Cancellation of bending-induced birefringence in single-mode fibers: application to Faraday sensors César D. Perciante and José A. Ferrari
We demonstrate that one can cancel the bending-induced linear birefringence in single-mode fibers by inducing a controlled anisotropy in a direction orthogonal to the bending plane. In particular, the controlled anisotropy can be generated by application of a lateral compressive stress on the fiber. This effect can be applied for the construction of birefringence-free fiber coils in Faraday sensor heads (e.g., in current sensors) to improve their sensitivity. © 2006 Optical Society of America OCIS codes: 060.2370, 260.1440.
1. Introduction
Many fiber-optic sensors (i.e., gyroscopes and temperature, pressure, and current sensors; see, e.g., Refs. 1 and 2 and the references therein) rely on the interference of two polarized light waves guided along single-mode optical fibers. Changes in the state of polarization of the light owing to the fiber’s birefringence may cause a fading of the detected signal in fiber-optic sensors. Linear birefringence in singlemode fibers can be induced by the geometry of the core (e.g., elliptical core fibers) or by mechanical stress through the elasto-optic effect. Taking the z direction along the fiber, the stress-induced birefringence will be given by  ⬅ 共2兾兲共␦nx ⫺ ␦ny兲, where is the vacuum wavelength of light and ␦nx,y are the changes in refractive indices for the x and y polarization directions, respectively. From the basic elasto-optic equations3– 6 one obtains  ⫽ ⫺共2兾兲C共x ⫺ y兲,
(1)
C ⬅ 共n3兾2E兲共p11 ⫺ p12兲共1 ⫹ 兲,
(2)
with
C. D. Perciante (
[email protected]) is with the Facultad de Ingeniería y Tecnologías, Universidad Católica del Uruguay, Avenada 8 de Octubre 2738, Montevideo, Uruguay. J. A. Ferrari is with the Instituto de Física, Facultad de Ingeniería, J. Herrera y Reissig 565, Montevideo, Uruguay. Received 28 June 2005; revised 20 October 2005; accepted 2 November 2005. 0003-6935/06/091951-06$15.00/0 © 2006 Optical Society of America
where x,y are the components of stress in the x and y directions, respectively, p11,12 are the strain-optical coefficients, E is Young’s modulus, is Poissons’s ratio, and n is the refractive index of the fiber core. For fused-silica fibers the numerical values are n ⫽ 1.458, p11 ⫽ 0.121, p12 ⫽ 0.270, ⫽ 0.17, and E ⫽ 7.3 ⫻ 1010 N兾m2, and thus the stress-optic coefficient has the value C ⬇ ⫺3.7 ⫻ 10⫺12 m2兾N. Bending-induced birefringence results from the (radial) compressive stress x that builds up in a bent fiber (see Fig. 1). In single-mode fibers, the core radius is small in comparison with the fiber’s outer radius (r), and thus the compressive stress on the core is given by4,5 x ⫽ ⫺共E兾2兲共r兾R兲2 ⫺ z共r兾R兲共2 ⫺ 3兲兾共1 ⫺ 兲, (3) where R is the radius of curvature of the bent fiber and z is the mean longitudinal (along the z direction) stress applied to the fiber. One of the techniques to suppress bending-induced birefringence is to subject the fiber to thermal processing to achieve relaxation of the internal stress.7,8 This processing is slow and difficult, and often it ends with the fiber’s breaking. Other techniques, e.g., twisting a fiber8 and the use of spun highbirefringence (hi-bi) fibers,9,10 have been discussed extensively in the literature. A different approach to reduce bending-induced birefringence is to produce a fiber with low elasto-optic constants, e.g., the flint glass fiber developed by cooperation between Tokyo Electric Power Company and Hoya Corporation.11 However, this fiber type is not currently available for sale. The purpose of this paper is to present some novel 20 March 2006 兾 Vol. 45, No. 9 兾 APPLIED OPTICS
1951
structing birefringence-free fiber coils. In the Section 3 we present some validation experiments, and in Section 4 we discuss the results. 2. Birefringence-Free Fiber Coils
When a fiber loop is subjected to external compressive stress 共y兲 in the y direction orthogonal to the bending plane (see Fig. 2), y will be related to the external force (F) applied in the y direction to the plates used to press the fiber, in the form y ⫽ ⫺
Fig. 1. Compressive (radial) stress x induced in a bent fiber.
ideas for the cancellation (compensation) of the bending-induced birefringence. We propose to induce a controlled anisotropy in the y direction to compensate for the anisotropy in the x direction produced by bending; thus the final result will be an isotropic bent fiber. To the best of our knowledge, in the literature there are no references to this procedure for cancellation of bending-induced birefringence. As we mentioned above, controlled anisotropy (birefringence) in an optical fiber can be produced by the core geometry or by application of stress. The negative sign in front of stress coefficient y in expression (1) means that the birefringence produced by bending-induced (compressive) stress x can be canceled by application of compressive stress y orthogonal to the (xz) bending plane (see Fig. 2). (Clearly, making y ⬇ x is a quite obvious way to cancel the bending-induced birefringence.) Although the possibility of inducing linear birefringence by pressing a fiber is well known and been applied to the construction of pressure sensors and polarization controllers,12,13 the sign of y is not of fundamental importance in those applications. On the contrary, for the application we propose, the negative sign in Eq. (1) is essential for canceling the linear birefringence. In Section 2 we investigate the possibility of con-
aF , lr
where l is the length of the fiber being pressed (i.e., for a closed loop l ⫽ 2R) and a is a dimensionless factor of the order of the unit. This factor depends on the deformation of the fiber’s plastic jacket from the circular geometry when it is laterally pressed, and also on the material of the plates. The idea of applying compressive stress in the y direction can be used to construct birefringence-free fiber coils. The compressive stress can be externally applied (as shown in Fig. 2) or, alternatively, it can be the result of internal stress applicators included in the fiber cladding near the core, as occurs in hibi fibers (e.g., bow-tie, polarization-maintaining and absorption-reducing, and elliptical-cladding fibers). In a typical hi-bi fiber the intrinsic compressive stress is perpendicular to the line joining the stressapplying parts,14 as shown in Fig. 3(a). Hi-bi fibers are characterized by the so-called beat length 共lb兲 that corresponds to a retardance of 2 共rad兲, so its intrinsic linear birefringence 共I兲 has the value
ⱍIⱍ ⫽ 2兾lb.
1952
APPLIED OPTICS 兾 Vol. 45, No. 9 兾 20 March 2006
(5)
Thus, if the stress-applying parts in the cladding are maintained in the x direction during the fiber coiling [see Fig. 3(b)], for a specific radius of curvature (R) the self-induced birefringence of the hi-bi fiber can cancel (compensate for) the bending-induced birefringence in the x direction. The birefringence compensation occurs for |B| ⫽ |I|, where B ⬅ ⫺共2兾兲Cx is the bendinginduced birefringence. Thus, assuming that z ⫽ 0, from Eqs. (1)–(5) we obtain
共2兾兲ⱍCⱍ共E兾2兲共r兾R兲2 ⫽ 2兾lb.
Fig. 2. Optical fiber subjected to external compressive stress in the y direction, i.e., orthogonal to the bending plane.
(4)
(6)
For example, using the typical values for fusedsilica fibers, E ⫽ 7.3 ⫻ 1010 N兾m2, |C| ⬇ 3.7 ⫻ 10⫺12 m2兾N, and r ⫽ 62.5 m, for ⫽ 0.633 m we found that a birefringence-free coil with radius R ⫽ 1 cm can be constructed with a fiber with beat length lb ⫽ 12 cm. Typical commercial hi-bi fibers have beat lengths of the order of millimeters. Thus the intrinsic birefringence of the hi-bi fiber required for producing a birefringence-free coil with radius R ⫽ 1 cm is 1–2 orders of magnitude smaller than the typical com-
Fig. 4. Experimental setup: QW, quarter-wave plate with its fast axis (F.A.) at 45 deg with respect to the polarization direction of the incident laser beam. PL, polarized He–Ne laser; P, rotating polarizer; D, photodetector.
birefringence 共1兾B兲共⭸B兾⭸T兲 for silica fiber is of the order of 5.7 ⫻ 10⫺4 K⫺1 (see, e.g., Ref. 15), while its thermal expansion coefficient is very small, 共1兾l兲 共⭸l兾⭸T兲 ⬇ 5 ⫻ 10⫺7 K⫺1, so the effect of the thermal expansion can be disregarded in expression (8). Now, as the residual birefringence is relatively small, to first order we can write B ⬇ ⫺I; then expression (8) can be rewritten as ⌬␦res ⬇
冉
冊
1 ⭸B 1 ⭸I  l ⌬T. ⫺ B ⭸T l ⭸T 共 B 兲
(9)
Therefore we conclude that, to minimize the effect of temperature on the residual retardance, the hi-bi fiber should have a temperature coefficient Fig. 3. (a) Cross section of a hi-bi fiber with stress-applying parts in the x direction, i.e. the compressive intrinsic stress is in the y direction. (b) Coiled hi-bi fiber with the y direction orthogonal to the (xz) bending plane.
mercial values. Although our discussion was centered on hi-bi fibers with stress-induced birefringence, it is important to remark that the same ideas will also work with hi-bi fibers in which the linear birefringence is induced by the core geometry (e.g., ellipticalcore fibers). The issue of the temperature dependence of the residual retardance in a fiber coil requires special discussion. It is convenient to consider the total induced birefringence 共兲 as the addition of two effects, bending- and intrinsically induced birefringence:  ⫽ B ⫹ I.
(7)
Then the variation of the residual retardance, ␦res 共 ⫽ l兲, with temperature 共T兲 will be
冉 冉
冊
冉 冊 冉 冊
⭸B ⭸I ⭸l l⌬T ⫹ 共B ⫹ I兲 ⌬T ⫹ ⭸T ⭸T ⭸T 1 ⭸B 1 I 1 ⭸l ⫽ Bl兲⌬T ⫹ ␦res ⌬T. ⫹ 共 B ⭸T B ⭸T l ⭸T (8)
⌬␦res ⬇
冊
The temperature coefficient of the bending-induced
冉
1 ⭸I 1 ⭸lb ⫽ I ⭸T lb ⭸T
冊
of the same order that the one that is due to bendinginduced birefringence of the fiber, 共1兾B兲共⭸B兾⭸T兲. If compensation for the temperature effects is not possible, it is desirable at least to minimize these effects as much as possible. For this purpose elliptical-core fibers are advantageous in that retardation changes with temperature are relatively small,16
冉
冊
1 ⭸I ⬇ ⫺2.2 ⫻ 10⫺4 K⫺1 . I ⭸T
In contrast, the temperature dependence of commercial hi-bi fibers with stress-induced birefringence is larger by roughly an order of magnitude. 3. Experimental Results A.
Experiments with Single Loops
We have carried out a series of validation experiments using single-mode fibers 共 ⫽ 633 nm, r ⫽ 62.5 m兲 with very low intrinsic birefringence (fiber type LB600-125P provided by Oxford Electronics). According to the manufacturer’s data, the intrinsic linear birefringence of this fiber is ⬍1 deg兾m. We measure the retardance 共l兲 with the device shown schematically in Fig. 4, which consists of a quarter-wave plate at the fiber input and a rotating polarizer at the output. The laser beam has its polar20 March 2006 兾 Vol. 45, No. 9 兾 APPLIED OPTICS
1953
Fig. 5. Retardance as a function of applied force for a fiber loop with radius R ⫽ 10 ⫾ 1 mm. The total birefringence is canceled for F ⫽ 3.1 N.
Fig. 6. Applied lateral force per unit length (F兾l) required for elimination of birefringence versus 1兾R2.
ization direction at 45° with respect to the quarterwave plate’s fast axis, so the light incident upon the fiber is circularly polarized. Then, the total retardance will be given by
approximately null. (Actually, in our experiments we pressed only a half of the complete loop, so l ⫽ R). From these measurements we obtained the value a ⫽ 0.63 ⫾ 0.02 for the dimensionless factor that appears in expression (4). With this mean value of a, the discrepancy between the measured stress and the theoretical values necessary to cancel the bending-induced birefringence was smaller than 8%.
冉
l ⫽ 兾2 ⫺ arc cos
Imax ⫺ Imin Imax ⫹ Imin
冊
共modulo 兾2兲,
where l is the total length of the fiber loops and Imax,min are the maximum and minimum light intensities after the rotating polarizer, measured at the photodetector. We studied the decrease of the retardance of fiber loops with radii ranging from 10 to 40 mm during the application of a lateral force (F). (Because the bending-induced birefringence decreases as 1兾R2, loops with larger radii of curvature will have less birefringence and thus are less interesting for our study.) Figure 5 shows representative results obtained for a fiber loop with radius of curvature R ⫽ 10 ⫾ 1 mm. The experimental data show that the total birefringence of the fiber loop decreases until cancellation with applied forces ranging from 1.4 to 3.1 N. If we assume that the relation between retardance and the applied force is linear, the mean dispersion of the measurements was of the order of 2°. Also, we measured the force per unit length 共F兾l兲 necessary to cancel the birefringence for several loop radii. Figure 6 shows the applied force per unit length 共F兾l兲 necessary to cancel the birefringence versus 1兾R2. The mean source of errors was the uncertainty in the measurement of the loop radii, which was ⫾1 mm. The experimental data shown in Fig. 6 are consistent, within the experimental errors; with a linear relation between 共F兾l兲 and 1兾R2, as expected from expressions (3) and (4) when the condition y ⬇ x is verified and the longitudinal tension 共 z兲 is 1954
APPLIED OPTICS 兾 Vol. 45, No. 9 兾 20 March 2006
B.
Experiments with Fiber Coils
The purpose of our experiments was to demonstrate that compensation for the bending-induced birefringence in fiber coils with tens of turns is practical. As we have no hi-bi fibers with the required (long) beat length as described in Section 2, we constructed a fiber coil by using a low-bi fiber wound on a hollow cylinder (aluminum former) and applied lateral pressure, using a spring 共S1兲 together with a nut (N) placed at the cylinder end (see Fig. 7). To ensure that the whole fiber was actually subjected to the same lateral compressive stress in a direction orthogonal to the bending plane of each loop, we coiled the fiber together with a (weak) spring 共S0兲 of rectangular cross section, such that each section of fiber lay between the planar walls of the spring. (Of course, there was some amount of residual inhomogeneous stress at the fiber coil ends.) Our coil had 33 fiber turns wound on a cylinder with radius of curvature R ⫽ 8.5 mm, so the total fiber length (l) involved in our experiments was I ⬇ 1.76 m. [We coiled the fiber by applying a moderate longitudinal tension z共r2兲 ⫽ 0.1 N during the coiling.] A fiber coil with these dimensions will have a bending-induced retardance [calculated from expressions (1)–(3)] l ⬇ 198.2 rad, which cannot be measured directly with the setup shown in Fig. 4 because
Fig. 7. Fiber coil on a hollow cylinder subjected to lateral compression. A spring (S0) of rectangular cross section was coiled together with the fiber to achieve lateral compression of the whole fiber. Spring S1 permits fine tuning of the lateral stress applied by a nut (N) placed at the cylinder end.
with this setup it is possible only to obtain the retardance modulo 兾2. An indirect way to estimate the order of magnitude of the absolute value of the retardance during pressing of the fiber coil it is to measure the Faraday rotation 共兲 of linearly polarized light launched into the fiber. For that purpose we introduced an electric conductor through the hollow cylinder over which the fiber was coiled (not shown in Fig. 7) and applied an electric current of amplitude i0 ⫽ 400 A through the conductor. Assuming that ⬍⬍ l, the expected Faraday rotation will be given by7
冋
⬇ N0
册
sin共l兲 Vi0, l
(10)
where N0 is the number of fiber turns, V is the Verdet constant of the fiber material 共V ⫽ 2.8 ⫻ 10⫺4 deg兾A for fused silica), and l is the total retardance of the coil. After lateral pressure was applied to the fiber coil, the best value obtained for the Faraday rotation was ⬇ 1.05 ⫾ 0.05 deg, which corresponds approximately to 8 ⫻ 10⫺5 deg兾A per fiber turn, and thus from expression (10) we conclude that sin共l兲兾l ⫽ 0.29. This value corresponds to total residual retardance l ⬇ 2.4 rad. It seems likely that this residual retardance was partially due to border effects, e.g., inhomogeneous stress at the first and the final turns of the coil. Despite the difference from the ideal value, it is important to remark that the measured retardance value with the coil subjected to pressure 共⬇2.4 rad兲 is of the order of 80 times smaller than the value expected without the application of lateral pressure 共⬇198 rad兲. 4. Conclusions
The cancellation of (compensation for) bendinginduced linear birefringence in optical fibers is of fun-
damental importance in improving the sensitivity of Faraday sensor heads. The purpose of this paper is to show that, in principle, one can cancel the bendinginduced birefringence by applying lateral compressive stress to the fiber. We have proposed inducing controlled anisotropy in the y direction to compensate for the anisotropy in the x direction produced by bending and thus to achieve an isotropic bent fiber. We have experimentally verified that it is possible to compensate for the bending-induced birefringence in single fiber loops and also in coils with tens of turns. A real-world application of birefringence-free fiber coils would require the use of hi-bi fibers to ensure the mechanical stability of the system. The constant amount of birefringence along a fixed direction can be provided by stress application of parts included in the fiber cladding or by the core geometry. The drawback of the proposed method is the necessity for drawing a fiber with a specific length beat. The advantage of the method (e.g., with respect to fiber annealing) is its easy implementation. We thank K. T. V. Grattan for his helpful advice during the writing of the paper. Also, we thank the referees for their useful comments. References 1. B. Lee, “Review of the present status of optical fiber sensors,” Opt. Fiber Technol. 9, 57–79 (2003). 2. M. Lopez-Higuera, ed., Handbook of Optical Fiber Sensing Technology (Wiley, 2002). 3. R. Ulrich, S. C. Rashleigh, and W. Eickhof, “Bending-induced birefringence in single-mode fibers,” Opt. Lett. 5, 273–275 (1980). 4. S. C. Rashleigh and R. Ulrich, “High birefringence in tensioncoiled single-mode fibers,” Opt. Lett. 5, 354 –356 (1980). 5. F. El-Diasty, “Multiple-beam interferometric determination of Poisson’s ratio and strain distribution profiles along the cross section of bent single-mode optical fibers,” Appl. Opt. 39, 3197– 3201 (2000). 6. H. Tai and R. Rogowski, “Optical anisotropy induced by torsion and bending in an optical fiber,” Opt. Fiber Technol. 8, 162–169 (2002). 7. D. Tang , A. H. Rose, G. W. Day, and S. M. Etzel, “Annealing of linear birefringence in single-mode fiber coils: application to optical fiber current sensors,” J. Lightwave Technol. 9, 1031– 1037 (1991). 8. A. H. Rose, Z. B. Ren, and G. W. Day, “Twisting and annealing optical fiber for current sensors,” J. Lightwave Technol. 14, 2492–2498 (1996). 9. R. Laming and D. Payne, “Electric current sensors employing spun highly birefringent optical fibers,” J. Lightwave Technol. 7, 2084 –2094 (1989). 10. A. H. Rose, P. G. Polynkin, and J. Blake, “Electro-optic Kerr effects in spun high-birefringent fiber current sensors,” in Proceedings of the 14th International Conference on Optical Fiber Sensors, A. G. Mignani, and H. C. Laferve, eds., Proc. SPIE 4185, 348 –351 (2000). 11. K. Kurosawa, “Optical current transducers using flint glass fiber as the Faraday sensor element,” in Proceedings of the 11th Optical Fiber Sensors Conference (Japan Society of Applied Physics, 1996), pp. 134 –139. 12. A. Bertholds and R. Dändliker , “High-resolution photoelastic pressure sensor using low-birefringence fiber,” Appl. Opt. 25, 340 –343 (1986). 20 March 2006 兾 Vol. 45, No. 9 兾 APPLIED OPTICS
1955
13. M. Johnson, “In-line fiber-optical polarization transformer,” Appl. Opt. 18, 1288 –1289 (1979). 14. T. Mitsui, “Development of a polarization-preserving opticalfiber probe for near-field scanning optical microscopy and the influences of bending and squeezing on the polarization properties,” Rev. Sci. Instrum. 76, 043703 (2005).
1956
APPLIED OPTICS 兾 Vol. 45, No. 9 兾 20 March 2006
15. Z. B. Ren, Ph. Robert, and P.-A. Paratte, “Temperature dependence of bent- and twist- induced birefringence in a lowbirefringence fiber,” Opt. Lett. 13, 62– 64 (1988). 16. K. Bohnert, P. Gabus, J. Nehring, and H. Brändle, “Temperature and vibration insensitive fiber-optic current sensor,” J. Lightwave Technol. 20, 267–276 (2002).
