# Nonparaxial Propagation of Bessel Correlated Vortex Beams in Free Space

## Abstract

**:**

## 1. Introduction

## 2. Cross-Spectral Density Function

_{coh}. A characteristic diffraction angle of a coherent beam in paraxial regime is defined by ${\theta}_{d}~\lambda /w$, while for an incoherent beam it is defined by the coherence length, i.e., ${\theta}_{d}~\lambda /{r}_{coh}.$ Note that for a nearly incoherent beam ${r}_{coh}\ll w$.

## 3. Coherent Mode Representation

#### 3.1. Nonparaxial Propagation in Free Space

_{0}]. Hence, any field in the initial plane z = 0 can be decomposed into these modal solutions.

#### 3.2. Modified Partially Coherent Vortex Bessel–Gauss Beams

_{f}in the plane z = 0.

_{f}. In contrast to the conventional LG modes, here, we consider generalized Laguerre–Gauss modes with spherical wavefrontss the eigenfunctions

## 4. Propagation of Coherent Vortex Bessel–Gauss Beams

#### 4.1. Bessel–Gauss Beam with l = 0

#### 4.2. Bessel–Gauss Beam with l = 1

#### 4.3. Nonparaxial Propagation and Focusing of a Gaussian Beam

_{f}= 1000 µm and R

_{f}= 100,000 µm at different propagation distances.

_{f}= 1000 µm. Note that tight focusing takes place if the wavefront curvature radius is not much different from the width of the incident beam. In this case, the nonparaxial effects become significant.

_{f}= 100 µm, we obtain a displacement of the focal plane by 4.7 µm (Figure 6a). For an incident Gaussian beam with a radius of the wavefront curvature R

_{f}= 50 µm, a displacement of the focal plane is 4.0 µm (Figure 6c). There is a significant difference in the axial intensity distributions in front of and behind the focus. Before the focus plane, there are significant oscillations in the field intensity. This asymmetry is caused by nonparaxiality. The beam intensity profile in the focus plane does not correspond to the Gaussian profile. It can be seen, that a noticeable sidelobe appears in the profile of the beam (Figure 6d). There is no sidelobe in the paraxial approximation. Note that these effects were also observed with nonparaxial focusing of light beams in a graded-index medium [36,37]. The observed effects may be important in optical trapping and manipulation of nanoparticles.

## 5. Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Intensity distributions of BG vortex beam with l = 0. w = 30 µm; (

**a**) ξ = 0.02; r

_{coh}= 100 µm; (

**b**) ξ = 0.80; r

_{coh}= 10 µm.

**Figure 2.**Intensity distributions of BG vortex beam with l = 1. w = 30 µm; (

**a**) ξ = 0.02; r

_{coh}= 100 µm; (

**b**) ξ = 0.80; r

_{coh}= 10 µm.

**Figure 3.**Intensity distributions of BG beam with l = 0; ${w}_{0}$ = 30 mm; w

_{B}= 20 mm; l = 0.63 µm. (

**a**) z = 0; (

**b**) z = 1000 µm; (

**c**) z = 2000 µm; (

**d**) z = 3000 µm.

**Figure 4.**Intensity distributions of BG vortex beam with l = 1. ${w}_{0}$ = 30 µm, w

_{B}= 20 µm, λ = 0.63 µm. (

**a**,

**b**) z = 0; (

**c**,

**d**) z = 500 µm; (

**e**,

**f**) z = 1000 µm.

**Figure 5.**Intensity distributions of Gaussian beam with w = 30 μm and λ = 0.63 μm at different distances: (

**a**) z = 0; R

_{f}= 1000 µm; (

**b**) z = 1000 µm, R

_{f}= 1000 µm; (

**c**) z = 2000 µm, R

_{f}= 1000 µm; (

**d**) z = 2000 µm, R

_{f}= 100,000 µm.

**Figure 6.**Intensity distributions of focused Gaussian beams in axial direction and radial direction at focus planes: (

**b**) z

_{f}= 95.3 µm; (

**d**) z

_{f}= 46.0 µm. Incident beams with w = 30 µm, λ = 0.63 µm and R

_{f}= 100 µm (

**a**,

**b**) and R

_{f}= 50 µm (

**c**,

**d**).

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Petrov, N.I.
Nonparaxial Propagation of Bessel Correlated Vortex Beams in Free Space. *Micromachines* **2023**, *14*, 38.
https://doi.org/10.3390/mi14010038

**AMA Style**

Petrov NI.
Nonparaxial Propagation of Bessel Correlated Vortex Beams in Free Space. *Micromachines*. 2023; 14(1):38.
https://doi.org/10.3390/mi14010038

**Chicago/Turabian Style**

Petrov, Nikolai I.
2023. "Nonparaxial Propagation of Bessel Correlated Vortex Beams in Free Space" *Micromachines* 14, no. 1: 38.
https://doi.org/10.3390/mi14010038