## Abstract

An accurate measurement of the plasma current is of paramount importance for controlling the plasma magnetic equilibrium in tokamaks. Fiber optic current sensor (FOCS) technology is expected to be implemented to perform this task in ITER. However, during ITER operation, the vessel and the sensing fiber will be subject to vibrations and thus to time-dependent parasitic birefringence, which may significantly compromise the FOCS performance. In this paper we investigate the effects of vibrations on the plasma current measurement accuracy under ITER-relevant conditions. The simulation results show that in the case of a FOCS reflection scheme including a spun fiber and a Faraday mirror, the error induced by the vibrations is acceptable regarding the ITER current diagnostics requirements.

© 2014 Optical Society of America

## 1. Introduction

The plasma current is an important parameter for controlling the plasma magnetic equilibrium in tokamaks [1]. It is therefore necessary to measure it accurately. In present-day devices, such measurements are performed using pick-up and Rogowski coils. The signal of these inductive sensors is proportional to the time derivative of the magnetic flux through the loop. Therefore, the current reconstruction involves time integration. The ITER quasi-stable operation combined with the presence of parasitic signals induced by radiation fields may result in significant errors when integrating the measurement signal. Fiber optic current sensors (FOCS) are an attractive alternative to the inductive sensors to measure the plasma current in ITER [2]. FOCS operation is based on the detection of the Faraday rotation experienced by a polarized light beam in an optical fiber when an external magnetic field is aligned with the propagation direction of the light. The rotation angle is proportional to the magnetic field and not to its variation, which makes FOCS suitable for measurements in steady-state conditions. In addition, FOCS presents advantages such as high frequency bandwidth, high linearity over a wide current range, sufficient radiation resistance and low installation volume. The principal sources of errors in normal environment are presented in [3]. However, during ITER operation, the vacuum vessel will vibrate as well as the optic fiber used for FOCS measurements. The vibrations may affect the polarisation properties of the fiber by inducing a time-varying modification of its birefringence and therefore constitute a potential source of errors affecting the accuracy of plasma current measurements. In this paper, we study the impact of such perturbations. Similarly to what has been done in [4], the Jones matrix formalism [5] has been applied to model the polarisation behaviour of the optical fiber. The fiber can be modelled as a concatenation of short length sections and each of them is described by a Jones matrix. The effect of vibrations was taken into account by inserting Jones matrices that modify the state of polarisation (SOP). The simulation parameters implementing the SOP perturbations and corresponding to the ITER operation were experimentally determined.

## 2. FOCS technology

The FOCS principle is based on the Faraday effect that characterizes the interaction between the polarization of a light beam traveling in a transparent medium and an external magnetic field aligned with the propagation axis [6]. If we consider a magnetic field along an optical path length *L*, the Faraday effect has for consequence to rotate the input polarization state (SOP, State of Polarisation) by an angle *θ _{H}* given by:

*V*is a constant of the material constituting the fiber named the Verdet constant and expressed in

*rad/*(

*m*.

*T*).

*L*is the length of the fiber wherein the magnetic field is applied. If the optical fiber is surrounding a current

*I*(Fig. 1), this current will induce a magnetic field in the fiber. The link between the SOP rotation and the current can be deduced from the Ampere’s theorem which links the projection of a magnetic field on a closed path

*c*with the value of the current that goes trough the closed path [7]:

*θ*of the SOP therefore allows determining the plasma current.

_{H}In practice, the intrinsic birefringence of the fiber will affect the measurement since it creates a extra unwanted modification of the SOP. This modification is represented in Fig. 1 by an extra rotation *θ _{L}* and a non-zero ellipticity of the output SOP. To decrease the influence of the intrinsic birefringence of the fiber, a Faraday mirror which reflects the light and turns the SOP by 90° can be incorporated in the FOCS set-up (Fig. 1) [8]. In this scheme, the influence of the linear birefringence is supposed to be compensated between the forward and the backward propagation due to the exchange of the fast and slow axis of the optical fiber. This ideal case is depicted in Fig. 1. The output SOP (SOP 2) stays linear and the total rotation of the SOP azimuth is 2

*θ*+

_{H}*π*/2. Nevertheless, in the presence of a non reciprocal effect (the Faraday effect is non reciprocal [9]), the compensation is not fully effective and an unwanted rotating component is created by the birefringence [7]. This phenomenon must be taken into account when designing a FOCS.

## 3. Requirements for current diagnostics in ITER

It is expected that during operation, the ITER plasma current will vary from 0 to 18 MA [1]. The ITER specifications providing the accuracy that is required for monitoring the plasma current is given in Table 1 [1]. The results obtained in this work will be compared to these specifications.

To simulate FOCS on ITER, some extra information needs to be taken into account:

- The perimeter of the ITER vessel, which is 28 m long.
- Due to the presence of radiation, the measurement apparatus must be placed at a distance of 10 m from the vessel. A link fiber must therefore be used to carry the light from the interrogating device to the sensing fiber.

## 4. Simulation principle

#### 4.1. Fiber modelling

In the simulation procedure, the fiber is considered as a cascade of small uniform fiber sections along which the polarization properties can be considered constant. Each section *i* can then be characterized by its local parameters: the linear birefringence *δ _{i}*, the linear birefringence axis

*q*and the circular birefringence

_{i}*ρ*. The Jones matrix

_{i}*J*of the section

_{i}*i*is given by [9, 10]:

*α*= cos(Δ

_{i}

_{i}*l*), ${\beta}_{i}=\frac{\delta}{2}\frac{\mathit{sin}({\mathrm{\Delta}}_{i}{l}_{\mathit{el}})}{{\mathrm{\Delta}}_{i}}$, ${\gamma}_{i}={\rho}_{i}\frac{\mathit{sin}({\mathrm{\Delta}}_{i}{l}_{\mathit{el}})}{{\mathrm{\Delta}}_{i}}$, $2{\mathrm{\Delta}}_{i}=2{\left({\rho}_{i}^{2}+\frac{{\delta}_{i}^{2}}{4}\right)}^{1/2}$,

_{el}*l*is the length of the fiber section

_{el}*i*and

*q*is the orientation angle of the local linear birefringence. The calculation of the SOP evolution along the fiber is performed by multiplying the corresponding Jones matrices. where

_{i}*J*is the Jones matrix of the section i and

_{i}*J*is the Jones matrix of the entire section. In presence of an axial magnetic field along the fiber, there is an extra non reciprocal circular birefringence

_{T}*ρ*=

_{H}*VB*[7] where

_{f}*V*is the Verdet constant of the fiber and

*B*the component of the magnetic field aligned with the fiber axis. The total circular birefringence

_{f}*ρ*is then composed of two contributions

*ρ*=

*ρ*+

_{f}*ρ*where

_{H}*ρ*the intrinsic birefringence of the fiber and

_{f}*ρ*the circular birefringence induced by the magnetic field.

_{H}After the Faraday mirror, the propagation direction in the fiber is changed. In the following, the Jones matrix for a section *i* is represented by
$\overrightarrow{{J}_{i}}$ for the forward direction (before the Faraday mirror) and by
$\overleftarrow{{J}_{i}}$ for the backward propagation. The matrix
$\overleftarrow{{J}_{i}}$ is obtained by changing the sign of *ρ _{f}* (

*ρ*and

_{H}*δ*remaining the same) [11].

It has been shown in [7] that the choice of the fiber type has a crucial effect on the measurement accuracy. The use of a spun fiber as proposed in [12] revealed to be the best choice for FOCS in the ITER environment [7]. Based on that assessment, we simulated a spun fiber with a beat length (corresponding to
$\frac{2\pi}{{\delta}_{i}}$) of 2 meter and a spun period of 1 cm which correspond to typical available specifications [13]. In the following, the spun fiber will be cut in sections of 1 mm in order to accurately take into account the spinning of the fiber. The fiber is considered as untwisted so that *ρ _{f}* = 0 everywhere.

#### 4.2. Vibrations modelling

A vibration is modeled by a local variation of the SOP represented by a Jones matrix
$\overrightarrow{{J}_{P}}$ for the forward propagation and a Jones matrix
$\overleftarrow{{J}_{P}}$ for the backward propagation. We give to the perturbation matrix
$\overrightarrow{{J}_{P}}$ the form of a point effect that is represented by an equivalent combination of a local linear birefringence *δ _{P}* and a local reciprocal circular birefringence

*ρ*. The matrix $\overrightarrow{{J}_{P}}$ is of the same form as in Eq. (3) but with :

_{P}It can then be shown that the backward matrix
$\overleftarrow{{J}_{P}}$ is simply equal to the transpose of
$\overrightarrow{{J}_{P}}$ [9]. The random nature of the perturbations is taken into account by a random choice of the parameters *δ _{P}*,

*ρ*and

_{P}*q*. As an hypothesis, we considered that the statistical distributions of

_{P}*δ*and

_{P}*ρ*are uniform. An experimental set-up has been built (see section 5) to evaluate the range for the parameters

_{P}*δ*and

_{P}*ρ*.

_{P}Since the perturbation matrices in the simulation are random, a statistical analysis is necessary to study the resulting plasma current accuracy. This is performed by repeating the simulation several times with different randomly chosen perturbations.

## 5. Experimental investigation of vibration effects

The current FOCS design for ITER considers that the sensing fiber will be inserted in a stainless steel tube attached to the vacuum vessel. Therefore, to estimate a realistic range for the *δ _{P}* and

*ρ*parameters implemented in the simulations, a standard single mode (SMF) fiber of 2m was inserted in a metallic tube (similar to the tube that will be used in FOCS on ITER). The experimental set-up is depicted in Fig. 2. A shaker was used to induce a sinusoidal vibration on the tube. The voltage applied on the shaker was taken so that the acceleration on the tube is close to the value expected in ITER (∼ 12m/s

_{P}^{2}). During our experiments, we chose the lowest frequency (30 Hz) allowed by our measurement apparatus. A defined state of polarization is sent into the fiber. The output SOP is then analyzed using a fast inline polarimeter. The variation of the SOP at the fiber output measured by the polarimeter is given on the Poincare sphere in Fig. 3. The estimated interval for the

*δ*and

_{P}*ρ*parameters were taken such as each of these two parameters individually generates an SOP variation equal to the maximum variation obtained in the experience when inserting a matrix $\overrightarrow{{J}_{P}}$ after a 2 m section of fiber. By estimating the parameters in this way, we considered the worst case. This estimation of the simulation parameters gave the following ranges : values

_{P}*δ*∈ [0,

_{p}*π*/140] and

*ρ*∈ [−

_{p}*π*/140,

*π*/140] (by definition

*δ*must be positive, which is not the case for

_{p}*ρ*that can be either positive or negative). The simulation results presented in the following of the paper were obtained by considering spun and low-bi fibers. The extra-birefringence created by the vibrations will have a stronger effect (larger SOP variation) for fibers characterized by a smaller intrinsic birefringence. In the case of spun fibers, the local beat length is smaller than in SMFs (larger birefringence). Therefore, the

_{p}*δ*and

_{p}*ρ*values will be smaller for a spun fiber. Considering

_{p}*δ*∈ [0,

_{p}*π*/140] and

*ρ*∈ [−

_{p}*π*/140,

*π*/140] therefore constitutes a worst case analysis. When using a low-bi fiber, the intrinsic birefringence is smaller than that in a SMF so that

*δ*∈ [0,

_{p}*π*/140] and

*ρ*∈ [−

_{p}*π*/140,

*π*/140] underestimate the vibration effect. This issue is discussed in section 6.

## 6. Simulation results

The variation range of the polarisation state induced by the vibrations and relevant for the ITER operation has been determined in the previous section for a two meters fiber. This range is relatively small. In the modelling, we assume that the perturbations along the entire fiber can be simulated by inserting a perturbation matrix every two meters with *δ _{P}* within the [0,

*π*/140] range and

*ρ*within the [−

_{P}*π*/140,

*π*/140] range. The perturbations at different locations are considered as non-correlated. The simulation program is constructed as follows:

- In ITER, the fiber around the vacuum vessel has a D-shape (see Fig. 4). The projection of the magnetic field on a simplified D-shape path is calculated for each value of the plasma current and with the hypothesis that the current flows symmetrically around the toroidal axis. Thereafter the birefringence caused by the Faraday effect is determined. A comparison between the D-shape and a circular shape (like in TORE SUPRA reactor) has been performed. As shown later, the results in those two cases were found to be very similar.
- The perturbation matrices in the Jones formalism are then generated to be further inserted every two meters. For each of them, random values uniformly distributed between 0 (−
*π*/140) and*π*/140 (*π*/140) are chosen for*δ*(_{P}*ρ*). The parameter q is taken randomly in the interval [0,2_{P}*π*] and is considered equal for all the perturbation matrices along the fiber. This choice was made to take into account the worst case. This is indeed the worst case because when q vary, the perturbation matrices partially compensate each others due to the change of the sign in the imaginary part of the perturbation matrix (see Eq. (3)). - The entire set-up including a Faraday-mirror, two link fibers and a sensing fiber is considered (Fig. 4). The sensing fiber is divided in 14 sections of 2 meters (the total length of the sensing fiber is indeed 28 m). The total matrix describing the entire FOCS system is calculated by the multiplication of all the matrices constituting the optical link (Fig. 5):$${J}_{\mathit{TOTAL}}=\overrightarrow{{J}_{lf}}\overrightarrow{{J}_{S1}}\overrightarrow{{J}_{P1}}\overrightarrow{{J}_{S2}}\cdots \overrightarrow{{J}_{P13}}\overrightarrow{{J}_{S14}}\overrightarrow{{J}_{E}}{J}_{FM}\overleftarrow{{J}_{E}}\overleftarrow{{J}_{S14}}\overleftarrow{{J}_{P13}}\cdots \overleftarrow{{J}_{S2}}\overleftarrow{{J}_{P1}}\overleftarrow{{J}_{S1}}\overleftarrow{{J}_{lf}}$$where
*J*is the Jones matrix describing the link fiber between the interrogating device and the sensing fiber (link fiber 1),_{lf}*J*is the Jones matrix describing the 2m fiber section_{Si}*i*,*J*is the Jones matrix describing the vibration effects on section_{Pi}*i*and*J*is the Jones matrix describing the link fiber between the sensing fiber and the Faraday mirror._{E}*J*is the Jones matrix of the Faraday mirror given by [7]: The output SOP is then given by_{FM}*SOP*=_{output}*J*·_{TOTAL}*SOP*. The spun of the fiber was taken into account by linearly varying the parameter_{input}*q*as a function of the distance. The parameter*q*undergoes a rotation of 2*π*every spun period. The fiber was cut in sections of 1*mm*to accurately take into account the spinning. As a consequence, each matrix*J*consists of a multiplication of 2000 Jones matrices (each describing 1mm of the fiber)._{Si} - The output SOP is elliptical (Fig. 6). This is due to the effect of the linear birefringence of the fiber and of the vibrations. The orientation of the major axis is determined from the
*SOP*vector obtained in the previous step. The rotation angle_{output}*θ*between the orientation of the input SOP and the output SOP major (longest) axis (see Fig. 6) is then calculated. In the ideal case where there is no perturbation and no intrinsic birefringence, the angle_{a}*θ*is related to the angle_{a}*θ*defined in section 2 by ${\theta}_{a}=2{\theta}_{H}+\frac{\pi}{2}$. Using Eq. (2), the measured plasma current_{H}*I*is deduced from: In the non ideal case, the FOCS still uses Eq. (11) to estimate the plasma current. This results to a measurement error that can be calculated by simulations._{m} - Step 2 and 4 are repeated 1000 times to estimate the statistics of the measured plasma current.

Two simulation examples are presented in Figs. 7 and 8. To obtain Figs. 7 and 8, the steps 1 to 5 have been performed for a current of 1 and 8 MA, respectively. The different values obtained at the end of step 5 are then plot on a histogram in order to get the statistics of the reconstructed current. These figures show that the distribution is non symmetric and depends on the current, which is mostly attributed to the presence of the Faraday mirror. For each value of the current, it is possible to determine the mean error and the corresponding variance.

Figures 9 and 10 respectively show the mean relative and absolute errors obtained on the plasma current. The one standard deviation are also represented. The red lines on the figures represent the ITER requirements (see section 3). Figure 9 allows comparing the values obtained by simulations with the ITER requirements defined in the 0–1 MA region while Fig. 10 allows comparing within the 1–18MA region. The simulation results show that the requirements are fulfilled and that the vibrations generate errors acceptables for plasma current measurements in ITER.

As shown in Figs. 11 and 12, the ITER requirements are not fulfilled in the case of an unspun fiber although the value of *δ _{P}* and

*ρ*have been underestimated. They have indeed been taken equal to the values obtained for a SMF (see section 5). The real case is characterized by a stronger effect (stronger SOP variation) induced by the vibrations. Nevertheless, the simulation results showed that even with this underestimation, the low-bi fibers do not satisfy the ITER requirements. Consequently, taking into account more appropriate values for

_{P}*δ*and

_{P}*ρ*will not change this conclusion. To obtain these figures, a beat length of 360m has been chosen, which is the best value for commercially available low-birefringent fibers [7].

_{P}Two extra simulations were performed to verify that the shape of the fiber loop around the vessel and the distance between two successive perturbation matrices (so far equal to 2 meter) do not significantly affect the results.

For evaluating the influence of the number of perturbation matrices, simulations were performed with ten times more matrices (distance between two successive perturbation matrices set to 20 cm). The results are depicted in Figs. 13 and 14. In this case, the ranges for *δ _{P}* and

*ρ*in the perturbation matrices were taken such as the multiplication of the matrices along a 2 meter fiber generate a scatter plot on the Poincaré sphere that covers the range of variation obtained in the experiment (Fig. 15). This simulation showed that the generated errors is of the same order of magnitude compared to the results obtained for the 2 m case and the vibrations generate errors that are still acceptable for plasma current measurements in ITER. A simulation with perturbation matrices inserted every 2cm was also performed and shows results similar to the previous case where the perturbation matrices were inserted every 20 cm.

_{P}Figure 16 compares the errors observed for a circular and a D-shape vessel. The result shows that the error is only weakly dependent of the fiber shape. In the case of a D-shape vessel, the projection of the magnetic field along the fiber is not constant We therefore conclude that variations of the magnetic field projection do not significantly increase the errors due to vibrations on the plasma current measurement.

Finally, we investigated the influence of the correlation properties between the perturbation matrices. No correlation was considered so far. We performed an extra simulation taking into account the opposite case: a total correlation between the perturbations matrices. All the perturbations matrices along the fiber were therefore considered identical. The results are presented in Figs. 17 and 18. The standard deviation takes larger values in this case but the obtained plasma currents are still within the ITER requirements.

## 7. Conclusion

In this paper, we studied the influence of vibrations on the plasma current measurement performed by a FOCS system. The FOCS system is based on the detection of the rotation of the state of polarisation induced by the Faraday effect. The simulated FOCS system includes a Faraday mirror in order to reduce the effect of the birefringence on the measurements. The Jones matrix formalism was used to simulate the modifications of the SOP at the fiber output. The effect of vibrations was taken into account by inserting Jones matrices with random parameters to alter the state of polarization every 2 meters. The range of the Jones matrices parameters variation was found from an experimental investigation. The sensing fiber length were taken to correspond to the expected ITER implementation. Two loop shapes for the sensing fiber (circular and D-shape) have been considered to verify the low sensitivity of the sensor to the loop shape. The accuracy of measurements using standard and spun fibers has been simulated and compared to the ITER requirements. It has been found that these requirements are respected for available spun fibers. On the contrary, in the case of standard fibers, the FOCS performances are expected to not satisfy the requirements. Two others simulations were performed to verify that the results remain similar when the perturbation matrices are correlated and when their number is increased.

## Acknowledgments

This work is partly supported by Fusion for Energy under grant F4E-GRT-294 and by ITER IO under ITA C55TD32FE. The views expressed in this work are the sole responsibility of the authors and do not necessarily reflect the views of the Fusion for Energy and the ITER Organization. Neither Fusion for Energy nor any person acting on behalf of Fusion for Energy is responsible for the use, which might be made of the information in this publication. The authors thank the support of the COST action TD1001 OFSeSa, as well as the support of the Federal Belgian Government through the Federal Public Service Economy, SMEs, Self-Employed and Energy.

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