GHAPTER 2 Wavegqldfl*g, amd Fabriaatiom One of the most important components in any optical fiber system is the optical fiber itself, since its transmission characteristics play a major role in determining the performance of the entire system. Some of the questions thal arise eortceming optical fibers are 1. What is the structure o[ an optical fiber? 2. How does light prspagate along a fiber? 3. Of what materials are fibers made? - 4. How is the fiber fabricated? 5. How are fibers incorporatod into cable structures'? 6. What is the signal loss or attenuafion mechanism in a fiber? 7. Why and to what degree does a signal get distorted as it travels along a fiber? The purpose of ttris chapter is to present some of the fundamental answers to the frst five questions in ordei to attain a good understanding of the physical stru€ture and waveguiding properties of optical fibers. Questions O anO Z are answered in Chapter 3. The discussions address both conventional silica and photonic crystal fibers. Since fiber optics technology involves the emission, traRsmission and detection of light, we begin our discussion by first considering the nature of light and then we shall review a few basic laws and definitions of optics.l Following a description of the structure of optical fibers, two methods are used to describe how an optical fiber guides light. The first approach uses the geometrical or ray optics concept of light reflection and refraction to provide an intuitive picture of the propagation mechanisms. In the second apprOaCh, light is treated as an electromagnetic wave which propagates along the optical fiber waveguide. This irnvolves solving Maxwell's equations subject to the cylindrical boundary conditions of the fiber. ,/ {tz.L ffi F The Nature of Light The concepts conceming the nature of light have undergone several variations during the history of physics. Until the early seventeenth century, it was generally believed that light consisted of a stream of minute particles that were emitted by luminous sources. These particles were pictured as travelling in straight lines, and it was assumed that they could penetrate transparent materials but were reflected from opaque ones. This theory adequately described certain large-scale optical effects, such as reflection and refraction, but failed to explain finer-scale phenomena, such as interference and diffraction. The correct explanation of diffraction was given by Fresnel in 1815. Fresnel showed that the approximately rectilinear propagation character of light could be interpreted on the assumption that light is a wave motion, and that the diffraction fringes could thus be accounted for in detail. Later, the work of Maxwell in 1864 theorized that light waves must be electromagnetic in nature. Furthermore, observation of polarization effects indicated that light waves are transverse (i.e., the wave motion is perpendicular to the dirqction in which the wave travels). In this wave or physical optics viewpoint, the electromagnetic waves radiated by a small optical source can be represented by a train of spherical wave fronts with the source at the center as shown in Fig. 2.1. A wave fronr is defined as the locus of all points in the wave train which have the same phase. cenerattyl6ffi wave fronts passing through either the maxima or the minima,of the wave, iuch as the peakor trough of a sine *uu", fo, exilpte. Thus, the wave fronts (also called phase fronts) are separated by one wavelength. Spherical wave fronts Plane wave fronts Point source fig. 2.1 Representatians oJ splrcrical ard plnne uatse Jronts and. their associaled" rays When the wavelength of the light is much smaller than the object (or opening) which it encounters, the wave fronts appear as straighi lines to this object or opening. In this case, the light wave can be represented as a plane wave, and its direction of tiavel iniicated by a light raf which is drawn "*-b"large-scale optical effects such perpendicular to the phase front. The light-ray concept allows as reflection and refraction to be analyzed by the simple geometrical process of ray tmcing. This view of optics is referred to as ray or geometrical optics. The concept oflight rays is very usefU because the rays show the direction of energy flow in the light beam. 4.1.1 ' Linear Polarization The electric or magnetic field of a train of plane linearly polarized waves traveling in a direction k can be represented in the general form A(x. r) = QiAoexpLi (at - k.x)l (2.1) withx=-rer+ley*zerrepresentingageneralpositionvectorandk=kq+fr.,,er+k.e.representingth€ wave propagation vector. Here, Ao is the maximum amplitude of the wave, a = 27tv, where y is the frequency of the light; the magnitude of the wavevector k is k = 2ttfi,, which is known as the wave propagation constant, with ), being the wavelength of the light; and e, is a unit vector lying parallel to an axis designated by j. il l'+ ,!K, F'tbers: Stnlciltres, Wat-teguiding, and Fabrication Note that the components of the actual (measurable) elecffomagnetic field represented by Eq. (2.1) are obtained by taking the real part of this equation. For example, if k = ker, and if A denotes the electric field E with the coordinate axei chosen such that €i = €*, then the real measurable electric field is given by (2.2) E r(2, t) = Re(E) = e{or cos(at - kz) '.:: which represents a plane wave that varies harmonically as it travels in the z direction. The reason for using.the exponential form is that it is more easily handled mathematically than equivalent expressions given in terms of sine and cosine. In addition, the rationale for using harmonic functions is that any waveform can be expressed in terms of sinusoidal waves using Fourier techniques. The electric and magnetic field distributions in a train of plane electromagnetic. waves at a given instant in time are,shown in Fig. ?.2, The wave-s are moving in the direction indicated b.y the vector k. Based on Maxwell's equatio:rs.!1 can be shown2 that E and H are both perpendicular to the- direction of propagation. This condition defines a plane wave, that is, the vibrations in the electricJield lle,parallel other at all points in the wave. Thus, the electric field forms a plane calledthe plane of vibration. io "i"f, Likewise all points in the magnetic field component of the wave lie in another plane of vibration. Furthermore, E and H are mutually perpendicular, so that E, H, and k form a set of orthogonal vectors. v l H.' Fyg. ]- 2.2 Etecffic and. mognetic-field.distrihttions in a train oJ plrute electromagnetic uaues The plane wave exar,nple given by Eq. (2.2) has its electric field vector always pointing in the. e, dir_ection. Such a wave iq linearly polarized with polarization vector er. A general state of polarization is described by considering another linearly polarized wave which is independent of the first wave and orthogonal to it. Let this wave be (2.3) Er(z t) erE6rcos(art kz + 6) = - where 6 is the relative phase difference between the waves. The resultant wave is E(z t) = &(2, r) + E,(i, t) (2.4) If 6 is zero or an integer multiple of 2n, the waves are in phase. Equation (2.4) is then also a linearly polarized wave with a polarization vector making an angle e = EoY- arctan (2.s) Eo* with respect to e, and having a magnitude n=(Ei,*rir)',' (2.6) This case is shown schematically in Fig. 2.3. Conversely, just as any two orthogonal plane waves can be combined into a linearly polarized wave, an arbifary linearly polarized *uue Le resolved into "an two independent orthogonal plane waves that are in phase. ffg. 2.3 Addition oJ J2.1.2 Elltptlcal hn tteaflU polilizd" uaDes twfiW a zeto W, relntil:e plwx fufueenthem and Clrcular polarlzatlon For general values of 6the wave given by Eq. (2.4) is elliptically polarized. The resultant field vector E and change its magnitude as a function of the angular frequency ar. From Eqs. (2.2) and (2.3) we can show that (see Prob. 2.4) for a general value of d, will both rotate (*1. [# I \H(+l c.s 6= sin2 d (2.7) which is the general equation of an ellipse. Thus, as Fig.2.4 shows, the endpoint of E will trace out an ellipse at a given point'irt space. The axis of the ellipse makes an angle a relative to the x axis given by tan2a= ZEu*Er, cosi 83, I (2.8) - 4n To get a bettor pieture of Eq. (2.1),let us elign the principal axis of the ellipse with the ttlz, 3ttl2, ..,, so that Bq. (2.7) becomes Q, or, equivalently, 6 = t a= t I (*\-[#l a" t.' _ ; r axis. Then =, (2.e) v t, EI ,,/ " e /r' f .t, \l t F, T E , J_=, Ey 0/) El{'. 2.4 Elliptlcally plarizedlightresultsJromthe addtfwnaJfimlircarlg polaritedwaues uequal atnplffude hadng a rwnzcro ptwse dtlference 6 behpeen them oJ This is the familiar equation of an ellipse with the origin at the center and semi-axes E6, and E*. When Es, * Eoy-,Eti.8tdthe relative phase difference d = *,tc/2 + 2mft, where m = A, +1, +2,..., then we have circularly polarized light.In this case, Eq. (2.9) reduces to 1*rtr=4 (2.10) which defines a circle. Choosing the positive sign for 6, Eq* (2.2) and (2.3) become (2.t1\ Er(2, t) = -erEr sin (ot - kz) (2.12) *ill In this case, the elrdpoint of,B face out a circle ata given point in space, as Fig.2.5 illustrates. To see this, consider an observer.loegted at some arbitrary point 2,.1 toward whom the wave is moving. For convenience, we a tt/k at t = A. Then, frorn Eqs (2.11) and (2.12) we have Er(z, t) * * erEo and En(2, t) = Q will pick this paint at z so that E lies along the negative.r axis as Fig. 2.5 shows. At a later time, say t = fina, the electric field vector has rotated througlr 90" and now lies aforrg the positive y axis at 2."s. Thus, as the wave moves toward the observer with inereasing time, the resultant electric field vector E rotates clockwise at an fig. 2.5 Addition of ht:o eqtnl-anplirude Uneorlg polarizcd wales u:ifh a relatilse ptwse differerrce 6 = r/2 + 2mn results in a fuht circularlg potafized. uaue angular frequency al. It makes one complete rotation as the wave advances through one wavelength. Such a light wave is right circularly polaized. If we choose the negative sign fur d, then the electric field vector is given by E = Eo[e, cos(ott - ke),+ e, sin(art - kz)1 (2.13) J Now E rotates countercIoclew[se and the wave is lefi circurarry potarized. J2.r.3 1 The 9uantum,e_ErG of Ltght The wave theory of light adequately accounts for all phenomena involving the transmission of light. However, in dealing wi*r the interaction of light and matter, such as occurs in dispersion and in the emission and absorption of light, 4either the particle theory nor the wave theory of light is appropriate. Instead, we must tu1n to quaryry theory, which indicates that optical radiafion has particle as well as wave properties. The particle naflile arises from the observation that light energy is always emitted or absorbed in discrete,urits,called-qaatrtu ot photons. In all experiments used ts show the existence of photons, the photon energy is fouq.d to depend only on the frequency v. This frequency, in tum, must be measured by observing a wave property of light. . The relationship be1*eeritro 6-. y E and the frequency v of a photon is given by E=hv i (2.14) j tr l A Fibers; Sfrttcfiires, Waoeguiding, and Fobricatian = 6.625 x 10-34 J.s is Planck's constant. When light is incident on an atom, a photon can its transfer energy to an electron within this atom, thereby exciting it to a higher energy level. In this process either all or none of the photon energy is imparted to the electron. The energy absorbed by the where h electron must be exactly equal to that required to excite the electron to a higher energy level. Conversely, an electron in an excited state can drop to a lower state separated from it by an energy fuv by emitting a photon of exactly this energy. l_ .,{k} Baslc Optical Laws and Definitions This section reviews some of the basic optics laws and definitions relevant to optical fiber transmission technology. These include Snell's law, the definition of the refractive index of a material, and the concepts of reflection, refraction, and polarization. ..{.2.t Refractlvo Index A fundamental optical parameter of a material is the refractive index (or index of refraction). In free spacealightwavetravelsataspeed c=3x 108m/s.Thespeedof lightisrelatedtothefrequency vand the wavelength 1. by c = vL Upon entering a dielectric or nonconducting medium the wave now travels at a speed.a, which is chmacteristic of the material and is less than c. The ratio of the speed of light in a vacuum to that in rnatter is,the index of refraction n of the material and is given by n=- c (2.1s) a Typical values of n are 1.00 for aA 1.33 for water, 1.45 for silica glass, and2.42 for diamond. 4.2.2 Refleotlon aud Re&actlon The concepts of reflection and refraction can be interpreted most easily by considering the behavior of light rays associated with plane waves traveling in a dielectric material. When a light ray encounters a boundary separating two different media, part of the ray is reflected back into the first medium and the remainder is bent (or refracted) as it enters the second material. This is shown in Fig.2.6 where n2 < ry. The bending or refraction of the light ray at the interface is a result of the difference in the speed of light in two materials that have different refractive indices. The relationship at the interface is known as Snell's law and is given by n, sin @1 =nzsinQz (2.16) n1 cos 0, (2.17) or, equivalently, as = where the angles are defined in Fig. 2.6. The angle surface is known as the angle of incid.ence. nZCoS 0Z @1 between the incident ray and the normal to the According to the law oi reflection, the angle g, at which the incident ray strikes the interface is exactly equal to the angle that the reflected ray makes with the same interface. In addition, the incident ray, the normal to the interface, and the reflected ray all lie in the same plane, which is perpendicular to the interface plane between the two materials. This plane is called the plane of incidence. When light ftaveling in a certain medium is reflected off an optically denser material (one with a higher refractive index), the process is referred ta as external reflection. Conversely, the reflection of light off of less optically dense material (such as light traveling in glass being reflected at a glass-air interface) is called internal reJlection. Material boundary ffg. 9.6 Rqfractun and. refuctton oJ a lfgft rag at a. fi:rrurlrd.Aourffiry -\o ,o As the ailgle of incidencc $r in an optically denser material becomes lryger, the refracted angle p2 approaehes rrl2. Beywd ttris point no refraction is possible and ttte light rays become totally iatemally reJlected. The condidons roquircd for total intemal reflection can be determined by using Snell's law lEq. (2.16)1. Consider Fig: 3.?, which shows a glass surfaee in air. A light ray gets bent toward *re glass surface as it leaves the glass in accordance with Snell's law. If the angle of incidence p1 is increased, a will eventually be reached where the light ray in air is parallel to the glass surface. This point is known as the cndq{ at itqcidence p. When the incidence angle is greater than the critical angle, the condition for total internal reflection is satisfied; that is, the light is totally reflected back into the glass with no light escaping from the glass surface. (This is an idealized situation.,In praetice,'dre&, is always some tunneling of optical energy through the interface. This can be explained in terms of the electromagnetic wave theory of light, which is presented in Sec. 2.4.) point Wk nz4 nl Incident r{y Hg. 2.7 I Retacted T r4y fi Refracted ray No Refracted ray Refl€cted rey Repes'erttuttorta{thesifrffilaryleand.totalir.temnlreflaf;iimataglfrs$-atrharlae As an example, csl*ide,r fte glass*eir interface shown in Fig. 2.7. Wherr dre light ray in air is parallel to the glass surfrse, dren 6 = Sb so that sin h = t. The critical angle in the glass is thus sin @" * n2 nl (2.18) I l 1.ffi on I i EXaWl,e ?.. Using a1 * I .48 for glass and n, = for aia 0" is about 42.5o Any light in the glass incident ttre inhrface at an angle Or greater than 42.5" is totdly reflected back into the glass. In addition, when light is totally internally reflected, a phese changp.d:opel*rs,in the reflectsd wave. This phase ehange depends on the angle 0, < rcl2 * @" 180 aecording to the \.\d" relationshipsl Ea tan (2.19a) *= 6o tan 1= n. I (l 120 5 o a Qv o o rtr; ai-l (2.tsb) sin 0, OA Aoo Here, dy and 6o are the phase.shifts of ttre electric-field wave component$ normal and parallel to the plane of incidence, resPectively, andn = n1ln2.Tltese phase shifts are shown in Fig. 2.8 for a glass-air interface (n = 1.5 and 0" = 42.50). The values range from zero immediately at the critical angle to nlL Q, when p, = 90o. - These basic optical principles used will now l0 20 fi fU. 2.8 be to illustrate how optical power is 30 40 50 (degrees) Phase shiJts oecurring Jrom the reJtection of waue comPonents narmal (dN,) and parallet (6r) to the platw of incidence transmitted along a fiber. u{,2.3 Polarlzatlon Components of Ught An ordinary lightwave consists of many Eansverse electromagnetic waves that vibrate in a variety of directions (i"e., in more than one plane) and is called unpolariryd light. However, one can represent aRy arbitrary direction of vibration as a combination of a parallel vibration and a perpendicular vibration, as shown in Fig. 2.9. Therofore, one can consider unpolarized light as consisting of two orthogonal plane Multitude of polar-ization components Parallel polarization component Perpeadicllar polarization component Fig. 2.9 Wfizaffn&representedasacombirwtionoJaparullelulbafanwrdaperpendtaiar uibration polarization components, one that lies in the plane of incidence (the plane containing the incident and reflected rays) and the other of which lies in a plane perpendicular to the plane of incidence. ThEse are the parallel polarization and the perpendicular polarizatior components, respectively. In the case when all the electric field planes of the differerit transverse waves are aligned parallel to each other, then the lightwave is linearly polarized. This is the simplest type of polarization, as Sec. 2.1.tr desffibes. Unpolarized light can be split into separate polarization components-either by reflection'off-of a nonmetallic surface or by refraction when the light passes from one rnateridl to another. ,A.s noted:in Fi$.2.6, when an unpolarized light beam ffavelling in air impinges on a nonmetallic surface srrch as glass; part of the beam is reflected li,nd part is refracted into the glass. A circled dot and an irrrow designate the par:ilel and perpendicular polarization components, respectively, in Fig. 2. 10. The reflected beam ispartially polarized arld at a specific angle (known us the Brewster's angle) the reflected light is completely perpendicularly polarized. The parallel component of the refracted beam is transmitted entirely into the glass, whereas the perpendicular component is only paniatly refracted. How much of the refracted light is polarized depends on the angle at which the light approaches the surface and on the material composition. Incident ray O € Parallel polarizatron @ Fig. 2.1O Perpendicularpolarization Partiallyrefractedperendicularpolarization Behnsiar oJ an unynloriz-ed Light beam at the interJace fuhoeen air and, a nonmetallic surJgce 2.2.4 Polarizatlon€ensitive Materials The polarization characteristics of light are important when examining the behavior of components such as optical isolators and light filters. Here we look at three polarization-sensitive materials or devices that are used in such coqltrrerents: These are polarizers, Faraday rotators, and birefringent crystals. OpfrcatFtbers: St* and blocks the A pol.arizer is a material or device that transmits only one polarization component ffansmission vertical a has that n:polarizer light other,For example, in thp case when unpolarized ",i*'5 A familiar device' the passes through cornponeflt polarization axis as shown in fig. 2.1i, only the vertiial polarized partially glare of the reduce to sunglassei the use of polfozing example of this a sunglasses, the property of "or""ti, polarization see the To .unfijn reflections from road or water surfaces-. the in filters polarization The sideways. head their ;;;"; gf glare spots will appear when users tilt glare spots when the head is held normally' sunglasses block out trr" porulir"a hght coming from these Polarizer Fig. 2.11 Onty,.the,udrticdl.phtization component passes through a oerticallg oriertted polarizer passing through it a device that rotates the state of polarization (SOP) of light A Faraday rotatoris by a specific amount. For examPle a popular device rotates the SQJ clockwise by45" or a quarter of a wavelength, as shown in Fig. 2.12.' ' This rotation is independent of the SOP of input light, but the rotation angle is different depending on the direction in which the light passes through the devicg. That is, the rotation prldccss'is'.'not reciprocal. In this proces-q. the SQJ of.the input light is maintained ffi?I 9;":otzitl,% ypig. For example, if the inpiit hlght'to a 45" Faraday rotator is linearly tridlarized in a vertical direction, then the rotated light 45o FaradaY rotator atuuic9 tlwtrgtare;1 claclcrtsise bg 45" potnrization ttw state oJ auanselength quanter oJ or a Z.LZ AFaradng rotator is at a 45" angle. The Faraday rotator material usuallv is an iron garnet (YIG) and the degree of angular rotation is proportional ;;tr;;,h"t;.y;A'j*;';r-H;;t'pot#r"a ury,,fr"nl" crystat .o"t u* ytttir* thickness of the deviie. to'the "" This means that A, ionit" t yrqcttve crystals have a properry c4lga (ouble reftaction. as shown in crystal the of axes 'ditei"nt-Jor! p"tp"naicular i*o the indices of refracrion arj slightly (SWP). The SWP walk-:offpolarfzer a ipatiat ai r:r.rimaterials is known Btr;i;;;; Fi;.';,fu:;-il;;;-"J.n * splits the lighr signal entering it into two orthogonally (perpcndicularly) polarized beams. One of the beams is called w ordinary ra! or oray, since it obeys Snell's law of refraction at the crystal surface. The second beam is called the extraordinary ray ot e-ray, since it refracts at an angle that deviates from the prediction of the standard forrn of Snell's law. Each of the two orthogonal polarization components thus is refracted at a different angle as shown For example, if inFig.2.l3. the incident unpolarized light arrives at an angle perpendieular !o the surface of Unpolarized incident ray the device, the o-ray can pass straight through the device whereas the e-ray component is deflected O at a slight angle so it follows a different path through the material. Thble 2.1 lists the ordinary index no and the extraordinary index n" of some common birefringent crystals that are used in optical communication components and gives some of their applications. Extraordinary ray polwization -<--> Q1di16ry ray polaization Ftg. z.fg A bire{ringent crystal splits the Light signal enterirg X into htso perpendiularlg polortzed beams Crystal name Symbol no ne Calcite CaCO3 1.658 1.486 Polarization controllers and beamsplitters Lithium niobate LiNbo3 2.286 2.2m Light signal modulators Rutile Tio2 2.616 2.903 Optical isolators and circulators Yttrium vanadale Yy04 1.945 2.149 Optical isolators, circuliators, and beam displacers Applications Befbre going intr: details on optical fiber characteristics, this section first presents a brief overview of the underlying concepts of optieal fiber modes and optical fiber configurations. The discussions in Sec. 2.3 tfuough 2.7 adfuess conventional optical fibers, which consist of solid dielectric stfl.rctwes. Section 2'8 describes (he structure of photonic crystal fibers, which can be created to have a variety of internal microstructures. Chapter 3 describes the operational characteristics of both categories of fibers. An optical fiber is a diefgefie waveguide that operates at optical frequencies. This fiber waveguide is normally cylindrical in forrn. It confines electromagnetic energy in the iorm of light to within its surfaces and guides the light in a direction parallel to its axis. The transmission propertier-of a, optical waveguide Optical Nbers: Stnrchrres, W@ an optical are dictated by its strucnyal characteristics, which have a major effect in detenirining how informationthe establishes signal is atrectea as it propagates along the fiber. The structure basically capacity of the nUer anO also influences the response of the waveguide to environmental "iryirg perturbations The propagation of light along awaveguide can be described in terms of a set of guided electromagnetic ot trapped waves called the modeiofthe iaveguide. These guided modes are referred to as the bound distributions that modes of the waveguide. Each guided mode is a patterruof elecffic and magnetic field is repeated along tf,e fiber at {ual intervals. Only a certain discrete number of modes are capable of be seen in Sec. 2.4, these modes are those electromagnetic waves propagating aloig the guide. As will it ui rutirfy the hornogJneous wave equation in the fiber and the boundary condition at the waveguide surfaces. Although many different configurations of the optical waveguide have been discussed in the literature,3 of refraction the most widely accepted structuri is the single solid dielectric cylinder of radius a and index by a solid is surrounded The core n 1 shown in Fig. Z.ti. fnis cylinder is known as the core of the fiber. cladding principle, a in dielectric cladiing which hasa refractive indexn2that is less than 21. Although, cladding The purposes' is not necessary fJr fight to propagate along the cdre ofthe fiber, it serves several mechanical adds it reduces ,"utt".ing tosi ttrat resutti from dielecnic discontinuities at the core surface, which it could strength to the fi6er, and it protects the core from absorbing surface contaminants with come in contact. ffg. 2. t4 khematlc oJ q 61rusentt:rrr;rt slticafiber strttcture, A ciraiar soltd" mre oJ reJractlue index n, is iunotnd.ed" bg a ctaddbg houing a reJractilre index flz 1 fl t. An elastic plast:a bt+ffer enmPstiates tle fifur In standard optical fibers the core material is highly pure silica glass (SiOr) and is surrounded by a glass cladding. Higher-loss plastic-core fibres with plastic claddings are also widely in use. In addition, irost fibers ai" enJapsrrtut"i in an elastic, abrasion-iesistant plastic- matenal. This material adds further iiregularities, strength to the fiber ana mechanically isolates or buffers the fibers from small geometrical scattering cause otherwise could distoiions, or roughnesses of adjacint surfaces. These perturbatitins into cables incorporated are fibers the losses induced by iandom microsicopic bends that can atite when or supported by other structures. types Variations in the material composition of the core give rise to the two commonly used fiber undergoes and throughout uniform th, refractive indeiof the core is rt o*n in Fig. 2.15. In the first case, "u.", unge (or step) at the cladding boundary. This is called a step'index fiber In the second an abrupt fiber' of the "t the center from the core refractive indei is made to'vary as a funciion of the radial distance This type is a graded-index fiber Both the step- and the graded-index fibers can be further divided into single-mode and multimode whereas classes. As the- name imp-lies, a single-mode fiber sustains only one mode of propagation, fibers multimode and multimode fibers contain many hundieds of modes. A few typical sizes of single- Multimode fibers offer several are given in Fig:2,15 to provide an-id1 of the dimensional scale. chapter 5, the larger core radii of advantages,compared'with'sirtgle-mode fibers, As we shall see in and facilitate the connecting multimode fibers make it,easier to launch optical power into the fiber be launched into a multimode fiber using together of similar fibers- Another advantage is ttratiigtrt can fibers must generally be excited with laser a light-emifiing-diode (LED) source, whereas single-mode laser diodes (as we shalldiscuss in chapter 4)' diodes. Although LEDs have less optical otltput poier than ciicuitry, and have lOnger lifetimes they-are easier to make, are less expensive, require less complex applications' than laser diodes; thus,making thern-more desirable in certain Fiber Cross Section and RaY Paths Inilex profile n2t ft Tlpical dimensions { 125 pm (cladding) -l2au-J--l_-_-- _T _T _lr Monomode stepindex fiber + _t _T nzt + inl I l--Es 8-12 Pm (core) 125-400 pm (cladding) 5G-200 pm (core) 125-140 Pm (cladding) A _I ) -,tr-:-o --'-- _r Multimode graded-index fiber -T 5G-100 Pm (core) rrg. 2.16 Ctr4partson of 9ou.:enaorwtstngte-mde andmiltilnode step-indexandgradd' hdex qPtioaljtbers we shall describe dispersion' A disadvantage of multimode fiters-rs tfrat they quffer from intermodal as follows' When an described be can dispersion intermodal Briefly, 3. this effect in detail io Ch;$er (or most) of the tiu*r, tr'"-optical power in-the gulse is {iryiuyte$over all at a slightly travels propagate ]n a multimode fiber modes of the fiber. E";[; th" modes tirut "an moaes in a'giverioptic"t to spread out in time as it travels along the flberpulse the Jiff"."nt times,'thus causing reduced by using a gradedt"i*rrkl-iiiiirtio" or tnterm.odnl distortion, can be larger bandwidths (data rate hav.e m99h inJex profile in a nber core. ft ir afo*. graded-index fibers t9 in single-mrcde possible trtel utpp.ino"inu".t. Even higher band-widths are transmission ffiffifi; ;ffiil;]ril; #;,ilffi;. hffi;i 'ill1" pltt:ffiJi3;f#Til:t ilJ:*tl ffi;il i.ri" i "apaulitie*i fibers, where intermodal dispersion effects are not present' 2.g.2 RaYB and Modee fiber can be r:p.reseltel by a superposition The elecuomagnetic light field that is guided along an optical of a seJ of simple electromagnetic field consists guided irodei of bound or tr,apped *S"*-iiJ "rtri"t" jl;rll::1rr;,l:: Opticat Nbers: Sbttctures, Wat)eguidirlig; orld' Fabricalian lffi A configurations. For monochromatic light fields of radian frequency @ amode traveling.in the positire z direcfron(i'e.,alongthefiteraxis)hasatimeandedependenie$ivenby , - ei@t-p) parameter The factor B is the z component of the wave propagation constant k = 2rl )' andis the main p of interest in describing fiber modes. For guided modes, can assume only certain discrete values, which are determined dom the requirement tha! the mode field must satisfy Maxwell's. equations and in the electric and magnetic field tounaary conditions at the core-cladding interface. This is described detail in $ec.2.4. Another method for theoretically srudying the propagation characteristics of light in an optical fiber is the geometrical optics or ray-trdcing approach. This method provides a good approximation to the aJceptance and guiding properties'of opical fibers when the ratio of the fiber radius to the wavelength ijarge. This is known asthi small-wavelength limit. Although thelaV anproach.is sgictly validgnlyin the zJro-wavelength limit, it is still relatively accurate and extremely valuable for nonzero wavelengths 11ght whgn the numbei of guideci modes'is large; that is, for multimode fibers. The advantage of the ray with the exact electromagnetic wave (modal) analysis, it gives a more direct approach iS that, "o*p-*qa Iight propagation charactbristics in an optical fiber. ofthe interpretation ^pirysical ray is vJrydifferent from that of a mode, let us see qualitatiVely what the a ligt* of c^oncept Sior" the relationship is between trr"*. rnL mathematical details of this relationship are beyond the scope of this (along the fiber book but can be found in the literature.6; A guided mode traveling in the z direction standingaxis) can be decomposed into a family of superimposed plane waves that collectively form a are such plane waves phases of the is, the That axis. fiber the transverieto pattern in the direction wave we can plane wave with any Since stationary. remains waves of set the collective of that the envelope plane waves of family the wave, of the phase front the to perpendicular is that ray associate a light panicular of this Eachtay ray congruence. a called of rays a set forms mode correspondinf to u iu.ti"ular a certain set travels in the fiber at the same imgle relative to the fiber axis. We note here that, since only congruences ray of the possible angles the fiber; in a eXist guided modes discrete number M of picture corresponding to fhese moa". at'e,also limited to the same number M. Although a simple ray upp"*r'to allow rays at any angle greater than the critical angle to propagate in a fibet the allowable quantized propagation ungtls reiult when the phase condition for standing waves is introduced into the picture. This is discussed further in Sec. 2.3.5. ray 'Iiespite the usefulness of the approximate geometrical optics method, a number of limitations and discrepancies exist between it and the exact modal analysis. An important case is the analysis of singlemode tr few-mode fibers, which must be dealt with by using electromagnetic theory. Problems involving In addition, coherence or interference phenomena must also be solved with an electromagnetic approach. is required. modes individual of distribution field the of knowledge when.a a modal analysis is necessary the analyzing or when mode individual of an excitation the analyzing w|en This arises, ior example, 3). in Chapter (which discuss shall we imperfections waveguide at power rnodes between coupling of enol1"r dir.."p*"y between the ray optics approach and the modal analysis occurs when an optical fiber is unifordybeniwith a constant;adius of curvature. As shown in Chapter 3, wave optics correctly predicts that every mode of the curved fiber experiences some radiation loss. Ray optics, on the other iland, erroneously predicts that some ray congruences can undergo total internal reflection at the curve 2.3.3 Steo-Inde= Flber Stnrcture We begin our discussion of light propagation in an optical waveguide by considering the step-index fiber. In practical step-index fibers the core ofradius a has a refractive index n1 which is typically equal to 1.48. This is surrounded by a cladding of slightly lower index n2, where nz= n{l - A) (2,.2o) The parameter A is called the core-cladding index difference or simply the index dffirence. Yaltrcs of n2 are chosen such that A is nominally 0.01. Typical values range from I to 3 percent for multimode fibers and from 0.2 to 1,0 percent for single-mode fibers. Since the core refractive index is larger than the cladding index, elecffomagnetic energy at optical frequencies is made to propagate along the fiber waveguide through internal reflection at the core-+ladding interface. Since the core size of multimode fibers is much larger than the wavelength of the light we are interested in (which is approximately ! pm} an intuitive picture of the propagation mechanism in an ideal multimode step-index optical waveguide.is most easily seen by a simple ray (geomenical) optics represeatation.Gll For simplicity, iq this analysls we shall consider only a particular ray belonging to a ray coRgnrence which represents a fiber mode. The two types of rays tfrat can propagate in a fiber are meridional rays and skew ruys Meridional rays are confined to the meridian planes of the fiber, which are the planes that contain the axis of symmetry of the fiber (the core axis). Since a given meridional ray lies in a single plane, its path is easy to track as it travels along the fiber. Meridional rays can be divided into two general classes: bound rays that are rapped in the core and propagate along the fiber axis according to the laws of geometrical riplics, and unbound rays that are refracted out of the fiber core. Skew rays Ne not confined to a single plane, but instead tend to follow a helical-type path along the fiber as shown in Fig. 2.16. These rays are rnore difficult to track as they travel along the fiber, since they do not lie in a single plane. Although skew rays constitute a major portion of the total number of guided rays, their analysis is not necessary to obtain a general picture of rays propagating in a fiber. The examination of meridionalrays will suffice for this purpose. However, a detailed inclusion of skew rays will change such expressionsas the light-acceptance ability of the fiberandpowerlosses of lighttraveling along a waveguide.6'lo Ray path prgjecled on fiber end face Ftg. 2.16 Rag optics representat'ron oJ skeu rags trarseling in a step-index optical@ are A greater power loss arises when skewrrays are included in the analyses, since f{r of the skew rays leaky rays that g"eometric optics pridic'ts to be trappedln the fiber are actually leaky rays.s't1".Jl':* along light fiavels the as attenuate and fiber *. o"nrv partialli cordned to the core of the circular optical alone' p].".r.ry thesry Uy be described cannot ,t" ffiif *uregoid.. This partial reflection of leaky rays by mode theory' Inste'ad, the ana{sis of radiation loss arising from these types of rays must be described This is explained further in Sec. 2-4. Refracted ray The meridional ray is shown in Fig. 2-17 Acceptance for a step-index fiber. The light ray enters the n2 Claddirrg cone fiber core from a medium of refractive index n at an angle 0s with respEct to thefrber axis and strikes the- coretladding ihterface at a n 1ill nt Core normal angle Q.If it strikes this interface at such eln angle that it is'totally iriternally nz Cladding reflected, then the meridional ray follows a Entrance rays zigzag path along the. fiber core, passing through the axis of the gUide after each Fig.z.LZ Meridional rag optics repre' sentation oJ the ProPagation reflection. From Snell's law, the minimum angle @6n that supports total internal reflection for the meridional ray is given bY mechanism in an ideat steP-index oPtical uatseguide flt Slll Q-;n = nl (2.2t) of the core and be Rays striking the core-+ladding interface at angles less than @r,n will refract out otBq' (2'21) lost in the chd-ing. sy apftying 3nefl's law to the air-fiber face boundary, the condition can be related to th".rn*i*urr entrance angle 0o, ** through the relationship (2.22) nsin 06..*= nlsing. =(nl -nl)t/' where 0" - fit2'-'O,.Thus,.thoSerayshaving entrance angles 0s less than 06, ru* will be totally internally reflected at the coie-cladding interface. rays: Equation (2.22) atsa defin'es the namerical aperture (NA) of a step-index fiber for meridional NA = n sin 06,-* = (d - nl)''' = n J^ (2.23) Eq. (2.20), The approximation on the right-hand side is valid for the typical case where A, as defined by it is angle, is much^liss than l. Since thE numerical aperture is related to the maximum acceptance sourcecalculate commonly used to describe the light acceptance or gathering cpalility of a frbeq and to aperture is a The numerical 5. in Chapter is detailed This Jffi"i"nii"r. power couplfrg to-fiber optical 0.14 to 0'50' dimensionless quantity whlchis less than unity, with values normally ranging from E x ampl e 2. 2 Cqnsider a rli*l+iinode silica fiber that has a core refractive index n, = 1,48 and a cladding index nz= 1'46' (a) / From Eq. (2-21) the critical angle is given by rta .o-r . --=sll 0. =sln-r nt 1.46 1.48 = g0.5o (b) From Eq. (2.23) the numerical aperture is NA = 0.242 (c) From Eq. (2.22) the acceptance angle in air is 00.* = sin-r NA = sin-r 0.'242 = 14" 2.3.5 ltrave Repre?-entation in a Dielectrlc Slat Wavegulde Refening to Fig.2.l7, !h" tpy thqory appears to allow rays at any angle O greater than the critical angle to propagate along ttie fibe1However, when the interference effect due to the phase of the plane wave associated with the iay is taken into account, it is seen that only waves at certain discrete *gf.* grau,"t than or equal to @. are capable ofpropagating along the fiber. To see this, let us consider wave propagation in an infinite dielectric slab waveguide of thickness d. lts refractive indeli n, ,is greater than the index z2 of the material above and below the slab. A wave will thus propagate in this guide through multiple reflections, provided that the angle of incidenbe with respect to the upper and lower surfaces satisfies the condition given in Eq. (2.221. Figure 2.18 shows the geometry of the waves reflecting at the material interfaces. Here, we consider two rays, designated ray and ray);2,.associated with the same wave. The ray,s arg incidgnt on the material interface at an angle 0 < 0" = ttlT - Q,. The ray paths in Fig. 2.18 are denoted by soiia fines and @, I The condition requiredfor wa.ve propagation in the dielectric slab is that all points on the same phase front of a plane wave must be in phase. This means that the phase change occurring in ray 1 when travelling frory no-rntl-fo poirrq B minui the phase chirnge in'ray')between point" C anab musi differ by an integer multiple of 2n As the wave travels through the material, it undergoes a phape shift given by 4 where kr = the propagation constant in the medium of refractive index n, k = k1ln1 is the free-space propagation constant s = the distance the wave has traveled in the material The phasb of the wave changes not only as the wave travels but also upon reflection from a dielectric interface, as described in Sec. 2.2. In going from point A to point B, ray Lffavels a distance sr = d/sin g in the material, and undbrgoes two phase changes 6at the reflection points. Ray 2 does not incur 4ny reflections in going from point C to point D, To, determiue its phasq change, first nole- !!ral the distance from point A to point D (dltan 07 - d tan 0. Thus, the distance between points C and D is .s2 = AD Phase . cos g= (cos2 g- sin2 is AD = 0; dtsin 0 front of dowriward.travelhng wave upward-travelling wave Fig. 2.18 LiQht uaite propogating along afiber warcguide. Ptnse chonges occir both as tlrc toaue tratsels tlvoughtttefiber medium and. atthe reflectian points Flbers; Shrrctures, WaDeguidW, and Fabrication The requirement for wave =!J wherem= 0, 1, 2,3, .... ** woeasan;;;i be written as (2.24a) Gr-s2) + Z6-2nm Substituting the expressions for s, and s2 into Eq' (2'24a) then yields L-[(cos2 e -sinz e)a l]. r, =ztcm ry{ sino l] )' [sin0 [ (2.24b) which can be reduced to 2ttndsin? L *6=fim (2.24c) plane of incidence' we have from Considering only electric waves with components normal to the F.q. (2.19a) that the phase shift upon reflection is -,-",*[tr1#ry] (2.25) be a decaying and not a growing The negative sign is needed here since the wave in the medium must yields (2.24c) wave. Substituting this expression into Eq. (2.26a) (2.25b) g Eq' (2'26) will propagate Thus, only waves that have those angles which satisfy the condition in the dielectric slab waveguide (see Prob. 2.13)' m in Mode Theory for Circular ltraveguides mechanism in a fiber, it is i";;, a more detailed understanding of the optical po\iler.propagation interface n"""rr*u to solve tntaxwett's equations-subject 6 the rylindrical boundary condition-s at thenumber of in a detail extensive in done been This has fiber. the ;;;;;; ,h" li"aaiiig of ;;;;;,,h"ia-fiir"" scope of this book, only a general outline of a w"ffi' rot u io*pf"t" traunent'is beyond the (but still complex) analysis will be given here' simplified " 2.4.1 we frst give a qualitative in ciicular optical tiefor" pi"."ntirg th" ;ari" ,noA" tt lbers, T 1"". "ory p]e.s:nts a.trie{ summary of the 1ec.2.4.2 Next, oimoa"t in'a-waveguide. ou"*la* if th. that thgse. who are not "oiq"po so 2.4.9, through 2.4.3 S""t. in il;i}|fi r;.ri . obiained from the analys"i (*) without loss of a star by designated sections those familiar with Maxwell', eqoations can skipover continuity. When solving Maxwell's equations for hollow metallic waveguides, only transverse electric (TE) modes and transverse magnetic (TM) modes are found. However, in optical fibers the core-cladding boundary conditions lead to a coupling between the electric and magnetic field components. This givei rise to hybrid modes, which makes optical waveguide analysis mori complex than metallic waveguide analysis. The hybrid modes are designated as IIE or EH modes, depending on whether the transverse electric field (the E field) or the transverse magnetic field (the H field) is larger for that mode. The two lowest-order modes are designated by tIE,, and TE61, where the subscriptsiefer to possible modes of propagation of the optical field. Although the theory of light propagation in optical fibers is well understood, a complete description of the guided and radiation modes is rather complex since it involves six-component hybrid electromagnetic fields that have very involved mathematical expressions. A simpli6"u6inte-z: of these expressions can be carried out, in practice, since fibers usually are constructed so that the difference in t]re cge and cladding indices of reiraction is very small; that ii, n, n2<< 1. With this assumption, only four field components need to be considered and their expressions become significantly simpler. The field components are called lineaily polarized (Lp) modes and are labeled *t"r" and m are ifr, j integers designating mode solutions. In this scheme for the lowest-order modei,'each Liro. mode is derived from an fIEr- mode and each LPr- mode comes from TE6r, TM6r, and FlEo_ modej.'Thus, the fundamental LPo, mode corresponds to an fIE, mode. Although the analysis required for even these simplifications is still fairly involved, this material is key to understanding the principles of optical fiber operation. In Secs. 2.4.3 through 2.4.9 we first will solve Maxwell's equations for a circular step-index waveguide and then will discribe the resulting solutions for some of the lower-order modes. 2.4.1 Ovenriew of Modes : Before we progress with a discussion of mode theory in circular optical fibers, let us qualitatively examine the appearance of modal fields in the planar dielectric slab waveguide shown in Fig. 2.I9. The this waveguide is a dielectric slab of index n, that is sandwiched between two dielectric layers "9T.ot which have refractive indices nzl nt. These surrounding layers are called the cladding. This represents the simplest fory 9f an optical waveguide and can serve as a model to gain an understanding of wave propagation in optical fibers. In fact, a cross-sectional view of the slab waveguide looks the same as the cross sectional view ofan optical fiber cut along its axis. Figure 2.19 shows the field patterns of several of the lower-order transverse electric (TE) modes (which are solutions of Maxwell's equations for the slab wavegu;6"2-to;. The order of amode is equal to the numberof field zeros across the guide. The order of the mode is also related to the angle that the ray congruence corresponding to this mode makes wjtlr the plane of the waveguide (or the axls of a fiber); ihat i{ the steeper ttL ,h" higi"r the order of the mode-. The plots $ow that the electric fields of the guided modls are not completeiy confined to the central dielectic slab (i.e., they do not go to zero at the guide-+ladding interface)-, bul-instead, they extend partially into the cladding. The field-s vary harmonicity in ttre guifrng regi;;';i;;il;"tive index n, and decay exponentially outside of this region. For low-order modes the fields are tightly concentrated near the center of the slab (or the axis of an Jptical fiber), with little penetration into it? li'-,aa:ig On the other hand, for higher-order modes tn" n"ta. are distributed-more toward tt of the guide " "ag". and penetrate farther into the cladding region. Solving Maxwell's equations shows that, in addition to supporting a finite number of guided modes, the optical fiber waveguide has an infinite continuum of radiaiion *rd"t that are not trapfed in the core and guided by the fiber but are still solutions of the same boundary-value problem. Theiadiation field ;ti;; A;;. OpfrmlFibers; "Stnrctures, Wauegutdtng, and. Fabrlcalion Evanescent tails extend into ilre cladding rE,/ \ 't/ \li Cladding z2 TEz I Exponential decay C.ore n1 Exponential Cladding z2 decay .l Mode type: Zeroth-order fE. 2.19 Electric field" dtsffibttttans Jor seueral of the lower-order guid.ed modes in a s?mmetical'slob usatseguide basically results from the optical power that is outside the fiber acceptance angle being refracted out of the core. Because of the finite radius of the cladding, some of this radiation gets trapped in the cladding, thereby causing cladding modes to appear. As the core and cladding modes propagate along the fiber, mode coupling occurs between the cladding modes and the higher-order core modes. This coupling occurs because the,elecfic fields of the guided core modes are not completely confined to the core but extend partially into the cladding (see Fig. 2.19) and likewise for the cladding modes. A diffusion of power back and forth between the core and cladding modes thus occurs; this generally results in a loss of power from the core modes. In practice, the cladding modes will be suppressed by a lossy coating which covers the fiber or they will scatter out of the fiber after traveling a certain distance because of roughness on the cladding surface. 8' 12' 13 i. in addition to bound and refracted modes, a third category of modes called lealq modess'6' present in optical fibers. These leaky modes are only partially confined to the core region, and aftenuate by continuously radiating their power out of the core as they propagate along the fiber. This power radiation out of the waveguide results from a quantum mechanical phenomenon known as the tunnel ffict.lts analysis is fairly lengthy and beyond the scope of this book. However, it is essentially based on the upper and lower bounds that the boundary conditions for the solutions of Maxwell's equations impose on the propagation factor p. A mode remains guided as long as B satisfies the condition n2kcB<nft t n, and n2 are the refractive indices of the core and cladding, respectively, and = 2xlL T\e boundary betwten truly guided modes and leaky modes is defined by the cutoff condition p = nrk. As soon as pbecomes smaller tltatn2k, power leaks out of the core into the cladding region. Leaky modes can carry significant anounts of optical power in short fibers. Most of these modes disappear after a few centimeters, but a few have sufficiently low losses to persist in fiber lengths of a kilometer. where An important pararneter connected with the cutoff condition is the V number defined by v=T@ ho -n?)v'= L *o (2.27) l This is a dimensionless number that determines.how many modes a fiber can support. Except for the lowest-order fIE,, mode, each mode can exist only for values of V that exceed a certain limiting value (with each mode having a different ylimi0. The modes are cut off when F = nzk rhis occurs when V <2.405. The IIEI mode has no cutoff and ceases to exist only when the core diameter is zero. This is the principle on which single-mode fibers are based. The details for these and other modes are given in Sec. 2.4.7. The Y number also can be used to express the number of modes M in amultimode fiber when V is of the total number of modes supported in a fiber is large.- For this case, an estimate "=+(Tlot-"n=l Since the field of 'a'guided mode partly into the cladding, (2.28) extends as shown in Fig. 2.19, afinal quantity of interest for a step-index fiber is the fractional power flow irithe core and cladjing for a given mode. As the Vnumber approaches cutoff for any particular mode, more of the power of that mode is in the cladding. At the cutoff point, the mode becomes radiative with all the optical power of .the mode residing in the cladding. Far from cutoff-that is, for large values of V-the fraction of the average optical power residing in the cladding can be estimated by P. a P iq the total. optical power in thg fiber. The d.etails for the power distribution between the core and the cladding of various LPrT modes are given in Sec. 2.4.9. Note that since M is proportion al to V2, floy.il chqdi$-decreases as-V increases. However, this increases the number of modei $:.pqY."t in the fiber, which is not desiiable for a high-bandwidth capability. whg,re tlt 2.4.g Marwell's H[uatlons* To analyze the optical waveguide we need to consider Maxwell's equations that give the relationships bgtween the electric qrd qagnetic, fiel4s. Assuming a linear, isotropic dielectrii material having no currents and free charges, these equations take the form2 12_30a) dt ' I :: .:t , .:. ., vxH-aD, (2.30b) V.D=0 V.B=0 (2.30c) dt Qsoa P.: .E, LnO n'= p-ff.Jhg,parametei e is the permittivity (or dielectric constant) and p is peungability of the mgdium-, ,. . . , re,I1tlo.lsliip deliniag thE wave plenomena o{ the electrolnagnetic fields can be derived from ."$Maxwell's equations.,Taking-th9 curl,ofEq,.1z.zoa1and making us'e-of E+ (2.300) yields whire v x (v x E) = -r$fvrn )*.ett#, Using the vector identity (see App. B), Vx(V"xE) =V(V'E)-V2E (2;3la) l and using \. (2.3Oc) (i.e., V'E = 0), Eq. (2.31a) becomes ^ ep VjE= a2E (2.3rb) ar, Similarly, by taking the curl otEq' (2.30b), it can be shown that VIH = a2H ,F# (2.31c) Equations (2.31b) and (2.31c) are the standard wave equations' 2.4.4 JyavegufdeEqridtfona* ' ', : 2.20. For this Consider electromagnetic waves propagating along the cylindrical fiber shown in Fig. axis of the the along lying axis (r, the with z is defined Q,- zl fiber a cylindrical ioordinate .y*t"functional a have will they axis, the along propagate z to are waveguide. If the electromugn"ii" waves dependence of the form E = Eo(r, Q) ei@t-Fzl (2.32a) H = IIo(r, Q) si(ot-Fzl (2.32b) the propagation which are harmonic in time r and coordinate z. Thelparameter B is the z component of at the corefields electromagnetic on thg conditions boundary vector and will be determined by the Maxwell's into are substituted (2.3h) and Eqs When 8.32b) cladding interface described in Sec. 2.4.6. (2.30a) curl equations, we have, from Eq. i[* + irB,a)=-''u'' iBE,.*= ialtHE, : (2.33a) Q33b) i[*,o,-*)= -ittaH, (2.33c) i[+. i'FHa)="'u' (2.34a) and, from F;q. (2.30b), ' .', , ,ioH,+*=-ieaEq By gliminating variables ,: (2.34b) f. *6,Hr,{" known' the tfat, le-lwritt9n ych -wP" For examp!", EOo: 11, can be E'r, er, H, and Hrcan be determined. these, equatrons Can remaining transverse components from'Eqs (Z.3ia) ana <Z'.i+i> so that the'bomponent 11, or E,, respectively, can be found in terms of E or .EI., Doing so Yields "iil;6 : Direction of wave ProPagation ' , F1rg.z.zO Cgtindrical coordinate system used Jor analAzing electromagnetic uaDe Proqagation in an oPtical E.=- fuer aH' ) ) p*+to,;'A +( qL[ d' (2.35a) (2.35b) "=-i(+*-u'*) ''=-il'* +*) (2.35c) ,,=--i(+*.,.*) where ( - afeP- fr' = ll - B'. Substitution of Eqs (Z.35ci and (2.3sd) (2354 into Eq. (2.34c) results in the wave-equation in cylindrical coordinates, 02E, +L?E n L dzn, +02E_=O ' dr2 rdr r2002 4 (2.36) and substitution of Eqs (2.35a) and (2.35b) into Eq. (2'33c) leads to Y!-*LaH, + I a2H z + a2H - =o (2.37) This It is interesting to note that Eqs. (2.36) and (2.37) eachcontain either only,. or 9nly.Il" arbitrarily chosen can be and uncoupled H are E and of components to imply that thJ longitudinal required is I/. a1d of E coupling general, in (2.37). However, (2.36i and p.*iA"i that they .utirfy Eqs appears field components described in Sec' 2'4'6' If the bV th" boundaryconditions oi the electromagnetic the field components, mode solutions can be to couplingletween blrraa.y conditions do not lead 0 ttrg mo{s3re calledtransverse electric otTE Er= wrr* ti. Hr= obtained in which either Er=0 or or TM modes. Hybrid modes exist if both ffansverse'magnetic ; catLd ;fr;,ffi *iin lr. b-u{"v are modes, depending on whether Hr-or E* EH 6r HE as a"Signated E- and H- ue no*"ro.-ft"'r" *gt The fact that the hybrid modes are field. transverse to the [t p""r*fyl *uk", a larger contribution than in the simpler case of hollow complex more analysis their p."r"nt in Lptical *ur"g,iid". makes found' are modes TM and metallic waveguides where only TE 2.4.5 Wave Equatlons for Step'Iqdcr Flbersr fiber.-A standard mathematical we now use the uuou"ffili*o find the guided modes in a step-index method' which separation-of-variables the use to is procedure for solving.q*,fur such as 4. iZ.:Ol assumes a solution of the for-m (2'38) E,= AFrQ)Fz@)FzQ)F+(t) Aswasalreadyassumed,thetime-andx.dependentfactorsaregivenby F'(z)F4Q) = si@t-Fz) In addltion, wave is sinusoidal in time and propagates in the z direction. since the ;;p;ent (2.3e) beg.ause of.the circular must not change when the coordinate symmetry of the waveguide, each fieH iy Zn. W" tttut assume a periodic function of the form The constant y can be positive in -- @with apeiodof 2n. V*ra, an integer since the fields must be periodic equation for r') \ -4ln n is increased (2.:40) * *r"0r",1',13;il". brtrtitriing Eq. (2.+O) into Eq. (2.38), the wave d2F, . t a!*( @ E [Eq' 2'36)] becomes =o e.4t) An exactly identical equation can which is a well-known differential equation for Bessel functions.2L26 be derived for Hr. a homogeneous core of refractive index n1 For the configuration of the step-index fiber we consider of index n2' The-reason for assuming an and radius a, which i. ,o,'ooraei by an infinite cladding hur" exponintialll dTuYlTg fields outside infinitely thick cladding'is'that the guided modes in the "or" outer boundary of the cladding'.In practice' the core and these must have insignificant values at the thick so that the guided-mode field does optical fibers are a"rigilJ *itr, aioailes. tlat are sufficiently tfr. .iuaaing To ge! an idea of the field patterns' the electric field not reach the outer tirra*y "f symmetical-slab waveguide were shown distributions for several of the lower-order guided rrioo"r in a in the guiding region of refractive index nl and decay in Fig. 2.l4.Thefieldsvary harmonically exponentially outside of this region. outside the core. For the inside region Equation (2.41) must now be-solved for the regions inside and 0, whereas on the outside the solutions the solutions for the guided modes must remain"finite as r -) are Bessel functions of the first kind of order must decay to zero u, , - -. Thus, for r < a the solutions with k1 = 2nntl h' The y. For these functions we Use the common designation J u@r) ' Hete, u2 = tq - P2 expressions for E and fI. inside the core are thus = AJr(ur) ,ivo'i@t-|d Hr(r < a) = BJu(ur) 'ivQ'l@tt-Fd Er(r < a) (2'42) (2'43) where A and B are arbitrary constants' modified Bessel functions of the second outside of the core the solutions to Eq. (2.41) are given by for Erand II. outside the core are kind, K,(wr), where i= p; -,7r*itf, t, = 2nnzlL. The e*presiions therefore = CKv(wr) eivoei@t-PzJ HzQ > a) = DKr(wr) Er(r > a) "lv0'i@t-92) with C and D being arbitrary constants' (2'44) (2'45) The definitions of J,(ur) and K,(wr) and yiirious rccursion.relations are given:in App e From the definition of themodifiedBgBse!function, itis seen that KJwr) -+ -. Sinia K,(wr) must go ? "*'aswr tozeroas1--1'*,itfollowsthatw)0.This,intum,impliesthatp2fo,whichrepre."ot u"rioffcondition. \tte cutoffcond.itionisthe poirit'atrirtrich a mode is no longerbound to the core region. A seeond condition onf 9an Ue !9{u9e0 from the behavior of ,I,(rr). Insid" thq core the parameter u-*u.t be real for F, ro be real, from which it follows that &1 > P. Th" permissible range of for bound solutions is therefore B n2k=kr{B<kr:'nrk (2.46) wheie ft = Zttl),is the free-space propagation consiant. 2.4,6 Modal Pquatio.n! , The solutions for pmust be determined from the boundary conditions. The tioundry conditions require that the tangential components Erand Erof E inside and outside of the dielectric interface at r a must = be the s4n1e, 4nd.simil.arl.y fgr the tangential components H, and I1., Consider first the tangential components of E. For the?'iomponent-*e have, from yq. (Z.iZ) at tn! inner cor.-"fuOairrgffi;;rry (Er= Er) and from fa. {2,U1at the outside of the boundary (8, = Er2), that Et - Ea= AJ,Q.n) - CK,(wa) = 0 The @ component is found froni:Eq , (2.35b). Inside the core the factor q2 is given by (2.47) (2.48) where t, = Znnrlh= aG*, while outside the core *2 = fr2 -14 (2.4e) withk, =21vr2ll=,Qur{w .substiarting,fos,Q.az) and(2.43)into Eq.r(2.35 b)tofindEq1, and, similarly, I I Eu - Eqr= - ,rtl o4 l J,@n) - naauti@n;] I .J where the prime indicates iiiffeientiation with respect to th€ argument ' Similarly, for the tangential components of H ii is readily *iio*n'thut, at Ht - Ha= Hqt - HE2= - BJv(ua) - DKu(wa) = l S , U:r* r,@n) + t,a,"rut ,{ual #l' + K, (w a) + c ot e 2w x i 1w l r = a, i (2.5t) (2.s2) a1f=o Equations (2.47), (2.5O), (2.5I), and (2.52) are a set,of four equations with four unknown coefficients, A, B, C, and D. A solution to these equations exists only if the determinant of these coefficients is zero: I Ju(ua) 0 4h@o) au' !@l! i,"1ua) 0 Ju@a) -t9 -Ku(wa) iol -\x"1wa1 w aw' u 0 ff t"o*t -# n:,r,^ tisa) x'"1*o1 -Ku(wa) {2"53) 4x"@o) aw' Evaluation of this tleGelininant-yields the following eigenvalue equation for B: Q.s4) where =JiQ. uJr(*\ '- Ko_.(Y?, lt,* wKr(wa) and '-'v Gre to the range given Upon solving Eq. (2.5a) for p, it will be found that only discrete values restricted which is transcendental (2.s4) complicated is a by .gguation ir.;;t #rri Jr"iio.'e-tttro"gh Eq. all the proviae will modq particular any for generally solved by numerical techniiues, its solution of modes lowest-order of the some fbr equation characteristics of that mode. we shall now consider this {. i; a step-index waveguidu. 2.4.? Mgdes h stcPI*{fT 4bers* Bessel functions' These To help describe the motbd, *e ,it{uu-{i*t examlne the behavior of.the /-type *.: :':Tlq to harmonic are plotted in Fig. Z.Zt- ioi'ttte fi*t tf,t"" orders. The J-type Bessel functigns fuilctions- Because functions since tfiey exhibitosct]ta$oqlrbetrar4or forreal k, as is the case for'siriusoidal These roots will value. given v of the oscillatory Uefravior*4; ther! will be rr roots of Eq. (2.54) for a TE,^,N* EfI,,, or Y[E* Schematics B,*, and the corresponding modes are either cross section of a stepof the transverse electric field patterns for the four lowest-order modes over the index fiber are shown inFig.2.22. for which v = 0' When For the dielectric fiber w"aveguide, all modes are hybrid modes except those y = 0 the right-hand side of Eq.-(2.54) vanishes and two different eigenvalue equations result. These are Q'55a) $s + ;'ffo= 0 i" J*igr"t"dbi or, using the relations for Ji and Ki in App' J{u.a) * Kt(wa) -O (2.ssb) uJo(ua) wKx(wa) . _ which corresponds to f80,, modes (E, = 0)' and ' '' tcff,"+*]flo=o ' k?JJua) (2.56a) (2.s6b) -o wKn(wa) (see Prob. 2'16). . q' which corresponds to TMo,, modes (H. = 0). The proof of this is left as an exercise *kZKJwa) uJo@a) When y* 0 the situation is more complex and numerical methods are needed to solve Eq. (2.5q exactly. However, simplified and highly accurate approximations based on the principle that the core and cladding refractive indices are nearly the same have been d"ir"6te,m'n. The condition that n, - nz 11 | was referred to by Gloge as giving riie to wedcly guided modes. A treafrnent of these derivations is given in Sec. 2.4.8. Let us examine the cutoff conditions for fiber modes. As was mentioned in relation to Eq. (2.46), a mode is referred to as being cut off when it is no longer bound to the core of the fiber, so that its field no longer decays on the outside of the core. The cutoffs for the e F1rg. 2.21 various modeq are found by solving Eq. (2.54) in the limit wz -+ O. This is, in general, fairly Voriation oJtlw EtesseljnctionJufu) Jor tlw fvst tlvee orders (v = O, 1, 2) ptotted. as afurrction oJ x complex, so that only thg results,la, 16 which will be given here. An important parameter connected with the cutoff condition called the V number or V parameter) defrnedby are listed in Thble 2-2, v2 = (u2 + * =(TI w2) @ is the normalized frequency V (also -,,il =(T\ -" (2.s7) which is a dimensionless number that determines how many modes a fiber can support. The number of modes that can exist in a waveguide as a function of V may be conveniently represented in terms of a normalized propagation constant b defined by20 Lowest-order mode HErr TEor W. 2.22 TM61 HEzr Fiber end- uiews oJ the trantsuerse electric field. uectors Jor the Jour lawest-order ndes in a step-indexfiber i Flbers; fuufrireE, Wat:qpfdng, and Fablcallon Mode Cutoff condition 0 TE6., TMs,, h(ua) i [IE1_, EH1_ t(ua) = Q >2 EHr tr{ua) = HE-, g.tlr,-,(,,) =#l'@a) [r; =0 Q ) a2w2 (Blk\z o=T=W_ _n4 A plot of D (in terms of Blk) as a function of V is shown in Fig. 2.33 for a few of the low-r:rder modes, This figure shows that each modecan exist only for values of Vthat exceed a certain limiting value, The modes are cut off when plk = n2. The [1811 mode has no cutoff and ceases to exist only when the core diameter is zero. This is the principle on which the single-mode fiber is based. By appropriately choosing a, fl1, ?lld n2 SO tbflt v=ff<"7 -n|1rtz <z.4os (2.s8) which is the value at which the lowest-order Bessel function /o = 0 (seo Fig. 2.21), allmodes except the are cut off [IE,, mode ' W. 2,29 Normalizedfreqwncy Y Plots oJthe propogollon cntstant (tnterms oJ F/k) os afinctlory^oJVJor aJew oJ the bwest-order mdes (Mdlfied wlth permisstonJrom Glagdl Example 2.3 A step-index fiber has a normalized frequency V = 26.6 al a 1300-nm wavelength. If the core radius is 25 trtrl let us find the nurnerical aperture. From NA=v Eq. t2.5'l) we have 1 =2a.6 l'3 =tj.zz 2tca %r (25) The parameter V can also be related to the number of modes M in a multi-mode fiber when M is large. An approximate relationship for step-index fibers can be derived from ray theory. A ray congruence will be accepted by the fiber if it lies within an angle 0defined by the numerical aperture as given inF4. Q.23): (2.59) NA=sin0=@l-njfz for practical The solid acceptance angle the For numerical apertures, sin 0 is small so that sin 0 = 0. fiber is therefore dl= = n@? n02 - (2.60) n?) For electromagnetic radiation of wavelength ,1, emanating from a laser or a waveguide, the number of modes per unit solid angle is given by 2AtA?, where A is the area the mode is leaving or entering.2s The area A in this case is the core cross section na2. Thefactor 2 comes from the fact that the plane wave can have two polarization orientations. The total number of modes M enteing the fiber is thus given by ,=#n=ffrf -C,=l (2.61) 2.4.8 Llnearly Polarized Modes* The exact analysis for the modes of a fiber is mathematically complex. However, a simpler but highty accurate approximation can be used, based on'the principle that in a typical step.in6sx fiber the difference between the indices of refraction of the core and cladding is very small; that is, A << 1; This is the basis of the weakly guiding fiber approximation.T'1e'2o'27In this approximation the electromagnetic field paffernsandthepropagationconstantsofthemodepatsFlE,*,,.andEH*,, marevery similar.Thisholds likewise for the tlree modes TE,., TM6., andHE2*. This can be seen from Fig. 2.23 with (v, m) = (0, I and(2,1) forthe modg groupings {[IErr], {TE.r, TMgr, [1821], {HE31, EHrr}, {HEr2}, {I{E4r, EH,}, and {TE6r, Tlld,z,lfi,,l. The result is that only four field components need to be considered instead of six, andthefielddescription is further simplifiedby theuse of caretesian instead of cylindricalcoordinates. When A << 1 we have that k? =kZ = B2 . Using these approximations, Eq. (2.50 becomes {" *.t{, = tY(i. #) (2.62) Thus, Eq. (2.55b) for TEo- modes is the same asE4. Q.56b) for 1}r4s. modes. Using the reculrence relations tor J i and Ki given in App. C, we get two sets of equations for Eq. (2.62) for the positive and negative signs. The positive sign yields Jrr(ua) , Kr*r(wa) uJr(ua) The solution of this equati'On gives a set of modes called the Eq. (2.62) we get ---;n (2.63) wKu(wa) EII modes. For the negative sign in Opdcal Ftbers; Stnrcfurres, Wat:eguiding, and Fabricqtion Ju;@a) _Ku-t(wa) =O uJu(ua) wKr(wa) or, alternatively, taking the inverse from Sec. C.1.2 and Sec. C,2.2, (2.64a) of Eq. (2.64a) and using the first expressions for Jr(ua) and Kr(wa) _uJu-z(ua) _ wKr-z(wa) Jua(ua) wKua(wa) (2.64b) This results in a set of modes called the HE modes. If we define a new parameter ; (t = .{v*r [,- I for TE and TM modes for EH modes (2.6s) for tIE modes then Eqs (2.55b) (2.63), and (2.&b) can be written in the unified form uJi-tfua) __wK,t(w:a) Kt(wa) J,(ua) (2.66) Equations (2.65) and(2.66) show that within the weakly guiding approximation all modes characterized by a iommon set ofj and m satisfy the same characteristic equation. This means that these modes are dlgenerate. Thus, if an WwL, **od" ir degenerate with an EHr-r, - mode (i.e., if tIE and EH modes of corresponding radial order m and equal circumferential order v form degenerate pairs), then any combination of an [IEv+1,. mode with an EHor,. mode will likewise constitute a guided mode of the fiber. Such degenerate modes arre called linearly polarized QP) modes, and are designated LPy, modes regardless of tn"ir TM, TE, EH, or IIE fietd configuration2o. The normalized propagation conltant b as a function of V is given for various LPrT modes inFig.2.24 general, we have the following: 1. Each LPo, mode is derived from an FIE,, mode. 2. BachLPl. mode comes from TE6r, TMs, andHB*modes. 3. Each LP,* mode (v>-Z) is from an fIE*,.. and an EH *r,. mode. The correspondence between the ten lowest LP modes (i.e., those having the lowest cutofffrequencies) and the raditional TM, TE, EH, and IIE modes is given in Table 2.3. This table also shows the number of degenerate modes. A very useful feature of the LP-mode designation is the ability to readily visualize a mode. In a complete set of modes only one electric and one magnetic field component are significant. The electric field vector E can be chosen to lie along an arbitrary axis, with the magnetic field vector H being perpendicular to it. In addition, there are equivalent solutions with the field polarities reversed. Since each of the two possible polarization directions can be coupled with either a cos j@ or a sin j@ azimuthal dependence, four discrete mode patterns can be obtained from a single LPrT label. As an example, the four possible electric and magnetic field directions and the corresponding intensity distributions for the LP,, mode are shown Fig.2.25. Figure 2.26a and2.26b illustrate how two LP1 modes are composed from the exact IIE2, plus TEel and the exact HE21 plus TMo, modes, respectively. d I i'0.6 d I -e E o.+ il "s W, 2,24 *to qf ttw Ctryaltqn,eanetar:f b aa a jncfinn qf {Reprofu*ed utith perrrdes tnn fum LP-mode designatlon Trditi onal - mo de Glqd' ) d e s i g nat i o n and number ofmodes LPor IIE11 x 2 LPrr TEs1, TMo1, LP,, EH11 LPoz V Jor uErtow l,Pyn.nrcrcs, Number of degenerate modes 2 I{F,r, x 2 x Z,HErrx? ex2 z,HEuxz TMo2, tfr.ux2 4 4 ) LPrr EH2, x 4 LP,, TEm, 4 LP*r EH31x2,HE,x2 4 LPzz EHr2 x 2,1#,rrx2 4 LPo: f-IErl x 2 7 LP,, Ellorx1, HE., x ? 4 2.4.9 Powcr Flos ta 6tcp-Ildo+ Flberg* A final quantity of interest for step-index fibers is the fractional power flow in the core and eladding for a given mode. As illustrated in Fig. 2.l9,the electromagnetic field for a given mode does not go to zero athe core-cladding interface, but changes from an oscillating form in the core to an exponcntial decay in the cladding. Thus, the electromagnetic energy of a guided mode is carried partly in the core and partly in the cladding. The further away a mode is from its cutoff frequency, the more concentrated its ir.rgy is in the core. As cutoff is approached, the field penetates farther into the cladding region and a gr"ater percentage of the energy uavels in the cladding. At cutoff ttre field no longer decays outside the core and the mode now becomes a fully radiating mode. Intensity distribution WW E vertically polarized @@ E hotizontally polarized ffg. 2.25 Jour psstble tratuuerse elecfficfreld otd magnetlcfreld. dlrectlorw and the corespondtng tnter.slfu dlsffibtfiIonsJor tlle LPn mde The The relative amounts of power flowing in the corc and the cladding ean be obtained by integrating the Poynting vectdf in the axial direction, I Re(ExH*).e S-= "22---t-..-^r-z 1 (2.67) over the fiber cross section. Thus, the powers in the core and cladding, respectively, are given by P"*= (2.68) P.uo (2.6e) il: fr'n .H;-EyH:)dodr = it: frr r@,H., - ErHi)dQdr weakly guided mode approximation, where the asterisk denotes the complex conjugate. Based on the which has an accuftrcy on the order of the index difference A betrveen the core and cladding, the relative core and cladding po*er. for a particular mode v are given byzo'ze t3@o) +=(, 5)l Jr*r(ua) Ju_1(ua) (2.70) and DD 'clad P - tl-- 'core (2.71) P'. where P is the total'power in'the mode v. The relationships between P"or" and P"66 are plofted in Fig. 2.27 interms of the fractional powers P"orc/P and P"1^olP for various LPrT modes. In addition, far from cutoff LPrr 1 the I average total power in the claddr4g has been derived for fibers in which rhany modes can propagate. Because of this large nurnber of modes, those few modes thatare appreciably close to cutoffcan be ignored to a reasonable approximation. The derivation assumes an i\ i I I HEzr incoherent source, such as a tungsten filament lamp or a light-emitting diode, which, in general, excites every fiber mode with the same amount of power. The total average cladding power is thus approximated by2o (rw) = o I P )orut 3 r-,,, ffiffi (2.72) (L6I) , M,isthe totahurnber of modes entering the fiber. fmmFig. Z.2l where, from E q. (b) and Eq. (2.72) itcan be seen that, since M is proportional to V2, the power flow in the cladding decreases as Vincreases. Fig; 2.26 Cornpasition oJ two lPrr,mdes trom exact modes and.their transuer.se electric field" and- E x ampl i 2; 4 As rur exafoplg'coniidef afiberhaving of 25 1tm, a core index of L,18, and A = 0.01. Atan operating wavelength of 0.84 1rm the value of Vis 39 and there we760 modes in the fiber. FromBq. (2.72), approximalely 5 percent of the power propagates in the cladding. ff Ais decreased to, say, 0.003 in ordei io discrease a core radius signal dispersion Gee Ctgp.t* 11 ,1hen2"41, modes propagare intensily distributions in the fiber and:about 9:percent of the power resides in the cladding. For the case of the single-mode fiber, considering the !Po, mgde (the IIE,, mode) inFig.2.27, it is seen that for V = I about 70 percent of the power propagates in the cladding,.whereas for V = 2.405, which is where the LP,, mode (the TEo, mode) begins, approximately 84 percent of the power is now within the core. I l I I I i: ,.t { Optical Ftbers: Strttctur'es, WW \ o rL 246 Fig. 2.27 l/ FYactionatpouerJlow tntlrc clndding oJastep-indexopticatfiber as afunctionoJ v. wlwnv'+ 7, the curue ntnnbers vmdesignate tlrc HDo*t,^arTdEHr-1,mmodes. For v = 7, tt:re curue numbers vm giue--the HEr*, TEo^, andTMo*modes (Reprodtrced' u:ith permis sion Jrorn Glry*o ) singte-mtde rlbers m Sl"gi"i"oa" fibers are constructed by letting the dimensions of the core diameter be a few wavelengths and the cladding; FronBq' (2'27) ir*i"ff, S-f Z) and by having small index di*erences between the core large variations or (2.S'A) with V = 2.4, it"ui b" ,"", that single-mode propagation is possible for fairly in practical A. However, differences index core-claddin-g andthe i, ,atues of the physical core size a percbnt, 1.0 0.2 and b.1YqP di=ffere19e.varie1 index designs of single-moa" nU"ii, ttre,core-cladding that is, model higher-order first the of just cqtoff below be ana Ihe core dlameter shpuld be chosen to the pm and of 3 t uY" may fiber single-mode a typical foi V .tightty less'than 2y'. For example, 1 :9l"-tudlus (2'27)l,this (2.57) (2.23) and pr". From Eqs lorEq' or o.a ut u *ufi"ngtrt ;;;#;i;d;"9f O1 yields V = 2.356. 2.5.1 Mode Field Diameter Formultimodefibersthecorediar.nQterandnumericalaperturearekeyparametersfordescribingthe propagating p.rop9-rtles. tn iingle-mo4e fibers the geometric distribution of light in the a fundamental Thus fibers. these of needed *hio.pr"dicting the perfoqrance characteristics ;fiffiil;rrio1 ifoOe is wtrat i, detemined' parameter of a single-rnode im"r is, tni *rd"A"ta diameter (Ym) This parameter can be of tty function is a ggtical source irom the mode.figid OlstUution of-the fundarnental fiber mode, and diameter is mode-field The fiber. wavelength, the core radius, *d.*re refraclive index profile of the the light all not fibers analogoils to the core diameter:in multimode fibers, except that in single-mode this,effect' illustrates 2.28 that piopagates through the fiber is carried in the iore (see Sec. 2.4). Figure For example, &t V = 2 only 13 percent of the optical power is confined to the core. This percentage increases for larger values of V and is less foi smaller-V values. The_MFD is an important parameter for single-mode ^. flber, since it is used to predict fiber properties such as 01L splice loss,.bending loss, cut-off wavelength, and waveguide dispersiou. Chapters 3 and 5 describe these pafameters and their effects on fiber performance. A variety of models have been p^roposed for characterizing and measuring the MFD.30-35 These include far_fieli scanning, near-field scanning, traflsvefse offset, variable aperture in the far field, knife-edge, and mask methods.30 The main consideration of all these methods is how to approximate the optical power distribution. A standard technique to find the MFD is to measure the far-field intensify distribunm R @) and rherr calculate the MFD using the Petermarin II Core diameter -+- e'LO equation32 I zll t'st" MFD= 2Ws=ZlJi:_, _ "1''' | r'trlrdr l_ )o ) where Zwo(called the spot size) isthe lQ33) Flg' the far-field distribution. Foicaleulation simpticity the exact field distribution can be fitted to a Gaussian function2r E(r)=Eoexp full width of (r,lw&) 2'28 e.i4) Mode nerd drryter Distribution oJ tight in a single-mde fiber ahoue its *taff tuauelength. For a Gaussian distrlbution the MFD is stuen wldrh aJ ttte optlcal pwer ii ii tiJ where r is the radius and E6 is the field at zeraradius, shown in Fig. Z.Z9.T\enthe MFD is given by the rlez widthof the optical power. as As described in Sec. 2.4.8, o-rdinary single-mode fiber there are actually two independent, -in.any degenerate propagation .o6.r3G3e. These mode-s are very ririi* but their polarization planes are o.tmosg't rhese may. be ehose-n arbitrarily ag the horizonial ,nJ it r."ririiviprj*zations as fnl shown in Fig' Z.}9.F;tther tfese two polarization modes constitutes" the fundamental tfi,, mode. 9f In general, the electric field9n9 of the light propagating along the fiber is a linear superposition of these two " polarization modes an{_depends olr the polarization of tfie fight at rhe Iaunehing pnint io;";dH;;." suppose we arbimmily ihoose one of the modes to have iirt un.u"rr" ;i";d."fi;il poiurtz"o atong the r direction and the:other independeat orthogonal, mode to be polarized in the y direcri* u, shown in Fig' 2'29' In ideal tbe.rs w.r$ perr""t rot3tional synmetry, trr* i*u modes are degenerate with equal propagation constants (krx.kr),a1d any polarizationstate iqiected inro the firer wil prlpaluie orctrungeo. In actual fibers there are iniperfections, such as asymmetrical lateral stresses, noncircular cores, and variations in refraefive-index profiles. These imperfections break the cireular symmetry of the ideal fht-.Td lift the degeneracy of the two modes. Tlre modes p.opugui. *ith different phase velocities, and the difference between their'effective refractive indices i* fiber birefring"n "ai"itrr" ", ; f, Vertical mode H-qr&oetal'mode Ftg. ! 2.2g ltj"l,o p oJmetuiaamentalfiEll B7= \ Equivalently, we may define &e birefringence n'- "ry in a sbqli-ndefiber (2.7s) n* as D F=ko(ny-n) constant. '' free,tpaee,propagation is the Znl)t whers h= modes are excited, then one will that both fibei.so into theis injected 11g[t If (2.76) be delayed in phase relative to the other as they Fopagate. When this phase difference is an integral multiple of 2n, the two modes will beat at this poini aIrd ;h" input polarization state will be reproduced. The length over which this beating occurs is'the fiber beat length, (2.77) Lp= 2nl7 Examole 2.5 A sinslemodeoptical fiberhas abeat length of 8 cm at 1300 nm. From Eqs. (2.15) to (2.77) we have that the modal birefringence i9 B.= r u, n..-n-=L) Lp altematively, I'3xlOlm 8x10.'m =1.63xr0-5 'B=4Lp 2n 0.08 m =7g.5m-r This indicates an intermediate-type fiber, since = I x 10-'1 (a typical highbirefringence fiber) to Bf = I x lO-E (a typical lowbirefringences can vary from B, birefringence fiber). Fiber Stnrcture ";ze # t Graded-index In the graded-index fiber dcslgn th1 core refractive index decreases c9nti1u91slY.yth incrlasine rad$ distance r from the'center-tif-ttiti:ltber,'but is generally constant in the cladding. The'most commonly io. the refractive-index variition in ihe core is the power law relationship *J.";r*.,ion i1' I II core axis, rc2 is the refrrctiveindex+f the cladding, and thedirrensionless ryram-eter crdefines the shape of the indeiprofile;Thei1gfuxdifference A for the eraded-jiidex fiber is gi@-by I n_"r'-"i 2"? if _flt-tt2 (2.7e) nl The approximation on the right-hand side of this equation reduces the expression for A to that of the step-index fiber given by Eq. (2,20). Thus, the same symbol is used in both cases. For d= e,Eq. (2.78) reduces to the step-index profile n(r) = yir. Determining the NA for graded-index fibers is more complex than for step-index fibers, since it is a function of position across the core end face. This is in contrast to the step-index fiber, where the NA is constant across the core. Geometrical optics considerations show that light incident on the fiber core at position r will propagate as a guided mode only if it is within the local numerical aperture NA(r) at that point. The local numerical aperture is defined as{ {tn'<rl-n:lt.2 =NA(o)f NA(r) = . |.0 -(r/a)" for rsa for r>0 (2.80a) where the axial numerical aperture is defined as NA(0) = [n2(o)-n?1Y2 =@? 0L 0 fig. Z.3O A comparbon profites -nilV'=n Jitr 0:4 ,, A.6 oJ tJtg ruunerical aperdres Jor 0.8 (2.80b) 1.0 fibers haDing uarious are index Thus, the NA of a graded-index fiber decreases from NA(0) to zero as r moves from the fiber axis to the core--cladding boundary. A comparison of the numerical apertures for fibers having various aprofiles is shown in Fig. 2.30. The number of bound modes in a graded-index fiber . iser M--do2kzn?L-ov' " a+2 ' a+22 (2.81) where lr = Zrcil" and the rigl+hand'approximation is derived using Eqs (2.23) and (Z:.Tt). As Sec. 3.4 describes. fiber manufacturers typically choose a parabolic refractive indcx profilc given by a = 2.O. ln i I { Optical Ftbers; Stnrcturres, Waueguiding, and. Fabricalian Mc= V214, which is half the number of modes supported by a step-index fiber (for which that has the same Vvalue, as Eq. (2.61) shows. this case, a,= *) I'iber Materials m In selecting materials for optical fibers, a number of requirements must be satisfied. For example: 1. It must be possible to make long, thin, flexible fibers from the material. 2. T\e material must be hansparent at a particular optical wavelength in order for the fiber to guide light efficiently. 3. Physically compatible materials that have slightly different refractive indices for the core and cladding must be available. Materials that satisfy these requirements are glasses and plastics. The majority of fibers ar9 made of glass consisting of either silica (SiOr) or a silicate. The variety of available glass fibers ranges from moderate-loss fibers with large cores used for short-transmission distances to very transparent (lowJoss) fibers employed in long-haul applications. Plastic fibers are less widely used because of their substantially higher attenuation than glass fibers. The main use of plastic fibers is in short-distance applications (several hundred meters) and in abusive environments, where the greater mechanical strength of plastic fibers offers an advantage over the use of glass fibers. 2.7.1 Glass Fibers Glass is made by fusing mixtures of metal oxides, sulfides, or selenides.4zaa 7X" resulting material is a randornly connected molecularnetworkrather than a well-defined ordered structure as found in crystalline materials. A consequence of this random order is that glasses do not have well-defined melting points. When glass is heated up from room temperature, it remains a hard solid up to several hundred degrees centigrade. As the temperatu e increases further, the glass gradually begins to soften until at very high temperatures it becomes a viscous liquid. The expression "melting temperature" is commonly used in glass manufacture. This term refers only to an extended temperature range in which the glass becomes fluid enough to free itself fairly quickly of gas bubbles. The largest category of optically transparent glasses from which optical fibers are made consists of the oxide glasses. Of these, the most common is silica (SiOr), which has a refractive index of 1.458 at 850 nm. To produce two similar materials that have slightly different indices of refraction for the core and cladding, either fluorine or various oxides (referred to as doponrs), such as BrOr, GeOr, or PrOr, are added to the silica. As .i I I 1 shown in Fig. 2.31 the' addition of GeO, or PrO, increases the refractive, index. whereas doping the silica with fluorine or BrO, decreases it. Since the cladding must have a lower index than the core, examples of fiber compositions are x o 1.48 GeO2 13 () H Pzos r.+o {) Bzo: F 1.44 sl0 t5 20 Dopant addition (mol %) IIg. 2.31 VarialioninreJracttue index as aJtutction dopW corwentration m srlica glass oJ 1. GeOr-Siq corc; SiG? cladding 2. Prq-SiO2 cere; SiO, cladding Bpr-SiQ 3. SiO2 core; 4. GeAz B2q-SiO, core; BrOr-SiQ cladding cladding Here, the notation GeO2-Siq2, f,at example, denotes a Ge$2-doped silica glass. The principal raw material for silica is high-purity sand. Glass composed of pure silica is referred to as eiiher silica glass, fased. rilic.c, or vitreoas sillca Some of its desirable properties are a resislance to deformation at temperatures as high as 1000oC, a high resistance to breakage from thermal shock because of its low thermal expansion. go@ chemical durability, and high fiansparency in both the visible and optic communication systems. Its high melting temperature is a infrared regions of interest to from a molten state. However, this problem is partially avoided glass prepared is if the disadvantage fikr when using vapor deposition techniques. 2.7.2 Active Glass Fibers a normally passive glass gives the resulting properties allow the material to perform material new optical and magnetic properties. These new passing i1.as-a1 Doping can be prasj through retardation on the light amplification, altenuation, anl carried out for silica, tellurite, and halide glasses. Two commonly used materials for fiber lasers are erbium and neodymium. The iolic coucentrations of the rare-earth elements are low (on the order of 0.005-0.05 mole percent) to avoid clustering effects. By examining the absorption and fluorescence specffa of these materials, one can use an optical source it an absoqption"wavelengh to excite elecfions to higher energy levels in the rare-earth *hieh "mits dopants. When these excited elecfons are stimulated by a photon to drop to lower energy levels, they emit light in a narrow optical speetrum at the fluorescence wavelengh. Chapter 11 discusses the applications of fibers doped with rare-earth elements to create optical amplifiers. Incoqporating rare-earth elements'(atomic numbers 57*?l) into Chqracteristic PMMA FAF PF PAF Core diameter 0.4 mm 0.125*0.30 mm Cladding diameter Numerical ap.rture l.0mm 0.25-O.60 mm A.2:5 0.20 Attenuafion Bandwidth 150 dB/km at 650 nm 2.5 Gbls over 200 rn < 40 dBlkm at 650-1300 nm 2.5 Gbls over 550 m 2.7.3 Ptastic Optieal Fibers The growing demand fordelivering high-speed services directly to the workstation has led fiberdevelopers to create high-bandwidth graded.index polymer (plastic) optical fibers (POF) for use in a customer premises.4s,ae The core of *rese'fibers is either polymethylmethacrylate or a perfluorinated polymer. These fibers are h€ee,,laSerr€d to.as PMMA FOF and PF POR respectively. Although they exhibit considerably greateroptieal signal :rttenuations than glass fibers, they are tough and durable. For example, since the modulus of these polymers is nemly two orders of magnitude lower than that of silica, even a fiber cable l-mmdiameter graded-index FOF is sufficiently flexible to be illtalled in conventional which larger, times 10-20 are plast'rc fibers of routes. Comparei with silica fibers, the core diameters Thus, efficiencies. coupling optical saerificing allows a relaxation of connectsr toletances wittrout and splices, connectors, fabricate to be used can inexpensive plastic injection-molding technologies transceivers. fibers' Table 2.4 gives sample characteristics of PMMA and PF polymer optical lz.e It Photonic CrY',rstal Flbers | --- E Irritially this In the early 1990s researchers envisioned and demonstrated a new optical fibe_r structura (PCF) mictostruatured or s photonic crystal as a known fiber beeame was calle,ia hotey fiber and later that the cladding fiber.5cv Trc aifrerence between this Rew structure and that of a conventional fiber is length of the entire the mn along which holes, air contain PCF of a and, in some cases, the core regions light-transmissioucharacteristics the define cladding core and the of fiber. Whereas the malerial prcrtrties which of conventional fibers, tte .tuctorat arrangement in a PCF creates an internal rnicrosm'lcture' the fiber' in offers another dimension of light control refractive index The size and spacing (knoJn as the plrcft) of the holes in the microstructure and the fiberc' The two photonic crystal of characteristics tigtrt-guialng of its constituent mate;ai determine fre transmission light The fibers. bandgap photonic pCn and TiUers categories are index-guiOing taslc a high-index it has since fiber, conventional in a to tlrai iimitar mechanism in ai index-caidiflg fb*r is index of the refrastive effiective PCF the for a However, core srilrounded by u tJ**irido* cladding. a phatonie bandCsp cladding depends on the wavelength and the size and pitch of the holes. In contrast, of'fibers guides class This cladding. structured micro by a fiberhuu-a hollow eore which is iunounded light by means of a photonic baodgap effect. 2.8.1 Index-Guldtng'PICF Figure 2.32 shows a two-dimensional cross-sectional end vierv' of,the iffucture of an index.guiding PCF- The fiber has a solid core that is iurrcunded by a eladding region which contains air hoies- amning along the length of the fiber- Ttre holes have a diameter d and apitch,i.' Although the core and the claddiagare made of the same material (forexample, pure silica), &e air holes lower the e{fective index of refraction in the ,;.adding region, since n = I for air and 1.45 for.si]iea" Consequently, the microsfircture arr{ngsment creates a stepindex optical fiber. The fact that the cors can be made of pure silic4 gives the PCF arnun'rbor of opemtional advantages Over convenlional hbers, which typically have a germaniumdoped silicacore. These include very low Solid high-index core Buffer coating oo\oooo ooo\\ oco ooo coo ooo ooo Claddiagwith #ooo"!3: Hole diameter d embedded holes Pitch A fE. 2.32 Cross-sectio nal end- uieu oJ tle sf-lcfile aJ an indexguiding plwtonic crystnlfikr Iosses, the ability to transmit high optical power levels, and a high resistance to darkening effects from nuclear radiation. The fibers can support single-mode operation over wavelengths ran^ging from 300 nm to more than 2000 nm. The mode-field area of a PCF can be greater than 300 pm2 compared to the 80 pm2 area of conventional single-mode fibers. This allows the PCF to ffansmit high optical power levels without encountering the nonlinear effects exhibited by standard fibers (see Chapter 12). 2.A.2 Photonic BandEiap Fiber Figure 2.33 shows a two-dimensional cross- sectional end view of the structure of a photonic bandgap (PBG) fiber. In contrast to an index-guiding PCR here the fiber has ffiooooooJ%\ f oo\ocoo ooo\,-----.ooo o o o (\ )o o o oooVooo \ ooocoo a hollow core that is surrounded by a cladding region which contains air holes running along the length of the fiber. Again the holes in the cladding region have a diameter d and a pitch A. The functional principle of a photonic bandgap fiber is analogous to the role of a periodic crystalline lattice in semicortductor, occupying a from which blocks electrons PBG fiber the hollow bandgap region. In a photonic bandgap core acts as a defect in the Kloororoo# a Cladding with m d embedded holes Pitch A structure, which creates a region in which the light can propagate. Hole diameter Ftg.2.33 Cross-secti,onal end uieu oJ the shttcfitre oJ a ptatonic bandgop fiber Fiber Fabrication Two basic techniquesss-S8 are used in the fabrication of all-glass optical waveguides. These are the vapor-phase oxidation process and the direct-melt methods. The direct-melt method follows traditional glass-making proceduies in that optical fibers are made directly from the molten state of purified co*ponents bf silicate glasses. In the vapor-phase oxidation process, highly pure vapors of metal halides particles. The particles are 1e.g., SiCto and GeCl) react with oxygen to form a white powder of SiO2 then colleited on the surface of a bulk glass by one of four different commonly used processes and are sintered (ffansformed to a homogeneous glass mass by heating without melting) by.one of a variety of techniques to form a clear glass rod or tube (depending on the process). This rod or tube is called a perform.It is typically around 1O-25 mm in diameter and 60-tr20 cm long. Fibers are made from the preformsz-ss by using the equipment shown in Fig. 2.34. The perform is precision-fed into a circular heater called the d.rawing furnace.Here the preform end is softened to the point where it can be drawn into a very thin filanient, which becomes the optical fiber. The turning speed of the takerrp drum at the bottom of the draw tower determines how fast the fiber is drawn. This, in turn, will determine the thickness of the fiber, so that a precise rotation rate must be maintained. An optical fiber thickness monitor is used in a feedback loop for this speed regulation. To protect the bare glass fiber from extemal contaminants, such as dust and water vapor, an elastic coating is applied to the fiber immediately after it is drawn. Precision feed mechanism Flg. 2.94 *hemotic oJ afiber-drau:ing apgrufi$ 2.9.1 Outside Vapor-Phase Oddatlon Workstu2 by the The frst fiber to have a loss of less than 20 dBikm was made at the Corning Glass First, a layer of 2'35. in Fig. is illustrated outside vapor-phase oxidation(OvPo) process. This method mandrel' The or graphite ceramic. SiO, particle, iutt"O a soot is deposited irom a burner onto a rotating is built up' By preform glass polou.s gtu.i soot adheres to this bait rod and, layer by layer, a cylindrical' process, the the deposition during vapor stream iroperly controlling the constituents of the metal halide preform' into the incorporated be can lmr .o-positions-and dimensions desired for the core and cladding gither step- or graded-index preforms can thus be made. vitrified Wren tfre Aeposition process is completed, the mandrel is removed and the porous tube is then perform is This clear preform. glass in a dry atmosphere at ; high temperaiure (above 1400') to a clear central Fig. 2.34.The in shown as subsequently mounted in aiber-drawing tower and made into a fiber, hole in the tube perform collapses during this drawing process' 2.9.2 Vapor-Phase Arlal DePosition process The OVpO process described in Sec. 2.9.1 is a lateral deposition method. Another OVPO-type the SiO2 is the vapor-phase axial deposition (VAD) method,63'fl illusfiated in Fig. 2.39.1n this.method, from purti"t". ur"ior*ed in the s,u*" *uy as described in the OVPO process. As these particles emerge as a seed. A porous the torches, they are deposited ontothe end surface of a silica glass rod which acts preform is grown in the axial direction by moving the rod upwald. The rod is also continuously rotated it is io maintairicylindrical symmetry of the particle deposition. As the porous preform moves upward, zone) localized transformed into a solid, transparent roilpreform by zone melting (heating in a narrow into a fiber by drawn be then can preform resultant The 2.:6. in Fig. with the carbon ring heater shown heating it in another furnace, as shown inEig.2.34. 8o6 step and graded-iniex fibers in eithir muitimode or single-mode varieties can be made by the occurs with VAD method. The advantages of the VAD method are (1) the preform has no central hole as O2 + metal halide vapors Bait rod (Mandrel) Soot preform Glass particles (a) Soot deposition Glass @ H co." )'{cludding (6) Soot preform preform [:Hffi.-i:"t*, uw l;."" vu L_ 67\ I o,u,. Fiber preform (c) Preform sintering (d) Fiber drawing cross section fig. 2.35 Basic steps in preparing a preJonnby the OVPO process. (a) Boiit rod. rotates and. fflrlrues back ardiorth under the burner to prodtrce a un!f,orm depositian oJ gtnss soot partictes along the rod; (b) Proflites can be step or graded index; (c) Follon:W deposition, the soot preJorm is sintered into a clEar gtass preforrn; (d) fiber-f dranunJrom the gtnsi pr.J"rnt (Reprodtrced. u:ith pirmissaiJrom Schuttz.ss) the OVPO process, (2) the preform can be fabricated in continuous lengths which can affect process costs and product yields and (3) the fact that the deposition chamber and the zone-melting ring heater are tightly connected to each other in the same enclosure allows the achievement of a clean Jnvironment. 2.9.3 Modified Chemical Vapor Deposition The modified ch,emicaL t'apor deposition TMCYD) process shown in Fig. 2.3'l was pioneered at Bell Laboratories,43'6s-6't and widely adopted elsewhere to produce very lowlloss graded-index fibers. The glass vapor particles, mising from the reaction of the constituent metal halide gases and oxygen, flow through the inside of a revolving silica tube. As the SiO2 particles are deposited, they are sintered to a clear glass layerby an oxyhydrogen torch which travels back and forth along the tube. When the desired thickness ofglass has been deposited, the vapor flow is shut offand the tube is heated strongly to cause it to collapse into a solid rod preforrn. The fiber that is subsequently drawn from this preform rod will have a core that consists of the vapor-deposited material and a ctaaOing that consisti of the original silica tube. 2.9.4 Plasma-Activated Chemlcal Vapor Deposltion Scientists at Philips Research invented the plasma-activated chemical vapor deposition (PCYD) process.6&70 As shown in Fig. 2.j8, the PCVD method is similar to the MCVD process in that deposition occurs within a silica tube. However, a nonisothermal microwave plasma operating at low pressure initiates the chemical reaction. Wjth the silica tube held at temperatures in the range of 1000-1200"C to reduce mechanical stresses in the growing glass films, a moving microwave resonator operating at 2.45 process deposits clear glass GHz generates a plasma inside the tube to activate the chemical reaction. This is required. When one has ,11ut"rld directly on ttre tube wall; there is no soot formation. Thus, no sintering just in the MCVD case' as Jefost"A the desired glass thickness, the tube is collapsed into a preform 2.g.5 Photolri" C"y"t"I Fibu" F"b"toti* The fabrication of a photonic crystal fib-er also is based on first creiting a preformTlja. The perform is made by rireans of an alray of hollow capillary silica tubes. To make a preform for an Pulling machine index-guiding fiber, the capillary tubes first are bundled in an array around a'solid silica rod. For a photonic bandgap fiber, the hollow core is established by leaving an emBty space at the center of the array. Following the array stacking rod Transparent preform Vessel processes, these configur'ations are fused iogether to create a preform and then made into Ring heater a fiber using a conventional optical fiber drawing tower. In the drawing process, the holes keep their original arrangement. This allows the creation of any type of array pattern (for example, close- Thermo- packed arrays, a single circle surrounding a large viewer iolid core, or a star-shaped hollow core) and hole shape, (for exampie,' circular,, hexagonal, r' or oval openings) in the final In addition to using silica, polymer and soft Glass particles fiber. glass materials, such asPMMA Reaction chamber and SF6 or SF57 Fl'g. Schott (lead silicate) glass, also have @n,u ed ?t-76. p'''t11,"." to make photonic cryi,tal fibetsf3' fiber types the prefeirm is made,by an:extrusion process. In an example extrusion'process a glass ::t - 2.gZ ffhematic oJ MCVD (ry\Wd ApparatusJortteV,\D (uapor ptwse anat depositinnJ process (Reprodrrced uilh PrmissionJrom lzausa and tnogaki,ffi A rcgo, IEEE) Soot deposit t Sintered glass Fig. 2.36 + Traversing burner ctvmical uaryr deposifar| p,rocess. @eproduced' uithpermiislbinfiomsclufifs) Exhaust Reactants (SiCl++ 02+ dopants) Plasma Glass layers <- -+ Moving microwave resonator fig. 2.38 $x,]lr:mattc.,"JrcVD @itnsriw-actilnted chemtcat uaryr deposilton) pnocess disk is heated until it is soft and tlren is pressed through a die. The sffucture of the die dbtermines the configuration of tfre fiber cross,section. The extrusion time typically is around two hours. After the preform is fabricated, fibers are made using standard optical fiber drawing equipment. m Mechaniesl Properties of Fibers In addition to the transmissioir preperties of optical waveguides, their mechanical characteristics play a very important role when fibers are used as the transmission medium in optical communication systemsTTFibers must be able to wi&stand the stresses and strains that occur during the cabling process and the loads induced during the installation-and.service of the cable. Dtring cable manufacture and installation, the loads applied to the fiber can be eitherimpulsive or gradually varying. Once the cable is in place, the service loads are usually slowly varying loads, which can arise from temperature variations or a general settling of the cable foliowing-:installation. Strength and static fatfgiare thg,two basic mechanical characteristics of glass optical fibers. Since the sight and sound of shattering glass are quite familiar, one intuitively suspects that glass is not a very stong material. However, the longitudinal breaking stress of pristine glass fibers is comparable to that of metal wires. The cohesive bond strength of the constituent atoms of a glass fiber governs its theoretical intrinsic strength. Maxiinum tensite strengths of 14 GPa (2 x 106 lb/in.l have been observed in shortgauge-length glass fibers. fhis is close to the 20-GPa tensile strength of steel wire. The difference b"t*""n giass-and metal is that, under an applied sEess, glass will extend elastically up to its breaking strength, whereas metals can be stretched plastically well beyond their true elastic range. Copper wires, 80. \ for example, can be elongated plastically by more than 2! percent betore they fractUre' For glass fibers, elongations of only about 1 percent are possible beforeifracture occurs. Irf practice, thekistence of stress concentrations at surface flaws or microcracks limits the median strength of long glass fibers to the 7ffi-to-3500-MPa (1-5 x 10s lb/in.2) range. The frlcture strength of (the one that a giv"en length-oiglass fiber is determined by the size and geometry of the severest flaw prido"". tfr! Urg"-st streJs concent ution) in the fiber. I hypothetical; physical flaw model !s sh9w1-i1 'fig. iSg. 1nis eiliptically shaped crack is generally referred to as a Grffith m_icrocracl|l-. Lt has a width w, a depth ,16, anda tip radius p. Thelsrength of the crack for silica fibers follows the relation (2.82) K=Y . \ : . I tno where the stress intensity factor K is given in terms of the stress o in megapascals applied k-1r-, ll to the fiber, the crack depth yin millimeters, and a dimensionless constant I that depends oD flaw geometry. For surface flaws, which tip radius p i___br-crack G . most critical in glass fibers, y = size From this equation the maximum crack allowable for a given applied stress level can *"the Applied stress d + be calculated. The maximuin values of K Fig. 2.39 A ttgpotletical mdel oJ a micrarack depend upon the glass composition but tend in ott oPticalfiber to be in the range 0.6{.9 MN/m3/2. Since an optical fiber generally contains many flaws tf,at have u r*ao-m, distribution of size, the fracture strength of a fiber must be viewed statistically. lf F (o,I) is defined as the cumulative probability that a fiber of length L will fail below a in the stress level o, then, under the aSsumption that the flaws are independent and randomly distributed we have flaw, most severe fiber and that the fracture will occur " at theF(o'L)=l-e-IN(A (2.83) used A widely than o. less a strength per with length unit where N(o) is the cutnulatiue nnnber of flaws form for N(o) is the bmpirical eipression N(o'= where rn, o6, and called Weibull h r(oY u[a.] (2.84) ue constants;related to the initial inert strength distribution. This leads to the expressions2 so- ': F(o.L)= 1_exp [-t*l;] (2.8s) A plot of the Weibull expression is shown in Fig. 2.4O for measurements formed on long-fiber samplesTs'83. Thesi data points were oUt i*A by Esting to destruction a large number of fiber samples. The fact that be drawn through the data indicates that the failures arise from a single type of flaw. a single .u*" "* ny dreru environmental control if,the fiber-drawing fumace, lengths of silica fiber greater than 50-km with a single failure disnibution can be fabricated. In contrast torstrerEith;.,which,relates.to instantaneous failure under an applied load, stetie fatigue relates to the slow growth of pre-existing flaws in the'glass fiber under humid,conditions.and tensile stress78. This gradual.flaw growth,can$es tre fiber to fail at a lower stress level than that which could be reached under a strength test. A flaw such as the one shown in Fig. 2,39 propagates through the fiber because of chemical erosion of the fiber material at the flaw tip. The prirnary cause of this erosion is the presenee of'water in the environment, which reduces the snengttr oi ttre Sio2,bonds in the glass: The speed of the growth reaction.is.increased,when the fiber,is put under sEess Certain fiber materials are more resistant to s{atiarfatigue than others,rr}itlr fused:silica being the most resista$t of the glasses in water. In general, coatings that are applied to the fiber immediately during the manufacturing process afford a good degree of proteetiron against environmental corrosions4. Another important factor !o consider is dynamic fatigue. When an optical cable is beihg installed in a duct, it experiences repeated sftess owing to surging effecB. The surging is caused by varying degrees of friction between the optical eable and the duct or guiding tool in a manhole on a curyed route. Varying sfiesses also. ar,ise in aerial cables that are set into ftansverse vibrafion'by the wirrd.lheoretical and experimental investigationsss'86 havs shown that the time to failure under these conditions is related Tensile strength 105 7 99.9 ee l- 95 I- 80 [- e0l- 7ol 60F 2 (I b/i#) 3 4 5 6789rc6) o Silicone-coated fibers o Ethylene-vinyl-acetatecoated fibers o 501401301- I 99 li il { \o o\ O( 95 90 srl 70 _d a _?ob sF = EI .80. rG rE I -2S rol- ol^t 60 50 o ,.44 k E o, o ,ol5 ctl 2?. JI o o 44 T 30 ? 20 I I 0.5 -5 I 10 0.2 ,a) -0;l 4 5 67 -1 , FE'2.4O A Wa:biil tgpe'ptat sha:ulrng t}e ewanafipe protubilifu tlwtfrbeirs oJ 2O-m and. l-lcrn lengttts uilt Jracfire at the.indicated applied.stress. (Reprodtrced with prmissiort Jrom Miller, Haft, Vroom. and. Bouderuffi ) are found from the cases of static to the maximum allowable stress by, the same lifetime parameters that stress and stress that increases at a constant rate' 87' 88 In an . A high assurance oi tiU". reliability can be provided by proof testing.33' -trt method' time'during9" opticalTiber is subjected to a tensile load greater than thaiexpected at 1ny ".fl: Empirical pais the proof test are rejected' manufacturing, instattatiln, and service. Any dbers which do not proportional to a power studies of slow crack growth show that the growth tate dfldt is approximately of the stress intensity factor; that is, 4ldt Here, '\ A and b are material constants and the (2.86) =tYu stress intensity factor glasses, b ranges between 15 and 50. If a proof iest stress op is applied for a time to is given by W. Q.82).For most then from Eq. (2.86) we have (2.87) B(o!-z -60-z)=ouo', -.-\ where q is the initial inert strength and' '=*(+\'# (288) to failure r, is found from Eq' when this fiber is subjected to a static stress or after proof testing, the time (2.86) to be ,("ur-' - ob-') = o! (2.8e) r, Combining Eqs (2.87) and (2'89) yields ,("!'' - o!-') = "urto +.ob, t, Qso) proof testing, we frst define N(r, o) to To find the failure probability F, of a fiber after a time r, after f"irgtf, which will fait in a time / under an applied stress o' Assuming be the number of flawsper *ii ttratN(q) >> N(q), then Solving Eq. (2.90) for, q and:substituting into Eq' (2'84), we have' from Eq' (2'91)' N(r, q) = r ll @i,, + o! t,) I n + oi'zfv<t-ztl* ej (2.e2) 1 found from Eq. (2-92) by setting The failure number N(tp, oe) Wr unit length during proof testing is o, = op and letting /, = 0, so that oootofo*our-r)!(b-z) N(te, cp) = 4 Os (2.e3) Letting N(t*, o*) = N,, the failurg probability F, for a fiber after it has been proof tested is given by F = l_e_Z(N,_ilp) (2.e4) I'S : SubstitutingEqs'(2:92) and (2.93) intoEq. (2_94), we have ml(b-2) F" = 1- exp where C = n/o,rt, , ["'{l['.#)# il (2.9s) and where we have ignored the term tul +--, (2.e6) This holds because typical values of the parameters in this term are oJop=0.3 0.4, tp= la s, b > 15, op= 35A MN/m2, and B = 0.05 - O.S 6aXim2;2s. The expression for F, given by Eq. (2.95) is valid only when the proof stress is unloaded immediately, which is not the case in actual proof testing of optical fibers. When the proof sfress is released within a finite duration, the C value should be rewritten as B o'pI, (2.e7) where yis a coefftcient of slow-crack-growth effect arising during the unloading period. Lz.+ ;, f iber Optic Cables In any practical application of optical waveguide technology, the fibers need to be incorporated in some type of cable structure8e-es 11t" cable structure will vary gieatly, depending on whether trre caufe is io be pulled into underground or intrabuilding ducts, buried directly in the ground, installed on outdoor poles, or submerged under water. Different cable designr ur" ,"quir"d for iach type of application, but certain fundamental cable design principles will apply in every case. The objectives of cable manufacturers have been that the optical fiber cables should be installible with the same equipment, installation techniques, and precautions as those used for conventional wire cables. This requires special cable designs because of the mechanical properties of glass fibers. ,2.ll,.l Cable Stnrctures "One important mechanical propefty is the maximlm allowable axial load on the cable, since this factor determines the length of cable that can be reliably installed. In copper cables the wires themselves are generally the principal load-bearing members of the cable, and elongations of more than 20 percent are possible without fracture. On the other hand,,extremely strong optical fibers tend to break at 4-percent elongation, whereas typical good-quality fibers exhibit long-length breaking elongations of about 0.5-1 .0 percent. Since static fatigue occurs very quickly at stress levels above 40 percent of the permissible Optlml Ftbers.' Stnrctures, Wat:qutdttg, and Fobrication elongation and very slowly below 20 percent, fiber elongations during cable manufacture and installation should be limited to 0.1 - 0.2 percent. Steel wire has been used extensively for reinforcing electric cables and also can be used as a strength member for optical fiber cables. For some applications it is desirable to have a completely nonmetallic In such construction. either to avoid the effects popular yarn is A yarns used. are synthetic highltensiie-strength and cases, plastic sfiength members yam generic family to a belonging nylon material syntiretic tough.yellow Kevhrb, which is soft but keep components, cable from other the fibers practices will isolate known as aramids. Good fabrication or flexed cable is when the freely to move the fibers and allow them close to the neutral axis of the-cable stretched. The generic cable configuration shown in Fig. 2.41 illusrates some common materials that are used in the optical fiber cabling process. Individual fibers or modules of bun9led fiber grouping9 an4 copper wires for encapsulate ,.-_rr---e tape and oilier w-rapping e qc&er that provides I tough is a components these fiber groupings together. Surrounding all these inside are not the fibers so that cable to the -crush resistance and handles any tensile stresses applied jacket also protects the fibers inside igainstjbrasion, rngl!gfe,-9!l-!9!vents, contairinantr. Tt" jacket type largely defines the applicatioEtrar:Gtics, for example, heavy-duty of@@t. i t an@r t*r"gam" outside-plant cables for direct-burial and aerial applications have much thicker and ou'ghei jaekets than lightduty indoor Outer sheath iables. i The two basic fiber optic cable structures ard the tight-buffered fiber cable design and the loose-tube cable configuration. Cables with tightbuffered fibers norninally are used Yam strength member indoors whenias the loose-tube structure Insulated copper conductors is intend6il-for long-haul outdoor applications. A ribbon cable is an Polyurethane/PVC j acket Buffered strength member Paper/plastic binding tape Basic fiber building block extension of the tight-buffered cable. In A tgptcat stx-Jiber cable created bg Flg. all cases the fibers themselves consist strandirg slx baslc fiber-build@ blrcks of the normally manufactured glass core around a cenlrol strerEthmember and cladding which is surrounded by a protective 250-pm diameter coating. As shown in Fig. 2.42, in the tight-buffered design each fiber is individually encapsulated within its own 900-pm diameter plastic buffer structure, hence the designation tight-buffered design. The 900-pm buffer is nearly four times the diameter and five times the thicliness of the 250-pm protective coating material. This construotion feature contributes to the excellent moisture and temperature performance of tight-buffered cables and also permits their direct termination with connectors. In a single-fiber module, a layer of amarid strength material surrounds the 900-pm fiber structure. This configuration then is encapsulated within a PVC outer jacket. lithe loose+ube cable c,onfi$uration one or more standard-coated fibers are enclosed in a thermoplastic tube that has an inner diameteiwhich is much larger than the fiber diameter as shown in'Fig. 2.43.T\e fibers in ttre tube are slightly longer than the cable itself. The purpose of this construction is to isolate 2.41 the fiber from any stretching of the surrounding cable structure caused by factors such as temperature changes, wind forces, or Glass fiber with cladding ice loading. The tube is filled with either a gel or a dry water-blocking material that act as buffers, enable the fibers to move freely within the tube, and prevent moisture from Fiber coating (250 pm) entering the tube. Yarn strength member Fiber buffer (900 pm) To facilitate the field operation of splicing cables that contain alxge number of fibers, cable designers devised the fiber-ribbon structure. As shown in Fig. 2.4!, tbe ribbon cable is an arrangement of fibers that are Outer cablejacket (e.9.,2.4 mm diameter) F79.2.42 aligned precisely next to each other and then are encapsulated in a plastic buffer orjacket to form a long continuous ribbon. The number of fibers in a ribbon tlpically ranges from four to twelve. These ribbons can be stacked on top ofeach other to form a densely packed arrangement of many fibers (e.g., 144 fibers) within a cable structure. There are many different ways in which to arrange fibers inside a cable. The particular arangement of fibers and the cable design itself need to take into account issues such as the physical environment, the services that the optical link will provide, and any anticipated maintenance and repair that may be needed. 2.Lt.2 Construction oJ a stmplex tightbufiered fiber cable mdule Thermop lastic tube ,z^\- ,r' \ /or,""\ I rrul ) Optical fiber (a) End view (b) Side view \v Fig. 2.43 Corrcept oJaloose-htbe cable corfigurafion Indoor Cable Designs Indoor cables can be used for interconnecting instruments, for distributing signals among office users, for connections to printers or servers, and for short patch cords in telecommunication equipment racks. The three main types are described here. Interconttect cable serves light-duty lowfiber count indoor applications such as fiber-tothe-desk links, patch cords, and point-to-point runs Coated fibers (250-pmouterdiameter) Plasticencapsulation Fta.2.iL4 Example oJ a hrelue-fiber ribbon cable module in conduits and trays. The cable is flexible, compact, and lightweight with a tight-buffered construction. A popular indoor cable type is the duplex cable, which consists of two fibers that are encapsulated in an outer PVC jacket. Fiber optic patch cords, also known as jumper cables, are short lengths (usually less than 2 m) of simplex or duplex cable with connectors on both ends. Thby are used to connect lightwave test equipment to a fiber patch panel or to interconnect optical transmission modules within an equipment rack. Bt1p1kout orfanout cabte consists of up to 12 tight-buffered fibers stranded around a central sfength where it is necessary to protect member. Such cables serve low- to medium-fiber-count applications on individual fibers individual jacketed fibers. The breakout cable allows easy installation of connectors separate in the cable. With such a cable configuration, routing the individually terminated fibers to pieces of equipment can be achieved easily. consists of,individual or small groupings of tight-buffered fibers stranded aI*rA *.ral strength mernber. This cable serves a wide range of network applications for sending trays, " and video signals. Disfibution'cables are designed for use in intra-building cable data, voice, groupings they enable that conduits, and loose placeiientin dropped-ceiling structures. A main feature is locations' (distributed) to various branched be to of fibers within thelable Distribution cable 2.11.3 Outd?or Cables i (, applications. Invariably Outdoor cable installations include aerial, duct, direct-burial, and underwater of outdoor cables are sizes and designs different Many these cables consist of a looseltube structure. and the particular used be will cable the which in available depending on the physical environment application. The two intended for mounting outside between buildings or on poles or towers' self-sapporting structures.T\e cable facility-supporting popular designs are the self-supportiog *a the poles without cable corfiains an internal strength mlmber that permits the cable to be strung between a separate ftst cable thefacility-supporting For using any additional support rnrih*i.*r for the cable. to this clipped or lashed is cable the then poles and wire or strength 111".u.i is stmng between the Acrial cableis member. direct-burial or underground-duct applications has one or more layers of steeljacketing as shown in Fig' 2'45' wire or steel-sheath protective armoring below a layer of polyethylene it from gnawing animals such as protects This not only providls additional str"ngth to the cable but also cables. For example, in the squirrels or Lunowing rodents, which-often cause damage to underground Annored cablefor Central strength member Loose-tube structure Wrapping and binding taPes Comrgated steel taPe Water-blocking comPound Inner polyethylene jacket Outer polyethylene jacket Ftg. 2.45 Exomple a4frguration oJ an armored outdmrfiber opdc cable y'it"g States the plains pocket gopher (Geomys busarius) can destroy unprotected cable that is buried less than 2 m (6 ft) deep- other cable components include a central strengttr member, binding tapes, and water-blocking materials: *6fu;;; tlndenoater cable,also known as submaine cable, isused in rivers, lakes, and ocean environments. since such cables normally are exposed to high water pressures, ttrey h";" ;;;h ;;e stringent requirements than underground cables. For example, as shown in Fig. 2.46 cablesthat can be used in rivers and lakes have various water-blocking layeis, one or .;;rd;"" m""r priv"iiyr"ne shearhs, and i. heavy outer armor jacket. ,cables thai run under the o."ari hur" further layers of armoring and contain coppej wires to provide electrical power for submersed optical arnplifiers or regenerators. Loose-tube structure Strength member (steel wires) Wrapping tape and moisture barrier Polyethylene inner sheath Wrapping tape and lead sheath Polyethylene central sheath Bedding layer (yam + asphalt) Galvanized steel wire armoring Outer protective sheath Ftg. 2-tl.6 Examprb co4figrtration oJ an wrderusaterfiber optic cabte 2.11.4 InstallattoT Mettrglds Workers can install optical fiber cables by plowing them directly into the ground, laying them in an outside trench, pulling or blowing them through-ducts (both indoor urairta"".iir'Jrrr". spaces, suspending them on poles, placing them underwater. Although each method has its own special handing procedures, they all-orneed to adhere to a common set of"precaurions. d;l;;lr;e avoiding sharp bends of the cable, minimizing stresses on the installed cabie, periodically allowin! extra cable slack along the cable t9llt f:t unexpected repairs, and avoiding exceisive pulling forces 6n the cable. fhe rrU-T has priblished a number of L-ieries re"o.*"rditions for the construction, installation, and protection of fiber optic cables in the outside plant. Table 2.5lists some of these recommendations-and states their application. W*q"AuW, *A OpficatFibers; Strurture+ Y' , Fe'ffi Recommpndation Name Description L.35, Installation of oqtical fiber cabks in the access network, Oct. 1998. Gives guidance for installing optical cables in ducts, on poles, and through direct burial. L.38, Use of trenchless.techniques for the construition of underground infrastructures for telecommanication cable installation, Sept. 1999. Describes underground drilling techniques for installing telecommunication cables without the need for L.39, Invesrtgafion af tle soil before using trenchkss techniryls, MaY 200O.' Describes soil-investigation techniques for obtaining information about the position of buried objects and the nature of the ground. L.42, Mini-trench instailarton techniques, Describes a technique for puuing duct-based underground optical cables in small trenches. This allows quicker installations, lower cost, and limited surface disruption' March 2003. L.43, Micro4rench irustallatioft techttique disruptive excavation or Plowing. Describes installing underground cables at a shallow depth in small grooves. s, March 2003. Describes air-assisted methods for installation of optical fiber cables in ducts. L.57, Air-assisted install.cttion of optical fiber cables, MaY 2003. $ pnoeLEMS where 2.1 Consider an electrib fietd fepresented by the expression *20t"-i*'", n= [tooei3o" ", Az * o"t"o'"r)rl'' where y is expressed in micromcters and the l. propagation cbnstanl lE'given in prn Find (a) the amplitude, (D) the wavelength, (c) the angular frequency, and (d) the'displacement time, = 0 and z = 4 [rm. 2.S Eonsider two plane waves X1 andX2 traveling \'-z1n the same direction. If they have the same frequency ar but different ar.nplitudes a, and phases 6,, then we c.att represent them by Xr = at cqs (o)r - 6,1) I .+.at ( : Xr= azcos'(ar- 4) . According to the principle ol superposition, the resultant wave X is si'rnply the sum of X, andX2. Show thatXgan be written in the form X= A,cos (at - 0) a? +al+Z.ararcos(6, -6r) and Express this as a measurable elecftic field as described by Eq. (2.2) at a frequency of 100 MHz. 2.2 A w aveis specified by y = $ cos 2.td2r - 0'82), = tan @= Cz.&rrtr,i.ally - q sin6, +arsin6, +arcos6, arcos6, polarized light can be represented t*o orttrogonal waves given in Eqs (2.2) and (2.3). Show that elimination of the (at - kz) dependence between them yields by th" (r.\ -(l\ [,ao.,J [r., J -rtnE, Eo, Eo, cos6=sin26 which is the equation of an ellipse making an angle a with the x axis, where a is given by Eq. (2.8). 2.s l-et E*= E* = I in Eg. (2.7). Using a comput€r or a graphilal calculator, write a program to tF) plot this equation for values of 6= (nal8, where n = 0, 1,2, ....'16. What does ttris show about $ the state of polarization as the angle dchanges? (b) Solve Eq. (2.26b) graphtcally to show that 2.6 Show that any linearly po'larized wave may be considered as the supelposition of left and there are three angles of incidence which satisfy this equation. What happens to the number of angles as the wavelength is decreased? right ciicularly polarized Waves which are in phase and have equal amplitudes and (c) frequencies. 2.7 Light traveling in aA sEikes,a glass plate at an angle 0, = 33o, where 0, is measured between the incoming ray and the glass surface. Upon striking the glass, part ofthe beam is reflected and part is refracted. If the refracted and reflected beams make an angle of 90' with each other, what is the.refractive index of the glass? What is the critical angle for this glass? 2A A point source of light is 12 cm below the surface of a large body of watei (n = 1.33 for water). What is the radius of the largest circle on the water surface through which the light can emerge? A 45"45"-90o prism is immersed in alcohol (n = 1.45). What is the minimum refractive index the prism must have if a ray incident normally on one of the short faces is to be totally reflected at the long face of the prism? 2.14 Denve the approximation of the right-hand side of Eq. (2.23) for A << 1. What is the difference in the approximate and exact expressions for the value of NA if n, = 1.49 and field cornponents given in Eqs (2.35a) to Q35A. Show that these expressions lead to Eqs (2.36) (2.46). 2.19 aperture of 0.20 supports approximately 1000 modes at an 850-nm wavelength. (a) What is the diameter of its core? (b) How many modes does the fiber support at 1320 nm? (c) How any modes does the fiber support at 1550 nm? (a) Determine the normalized frequency at ' 25-pm core radius, /21 = 1.48, arfi n = 1.46. (b) How many modes propagate in this fiber at 820 nm? (c) How many modes propagate in,hiff"., polarized light is given by o _\cos(p2-n2cos(pl ,,, -i"*rpr*i@6 Show that the condition R : 0 for the nrewster angle occurs when tan et = ny'i,rh. Calculate the numerical aperture of a stepindex fiber having \ = 1.48 and n2 = 1.46. What is the maximum entrance angle 00.,,,o 820 nm for a step-index fiber having a the outer medium is air with at" 2.13 Consider a dielectric slab having a thickness d = l0 mm and index of reAaction n, = 1.50. l,et the medium'above and below the slab be air, in which flz = I. Let the wavelength be ,t = 10 mm (equal to the thickness of the' i , waveguide). (a) What is the critical angle for the slab waveguide? (2.37). 2.18 A step-index multimode fiber with a numerical The reflection coefficient R, for parallel if nd 2.16 Show that for y = 0, Eq. (2.55b) corresponds to TEo, modes (E. = 0) and that Eq. (2.56b) corresponds to TMo* modes (H, = 0). Z.fi Yenty that qr = 4. = F' when A << 1, where k, and k, ue thecore and cladding propagation constants, respectively, as defined in Eq. Show that the critical angle at ah interface for this fiber n = 1.00? 1.49t the expressions in Eqs (2.33) and (2.34) derived from Maxwell's curl equations, derive the radial and hansverse electric and magnetic between doped silica with nr = 1.460 and pure silica with nz= 1.450 is 83.3". 2.12 nr= 2.15 Using l32O nm? .._ ._r (d) How many modes propagate in this fiber at 1550 nm? (e) What percent of the optical power flows in the cladding in each case? : 2.20 Consider a fiber with a 25-pm core radius, a core index nr = 1.48, and A = 0.01. 1320 nm, what is the value of Vand how many modes propagate in the fiber? (a) If ),= OpticatF-ibers: S : (b) What percent of the optical power flows c6re in the cladding? (c) If fiber support and what fraction of the optical Power flowsin the cladding? Show that for a solid rod preform this is represented bY the exPression operation at 1320 nm of a step-index fiber with ,rl = t.+SO and nr= 1'47-8' What are the numerical aperture and maximum acceptance angle of this fiber? . L22 Amanufacturer wishes ts make a silica-core' step-index fiber with' V =.75 and 11ume1^9al aperture NA = 0.30 to be used at 820 nm' If n, ,=sfzT \d . ' = t.+Sg, what should the core 23 in cm/min for a 9-mm-diameter pteforyJ 2.30 A silica tube with inside and outside radii of 3 and 4 mm, respectively, is to have a certain thickness of glass deposited on the inner a silica-core (nr = 1'458), single-mode fibel to operate at 1300 nm. Suppose the fibe-r we-select from *ris curve has a 5-Hn core radius' Is this fiber still single-mode at820 um? Which modes exist in the fiber at 820 nm? o1o.ss + i-6lg'v3i2 +,2'w9 u4) glass surface. What should the thickness of this diameter a core having a fiber deposition be if Wo ^ What range of birefringent i'efractive-index differences does this correspond to for I = 1300 nm? Plotthe refractive-inclex piofiles framnlto n2 ( gradedas a function of radiat Cistance r a for index fibers thatliirre dvalues of |,2,4'8'ar;rd (step index), Atsru* the fters lloyg a25' - Prn co; radius,'n, = ,l'48, andA 19:01' 2.2i1 Czlculatethe numbero-f modes at 820 nm and 1.3 pm in agraded-index'fibe1-having a parabolic-index profile (a'= 2) ;' a-25-*m core iadius, nr = 1'48,'and nr l'1'46' How does 2;2E Calculate the numeri,cal apertures of (a) 'index of and an outer cladding diameter of be drawn from this perform using pm to is 125 the MCVD fabrication Process' 2.31 (a) The density of fused silica is2'6 g/cm3' How many grcms are needed for a l-km- ofiO pm and plot E(r)t4swith rran^ghrg from "Juftut" 0 to 3 for values of V = 1 .0, 1'4' 1 '8, 2'2, 2'6' and 3.0. Herc a is the fiber ra.{ius'-2.25 Commonty availaUte single-modefibers have beat lengtirs in the range 10 cm < I,<.2 m' 2.26 andfiber-drawspeeds,respectively'Atypical drawing speed is l.2mls for a 125-pm outerdiameter fib"r. Whut is the preform feed-rate Draw adesign curve of the fractio'ml reft activeindex difference A versus thecore radius a for fro= diameters, and S and s are the preform feed ' size and 2.24 tJsing the following approximation for ) where D and d arb the preform and fiber cladding index be? 2 1.405). satisfi id under steady-state drawing conditiors' ttre core radius necessary'for single-mode , a 2.29 When a preform is drawn into a fiber' the principle of conservation of mass must be to A = 0.003,rhow mamY modes does the 2.21 Find 6r= (n, = ttre core--cladding dffierence is reduced step-index fiber having a silica 1.458) and a silicone resin cladding fl2= 1.4g, (b) a i, = l'60 and a cladding index of long 5O-Pm-diameter fiber core? (b) If the core material is to be deposited ' inside of a glass tube at- a 0'5-g/min deposition rate, how long does it take to make the Preform for this fiber? 2.32 Dwing fabrication of optical fibers' dust particles incorporated into the fiber surface are prime examples of surface flaws ttrat can lead io reduced-fiber strongth' What size particles are tolerable ' if dust a glass fiber having a i0-N/mm3/2 stress intensity factor is to withstand a 7oo-MN/m? stress? 2.33 Static fatigue in a glass fiber refers to the condition *h"t" u fiber is stressed to a level o,, which is much less than the fracturestress aJsociated withthe weakest flaw' Initially' the fiber will not fail but, with time, cracks in the fiber will grcw as a result of chemical erosion at the crack tip. One model for the growth rate of a crack of depth 7 assumes a relation of the form given in Eq. (2.86)' l ) 1 (c) Using this equation,: s[row that the time required for a crack ofinitial depth 1,to grow to its failure size t = 26ris 2r,n @&6flx!'-u'' (b) For long, static fatigue times of 20 years), K?-u .. K|* for large values of b. Show that under this condition the failure time is given by - x(i-t"21 2K:-', (b (on the order -2) Ao'Y' $ nnn'nRENCES 1. See any general physics book or introductory optics book; for example: (a) D.Halliday, R. Resnick, and J. Walker, F undame nt al s of P hy s i c s, W iley, Hoboken, NJ, 7th ed.,2004. (D) E. Hecht, Optics,Addison-Wesley, Boston, 4th ed.,2002. (c) F. A. Jenkins and H. E. White, Funda- mentals 2. of Optics, McGraw-Hill, Burr Ridge, tL, 4th ed., 2001. See any introductory electromagnetics book; for example: (a) W.H. Hayt,Jr. andJ. A. Buck,Engineering Ele ctroma gne ti c s, McGraw-Hill, Burr Ridge, IL,4thed., 2([6. (b) F. T. Ulaby, Fundamentals of Applied Ele ctromagnetics, Prentice Hall, Upper Saddle River, NJ,5th ed.,2007. (c) J. 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