# Propagation of a Partially Coherent Bessel–Gaussian Beam in a Uniform Medium and Turbulent Atmosphere

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Basic Definitions

## 3. Basic Relations

## 4. Coherence Degree of Bessel Beams

#### 4.1. Coherence Degree of Partially Coherent Bessel Beams in a Uniform Medium

#### 4.2. Coherence Degree of Coherent Bessel Beams in a Turbulent Atmosphere

#### 4.3. Coherence Degree of Partially Coherent Bessel Beams in a Turbulent Atmosphere

- (1)
- Factors affecting the coherence of the vortex Bessel beam, partial coherence of the optical radiation, and random inhomogeneities of a turbulent atmosphere exert not an additive, but a multiplicative distorting effect (see Equation (14) or Equation (10)).
- (2)
- As a partially coherent vortex Bessel beam propagates in a turbulent atmosphere, we could expect the formation of two ring dislocations: one due to atmospheric turbulence and another due to the partial coherence of the radiation source. However, only one ring dislocation of coherence degree of vortex Bessel beam is actually observed, because these factors affect optical radiation simultaneously.
- (3)
- A ring dislocation of the vortex Bessel beam is formed at any values of the optical thickness of a uniform medium for the partially coherent radiation ${q}_{c}$.
- (4)
- As the optical thickness of a uniform medium for the partially coherent radiation ${q}_{c}$ increases, the ring dislocation of the partially coherent vortex Bessel beam grows in size, while its dependence on the optical thickness of a turbulent atmosphere $q$ reduces.
- (5)
- However, if the optical thickness of a uniform medium for the partially coherent radiation is greater than the unity (${q}_{c}\ge 1.0$), a ring dislocation of the coherence degree is formed at low levels of the coherence degree ${\mu}_{\mathrm{vbb}}\left(x,\rho \right)$ and, thus, no longer exerts a significant effect on the coherence level of the partially coherent vortex Bessel beam as a whole.
- (6)
- (7)
- At a low level of the initial coherence of optical radiation (${q}_{c}\ge 1.0$), the effect of atmospheric turbulence on the coherence degree ${\mu}_{\mathrm{vbb}}\left(x,\rho \right)$ manifests itself only in the area of strong radiation fluctuations due to turbulence ($q\ge 1.0$) (see Figure 3c,d).

## 5. Integral Coherence Scale of Bessel Beams

## 6. Integral Coherence Scale of Bessel–Gaussian Beams

## 7. Discussion

- (1)
- The main factors affecting the coherence of partially coherent vortex Bessel–Gaussian beams propagating in turbulent atmosphere are atmospheric turbulence and the partial coherence of the source of optical radiation. These factors exert not an additive, but a multiplicative distorting effect on optical radiation.
- (2)
- The partial coherence of the source of optical radiation has a greater influence in the region of weak intensity fluctuations due to atmospheric turbulence than in the region of strong fluctuations.
- (3)
- An optical vortex (helical wavefront of the beam) plays a certain role in the transition region from weak to strong wave intensity fluctuations due to atmospheric turbulence.
- (4)
- Geometrical parameters of the Bessel and Gaussian components of partially coherent vortex Bessel–Gaussian beams propagating in a uniform medium or a turbulent atmosphere have a smaller effect on the coherence of these beams than the above factors do.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Coherence degree ${\mu}_{\mathrm{vbb}}\left(x,\rho \right)$ of partially coherent vortex Bessel beams propagating in a uniform medium at $\tilde{\beta}=1.0$ for different values of the topological charge of the vortex beam $m$: (

**a**) $m=1$; (

**b**) $m=2$; (

**c**) $m=3$. The white curves represent the coordinates of a ring dislocation.

**Figure 2.**Coherence degree ${\mu}_{\mathrm{vbb}}\left(x,\rho \right)$ of coherent vortex Bessel beams propagating in a turbulent atmosphere at $\tilde{\beta}=1.0$ for different values of the topological charge of the vortex beam $m$: (

**a**) $m=1$; (

**b**) $m=2$; (

**c**) $m=3$. The white curves represent the coordinates of a ring dislocation.

**Figure 3.**Coherence degree ${\mu}_{\mathrm{vbb}}\left(x,\rho \right)$ of partially coherent vortex Bessel beams propagating in a turbulent atmosphere at $\tilde{\beta}=1.0$ and $m=1$ for different values of the dimensionless parameter ${q}_{c}$: (

**a**) ${q}_{c}=0.01$; (

**b**) ${q}_{c}=0.1$; (

**c**) ${q}_{c}=1.0$; (

**d**) ${q}_{c}=10.0$.

**Figure 4.**Integral coherence scale ${\rho}_{m\hspace{0.17em}\mathrm{vbb}}$ of partially coherent vortex Bessel beams propagating in a turbulent atmosphere at different source coherence levels ${q}_{c}$ for four values of the topological charge of the vortex beam $m$: (

**a**) $m=1$; (

**b**) $m=2$; (

**c**) $m=3$; (

**d**) $m=4$.

**Figure 5.**Integral coherence scale ${\rho}_{m\hspace{0.17em}\mathrm{vbb}}$ of partially coherent vortex Bessel beams propagating in a turbulent atmosphere at different turbulence level $q$ in the propagation medium for four values of the topological charge of vortex beam $m$: (

**a**) $m=1$; (

**b**) $m=2$; (

**c**) $m=3$; (

**d**) $m=4$.

**Figure 6.**Integral coherence scale ${\rho}_{m\hspace{0.17em}\mathrm{vbb}}$ of partially coherent vortex Bessel beams ($m=1$) propagating in a turbulent atmosphere at the different source coherence ${q}_{c}$ for four values of the Bessel beam parameter $\tilde{\beta}$: (

**a**) $\tilde{\beta}=0.5$; (

**b**) $\tilde{\beta}=1.0$; (

**c**) $\tilde{\beta}=2.0$; (

**d**) $\tilde{\beta}=4.0$.

**Figure 7.**Ratio of the integral coherence scales of partially coherent vortex Bessel–Gaussian ${\rho}_{m\hspace{0.17em}\mathrm{vbgb}}$ and Gaussian ${\rho}_{m\hspace{0.17em}\mathrm{gb}}$ beams with $\tilde{\beta}=1.0$ and $m=1$ at different values of the Fresnel number of the transmitting aperture ${\Omega}_{0}$ for three source coherence levels ${q}_{c}$: (

**a**) ${q}_{c}=0$; (

**b**) ${q}_{c}=0.1$; (

**c**) ${q}_{c}=1.0$; (

**d**) ${q}_{c}=10.0$.

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**MDPI and ACS Style**

Lukin, I.; Lukin, V.
Propagation of a Partially Coherent Bessel–Gaussian Beam in a Uniform Medium and Turbulent Atmosphere. *Photonics* **2024**, *11*, 562.
https://doi.org/10.3390/photonics11060562

**AMA Style**

Lukin I, Lukin V.
Propagation of a Partially Coherent Bessel–Gaussian Beam in a Uniform Medium and Turbulent Atmosphere. *Photonics*. 2024; 11(6):562.
https://doi.org/10.3390/photonics11060562

**Chicago/Turabian Style**

Lukin, Igor, and Vladimir Lukin.
2024. "Propagation of a Partially Coherent Bessel–Gaussian Beam in a Uniform Medium and Turbulent Atmosphere" *Photonics* 11, no. 6: 562.
https://doi.org/10.3390/photonics11060562