FIBEROPTIC SENSOR TECHNOLOGY HANDBOOK

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”’’-” [--::= ---- \ .! ( ACCEPTANCE & CONE J-’’” Fig. 2.8 The acceptance cone for a ~tep~i~~$x optical fiber. (N. A.=sinec=(nl ‘n2 ) end surface, where the refractive index of air, no7 IS equal to unity, there is another critical angle,3c , such that all ,the light contained within the cone of half angle Clc , will be trapped within the fiber. Applying Snell’s Law, this time in the sine form because ec ‘ is the angle between the cone edge (elements) and the normal to the surface of incidence (end face of fiber), the ratip of the sine of ec within the core to the sine of ec in air is equal to the refractive index n. of air divided by the refractive index nl of the core. Thus, since n. = 1: (sinoc)/no = sinec = (sin9c’)/nl (2.8) At the core-cladding interface within the fiber it was shown that Cosoc = n2/nl Eq. (2.3). Combining Eqs. (2.3) and (2.8) and using the Pythagorean theorem: sin28c + c0s20c=l=(sin20c’ )/n12 + n22/n12 (2.9) Transposing: (5in2ec’ )/n12=l-n22/n12=(n12-n22)n12 (2.10) Thus, from the defini~ion that the numerical aperture is equal to the sin ec” : As 2 1/2 N.A. = sinoc’ = (n12 - n2 ) (2.11) defined in Eq. (2.2) above: Solving for n2: A = (nl - n2)/nl (2.12) n2 = nl (1 - A) (2.13) Substituting Eq. (2.13) in Eq. (2.11) and simplifying: N.A. = nl(2A - A2)1/2 (2.14) If A2 !9’ ~ A Light rays in a step-index fiber core. Two rays are ,showo in Fig. 2.9. One enters from air at an angle, 0 , such that on intersecting the core-cladding interface, it makes an angle less than the critical angle, 13c, with the interface. This ray will be totally internally reflected and will be trapped in the core, so that it propagates with minimal loss through the fiber core. A second ray, incident from air at a larger angle, intersects the core-cladding interface at an angle greater than tic. At each reflection part is reflected back into the core and part is transmitted into the cladding. Such a ray is strongly attenuated, rapidly decreasing in intensity as it propagates along the core of the fiber. The Eqs. (2.8), (2.11) and (2.15) show that the critical angle at the core-cladding interface for total internal reflection in radian measure for the trappin of light in the core is approximately equal to (ti) f 12 when A2
6=0 REFERENCE PLANE that propagate along the axis of the guide each with a discrete velocity. ,> = (J REFERENCE LI z In order to characterize light propagation in a step-index optical fiber it is convenient to use a parameter, usually referred to as the waveguide V-parameter (V-value). It is defined by the equation: V = 2ma(nl and therefore from Eq. (2.11): 2- 2 112/i n2 ) o (2.18) V = 2~a(N.A.)/& (2.19) Fig. 2.10 The cylindrical coordinate system for expressing lightwave propagation in an optical fiber. The point P is designated as p, 0, z. The Z-axis of this coordinate system is taken to be the central axis of symmetry of the waveguide. The z-coordinate is the distance from a reference plane designated as Z. (zEO). A spatial point in a fiber is located by defining a radius coordinate, p, as the distance radially from the Z-axis; an azimuthal (angular) coordinate, $, measured from an arbitrary reference plane ($=0); and a value of z. This coordinate system is commonly known as the cylindrical coordinate system. In the simple case where the refractive index depends only on the radial coordinate, the solution to the wave equation, derived from Maxwell’s equations for the electric fields, can be expressed as the product of two functions as follows: where a is the core radius; N.A. is the fiber numerical aperture, a function of the refractive indices of the core and cladding; and i. is the wavelength of the incident light in a vacuum. (The wavelength of a lightwave in a vacuum is nearly equal to its wavelength in air.) These are the main parameters needed to describe light propagation in a step-index optical fiber. The V-parameter may be designated as the light propagation characteristic of an optical fiber. The larger the V- value the larger the number of modes (different discrete waves) the fiber can support, i.e., allow to propagate. The predictions or conclusions that may be drawn from the wave theory of light propagation in fibers may be summarized in graphical form. As already mentioned, only particular (discrete) solutions of the wave equation exist. These correspond to discrete waves propagating along the axis of the guide with particular velocities. Some of the characteristics of these particular (allowed) modes are shown in Fig. 2.11. The kn, E(p, $,z,t) = E(p, $)e-j(~t-W) (2.16) = E(p, $)sin(ut-&) (2.17) The variable t is the time measured from a time reference of to. The first is an amplitude factor E(P ,$) that depends on the radius vector P and the azimuthal (angular) coordinate $. The second, which can be expressed as complex exponential or sinusoidal function, indicates that the electric fields are sinusoidal waves in time and in space. The angular frequency is u= 2nf where f is the optical frequency. The B is the propagation constant, defined as the refractive index, n, times the z component of the wave vector k, where kn= 2h/io, and i. is the optical wavelength in a vacuum at frequency f. Thus, optical energy transfer in dielectric media takes place in the form of wave propagation along the axis of the guide. The absolute value of k is also called the wave number. 2.1.3 Propagation Modes When the geometric boundary conditions at the core-cladding interface are introduced only particular (discrete) solutions of the wave equations are permitted. Only these values can exist, each designated by a value for i, for the amplitude factor Ei(P,$) and cor - responding discrete values for the propagation constant ~. The velocity of propagation of each allowed wave, i.e. mode, along the axis of the waveguide is given by the ratio of the angular frequency divided by the propagation constant Bi, of a particular wave designated by the subscript. Thus, the various allowed solutions represent discrete waves with discrete amplitudes 2-5 kn2 V=? G wAVEGUIDE PARAMETER Fig. 2.11 The propagation constant, i3, and the velocity of various modes as a function of the V-parameter of an optical fiber. curves show the allowed values of the propagation constant & as a function of the V-parameter (V-value). Each curve corresponds to a particular allowed solution of the wave equation. This graph indicates that the allowed values of %for the various solutions are in between knl and kn2, corresponding to the wave numbers in the core and in the cladding, respectively. Since the wave (phase) velocity in the Z-direction is given by the quantity m/6 the curves in Fig. 2.11 also show the phase velocity versus the waveguide V-parameter (Vvalue). Thus, the velocity of the allowed waves (modes) represented by the different curves is seen to range from the higher velocity in the cladding (lower ordinate, higher value) to the lower velocity in the core (upper ordinate, lower value) as the waveguide V-parameter increases. Thus each curve in Fig. 2.11 corresponds to an allowed solution of the wave equation applied to dielectric waveguides. The waveguide V-para-
Page 1 and 2: FIBEROPTIC SENSOR TECHNOLOGY HANDBO
Page 3 and 4: FOREWORD THE FIBEROPTIC SENSOR TECH
Page 5 and 6: 4.2 Phase 4.2.1 4.2.2 4.2.3 4.2.4 4
Page 7 and 8: CHAPTER 1 INTRODUCTION 1.1 BACKGROU
Page 9 and 10: CHAPTER 2 FIBEROPTIC SENSOR COMPONE
Page 11: fleeted each time it is incident on
Page 15 and 16: in the right center of Fig. 2.13. T
Page 17 and 18: The vertical height of the various
Page 19 and 20: 60 40 20 10.C ~8 x z 6 n u 4 % c 62
Page 21 and 22: : 1.5 z o GeOz 0 ~ =1.49 u w > ~1.4
Page 23 and 24: tension it collapses to eliminate t
Page 25 and 26: MAXIMUM TENSILE STRENGTH (GN/m2 = l
Page 27 and 28: mately a 0.5 electron-volt increase
Page 29 and 30: I in one portion of the recombinati
Page 31 and 32: crated electron-hole paira that are
Page 33 and 34: sink fiber. Likewise angular misali
Page 35 and 36: fused together. The ends are bent a
Page 37 and 38: A laboratory beam splitter set-up u
Page 39 and 40: aged multichannel unit. High-streng
Page 41 and 42: CHAPTER 4 LIGHTWAVES IN FIBEROPTIC
Page 43 and 44: tor addition indefinitely, as indic
Page 45 and 46: modes. Since at point (a) the two e
Page 47 and 48: as a number of wavelengths. An expr
Page 49 and 50: 100 meters of fiber, length changes
Page 51 and 52: the aingle-sideband phaae noiae int
Page 53 and 54: sets up periodic spatial fluctuatio
Page 55 and 56: ient configuration in the lower rig
Page 57 and 58: sors or a pressure gradient sensor.
Page 59 and 60: nickel H. = 3.6 oersteds and for a
Page 61 and 62: optical source of constant intensit
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in a straight section of the fiber
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A simplified design of the cladding
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Referring once more to Fig. 5.35, i
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The path length AL traveled by ligh
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5.4.9 Fiberoptic Rotation-Rate Sens
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a modulation scheme, the output of
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Fig. 5.60 lists several sources of
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CHAPTER 6 FIBEROPTIC SENSOR ARRAYS
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method of sensor energization will
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1 The preceding discussion applied
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SIMPLEX FIBEROPTIC TRANSMITTER FIBE
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MAXIMUM ALLOWABLE RISETIME = 0.7/RN
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PROCESSING STATION TRANSMISSION wAV
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fiberoptic cable (optical cable), a
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angstrom. A unit of length equal to
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core. The central primary light-con
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diffraction. 1. The procesa by whic
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fiberoptic. Pertaining to optical f
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methods of coupling power outside t
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M Mach-Zehnder interferometer. An i
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N nanometer. One thousandth of a mi
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velocity is also the velocity at wh
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where nl and n2 are the reciprocals
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slab dielectric optical waveguide.
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v vacancy defect. In the somewhat o
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