraction integral can be used to calculate the near-field behavior of light when it interacts with apertures, lenses or refractive elements using [ 13 ]
Uðx; yÞ ¼ 1 ZZ 1
Uðx 0; y 0 Þe �ikr cos h dx 0 dy 0; ð1Þ ikz �1 r
where U( x, y) is the complex amplitude of the diffracted field at the observation plane, U( x ', y ') is the complex amplitude of the source field, k is the wavelength of the light, z is the distance from the source to the observation plane, x ', y ' are the coordinates of the source qfield ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
, x, y are the coordinates of the observation point, r ¼ ðx � x 0 Þ 2 þðy � y 0 Þ 2 þ z 2 is the distance between the observation point and the point on the source plane, k = 2p / k is the wave number and h the angle between the wave vector and the z-axis with which the source field arrives at the image plane. The near-field zone can include any propagation distance between the evanescent field region and several wavelengths. The Rayleigh-Sommerfeld diffraction integral may be challenging to solve in a closed-form solution, but it can be solved numerically using methods such as the fast Fourier transform( FFT) [ 14 ] or assuming suitable boundary conditions to simplify the calculations [ 15 ].
In our case, the solution of the Rayleigh-Sommerfeld diffraction integral is calculated by a plane wave decomposition without the use of any simplifications apart from the assumptions described above. As the propagation distance is in a range of Dz k, we can not use the Fresnel approximation. The plane wave decomposition for the free-space propagation is calculated via the transfer function in the frequency domain
" rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# 1
Hðm x; m y Þ¼exp i 2p z k � 2 m2 x � m2 y
; ð2Þ
where m x and m y are the spatial frequencies, k the wavelength of the light and Dz the propagation distance [ 13 ]. As a result of the very small propagation distances and thus the evanescent waves we are dealing with, no cutoff frequency occurs as in the case of propagation with Dz k. The transfer function does therefore not only account for the propagation of far field modes but also evanescent field components. Determining the transfer function in frequency space makes it easy to choose the field components that are taken into account in the calculation by filtering space frequencies using high-pass, low-pass, or band-pass filters [ 13 ]. This procedure enables us to calculate and evaluate only the propagation of evanescent field components. Furthermore, this procedure can be used for the evaluation of the optical properties of the simulated structures, especially for an optimization of the shape of the surface. However, there are limitations compared to FDTD or Mie theorybased simulations, because back scattering effects are not considered in scalar diffraction theory and thus cannot be described in the proposed approach.
2.2 Calculation of wave propagation using 2D-slicing of 3D-structures
In order to compute the effect of a 3-dimensional( 3D) dielectric micro structure on a wavefront with scalar diffraction theory methods, the model domain is divided into separate layers with a layer thickness Dz k. Figure 1( a) shows a schematic illustration of the segmentation of the 3D simulation environment into M layers( b) and the calculation sequence for a layer with 2 different refractive indices n 0 and n 1.
The numerical calculation of the Rayleigh-Sommerfeld diffraction integral subsequently determines the effect of these layers on the wavefront. The effect on the wavefront is determined for all refractive indices separately and then the resulting wavefront for the entire layer is compiled according to the regions with different refractive indices. For each occurring refractive index, the propagation over the distance Dz is determined by free space propagation for this layer. Then the result of this propagation is assigned to the area inside the layer that corresponds to this refractive index. The procedure is repeated for all refractive