# Fabrication of Silica Optical Fibers: Optimal Control Problem Solution

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## Abstract

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## 1. Introduction

^{−7}wt%, which virtually eliminates losses associated with the absorption of light in the range of 0.63 to 1.55 μm [6]. The production of such pure oxides is considered because of the use of especially pure starting halides [7] (in which the concentration of impurities of these metals is at a level of 10

^{−7}–10

^{−8}wt%, and hydrogen-containing compounds at a level of 10

^{−4}–10

^{−7}wt%) [8]. Additional purification occurs during their evaporation, since the “coloring” metal halides have a significantly higher boiling point than the above-listed halides and, accordingly, lower saturated vapor pressure at the working temperature of the bubblers. The vapor-phase methods for the preparation of preforms by the nature of the formation and deposition of the resulting silicon oxides and alloying components are divided as follows [9]:

- MCVD is the modified chemical vapor deposition;
- OVD is the outside vapor deposition;
- VAD is the vapor axial deposition;
- plasma-chemical methods (PMCVD, PCVD, etc.).

#### Motivation

- (a)
- To control and manage the allowing processes by the MCVD method and the drawing of optical fibers not by measuring the temperature and fiber diameter at only one point, as it is done now, but by measuring these parameters over extended sections (by length). In other words, to conduct a distributed observation;
- (b)
- To control the process not with the help of PID controllers, as it is done now, but based on the theory of the optimal control of distributed systems.

## 2. Distributed Optimal Control of the Heat Source in Vapor Deposition Processes

#### 2.1. One-Dimensional Mathematical Model of MCVD Process

_{max}is the burner intensity (power), t is time, and z is a spatial variable.

#### 2.2. Solution of the Distributed Optimal Control Problem

_{max}, speed of his movement v(ξ), and shape parameter (heat flux width) H, for which the temperature of the silica tube is kept in the vicinity of a given state. We formulate the optimal control problem considering the linearization of the initial model and the further transition to the research of temperature deviations. Consider the problem of distributed control and a distributed monitoring system (11), (12) [19]. Let the function $\Delta q$ from the right side of the equation be the control function (further noted as $\Delta u$), and observation is a function of the state of the system $\Delta \Theta $ at every point in space and time. As a control space, we consider the space $U={L}_{2}\left(\left(0,\tau \right)\times \left(0,L\right)\right)$, and as a space of solutions consider space $\mathrm{\Omega}=\left(0,\tau \right)\times \left(0,L\right)$, where $\tau $ is the process control time. Notice that the value $\tau $ generally does not match the value T, defined above.

#### 2.3. General Results

^{−8}).

_{max}is found, which is necessary to stabilize the process within the estimated time (0, τ). Note that the stabilizing control found will only act for the specified time period, after which the process repeats. A new temperature measurement takes place, the optimal control problem is solved again, the heat source power is corrected again, etc.

_{max}was calculated according to the found values $\Delta {u}_{opt}(t,z)$. The approximation result for the control time τ = 3 [s] is shown in Figure 7.

## 3. Boundary Optimal Control of the Heat Source in Vapor Deposition Processes

#### 3.1. Two-Dimensional (Axisymmetric) Mathematical Model of MCVD Process

#### 3.2. Solution of the Boundary Optimal Control Problem

_{max}. This control will be valid only for the time interval (0, τ), after which a new temperature measurement takes place, etc. The objective of optimal control is to find a level of increase (decrease) in the power of the heat source at which the difference between the current temperature distribution and the program would be minimal. As a simulation object, we take a pipe made of synthetic silica with an external radius of R = 14 mm, an internal radius of r

_{0}= 12 mm, and a length of L = 1000 mm. Using a scanning pyrometer, we will measure the temperature. Figure 8 shows the profile of desired and actual (obtained by the measurement) temperature.

_{max.}We will perform calculations using the COMSOL Multiphysics software product, as well as the MatLab and Maple software packages. To calculate the optimality system, we will use the simple iteration method with a constant control time τ and control cost σ. As a result of solving the optimality system (29), we obtained the temperature and power distribution of the burner over the time interval (0, τ).

_{max}. As noted earlier, the thermal flux of the MCVD flare is described by the Gauss function (1). In this regard, the burner physically cannot produce the heat flux described in Figure 10b, and the function Δu must be approximated by a function that describes the shape of the heat source. As a result of approximation, we obtain the following law of change q

_{max}from the time (Figure 11).

#### 3.3. Adjustment and Experimental Determination of the Parameters of a Moving Exposure Source

_{max}and H so that when they are substituted into Equations (22)–(27), the solution of the latter would give the temperature distribution on the outer surface of the pipe, which coincides with the experimentally obtained values. We assume that we know the law of motion of the source, i.e., the change in speed v(t) is known in advance (Equation (1)).

_{max}, form parameter H and source moving speed. We also know that when the flow rate of the burner hydrogen changes, the shape and power of the source changes. Figure 13 presents a comparison of simulation results and experimental data.

_{max}and H. The experiments showed that, in the researched range, the parameters q

_{max}and H are linearly dependent on hydrogen consumption (Figure 14 and Figure 15, as well as Equations (31) and (32)).

_{max}= 5354.6 × Q

_{H2}+ 950,201,

_{H2}+ 0.0001,

_{max}= 5354600 × H + 949665.54

_{max}on dispersion (width) H. Returning to the optimal control problem, we obtain the distribution of the increment in the hydrogen flow rate from the process time (Figure 16).

_{opt}are presented. The distribution of the increment of hydrogen flow from the control time is obtained.

## 4. Optimal Control of Silica Optical Fiber Drawing

#### 4.1. Optimal Control Problem: A Simplified Mathematical Model of Fiber Drawing

- Searching for a stationary solution of the system (34) (i.e., definition of functions $\overline{R}\left(z\right)$ and $\overline{V}\left(z\right)$);
- Determination of a function $\beta \left(z\right)$ depending on stationary states;
- Searching for a solution of the optimality system (47) and finding the optimal control function ${\tilde{u}}_{0}\left(t\right)$ (the numerical solution of the optimality system (47) was implemented using the Comsol Multiphysics modeling package);
- Result analysis.

^{3}, µ = 10,000 Pa·s, L = 0.3 m. The function of the fiber radius deviation from its stationary state is shown in Figure 18 (line a). The deviation is specified in the form of an upward-convex function where the maximum is 10%.

## 4.2. Optimal Control Problem for the Optical Fiber Drawing in a Formulation That Takes into Account Surface Tension and Gravity Forces

- Searching for a stationary solution. These functions were found using the solution of system, which describes the stationary state of drawing process [23];
- Finding the functions ${\alpha}_{1}\left(z\right)$, ${\alpha}_{2}\left(z\right)$, ${\beta}_{1}\left(z\right)$, and ${\beta}_{2}\left(z\right)$, depending on stationary states;
- Solving of the optimality system (52) and finding the optimal control function ${\tilde{u}}_{0}\left(t\right)$;
- Result analysis.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 4.**Surface graphs: (

**a**) deviation of the silica temperature ΔΘ(t, z); (

**b**) optimal control Δu

_{opt}(t, z) at τ = 3 [s].

**Figure 5.**The dependence of the temperature distribution $\Delta \Theta (\tau ,z)$ on the value of the parameter $\sigma $.

**Figure 6.**Temperature dependence $\Delta \Theta (t,z)$ from time to time in various time modes control τ; z coordinate is fixed by z = L/2 (m).

**Figure 13.**Comparison of the results of numerical simulation and experiment with a moving heat source. v(t) = 70 (mm/min); Q

_{H2}= 31.5 [l].

**Figure 15.**The dependence of the dispersion (width) H on the consumption Q

_{H2}with a mobile burner.

**Figure 20.**Comparison of the deviation functions of the resulted fiber radii for two drawing modes: $\tilde{u}=0$ (line 1); the optimal value of the control function (line 2).

**Figure 24.**(

**a**,

**b**) Comparison of the deviation functions of the resulted fiber radii for two different drawing modes.

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**MDPI and ACS Style**

Pervadchuk, V.; Vladimirova, D.; Gordeeva, I.; Kuchumov, A.G.; Dektyarev, D.
Fabrication of Silica Optical Fibers: Optimal Control Problem Solution. *Fibers* **2021**, *9*, 77.
https://doi.org/10.3390/fib9120077

**AMA Style**

Pervadchuk V, Vladimirova D, Gordeeva I, Kuchumov AG, Dektyarev D.
Fabrication of Silica Optical Fibers: Optimal Control Problem Solution. *Fibers*. 2021; 9(12):77.
https://doi.org/10.3390/fib9120077

**Chicago/Turabian Style**

Pervadchuk, Vladimir, Daria Vladimirova, Irina Gordeeva, Alex G. Kuchumov, and Dmitrij Dektyarev.
2021. "Fabrication of Silica Optical Fibers: Optimal Control Problem Solution" *Fibers* 9, no. 12: 77.
https://doi.org/10.3390/fib9120077