14-butter1978 | Optical Fiber | Optics

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Fiber Optic Strain Guage
   iber optics strain gauge C. D. Butter and G. B. Hocker Honeywell Corporate Material Sciences Center, Bloom-ington, Minnesota 55420. Received  11  May 1978. 0003-6935/78/0915-2867 0.50/0. © 1978 Optical Society of America. We have demonstrated a fiber optics strain gauge measuring strains of <0.4 × 10 -6 . It utilizes the change in optical pathlength caused by strain in a fiber. Strain is measured as motion of fringes in an optical interference pattern. Optical phase shifts per unit strain per unit fiber length of about 1.2 ×  10 7  m -1  are predicted by theory and observed in experiments. If light from a laser having sufficient coherence length is launched into two single-mode fibers of approximately the same length, the light issuing from the other ends can be made to interfere and produce visible fringes. If the optical path-length through one fiber is changed with respect to the optical path through the other, the fringes will shift; the amount of fringe shift is proportional to the relative change in optical paths. Thus, by observing the motion of the fringes, we can determine the optical pathlength changes that occur. Introducing different strains in the two fibers causes a difference in optical pathlengths and, hence, motion of the fringes. This phenomenon serves as a sensitive fiber-optic strain gauge. If one or both fibers are attached to a structure which is then strained under load, the strain in that structure can be determined from the motion of the optical interference fringes. To calculate the expected fringe shift due to longitudinal strain, let the section of single-mode fiber to be strained be of length L with its axis in the  x  direction. The propagation constant of the mode in the fiber is designated by  β, its free-space propagation constant  is  k 0 ,  the fiber core index is n, the core diameter is  D,  and Poisson's ratio for the fiber material is  μ. The phase of the light wave after going through this fiber section is    =  βL. Straining the fiber in the axial direction by an amount  ε  changes this phase by an amount A differential phase change between two fibers of  2π  causes the interference pattern to shift by one fringe. The first term in Eq. (1) represents the physical change of length produced by the strain and is simply βΔL = βεL. The second term, the change in   due to a change in  β, can come about from two effects: the strain-optics effect whereby 15 September 1978 / Vol. 17 No. 18 / APPLIED OPTICS 2867  the strain changes the refractive index of the fiber, and a waveguide mode dispersion effect due to a change in fiber diameter  D  produced by longitudinal strain: Although β = n eff k 0 , where the effective index n eff  lies between core and cladding indices, these indices typically only differ by the order of 1 so w can use β   nk 0 .  Thus,  dβ/dn =  k 0   β/n. The strain-optics effect.appears;as a change in theioptical indicatrix: where  S j  is the strain vector, p ij  is  the strain-optic tensor, and the subscripts are in the standard contracted notation. For longitudinal strain in the  x  direction as considered here, the strain vector is For a homogeneous isotropic medium, p ij  has only two numerical values, designated p 11  and  p 12 ,  and the change in the optical indicatrix in the  y  and  z  directions  i =  2,3) is just Therefore, light propagating in the  x  direction sees an index change: The last term in Eq. (2) represents the change in the waveguide mode propagation constant due to a change in fiber diameter. The change in diameter is just ΔD =  μεD.  The dβ/dD  term can be evaluated using the normalized parameters 6 and  V  describing the waveguide mode. 1  It can be shown that  dβ/dB  =  V 3 /2βD 3 )db/dV,  where  db/dV  is the slope of the  b  V  dispersion curve at the point which describes the waveguide mode. This term will be shown to be negligible. Combining these expressions,  we  can write the phase change per unit stress per unit fiber length as For typical glasses,  n =  1.5,  μ =  0.25, and p 11   P 12   0.3. In the single-mode region of the  b V  dispersion curve,  V   2 5  and  db/dV   0.5. For operation at  λ = 0.63  × 10 -6  m with a diameter  D  = 2 × 10 -6  m, we find We note that the effect of waveguide mode dispersion is negligible, and so that term can be dropped, giving the simplified expression: We have constructed an apparatus in which  we  can conve-2868  APPLIED OPTICS  / Vol.  17, No.  18 /  15  September 1978 Fig. 1. Apparatus for using optical fibers to measure strain in cantilever bar. niently strain one single-mode fiber with respect to another in a known manner. The apparatus, shown in Fig. 1, consists of three groups of parts. The first group is composed of a beam splitter, two adjustable microscope objective lenses, and fixtures for the input ends of the single-mode fibers. The purpose of this group  is  to split the laser beam into two beams of approximately equal intensity and to focus those beams onto the input ends of the fibers. The second group has a 30-cm cantilever bar with two parallel lengthwise grooves on one side, a screw to displace the free end of the cantilever, and a dial gauge to measure the displacement. The final group is a fixture for the exit ends of the fibers and a screen on which to observe the interference fringes. One of the two fibers extends from the input fixture along one of the grooves in the bar from the fixed end to the free end, loops back along the same groove, and then to the exit fixture. The other fiber follows a similar path from the input to output fixtures along the other groove on the side of the cantilever. Both fibers are cemented into their respective grooves. The ends of the two fibers are even with each other, with the light from them forming interference fringes on the screen. When the screw is turned, displacing the end of the cantilever bar in the plane containing the two grooves, one fiber will be longitudinally stretched, while the other will be compressed along the sections cemented into the grooves, and the interference fringes on the screen are observed to move. We tested our device by counting the fringes passing a reference mark as the screw was turned. When ten fringes had passed the reference mark, the screw was stopped and the displacement read from a dial gauge. This procedure was repeated until 25 fringes had passed, a displacement of slightly over  2 5  mm. The screw was then backed off, and the same fringe counting procedure followed until the cantilever returned to its original position. The displacement factor, defined as total displacement divided by the number of fringes  N  that have passed the  ref- erence mark, is plotted in Fig.  2  as a function of  N.  The two sets of points correspond to increasing and decreasing displacement. The average displacement factor of this device is 0.0004049 in./fringe = 10.28  μm/fringe. The variation in displacement factor with  N  is probably indicative of the accuracy and reading error in the dial gauge. Cantilever beam displacement  d  can be converted to strain in the fiber by using Ref. 2 to derive  Fig.  2.  Displacement factor (total displacement   number of fringes) vs number of fringes  N  for apparatus shown in Fig. 1. where  ε x)  is the strain at a distance  x  from the free end of the cantilever, l is the cantilever length, and  a  is the distance from the neutral axis. The average value of strain is found by integration to be In our experiment l = 30 cm and  a =  ±0.25 cm, so the average strain in each fiber for a displacement  d  = 10.28 μm corresponding to one fringe is ε = ±4.35  × 10 -7 . This causes a phase shift of  2Π  rad between the two fibers, each with a total strained length of 60 cm, so the experimentally measured value of Δ  / εL is 1.20  × 10 7  m -1 , in excellent agreement with the value predicated by theory in Eqs. (8) and (9). In conclusion, we have used a fiber optics technique to measure strains of <0.4 × 10 -6  in a cantilever beam. The expected fringe shift per unit strain per unit fiber length was calculated and found to be in excellent agreement with experimental results. References 1. G.  B.  Hocker and W. K. Burns, IEEE. J. Quantum Electron. QE- 11,  270 (1975); D. B. Keck, in  Fundamentals of Optical Fiber Communications,  M. K. Barnoski, Ed. (Academic, New York, 1976), Chap. 1. 2.  S. Timoshenko,  Strength of Materials  (Van Nostrand, New York, 1958). 15 September 1978 / Vol. 17 No. 18 / APPLIED OPTICS 2869
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