Manipulating the incident wavefront in biomedical applications to enhance the penetration depth and energy delivery in scattering media such as biological tissue has gained a lot of attention in recent years. However, focusing inside scattering media and examining the electromagnetic field inside the medium still is an elaborate task. This is where electromagnetic field simulations that model the wavefront shaping process can help us understand how the focal near field evolves at different depths. Here we use a two-step beam synthesis method to simulate the scattering of complex incident wavefronts by well-characterized media. The approach uses plane wave electromagnetic near-field solutions in combination with an angular spectrum approach to model different light beams. We apply this approach to various two-dimensional scattering media and investigate the focus intensity over depth while scanning with and without phase optimization. We find that the scanned non-optimized beams have two regions characterized by exponential decays. The absolute progression of the focus intensity over depth for phase-optimized beams using all channels can be described by solutions of the radiative transfer theory. Furthermore, the average enhancement factor over depth of the phase-optimized focus intensity compared to that without optimization is investigated for different numerical apertures and scattering media. Our results show that, albeit the incident beam is diffusively scattered, the theoretical enhancement for a large number of optimization channels cannot be reached due to correlations between the channels. An increase in focus depth and an increase in the numerical aperture reduces the difference between the expected theoretical and simulated enhancement factors.

In this work, the image formation in a confocal laser scanning microscope (CLSM) is investigated for custom-made multi-cylinder phantoms. The cylinder structures were fabricated using 3D direct laser writing and consist of parallel cylinders with radii of 5 and 10 $\mathsf{\mu }\mathrm{m}$ for the respective multi-cylinder phantom, with overall dimensions of about $200\times 200\times 200 \mathsf{\mu }{\mathrm{m}}^3$. Measurements were performed for different refractive index differences and by varying other parameters of the measurement system, such as pinhole size or numerical aperture (NA). For theoretical comparison, the confocal setup was implemented in an in-house developed tetrahedron-based and GPU-accelerated Monte Carlo (MC) software. The simulation results for a cylindrical single scatterer were first compared with the analytical solution of Maxwell’s equations in two dimensions for prior validation. Subsequently, the more complex multi-cylinder structures were simulated using the MC software and compared with the experimental results. For the largest refractive index difference, i.e., air as the surrounding medium, the simulated and measured data show a high degree of agreement, with all the key features of the CLSM image being reproduced by the simulation. Even with a significant reduction in the refractive index difference by the use of immersion oil to values as low as 0.005, a good agreement between simulation and measurement was observed, particularly with respect to the increase in penetration depth.

We study the Lambertian illumination of dielectric media with monochromatic light by numerically solving Maxwell's equations in three dimensions. We compute the average energy density inside spherical and cubic objects with a uniform refractive index. The Lambertian illumination of objects with uniform refractive index reproduces the known enhancement of the average energy density equal to the third power of the refractive index if the surface is sufficiently textured. Furthermore, inhomogeneous refractive index distributions with scattering areas much smaller than the wavelength, of the order of the wavelength, and larger than the wavelength are studied. The average energy density inside these media is compared to effective medium theories. Our investigation shows that the average energy density enhancement inside scattering media with scatterers much smaller than the wavelength can be modeled accurately with effective medium approximations.

Wavefront shaping is a well-known method of restoring a focus deep within scattering media by manipulating the incident light. However, the achievable focus enhancement depends on and is limited by the optical and geometrical properties of the medium. These properties contribute to the number of linearly independent transmission channels for light propagating through the turbid medium. Correlations occur when the number of incident waves coupled into the scattering medium exceeds this finite number of transmission channels. This paper investigates the wavefront shaping of such correlated electric fields. The influence of the observed correlations persists even though the average electric field distribution at positions in the focal plane follows a circular complex Gaussian. We show that correlations of the transmitted electric fields reduce the achievable intensity enhancement, even deep in the turbid medium. The investigations are carried out using a Monte Carlo algorithm. It is based on the speckle statistics of independent waves and introduces correlations of neighbouring electric fields via a Cholesky decomposition of the covariance matrix. Additional investigations include scenarios where the electric fields are not completely randomized, such as for ballistic or insufficiently scattered light. Significant contributions from such little-scattered light are observed to reduce the intensity enhancement further. Data from simulations solving Maxwell’s equations are compared with the results obtained from the Monte Carlo simulations for validation throughout this paper.

Volumetric scattering prevents imaging modalities in biomedical optics from imaging deep inside tissue. The optimization of the incident wavefront has the potential to improve these imaging modalities. To investigate the optimization and light propagation of such beams inside scattering media rigorously, full-vectorial simulations based on solutions of Maxwell’s equations are necessary. In this publication, we present a versatile two-step beam synthesis method to efficiently simulate the scanning and phase optimization of a focused beam inside a static scattering medium. We present four different approaches to the phase optimization of the energy density and the absolute value of the Poynting vector. We find that these quantities have two regions with different, almost exponential decays over depth for a non-optimized beam. Optimization by conjugating the phase of the projected electric field in various directions at the focus shows an improvement below a certain penetration depth. Seeking global solutions to the optimization problems reveals an even better enhancement in the energy density and the absolute value of the Poynting vector in the focus. For Poynting vector optimization, the differences between the presented optimization approaches are more significant than for the energy density. With the presented method, it is possible to efficiently simulate different imaging methods improved by wavefront shaping to investigate their possible penetration depths.