1. Introduction
In the International Thermonuclear Experimental Reactor (ITER) project, the plasma current is one of the most important parameters to be monitored for ensuring plasma stability and machine protection. For this measurement, inductive sensors have been utilized as a common method [
1]. However, these sensors are based on the measurement of the time derivate of a magnetic field, which can induce measurement drift due to the long steady state operation of the ITER since integrators are used to retrieve the plasma current [
2,
3]. A fiber optic current sensor (FOCS) measures the plasma current by means of the Faraday effect [
4,
5]. The current flowing inside the fiber loop is determined by measuring the rotation angle of the light polarization state without involving integration. Therefore, an FOCS, as a non-inductive sensor, is appropriate for monitoring long steady state plasma pulses and is planned to be installed in the ITER.
The sensing fiber of the FOCS will be installed on the external surface of the vacuum vessel and will be connected by optical fibers to the optical devices installed in the cubicle area for sensor operation and polarization state measurement. During plasma operation, sensor accuracy can be degraded due to the temperature changes and vibrations induced on the fibers [
6,
7]. These effects induce additional birefringence that changes the polarization properties of the fiber. It is well known that using a Faraday mirror can compensate for the unwanted reciprocal effect of the fibers [
8,
9]. However, this compensation is not perfect when the non-reciprocal Faraday effect coexists [
10,
11]. The influence of the linear birefringence can be drastically reduced by using spun fibers [
12,
13]. Since the Faraday effect also exhibits temperature dependence, an investigation was also conducted to determine whether the effect of temperature changes is significant. A previous study [
7] showed that a ratio of the intrinsic beat length (
) to the spun period (
) greater than 10 is required to satisfy the plasma current measurement accuracy required for the ITER when considering the temperature range undergone by the fibers. The effect of vibrations on the FOCS accuracy was also investigated when considering vibrations applied to the ITER’s vacuum vessel (VV) [
6]. However, with the ITER FOCS project’s progress, it became clear that a more realistic vibration analysis should also include vibrational properties of other parts of the reactor. In particular, it is essential to consider vibrations applied on the cryostat bridge along which the fibers are placed in a metal tube having a flexible helical shape.
In this paper, we investigate the FOCS accuracy changes due to the vibration-induced polarization perturbations caused by the presence of a helical structure. The vibration effects were analyzed by monitoring the polarization state change obtained when applying vibrations to a miniaturized bridge mock-up for both low- and high-birefringence spun fibers. Structural analysis performed at the ITER indicates that the maximum vibration acceleration and displacement were 76.81 m/s
2 and 16 mm, respectively, when pulse-like vibrations were applied in an accidental situation such as a seismic event [
14]. The ITER is currently under construction, and the first plasma shot is planned for 2025. However, considering the inevitable nature of the bridge vibration effects, it is very important to assess the performance of the FOCS in their presence in terms of satisfying the ITER’s required accuracy. Since the ITER is not yet operational, and as there is no other practical way of imitating the ITER environment, only a simulation approach can be undertaken. An optical model based on the Jones formalism, in which the experimental data were included, was developed to evaluate the FOCS accuracy when such vibrations were taken into account. The simulation results show that the vibration parameters affected the FOCS accuracy in different ways and that a spun fiber with a low intrinsic birefringence and a small spin pitch is required to fulfill the ITER specifications [
1].
2. FOCS Configuration for the ITER and Optical Modeling
Figure 1a shows the FOCS configuration to be installed in the ITER. A laser, a state of polarization (SOP) controller, an SOP analyzer, a fiber circulator, and a Faraday mirror (FM) are installed in the cubicle area. Light generated by the laser source passes the SOP controller and the circulator and propagates via the fiber bundle down the spun fiber (yellow cable in
Figure 1a) installed in the tokamak area. During the plasma operation, plasma current flows inside the VV, and the light propagating in the spun fiber around the VV undergoes an SOP rotation due to the Faraday effect induced by the magnetic field. This rotation is doubled because of the roundtrip propagation ensured by the Faraday mirror (FM) installed in the cubicle. The reflected light wave is directed into the SOP analyzer via the circulator. The SOP analyzer measures the polarization state of light and provides the corresponding Stokes parameters [
15], which allows computing the Faraday rotation angle.
In the tokamak area, a spun fiber is placed in two different regions: along the bridge structure and around the VV. The bridge structure subject to vibrations is placed between the cryostat wall and the VV, as shown in
Figure 1b. In this bridge structure, a spun fiber is placed inside five-turn metal tubes of a helical shape, which are attached under the bridge with flexible sticks. The number of turns was limited due to the limited space available on the bridge structure for installing the metal tube-fixing parts. Since the machine has a large temperature change from −180 to 200 °C, the bridge structure may have considerable thermal expansion or contraction. To prevent the internal fibers from breaking due to thermal deformation, the metal tube for fiber installation is designed in a helix shape. This allows the fibers to withstand any expansion or contraction that may occur. In the VV section, the spun fibers are placed inside a metal tube attached to the outside of the D-shaped VV. The FOCS measurement accuracy needs to be evaluated by analyzing the polarization state change of the lightwave passing through the spun fiber, whose modeling takes into account both the vibration effect in the bridge structure and the Faraday effect in the whole tokamak area, as shown in
Figure 1c. Because of the roundtrip propagation induced by the presence of the Faraday mirror, the Jones vector of the output polarization state (
, SOP at the bridge section input after roundtrip propagation; see
Figure 1c) during plasma operation can be expressed as follows:
where
is the SOP at the bridge section input,
is the forward Jones matrix of the spun fiber installed around the VV,
and
are the forward Jones matrices of the spun fiber installed in the 1st and 2nd bridge sections, respectively,
is the Jones matrix of the Faraday mirror.
,
, and
are the backward Jones matrices of the spun fiber installed in the bridge and VV sections, and
and
are the forward and backward fiber bundle Jones matrices, respectively. Thanks to the FM, both
and
can be neglected. Since the polarization properties are reciprocal in the fiber bundle section, the FM compensates for their effect [
8]. The equation can then be rewritten as
where
,
, and
correspond to a spun fiber section whose Jones matrix can be generally expressed as a retarder-rotator pair, yielding [
12]
where
l is the spun fiber length,
the retardance,
is the angle of the retarder’s fast eigenmode, and
is the rotation angle of the rotator. We have [
12]
where
,
is the intrinsic local linear birefringence of the spun fiber,
is the spin rate (radians per meter),
f is the Faraday effect-induced rotation angle per unit length,
is the initial orientation of the local slow axis of the fiber, and
and
are integers. The Jones matrix of each spun fiber section can be deduced from the expression of
, taking into account the spun fiber parameters (
,
l,
f, and
) as summarized in
Table 1 and explained in the next paragraph.
Since one long spun fiber is installed over the entire tokamak area without breaking, the initial orientation (
) for each section is equal to the accumulated spinning effect (
) from the previous spun fiber. Note that we assumed that there were no additional twists present. This is because the fibers being considered had a very short spinning period of 5 mm, and we believed that a small amount of twisting would not significantly affect the results. The spun fiber length to be installed in the bridge section (
) was 6 m, and the perimeter of the loop around the VV (
) was 28 m. The spin rate (
) in forward propagation was set to
. To define the Faraday effect-induced rotation (
f) for each section, the shape of the VV and bridge could be assumed to be a circle (comparison with a D shape showed only a small difference, as investigated in [
6]) and a straight line, as shown in
Figure 2.
Assuming the plasma current flows along an infinite straight line, the Faraday effect-induced rotation per unit of length along the spun fiber installed around the VV (
) is uniformly given by
where
is the plasma current,
V is the Verdet constant of the silica material (0.7 rad/MA at 1550 nm) [
16]. On the other hand, the Faraday effect-induced rotation along the bridge part (
) is not uniform and depends on the position in the bridge (
x), given by
where
r is the radius of the VV (
). With these spun fiber parameters, the Jones matrix of the spun fiber in the VV section (
) can be defined from the spun fiber model with its initial orientation (
) of
, length (
l) of
, and Faraday effect-induced rotation (
f) of
. On the other hand, since the magnetic field forming along the bridge is not uniform, the Jones matrices of the bridge section (
and
) can be expressed by cascading small-segment matrices:
where
n is the total number of segments and
and
are the
ith segment of spun fiber in the 1st and 2nd bridge sections, respectively. We set the number of segments (
n) in the bridge section to 1000 so that the magnetic field difference between the segments was small enough to avoid the 2
ambiguity that might arise from the arctangent calculation in Equation (
5). Choosing
also allowed considering the magnetic field constant along each segment of the bridge. Accordingly, in the
ith segment, the Faraday effect-induced rotation per unit of length in both bridge sections
and
is given by
where
is the segment length of the sliced spun fiber in the bridge section
. The Jones matrix models for the backward propagation (
,
, and
) can be defined in the same way. Let us note that for the backward direction, we used a coordinate system keeping the same
x and
y axes as for the forward case and reversed the
z axis [
10]. Under this convention, the Jones vector that represents the polarization state after reflection (backward propagation) is the same as the Jones vector of the polarization state incident to the reflection point (forward propagation). The initial rotation (
) is redefined in the same way by considering the accumulated spinning effect. The sign of the spin rate is reversed (
), but the sign of the Faraday rotation (
f) does not change, since the effect is non-reciprocal. In this coordinate system, the Jones matrix of the Faraday mirror is given by
where
is the rotation angle induced by the FM (i.e., 90°).
The metal tubes installed under the bridge are designed to be flexible in order to accommodate thermal expansion and contraction, which means that the vibration will easily be transmitted along the tubes. In this scenario, it can be difficult to accurately model the behavior of the vibration effect in the helix-shaped structure along the optical path. To address this, we used a Jones matrix of the vibration effect (
) to represent the accumulated polarization change that occurs when vibration is applied to the helix as measured in the experiment, which is described in
Section 4. The Jones matrix, which models the vibration effect, was inserted multiple times along the light path of the 1st and 2nd bridge sections (
,
,
, and
), as detailed in
Section 3 and
Section 4. We will consider that each (
) matrix has the form of a retarder-rotator pair given by
The values of the three parameters (
R,
, and
) defining
will be chosen according to the experiments, as described in
Section 5.
Finally, by calculating the current-induced rotation angle of the output SOP (
) from
in Equation (
1), the plasma current can be determined as follows [
16]:
3. Polarization State Measurement Set-Up for Vibration Effects
To evaluate the scale of vibration-induced SOP changes in a spun fiber, we prepared a one-turn, helix-shaped stainless-steel tube which represented a part of the periodic structure from the ITER design. A lo-bi spun fiber (SLB1250, Fibercore) was first inserted in the tube and spliced with two standard single-mode fiber (SMF) pigtails for light coupling. This lo-bi spun fiber had a spin period (
=
) of 5 mm. The linear beat length (
=
) of this spun fiber was not specified but was expected to be several meters, so it was assumed to be 1 m [
17]. Therefore, when the FOCS was configured with this fiber, we expected to meet the ITER specification thanks to its sufficiently large
/
value (≫10) [
18]. To apply vibrations on this structure, a shaker was placed in the middle of the metal tube, and an acceleration sensor was attached to monitor the applied acceleration. To monitor the SOP changes, a laser source (81940A, Agilent), an SOP controller (PSY-201, Luna), and an SOP analyzer (POD-201, Luna) were connected to the spun fiber, as shown in
Figure 3. The SOP controller was used to control the input polarization state. The polarization dependence of the vibration effect was investigated through measuring the change in the azimuth angle of the linear input SOP. However, because of the SMFs spliced at both ends of the spun fiber, the SOP at the spun fiber input was different from the SOP defined at the SOP controller, and the SOP analyzer would show the spun fiber’s output SOP with an unknown shift. This detrimental effect was taken into account in the analysis of the measurement data presented in
Section 4. The same measurement was also performed with a hi-bi spun fiber (SHB1250, Fibercore), which had a smaller beat length (
) of 9.6 mm and a spin period (
) of 4.8 mm. Though this hi-bi spun fiber’s
/
was so small (∼2) that it would not satisfy the ITER specifications, it was worth analyzing the vibration effect because hi-bi spun fibers are known to be less sensitive to external perturbations (e.g., bendings) due to their large linear intrinsic birefringence, owing to which sensing coils with small diameters are allowed [
19,
20].
Vibrations applied during the experiment corresponded to the worst-case scenario (such as a seismic event). The structural vibration analysis performed on the bridge structure of the ITER showed that the maximum predicted vibration was characterized by acceleration of 76.81 m/s
2 with a 16 mm displacement when a pulse-like signal (seismic event) was applied [
14]. Obviously, vibrations during normal operation are much lower than this value, but it is worth making conservative assumptions to predict the worst-case scenario. A frequency modal analysis of the structure design was also performed in [
14] thanks to the bridge modeling and showed that its frequency response was in the range of 10–30 Hz. Therefore, we assumed the maximum values of the acceleration, displacement, and frequency range to be 80 m/s
2, 20 mm, and 10–30 Hz, respectively. According to the ITER report, the maximum displacement of 16 mm can occur in the middle of the metal tube attached under the bridge structure. We therefore fixed both ends of the metal tube and applied vibration to the middle of the tube by using a shaker (TIRA S51075). In normal operation, the vibration waveform will be close to a sinusoidal signal. However, when a sinusoidal vibration of 80 m/s
2 and 20 mm was applied, the vibration frequency is fixed at 20 Hz because of the second derivative relationship between the acceleration and displacement. We therefore separately investigated the effect of displacement and acceleration by keeping constant one of them and changing the signal frequency. To control these vibration parameters, a sinusoidal voltage signal (obtained from a function generator) with a frequency ranging between 10 and 30 Hz was applied to the shaker via an electrical amplifier, and the applied vibration was monitored by an acceleration sensor attached to the shaker as shown in
Figure 4a.
Two different input conditions (look-up table) were prepared to find out which parameter (acceleration or displacement) played the most significant role in the SOP variation. First, we investigated the effect of displacement by applying a constant acceleration, and secondly, we investigated the effect of acceleration by applying a constant displacement, as shown in
Figure 4b and
Figure 4c, respectively, where the corresponding amplitudes of the sinusoidal signal applied to the electrical amplifier are displayed.
5. Optical Modeling of Vibrations and Calculation of the Measurement Accuracy
To evaluate the bridge vibration effect on the FOCS measurement accuracy by simulation, we inserted the vibration matrices in the Jones modeling of the bridge spun fiber sections. Since there were five turns of a helical tube in the bridge area, each Jones matrix of the bridge section (
,
,
, and
) was divided into five subsection matrices (
,
,
, and
,
):
Then, five vibration matrices (
and
,
) were inserted into the middle of each subsection of the bridge model as shown in
Figure 8.
The complete bridge models (
,
,
, and
), including the vibration effect could then be redefined with vibration matrices (
and
) by inserting each vibration matrix into the middle of each subsection model (
,
,
, and
,
):
where
n is the number of spun fiber segments for each subsection in the bridge section,
j is the subsection number and
,
,
, and
are the
ith segment of spun fiber matrix in the 1st and 2nd bridge section as defined in
Section 2. For each vibration matrix (
and
), the parameters (
R,
, and
) were randomly generated with a uniform distribution in experimentally determined vibration-induced SOP change. The angle of the optic axis (
) was picked up in the range of [
,
], and both linear (
R) and circular (
) global birefringence were picked up in the range of [
,
]. The vibration matrices in the backward propagation are denoted by
, which is equal to the transposed matrix of the forward vibration matrices (
).
With the complete Jones matrix modeling, including the vibration effect, the FOCS accuracy could be evaluated by the simulation of Equation (
1). The accuracy estimation was performed by using the Monte Carlo approach as in [
6]. Simulations were performed 1000 times for each spun fiber (lo-bi and hi-bi), and the vibration matrices were randomly redefined for each run. The mean FOCS measurement errors obtained for the lo-bi and hi-bi spun fibers are shown in
Figure 9.
The error bar corresponds to an interval given by
(twice the standard deviation of the measurement error from the result of 1000 simulations). Since the lo-bi spun fiber used in the simulations had a sufficiently high
/
(200), the estimated error for the lo-bi spun fiber case was maintained within
(inset of
Figure 9), satisfying the ITER specifications (dashed red curve [
18]). However, when using a hi-bi spun fiber having a small
/
(∼2), the mean error was higher (>1% at 1 MA) and no longer fulfilled the ITER requirements. Furthermore, the standard deviation of vibration-induced errors was also higher than that in the case of the lo-bi spun fiber, which gave rise to additional uncertainty in the FOCS measurement. Hence, when vibration-induced SOP variation is expected, it is helpful to use a lo-bi spun fiber having a high
/
value to reduce the error in the measurement [
21].