Abstract

We investigate the various contributions to the temperature dependence of an interferometric fiber-optic current sensor employing spun highly-birefringent sensing fiber, in particular, the contributions from the fiber retarder at the fiber coil entrance, the spun fiber’s birefringence, and the Faraday effect. We theoretically and experimentally demonstrate that an appropriately designed retarder inherently compensates the temperature dependence of the fiber birefringence and the Faraday effect. We demonstrate insensitivity to temperature to within ± 0.2% between −40 and + 85 °C. Furthermore, we analyze the influence of the retarder parameters on the linearity of the recovered magneto-optic phase shift vs. current and determine a set of parameters that results in a perfectly linear relationship.

© 2016 Optical Society of America

1. Introduction

Fiber-optic current sensors (FOCS) based on the Faraday effect in a coil of fused silica fiber have unique advantages over conventional inductive current transformers in electric power transmission systems. In particular, the weight and size of FOCS are dramatically reduced so that the sensors can be easily integrated into other primary equipment of high voltage substations such as circuit breakers. Furthermore, the current is measured with higher fidelity and larger bandwidth due to the absence of magnetic saturation (e.g., at fault currents). The digital output of FOCS is directly compatible with modern digital substation control and protection systems. Other benefits include inherent galvanic separation of electronics from high voltage and high safety of operation (e.g., no electrical safety hazards or risk of explosive failures). Fiber-optic current sensors have been under investigation since the 1970s [1,2]. A particularly robust sensor configuration which has paved the way towards commercial applications employs a reflective fiber-coil and a non-reciprocal phase modulation interrogation scheme adopted from fiber-optic gyroscopes [3–5]. First commercial applications of fiber-optic current sensors have included the measurement of high direct current in the electro-winning industry (for example aluminum smelters) [6] where the sensors have gained significant market share since their introduction in 2005.

The application of fiber-optic current sensors in electrical power transmission as well as industrial process control demands high measurement accuracy (often within ± 0.2% or even ± 0.1%) over a wide temperature range, e.g., from −40 °C to + 85 °C. For comparison, the Faraday effect (Verdet constant) in silica fiber varies by almost 1% over this range [5]. Since conventional temperature measurements are not easily implemented on high voltage potential, methods of inherent temperature compensation are preferred whereby different temperature-dependent contributions to the sensor signal cancel each other. We have shown earlier for coils from low-birefringent fiber that the fiber retarder at the coil entrance which generates the left and right circularly polarized light waves in the sensing fiber coil can be prepared such that it compensates the temperature dependence of the Faraday effect. The method was demonstrated for both, coils with negligible bend-induced birefringence (thermally annealed coils and fiber loops of large diameter [5,6]) and coils with moderate amounts of bend-induced birefringence [7,8]. The fiber coils were operated in an interferometric sensor configuration (Sagnac or in-line interferometers) with interrogation based on non-reciprocal phase modulation.

In this work, we extend the method of inherent temperature compensation to spun highly birefringent fiber coils [9–17]. Spun birefringent fiber is produced by spinning the preform of a linearly birefringent fiber in the drawing process so that the principal fiber axes rotate along the fiber. Such a fiber is elliptically birefringent, i.e., the fiber has elliptically polarized eigenmodes. Depending on the parameters of the spun fiber, these modes can be very close to circularly polarized modes. The elliptical birefringence quenches to a large extent disturbing linear birefringence from external sources, while maintaining the fiber’s sensitivity to the Faraday effect. As a result, the fiber coils are largely insensitive to bend-induced birefringence or birefringence due to stress from the fiber coating and other packaging related stress. This is a significant advantage over coils from low birefringent fiber where sophisticated packaging methods are needed in order to attain stable operation over extended temperature ranges and bend-induced birefringence typically limits the number of fiber loops to a few [5, 6]. However, the birefringence of a spun highly birefringent fiber commonly varies with temperature, which can result in a significant extra temperature dependence of the sensor, in addition to the temperature dependence of the Faraday effect.

In the following, we theoretically and experimentally investigate the temperature dependence of reflective spun fiber coils in an interferometric current sensor with non-reciprocal phase modulation. We determine how the spun fiber parameters, in particular, the embedded linear birefringence (linear beat length), the spin pitch, and the temperature dependence of the beat length, affect the sensor scale factor as a function of temperature. We show how the parameters of the fiber retarder at the fiber coil entrance (retardation and orientation with respect to the spun fiber axes) must be chosen for a temperature-insensitive sensor signal. In particular, we show that there are sets of retarder parameters for which the temperature dependence of the beat length does not enter into the sensor signal. In the regime of small magneto-optic phase shifts the sensing coil then behaves like an ideal coil free of any linear birefringence. It turns out that the apparent magneto-optic phase shift of a temperature-compensated sensor increases slightly nonlinearly with the applied current. We also determine sensor parameters that result in a perfectly linear relationship, however, at the expense of some remaining temperature dependence.

2. Sensor configuration

The sensor configuration is depicted in Fig. 1(a) [3,5]. The broadband output from a superluminescent light emitting diode (SLED, center wavelength λ ≈1310 nm, full width at half maximum Δλ ≈40 nm) is polarized and then excites, at a 45°-splice, the two orthogonal polarization states of a polarization maintaining (PM) fiber with equal amplitudes. In the basic configuration, a quarter-wave retarder (QWR) consisting of a piece of elliptical-core fiber (beat length around 6 mm) at the entrance to the sensing fiber coil converts the orthogonal linear polarization states into left and right circularly polarized light waves. The two light waves are reflected by a metallic coating on the far end of the fiber coil, retrace the optical path with swapped polarization states (the polarization swapping is a combined effect of reflection and double passage through the QWR), and finally interfere at the fiber polarizer. During their round trip through the fiber coil, the circularly polarized light waves accumulate a magneto-optic phase shift Δϕ given by

Δϕ=4NVI.
Here, N is the number of loops of the fiber coil, V the Verdet constant of the sensing fiber (1.0 × 10−6 rad/A at 1310 nm), and I the current. Equation (1) is valid for an ideal sensor, i.e., a sensing fiber free of linear birefringence (e.g., a thermally annealed fiber coil [5]) and a perfect quarter-wave retarder. The magneto-optic phase is retrieved in a closed-loop non-reciprocal phase modulation circuit, adopted from fiber-optic gyroscopes [18], which includes an integrated-optic lithium niobate birefringence modulator. Note that as a result of the polarization swapping the differential roundtrip group delay that the two polarization modes accumulate in the PM fiber is zero. Further note that changes in insertion loss, e.g., with temperature, or mirror reflectivity do not affect the retrieved magneto-optic phase shift.

 

Fig. 1 (a) Sensor configuration, SLED: superluminescent light emitting diode, PD: photodiode. (b) Relative orientation of slow axes of PM lead fiber, retarder, and spun fiber.

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The evolution of the polarization states of the two light waves during their roundtrip through the ideal sensor can be described by

VLP(HLP)QWRRCP(LCP)mirrorLCP(RCP)QWRHLP(VLP).
HLP and VLP denote the linearly polarized light states with polarization directions parallel to the fast or slow axis of the PM fiber lead [here, the two axis directions are referred to as horizontal (H) and vertical (V)], and RCP and LCP denote the right and left circularly polarized light states in the sensing fiber.

The temperature dependence of the retardation ρ of the fiber retarder can be employed to compensate for the temperature dependence of the Verdet constant, (1/V)·dV/dT = 0.7 × 10−4 °C−1 [5]. To this end, the retarder is detuned by an amount ε from quarter-wave retardation so that the light waves in the sensing fiber assume elliptical polarization states. The elliptical states can be described as superposition of left and right circularly polarized light waves with amplitudes that are modified, compared to the case of a perfect QWR, by terms cos (ε/2) and sin (ε/2):

VLPretardercosε2RCP+isinε2LCP,HLPretardercosε2LCPisinε2RCP.
Accordingly, two pairs of left and right circularly polarized light are travelling back and forth in the sensing fiber. The retarder converts these two pairs returning from the coil into four pairs of orthogonal linearly polarized light waves in the PM fiber that finally interfere at the polarizer. Accordingly, the evolution of light polarization in case of a retarder different from an exact QWR deviates from the ideal case described by Eq. (2). In the following, the light states included in Eq. (2) are denoted as primary whereas additional light states that do not appear in Eq. (2) are denoted as secondary states [e.g., the light states weighted with sin (ε/2) in Eq. (3)]. Compared to the case of a true quarter-wave retarder, the apparent magneto-optic phase shift for 4NVI << 1 is enhanced by a factor 1/cos ε [4, 5]:
Δϕ=4NVI/cosε.
The enhancement arises from the fact that the two coherently interfering pairs of primary and secondary orthogonal waves have a relative phase difference of 180° and experienced differential magneto-optic phase shifts of opposite sign [5]. (Note: Two of the three pairs of secondary waves return with unswapped polarization directions and give rise to incoherent background light that does not contribute to the signal). For a given temperature dependence of the retardation ρ, a proper choice of ρ = 90° + ε allows balancing the temperature dependence of the Faraday effect and of other potential contributions with the temperature dependent contribution of ρ [5].

3. Sensor employing spun highly birefringent fiber

A spun highly birefringent fiber is characterized by the embedded linear beat length LLB and the spin pitch length p. The spin ratio is defined as x = 2 LLB/p. For spun fibers that are appropriate for current sensing, the linear beat length exceeds the spin pitch [9]. The optical circuit including the spun fiber can be theoretically investigated using the Jones matrix formalism. Neglecting bend-induced birefringence (which is justified for the coil diameter of 170 mm used below), the analytical expression of the Jones matrix of a spun fiber is obtained by integration of the corresponding coupled mode equation [19].

In the absence of a magnetic field, the eigenmodes of a spun fiber, defined in the reference frame of local birefringence axes rotating along the spun fiber, are denoted as RCP* and LCP*, and are here represented as a (not normalized) superposition of left and rightly circular polarized waves:

RCP*RCP+LLBLEBe2iα0LCP,LCP*LCPLLBLEBe+2iα0RCP.,
The quantity α0 represents the angle between the slow axis of the fiber retarder and the slow spun fiber axis at the fiber’s front facet and is positive in the rotation direction of the spun fiber’s principal axes [see Fig. 1(b)]. The denominations RCP* and LCP* are to express that these states are close to circular light polarization for x > 1, i.e., if (LLB/LEB) = 1+x2xis small; LEB is commonly referred to as elliptical beat length of the spun fiber [9, 13]. A short elliptical beat length favors the preservation of the elliptical modes under external perturbations, similarly to a short linear beat length favoring the preservation of the linear polarization modes in an unspun PM fiber [13].

During their propagation through the spun sensing fiber coil, the eigenmodes of the spun fiber experience a mode dispersion per unit length of λ/(cLEB) due to the birefringence of the spun fiber [10]. On the return path, the light states reflected by the end mirror with swapped polarizations do no longer exactly match the eigenmodes of the Jones matrix of the spun fiber [because of the sign change in Eq. (5)]. Accordingly, secondary light states are created upon reflection for which the spun fiber’s polarization mode dispersion is not compensated on the return path, but doubles. Hence, for short sensing fiber lengths (modal group delay smaller or comparable to the coherence length of the light source), the sensor signal is overlaid by signal oscillations as a function of temperature (due to the temperature dependence of the birefringence) or the fiber length; the normalized amplitude of these unwanted signal oscillations calculates to

K=x2expπ24ln2(2lΔλLEBλ)2.
In the case of a broadband light source and sufficiently long spun fiber (group delay significantly exceeding the coherence length of the light source), signal oscillations with temperature disappear [9, 13] and those secondary light states become apparent as an incoherent background in the detector signal and a reduced fringe visibility compared to the ideal case described by Eq. (2). Also, the signal becomes independent of the exact fiber length (as long as the number of fiber windings N is kept constant).

A retarder orientated at α0 = ± 45° with a retardation deviating by ε = cot−1 x from perfect quarter-wave retardation maps the incoming orthogonal linear polarization modes of the PM fiber one-to-one onto the elliptical eigenmodes of the spun fiber since in this case itanε2=±LLB/LEBe2iα0 [see Eqs. (3) and (5)]:

VLP(HLP)α0=±45°,ε=cot-1xRCP*(LCP*).
Accordingly, this configuration resembles the case of a non-birefringent sensing fiber with exact quarter-wave retardation described by Eq. (2). For all other choices of α0 and ε, also secondary light states, accumulating opposite magneto-optic phase shift, are created upon injection of light from the retarder into the spun sensing fiber, resulting in an additionally reduced interferometer visibility and an enhancement of the apparent magneto-optic phase shift (according to the discussion in Section 2). In contrast, retarder parameters made according to Eq. (7) result in the highest fringe visibility and lowest apparent magneto-optic phase shift.

We calculate the Jones matrix for the entire sensor system with non-reciprocal phase modulation (using the Jones matrices for spun fiber from [19]) to derive an expression for the modulated detector signal. The apparent magneto-optic phase shift Δϕ can be retrieved from the detector signal as shown in [5, 18]. We find:

Δϕ=4NVI1+[sin(2α0)/x]2cos(ε+tan1[sin(2α0)/x]).

Here, it is assumed that the Faraday phase shift is small (4NVI << 1) and the fiber is sufficiently long to prevent the mentioned sensitivity oscillations (l >> LLBλ/Δλ). Apart from Section 5 where also high magneto-optic phase shifts are considered, all calculations and experiments are conducted with parameters such that these two assumptions are valid.

For the special retarder orientations α0 = ± 45°, Eq. (8) reduces to:

Δϕα0=±45°=4NVI1+x2cos(ε±cot-1x).
For α0 = 0°, 90°, Eq. (8) becomes:

Δϕα0=0°,90°=4NVI/cosε.

Note that, in this case, the detected magneto-optic phase shift is the same as in case of a sensor with ideal non-birefringent sensing fiber [Eq. (4)]. In particular, it is interesting to note that for α0 = 0°, 90° the sensor signal is not affected by the variation of the linear beat length LLB with temperature. Still, in this case, in contrast to the case of a low-birefringent sensing fiber described by Eq. (4), there are also secondary states excited for ε = 0° and the interferometer visibility is reduced with respect to the ideal case.

Figure 2 shows the normalized signal, i.e., the apparent magneto-optic phase shift Δϕ relative to the phase shift of an ideal sensor according to Eq. (1) as a function of the retarder length for retarder orientations of α0 = 0° and ± 45°. The assumed fiber parameters are given in Table 1 (fiber A). For α0 = ± 45°, the minima of the parabola are reduced by a factor of 1+x2 and are shifted in the horizontal direction by ± cot−1x compared to the cases of α0 = 0°, 90° [or the case of a non-birefringent fiber, Eq. (4)]. A retardation of 90° ± cot−1x yields injection of pure eigenmodes into the fiber as discussed above [see Eqs. (3), (5), and (7)].

 

Fig. 2 Calculated normalized magneto-optic phase shift vs. deviation from quarter-wave retardation for different choices of the spun fiber’s angular orientation (fiber A, see Table 1).

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Tables Icon

Table 1. Linear Beat Length and Spin Pitch of the Spun Highly Birefringent Fibers Used for This Work

4. Inherent temperature compensation for current sensors employing spun fiber

From Eq. (8), it is obvious that the retarder’s contribution to the sensor’s overall temperature dependence can be controlled by appropriately adjusting α0 and ε. Further temperature dependent contributions stem from the Verdet constant and from the spun fiber’s birefringence. The contribution of the birefringence itself depends on the choice of α0 and ε [see Eq. (8)]. Figure 3(a) shows for a temperature compensated sensor the calculated individual temperature contributions from the retarder, the Verdet constant, and the spun fiber’s birefringence as well as their sum. With α0 = 45°, beat length and pitch according to fiber A (see Table 1), and the measured temperature coefficients of the retarder and the fiber birefringence (/(ρdT) = −3.2 × 10−4 °C−1 and dLLB/(LLBdT) = 1.0 × 10−3 °C−1), the room temperature retardation ρRT must be set to ρRT = 107.2° (ε = 17.2°) in order to achieve temperature compensation. Figure 3(b) depicts the proper choice of ε for temperature compensation as a function of the spinning ratio x for the cases of α0 = 0°, ± 45°, and 90°. Note that the solution for α0 = 0°, 90° (black line) is independent of the spinning ratio since the birefringence of the spun fiber does not contribute to the overall temperature dependence.

 

Fig. 3 (a) Calculated normalized magneto-optic phase shift vs. temperature for a temperature-compensated sensor (black). The other lines depict the individual contributions from the Verdet constant (green), the spun fiber’s birefringence (orange), and the fiber retarder (blue), see text for parameters. (b) Retarder detuning ε required for temperature compensation vs. spin ratio [p = 2.9 mm].

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The experimental data of Fig. 4 confirms the relationship of Eq. (8). Figure 4(a) depicts for spun fiber A the contribution of the retarder to the temperature dependence of the sensor signal, i.e., the term d(lnΔϕ)/dTretarder as a function of the orientation angle α0 of a straight spun fiber (red squares). Qualitatively, we observe that the orientation of the spun fiber with respect to the retarder is an important sensor parameter. The spun fiber orientation has a significant influence of the sensor’s temperature dependence and we can even demonstrate a sign change of d(lnΔϕ)/dTretarder. By contrast, in case of low-birefringent sensing fiber, the retarder orientation has no influence for a straight fiber (and only must be taken into account if the fiber is bent to a coil [6]). In addition, the data is in good agreement with numerical results (black line). The theoretical curve was calculated with the fiber parameters given in Table 1 (fiber A), a retarder detuning angle ε(20 °C) = −2.5°, and a retarder temperature dependence of /(ρdT) = −1.6 × 10−4 °C−1. Here, ε and /(ρdT) were treated as adjustable parameters and, as only relative changes of α0 were experimentally accessible, the absolute value of α0 was adjusted to the theoretical expectations by adding an offset. (In this particular experiment, there was a short section of low-birefringent fiber spliced to the retarder to manufacture the retarder as described in [5]. The same retarder could then be used for different spun fiber orientations. However, this hindered knowledge of absolute values for α0 and also experimental determination of ε with an accuracy better than within a few degrees.) The retrieved value for /(ρdT) is a factor of 2 smaller than measured for long undisturbed sections of the same elliptical-core fiber (−3.2 × 10−4 °C−1). In part, the deviation may be explained by modifications of the retarder fiber properties caused by the splice procedure.

 

Fig. 4 (a) Contribution of the retarder, d(lnΔϕ) /dTretarder, to the temperature dependence of the sensor signal as a function of the spun fiber orientation angle α0, experimental: red squares, calculated: black line. (b) Normalized sensor signal as a function of the temperature of the retarder (sensing fiber at room temperature, blue squares) and of the temperature of the entire fiber coil (retarder and sensing fiber, black data points); the red line is a linear fit (sensor parameters: fiber type B, α0 = 0°, ε = 0°). (c) Same as in (b), but with ε ≈7.5° so that sensor signal is temperature compensated to within < ± 0.2%.

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We have fabricated temperature-insensitive current sensors with various types of spun fiber and different choices of the fiber orientation α0. Note that there is a small uncertainty of α0 introduced by aligning the spun fiber with respect to the retarder: While the PM fiber orientation angle and thus the retarder orientation can be precisely determined before splicing by means of polarizers, the angular orientation of the spun fiber is set by eye using an end view of the fiber’s facet. The estimated error is within ± 2.5° for elliptical-core spun fiber and less for spun fibers with stress bodies such as bow-tie spun fiber. This uncertainty of α0 results in turn in an uncertainty of the retarder contribution to the overall temperature-dependence of the sensor and, accordingly, in an uncertainty of the optimum retarder length for temperature compensation. It is obvious from Fig. 4(a) that errors in α0 are least critical at α0 = ± 45°. In practice, accuracy of angular orientation within ± 2.5° is sufficient for the targeted sensor accuracy at all angles α0, though. In the following, typical results are shown for a spun elliptical-core fiber (fiber B in Table 1) and α0 = 0°. According to Eq. (10), the temperature sensitivity for this sensor originates from the Verdet constant and the retarder, but not from the spun fiber’s birefringence. Figure 4(b) depicts the signal versus temperature of an uncompensated sensor with an exact quarter-wave retarder (ε = 0, black data points), showing a slope of dΔϕ /(Δϕ dT) ≈0.7 × 10−4 °C−1 (fit, red line) in excellent agreement with values reported for the temperature dependence of the Verdet constant in fused silica fiber [6]. The blue squares in Fig. 4(b) show the sensor signal vs. the retarder temperature (with the spun fiber kept at room temperature). It is obvious that, as expected for ε = 0, the retarder does not contribute to the temperature dependence. Figure 4(c) shows corresponding results for a temperature compensated sensor (ε ≈7.5°). The retarder contribution (experiment: blue squares, fit: red line) is now opposite to that of the Verdet constant and as a result the sensor signal becomes insensitive to temperature within < ± 0.2% from −40 to + 85 °C (black data points).

5. Sensor response at large magneto-optic phase shifts

It is known that the apparent magneto-optic phase shift of a sensor using low-birefringent sensing fiber and a retarder that deviates from perfect quarter-wave retardation (e.g., for the purpose of temperature compensation) exhibits some nonlinearity as a function of current [6–8]. In addition to the retarder, bend-induced birefringence in the fiber coil also influences the signal vs. current relationship. The nonlinearity is again the result of the interference of primary and secondary pairs of orthogonal light waves as described in Section 2. Similar effects also occur in sensors employing spun fiber.

In the remainder of the paper, we theoretically and experimentally examine the nonlinearity of sensors with spun fiber coils and show that the retarder can be prepared in a way that non-linearity can be avoided. To experimentally examine magneto-optic phase shifts up to ± π, a sensor was fabricated with 385 spun fiber windings with a diameter of 165 mm (fiber C in Table 1). The total fiber length was thus about 200 m. The retarder was set so that the signal was temperature compensated at low magneto-optic phase shifts (Δϕ << 1, α0 = 0°). Figure 5(a) shows the measured signal as a function of current, i.e., vs. the terms 4NVI and NI, for coil temperatures of −20 °C (black squares) and 70 °C (red squares). The solid lines are calculated assuming a retardation of the fiber retarder of ρ = 96° [ε = 6°, /(ρdT) = −3.2 × 10−4 °C−1], the fiber parameters from Table 1 [dLLB/(LLBdT) = 1.0 × 10−3 °C−1], cross coupling of 26 dB at splices in the PM fiber lead (cross-coupling at PM fiber splices also leads to the generation of secondary light states in the sensing fiber), and a Verdet constant of V = 1.055 μradA−1 in good agreement with literature values [20] with temperature dependence of dV /(VdT) = 0.7 × 10−4 °C−1. At a magneto-optic phase shift of 180°, the signal deviates from a linear relationship by several percent. Note also that the temperature compensation works only in the regime of small magneto-optic phase shifts. This is however not a limitation for most practical applications since for high-voltage substation applications the sensors are prepared such that the rated currents correspond to the low current regime of the sensor. The high current range is reserved for short-time fault currents. Fault currents can be 1-2 orders of magnitude higher than the rated currents but need to be measured only with relatively moderate accuracy.

 

Fig. 5 (a) Normalized sensor signal as a function of applied current for a temperature-compensated sensor with 385 fiber windings at temperatures of −20 °C (black squares) and + 70 °C (red squares), the lines represent corresponding theoretical data. (b) Normalized sensor signal as a function of applied current for ε = 0° (black line) and α0 = ± 45° and ε = ± cot−1 x (red line).

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The non-linearity in the signal can be eliminated with a retarder that maps the incoming pair of orthogonal linearly polarized waves one-to-one on the left and right elliptical eigen-modes of the spun fiber. As argued in Section 3, this can be achieved by setting the spun fiber’s angular orientation to α0 = ± 45° and the retarder to 90° cot−1 x. Figure 5(b) depicts for this case the calculated normalized sensor signal Δϕ/(4NVI) (red line, parameters of fiber A, Table 1) showing negligibly small non-linearity. For comparison, the calculation was also conducted for a perfect quarter-wave retarder (ε = 0°, black line) which does result in a nonlinear signal due to the generation of secondary light states, contrary to the case of a non-birefringent sensing fiber with such a retarder. It should be noted that a linear sensor output excludes inherent temperature compensation and requires an extra temperature measurement if compensation is necessary. Note that, at an apparent magneto-optic phase shift of 180° × 1+x2 = 1.04 × 180° where the black and red lines cross in Fig. 5(b), the coherent primary and secondary light states of each polarization are in phase. The signal is then the same for all choices of α0 and ε.

6. Conclusion

In conclusion, we have experimentally and theoretically demonstrated that the various contributions to the temperature dependence of an interferometric current sensor with spun highly birefringence fiber can be balanced such that the sensor signal becomes independent of temperature. A residual variation of the sensor signal as a function of fiber coil temperature of less than ± 0.2% was achieved between −40 and + 85 °C in the regime of small magneto-optic phase shifts (this range is the normal operating range of the sensor and relevant for metering applications whereas the range of large phase shifts is reserved for fault currents; typically, measurement of the latter requires only accuracy to within ± 5%). Hence, the sensor can meet the stringent accuracy requirements of electric power transmission systems without the need of an extra temperature sensor on high voltage. Furthermore, we have determined design rules for sensors with spun fiber that result in a perfectly linear signal as a function of current.

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References

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  1. A. M. Smith, “Polarization and magnetooptic properties of single-mode optical fiber,” Appl. Opt. 17(1), 52–56 (1978).
    [Crossref] [PubMed]
  2. A. Papp and H. Harms, “Magnetooptical current transformer. 1: Principles,” Appl. Opt. 19(22), 3729–3734 (1980).
    [Crossref] [PubMed]
  3. G. Frosio and R. Dändliker, “Reciprocal reflection interferometer for a fiber-optic Faraday current sensor,” Appl. Opt. 33(25), 6111–6122 (1994).
    [Crossref] [PubMed]
  4. S. X. Short, J. U. de Arruda, A. A. Tselikov, and J. N. Blake, “Elimination of birefringence induced scale factor errors in the in-line Sagnac interferometer current sensor,” J. Lightwave Technol. 16(10), 1844–1850 (1998).
    [Crossref]
  5. K. Bohnert, P. Gabus, J. Nehring, and H. Brändle, “Temperature and vibration insensitive fiber-optic current sensor,” J. Lightwave Technol. 20(2), 267–276 (2002).
    [Crossref]
  6. K. Bohnert, P. Gabus, J. Nehring, H. Brändle, and M. G. Brunzel, “Fiber-optic current sensor for electrowinning of metals,” J. Lightwave Technol. 25(11), 3602–3609 (2007).
    [Crossref]
  7. G. M. Müller, A. Frank, M. Lenner, K. Bohnert, P. Gabus, and B. Guelenaltin, “Temperature compensation of fiber-optic current sensors in different regimes of operation,” in Proceedings of Photonics Conference (IEEE, 2012), 745–746.
    [Crossref]
  8. R. Wüest, A. Frank, S. Wiesendanger, P. Gabus, U. E. Meier, J. Nehring, and K. Bohnert, “Influence of residual fiber birefringence and temperature on the high-current performance of an interferometric fiber-optic current sensor,” Proc. SPIE 7356, 73560K (2009).
    [Crossref]
  9. R. I. Laming and D. N. Payne, “Electric current sensors employing spun highly birefringent optical fibers,” J. Lightwave Technol. 7(12), 2084–2094 (1989).
    [Crossref]
  10. A. J. Barlow, J. J. Ramskov-Hansen, and D. N. Payne, “Birefringence and polarization mode-dispersion in spun single-mode fibers,” Appl. Opt. 20(17), 2962–2968 (1981).
    [Crossref] [PubMed]
  11. L. Li, J. R. Qian, and D. N. Payne, “Current sensors using highly birefringent bow-tie fibres,” Electron. Lett. 22(21), 1142–1144 (1986).
    [Crossref]
  12. I. G. Clarke, “Temperature-stable spun elliptical-core optical-fiber current transducer,” Opt. Lett. 18(2), 158–160 (1993).
    [Crossref] [PubMed]
  13. Q. Guo, J. R. Qian, and L. Li, “Spun linear birefringence fibres and their sensing mechanism in current sensors with temperature compensation,” IEE Proc., Optoelectron. 141(6), 373–380 (1994).
    [Crossref]
  14. P. Polynkin and J. Blake, “Polarization evolution in bent spun fiber,” J. Lightwave Technol. 23(11), 3815–3820 (2005).
    [Crossref]
  15. V. P. Gubin, V. A. Isaev, S. K. Morshnev, A. I. Sazonov, N. I. Starostin, Y. K. Chamorovsky, and A. I. Oussov, “Use of Spun optical fibres in current sensors,” Quantum Electron. 36(3), 287–291 (2006).
    [Crossref]
  16. Y. K. Chamorovskiy, N. I. Starostin, M. V. Ryabko, A. I. Sazonov, S. K. Morshnev, V. P. Gubin, I. L. Vorob’ev, and S. A. Nikitov, “Miniature microstructured fiber coil with high magneto-optical sensitivity,” Opt. Commun. 282(23), 4618–4621 (2009).
    [Crossref]
  17. M. Aerssens, F. Descamps, A. Gusarov, P. Mégret, P. Moreau, and M. Wuilpart, “Influence of the optical fiber type on the performances of fiber-optics current sensor dedicated to plasma current measurement in ITER,” Appl. Opt. 54(19), 5983–5991 (2015).
    [Crossref] [PubMed]
  18. H. Lefèvre, The Fiber-Optic Gyroscope (Artech House, 1993).
  19. R. Dändliker, “Rotational Effects of Polarization in Optical Fibers,” in Anisotropic and Nonlinear Optical Waveguides, C. G. Someda and G. I. Stegeman, eds. (Elsevier, Amsterdam, 1992).
  20. A. H. Rose, S. M. Etzel, and C. M. Wang, “Verdet Constant Dispersion in Annealed Optical Fiber Current Sensors,” J. Lightwave Technol. 15(5), 803–807 (1997).
    [Crossref]

2015 (1)

2009 (2)

Y. K. Chamorovskiy, N. I. Starostin, M. V. Ryabko, A. I. Sazonov, S. K. Morshnev, V. P. Gubin, I. L. Vorob’ev, and S. A. Nikitov, “Miniature microstructured fiber coil with high magneto-optical sensitivity,” Opt. Commun. 282(23), 4618–4621 (2009).
[Crossref]

R. Wüest, A. Frank, S. Wiesendanger, P. Gabus, U. E. Meier, J. Nehring, and K. Bohnert, “Influence of residual fiber birefringence and temperature on the high-current performance of an interferometric fiber-optic current sensor,” Proc. SPIE 7356, 73560K (2009).
[Crossref]

2007 (1)

2006 (1)

V. P. Gubin, V. A. Isaev, S. K. Morshnev, A. I. Sazonov, N. I. Starostin, Y. K. Chamorovsky, and A. I. Oussov, “Use of Spun optical fibres in current sensors,” Quantum Electron. 36(3), 287–291 (2006).
[Crossref]

2005 (1)

2002 (1)

1998 (1)

1997 (1)

A. H. Rose, S. M. Etzel, and C. M. Wang, “Verdet Constant Dispersion in Annealed Optical Fiber Current Sensors,” J. Lightwave Technol. 15(5), 803–807 (1997).
[Crossref]

1994 (2)

Q. Guo, J. R. Qian, and L. Li, “Spun linear birefringence fibres and their sensing mechanism in current sensors with temperature compensation,” IEE Proc., Optoelectron. 141(6), 373–380 (1994).
[Crossref]

G. Frosio and R. Dändliker, “Reciprocal reflection interferometer for a fiber-optic Faraday current sensor,” Appl. Opt. 33(25), 6111–6122 (1994).
[Crossref] [PubMed]

1993 (1)

1989 (1)

R. I. Laming and D. N. Payne, “Electric current sensors employing spun highly birefringent optical fibers,” J. Lightwave Technol. 7(12), 2084–2094 (1989).
[Crossref]

1986 (1)

L. Li, J. R. Qian, and D. N. Payne, “Current sensors using highly birefringent bow-tie fibres,” Electron. Lett. 22(21), 1142–1144 (1986).
[Crossref]

1981 (1)

1980 (1)

1978 (1)

Aerssens, M.

Barlow, A. J.

Blake, J.

Blake, J. N.

Bohnert, K.

R. Wüest, A. Frank, S. Wiesendanger, P. Gabus, U. E. Meier, J. Nehring, and K. Bohnert, “Influence of residual fiber birefringence and temperature on the high-current performance of an interferometric fiber-optic current sensor,” Proc. SPIE 7356, 73560K (2009).
[Crossref]

K. Bohnert, P. Gabus, J. Nehring, H. Brändle, and M. G. Brunzel, “Fiber-optic current sensor for electrowinning of metals,” J. Lightwave Technol. 25(11), 3602–3609 (2007).
[Crossref]

K. Bohnert, P. Gabus, J. Nehring, and H. Brändle, “Temperature and vibration insensitive fiber-optic current sensor,” J. Lightwave Technol. 20(2), 267–276 (2002).
[Crossref]

G. M. Müller, A. Frank, M. Lenner, K. Bohnert, P. Gabus, and B. Guelenaltin, “Temperature compensation of fiber-optic current sensors in different regimes of operation,” in Proceedings of Photonics Conference (IEEE, 2012), 745–746.
[Crossref]

Brändle, H.

Brunzel, M. G.

Chamorovskiy, Y. K.

Y. K. Chamorovskiy, N. I. Starostin, M. V. Ryabko, A. I. Sazonov, S. K. Morshnev, V. P. Gubin, I. L. Vorob’ev, and S. A. Nikitov, “Miniature microstructured fiber coil with high magneto-optical sensitivity,” Opt. Commun. 282(23), 4618–4621 (2009).
[Crossref]

Chamorovsky, Y. K.

V. P. Gubin, V. A. Isaev, S. K. Morshnev, A. I. Sazonov, N. I. Starostin, Y. K. Chamorovsky, and A. I. Oussov, “Use of Spun optical fibres in current sensors,” Quantum Electron. 36(3), 287–291 (2006).
[Crossref]

Clarke, I. G.

Dändliker, R.

de Arruda, J. U.

Descamps, F.

Etzel, S. M.

A. H. Rose, S. M. Etzel, and C. M. Wang, “Verdet Constant Dispersion in Annealed Optical Fiber Current Sensors,” J. Lightwave Technol. 15(5), 803–807 (1997).
[Crossref]

Frank, A.

R. Wüest, A. Frank, S. Wiesendanger, P. Gabus, U. E. Meier, J. Nehring, and K. Bohnert, “Influence of residual fiber birefringence and temperature on the high-current performance of an interferometric fiber-optic current sensor,” Proc. SPIE 7356, 73560K (2009).
[Crossref]

G. M. Müller, A. Frank, M. Lenner, K. Bohnert, P. Gabus, and B. Guelenaltin, “Temperature compensation of fiber-optic current sensors in different regimes of operation,” in Proceedings of Photonics Conference (IEEE, 2012), 745–746.
[Crossref]

Frosio, G.

Gabus, P.

R. Wüest, A. Frank, S. Wiesendanger, P. Gabus, U. E. Meier, J. Nehring, and K. Bohnert, “Influence of residual fiber birefringence and temperature on the high-current performance of an interferometric fiber-optic current sensor,” Proc. SPIE 7356, 73560K (2009).
[Crossref]

K. Bohnert, P. Gabus, J. Nehring, H. Brändle, and M. G. Brunzel, “Fiber-optic current sensor for electrowinning of metals,” J. Lightwave Technol. 25(11), 3602–3609 (2007).
[Crossref]

K. Bohnert, P. Gabus, J. Nehring, and H. Brändle, “Temperature and vibration insensitive fiber-optic current sensor,” J. Lightwave Technol. 20(2), 267–276 (2002).
[Crossref]

G. M. Müller, A. Frank, M. Lenner, K. Bohnert, P. Gabus, and B. Guelenaltin, “Temperature compensation of fiber-optic current sensors in different regimes of operation,” in Proceedings of Photonics Conference (IEEE, 2012), 745–746.
[Crossref]

Gubin, V. P.

Y. K. Chamorovskiy, N. I. Starostin, M. V. Ryabko, A. I. Sazonov, S. K. Morshnev, V. P. Gubin, I. L. Vorob’ev, and S. A. Nikitov, “Miniature microstructured fiber coil with high magneto-optical sensitivity,” Opt. Commun. 282(23), 4618–4621 (2009).
[Crossref]

V. P. Gubin, V. A. Isaev, S. K. Morshnev, A. I. Sazonov, N. I. Starostin, Y. K. Chamorovsky, and A. I. Oussov, “Use of Spun optical fibres in current sensors,” Quantum Electron. 36(3), 287–291 (2006).
[Crossref]

Guelenaltin, B.

G. M. Müller, A. Frank, M. Lenner, K. Bohnert, P. Gabus, and B. Guelenaltin, “Temperature compensation of fiber-optic current sensors in different regimes of operation,” in Proceedings of Photonics Conference (IEEE, 2012), 745–746.
[Crossref]

Guo, Q.

Q. Guo, J. R. Qian, and L. Li, “Spun linear birefringence fibres and their sensing mechanism in current sensors with temperature compensation,” IEE Proc., Optoelectron. 141(6), 373–380 (1994).
[Crossref]

Gusarov, A.

Harms, H.

Isaev, V. A.

V. P. Gubin, V. A. Isaev, S. K. Morshnev, A. I. Sazonov, N. I. Starostin, Y. K. Chamorovsky, and A. I. Oussov, “Use of Spun optical fibres in current sensors,” Quantum Electron. 36(3), 287–291 (2006).
[Crossref]

Laming, R. I.

R. I. Laming and D. N. Payne, “Electric current sensors employing spun highly birefringent optical fibers,” J. Lightwave Technol. 7(12), 2084–2094 (1989).
[Crossref]

Lenner, M.

G. M. Müller, A. Frank, M. Lenner, K. Bohnert, P. Gabus, and B. Guelenaltin, “Temperature compensation of fiber-optic current sensors in different regimes of operation,” in Proceedings of Photonics Conference (IEEE, 2012), 745–746.
[Crossref]

Li, L.

Q. Guo, J. R. Qian, and L. Li, “Spun linear birefringence fibres and their sensing mechanism in current sensors with temperature compensation,” IEE Proc., Optoelectron. 141(6), 373–380 (1994).
[Crossref]

L. Li, J. R. Qian, and D. N. Payne, “Current sensors using highly birefringent bow-tie fibres,” Electron. Lett. 22(21), 1142–1144 (1986).
[Crossref]

Mégret, P.

Meier, U. E.

R. Wüest, A. Frank, S. Wiesendanger, P. Gabus, U. E. Meier, J. Nehring, and K. Bohnert, “Influence of residual fiber birefringence and temperature on the high-current performance of an interferometric fiber-optic current sensor,” Proc. SPIE 7356, 73560K (2009).
[Crossref]

Moreau, P.

Morshnev, S. K.

Y. K. Chamorovskiy, N. I. Starostin, M. V. Ryabko, A. I. Sazonov, S. K. Morshnev, V. P. Gubin, I. L. Vorob’ev, and S. A. Nikitov, “Miniature microstructured fiber coil with high magneto-optical sensitivity,” Opt. Commun. 282(23), 4618–4621 (2009).
[Crossref]

V. P. Gubin, V. A. Isaev, S. K. Morshnev, A. I. Sazonov, N. I. Starostin, Y. K. Chamorovsky, and A. I. Oussov, “Use of Spun optical fibres in current sensors,” Quantum Electron. 36(3), 287–291 (2006).
[Crossref]

Müller, G. M.

G. M. Müller, A. Frank, M. Lenner, K. Bohnert, P. Gabus, and B. Guelenaltin, “Temperature compensation of fiber-optic current sensors in different regimes of operation,” in Proceedings of Photonics Conference (IEEE, 2012), 745–746.
[Crossref]

Nehring, J.

R. Wüest, A. Frank, S. Wiesendanger, P. Gabus, U. E. Meier, J. Nehring, and K. Bohnert, “Influence of residual fiber birefringence and temperature on the high-current performance of an interferometric fiber-optic current sensor,” Proc. SPIE 7356, 73560K (2009).
[Crossref]

K. Bohnert, P. Gabus, J. Nehring, H. Brändle, and M. G. Brunzel, “Fiber-optic current sensor for electrowinning of metals,” J. Lightwave Technol. 25(11), 3602–3609 (2007).
[Crossref]

K. Bohnert, P. Gabus, J. Nehring, and H. Brändle, “Temperature and vibration insensitive fiber-optic current sensor,” J. Lightwave Technol. 20(2), 267–276 (2002).
[Crossref]

Nikitov, S. A.

Y. K. Chamorovskiy, N. I. Starostin, M. V. Ryabko, A. I. Sazonov, S. K. Morshnev, V. P. Gubin, I. L. Vorob’ev, and S. A. Nikitov, “Miniature microstructured fiber coil with high magneto-optical sensitivity,” Opt. Commun. 282(23), 4618–4621 (2009).
[Crossref]

Oussov, A. I.

V. P. Gubin, V. A. Isaev, S. K. Morshnev, A. I. Sazonov, N. I. Starostin, Y. K. Chamorovsky, and A. I. Oussov, “Use of Spun optical fibres in current sensors,” Quantum Electron. 36(3), 287–291 (2006).
[Crossref]

Papp, A.

Payne, D. N.

R. I. Laming and D. N. Payne, “Electric current sensors employing spun highly birefringent optical fibers,” J. Lightwave Technol. 7(12), 2084–2094 (1989).
[Crossref]

L. Li, J. R. Qian, and D. N. Payne, “Current sensors using highly birefringent bow-tie fibres,” Electron. Lett. 22(21), 1142–1144 (1986).
[Crossref]

A. J. Barlow, J. J. Ramskov-Hansen, and D. N. Payne, “Birefringence and polarization mode-dispersion in spun single-mode fibers,” Appl. Opt. 20(17), 2962–2968 (1981).
[Crossref] [PubMed]

Polynkin, P.

Qian, J. R.

Q. Guo, J. R. Qian, and L. Li, “Spun linear birefringence fibres and their sensing mechanism in current sensors with temperature compensation,” IEE Proc., Optoelectron. 141(6), 373–380 (1994).
[Crossref]

L. Li, J. R. Qian, and D. N. Payne, “Current sensors using highly birefringent bow-tie fibres,” Electron. Lett. 22(21), 1142–1144 (1986).
[Crossref]

Ramskov-Hansen, J. J.

Rose, A. H.

A. H. Rose, S. M. Etzel, and C. M. Wang, “Verdet Constant Dispersion in Annealed Optical Fiber Current Sensors,” J. Lightwave Technol. 15(5), 803–807 (1997).
[Crossref]

Ryabko, M. V.

Y. K. Chamorovskiy, N. I. Starostin, M. V. Ryabko, A. I. Sazonov, S. K. Morshnev, V. P. Gubin, I. L. Vorob’ev, and S. A. Nikitov, “Miniature microstructured fiber coil with high magneto-optical sensitivity,” Opt. Commun. 282(23), 4618–4621 (2009).
[Crossref]

Sazonov, A. I.

Y. K. Chamorovskiy, N. I. Starostin, M. V. Ryabko, A. I. Sazonov, S. K. Morshnev, V. P. Gubin, I. L. Vorob’ev, and S. A. Nikitov, “Miniature microstructured fiber coil with high magneto-optical sensitivity,” Opt. Commun. 282(23), 4618–4621 (2009).
[Crossref]

V. P. Gubin, V. A. Isaev, S. K. Morshnev, A. I. Sazonov, N. I. Starostin, Y. K. Chamorovsky, and A. I. Oussov, “Use of Spun optical fibres in current sensors,” Quantum Electron. 36(3), 287–291 (2006).
[Crossref]

Short, S. X.

Smith, A. M.

Starostin, N. I.

Y. K. Chamorovskiy, N. I. Starostin, M. V. Ryabko, A. I. Sazonov, S. K. Morshnev, V. P. Gubin, I. L. Vorob’ev, and S. A. Nikitov, “Miniature microstructured fiber coil with high magneto-optical sensitivity,” Opt. Commun. 282(23), 4618–4621 (2009).
[Crossref]

V. P. Gubin, V. A. Isaev, S. K. Morshnev, A. I. Sazonov, N. I. Starostin, Y. K. Chamorovsky, and A. I. Oussov, “Use of Spun optical fibres in current sensors,” Quantum Electron. 36(3), 287–291 (2006).
[Crossref]

Tselikov, A. A.

Vorob’ev, I. L.

Y. K. Chamorovskiy, N. I. Starostin, M. V. Ryabko, A. I. Sazonov, S. K. Morshnev, V. P. Gubin, I. L. Vorob’ev, and S. A. Nikitov, “Miniature microstructured fiber coil with high magneto-optical sensitivity,” Opt. Commun. 282(23), 4618–4621 (2009).
[Crossref]

Wang, C. M.

A. H. Rose, S. M. Etzel, and C. M. Wang, “Verdet Constant Dispersion in Annealed Optical Fiber Current Sensors,” J. Lightwave Technol. 15(5), 803–807 (1997).
[Crossref]

Wiesendanger, S.

R. Wüest, A. Frank, S. Wiesendanger, P. Gabus, U. E. Meier, J. Nehring, and K. Bohnert, “Influence of residual fiber birefringence and temperature on the high-current performance of an interferometric fiber-optic current sensor,” Proc. SPIE 7356, 73560K (2009).
[Crossref]

Wüest, R.

R. Wüest, A. Frank, S. Wiesendanger, P. Gabus, U. E. Meier, J. Nehring, and K. Bohnert, “Influence of residual fiber birefringence and temperature on the high-current performance of an interferometric fiber-optic current sensor,” Proc. SPIE 7356, 73560K (2009).
[Crossref]

Wuilpart, M.

Appl. Opt. (5)

Electron. Lett. (1)

L. Li, J. R. Qian, and D. N. Payne, “Current sensors using highly birefringent bow-tie fibres,” Electron. Lett. 22(21), 1142–1144 (1986).
[Crossref]

IEE Proc., Optoelectron. (1)

Q. Guo, J. R. Qian, and L. Li, “Spun linear birefringence fibres and their sensing mechanism in current sensors with temperature compensation,” IEE Proc., Optoelectron. 141(6), 373–380 (1994).
[Crossref]

J. Lightwave Technol. (6)

Opt. Commun. (1)

Y. K. Chamorovskiy, N. I. Starostin, M. V. Ryabko, A. I. Sazonov, S. K. Morshnev, V. P. Gubin, I. L. Vorob’ev, and S. A. Nikitov, “Miniature microstructured fiber coil with high magneto-optical sensitivity,” Opt. Commun. 282(23), 4618–4621 (2009).
[Crossref]

Opt. Lett. (1)

Proc. SPIE (1)

R. Wüest, A. Frank, S. Wiesendanger, P. Gabus, U. E. Meier, J. Nehring, and K. Bohnert, “Influence of residual fiber birefringence and temperature on the high-current performance of an interferometric fiber-optic current sensor,” Proc. SPIE 7356, 73560K (2009).
[Crossref]

Quantum Electron. (1)

V. P. Gubin, V. A. Isaev, S. K. Morshnev, A. I. Sazonov, N. I. Starostin, Y. K. Chamorovsky, and A. I. Oussov, “Use of Spun optical fibres in current sensors,” Quantum Electron. 36(3), 287–291 (2006).
[Crossref]

Other (3)

H. Lefèvre, The Fiber-Optic Gyroscope (Artech House, 1993).

R. Dändliker, “Rotational Effects of Polarization in Optical Fibers,” in Anisotropic and Nonlinear Optical Waveguides, C. G. Someda and G. I. Stegeman, eds. (Elsevier, Amsterdam, 1992).

G. M. Müller, A. Frank, M. Lenner, K. Bohnert, P. Gabus, and B. Guelenaltin, “Temperature compensation of fiber-optic current sensors in different regimes of operation,” in Proceedings of Photonics Conference (IEEE, 2012), 745–746.
[Crossref]

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Figures (5)

Fig. 1
Fig. 1 (a) Sensor configuration, SLED: superluminescent light emitting diode, PD: photodiode. (b) Relative orientation of slow axes of PM lead fiber, retarder, and spun fiber.
Fig. 2
Fig. 2 Calculated normalized magneto-optic phase shift vs. deviation from quarter-wave retardation for different choices of the spun fiber’s angular orientation (fiber A, see Table 1).
Fig. 3
Fig. 3 (a) Calculated normalized magneto-optic phase shift vs. temperature for a temperature-compensated sensor (black). The other lines depict the individual contributions from the Verdet constant (green), the spun fiber’s birefringence (orange), and the fiber retarder (blue), see text for parameters. (b) Retarder detuning ε required for temperature compensation vs. spin ratio [p = 2.9 mm].
Fig. 4
Fig. 4 (a) Contribution of the retarder, d(lnΔϕ) /dTretarder, to the temperature dependence of the sensor signal as a function of the spun fiber orientation angle α0, experimental: red squares, calculated: black line. (b) Normalized sensor signal as a function of the temperature of the retarder (sensing fiber at room temperature, blue squares) and of the temperature of the entire fiber coil (retarder and sensing fiber, black data points); the red line is a linear fit (sensor parameters: fiber type B, α0 = 0°, ε = 0°). (c) Same as in (b), but with ε ≈7.5° so that sensor signal is temperature compensated to within < ± 0.2%.
Fig. 5
Fig. 5 (a) Normalized sensor signal as a function of applied current for a temperature-compensated sensor with 385 fiber windings at temperatures of −20 °C (black squares) and + 70 °C (red squares), the lines represent corresponding theoretical data. (b) Normalized sensor signal as a function of applied current for ε = 0° (black line) and α0 = ± 45° and ε = ± cot−1 x (red line).

Tables (1)

Tables Icon

Table 1 Linear Beat Length and Spin Pitch of the Spun Highly Birefringent Fibers Used for This Work

Equations (10)

Equations on this page are rendered with MathJax. Learn more.

Δϕ=4NVI.
VLP(HLP) QWR RCP(LCP) mirror LCP(RCP) QWR HLP(VLP).
VLP retarder cos ε 2 RCP+isin ε 2 LCP, HLP retarder cos ε 2 LCPisin ε 2 RCP.
Δϕ=4NVI/cosε.
RC P * RCP+ L LB L EB e 2i α 0 LCP, LC P * LCP L LB L EB e +2i α 0 RCP. ,
K= x 2 exp π 2 4ln2 ( 2lΔλ L EB λ ) 2 .
VLP(HLP) α 0 = ±45°, ε = cot -1 x RC P * (LC P * ).
Δϕ= 4NVI 1+ [ sin( 2 α 0 )/x ] 2 cos( ε+ tan 1 [ sin( 2 α 0 )/x ] ) .
Δ ϕ α 0 =±45° = 4NVI 1+ x 2 cos( ε± cot -1 x ) .
Δ ϕ α 0 =0°,90° =4NVI/cosε.

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