rus
Issue #7/2022
Ya. V. Przhiyalkovskiy, N. I. Starostin, S. K. Morshnev, A. I. Sazonov
Influence of Strong Bending Birefringence in the Spun Fiber on Excess Noise Suppression in the Fiber Current Sensor
Influence of Strong Bending Birefringence in the Spun Fiber on Excess Noise Suppression in the Fiber Current Sensor
DOI: 10.22184/1993-7296.FRos.2022.16.7.552.563
We have considered the influence of bending birefringence occurred in the spun fiber of a fiber-optic current sensor based on the Faraday effect, on the efficiency of excess noise suppression. It is shown that when the radius of spun fiber winding is small, the secondary waves arise in the fiber. Such waves eventually distort the radiated spectrum at the sensor output and therefore cause a decrease in the noise suppression efficiency. The proposed theoretical model of this effect is confirmed experimentally for the spun fiber winding radius of 5 mm.
We have considered the influence of bending birefringence occurred in the spun fiber of a fiber-optic current sensor based on the Faraday effect, on the efficiency of excess noise suppression. It is shown that when the radius of spun fiber winding is small, the secondary waves arise in the fiber. Such waves eventually distort the radiated spectrum at the sensor output and therefore cause a decrease in the noise suppression efficiency. The proposed theoretical model of this effect is confirmed experimentally for the spun fiber winding radius of 5 mm.
Теги: excess noise faraday effect fiber-optic current sensor spun fiber spun-волокно волоконно-оптический датчик тока избыточный шум эффект фарадея
Influence of Strong Bending Birefringence in the Spun Fiber on Excess Noise Suppression in the Fiber Current Sensor
Ya. V. Przhiyalkovskiy, N. I. Starostin, S. K. Morshnev, A. I. Sazonov
Kotelnikov Institute of Radioengineering and Electronics (Fryazino branch), the Russian Academy of Sciences, Fryazino, Moscow region, Russia
We have considered the influence of bending birefringence occurred in the spun fiber of a fiber-optic current sensor based on the Faraday effect, on the efficiency of excess noise suppression. It is shown that when the radius of spun fiber winding is small, the secondary waves arise in the fiber. Such waves eventually distort the radiated spectrum at the sensor output and therefore cause a decrease in the noise suppression efficiency. The proposed theoretical model of this effect is confirmed experimentally for the spun fiber winding radius of 5 mm.
Keywords: fiber-optic current sensor, Faraday effect, spun fiber, excess noise
Received on: 17.06.2022
Accepted on: 27.07.2022
1. Introduction
Recent advances in the development of fiber-optic electric current sensors (FOCS) based on the magneto-optical Faraday effect have dramatically increased their efficiency for metrological purposes that has led to a significant expansion of their scope of application. In particular, with due regard to the low inertia of the Faraday effect (~10–9 s), such sensors are of interest for measuring impulse currents [1, 2], for example, in the linear pulsed electron accelerators. The modern FOCSs are usually based on the basis of a low-coherence reflective interferometer [3]. They use a fiber with a high linear birefringence (BR) and a spiral structure of the BR axes (spun fiber) as a detecting element [4].
In practice, the spun sensor fiber is wound on coils with a radius of at least 10 cm, while the effect of the bending BR on the polarization dynamics is negligible. Having considered that the beat length of the built-in linear BR is usually several times greater than the step length of the BR helical structure, the radiation polarization in the spun fiber remains close to the circular one that ensures high performance of the interferometer. The use of a small winding radius, up to several millimeters, makes it possible to significantly reduce the linear length of the detecting fiber with a constant number of turns (and, accordingly, the sensor sensitivity). In this case, the transit radiation time is decreased leading to an increased in the temporal detector resolution, this is especially important for measuring the pulsed currents [2]. However, at such small winding radii, the bending BR becomes quite significant and substantially changes the evolution of the polarization state [5]. The experiment shows that this leads not only to a decrease in the magneto-optical detector’s sensitivity, but also to a decrease in the interference pattern visibility, since the incoherent light component in the interferometer path is increased and its spectrum is distorted. Ultimately, all these factors have a negative impact on the signal-to-noise ratio. This effect was first revealed in [6], but no analytical consideration was performed.
One way to increase the signal-to-noise ratio in the low-coherence optical sensors is to suppress excess noise caused by beating of the spectral components of low-coherence light [7]. To implement this method, a reference optical channel is built into the detector’s circuit, the excess noise of which is subtracted from the measurement channel signal. This method has already become widespread in the fiber optic gyroscopy [8, 9]. In particular, in [9], the authors study the influence of differences in the emission spectra in the optical channels of a fiber-optic Sagnac gyroscope on the excess noise suppression. As for the optical current sensors, this method was first proposed in [10]. It was found in [6] that when winding a spun fiber with a small radius, the efficiency of excess noise suppression is decreased, and the direct cause of this effect is the optical spectrum distortion. However, no more detailed analysis was provided.
The purpose of this paper is to systematically study the features of excess noise suppression in the FOCS based on a low-coherence reflective interferometer, the sensitive spun fiber of which is wound with a small radius.
2. Theory
2.1. Fiber-optic Current Sensor Based on the Faraday Effect
The FOCS operating principle with a two-channel detection system is shown in Fig. 1 and described below. The unpolarized radiation generated by a superluminescent source is passed through a series-connected polarizer, PM coupler (the polarizer axis is oriented with one of the BR axes of the coupler’s input fiber) and then is injected into a spun fiber surrounding a current-carrying conductor. Further, the light propagates in the spun fiber in the forward direction, is reflected from the mirror at the end of the fiber, and passes through the optical path in the opposite direction. When the fiber passes in both directions, a phase shift occurs between the circularly polarized radiation components due to the Faraday effect [3]:
, (1)
where the contour integral of the magnetic field B induced by the measured current im is taken along the spun fiber path, and the second equality is applicable in the case of a closed loop. In this case, V is a Verdet constant of the fiber material, N is the number of fiber turns around the current-carrying conductor.
In order to increase the detector sensitivity to the low currents and provide the ability to distinguish the measured current direction, the operating point of the interferometer is shifted by a constant angle π / 2 using a Faraday rotator (that can also be achieved by a modulation detection circuit [3]). As a result, the intensity of the light incident on the photodetector will have the following form
, (2)
that will lead to the photocurrent generation i = ηI, where η is the detector efficiency factor.
As for the photodiode output signal noise δi, its main sources include the photon and excess noise [8]:
. (3)
The first term in (3) corresponds to the photon noise, and the second term corresponds to the excess noise, due to the beating of the spectral light components when using the low-coherence radiation sources. In this case, Δν is the optical spectrum width, B is the band of the photodetector and amplifier, and e is the electron charge. It should be noted that due to the linear dependence of the excess noise on the radiation intensity, the photon noise correlates to the excess noise as 10–1 even at a radiation power of ~10 μW, therefore, in practice its influence can be neglected.
The excess noise at the output of the detector’s signal channel can be subtracted from the desired signal using the noise signal of the reference photodetector connected to the 2 output of the PM coupler [6, 10] (Fig. 1). Indeed, since the excess noise is received from a common radiation source, the output current noise of the signal and reference photodetectors δi1 and δi2 are correlated. It is important that when subtracting the output currents of the photodiodes, the desired interference signal will not be subtracted: in the reference channel, the light does not pass through the polarizer in the opposite direction, so there is no interference signal. The excess noise suppression ratio κ is determined by the formula
, (4)
where the angle brackets denote the time averaging operation. It should be noted that for the most effective noise reduction by this method, the output noise must be synchronized and equalized in terms of power: . In this case, based on (4) we obtain that the excess noise suppression coefficient depends on the correlation of output currents as follows:
. (5)
2.2. Polarized Radiation Propagation in the Spun Fiber
We will consider the main features of the polarized light propagation in the FOCS spun fiber. The polarization properties of spun fibers are currently quite well studied [5, 11] and are determined by the BR in the fiber and the spin pitch. The built-in linear BR is specified by the beat length Lb,i = 2π / βi, where βi = k0(nx – ny) is the difference between the propagation constants of linearly polarized modes for a small fiber layer, k0 = 2π / λ, and the spin pitch Ls is determined by the spatial rotation frequency of the axes . The bending BR is characterized by the spatial frequency βe or the relevant beat length [12]:
, (6)
where the coefficient depends on the strain tensor pij components of the fiber material and Poisson’s ratio νp (for quartz Cs ≈ 1,1 · 106 m‑1 at a wavelength λ = 1,55 µm), R is the bending radius, r is the fiber cladding radius, is the average value of the fiber refractive index. The circular BR will be further specified by the difference between the propagation constants of the circularly polarized modes α.
The monochromatic light waves propagating in the spun fiber in the forward (f) and backward (b) directions as a first approximation can be decomposed into the elliptically polarized modes (hereinafter referred to as modified helical modes) [5]:
,
, (7)
where
(8)
are the polarization vectors of these modes. In this case, the vectors
(9)
recorded in the basis of circular polarizations (the upper and lower components represent the left and right circular polarizations, respectively) correspond to the polarization modes of the rectilinear spun fiber. The coefficients
(10)
describe deviation of the helical modes of a straight spun fiber from the circularly polarized modes, and the coefficients
,
(11)
describe the difference between the polarization modes of a bent spun fiber and the relevant modes of a straight spun fiber in a similar way, with the approximate equality is valid for small σ. ϕ denotes the angle at which the bending plane is inclined relative to the slow fiber axis at its beginning. The modes 1 and 3 are slow and have a refractive index , while the modes 2 and 4 are fast and have a refractive index , where
. (12)
The polarization states of the modes are elliptical, thus, their sensitivity to the Faraday effect is decreased with a factor that is hereinafter referred to as the relative response factor.
2.3. Spectrum Deformation in the FOCS with Small-radius Spun Fiber Winding
We will use the above formal description to consider the propagation of light waves in a reflective FOCS shown in Fig. 1. According to the circuit, an unpolarized light wave is passed through a polarizer and acquires linear polarization
, (13)
oriented with an angle θ, where el,r are the basis vectors of circular polarizations. When entering a spun fiber, such a wave excites both polarization modes 1 and 2 (see (7)), which amplitudes C1 and C2 are based on the initial condition Ef(0) = Ep:
, (14)
where the parentheses indicate the dot product and the subscript «0» means that the polarization vectors should be taken at the beginning of the fiber (z = 0).
The amplitudes of the reverse waves arising due to the light reflection from the mirror at the end of the fiber are calculated on the basis of the boundary condition Ef(L) = Eb(L). For an incident wave with the mode 1, this condition implies that at the initial fiber point, the electric field of the reflected light is equal to the following
, (15)
and for an incident wave, having the mode 2 respectively, it is equal to the following
, (16)
where the equalities , and were used. The reflected mode amplitudes are
(17)
where the index «L» indicates that the polarization vectors should be taken at z = L. The forward calculation of these products leads to the following formula
(18)
We can note that each mode of light incident on the mirror in general excites both modes of light propagating in the opposite direction, the phase shift between which to the beginning of the fiber is Ω~L. This reflected light structure is due to the fact that the polarization modes of the spun fiber are elliptically polarized.
We will consider in more detail how the waves arising after the light reflection from the mirror in (15) and (16) at are related. If the winding radius is large enough so that the bending BR can be neglected, the polarization modes (9) are almost circular, and the ratio between the mode amplitudes is . Therefore, the almost complete 1 → 4 and 2 → 3 mode conversion occurs in the case of mirror reflection. If the fiber is wound on a coil, then the ratio between these amplitudes becomes approximately . The essential point is that the parameter value can be comparable to 1 for the sufficiently small winding radii. Thus, for example, for a spun fiber with Ls = 3 mm, Lb, i = 9 mm and wound with a radius R = 5 mm, the value reaches 0.5 for a fiber with a diameter of 80 μm and 1.3 for a fiber with a diameter of 125 μm. This means that the polarization modes of the bent spun fiber (8) have elliptical polarization states that are far from the circular ones, so that the secondary waves generated in the case of mirror reflection cannot be ignored.
When combining (15) and (16), we obtain the following: when a spun fiber is excited by linearly polarized light, its electric field after passing through the fiber in the forward and reverse directions will be equal to the following
. (19)
Further, when the light is passed through the polarizer, only the component (Eb(0), Ep)Ep that is parallel to the polarizer axis, enters the photodetector. As a result of direct calculations and based on (13) and (19), we obtain the final expression for the intensity
, (20)
where the values and U are constants determined by the input linear polarization angle θ, the spun fiber parameters and its bending. It can be seen from (20) that two more terms are added to the usual interference term, due to the occurrence of secondary waves when the light is reflected from the mirror. It is important to note that the amplitudes I1 and I2 of these terms obviously depend on the bending radius and are increased with the decreasing radius. Figure 2 shows an example of such a dependence calculated for a fiber with a diameter of 80 μm and a length L = 11 m, having the parameters Ls = 3 mm and Lb = 8.9 mm at a wavelength λ = 1.55 μm. The angle values are taken as θ = ϕ = 45° in accordance with the experimental data provided below.
We will consider how the detected light (20) intensity is changed depending on the wavelength. Since the Ω~L phase depends on the wavelength (since the parameter Ω~ includes the wavelength-dependent beat lengths), the last two terms in (20) vary over the spectral radiation range. In the case of a small bending radius of the fiber , the last terms in (20) become significant. This means that the intensity of light incident on the photodetector (normalized to the initial light source intensity) significantly depends on the wavelength, and the shape of the intensity spectrum is noticeably deformed. An example of such a spectrum deformation is provided in the section with the experiment description. It should be noted that if the bending radius is large, then and these terms are small. In this case, the intensity is almost independent of the wavelength, and the light spectrum shape remains almost unchanged. This fact explains why the optical spectrum is reconstructed in a sensitive spiral-wound FOCS coil [6], where the winding radius is large at the beginning and end of the fiber.
3. Experiment
For experimental confirmation of the theoretical model, an experimental setup was prepared, as shown in Fig. 1. A superluminescent fiber source based on Er / Yb doped fiber (IPG Photonics) with a power of 30 mW, a wavelength of λ = 1.55 µm and a spectral width of ∆λ = 20 nm was used as a radiation source. The light power in both channels was balanced using an attenuator and amounted to 250 μW. The light was detected using the broadband photodetectors with high sensitivity η ≈ 1 A / W, thus, the final bandwidth of the electrical detection circuit was limited only by the amplifier and amounted to B = 3.5 MHz. The spun fiber had a beat length Lb,i = 9 mm and the helical structure step Ls = 3 mm (σ = 0.17), the total fiber length was L = 11 m.
During the experiment, the spectra of optical radiation incident on the photodetectors of the sensitive and reference detector’s channels, as well as the electrical signals from the outputs of both photodetectors, were obtained. The continuous line in Fig. 3 describes the light spectrum in the measurement channel of the detector when winding the spun fiber with a radius of 5 mm. It can be seen that the spectrum contains oscillations with the periods differing by a factor of two, in accordance with the above theoretical formula (20) (the calculation is shown by a dotted line in Fig. 3). The values of the parameters used in the calculation were Lb,i = 8.9 mm, Ls = 3 mm, L = 11 m, the winding radius R = 5 mm, and the input polarization angles θ and the winding plane slope ϕ were selected to provide the better dependence on the experimental data.
Figure 4 shows the measurement results of the excess noise suppression coefficient for this sensitive coil, as well as the theoretical dependence (5). The correlation calculations using the output signals are indicated by squares, and the spectra correlation calculations are indicated by circles. Moreover, the figure demonstrates the suppression coefficients for winding with a large radius of 100 mm and with a radius of 5 mm for helical winding of the initial and final sections of the fiber [6, 10]. The best suppression at the level of 15.3 dB is achieved for a radius of 100 mm, when the bending BR is insignificant and the light polarization state along the entire fiber length is close to the circular one. For winding with a radius of 5 mm, the excess noise suppression ratio is reduced to a value of 6 dB in accordance with the above theory. In the third coil, almost all of the fiber was wound with a radius of 5 mm, however, the segments with the length of 1 m at the beginning and end of the fiber were helically wound with a smooth change in radius [10]. With this winding configuration, the radiation polarization on the mirror is restored to almost circular one (as in a straight spun fiber). In this case, when light is reflected, an almost complete conversion of polarization modes into the orthogonal ones is provided, therefore I1 ≈ 0 and I2 ≈ 0 are obtained in (20). As a result, the optical spectrum is restored and the suppression is increased to a value of 14.8 dB.
4. Conclusion
In this paper, we have considered the influence of bending birefringence in a spun fiber used as a sensitive element of a fiber-optic current sensor based on the Faraday effect on the excess noise suppression efficiency. It is shown that, at a small winding radius, the secondary waves occur in the fiber, the attenuation of which on the detector polarizer depends on the wavelength. Due to this fact, the radiation spectrum is deformed leading to a decrease in the excess noise suppression efficiency. The effect was demonstrated during an experiment, when reducing the detector’s spun fiber winding radius to 5 mm resulted in a 9.3 dB reduction in the suppression ratio. The developed model explains the observed decrease in the suppression efficiency, and also provides a theoretical basis for increasing the excess noise suppression efficiency in the case of helical winding of the initial and final fiber segments [10].
The paper was prepared within the framework of the state task of the Kotelnikov Institute of Radioengineering and Electronics of the Russian Academy of Sciences.
Ya. V. Przhiyalkovskiy, N. I. Starostin, S. K. Morshnev, A. I. Sazonov
Kotelnikov Institute of Radioengineering and Electronics (Fryazino branch), the Russian Academy of Sciences, Fryazino, Moscow region, Russia
We have considered the influence of bending birefringence occurred in the spun fiber of a fiber-optic current sensor based on the Faraday effect, on the efficiency of excess noise suppression. It is shown that when the radius of spun fiber winding is small, the secondary waves arise in the fiber. Such waves eventually distort the radiated spectrum at the sensor output and therefore cause a decrease in the noise suppression efficiency. The proposed theoretical model of this effect is confirmed experimentally for the spun fiber winding radius of 5 mm.
Keywords: fiber-optic current sensor, Faraday effect, spun fiber, excess noise
Received on: 17.06.2022
Accepted on: 27.07.2022
1. Introduction
Recent advances in the development of fiber-optic electric current sensors (FOCS) based on the magneto-optical Faraday effect have dramatically increased their efficiency for metrological purposes that has led to a significant expansion of their scope of application. In particular, with due regard to the low inertia of the Faraday effect (~10–9 s), such sensors are of interest for measuring impulse currents [1, 2], for example, in the linear pulsed electron accelerators. The modern FOCSs are usually based on the basis of a low-coherence reflective interferometer [3]. They use a fiber with a high linear birefringence (BR) and a spiral structure of the BR axes (spun fiber) as a detecting element [4].
In practice, the spun sensor fiber is wound on coils with a radius of at least 10 cm, while the effect of the bending BR on the polarization dynamics is negligible. Having considered that the beat length of the built-in linear BR is usually several times greater than the step length of the BR helical structure, the radiation polarization in the spun fiber remains close to the circular one that ensures high performance of the interferometer. The use of a small winding radius, up to several millimeters, makes it possible to significantly reduce the linear length of the detecting fiber with a constant number of turns (and, accordingly, the sensor sensitivity). In this case, the transit radiation time is decreased leading to an increased in the temporal detector resolution, this is especially important for measuring the pulsed currents [2]. However, at such small winding radii, the bending BR becomes quite significant and substantially changes the evolution of the polarization state [5]. The experiment shows that this leads not only to a decrease in the magneto-optical detector’s sensitivity, but also to a decrease in the interference pattern visibility, since the incoherent light component in the interferometer path is increased and its spectrum is distorted. Ultimately, all these factors have a negative impact on the signal-to-noise ratio. This effect was first revealed in [6], but no analytical consideration was performed.
One way to increase the signal-to-noise ratio in the low-coherence optical sensors is to suppress excess noise caused by beating of the spectral components of low-coherence light [7]. To implement this method, a reference optical channel is built into the detector’s circuit, the excess noise of which is subtracted from the measurement channel signal. This method has already become widespread in the fiber optic gyroscopy [8, 9]. In particular, in [9], the authors study the influence of differences in the emission spectra in the optical channels of a fiber-optic Sagnac gyroscope on the excess noise suppression. As for the optical current sensors, this method was first proposed in [10]. It was found in [6] that when winding a spun fiber with a small radius, the efficiency of excess noise suppression is decreased, and the direct cause of this effect is the optical spectrum distortion. However, no more detailed analysis was provided.
The purpose of this paper is to systematically study the features of excess noise suppression in the FOCS based on a low-coherence reflective interferometer, the sensitive spun fiber of which is wound with a small radius.
2. Theory
2.1. Fiber-optic Current Sensor Based on the Faraday Effect
The FOCS operating principle with a two-channel detection system is shown in Fig. 1 and described below. The unpolarized radiation generated by a superluminescent source is passed through a series-connected polarizer, PM coupler (the polarizer axis is oriented with one of the BR axes of the coupler’s input fiber) and then is injected into a spun fiber surrounding a current-carrying conductor. Further, the light propagates in the spun fiber in the forward direction, is reflected from the mirror at the end of the fiber, and passes through the optical path in the opposite direction. When the fiber passes in both directions, a phase shift occurs between the circularly polarized radiation components due to the Faraday effect [3]:
, (1)
where the contour integral of the magnetic field B induced by the measured current im is taken along the spun fiber path, and the second equality is applicable in the case of a closed loop. In this case, V is a Verdet constant of the fiber material, N is the number of fiber turns around the current-carrying conductor.
In order to increase the detector sensitivity to the low currents and provide the ability to distinguish the measured current direction, the operating point of the interferometer is shifted by a constant angle π / 2 using a Faraday rotator (that can also be achieved by a modulation detection circuit [3]). As a result, the intensity of the light incident on the photodetector will have the following form
, (2)
that will lead to the photocurrent generation i = ηI, where η is the detector efficiency factor.
As for the photodiode output signal noise δi, its main sources include the photon and excess noise [8]:
. (3)
The first term in (3) corresponds to the photon noise, and the second term corresponds to the excess noise, due to the beating of the spectral light components when using the low-coherence radiation sources. In this case, Δν is the optical spectrum width, B is the band of the photodetector and amplifier, and e is the electron charge. It should be noted that due to the linear dependence of the excess noise on the radiation intensity, the photon noise correlates to the excess noise as 10–1 even at a radiation power of ~10 μW, therefore, in practice its influence can be neglected.
The excess noise at the output of the detector’s signal channel can be subtracted from the desired signal using the noise signal of the reference photodetector connected to the 2 output of the PM coupler [6, 10] (Fig. 1). Indeed, since the excess noise is received from a common radiation source, the output current noise of the signal and reference photodetectors δi1 and δi2 are correlated. It is important that when subtracting the output currents of the photodiodes, the desired interference signal will not be subtracted: in the reference channel, the light does not pass through the polarizer in the opposite direction, so there is no interference signal. The excess noise suppression ratio κ is determined by the formula
, (4)
where the angle brackets denote the time averaging operation. It should be noted that for the most effective noise reduction by this method, the output noise must be synchronized and equalized in terms of power: . In this case, based on (4) we obtain that the excess noise suppression coefficient depends on the correlation of output currents as follows:
. (5)
2.2. Polarized Radiation Propagation in the Spun Fiber
We will consider the main features of the polarized light propagation in the FOCS spun fiber. The polarization properties of spun fibers are currently quite well studied [5, 11] and are determined by the BR in the fiber and the spin pitch. The built-in linear BR is specified by the beat length Lb,i = 2π / βi, where βi = k0(nx – ny) is the difference between the propagation constants of linearly polarized modes for a small fiber layer, k0 = 2π / λ, and the spin pitch Ls is determined by the spatial rotation frequency of the axes . The bending BR is characterized by the spatial frequency βe or the relevant beat length [12]:
, (6)
where the coefficient depends on the strain tensor pij components of the fiber material and Poisson’s ratio νp (for quartz Cs ≈ 1,1 · 106 m‑1 at a wavelength λ = 1,55 µm), R is the bending radius, r is the fiber cladding radius, is the average value of the fiber refractive index. The circular BR will be further specified by the difference between the propagation constants of the circularly polarized modes α.
The monochromatic light waves propagating in the spun fiber in the forward (f) and backward (b) directions as a first approximation can be decomposed into the elliptically polarized modes (hereinafter referred to as modified helical modes) [5]:
,
, (7)
where
(8)
are the polarization vectors of these modes. In this case, the vectors
(9)
recorded in the basis of circular polarizations (the upper and lower components represent the left and right circular polarizations, respectively) correspond to the polarization modes of the rectilinear spun fiber. The coefficients
(10)
describe deviation of the helical modes of a straight spun fiber from the circularly polarized modes, and the coefficients
,
(11)
describe the difference between the polarization modes of a bent spun fiber and the relevant modes of a straight spun fiber in a similar way, with the approximate equality is valid for small σ. ϕ denotes the angle at which the bending plane is inclined relative to the slow fiber axis at its beginning. The modes 1 and 3 are slow and have a refractive index , while the modes 2 and 4 are fast and have a refractive index , where
. (12)
The polarization states of the modes are elliptical, thus, their sensitivity to the Faraday effect is decreased with a factor that is hereinafter referred to as the relative response factor.
2.3. Spectrum Deformation in the FOCS with Small-radius Spun Fiber Winding
We will use the above formal description to consider the propagation of light waves in a reflective FOCS shown in Fig. 1. According to the circuit, an unpolarized light wave is passed through a polarizer and acquires linear polarization
, (13)
oriented with an angle θ, where el,r are the basis vectors of circular polarizations. When entering a spun fiber, such a wave excites both polarization modes 1 and 2 (see (7)), which amplitudes C1 and C2 are based on the initial condition Ef(0) = Ep:
, (14)
where the parentheses indicate the dot product and the subscript «0» means that the polarization vectors should be taken at the beginning of the fiber (z = 0).
The amplitudes of the reverse waves arising due to the light reflection from the mirror at the end of the fiber are calculated on the basis of the boundary condition Ef(L) = Eb(L). For an incident wave with the mode 1, this condition implies that at the initial fiber point, the electric field of the reflected light is equal to the following
, (15)
and for an incident wave, having the mode 2 respectively, it is equal to the following
, (16)
where the equalities , and were used. The reflected mode amplitudes are
(17)
where the index «L» indicates that the polarization vectors should be taken at z = L. The forward calculation of these products leads to the following formula
(18)
We can note that each mode of light incident on the mirror in general excites both modes of light propagating in the opposite direction, the phase shift between which to the beginning of the fiber is Ω~L. This reflected light structure is due to the fact that the polarization modes of the spun fiber are elliptically polarized.
We will consider in more detail how the waves arising after the light reflection from the mirror in (15) and (16) at are related. If the winding radius is large enough so that the bending BR can be neglected, the polarization modes (9) are almost circular, and the ratio between the mode amplitudes is . Therefore, the almost complete 1 → 4 and 2 → 3 mode conversion occurs in the case of mirror reflection. If the fiber is wound on a coil, then the ratio between these amplitudes becomes approximately . The essential point is that the parameter value can be comparable to 1 for the sufficiently small winding radii. Thus, for example, for a spun fiber with Ls = 3 mm, Lb, i = 9 mm and wound with a radius R = 5 mm, the value reaches 0.5 for a fiber with a diameter of 80 μm and 1.3 for a fiber with a diameter of 125 μm. This means that the polarization modes of the bent spun fiber (8) have elliptical polarization states that are far from the circular ones, so that the secondary waves generated in the case of mirror reflection cannot be ignored.
When combining (15) and (16), we obtain the following: when a spun fiber is excited by linearly polarized light, its electric field after passing through the fiber in the forward and reverse directions will be equal to the following
. (19)
Further, when the light is passed through the polarizer, only the component (Eb(0), Ep)Ep that is parallel to the polarizer axis, enters the photodetector. As a result of direct calculations and based on (13) and (19), we obtain the final expression for the intensity
, (20)
where the values and U are constants determined by the input linear polarization angle θ, the spun fiber parameters and its bending. It can be seen from (20) that two more terms are added to the usual interference term, due to the occurrence of secondary waves when the light is reflected from the mirror. It is important to note that the amplitudes I1 and I2 of these terms obviously depend on the bending radius and are increased with the decreasing radius. Figure 2 shows an example of such a dependence calculated for a fiber with a diameter of 80 μm and a length L = 11 m, having the parameters Ls = 3 mm and Lb = 8.9 mm at a wavelength λ = 1.55 μm. The angle values are taken as θ = ϕ = 45° in accordance with the experimental data provided below.
We will consider how the detected light (20) intensity is changed depending on the wavelength. Since the Ω~L phase depends on the wavelength (since the parameter Ω~ includes the wavelength-dependent beat lengths), the last two terms in (20) vary over the spectral radiation range. In the case of a small bending radius of the fiber , the last terms in (20) become significant. This means that the intensity of light incident on the photodetector (normalized to the initial light source intensity) significantly depends on the wavelength, and the shape of the intensity spectrum is noticeably deformed. An example of such a spectrum deformation is provided in the section with the experiment description. It should be noted that if the bending radius is large, then and these terms are small. In this case, the intensity is almost independent of the wavelength, and the light spectrum shape remains almost unchanged. This fact explains why the optical spectrum is reconstructed in a sensitive spiral-wound FOCS coil [6], where the winding radius is large at the beginning and end of the fiber.
3. Experiment
For experimental confirmation of the theoretical model, an experimental setup was prepared, as shown in Fig. 1. A superluminescent fiber source based on Er / Yb doped fiber (IPG Photonics) with a power of 30 mW, a wavelength of λ = 1.55 µm and a spectral width of ∆λ = 20 nm was used as a radiation source. The light power in both channels was balanced using an attenuator and amounted to 250 μW. The light was detected using the broadband photodetectors with high sensitivity η ≈ 1 A / W, thus, the final bandwidth of the electrical detection circuit was limited only by the amplifier and amounted to B = 3.5 MHz. The spun fiber had a beat length Lb,i = 9 mm and the helical structure step Ls = 3 mm (σ = 0.17), the total fiber length was L = 11 m.
During the experiment, the spectra of optical radiation incident on the photodetectors of the sensitive and reference detector’s channels, as well as the electrical signals from the outputs of both photodetectors, were obtained. The continuous line in Fig. 3 describes the light spectrum in the measurement channel of the detector when winding the spun fiber with a radius of 5 mm. It can be seen that the spectrum contains oscillations with the periods differing by a factor of two, in accordance with the above theoretical formula (20) (the calculation is shown by a dotted line in Fig. 3). The values of the parameters used in the calculation were Lb,i = 8.9 mm, Ls = 3 mm, L = 11 m, the winding radius R = 5 mm, and the input polarization angles θ and the winding plane slope ϕ were selected to provide the better dependence on the experimental data.
Figure 4 shows the measurement results of the excess noise suppression coefficient for this sensitive coil, as well as the theoretical dependence (5). The correlation calculations using the output signals are indicated by squares, and the spectra correlation calculations are indicated by circles. Moreover, the figure demonstrates the suppression coefficients for winding with a large radius of 100 mm and with a radius of 5 mm for helical winding of the initial and final sections of the fiber [6, 10]. The best suppression at the level of 15.3 dB is achieved for a radius of 100 mm, when the bending BR is insignificant and the light polarization state along the entire fiber length is close to the circular one. For winding with a radius of 5 mm, the excess noise suppression ratio is reduced to a value of 6 dB in accordance with the above theory. In the third coil, almost all of the fiber was wound with a radius of 5 mm, however, the segments with the length of 1 m at the beginning and end of the fiber were helically wound with a smooth change in radius [10]. With this winding configuration, the radiation polarization on the mirror is restored to almost circular one (as in a straight spun fiber). In this case, when light is reflected, an almost complete conversion of polarization modes into the orthogonal ones is provided, therefore I1 ≈ 0 and I2 ≈ 0 are obtained in (20). As a result, the optical spectrum is restored and the suppression is increased to a value of 14.8 dB.
4. Conclusion
In this paper, we have considered the influence of bending birefringence in a spun fiber used as a sensitive element of a fiber-optic current sensor based on the Faraday effect on the excess noise suppression efficiency. It is shown that, at a small winding radius, the secondary waves occur in the fiber, the attenuation of which on the detector polarizer depends on the wavelength. Due to this fact, the radiation spectrum is deformed leading to a decrease in the excess noise suppression efficiency. The effect was demonstrated during an experiment, when reducing the detector’s spun fiber winding radius to 5 mm resulted in a 9.3 dB reduction in the suppression ratio. The developed model explains the observed decrease in the suppression efficiency, and also provides a theoretical basis for increasing the excess noise suppression efficiency in the case of helical winding of the initial and final fiber segments [10].
The paper was prepared within the framework of the state task of the Kotelnikov Institute of Radioengineering and Electronics of the Russian Academy of Sciences.
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