# Depolarization of Light in Optical Fibers: Effects of Diffraction and Spin-Orbit Interaction

## Abstract

**:**

## 1. Introduction

## 2. 2D Polarization

#### 2.1. Rotation of Polarization Plane in a Graded Index Fiber

_{0}≠ 0, y

_{0}= 0, p

_{x}

_{0}= 0, and p

_{y}

_{0}= 0, we obtain the following expressions for the beam trajectory and beam width [35]:

_{0}≠ 0, y

_{0}= 0, p

_{x0}= 0, and p

_{y0}≠ 0 have the form [35]:

_{y}

_{0}= 0) rotates in the direction of circulation of the polarization vector. A similar rotation takes place for the sagittal rays (p

_{y}

_{0}≠ 0). The rotation of the meridional ray and the additional rotation of the sagittal rays increase linearly with distance:

_{0}, the evolution of the beam width is defined by the expressions [43]:

_{0}is the radius of the fundamental mode.

_{0}π/(2ω), the beam width takes the minimum value:

#### 2.2. Depolarization of Light

#### 2.2.1. Linear Polarization

_{0}is the total intensity of the incident beam.

^{2}and substituting the solutions in Equation (7), we obtain the following expression for the value, determining the depolarization [16,17]:

_{0}≠ 0, y

_{0}= 0, p

_{x}

_{0}= 0, p

_{y}

_{0}≠ 0.

_{0}= 0, p

_{y}

_{0}= 0), we obtain oscillations in the degree of polarization that are purely diffractive in origin. The effect of periodic retrieval of the initial degree of polarization should be observed in a single-mode isotropic optical fiber, when the fundamental mode of the optical waveguide is considered.

_{0}, including the meridional rays and the radius-preserving (${p}_{y0}={x}_{0}\omega $) spirally twisted sagittal rays. It is clear that the degree of polarization decreases inversely with the square of the distance. The decrease in the degree of polarization becomes more pronounced as the displacement or inclination of the beam relative to the fiber axis increases. The depolarization length of the meridional rays is 1.4 times longer than the depolarization length of the sagittal rays.

_{0}, where the distance z

_{0}is determined by the accuracy of the solution of Equation (8). In our case, this distance is equal to

^{−1}has the form [13,14]:

_{0}= 0, p

_{y}

_{0}= 0):

_{01}and LP

_{21}can be estimated at 0.7.

#### 2.2.2. Circular Polarization

^{2}[16,17]:

^{−1}may be defined as

_{y}

_{0}= x

_{0}ω) are considered here. The degree of polarization decreases with increasing distance, similar to the degree of polarization of a linearly polarized beam. A similar result occurs in the case of left-handed polarized light.

#### 2.3. Berry Phase and Degree of Polarization

_{y}

_{0}= n

_{0}sinθ) of the beam to the waveguide axis or with the axis displacement x

_{0}for sagittal rays with constant radius of twisting (${p}_{y0}={x}_{0}\omega $) (Figure 5a,b). The calculated results for the angle of rotation of the polarization plane are in good agreement with the experimental data [13]. Note that the cubic dependence of the rotation angle of the polarization vector on the angle between the propagation direction and the axis of the fiber with a step profile of the refractive index was obtained in [37].

## 3. 3D Polarization

_{z}= 0, many problems of near-field optics, optical microscopy, data storage, and scattering need to be analyzed, taking into account the longitudinal component of the field E

_{z}. Strongly focused beams contain a non-zero longitudinal component of the field ${E}_{z}\ne 0$, so we should consider 3 × 3 coherence matrices to describe them. The Stokes 2 × 2 formalism was extended to 3 × 3 coherence matrices, and the degree of polarization for arbitrary electromagnetic fields was introduced in [58,59,60,61]. The evolution of three-dimensional electromagnetic fields in a graded index medium is studied in [25,26,27,28,62]. The coherence matrix is represented in terms of spherical or rectangular tensors, similar to the Gell-Mann matrices. The evolution of the coherence matrix is defined in terms of the Hamiltonian obtained from the three-component Maxwell field equations.

#### 3.1. Vector Laguerre–Gauss (LG) Beams in a Graded-Index Fiber

#### Simulation Results

#### 3.2. The Splitting of the Degenerate Modes

#### 3.3. Simulation of Delay Time and Group Delay Splitting

#### 3.4. Vector and Tensor Polarization Degrees of Light

_{KQ}:

_{v}and D

_{t}are determined by the possible values of the components t

_{1Q}and t

_{2Q}, which can be determined from the condition of positivity property of the diagonal elements of the density matrix (${\rho}_{11}\ge 0$, ${\rho}_{22}\ge 0$, ${\rho}_{33}\ge 0$), and the normalization condition for the total intensity.

_{v}is the length of the vector OP (Figure 11). If we determine the vector polarization in the direction of the z-axis, the components ${t}_{1\pm 1}$ disappear. In the case of the direction of the polarization vector along the principal axis of the symmetric second-rank tensor formed by $\langle {\widehat{T}}_{2Q}\rangle $, the components ${t}_{2\pm 1}$ also disappear. The possible states of tensor polarization cannot be represented by a spherical area; these points form a cone inside a sphere with a radius of $\sqrt{2}$, and the permissible values of the degree of tensor polarization are $0\le {D}_{t}\le 1$ (Figure 12).

_{t}= 0), i.e., the vector polarized beam is only partially polarized (Figure 11). This indicates that it cannot be represented by a pure polarization state. However, there is a completely polarized beam that does not have a preferred direction of the polarization vector (${t}_{11}={t}_{1-1}={t}_{10}=0$), and this light beam is a pure tensor polarized beam (D = D

_{t}= 1). This corresponds to the point ${t}_{20}=-\sqrt{2}$ at the top of the cone (point A in Figure 12). As can be seen from Figure 12, there are completely polarized states consisting of vector and tensor components of polarization on the circle abcd of the cone basis (for example, points a and c with ${t}_{10}=\pm \sqrt{3/2}$ and ${t}_{20}=\sqrt{1/2}$). The points on the circle abcd correspond to circularly and linearly polarized beams and elliptically polarized states.

_{1}= 0.7, λ

_{2}= 0.2, λ

_{3}= 0.1 and ${\tilde{\lambda}}_{1}=0.65,{\tilde{\lambda}}_{2}=0.343,{\tilde{\lambda}}_{3}=0.007$ have the same values of the total degree of polarization D = 0.56. However, these matrices have different degrees of vector and tensor polarization: D

_{v}= 0.52, D

_{t}= 0.2, and ${\tilde{D}}_{v}=$ 0.56, ${\tilde{D}}_{t}=$ 0.0145, respectively.

_{z}→ 0, and all tensor components of the second-rank are transformed into vector components. This means that the 2D formalism can be applied to describe the field in the far zone.

## 4. Discussion

_{0}of the beam and the wavelength $\lambda $. From Equations (30) and (37), it is possible to determine that the depolarization length corresponds to a 50% decrease in the degree of polarization:

^{−1}(a fiber with a radius r

_{0}= x

_{0}= 25 µm and a relative difference in the refractive index $\Delta \approx 6.9\cdot {10}^{-3}$) and the wavelength $\lambda $ = 0.63 µm, we get the resulting distance ${l}_{d}\approx 8.3$ cm. Such depolarization was experimentally observed in an isotropic multimode optical fiber [12]. Depolarization is enhanced by increasing the axial displacement of the incident beam or the angle of its inclination to the waveguide axis. Since rays with a large axial displacement x

_{0}correspond to high-order fiber modes, the depolarization of modes with large numbers will be stronger compared to low-order modes. Therefore, in coherent communication systems, it is better to use single-mode fibers without birefringence, where the degree of polarization is retrieved with a period $z=\pi /\omega $ on propagation. This length in graded-index optical fibers is equal to $z\approx $ 0.5 mm.

_{0}→ 0, it can be interpreted as the result of the interaction of polarization (spin) and trajectory (orbital angular momentum). The degree of polarization of both linearly and circularly polarized beams decreases with distance according to the quadratic law. The depolarization of the meridional rays is less than that of the sagittal rays. For right- and left-handed circularly polarized light, the effect of asymmetry with respect to the sign of the twist of the sagittal ray trajectory is observed. Depolarization is enhanced by increasing the axial displacement of the beam, the gradient parameter of the fiber and the radiation wavelength.

## 5. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**The degree of linear polarization as a function of distance for both meridional and sagittal rays (solid curves) and only for meridional rays (dashed curves). (1) x

_{0}= 15 μm; (2) x

_{0}= 5 μm.

**Figure 3.**The values $F={10}^{5}\left(4\mathrm{det}J/S{p}^{2}J\right)$ as a function of the distance at x

_{0}= 15 μm for both meridional and sagittal rays (solid curve 1) and only for meridional rays (dashed curve 2).

**Figure 4.**The values $F={10}^{5}\left(4\mathrm{det}J/S{p}^{2}J\right)$ as a function of distance for x

_{0}= 15 μm; curve 1—sagittal ray with positive helicity (${p}_{y0}={x}_{0}\omega $); curve 2—sagittal ray with negative helicity (${p}_{y0}=-{x}_{0}\omega $); curve 3—meridional ray.

**Figure 5.**The dependence of the angle of rotation of the polarization plane on (

**a**) tilt angle ${\theta}_{0}$ of the beam relative to the fiber axis; (

**b**) offset ${x}_{0}$ in a graded-index optical fiber with the parameters $\omega =7\cdot {10}^{-3}$ μm

^{−1}, n

_{0}= 1.5, and the length z = 5 cm.

**Figure 6.**Dependence of the rotation angle of the polarization plane on the tilt angle of the beam relative to the axis of an optical fiber of length 7.5 cm; theoretical (solid curve) and experimental [13] values (triangles).

**Figure 7.**Field amplitudes of even modes $|00\rangle ,|20\rangle ,|22\rangle $ (

**a**) and odd modes $|11\rangle ,|31\rangle $ (

**b**).

**Figure 8.**Intensity distributions of the transverse electric field component (left column) and the longitudinal electric field component (right column) in the focal planes ${z}_{f}=331\mathsf{\mu}m$ (

**a**–

**d**) and ${z}_{f}=333\mathsf{\mu}m$ (

**e**–

**h**): (

**a**,

**b**) $l=0$; (

**c**,

**d**) $l=0$ —3D intensity patterns; (

**e**,

**f**) $l=1$; (

**g**,

**h**) $l=-1$.

**Figure 9.**Dependence of the delay time (

**a**) and group delay splitting (

**b**) on the radial and azimuthal indices, accordingly, z = 1 km, ${n}_{0}=1.5$.

**Figure 10.**Dependence of the mode splitting (

**a**) and degeneracy lifting (

**b**) on the azimuthal index. $p=0,{n}_{0}=1.5$.

**Figure 11.**Geometrical representation of possible vector polarization states. The vector polarization state is determined by the point P in the sphere and the degree of vector polarization by the length of vector OP: ${D}_{v}={R}_{v}/\sqrt{2}$, R

_{v}= OP. The maximum radius of the sphere is ${R}_{v}^{\mathrm{max}}=\sqrt{3/2}$.

**Figure 12.**Admissible values of combined vector and tensor polarization states are defined by the point P in a cone inserted into a sphere with the radius $\sqrt{2}$. The cross-section aAc of the cone is the equilateral triangle.

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**MDPI and ACS Style**

Petrov, N.I.
Depolarization of Light in Optical Fibers: Effects of Diffraction and Spin-Orbit Interaction. *Fibers* **2021**, *9*, 34.
https://doi.org/10.3390/fib9060034

**AMA Style**

Petrov NI.
Depolarization of Light in Optical Fibers: Effects of Diffraction and Spin-Orbit Interaction. *Fibers*. 2021; 9(6):34.
https://doi.org/10.3390/fib9060034

**Chicago/Turabian Style**

Petrov, Nikolai I.
2021. "Depolarization of Light in Optical Fibers: Effects of Diffraction and Spin-Orbit Interaction" *Fibers* 9, no. 6: 34.
https://doi.org/10.3390/fib9060034