Unit Iintroduction

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UNIT I INTRODUCTION

Introduction:Communication is defined as the transfer of information from one point to another. The

function of communication system is to convey signal from information source to destinationover transmission medium. Communication system consists of transmitter/ modulator linked toinformation source, transmission medium and receiver/demodulator at destination point.

General system:An optical fiber communication system is similar in basic concept to any type of

communication system. The function of general communication system is to convey the signalfrom the information source over the transmission medium to the destination. Thecommunication system therefore consists of a transmitter or modulator linked to the informationsource, the transmission medium, and a receiver or demodulator at the destination point. Thetransmission medium can consist of a pair of wires, a coaial cable or a radio link through free

space down which the signal is transmitted to the receiver, where it is transformed into theoriginal electrical information signal !demodulated" before being passed to the destination.

#n any communication system there is a maimum permitted distance between thetransmitter and the receiver beyond which the system effectively ceases to give intelligiblecommunication. $or long haul applications these factors necessitate the installation of repeatersor line amplifiers at intervals, both to remove signal distortion and to increase signal level beforetransmission is continued down the link. The optical source which provides the electrical%opticalconversion may be either a semiconductor laser or light&emitting diode !'()". The transmissionmedium consists of an optical fiber cable and the receiver consists of an optical detector whichdrives a further electrical stage and hence provides demodulation of the optical carrier.

*hotodiodes ! p % n, p % i % n or avalanche" and, in some instances, phototransistors and photoconductors are utili+ed for the detection of the optical signal and the optical%electricalconversion.


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The laser drive circuit directly modulates the intensity of the semiconductor laser with theencoded digital signal. ence a digital optical signal is launched into the optical fiber cable. Theavalanche photodiode !A*)" detector is followed by a front&end amplifier and e-uali+er or filter to provide gain as well as linear signal processing and noise band width reduction. $inally, thesignal obtained is decoded to give the original digital information.

eed for optical fiber communication:" eed of low loss transmission medium in long haul communication system.0" eed of compact and less weight transmitter and receiver.1" eed of increased span of transmission.2" eed of increased bit rate distance product.Advantages of optical fiber communication:

Communication using an optical carrier wave guided along a glass fiber has anumber of etremely attractive features, several of which were apparent when the techni-ue wasoriginally conceived. The advances in the technology to date have surpassed even the mostoptimistic predictions, creating additional advantages. ence it is useful to consider the meritsand special features offered by optical fiber communications over more conventional electrical

communications.a" 3ider bandwidth and greater information capacity:

The optical carrier fre-uency in the range 41 to45 + !generally in the near infrared around 42 + or 46 G+" yields a far greater potential transmission bandwidth than metallic cable systems !i.e. coaial cable bandwidth typically around04 7+ over distances up to a maimum of 4 km" or even millimeter wave radiosystems !i.e. systems currently operating with modulation bandwidths of844 7+over a few hundreds of meters". Although the usable fiber bandwidth will be etendedfurther towards the optical carrier fre-uency, it is clear that this parameter is limited by the use of a single optical carrier signal. ence a much enhanced bandwidthutili+ation for an optical fiber can be achieved by transmitting several optical signals,each at different center wavelengths, in parallel on the same fiber.

!b" 9mall si+e and weight.$iber cable diameter is not greater than diameter of human hair so it re-uires less

storage space. Glass or plastic cables are much lighter than copper cables.!c" 'ower losses.

ptical fiber cable has less signal attenuation over long distance!;4.0d


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!d" =epeater spacing.=epeater is not an amplifier. #t is a regenerator of original signal. #n optical fiber

signal can be transmitted without repeaters with a maimum data rate of 6Gbps over adistance of kms.

!e" 9ignal security.

$iber cables cannot be tapped as easily as electrical cables. 9o it is used for military, banking and general data transmission.!f" (nvironmental #mmunity.

ptical cables are more resistant to environmental etremes !ie weather variations" than metallic cables. They also operate over wider temperature range andare less affected by corrosive li-uids and gases.

!g" 'ow cost and ease of maintenance.Cost of fiber optic cable is approimately same as metallic cables.

!h" 9ystem reliability.The predicted life time of optical components are 04 to 14 years.

!i"'ong distance transmission.

ptical fiber have less attenuation so used for long distance communication. !>"9afe and easy installation. ptical fiber cable is safer and easier to install and maintain. The small si+e andlight weight makes installation easier.

)isadvantages:" igh initial cost.

#nitial cost for installation of optical fiber is very high.0" 7aintenance and repairing cost.

7aintenance and repairing is difficult and epensive in optical fibers.1" ?oining and test procedure.

As fiber cable is small in si+e, fiber >oining process is very costly and re-uiresskilled man power.

2" Tensile stress.$iber cables are more susceptible to buckling, bending and tensile stress than

copper cables.6" $iber losses.

9cattering, dispersion, attenuation and reflection are the losses in optical fiber cables.

Applications:" @sed in telecommunications, instrumentation, cable T network! CAT" and data

transmission and distribution.0" @sed in telephone system because of small si+e and large information carrying

capacity.1" @sed for transmitting digital data generated by computers between C*@ and

peripherals, between C*@ and memory and between C*@Bs

Ray theory transmission introduction:The optical fiber has a transparent core with a refractive inde n surrounded by a

transparent cladding of slightly lower refractive inde n0. The cladding supports the waveguidestructure while also, when sufficiently thick, substantially reducing the radiation loss into the


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surrounding air. #n essence, the light energy travels in both the core and the cladding allowing theassociated fields to decay to a negligible value at the cladding%air interface.

Total internal reflection:=efractive inde of a medium is defined as the ratio of the velocity of light in a

vacuum to the velocity of light in the medium.DA ray of light travels more slowly in an optically dense medium than in less dense

medium. As a ray of light incident on interface between 0 dielectrics of differing refractiveindices then refraction occurs which is shown in fig.

Consider a ray of light incident on interface between 0 dielectrics of differingrefractive indices, so refraction occurs. The ray approaching interface is propagating in dielectricof refractive inde n at an angle E to the normal at the surface of interface. n the other handside, interface has refractive inde n0 at an angle E0 to the normal. Then n0 is less than n and E0is greater than E. The angle of incidence and refractive indices are related by snellBs law of refraction

nsin E Fn0sin E0


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sin E F !n0sin E0 "/nsin E/sin E0 F n0/n

Above figure shows that small amount of light is reflected back into originating dielectricmedium !partial internal reflection". As n is greater than n0, angle of refraction is 4o andrefracted ray emerges parallel to interface between dielectrics the angle of incidence must be less

than 4o

. This is the limiting case of refraction and the angle of incidence is now known ascritical angle Ec.

sinEc Fn0/n

At angle of incidence greater than critical angle the light is reflected back into originatingdielectric medium !total internal reflection" with high efficiency !around .H".

Total internal reflection occurs at the interface between 0 dielectrics of differingrefractive indices when light is incident on dielectric of lower inde from dielectric of higher inde and angle of incidence of the ray eceeds critical value.D

Acceptance angle:The maimum angle !Ia" to the ais at which light may enter the fiber in order to be

propagated is known as acceptance angle.D3hile seeing the propagation of light in an optical fiber through total internal reflection at

core cladding interface, it is useful to enlarge upon geometric optics approach with reference tolight rays entering the fiber. nly rays with sufficiently shallow gra+ing !ie with an angle to thenormal greater than Ec " at core cladding interface are transmitted by total internal reflection, soall rays entering the fiber core will not continue to be propagated down its length.

Consider that a ray is entering the fiber core at an angle Ia to the fiber core. #t is refractedat air core interface before transmitting to core cladding interface at E c. 9o we conclude that therays which are incident into fiber core at an angle greater than Ia will be transmitted to corecladding interface at an angle less than Ec and it will not be totally internally reflected.

ow consider light ray < is incident into fiber core at an angle greater than I a. #tis refracted into cladding and eventually lost by radiation. 9o rays which are to be transmitted by total internal refection should incident on core at acceptance angle only defined by conicalhalf angle Ia.

umerical aperture: umerical aperture !A" gives the relationship between acceptance angle and

refractive indices of core, cladding and air.


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Consider a light ray incident on core at an angle I which is less than Ia. Therefractive indices of air is no, core is n and cladding is n0. n is slightly greater than n0. owconsider the incident ray at fiber core to be normal to ais and refraction at air core interfaceusing snellBs law


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9kew rays:#n meridional rays, some will propagate in optical waveguide and some rays are

transmitted without passing through fiber ais. This out number ray follow helical path via fiber and are called skew rays. The skew ray path can not be visuali+ed in 0) but observed in helical path, that is traced through fiber gives a change in direction of 0J at each reflection where J is theangle between pro>ection of ray in 0) and radius of fiber core at point of reflection.


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3hen input light is non uniform, skew rays will tend to have smoothing effect ondistribution of light as it is transmitted, giving non uniform output. Amount of smoothing dependon number of reflection caused by skew rays. Consider the skew ray is incident on fiber core atan angle Is at point A. This ray is refracted in air core interface before reaching point < in same plane. The angle of incidence and refraction is E at point < which is greater than Ec. #f 0

perpendicular planes via ray path A< are considered then J is the angle between core radius and pro>ection of ray on to plane


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$or meridional rays JF and IasF Ia. 9o Ia maimum conical half angle for acceptance of meridional ray defines the minimum input angle for skewrays. 9kew rays propagate only in annular region near outer surface of core, do notutili+e core fully for transmission.

Electromagnetic mode theory of optical propagation:(lectromagnetic waves:

An improved model for propagation of light in optical fiber is electromagnetictheory and is provided by 7awellBs e-uations. $or medium with +ero conductivity, vector relationship is given by electric field (, magnetic field , electric flu density ) and magneticflu density


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7odes in *lanar guide:The simplest form of optical waveguide is planar guide which consists of a slab of

dielectric with refractive inde n sandwiched between two regions of lower refractive inden0.The conceptual transition from ray to wave theory may be aided by consideration of a planemonochromatic wave propagating in the direction of the ray path within the guide, it is shown infig.

The refractive inde within guide is n, optical wavelength is reduced to K/n and

vacuum propagation constant is increased to nk.3hen I is the angle between the wave propagation vector or the e-uivalent ray and the guide ais, the plane wave can be resolved intotwo component plane waves propagating in the z and x directions, as shown in $igure 0.L!a". Thecomponent of the phase propagation constant in the z direction M z is given by:

M z F nk cos I !0.12"The component of the phase propagation constant in the x direction M x is:

M xF nk sin I !0.16"


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The component of the plane wave in the x direction is reflected at the interface between the higher and lower refractive inde media. 3hen the total phase change after twosuccessive reflections at the upper and lower interfaces !between the points P and Q" is e-ual to0mN radians, where m is an integer, then constructive interference occurs and a standing wave isobtained in the x direction which is shown in fig 0.L!b". evertheless, the optical wave is

effectively confined within the guide and the electric field distribution in the x direction doesnot change as the wave propagates in the z direction. The sinusoid ally varying electric field inthe z direction is also shown in $igure 0.L!b". The stable field distribution in the x directionwith only a periodic z dependence is known as a modeD

ence the light propagating within the guide is formed into discrete modes, eachtypified by a distinct value of I. These modes have a periodic z dependence of the form ep!O >M z z " where M z becomes the propagation constant for the mode as the modal field pattern isinvariant ecept for a periodic z dependence. 3e denote the mode propagation constant by M,where M F M z . #f we now assume a time dependence for the monochromatic electromagnetic lightfield with angular fre-uency P of ep!>Pt ", then the combined factor epQ >!Pt O M z "R describes a

mode propagating in the z direction.


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numbers are incorporated into this nomenclature by referring to the T( m and T7 m modes.3hen the total field lies in the transverse plane, transverse electromagnetic !T(7" waves eistwhere both E z and H z are +ero. owever, although T(7 waves occur in metallic conductors!e.g.coaial cables" they are seldom found in optical waveguides.

*hase and group velocity:

$or plane waves the constant phase points form a surface which is referred to as awave front. As a monochromatic light wave propagates along a waveguide in the z directionthese points of constant phase travel at a phase velocity p given by:

where P is the angular fre-uency of the wave. owever, it is impossible in practice to produce perfectly monochromatic light waves, and light energy is generally composed of a sum of planewave components of different fre-uencies. 9ometimes group of waves with closely similar fre-uencies propagate so that their resultant forms a packet of waves. The formation of such awave packet resulting from the combination of two waves of slightly different fre-uency propagating together is illustrated in $igure 0.4.This wave packet does not travel at the phasevelocity of the individual waves but is observed to move at a group velocity g given by:

The group velocity is of greatest importance in the study of the transmissioncharacteristics of optical fibers as it relates to the propagation characteristics of observable wavegroups or packets of light. #f propagation in an infinite medium of refractive inde n isconsidered, then the propagation constant may be written as:

3here c is the velocity of light in free space. (-uation !0.1L" follows from (-s !0.11" and!0.12"where we assume propagation in the z direction only and hence cos I is e-ual to unity. @sing (-.!0.15" we obtain the following relationship for the phase velocity:


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Cylindrical fier:

7ode: The eact solution of 7awellBs e-uations for a cylindrical homogeneous coredielectric waveguide involves much algebra and yields a comple result . Although the presentation of this mathematics is beyond the scope of this tet, it is useful to consider theresulting modal fields. #n common with the planar guide, T( !where E z F 4"and T7 !where H z F4" modes are obtained within the dielectric cylinder. The cylindrical waveguide, however, is bounded in two dimensions rather than one. Thus two integers l and m, are necessary in order tospecify the modes, in contrast to the single integer ! m"re-uired for the planar guide. $or thecylindrical waveguide we therefore refer to T( lm and T7lm modes. These modes correspond tomeridional rays traveling within the fiber. owever, hybrid modes where E z and H z are non+eroalso occur within the cylindrical waveguide. These modes, which result from skew ray

propagation within the fiber, are designated (lm and (lm depending upon whether thecomponents of H or E make the larger contribution to the transverse !to the fiber ais"field. Thusan eact description of the modal fields in a step inde fiber proves somewhat complicated.owever, as U in weakly guiding fibers is very small, then (%( mode pairs occur which havealmost identical propagation constants. 9uch modes are said to be degenerate. The super positions of these degenerating modes characteri+ed by a common propagation constantcorrespond to particular '* modes regardless of their (, (,T( or T7 field configurations.This linear combination of degenerate modes obtained from the eact solution produces a usefulsimplification in the analysis of weakly guiding fibers. The relationship between the traditional(, (, T( and T7 mode designations and the '*lm mode designations is shown in Table 0..


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The mode subscripts l and m are related to the electric field intensity profile for a particular '* mode. There are in general 0l field maima around the circumference of the fiber core and m field maim along a radius vector. $urthermore, it may be observed from Table 0.that the notation for labeling the ( and ( modes has changed from that specified for the eactsolution in the cylindrical waveguide mentioned previously.

#t may be observed from the field configurations of the eact modes that the field strengthin the transverse direction ! E x or E y" is identical for the modes which belong to the same '*mode. ence the origin of the term Vlinearly polari+edB. @sing (-. !0.1" for the cylindricalhomogeneous core waveguide under the weak guidance conditions outlined above, the scalar wave e-uation can be written in the form

3here Ψ is the field !E or H", n is the refractive inde of the fiber core, k is the propagationconstant for light in a vacuum, and r and φ are cylindrical coordinates. The propagation constantsof the guided modes β lie in the range:

3here n0 is the refractive inde of the fiber cladding. 9olutions of the wave e-uation for thecylindrical fiber are separable, having the form:

where in this case Ψ represents the dominant transverse electric field component. The periodicdependence on φ following cosl φ or sin l φ gives a mode of radial order l . ence the fiber supports a finite number of guided modes of the form of (-. !0.51".#ntroducing the solutionsgiven by (-. !0.51" into (-. !0.5" results in a differential e-uation of the form:

$or a step inde fiber with a constant refractive inde core, (-. !0.52" is a


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The sum of the s-uares of # and $ defines a very useful -uantity which is usually referred to asthe normali+ed fre-uency % where:

#t may be observed that the commonly used symbol for this parameter is the same as that

normally adopted for voltage. owever, within this chapter there should be no confusion over this point. $urthermore, using (-s !0.L" and !0.4" the normali+ed fre-uency may be epressedin terms of the numerical aperture &' and the relative refractive inde difference∆, respectively,as:

The normali+ed fre-uency is a dimensionless parameter and hence is also sometimes

simply called the % number or value of the fiber. #t combines in a very useful manner theinformation about three important design variables for the fiber: namely, the core radius a, therelative refractive inde difference ∆ Wand the operating wavelength K. #t is also possible to definethe normali+ed propagation constant for a fiber in terms of the parameters of (-. !0.5L" so that:

=eferring to the epression for the guided modes given in (-. !0.50", the limits of M are

n0k and nk , hence must lie between 4 and .#n the weak guidance approimation the fieldmatching conditions at the boundary re-uire continuity of the transverse and tangential electricfield components at the core%cladding interface !at r F a". Therefore, using the


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guided modes rather than radiation modes. Alternatively, as β is increased above n0k , less power is propagated in the cladding until at βnk all the power is confined to the fiber core. As indicated previously, this range of values for β signifies the guided modes of the fiber.


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The optical fiber considered in the preceding sections with a core of constant refractive inde nand a cladding of a slightly lower refractive inde n0 is known as step inde fiber. This is becausethe refractive inde profile for this type of fiber makes a step change at the core%claddinginterface, as indicated in $igure 0.0, which illustrates the two ma>or types of step inde fiber.

The refractive inde profile may be defined as:

in both cases. $igure 0.0!a" shows a multimode step inde fiber with a core diameter of around64 Xm or greater, which is large enough to allow the propagation of many modes withinthe fiber core. The single&mode step inde fiber has the distinct advantage of low intermodaldispersion!broadening of transmitted light pulses", as only one mode is transmitted, whereas withmultimode step inde fiber considerable dispersion may occur due to the differing groupvelocities of the propagating modes . This in turn restricts the maimum bandwidth attainablewith multimode step inde fibers, especially when compared with single&mode fibers. owever,for lower bandwidth applications multimode fibers have several advantages over single&modefibers. These are:

!a" the use of spatially incoherent optical sources !e.g. most light&emitting diodes"which

cannot be efficiently coupled to single&mode fibersY!b" larger numerical apertures, as well as core diameters, facilitating easier coupling to

optical sourcesY!c" lower tolerance re-uirements on fiber connectors.

7ultimode step inde fibers allow the propagation of a finite number of guided modesalong the channel. The number of guided modes is dependent upon the physical parameters!i.e.relative refractive inde difference, core radius" of the fiber and the wavelengths of the


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transmitted light which are included in the normali+ed fre-uency % for the fiber. #t was indicated before that there is a cutoff value of normali+ed fre-uency % c for guided modes below whichthey cannot eist. owever, mode propagation does not entirely cease below cutoff. 7odes may propagate as unguided or leaky modes which can travel considerable distances along the fiber. evertheless, it is the guided modes which are of paramount importance in optical fiber

communications as these are confined to the fiber over its full length. #t can be shown that thetotal number of guided modes or mode volume ( s for a step inde fiber is related to the % valuefor the fiber by the approimate epression:

which allows an estimate of the number of guided modes propagating in a particular multimodestep inde fiber.

#n an ideal multimode step inde fiber with properties !i.e. relative inde difference, corediameter" which are independent of distance, there is no mode coupling, and the optical power launched into a particular mode remains in that mode and travels independently of the power launched into the other guided modes. Also, the ma>ority of these guided modes operate far fromcutoff, and are well confined to the fiber core. Thus most of the optical power is carried in thecore region and not in the cladding. The properties of the cladding !e.g. thickness" do nottherefore significantly affect the propagation of these modes.

Graded inde fibers:Graded inde fibers do not have a constant refractive inde in the core but a

decreasing core inde n!r " with radial distance from a maimum value of n at the ais to a


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constant value n0 beyond the core radius a in the cladding. This inde variation may berepresented as:

3here ∆ Wis the relative refractive inde difference and α is the profile parameter which gives thecharacteristic refractive inde profile of the fiber core. (-uation !0.86" which is a convenientmethod of epressing the refractive inde profile of the fiber core as a variation of α, allowsrepresentation of the step inde profile when α∞, a parabolic profile when α0 and a triangular profile when α.

The graded inde profiles which at present produce the best results for multimode optical propagation have a near parabolic refractive inde profile core with α≈0. $ibers with such coreinde profiles are well established and conse-uently when the term Vgraded indeB is usedwithout -ualification it usually refers to a fiber with this profile.

A multimode graded inde fiber with a parabolic inde profile core is illustrated in $igure0.01. #t may be observed that the meridional rays shown appear to follow curved paths throughthe fiber core. @sing the concepts of geometric optics, the gradual decreasing refractive indefrom the center of the core creates many refractions of the rays as they are effectively incident ona large number or high to low inde interfaces. This mechanism is illustrated in $igure 0.02where a ray is shown to be gradually curved, with an ever increasing angle of incidence, until the


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conditions for total internal reflection are met, and the ray travels back towards the core ais,again being continuously refracted.

7ultimode graded inde fibers ehibit far less intermodal dispersion than multimode stepinde fibers due to their refractive inde profile. Although many different modes are ecited inthe graded inde fiber, the different group velocities of the modes tend to be normali+ed by theinde grading. Again considering ray theory, the rays traveling close to the fiber ais haveshorter paths when compared with rays which travel into the outer regions of the core. owever,the near aial rays are transmitted through a region of higher refractive inde and therefore travelwith a lower velocity than the more etreme rays. This compensates for the shorter path lengthsand reduces dispersion in the fiber. These travel for the most part in the lower inde region atgreater speeds, thus giving the same mechanism of mode transit time e-uali+ation. ence,multimode graded inde fibers with parabolic or near&parabolic inde profile cores havetransmission bandwidths which may be orders of magnitude greater than multimode step indefiber bandwidths.

The parameters defined for step inde fibers !i.e. &', Z, % " may be applied to graded

inde fibers and give a comparison between the two fiber types. owever, it must be noted thatfor graded inde fibers the situation is more complicated since the numerical aperture is afunction of the radial distance from the fiber ais. Graded inde fibers, therefore, accept lesslight than corresponding step inde fibers with the same relative refractive inde difference.(lectromagnetic mode theory may also be utili+ed with the graded profiles. Approimate fieldsolutions of the same order as geometric optics are often obtained employing the 3[< methodfrom -uantum mechanics after 3ent+el, [ramers and


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where ) and * are assumed to be real functions of the radial distance r . 9ubstitution of (-. !0.85"into the scalar wave e-uation of the form given by (-. !0.5"!in which the constant refractiveinde of the fiber core n is replaced by n!r "" and neglecting the second derivative of )i!r " withrespect to r provides approimate solutions for the amplitude function )i!r " and the phasefunction * !r ". ence the caustics define the classical turning points of the light ray within thegraded fiber core. These turning points defined by the two caustics may be designated asoccurring at r r and r r 0.The result of the 3[< approimation yields an oscillatory field in theregion r r r 0between the caustics where:

The 3[< method does not initially provide valid solutions of the wave e-uation in thevicinity of the turning points. $ortunately, this may be amended by replacing the actual refractiveinde profile by a linear approimation at the location of the caustics. The solutions at theturning points can then be epressed in terms of ankel functions of the first and second kind of order / 1.This facilitates the >oining together of the two separate solutions described previously


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for inside and outside the interval r Wr r 0. Thus the 3[< theory provides an approimateeigenvalue e-uation for the propagation constant MWof the guided modes which cannot bedetermined using ray theory. The 3[< eigenvalue e-uation of which MWis a solution is given by:

where the radial mode number m W, 0, 1 . . . and determines the number of maima of theoscillatory field in the radial direction. This eigenvalue e-uation can only be solved in a closedanalytical form for a few simple refractive inde profiles. ence, in most cases it must be solvedapproimately or with the use of numerical techni-ues. $inally the amplitude coefficient + may be epressed in terms of the total optical power P G within the guided mode. Considering the power carried between the turning points r and r 0 gives a geometric optics approimation of:

where:

The properties of the 3[< solution may by observed from a graphical representation of the integrand given in (-. !0.8L". This is shown in $igure 0.05, together with the corresponding3[< solution. $igure 0.05 illustrates the functions !n0!r "k 0 WM 0" and !l 0/r 0". The two curvesintersect at the turning points r Wr and r Wr 0. The oscillatory nature of the 3[< solution between the turning points !i.e. when l 0/r 0 Wn0!r "k 0 WM 0" which changes into a decayingeponential !evanescent" form outside the interval r Wr Wr 0 !i.e. whenl 0/r 0 Wn0!r "k 0 WM 0" can

also be clearly seen.#t may be noted that as the a+imuthal mode number l increases, the curve !l 0/r 0" moves

higher and the region between the two turning points becomes narrower. #n addition, even whenl is fied the curve !n0!r "k 0 WM 0" is shifted up and down with alterations in the value of the propagation constant M. Therefore, modes far from cutoff which have large values of MWehibitmore closely spaced turning points. As the value of MWdecreases below n0k ,!n0!r "k 0 WM 0" is nolonger negative for large values of r and the guided mode situation depicted in $igure 0.05changes to one corresponding to $igure 0.08. #n this case a third turning point r Wr 1 is createdwhen at r Wa the curve !n0!r "k 0 WM 0" becomes constant, thus allowing the curve ! l 0/r 0" to drop below it. 7oreover, for r Wr 1the field resumes an oscillatory behavior and therefore carries power away from the fiber core. @nless mode cutoff occurs at MW Wn0k , the guided mode is nolonger fully contained within the fiber core but loses power through leakage or tunneling into thecladding. This situation corresponds to the leaky modes mentioned previously.

The 3[< method may be used to calculate the propagation constants for the modes in a parabolic refractive inde profile core fiber where, following (-. !0.86":


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#t is interesting to note that the solution for the propagation constant for the variousmodes in a parabolic refractive inde core fiber given in (-. !0." is eact even though it wasderived from the approimate 3[< eigenvalue e-uation !(-. !0.L6"". owever, although (-.!0." is an eact solution of the scalar wave e-uation for an infinitely etended parabolic profilemedium, the wave e-uation is only an approimate representation of 7awellBs e-uation.$urthermore, practical parabolic refractive inde profile core fibers ehibit a truncated parabolicdistribution which merges into a constant refractive inde at the cladding. ence (-. !0." is noteact for real fibers. (-uation !0." does, however, allow us to consider the mode number planespanned by the radial and a+imuthal mode numbers m and l . This plane is displayed in $igure0.0Lwhere each mode of the fiber described by a pair of mode numbers is represented as a pointin the plane. The mode number plane contains guided, leaky and radiation modes.

$or each pair of mode numbers m and l the corresponding mode field can have a+imuthal modedependence cosl φ Wor sin l φ Wand can eist in two possible polari+ations. ence the modes are saidto be fourfold degenerate. #f we define the mode boundary as the function m W f !l ", then the totalnumber of guided modes ( is given by:

as each representation point corresponding to four modes occupies an element of unit area in themode plane. (-uation !0.0" allows the derivation of the total number of guided modes or modevolume ( g supported by the graded inde fiber. #t can be shown that:


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$urthermore, utili+ing (-. !0.84", the normali+ed fre-uency % for the fiber when W∆ ≪ is

approimately given by:

9ubstituting (-. !0.2" into (-. !0.1", we have:

ence for a parabolic refractive inde profile core fiber !α W W0", ( g ≈% 0/2, which is half thenumber supported by a step inde fiber !α W W∞" with the same % value.

9ingle&mode fibersThe advantage of the propagation of a single mode within an optical fiber is that the

signal dispersion caused by the delay differences between different modes in a multimode fiber may be avoided. 7ultimode step inde fibers do not lend themselves to the propagation of asingle mode due to the difficulties of maintaining single&mode operation within the fiber whenmode conversion !i.e. coupling" to other guided modes takes place at both input mismatches andfiber imperfections. ence, for the transmission of a single mode the fiber must be designed toallow propagation of only one mode, while all other modes are attenuated by leakage or absorption. $ollowing the preceding discussion of multimode fibers, this may be achievedthrough choice of a suitable normali+ed fre-uency for the fiber. $or single&mode operation, onlythe fundamental '*4 mode can eist. ence the limit of single&mode operation depends on thelower limit of guided propagation for the '* mode. The cutoff normali+ed fre-uency


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for the '* mode in step inde fibers occurs at % c W0.246. Thus single&mode propagation of the'*4 mode in step inde fibers is possible over the range:

as there is no cutoff for the fundamental mode. #t must be noted that there are in fact two modes

with orthogonal polari+ation over this range, and the term single&mode applies to propagation of light of a particular polari+ation. Also, it is apparent that the normali+ed fre-uency for the fiber may be ad>usted to within the range given in (-. !0.5" by reduction of the core radius, and possibly the relative refractive inde difference following (-. !0.84", which, for single&modefibers, is usually less than H.

#t is clear from (ample 0.5 that in order to obtain single&mode operation with amaimum % number of 0.2, the single&mode fiber must have a much smaller core diameter thanthe e-uivalent multimode step inde fiber !in this case by a factor of 10". owever, it is possibleto achieve single&mode operation with a slightly larger core diameter, albeit still much less thanthe diameter of multimode step inde fiber, by reducing the relative refractive inde difference of the fiber. ointing, and the reduced relativerefractive inde difference presents difficulties in the fiber fabrication process.


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Graded inde fibers may also be designed for single&mode operation and some specialistfiber designs do adopt such non step inde profiles. owever, it may be shown that the cutoff value of normali+ed fre-uency % c to support a single mode in a graded inde fiber is given by:

Therefore, as in the step inde case, it is possible to determine the fiber parameters which givesingle&mode operation.

#t may be noted that the critical value of normali+ed fre-uency for the parabolic profilegraded inde fiber is increased by a factor of √0 on the step inde case. This gives a corediameter increased by a similar factor for the graded inde fiber over a step inde fiber with thee-uivalent core refractive inde !e-uivalent to the core ais inde" and the same relativerefractive inde difference. The maimum % number which permits single&mode operation can be increased still further when a graded inde fiber with a triangular profile is employed.


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A further problem with single&mode fibers with low relative refractive inde differencesand low % values is that the electromagnetic field associated with the '*4 mode etendsappreciably into the cladding. $or instance, with % values less than .2, over half the modal power propagates in the cladding. Thus the eponentially decaying evanescent field may etendsignificant distances into the cladding. #t is therefore essential that the cladding is of a suitable

thickness, and has low absorption and scattering losses in order to reduce attenuation of themode. (stimates show that the necessary cladding thickness is of the order of 64 Xm to avoid prohibitive losses !greater than d< km \" in single&mode fibers, especially when additionallosses resulting from micro bending are taken into account. Therefore, the total fiber cross&section for single&mode fibers is of a comparable si+e to multimode fibers. Another approach tosingle&mode fiber design which allows the % value to be increased above 0.246 is the 3 fiber. #f the undesirable higher order modes are ecited or converted to have values of propagationconstant β W Wkn1, they will leak through the barrier layer between a and a0 !$igure 0.0" into theouter cladding region n1. Conse-uently these modes will lose power by radiation into the lossysurroundings. This design can provide single&mode fibers with larger core diameters than can theconventional single&cladding approach which proves useful for easing >ointing difficultiesY 3

fibers also tend to give reduced losses at bends in comparison with conventional single&modefibers.$ollowing the emergence of single&mode fibers as a viable communication medium in

L1, they -uickly became the dominant and the most widely used fiber type withintelecommunications. 7a>or reasons for this situation are as follows:

. They ehibit the greatest transmission bandwidths and the lowest losses of the fiber transmission media.

0. They have a superior transmission -uality over other fiber types because of the absenceof modal noise.

1. They offer a substantial upgrade capability !i.e. future proofing" for future widebandwidth services using either faster optical transmitters and receivers or advanced

transmission techni-ues !e.g. coherent technology".2. They are compatible with the developing integrated optics technology.6. The above reasons to 2 provide confidence that the installation of single&mode fiber

will provide a transmission medium which will have ade-uate performance such thatit will not re-uire replacement over its anticipated lifetime of more than 04 years.

3idely deployed single&mode fibers employ a step inde !or near step inde" profiledesign and are dispersion optimi+ed !referred to as standard single&mode fibers, for operation inthe .1 Xm wavelength region. These fibers are either of a matched cladding !7C" or adepressed&cladding !)C" design, as illustrated in $igure 0.14. #n the conventional 7C fibers, theregion eternal to the core has a constant uniform refractive inde which is slightly lower thanthe core region, typically consisting of pure silica.

Cutoff wavelength#t may be noted by rearrangement of (-. !0.84" that single&mode operation onlyoccurs above a theoretical cutoff wavelength Kc given by:


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3here % c is the cutoff normali+ed fre-uency. ence Kc is the wavelength above which a particular fiber becomes single&moded. )ividing (-. !0.L" by (-. !0.84" for the same fiber weobtain the inverse relationship:

Thus for step inde fiber where % c W0.246, the cutoff wavelength is given by :

An effective cutoff wavelength has been defined by the #T@&T which is obtained from a 0m length of fiber containing a single 2 cm radius loop. This definition was produced becausethe first higher order '* mode is strongly affected by fiber length and curvature near cutoff.

7ode&field diameter and spot si+e:7any properties of the fundamental mode are determined by the radial etent of its

electromagnetic field including losses at launching and >ointing, micro bend losses, waveguidedispersion and the width of the radiation pattern. Therefore, the 7$) is an important parameter for characteri+ing single&mode fiber properties which takes into account the wavelength&dependent field penetration into the fiber cladding. #n this contet it is a better measure of thefunctional properties of single&mode fiber than the core diameter. $or step inde and graded!near parabolic profile" single&mode fibers operating near the cutoff wavelength K c, the field iswell approimated by a Gaussian distribution. #n this case the 7$) is generally taken as the

distance between the opposite /e W4.18 field amplitude points and the power /e0

W4.16 points in relation to the corresponding values on the fiber ais, as shown in $igure 0.1. Another parameter which is directly related to the 7$) of a single&mode fiber is the spot

si+e !or mode&field radius" ω4. ence 7$) W0 ω4 , where ω4 is the nominal half width of theinput ecitation !see $igure 0.1". The 7$) can therefore be regarded as the single mode analogof the fiber core diameter in multimode fibers. owever, for many refractive inde profiles andat typical operating wavelengths the 7$) is slightly larger than the single&mode fiber corediameter. ften, for real fibers and those with arbitrary refractive inde profiles, the radial field


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distribution is not strictly Gaussian and hence alternative techni-ues have been proposed.owever, the problem of defining the 7$) and spot si+e for non&Gaussian field distributions isa difficult one and at least eight definitions eist. 7oreover, good agreement has been obtainedusing this definition for the 7$) using different measurement techni-ues on arbitrary indefibers.

(ffective refractive inde:The rate of change of phase of the fundamental '*4 mode propagating along a

straight fiber is determined by the phase propagation constant β. #t is directly related to thewavelength of the '*4 mode K4 by the factor 0π, since β Wgives the increase in phase angle per unit length. ence:

7oreover, it is convenient to define an effective refractive inde for single&mode fiber,sometimes referred to as a phase inde or normali+ed phase change coefficient neff , by the ratio of the propagation constant of the fundamental mode to that of the vacuum propagation constant:

ence, the wavelength of the fundamental mode K 4 is smaller than the vacuum wavelength KW bythe factor /neff where:

#t should be noted that the fundamental mode propagates in a medium with a refractive inden!r " which is dependent on the distance r from the fiber ais. The effective refractive inde cantherefore be considered as an average over the refractive inde of this medium.

3ithin a normally clad fiber, not depressed&cladded fibers, at long wavelengths !i.e. small% values" the 7$) is large compared to the core diameter and hence the electric field etends far into the cladding region. #n this case the propagation constantβ Wwill be approimately e-ual to n0k !i.e. the cladding wave number" and the effective inde will be similar to the refractive inde of the cladding n0. *hysically, most of the power is transmitted in the cladding material. At short


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wavelengths, however, the field is concentrated in the core region and the propagation constantβ Wapproimates to the maimum wave number nlk . $ollowing this discussion, and as indicated previously in (-. !0.50", then the propagation constant in single&mode fiber varies over theinterval n0k Wβ W Wnk . ence, the effective refractive inde will vary over the range n0 Wneff Wn.#n addition, a relationship between the effective refractive inde and the normali+ed propagation

constant defined in (-. !0.8" as:

The dimensionless parameter which varies between 4 and is particularly useful in thetheory of single&mode fibers because the relative refractive inde difference is very small, givingonly a small range for β. 7oreover, it allows a simple graphical representation of results to be

presented as illustrated by the characteristic shown in $igure 0.10 of the normali+ed phaseconstant of β Was a function of normali+ed fre-uency % in a step inde fiber. #t should also be notedthat !% " is a universal function which does not depend eplicitly on other fiber parameters.

Group delay and mode delay factor:The transit time or group delay ]g for a light pulse propagating along a unit length

of fiber is the inverse of the group velocity νg . ence:

The group inde of a uniform plane wave propagating in a homogeneous medium has beendetermined following (-. !0.24" as:

owever, for a single&mode fiber, it is usual to define an effective group inde & ge by:


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where νg is considered to be the group velocity of the fundamental fiber mode. ence, thespecific group delay of the fundamental fiber mode becomes:

7oreover, the effective group inde may be written in terms of the effective refractive inde neff defined in (-. !0.40" as:

#t may be noted that (-. !0.4" is of the same form as the denominator of (-. !0.24" which givesthe relationship between the group inde and the refractive inde in a transparent medium!planar guide". =earranging (-. !0.8", β Wmay be epressed in terms of the relative indedifference ∆ Wand the normali+ed propagation constant by the following approimate epression

$urthermore, approimating the relative refractive inde difference as !n & n0"/n0, for a weakly

guiding fiber where ∆ ≪ , we can use the approimation:

where & g and & g0 are the group indices for the fiber core and cladding regions respectively.9ubstituting (-. !0." for β Winto (-. !0.48" and using the approimate epression given in (-.

!0.0", we obtain the group delay per unit distance as:

The dispersive properties of the fiber core and the cladding are often about the same andtherefore the wavelength dependence of ∆ Wcan be ignored. ence the group delay can be writtenas:

The initial term in (-. !0.2" gives the dependence of the group delay on wavelengthcaused when a uniform plane wave is propagating in an infinitely etended medium with arefractive inde which is e-uivalent to that of the fiber cladding. owever, the second termresults from the wave guiding properties of the fiber only and is determined by the mode delayfactor d!% "/d% , which describes the change in group delay caused by the changes in power distribution between the fiber core and cladding. The mode delay factor is a further universal


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parameter which plays a ma>or part in the theory of single mode fibers. #ts variation withnormali+ed fre-uency for the fundamental mode in a step inde fiber is shown in $igure 0.11.

The Gaussian approimation:The field shape of the fundamental guided mode within a single&mode step inde

fiber for two values of normali+ed fre-uency is displayed in $igure 0.12. As may be epected, it

has the form of a


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where the radius parameter r 0 W x0 ^ W y0, ω4 is a width parameter which is often called the spot si+eor radius of the fundamental mode and β Wis the propagation constant of the guided mode field.

The factor preceding the eponential function is arbitrary and is chosen for normali+ation purposes. #f it is accepted that (-. !0.8" is to a good approimation the correct shape , then the parameters β Wand ω4 may be obtained either by substitution or by using a variational principle.@sing the latter techni-ue, solutions of the wave e-uation, (-. !0.5", are claimed to befunctions of the minimum integral:

where the asterisk indicates comple con>ugation. The integration range in (-. !0.L" etendsover a large cylinder with the fiber at its ais. 7oreover, the length of the cylinder - is arbitrary

and its radius is assumed to tend towards infinity. @se of variational calculus indicates that thewave e-uation (-. !0.5" is the (uler e-uation of the variational epression given in (-.!0.L". ence, the functions that minimi+e ! satisfy the wave e-uation. $irstly, it can be shownthat the minimum value of ! is +ero if φ Wis a legitimate guided mode field. 3e do this by performing a partial integration of (-. !0.L" which can be written as:

where the surface element d s represents a vector in a direction normal to the outside of thecylinder. owever, the function φ Wfor a guided mode disappears on the curved cylindrical surface

with infinite radius. #n this case the guided mode field may be epressed as:

#t may be observed from (-. !0.04" that the z dependence is limited to the eponential functionand therefore the integrand of the surface integral in (-. !0." is independent of z . Thisindicates that the contributions to the surface integral from the two end faces of the cylinder aree-ual in value, opposite in sign and independent of the cylinder length. Thus the entire surfaceintegral goes to +ero. 7oreover, when the function φ Wis a solution of the wave e-uation, thevolume integral in (-. !0." is +ero and hence ! is also e-ual to +ero. The variationalepression given in (-. !0.L" can now be altered by substituting (-. !0.04". #n this case the

volume integral becomes an integral over the infinite cross section of the cylinder !i.e. the fiber"which may be integrated over the length coordinate z . #ntegration over z effectively multiplies theremaining integral over the cross&section by the cylinder length - because the integrand isindependent of z . ence dividing by - we can write:


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where the operator ∇t indicates the transverse part !i.e. the x and y derivatives" of ∇. 3e havenow obtained in (-. !0.0" the re-uired variational epression that will facilitate thedetermination of spot si+e and propagation constant for the guided mode field. The latter parameter may be obtained by solving (-. !0.0" for β0 with ! W4, as has been proven to be thecase for solutions of the wave e-uation. Thus:

(-uation !0.00" allows calculation of the propagation constant of the fundamental mode if thefunction φ Wis known. owever, the integral epression in (-. !0.00" ehibits a stationary value

such that it remains unchanged to the first order when the eact mode function is substituted by a slightly perturbed function. ence a good approimation to the propagation constant can beobtained using a function that only reasonably approimates to the eact function. The Gaussianapproimation given in (-. !0.8" can therefore be substituted into (-. !0.00" to obtain:

Two points should be noted in relation to (-. !0.01". $irstly, following 7arcuse thenormali+ation was picked to bring the denominator of (-. !0.00" to unity. 9econdly, thestationary epression of (-. !0.01" was obtained from (-. !0.00" by assuming that therefractive inde was dependent only upon the radial coordinate r . This condition is, however,satisfied by most common optical fiber types. $inally, to derive an epression for the spot si+e ω4we again make use of the stationary property of (-s !0.00" and !0.01". ence, if the Gaussianfunction of (-. !0.8" is the correct mode function to give a value for ω4, then β0 will not alter if ω4 is changed slightly. This indicates that the derivative of β0 with respect to ω4 becomes +ero!i.e. dβ0/dω4 W4". Therefore, differentiation of (-. !0.01" and setting the result to +ero yields:

(-uation !0.02" allows the Gaussian approimation for the fundamental mode within single&mode fiber to be obtained by providing a value for the spot si+e ω4. This value may be utili+ed in

(-. !0.01" to determine the propagation constant β.$or step inde profiles it can be shown thatan optimum value of the spot si+e ω4 divided by the core radius is only a function of thenormali+ed fre-uency % . The optimum values of ω4/a can be approimated to better than Haccuracy by the empirical formula:


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The approimate epression for spot si+e given in (-. !0.05" is fre-uently used to determine the parameter for step inde fibers over the usual range of K / Kc !i.e. 4.L to .".

The accuracy of the Gaussian approimation has also been demonstrated for graded inde fibers,having a refractive inde profile given by (-. !0.85" !i.e. power&law profiles in the core region".

3hen the near&parabolic refractive inde profile is considered !i.e. α W W0" and the s-uare&lawmedium is assumed to etend to infinity rather than to the cladding where n!r " Wn0, for r ≥ Wa !(-.!0.85"", then the Gaussian spot si+e given in (-. !0.02" reduces to:

$urthermore, the propagation constant becomes:

#t is interesting to note that the above relationships for ω4 and β Win this case are identical to thesolutions obtained from eact analysis of the s-uare&law medium. umerical solutions of (-s!0.01" and !0.02" are shown in $igure 0.16 !dashed lines" for values of α Wof 5 and ∞ Wfor profileswith constant refractive indices in the cladding region. #n this case (-s !0.01" and !0.02"cannot be solved analytically and computer solutions must be obtained. The solid lines in $igure0.16 show the corresponding solutions of the wave e-uation, also obtained by a direct numerical


38/40

techni-ue. These results for the spot si+e and propagation constant are provided for comparisonas they are not influenced by the prior assumption of Gaussian shape.

The Gaussian approimation for the transverse field distribution is very much simpler than the eact solution and is very useful for calculations involving both launching efficiency atthe single&mode fiber input as well as coupling losses at splices or connectors. #n addition, for

single&mode fibers with homogeneous cladding, the true field distribution is never eactlyGaussian since the evanescent field in the cladding tends to a more eponential function for which the Gaussian provides an underestimate. owever, for the calculations involving claddingabsorption, bend losses, crosstalk between fibers and the properties of directional couplers, thenthe Gaussian approimation should not be utili+ed. or simplicity of the Gaussian approimation, in which essentially one parameter !the spot si+e" defines the radialamplitude distribution, because they necessitate two parameters to characteri+e the samedistribution.

(-uivalent step inde methods:Another strategy to obtain approimate values for the cutoff wavelength and spotsi+e in graded inde single&mode fibers !or arbitrary refractive inde profile fibers" is to definean e-uivalent step inde !(9#" fiber on which to model the fiber to be investigated. ariousmethods have been proposed in the literature which commence from the observation that thefields in the core regions of graded inde fibers often appear similar to the fields within stepinde fibers. ence, as step inde fiber characteristics are well known, it is convenient to replacethe eact methods for graded inde single&mode fibers by approimate techni-ues based on stepinde fibers. #n addition, such (9# methods allow the propagation characteristics of single&modefibers to be represented by a few parameters. 9everal different suggestions have been advancedfor the choice of the core radius a(9#, and the relative inde difference ∆(9#, of the (9# fiber which

lead to good approimations for the spot si+e !and hence >oint and bend losses" for the actualgraded inde fiber. They are all conceptually related to the Gaussian approimation in that theyutili+e the close resemblance of the field distribution of the '* 4 mode to the Gaussiandistribution within single&mode fiber. An early proposal for the (9# method involvedtransformation of the basic fiber parameters following:

where the subscript s is for the (9# fiber and , / are constants which must be determined.owever, these (9# fiber representations are only valid for a particular value of normali+edfre-uency % and hence there is a different , / pair for each wavelength. The transformation can be carried out on the basis of either compared radii or relative refractive inde differences.

$igure 0.15 compares the refractive inde profiles and the electric field distributions for twograded inde fibres !α W W0, 2" and their (9# fibers. #t may be observed that their fields differ slightly only near the ais.

An alternative (9# techni-ue is to normali+e the spot si+e ω4 with respect to an optimumeffective fiber core radius aeff . This latter -uantity is obtained from the eperimentalmeasurement of the first minimum !angle θmin" in the diffraction pattern using transverseillumination of the fiber immersed in an inde&matching fluid. ence:


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where k W0 π / K. #n order to obtain the full comparison with single&mode step inde fiber, theresults may be epressed in terms of an effective normali+ed fre-uency % eff which relates the

cutoff fre-uencies/wavelengths for the two fibers:

The techni-ue provides a dependence of ω4/aeff on % eff which is almost identical for a reasonablywide range of profiles which are of interest for minimi+ing dispersion !i.e. .6 W% eff W0.2". Agood analytical approimation for this dependence is given by:

=efractive inde profile&dependent deviations from the relationship shown in (-. !0.10" arewithin ± W0H for general power&law graded inde profiles.

=efractive inde profile&dependent deviations from the relationship shown in (-. !0.10" arewithin W±0H for general power&law graded inde profiles. The Gaussian approimation and givenin (-. !0.06". An alternative formula which is close to (-. !0.06" is provided by 9nyder as:

owever, it is suggested that the epression given in (-. !0.11" is probably less


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accurate than that provided by (-. !0.06". A cutoff method can also be utili+ed to obtain the (9# parameters. #n this case the cutoff wavelength Kc and spot si+e ω4 are known. Therefore,substituting % W0.246 into (-. !0.06" gives:

Then using (-. !0.84" the (9# relative inde difference is:

where n is the maimum refractive inde of the fiber core.

Alternatively, performing a least s-uares fit on (-. !0.06" provides Vbest valuesB for the(9# diameter !0a(9#" and relative inde difference !∆(9#". #t must be noted, however, that these best values are dependent on the application and the least s-uares method appears most useful inestimating losses at fiber >oints. #n addition, some work has attempted to provide a moreconsistent relationship between the (9# parameters and the fiber 7$). verall, the concept of the (9# fiber has been relatively useful in the specification of standard 7C and )C fibers bytheir e-uivalent a(9# and ∆(9# values. @nfortunately, (9# methods are unable accurately to predict7$)s and waveguide dispersion in dispersion&shifted and dispersion&flattened fibers.

Documents

Unit Iintroduction