Centro de Investigación Cientıfica y de Educación Superior de Ensenada, Carretera Tijuana-Ensenada km. 107,. Apartado Postal 2732, Ensenada, B.C., Mexico.
June 15, 1997 / Vol. 22, No. 12 / OPTICS LETTERS
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Ultralow-birefringence measurement in optical fibers by the twist method A. Rodr´ıguez, A. V. Khomenko, R. Cort´es, and A. Garc´ıa-Weidner Centro de Investigacion ´ Cient´ıfica y de Educacion ´ Superior de Ensenada, Carretera Tijuana-Ensenada km. 107, Apartado Postal 2732, Ensenada, B.C., Mexico Received January 13, 1997 We describe a simple method for measuring ultralow birefringence in optical f ibers. It allows one to measure birefringence in the range of 4 3 1026 – 1.25 3 1028 , which corresponds to the 0.25 – 80-m range of the polarization beat length at the wavelength of 1 mm. A f iber section of a length shorter than the polarization beat length can be used for measurement. The measuring procedure involves measurement of the light intensity only and does not require an analysis of the light-polarization state. Experimental results for an optical f iber with a beat length of 1.9 m are presented. 1997 Optical Society of America
It is diff icult to expect that birefringence could be the same along an optical fiber that is many beat lengths long when birefringence is ultralow, Dn , 1026 , and the beat length is longer than 0.5 m. Hence a section of a fiber that is shorter than one beat length should be used for precise measurement of birefringence. This makes it impossible to use many of the proposed experimental techniques to measure fiber birefringence, because a section of fiber with a length of several beat lengths has to be used for measurement.1,2 In our opinion a twist method is most suitable in this case in spite of the fact that this method was basically developed for measuring in highly birefringent fibers.3 – 5 Here we present a new modification of the twist method, which allows one to measure the birefringence of fibers in the range of 4 3 1026 –1.25 3 1028 . The method involves only the measurement of light-intensity dependence on fiber twist and does not require analysis of the polarization state as is necessary in other modifications of the twist method. Let us consider a linear birefringent, twisted, singlemode optical fiber illuminated by a monochromatic light through a linear polarizer. The light emitted from the output end of the fiber passes through a linear polarization analyzer and is incident upon a photodetector. The twisted fiber has elliptical birefringence that results from superposition of intrinsic linear and twist-induced circular birefringences.6 We consider the case in which a uniform twist is applied to a fiber of an uniform linear birefringence. Using a matrix-operator formalism,7 we calculate the output intensity as ˆ GEj ˆ 2, (1) Iout jEout j2 jA ˆ and G ˆ represent the outputwhere matrix operators A polarization analyzer and the twisted fiber, respectively: É É cos2 u sin u cos u ˆ A , sin u cos u sin2 u Ø Ø Ø Ø a Db Ø Ø Ø cos Sz 2 i Ø sin Sz sin Sz Ø Ø , 2S S ˆ Ø Ø GØ Db a Ø Ø Ø sin Sz cos Sz 1 i sin Sz 2 Ø Ø S 2S (2) 0146-9592/97/120877-03$10.00/0
where Db b2 2 b1 is the phase mismatch per unit length between two linearly polarized eigenmodes with propagation constants b1 and b2 ; Db 2pDnyl, where Dn is the linear birefringence; a is the twist-induced optical activity; S is the elliptical birefringence of the fiber, S fsDby2d2 1 a 2 g1/2 ; and z is the fiber length. We associate the coordinate system with the local fast and slow axes of linear birefringence.6 In this case a s2 2 gdty2, where the photoelastic coeff icient g , 0.14 –0.16 for silica fibers5,6 and t is the rate of fiber twist (in radians per meter). The angle u is the angular position of the analyzer from the principal axis of the fiber. In our experimental setup the input end of the fiber is rotated to produce a fiber twist. Hence the input light polarization depends on the twist of the fiber, and the input light amplitude can be written as Ç Ç cos tz . E (3) 2sin tz If g, l, and u are known parameters, according to Eqs. (1) –(3) the output intensity I can be considered a function of two variables, namely, the rotation of the input end of the fiber tz and the product of linear birefringence and the fiber length Dnz. Figure 1 shows the output intensity as a function of the fiber input-end rotation tz for different values of the Dnz product. These dependencies are calculated numerically by use of Eqs. (1) –(3) and g 0.16 and l 0.725 mm. Only one polarization eigenmode is excited when the fiber is not twisted, t 0, and the analyzer has the crossed position u 90±. When the twist is weak, jaj ,, jbj, the fiber maintains the linear polarization state of the light when only one polarization mode is excited. In this case the outputintensity dependence on the fiber twist has a period of half a turn and does not depend on linear birefringence [Fig. 1(a)]. When the fiber twist is strong and jaj .. jbj, twistinduced birefringence is the dominant effect. The output light has a linear polarization state that rotates at the rate gt owing to induced circular birefringence [tz . 1 in Figs. 1(e) and 1(f ), tz . 2 in Fig. 1(d), and tz . 4 in Fig. 1(c), and tz . 8 in Fig. 1(b)]. For this range of the fiber twist, the dependencies shown in Fig. 1 have the period gty2. 1997 Optical Society of America
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Fig. 1. Dependencies of the output intensity on the rotation of fiber end: (a) Dnz 1024 , ( b) 3.2 3 1026 , (c) 1.6 3 1026 , (d ) 8 3 1027 , (e) 2 3 1027 , (f ) Dn 0.
In the case of a medium-twist regime with jaj , jDbj [tz , 1 in Figs. 1(d) and 1(e), tz , 2 in Fig. 1(c), and tz , 4 in Fig. 1(b)], the maximum intensity occurs after each half-turn of the input end of the fiber. In the meantime the intensity at the maxima varies from one maximum to the other. The shape of the intensity dependence on the fiber-end rotation is unique for each value of the Dnz product. Thus the linear birefringence of an optical fiber can be determined in the medium-twist regime. Let us restrict the fiber-end rotation by three turns in one direction and enumerate the maxima starting from the zero-twist point. The first maximum occurs when the rotation of the fiber end is equal to a quarter turn of the fiber if any maximum appears in the range of the twist that we used. Each following maximum occurs after approximately a half-turn. Figure 2 shows the dependencies of the normalized intensities at the first six maxima on the Dnz product. The dependence of the output intensity on the rotation of the fiber end I stzd does not have any maxima in the range jtzj , 3 turns if Dnz , 2.5 3 1028 m. Hence the curve corresponding to the first maximum in Fig. 2 starts at Dnz ø 2.5 3 1028 m. The second maximum can be observed when Dnz . 2.25 3 1027 m, and the appropriate curve in Fig. 2 starts at Dnz ø 2.25 3 1027 m, and so on. When Dnz increases, each curve in Fig. 2 reaches a maximum and then oscillates with decreasing amplitude. To avoid ambiguity in the determination of linear birefringence one should decrease the
number of maxima by decreasing the fiber length so that the Dnz product is smaller than 1.5 3 1026 m. When the experimental dependence of the intensity on the fiber-end rotation is measured, it is easy to find which maximum intensity has almost linear dependence on the Dnz product. One should obtain this from comparison of the experimental dependence and the data in Fig. 2. Then the comparison of the normalized intensity at this maximum and the appropriate curve in Fig. 2 give the Dnz product. With the known fiber length z the linear birefringence Dn can be calculated. To estimate a birefringence range that can be measured with the proposed method, we consider the minimum value of the Dnz product, which allows us to observe the maximum of the intensity at the quarter-turn of the fiber end, that is, 2.5 3 1028 m, as was mentioned above. The fiber length of 2 m seems to be realistic for use with the proposed method. Hence the linear birefringence of a fiber of 1.25 3 1028 that corresponds to the polarization beat length of 80 m at l 1 mm can be detected. The value of the Dnz product of 1.5 3 1026 m allows one to avoid ambiguity in measuring birefringence and can be considered the maximum value of this product. Together with the maximum twist rate, which can be applied nondestructively to an optical fiber, the maximum value of the Dnz product gives another limit for the birefringence range. Our experiments show that maximum twist rate is somewhere between 40 and 100 turnsym. To apply our measuring method, we need to rotate a fiber end for five turns. Hence, a fiber section as short as approximately 0.25 m can be used for measurements without fiber damage, and fiber birefringence of 4 3 1026 can measured, which corresponds to a polarization beat length of 0.25 m. When we are measuring the birefringence of a single-mode fiber by twisting, the fixed parts at the fiber ends are not twisted. It was proposed that the three-section model of optical fibers be used to consider the effects associated with untwisted sections of the fiber.4,6 In our case the polarization analyzer selects one of two polarization modes. Hence the untwisted
Fig. 2. Light intensity in different maxima as a function of Dnz product: solid curves, the fiber is twisted along the entire length; dashed curves, the length of the twisted fiber section is 0.25 m, and an untwisted section of 0.05 m is at the fiber input end. From left to right, the pairs of curves are the first, second, third, fourth, fifth, and sixth maxima.
June 15, 1997 / Vol. 22, No. 12 / OPTICS LETTERS
Fig. 3. Experimental setup: L, lamp; A, analyzer; P, polarizer; S, spectrograph. F, optical fiber; O1, O2, lenses; MO1, MO2, microscope objectives.
Fig. 4. Experimental dependence of the output intensity on the fiber-end rotation. Curve 1, fiber length 0.915 m; curve 2, fiber length 0.515 m.
fiber section near the output fiber end does not affect the measurement result, and we can use our twosection model. The dashed curves in Fig. 2 show the results of calculations when the matrix operator that represents the untwisted fiber section near the input end of the fiber is added to Eq. (1). One can see that the first maximum is not affected by the presence of the untwisted sections. At the same time the other curves are displaced. For a 0.05-m length of untwisted section and a 0.25-m length of twisted section displacement of the curves leads to inaccurate fiber-birefringence determination, which reaches 3% when the fourth maximum is used for birefringence measurement. For experimental verification of the proposed method we measured the birefringence of a lowbirefringence single-mode optical fiber fabricated by Oxford Electronics, Ltd. Fiber sections of different lengths with untwisted ends of 0.05 m were used in the experiments. White light emitted from a 175-W xenon lamp, L, passes through polarizer P and is launched into the optical fiber (Fig. 3). The input end of the fiber is fixed upon a motorized rotation stage, which allows one to twist the fiber. Light emitted from the fiber passes linear polarization analyzer A and is launched into CCD spectrograph S. A computercontrolled CCD spectrograph gives us the ability to observe an output light spectrum in real time. The output-intensity dependencies on the rotation of the input end of the fiber were simultaneously recorded for as many as 12 wavelengths.
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During the measurement procedure the polarizer and analyzer are adjusted in a crossed position after the optical fiber is removed from the experimental setup. Then optical fiber is installed and the analyzer is aligned along one of the principal optical axes of the fiber by simultaneous rotation of analyzer and polarizer toward a position that gives a symmetric dependence of the output intensity on the fiber twist with respect to positive and negative twists. In our experiments this task is usually fulfilled during three or five analyzer rotations with the proper intensity recordings. To conf irm the feasibility of the method, we measured the birefringence of the same fiber several times. The fiber was cut after each measurement, thus each measurement corresponds to a different fiber length. Figure 4 shows the experimental dependence of the output intensity recorded at l 0.725 mm normalized by the maximum intensity I0 . The deviations of the experimental curves from the theoretical ones reached 10%. We assume that these deviations are caused mainly by the intensity variations associated with the rotation of the input end of the fiber. These variations are a main source of measurement errors. The curves in Fig. 4 have their maxima nearest to the zero-twist point, which is equal to 0.73 for curve 1 and 0.36 for curve 2. According to the data presented in Fig. 2, these values correspond to Dnz products of 3.6 3 1027 and 2.0 3 1027 m, respectively. Hence the fiber birefringence is s3.8 6 0.1d 3 1027 , and the polarization beat length is 1.9 m. Note that the measured polarization beat length is four times longer than the length of the fiber. In conclusion, we have presented a method for measuring ultralow birefringence in fibers with polarization beat lengths in the range of 0.25–80 m by fiber twist. To our knowledge, the ultralow birefringence of this range cannot be measured with other methods. The measuring procedure is simple, and a section of the fiber that is shorter than a polarization beat length is needed for measuring. Our experimental setup allows one to measure birefringence in a broad spectral range. Experimental verification of the proposed method has been provided. The authors gratefully acknowledge support of this work by the Direcci´on Adjunta de Investigacio´ n Cientif´ıcayConsejo Nacional de Ciencia y Tecnolog´ıa (M´exico). References 1. R. Calvani, R. Caponi, and F. Cisternino, J. Lightwave Technol. 7, 1187 (1989). 2. T. Okoshi, S. Ryu, and K. Emura, Opt. Commun. 2, 134 (1981). 3. M. Monerei and P. Lamouler, Electron. Lett. 17, 252 (1981). 4. Sh. Huang and Z. Lin, Appl. Opt. 24, 2355 (1985). 5. T. Z. Wolinski and W. J. Bock, IEEE Trans. Instrum. Meas. 44, 708 (1995). 6. R. Ulrich and A. Simon, Appl. Opt. 18, 2241 (1979). 7. A. Yariv and J. F. Lotspeich, J. Opt. Soc. Am. 72, 273 (1982).
