# Full-Vectorial Light Propagation Simulation of Optimized Beams in Scattering Media

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## Abstract

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## 1. Introduction

## 2. Simulation Method

#### 2.1. Two-Step Beam Synthesis Method

#### 2.2. Phase Optimization Techniques

#### 2.3. Scattering Medium and System Specifications

## 3. Results

#### 3.1. Scattering of a Focused Beam by the Turbid Medium

#### 3.2. Phase Optimization to Enhance the Focus Energy Density

#### 3.3. Phase Optimization to Enhance the Absolute Value of the Focus Poynting Vector

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Illustration of the focusing system with the optical axis along the z direction. The incident light is characterized in k-space by the normalized coordinates ${s}_{x}$ and ${s}_{y}$ with its angular spectrum ${\mathit{E}}_{\mathrm{inc}}\left(\mathit{s}\right)$. The in- and out-of-plane polarizations of ${\mathit{E}}_{\mathrm{inc}}\left(\mathit{s}\right)$ with respect to the meridional plane (plane rotated by $\varphi $ around the optical axis) can be calculated with the normal vectors ${\mathit{n}}_{\varphi}$ and ${\mathit{n}}_{\mathit{\rho}}$. The aplanatic lens refracts the incident light at the Gaussian reference sphere. The resulting electric field vector ${\mathit{E}}_{\infty}\left(\mathit{s}\right)$ for the plane wave propagating in direction $\mathit{s}$ is calculated with the normal vectors ${\mathit{n}}_{\varphi}$ and ${\mathit{n}}_{\mathit{\theta}}$. The refracted light then propagates with angle $\theta $ relative to the optical axis to the focus region.

**Figure 2.**(

**a**) Scattering medium composed of cubic scatterers (this plot and all subsequent three-dimensional plots were produced with Makie [34]) with ${f}_{\mathrm{V}}=0.3$ and normalized dimensions ${L}_{x}=41.7\lambda $, ${L}_{y}=41.7\lambda $, and ${L}_{z}=20\lambda $. The x-, y-, and z-axis are normalized by the wavelength $\lambda $. (

**b**) Determination of ℓ by the decay of the coherent intensity (blue line) and collimated transmission simulations (orange markers). The black line shows the obtained exponential law with $\ell =1.05\lambda $ and the gray shaded area shows the region of the medium.

**Figure 3.**Equidistant distribution of the 1009 sampling points over the numerical aperture with NA = 0.45 (gray region).

**Figure 4.**Energy density distribution of a beam focused at $z=15\lambda $. Panel (

**a**) shows $w\left(\mathit{r}\right)/{w}_{0}$ of the focused beam in vacuum and panel (

**b**) when the scattering medium is present. The color for energy density values $<0.01$ is made transparent for better visibility.

**Figure 5.**The absolute value of the Poynting vector of a beam focused at $z=15\lambda $. Panel (

**a**) shows $\left|\Re \left({\mathit{S}}_{\mathrm{foc}}\left(\mathit{r}\right)\right)\right|/\left|\Re \left({\mathit{S}}_{0}\right)\right|$ of the focused beam in vacuum and panel (

**b**) when the scattering medium is present. For values $<0.01$, the color is transparent for better visibility.

**Figure 6.**Energy density distribution of phase-optimized beams at $z=15\lambda $. Panel (

**a**) shows the optimized phase pattern depicted in the left plot for 137 optimization channels and the resulting normalized energy density distribution $w\left(\mathit{r}\right)/{w}_{0}$ after applying the optimized phase pattern. Panel (

**b**) shows the optimized phase pattern depicted in the left plot for 1009 optimization channels and $w\left(\mathit{r}\right)/{w}_{0}$ after applying the optimized phase pattern in the right plot. Values of $w\left(\mathit{r}\right)/{w}_{0}<0.003$ (

**a**) and <$0.008$ (

**b**) are transparent for better visibility.

**Figure 7.**(

**Top**): Average focus energy density for the non-optimized beam scanned over depth z and phase-optimized beams obtained for different optimization methods (line style) and with different numbers of optimization channels (colors). The gray lines depict exponential decays. (

**Bottom**): Enhancement between the scanned and phase-optimized average energy density in the focus over depth. The shaded areas show the interval between the expected enhancement from the scalar theory and one third of it.

**Figure 8.**Distribution of the absolute values of the Poynting vector of phase-optimized beams at $z=15\lambda $. Panel (

**a**) shows the optimized phase pattern depicted in the left plot for 137 optimization channels and the resulting $\left|\Re \left(\mathit{S}\left({\mathit{r}}_{\mathrm{foc}}\right)\right)\right|/\left|\Re \left({\mathit{S}}_{0})\right)\right|$ after applying the optimized phase pattern. Panel (

**b**) shows the optimized phase pattern depicted in the left plot for 1009 optimization channels and the resulting $\left|\Re \left(\mathit{S}\left({\mathit{r}}_{\mathrm{foc}}\right)\right)\right|/\left|\Re \left({\mathit{S}}_{0})\right)\right|$ after applying the optimized phase pattern. The applied phase shifts are shown in the inset. Values of the normalized absolute value of the Poynting vector <0.002 (

**a**) and <$0.004$ (

**b**) are transparent for better visibility.

**Figure 9.**

**Top plot**: Average absolute value of the Poynting vector in the focus for the non-optimized beam scanned over depth z and phase-optimized beams obtained for different optimization methods (linestyle) and with different numbers of optimization channels (colors). The gray lines depict exponential decays.

**Bottom plot**: Enhancement between the scanned and phase-optimized average absolute value of the Poynting vector in the focus over depth.

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**MDPI and ACS Style**

Ott, F.; Fritzsche, N.; Kienle, A.
Full-Vectorial Light Propagation Simulation of Optimized Beams in Scattering Media. *Photonics* **2023**, *10*, 1068.
https://doi.org/10.3390/photonics10101068

**AMA Style**

Ott F, Fritzsche N, Kienle A.
Full-Vectorial Light Propagation Simulation of Optimized Beams in Scattering Media. *Photonics*. 2023; 10(10):1068.
https://doi.org/10.3390/photonics10101068

**Chicago/Turabian Style**

Ott, Felix, Niklas Fritzsche, and Alwin Kienle.
2023. "Full-Vectorial Light Propagation Simulation of Optimized Beams in Scattering Media" *Photonics* 10, no. 10: 1068.
https://doi.org/10.3390/photonics10101068