- 13.1 INTRODUCTION. - 13.2 LIGHT PROPAGATION THROUGH OPTICAL FIBRES. - Figure 13.1 shows the basic construction of an optical fibre. - n 0 sin θ 0 = n 1 sin θ 1 (13.1) where n 0 is the refractive index of the surrounding medium. - Figure 13.1 Basic construction of an optical fibre. - which combined with Equations (13.1) and (13.2) gives. - N A = n 0 sin θ a = n 2 1 − n In practice, coupling of the light into the fiber can be accomplished with the help of a lens, see Figure 13.2(a) or by putting the fibre in close proximity to the light source and. - Figure 13.2 Coupling of light into a fibre by means of (a) a lens and (b) index-matching liquid. - When using the method in Figure 13.2(a), it is important to have the angle of the incident cone less than θ a to get maximum coupling efficiency.. - where d is the waveguide diameter and θ is the angle of the incident beam. - Figure 13.3 A conducting slab waveguide. - Figure 13.4 Electric field distribution of the lowest-order guided transversal modes in a dielectric slab waveguide. - (13.10). - (13.11). - N A (13.12). - 13.3 ATTENUATION AND DISPERSION. - Figure 13.5 Attenuation in a silica glass fibre versus wavelength showing the three major wave- length regions at which fibre systems are most practical. - Figure 13.5 shows the attenuation as a function of wavelength for silica glass fibres. - (13.13) In dispersive media a light pulse propagates at the group velocity (Senior 1985) defined by. - dβ (13.14). - λ (13.15a). - λ (13.15b). - dλ (13.16) This gives. - c (13.17). - dλ 2 (13.18). - Figure 13.6 (a) Refractive index versus wavelength for SiO 2 glass and (b) The second derivative of the curve in (a). - Mλ (13.19). - Because of this, d 2 n/dλ 2 = 0 at λ 0 as seen from the curve of the second derivative in Figure 13.6(b). - 13.4 DIFFERENT TYPES OF FIBRES. - Another construction than the step-index (SI) fibre sketched in Figure 13.1 is the so- called graded-index (GRIN) fibre. - Figure 13.7 Graded index fibre: (a) refractive index profile. - Table 13.1 shows representative numerical values of important properties for the various fibres. - Table 13.1 (From Palais, J. - Figure 13.8 Light-duty, tight-buffer fibre cable (Siecor Corporation). - 13.5 FIBRE-OPTIC SENSORS. - Figure 13.9 shows some typical examples of fibre-optic sensors. - In Figure 13.9(a) a thin semiconductor chip is sandwiched between two ends of fibres inside a steel pipe.. - Figure 13.9 (a) Fibre-optic temperature sensor and (b) Fibre-optic pressure sensor. - Figure 13.9(b) shows a simplified sketch of a pressure-sensing system. - Figure 13.10 shows the principle of a class of fibre-optic sensors based on interferome- try. - Figure 13.11 shows a special application of optical fibres. - In Figure 13.11(a) two fibre bundles, A and B, are mixed together in a bundle C in such a way that every second fibre in the cross-section of C comes from, say, bundle A. - Figure 13.11(b) shows two neighbouring fibres, A and B. - Figure 13.10 Interferometric fibre-optic sensor. - Figure 13.11. - A curve describing the relation between the received light intensity I B versus the distance l therefore will look like that given in Figure 13.11(c). - The fibre bundles coupled together as in Figure 13.11(a) thus can be used as a non- contact distance sensor (Cook and Hamm 1979. - By placing the end of bundle C close to a surface, the detector will give a signal which is proportional to the distance from the fibre end to the surface, as long as one is working within the linear portion of the left flank of the curve in Figure 13.11(c). - A system based on wavelength division multiplexing is sketched in Figure 13.12. - Figure 13.12 Multiplexing of fibre-optic sensors (accelerometers) using WDM. - 13.6 FIBRE-BRAGG SENSORS. - n c + n cos qz (13.20). - n is the amplitude of the refractive index variations and d is the grating period. - Equation (13.20) could also describe the refractive index variations through the depth of a volume hologram. - Let us start by analysing the situation shown in Figure 13.13. - If r = (dr/dz)z is the incremental complex amplitude reflectance of the layer at position z, the total amplitude reflectance for the overall length L (see Figure 13.13) is. - Figure 13.13 Reflections from planar layers. - dz dz (13.21). - 2n sin 2 θ n (13.22). - 2n sin 2 θ n (13.23). - Using Equations (13.20) and (13.22) we obtain. - 2n sin 2 θ n sin qz = r sin qz (13.24) where. - 2n sin 2 θ n (13.25). - Finally we substitute Equation (13.24) into (13.21) to find r = r. - −L/2 e i(2k sin θ+q)z dz (13.26) By evaluating the first integral we get. - r = ir 2 L sinc (q − 2k sin θ ) L 2π (13.27) The sinc-function has its maximum value of 1 when its argument is zero, i.e. - λ sin θ (13.28). - 2d (13.29). - k r = k + q (13.30). - The function in Equation (13.27) drops sharply and reaches its first zero when (q − 2k sin θ ) L. - 2 = π (13.31). - 2L (13.32). - This is also the reason why we evaluated only the first integral of Equation (13.26). - We get the answer by putting θ = 90 ◦ into Equation (13.29), giving λ B = 2d, where λ B is the Bragg wavelength inside the fibre. - From Equation (13.32) we find the wavelength of the first minimum to be. - L − d (13.33). - L (13.34). - Therefore when a wave u i (λ i ) is incident from the left into a fibre as in Figure 13.14, light with a narrow bandwidth is reflected from the refractive index variations, maximum reflectance occurring at the Bragg wavelength λ B . - The unreflected light is transmitted, and a typical intensity profile for the transmitted light is shown in Figure 13.15.. - Figure 13.14 Reflection and transmission from an FBG. - Figure 13.15 Transmission spectrum for a moderate reflectivity fibre grating. - 0.53 µ m, Equation (13.34) gives λ = 1.12 ˚ A, which compares quite well with the experimental value of 1.75 ˚ A.. - As mentioned in Section 13.3, a light pulse sent through a fibre will broaden due to different velocities for the different wavelengths.. - (13.35). - Figure 13.16 Bragg grating thermal sensitivity at elevated temperature. - λ B = λ B [1 − p e ]ε ≈ 0.78ε (13.36). - (13.37) p 11 and p 12 are called the fibre Pockel’s coefficients and µ is the Poisson ratio. - Figure 13.16 shows λ B as a function of temperature at λ B = 1556 nm. - 13.1 Consider an optical fibre designed to guide monochromatic light of λ = 400 nm. - The diameter of the core is 10 µ m. - (a) the allowable angles of the incident beams;. - (b) the largest allowable angle of the incident beam;. - (c) the actual acceptance angle of the fibre;. - (d) the numerical aperture of the fibre.. - 13.2 The refractive index of a single-mode fibre core and its cladding are n 1 = 1.5 and n 2 = 1.47, and the wavelength of the light is λ = 600 nm. - 13.3 (a) Find the amount of pulse spreading in pure silica for an LED operating at 820 nm and having a 20 nm spectral width. - 13.4 Due to the pulse spread τ , the wavelength components of a sinusoidally modulated beam will have different transit times. - 13.5 The number of modes in a step-index fibre is given by N = 2. - 13.6 Consider the non-contact fibre-optic sensor illustrated in Figure 13.11. - (a) Show that the received intensity I B is indeed like that sketched in Figure 13.11(c)
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