# Phase Dislocations in Hollow Core Waveguides

## Abstract

**:**

## 1. Introduction

## 2. Role of Losses in Vortex Formation in Hollow Core Waveguides

_{11}mode) in the air core (Figure 2a) are described by the same Bessel functions as in the case of a step-index fiber (1). Assuming the fundamental air core mode has a complex propagation constant $\beta ={\beta}^{(\mathrm{Re})}-i{\beta}^{(\mathrm{Im})}$, where ${\beta}^{(\mathrm{Im})}<<{\beta}^{(\mathrm{Re})}$ as it often happens in practice, the function argument $u$ in (1) can be represented as $u\approx {u}^{(\mathrm{Re})}+i{u}^{(\mathrm{Im})}$, where ${u}^{(\mathrm{Re})}=a\sqrt{{k}_{1}^{2}-{\beta}^{(\mathrm{Re})2}}$ and ${u}^{(\mathrm{Im})}=a{\beta}^{(\mathrm{Re})}{\beta}^{(\mathrm{Im})}$. This expansion of the function argument is possible since, for example, for the fundamental air core mode shown in Figure 2a $\beta =\frac{2\pi}{\lambda}\left({n}_{eff}^{(\mathrm{Re})}-i{n}_{eff}^{(\mathrm{Im})}\right)=\frac{2\pi}{\lambda}\left(0.99958-i1.5e-5\right)$. In this case, it is clear that $\phi $- and $r$-projections of the electric and magnetic fields in the air core will have both imaginary and real parts (3). For example, near the origin, the axial components of the electric and magnetic fields of the fundamental air core mode can be represented using an asymptotic of the Bessel function of ${J}_{1}(q)$ at $q\to 0$:

## 3. Impact of the Core-Cladding Boundary Shape on the Vortex Formation. Problem of Loss Reduction

## 4. Discussion

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Schematic representation of the cross section of a waveguide in a Cartesian and cylindrical coordinate system; (

**b**) Schematic representation of the refractive index profiles for a waveguide that localizes radiation according to the principle of total internal reflection (step-index fiber) (left) and an air core waveguide with leaky modes (right).

**Figure 2.**Axial component of the Poynting vector (in color) and the streamlines of transverse component of the Poynting vector of fundamental air core modes (HE

_{11}) (black lines) at a wavelength of 3.39 µm: (

**a**) air hole in an infinite layer of dielectric (refractive index n = 1.45) with an air core diameter of D = 90 µm; (

**b**) air core hexagon in an infinite layer of a dielectric (refractive index n = 1.45) with the same effective mode area of the fundamental air core mode as the one shown in (

**a**).

**Figure 3.**Phase distributions of the axial components of electric fields E

_{z}for the air core waveguides shown in Figure 2: (

**a**) the air hole in the infinite layer of a dielectric; (

**b**) hollow core hexagon in the infinite layer of a dielectric. Thin red and white lines correspond to the condition that the real (red) and imaginary (white) parts of the axial component of electric field being equal to zero. The phase changes from −π (dark blue) to π (dark red).

**Figure 5.**Radial projection of the transverse component of the Poynting vector of the fundamental air core mode at a wavelength of 3.39 µm for: (

**a**) a capillary with a wall thickness of 1 µm (refractive index n = 1.45) with an air core diameter of D = 90 µm; (

**b**) hollow hexagon with a wall thickness of 1 µm (refractive index n = 1.45). Black lines are streamlines of the transverse component of the Poynting vector of the fundamental air core mode. The effective mode areas are the same in both cases.

**Figure 6.**Phase distribution of the axial electric field component at a wavelength of 3.39 µm for: (

**a**) a capillary with a wall thickness of 1 µm (

**b**) hollow hexagon with a wall thickness of 1 µm. Black lines are streamlines of the transverse component of the Poynting vector of the fundamental air core mode. Thin red and white lines correspond to the condition that the real (red) and imaginary (white) parts of the axial component of electric field being equal to zero. The phase changes from −π (dark blue) to π (dark red).

**Figure 7.**Hollow core waveguides formed by six capillaries with an outer diameter of 20 µm and a wall thickness of 1 µm inserted into the corners of the hollow core hexagon (Figure 2b): (

**a**) radial projection of the transverse component of the Poynting vector of the fundamental air core mode at a wavelength of 3.39 µm; (

**b**) phase distribution of the axial electric field component of the fundamental air core mode at a wavelength of 3.39 µm. Thin red and white lines correspond to the condition that the real (red) and imaginary (white) parts of the axial component of the electric field being equal to zero. The phase changes from −π (dark blue) to π (dark red). The refractive index of the capillaries is 1.45.

**Figure 8.**Phase distribution of the axial component of the electric field of the fundamental air core mode in the cladding capillary wall at a wall thickness: (

**a**) d = 1 µm (the longest wavelength transmission band); (

**b**) d = 2.42 µm. The phase changes from −π (dark blue) to π (dark red), with light green corresponding to zero. Thin red and white lines correspond to the condition that the real (red) and imaginary (white) parts of the axial component of electric field being equal to zero. For (

**a**) red and white lines superimpose on the inner border of the cladding capillary. Other parameters are the same as in Figure 7.

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Pryamikov, A. Phase Dislocations in Hollow Core Waveguides. *Fibers* **2021**, *9*, 59.
https://doi.org/10.3390/fib9100059

**AMA Style**

Pryamikov A. Phase Dislocations in Hollow Core Waveguides. *Fibers*. 2021; 9(10):59.
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**Chicago/Turabian Style**

Pryamikov, Andrey. 2021. "Phase Dislocations in Hollow Core Waveguides" *Fibers* 9, no. 10: 59.
https://doi.org/10.3390/fib9100059