# Generation of Vortex Optical Beams Based on Chiral Fiber-Optic Periodic Structures

^{*}

## Abstract

**:**

## 1. Introduction

_{11}{TE

_{01}, HE

_{21}, TM

_{01}} (>2.1 × 10

^{−4}) is numerically and experimentally confirmed in this fiber in the whole C-band as well as the possibility of OAM transmission of +/−1 orders for distances up to 1.1 km. The authors also provide simple design optimization rules for redefining fiber parameters.

_{2}-laser irradiation, an LPFG with strong period deformation is achieved in the 4MF. Based on this LPFG, one can directly convert the fiber fundamental mode (LP

_{01}) to the high-order LP core mode (LP

_{21}) with an efficiency of 99.7% and then transform the LP

_{21}mode into a high-order vortex mode (±2 order).

## 2. The Theoretical Model for a Chiral (Vortex) Fiber Bragg Grating (ChFBG)

_{0}mode into OAM

_{1}; therefore, the theoretical model of the ChFBG is based on the aforementioned coupled modes theory, considering ChFBG as a type of a mode-coupling device. It is known that the existence of vortex modes requires a few-mode fiber-optic regime since any non-zero order OAM mode is represented by a superposition of the degenerated TE (transverse electric) and TM (transverse magnetic) modes [23]. Thus, since G.652 fiber is multimode in the first spectral window only, the operation in the optical C-band (1530–1565 nm) requires special few-mode fibers, e.g., the step-index FMF by the Optical Fiber Solutions (OFS) company. For the fiber considered, the approximation of a weakly guiding fiber is applicable, i.e., a fiber in which the difference between the refractive indices of the core and the cladding is less than 1% (note that the most commercially available fibers are considered as weakly guiding). In this case, the apparatus of Bessel functions and LP-modes, mentioned in previous section, can be used to describe the mode composition (including OAM modes) of the optical field propagating through the optical fiber [23]. Hence, using the weakly guiding fiber approximation and Bessel-function formalism, the full-fiber OAM-generation problem statement can be formulated as follows: it is necessary to develop a grating that transforms the zero-order OAM mode (with flat wavefront—LP

_{01}):

_{co}is the core radius (since the grating only exists in the fiber core), J

_{01}and J

_{11}are Bessel functions of 01 and 11 orders, respectively. The parameters u

_{01}and u

_{11}will be described further in the text.

_{0}is the incident radiation—Gaussian-like incoming field—LP

_{01}(OAM = 0); b

_{0}is the reflected radiation—the aforementioned superposition ${\mathrm{LP}}_{11}^{e}+i{\mathrm{LP}}_{11}^{o}$ (OAM = 1); and a

_{1}is radiation passed through the grating; b

_{1}is the radiation incident on the grating from the receiver side; in the general case, radiation b

_{1}arises due to inevitable Rayleigh scattering or in case of duplex communication. Such radiation provides the generation of a reflected signal in the b

_{1}direction, which is an undesirable circumstance, since additional low-power radiation will appear in the grating, i.e., performing additional noise and re-reflections. One can easily remove undesirable b

_{1}by placing an optical isolator behind the grating (Figure 1b).

^{−1}. According to the matrix-converting rule and taking into account that determinant of the F matrix is unitary, we obtain:

_{1}= 0 (no backward radiation), it can be found from (3) that the equations for coupling modes in the case of a ChFBG remain the same as for a regular FBG, differing in the coupling coefficient only, therefore, with respect to [27] we can obtain:

_{0}is defined below and depending on Λ—the grating period, defined as Λ = λ

_{B}/2n

_{0}, where λ

_{B}is the Bragg reflection wavelength; and β

_{01}and β

_{11}are the phase coefficients of the incident (OAM = 0) and reflected modes (OAM = 1), respectively. The parameter γ is defined as ${\mathsf{\gamma}}^{2}={k}_{ab}^{2}-\mathsf{\Delta}{\mathsf{\beta}}^{2}$, while k

_{ab}is the complex overlap integral (which specifies the coupling coefficient between modes) determined by the inhomogeneity of the refractive index $\mathsf{\delta}n$ [28]:

_{0}= 2π/y(z), y(z) is the chirp function (Figure 3), used to make the ChFBG broadband and which in the general case has an arbitrary form.

_{0}z (without chirp—y(z) = Λ); g(z) is the apodization function, used to narrow down the reflection spectrum and change its shape; in our case g(z) is equal to 1 (there is no apodization), f(r) is the radial function, determining the radial refractive index perturbation (because the grating is uniform in the azimuth), since the degree of mode coupling depends on k

_{ab}according to (5).

_{0}mode to the OAM

_{1}mode, the overlap integral between these modes must be non-zero, and in the case of orthonormal signals, it should be close to 1 (which means 100% energy transfer from the OAM

_{0}mode to the OAM

_{1}mode). In other words, one needs to make orthogonal modes non-orthogonal to increase power coupling between them. Based on these considerations, it can be shown that the radial function f(r) can be defined as follows:

_{01}and u

_{11}are the roots of the characteristic equation for a particular type of fiber in which the grating is written; these factors can be calculated easily for any step-index fiber. Figure 4 shows the shape of the function (8), which describes one period of the considered grating; the corresponding refractive index profile of one finger of the ChFBG is shown in Figure 5. This function, according to (6), theoretically provides an absolute magnitude of the overlap integral equal to 1.

^{−3}(which will be explained below), it can be defined as:

_{01}mode to OAM

_{1}mode. In this case, m should be equal to 1. Note that the reflection coefficient w, according to (4), upon apodization has a dependence on z, i.e., w = w(z). The amplitude of the reflected mode in the case of grating apodization can be defined as [28]:

_{i}, and the resulting field can be expressed as a product of the matrices:

## 3. Numerical ChFBG Analysis and Fiber Design

_{01}and LP

_{11}satisfy the cut-off condition. Practically, the cut-off wavelength can be shifted by doping GeO

_{2}core of the fiber. An alternative approach is to slightly increase both the diameter of the fiber core and Δn. In turn, strongly induced chirality will allow these three modes to be “folded” into the desired 1st order OAM. The induced chirality by the aforementioned drawing process modification adds to the desired mode coupling.

## 4. Sensor Application

_{e}is the strain optic coefficient, α

_{Λ}is the thermal expansion coefficient (which is 0.55 × 10

^{−6}/°C for silica), α

_{n}is the thermo-optic coefficient (1.05 × 10

^{−5}/°C for silica) [37], ε = ΔL/L is strain, L is ChFBG length, ΔL and ΔT are length and temperature increments, respectively. It should be noted that, in this case, the chirality of the grating does not depend on the wavelength in terms of the topological charge—OAM order (i.e., when the wavelength changes due to heating or stretching of the ChFBG, its chiral nature will be conserved; therefore, the OAM order does not depend on external factors). Since the distance between any two adjacent point of grating fingers with the same spatial phase $\mathsf{\Delta}\mathsf{\phi}$ remains unchanged and equal to Λ, environment changes will lead to proportional changes in grating period along the whole ChFBG. The proposed ChFBG, as well as chiral fiber gratings [38], can be used as sensors for temperature, pressure, etc. In [38] it is proven experimentally that chiral fiber gratings can act like sensors. However, in contrast to reference [38], in which authors use high-temperature silica fiber, the proposed ChFBG is supposed to be written in a conventional silica fiber with the temperature parameters listed under expression (12). Thus, we obtain the following dependences for the reflection wavelength on temperature (Figure 11). The results obtained are in good agreement with classical FBG sensors [24]. In contrast to classical FBGs, the proposed grating is not only an effective sensor of physical parameters, but also has an important advantage—invariance of the OAM order (chirality) of the reflected mode to changes in the physical parameters, since as was mentioned above grating chirality remains constant with changes in grating period caused by temperature or/and strain. Thus the main advantage of the proposed solution over the device presented in [38] or in [24] is that it can also replace (or supplement) the complex arrays of fiber Bragg gratings in so-called addressed sensor systems.

## 5. Conclusions

_{1}to OAM

_{0}mode conversion. Moreover, based on the analysis of the disadvantages of existing methods for optical vortex signal generation, a new optical fiber is proposed, which is a chiral microstructured optical fiber that mimics a ring-core structure and allows the formation of first-order OAM modes. Since the cross-section of the fiber is a periodic structure (formed by capillaries), it can be considered as a photonic crystal and, therefore, as a spatial-phase filter for the generation of vortices with the desired order. As a vortex-maintaining fiber, we propose a strongly chiral multimode fiber of size 100/125 μm with low modal dispersion that allows the vortex structure of the optical field propagating over relatively large distances to be maintained. These novel fibers can be applied in future communication systems, e.g., in the radio-over-fiber backbone infrastructure. Thus, in this paper, a full-fiber vortex generation method and vortex maintaining fiber have been proposed. New application of the results obtained in this work as address sensors with an address in the form of a constant topological charge in multi-sensor systems for monitoring various physical fields is also proposed.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The schematic representation of the chiral (vortex) fiber Bragg grating (ChFBG): (

**a**) in the form of a four-pole circuit; (

**b**) with isolator for reflection mitigation.

**Figure 4.**(

**a**) The shape of the radial function f(r) according to (8), describing the transverse profile of the refractive index of the grating finger; (

**b**) 3D image of a homogeneous finger with a given profile (in case of non-chiral regular grating, normalized units).

**Figure 9.**Relation of the reflection coefficient and spectrum of the ChFBG: (

**a**) refractive index versus the number of periods (fingers), N; (

**b**) width of the reflection spectrum vs. N of the grating at different values of the amplitude of the refractive index induced modulation.

**Figure 10.**(

**a**) The relative Δn between core and cladding for the proposed fiber. One can obtain a ring-structured profile of refractive index; (

**b**) the proposed micro-structured fiber with ring-shape hexagonal geometry [34].

**Figure 11.**(

**a**) ChFBG wavelength as a function of temperature change; (

**b**) ChFBG transmission as a function of temperature change for three temperature increments: 0 °C, 40 °C, and 160 °C.

**Figure 12.**Schematic example of an addressed sensor system based on an array of single-frequency ChFBGs: A coherent optical signal from a laser at a wavelength λ

_{0}, which is a central wavelength for all gratings, enters an array of three ChFBGs, which form signals with orbital angular momentums OAM

_{−1}, OAM

_{+1}and OAM

_{0}. The signals from the sensors can be separated using a mode splitter and directed to further processing (the photodetector is not shown in the figure).

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**MDPI and ACS Style**

Gizatulin, A.; Meshkov, I.; Vinogradova, I.; Bagmanov, V.; Grakhova, E.; Sultanov, A.
Generation of Vortex Optical Beams Based on Chiral Fiber-Optic Periodic Structures. *Sensors* **2020**, *20*, 5345.
https://doi.org/10.3390/s20185345

**AMA Style**

Gizatulin A, Meshkov I, Vinogradova I, Bagmanov V, Grakhova E, Sultanov A.
Generation of Vortex Optical Beams Based on Chiral Fiber-Optic Periodic Structures. *Sensors*. 2020; 20(18):5345.
https://doi.org/10.3390/s20185345

**Chicago/Turabian Style**

Gizatulin, Azat, Ivan Meshkov, Irina Vinogradova, Valery Bagmanov, Elizaveta Grakhova, and Albert Sultanov.
2020. "Generation of Vortex Optical Beams Based on Chiral Fiber-Optic Periodic Structures" *Sensors* 20, no. 18: 5345.
https://doi.org/10.3390/s20185345