# Asymmetries Caused by Nonparaxiality and Spin–Orbit Interaction during Light Propagation in a Graded-Index Medium

## Abstract

**:**

## 1. Introduction

## 2. Basic Equations

_{0}is the refractive index on the waveguide axis, $\omega ={\left(2\u2206\right)}^{1/2}{n}_{0}/a$ is the gradient parameter, $\u2206=\frac{{n}_{0}^{2}-{n}^{2}\left(a\right)}{2{n}_{0}^{2}}$, $a$ is the fiber radius, $r=\sqrt{{x}^{2}+{y}^{2}}$.

**σ**is the spin angular momentum.

## 3. Simulation Results

#### 3.1. Effect of Nonparaxiality on the Beam Width and Axial Intensity Distribution

#### 3.2. Effect of Spin–Orbit Interaction on the Intensity Distribution

#### 3.3. Effect of Spin–Orbit Interaction on the Speed of Vortex Beams in Optical Fiber

## 4. Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Beam width change with distance. l = 0, σ = 0, $\lambda =0.63\mathsf{\mu}\mathrm{m},{n}_{0}=1.5$. Dashed line—paraxial approximation. (

**a**) ${a}_{0}=45$ µm; (

**b**) ${a}_{0}=90$ µm.

**Figure 2.**Intensity distributions of the transverse electric field in the axial direction. l = 0, σ = 0, $\lambda =0.63\mathsf{\mu}\mathrm{m},{n}_{0}=1.5,{a}_{0}=45$ µm. (

**a**) Nonparaxial; (

**b**) paraxial approximation; (

**c**) intensity profiles at a second focus plane: black line—nonparaxial, red line—paraxial; (

**d**) intensity profiles at a third focus plane: black line—nonparaxial, red line—paraxial.

**Figure 3.**Intensity distributions of the transverse electric field (

**left column**) and the longitudinal electric field component (

**right column**) for the circularly polarized incident beam with zero radial index in the focal plane ${z}_{f}=331\mathsf{\mu}\mathrm{m}$: (

**a**,

**b**) $l=1,\sigma =1$; (

**c**,

**d**) $l=-1,\sigma =1$.

**Figure 4.**Intensity distributions of the transverse electric field (

**left column**) and the longitudinal electric field component (

**right column**) for the circularly polarized incident beam with zero radial number in the focal plane ${z}_{f}=331\mathsf{\mu}\mathrm{m}$: (

**a**,

**b**) $l=1,\sigma =1$; (

**c**,

**d**) $l=1,\sigma =-1$.

**Figure 5.**Intensity distributions of the transverse electric field component (

**left column**) and the longitudinal electric field component (

**right column**) for the circularly polarized incident beams with zero radial number in the focal plane ${z}_{f}=331\mathsf{\mu}\mathrm{m}$: (

**a**,

**b**) $l=0,\sigma =1$; (

**c**,

**d**) $l=2,\sigma =-1$.

**Figure 6.**Intensity distributions of the transverse electric field component (

**left column**) and the longitudinal electric field component (

**right column**) for the circularly polarized incident beams with zero radial number in the focal plane ${z}_{f}=331\mathsf{\mu}\mathrm{m}$: (

**a**,

**b**) $l=1,\sigma =1$; (

**c**,

**d**) $l=3,\sigma =-1$.

**Figure 7.**Intensity distributions of the transverse electric field component (

**left column**) and the longitudinal electric field component (

**right column**) for the circularly polarized incident beams with nonzero radial number p = 1 in the focal plane ${z}_{f}=331\mathsf{\mu}\mathrm{m}$: (

**a**,

**b**) $l=1,\sigma =1$; (

**c**,

**d**) $l=1,\sigma =-1$.

**Figure 8.**Delay times as a function of radial (

**a**) and azimuthal (

**b**) indices, accordingly, z = 1 km, ${n}_{0}=1.5$, σ = 0.

**Figure 9.**Relative delay times as a function of topological charge: z = 1 km, ${n}_{0}=1.5$; 1—delay between beams with σ = −1 and σ = 1, $\u2206\tau ={\tau}_{-1}-{\tau}_{1}$; 2—$\u2206\tau ={\tau}_{0}-{\tau}_{1}$; 3—$\u2206\tau ={\tau}_{0}-{\tau}_{-1}$; 4—$\u2206\tau ={\tau}_{1}-{\tau}_{-1}$. (

**a**) $\omega =8\xb7{10}^{-3}$ µm

^{−}

^{1}; (

**b**) $\omega ={2.2\xb710}^{-2}$ µm

^{−}

^{1}. Subindex in ${\tau}_{\sigma}$ corresponds to the spin angular momentum σ = 0, 1. −1.

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**MDPI and ACS Style**

Petrov, N.I.
Asymmetries Caused by Nonparaxiality and Spin–Orbit Interaction during Light Propagation in a Graded-Index Medium. *Symmetry* **2024**, *16*, 87.
https://doi.org/10.3390/sym16010087

**AMA Style**

Petrov NI.
Asymmetries Caused by Nonparaxiality and Spin–Orbit Interaction during Light Propagation in a Graded-Index Medium. *Symmetry*. 2024; 16(1):87.
https://doi.org/10.3390/sym16010087

**Chicago/Turabian Style**

Petrov, Nikolai I.
2024. "Asymmetries Caused by Nonparaxiality and Spin–Orbit Interaction during Light Propagation in a Graded-Index Medium" *Symmetry* 16, no. 1: 87.
https://doi.org/10.3390/sym16010087