# Eye Shielding against Electromagnetic Radiation: Optimal Design Using a Reduced Model of the Head

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Human Body Models

_{1}= 484 mm, T

_{1}= 277 mm, and H

_{1}= 300 mm. Computer simulation of the body’s exposure to EM waves of a 3.5 GHz frequency, with the use of this model, requires 1.5 GB of computer memory. Moreover, it is quite time-consuming and takes about 6 min on a computer equipped with a graphics card with a GPU computing unit having 16 GB of memory. Such a long simulation time can be a limitation in the design process of a shielding structure, which requires many iterations. As it was planned to use an optimization algorithm in this research, we aimed to reduce the time required for the simulation that utilized the head model.

_{2}= 147 mm, T

_{2}= 70 mm, and H

_{2}= 204 mm. While it preserves the shape of the face and the tissues properties, it was used to investigate whether limiting the size of the heterogenous model without changing its electric properties can result in faster simulations with acceptable accuracy in reference to the full model. To investigate this, a numerical experiment was performed in which the human head was exposed to an EM wave of linear vertical polarity and 3.5 GHz frequency. The wave was propagating from the ϕ = 90° direction, and the amplitude of the electric field component was equal to 87 V/m. Table 2 presents a comparison of the Poynting vector module (S) obtained for the 4 points located in the layers of the eye for the reference model (heterogeneous, full size) and the cutout model, respectively.

^{3}. The cutout model was limited to an area of 147 × 204 × 70 and thus contained only 2.1 million voxels. It allowed for the reduction of the simulation time by 40%. Unfortunately, however, the results of the simulation of power density in the eyeball area obtained with the use of the whole head model differed by even 30 times from the results obtained with the cutout model.

#### 2.2. Optimal Synthesis of the Simplified Model of the Face

_{3}= 147 mm, T

_{3}= 70 mm, and H

_{3}= 204 mm, and it contained only 2.1 million voxels of 1 mm size.

- Searching for the best shape of an object among an admissible set of shapes;
- Searching for the best location of an object, given its shape;
- Searching for the material property values, given the shape and location of an object;
- Searching for the part of the boundary that is not accessible to the physical measurement, given the remaining part of the boundary;
- Free boundary problems.

_{o}

_{,}and the value of its elements was proportional to the feasible range of the corresponding design variable. Vector d, which drives the search, was updated according to the prescribed rate of success in improving the objective function; that is where the self-adaptation of the strategy parameter comes in. d itself undergoes a modification, which is ruled by a randomized process. In fact, given the correction rate $q\in \left(0,1\right)$, considering the kth iteration, ${d}_{k+1}={q}^{-1}\xb7{d}_{k}$ (or ${d}_{k+1}=q\xb7{d}_{k}$) is set to force a larger (or smaller) standard deviation of Gaussian distribution associated with x in the next iteration, respectively. In other words, the solution vector x and the standard deviation vector d are both subject to random mutation. In a basic, cost-effective (1 + 1) implementation, the operator of selection allows for the best individual out of parent m and offspring x to survive to the next generation. In other words, an offspring individual is selected to survive if and only if it is better, or at least non-worse, than the parent individual against objectives and constraints. This way, given an initial point, there is a non-zero probability that the optimization trajectory eventually leads to a point close to the optimal solution point. The algorithm converges when the ratio of the largest value of d vector elements to the corresponding element of the initial standard deviation vector d

_{o}is smaller than a prescribed search tolerance.

_{0}is the hardware-dependent time necessary to run a single solution of the direct problem associated with the optimization problem, n

_{i}is the number of convergence iterations for a prescribed search accuracy, n

_{p}is the number of evolving solutions (in our case, n

_{p}= 1), and n

_{c}is the number of constraints.

_{k}—electrical permittivity of the kth model layer, σ

_{k}—electrical conductivity of the kth model layer, and k = 1, 3; layers refer to the region of skin, muscle, and eye sclera, respectively. The results of this identification process are non-physical, meaning that there is such material in the human body that results from the identification process. As we are designing the equivalent model, the materials are made to model the interaction of EM waves in a similar way to the human body, but only for the simulation of power density in the region of the eyes.

_{1}that was minimized was the largest difference between the data obtained from the reference model (heterogeneous model of the whole head) and the simplified model. The Poynting vector modules obtained at 4 test points located in areas of different eye tissues (see Figure 3) were compared.

- S
_{ref}—Poynting vector values obtained from the reference model (W/m^{2}) - S
_{simp}—Poynting vector values obtained from the simplified model (W/m^{2}) - n—Number of test points
- k—Layer index

_{k},σ

_{k}) that minimize (1) subject to suitable constraints, for instance 1 < ε

_{k}< 100, 1 < σ

_{k}< 100. In Figure 6, the procedure of the identification of the head model parameters is presented. The optimization algorithm was implemented in a Matlab environment, while the evaluation of the objective function was performed with the XFdtd program.

_{1}was 5.64, and the final value obtained as a result of 47 iterations was 1.58. It is the value of the maximum difference between the power density obtained with the reference and the simplified model, and this discrepancy is acceptable for the purpose of a shielding structure design. Due to the limited size of the simplified model, it uses only 2.1 million voxels, thanks to which the calculation time was reduced by about 50% in relation to the whole head model. The simplified model developed in this way was used in further research to design the shielding structure.

#### 2.3. Shielding Structure Optimal Design

_{2}, σ), where σ is the conductivity of the material, were synthesized using the EStra automatic optimization algorithm and Remcom XFdtd program. The objective function f

_{2}minimized in the optimization process was the maximum value of the power density in the eye area, similarly obtained to the previous case for 4 points inside the eye.

- S
_{Ez}—Vector values in the eye area for vertical polarization (W/m^{2}) - S
_{Ex}—Poynting vector values in the eye area for horizontal polarization (W/m^{2}) - L—Length of vertical wire in the shielding structure (mm)
- L
_{2}—Length of horizontal wire in the shielding structure (mm) - σ—Conductivity of the material (S/m)

_{2}, σ) minimizing (2) subject to suitable constraints. The geometrical constraints were resulting from the location of the structure in the proximity of the face. The length of vertical elements could change from 10 mm to 70 mm; in the case of horizontal elements, the length could change from 10 mm to 50 mm. The material conductivity could change from 1 S/m to 900 S/m. Figure 8 presents the procedure of the shield design. Again, the optimization algorithm was implemented in a Matlab environment, while the evaluation of the objective function relied on the XFdtd program. At each iteration, two simulations of the shielding structure were performed, located in front of a simplified model of the face. One field analysis was run for the vertical polarization of EM waves, and another one for the horizontal polarization. In both cases, the frequency was equal to 3.5 GHz, and the amplitude of the electric field component was equal to 87 V/m. The greatest overall value of the Poynting vector was sent to the EStra algorithm as the updated value of the objective function.

## 3. Results

_{start}= 10 mm, L

_{2start}= 10 mm, and σ

_{start}= 300 S/m, respectively. The initial value of the objective function to minimize was f

_{2start}= 20.56 W/m

^{2}. As a result of 132 iterations, the value of the objective function was reduced to f

_{2stop}= 4.44 W/m

^{2}. The optimization process took 10 h and 8 min. The parameters of the optimized structure are presented in Table 5. The history of the optimization process is presented in Figure 9. In Figure 10 and Figure 11, the history of the two design parameters along the optimization procedure is presented, respectively.

^{2}for the vertical polarization and 6.5 W/m

^{2}for the horizontal polarization. After applying the synthesized structure, it was reduced to 0.98 W/m

^{2}for the vertical polarization and 0.9 W/m

^{2}for the horizontal polarization. Additionally in this experiment, the assumed amplitude of the electric field intensity of the incident wave was 87 V/m.

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Heterogeneous model of head in the XFdtd program: (

**a**) model view; (

**b**) horizontal plane cross-section at the level of the eyeballs.

**Figure 3.**The 4 test points located in the layers of the eye for the reference model. Eyelid (A), muscle (B), lens (C), and eye vitreous (D).

**Figure 5.**Simplified model of a human face. (

**a**) Horizontal plane cross-section at the level of the eyeballs. (

**b**) Front view of the face.

**Figure 7.**Geometric design variables (L, L

_{2}) and fixed parameters (R, D) of the shielding structure.

**Figure 10.**History of the optimization process. The changes in material conductivity over iterations.

**Figure 13.**Power density distribution without shield (vertical polarization); reference value 0 dB = 40 W/m

^{2}. The regions in the circle are the areas of the eye.

**Figure 14.**Power density distribution with shield (vertical polarization); reference value 0 dB = 40 W/m

^{2}. The regions in the circle are the areas of the eye.

**Figure 15.**Power density distribution without shield (horizontal polarization); reference value 0 dB = 40 W/m

^{2}. The regions in the circle are the areas of the eye.

**Figure 16.**Power density distribution with shield (horizontal polarization); reference value 0 dB = 40 W/m

^{2}. The regions in the circle are the areas of the eye.

**Table 1.**The comparison of the Poynting vector values [W/m

^{2}] at the test points, depending on the angle of the EM wave in the x–y plane.

Test Point Location | EM Wave at ϕ = 90° (EM from the Front) | EM Wave at ϕ = 67.5° | EM Wave at ϕ = 45° |
---|---|---|---|

Eyelid (A) | 12.5 | 5.9 | 1.37 |

Muscle (B) | 8.7 | 3.65 | 1.31 |

Lens (C) | 10.5 | 4.84 | 2.06 |

Eye vitreous (D) | 7.96 | 3.46 | 0.5 |

Reference Model | Cutout Model | |
---|---|---|

Number of voxels [mln] | 40.2 | 2.1 |

Memory required [MB] | 1500 | 264 |

Simulation time [s] | 332 | 109 |

S(Eyelid) [W/m^{2}] | 12.5 | 66 |

S(Muscle) [W/m^{2}] | 8.7 | 179 |

S(Eye Sclera) [W/m^{2}] | 10.5 | 68 |

S(Eye Vitreous) [W/m^{2}] | 7.96 | 351 |

**Table 3.**The material properties of the simplified model of the face: initial values and final values.

Skin Region | Muscle Region | Eye Sclera Region | |
---|---|---|---|

Initial values | ${\epsilon}_{1}$ = 4; ${\sigma}_{1}$ = 0.0002 | ${\epsilon}_{2}$ = 4; ${\sigma}_{2}$ = 0.2 | ${\epsilon}_{3}$ = 4; ${\sigma}_{3}$ = 0.5 |

Final values | ${\epsilon}_{1}$ = 8.93; ${\sigma}_{1}$ = 0.57 | ${\epsilon}_{2}$= 7.09; ${\sigma}_{2}$ = 0.44 | ${\epsilon}_{3}$ = 2.01; ${\sigma}_{3}$ = 1.59 |

Test Point Location | Power Density (Reference Model) [$\frac{\mathbf{W}}{{\mathbf{m}}^{2}}$] | Power Density (Simplified Model) [$\frac{\mathbf{W}}{{\mathbf{m}}^{2}}$] |
---|---|---|

Eyelid (A) | 12.5 | $11.49$ |

Muscle (B) | 8.7 | 10.49 |

Lens (C) | 10.5 | 8.97 |

Eye vitreous (D) | 7.96 | 6.38 |

Parameter Name | Value |
---|---|

Vertical length L [mm] | 66 |

Horizontal size L_{2} [mm] | 46 |

Conductivity σ [S/m] | $635$ |

Test Point Location | Power Density without Shield [$\frac{\mathbf{W}}{{\mathbf{m}}^{2}}$] | Power Density with Shield [$\frac{\mathbf{W}}{{\mathbf{m}}^{2}}$] |
---|---|---|

Eyelid | 12.5 | $0.79$ |

Muscle | 8.7 | 0.78 |

Lens | 10.5 | 0.98 |

Eye vitreous | 7.96 | 0.94 |

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**MDPI and ACS Style**

Kawecki, J.; Januszkiewicz, Ł.; Di Barba, P.; Kropidłowski, K.
Eye Shielding against Electromagnetic Radiation: Optimal Design Using a Reduced Model of the Head. *Electronics* **2023**, *12*, 291.
https://doi.org/10.3390/electronics12020291

**AMA Style**

Kawecki J, Januszkiewicz Ł, Di Barba P, Kropidłowski K.
Eye Shielding against Electromagnetic Radiation: Optimal Design Using a Reduced Model of the Head. *Electronics*. 2023; 12(2):291.
https://doi.org/10.3390/electronics12020291

**Chicago/Turabian Style**

Kawecki, Jarosław, Łukasz Januszkiewicz, Paolo Di Barba, and Karol Kropidłowski.
2023. "Eye Shielding against Electromagnetic Radiation: Optimal Design Using a Reduced Model of the Head" *Electronics* 12, no. 2: 291.
https://doi.org/10.3390/electronics12020291