Tài liệu Đo lường quang học P13 pdf

13 Fibre Optics in Metrology 13.1 INTRODUCTION With a carrier frequency of some 10 14 Hz, light has the potential of being modulated at much higher frequencies than radio waves. Since the mid-1960s the idea of communication through optical fibres has developed into a vital branch of electro-optics. Great progress has been made and this is now an established technique in many communication systems. From the viewpoint of optical metrology, optical fibres are an attractive alternative for the guiding of light. An even more important reason for studying optical fibres is their potential for making new types of sensors. 13.2 LIGHT PROPAGATION THROUGH OPTICAL FIBRES More extensive treatments on optical fibres can be found in Senior (1985), Palais (1998), Keiser (1991) and Yu and Khoo (1990). Figure 13.1 shows the basic construction of an optical fibre. It consists of a central cylindrical core with refractive index n 1 , surrounded by a layer of material called the cladding with a lower refractive index n 2 . In the figure a light ray is incident at the end of the fibre at an angle θ 0 to the fibre axis. This ray is refracted at an angle θ 1 and incident at the interface between the core and the cladding at an angle θ 2 . From Snell’s law of refraction we have n 0 sin θ 0 = n 1 sin θ 1 (13.1) where n 0 is the refractive index of the surrounding medium. From the figure, we see that θ 1 = π 2 − θ 2 (13.2) If θ 2 is equal to the critical angle of incidence (cf. Section 9.5), we have sin θ 2 = n 2 n 1 (13.3) Optical Metrology. Kjell J. G ˚ asvik Copyright  2002 John Wiley & Sons, Ltd. ISBN: 0-470-84300-4 308 FIBRE OPTICS IN METROLOGY n 0 q 0 n 2 n 1 q 1 q 2 Lost ray Cone of acceptance Figure 13.1 Basic construction of an optical fibre which combined with Equations (13.1) and (13.2) gives θ 0 ≡ θ a = sin −1     n 2 1 − n 2 2 n 0    (13.4) For θ 0 <θ a the light will undergo total internal reflection at the interface between the core and the cladding and propagate along the fibre by multiple reflections at the interface, ideally with no loss. For θ 0 >θ a some of the light will transmit into the cladding and after a few reflections, most of the light will be lost. This is the principle of light transmission through an optical fibre. The angle θ a is an important parameter when coupling of the light into a fibre, usually given by its numerical aperture NA: NA = n 0 sin θ a =  n 2 1 − n 2 2 (13.5) 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 Laser beam 2q a Cladding Core (a) (b) LED Cladding Core Index matching liquid Figure 13.2 Coupling of light into a fibre by means of (a) a lens and (b) index-matching liquid LIGHT PROPAGATION THROUGH OPTICAL FIBRES 309 linking them with an index-matching liquid to reduce reflection losses, Figure 13.2(b). 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. The above description of light propagation through an optical fibre is not fully complete. To gain better understanding, the fibre must be treated as a waveguide and the electro- magnetic nature of the light must be taken into account. If a waveguide consisting of a transparent layer between two conducting walls is considered, the electric field across the waveguide will consist of interference patterns between the incident and reflected fields, or equivalently, between the incident field and its mirror image, see Figure 13.3. The path-length difference l between these fields is seen from the figure to be l = d sin θ(13.6) where d is the waveguide diameter and θ is the angle of the incident beam. From the boundary conditions for such a waveguide we must have destructive interference at the walls, i.e. the path-length difference must be equal to an integral number of half the wavelength: l = m λ 2 (13.7) which gives sin θ = mλ 2d (13.8) where m is an integer. Thus we see that only certain values of the angle of incidence are allowed. Each of the allowed beam directions are said to correspond to different modes of wave propagation in the waveguide. The field distribution across the waveguide for the lowest-order guided modes in a planar dielectric slab waveguide are shown in Figure 13.4. This guide is composed of a dielectric core (or slab) sandwiched between dielectric claddings of lower refractive index. As can be seen, the field is non-zero inside the Incident beam d sin q Reflected beam Conducting wall Conducting wall Mirror image of reflected beam Mirror image of incident beam q d Figure 13.3 A conducting slab waveguide 310 FIBRE OPTICS IN METROLOGY TE 0 n 2 n 1 TE 2 Cladding Core Cladding TE 1 Figure 13.4 Electric field distribution of the lowest-order guided transversal modes in a dielectric slab waveguide cladding. This is not in contradiction with the theory of total internal reflection (see Section 9.5) which predicts an evanescent wave decaying very rapidly in the cladding material. The lowest number of modes propagating through the waveguide occurs when the angle of incidence is equal to θ a . Then (assuming n 0 = 1forair) sin θ a = mλ 2d =  n 2 1 − n 2 2 (13.9) or d λ = m 2  n 2 1 − n 2 2 (13.10) To have only the lowest-order mode (m = 0) propagating through the waveguide, we therefore must have d λ < 1 2  n 2 1 − n 2 2 (13.11) An exact waveguide theory applied to an optical fibre is quite complicated, but the results are quite similar. The condition for propagating only the lowest-order mode in an optical fibre then becomes d λ < 2.405 2π  n 2 1 − n 2 2 = 2.405 2π(NA) = 0.383 NA (13.12) A fibre allowing only the lowest-order mode to propagate is called a single-mode fibre, in contrast to a multimode fibre which allows several propagating modes. 13.3 ATTENUATION AND DISPERSION That light will propagate through a fibre by multiple total internal reflections without loss is an idealization. In reality the light will be attenuated. The main contributions to attenuation is scattering (proportional to λ −4 ) in the ultraviolet end of the spectrum and absorption in the infra-red end of the spectrum. Therefore it is only a limited part of ATTENUATION AND DISPERSION 311 First window Total loss Rayleigh scattering Second window OH absorption peak Third window 800 900 1000 1100 1200 1300 1400 1500 1600 1700 0 0.5 1.0 1.5 2.0 2.5 3.0 Wavelength (nm) Attenuation (dB/km) 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. (From Palais, J. C. (1998) Fiber Optic Communications (4th edn), Prentice Hall, Englewood Cliffs, N.J.) Reproduced by permission of Prentice Hall Inc.) the electromagnetic spectrum where fibre systems are practical. Figure 13.5 shows the attenuation as a function of wavelength for silica glass fibres. Here are also shown the three major wavelength regions at which fibre systems are practical. These regions are dictated by the attenuation, but also by the light sources available. Another source of loss in fibre communication systems is dispersion. Dispersion is due to the fact that the refractive index is not constant, but depends on the wavelength, i.e. n = n(λ). In fibre systems one talks about material dispersion and waveguide dispersion. Here we will briefly mention material dispersion. That the refractive index varies with wavelength means that a light pulse from a source of finite spectral width will broaden as it propagates through the fibre due to the different velocities for the different wavelengths. This effect has significant influence on the information capacity of the fibre. The parameter describing this effect is the pulse spread per unit length denoted τ/L where τ is the difference in travel time for two extreme wavelengths of the source’s spectral distribution through the length L.Thisgives τ L =   1 ν g  (13.13) In dispersive media a light pulse propagates at the group velocity (Senior 1985) defined by ν g = dω dβ (13.14) With the relations ω = kc = 2πc λ (13.15a) β = kn = 2πn λ (13.15b) 312 FIBRE OPTICS IN METROLOGY we get 1 ν g = dβ dω = dβ dλ dλ dω =  −λ 2 2πc  2π  1 λ dn dλ − n λ 2  = 1 c  n − λ dn dλ  (13.16) This gives τ L =   1 ν g  =   n − λdn/dλ c  (13.17) The pulse spread per unit length per wavelength interval λ becomes τ Lλ = d dλ  1 ν g  = d dλ  n c − λ c dn dλ  =− λ c d 2 n dλ 2 (13.18) Refractive index 1.45 n l 0 (a) 0 (b) d n 2 / dl 2 l 0 Wavelength Wavelength Figure 13.6 (a) Refractive index versus wavelength for SiO 2 glass and (b) The second derivative of the curve in (a) DIFFERENT TYPES OF FIBRES 313 The material dispersion is defined as M = (λ/c)(d 2 n/dλ 2 ). The pulse spread per unit length then can be written as τ L =−Mλ (13.19) The refractive index for pure silicon dioxide (SiO 2 ) glass used in optic fibres has the wavelength dependence shown in Figure 13.6(a). At a particular wavelength λ 0 ,thereis an inflection point on the curve. Because of this, d 2 n/dλ 2 = 0atλ 0 as seen from the curve of the second derivative in Figure 13.6(b). For pure silica, the refractive index is close to 1.45 and the inflection point is near λ 0 = 1.3 µm. Therefore this wavelength is very suitable for long distance optical fibre communication. 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. It has a core material whose refractive index varies with distance from the fibre axis. This structure is illustrated in Figure 13.7. As should be easily realized, the light rays will bend gradually and travel through a GRIN fibre in the oscillatory fashion sketched in Figure 13.7(d). As opposed to an SI fibre, the numerical aperture of a GRIN fibre decrease with radial distance from the axis. For this reason, the coupling efficiency is generally higher for SI fibres than for GRIN fibres, when each has the same core size and the same fractional refractive index change. Conventionally, the size of a fibre is denoted by writing its core diameter and then its cladding diameter (both in micrometers) with a slash between them. Typical dimensions (a) (b) (c) (d) 2 a r n 2 n n 1 a 0 r z n ( r ) n 2 2 a Figure 13.7 Graded index fibre: (a) refractive index profile; (b) end view; (c) cross-sectional view; and (d) ray paths along a GRIN fibre 314 FIBRE OPTICS IN METROLOGY of SI fibres are 50/125, 100/140 and 200/230 and typical dimensions of multimode GRIN fibres are 50/125, 62.5/125 and 85/125. SI fibres have three common forms: (1) a glass core cladded with glass, (2) a silica glass core cladded with plastic (termed plastic-cladded silica (PCS) fibres), and (3) a plastic core cladded with another plastic. All-glass fibres have the lowest losses and the smallest pulse spreading, but also the smallest numerical aperture. PCS fibres have higher losses and larger pulse spreads and are suitable for shorter links, normally less than a few hundred metres. Their higher NA increase the coupling efficiency. All-plastic fibres are used for path lengths less than a few tens of meters. Their high NA gives high coupling efficiency. Single-mode fibres have the highest information capacity. GRIN fibres can transmit at higher information rates than SI fibres. Table 13.1 shows representative numerical values of important properties for the various fibres. Somewhat different characteristics may be found when searching the manufacturers’ literature. Table 13.1 (From Palais, J. C. (1998) Fiber Optic Communication (4th edn), Prentice Hall, Engle- wood Cliffs, New Jersey). Reproduced by permission Description Core Diameter (µm) NA Loss (dB/km) (τ/L) (ns/km) Source Wavelength (nm) Multimode Glass SI 50 0.24 5 15 LED 850 GRIN 50 0.24 5 1 LD 850 GRIN 50 0.20 1 0.5 LED, LD 1300 PCS SI 200 0.41 8 50 LED 800 Plastic SI 1000 0.48 200 – LED 580 Single mode Glass 5 0.10 4 <0.5 LD 850 Glass 10 0.10 0.5 0.006 LD 1300 Glass 10 0.10 0.2 0.006 LD 1550 Polyurethane, 3.8 mm Kevlar, 2 mm Hytrel secondary buffer, 1 mm Silastic primary buffer, 0.4 mm Fibre, 0.23 mm Figure 13.8 Light-duty, tight-buffer fibre cable (Siecor Corporation). The dimensions given are the diameters. (From Palais, J. C. (1998) Fiber Optic Communications (4th edn), Prentice Hall, Englewood Cliffs, N.J.) Reproduced by permission of Prentice Hall FIBRE-OPTIC SENSORS 315 The amount of protection against the environment of a fibre varies from one application to another. Various cable designs have been implemented. A representative light-duty cable is sketched in Figure 13.8. This cable weighs 12.5 kg/km and can withstand a tensile load of 400 N during installation and can be loaded up to 50 N in operation. Fibre-optic communications developed very quickly after the first low-loss fibres were produced in 1970. Today, over 10 million km of fibre have been installed worldwide, numerous submarine fibre cables covering the Atlantic and Pacific oceans and many other smaller seas are operational. In addition, installation of fibre-optic local area networks (LANs) is increasing. 13.5 FIBRE-OPTIC SENSORS Over the past few years, a significant number of sensors using optical fibres have been developed (Kyuma et al. 1982; Culshaw 1986; Udd (1991, 1993)). They have the potential for sensing a variety of physical variables, such as acoustic pressure, magnetic fields, temperature, acceleration and rate of rotation. Also sensors for measuring current and voltage based on polarization rotation induced by the magnetic field around conductors due to the Faraday effect in optical fibres have been developed. It should also be mentioned that a lot of standard optical equipment has been redesigned using optical fibres. The Laser Doppler velocimeter is an example where optical fibres have been incorporated to increase the versatility of the instrument. 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. The light is coming through the fibre from the left and is partly absorbed by the semi- conductor. This absorption is temperature-dependent and the amount of light detected at the end of the fibre to the right is therefore proportional to the temperature and Optical fibre Stainless holder Semiconductor absorber Optical fibre (a) Pressure plate Fibre To detector Input light (b) Figure 13.9 (a) Fibre-optic temperature sensor and (b) Fibre-optic pressure sensor 316 FIBRE OPTICS IN METROLOGY we have a fibre-optic temperature sensor. Figure 13.9(b) shows a simplified sketch of a pressure-sensing system. The optical fibre is placed between two corrugated plates. When pressure is applied to the plates, the light intensity transmitted by the fibre changes, owing to microbending loss. Such systems have also been applied as hydrophones and accelerometers. Figure 13.10 shows the principle of a class of fibre-optic sensors based on interferome- try. The fibres A and B can be regarded as either arm in a Mach–Zehnder interferometer. The detector will record an intensity which is dependent on the optical path-length differ- ence through A and B. When, for example, fibre A is exposed to loads such as tension, pressure, temperature, acoustical waves, etc., the optical path length of A will change and one gets a signal from the detector varying as the external load. 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. Fibre A emits a conical light beam. Fibre B will receive light inside a cone of the same magnitude. If a plane surface is placed a distance l in front A B Light source Detector Figure 13.10 Interferometric fibre-optic sensor (a) (b) (c) C A B I B l AB l Figure 13.11