rus
Issue #6/2018
S.K.Morshnev, V.P.Gubin, N.I.Starostin, Ya.V.Prshiyalkovsky, A.I.Sazonov
Beat Length Measurement in Birefringence Optical Fibers
Beat Length Measurement in Birefringence Optical Fibers
Birefringence optical fibers (OF) appeared in 1978–1983 together with the technologies of thermoelastic inserts in the workpieces for drawing optical fibers. This article deals with two basic methods of measurement of embedded BOFs: spectral method and elastic twisting method. Three OFs types, drawn from the workpieces with embedded linear birefringence are taken as objects of study: HiBi or also PM- (polarization maintaining) fibers; fibers, drawn from the same workpieces, but with the rotation of the workpiece – spun-fiber; LoBi-fibers, which by the method are referred to as spun-fibers. The technique of measuring the beat length of the embedded BOFs is described in detail.
The paper analyzes two known methods for measuring the characteristics of an embedded birefringence of special optical fibers. The features of beat length measurement of the embedded birefringence for each method and different types of special fibers are shown. Advantages, disadvantages and limitations for each of the methods are pointed.
The paper analyzes two known methods for measuring the characteristics of an embedded birefringence of special optical fibers. The features of beat length measurement of the embedded birefringence for each method and different types of special fibers are shown. Advantages, disadvantages and limitations for each of the methods are pointed.
Теги: beat length birefringence optical fiber elastic twisting phase delay spectral method двулучепреломляющий волоконный световод длина биений спектральный метод упругое кручение фазовая задержка
1. INTRODUCTION
The phenomenon of linear birefringence manifests itself at various propagation velocities of two linearly polarized waves: ordinary and extraordinary. In an optical fiber, both waves propagate along its axis. Symmetry of elastic stresses or geometric symmetry of inserts of various materials in the design of birefringence fibers allows us to distinguish two orthogonal axes of birefringence – fast and slow. A wave with an electric vector parallel to the fast axis will propagate faster than a wave with an electric vector directed along the slow axis, i. e., the refraction index (RI) of the first wave will be smaller than that of the second wave. The axes of the linear birefringence are x and y, and the RI of the waves with linear polarizations along these axes are respectively nx and ny. The linear BOFs value is measured by the difference nx – n.
The birefringence optic fibers include fibers drawn from the workpieces with embedded linear birefringence: HiBi-fibers [1–7], and drawn from the same workpieces, but with the rotation of the workpiece: spun-fibers [11–13]. HiBi-fibers are also referred to as the PM- (polarization maintaining) fibers, i. e., fibers that retain polarization. The so called LoBi-fibers [8–10] stand somewhat separately which by the manufacturing method (workpiece rotation) are referred to as spun-fibers. We believe, these fibers deserve a thorough consideration, since all possible for liquidation of the original embedded linear birefringence was done principally in these fibers [8].
Birefringence optical fibers appeared in 1978–1983 together with the technologies of thermoelastic inserts into the workpieces for drawing optical fibers. Currently, these fibers are widely used in various sensors of physical quantities.
There are Hi-Bi-fibers of the following types: 1) with an elliptical core [1, 2]; 2) with elliptical shell [3, 4], 3) "Panda" with round and thermoelastic inserts [5, 6]; 4) "Bow tie" with trapezoidal inserts [7]; 5) combinations of types are possible. Spun-fibers are obtained by drawing with rotation of the workpiece for any HiBi-fiber [11–13]. Finally, drawing with rotation of the workpiece without any artificial birefringence and measures are taken to obtain a round core and its position exactly in the fiber center gives a LoBi-fiber [8–10].
All these fibers require measurements of the embedded linear birefringence. This parameter of HiBi-fibers is one of the main – it is related to the property of maintaining the polarization of radiation in these fibers. The spiral structure of the axes of embedded linear birefringence of the spun-fiber ensures maintenance of the direction of rotation of the electric vector of elliptical states necessary to accumulate the Faraday effect with the fiber length, and also allows the fiber to resist the harmful influence of bends. The presence of linear birefringence in a fiber of the LoBi-fibers is a parasitic effect, which also requires measurement. The purpose of this paper is to consider and analyze two basic methods for measuring the embedded birefringence of the above optical fibers (OFs) – the spectral method and the elastic twisting method.
2. THE SPECTRAL METHOD
2.1.
Experimental setting and methods of measurement
Light radiation propagates in birefringence fibers, as in uniaxial birefringence crystals along the direction of the perpendicular optical axis of the crystal. It is known that, in the most general form, a wave with an electric vector that does not coincide with any of the birefringence axes cannot excite a wave of the same polarization in the crystal. There are two waves: ordinary and extraordinary, generally propagating along different directions [14]. However, when the wave vector of the incident wave is oriented along the direction of the perpendicular optical axis of the crystal, the directions of propagation of ordinary and extraordinary waves coincide, they differ in orthogonal polarizations (the ordinary wave is polarized perpendicular to the optical axis, extraordinary – along the optical axis) and, respectively, in the propagation velocities (refraction indices nx and ny). In a birefringence fiber, the direction of propagation coincides with the fiber axis.
Linear birefringence is determined by the difference in refractive indices nx – ny of two linearly polarized waves whose electric vectors are oriented along the x and y of the indicated birefringence. On the length z of the fiber, the phase difference Δϕ between the waves of orthogonal polarizations reaches the value:
Δϕ = k0 · (nx – ny) · z = (2π / λ0) · (nx – ny) · z,
where k0 is wave vector of the wave, λ0 is wavelength in vacuo. The beat length Lb is more often characteristic of birefringence, i. e. fiber length z = Lb, at which the phase difference reaches Δϕ (z = Lb) = 2π:
Lb = λ0 / (nx – ny).
Against the background of various ways of measuring the beat lengths Lb [15–19], one can single out the spectral method popular because of its apparent simplicity [20, 21]. Figure 1 shows a scheme of the arrangement for such measurements. The broadband radiation (width Δλ ~ 80 nm at the level of –60 dbm) of source 1 (erbium superluminescent source) passes fiber optic polarizer 2 and passes through welded joint 3 to measured fiber 4. Welded connection 3 is designed in such a way that the electric wave vector at the output of polarizer 2 at an angle of 45° to the axes of the linear birefringence embedded in measured fiber 4 (45° welded connection). As a result, two linearly polarized waves propagating in the HiBi-fiber have approximately equal intensities and orthogonal polarizations. To observe the interference of these waves, we also need a fiber-optic polarizer-analyzer 6 also connected through 45° welded connection 5. Polarizer 6 output is connected to the spectrum analyzer 7. Figure 2 shows a typical interference spectrum observed on the screen of spectroanalyser 7:
I = I0 · [1 + cos(Δϕ)] = I0 · [1 + cos[(2π / λ0) · (nx – ny) · z]].
The logarithm of the intensity I is plotted along the ordinate axis. If we do not take into account the change in the RI with the wavelength, i. e., nx – ny = const, then the beat length is calculated by the formula [20]:
Lb = (Δλ / λ0) · z. (1)
The beat length of the HiBi-fiber can thus be determined by knowing the distance from the spectrum Δλ between two adjacent minima in the spectrum of the interference pattern, the wavelength λ0 of one of the minima, and the length of the measured segment z of the fiber:
2.2. HiBi-fibers.
The above birefringence measurement method [20] was criticized in [21] on the grounds that it does not take into account the dispersion of the refractive indices, at least in the wavelength range ~Δλ. Indeed, taking into account the dispersion of the RI, we obtain [21]:
(Δλ / λ) · z = λ{(nx – ny) – [d(nx – ny) / dλ] · λ}–1 = Lbgr. (2)
The quantity in curly brackets, by definition, is a group birefringence (with a beat length Lbgr). Thus, the formula (1) and the proposed spectral method allows to determine beat length in Hibi-fibers only of the group linear birefringence. To determine the beat length of the phase birefringence, additional studies of the dispersion of the RI nx and ny in the spectral range λ ± Δλ are required.
For example, a segment of HiBi-fiber z = 1,0 m is placed in the measurement scheme of Fig.1. Working wavelength λ0 = 1 550 nm, experimental spectral interval between two minima Δλ = 14 nm. The group beat length according to formula (1) Lbgr = 9,0 mm. Dispersions of the RI nx and ny should be measured separately to obtain a "phase" birefringence. The error in the determination of Δλ in the spectroanalyzer is ±0.02 nm, which makes it possible to measure the beat length up to Lbgr ~ 0.1 mm. The limitation of weak birefringence in the circuit in Fig. 1 is the width of the source spectrum and the length of the fiber. In our case, the maximum interval between two minima Δλmax ~ 40 nm, the maximum length of HiBi-fiber, limited coherence length of the radiation source, z ~ 6 m. Indeed, the homogeneous width of the source spectrum is δλ ~ 2 nm ≈ 10% of the inhomogeneous width due to various transitions between Er+3 levels. The length of the coherence (train length) in this case is ~ 1, 2 mm: lkog = λ2 / δλ ≈ 1.2 mm, and the maximum possible fiber length z max at nx – ny ~ 2 · 10–4:
zmax = l kog / (nx· – n y) ≈ 6 m.
Assuming the above values: zmax ~ 6 m, Δλ ~ 40 nm, for the working wavelength λ0 = 1550 nm from the formula (1) we obtain Lbgr = 150 mm. Thus, according to our estimates, the spectral method with the erbium superluminescent source can be measured by "group" beat length linear birefringence in HiBi-fibers in the range of 150 mm ≤ Lbgr ≤ 0.1 mm.
2.3. Spun-fibers.
Spectral measurements of embedded linear birefringence in the spun-fibers is carried out similar to measurements of HiBi-fibers with the setup shown in Figure 1, but the processing of the results should be carried out by other formulas. Indeed, the nature of light propagation in the spun-fibers is substantially different from that considered above. The initially linearly polarized light during propagation in the spun-fiber has a plane of polarization is rotatable, and periodic changes of ellipticity angle are observed. An illustration of this rotation can be given in Fig.3, which shows the evolution of polarization states as the light propagates in the spun-fiber. Figure 3 shows the Poincare spheres, where the longitude of the place is a doubled azimuth angle (the change in the azimuth angle is the rotation of the plane of polarization), and the latitude is a doubled ellipticity angle of the light radiation. Comparing Fig. 3a and Fig. 3b, it’s clear that the magnitude of rotation of the plane of polarization depends on the value of the embedded linear birefringence [22]. The maxima and minima observed pattern (such as Figure 2) are now dependent on the orientation of the electric vector of the wave with respect to the orientation of the analyzer transmittance plane. If we do not take into account the variance of the RI, we obtain a formula for determining the beat length of the embedded birefringence from the experiment [23]:
L2b exp = (Δλ / 2λ) · Ltw · z. (3)
Beat length of the embedded linear birefringence of the spun-fiber can thus be determined by knowing the distance from the spectrum Δλ between two adjacent minima in the spectrum, the wavelength λ0 of one of the minima, the pitch of the spiral structure Ltw of the axes of linear birefringence embedded into spun-fiber, and the length of the measured interval z of the spun-fiber.
As shown in [22], the experimental value of the beat length Lb exp is the geometric mean of the group Lbgr and phase Lbph the beat length:
L2b exp = Lbgr · Lbph = (Δλ / 2λ0) · Ltw · z. (4)
Spun-fiber is drawn from the workpiece with a sufficiently strong linear birefringence, exposing the workpiece to a rapid rotation to provide one turn of the workpiece on the drawing length equal to the pitch of the helical structure Ltw. In principle, you can draw HiBi-fiber from the same workpiece, if you do not rotate the workpiece. In this case, as shown above, the spectral method can determine the group of the beat length of HiBi-fiber Lbgr, experimental beat length of spun-fiber Lb exp and, assuming that when drawing, the beat length of the linear birefringence embedded into the fiber does not change, according to the formula (4), it is possible to determine the phase beat length L bph.
For example, the segment of spun-fiber z = 10 m is placed in the measurement scheme shown in Fig. The working wavelength λ0 = 1550 nm, the experimental spectral interval between two minima Δλexp = 10 nm, the pitch of the spiral structure of the axes of the linear birefringence Ltw = 3 mm. The experimental beat length according to formula (3) Lb exp = 10,3 mm. Let the study of the HiBi-fiber drawn from the same workpiece but without rotation, by the same spectral method has given a value of group beat length of Lbgr = 9,6 mm, then the phase length of the embedded linear birefringence of the spun-fiber according to formula (4) will be Lbph = 11 mm. As it was shown in [22], in temperature dependences Lbgr (T) and Lbph (T) have the same temperature coefficients (slopes of the plots), which allows us to hope that during the drawing, the embedded birefringence does not change.
The rotation of the polarization plane is due to the lag of one circularly polarized mode relative to the other as the light propagates along the spun-fiber. This delay ϕL – ϕR leads to a rotation of the polarization plane of the total mode at the fiber output by an angle of δθ equal to half phase delay δθ = (ϕL – ϕR) / 2. The refractive index difference nL – nR of the left-polarized and right-polarized modes is determined by the formula [25]:
nL – nR = λ · Ltw / (2Lb)2.
For spun-fibers with parameters Ltw = 3 mm; Lb = 10 mm difference nL – nR ~ 10–5, which is an order of magnitude smaller than the linear birefringence value nx – ny ~ 2 · 10–4 for the same parameters. Therefore, for the same source coherence length lcoh = 1.2 mm, as in 2.2, the maximum possible fiber length zmax:
zmax = lког / (nL – nR) ≈ 120 м.
Assuming the values: zmax ~ 120 m, Δλ ~ 40 nm, for the working wavelength λ0 = 1550 nm from formula (4) we obtain the upper limit Lb exp = 70 mm. Minimum value Lb exp, as in 2.2, is determined by the resolving power of the spectroanalyzer ±0.02 nm, which gives, for a fiber length z = 1 m by formula (4) the limitation from below Lb exp = 0.02 mm. Thus, the spectral method with an erbium superluminescent source can be used for measurement of "mixed" linear birefringence beat length in the spun-fibers in the range of 70 mm ≤ Lb exp ≤ 0.02 mm.
2.4. LoBi-fibers
In the workpieces for the LoBi-fibers, all the possible causes of birefringence are thoroughly removed: the tubes for the workpiece are "ideally" round, the deposition of the substances forming the core and the collapse of the workpiece are made with homogeneous heating along the azimuth of the workpiece, the core is formed precisely along the center of the workpiece, the axis of rotation when drawing LoBi-fiber is exactly along the center of the workpiece, etc. Our [24], however, showed that LoBi-fibers made from our workpieces and foreign workpieces firms had weak birefringence with a beat length of up to Lb ~ 80 mm. In the foreign-made LoBi-fibers available to us, embedded linear birefringence was of the same order. We believe that it is associated with a very small misalignment of the axis of rotation when drawing with the center of the core in the workpiece. In this sense, real LoBi-fibers can be referred to spun-fibers with a very weak embedded linear birefringence. This makes it possible to apply the spectral method to LoBi-fibers with the calculations and according to formula (4) and the limitations in 2.3: 70 mm ≤ Lb exp ≤ 0,02 mm.
A weak birefringence creates additional difficulties in the use of the spectral method. To measure large beating lengths, long lengths of measured LoBi fibers ~ 100 m are required. Winding such segments even over relatively large radii will lead to an additional, induced by bending, linear DLB, which will be measured as an embedded linear SLP, since the embedded BOFS itself is small.
3.
THE METHOD OF ELASTIC TWISTING OF A FIBER AROUND AN AXIS
3.1.
Experimental setting of elastic twisting and measurement technique
The method of elastic twisting around the fiber axis makes it possible to measure the phase beat length of the birefringence embedded in the fiber. During twisting, up to 10ч15 revolutions, the influence of the circular birefringence formed is negligible and does not interfere with the relatively simple interpretation of the results through the phase delay Rtd between modes differing in orthogonal polarizations. They are fast and slow waves in the HiBi-fibers, and in the spun- fibers, this is the phase delay between linearly polarized modes, into which an elliptically polarized state resulting from the propagation of light through the spun fiber can be propagated (see Fig. 3). The angle of phase delay Rtd is equal to doubled angle of ellipticity Rtd = 2δ = 2 · arc tg(b / a).
The scheme of the experimental setting is shown in Fig. As a light source, semiconductor laser 1 is used that generates radiation with a wavelength l = 1.55 microns, fed from current stabilizer 2. Lens 3 generates a parallel light beam interrupted by mechanical modulator 4, and passing through linear polarizer 5. Rotating polarizer changes azimuthal angle a relative to the position of axes of linear birefringence at the OF inlet. Lens 6 with alignment table 7 focuses radiation at the input end of test fiber 8 fixed to platform 9 of the tension device using optical adhesive 10. The tension is transmitted through a thread let through block 11, and its magnitude is regulated by load 12 (~ 0.4 N). The OF rotation around the fiber axis is provided by rotating mechanism 13, on the platform of which the second end of the fiber is fixed with optical glue 10 (the length of the elastically twisted fiber segment is z). Radiation supplied from the OF output passes analyzer 14 and is registered with photodiode 15. The modulated photocurrent is supplied to synchronous amplifier 16, the signal value is recorded with computer 17.
The phase delay Rtd is determined by the formula:
cos Rtd = (Imax – Imin) / (Imax + Imin), (5)
where Imax, Imin are the intensities of radiation received by the rotation of analyzer 14, characterizing ellipticity of light output in the absence of sun dichroism fibers.
As shown in [25], the general formula for the phase delay Rtd, applicable to all types of birefringence fibers, is:
, (6)
where:
Δβ = 2π / Lb is slew rate of the phase delay with a length z of fiber between the waves of orthogonal linear polarizations of embedded linear birefringence with the beat length Lb;
γ0 = 2π / LC is similarly, the rate of increase of the phase delay between the waves of orthogonal circular polarizations due to a circular birefringence with the beat length LC;
ξ0 = 2π / Ltw is angular velocity of the axes of the embedded linear birefringence in the helical structure with a spiral pitch Ltw;
ξ = ξ0 ± ϕ / z; ±ϕ is the angle of elastic twisting of a fiber sample with length z (the sign is chosen with respect to the twisting direction in the spiral structure);
γ = ϕ / µz is considered circular birefringence that occurs with elastic twisting of the fiber around its axis [26], µ = 6,85;
α is the azimuth angle of the input polarization state;
, (7)
Ω is spatial frequency.
As can be seen from formula (7), the elastic twisting angle ϕ enters the spatial frequency Ω and provides periodic variation of the phase delay Rtd in the process of elastic twisting. Moreover, as shown by Rashley [26], elastic twisting causes a weak circular birefringence:
γ = ϕ / (µz), (8)
where µ = 6.85 for quartz fibers. Physically, this means that in the fiber with length z circular birefringence beat length LC = z can be obtained, only making in the fiber length LC of about 7 turns around its axis. As the experiment [25] shows, the effect of the circular birefringence induced by twisting becomes noticeable starting from ~20 turns around the fiber axis at a specimen length z = 1 m. Formulas (5) – (8) are derived for monochromatic light, so the beat length here is phase.
The measurement procedure is as follows. By rotating input polarizer 5, an arbitrary angle α is set with respect to the axes of the birefringence of the studied fiber. Rotation of analyzer 14 produces the maximum Imax and minimum Imin values of the intensities of the elliptical polarization at the fiber output. Calculations are made using formula (5) cos Rtd. Then, input polarizer 5 is rotated by a small angle and using analyzer 14 an attempt is made to obtain the maximum value cos Rtd. When the value of cos Rtd ~ 1 is reached, it can be argued that the azimuth angle is α = 0, i. e., at the input, light is polarized along one of the fiber birefringence axes. By counting from this position of polarizer 5, any desired value of the azimuth angle α can be obtained. Further measurement procedures differ for different birefringence fibers: HiBi-, LoBi- and spun- fibers.
3.2. HiBi- fiber
For application of the method to HiBi-fibers it is necessary to modify formula (6). These fibers are drawn without rotation of the workpiece, therefore, ξ0 = 0. Therefore, in the expression for the new spatial frequency Ω, Δβ / 2 becomes significantly larger than the second component:
, (9)
indeed, Δβ ~ 2π / (2ч10 mm), and ϕ / z ~ N · 2π / (1 m), which even at 10 turns of the elastic correction satisfies condition (9). Furthermore, to satisfy condition (9), we can always increase the length z of the measured fiber. Up to components of the second order of smallness, the spatial frequency Ω ≈ Δβ / 2. Next, we choose the azimuth angle α = 0. The expression (6) is simplified:
.
The function sin Rtd linearly depends on the twisting angle ϕ. Fig. 5 shows plot of dependence Rtd (ϕ) calculated by accurate formula (6). It can be seen that the maximum of the oscillations due to the expression 1 – cos 2Ωz lie on one straight line.
The tangent of the slope angle tgδ of the straight line passing through the maxima Rtdmax(ϕ) makes it possible to determine the beat length of a linear birefringence embedded into the HiBi-fiber:
.
The measurement procedure is as follows. The HiBi- fiber is installed on the setting (Figure 4) and is fixed with optical adhesive. Tensile load of ~0.3 ч 0.5 N is applied. Before rotation, the position of the azimuth angle α = 0 is determined as in 3.1. This corresponds to the case os Rtd = 1, Rtd = 0. The fiber is rotated around its axis with a pitch of 10°ч20°. Without changing the position of the input polarizer and by rotating analyzer 14, the values Imax and Imin are determined according to the signal received by the photodetector and according to formula (5) the phase delay Rtd is determined. The graph such as shown in Fig. 5 is plotted. According to the detected maxima a straight line is plotted, the slope of which, knowing the length of the measured fiber, is used to obtain beat length Lb of the birefringence embedded in HiBi-fiber.
3.3. Spun-fiber
Formula (6) was derived for spun-fibers [25]. Compared with the formula for HIBi- fibers, it takes into account the rotation of the axes of the birefringence forming a spiral structure: ξ = ξ0 + ϕ / z; ξ0 = 2π / Ltw. Moreover, the pitch of the spiral structure is usually short Ltw ~ 3ч4 mm, and the beat length of the embedded birefringence is 3 ч 4 times larger. Thus, the speed of spatial rotation ξ0 exceeds not only the fiber twisting speed along the axis, but also the rate of phase delay rise of orthogonal linearly polarized waves Δβ. Therefore, the elastic twisting is added to (or subtracted from) the rotation axes of birefringence obtained when drawing fibers, as a small additive. This is the reason for the relatively simple behavior of the measured phase delay in the spun-fiber. As can be seen from Fig. 6, with any value of the azimuth angle α function Rtd(ϕ) represents a sine wave of the same amplitude but with the addition of varying constant component. By measuring the amplitude of the sinusoid from peak to peak, we can determine the beat length Lb embedded into the spun-fiber of the linear birefringence. Indeed, it follows from formula (6) in the interval of the first turns of elastic twisting (one can neglect the components with φ / z) at the azimuthal angle α = 0о:
, (10)
and at an azimuth angle α = 45°:
.
As has been said, the pitch of helical structure Ltw of the spun-fibers is 3–4 times less than the beat length Lb of the embedded birefringence, so the second component may be neglected in the square brackets, and the coefficients of the harmonic components become equal. In the experiment, the difference in amplitudes for various azimuthal angles α is practically not observed [25]. For the convenience of measurements, it is desirable to have a theoretical dependence of the maximum value of Rtdmax at α = 0о from the beat length Lb, calculated from (6) for the known value of the spiral structure pitch Ltw. The plot of this dependency is given in Fig. 7 (solid line) for Ltw = 3 , z = 1 m. It can be seen that the function Rtdmax(Lb) is ambiguous, and additional information is required to accurately determine the phase beat length (e. g., based on the spectral measurement method, see above). Ambiguity can also be eliminated if there is a fiber with a different twisting pitch Ltw., e. g., with a lower speed, but it is drawn from the same very workpiece. Figure 7 also shows dependence Rtdmax(Lb) for Ltw = 5 mm (dotted line). It is seen that for large values of Lb (right side of the plot), the values of Rtdmax(Lb) increase with a transition from Ltw = 3 mm, to Ltw = 5 mm, whereas for small values Lb (the left part of the plot), these values decrease. Thus, with respect to Rtdmax(Lb) for two fibers, one cannot eliminate the ambiguity of the plot.
The measurement procedure is as follows. A spun-fiber is placed on the machine (Figure 4) and is fixed with optical adhesive. The tensile load ~0,3ч0,5 N is applied. Before rotation, the position of the azimuth angle α = 0 is determined in accordance with 3.1. This corresponds to the case cos cos Rtd = 1, Rtd = 0. The fiber is rotated around its axis with a pitch of of 10°ч20°. Without changing the position of the input polarizer, and by rotating analyzer 14, the values Imax and Imin are determined according to the signal received by the photodetector and according to formula (6, 10) the phase delay Rtd is determined. A graph such is shown in Fig. 7 is plotted. The amplitude of the oscillations from peak to peak is used to obtain the value Rtdmax. According to the theoretical dependence such as shown in Fig. 7 the beat length Lb is determined.
3.4. LoBi- fibers
The measurement procedure is the same as in 3.3 for spun-fibers. Since at first it was expected that LoBi-fibers can have very large beat lengths (Lb ~ 1 m), it is necessary to evaluate the possibility of measuring the maximum value of the beat length by elastic twisting. The signal-to-noise ratio in our synchronous detection scheme did not exceed S / N = 5 · 105. The maximum ratio of Imax / Imin is limited by signal to noise ratio Imax / Imin ≤ S / N. Substituting in formula (5), we obtain the minimum value of phase delay Rtdmin = 0,6°. which corresponds to Lb ~ 300 mm. However, a great sensitivity was not needed.
Studies of LoBi-fibers show [24] that the phase delay Rtd, measured without elastic twisting, is Rtd ≈ 2° ч 4° for many companies that carefully excluded the causes of parasitic birefringence. Rtdmax ≈ 2°, according to formula (6), corresponds to the beat length Lb ≈ 80 mm and Сos Rtd = 0,9994. Thus, it can be argued that during the drawing process, LoBi-fibers acquire a significant embedded linear birefringence.
CONCLUSION
Physical bases and areas of applicability of two basic methods for measuring the embedded birefringence of optical fibers are considered: the spectral method and the elastic twisting method. The application of these methods to three OF types has been demonstrated: OFs maintaining linear polarization (HiBi-fibers), OFs maintaining the direction of rotation of the electric field vector (spun-fibers), and OFs with the rotation of the workpiece, in which artificial birefringence is minimized (LoBi-fibers). The technique of measuring the beat length of the embedded birefringence is described in detail. Areas of applicability of the methods are considered.
The phenomenon of linear birefringence manifests itself at various propagation velocities of two linearly polarized waves: ordinary and extraordinary. In an optical fiber, both waves propagate along its axis. Symmetry of elastic stresses or geometric symmetry of inserts of various materials in the design of birefringence fibers allows us to distinguish two orthogonal axes of birefringence – fast and slow. A wave with an electric vector parallel to the fast axis will propagate faster than a wave with an electric vector directed along the slow axis, i. e., the refraction index (RI) of the first wave will be smaller than that of the second wave. The axes of the linear birefringence are x and y, and the RI of the waves with linear polarizations along these axes are respectively nx and ny. The linear BOFs value is measured by the difference nx – n.
The birefringence optic fibers include fibers drawn from the workpieces with embedded linear birefringence: HiBi-fibers [1–7], and drawn from the same workpieces, but with the rotation of the workpiece: spun-fibers [11–13]. HiBi-fibers are also referred to as the PM- (polarization maintaining) fibers, i. e., fibers that retain polarization. The so called LoBi-fibers [8–10] stand somewhat separately which by the manufacturing method (workpiece rotation) are referred to as spun-fibers. We believe, these fibers deserve a thorough consideration, since all possible for liquidation of the original embedded linear birefringence was done principally in these fibers [8].
Birefringence optical fibers appeared in 1978–1983 together with the technologies of thermoelastic inserts into the workpieces for drawing optical fibers. Currently, these fibers are widely used in various sensors of physical quantities.
There are Hi-Bi-fibers of the following types: 1) with an elliptical core [1, 2]; 2) with elliptical shell [3, 4], 3) "Panda" with round and thermoelastic inserts [5, 6]; 4) "Bow tie" with trapezoidal inserts [7]; 5) combinations of types are possible. Spun-fibers are obtained by drawing with rotation of the workpiece for any HiBi-fiber [11–13]. Finally, drawing with rotation of the workpiece without any artificial birefringence and measures are taken to obtain a round core and its position exactly in the fiber center gives a LoBi-fiber [8–10].
All these fibers require measurements of the embedded linear birefringence. This parameter of HiBi-fibers is one of the main – it is related to the property of maintaining the polarization of radiation in these fibers. The spiral structure of the axes of embedded linear birefringence of the spun-fiber ensures maintenance of the direction of rotation of the electric vector of elliptical states necessary to accumulate the Faraday effect with the fiber length, and also allows the fiber to resist the harmful influence of bends. The presence of linear birefringence in a fiber of the LoBi-fibers is a parasitic effect, which also requires measurement. The purpose of this paper is to consider and analyze two basic methods for measuring the embedded birefringence of the above optical fibers (OFs) – the spectral method and the elastic twisting method.
2. THE SPECTRAL METHOD
2.1.
Experimental setting and methods of measurement
Light radiation propagates in birefringence fibers, as in uniaxial birefringence crystals along the direction of the perpendicular optical axis of the crystal. It is known that, in the most general form, a wave with an electric vector that does not coincide with any of the birefringence axes cannot excite a wave of the same polarization in the crystal. There are two waves: ordinary and extraordinary, generally propagating along different directions [14]. However, when the wave vector of the incident wave is oriented along the direction of the perpendicular optical axis of the crystal, the directions of propagation of ordinary and extraordinary waves coincide, they differ in orthogonal polarizations (the ordinary wave is polarized perpendicular to the optical axis, extraordinary – along the optical axis) and, respectively, in the propagation velocities (refraction indices nx and ny). In a birefringence fiber, the direction of propagation coincides with the fiber axis.
Linear birefringence is determined by the difference in refractive indices nx – ny of two linearly polarized waves whose electric vectors are oriented along the x and y of the indicated birefringence. On the length z of the fiber, the phase difference Δϕ between the waves of orthogonal polarizations reaches the value:
Δϕ = k0 · (nx – ny) · z = (2π / λ0) · (nx – ny) · z,
where k0 is wave vector of the wave, λ0 is wavelength in vacuo. The beat length Lb is more often characteristic of birefringence, i. e. fiber length z = Lb, at which the phase difference reaches Δϕ (z = Lb) = 2π:
Lb = λ0 / (nx – ny).
Against the background of various ways of measuring the beat lengths Lb [15–19], one can single out the spectral method popular because of its apparent simplicity [20, 21]. Figure 1 shows a scheme of the arrangement for such measurements. The broadband radiation (width Δλ ~ 80 nm at the level of –60 dbm) of source 1 (erbium superluminescent source) passes fiber optic polarizer 2 and passes through welded joint 3 to measured fiber 4. Welded connection 3 is designed in such a way that the electric wave vector at the output of polarizer 2 at an angle of 45° to the axes of the linear birefringence embedded in measured fiber 4 (45° welded connection). As a result, two linearly polarized waves propagating in the HiBi-fiber have approximately equal intensities and orthogonal polarizations. To observe the interference of these waves, we also need a fiber-optic polarizer-analyzer 6 also connected through 45° welded connection 5. Polarizer 6 output is connected to the spectrum analyzer 7. Figure 2 shows a typical interference spectrum observed on the screen of spectroanalyser 7:
I = I0 · [1 + cos(Δϕ)] = I0 · [1 + cos[(2π / λ0) · (nx – ny) · z]].
The logarithm of the intensity I is plotted along the ordinate axis. If we do not take into account the change in the RI with the wavelength, i. e., nx – ny = const, then the beat length is calculated by the formula [20]:
Lb = (Δλ / λ0) · z. (1)
The beat length of the HiBi-fiber can thus be determined by knowing the distance from the spectrum Δλ between two adjacent minima in the spectrum of the interference pattern, the wavelength λ0 of one of the minima, and the length of the measured segment z of the fiber:
2.2. HiBi-fibers.
The above birefringence measurement method [20] was criticized in [21] on the grounds that it does not take into account the dispersion of the refractive indices, at least in the wavelength range ~Δλ. Indeed, taking into account the dispersion of the RI, we obtain [21]:
(Δλ / λ) · z = λ{(nx – ny) – [d(nx – ny) / dλ] · λ}–1 = Lbgr. (2)
The quantity in curly brackets, by definition, is a group birefringence (with a beat length Lbgr). Thus, the formula (1) and the proposed spectral method allows to determine beat length in Hibi-fibers only of the group linear birefringence. To determine the beat length of the phase birefringence, additional studies of the dispersion of the RI nx and ny in the spectral range λ ± Δλ are required.
For example, a segment of HiBi-fiber z = 1,0 m is placed in the measurement scheme of Fig.1. Working wavelength λ0 = 1 550 nm, experimental spectral interval between two minima Δλ = 14 nm. The group beat length according to formula (1) Lbgr = 9,0 mm. Dispersions of the RI nx and ny should be measured separately to obtain a "phase" birefringence. The error in the determination of Δλ in the spectroanalyzer is ±0.02 nm, which makes it possible to measure the beat length up to Lbgr ~ 0.1 mm. The limitation of weak birefringence in the circuit in Fig. 1 is the width of the source spectrum and the length of the fiber. In our case, the maximum interval between two minima Δλmax ~ 40 nm, the maximum length of HiBi-fiber, limited coherence length of the radiation source, z ~ 6 m. Indeed, the homogeneous width of the source spectrum is δλ ~ 2 nm ≈ 10% of the inhomogeneous width due to various transitions between Er+3 levels. The length of the coherence (train length) in this case is ~ 1, 2 mm: lkog = λ2 / δλ ≈ 1.2 mm, and the maximum possible fiber length z max at nx – ny ~ 2 · 10–4:
zmax = l kog / (nx· – n y) ≈ 6 m.
Assuming the above values: zmax ~ 6 m, Δλ ~ 40 nm, for the working wavelength λ0 = 1550 nm from the formula (1) we obtain Lbgr = 150 mm. Thus, according to our estimates, the spectral method with the erbium superluminescent source can be measured by "group" beat length linear birefringence in HiBi-fibers in the range of 150 mm ≤ Lbgr ≤ 0.1 mm.
2.3. Spun-fibers.
Spectral measurements of embedded linear birefringence in the spun-fibers is carried out similar to measurements of HiBi-fibers with the setup shown in Figure 1, but the processing of the results should be carried out by other formulas. Indeed, the nature of light propagation in the spun-fibers is substantially different from that considered above. The initially linearly polarized light during propagation in the spun-fiber has a plane of polarization is rotatable, and periodic changes of ellipticity angle are observed. An illustration of this rotation can be given in Fig.3, which shows the evolution of polarization states as the light propagates in the spun-fiber. Figure 3 shows the Poincare spheres, where the longitude of the place is a doubled azimuth angle (the change in the azimuth angle is the rotation of the plane of polarization), and the latitude is a doubled ellipticity angle of the light radiation. Comparing Fig. 3a and Fig. 3b, it’s clear that the magnitude of rotation of the plane of polarization depends on the value of the embedded linear birefringence [22]. The maxima and minima observed pattern (such as Figure 2) are now dependent on the orientation of the electric vector of the wave with respect to the orientation of the analyzer transmittance plane. If we do not take into account the variance of the RI, we obtain a formula for determining the beat length of the embedded birefringence from the experiment [23]:
L2b exp = (Δλ / 2λ) · Ltw · z. (3)
Beat length of the embedded linear birefringence of the spun-fiber can thus be determined by knowing the distance from the spectrum Δλ between two adjacent minima in the spectrum, the wavelength λ0 of one of the minima, the pitch of the spiral structure Ltw of the axes of linear birefringence embedded into spun-fiber, and the length of the measured interval z of the spun-fiber.
As shown in [22], the experimental value of the beat length Lb exp is the geometric mean of the group Lbgr and phase Lbph the beat length:
L2b exp = Lbgr · Lbph = (Δλ / 2λ0) · Ltw · z. (4)
Spun-fiber is drawn from the workpiece with a sufficiently strong linear birefringence, exposing the workpiece to a rapid rotation to provide one turn of the workpiece on the drawing length equal to the pitch of the helical structure Ltw. In principle, you can draw HiBi-fiber from the same workpiece, if you do not rotate the workpiece. In this case, as shown above, the spectral method can determine the group of the beat length of HiBi-fiber Lbgr, experimental beat length of spun-fiber Lb exp and, assuming that when drawing, the beat length of the linear birefringence embedded into the fiber does not change, according to the formula (4), it is possible to determine the phase beat length L bph.
For example, the segment of spun-fiber z = 10 m is placed in the measurement scheme shown in Fig. The working wavelength λ0 = 1550 nm, the experimental spectral interval between two minima Δλexp = 10 nm, the pitch of the spiral structure of the axes of the linear birefringence Ltw = 3 mm. The experimental beat length according to formula (3) Lb exp = 10,3 mm. Let the study of the HiBi-fiber drawn from the same workpiece but without rotation, by the same spectral method has given a value of group beat length of Lbgr = 9,6 mm, then the phase length of the embedded linear birefringence of the spun-fiber according to formula (4) will be Lbph = 11 mm. As it was shown in [22], in temperature dependences Lbgr (T) and Lbph (T) have the same temperature coefficients (slopes of the plots), which allows us to hope that during the drawing, the embedded birefringence does not change.
The rotation of the polarization plane is due to the lag of one circularly polarized mode relative to the other as the light propagates along the spun-fiber. This delay ϕL – ϕR leads to a rotation of the polarization plane of the total mode at the fiber output by an angle of δθ equal to half phase delay δθ = (ϕL – ϕR) / 2. The refractive index difference nL – nR of the left-polarized and right-polarized modes is determined by the formula [25]:
nL – nR = λ · Ltw / (2Lb)2.
For spun-fibers with parameters Ltw = 3 mm; Lb = 10 mm difference nL – nR ~ 10–5, which is an order of magnitude smaller than the linear birefringence value nx – ny ~ 2 · 10–4 for the same parameters. Therefore, for the same source coherence length lcoh = 1.2 mm, as in 2.2, the maximum possible fiber length zmax:
zmax = lког / (nL – nR) ≈ 120 м.
Assuming the values: zmax ~ 120 m, Δλ ~ 40 nm, for the working wavelength λ0 = 1550 nm from formula (4) we obtain the upper limit Lb exp = 70 mm. Minimum value Lb exp, as in 2.2, is determined by the resolving power of the spectroanalyzer ±0.02 nm, which gives, for a fiber length z = 1 m by formula (4) the limitation from below Lb exp = 0.02 mm. Thus, the spectral method with an erbium superluminescent source can be used for measurement of "mixed" linear birefringence beat length in the spun-fibers in the range of 70 mm ≤ Lb exp ≤ 0.02 mm.
2.4. LoBi-fibers
In the workpieces for the LoBi-fibers, all the possible causes of birefringence are thoroughly removed: the tubes for the workpiece are "ideally" round, the deposition of the substances forming the core and the collapse of the workpiece are made with homogeneous heating along the azimuth of the workpiece, the core is formed precisely along the center of the workpiece, the axis of rotation when drawing LoBi-fiber is exactly along the center of the workpiece, etc. Our [24], however, showed that LoBi-fibers made from our workpieces and foreign workpieces firms had weak birefringence with a beat length of up to Lb ~ 80 mm. In the foreign-made LoBi-fibers available to us, embedded linear birefringence was of the same order. We believe that it is associated with a very small misalignment of the axis of rotation when drawing with the center of the core in the workpiece. In this sense, real LoBi-fibers can be referred to spun-fibers with a very weak embedded linear birefringence. This makes it possible to apply the spectral method to LoBi-fibers with the calculations and according to formula (4) and the limitations in 2.3: 70 mm ≤ Lb exp ≤ 0,02 mm.
A weak birefringence creates additional difficulties in the use of the spectral method. To measure large beating lengths, long lengths of measured LoBi fibers ~ 100 m are required. Winding such segments even over relatively large radii will lead to an additional, induced by bending, linear DLB, which will be measured as an embedded linear SLP, since the embedded BOFS itself is small.
3.
THE METHOD OF ELASTIC TWISTING OF A FIBER AROUND AN AXIS
3.1.
Experimental setting of elastic twisting and measurement technique
The method of elastic twisting around the fiber axis makes it possible to measure the phase beat length of the birefringence embedded in the fiber. During twisting, up to 10ч15 revolutions, the influence of the circular birefringence formed is negligible and does not interfere with the relatively simple interpretation of the results through the phase delay Rtd between modes differing in orthogonal polarizations. They are fast and slow waves in the HiBi-fibers, and in the spun- fibers, this is the phase delay between linearly polarized modes, into which an elliptically polarized state resulting from the propagation of light through the spun fiber can be propagated (see Fig. 3). The angle of phase delay Rtd is equal to doubled angle of ellipticity Rtd = 2δ = 2 · arc tg(b / a).
The scheme of the experimental setting is shown in Fig. As a light source, semiconductor laser 1 is used that generates radiation with a wavelength l = 1.55 microns, fed from current stabilizer 2. Lens 3 generates a parallel light beam interrupted by mechanical modulator 4, and passing through linear polarizer 5. Rotating polarizer changes azimuthal angle a relative to the position of axes of linear birefringence at the OF inlet. Lens 6 with alignment table 7 focuses radiation at the input end of test fiber 8 fixed to platform 9 of the tension device using optical adhesive 10. The tension is transmitted through a thread let through block 11, and its magnitude is regulated by load 12 (~ 0.4 N). The OF rotation around the fiber axis is provided by rotating mechanism 13, on the platform of which the second end of the fiber is fixed with optical glue 10 (the length of the elastically twisted fiber segment is z). Radiation supplied from the OF output passes analyzer 14 and is registered with photodiode 15. The modulated photocurrent is supplied to synchronous amplifier 16, the signal value is recorded with computer 17.
The phase delay Rtd is determined by the formula:
cos Rtd = (Imax – Imin) / (Imax + Imin), (5)
where Imax, Imin are the intensities of radiation received by the rotation of analyzer 14, characterizing ellipticity of light output in the absence of sun dichroism fibers.
As shown in [25], the general formula for the phase delay Rtd, applicable to all types of birefringence fibers, is:
, (6)
where:
Δβ = 2π / Lb is slew rate of the phase delay with a length z of fiber between the waves of orthogonal linear polarizations of embedded linear birefringence with the beat length Lb;
γ0 = 2π / LC is similarly, the rate of increase of the phase delay between the waves of orthogonal circular polarizations due to a circular birefringence with the beat length LC;
ξ0 = 2π / Ltw is angular velocity of the axes of the embedded linear birefringence in the helical structure with a spiral pitch Ltw;
ξ = ξ0 ± ϕ / z; ±ϕ is the angle of elastic twisting of a fiber sample with length z (the sign is chosen with respect to the twisting direction in the spiral structure);
γ = ϕ / µz is considered circular birefringence that occurs with elastic twisting of the fiber around its axis [26], µ = 6,85;
α is the azimuth angle of the input polarization state;
, (7)
Ω is spatial frequency.
As can be seen from formula (7), the elastic twisting angle ϕ enters the spatial frequency Ω and provides periodic variation of the phase delay Rtd in the process of elastic twisting. Moreover, as shown by Rashley [26], elastic twisting causes a weak circular birefringence:
γ = ϕ / (µz), (8)
where µ = 6.85 for quartz fibers. Physically, this means that in the fiber with length z circular birefringence beat length LC = z can be obtained, only making in the fiber length LC of about 7 turns around its axis. As the experiment [25] shows, the effect of the circular birefringence induced by twisting becomes noticeable starting from ~20 turns around the fiber axis at a specimen length z = 1 m. Formulas (5) – (8) are derived for monochromatic light, so the beat length here is phase.
The measurement procedure is as follows. By rotating input polarizer 5, an arbitrary angle α is set with respect to the axes of the birefringence of the studied fiber. Rotation of analyzer 14 produces the maximum Imax and minimum Imin values of the intensities of the elliptical polarization at the fiber output. Calculations are made using formula (5) cos Rtd. Then, input polarizer 5 is rotated by a small angle and using analyzer 14 an attempt is made to obtain the maximum value cos Rtd. When the value of cos Rtd ~ 1 is reached, it can be argued that the azimuth angle is α = 0, i. e., at the input, light is polarized along one of the fiber birefringence axes. By counting from this position of polarizer 5, any desired value of the azimuth angle α can be obtained. Further measurement procedures differ for different birefringence fibers: HiBi-, LoBi- and spun- fibers.
3.2. HiBi- fiber
For application of the method to HiBi-fibers it is necessary to modify formula (6). These fibers are drawn without rotation of the workpiece, therefore, ξ0 = 0. Therefore, in the expression for the new spatial frequency Ω, Δβ / 2 becomes significantly larger than the second component:
, (9)
indeed, Δβ ~ 2π / (2ч10 mm), and ϕ / z ~ N · 2π / (1 m), which even at 10 turns of the elastic correction satisfies condition (9). Furthermore, to satisfy condition (9), we can always increase the length z of the measured fiber. Up to components of the second order of smallness, the spatial frequency Ω ≈ Δβ / 2. Next, we choose the azimuth angle α = 0. The expression (6) is simplified:
.
The function sin Rtd linearly depends on the twisting angle ϕ. Fig. 5 shows plot of dependence Rtd (ϕ) calculated by accurate formula (6). It can be seen that the maximum of the oscillations due to the expression 1 – cos 2Ωz lie on one straight line.
The tangent of the slope angle tgδ of the straight line passing through the maxima Rtdmax(ϕ) makes it possible to determine the beat length of a linear birefringence embedded into the HiBi-fiber:
.
The measurement procedure is as follows. The HiBi- fiber is installed on the setting (Figure 4) and is fixed with optical adhesive. Tensile load of ~0.3 ч 0.5 N is applied. Before rotation, the position of the azimuth angle α = 0 is determined as in 3.1. This corresponds to the case os Rtd = 1, Rtd = 0. The fiber is rotated around its axis with a pitch of 10°ч20°. Without changing the position of the input polarizer and by rotating analyzer 14, the values Imax and Imin are determined according to the signal received by the photodetector and according to formula (5) the phase delay Rtd is determined. The graph such as shown in Fig. 5 is plotted. According to the detected maxima a straight line is plotted, the slope of which, knowing the length of the measured fiber, is used to obtain beat length Lb of the birefringence embedded in HiBi-fiber.
3.3. Spun-fiber
Formula (6) was derived for spun-fibers [25]. Compared with the formula for HIBi- fibers, it takes into account the rotation of the axes of the birefringence forming a spiral structure: ξ = ξ0 + ϕ / z; ξ0 = 2π / Ltw. Moreover, the pitch of the spiral structure is usually short Ltw ~ 3ч4 mm, and the beat length of the embedded birefringence is 3 ч 4 times larger. Thus, the speed of spatial rotation ξ0 exceeds not only the fiber twisting speed along the axis, but also the rate of phase delay rise of orthogonal linearly polarized waves Δβ. Therefore, the elastic twisting is added to (or subtracted from) the rotation axes of birefringence obtained when drawing fibers, as a small additive. This is the reason for the relatively simple behavior of the measured phase delay in the spun-fiber. As can be seen from Fig. 6, with any value of the azimuth angle α function Rtd(ϕ) represents a sine wave of the same amplitude but with the addition of varying constant component. By measuring the amplitude of the sinusoid from peak to peak, we can determine the beat length Lb embedded into the spun-fiber of the linear birefringence. Indeed, it follows from formula (6) in the interval of the first turns of elastic twisting (one can neglect the components with φ / z) at the azimuthal angle α = 0о:
, (10)
and at an azimuth angle α = 45°:
.
As has been said, the pitch of helical structure Ltw of the spun-fibers is 3–4 times less than the beat length Lb of the embedded birefringence, so the second component may be neglected in the square brackets, and the coefficients of the harmonic components become equal. In the experiment, the difference in amplitudes for various azimuthal angles α is practically not observed [25]. For the convenience of measurements, it is desirable to have a theoretical dependence of the maximum value of Rtdmax at α = 0о from the beat length Lb, calculated from (6) for the known value of the spiral structure pitch Ltw. The plot of this dependency is given in Fig. 7 (solid line) for Ltw = 3 , z = 1 m. It can be seen that the function Rtdmax(Lb) is ambiguous, and additional information is required to accurately determine the phase beat length (e. g., based on the spectral measurement method, see above). Ambiguity can also be eliminated if there is a fiber with a different twisting pitch Ltw., e. g., with a lower speed, but it is drawn from the same very workpiece. Figure 7 also shows dependence Rtdmax(Lb) for Ltw = 5 mm (dotted line). It is seen that for large values of Lb (right side of the plot), the values of Rtdmax(Lb) increase with a transition from Ltw = 3 mm, to Ltw = 5 mm, whereas for small values Lb (the left part of the plot), these values decrease. Thus, with respect to Rtdmax(Lb) for two fibers, one cannot eliminate the ambiguity of the plot.
The measurement procedure is as follows. A spun-fiber is placed on the machine (Figure 4) and is fixed with optical adhesive. The tensile load ~0,3ч0,5 N is applied. Before rotation, the position of the azimuth angle α = 0 is determined in accordance with 3.1. This corresponds to the case cos cos Rtd = 1, Rtd = 0. The fiber is rotated around its axis with a pitch of of 10°ч20°. Without changing the position of the input polarizer, and by rotating analyzer 14, the values Imax and Imin are determined according to the signal received by the photodetector and according to formula (6, 10) the phase delay Rtd is determined. A graph such is shown in Fig. 7 is plotted. The amplitude of the oscillations from peak to peak is used to obtain the value Rtdmax. According to the theoretical dependence such as shown in Fig. 7 the beat length Lb is determined.
3.4. LoBi- fibers
The measurement procedure is the same as in 3.3 for spun-fibers. Since at first it was expected that LoBi-fibers can have very large beat lengths (Lb ~ 1 m), it is necessary to evaluate the possibility of measuring the maximum value of the beat length by elastic twisting. The signal-to-noise ratio in our synchronous detection scheme did not exceed S / N = 5 · 105. The maximum ratio of Imax / Imin is limited by signal to noise ratio Imax / Imin ≤ S / N. Substituting in formula (5), we obtain the minimum value of phase delay Rtdmin = 0,6°. which corresponds to Lb ~ 300 mm. However, a great sensitivity was not needed.
Studies of LoBi-fibers show [24] that the phase delay Rtd, measured without elastic twisting, is Rtd ≈ 2° ч 4° for many companies that carefully excluded the causes of parasitic birefringence. Rtdmax ≈ 2°, according to formula (6), corresponds to the beat length Lb ≈ 80 mm and Сos Rtd = 0,9994. Thus, it can be argued that during the drawing process, LoBi-fibers acquire a significant embedded linear birefringence.
CONCLUSION
Physical bases and areas of applicability of two basic methods for measuring the embedded birefringence of optical fibers are considered: the spectral method and the elastic twisting method. The application of these methods to three OF types has been demonstrated: OFs maintaining linear polarization (HiBi-fibers), OFs maintaining the direction of rotation of the electric field vector (spun-fibers), and OFs with the rotation of the workpiece, in which artificial birefringence is minimized (LoBi-fibers). The technique of measuring the beat length of the embedded birefringence is described in detail. Areas of applicability of the methods are considered.
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