# Application of Wigner Distribution Function for THz Propagation Analysis

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Review of Simulation Methods

#### 1.2. Purpose

^{17}space-mesh points and 10

^{6}time steps.

## 2. Wigner Methodology

**r**

_{1}and

**r**

_{2}. The brackets for time or spatial averaging and the asterisk * means a complex conjugate. In our case, the MI describes the modified correlation between the EM field oscillations at two points

**r**

_{1}and

**r**

_{2}. Note that $\Gamma \left(r,r\right)=\langle |f\left(r\right){|}^{2}\rangle $ corresponds to the intensity distribution

**I**(

**r**)

**,**and so can be measured.

**W**(

**r, p**) is the amplitude of a ray passing through the point

**r**with a frequency

**p**(i.e., direction q).

**I**) is defined by the Wigner distribution

**E**) we obtain the set of GO rays

**I**$(x,y,{k}_{x},{k}_{y}$). The signal propagation in the space after the aperture is represented by the propagation of the GO rays [34].

## 3. Results

#### 3.1. Previous Investigations on EM-Field Distribution on the Aperture

#### 3.1.1. Previous Investigations of the THz Radiation

#### 3.1.2. FEL Simulations

#### 3.2. WDF Simulations

_{x}, k

_{y}

_{,}and the WDF is a complex function. The

**E**Field distribution at the output of a waveguide is demonstrated in Figure 5a together with the test signal (the window function) in Figure 5b.

_{x}_{x}and k

_{y}angles is very small and therefore is negligible in the energy dispersion. This indicates that at this specific frequency the radiation goes forward along the z-axis. Consequently, this is the desired working frequency. At this frequency, most of the radiation is concentrated around the electron beam that is in the middle of the profile. In reality, a frequency of synchronization of the electron with an undulator is expressed.

_{x}= 0, k

_{y}= 0) we got a ‘perfect triangle’ signal (Figure 7b), which perfectly matches the theory and analytical calculation. At this time, the code is not published and will be used for future research.

## 4. Discussion

**E**field at a central frequency of 2.89 THz. In this frequency the energy is maximal and the field consists of three main modes, as was mentioned above. The EM field representation mentioned may be easily expanded to more mode representations as well as an addition to the TM modes to complete the

_{x}**E**field. Note that the contribution of TM-Modes is negligible. One may also expand the research by adding an E

_{x}_{y}field (containing all its components) and examining all changes due to its contribution. To do this, all the steps done for the E

_{x}field must be repeated. Pre-preparations have shown that the contribution of the E

_{y}field is relatively small. Indeed, it is also possible to extend to the time component of the WDF, which is a basis for a study in future research.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 4.**Spatial profile of the output radiation field: (

**a**) f = 2.5 (THz), (

**b**) f = 2.65 (THz), (

**c**) f = 2.89 (THz). As can be seen at frequency 2.89 (THz), the energy is concentrated in the middle of the aperture and is maximum in size. Whereas, at a frequency of 2.65 (THz), the energy is smaller and what is important is that it is concentrated on the sides. At 2.5 (THz) the energy is even smaller. Since the concentration is on the sides and there is almost nothing in the middle of the aperture, we will perform the calculations for the frequency 2.89 (THz).

**Figure 5.**(

**a**) Spatial profile of the output radiation field at the working frequency of 2.89 (THz) and the test signal and (

**b**) the window function (test signal).

**Figure 6.**WDF from the center of the aperture—W

_{x}(x = 0, y = 0). (

**a**) Of the radiation field at 2.89 (THz). (

**b**) Of the window function. The energy is concentrated in the middle of the aperture.

**Figure 7.**WDF from the entire aperture, only with rays propagating parallel to the z-axis, W

_{x}(k

_{x}= 0, k

_{y}= 0). (

**a**) Of the radiation field at 2.89 [THz]. (

**b**) Of the window function.

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**MDPI and ACS Style**

Gerasimov, M.; Dyunin, E.; Gerasimov, J.; Ciplis, J.; Friedman, A.
Application of Wigner Distribution Function for THz Propagation Analysis. *Sensors* **2022**, *22*, 240.
https://doi.org/10.3390/s22010240

**AMA Style**

Gerasimov M, Dyunin E, Gerasimov J, Ciplis J, Friedman A.
Application of Wigner Distribution Function for THz Propagation Analysis. *Sensors*. 2022; 22(1):240.
https://doi.org/10.3390/s22010240

**Chicago/Turabian Style**

Gerasimov, Michael, Egor Dyunin, Jacob Gerasimov, Johnathan Ciplis, and Aharon Friedman.
2022. "Application of Wigner Distribution Function for THz Propagation Analysis" *Sensors* 22, no. 1: 240.
https://doi.org/10.3390/s22010240